Physica status solidi / A.: Volume 54, Number 1 July 1979 [Reprint 2021 ed.] 9783112551127, 9783112551110

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plxysica status solidi (a)

ISSN 0031-8965 * VOL. 54 • NO. 1 • JULY 1979

Classification Schemc 1. Structure of Crystalline Solids 1.1 Perfectly Periodic Structure 1.2 Solid-State Phase Transformations 1.3 Alloys. Metallurgy 1.4 Microstructure (Magnetic Domains See 18; Ferroelectric Domains See 14.4.1) 1.5 Films 1.6 Surfaces 2. Non-Crystalline State 3. Crystal Growth 4. Bonding Properties 5. Mössbauer Spectroscopy 6. Lattice Dynamics. Phonons 7. Acoustic Properties 8. Thermal Properties 9. Diffusion 10. Defect Properties (Irradiation Defects See 11) 10.1 Metals 10.2 Non-Metals 11. Irradiation Effects (X-Ray Diffraction Investigations See 1 and 10) 12. Mechanical Properties (Plastic Deformations See 10) 12.1 Metals 12.2 Non-Metals 13. Electron States 13.1 Band Structure 13.2 Fermi Surfaces 13.3 Surface and Interface States 13.4 I m p u r i t y and Defect States 13.5 Elementary Excitations (Phonons See 6) 13.5.1 Excitons 13.5.2 Plasmons 13.5.3 Polarons 13.5.4 Magnons 14. Electrical Properties. Transport Phenomena 14.1 Metals. Semi-Metals 14.2 Superconductivity. Superconducting Materials and Devices 14.3 Semiconductors 14.3.1 Films 14.3.2 Surfaces a n d Interfaces 14.3.3 Devices. Junctions (Contact Problems See 14.3.4) 14.3.4 High-Field Phenomena, Space-Charge Effects, Inhomogeneities, Injected Carriers (Electroluminescence See 20.3; Junctions See 14.3.3) 14.4 Dielectrics 14.4.1 Ferroelectrics 15. Thermoelectric and Thermomagnetic Properties 16. Photoconductivity. Photovoltaic Effects 17. Emission of Electrons and Ions 17.1 Field Emission Microscope Investigations 18. Magnetic Properties 18.1 Paramagnetic Properties 18.2 Ferromagnetic Properties 18.2.1 Ferromagnetic Films 18.3 Ferrimagnetic Properties 18.4 Antiferromagnetic Properties (Continued on cover three)

Classification Scheme — Continued 19. Magnetic Resonance 20. Optical Properties 20.1 Spectra. Optical Constants (X-Ray Spectra See 20) 20.2 Lasers 20.3 Luminescence (X-Ray Spectra See 20) 21. Metals. Alloys 21.1 Transition Metals (Group Sc to Zn) and Their Alloys 21.1.1 Fe and Fe Alloys 21.2 Alkali Metals 21.3 Alkali-Earth Metals 21.4 Rare-Earth Metals 21.5 Actinides 21.6 Noble Metals 21.7 Semi-Metals (Semiconducting Alloys 22.7) 22. Semiconductors and Ionic Crystals 22.1 Elements 22.1.1 Germanium 22.1.2 Silicon 22.1.3 Group VI Elements 22.2 I I I - V Compounds 22.2.1 Arsenides 22.2.2 Phosphides 22.2.3 Antimonides 22.2.4 Mixed Crystals 22.3 IV-IV, II-V, and I I I - V I Compounds 22.3.1 Mixed Crystals 22.4 I I - V I Compounds 22.4.1 Sulphides 22.4.2 Selenides 22.4.3 Tellurides 22.4.4 Mixed Crystals 22.5 Hal ides 22.5.1 Silver Halides 22.5.2 Alkali Halides 22.5.3 Alkali-Earth Halides 22.5.4 Mixed Crystals 22.6 Simple Oxides (CdO, ZnO See 22.4) 22.6.1 Mixed Crystals 22.7 Semiconducting Intermetallic Compounds 22.7.1 Mixed Crystals 22.8 Tertiary and Higher Compounds 22.8.1 Salts of Oxy-Acids (Carbonates, Phosphates, Silicates Including Mica etc.) 22.8.2 Spinels, Garnets, Ferrites 22.9 Organic Semiconductors 23. Solidified Gases

Attention The preceding classification scheme is closely akin to the scheme used by Physics Abstracts to classify the various fields of solid state physics. A paper will either be classified under one or more of the subsections or, if none of these are relevant, it will just appear under the general section heading (i. e. sections and subsections are equivalent in this sense). The subject classification given at the head of each paper begins with the main subject matter of thfe paper, and this is followed, if necessary, by figures referring to subsidiary subject matter and the type of solids investigated. The papers published in this issue are listed in the order of their subject classification at the end of the table of contents.

82914

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physica status solidi (a) applied research

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Volume 54 • Number 1 • Pages 1 to 436, K1 to K84, and A l to A8 July 16, 1979 PSSA 54(1) 1—436, K l — K 8 4 , A l — A 8 (1979) ISSN 0031-8965

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Contents Review Article R . S. BRAZIS, J . K . F U R D Y N A , a n d J . K .

POZELA

Microwave Effects in Narrow-Gap Semiconductors (II)

II

Original Papers E . V . SUVOROV, O . S . G O R E L I K , V . M . K A G A N E R , a n d V . L .

S.

LANYI

IXDENBOM

Shape of Extinction Fringes and Determination of the Burgers Vector in Single-Crystal Interferometric Techniques

29

Contact-Limited Conduction in V 2 0 5 Single Crystals

37

F . X . C H U K H O V S K I I a n d A . M . A I U S T A M VAN

A Dynamical Analytical Theory of the Dislocation Image Contrast in Electron Transmission Microscopy M . G . GRIMALDI, P . BAERI, S. U . CAJIPISANO, G . FOTI, a n d E .

RIMINI

Laser Induced As Profile Broadening in Amorphous Silicon Layers J . G. KLIAVA, Z. A . KONSTANTS, J . J . PUUANS et A . I.

. . .

and

Manganese from

C.

79

FISCHBECK

Optically Stimulated Exoelectron Emission from Potassium Single Crystals K.

67

ATJLEYTXER

Soft X-Ray Emission Spectra of Aluminium Gd(Ali ^ j M i i j ) , Intermetallic Compounds G. M. RENFRO a n d H . J .

61

SHUTOV

An Analysis of the Dispersive Charge Transport in Vitreous 0.55As 2 S 3 : 0.45 Sb 2 S 3 A . SLEBARSKI, K . LAWXICZAK, a n d J .

55

DIMANTE

R P E de Mn ! + dans les polyeristaux de basse symmetric. Le metaphosphate de magnesium

V . I . A R K I I I P O V , M . S . I O V U , A . 1. R U D K X K O , a n d S . D .

45

Chloride 85

ROZWADOWSKA

KOXAK

Preparation and DC Conductivity of the Low-Temperature Glassy Carbon

93

Changes of Optical Properties at an Incommensurate-Commensurate Phase Transition in (NH 4 ) 2 BeF4 Crystals

99

E . KRATOCHVILOVA, V . ROSKOVEC, a n d M . NEKVASIL

Magnetocrystalline Anisotropv of Neodymium-Substituted Y t t r i u m - I r o n Garnet " M . MASZKIEWICZ, B . MRYGON, a n d K .

WENTOWSKA

Rounding of Specific Heat of Nickel near the Critical Temperature . . . . l

105

Ill

Contents

4

H . SZYMCZAK a n d N .

-

TSUYA

Phenomenological Theory of Magnetostriction and Growth-Induced Anisotropy in Garnet Films

117

K . D . GLINCITUK. A . V . PROKHOROVICH, a n d V . I . V O V N E X K O

Study of Non-Linear Extrinsic Luminescence in GaAs (III)

121

V . B . S H A R A N , D . N . S . SRIVASTWA, a n d S . C. S E N

On t h e Origin of the 375 nm Emission Band in KC1:T1 E . V . KAPITANOV a n d E . N .

129

YAKOVLEV

Mossbauer S t u d y of Phase Transitions under High Hydrostatic Pressures (II)

139

F . F . LAVRENTEV, 0 . P . SALITA, a n d S . V . SOKOLSKI

J u m p - L i k e Plastic Deformation and Dislocation Structure Instability in Zinc Crystals with Forest Dislocations H . KUPFER a n d A. A.

E.

NEJIBACH

145

MANUEL

Summation and Saturation Behaviour of the Volume Pinning Force in Neutron Irradiation V3Si

153

Analysis of the E x t r a o r d i n a r y Viscous Magnetic Domain Wall Motion in NiFe Films

167

N . M . G H O N I E M a n d D . D . CHO

The Simultaneous Clustering of Point Defects during Irradiation M. PASEMANN a n d P.

N . RIHON

171

WERNER

Identification of Small Defects in Silicon

179

Field Emission Microscopy f r o m Glass-Coated Tips. Observation of FowlerNordheim Emission Regime and of Periodic Current Oscillations

189

I . V . GRIDNEVA, Y U . V . MILMAN, V . I. TREFILOV, a n d S . I . CHUGUNOVA

A.

VAN CALSTER

R . HERZ a n d H .

Analysis of Dislocation Mobility under Concentrated Loads at Indentations of Single Crystals

195

Barrier-Limited Conductivity in Thin Semiconducting Films

207

KRONMULLER

The Parasusceptibility of Dysprosium N . TOYODA, M . MIHARA, a n d T.

217

HARA

I m p u r i t y Effect on the Formation of Terraces in GaAs L P E Growth.

. .

225

V . M . F R I D K I N , R . N I T S C H E , N . K O R C H A G I N A , N . A . KOSONOGOV, R . M A G O M A D O V , A . I . R O D I N , and

K . A.

VERKHOVSKAYA

Photoferroelastic Phenomena in S b 5 0 7 I Crystals S . F . D U B I N I N , S. G . TEPLOUCHOV, a n d S. K .

231

SIDOROV

Atomic Structure of the Invar F e - N i Alloys

239

Contents

5

M . M U T H A R E D D Y , K . SOMAIAH, a n d V . H A B I

D. L. KIBK

BABU

Thermoluminescence of B a P B r Crystals

245

The Relationship between the Defect Structure and the Intrinsic Ionic Conductivity of Monocrystalline Sodium Chloride

251

S. P . S. BADWAL a n d H . J . DE

BBUIN

Electrode Kinetics at the Pt/Yttria-Stabilized Zirconia Interface by Complex Impedance Dispersion Analysis F . M O N T H E I L L E T a n d J . M . HAT; DIN

261

Coherent Precipitation near Dislocations. A Theoretical Analysis . . . .

271

B . KTJHLOW

Lichtbeugung an magnetischen Bubble-Bereichsgittern in substituiertem Yttrium-Eisengranat

281

H. H. WAWEA

On the Role of Griineisen's Elastic Parameters in the Theory of Polycrystal Elasticity. A Critical Assessment

291

V. A. SOLOVEV

The "Superdislocation" Model of a Plastic Zone near a Wedge-Shaped Crack

297

C. HOLSTE, G . K . SCHMIDT, a n d R .

TOBBER

A Quantitative Estimation of Specific Internal Stress Fields in Cyclically Deformed Nickel

G. KUHNEL, W . SIEGEL, a n d E .

G. FUCHS

305

ZIEGLER

Deep Levels in ZnSiP 2 Determined by Schottky TSC Measurements . . .

315

Pinning Mechanism in Superconducting NbTi Alloys at High Magnetic Fields

321

J . A . GBOENINK, C. HAKFOOET, a n d G . BLASSE

R . S. POPOVIC

The Luminescence of Calcium Molybdate

329

Analytical Approximations of Non-Degenerate Schottky Diode CurrentVoltage Characteristics

337

P. HEGEDUS and S. KOLNIK

Some Invariance Relations for Acoustic Waves in Crystals Deformed by Hydrostatic Pressure Y . MOTOHASHI, M . G . BLANCHIN, E . VICABIO, G . F O N T A I N E , a n d S .

OTAKE

Elastic Parameters, Elastic Energy, and Stress Fields of Dislocations in T i 0 2 Rutile Crystals I . POP, E . R U S , M . COLDEA, a n d 0 .

349

355

POP

Knight Shift and Magnetic Susceptibilities of Intermetallic Compounds PrCu 4 and PrCu 5

365

6

Contents

V . S . SIVASANKAR a n d P . W .

V.

G.

KOHN

WHIPPEY

Thermoluminescence and Optical Absorption of X-Ray Irradiated N a B r 0 3

369

On the Theory of the Bragg Reflection in the Case of Multiple X-Ray Diffraction

375

J . H A J T 6 a n d P . J . S. E W E N

Natural Optical Activity and Related Phenomena in As 2 S 3 Glasses. . . .

385

P . C. MATHUR, P . 0 . T E N E J A , K . V . KRISHNA, a n d A . L . DAWAR

C.

B.

CARTER

The Effect of Non-Stoichiometric Excess of In on the Electrical Transport Properties of Polycrystalline p-Type InSb Thin Films

391

The Extension of Jogs on Dissociated Dislocations in F.C.C. Metals . . .

395

J . H . BASSON, H . BOOYENS, a n d C. A . B . BALL

The Nucleation and Generation of Dislocation Half-Loops in I I I - V Compounds

407

B . V . R . CHOWDARI a n d Y . RAVI SEKHAR

Electrical Conductivity and Thermal Depolarisation Studies of C r O l - and Cr 2 0?" Doped K D P and A D P Crystals

413

P . H . N G U Y E N , C. BOUTRIT, B . L E P L E Y , a n d S . RAVELET

T.

H. O'DELL

A Method for Determining Interfacial Parameters of MIS Schottky Barriers

421

Domain Wall Mass in Magnetic Bubble Eilms under Larger In-Plane Fields

429

Short Notes V . V . GOLUBKOV, V . I . PETROV. a n d G . V . SPIVAK

Dynamics of Xéel Wall Transitions into Cross-Tie Walls

K1

N . D . STOJADINOVIC

Conditions for the Creation of Dislocations by Diffusion of Phosphorus into Silicon

Ko

J . K O S H Y a n d C . C . OGBONNA

Etching of Synthetic Barium Fluoride Single Crystals

Kll

G . D L U B E K a n d 0 . BRÜMMER

The Composition Dependence of the Effective Vacancy Formation Enthalpy in Copper-Rich Brass by Means of Positron Annihilation

K13

C. TEMPLIER e t J . DELAFOND

Etude par diffusion centrale des rayons X de la démixtion d'un alliage Al-Zn 22 at % K17

Contents

7

J . SALM, K . S T E E N B E C K , a n d E . S T E I N B E I S S

Current—Voltage Characteristic of Reactive Sputtering with Element Targets

K23

N . ROMEO, D . SEURET, a n d F . F I R S T

Properties of t h e ZnSe/ZnTe Heteroj unction Prepared by a Multi-Source Evaporation of ZnTe: Sb on ZnSe Single Crystals K27 G. A . HASSAN

P . LUKAC

Effect of the Type of Pre-Torsional Deformation on the Creep R a t e s of an Aluminium-Silicon-Zinc Alloy

K31

The Critical Resolved Shear Stress of Mg-Cd Single Crystals

K33

V . P . LEDOVSKOI, N . A . P O T S A R , V . L . L A S K A , a n d E . S E I D O W S K I

Superconducting Properties of Nb Films Deposited in a Quasi-Closed Volume

K37

D . WEAIRE a n d F . WOOTEN

Interpretation of Structural Anomalies in Hydrogenated Amorphous Silicon K41 C . M . VAN B A A L , R . F A S T E N A U , a n d P . P E N N I N G

The Probability of Visiting a New Site in a R a n d o m Walk on a B.C.C. Lattice . . . . . ' V . A . ZHUKOV, UKIIIN

V . V . SAVYALOV,

B . V . KOBA,

V . L . LITVINOV,

V . P . SADIKOV,

and

K45

X. A.

Electrical Properties of Silicon, Irradiated by Neutrons and y-Quanta, in Strong Electric Fields

K49

A . E . K A R K I X , V . E . ARKHIPOV, V . A . MARCHENKO, a n d B . N . GOSHCHITSKII

Electrical Resistivity of V3Si and Nb 3 Sn under Neutron R a d i a t i o n . . . .

K53

H . - J . BURJIEISTER a n d H . HERMANN

A Simple Model for an Inhomogeneous Dislocation Structure

K59

P . K . R E D D Y a n d S. R . JAWALEKAR

Dielectric Properties of Tantalum Oxynitride Films

K63

P . R . VAYA a n d D . KAKATI

Improvement of Breakdown Characteristics of a Planar Microwave P - I - N Diode by the Bevelling Technique

K67

A . CIZEK, I . Y A . D E K H T Y A R , a n d A . I . MOSKALEVSKII

Investigations of Anomalous Passing of Positrons through Si Crystals

. . K71

R . KRISHNAN, M . TESSIER, a n d J . SZTERN

Magnetic Properties of Amorphous Fe 38 Ni 38 B 24 Alloy

K77

M . S. A . SWEET a n d D . URQUHART

Shallow Traps in ZnS Single Crystals

K81

Prc-Printed Titles of papers to be published in the next issues of physica s t a t u s solidi (a) and physica s t a t u s solidi (b)

Al

Contents Systematic List Subject classification:

Corresponding papers begin on the following pages (pages given italics refer to the principal subject classification):

1 1.2 1.3 1.4 1.6 2 3 5 7

117, 375 99,139 239, K17 45, 179, 271 225, K l l , K23 55,67, 93, 385, K41, K77 K23 139 349

8

Ill

9 10 10.1 10.2 11 12 12.1 12.2 13 13.3 13.4 14.2 14.3 14.3.1 14.3. 2 14.3.3 14.3.4 14.4 14.4.1 16 17 17.1 18.1 18.2 18.2.1 18.3 19 20 20.1 20.3 21 21.1 21.1.1

55, K5 45, 297, K45, K59 145, 171, 305, 321, 395, K13, K33 29, 85, 179,195, 225, 245, 251, 271, 355, 407, K5, K71 55, 153, 171, 369, K49, K53 349, K59 291, K31 195, 231, 355 11 261 121, 129, 207, 315, 329, 413, K81 153, 321, K37, K53 67, 315, 413 93, 207, 391 421 337, 421, K27, K67 37, 261, 337, K49 189, 251, K63 99 231 85 189 11,365 217, 239, K77 167, 429, K1 105, 117, 281 11,61, 365 11, 79 99, 129, 245, 385 121, 129, 245, 329, 369, K81 171, 321, 395, K13, K17, K31, K33, K37, K53 I l l , 145, 167, 239, 291, 305, 365, K13, K17 167, 239, K l , K77

10 21.4

Contents 79, 217, 365

21.6

261

22 22.1. 1 22.1.2 22.2 22.2.1 22.2.2 22.2.3 22.2. 4 22.4. 1 22.4.2 22.4. 3 22.4. 4 22.5 22.5.2 22.5. 3 22.5.4 22.6 22.7 22.8 22.8.1 22.8.2 22.9

67, 337, 385, K63 195 29, 55, 179, 195, 375, K5, K23, K41, K49, K67, K71 407 11, 121, 195, 225 421 11, 195, 391 11 K81 207, K27 11, K27 11 129 85, 251 Kll 245 37, 261, 355 139 231, 315 61, 99, 329, 369, 413 105, 117, 271, 281 93

Contents of Volume 54 Continued on Page 489

Review

Article

phys. stat. sol. (a) 54, 11 (1979) Subject classification: 19; 13; 18.1; 20; 22.2.1; 22.2.3; 22.2.4; 22.4.3; 22.4.4 Institute of Semiconductor Physics, Academy of Sciences of the Lithuanian SSR, Vilnius,1) (a) and Department of Physics, Purdue University, W. Lafayette Indiana2)

(b)

Microwave Effects in Narrow-Gap Semiconductors (II)3)4) By R . S . BRAZIS (a), J . K . F U R D Y S A ( b ) , a n d J . K . POZELA ( a )

5. Small

Contents spheres,

ellipsoids,

and pon

ders

5.1 Spheres, ellipsoids, a n d disks 5.2 P o w d e r s 6. Electron

spin resonance

in narrow-gap

semiconductors

6.1 Helicon-excited spin resonance of p a r a m a g n e t i c impurities 6.2 Conduction-electron spin resonance in N G S 7. Concluding References

(Part

remarks II)

5. Small Spheres, Ellipsoids, and Powders T h e m a t h e m a t i c a l analysis used so f a r applies t o p l a n e waves n o r m a l l y incident either on a n infinite half-space or on a n infinite slab of thickness d. T h e a p p l i c a t i o n of t h e results for these u n b o u n d e d geometries t o f i n i t e samples, a l t h o u g h f r e q u e n t l y well justified, is s t r i c t l y speaking an a p p r o x i m a t i o n . T h e r e are no e x a c t a n a l y t i c solutions for t h e interaction of a n electromagnetic wave w i t h a g y r o t r o p i c s a m p l e which is f i n i t e in all t h r e e dimensions. Nevertheless, some v e r y good a p p r o x i m a t i o n s are available for t h e case of a g y r o t r o p i c sphere a n d , in certain m o r e restrictive limits, also for t h e g y r o t r o p i c ellipsoid. T h e m a j o r difficulties of t h i s classical p r o b l e m were in f a c t only r e c e n t l y overcome [83, 28], T h e s e results h a v e been applied t o s t u d y N G S in t h e f o r m of isolated spheres a n d disks in a m i c r o w a v e waveguide, a n d also can serve as a p o i n t of d e p a r t u r e in analyzing t h e response of a g r a n u l a r m e d i u m (powder) t o m i c r o w a v e e x c i t a t i o n . T h e r e are t w o p a r a m e t e r ranges of i n t e r e s t : 1. T h e R a y l e i g h limit, in which t h e field of t h e w a v e is u n i f o r m b o t h o u t s i d e a n d within t h e b o u n d e d specimen. I n t h i s case, solutions for wave i n t e r a c t i o n exist for ellipsoidal samples, of which t h e sphere is a special case [28, 86, 87]. 2. T h e dipole limit in which t h e wave field is a s s u m e d uniform outside t h e sample, b u t can be a r b i t r a r y within t h e sample, i.e. t h e s a m p l e can effectively contain m a n y w a v e l e n g t h s of p r o p a g a t i n g n o r m a l modes. This case has been only r e c e n t l y solved for t h e spherical g e o m e t r y [83]. Vilnius 232600, USSR. ) W. Lafayette, Indiana 47907, USA. ) Work supported by the program of scientific exchange between the Academy of Sciences of the USSR and the US National Academy of Sciences. 4 ) P a r t I see phys. stat. sol. (a) 53, 11 (1979). 2

3

12

R . S . BRAZIS, J . K . FURDYNA, a n d J . K . POZELA

5.1 Spheres,

ellipsoids, and

disks

5.1.1 Magnetoplasma resonance in the Rayleigh limit Consider first the interaction of an isotropic ellipsoidal sample (described b y scalar x) with t h e electric field of an electromagnetic wave. A good discussion of the response of isotropic spheres and a definition of basic concepts are given in monographs [84, 85]. I f the electric field of the incident wave (outside the sample) is assumed parallel to one of t h e principal axes of the ellipsoid, the internal field i? in is uniform in the Rayleigh limit, and is related to b y [88] Ein = E

- L ~ £o

1

E L

1 + L(x -

= E1 — L(x — 1) E,n

1

1)

L

=

(105)

E L

y. -¡- yL'

where P = e0(y. — 1) Ein is t h e polarization of the ellipsoid medium, L is t h e depolarizing f a c t o r of t h e ellipsoid for the direction of Elt and yL = (1 ¡L) — 1. F o r an oblate ellipsoid of revolution with the rotation axis perpendicular to the microwave electric field E t h e depolarizing factor (denoted for this field orientation as Lj_) can be expressed in terms of the m a j o r axis a and t h e minor axis b [88]: = — 11 — — ' „

e2

(106)

(e — a r c t a n e)

where e = |/l — (^/a)2. W h e n the rotation axis is parallel to the microwave electric field E t then the depolarizing factor is L^: Ln = 1 -

2LX

F o r a sphere L\\ =

.

(107)

— L = 1/3. T h e dipole moment p excited b y

V y. — 1 p = | p d F = fo4, ^ E , , L y. + yL ' v

is then (108)

where V is the volume of the ellipsoid. T h i s result can be extended to gyrotropic ellipsoids with the dc magnetic field parallel t o t h e minor axis by means of a polarizability tensor a ( e ) . T h e electric dipole moment can then be written

(109)

where V--. (HO) * II + 7|| and the subscript s can be + 1 or —1 corresponding to t h e principal modes of excitation given b y equations (52). T h e power absorption associated with the s-th mode is given b y ' - T -

I

m

-M

WTrh^f-

( 1 1 1 >

F o r a single-component plasma and the c o n s t a n t - r approximation, t h e above expression for t h e power absorbed can be reformulated in microscopic terms b y

13

Microwave Effects in Narrow-Gap Semiconductors (II)

substituting expressions (52) into (111). We t h e n obtain [86, 87] /', = \E S ?

2

L2

.

,2 - — . -

(r.{

+

y

L

y

/ 1

+

- -

.

(112)

w g ft)

—

\

—

—

SO)..

(')

where «)„ = ) / n e l [ m * ' e ( y . + -y^)] is the effective plasma frequency for the ellipsoid. The interaction exhibits a resonance in Ps when 2

0

]

2

ft) - —•~ - s w c = 0 .

(113).

ft)

Note t h a t this condition can also be obtained by including the restoring depolarization fields directly into the microscopic equation of motion of the electrons, as was done by Dresselhaus et al. [89] in connection with cyclotron resonance in Ge. I t is instructive to consider the evolution of this resonance as a>0 increases. F o r low carrier concentration (ft)0 < o) the resonance occurs in the CRA, or « = + 1 polarization. At co0 = ft), the resonance occurs at B = 0 a n d corresponds to the plasma oscillation associated with the sample geometry. For high carrier concentration (ft)0 o>) the resonance appears in the C R I (s = —1) polarization, with the resonance field increasing to very high fields as co0 increases. The resonance m a y be viewed alternatively as either plasma-shifted cyclotron resonance [89], or as cyclotron-shifted plasma resonance. W e shall refer to it simply as "magnetoplasma resonance". F o r spheres (yL = 2) and for parameters typical for narrow-gap semiconductors, the condition ft)0 ai is satisfied at microwave (and lower) frequencies. I t can be seen from (113) t h a t , for w0 to, magnetoplasma resonance in a sphere occurs in the C R I polarization at field B = n e l [ u ) F ( x + 2)]. This occurs near the " m a g n e t o plasma cut-off" field given b y equation (72) corresponding to = 0. W i t h t h e exception of very p u r e InSb, satisfying these conditions ( B = „B s p h ere or B = . B m p c ) in NGS requires extremely high magnetic fields because of t h e high carrier concentration typical of these materials. As can be seen from (113), the field at which resonance occurs, s p h c r e

B = Bcb.-

,

^ +

0

l

(114)

YL)

can be significantly lowered by increasing yL, i.e. b y going to oblate ellipsoids with very high axial ratio, for which t h e depolarizing factor is very small [90]. Fig. 17 illustrates the magnetoplasma resonances in a sphere a n d in a thin disk. Fig. 18 represents the comparison of the experimental results obtained on disks with t h e theoretical predictions for an ellipsoid of revolution. The extent of agreement between the theory a n d t h e experiment is r a t h e r surprising. These results show t h a t an effective depolarizing factor can be ascribed to the disk geometry, a n d t h e f a c t o r is given by the depolarizing factor of an ellipsoid of revolution having t h e same axial ratio. I t is worth noting t h a t for large yL, the resonances occur a t large negative values of for which the waves are evanescent in the infinite medium. Such resonances then provide a probe of this otherwise inaccessible region. The resonance condition is size-independent in t h e Rayleigh limit. I t was shown b y solving the dynamic b o u n d a r y value problem for a small gyrotropic sphere [28] t h a t equation (111) is correct to terms of the order of -j- (2jra/A0). Magnetoplasma resonance in n-type I n S b spheres was studied experimentally b y several authors [28, 89 to 92]. Typical microwave absorption d a t a are shown in Fig. 19, illustrating t h e relative independence of this effect of size [28, 91]. H e r e t h e sample

14

R . S. B r a z i s , J . K . F u r d y n a , a n d J . K . P o z e l a

» 1.0

-L. 0.8

fflff 04 0.2

Fig. 17

Fig. 18

Fig. 17. Magnetoplasma resonances observed in transmission (T) and reflection (R) of the CRI wave for a small disk and a sphere prepared from the same crystal of n-InSb (n = 1.18 X 1015 c m - 3 , T = 65 K, / = 35 GHz). Both samples are placed symmetrically on the axis of a circular waveguide and separated from each other by approximately 1 cm. Sphere diameter d = 0.793 mm, disk diameter a = 2.0 mm (bja = 0.024) (after [90]) Fig. 18. Experimental dependence of the magnetoplasma resonance field B x on the thickness-todiameter ratio for various cylindrical samples of n-InSb (points). The theoretical dependence of Br on axial ratio b/a calculated for ellipsoids of revolution is given b y the solid Curve, ^sphere i® the resonance field for the sphere geometry measured for each sample series of a given diameter. A rectangular sample is also included for comparison ( • ) (after [90]) a (mm)

O

2.74 1.92 1.38 1.29 2.00 2.40

n ( 1 0 u cm" 1.7 2.3 2.5 2.8

11.8

2.5

was placed in a microwave cavity and excited by a linearly polarized electric field E transverse to B, and a linearly polarized magnetic field B(a>) parallel to B. As pointed out later, the low-field non-resonant hump is associated with the ac magnetic field B(a>) excitation. As in the case of spheres, the above microwave interactions in disks can be used for the purpose of a quantitative characterization of materials. There are several advantages to the disk geometry. First, the disk is one of the easiest bounded geometries to prepare. This geometry is natural for epitaxial layers, and materials crystallizing in the form of thin platelets. 8econd, in samples involving high carrier concentrations, (e.g. n 1016 cm - 3 , typical of NGS), where microwave magnetoplasma resonance in spheres occurs at exceedingly high fields, the resonance field can be significantly lowered by using a thin disk geometry, as seen in Fig. 17. Thus the response function of a material in the otherwise inaccessible evanescent range can be "mapped o u t " by observing the magnetoplasma resonances in disks of successively decreasing axial ratio. Magnetoplasma resonance, which depends on y.x, is of considerable importance as a possible tool for direct determination of the static dielectric constant in zero-gap materials (HgTe, HgSe, a-Sn). The ability to position the resonance a t an arbitrary magnetic field by controlling the sample geometry offers the possibility of investigating the magnetic-field dependence of the static dielectric constant, predicted for zero-gap materials [30].

Microwave Effects in Narrow-Gap Semiconductors (II)

15

Fig. 19. Microwave absorption data for E _]_ B, B(a>) || li observed in a series of n-type I n S b spheres of various sizes, obtained by successive etching of a single sample (» = 1.7 X 10 14 c m - 3 , T = 77 K , / = 35 GHz). The solid curve represents experimental data, the large peak corresponds to magnetoplasma resonance. For details of the fitting procedure (dashed curve) see [28]

W e have concerned ourselves primarily with the one-electron Drude model with scalar effective mass. When the Drude model is inadequate (as for intrinsic or manyvalley NGS), equations (111) and (112) are no longer equivalent. Equation (111) should then be used directly, with the resonance condition given by X.=

- V L .

(115)

This condition is applicable also to NGS with anisotropic carrier effective mass provided that the rotation axis of the ellipsoidal sample and the dc magnetic field are parallel to a high-symmetry crystallographic axis. Equations (111) and (115) were applied to the study of Rayleigh electric dipole resonances in many-valley semiconductors, and in semiconductors with more than one type of carriers [93]. 5.1.2 Dimensional resonances in the dipole limit and their relation to the Fabry-Perot resonances in an infinite slab When the sample size becomes comparable to the internal wavelength of a given mode, so that \qa\ ig 1, dimensional resonances can take place. The condition \qa\ ^ 1 implies that the Rayleigh limit is violated, although the dipole limit may very well hold. In practice this can be easily satisfied for the CRA mode, particularly in the helicon region defined by inequalities (51). The problem is now inherently dynamic (non-static) and the corresponding boundary value problem of a gyrotropic sphere for arbitrary qa escaped rigorous solution until 1974. Even now the available formulation is only tractable by numerical methods [83]. Dimensional resonances occur in both magnetic and electric multipole excitations of the sample, with the magnetic dipole resonances becoming dominant as the sample size decreases. According to numerical studies, in the parameter range \ \x±i\ the principal dipole resonances for a sphere occur approximately when d j xs = 4N^N

+ i j

,

where d is the sphere diameter, 1 1 , 2, 3, ... \ —1, —2, —3, ...

for electric resonances, for magnetic resonances.

(116)

16

R. S. Brazis, J . K. Furdyna, and J . K. Pozela

This behaviour can be viewed qualitatively as a series of Fabry-Perot-like resonances, associated with " f i t t i n g " of CRA (s = + 1 ) or C R I (s = —1) wavelengths inside the sphere diameter. Note that for an infinite plane-parallel slab of thickness d in the Faraday configuration, i.e. for B [| (¡¡s [| n, where qs is the wave vector for the s-th mode and n is the normal to the surface of the slab, the conditions for Fabry-Perot resonance are qsd =

Nti\

N=

1,2,3,....

(117)

Substituting (17) or (52) in (117) and taking into account (9) and (72) we obtain [90] 5.MP0 + Bvii -/i /*,,,

_ n*c*N* y..,„*

1 ,/*

' *

( U 8 )

Equation (118) can be represented by a straight line in coordinates (By LPC + + -Bck)i(B — Bv|{) versus d~2. This equation describes in a unified manner the resonance conditions in slabs for arbitrary magnetic field, including the helicon resonances in the CRA polarization, as well as resonances occurring at fields exceeding jBju'c which appear in both CRA and CRI polarizations. The advantage of this formulation is that it applies for arbitrary electron effective mass and concentration (which enter the expression implicitly through the characteristic fields B^yi and .B.m ;>(•)• For a sphere we obtain from (116) By, PC + Bm

_

/,',,.

~

N(N

+

d*

4) 1

'

The above resonances iiianifest themselves as absorption peaks when a plane electromagnetic wave is incident. Note that the lowest-order resonance which dominates the interaction is a magnetic one. The Fabry-Perot-like dipole resonances observed for spheres mounted in a waveguide in a styrofoam holder and excited with CRA polarization are illustrated in Fig. 20. Note the strong dependence of dimensional resonances on the sphere dimension, while the magnetoplasma resonance is insensitive to the radius of the sphere. Similar transmission spectra were observed also for nearly ellipsoidal samples [90], The resonances can be classified as electric or magnetic dipole resonances, marked ~E(N, 0) and M(iV, 0), respectively, where N is the order of the resonance. (The labeling scheme is the same as that used in [29] and [83].) Selection of the resonances as electric or magnetic is obtained by observing the reflected signal when the sample section is terminated with a short circuit: When the small sample is located at an

d'0.465mm W)

\ /0J2b \jM(\0) 0.985 1?Q

/

l

^M12.01 EHO)211

i

l

I

1

1 1

Fig. 20. CRA wave transmission curves for small disk samples of n-InSb (n = 2.8 X 1014 cm" 3 , / = 35 GHz, CRA) of several thicknesses but the same diameter a = 1.29 mm, placed on the axis of circular waveguide. Each curve is shifted arbitrarily along the vertical axis for clarity. The minima are found to be electric and magnetic dipole resonances marked E(JV, 0) and M(N, 0), respectively (after [90]; for the resonances in spheres see [29])

17

Microwave Effects in Narrow-Gap Semicondcutors (II)

Fig. 21. Normalized magnetic field for Fabry-Perot-type resonances occurring a t B >- Bqh i n t h e CRA polarization and a t B > Bjii'c i n t h e C R I polarization as a function of sample thickness. Experimental points obtained on various disk samples • , O) and rectangular 1 to 7 ( • , V, •> A, samples 8, 9 (n, x) of n-InSb fall in t h e shaded regions between two straight lines, one of which (labeled N) represents t h e F a b r y - P e r o t resonance conditions for an infinite slab and t h e other [labeled M(AT, 0) 2 2 or E (AT,0)] represents the conditions of t h e d' (mm ) magnetic or electric dipole resonances for t h e sphere of diameter d; frequency / = 35.0 GHz, .BCR = 0.0175 T. F o r details see [90]

electric node, the absorption peaks marked E (electric dipole resonances) disappear in the reflected signal; and, similarly, the magnetic dipole resonances (M) disappear when the sample is at a magnetic node. The same behaviour was observed for a rectangular bar [94], The experimental and theoretical study of the standing-wave resonance conditions is summarized in Fig. 21. Theoretical conditions for successive Fabry-Perot resonances in a semi-infinite plate (marked N = 1, 2, ...) and successive magnetic and electric resonances in spheres, obtained using (118) and (119) are given by the straight lines. Note that all experimental points for disk samples fall in a region between a FabryPerot resonance condition for an infinite plane-parallel slab and a sphere resonance. More specifically, magnetic resonances of the disks occur between odd-N Fabry-Perot resonances in slabs and magnetic resonances in spheres; electric resonances in disks lie between even-N slab resonances and electric sphere resonances. The relationship of the disk resonances to the two limiting cases is worth emphasizing : While the resonance condition for a disk is very close to t h a t for an infinite slab of the same thickness, the nature of excitation (electric or magnetic) is qualitatively identical to that of the sphere. On the other hand, a close analogy exists between the dipole resonances of the small disks and the slab resonances described by equations (91), (92) or (97), (98). I t is clear from the behaviour of k that at fields corresponding to the helicon approximation, (51), the resonance position for a sphere measures essentially n. At higher fields, close to and above the magnetoplasma edge, the resonance conditions depend on both n and x v The resonance widths are primarily sensitive to the mobility and the free-carrier statistics. These effects can be taken into consideration via computer fitting. In addition we note that the theory for a sphere predicts a number of weak satellites associated with both the electric and magnetic dipole resonances [83]. 5.1.3 Magnetic dipole resonance in the Rayleigh

limit

As the sphere radius decreases toward the Rayleigh limit, the dominant magnetic dimensional resonance gradually transforms into a size-independent resonance. I t was recently shown [28] that the magnetic dipole moment m of a small gyrotropic sphere can be written in terms of the magnetic polarizability tensor a (m) as /m+1\ m_x

\mz / 2

physica (a) 54/1

M»> 0 =

o

\0

0

oL^O

0

\

AB,A '

-B-i

o£»>/ \B,

-

)

(120)

18

R . S . BRAZIS, J . K . F U R D Y N A , a n d J . K .

where

POZELA

^-MtJ- - . 7 ^ 1

~

=

2 y.s/.zz xs

+

:

.

-21 b

¿a

"J

2x+1y._1

=

.

U-")

+

and the indices « = + 1 , —1, and (zz) correspond to the principal modes of excitation described previously. W h e n the dimensional corrections in (121) can be neglected, the response of the sample varies simply as v.. Thus, in the small-sphere limit, the resonance condition for t h e transverse circularly polarized magnetic dipole excitation is < + *« = 0 .

(123)

I t is easily shown t h a t for oip/w [95 to 97J

2

1 a n d on

1 this predicts a resonance when

K | « 2co

(124)

independent of n and a. The existence of a resonance at w(. = 2m was first suggested on the basis of a q u a n t u m model by F u r d y n a and Mycielski [98], This f e a t u r e is especially important for XGS because it opens the possibility of determining m * in materials with high electron concentration, which was not possible in conventional

2

Jo i; 6 i 3

2 7 0 5 k 3 I

7 0

0.2 0.1 B 0 m — -

0

0.2

OA

0

0.2

OA s „ m — -

Fig. 22. Microwave absorption d a t a for li{oj) _[_ I t , E\\ Ji for the same InSb material, frequency, and temperature as t h a t in Fig. 19. Experimental d a t a (solid curve) illustrate the Rayleigh magnetic-dipole resonance. The dashed lines show the theoretical fit obtained with a single set of parameters. For details see [28]

Microwave E f f e c t s in N a r r o w - G a p Semiconductors (II)

19

microwave cyclotron resonance experiments (electric dipole interaction) because of the magnetoplasma shift discussed above. The Rayleigh magnetic dipole resonance was observed experimentally by E v a n s and coworkers [91, 99, 100], and discussed using a semiquantitative, empirical model. A rigorous classical macroscopic analysis was carried out in [28], resulting basically in equations (120) to (122). An illustration of the fit of theory to experiment is shown in Fig. 22, showing magnetic dipole absorption for a linearly polarized microwave magnetic field transverse to the dc field B. The electric field was polarized parallel to B, and did not elicit magnetic-field-dependent response. The data and t h a t in Fig. 19 were obtained on the same set of spheres, and the dashed curves represent theoretical fits using a single set of parameters in both figures. The low-field nonresonant hump in Fig. 22 corresponds to magnetic dipole absorption associated with a [ f \ i.e. excited by Bz(o>). The theory also verified the experimental observations that the magnetic dipole resonance varies as the fifth power of the radius, is excited by CRA polarization, and in the small-size limit is independent of n and r [91]. Note also the low-field hump excited by Bz. In the small-size limit this response is related to the Voigt dielectric constants (cf. equation (122)), but again evolves into a dimensional resonance as the sample size is increased. 5.2

Powders

Microwave techniques can also be used to study electrical properties of semiconductors in powder form. An exact solutionis, needless to say, lacking because of the complexity of the problem, which consists of two difficult steps: (i) the response of a single particle and (ii) summing up over the contributions of all grains. One must therefore resort to rather drastic approximations. The simplest approach, which nevertheless offers some rather useful features, is to assume that the grains are spherical and noninteracting. The powder response can then be discussed in terms of the foregoing discussion, and will manifest absorption related to the magnetoplasma and dimensional resonances in spheres. Characteristic transmission spectra of circularly polarized microwaves through unsieved powders of InSb, InAs, and HgTe are shown in Fig. 23 [101 to 103]. All three materials satisfy helicon conditions (51) except near B = 0. The striking feature of the data is that the powder is opaque to the CRA polarization and relatively transparent to the CRI wave, in contrast to helicon transmission through slabs. The mechanism for CRA absorption is the dimensional magnetic dipole resonance in powder grains, corresponding to N = 1 in (116). Because of the wide distribution of grain sizes in unsieved powders, there are CRA resonances occurring throughout the magnetic

Fig. 23. T r a n s m i s s i o n of circularly polarized microwaves t h r o u g h unsieved powders of n - t y p e a) I n A s (n = 9.6 X 10 16 cm" 3 ), b) I n S b (o.l x 1016 cm" 3 ), and c) H g T e (1 x 1 0 " cm" 3 ) a t 3 5 . 0 G H z and T = 4.2 K in F a r a d a y geometry. H e r e positive fields correspond to C R I , n e g a t i v e ones t o CRA. N o t e q u a n t u m oscillations of t h e t r a n s m i t t e d signal, seen especially well in t h e I n A s d a t a (after [101 t o 103])

2*

R. S. Brazis, J. K. F u r d y n a , and J. K. Pozela

20

field range. I n size-fractioned powders satisfying helicon conditions, CRA t r a n s mission shows a n a r r o w m a g n e t i c dipole resonance line [103], as shown in Fig. 24. This p o w d e r resonance is in reasonable a g r e e m e n t w i t h (116), a n d can be used t o estim a t e t h e electron c o n c e n t r a t i o n in p o w d e r grains. E l e c t r o n c o n c e n t r a t i o n s corresponding to Fig. 23 a r e too high for m a g n e t o p l a s m a cut-off (equation (72)) t o be observed in t h e available field range. T h u s t h e p o w d e r is relatively t r a n s p a r e n t t o t h e C R I polarization because n e i t h e r dimensional nor m a g n e t o p l a s m a resonances are p r e s e n t . I t was shown a b o v e t h a t t h e power a b s o r b e d b y a C R I - e x c i t e d sphere satisfying conditions (51) is essentially p r o p o r t i o n a l t o t h e t r a n s v e r s e dc m a g n e t o - c o n d u c t i v i t y a x x [103]. I n t h e region of orbital q u a n t i z a t i o n , a x x displays S h u b n i k o v - d e H a a s oscillations, a n d t h i s is seen p a r t i c u l a r l y clearly in the p o w d e r transmission d a t a for I n A s in Fig. 23. T h e observed q u a n t u m oscillations in I n S b a n d I n A s p o w d e r s give excellent a g r e e m e n t w i t h single crystal d a t a a n d with t h e o r y [103]. Fig. 25 illustrates t h e b e h a v i o u r of m a g n e t o p l a s m a resonance in I n S b p o w d e r of lower carrier c o n c e n t r a t i o n , i n v e s t i g a t e d as a f u n c t i o n of t e m p e r a t u r e in b o t h t h e extrinsic a n d intrinsic ranges (see [104] a n d l i t e r a t u r e cited therein). T h e r e s o n a n t field B t is relatively c o n s t a n t while t h e sample r e m a i n s extrinsic, a n d in t h e intrinsic t e m p e r a t u r e range, as n increases, Br shifts to higher fields. This is q u a l i t a t i v e l y in a g r e e m e n t with t h e e x p e c t e d behaviour of t h e resonance field for a n isolated sphere, cf. (114). Q u a n t i t a t i v e l y , resonances in t h e p o w d e r d a t a of Fig. 25 occur a t fields a p p r o x i m a t e l y 4 0 % below t h e values p r e d i c t e d b y (114). T h i s is a result of p a r t i c l e particle i n t e r a c t i o n which is not f u l l y u n d e r s t o o d . A q u a n t i t a t i v e model of wave p r o p a g a t i o n in powders, including g r a i n - g r a i n interaction, is yet t o be developed. T h e s y s t e m a t i c , reproducible b e h a v i o u r observed in X G S p o w d e r s a n d its s e m i q u a n t i t a t i v e i n t e r p r e t a t i o n in t e r m s of t h e i n d e p e n d e n t sphere model a p p e a r encouraging in this c o n t e x t . W h e n u n d e r s t o o d m o r e fully, e x p e r i m e n t s with powders, in a d d i t i o n to t h e i r interest as such, will also offer a new a p p r o a c h to surface problems.

gjtrN

0

M 1 II (\ M 1 1 I I 1 -—CRA

c M y r —

b i i i i ii 11 i i i i i -12.0-S.0-i0 0 iO 80 no

0

[12

Fig. 24

04

OS

0.8

1.0 BIT)

B(T) -

Pig. 25

Pig. 24. Comparison of transmission of circularly polarized microwaves through fine and coarse powders obtained from the same crystal of InSb (n = 1.5 X 1017 cm" 3 , T = 4.2 K , / = 35.0 GHz), a) d < 40 ¡xm, b) 40 (xm < d < 250 ¡xm (after [101 to 103]) Fig. 25. Absorption of CRI-polarized microwaves in InSb powder in the immediate vicinity of magnetoplasma resonance at various temperatures (reD = 7.3 X 1013 cm - 3 , / = 35.0 GHz, t sb sb 2 mm, 250 ¡xm < d < 295 |im (after [104])

21

Microwave Effects in Karrow-Gap Semiconductors (II) 6. E l e c t r o n Spin R e s o n a n c e in Narrow-Gap Semiconductors 6.1 llelicon-excited

spin resonance

of paramagnetic

impurities

The study of electron paramagnetic resonance (EPR) of localized magnetic impurities in conducting media is greatly complicated by lack of sufficient penetration of the microwave signal into the sample interior. This difficulty can be circumvented, however, when the electrical parameters in the vicinity of E P R satisfy the conditions for helicon wave propagation, equation (51). Electromagnetic waves can then j)enetrate the medium in the form of circularly polarized helicons and can resonantly interact with the paramagnetic ions contained within the bulk. These conditions will be satisfied when the free charge carriers have a relatively small effective mass, so that their cyclotron frequency |co(.j exceeds the Larmor frequency |coL| of the localized magnetic moments. Furthermore, the free carrier collision timer must be long enough so that |ft)(.r| 1 in the E P R vicinity. Finally, the sense of helicon polarization must be compatible with the sense of precession of the localized magnetic moments. In the case of paramagnetic ions, the sense of precession is identical with the sense of polarization of helicons propagating in a gas of conduction electrons. These conditions are easily satisfied in the case of XGS containing paramagnetic impurities. It is readily shown that in the Faraday geometry, propagation in such a medium is described by [105, 106] KV2(1 + X±)Vi -

q± = ~

(125)

where are the transverse components of the dynamic paramagnetic susceptibility of the spin system in the coordinate frame (7). Magnetic susceptibilities % ± are conveniently expressed by the Bloch model (see, e.g. [107]) 1 — t(U) + L|), which for microwave frequencies and g = 2 corresponds to fields of the order of several kG. In this region narrow-gap semiconductors typically satisfy the helicon conditions (51) owing to their relatively high carrier concentration (n > 10 15 cm - 3 ), small effective mass, and high electron mobility. Assuming weak paramagnetism 1), we can readily show that kT). T h i s t y p i c a l l y requires fields in excess of 0.1 T a n d t h u s millimeter a n d submillimeter waves. U n f o r t u n a t e l y e x p e r i m e n t a l t e c h n i q u e s in this f r e q u e n c y r a n g e usually do n o t m e e t t h e exacting sensitivity requirem e n t s of t h e e x p e r i m e n t s . A convenient a n d v e r y sensitive t e c h n i q u e consists of illuminating t h e sample w i t h microwaves, b u t m o n i t o r i n g t h e sample resistivity [125], I n t h i s case t h e changes in c o n d u c t i v i t y are observed b y placing t h e s a m p l e in t h e vicinity of a n i n d u c t o r in a t a n k circuit, so t h a t c o n d u c t i v i t y changes caused b y absorption of microwaves m a n i f e s t themselves as changes of Q of t h e t a n k circuit. T h e sample is, in effect, a m i c r o w a v e d e t e c t o r w i t h response p e a k s at t h e a b s o r p t i o n maxim u m , i.e. a t o) = |wjJThe m e t h o d exploits specific p r o p e r t i e s of I n S b , i.e. t h e f a c t t h a t a relatively small a b s o r p t i o n of power h e a t s electrons sufficiently so t h a t their mobilities (and t h u s t h e c o n d u c t i v i t y ) show a d e t e c t a b l e change. I n t h e case of E S R , t h e power a b s o r b e d t r a n s f o r m s i n t o electron kinetic energy b y s p i n - o r b i t coupling. T h e f e a t u r e s necessary t o observe t h i s in I n S b are characteristic of o t h e r XGS, a n d t h e t e c h n i q u e should p r o v e a p o w e r f u l tool in t h e s t u d y of these m a t e r i a l s in general. 7. Concluding Remarks T h e a b o v e discussion enables us to conclude t h a t t h e microwave m a g n e t o r e f l e c t i o n a n d m a g n e t o t r a n s m i s s i o n , as well as t h e small-particle a b s o r p t i o n , p r o v i d e wide opport u n i t i e s for a n electrical c h a r a c t e r i z a t i o n of X G S . Such p a r a m e t e r s as t h e carrier c o n c e n t r a t i o n n, m o b i l i t y ¡1, effective m a s s TO*, lattice dielectric c o n s t a n t y.lt etc. can be d e t e r m i n e d b o t h in " s e m i - i n f i n i t e " ingots a n d in small disk-shaped or spherical samples. X o t e t h a t t h e t y p i c a l m i c r o w a v e s p e c t r o m e t e r shown in Fig. 3, o p e r a t i n g a t a single f r e q u e n c y of 35 G H z covers a wide r a n g e of p a r a m e t e r s (e.g. n = 10 11 t o

Micro-nave Effects in A'arrow-Gap Semiconductors (II)

25

10 18 c m - 3 ) a n d sample shapes. Moreover, microwave helicons can be used for determining t h e inhomogeneity in t h e distribution of carrier concentration in XGS [126, 127], F u r t h e r m o r e , MIIz magnetoplasma wave methods combined b o t h with h y d r o s t a t i c and uniaxial pressure techniques are successfully used for t h e s t u d y of b a n d - s t r u c t u r e effects in many-valley semiconductors [128], E f f e c t s arising from the non-ellipsoidal b a n d model were recently observed at microwave frequencies [49]. Another interesting trend in the microwave studies of XGS is t h e wave propagation in periodic multiplayer structures. Stop-bands a n d pass-bands were observed in I n S b structures [129] in accordance with theory [10]. These effects are of potential importance for the s t u d y of superlattices and growth inhomogeneities in XGS. Other excitation types are studied at microwaves, e.g. t h e magnetoplasma surface polariton excitation using a t t e n u a t e d total reflection [130, 131] or the cyclotron waves [132] which are of potential interest for the XGS diagnostics. Gelmont et al. [133] predicted a new t r a n s p a r e n c y window in zero-gap semiconductors a t frequencies below the plasma frequency in the ordinary Voigt configuration using the q u a n t u m approach. Recent results on the guided microwave interactions with b o u n d e d semiconductors are given in [134 to 137]. Microwave resonances in gyroelectric (semiconductor) spheroids were recently studied [138] employing t h e m a g n e t o s t a t i c theory developed for ferromagnetics. On t h e other hand, the ideas of microwave interaction with a semiconductor sphere were applied to the study of resonances in a ferromagnetic sphere [139]. The general problem of electromagnetic wave interaction with ferromagnetic semiconductor spheroids in a dc magnetic field still awaits solution. Finally, the submillimeter studies of XGS should be mentioned. Resonance submillimeter spectroscopy of H g i ^ C d ^ T e was used b y Guldner et al. [140]. Such imp o r t a n t p a r a m e t e r s as the L a n d e factor, the effective mass including polaron effects, etc. were measured in this region, and reviewed b y Lax [112], B u t t o n [141], L a n d w e h r [142], Gershenzon [143], a n d others. References (Part II) References [1] to [82] see phys. stat. sol. (a) 53, 11 (1979). [83] G. W . F O R D and S . A. W E R N E R , Phys. Rev. B 8, 3702 (1973). [84] H . C. VAN DE HÜLST, Light Scattering by Small Particles, Wiley, Xew York 1957. [85] M. KEEKER, The Scattering of Light, Academic Press, New York 1969. [86] F. L. GALEENER, Phys. Rev. Letters 22, 1292 (1969). [ 8 7 ] B . L A X and J . G . M A V R O I D E S , Solid State Phys. 1 1 , 2 6 1 ( 1 9 6 0 ) . [88] L . D . L A N D A U and E. M. L I F S H I T Z , Electrodynamics of Continuous Media, Addison-Wesley, Reading (Mass.) 1960 (p. 299), cf. also [46], [89] G. DRESSELHAUS, A . F . K I P , a n d C. KITTEL, P h y s . R e v . 1 0 0 , 6 1 8 (1955). [ 9 0 ] R . S. BRAZIS [91] T . A . EVANS

and J . and J .

appl. Phys. 4 8 , 2 9 2 7 ( 1 9 7 7 ) . Phys. Rev. B 8 , 1 4 6 1 ( 1 9 7 3 ) .

K . FURDYNA, J . K . FURDYNA,

[92] J . R . DIXON, u n p u b l i s h e d .

[93] J . K . FURDYNA, Postepy Fiz. 25, 67 (1974). [94] E. A. N E I F E L D and A. B. D A V Y D O V , Fiz. Tekh. Poluprov. 5, 1254 (1971) (Soviet Phys. Semicond. 5, 1107 (1971)). [95] J . K . F U R D Y N A and J . M Y C I E L S K I , Bull. Amer. Phys. Soc. Ser. I I , 1 9 , 246 (1974). [ 9 6 ] R. S . M A R K I E W I C Z , J . P. W O L F E , and C . I » . J E F F R I E S , Phys. Rev. Letters 3 2 , 1 3 5 7 ( 1 9 7 4 ) . [97] R. S. MABKIEWICZ, Phys. Rev. B 10, 1766 (1974). [98] J . K . F U R D Y N A and J . M Y C I E L S K I , Proc. Topical Conf. I I — VI Semiconductor Compounds, Jaszowiec 1973, I F - I ' A X , Warsaw 1974 (p. 178). [99] F . L . GALEENER, T . A . EVANS, a n d J . K . FURDYNA, P h y s . R e v . L e t t e r s 2 9 , 7 2 8 (1972).

[100] T. A. E V A N S , F . L. G A L E E N E R , a n d J . K. F U R D Y N A , Proc. I n t e r n a t . Conf. Physics of Semiconductors, Warsaw 1972, P W N , Warsaw, 1972 (p. 357).

26

R . S . BEAZIS, J . K . FUEDYNA, a n d J . K . POZELA

[101] J. K. FURDYNA and F. L. GALEENER, Proc. Internat. Conf. Semiconductor Physics, Moscow 1968, Vol. 2, Nauka, Leningrad 1968 (p. 870). [102] F. L. GALEENEE and J. K. FURDYNA, Appi. Phys. Letters 14, 163 (1969). [103] F. L. GALEENER a n d J . K . FURDYNA, P h y s . Rev. B 4, 1953 (1971).

[104] K. K. CHEN and J. K. FURDYNA, J. appi. Phys. 49, 3363 (1978). [105] R. T. HOLM and J, K. FURDYNA, Solid State Commun. 15, 1459 (1974). [106] R . T . HOLM a n d J . K . FURDYNA, P h y s . R e v . B 1 5 , 8 4 4 (1977).

[107] C. P. SLIGHTER, Principles of Magnetic Resonance, Harper and Row, New York 1963. [108] D. P. MULLIN, J. R. DIXON, JR., and J. K. FURDYNA, Proc. III. Internat. Conf. Phys. Narrow-Gap Semicond., Warsaw 1977, PVVN, Warsaw 1977. [109] K . LEIBLER. W . GIEIAT, K . CH^CINSKI, a n d Z . WILAMOWSKI, COLL. A m p è r e 1 6 , 9 8 7 ;

993

(1970). [110] K . LEIBLER, W . GIRIAT, Z . WILAMOWSKI, a n d R . IWANOWSKI, p h y s . s t a t . sol. (b) 4 7 , 4 0 5

(1971). [111] K . LEIBLER, W . GIRIAT, K . CH^CINSKI, Z. WILAMOWSKI, a n d R . IWANOWSKI, p h y s . s t a t . sol.

(b) 55, 447 (1973); Praca 1F-PAX 11 and 12, Warsaw (1971). [112] B. LAX, Proc. Internat. Symp. Microwave Diagnostics of Semiconductors, Porvoo (Finland) 1977, R. Paananen, Swedish Acad. Engineering Sciences in Finland, Rep. Mo. 31, Helsinki, 1 9 7 7 (p. 1). [113] L . M . ROTH, B . LAX, a n d S. ZWEEDLING, P h y s . R e v . 1 1 4 , 9 0 ( 1 9 5 9 ) .

[114] R . BOWERS a n d Y. YAFET, P h y s . R e v . 115, 1165 (1959).

[115] Y. YAFET, Phys. Rev. 115, 1172 (1959). [116] B . LAX, J . G. MAVROIDES, H . J . ZEIGER, a n d R . J . K E Y E S , P h y s . R e v . 1 2 2 , 3 1 ( 1 9 6 1 ) .

[117] W . ZAWADZKI, P h y s . L e t t e r s 4, 190 (1963). [118] W. ZAWADZKI, phys. s t a t . sol. 3, 1421 (1963). [119] P . KACMAN a n d W . ZAWADZKI, p h y s . s t a t . sol. (b) 4 7 , 6 2 9 (1971).

[120] G. BEMSKI, P h y s . Rev. L e t t e r s 4, 62 (1960).

[121] M. GUERON, Proc. Internat. Conf. Semiconductor Physics, Paris 1964, Dunod, Paris 1964 (p. 433). [122] R . A . ISAACSON, P h y s . R e v . L(I!(, 3 1 2 (1968).

[123] J. KONOPKA, Phys. Letters A 2«, 29 (1967). [124] D. KAPLAN and J . KONOPKA, Proc. Internat. Conf. Semiconductor Physics, Moscow 1968, Nauka, Leningrad 1968 (p. 1146). [125] J . KONOPKA, P h y s . R e v . L e t t e r s 24, 666 (1970). [126] A. LAURINAYICHIUS a n d Y u . POZHELA, P r i b . i Tekh. E k s p e r . 4, 241 (1976).

[127] A. LAURINAYICHIUS, YU. POZHELA, and A. VITKUS, Proc. Internat. Symp. Microwave Diagnostics of Semiconductors, Porvoo (Finland) 1977, R. Paananen, Swedish Acad. Engineering Sciences in Finland, Rep. No. 31, Helsinki 1977 (p. 250). [128] Y u . K . POZHELA a n d R . B. TOLUTIS, i n : E l e k t r o n y v P o l u p r o v o d n i k a k h , Vol. I , E d . Y u . K .

POZHELA, Mintis, Vilnius 1978. [129] R . S. BRAZIS, A . S. MIRONAS, a n d Y u . K . POZHELA, L i t o v s k i i F i z . S b o r n i k , 1 4 , 9 5 (1974)

(Soviet Phys. - Collection 14, 54 (1974)). [130] V. AMBRAZEVIÒIENÉ a n d R. BRAZIS, Solid State C o m m u n . 18, 415 (1975).

[131] V. AMBEAZEVIFIENE and R, BEAZIS, Fiz. Tekh. Polpurov. 12, 1114 (1978) (Soviet Phys.

-

S e m i c o n d . 12, 661 (1978)).

[132] R. HEREMANN, G. OELGAET, and R. STEGMANN, Wiss. Z. Humboldt-Univ. Berlin, Math.naturwiss. R., 25, 3 (1976). [133] B . L . GELMONT, V . I . IVANOV-OMSKII, a n d E . X . UKEAINTSEV, p h y s . s t a t . sol. (b) 87, 3 8 1

(1978). [134] Y u . G . AEAPOV a n d A . B . DAVYDOV, D e f e k t o s k o p i y a 1 1 , 6 3 (1978). [135] A . B . DAVYDOV, V . A . ZAKHAEOV, a n d I . E . PODCHINYONOV, F i z . t v e r d . T e l a 19, 1 6 7 6 (1977)

(Soviet Phys. - Solid State 19, 978 (1977)). [136] D. A. USANOV, V. B. FEKLISTOV, and A. Yu. DEEYAGIN, II. Vsesoyuz. Symp. Millimetrovym i Submillimetrovym Volnam, Kharkov 1978, Vol. I, Inst. Radiophys. Electronics, Acad. Sci. USSR (p. 139). [137] L . 1. KATS, V . V . POPOV, a n d R , M . REVZIN, R a d i o t e k h n i k a i E l e k t r o n i k a 2 2 , 1 1 0 7 ( 1 9 7 7 ) .

Microwave Effects in Narrow-Gap Semiconductors (II)

27

[ 1 3 8 ] A. B . DAVYDOV, B . B . PONIKAROV, a n d G. L . SHTRAPENIN, F i z . T e k h . P o l u p r o v .

12,

866

(1978) (Soviet Phys. — Semicond. 12, 511 (1978)). [139] R. S. BRAZIS and J. K. FURDYNA, Phys. Rev. B IB, 3273 (1977). [ 1 4 0 ] Y . GULUNER, C. RIGAUX, A . MYCIELSKI, a n d Y . COUDER, p h y s . s t a t . sol. (b) 81, 6 1 5 ( 1 9 7 7 ) ; 8 2 , 149 (1977).

[141] K. J. BUTTON, Lectures Xotes Internat. Conf. Application of High Magnetic Fields in Semiconductor Physics, Würzburg 1976, Phys. Inst. Univ. Würzburg (p. 326). [142] G. LANDWEHR, II. Internat. Conf. and Winter School on Submillimeter Waves and Their Applications, 1976, Conf. Digest, Ed. S. PERKOWITZ, Inst. Electrical and Electronics Engineers, Inc. 1976 (p. 146); cf. also J . Opt. Soc. Amer. 67, 922 (1977). [143] E. M. GERSHENZON, Proc. Internat. Symp. Microwave Diagnostics of Semiconductors, Porvoo (Finland) 1977, R. Paananen, Swedish Acad. Engineering Sciences in Finland, Rep. Xo. 31, Helsinki 1977 (p. 242). (Received

Note added

in

October 19,

1978)

proof

A new theory of scattering of electromagnetic waves by a gyrotropic sphere [1] has appeared during the preparation of this paper. The theory is a generalization of the classic Mie formulation for dielectric spheres, and has direct relevance to the interaction of electromagnetic waves with spherical XGS samples. A series of microwave applications of the theory, such as helicon-like dimensional resonances in semiconductors, cyclotron resonances, and Alfven resonances in electron-hole droplets, are presented [2], Furthermore, new experimental results on the propagation of cyclotron waves in the electronhole system of bismuth at frequencies up to 300 GHz have been published [3]. It is also necessary to mention earlier work on the theoretical [4] and experimental [5] evidence for the existence of a new type of wave propagation in metals, semimetals and semiconductors: the slow electromagnetic surface waves. These waves are of potential interest as a diagnostic tool for XGS. Finally, we would like to mention a recent book [6] devoted to growth, structure and transport properties of XGS. We hope that the reader will bear with us in our dual endeavor to give as complete a presentation as possible while confining the volume of the paper to reasonable limits, in consequence of which we had to leave unmentioned many other sources in this rapidly developing field.

[1] S. W. FORD and S. A. WERNER, Phys. Rev. B 18, 6752 (1978). [2] J. R, DIXON, JR. and J . K. FURDYNA, Phys. Rev. B 18, 6770 (1978). [3] W . BRAUNE, J . LEBECH a n d K . SAERMARK, p h y s . s t a t . sol. (b) 8 7 , 5 2 7 ( 1 9 7 8 ) .

[4] S. I. KHANKINA and V. M. YAKOVENKO, Fiz. tverd. Tela 9, 2943 (1967) (Soviet Phys. - Solid State 9, 2313 (1968)). [5] V. I. BAIBAKOV and V. X. DATSKO, Fiz. tverd. Tela 15, 1616 (1973) (Soviet Phys. - Solid S t a t e 15, 1084 (1973)).

[6] D. R. LOVETT, Semimetals and Xarrow-Bandgap Semiconductors, Pion Limited, London 1977.

Original

Papers

phys. s t a t . sol. (a) 54, 29 (1979) S u b j e c t classification: 10.2; 22.1.2 Institute Institute

of Solid State Physics, Chernogolovka1) (a) and of Crystallography, Moscow2) (b) Academy of Sciences of the

USSR

Shape of Extinction Fringes and Determination of the Burgers Vector in Single-Crystal Interferometric Techniques By E . V . SUVOEOV ( a ) , 0 . S . GORELIK ( a ) , V . M . K A C A X E R

(b),

a n d Y . L . INDENBOM (b)

An analysis of t h e shape of e x t i n c t i o n contours on section t o p o g r a p h s of dislocations in singular position, for high a n d low d i f f r a c t i v e power of a dislocation is given. A comparison of e x p e r i m e n t a l d a t a w i t h t h e conclusions of t h e t h e o r y enables t o v e r i f y q u a n t i t a t i v e l y t h e earlier developed concepts of f o r m a t i o n of t h e d y n a m i c image of dislocations. Methods are suggested t o d e t e r m i n e t h e Burgers vectors t o a n accuracy of 6 % , based on analyzing t h e shape of interference fringes. ripOBO,"[HTCH a i i a a H 3 (JlOpMBI 3KCTIHIKUM0HIIMX KOlITypOB Iia CeKHHOIIHhlX TOIIOrpaMMaX g n c a o i ; a n n i l B OCOOOM RIOJIOIKCHHII n p i i M a j i t i x H 6 o . i i . n i n x SHANEIIMFIX ,T;n({>paKUMonuon MomnocTH anc,TioKai;nii. C p a B i i e i m e 3KcnepiiMeiiTaJii>in,ix j r a m i b i x c BLiBtuaMH T e o p i i n ;;acT B03M0/Kii0CTb KojiH^ecTBeimoil npoBepKii paaBHTWx p a n e e npeacTaBJieiiHii o (j)opMiipoBamin a n n a M H i e c K o r o H3o6pa5Keimji .iiic.ioHauHi't. r i p e j j i a r a i O T C H ociiOBamiBie n a a i i a a H 3 e (¡lopMjbi n o j i o c w e T o a t i o n p e a e n e i i H H BCJIHHHIII.I BT'KTopa B i o p r e p c a , o o a a ;iaiOLHHC TOy = /?i>f/2

Pig. 6. The dependence of the ratio of free to trapped charge 6 and of the concentration of thermally generated charges n 0 on the thickness of insulating layers calculated assuming SCL conduction through the surface barriers in (001) V„0 5 with A1 electrodes (cf. Fig. 4)

10 6

10' 5

10 *

io 3 10 2 slcm)—-

42

S. Lanyi Fig. 7. lg a vs. E1I2 dependence for (001) V„05 with A1 electrodes (of. Fig. 4)

7

[23]. A plot of the logarithm of the conductance versus E l l' 2 is seen in F i g . 7 for a coriented sample 0.4 cm thick with A1 electrodes. A least-squares fit yields the coefficient /? by a factor of 76 greater than expression (3), which would mean t h a t t h e effective thickness of the high-resistivity surface layer is 1.3 ¡j.m for the P F effect or 0.34 u.m for the anomalous P F effect. T h e capacitance of a capacitor with such a plate spacing and F-:R = 5 is approximately 3.3 X 10~ 9 and 1.3 X 10~ 8 F / c m 2 , respectively. However, the space charge should reside inside the surface barriers and the plate spacing of an equivalent capacitor should be equal to t h e distance from electrode to the centre of the space charge, if it is sufficiently mobile, or to some point, up to which it is able to respond to the ac signal of a given frequency. This can give a few times greater capacitance in reasonable agreement with 2C c estimated from t h e frequency dependence of the conductance. T h e concept of field-enhanced emission in t h e surface barriers is consistent with the idea of the bulk conductance b y small polarons, as the interaction of t h e polaron with the ionic lattice is Coulombic. T h e validity of expression (2) requires a small overlap of neighbouring potential wells [24], i.e. large separation of e m p t y sites. T h i s can arise by such a bending of the polaron band near the electrodes t h a t relatively few states in the band tail or in the forbidden gap remain available to thermal excitation. T h u s we can draw the following picture of the nature of the surface barriers: E l e c trons injected into t h e crystal from a low work function metal cause a downward band bending. Since the conduction in t h e crystal takes place in a narrow probably polaron-like band [5, 7, 12, 13], insulating layers would have to arise near the electrodes. T h e transport through these layers m a y be explained either as space charge limited conduction in presence of traps or b y field enhanced hopping between Coulombic sites. T h e thickness of the barriers deduced from the interface capacitance should reach tenths of ¡j.m. T h i s size supports the latter possibility since t h e conduction should occur outside t h e conductivity band, hence S C L conduction is not probable to a long distance as far as the effective mass of the electrons is as large as in the bulk of the crystal. However, a coexistence of both effects might be possible. Acknowledgements T h e author t h a n k s D r . G. F . Dobrzhanskii for supplying the V 2 0 5 single crystals and Dr. G. Vlasak for preparation of electrodes. References [1] S. Kachi, T. Takada, and K. Kosuge, J . Phys. Soc. Japan 18, 1839 (1963). [2] T. S. Zolian and A. R. R e g e l , Fiz. tverd. Tela 6, 1520 (1964). [3] I. B. P a t r i n a and V. A. Ioefe, Fiz. tverd. Tela 6, 3227 (1964).

Contact-Limited Conduction in V 2 0 5 Single Crystals

43

[4] T . ALLERSMA, R . HAKIM, T . N . KENNEDY, a n d J . D . MACKENZIE, J . c h e m . P h y s . 4 6 , 154

(1967). [5] D . S. VOLZHENSKII a n d M. V . PASHKOVSKII, F i z . t v e r d . T e l a 11, 1 1 6 8 (1969).

[6] L. D. KIST.OVSKII and A. A. ABDULLAEV, Optika i Spectroskopiya 29, 737 (1970). [7] P. NAGELS and M. DENAYER, Proc. 10th Internat. Conf. Phys. Semiconductors, Cambridge 1970 (p. 321). [8] J. H. PERLSTEIN, J. Solid State Chem. 3, 217 (1971). [9] A. B. SCOTT, J. C. MCCULLOCH, and K. M. MAR, Proc. 2nd Internat. Conf. Conduction in Low Mobility Materials, Eilat 1971 (p. 107). [10] J . HAEMERS, E . BAETENS, a n d J . VENNIK, p h y s . s t a t . sol. (a) 2 0 , 381 (1973). [11] X . KENNY, C. R . KANNEWURF, a n d D . H . WHITMORE, J . P h y s . C h e m . S o l i d s 2 7 , 1 2 3 7 ( 1 9 6 6 ) .

[12] A. A. VINOGRADOV and A. I. SHELIKH, Fiz. tverd. Tela 13, 3310 (1971). [13] V. A . IOFFE a n d I . B . PATRINA, p h y s . s t a t . sol. 4 0 , 3 8 9 (1970).

[14] I. M. CHERNENKO and A. I. Ivox, Fiz. tverd. Tela 16, 2130 (1974). [15] A . A . ABDULLAEV, L . M . BELIAEV, I . V . VINAROV, G. F . DOBRZHANSKII, a n d LEVICH, K r i s t a l l o g r a f i y a 1 4 , 1095 (1969).

[16] [17] [18] [19] [20]

R . G. YANKE-

S. LÄNYI, J. Phys. Chem. Solids 36, 775 (1975). S. LAXYI, prepared for publication. R. MEAUDRE and M. MEAUDRE, phys. stat. sol. (a) 37, 633 (1976). C. A. MEAD, Solid State Electronics 9, 1023 (1966). R. H. WILLIAMS, V. MONTGOMERY, and R. R. VARMA, J. Phys. C l l , L735 (1978).

[21] A . ROSE, P h y s . R e v . 97, 1538 (1955).

[22] M. A. LAMPERT, R e p . Progr. P h y s . 27, 329 (1964),

[23] J. G. SIMMONS, Phys. Rev. 155, 657 (1967). [24] A. E. OWEN and J. M. ROBERTSON, I E E E Trans. Electron Devices 20, 105 (1973). (Received

February

23,

1979)

F . N.

CHUKHOVSKII

and

A.

M.

AEUSTAMYAN:

Theory of Dislocation I m a g e Contrast

45

phys. s t a t . sol. (a) 54, 45 (1979) S u b j e c t classification: 1.4 a n d 10 Institute of Crystallography, and Erevan State University

Academy (b)

of Sciences of the USSR,

Moscow

(a)

A Dynamical Analytical Theory of the Dislocation Image Contrast in Electron Transmission Microscopy By F. X.

CHUKHOVSKII

(a) a n d

A. M. ARUSTAMYAN

(b)

A dynamical t h e o r y of dislocation contrast in electron t r a n s m i s s i o n microscopy (DCETM) is discussed. A d i a g r a m m e t h o d for t h e solution of t h e H o w i e - W h e l a n e q u a t i o n s in t h e strongly d e f o r m e d region of a crystal close t o t h e dislocation core is developed. T h e solution is based on t h e general scattering m a t r i x (S-matrix) theory. T h e complete analytical description of t h e dislocation image p i c t u r e as a whole is proposed. I t is shown t h a t in t h e regions of slight a n d strong d e f o r m a t i o n s t h e quasi-classical (eikonal) a n d t h e anti-classical a p p r o x i m a t i o n s , respectively, are in good accordance w i t h t h e well-known numerical results. T h e ranges of t h e applicability of each analytical a p p r o a c h are established a n d t h e general p r o p e r t i e s of t h e dislocation image profiles are explained. E i n e d y n a m i s c h e Theorie des V e r s e t z u n g s k o n t r a s t s im E l e k t r o n e n - T r a n s m i s s i o n s m i k r o s k o p (DCETM) wird d i s k u t i e r t . Dazu wird eine D i a g r a m m e t h o d e f ü r die Lösung der H o w i e - W h e l a n Gleichungen in dem s t a r k d e f o r m i e r t e n Bereich eines Kristalls in der N ä h e des Versetzungskerns entwickelt. Die Lösung basiert auf d e r allgemeinen S t r e u m a t r i x (S-Matrix)-Theorie. E i n e vollständige analytische Beschreibung des Versetzungsabbildes als ganzes wird vorgeschlagen. E s wird gezeigt, d a ß in d e n Gebieten leichter u n d starker D e f o r m a t i o n e n die quasiklassische (Eikonal) bzw. die antiklassische N ä h e r u n g sich in guter Ü b e r e i n s t i m m u n g m i t d e n g u t b e k a n n t e n n u m e r i schen Ergebnissen b e f i n d e n . Die Anwendungsbereiche jeder a n a l y t i s c h e n N ä h e r u n g w e r d e n festgestellt u n d die allgemeinen E i g e n s c h a f t e n der Versetzungsbildprofile e r k l ä r t .

1. Introduction A dynamical theory of diffraction provides a q u a n t i t a t i v e guide t o t h e interpretation of transmission electron micrographs of distorted crystals. I t is widely used for t h e discussion of diffraction contrast effects due to strains a r o u n d defects of a crystal lattice. A n u m b e r of theoretical papers [1 to 5] are devoted to computer simulation of single defect images to solve t h e diffraction contrast problem b y numerical methods. However, so f a r t h e straightforward transition f r o m t h e dynamical equations to numerical results does not permit a comparable physical insight t o describe analytically the features of the defect image contrast, in particular t h e contrast close t o the dislocation core. The analytical approach to the problem in question was firstly proposed b y Wilkens [6] (see also [7]) using the geometric optics (eikonal) approximation (so-called "modified Bloch w a v e " theory). The most important conclusion obtained in [7] is t h a t t h e a s y m m e t r y of the long-range contrast on two sides of the dislocation images is explained b y the phase contrast due to the quasi-classical electron scattering in slightly deformed regions of a crystal outside t h e dislocation core. I n the present paper the consequent dynamical t h e o r y of diffraction contrast in electron transmission microscopy (DCETM) is proposed on t h e basis of t h e general

46

F . X . CHUKHOVSKII a n d A . M .

ARUSTAMYAN

formalism of the scattering matrix (¿^-matrix) [8 to 10]. I t opens the possibility to investigate the physically different approaches, the quasi-classical and anti-classical ones, for the description of the electron propagation through slightly and strongly deformed regions of a crystal, respectively, and to determine the applicability ranges of each theoretical approximation. The principal conclusion is that the anti-classical approach describes D C E T M taking into account the interband scattering involved, whereas the quasi-classical one quantitatively explains the "background contrast" of the image profiles outside the dislocation core, as it must be from physical speculations. 2. S-Matrix Formalism The dynamical scattering of an electron in the imperfect absorbing crystal is described by the Howie-Whelan equations [2] (assuming a centre of symmetry which does not matter for the problem under consideration) d0 m , — - = — 0 0n Hz £, 0 uz dz

:

•>7¡

'P,j exp (2jtisz -f 2TcigR) , (2.1)

171 = J- 0g + —

exP

(

—

—

2TtigR) ,

where the atomic displacements from the perfect crystal positions at depth z in the crystal are given by R(z). Hereafter all notations are taken from [3]. Substitution of 0O = o exp (insz + wrz/|0), , 225 (1978).

R . K . KLABES,

M . M . ZARIPOV, R . M . BAYAZITOV,

U. KREISSIG,

and M. F .

and

GALYATU-

[ 5 ] D . H . AUSTON, C. M . SURKO, T . N . C. V E N K A T E S A N , R . E . S H U S H E R , a n d J . A . GOLOVCHENKO,

Appl. Phys. Letters 3», 437 (1978).

[6]

H . - D . G E I L E R , G . GÖTZ, K . - D . K L I N G E , a n d N . T R I E M , p h y s .

s t a t . sol. (a) 4 1 , K 1 7 1

[ 7 ] P . B A E R I , S . U . CAMPISANO, G . F O T I , a n d E . R I M I N I , A p p l . P h y s . L e t t e r s 3 2 , 1 3 7

(1977).

(1978).

[8] A . V . DVURECHENSKII, G. A . KACHURIN, a n d A . K H . A n t o n e n k o , R a d i a t . E f f . 3 7 , 1 7 9

(1978).

[9] G. K . CELLER, J . M. POATE, and L. C. KIMERLINGH, Appl. Phys. Letters 32, 464 (1978). [ 1 0 ] G . FOTI, E . RIMINI, a n d S . U . CAMPISANO, p h y s . s t a t . sol. (a) 4 7 , 5 3 3 ( 1 9 7 8 ) . [ 1 1 ] P . B A E R I , S . U . CAMPISANO, G . F O T I , a n d E . R I M I N I . P h y s . R e v . L e t t e r s 4 1 , 1 2 4 6

[ 1 2 ] M . H . BKODSKY,

(1978).

R . S . T I T L E , K . W E I S E R , a n d G . D. P E T T I T , P h y s . R e v . B 1 , 2 6 3 2

[13] J . S. WILLIAMS, Nuclear Instrum. and Methods 126, 205 (1975).

[ 1 4 ] M . M . ZARIPOV, 1. B . KHAIBULLIN, a n d E . 1. SHTYRKOV, S o v i e t P h y s . -

(1976).

(1970).

Uspekhi 19,

1032

[ 1 5 ] P . BAERI, S . L T . CAMPISANO, G. FOTI, a n d E . RIMINI, J . a p p l . P h y s . 5 0 , 7 8 8 ( 1 9 7 9 ) . [ 1 6 ] T . E . S E I D E L , G . A . P A S T E U R , a n d J . C. C. TSAI, A p p l . P h y s . L e t t e r s 2 9 , 6 4 8

(1976).

[17] A. GOLDSMITH, Handbook of Thermophysical Properties of Solid Materials, Vol. I , Mac MilIan Co., New York 1961. [18] C. Y . Ho, R . W. POWELL, and P E . SILEY, Reference Data on the Thermal Conductivity of the Elements, National Standard Reference Data Series, N. B . S., 1969. [19] B. SMITH, Ion Implantation Range Data, Learned Information Ltd., Oxford 1977. [20] B . W. ARDEN and K . N. ASTILL, Numerical Algorithms — Origins and Applications, AddisonWesley, London 1970 (p. 280). [21] C. S. FULLER and J . A. DITZEENBERGER, J . appl. Phys. 27, 544 (1956). [22] J . 0 . MCCALDIN, J . Vacuum Sei. Technol. 11, 910 (1974). (Received

February

26,

1979)

J . G. KLIAVA et al.: R P E de Mn 2+ dans les polycristaux de basse symétrie

61

phys. stat. sol. (a) 54, 61 (1979) Subject classification: 19; 22.8.1 Institut de Physique des Solides, Université d'Etat de Lettonie,

RPE de Mn

2+

Riga

dans les polycristaux de basse symétrie

Le métapliosphate de magnésium

Par J . G . KLIAVA, Z. A . KONSTANTS, J . J . PITRANS e t A . I . DIMANTE On a analysé les spectres de R P E de Mn 2+ dans les polycristaux pour le hamiltonien de spin X , = g(¡IIS + D (s* - ~ (8 + l ) j

E{Sl - £$) H- ASIpour

|2)| < g^H,\E\ < gpH,\A\
' =

ON

(N\V* —

(12) NFV*)

K0,

and in the latter one

A« 3 T = i + T , (13) where R^ and P™S are the spontaneous values of these variables in the domain-like structure (the domain walls are assumed to be very thin: The linear electrooptic effect is allowed by symmetry in the phase I I I only (for details see [15]). In the phases I and I I only the quadratic electrooptic effect can be

An Incommensurate-Commensurate Phase Transition in (NH 4 ) 2 BeF 4 Crystals

103

measured. Using relation (9) we get from (10) in t h e t e m p e r a t u r e range close to T (14) a n d in phase I Z =

R

(15)

(Xv) 2 •

4. Survey of Experimental Results Our published experimental d a t a on the birefringence An 31 [15] a n d A n i s t r a t o v ' s ones on the quadratic electrooptic effect Z [14] will be used for a discussion. As the spontaneous birefringence Aii|i was not explicitly published in [15], it is plotted in Fig. 1. These d a t a have been obtained b y subtracting the linearly extrapolated t e m p e r a t u r e dependence of An 31 in phase I from d a t a for phases I I a n d I I I . T h e mark a t Tc indicates the t e m p e r a t u r e of the ferroelectric P T obtained f r o m our electric and electooptic measurements. Tl designates the t e m p e r a t u r e of t h e IC phase transition according to the specific heat measurements [11]. Our t e m p e r a t u r e scale is shifted b y 2 K u p w a r d s in comparison with other works [1 to 3], which is within t h e accuracy of thermometer calibration. The measurements were carried out on cooling. I t can be seen on Fig. 1 t h a t Awfi is (within experimental error) a continuous function of t e m p e r a t u r e in the vicinity of Tc; it agrees well with results of other a u t h o r s [13, 14, 16]. On t h e other hand, the first derivative dAwli/dT 7 is discontinuous a t Tc. T h e tail a t t e m p e r a t u r e s T Tl is a t t r i b u t a b l e p a r t l y to I C phase f l u c t u a t i o n s a n d p a r t l y to a blurring due to mechanical local strains. As for the q u a d r a t i c electrooptic effect, A n i s t r a t o v found t h a t t h e coefficient Z has an anomaly near Tc. R was found t o be practically independent of t e m p e r a t u r e in t h e phases I a n d I I . 5. Discussion and Conclusions E q u a t i o n s (11), (12), a n d (13) form the basis for t h e i n t e r p r e t a t i o n of the spontaneous values Aw3i(T) in the phases I I a n d I I I ; t h e y contain t h e electrooptic contribution, t h e elastooptic one, a n d the contribution of T P ' s . Let us first deal with the phase I I I for which sufficient d a t a on spontaneous quantities are available. The electrooptic a n d elastooptic contributions can be estimated from coefficients Rjj a n d py, which are measurable in phase I , and spontaneous values Ps and w'f; the contribution of T P ' s can be found b y subtracting these f r o m t h e measured Aw-31. I t was shown in [14, 15] t h a t t h e electrooptic contribution to Awîji is very small, only 0.5%, a n d the elastooptic one could account for t h e whole spontaneous optical effect. U n f o r t u n a t e l y , there are no d a t a on t h e sign o f p i j ; therefore two situations can occur: (i) The elastooptic contribution has t h e same sign as Aw.31; t h e n t h e P T contribution is small, (ii) The elastooptic contribution has t h e opposite sign to A?4'f ; then T P ' s play a prevailing role in the origin of t h e spontaneous birefringence. This problem can be solved b y means of electrooptic measurements a n d will be discussed f u r t h e r .

0I W

L 175

L 180

m ÏÏK1

190

•

Fig. 1. Temperature dependence of the spontaneous birefringence A»| x in the vicinity of both phase transitions

C. KOXÀK: A n I n c o m m e n s u r a t e - C o m m e n s u r a t e P h a s e Transition in (NH 4 ) 2 BeF 4

104

L e t u s discuss t h e I C phase. S t a r t i n g f r o m (10) a n d t a k i n g into a c c o u n t t h e results (3), (4) a n d (6), (7) one arrives at t h e conclusion t h a t t h e relative c o n t r i b u t i o n s t o A»™ close to T c a r e t h e same as in p h a s e I I I . T h e e x p e r i m e n t a l f a c t t h a t A M J I is a c o n t i n u o u s f u n c t i o n of t e m p e r a t u r e at Tc allows us to d r a w t h e conclusion t h a t |i>s | = |rj"| a t Tc. I n o t h e r words, t h e I C - C p h a s e t r a n s i t i o n in A F B should be n e a r l y of second order. E q u a t i o n s (14) a n d (15) can be used to discuss t h e t e m p e r a t u r e d e p e n d e n c e of Z in p h a s e I a n d I I , respectively. T h e e x p e r i m e n t a l result Z = Ry* in b o t h p h a s e s leads to t h e conclusion t h a t co' == 0 in (14). T h u s t h e case (i) m e n t i o n e d a b o v e could b e t t e r explain all optical d a t a k n o w n u p to now. As for t h e a n o m a l y of dAw^/dT 1 a t Tc, it can be q u a l i t a t i v e l y explained b y generalized P i p p a r d relations [19], considering t h e specific heat d a t a showing a n a n o m a l y a t Tc [1], Since t h e coefficients of t h e r m a l expansion a n d t h u s t h e first d e r i v a t i v e s of r e f r a c t i v e indices should b e h a v e at P T like t h e specific h e a t , t h e a n o m a l y m e n t i o n e d a b o v e p r e s e n t s no m y s t e r y . I n t h e a u t h o r ' s opinion t h e most interesting result is t h e e q u a l i t y |)"s | = |r,™| at Tv. U n f o r t u n a t e l y we h a v e no complete set of optical e x p e r i m e n t a l d a t a for o t h e r I C - C p h a s e t r a n s i t i o n s t o look w h e t h e r t h i s result is accidental or not. Nevertheless, measu r e m e n t s of t h e s p o n t a n e o u s d e f o r m a t i o n can be discussed in t h e same w a y as t h e s p o n t a n e o u s optical d a t a using e q u a t i o n s (4) a n d (7). V e r y interesting e x p e r i m e n t a l d a t a on s p o n t a n e o u s d e f o r m a t i o n were published for K 2 S e 0 4 crystals [20]. Similar as in our m e a s u r e m e n t s , usa was f o u n d to be a c o n t i n u o u s f u n c t i o n of T at Tc, which gives t h e same e q u a l i t y for K 2 S e 0 4 . Acknoicledgements

T h e a u t h o r wishes t o express his t h a n k s t o D r . V. D v o r a k for s t i m u l a t e d discussions, D r . B . Brezina for p r o v i d i n g him with crystals of A F B , a n d D r . J . P e t z e l t for critically reading the manuscript. References [ 1 ] B . A . STKUKOV, T . L . SKOMOROKHOVA, V . A . KOPTSIK, A . A . BOIKO, a n d A . X . JZRAII. I: N KO, K r i s t a l l o g r a f i y a 18, 143 (1973). [ 2 ] R . P E P I X S K Y a n d J . JOXA, P h y s . R e v . 1 0 5 , 3 4 4 ( 1 9 5 7 ) . [ 3 ] S . H I S H I N O , K . VEDAM, J . OKAYA, a n d R . P E P I X S K Y , P h y s . R e v . 1 1 2 , 4 0 5 ( 1 9 5 8 ) . [ 4 ] Y . MAKITA a n d Y . YAMAUCHI, J . P h y s . S o c . J a p a n 3 7 , 1 4 7 0 ( 1 9 7 4 ) . [5] M . IIZUMI a n d K . G E S I , S o l i d S t a t e C o m m u n . 2 2 , 3 7 ( 1 9 7 7 ) . [6] A . P . LEVAXYUK a n d D . G . SAXXIKOV, Z h . e k s p e r . t e o r . F i z . 5 5 , 2 5 6 ( 1 9 6 8 ) . [ 7 ] A . P . LEVAXYUK a n d D . G . SAXXIKOV, F i z . t v e r d . T e l a 1 8 , 4 2 3 ( 1 9 7 6 ) .

[8] [9] [10] [11] [12]

L. D. LANDAU a n d E . M. LIFSHXTS, Statistical Physics, P e r g a m o n Press, 1959 (p. 444). V. DVORAK a n d Y. ISHIBASHI, J . P h y s . Soc. J a p a n 45, 775 (1978). Y. ISHIBASHI a n d V. DVORAK, J . P h y s . Soc. J a p a n 44, 32 (1978). H . SHIBA a n d Y. ISHIBASHI, J . P h y s . Soc. J a p a n 44, 1592 (1978). J . D. AXE, Proc. Conf. N e u t r o n Scattering, G a t l i n b u r g (Tenn.) 1976.

[ 1 3 ] B . A . STRUKOV, K r i s t a l l o g r a f i y a , 6 3 5 ( 1 9 6 1 ) .

[14] A. T. AXISTRATOV, Rev. Phys" appl. 7, 77 (1972). [15] [16] [17] [18]

C. A. C. J.

KONAK a n d J . MATRAS, C z e c h . J . P h y s . B2»i, 5 7 7 ( 1 9 7 6 ) . T . AXISTRATOV a n d S . V . MELXIKOVA. K r i s t a l l o g r a f i y a 1 8 , 1 2 8 9 ( 1 9 7 3 ) . KONAK, C z e c h . J . P h y s . B 2 5 , 1 0 7 3 ( 1 9 7 5 ) . FOUSEK a n d M . GLOGAROVA, S o l i d S t a t e C o m m u n . 1 " , 9 7 ( 1 9 7 5 ) .

[19] V. JAXOVEC, IV. l n t e r n a t . f l e e t i n g Ferroelectricity, Leningrad, September 1977 (to be published in Ferroelectrics (1979)). [ 2 0 ] SH. SIOZAKI, A . SAWADA, Y . ISHIBASHI, a n d Y . TAKAGI, J . P h y s . S o c . J a p a n 4 2 , 3 5 3 ( 1 9 7 7 ) .

(Received

March 22,

1979)

E . KRATOCHVILOVA et, al.: Magnetocrystalline Anisotropy of N d - S u b s t i t u t e d Y I G

105

phys. s t a t . sol. (a) 54, 105 (1979) Subject classification: 18.3; 22.8.2 Institute

of Physics,

Czechoslovak

Academy

of Sciences,

Prague

Magnetocrystalline Anisotropy of Neodymium-Substituted Yttrium-Iron Garnet By

E. KRATOcnviLOVA,

V . ROSKOVEC 1 ),

and Y.

XKKYASIJ,

An experimental s t u d y of t h e low-temperature magnetocrystalline anisotropy inNdo.:u:sY2.()5rFu 5 0 12 is presented. The m e t h o d chosen a t 4.2 K is t h e d e t e r m i n a t i o n of a n i s o t r o p y c o n s t a n t s f r o m magn e t i z a t i o n curves, s u p p l e m e n t e d b y t o r q u e m e a s u r e m e n t s at t e m p e r a t u r e s u p to 77 K . T h e resulting values of t h e anisotropy c o n s t a n t s are compatible w i t h t h e l i t e r a t u r e d a t a o b t a i n e d b y different m e t h o d s a n d on samples w i t h widely differing n e o d y m i u m concentration, confirming t h u s t h e essentially a d d i t i v e c h a r a c t e r of t h e r a r e - e a r t h s c o n t r i b u t i o n . Special a t t e n t i o n is paid t o t h e a s s u m p t i o n s used w h e n evaluating t h e e x p e r i m e n t a l d a t a . E b i n a aKciiepnMeiiTani.no n c c a e ; i o B a i i a MarHHTOKpncTaaanueci;aH aiiM30TponMH r p a n a T a Xdo.3MY2,85iFc s O i a . II])h 4 , 2 lv f)bi.:iH KoiicTaHTbi aiiH3t>Tpoiinn o n p e ^ e a e i n . i iia KpiiBbix iiaMariin l iHBaiiHH, i i p n T e M n e p a x y p a x a o 77 K BpamaTeai.iibiM MeTo;joM. P e 3 y a b T a T b i coraacaioTCH c aaniibiMH B j i H T e p a T y p e , iio.iyieiniMMii a p y i i i M i i MeTo.uiMii n a ofipa.'iuax B LUHpoHOM a n a n a 3 0 i i e c o a e p w a n i i n l i e o a i o i a . TaniiM o 6 p a 3 o \ i noaTBepi+uaeTCH aaiiTHBiibiii x a p a K T e p B H J i a j a pe^KoseMeai.iibix HOIIOB k aiiH30Tp0iiiiH. B JHCKVCCHM p a 3 o 6 p a i i b i iieKOTopbie n p e a a o s « e i i H H ae?KaiHHe B ocuoBe b b i h h c a e r i h h KoncTaiiT aniMOTponiiH n a 3KcnepiiMeiiTaabHbix a a i m b i x . 1.

Introduction

The magnetocrystalline anisotropy of the garnet system Xd :c Y 3 _. r Fe 5 0 12 was studied by several authors [1 to 4], Volkova and Rajcis [1] explored this system for 0 x 0.09 in the temperature range 4.2 to 77 K, Krishnan and Rivoire [2] for x = 0.002, 0.006, and 0.01 at 77 K, both using the resonance method. The torque measurements in magnetic field up to 7 T were performed by Egami and co-workers [3, 4] for x = 1.0, 1.3, 1.6, and 1.8 at 77 K. An advantage of the measurements carried out on the samples with low concentration of rare-earth ions is the possibility to achieve magnetic saturation even at liquid helium temperatures by readily accessible magnetic fields. A disadvantage appears to be the difficulty in the determination of low x, making uncertain the conclusions concerning the contribution of the rare-earth ions to the magnetocrystalline anisotropy. The aim of this paper is to determine the constants of magnetocrystalline anisotropy at low temperatures for a sample with higher content of neodymium ions, for which a standard chemical analysis is suitable. The method chosen at 4.2 K was the determination of anisotropy constants from magnetization curves (-6nlax = 5 T), supplemented by torque measurements (Bmiix = 1.5 T) in the temperature range up to 77 K. In order to achieve saturation by our magnet we have chosen the Xd concentration in the sample x = 0.3. N o w at Research I n s t i t u t e for M a t h e m a t i c a l Machines, Loretanske n a m . 3, 11855 P r a g u e 1, Czechoslovakia.

106

E . KRATOCHVILOVA, V. ROSKOVEC, and V. NEKVASIL

As it is shown in Section 2 of this paper, we were able to determine: 1. the constants K1 and K2 at 4.2 K from the magnetization curves measured in directions [001], [110], and [111]; 2. the values of (Kx -j- K2) in the temperature range 4.2 to 50 K from the straight parts of the "unsaturated" torque curves in the vicinity of the easy direction [111]; 3. the constants K1 and K2 by Fourier analysis of the "saturated" torque curves observed at 77 K . In Section 3 we discuss our results with emphasis on the assumptions used in the evaluation of the experimental data. 2. Experimental Methods and Results Single crystals of the ferrimagnetic garnet of the composition Nd 0 343Y2.657F'e5012 were prepared by the standard flux method. Two spherical samples were ground, then polished and oriented by the X-ray method in the (110) plane. One of the samples (diameter d = 2.87 mm) was used in the torque measurements, the other one (d = = 2.29 mm) in magnetization studies. The magnetization curves at 4.2 K were determined by the induction method on an apparatus with a ballistic galvanometer, in directions [001], [110], and [111]. Magnetic fields up to 5 T were produced by a superconducting solenoid. The anisotropy constants were obtained from the magnetization curves shown in Fig. 1 by the following procedure. I t is well known [5] that as long as the magnetization process in the cubic crystal consists of the rotation of the magnetization vector only, the anisotropy constants K x and K2 are given by the relations K, = 4( JF110 -

H'10o) .

K2 = 2 7 ( l F m -

W100) - 36( Wm -

Wm) ,

(1)

where W^i is the work required to turn the magnetization from the easy direction to the direction [hkl]. In real conditions the Bloch wall displacements take place at the beginning of the magnetization process. Assuming that their contribution does not depend strongly on the applied field direction we can approximate the differences W110 — W100 and WU1 — TF100 by the differences of the total magnetization work in the corresponding directions. These differences are equal to the area between the magnetization curves and were obtained by numerical integration. This way we got the anisotropy constants K x = —2.47 X 105 J m" 3 and K 2 = —4.22 X 10 5 J m" 3 . We note that the results of the magnetization measurements do not contradict the above assumption about the nature of the magnetization process. I f the remanent magnetizations M m 0 and M 110 in the directions [100] and [110], respectively, result from the rotation of the magnetization vector from the easy direction, we have Mlm = Min cos 54.7° = Mln 0.5773 , "1 M 110 = M n l cos 35.3° = M U 1 0.8165 . j For the experimental value of the saturation magnetization M n l = 193 kA m

^ -1

and

F i g . 1. Magnetization curves of Ndo.343Y2.657Fe 5 0 1 2 a t 4.2 K . O [100], x [110], A [111], • extrapolated value of M for Ba = 0

107

Magnetocrystalline Anisotropy of Nd-Substituted Y t t r i u m - I r o n Garnet

t h e extrapolated values of the remanent magnetization M100 = 1 1 1 kA m = 159 kA m - 1 (see Fig. 1), we obtain the ratios Mu

Mn

= 0.575

1

0.824

and

and Mv

(3)

which satisfactorily agree with (2). The torque measurements were performed on the compensated magnetometer in the magnetic field 1.5 T at temperatures 4.2 to 77 K . T h e anisotropy constants Kx and K2 of a cubic crystal are related to the torque in (110) plane by the equation (2 sin 2()9 + 3 sin 4 Tc only. A comparison of t h e e x p e r i m e n t a l d a t a v e r y n e a r Tr w i t h t h e t h e o r y is given on Fig. 3 for t h e case of t h e " b e s t " sample 3. Our t h e o r e t i c a l d e p e n d e n c e of specific h e a t versus t e m p e r a t u r e n e a r Tc contains only one f i t t i n g p a r a m e t e r A, which h a s t h e m e a n i n g of t h e m e a n value of t h e m a g n e t i c m o m e n t a t t h e origin of t h e fluctuation. I t a p p e a r s t h a t t h e results o b t a i n e d for t h e f l u c t u a t i o n model of a f e r r o m a g n e t a r e in m u c h b e t t e r q u a l i t a t i v e a g r e e m e n t with t h e e x p e r i m e n t a l d a t a t h a n t h o s e b a s e d on t h e m e t h o d of critical e x p o n e n t s . I n p a r t i c u l a r , t h e effect of r o u n d i n g was p r e d i c t e d as a n intrinsic p r o p e r t y of t h e p h a s e t r a n s i t i o n . As we believe, t h i s m e a n s t h a t o u r f l u c t u a t i o n model of t h e p h a s e t r a n s i t i o n m a y be generally a d e q u a t e t o t h e physical reality c o m m o n l y observed e x p e r i m e n t a l l y . References [1] I I . BROUT, P h y s . R e v . 1 1 5 , 8 2 4 ( 1 9 5 9 ) . L2] M. E . FISHER, P h y s . R e v . 17, 2 5 7 ( 1 9 6 8 ) .

M. E . FISHER a n d P. E . SCESNEY, P h y s . Rev. A 2, 825 (1970). [3] E . BREZIN, J . C. L E GUILLOUI, a n d J . ZINN-JUSTIN, P h y s . L e t t e r s A 47, 2 8 5 ( 1 9 7 4 ) .

[4] [5] [6] [7]

B. M. MCCOY and T. T. Wu, Phys. Rev. 17(5, 631 (1968). I. ISYOZI and S. MIYAZIJIA, Progr. theoret. Phys. (Kyoto) 3»>, 1083 (1905). A. B. HARRIS, J. Phys. C 7, 1671 (1974). T. C. LUBENSKY, Phys. Rev. B 11, 3573 (1975).

[8] G. GRINSTEIN a n d A. LUTIIER, P h y s . Rev. B 13, 1329 (1976). [9] L . J . SCHOWALTER, M . B . SALAMON, C. C. TSUEI, a n d R . A . CRAVEN, S o l i d S t a t e C o n u n u n . 2 4 , 5 2 5 (1977). [10] D . P . LANDAU, B . E . K E E N , B . SCHNEIDER, a n d W . P . WOLIHbix 3Jiei;TpoiiHbix n e p e x o a o B b u e i i T p a x aioMHHecnenuHH, cooTBeTCTBeniio. OcHOBHbie iepTbi iiejimieiiHoro ripiiMecnoro M.i.iyMeiiHH (cyó.'iHiietiHoe BospacTaime H H T C H C H B H O C T H nsjiyHemiH c HiiTencHBHocTi.io B03OyjKaeHHH, yMeiibiiieHHH BuyTpeHiieii KBaiiTOBOü 3(J)({)eHTHBHOCTH h BpeMenii 3aTyxaiinji H3JiyleiIHH Iipn IIOBbimeilHH ypOBIIH B036y>KaeHHH) yjOBJieTBOpHTCJIIiHO 0ÑT>HCIIHI0TCH Moaejibio, yiHTbiBaiomeñ y B e j n m e i w e CHopocTH 6e3bi3Jiynoro H3MeneiinH npii ocBemenHH 3anojiiieiiHH abipnaMii ;ie(])enTOB, accouHiipoBaiiHbix c HayiiaeMi>iMn UeHTpaMH JlIOMHIieCUeHUHH. List of Symbols c

Cpr. pdi cps h o l e c a p t u r e c o e f f i c i e n t s b y r a d i a t i v e c e n t r e s , d e f e c t s , s - c e n t r e s , r e s p e c tively; cni a n d cn2 r a d i a t i v e a n d n o n - r a d i a t i v e e l e c t r o n c a p t u r e c o e f f i c i e n t s b y r a d i a t i v e centres, respectively; c n( i e l e c t r o n c a p t u r e c o e f f i c i e n t b y d e f e c t s ; Ec a n d Ev e n e r g i e s of t h e b o t t o m of t h e c o n d u c t i o n a n d v a l e n c e b a n d s , r e s p e c t i v e l y ; e r a n d e d e n e r g y p o s i t i o n s of r a d i a t i v e c e n t r e s a n d d e f e c t s , r e s p e c t i v e l y ; / d a n d 1 — / a p r o b a b i l i t y f o r t h e d e f e c t t o b e filled b y a n e l e c t r o n a n d b y a h o l e , respectively; / r e l e c t r o n o c c u p a t i o n of r a d i a t i v e c e n t r e s ; g'T f r a c t i o n of c h a r g e c a r r i e r s r e c o m b i n i n g via r a d i a t i v e c e n t r e s ; k absorption coefficient for the exciting light; L i n t e n s i t y of t h e e x c i t i n g l i g h t ; J

) P a r t i and I I see phys. stat. sol. (a) 48, 593 (1978) and 51, 645 (1979), respectively.

K. D. G l i n c h u k , A. V. Phokhorovicii. and V. I. Vovnenko

122

n 0 and 8n equilibrium and excess free electron concentration; n = n0 + 8n electron concentration; 8p excess free hole concentration; N f ' concentration of pairs which include the radiative centre filled by an electron a n d the defect filled b y a hole; 8N~ concentration of pairs which include the radiative centre filled by a hole and t h e defect filled by an electron; N a n d 8N° concentration of pairs in which both constituents are filled by electrons and holes, respectively; N = i V f ' + N--' + 8N~ + SA7° pair concentration; N's concentration of deep s-centres; / r l and 7 r 2 rates of radiative electron transitions via 8N~ and 8N° pairs per unit volume, respectively; I T = 7 r i + 7 r 2 extrinsic emission intensity; r separation between the radiative centre and t h e defect ; i, duration of a light pulse; w rl and uv¿ rates of electron transitions via 8N~ and 8N° respectively;

pairs per unit volume,

wr = u r l + wr2 rate of electron transitions via radiative centres; u m i and Mpli rates of electron and hole capture b y defects, respectively; u s and u A rates of excess carrier capture by s-centres and defects, respectively; r¡ internal quantum efficiency of the extrinsic emission; r emission decay time. 1. Introduction I t is known t h a t in semiconductors at high excitation intensities one can observe a non-linear extrinsic luminescence (a non-linear increase in the extrinsic luminescence intensity 7 r with exciting power L) [1, 2]. As t h e extrinsic emission intensity is 7 r = = r¡g[L, the luminescence non-linearity could be due to an excitation dependence of t h e fraction of charge carriers g'T, recombining via radiative centres [1], or of the quant u m efficiency r¡ of the extrinsic emission [2]. 2 ) T h e emission non-linearity caused b y t h e excitation dependence of g'r was investigated in detail previously and is well understood now [1, 2]. On the contrary, scattered data exist now about the emission non-linearity caused by the excitation dependence of r¡, so further a t t e m p t s are needed to study it [2 to 4]. T h e origin of this non-linearity is the following: it is caused by a non-linear change with excitation in the rate of non-radiative electronic transitions in radiative centres [2]. T h i s is due to the f a c t t h a t not only radiative, but non-radiative electronic transitions in radiative centres could exist, too [2 to 4]. These radiationless transition are of Auger type in which the energy available is imparted by collisions to other localized carriers — electrons (GaAs [2, 3, 5, 6], CdSe [7]) or holes (CdP 2 [4], C a P [8 to 10]). T h e n excitation could non-linearly change the rate of Auger transitions involving traps because of a considerable change in the concentration of localized carriers which are absorbing the energy released by impact ionization [2], Earlier [2] we showed t h a t a strong superlinear extrinsic emission could be observed in n - t y p e G a A s crystals due to a decrease with excitation in the concentration of localized electrons which could absorb the energy released b y a recombining electron b y impact 2 ) The >;-value shows which fraction of charge carriers of those recombining via radiative rcentres emits photons, i.e. it is the ratio of the rate of radiative recombination I c via r-centres to the sum rate of radiative and non-radiative recombination uT via r-centres — t] = IT¡ur.

123

Study of Non-Linear Extrinsic Luminescence in GaAs (III)

ionization. T h i s leads t o a decrease with excitation in t h e r a t e of n o n - r a d i a t i v e (Auger) t r a n s i t i o n s in deep r a d i a t i v e c e n t r e s ; as a result a n e x c i t a t i o n increase in t h e q u a n t u m efficiency i] a n d t h e emission d e c a y t i m e r arises [2]. I n t h e p r e s e n t work we will show t h a t a s t r o n g sublinear extrinsic emission caused b y a n increase with e x c i t a t i o n in t h e r a t e of n o n - r a d i a t i v e (Auger) t r a n s i t i o n s in deep r a d i a t i v e centres could also be observed in n - t v p e GaAs crystals. This is caused b y a n increase with e x c i t a t i o n in t h e c o n c e n t r a t i o n of localized holes which could a b s o r b t h e energy released b y a reconibining electron b y i m p a c t ionization; as a result a n e x c i t a t i o n decrease in rj a n d T arises. 2. Sublinear Emission Due to Auger Recombination of Free Electrons (Theory) 2.1 Recombination

model.

Kinetic

equations

L e t us f i n d t h e s t e a d y - s t a t e extrinsic emission i n t e n s i t y /,. versus exciting power L (it generates kL h o l e - e l e c t r o n p a i r s per unit volume) for a n n - t y p e GaAs crystal which c o n t a i n s t h e r a d i a t i v e r - c e n t r e associated with t h e defect (this is t h e real s t r u c t u r e of deep luminescence centres s t u d i e d below [5, 6]) (Fig. 1). If t h e defect is filled b y a hole r a d i a t i v e (with t h e coefficient cnl) a n d n o n - r a d i a t i v e (with coefficient c n2 ) f r e e elect r o n t r a n s i t i o n s t o r a d i a t i v e centres are possible (Fig. 1). I n t h e first case p h o t o n emission of energy hv occurs. I n t h e second case t h e energy released b y a r e c o m b i n i n g free electron is i m p a r t e d b y i m p a c t collision t o a localized hole of t h e associated defect, so a n o t h e r free hole a p p e a r s in t h e valence b a n d (Fig. 1). If t h e defect is filled b y a n electron, t h e n only t h e r a d i a t i v e electronic t r a n s i t i o n exists (the energy released b y a recombining f r e e electron is not sufficient for t h e i m p a c t ionization of a defect elect r o n as E q — f r 8N°(Ni'))

a n d J r ~ [c„i/(c n l +

cn2)]L

a t high excitation intensities (un ul2 ^ ux, In /r2 « Ir, SN°(N^') 8N~(N ')) a n d IT ~ L& {¡3 1) a t i n t e r m e d i a t e excitation intensities. A considerable change in a ratio 8N~I8N° ~ N^'/Nf' ^ / d / ( l — fd) with t h e exciting power occurs d u e t o a change in fd with L (see e q u a t i o n (7)): / a ~ 1 a t low excitation intensities (c m i» 0 > > c pli Sp) a n d fd 1 a t high excitation intensities (cpd Sp > c n d » 0 a n d c p r c „ d « 0 a n d c,,(i > c p r if c n 2 e x c i t a t i o n intensities will be f o l l o w e d b y a sublinear emission at i n t e r m e d i a t e e x c i t a t i o n intensities a n d then one again o b s e r v e s a linear emission at high e x c i t a t i o n intensities. 2.3 Excitation emission

dependence

of the

extrinsic

A s f o l l o w s f r o m (2), (3), the internal q u a n t u m e f f i c i e n c y o f t h e e x t r i n s i c studied r] = IJUj. is g i v e n b y

emission

c n l 8N~

r¡

=

Cnl 8N-

=

/d +

+ c nl

c

"

+

of the internal

c,„ 8Na

N-

c, l2 ) 8N0

(cnl +

N

quantum

'

cnl c,a +

efficiency

A c ll2

T

f'_

N

••(!-/«.)•

W

c nl

~r cn2 A s one can see if radiationless transitions in r a d i a t i v e centres c o u l d exist (c Il2 cnl), then t h e e x c i t a t i o n d e p e n d e n c e o f r¡ could be o b s e r v e d ( i f c n 2 < ^ c n ] , then r¡ « 1 at all e x c i t a t i o n intensities). R e a l l y in this case a decrease in / d w i t h L leads t o a decrease in Tj — t h e q u a n t u m e f f i c i e n c y r¡ ^ 1 at fd as 1 ( l o w e x c i t a t i o n intensities 8N~ 8N°) a n d r¡ = c n i/(c n l + c l l 2 ) at fa - 0 at t í¡ as c ps , c pr c n l 2 , a n d ms w r , uA [ 6 ] ) . T h e n as f o l l o w s f r o m (2), (3), 8N~(t) = SA T "(í¡) e x p ( - c n l » 0 i ) a n d 8N°(t) = SN0^) X X e x p [ — ( c „ i + c n 2 ) n0t)] a n d the emission d e c a y is h

=

8N~(t¡)

exp ^ _

i-j +

8N°(t¡)

exp ^ - -

t

c„iw0.

(9)

w h e r e the emission d e c a y t i m e s x1 a n d r 2 a r e : Tj =

(CmWo)-1 ,

r2 =

[(c„i +

c n 2 ) Wo]- 1 .

(10)

I f n o n - r a d i a t i v e ( A u g e r ) t r a n s i t i o n s in r a d i a t i v e centres exist (cn2 < ; c n l ) , an increase in e x c i t i n g p o w e r w i l l l e a d t o an decrease in r f r o m xx at l o w e x c i t a t i o n intensities (8N~(t¡)

>

SA r o (i 1 )) t o r 2 at high e x c i t a t i o n intensities ( S A 7 0 ^ ) >

a v a r i a t i o n o f the SN-^/BN0^ 2.5 Concluding

N^'/N^'

~ f j {1 -

S i V - ( i j ) ) , i.e.

with

/d)) ratio.

remarks

A s one can see f r o m ( 6 ) t o (10) t h e sublinear Ir(L) d e p e n d e n c e a n d t h e decrease o f r] a n d x w i t h L a r e the consequences o f simultaneous r a d i a t i v e a n d n o n - r a d i a t i v e elect r o n i c transitions in r a d i a t i v e centres caused b y a s t r o n g association b e t w e e n t h e r a d i a t i v e centre a n d t h e d e f e c t (c n 2 ^ c „ i ) . I f t h e association b e t w e e n t h e r a d i a t i v e centre a n d t h e d e f e c t is w e a k ( c n 2 < ^ c n l ) , i.e. t h e r e are p r a c t i c a l l y no r a d i a t i o n l e s s transitions in r a d i a t i v e centres ( o n l y r a d i a t i v e t r a n s i t i o n s e x i s t ) , then, as f o l l o w s f r o m (6) t o (10), o n l y linear e x t r i n s i c luminescence a n d no e x c i t a t i o n d e p e n d e n c e s o f rj a n d x will be o b s e r v e d . So the characteristics o f non-linear emission caused b y a non-linear i n t e n s i t y d e p e n d e n c e o f radiationless t r a n s i t i o n s in r a d i a t i v e centres, d e p e n d 3 ) Cpd > c p r means that the defects could carry a recombination f l o w more e f f i c i e n t than the r a d i a t i v e centres do, especially at low e x c i t a t i o n intensities (ui v a l u e s (N~' + i V f ' a ; A r ) > 8N° + + 8N~, g'T ¿is const 4= f(L) a n d 8n n0, N) [1, 11]. L i n e a r e x c i t a t i o n d e p e n d e n c e s o f t h e i n t r i n s i c emission i n t e n s i t y 7 C V c o n f i r m t h e f a c t t h a t n o n - r a d i a t i v e c e n t r e s do n o t s a t u r a t e a n d t h a t a t t h e e x c i t a t i o n s u s e d 8n did not e x c e e d n 0 ( F i g . 2) [1], T h e ^ - v a l u e s were f o u n d f r o m t h e a n a l y s i s o f I r j L d e p e n d e n c e s a n d t h e r - v a l u e s f r o m t h e a n a l y s i s o f t h e emission d e c a y e x c i t e d b y a pulse light with ti 3 ¡¿s s u p e r i m p o s e d on a c o n s t a n t b i a s light [2], 4 . Results Fig. 2 and 3 illustrate the main typical characteristics of non-linear extrinsic luminesc e n c e f o r t h e 0 . 9 4 , 1 . 0 e V emission b a n d s in n - t y p e G a A s w i t h i n c r e a s i n g e x c i t a t i o n i n t e n s i t y . D a t a ( e x t r i n s i c emission i n t e n s i t y i n t e r n a l q u a n t u m e f f i c i e n c y i], a n d e m i s s i o n d e c a y t i m e r versus e x c i t i n g p o w e r ) a r e given f o r c r y s t a l s in w h i c h t h e role o f t h e n o n - r a d i a t i v e t r a n s i t i o n s in 0 . 9 4 a n d 1 . 0 e V r a d i a t i v e c e n t r e s differs c o n s i d e r a b l y — f r o m s t r o n g (c„i/(c,u + «„•>) ~ 0 . 3 ) t o w e a k (c n l /(c„i + c„.>) ~ 1). T h e f o l l o w i n g m a i n f e a t u r e s d e s e r v e special a t t e n t i o n : I f s t r o n g r a d i a t i o n l e s s t r a n s i t i o n s in r a d i a t i v e c e n t r e s e x i s t ( c n 2 i g c n l ) , t h e n a) t h e e x t r i n s i c e m i s s i o n i n t e n s i t y I T c o u l d v a r y subl i n e a r l y w i t h e x c i t i n g power ( 7 r ~ w h e r e /? = 0 . 5 a n d 0 . 7 ) 4 ) ; b) t h e i n t e r n a l q u a n t u m e f f i c i e n c y r¡ d e c r e a s e s with e x c i t i n g p o w e r L u n t i l it r e a c h e s a p l a t e a u v a l u e less t h a n u n i t y ; c) t h e emission d e c a y t i m e r d e c r e a s e s w i t h e x c i t i n g p o w e r a t a d e f i n i t e L (quantajcmh) 10"

1

1

— 10"

/

10'

/

i 10"

101 4

/

J&

§

¿ Y

i 1011

1o11

-

w'

r

10°

10

L lquonla/cm2s)

= 0.5 to 0.9 could be observed

Fig. 2. 77 K dependence of the extrinsic (1 to 3) and intrinsic (1' to 3') emission intensities on the exciting light intensity for n-type GaAs (ra0 10 1 ' c m - 3 ) with weak and strong non-radiative electronic transition in radiative centres o, A 0.94 eV a n d « , A 1.0 eV; (1), (1') CniRcm

-

cn2) =

1, ( 2 ) , ( 2 ' ) 0 . 5 , ( 3 ) , ( 3 ' )

0.33

Study of Xon-Linear Extrinsic Luminescence in GaAs (III)

127

Fig. 3. 77 K dependence of the internal quantum efficiencies (O, • , A, A) and emission decay times (c, V) on the exciting light intensity for n-type GaAs (n0 as 1017 cirr 3 ) with weak and strong non-radiative electronic transitions in radiative centres, o , • , A, V hvm 0.94 eV; A 1.0 eV; (1) c n l /( C l l i + c„ 2 ) = 1; (2) 0.5 ; (3) 0.33 ; •//(0) and r(0) are measured at i « 10 l s quanta/cm 2 s 10"

10"

L /quanta¡em's) -

i n t e r v a l o f L v a l u e s . I f r a d i a t i o n l e s s t r a n s i t i o n s in r a d i a t i v e c e n t r e s a r e w e a k e n o u g h (c„2 c„i), t h e n (i) / r c h a n g e s l i n e a r l y with£(/3 = 1 ) ; (ii) r¡ a n d x do n o t d e p e n d on L. 5. Discussion All t h e d a t a o b s e r v e d a g r e e well w i t h t h e o r y (see S e c t i o n 2) a n d a r e e x p l a i n e d b y t h e s a m e a r g u m e n t s d i s c u s s e d a b o v e , i.e. t h e o b s e r v e d s u b l i n e a r e x c i t a t i o n d e p e n d e n c e o f t h e 0 . 9 4 , 1.0 e V e m i s s i o n b a n d s in n - t y p e G a A s is due t o t h e n o n - l i n e a r i n t e n s i t y d e p e n d e n c e o f t h e r a d i a t i o n l e s s e l e c t r o n t r a n s i t i o n s in 0 . 9 4 a n d 1 . 0 e V r a d i a t i v e c e n t r e s . T h e p i c t u r e is t h e following. E x c i t a t i o n g r a d u a l l y i n c r e a s e s t h e l o c a l i z e d h o l e c o n c e n t r a t i o n on t h e d e f e c t l e v e l s a s s o c i a t e d w i t h r a d i a t i v e c e n t r e s . T h i s r e s u l t s in an i n c r e a s e o f t h e r a t e o f n o n - r a d i a t i v e e l e c t r o n i c t r a n s i t i o n s in r a d i a t i v e c e n t r e s as t h e c o n c e n t r a t i o n o f l o c a l i z e d holes w h i c h could a b s o r b t h e e n e r g y r e l e a s e d b y a r e c o m b i n i n g free e l e c t r o n b y collision is g r e a t l y i n c r e a s e d . All t h i s l e a d s t o a s u b l i n e a r e x t r i n s i c emission, t o a d e c r e a s e in r¡ a n d r with e x c i t a t i o n ( F i g . 2, 3 ) . A t s u f f i c i e n t l y h i g h e x c i t a t i o n i n t e n s i t i e s all d e f e c t levels a r e m o s t l y filled b y holes. I n t h i s c a s e I v a g a i n l i n e a r l y c h a n g e s w i t h L a n d 7] ( < [ 1 ) a n d r does n o t d e p e n d on L ( F i g . 2, 3 ) . I f n o n r a d i a t i v e t r a n s i t i o n s in r a d i a t i v e c e n t r e s a r e w e a k e n o u g h (c n 2 cnl), then only linear l u m i n e s c e n c e is o b s e r v e d a t all e x c i t a t i o n i n t e n s i t i e s ( F i g . 2, 3). N a t u r a l l y , t h e m i c r o c h a r a c t e r i s t i c s o f r a d i a t i v e c e n t r e s could differ in v a r i o u s c r y s t a l s ; a s a r e s u l t , v a r i o u s emission s u b l i n e a r i t i e s could b e o b s e r v e d (/3 = 0 . 5 t o 0 . 9 ) . 5 ) T h e v a l i d i t y o f t h e a s s u m p t i o n s o f t h e t h e o r y , see S e c t i o n 2 . 1 (low c m l v a l u e s , c p(1 > cpr > c.ii.2 > c11(i, weak d e p e n d e n c e o f c„i, c p r , c ll(l , cpg'T =t= =j= f{L), fT /d) is c o n f i r m e d in e x p e r i m e n t . R e a l l y t h e o b s e r v e d c o i n c i d e n c e o f t h e r¡(L) a n d r ( L ) d e p e n d e n c e s a n d no e x c i t a t i o n d e p e n d e n c e s o f r¡ a n d T, i f r a d i a t i o n l e s s tran_ sitions a r e a b s e n t , t e s t i f y , as one c a n e a s i l y see, t h e v a l i d i t y o f t h e a b o v e a s s u m p t i o n s . 6 ) 5 ) GaAs heat treatment generating many types of efficient local centres usually promotes the appearance of a variety of nonlinear emission effects (a change of /J-values) due to radiationless transitions in radiative centres [131. 6 ) If coefficients e , c i depend on a pair charge (hole capture coefficients c , c~ by JVf', N^' pr n pr r pairs and electron capture coefficients c n i, cj^ by 8N~, 8Aro pairs differ), i.e. / r l = C n i» 0 8N~ = c =N-'&p = u r i ,

In = c'nln0 = [ci'il/(cnl + c ni)] Cpr^'l' SP = [ c ni/( c nl + cn2)] «r2 , then ((¡). (8). (10) are changed for the following: c pr/d

Cnl

-f

Cnl

c pi/il

n Tl

=

-

= (c„i» 0 )

--Cpr(l - / d ) Cn2 Cn2

cp~r/d 1

,

.Y 8 p ,

(H)

Cpr*1 - / d )

+ Cprf1 —/d) r 2 = [(c,' a + c„ 2 ) m,,]-1 .

(12)

(13)

So in this case at Cpr c~r the sublinear Ir(L) emission could be observed even if non-radiative transitions in radiative centres are absent (cn2 c'nl, i.e. tj s» 1) (the 7 r (L)-sub-linearity is due to excitation changes in g'r = [cpriV/d + CprAT(l — /d)]/cps- v s )- ® u t here no coincidence of I I j L = g'r(L) and t(L) dependences will be observed.

128

K. D.

GLINCHCK

et al.: Non-Linear Extrinsic Luminescence in GaAs (III)

From the r¡- and r-values found follows that cn2 ^ 10 13 cm 3 /s. Such c n2 -values are also predicted theoretically for Auger transitions involving traps for the pair radius r = 10"6 cm [12], Comparing the above results and those given in [2] one sees that a variety of IT(L) non-linearities due to non-linear Auger transitions in radiative centres could he observed in GaAs crystals (1 Js /? = 0.5 to 1.3). 7 )This is due to the fact that deep radiative centres could be associated with different defects [5, 6]. In conclusion we note that the data obtained confirm the origin of non-radiative transitions as Auger transitions involving traps (radiationless multiphonon emission should not show any excitation dependence of r¡, x) and are important for solving the complex problem of non-radiative electronic transitions in radiative centres [8, 14, 15]. References [1] K. D. G L I N C I I U K , A. V. sol. (a) 48, 593 (1978). [2] [3]

[4] [5]

[6] [7] [8] [9] 110]

[11] £12] [13] [14]

PROKHOROVICH,

V. E.

RODIONOV.

and

Y.

I.

YOVNENKO,

phys. slat,

and Y . I . Y O V N E N K O , phys. stat. sol. (a) 51. 6 4 5 ( 1 9 7 9 ) . and M . K . S C H E I N K J I A N , phys. stat. sol. (a) 21, 3 6 9 ( 1 9 7 4 ) ; 31». 7 1 7 ( 1 9 7 7 ) . W. M A S D E J I A and N. J . T R A P P E N I E R S . Physica B (Utrecht) 76, 123 (1974). Yu. G A C H E G O R . I . V . I S M E S T I E R , and G . B . S O I F E R , Radiospectroscopiya Xr. 1 0 . Perm State Univ. Publ., Perm 1976 (p. 33). K. D. G L I X C H U K , A. V. P R O K H O R O V I C I I , V. E. R O D I O X O V , and V. L V O V X E X K O , phys. stat. sol. (a) 41, 659 (1977). > L K. S H E I N K M A N , A. V. L U B C H E N K O , and A. I. F E D O K O V , Fiz. Tekh. Poluprov. 11. 955 (1977). A . A . B E R G H and P. J . D E A X , Proc. I E E E ( ¡ 0 , 1 5 6 ( 1 9 7 2 ) . J . C . T S A X G , P. J . D E A N , and P. T . L A X D S B E R G , Phys. Rev. 1 7 8 , 8 1 4 ( 1 9 6 8 ) . J . M . D I S H M A N , Phys. Rev. B 3 , 2 5 8 8 ( 1 9 7 1 ) . D. F. B A I S A , A. V. B NO D A R , A. Y A . G O R D O N , and S . Y. M A L T S E V , Ukr. fiz. Zh. 2 4 . 128 (1979). E. I. T O L P I G O , K. B . T O L P I G O , and M. K. S H E I N K M A N , Fiz. Tekh. Poluprov. 8, 509 (1974). K. D. G L I N C H U K , A. V. P R O K H O R O V I C I I , and V. I. Y O V N E N K O , Fiz. Tekh. Poluprov. 10, 2167 (1976). P. T. L A N D S B E R G , phys. stat. sol. 4 1 , 457 (1970). P. T. L A N D S B E R G and M. J . A D A M S , J . Lum. 7, 3 (1973); Proc. Royal. Soc. London, A 3 3 4 , 523 (1973) K . D . G L I X C H U K , A . V . PROKHOROVICII.

Y U . V . VOROBEV

[ 1 5 ] C. H . HENRY, J . L u m . 1 2 / 1 3 , 4 7 (1976).

(Received April 12, 1979)

') Xote here that as for the discussed case the emission decay is:

It(t) = In

exp

exp

I r2

1 — exp

then the transient (measured at t = ti 1 MeV)

I vrrmW

io's w"" f> lari2)

U p to fluenees of 3 X 1017 e m - 2 the expected, but small decrease of T(, with increasing fluence is not observed. On the contrary, Tt. shows a maximum of 16.82 Iv a t 5 X 1016 c m - 2 . As described in Section 2.1 this behaviour m a y be correlated with t h e stabilization of the cubic phase by the irradiation induced defects. Comparing the samples SO, SI, and S2 in Fig. 3 with t h e samples of the group showing the much lower R R R of 18, but a similar T(. of 16.70 K, one can conclude t h a t a clear correlation between T t . and R R R — often used in literature — does not exist in spite of the single-phase state of these materials. This correlation seems to be valid only f a r below the optimal transition t e m p e r a t u r e where the specimen is in a highly disordered state. •i.2 Critical

current

measurements

antl volume

pinning

force

T h e radial distribution of t h e magnetic induction was measured by means of an ac method developed by Rollins et al. [12], The measurement was performed as a f u n c tion of the t e m p e r a t u r e a n d of the external field directed parallel to the cylinder axis of the sample. One important a d v a n t a g e of this technique as compared to integral measurements is the recognition of surface pinning, which otherwise is often correlated erroneously with the microstructure in t h e bulk. T h e annealed a n d the irradiated samples did not show an enhanced pinning in the surface region. The critical current density was calculated from the slope of the local magnetic induction versus radial position. This was done always at the same distance about 50 ii.m from the surface of the samples. The relative error of the determination is between 5 % a n d 10% for a reduced field h ^ 0.1 and less t h a n 5 % for b > 0 . 1 . The absolute error may be twice as large as the relative one. Small differences in the values of at the same field and the same t e m p e r a t u r e appeared for different p a t h s taken to arrive at the state (B, T). This p r o p e r t y was observed at low fields for the samples SI, S2, and P2 and is in full accordance with

F i g . 4. Critical c u r r e n t d e n s i t y j(. v e r s u s m a g n e t i c f i e l d B a t r e d u c e d t e m p e r a t u r e 0.84 for t h e s a m p l e s SO ( • ) , SI (V). S2 (n), S3 (O), S4 (A), So (A), a n d P 2 ( • )

SfD —

158

H . KUPFER a n d A . A. MANUEL

Fig. 5. Critical current density jt. versus magnetic field at 4.2 K for the sample S3

t h e e x p l a n a t i o n o f t h e ' h i s t o r y e f f e c t ' given b y K i i p f e r a n d G e y f 13]. All m e a s u r e m e n t s d i s c u s s e d in t h i s p a p e r were p e r f o r m e d in t h e following m a n n e r : F i r s t t h e t e m p e r a t u r e was a d j u s t e d , a n d t h e n t h e field was swept from t h e n o r m a l s t a t e t h r o u g h Hc2 t o t h e desired v a l u e . F i g . 4 shows t h e c r i t i c a l c u r r e n t d e n s i t y o f t h e single c r y s t a l s a n d o f P 2 in d e p e n d e n c e on t h e m a g n e t i c field a t a r e d u c e d t e m p e r a t u r e t = TjTK of 0.84. T h e specimens S I , S 2 , a n d P 2 with a f l u e n c e less t h a n 1 0 1 8 c u r 2 show a p r o n o u n c e d p e a k a t high fields. T h i s p e a k e f f e c t is discussed in S e c t i o n 4 . I n F i g . 5 t h e c r i t i c a l c u r r e n t d e n s i t y is p l o t t e d u p t o 13 T a t 4 . 2 K f o r t h e s a m p l e S 3 . R e m a r k a b l e a r e t h e v e r y high v a l u e s o f /t. e x c e e d i n g 1 0 6 A c m - 2 a t 7 T . T h e s e v a l u e s a r e o b t a i n e d o n l y b y n e u t r o n i r r a d i a t i o n , l e a v i n g t h e s a m p l e in a m u c h b e t t e r c r y s t a l l i n e s t a t e t h a n s p e c i m e n s p r o d u c e d b y diffusion t e c h n i q u e s , s p u t t e r i n g , or e v a p o r a t i o n , which show lower c r i t i c a l c u r r e n t s . T h i s m e a n s , if t h e r e is no i n t r i n s i c s a t u r a t i o n o f t h e p i n n i n g m e c h a n i s m , it should b e p o s s i b l e t o get m u c h h i g h e r c u r r e n t s a n d a s m a l l e r r e d u c t i o n o f t h e o t h e r c r i t i c a l p a r a m e t e r s b y a m o r e careful p r e p a r a t i o n .

Fig. 0. Reduced volume pinning force Fp/F™AX versus reduced field b — BjHto at the reduced temperatures 0.97 (x), 0.92 (A), 0.84 (o), 0.76 (A), 0.66 ( c ) , and 0.53 (V), a) S I , b) S2, c) S3, d) S4

Summation and Saturation Behaviour of the Volume Pinning Force in V 3 Si

159

Fig. 7. Maximum volume pinning force F™ax versus H{-2 for the samples S I (•), S2 (•). S3 (•), S4 (A), and S5 (A)

Fig. 6 shows the reduced volume pinning force F ^ F ™ * * ) - 1 in dependence on the reduced field at different temperatures. I n the following we list only some remarkable properties of these pictures; a comparison with theory is made in Section 4. All samples show a scaling behaviour Fv ~ H%(T) f(b) with satisfying accuracy. W i t h decreasing temperature the maximum of the volume pinning force is slightly shifted to higher fields and the width at half maximum of the F v (F™ A *)~ l function decreases. This behaviour is emphasized by sample S2. The origin of this deviation from the temperature scalign is the paramagnetically limited H a , which is used for the calculation of the reduced field. The paramagnetic effect vanishes for temperatures close to T c [14]. This is the reason why a fairly good scaling is observed in the measured temperature region. An additional deviation from the scaling law is observed at reduced temperature t 0.9 and at low fields in the samples S I and S2. This is due to the unaffected jc at low fluences in the field region below the maximum of je, as can be recognized in Fig. 4. B u t the temperature dependence of Fp in SO is different from that induced by irradiation. Therefore the ratio of these two mechanisms varies with temperature. Fig. 4 and 6 show that the dependence of jt„ respectively of Fp, on the field is approximately unchanged for fluences above 10 18 c m - 2 in the samples S3, S4, and S5. The absolute value of F decreases for fluences larger than 4 X 10 18 c m - 2 compared at the same reduced temperature. The exponent n of the scaling law is taken from a lg-lg plot of F™'dx versus Hc2 (Fig. 7). The values are between 2.3 (S3) and 2.8 (SI). T h e results at the lowest temperature for the specimens S4 and S5 are not considered. Probably the influence of the paramagnetically limited Hk2 is more pronounced in these samples. This is not in contradiction to the apparent better temperature scaling of these samples compared to S I and S2, because their maximum shows a smaller curvature, i.e., they are less sensitive to a too small //,.2. 4 . Discussion

In this section the elementary pinning force /p is calculated (4.1), and then we enter into a discussion of the summation of these elementary pinning forces (4.2), the dependence of the volume pinning force on the concentration of the pinning centres (4.3), the peak effect and the satuiation behaviour (4.4), and the threshold criterion (4.5). 4.1

Calculation

of the elementary

pinning

force

For an introduction we briefly discuss the possible influence of the disordered regions (agglomeration of antisite defects) on jc. The threshold behaviour of Tc at high fluences (Fig. 3) has been attributed to these defects by Pande [9, 10]. According t o a fairly good agreement between this theory and the experiments, these disordered regions may influence Tc, but they are ineffective as pinning centres because their size is smaller than the coherence length. Further, the defects may cause only a very small elastic interaction with the flux line lattice because they have the same lattice parameter as the matrix [10]. This is confirmed by the different dependence of Tc

H. Kupfer and A. A. Mantki,

160

and / c on the fluence, on the irradiation temperature, and on the post-irradiation annealing. This means that Te and jc are influenced by different kinds of defects: jt. is predominantly governed by loops and only indirectly by y. and Te, which are taken into account in the following considerations. For the calculation of the elementary pinning force / between a dislocation loop and a flux line, we have to know /7C and y. Neglecting the strong coupling of V 3 Si, the relation //,('»)

211 I 7 '/',

(1)

is used. The electronic heat capacity coefficient y and its dependence on the fluence was taken by interpolation from the data of Viswanathan and Caton [5]. These data measured on very similar specimens seem to be more reliable than those calculated from the Maki parameter and the normal-state resistivity. Using He and &H v *l&T\r..T v from Table 1, we calculate -(d/7(.2/dT|r

y

,)T(.

2 |'2 //,((») This equation is valid in the temperature range of our experiment, where the paramagnetic limitation can be neglected. The data are given in Table 2. With H(.(0) and y. we estimate the elementary pinning interaction as follows: After Kramer [1], the first-order elastic interaction corresponds to the predominant elementary pinning force between a dislocation loop and a flux line. Using the equations in Section 2.1 and Fig. 4 in reference [1], we obtain the maximum of this elementary pinning force. Values of the elastic moduli have been taken from Testardi [15] at T y for a non-transforming sample. After Guha et al. [16], fluences up to 3.5 X 10 18 c m - 2 do not affect these moduli very much. For this reason, the same values are used for all specimens. In order to get the maximum elementary pinning force, we did not use the soft acoustic shear mode (C u — C12)j2, but C4i for the calculation. Using equation (1) the simplification was made that the relative volume dependence of H^ was assumed to be primarily given by the volume dependence of or T,., respectively, and not by the volume dependence of y. This is justified since the resistance is constant below 25 K , and thus the phonon influence on the mean free path of the electrons, and there fore on y, can be neglected. With the pressure dependence of 3 X 1 0 - 5 K b a r - 1 from Smith et al. [17], the elementary pinning force was estimated for a loop of a root mean square diameter of 170 A. The values of / p for b = 0.5 and t = 0.84 are given in Table 2. 4.2 The stiturnation

of the elementary

pinning

forces

Comparing the mean diameter of the loops and the interaction length £ with the mean spacing of the flux line lattice, fp can be regarded as a point force: In the temperature region 0.66 Si t 0.92 the coherence length varied between 50 and 120 A, whereas the minimum flux line distance is 140 A at He2 (t = 0.66) and 300 A at Hci (t = 0.92). The mean distance between the loops is about 1700 A for sample S5 with the highest fluence. As this distance is much larger than the interaction length, it is obvious that all samples are in the dilute limit, where neighbouring centres do not interact. Therefore a proportionality between the volume pinning force and the concentration of pins is expected [18, 19]. In the following we shall compare our results with different summation models. WTe discuss the simple direct summation of the elementarv pinning forces [20], = Nfp ~ (1 -

b) (1 - t*) ,

(3)

Summation and Saturation Behaviour of the Volume Pinning Force in V3Si

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physica ( a ) 54/1

161

H. K u p f e r and A. A. Manuel,

162

t h e statistical s u m m a t i o n of L a b u s c h [21], which t a k e s into consideration t h e rigidity of t h e f l u x line lattice, P

1 / B \2'3 / I = — - t f / fP — J —

8 in

W

+

VC 6 6 C 4 4

1

_

iC

u

\

)

Cj

— 1/ft i point defect-straight dislocation bias factors, ( c u r 2 ) straight dislocation density, 7 ) v j ( c m - 2 s - 1 ) point defect diffusion coefficients, Z>2v (cm - 2 s^1) divacancy diffusion coefficient [6], i ) 2 v = V » 2 exp ( - - f ! ) , kT j'

(8)

migration energy of a divacancy and C|v (at/at/s) divacancy thermal fractional concentration [6],

174

N . M . GHONIEM a n d D . D . CHO

The set of r a t e equations (6) are derived b y considering the balance between various production and destruction processes for a certain cluster size. A detailed description of this derivation is given in [7]. The analytical solution of this set of equations can be very complicated, a n d would have to involve numerous approximations, therefore a numerical approach is followed. 3 .

M e t h o d

o f

S o l u t i o n

Two distinct problems arise when one a t t e m p t s to numerically solve a large n u m b e r of coupled rate equations such as t h a t described in the previous section. The first complication is due to the time scales of the problem while the second stems from handling a large set of equations. Some reaction rates are very fast while others are very slow giving rise to a stiff system of ordinary differential equations. For stability requirements, most of the conventional methods limit the integration time step to small values. This can considerably increase the computing time, a n d is impractical for solving our system. On the other hand, storage requirements can seriously discourage handling large systems of equations. F o r t u n a t e l y , t h e Gear [8] computer package was designed to handle both large a n d stiff systems of ordinary differential equations (ODEs). The r a t e equations described in the previous section have been numerically integrated using the G E A R computer package. U p to 200 rate equations were solved with no storage problems on the I B M 360/90 computer. On the other hand, a total of 178 CPU seconds were required to integrate 100 rate equations simultating the irradiation behavior of 316 stainless steel at 450 °C, a dose rate of 10~6 dpa/s and a dislocation density of 109 cm/cm 3 . The total number of rate equations required to give information on the system has been arbitrarily chosen by various investigators [1, 2, 9]. I n this work, we use a conservation principle for irradiation produced defects to determine t h e necessary number of equations. For interstitial clusters, a n d a t a n y irradiation time t*, the integrated rate of interstitial accumulation should be equal to the t o t a l n u m b e r of interstitials in loops; or, -Y i* { / \.K\(X — 1>0

Is

— K\(x,t')

,t = 2 0

C v (] [7] [8] [9] [10] [Ill

nuclear Hater. (Amsterdam) 7 5 . 7 7 ( 1 9 7 8 ) . and D . 1 . P O T T E R . Proc. IX. Internat. Symp. Eff. Rad. Structural Mat.. Richland, WA 1978. U . G O S E L E , J. nuclear Mater. (Amsterdam) 78. 83 (1978). W. G . W O L F E R and M. A S H K I N . J. appl. Phys. 4«. 5 4 7 ( 1 9 7 5 ) . D. R . O L A N D E R , Fundamental Aspects of Nuclear Fuel Elements. X ' T T S publication. Tennesse 1 9 7 6 (p. 2 1 5 ) . A. C. D A M A S K and G . D I E N E S . Phys. Rev. 120, 99 (19(50). X. M. G H O N I E M and D. D. C H O . Univ. Calif. Eng. Rep.. UCLA-EXG-7845. Los Angeles 1978. A. C. H I X D M A R C H . Lawrence Livermore Lab. Rep., UC1D-30001, Rev. 3. Livermore 1974. M. R. H A Y N S . J. nuclear Mater. (Amsterdam) 5K, 267 (1975). H. R. B R A G E R and J. L. S T R A A L S U N D , J. nuclear Mater. (Amsterdam) 4(>, 134 (1973). X . M. G H O N I E M and G . L . K U L C I N S K I , Radiat. Eff.. in the press. R.

A.

JOHNSON. J .

B . O. HALL

(Received March 7, 1979)

M. P a s e j i a x x a n d P . W e r k e r : I d e n t i f i c a t i o n of Small D e f e c t s in Silicon

17»

p h y s . s t a t . sol. (a) »4, 179 (1979) S u b j e c t classification: 1.4; 10.2; 22.1.2 Institut für Festkörperphysik und Elektronenmikroskopie der Akademie der Wissenschaften der DDK, Halle (Saale)

Identification of Small Defects in Silicon By M. J ' a s k m a w a n d I \

\Vr:U\kr

I n silicon m o n o e r y s t a l s g r o w n b o t h b y Czochralski a n d z o n e - f l o a t i n g m e t h o d small d e f e c t s (diame t e r a b o u t 100 A) in s u r f a c e - n e a r layers, w i t h t y p i c a l b l a c k - w h i t e c o n t r a s t in t h e d y n a m i c a l image in t h e t r a n s m i s s i o n e l e c t r o n microscope are o b s e r v e d . T h e i d e n t i f i c a t i o n is c a r r i e d o u t b y m e a n s of d i f f r a c t i o n c o n t r a s t a n a l y s i s a n d high r e s o l u t i o n l a t t i c e p l a n e imaging. A c c o r d i n g t o t h i s a n a l y s i s t h e d e f e c t s are f o u n d t o b e spherical a m o r p h o u s p a r t i c l e s w h i c h c o m p r e s s t h e silicon m a t r i x d u e t o a n i n t e r n a l s t r a i n e s i 0.03 t o 0.04. T h r e e possibilities of t h e s t r u c t u r e of t h e s e i n t e r s t i t i a l t y p e clusters are discussed. I n e i n k r i s t a l l i n e m Silizium, das sowohl m i t t e l s C z o c h r a l s k i v e r f a h r e n als a u c h tiegelfrei g e z ü c h t e t w o r d e n war. w e r d e n kleine D e f e k t e ( D u r c h m e s s e r ca. 100 A) in o b e r f l ä c h e n n a h e n S c h i c h t e n m i t t y p i s c h e m S c h w a r z - W e i ß - K o n t r a s t in d e r d y n a m i s c h e n A b b i l d u n g im T r a n s m i s s i o n s e l e k t r o n e n m i k r o s k o p b e o b a c h t e t . Die I d e n t i f i z i e r u n g erfolgt m i t t e l s B e u g u n g s k o n t r a s t a n a l y s e u n d hocha u f l ö s e n d e r X e t z e b e n e n a b b i l d u n g . D a n a c h sind diese D e f e k t e s p h ä r i s c h e a m o r p h e Teilchen, d i e die Silizium-Matrix m i t e i n e r S p a n n u n g v o n e a ; 0,03 bis 0,04 k o m p r i m i e r e n . Drei Möglichkeiten f ü r die S t r u k t u r dieser „ i n t e r s t i t i a l " - T v p Cluster w e r d e n d i s k u t i e r t . 1. I n t r o d u c t i o n

During the last few years an increasing interest has been directed to the investigation of microdefects in as-grown silicon. Especially the characterization of the so-called swirl defects in zone-floated silicon was the aim of a lot of papers (seee.g. [1, 2]). I n t h e case of the A-defects the identification was successful by means of high-voltage electron microscopy in connection with a special sample preparation [3]. I t was proved t h a t A-swirls are extrinsic prismatic dislocation loops with b — ^-, which are decorated with impurities in most cases. Concerning the B-defects, u p to now the att e m p t s for identification failed a n d it is not clear whether t h e y are vacancy or interstitial clusters or if there is some impurity precipitation, too. A quite different behaviour was found in Czochralski crystals. H e r e the d o m i n a n t microdefects are due to microsegregation (see e.g. [4]). Extensive studies have been carried out to characterize the behaviour of oxygen in Czochralski-grown silicon a n d it seems to be obvious that there is some precipitation of a silicon oxide [5, 6]. W h e t h e r there are defects due to thermal point defects of the silicon in Czochralski material, too, comparable to swirl defects has not been decided. The investigations of microdefects in silicon are very interesting because they t h e m selves can influence the electrical parameters of electronic devices or some more extended defects can be generated at the microdefects during the device m a n u f a c t u r i n g process. I n this work we made an a t t e m p t to identify very small defects (diameter a b o u t 100 Á) with a typical black-white contrast in T E M micrographs which we have f o u n d both in Czochralski and in zone-floated silicon crystals. As will be shown later, it has been proved to be a d v a n t a g e o u s to define the surface density of the defects. I n the as-grown state t h e surface density of the defects was very low (10 2 to 10 3 /cm 2 ), b u t a f t e r the thermal processes of t h e device m a n u f a c t u r i n g t h e density increased and it reached a value of about 10 5 to 10 6 /cni 2 in MOS devices. 12*

180

M . PASEMANN a n d P .

WERNER

B u t hero we f o u n d a d i f f e r e n c e in t h e t w o s t a r t i n g m a t e r i a l s . T h e a b o v e - m e n t i o n e d v a l u e is o n l y v a l i d f o r C z o c h r a l s k i s t a r t i n g m a t e r i a l , in z o n e - f l o a t e d m a t e r i a l t h e d e f e c t d e n s i t y is s m a l l e r . T h e d e f e c t s o c c u r in ( 1 1 1 ) , ( 1 0 0 ) , a n d ( 1 1 0 ) o r i e n t e d silicon wafers a n d t h e y o c c u r in s p e c i m e n s w h i c h were p r e p a r e d f r o m s e c t i o n s cut p e r p e n d i c u l a r t o t h e p l a n e o f t h e wafers. W e o b s e r v e d t h e m a f t e r c h e m i c a l t h i n n i n g f r o m one side, t h e o t h e r side being p r o t e c t e d a g a i n s t t h e c h e m i c a l r e a g e n t , a n d a f t e r c h e m i c a l t h i n n i n g f r o m b o t h sides of the specimens. R i i h l e et al. [7] s t u d i e d e x t e n s i v e l y b l a c k - w h i t e c o n t r a s t d e f e c t s in i r r a d i a t e d c o p p e r foils. T h e y r e f e r r e d t o t h e p o s s i b i l i t y o f t h e f o r m a t i o n o f such d e f e c t s due t o t h e p r e p a r a t i o n p r o c e s s f o r t r a n s m i s s i o n e l e c t r o n m i c r o s c o p y . F o r e x c l u d i n g such an e x p l a n a t i o n f o r t h e d e f e c t s d e s c r i b e d here we p r e p a r e d s o m e s a m p l e s b y ion b e a m t h i n n i n g a n d got t h e s a m e d e f e c t s with t h e s a m e d e n s i t y as a f t e r c h e m i c a l t h i n n i n g . S o we m u s t c o n c l u d e , t h e y a r e i n d e p e n d e n t o f t h e p r e p a r a t i o n process. F o r i d e n t i f y i n g t h e d e f e c t s o b s e r v e d f i r s t l y we u s e d t h e d i f f r a c t i o n c o n t r a s t a n a l y s i s b o t h in a h i g h - v o l t a g e e l e c t r o n m i c r o s c o p e ( 1 0 0 0 k V a c c e l e r a t i n g v o l t a g e ) a n d f o r b e t t e r r e s o l u t i o n in a . J E M 1 0 0 C h i g h - r e s o l u t i o n e l e c t r o n m i c r o s c o p e e q u i p p e d with a top entry goniometer stage. S e c o n d l y we u s e d t h e l a t t i c e p l a n e i m a g i n g m e t h o d a t t h e J E M 1 0 0 C m i c r o s c o p e with t h e s t a n d a r d s t a g e . 2. Diffraction Contrast Analysis 2.1 Dynamical

and qtiasi-h-inernatical

images

T h e e x p e r i m e n t s d e s c r i b e d here a r e b a s e d on t h e p a p e r s b y R i i h l e et al. [7, 8] a n d Chik et al. [9]. T h e r e f o r e , we will g e n e r a l l y use t h e i r n o m e n c l a t u r e . F o r e x p l a n a t i o n we w a n t t o refer t o t h e i r w o r k s where t h e c a l c u l a t i o n s a n d t h e n e c e s s a r y e x p e r i m e n t a l cond i t i o n s a r e d e s c r i b e d a n d p r o v e d in a v e r y c o m p r e h e n s i v e m a n n e r : 1. I n t h e d y n a m i c a l case, i.e. ic = 0 t h e d e f e c t s h a v e a clear b l a c k - w h i t e c o n t r a s t • 1). 2 . U s i n g k i n e m a t i c a l c o n d i t i o n s , i.e. iv ^ > 0 . 6 t h e d e f e c t s a p p e a r a s b l a c k d o t s (Fig. 2 a ) and under weak-beam conditions as white dots (Fig. 2 b ) . 3. I n b r i g h t - f i e l d i m a g e s t h e b l a c k - w h i t e v e c t o r I is a l w a y s p a r a l l e l or a n t i p a r a l l e l t o t h e d i f f r a c t i o n v e c t o r g a p p l i e d ( F i g . 1). T h i s m e a n s t h e d e f e c t s h a v e a s p h e r i c a l d i s t o r t i o n field. 4. I n b r i g h t - f i e l d i m a g e s t h e a m o u n t o f d e f e c t s with I a n t i p a r a l l e l t o g was h i g h e r t h a n t h a t w i t h I p a r a l l e l t o g. 5. I n d a r k - f i e l d i m a g e s I was a l w a y s a n t i p a r a l l e l t o g ( F i g . 3). T h i s m e a n s e i t h e r t h e d e f e c t s h a v e a l a r g e misfit (in t h i s case t h e A s h b v - B r o w n rule [ 1 1 ] c a n b e applied) or t h e d e f e c t s h a v e a small m i s f i t a n d m u s t show d e p t h o s c i l l a t i o n s . I n t h e case o f d e p t h o s c i l l a t i o n s t h e v e c t o r 1 c h a n g e s its sign e v e r y h a l f e x t i n c t i o n d i s t a n c e . T h e r e fore, in t h i s c a s e t h e d e f e c t s would lie in t w o l a y e r s , e a c h h a v i n g t h e s a m e d i s t a n c e f r o m b o t h foil s u r f a c e s (for e x p l a n a t i o n see [8]). T o decide b e t w e e n t h e s e t w o p o s s i b i l i t i e s one c a n e s t i m a t e t h e size o f t h e p a r t i c l e s a n d t h e a m o u n t o f t h e i n t e r n a l s t r a i n . A c c o r d i n g t o a d i a g r a m given b y C h i c k et al. [9J c o n c e r n i n g t h e c o n t r a s t b e h a v i o u r o f s p h e r i c a l p a r t i c l e s o n e c a n t h e n f i n d out w h i c h region is v a l i d f o r t h e d e f e c t s o b s e r v e d . 2.2 Size of the defect

and estimation

of the internal

strain

A c c o r d i n g t o t h e p a p e r o f A s h b y a n d B r o w n [ 1 1 ] o n e c a n d e t e r m i n e t h e size o f t h e p a r t i c l e s f r o m t h e i r d y n a m i c a l i m a g e s . A m o r e a c c u r a t e m e t h o d is t o u s e t h e k i n e m a t i c a l or t h e w e a k - b e a m i m a g e o f t h e d e f e c t s ( F i g . 2 ) . W e f o u n d t y p i c a l sizes b e t w e e n a b o u t 70 and 100 A diameter.

Identification of Small Defects in Silicon

181

Fig. 1. Black-white contrast defects in silicon. The same place is imaged under dynamical brightfield conditions with different {/-vectors, showing t h a t the black-white vector I is always parallel or antiparallel to g

One needs these sizes of t h e particles for an estimation of t h e internal strain in t h e matrix. This can be only very rough because of t h e influence of t h e surface which is very critical in our case. According to Ashby a n d Brown [11] t h e following equation for the evaluation of the internal strain e can be u s e d : image width

h

egr\} \

/

l° If J' g

where r 0 is the particle radius, the extinction distance, g t h e diffraction vector a p plied. As to t h e problem how to measure the special image width we refer to t h e original work [11].

182

M. Pasemann and P. Werner

Fig. 2. a) Kinematical image of the defects (w > 0.6); b) weak-beam image of the defects, (202) excited, imaged with (202)

Using t h e following values, g = (220), r 0 = 50 A , = 757 A , a n d a n image w i d t h of 150 A for 5 0 % intensity, we got a n internal stress p a r a m e t e r of e = 0.023. Using a n o t h e r vector, g = (111), l g = 602 A , a n d a n image width of 185 A for 5 0 % i n t e n s i t y , we got a value of e = 0.04. As these values can only be t a k e n as a clue, a n o t h e r m e t h o d of estimation was used as described in Section 3. 2.3 Depth

and nature

of the

defects

Using these values for r0 a n d e one f i n d s out t h a t t h e defects lie in t h e region of d e p t h oscillations (see [9]). B u t we only observed defects in t h e first layers of t h e d e p t h oscillation, i.e. down t o a d e p t h of m a x i m u m 0.3£ 9 f r o m b o t h surfaces. For an accurate depth measurement we tried t o c a r r y out stereo e x p e r i m e n t s a n d m a r k e d t h e surface of t h e silicon samples b y gold e v a p o r a t i o n , as described b y Diepers a n d Diehl [10] (Fig. 4). Fig. 4 a shows a specimen region in d y n a m i c a l d i f f r a c t i o n c o n t r a s t , a n d in Fig. 4 b one sees t h e same region u n d e r quasi-kinem a t i c a l conditions which a r e necessary for a n a c c u r a t e m e a s u r e m e n t in t h e stereo experiment. I n fact, using these d e p t h m e a s u r e m e n t s we only f o u n d defects down to d e p t h s of

Fig. 3. Dynamical dark-field image of the defects showing t h a t the black-white vector I is always antiparallel to g

183

Identification of Small Defects in Silicon

SP # it

JS;

*

j&

-Z^rSM '

I Sfij§| 3i , ' r ,r f SnSiii. T H k « "dPE

'

QJtt.m

--.I-

t

Pig. 4. Gold evaporated silicon sample with black-white contrast defects imaged a) under dynamical, b) under kinematical bright-field conditions

0.3£„ distance from both surfaces. Within these two layers the defects were lying at different distances from the surfaces. So we can state that the defects observed are situated within surface-near regions down to a maximum depth of about 230 A from both specimen surfaces (0.3fj, for (220) at 100 kV » 230 A). Another effect is worth mentioning. We observed a higher density of defects at that side of the sample which is turned towards the electron source. If one takes a photograph of a special place of the specimen and then turns the specimen round and takes a micrograph at the same specimen position, one finds more defects than before. This means, some defects must have been generated due to the electron beam. To summarize the observed black-white contrast effects one can state: As in darkfield images the defects always have an ¿-vector antiparallel to g and the defects are found to lie in the first layers of the depth oscillation they must be spherical particles of the interstitial type. 2.4 Some

remarks

concerning

the elastic

anisotropy

Lepski [12] calculated the figures of black-white contrast of small coherent particles in a matrix and took into consideration the elastic anisotropy of the matrix. As different distortions exist in the different crystallographic directions there is also an influence of the black-white contrasts on the contrast figures. The contrast behaviour is determined by the parameter of the elastic anisotropy (d — c)/( 1 + a), where d =

1 + a — ca

— 1 /, /C\ s |/l-(T)(l+«)

c12 — cn + 2 clt

c = —

44

C

-,

c12 a = — ;

11

C

Cjj being the elastic constants. If for a given substance this parameter differs very much from unity, then the blackwhite contrasts exhibit a contrast connected with the elastic anisotropy of the matrix. Lepski distinguishes between substances where c ]> 0 and c < 0, respectively. For c > 0 (e.g. iron) the displacements have maxima along the oriented platinum emitter. S o m e e x p e r i m e n t s were also carried out with tungsten tips coated with relatively thick layers of tungsten glass. X o electron emission was measured at applied v o l t a g e s u p to 30 kV. In our view, the fact that no emission is observed from metallic tip« covered by thick layers of glass can be explained a s follows. Electrons a r e first fieldinjected into the dielectric but. they a r e rapidly t r a p p e d by defects. High-field conduction mechanisms such a s those occurring in sandwich structures [3], for e x a m p l e Poole-Frenkel process where conduction is controlled by field-enhanced thermal emission of electrons from discrete t r a p s levels into the conduction b a n d of the insulator, cannot occur because the field strength decreases rapidly with d i s t a n c e from

•

r

F i g . 5. Field-electron emission image of a glass-coated platinum tip a f t e r heating in v a c u u m (2200 V)

Field-Emission Microscopy from Glass-Coated Tips

» -215

193

Fig. 6. Fowler-Nordheim plot for emission from a platinum tip coated with a thin layer of glass (77 K)

HO 4] HL vuno'^v the cathode surface (this is a consequence of the geometrical configuration in fieldemission microscopy [1, 6]). On the other hand, a direct tunneling from the metal to vacuum is not possible because the glass layers are too thick. Platinum tips coated with thin layers 0.1 fim) of glass have also been studied. Here, an electron emission was measured at applied voltages of the order of 3 kV; the field-electron image consisted of bright spots located on the fluorescent screen without any symmetry. As shown in Fig. 6, the current-voltage characteristic is in rather good agreement with the Fowler-Nordheim equation which establishes the following relation between the field-emitted current (i) and the applied voltage (U) when some correction factors due to image forces are neglected [1, 5]: i = A ^ e x p ^ ^ ,

(1)

where A and B represent coefficients depending essentially on the height of the potential barrier at the metal-dielectric interface and on the proportionality factor k between the field strength F at the emitter surface and U (F = kU). As expected from (1), it was found that the slope of Fowler-Nordheim plots does not vary when the tip temperature is changed by putting various refrigerants (ice, liquid nitrogen, liquid helium) in the cold finger. From these results, it seems well t h a t emission is controlled by tunneling at the platinum-glass interface, while bulk effects within glass such as space charge and electron trapping effects do not play an important role. At higher voltages, an oscillatory behaviour of current was sometimes observed. Fig. 7 a gives an experimental plot of current versus time at constant applied voltage (2700 V). Each current oscillation consists of a rapid increase to a maximum current, followed by a decrease to a minimum current. As Fig. 7 b reveals, the frequency of the oscillations increases with the applied voltage.

0

20

40

60

Fig. 7. Current oscillations from a glass-coated platinum tip (T — 77 K) 13

physica (a) 54/1

N. RIHON: Field-Emission Microscopy from Glass-Coated Tips

194

Similar c u r r e n t f l u c t u a t i o n s h a v e been m e a s u r e d in sandwich s t r u c t u r e s with several dielectrics, p a r t i c u l a r l y S i 0 2 [11] a n d organic p o l y m e r s [12]. T h e y a r e p r o b a b l y associated with t r a n s i t i o n s f r o m a low c o n d u c t i v i t y s t a t e t o a high c o n d u c t i v i t y s t a t e . Since t h e discovery of t h e Ovshinsky effect in chalcogenide glasses [13], t h e switching b e h a v i o u r of dielectrics s u b m i t t e d to v e r y high fields ( ^ 1 0 7 V / c m ) h a s received a g r e a t a t t e n t i o n f r o m physicists as well as f r o m engineers [14]. Several i n t e r p r e t a t i o n s based on electronic i n t e r f a c e states, ionic displacements, t h e r m a l processes, polarization a n d space charge effects h a v e been suggested. I t m i g h t be t h a t t h e current oscillations described here result f r o m t h e r m a l devitrification of glass; t h e t r a n s i t i o n f r o m insulating s t a t e to c o n d u c t i n g s t a t e would be related to al ocal g l a s s - c r y s t a l p h a s e t r a n s i t i o n d u e to h e a t i n g b y J o u l e ' s effect in t h i n channels of glass ( f i l a m e n t a r y cond u c t i o n [15]). I n conclusion, it was shown t h a t , in convenient e x p e r i m e n t a l conditions, fieldemission microscopy f r o m metallic tips coated with glass can give valuable i n f o r m a tion a b o u t t h e b e h a v i o u r of glass s u b m i t t e d to v e r y high electric fields. V a r i o u s emission regimes were observed. While no emission occurs w h e n t h e glass layer thickness is g r e a t e r t h a n a b o u t 1 pun, p u r e F o w l e r - N o r d h e i m field emission a n d bulk effects in glass h a v e been r e p o r t e d in t h e case of t h i n glass coatings. Acknowledgements

T h e a u t h o r wishes to g r a t e f u l l y acknowledge Prof. J . C. P . Mignolet for his s t i m u l a t i n g interest in t h i s work a n d for helpful discussions. H e is v e r y m u c h i n d e b t e d t o " L e F o n d s de la R e c h e r c h e F o n d a m e n t a l e Collective" for financial s u p p o r t to t h e laborat o r y a n d to " L e s Services de P r o g r a m m a t i o n de la P o l i t i q u e Scientifique, Centre de Catalyse, U n i v e r s i t é de Liège", for a research g r a n t . T h e skilful technical assistance of G. Quoilin a n d M. F e d y n i a k is also greatly a p p r e c i a t e d . References [1] R . GOJIER, A c c o u n t s c h e m . R e s . 5 , 4 1 ( 1 9 7 1 ) .

[2] M. A. LAMPERT and P. MARK, Current Injection in Solids, Academic Press. XCW York 1970. [3] J . J . O'DWYER, The Theory of Electrical Conduction and Breakdown in Solid Dielectrics. Oxford University Press, 1973. [ 4 ] R . M . ANDERSON a n d D . R . K E R R , J . a p p l . P h y s . 4 8 , 4 8 3 4 ( 1 9 7 7 ) . [ 5 ] L . AY. SWANSON a n d A . E . B E L L , A d v . E l e c t r o n . E l e c t r o n P h y s . 3 2 , 1 9 3 ( 1 9 7 3 ) . [ 6 ] R . COELHO a n d P . SIHILLOT, R e v . g é n . E l e c t r . 71», 2 9 ( 1 9 7 0 ) .

[7] [8] [9] [10]

B. HALPERN and R. GOSIER, J . chem. Phys. 51, 1031, 3043 (1969). H. ROYXET and J . C. P. MIGNOLET, Bull. Soc. Roy. Sci. Liège »/, 274 (1969). J. MARIES, J . C. P. MIGNOLET, and X. RIHON, B u l l Soc. Roy. Sci. Liège 11/12, 614 (1970). E.W.MÙLLER and T. T. TSONG, Field Ion Microscopy — Principles and Applications, Elsevier Publ. Co., Amsterdam 1969. [11] M. SHATZKES, M. AV-RON, and R. M. ANDERSON, J . appl. Phys. 45, 2065 (1974). [12] X. SWAROOP, J . appl. Phys. 42, 863 (1971). [ 1 3 ] S. R . OVSHINSKY, P h y s . R e v . L e t t e r s 2 1 , 1 4 5 0 ( 1 9 6 8 ) .

[14] R. GOFFAUX, Conf. Ageing Phenomena in Solid Dielectrics and Methods of Their Testing, Wroclaw, 16 to 19 September 1975; Bull. Sci. A.I.M. Université de Liège 4, 299 (1975). [15] H. J . STOCKER, Appl. Phys. Letters 15, 55 (1969). (Received

March

19,

1979)

I . V. GRIDNEVA et a l . : Dislocation Mobility under Concentrated Loads

195

phys. stat, sol. (a) 5 4 , 195 (1979) S u b j e c t classification: 10.2; 12.2; 22.1.1; 2 2 . 1 . 2 ; 2 2 . 2 . 1 ; 22.2.3 Institute

of Materials

Science,

Academy

of Sciences

of the Ukrainian

SSB,

Kiev

Analysis of Dislocation Mobility under Concentrated Loads at Indentations of Single Crystals By I. V .

GRIDNEVA, Y U .

V . MILMAN, V .

1. TREKU.OV,

and

S. I.

CHLGUKOVA

An approximate solution is given for the dependence of the dislocation track length, I, around a microhardness indentation in single crystals on the load applied to the indenter (P), t h e temperature (T), and the duration under load ( 2 ) . E s wird eine Xäherungslösung für die Abhängigkeit der Versetzungslänge l um einen Mikrohärteeindruck in Einkristallen von der angewandten L a s t auf den Stempel ( P ) , der T e m p e r a t u r ( T ) und der Dauer der Last (< Jangegeben: l~P'»!(2>n-, i) ( i/(2«n-i) e x p [ — U/kT(2m -f- 1)]. E s wird die Annahme gemacht, daß Versetzungen sich vom Stempel in ebenen Anordnungen bewegen und daß die Versetzungsgeschwindigkeit v ~ Tm exp (— U/kT) ist. wobei T eine effektive Spannung und U die Aktivierungsenergie ist. In einem definierten Bereich von P a r a m e t e r n s t i m m t die Gleichung befriedigend mit Literaturwerten überein und solchen, die in der vorliegenden Arbeit erhalten werden, und kann zur Berechnung der Versetzungsbeweglichkeit aus der Länge des Versetzungspfades benutzt werden. Der W e r t des Verhältnisses l/d (d ist die Eindruckdiagonale) ist für Ionenkristalle konstant, hängt jedoch von der T e m p e r a t u r und L a s t im F a l l der kovalenten K r i s t a l l e ab. I ist 1), jedoch wie 1-ts SK const für m i t der Flußstärke r s wie h3 = const für kovalente Kristalle (m Ionenkristalle (m ^ > 2 ) verknüpft. 1.

Introduction

T h e i n d e n t a t i o n a n d e t c h i n g of single c r y s t a l s w i t h t h e a i m t o r e v e a l d i s l o c a t i o n s l e a d s u s u a l l y t o t h e a p p e a r a n c e o f s o - c a l l e d d i s l o c a t i o n " r o s e t t e s " , t h a t is a s h a r p i n c r e a s e in t h e d i s l o c a t i o n d e n s i t y a r o u n d t h e i m p r e s s i o n w i t h a s y m m e t r y c e n t r e in the very point. T h e c h a r a c t e r of t h e dislocation distribution in t h e r o s e t t e a n d t h e l e n g t h o f disl o c a t i o n t r a c k s h a v e b e e n u s e d t o e s t i m a t e t h e d i s l o c a t i o n m o b i l i t y in c r y s t a l s a n d to evaluate strength properties of crystals. Investigations on semiconductors with a d i a m o n d l a t t i c e h a v e r e v e a l e d d i s l o c a t i o n l o o p s e x p a n d i n g f r o m a s u r f a c e in { 1 1 1 } p l a n e s [ 1 ] . I f t h e h a r d n e s s is m e a s u r e d o n t h e ( 1 1 1 ) p l a n e t h e g l i d e p l a n e s i n t e r s e c t i t in t h e < 1 1 0 > d i r e c t i o n a n d t h e e t c h i n g shows t h e e x i s t e n c e of d i s l o c a t i o n s ( t y p i c a l l y 6 0 ° ) on t h e surface a s " r a y s " spreading f r o m t h e impression ( F i g . 1). D i s l o c a t i o n r o s e t t e s h a v e b e e n s t u d i e d o n G e [1 t o 6 ] , Si [ 3 , 7 ] , a n d A m B v - t y p e c r y s t a l s [8, 10]. Most of t h e a b o v e - c i t e d work deals with t h e effect of t h e t e m p e r a t u r e o n t h e i n d e n t a t i o n t r a c k , I. 13»

196

I . V . G r i d x k v a . Y r . V . J I i l m a x , V . I . T r e f i l o v . a n d S. I. C h u c u x o v a

F i g . 1. A t y p i c a l d i s l o c a t i o n r o s e t t e a r o u n d a m i e r o h a r d n e s s i m p r e s s i o n 011 a Ge single c r y s t a l . I n d e n t e r b a d P = 2.34 X . T = 750 ° C (50 x )

I t is shown in [6, 8, 9] that I is an exponential function of temperature which grows with the stress P applied to the indenter as I ~ P", where n differs only slightly from T 1.

W e shall also p u t xh = 0 since dislocation generation near the indenter in the region of high stress concentration occurs without difficulty.

Making use of (1), (2), (4) we get ANP r,

(5)

'

where A = l

,

3 „/ ^ c o s ,

+

T

+

l

sin a \ - _ j .

If t h e dislocation m o t i o n is controlled b y a single t h e r m a l l y a c t i v a t e d process with s y m m e t r i c a l p o t e n t i a l barrier, t h e dislocation velocity v in t h e presence of t h e stress r is a p p r o x i m a t e l y given b y t h e relation [24] =

v

o

CX

P

(6)

H e r e v0 is a f r e q u e n c y f a c t o r , V t h e a c t i v a t i o n energy, a n d V t h e a c t i v a t i o n volume. I n some cases t h e dislocation velocity m a y be c o n v e n i e n t l y described as v = B(^Y exp(-^L), Jo/ * \ kTj'

(7)

where r 0 = 10 7 N / m 2 (kp/mm 2 ), B a n d m are m a t e r i a l p a r a m e t e r s . Comparison of (6) a n d (7) shows t h a t m m a y d e p e n d on t e m p e r a t u r e ; it grows w i t h decreasing t e m p e r ature. F o r covalent crystals t h e m-value f o u n d in some e x p e r i m e n t s a t high t e m p e r a t u r e s differs weakly f r o m u n i t y (see [25]), b u t a t r o o m t e m p e r a t u r e a v e r y s t r o n g v(r) dep e n d e n c e is c h a r a c t e r i s t i c of ionic crystals a n d t h e m-value m a y reach 20 to 30 [26, 27]. An i n t e r m e d i a t e position is t a k e n b y r e f r a c t o r y b.c.c. t r a n s i t i o n m e t a l s (Cr, Mo, W , N b ) f o r which direct velocity m e a s u r e m e n t s a t 20 °C h a v e given m 5 t o 6 [20] while a c c o r d i n g to t h e velocity d e p e n d e n c e of t h e yield s t r e n g t h t h e m-value m a y be 7 t o 20 [28]. S u b s t i t u t i n g x f r o m (5) into (7) we get dr dt

B(AN)"< P'" To»'2"'

ex

I U \ Pr \ — T7F kT •

(8)

On i n t e g r a t i n g we f i n d J r2'" dr = / B(ANr

^ J

exp ( - A j

di

.

(9)

D u r i n g i n t e g r a t i o n we ignore t h e t i m e r e q u i r e d to load a n d u n l o a d t h e i n d e n t e r as c o m p a r e d t o t h e t i m e w h e n t h e i n d e n t e r is u n d e r load, t. T h e size of t h e impression diagonal, d, is also neglected c o m p a r e d t o I. I t follows f r o m (9) t h a t I=

Cpml(2m

•; 1) (1/(2»,+ 1)

where C = [(2m + 1) S]i/(2»«

e x p

kT (2m + 1)

Mjy\»

< 1 ,

(4.13a)

.

(4.13b)

1

In the first case the integrand can be developed into the following Taylor series: e" (e* - 1)2

1 y*

1 , 1 12 ^ 240

if-....

(4.14)

We shall only take into account the first and the second term, the other terms can be neglected. Then the resulting expression for %v is found to be X P = i"o

n(ehj2m)4a*cTc

)

kBTn

2 I Af + g/i0 (eh12m) H

1 k\\TR Ae + gfji0(ehl2m) H + (4/3) kuT

Ij.07i(ehl2m)2 1 36a 2 c& B T

(4.15)

In the second ease the integrand can be represented as a power series of 00 ey e~'J — = = Y n (e" — l) 2 (1 — e _ ! / ) 2 n h

For

.

(4 16)

we find P

»=i

(e*/2ffl). iT 8 aickiiTu \Tr pi

X erf ! j / T „ )

_M)TC

e^«-«'.W^mn-uT .

x

-

(4.17)

^ 1 Now we have to investigate under which conditions (4.15) or (4.17) holds. In our measurements we used fields up to 2.4 X 10 6 A/111. The values for Af can be determined from infrared resonance measurements [5, 6], or can be calculated by using the following formula [4]:

where k2 and k% mean the macroscopic anisotropy constants. We find that (4.15) is valid for T > 22 K , whereas (4.17) has to be used for a discussion of %v in the temperature range below 22 K .

R. Hekz and H. K r o n m u l l e r

222 5. Discussion of Results 5.1 Field

dependence

F o r t h e discussion of t h e field d e p e n d e n c e of yp for c o n s t a n t t e m p e r a t u r e we rewrite (4.15) in a more comprehensive f o r m neglecting t h e second a n d t h i r d t e r m s : = , I iZeff

with He„•

=

Han

(5-D

+

H

=

A £

gUffinjzm

- + H .

(5.2)

If we c o m p a r e t h e values of t h e a n i s o t r o p v field H A n with t h e available field s t r e n g t h s , p r o d u c e d by our s u p e r c o n d u c t i n g coil, we f i n d t h a t H a n ^ H . Therefore, f r o m a theoretical p o i n t of view no significant field d e p e n d e n c e of y p should b e observed. Let u s now r e g a r d our susceptibility m e a s u r e m e n t s . I n Fig. 1 we h a v e r e p r e s e n t e d a log-log d i a g r a m for y E in dependence on t h e applied m a g n e t i c field, w i t h t h e t e m p e r a t u r e as a p a r a m e t e r . W e clearly can a n a l y s e t h a t yl: obeys a H~0A7 law in t h e high-field region. F r o m t h i s result we m a y conclude t h a t t h e spin waves only feel a f r a c t i o n of t h e t h e o r e t i c a l value Ae. Consequently t h e applied field H has t o be considered as t h e d o m i n a n t t e r m in t h e s u m (Hiin + H). T h i s could be possible if we a s s u m e t h a t t h e spin wave excitations correspond to spins oscillating on elliptical cones a r o u n d t h e a-axis, where t h e oscillation a m p l i t u d e within t h e basal p l a n e s is m u c h larger t h a n in «-direction. I n Section 4 we f o u n d t h a t (4.15) should be valid only for T > 22 K . If we now a s s u m e t h a t t h e effective value of Af is m u c h smaller t h a n p r e d i c t e d b y (4.18), t h e n condition (4.13a) will be valid also for lower t e m p e r a t u r e s t h a n T = 22 K , a n d as 3 a consequence t h e d e p e n d e n c e of yp should be m e a s u r e d also for T 22 K . Our m e a s u r e m e n t s confirm t h i s e x p e c t a t i o n . W e also can i n t e r p r e t t h e t e m p e r a t u r e d e p e n d e n c e of t h e beginning of t h e field range, where t h e p a r a e f f e c t is d o m i n a t i n g . As t h e energy gap shows a s t r o n g t e m p e r a t u r e dependence, Ae decreases with increasing t e m p e r a t u r e , i / a n also becomes smaller for higher t e m p e r a t u r e s . Therefore, for higher t e m p e r a t u r e s t h e relation yp ~ H~oi~ becomes valid a t lower fields. Our m e a s u r e m e n t s r e v e a l t h i s b e h a v i o u r for T Tc, where our calculations should be valid. 5.2 Temperature

dependence

I n Fig. 4 our e x p e r i m e n t a l results for t h e t e m p e r a t u r e d e p e n d e n c e of y p , m e a s u r e d a t a field of 2.4 X 10 6 A / m , a r e c o m p a r e d with our theoretical results as o b t a i n e d

J 11)

Fig. 4. Comparison between the theoretical and experimental temperature dependences of the parasusceptibility of Dy at a field H = 2 . 4 x l 0 6 A / m . x Experiment, measured values; (1) theoretical curve (At = 0); (2) theoretical curve ( \ e = 0)

The Parasusceptibility of Dysprosium

223

for t h e expressions (4.15) a n d (4.17). I n one ease (curve 1) t h e energy gap Ae as d e f i n e d b y (4.18) was t a k e n f u l l y i n t o a c c o u n t . I n t h e t e m p e r a t u r e r a n g e T > 22 K %v was d e t e r m i n e d b y (4.15), a n d for T < 22 K e q u a t i o n (4.17) was a p p l i e d considering o n l y t h e first t e r m (n = 1) of t h e sum. I n t h e o t h e r case (curve 2) we a s s u m e d Ae t o be negligible. I n t h i s case (4.15) was valid in t h e whole t e m p e r a t u r e r a n g e T Tc. If t h e two theoretical curves are c o m p a r e d w i t h t h e e x p e r i m e n t a l curve, it b e c o m e s obvious t h a t t h e theoretical values of for Ae #= 0 are m u c h smaller t h a n t h e m e a s u r ed values. On t h e o t h e r h a n d , if we consider t h e calculated c u r v e for Ae = 0, we f i n d a v e r y good correspondence with our e x p e r i m e n t a l curve in t h e t e m p e r a t u r e r a n g e b e t w e e n 40 a n d 75 K . Below T = 40 K , however, t h e e x p e r i m e n t a l values d e v i a t e significantly f r o m a linear t e m p e r a t u r e d e p e n d e n c e of At i " - + 0 K the parasusceptibility e x t r a p o l a t e s t o a c o n s t a n t value of t h e order of m a g n i t u d e £ p (0) » « 3 X 10" 3 . A similar b e h a v i o u r previously was f o u n d for Gd single c r y s t a l s [3]. An i n t e r p r e t a t i o n of t h i s peculiar t e m p e r a t u r e d e p e n d e n c e of could n o t be given so f a r . I t will be t h e object of f a r t h e r work t o i n v e s t i g a t e t h e effect of m o r e realistic spin wave spectra on t h e t e m p e r a t u r e d e p e n d e n c e of y v . T h e deviation f r o m t h e linearity for T 75 K , i.e. in t h e v i c i n i t y of t h e Curie t e m p e r a t u r e Tc = 85 K is n o t astonishing, because in t h i s t e m p e r a t u r e r a n g e spin wave t h e o r y no longer holds. Acknowledgement

T h e a u t h o r s g r a t e f u l l y acknowledge t h e i n t e r e s t a n d h e l p f u l r e m a r k s of P r o f . D r . A. Seegei d u r i n g t h e p r e p a r a t i o n of t h i s p a p e r . References [1] R . HERZ a n d H . KROXMULLER. p h y s . s t a t . sol. (a) 47, 4 5 1 ( 1 9 7 8 ) .

[2] T. HOLSTEIN a n d H . PRIMAKOFF, P h y s . R e v . 58, 1098 (1940). [3] H . - J . HOF, H. KRONMULLER, a n d B. ROTHFTJSS, phys. stat. sol. (b) 71, 585 (1975).

[4] B. R. COOPER, Solid State Phys. 21, 393 (1968). [5] H . S. MARSH a n d A . J . SIEVERS, J . a p p l . P h y s . 4 0 , 1 5 6 3 ( 1 9 6 9 ) .

[6] A. J. SIEVERS, J. appl. Phys. 41, 980 (1970). (Received

May

10,

1979)

N. TOYODA et al.: Impurity Effect on the Formation of Terraces in GaAs Growth

225

phys. stat. sol. (a) 54, 225 (1979) Subject classification: 1.0; 10.2; 22.2.1 Matsushita

Research

Institute

Tokyo,

Inc.,

Ikuta,

Tama-ku,

Kawasaki

Impurity Effect on the Formation of Terraces in GaAs L P E Growth By X . TOYODA. M. MIIIARA, a n d T . HARA Terraces on the surfaces of GaAs L P E layers grown from Sn or Ge doped solutions are studied. The concentration of Sn (Xi; n ) or Ge (^Q,.) in the growth solution is varied from 0.01 to 15 a t % . The terraces sensitively depend on the doping concentration. The width and height of terraces decrease with increasing -Xgn or -X(j e . Though the effects of Sn and Ge on the terrace formation differ when compared as a function of or they agree well when compared as a function of the concentration of Sn (Nan) or Ge (AT(je) in grown layers. E s werden Terrassen auf den Oberflächen von GaAs-LPE-Schichten, die aus mit Sn oder Ge dotierten Lösungen gezüchtet wurden, beobachtet. Die Konzentration von Sn oder Ge (Zjjg) in der Züchtungslösung wird von 0,01 bis 15 A t % variiert. Die Terrassen hängen empfindlich von der Dotierungskonzentration ab. Die Breite und Höhe der Terrassen nehmen mit zunehmendem -X" sn oder JTQ6 ab. Obwohl die Einflüsse von Se und Ge auf die Terrassenbildung als Funktion von -Xgn oder verschieden sind, stimmen sie gut in Abhängigkeit von den Konzentrationen von Sn (A7sn) oder Ge (A~oe) in den gewachsenen Schichten überein.

1. Introduction

A characteristic pattern called terraces (waves) usually appears on t h e surfaces of GaAs [1 to 5], G a P [6 to 8], I n P [9 to 12], and GaAlAs [13 to 17] L P E layers. These terraces lead to uneven surfaces and adversely affect the device performance. T h e origin of terraces (waves) has been studied extensively. Previously terraces have been a t t r i b u t e d to t h e instability of growth solution b y constitutional supercooling [1 to 3]. R e c e n t l y some workers [4, 6 to 8, 12] have shown t h a t terraces disappear when the growth is made on substrates a c c u r a t e l y oriented to the low-index planes, and have concluded t h a t the origin of terraces is the substrate misorientation from low-index planes. R o d e [18, 19] has suggested the correlation between terraces and surface reconstruction. Small a n d P o t e m s k i [15] have indicated that terraces should be distinguished from waves: Terraces are equilibrium forms, which would exist in an isothermal situation at zero growth rate, while waves are not present at zero growth rate. Terraces sensitively depend on various growth conditions. I t has been shown t h a t terraces are affected by the supercooling of t h e growth solution [5, 9, 10] and disappear in the growth from the solution initially supercooled beyond a critical value [5]. K o b a y a s h i and F u r u k a w a [13] have reported t h a t t h e temperature fluctuation of t h e furnace leads to terraces. Non-uniform wetting caused b y residual surface oxide also affects the formation of terraces [11, 16, 17], especially in t h e growth of reactive materials. Furthermore, terraces can be affected b y other growth conditions such as growth temperature, cooling rate, and impurity. However, few studies h a v e examined their effects on the terrace formation. Present name: Tokyo Research Laboratory, Matsushita Electronics Corp., 4896 Tama-ku, Kawasaki, J a p a n . 15

physica (a) 54/1

Ikuta,

N. Toyoda, M. Mihaba, and T. Haba

226

I t is t h e p u r p o s e of t h i s p a p e r t o e x a m i n e how t h e f o r m a t i o n of terraces is a f f e c t e d b y t h e c o n c e n t r a t i o n of doping impurities in t h e g r o w t h solution. 2. Experimental Procedure T h e g r o w t h was p e r f o r m e d in a horizontal sliding boat m a d e of h i g h - p u r i t y g r a p h i t e [20]. Before t h e g r o w t h r u n s t h e boat a s s e m b l y was b a k e d in v a c u u m at 1100 °C, a n d f u r t h e r u n d e r flowing P d - d i f f u s e d H 2 gas a t 900 °C for 24 h. A g r o w t h solution was p r e p a r e d as follows: T h e h i g h - p u r i t y Ga m e t a l (99.9999%) a n d t h e G a A s source material (undoped p o l y c r y s t a l s w i t h N D — NA = 1 X 10 16 c m - 3 ) were loaded in t h e b o a t , a n d b a k e d u n d e r H 2 gas flow a t 800 °C for 24 h to p r e p a r e a homogeneous solution. T h i s b a k i n g also r e m o v e d t h e residual oxides on t h e Ga metal, a n d led t o a h i g h - p u r i t y g r o w t h solution. T h e layers grown f r o m t h e u n d o p e d solution were n - t y p e a n d t h e net d o n o r c o n c e n t r a t i o n was below 5 X 10 14 c m - 3 . I n t h e d o p i n g e x p e r i m e n t , Sn or Ge, which a r e r e p r e s e n t a t i v e donor a n d a c c e p t o r i m purities, respectively, was a d d e d in t h e a m o u n t of 0.01 t o 15 a t % . T h e s u b s t r a t e s were Si-doped GaAs (ND — NA = 1 X 10 18 c n r 3 ) o r i e n t e d 0.1° off t h e (100) p l a n e t o w a r d t h e

500

700 900 1100

p h y s i c a ( a ) 54/1

1300 TIKI -

242

S. F . D U B I N I N , S. G . TEPLOUCHOV, a n d S. K .

SIDOROV

wave vectors (1) are not rational fractions of the basic vectors of the reciprocal lattice, the formation of the incommensurable atomic structure takes place in alloys with 65.5 C 70 a t % F e above the t e m p e r a t u r e of the martensite transformation, J)/,, while at C > 65.5 a t % F e (see Fig. 2 a and b) the structural phase transition takes place. J u s t in this region of composition the changes of the physical properties of the alloys have critical character [3]. As an example, Fig. 2c shows the concentration dependence of the paraprocess susceptibility, obtained in [3] at 225 K . I t is seen from the figure t h a t the j u m p of the susceptibility and the appearance of t h e satellites t a k e place approximately at the same concentration of iron in the alloy. 3.2

On the

nature

of the

superstructural

reflections

As in this paper the structure of the quenched F e - X i alloys is studied, it is hard to assume t h a t the observed superstructure can be connected with the development of static waves of composition or the order degree. It is left to consider that the origin of the satellites is caused either by the modulated change of the scattering amplitude connected with the regular redistribution of the charge (spin) density of the atoms in the alloy, or by the modulation of the scattering phase due to the static displacements of the atoms or the spin density from the sites of the intitial f.c.c. lattice according, for example, to the sine law. The neutron scattering p a t t e r n s in these two cases are different and they were described in detail in the review b y Axe [5]. According to [5], in the first case the neutron p a t t e r n corresponds to the sinusoidal or helicoidal structure whose intensities of the satellites decrease proportionally to the square of the magnetic form factor with increasing value of \q>. I n the second case, the main properties of the diffraction picture are determined by the structural scattering amplitude F(q)

= f AR"(eh

• q)"

2

e x p (2mk0

• H) %

+ 2xk0

+

«*,)

,

where / is the mean (per atom) nuclear-magnetic scattering amplitude, AR the displacement amplitude, eh the polarization vector, v the number of atoms in the elem e n t a r y cell, />•„ the commensurable vector of the structure, ti the radius vector of the R lattice site, nlit the incommensurable vector of order n of the i - t h star. From (2) three most peculiar features of the picture of neutron (X-ray) scattering by a crystal, in which the modulation of the scattering phase takes place, follow: the increase of the satellite intensity with increasing \q\, the rules of cancellation of the satellites are determined by the factor (ek • q), the presence of the higher "diffraction harmonics" on the neutron (X-ray) p a t t e r n by nk,. Before the classification of the satellites according to the listed features it is appropriate to s t u d y the diffraction p a t t e r n of the austenite phase at 300 K of a 67.7 a t % F e monocrvstal sample, being cooled before to 4.2 K . I n this case the satellite intensities 020 A/2

220AJ2

000

200X12

Tig. 4. Disposition of the satellites in the reciprocal lattice

Atomic S t r u c t u r e of t h e I n v a r F e - X i Alloys

243

Fig. 5. S t a t e diagram of t h e invar alloys (solid line M s )

should considerably increase due to the increase of stresses in the alloy because of the partial precipitation of t h e martensite. Really, a great number of strong coherent s u p e r s t r u c t u r a l reflections were revealed on the neutron p a t t e r n of t h e alloy. It is convenient to show the distribution of the reflections in the reciprocal lattice. Fig. 4 shows the positions of the reflections in the reciprocal lattice by circles. The diameter of a circle corresponds to the angular size of the peak at half of its height. T h e satellites on the figure are roughly divided according to their sizes into three groups. The intensities of the peaks having dark outline are not less t h a n the intensity (220)/.ya- By the light outline there are drawn the reflections whose intensities are approximately three times less then (220)a/2- By the dashed line there are drawn the reflections one order weaker t h a n (220);./2. The space distribution of the reflections in the reciprocal lattice is symmetrical t o t h e (000) site and to the (-•!-, 4 , 0) position. The satellites are distributed in groups near the directions [110] and [ 110], connected b y one a n o t h e r with the f o u r t h - o r d e r axis. From the figure it is seen t h a t the satellite k2 lies very close (|5| = 0.04) to the (-J-; \ ; 0) position and hence it is necessary to be connected with long-wave (/. 100 A) modulations of the structure /»•„ = (-j-> 0). W i t h t h e precipitation of the martensite in a sample the coordinates of the k1 reflection did not change, however, due to the increase of the intensity as a whole the extra reflections = lfc.,1 in the direction a = [320] appear to be noticeable on the neutron p a t t e r n . It is interesting t h a t in the presence of the martensite in a sample the reflection (1 -4- '2 decrease with increasing scattering vector, the intensities k^' in the vicinity of 0) are g r e a t e r t h a n in (-J-; 0). The presence of the higher diffraction harmonics on the n e u t r o n p a t t e r n a n d the different character of the angular dependence of the satellite intensities can be understood if one assumes t h a t the origin of the satellites k?}' is connected with t h e existence of sinusoidal static waves of atomic displacements in the alloy a n d the origin of k} a n d k2 is connected mainly with the modulated static redistribution of the spin (charge) d e n s i t y of atoms. I t is reasonable t o consider now t h a t the n a t u r e of t h e superstructure in the alloys without martensite in the region Ms T Tc is analogous. This means t h a t Jv0 = = (t> T> 0) is also the point of s y m m e t r y of the stars {/Cj}, {k2}, {k3}. As f a r a s t h e satellites klt k2 appear below Tc, it is a p p a r e n t l y t h e spontaneous magnetostriction 16*

244

S. F . DUBININ et al.: Atomic S t r u c t u r e of t h e I n v a r F e - X i Alloys

t h a t gives rise to t h e redistribution of t h e spin d e n s i t y . T h e n t h e decrease of t h e intensity of h\ at room t e m p e r a t u r e , when C 67.7 a t % Fe, becomes u n d e r s t a n d a b l e (Fig. 2). T h i s is connected with t h e increase of t h e reduced t e m p e r a t u r e T = 3 0 0 ¡ T c (since Tv decreases) which a p p r o a c h e s u n i t y in t h e 7 0 % F e alloy. T h e d e p e n d e n c e of />j a n d k2 u p o n t h e e x t e r n a l magnetic field also gives evidence to t h e striction model. Their intensities at 3 / (j increase t h r e e t i m e s since in t h e n o n - m a g n e t i z e d s t a t e only •i of t h e m a g n e t i c d o m a i n s contain klt k2 in t h e scattering plane. A b o v e Tc only t h e k3 reflection r e m a i n s w i t h t h e centre of s y m m e t r y k0 = 0. I t s i n t e n s i t y d e p e n d s upon q a n d becomes s u b s t a n t i a l only in t h e position (1 + b) (-§-;-§-; 0). T h e studies of t h e satellite intensities, belonging t o all t h e r a y s of t h e {/.'3} s t a r , allowed t o conclude t h a t t h e y are connected with t h e longitudinal m o d u l a t e d displacements of a t o m s f r o m t h e sites of t h e f.c.c. lattice. 3.3

State

diagram

T h e a b s e n c e of hysteresis of t h e physical p r o p e r t i e s in t h e region of critical compositions a n d t e m p e r a t u r e s allows to s t a t e t h a t t h e i n c o m m e n s u r a b l e p h a s e t r a n s i t i o n s in t h e i n v a r F e - X i alloys are of t h e second t y p e . T h i s fact allows to d r a w t h e p h a s e d i a g r a m of t h e a t o m i c s t r u c t u r e of t h e alloys (Fig. 5). F o r t h e c o n s t r u c t i o n of t h e s t a t e d i a g r a m t h e r e were also used d a t a f r o m [3, 6], in which m e a s u r e m e n t s of t h e magnetic susceptibility %{C, T) h a v e been carried out in a wide interval of compositions a n d t e m p e r a t u r e s . T h e v a l i d i t y of this follows f r o m Fig. 3 where it is seen t h a t t h e a n o m aly of %(T) corresponds to Tt, t h e t e m p e r a t u r e where t h e s u p e r s t r u c t u r e vanishes. T h e discovered i n c o m m e n s u r a b l e p h a s e s are d e n o t e d b y y ' (/»'„ = 0; {/>';!}) a n d y"(/«'0 = >' 0), {kj}, {k2}, {k3}) a n d t h e y are a p p a r e n t l y connected with t h e loss of s t a b i l i t y of t h e f.c.c. s t r u c t u r e . T h e region of their existence is limited by t h e d a s h - d o t t e d line in t h e figure. I n t h e beginning t h e y ' - p h a s e is created with decreasing t e m p e r a t u r e . Below t h e Curie t e m p e r a t u r e ( T c ( C , T) is shown b y t h e d a s h e d line) t h e i n c o m m e n s u r a b l e y " - s t r u c t u r e , t h e s y m m e t r y of which is lower t h a n y', is created. As f a r as t h e intensities of t h e satellites klt k2, k3 are m o d u l a t e d b y t h e m a g n e t i z a t i o n , it is n a t u r a l to consider t h a t t h e invar p r o p e r t y of t h e iron-nickel alloys is caused b y t h e peculiarities of t h e t e m p e r a t u r e d e p e n d e n c e of t h e y " - p h a s e . I n t h e lower p a r t of Fig. 5 t h e Xeel t e m p e r a t u r e T x is shown by t h e d a s h e d line. Below t h i s t e m p e r a t u r e a non-eollinear m a g n e t i c s t r u c t u r e , which is d e t e r m i n e d b y t h e superposition of t h e wave vectors k = 0 a n d k —- (-.'-; 0), is created in t h e alloys. T h e m a g n e t i c s t r u c t u r e of t h e invar F e - X i alloys will Vie published in detail elsewhere. Acknoicletlgetiienttt

T h e a u t h o r s wish to t h a n k Y u . A. I z y u m o v a n d V. E . Xaish for useful discussions a n d L. V. Smirnov for growing t h e excellent monocrvstals. References [1]

V . G . YEERARAUHAYAN,

C. F . E A G E X ,

H . R . HARRISON,

and

P. G. WINCHELL,

J.

appl.

Phys.

47, 4768 (1976). [ 2 ] S. F . D U B I N I N , S . G . TEPLOCCHOV. a n d S. K . SIDOROV, Z h . e k s p e r . t e o r . F i z . , P i s m a 2tt, 1 7 7

(1977). [ 3 ] G . T . DUBOVKA, p h y s . s t a t . sol. ( a ) 4 1 . K 6 3 ( 1 9 7 7 ) . [4-J Y u . D . T Y A P K I N . V . G . P U S H I N . R . R . R O M A N O V A , a n d X . X . B U I N O V , F i z . M e t a l l o v i

Metalb-

vedenie 41, 1040 (1976). [5] J . D. AXE, Proc. Conf. X e u t r o n Scattering, Vol. 1, G a t l i n b u r g (Tenn.) 1976 (p. 353). [ 6 ] V . 1. TCHECHERNIKOY, Z h . e k s p e r . t e o r . F i z . 4 2 , 9 5 6 ( 1 9 6 2 ) .

(Received

January

23,

1979)

M . MUTHA R E D D Y e t a l . : T h e r m o l u m i n e s c e n c e o f B a F B r

Crystals

245

phys. stat. sol. (a) 54, 245 (1979) Subject classification: 20.3; 10.2; 20.1; 22.5.4 Solid State and Materials Science Laboratories, Department of Physics, Osmania University, Hyderabad1)

Thermoluminescence of BaFBr Crystals By M . M U T H A R K D D Y , K . SOMAIAH, a n d H . V A R I

BABU

Thermoluminescence and optical absorption spectra of X - and y-irradiated crystals are studied. Optical absorption studies reveal two absorption bands around 455 and 480 nm, a well defined peak at 580 nm, and a broad band at about 810 nm. The glow curves obtained are resolved by the thermal cleaning technique. Four glow peaks are obtained around 338, 369, 408, and 465 K , respectively. The peaks are analyzed using first-order kinetics. The trap depth and frequency factor of the above peaks are calculated using various methods. The 338 K peak is attributed to the detrapping of electrons from impurities and subsequent recombination with holes and whereas the other three peaks perhaps arise from the ionization of two types of F-centres and coagulation centres. Thermolumineszenz und optische Absorptionsspektren von Röntgen- und y-bestrahlten Kristallen werden untersucht. Die Untersuchungen der optischen Absorption ergeben zwei Absorptionsbanden um 455 und 480 nm, ein wohldefiniertes Maximum bei 580 nm und eine breite Bande bei etwa 810 nm. Die erhaltenen Glowkurven werden mit thermischer Ausheiztechnik aufgelöst. Vier Glowmaxima werden bei etwa 338, 369, 408 und 465 K erhalten. Die Maxima werden mit Kinetik erster Ordnung analysiert. Die Haftstellentiefe und der Frequenzfaktor der oben erwähnten Maxima werden mit verschiedenen Methoden berechnet. Das Maximum bei 338 K wird der Befreiung von Elektronen aus Störstellen und der nachfolgenden Rekombination mit Löchern zugeschrieben, während die anderen drei Maxima wahrscheinlich aus der Ionisation von zwei Arten von F-Zentren und Agglomerationszentren herrühren. 1. Introduction Crystals of mixed dihalides of B a , Sr, and P b constitute an important class of partially i o n i c c o m p o u n d s h a v i n g i n t e r e s t i n g o p t i c a l p r o p e r t i e s [ 1 ] . A l l o f t h e m c r y s t a l l i z e in t h e t e t r a g o n a l s p a c e g r o u p P 4 / n m m [2 t o 4 ] . F - c e n t r e s in B a F C l w e r e s t u d i e d o p t i c a l l y b y N i c k l a u s a n d F i s c h e r [5], a n d m o r e r e c e n t w o r k b y Y u s t e et a l . [6J h a s c o v e r e d t h e optical absorption a n d electron spin resonance of b o t h B a F C l a n d S r F C l . I n alkali halides, the F - a b s o r p t i o n b a n d arises from t h e electronic transition from an s-like ground state to a triply degenerate p-like state. This degeneracy can be removed b y r e p l a c i n g o n e o f t h e n e i g h b o u r i n g i o n s b y a d i f f e r e n t a l k a l i ion w h i c h is t h e w e l l - k n o w n F A - c e n t r e [7]. I n t h e t e t r a g o n a l B a F C l b o t h a n i o n i c s i t e s , f l u o r i n e a n d c h l o r i n e [ 5 ] h a v e t e t r a g o n a l s y m m e t r y . S o t w o t y p e s of F - c e n t r e s F ( F " ) a n d F(C1~) are possible a n d t h e i r e x c i t e d p - l i k e s t a t e w o u l d b e split i n t o a s i n g l e t a n d a d o u b l e t s t a t e . Y u s t e e t a l . [6] h a v e d e s i g n a t e d t h e t w o t y p e s o f F - c e n t r e s a s I - a n d J - c e n t r e s . B y t a k i n g i n t o a c c o u n t t h e a n a l o g y b e t w e e n t h e p o s i t i o n o f t h e I 2 - b a n d (/. = 5 5 0 n m , E _[_ (J) in B a F C l a n d t h a t o f t h e F - b a n d in B a F 2 , N i c k l a u s a n d F i s c h e r [5] c o n c l u d e d t h a t I - a n d J - c e n t r e s might be a t t r i b u t e d to F ( F ~ ) a n d F(C1~) centres, respectively. On t h e o t h e r h a n d , a t h e o r e t i c a l s t u d y o f F - c e n t r e s in B a F C l [ 8 ] h a s led t o a n a t t r i b u t i o n o p p o s i t e to that of Nicklaus and Fischer. Hyderabad-500007, India.

246

M . M U T H A R E D D Y , K . SOMAIAII, a n d

H . VAEI

BABU

T h e r m o l u m i n e s c e n c e a n d optical a b s o r p t i o n h a v e been extensively used in t h e p a s t to s t u d y colour centres in alkali halide crystals. I n t h i s p a p e r we present t h e results of t h e r m o l u m i n e s c e n c e a n d optical a b s o r p t i o n of X - a n d y - r a y i r r a d i a t e d B a F B r cryst a l s a n d also t h e values of t h e f u n d a m e n t a l p a r a m e t e r s like t r a p d e p t h (E) a n d freq u e n c y f a c t o r (P 0 ) o b t a i n e d f r o m t h e r m o l u m i n e s c e n c e d a t a . 2. Experimental Single crystals of B a F B r were grown by t h e f l u x m e t h o d , t h e details of which were p u b l i s h e d elsewhere [9]. T r a n s p a r e n t crystals free f r o m traces of u n r e a c t e d mass were used in t h e present s t u d y . T h e crystals were X - i r r a d i a t e d a t room t e m p e r a t u r e (RT). F o r t h i s p u r p o s e a n X - r a y t u b e w i t h copper t a r g e t o p e r a t e d a t 30 kV a n d 10 mA was used. T h e c r y s t a l s were a l w a y s k e p t a t a distance of 2 cm f r o m t h e t a r g e t . T h e e m i t t e d light d u r i n g h e a t ing of these samples was measured using an E M I 6256 S p h o t o m u l t i p l i e r t u b e . T h e o u t p u t of t h e p h o t o m u l t i p l i e r t u b e was given t o a dc amplifier a n d was recorded finally b y a D I G I L O G 200 XY recorder. T h e sample was h e a t e d a t t h e r a t e of 0.5 K / s a n d t h e t e m p e r a t u r e of t h e sample was m e a s u r e d b y a c a l i b r a t e d c h r o m e l - a l u m e l t h e r m o c o u p l e fixed v e r y n e a r t h e sample. T h e optical a b s o r p t i o n spectra of y - i r r a d i a t e d B a F B r crystals were recorded on a Cary-14(R) recording s p e c t r o p h o t o m e t e r , t h e dose r a t e y - r a y s used being 0.325 M R / h . 3. Results 3.1 Glotv

curves

Fig. 1 a a n d b show t h e glow curves of B a F B r , X - i r r a d i a t e d for 40 min a t R T a n d a f t e r t h e r m a l cleaning of t h e first p e a k , respectively. F o u r p e a k s a t a b o u t 338, 369, 408, a n d 465 K were o b t a i n e d . T o resolve t h e p e a k s a t h e r m a l cleaning t e c h n i q u e was employed [10]. I n t h i s m e t h o d , t h e i r r a d i a t e d sample was h e a t e d u p t o a b o u t 340 Iv a t t h e r a t e of 0.5 K / s a n d t h e n rapidly cooled t o R T . T h e sample was n e x t h e a t e d u p t o a b o u t 370 K a t t h e s a m e r a t e a n d again r a p i d l y cooled to R T . T h i s process was r e p e a t ed for t h e t h i r d p e a k , too. F i n a l l y t h e glow c u r v e was recorded f r o m R T to a b o u t 540 K a n d t h e curves o b t a i n e d b y this process a r e shown in Fig. 2. I t was also observed t h a t annealing a t R T a f t e r y- or X - i r r a d i a t i o n for t h r e e or more d a y s resulted in t h e elimination of t h e first p e a k (338 K ) . Fig. 3 shows t h e glow curve of a y - i r r a d i a t e d B a F B r crystal a n n e a l e d for 25 d.

303 363

423

433

TIKI-

303

363

423

483

TIKI -

Fig. 1. a) Shows t h e glow curve of B a F B r crystal ( X - i r r a d i a t e d for 40 min). b) Glow curve recorded b y t h e r m a l cleaning t e c h n i q u e ( X - i r r a d i a t e d for 40 min). T h e r m a l cleaning was applied t o t h e p e a k a t 338 K a n d a f t e r w a r d s t h e curve was recorded continuously f r o m R T to a b o u t 540 K

Thermoluminescence of B a F B r Crystals

Fig. 2

247

Fig. 3

Fig. 2. Glow curves of B a F B r crystal resolved into component peaks by the thermal cleaning technique Fig. 3. Glow curve of BaFBr crystal (y-irradiated for 10 h) annealed at R T for 25 d

The optical absorption spectra of BaFBr immediately after y-irradiation at R T are shown in Fig. 4a. Two absorption bands around 455 and 480 nm, a well-defined peak at 580 nm, and a broad band at about 810 nm were observed. I t is also observed that annealing at R T for 25 d after y-irradiation resulted in the elimination of the 580 nil deak in the absorption spectra. Fig. 4b shows the absorption spectra BaFBr y-irradiated and annealed for 25 d. 3.2 Analysis

of gloic

curves

The glow curves obtained in the present investigation have been analyzed by the following methods. The calculated values of trap depth and frequency factor are tabulated in Table 1. (i) The Halperin-Branner formula [11] refined by Chen [12] for first-order kinetics and constant P0 reads as E=

1.52 — ? - ( l -

1.58/1) - 3.16A:Tg

(1)

Fig. 4. a) Optical absorption spectra of B a F B r crystal (y-irradiated) immediately after irradiationb) Optical absorption spectra of B a F B r crystal (y-irradiated) annealed at R T for 25 d

248

M. Mutha Rïddy, K. Sojiaiah, and H. Vari Babu

Thermoluminescence of B a F B r Crystals

249

a n d t h e f r e q u e n c y f a c t o r P0 is given b y P„

E

( E

\

ir=vn°**W}

is t h e t o t a l h a l f - i n t e n s i t y width. E q u a t i o n (4) is used only for t h e glow p e a k o b t a i n e d a t 465 K . 4. Discussion I m p u r i t i e s and o t h e r i m p e r f e c t i o n s give rise to t r a p s a t different d e p t h s in t h e f o r b i d den energy region [14]. W h e n t h e crystals are X - or y - i r r a d i a t e d , t h e ionizing r a d i a t i o n p r o d u c e s m a n y free electrons a n d these m a y be t r a p p e d a t t h e impurities a n d also o t h e r imperfections like vacancies. E l e c t r o n s t r a p p e d a t anion vacancies give rise to F - c e n tres. B a F B r crystals exhibit phosphorescence d e c a y a n d it is generally expected t h a t in t h e initial stages of decay, t h e c o m p a r a t i v e l y shallow t r a p s will be e m p t i e d giving rise to t h e luminescence light o u t p u t , whereas t h e light o u t p u t due to e m p t y i n g of c o m p a r a t i v e l y deeper t r a p s will occur a f t e r a sufficiently long time. Fig. 3 shows t h e glow curve of a B a F B r crystal y - i r r a d i a t e d for 10 h (0.325 M R / h ) a n d a n n e a l e d for 25 d. I t is observed t h a t annealing a t R T resulted in t h e elimination of t h e p e a k obt a i n e d a r o u n d 338 K. Therefore, t h e peak o b t a i n e d at a b o u t 338 K p e r h a p s arises d u e to t h e d e t r a p p i n g of electrons f r o m impurities a n d s u b s e q u e n t r e c o m b i n a t i o n with holes. As shown in Fig. 4 b optical a b s o r p t i o n spectra also exhibit t h e same b e h a v i o u r . T h e a b s o r p t i o n peak at a b o u t 580 n m was a b s e n t when t h e a b s o r p t i o n spectra were recorded a f t e r t h e samples (y-irradiated) were annealed a t R T for 25 d. T h e o t h e r t h r e e p e a k s might be due to t h e bleaching of t h e two t y p e s of F - c e n t r e s a n d some o t h e r coagulation centres. Until detailed investigations are m a d e it is very difficult to say which glow p e a k corresponds to which t y p e of centre. T h e geometrical f a c t o r fi = bjo> is f o u n d t o be less t h a n 0.52 s u p p o r t i n g t h e application of f i r s t - o r d e r kinetics. Acknowledgement

T h e a u t h o r s wish to t h a n k Prof. K . V. K r i s h n a R a o , H e a d , D e p a r t m e n t of Physics, Osmania U n i v e r s i t y , for his e n c o u r a g e m e n t in t h i s work. References [1] J . L. S O M M E R D I J K , J . M. P. J . V E R S T E G E N , and A. B R I L , J . Lum. 8, 502 (1974). [2] ASTM Card-Index 3-0304. [3] L. K. F R E V E L , H . W. R I N N , and M. C. A N D E R S O N , I n d . engng. Chem. 18, 83 (1965). [ 4 ] T . N . B H A T , H . L . B H A T , A . H . R A M A R A O , M . R . S R I N I V A S A N , and P . S . N A R A Y A N , Curr. Sei. India 4 7 , 2 0 4 ( 1 9 7 8 ) . [ 5 ] E . N I C K L A U S and F . F I S C H E R , phys. stat. sol. (b) 5 2 , 4 5 3 ( 1 9 7 2 ) .

250

M. MUTHA REDD Y et al.: Thermoluminescence of B a F B r Crystals

[6] [7J [8] [9] [10] [11]

M . Y U S T E , L. TAUREL. M . RAIIMANI, a n d D . LEMOYNE, J . P h y s . C h e m . S o l i d s 3 7 . 9 6 1 ( 1 9 7 6 ) . H . OKHUBA a n d T . UCHIDA. J . P h y s . S o c . J a p a n 1 5 , 2 1 1 4 ( 1 9 6 0 ) . S . LEFRANT a n d H . H . H A R K E R , S o l i d S t a t e C o m m u n , l i t , 8 5 3 ( 1 9 7 6 ) . K . SOMATAH a n d V . H A R I BABU, I n d i a n J . p u r e a p p l . P h y s . 1 4 . 7 0 2 ( 1 9 7 6 ) . X . M . GUPTA, J . M . LUTHRA, a n d J . SHAXKAR, I n d i a n J . p u r e a p p l . P h y s . 1 1 , 6 8 4 ( 1 9 7 3 ) . A. HALPERIN a n d A. A. BRASSER, P h y s . R e v . 117, 4 0 8 (1960).

[12] R. CHEN, J . appl. Phys. 40, 570 (1969). [ 1 3 ] J . J . RANDALL a n d M . H . F . W I L K I N S , P r o c . R o y . S o c . A 1 8 4 , 3 6 6 ( 1 9 4 5 ) .

[14] K. V. RAO and SUJATA ROY, Solid State Commun. 20, 941 (1976). (Received

March

7,

1979)

D . L . KIEK: T h e D e f e c t S t r u c t u r e a n d t h e I n t r i n s i c I o n i c C o n d u c t i v i t y of X a C ]

251

p h y s . s t a t . sol. (a) 54, 251 (1979) S u b j e c t c l a s s i f i c a t i o n : 10.2 a n d 14.4; 22.5.2 Department

of Electrical

and Electronic

Engineering,

University

of

Nottingham1)

The Relationship between the Defect Structure and the Intrinsic Ionic Conductivity of Monocrystalline Sodium Chloride By I). L . KIRK A s t u d y is m a d e of t h e r e l a t i o n s h i p b e t w e e n t h e i n t r i n s i c ionic c o n d u c t i v i t y of m o n o c r y s t a l l i n e s o d i u m c h l o r i d e a n d t h e d e f e c t s t r u c t u r e of t h e m a t e r i a l . C o n t r o l l e d d e f o r m a t i o n p l a c e s v a r i o u s d i s l o c a t i o n d e n s i t i e s w i t h i n t h e solid. C o n d u c t i v i t y s t u d i e s a r e m a d e p r i o r t o a n d f o l l o w i n g t h e d e f o r m a t i o n . T h e a n n e a l i n g b e h a v i o u r of t h e d e f e c t s t a t e a n d t h e i n t r i n s i c c o n d u c t i v i t y also is followed. E s werden die Beziehungen zwischen der ionischen Intrinsicleitfähigkeit von einkristallinem Xatriumchlorid und der Defektstruktur des Materials untersucht. Eine gesteuerte Deformation erzeugt verschiedene V e r s e t z u n g s d i c h t e n innerhalb des F e s t k ö r p e r s . Vor u n d n a c h der Deformation werden Leitfähigkeitsuntersuchungen durchgeführt. Das T e m p e r u n g s v e r h a l t e n der Defekte u n d der Intrinsicleitfähigkeit wird ebenfalls verfolgt.

1. Introduction T h e h i g h - t e m p e r a t u r e ionic c o n d u c t i v i t y of t h e simple rock salt s t r u c t u r e s has been a topic inviting m u c h e x p e r i m e n t a l investigation over m a n y years. T h e simple ionic s t r u c t u r e s of p o t a s s i u m a n d sodium chlorides present h i g h - t e m p e r a t u r e A r r h e n i u s c o n d u c t i v i t y plots t h a t accord well with models involving t h e f o r m a t i o n a n d movem e n t of intrinsic a n d extrinsically created point defects. Such models a n d e x p e r i m e n t a l d a t a were critically reviewed in earlier years b y L i d i a r d [1]. I n t h e past d e c a d e even more a t t e n t i o n has been focussed on t h e detailed b e h a v i o u r of t h e intrinsic regions of t h e c o n d u c t i v i t y b e h a v i o u r a n d in p a r t i c u l a r on the so called stage I of t h e A r r h e n i u s plot. Much effort h a s been spent in e x a m i n i n g t h e n o n - l i n e a r i t y of t h e region a n d t h e associated p o i n t defect m e c h a n i s m s t h a t control t h e h i g h - t e m p e r a t u r e c o n d u c t i v i t y . T h e most t h o r o u g h l y i n v e s t i g a t e d a n d discussed m a t e r i a l s h a v e been t h e simple monocrystalline solids of sodium a n d p o t a s s i u m chloride. A d e f i n i t i v e e x p l a n a t i o n of t h e c u r v a t u r e of stage I of these solids was originally p u t f o r w a r d b y B e a u m o n t a n d J a c o b s [2] who suggested t h a t t h e ionic conduction, at high t e m p e r a t u r e s , was d e t e r m i n e d b y t h e f o r m a t i o n a n d motion of anion a n d cation vacancies. T h e model proposed was t e s t e d b y c o m p u t e r f i t t i n g of o b s e r v a t i o n t o t h e A r r h e n i u s plot p r e d i c t e d b y t h e model, with o p t i m i z a t i o n of t h e f i t t i n g to yield t h e critical p a r a m e t e r s of t h e model. Since t h i s work m u c h a t t e n t i o n has been focussed on t h i s p a r t i c u l a r line of a p p r o a c h . H o w e v e r t h e m e t h o d of f i t t i n g a model w i t h u p t o eight v a r i a b l e p a r a m e t e r s to e x p e r i m e n t a l d a t a imposes a severe test for t h a t d a t a a n d considerable v a r i a t i o n s in b o t h t h e o b s e r v a t i o n s m a d e a n d t h e results o b t a i n e d f r o m t h e f i t t i n g p r o c e d u r e s h a v e been r e p o r t e d . R e c e n t l y a n i m p o r t a n t article b y M u r t h y a n d P r a t t [3] h a s reviewed some of t h i s previous work a n d has d e m o n s t r a t e d t h e possible weaknesses of t h i s m e t h o d of a p p r o a c h . Nottingham NG7 2RD, Great Britain.

252

D. L. KIEK

I n m a n y of t h e p r e v i o u s studies little or n o consideration has been m a d e of t h e mechanical s t a t e of t h e solid or of t h e m a n n e r in which t h e mechanical s t a t e m a y i n t e r a c t w i t h t h e p o i n t defect configuration t o influence t h e resulting c o n d u c t i v i t y of t h e m a t e r i a l . T h e aim of t h e present article is t o r e p o r t u p o n an e x a m i n a t i o n of t h i s p a r t i c u l a r p o i n t . T h e work d e m o n s t r a t e s q u i t e clearly t h a t t h e dislocation c o n t e n t of an ionic solid can influence q u i t e m a r k e d l y t h e m a g n i t u d e a n d t e m p e r a t u r e dependence of t h e intrinsic c o n d u c t i v i t y of monocrystalline sodium chloride. 2. Experimental Consideration T h e m e a s u r e m e n t s of ionic c o n d u c t i v i t y were p e r f o r m e d u p o n n o m i n a l l y p u r e , monocrystalline samples of sodium chloride. T h e y were grown f r o m t h e melt using a modified K y r o p o u l o s technique. T h e specimens utilized a c o n d u c t i v i t y rig t h a t has been described previously [4], This used a n ac m e a s u r e m e n t of t h e ionic c o n d u c t i v i t y , in a n a t m o s p h e r e of spectroscopically p u r e argon, with platinium/eolloidal g r a p h i t e elect r o d e s m a k i n g i n t i m a t e contact with t h e surface of t h e specimen. T h e c o n d u c t i v i t y a n d capacitance were m e a s u r e d a t an a n g u l a r f r e q u e n c y of 10000 r a d s _ 1 (1592 Hz). C o n d u c t i v i t y m e a s u r e m e n t s were also t a k e n over a r a n g e of frequencies f r o m 50 H z to 20 k H z to ensure t h a t t h e r e was no significant dispersion of t h e c o n d u c t i v i t y values. T h e mechanical s t a t e of t h e specimens was varied by increasing t h e dislocation c o n t e n t of t h e solid. T h i s was achieved with controlled d e f o r m a t i o n of t h e specimens in a n I n s t r o n t e s t i n g machine. A typical g e o m e t r y of t h e specimens investigated is displayed in Fig. 1, along with t h e load-compression characteristic of t h e material. T h e elastic, stage I easy glide a n d stage I I h a r d e n i n g of t h e material are readily discernible. T h e slip planes a n d slip b a n d s of t h i s material are also shown along with a n indication of t h e geometric axes along which t h e d e f o r m a t i o n was applied. Specimens which were utilized in t h e c o n d u c t i v i t y studies were d e f o r m e d into stage I a n d I I of t h e d e f o r m a t i o n characteristic. T h e a p p r o p r i a t e p e r c e n t a g e d e f o r m a t i o n associated with t h e defined g e o m e t r y of specimen used for c o n d u c t i v i t y i n v e s t i g a t i o n are also indicated. T h e dislocation c o n t e n t of t h e material a f t e r t h e controlled deform a t i o n was s t u d i e d with optical microscopy. This dislocation c o n t e n t was e x a m i n e d b y etch pit studies. T h e e t c h a n t s a n d m e t h o d o l o g y used to reveal t h e dislocations h a v e been described previously in an article by D a v i d g e a n d P r a t ' [5]. 3. Experimental Observations T h e h i g h - t e m p e r a t u r e ionic c o n d u c t i v i t y of a n o m i n a l l y pure monocrystalline sample of sodium chloride is p r e s e n t e d in Fig. 2. T h e d a t a p o i n t s were o b t a i n e d f r o m cond u c t i v i t y d e t e r m i n a t i o n s p e r f o r m e d u p o n f o u r d i f f e r e n t specimens cleaved f r o m t h e same t r a n s v e r s e platelet of a crystalline boule of p u r e m a t e r i a l . T h e specimens were u n a n n e a l e d a f t e r cleavage. T h e r e was excellent reproducibility which did not require 25 i

0

0.08

0.15

Oil

0.32 0A0 compression (mm)

0

Fig. 1. A load-extension characteristic of the monocrystalline specimens of sodium chloride utilized in the present study. Compression was applied in the . 7 5 9 ( 1 9 6 4 ) . TAYLOR a n d P . L . PRATT, P h i l . M a g . 4 , 6 6 5 ( 1 9 5 9 ) . W . D R E Y F U S a n d A . S . NOWICK, P h y s . R e v . 12, 1 3 6 7 ( 1 9 6 2 ) .

[8] K . THARMALINGHAM. J . P h y s . Chem. Solids 25, 225 (1964). [ 9 ] J . F . LAURENT a n d J . F . B E N A R D , J . P h y s . C h e m . S o l i d s 7, 2 1 8 ( 1 9 5 8 ) .

[10] H . C. GRAIIAJI. Dissertation, Ohio S t a t e University, 1965. [ 1 1 ] R . L . MOMENT a n d R . B . GORDON, J . a p p l . P h y s . 3 5 , 2 4 8 9 ( 1 9 6 4 ) .

[12] A. R. ALLNATT a n d P. W. M. JACOBS, Trans. F a r a d a y Soc. 58, 116 (1962). [13] R . G. FULLER, P h y s . R e v . 142, 5 2 4 (1966).

[14] D. L. KIRK, Ph.D* Thesis. U n i v e r s i t y of L o n d o n , 1968. [ 1 5 ] L . \ V . BARK, J . A . MORRISON, a n d P . A . SCHROEDER, J . a p p l . P h y s . 3 6 . 6 2 4 ( 1 9 6 5 ) .

(Received

January

15,

1079)

S. P. S. B a d w a l and H. J . de B r u i n : Complex Impedance Dispersion Analysis

261

phys. stat. sol. (a) 54, 261 (1979) Subject classification: 13.3 and 14.3.4; 21.6; 22.6 School of Physical Sciences,

The Flinders

University of South Australia, Bedford

Park1)

Electrode Kinetics at the Pt/Yttria-Stabilized Zirconia Interface by Complex Impedance Dispersion Analysis By S. P. S. B a d w a l and H. J. d e B k u i n Low frequency impedance dispersion analysis, attributed to polarization at the electrode-electrolyte interface, exhibits a strong frequency dependence of the equivalent parallel resistor and capacitor. Such response for Pt/YSZ/Pt cells is analysed as a Cole-Cole complex impedance-admittance dispersion with a relaxation distribution. A physical interpretation based on the results obtained and observations of reactions between noble metals and ceramic oxides, suggests the creation of a complex polarized layer at the electrode-electrolyte interface involving a proliferation of cationic species as the possible cause of irreversible energy losses in the Faradaic reactions. Their kinetics is found to change with the oxygen partial pressure of the atmosphere. Die Dispersionsanalyse der niederfrequenten Impedanz, die der Polarisation an der ElektrodenElektrolyt-Grenzfläche zugeschrieben wird, zeigt eine starke Frequenzabhängigkeit des äquivalenten, parallelen Widerstands und der Kapazität. Ein derartiger Response für Pt/YSZ/PtZellen wird als komplexe Cole-Cole-Impedanz-Admittanzdispersion mit einer Relaxationsverteilung analysiert. Eine physikalische Interpretation, die auf den erhaltenen Ergebnissen beruht sowie auf Beobachtungen der Reaktionen zwischen Edelmetallen und keramischen Oxiden, weist auf die Bildung einer komplexen Polarisationsschicht an der Grenzfläche Elektrode-Elektrolyt hin, die eine Anregung der Kationenarten als möglichen Grund für die irreversiblen Energieverluste in den Faraday-Reaktionen einschließt. Es wird gefunden, daß deren Kinetik sich mit dem Sauerstoffpartialdruck der Atmosphäre ändert. 1. Introduction

Interest in high-temperature electrolytic cells has concentrated largely on the study of the solid electrolytes with emphasis on transport numbers, concentrations of charge carrying species, electronic and ionic defects and their mobilities and the effects of oxygen partial pressure, dopant concentration, impurities, grain boundaries, and second phase precipitation on conductivity [1 to 3]. To evaluate the performance and application of high-temperature electrolytic cells, a knowledge of charge transfer phenomena at the electrode-electrolyte interface is essential. Y e t a minority of authors [4 to 8] have addressed themselves to this problem. Our recent observations on reactions between noble metals and refractory oxides [9, 10] must have an important bearing on the interpretation of charge transfer data. Until then the chemical inertia of noble metal electrodes in cells operating at high temperatures in oxidizing atmospheres had been tacitly assumed. In this paper we apply complex impedance-admittance dispersion analysis to cells of the type Pt, P o J Y i - x Z r x O 1 5 + 0 5 x IPt, P 0 z (for x = 0.818 and x = = 0.852, to study the reaction between P t and an ionically conducting ceramic oxide. As a basis we use our recent analysis [11] of charge transfer and polarization events at the electrode-electrolyte interface which gives rise to the non-ideal behaviour of such solid state galvanic cells. I f the only event at a noble metal electrode-solid electrolyte interface were the Faradaic reaction, -i- 0 2 + 2e~ = O 2 - , following a standard kinetic sequence in which Bedford Park, 5042, South Australia.

262

S. P . S. BADWAL a n d

H . J . DE

BRUIN

a slow s t e p d o m i n a t e s t h e r e a c t i o n r a t e , t h e n in t h e complex i m p e d a n c e p l a n e one would expect an ideal semicircle whose r a d i u s is a m e a s u r e of t h e r a t e c o n s t a n t for t h e r e a c t i o n . I n solid s t a t e cells at low frequencies (5 H z to 5 k H z ) a n d high t e m p e r a t u r e s (600 t o 1000 CC) t h e observed semicircles are s u b s t a n t i a l l y depressed (15 to 45°) below t h e real axis. T h e a c t u a l kinetics of t h e charge t r a n s f e r is little influenced b y t h e properties of t h e g r a i n b o u n d a r i e s in t h e electrolyte, b u t more so b y t h e oxygen m i g r a t i o n t h r o u g h t h e b u l k [12]. E l e c t r o d e p r e - t r e a t m e n t t o affect t h e reaction m e c h a n i s m by c h a n g i n g t h e d e n s i t y of t r i p l e - p h a s e - c o n t a c t s [13, 14] has a minor influence on t h e depression of t h e arc. I n general physical variables imposed on t h e interface cannot elim i n a t e it [15 t o 17] a n d chemical interference should be considered as a possible cause for t h e observed i m p e d a n c e b e h a v i o u r . F a r a d a i c side reactions can t h e n be seen t o c r e a t e t i m e d e p e n d e n t c a p a c i t o r s a n d resistors which give rise to t h e observed depressions of t h e semicircles a s proposed b y several a u t h o r s [18 to 22], I t implies t h a t M a c D o n a l d ' s c o n d i t i o n of single species of mobile charge of either p o l a r i t y will bo d i f f i c u l t t o achieve in such solid cells [15, 16]. O u r earlier model to q u a n t i s e t h e non-ideal b e h a v i o u r of t h e i m p e d a n c e arcs [11, 18] s u g g e s t s t h e existence of a complex capacitive layer in t h e solid electrolyte surface in which p o l a r i z a t i o n involves a multiplicity of charge carriers of b o t h polarities. T h e composition of t h e layer a n d hence its c a p a c i t a n c e changes w i t h f r e q u e n c y . Loss due t o p a r a s i t i c short-circuiting reactions also increases with f r e q u e n c y , as t h e thickness of t h e c a p a c i t o r decreases. T h e m u l t i p l i c i t y of charge carriers inherent ly causes a r a n g e of elect r o n t r a n s f e r reactions with d i f f e r e n t r a t e s reso n a t i n g a t different frequencies. T h i s t o g e t h e r with t h e changes in loss p h e n o m e n a gives rise to t h e variable resistor in t h e equivalent h'C circuit. 2. Experimental T h e i m p e d a n c e was m e a s u r e d for cells consisting of t w o identical parallel p l a t i n u m elect r o d e s in contact with discs of 8 or 10"'/ 0 Y 2 0 3 stabilized Z r 0 2 (YSZ). M e a s u r e m e n t s were m a d e in p u r e o x y g e n , air. or argon between 450 a n d 1000 °C in t h e f r e q u e n c y i n t e r v a l 5 H z to 10 M H z . YSZ discs a p p r o x i m a t e l y 10 m m 0 , between 1.5 a n d 4 mm thick a n d d e n s i t y 8 9 % of t h e theoretical, were o b t a i n e d f r o m Ceramic F a b r i c a t o r s L t d . Australia a n d polished with various grades of silicon carbide p a p e r s , diam o n d p a s t e s (down t o 1 to 2 fi.ni), a n d a l u m i n a powder (0.1 to 0.3 ¡¿m). E l e c t r o d e s were m a d e

Fig. 1. Experimental cell assembly. 1 Alumina tube, 2 specimen, 3 alumina support, 4 square hole cut in alumina tube (1), 5 stainless steel shield, 6 alumina sheathing, 7 thermocouple, 8 alumina rod for applying slight compressive stress to cell components, 9 stainless steel cage, 10 brass structural members

Electrode Kinetics by Complex Impedance Dispersion Analysis

263

from Johnson Matthey platinum paste N 758. A uniform thickness of the electrodes was obtained by repeated application of platinum paste followed by firing at 1000 °C. Experimental details have been described elsewhere [19]. I n s u m m a r y : oxygen partial pressures were measured using a CoO wire probe [23]. A Hewlett-Packard impedance meter, Model 4800A (frequency range 5 Hz to 500 kHz), and an rf vector impedance meter, Model 4815A (frequency range 500 k H z to 108 MHz) were used to measure amplitude and phase angle within + 5 % . The signal applied to the cell was usually 2.7 mVr.m.s. Impedance of various RC circuits made up of standard resistors and capacitors was compared with the simulated values to estimate the inductive effects in the shielded leads and to calibrate the impedance meters. All experimental equipment was operated inside an doubly shielded Faraday cage. The furnace assembly was modified for atmospheric control around the cell (Fig. 1). Initially the temperature of the sample was raised to 1000 °C and the system allowed to equilibrate with the surrounding gas atmosphere for several days. ¡A dc current of 100 to 150 mA was passed for 1 h in either direction to stabilize the electrodes. The impedance was then measured at intervals of 50 to 60 °C after thermal equilibrium had been established. When the minimum temperature had been reached the experiments were repeated for an increasing temperature cycle. Most measurements were made between 5 Hz to 500 k H z during the off cycle of the temperature controller. 3. Results and Discussion The impedance arc due to Faradaic dispersion at the electrode-electrolyte interface was generally observed between 5 Hz and 5 kHz. I t s length varied substantially with oxygen pressure. Warburg impedance was only observed (Fig. 2 a and b) for experiments in argon (Pq., = 3 X 10" 6 atm), but may be operative in air and oxygen at fractional frequencies outside the range available to us. The centres of the Faradaic arcs were invariably depressed substantially below the real axis; those for space charge phenomena a t the grain boundaries in the electrolyte considerably less and those for the bulk conductivity hardly at all (Fig. 2 c). For thick electrolyte discs ( > 2 mm) i overlap of relaxation was observed, par^ J • * *g'z* * * * • , ticularly at intermediate temperatures. " 500hz Analysis of the d a t a for such cells was • ¿"500Hz difficult and led to large relative errors. | 5\kHz^ i i i i ; i i a Such d a t a were generally not processed. 12 18 2{ 30 Using the Cole-Cole analysis for disper6 2'IQ) --»- s ion data [24], similar to that by Raistrick i 500Hz '»5kHz

5 H i f 50 Hz 0.04

0.12

0.16 6 (Q~'

Fig. 2. a) Impedance and b) a d m i t t a n c e response for a P t / Y S Z / P t cell a t 996 °C (in argon including a Warburg impedance), c) Impedance response of the same cell in 0 2 a t 457.4 °C showing the depressions for three relaxation p h e n o m e n a : Faradaic reaction (right), space charge (middle), and bulk conductivity (left)

264

S. P . S. B a d w a l and H. J . de B r u i n

Cv •z'

Fig. 3. Equivalent circuits and their dispersion in the impedance and admittance planes, a) Angle of depression 0 —f(a/b), relaxation r 0 ; variable ; b) 0 = f(a/b), relaxation variable; c) 0 = /(a/b), relaxation M0Cym/cos 0%

d) 0 =f(a/b),relaxation RoCvm/eos0; e)

0

=

/(Ceo,

djb),

Rut-\-ao) i "-i/2\2 r ,w r 1/'!)2+(o (Bct + ao)

-- 1/2)2 g +

-\PM 1'-)2 +

aoy

>

- , + boj'1 + O J C . {aa> 1/2)2

(7)

This only slightly modified the values of the circuit parameters (Table 1) obtained above and values of C1^ obtained were very small and lay between 0 to 10 a F . Furthermore the relaxation time t~1 = com = cos & j C s m R a . In Fig. 4 are shown the simulated impedance responses for circuit 3f. For these carves R0, n, Rcl, Cand alb were identical, but the individual values of a and b were increased by an order of magnitude. This resulted in rotation of the axis by 2°, 16.8°, and 24°, respectively. Inversely non-linear regression analysis of these curves was in-

Fig. 4. Impedance response for the circuit 3f for the conditions:

Cco (|iF) Rei (Ü)

Jf 0 (Q) • A •

24.0 24.0 24.0

0.72 0.72 0.72

9.33 x 109.33 x 10" 9.33 X 10"

2.0 X 10" 2.0 x 10" 2.0 X 10"

20 20 20

6.0 6.0 6.0

266

S. P . S. BADWAL a n d H . J. DE BRUIN

35.3° 350 Fig. (>

Fig. 5 15 10 -

20 Hz lOOHZn.

5 _500Hz 7kHz^ WiOOHz 5kHzj& H0\kHz "•28A0 20 2S°2S£o

X

\

V

* \

•

\

\

30

Fig. 7 Fig. 5. Cell's impedance in oxygen (symbols). The s i m u l a t e d curves (solid lines) h a v e been calc u l a t e d using circuit d in Fig. 3 for t h e values of circuit p a r a m e t e r s in Table 1. • T = 99G.8, • 947.0, • 902.2, • 850.7, A 799.2 °C Fig. 6. Cell's i m p e d a n c e in air. A T = 993.2, A 946.1, O 900.2, • 845,5. v 813.3, • 779.7 °C Fig. 7. Cell's impedance response in argon. The dashed lines were s i m u l a t e d using circuit in Fig. 3 d (or = 0) a n d t h e solid lines t h a t in 3g. A T = 99C.0, A 949.5, o 902.0, • 852.0, v 800.2, • 729.8 °C'

sensitive to the difference between circuits d and f in Fig. 3. It did not provide a n accurate estimate of C^ especially when values of b and a were large. However, t h e angle of depression of 24° in the last case is very close to t h e expected value of 25° for C x = 0. T h e m a g n i t u d e of C ^ is too small to influence the F a r a d a i c dispersion a n d can therefore not be derived from such low frequency d a t a in contrast to the analysis of high frequency arcs. The d a t a in Fig. 5 a n d 6 are representative of t h e typical impedance response in pure oxygen and air, respectively. The simulated curves using t h e circuit of Fig. 3 d are superimposed, showing t h a t agreement is good within the experimental error. The results in argon under the influence of a W a r b u r g impedance in series with B 0 required a n analysis according to the circuit in F i g . 3g, as shown in Fig. 7.

267

Electrode Kinetics by Complex Impedance Dispersion Analysis

© «

© ©

© © © © © ©

©

X X X X X

X X X X X X

O0 h; CO Li Ï . t^ -< « -c -i

« « ^ » - ^ oc: N e-i f î x w «

I !

CC 30 C PÏ t-^ Oî 02

o

o

-

(SXi/dp0l)r

+

+

0

ifc. t

i?e l

-

0

t0

T h e physical model proposed here, which in its simplest form is represented by t h e Faradaic equivalent circuit in Fig. 3 f recognizes the presence of the ideal charge transfer resistance and double layer capacitance, as well as the frequency dependent compo-

Electrode Kinetics by Complex Impedance Dispersion Analysis

269

nents due to the complexity of events at the interface between the solid ionic conductor and the metal electrode. As suggested elsewhere [11, 18], the proliferation of charged species near the interface involves noble metal cations a s well a s those of the electrolyte in various stages of oxidation and reduction. The anionic species are qualitatively not so varied, but certainly not unique, a s postulated in MacDonald's impedance dispersion theory [15]. The ionic mobilities in the solid electrolyte are slow compared with aqueous systems and consequently the composition and dimensions of the near-interface polarized region will vary with frequency. Not surprisingly, loss phenomena in this region are dependent on frequency as the simple charge transfer in the F a r a d a i c reaction is complicated by parasitic electron transfers between the host of cationic and anionic species whose compositions and concentrations are a function of the frequency (and amplitude) of the applied potential. The interface between two solids is not easily reproduced from specimen to specimen. Surface roughness, and the presence of impurities complicates reproducibility of results. So do segregation-combination of defect pairs in the surface layers, plastic deformation and residual stresses. Warburg impedance does not affect the non-ideal behaviour of electrode reactions [11] and is only significant a t very low oxygen partial pressures and very low frequencies in fluorite related oxide electrolytes. The distribution of relaxations expressed by the depression of the centre of arc in the Cole-Cole analysis is a mathematical model which relates only the product RC to frequency. Y e t these quantities are physically distinct with a resistance related to a reaction rate and the concentrations and diffusion constants of reactants and products, while a capacitance is a measure of the degree of polarization a t interfaces. Furthermore it is not difficult to fit a Cole-Cole arc to d a t a generated on the basis of equivalent circuits involving a limited number of nested and/or series RC circuits each representing individual steps in competing F a r a d a i c reactions. The analysis is a s yet not a d e q u a t e to unraffle the complexity of the reaction kinetics causing the low frequency impedance dispersions but it is based on a physical interpretation a n d supports earlier observations on the reactions taking place at the noble-metal-ceramic interface at elevated temperatures. 4. Conclusion Polarization phenomena at the electrode-electrolyte interface, other than those due to the double layer and F a r a d a i c charge transfers, are clearly significant. The variable capacitance is large compared with the constant value of the double layer and becomes a m a x i m u m under dc conditions. T h e d a t a obtained here support our earlier suggestion that the proliferation of cationic species near the interface contributes to the creation of such a variable polarized layer with parasitic charge transfer losses. These can contribute to irreversible energy losses, such as in high temperature fuel cell operations. Acknowledgements

We gratefully acknowledge the computational help of Mr. C. K l a u b e r a n d Mr. T . D. Ainsworth. This project was supported by a grant from the Australian Research Grants Commission. References [1] S . H . CHU a n d M. A . SEITZ, J . S o l i d S t a t e C h e m . 2 3 , 2 9 7 (1978). [2] T . H . ETSELL a n d S . N . FLENGAS, C h e m . R e v . 70, 339 (1970).

[3] L. HEYNE, Nat. Bur. Standards Spec. Publ. No. 296, 149 (1968).

[4] T . H . ETSELL a n d S . N . FLENGAS, J . E l e c t r o c h e m . S o c . 1 1 8 , 1 8 9 0 (1971). [5] R . J . BROOK, W . L . PELZMANN, a n d F . A. KROGER, J . E l e c t r o c h e m . S o c . 1 1 8 , 185 (1971).

270

S. P. S. BADWAL and H. J . DE BRUIN: Complex Impedance Dispersion Analysis

[ 6 ] H . YANAGIDA, R . J . BROOK, a n d F . A . K R O G E R , J . E l e c t r o c h e m . S o c . , 1 1 7 , 5 9 3 ( 1 9 7 0 ) . [ 7 ] R . L . ZAHRADNIK, J . E l e c t r o c h e m . S o c . 1 1 7 , 1 4 4 3 ( 1 9 7 0 ) . [ 8 ] M . K L E I T Z , P . F A B R Y , a n d E . SCHOULER, i n : F a s t I o n T r a n s p o r t i n S o l i d s , E d . \ Y . VAN* GOOL,

North-Holland, Amsterdam 1973 (p. 439). [ 9 ] H . J . DE B R U I N , A . F . MOODIE, a n d C. E . W A R B L E , J . M a t e r . S c i . 7, 9 0 9 ( 1 9 7 2 ) .

[10] H. J . DE BRUIN, N a t u r e '272 (5655), 712 (1978). [ 1 1 ] H . J . DE B R U I N a n d S . P . S . BADWAL, J . A u s t r a l . C e r a m . S o c . 1 4 , 2 0 ( 1 9 7 9 ) . [12] P . FABRY a n d M . KLEITZ, J . E l e c t r o a n a l . C h e m . 57, 165 (1974).

[13] D. O. RALEIGH, Electroanalytical Chemistry, A Series of Advances, Vol. 6, Ed. A. J . BARD, Marcel Dekker, Inc., New York 1973. [14] S. PIZZINI, in: Fast Ion Transport in Solids, Ed. W. VAN GOOL, North-Holland, Amsterdam 1 9 7 3 (p. 4 6 1 ) . [15] J . R. MACDONALD, in: Electrode Processes in Solid State Ionics, Ed. M. KLEITZ and J . DUPUY, Reidel, Dordrecht 1976 (p. 149). [ 1 6 ] D . R . FRANCESCHETTI a n d J . R . MACDONALD, J . E l e c t r o a n a l . C h e m . 8 2 , 2 7 1 ( 1 9 7 7 ) .

[17] R. J . FRIAUF, J . chem. Phys. 22, 1329 (1954). [18] [19] [20] [21] [22]

H . J . DE B R U I N a n d S . P . S . BADWAL, p h y s . s t a t . s o l . ( a ) 4 9 , K 1 8 1 ( 1 9 7 8 ) . S. P . S . BADWAL a n d H . J . DE B R U I N , A u s t r a l . J . C h e m . 3 1 , 2 3 3 7 ( 1 9 7 8 ) . A . K . JONSCHER, p h y s . s t a t . s o l . ( a ) 3 2 , 6 6 5 ( 1 9 7 5 ) . I . D . RAISTRICK, CHUN H O , a n d R . A . HUGGINS, J . E l e c t r o c h e m . S o c . 1 2 3 , 1 4 6 9 ( 1 9 7 6 ) . I . D . RAISTRICK, CHUN H o , Y A W W E N H U , a n d R . A . HUGGINS, J . E l e c t r o a n a l . C h e m . 319 (1977).

7",

[23] A. DUQUESNOY and F. MARION, Bull. Chim. Fr. 77 (1964). [24] K. S. COLE and R. H. COLE, J . chem. Phys. 9, 341 (1941). (Received

March

2,

1979)

F. M o n t h e i l i . k t and J . M. H a t t d i n : Coherent Precipitation near Dislocations

271

phys. stat, sol. (a) 54, 271 (1979) Subject classification: 1.4 and 10.2; 22.8.2 Centre de Mise en Forme des Matériaux, Ecole des Mines de Paris, Sophia Antipolis,

Valbonne1)

Coherent Precipitation near Dislocations A Theoretical Analysis

By F . M o n t h e i l l e t a n d J . M. H a u d i n A simple theoretical model is developed for a dislocation decorated by a cylindrical precipitate, which makes it possible to derive the energy change A IF associated with the precipitation process versus radius R of the precipitate. The distance a between the cylinder axis and the dislocation line is taken so as to minimize A IF. When the chemical and elastic energy terms are simultaneously considered, AJF(-R) is obtained by numerical computation. However, when the chemical energy terms are negligible, it is possible to calculate analytically the minimum of A W(a, R). In the former case, the precipitate grows until complete depletion of solute atoms in the matrix, after a very small critical radius is reached. In the latter case, growth leads to equilibrium radius and location. I n both the cases, decoration of the dislocation is more likely to happen than homogeneous precipit a t i o n of coherent spheres. ?vous proposons un modèle théorique simple de dislocation décorée par un précipité cylindrique permettant de calculer la variation d'énergie AH' associée au processus de précipitation en fonction du rayon R du précipité. La distance a entre son axe et la ligne de dislocation est choisie de façon à minimiser A IF. Lorsque les termes d'énergie chimique et élastique interviennent simultanément, la courbe AIF(-R) est obtenue numériquement. Par contre, dans le cas où les termes d'énergie chimique sont négligeables, on peut calculer analytiquement le minimum de AlF(a, R). Dans le premier cas, le précipité croît jusqu'à épuisement des atomes de soluté, après avoir franchi un rayon critique de germination voisin de zéro. Dans le second cas, la croissance aboutit à vin rayon et une position d'équilibre. Dans les deux cas, la décoration de la dislocation est énergétiquement plus favorable que la précipitation homogène de sphères dans la matrice. 1. Introduction I n m a n y alloys s u b m i t t e d to a n ageing t r e a t m e n t a f t e r annealing, p r e c i p i t a t i o n o c c u r s o n or n e a r d i s l o c a t i o n s . T h i s " d e c o r a t i o n " i n d u c e s s p e c i f i c d i f f r a c t i o n c o n t r a s t s in e l e c t r o n m i c r o s c o p y : i m a g e s consist of s e v e r a l a l t e r n a t i v e l y d a r k a n d b r i g h t f r i n g e s , s i m i l a r t o t h o s e of c o h e r e n t p r e c i p i t a t e s d i s c u s s e d b y A s h b y a n d B r o w n [1], S u c h p r e c i p i t a t i o n of c a r b i d e s [2, 3], n i t r i d e s [4, 10], or i n t e r m e t a l l i c c o m p o u n d s [5] o n dislocations has been previously reported. Only a few a t t e m p t s h a v e been m a d e t o justify theoretically these phenomena: the free energy change associated with decorat i o n h a s b e e n c a l c u l a t e d b y C a h n [6] a n d D o l l i n s [4]. T h e f o r m e r a s s u m e d i n c o h e r e n t p r e c i p i t a t i o n , a n d t h e l a t t e r c o n s i d e r e d s p h e r i c a l or d i s k - s h a p e d c o h e r e n t p r e c i p i t a t e s . H o w e v e r , in b o t h cases s o m e t e r m s h a v e b e e n n e g l e c t e d in e l a s t i c e n e r g y c a l c u l a t i o n s , a n d t h e r e s u l t s c o u l d b e o b t a i n e d o n l y b y n u m e r i c a l c o m p u t a t i o n . D u n d u r s [11] h a s c a l c u l a t e d t h e e q u i l i b r i u m p o s i t i o n s of a s c r e w o r e d g e d i s l o c a t i o n n e a r o r i n s i d e a cylindrical precipitate. This rather sophisticated derivation only takes into a c c o u n t t h e d i f f e r e n c e b e t w e e n e l a s t i c c o n s t a n t s of m a t r i x a n d s e c o n d p h a s e . T h e r e f o r e , a l t h o u g h precise results are given, practical inferences c a n n o t be d r a w n concerning t h e d e c o r a t i o n of d i s l o c a t i o n s in a l l o y s . !) 06560 Valbonne, France.

272

F . MONTHEILLET a n d J . M .

HAUDIN

I n this work, a simple model is developed from experimental observations, which is able to t a k e into account all physical relevant parameters. I t is further shown, in t h e case of intermetallic compounds, that the problem can be solved merely by analytical calculation. 2. Energy Change Associated with Decoration 2.1

The

model

I n most cases, precipitates on dislocations are likely to be coherent with t h e matrix, a t least in the first stages of precipitation. Accordingly, the precipitation process consists in substituting a number of atoms in the matrix for an equal number of different atoms, which will form t h e second phase (Fig. 1). T h e corresponding displacement field is assumed to be U = U

B

+

UP ,}

(1)

where t / D and J7 P are the displacement fields of a dislocation without precipitate and of a single coherent precipitate, respectively. These displacements will be calculated in isotropic linear elasticity theory. W h e n the m a t r i x and second phase have the same elastic constants, U is t h e exact solution of the problem as resulting from superposition of two equilibrium states. However, (1) is only expected to give a rather good approximation of the exact displacement field in general; corrections could then be brought b y using image forces [8]. A t the beginning of the precipitation process, the solute atoms diffuse in the crystal towards the region of minimum or maximum hydrostatic pressure, according to whether they are bigger or smaller than the matrix atoms (Cottrell effect). T h e map of hydrostatic pressure of an edge dislocation (Fig. 2) shows that precipitation is e x p e c t e d to start on t h e 0y s y m m e t r y axis. I n agreement with numerous experimental observations [5], supported by computer simulations [5, 7], it is supposed t h a t t h e p r e c i p i t a t e at a n y given moment is a cylinder of radius R and of large length t h a t can be taken as infinite. I t s axis lies parallel to dislocation line D at an algebraic distance a. T h u s the displacement field U remains unchanged by a n y translation parallel to D . T h e strains t a k e place only in the 0xy plane normal to D. F u r t h e r m o r e , if the diffusion rate is high enough, a will minimize t h e t o t a l energy of the system for each radius R. I t would be desirable to calculate the change AG of free enthalpy in t h e precipitation process. However, the transformation takes place under constant temperature and pressure and furthermore changes of entropy and volume can be reasonably neglected. T h u s we can merely use the energy change AW = J F D D — WB , where

Fig. 1. Formation of a coherent precipitate near a dislocation, a) Dislocation without precipitate, b) decorated dislocation. The second phase is constituted here by only one atomic species

273

Coherent Precipitation near Dislocations

[O

Ô" a

—

R* =

—

2710

\a*\

—

^^

|o*|

0)

Mm-

Fig. 2 4

200

-

180 g 160 1W 120 100

\\ \

80 60

0 20 W 60 80 wo 11=76%

no

10

thmn)-

Fig. 3

- , , , MU

0 20 W 60 80 i)"76 %

W ft.win) -

Fig. 4

Fig. 1. Shear modulus G of 76% cold-drawn electrolytic copper which subsequently has been annealed at temperatures from 200 K till 700 K . Measurements were performed at room temperature. Points I and II f r o m Griineisen. All values of G are corrected for small texture [14] Fig. 2. Young's modulus E of the specimens of Fig. 1 Fig. 3. Electrical conductivities y. of the specimens of Fig. 1 .x = 53: Griineisen's specimen Fig. 4. Viekers hardness VH (30/30) of the specimens of Fig. 1

294

H. H. WAWRA

700 K leads to a complete recrystallization of the specimen during which practically 3 ) all structural defects and also some texture which may he present in the cold-drawn specimen, disappear. The shear modulus then assumes its maximum value which turns out to be that of Bradfield. In a certain sense, the polycrystal is now in an ideal state. It is hard to say quantitatively how much the disappearing of the texture during the recrystallization contributes to the modulus effect. That this contribution is small can be seen as follows: A < l l l > - t e x t u r e in copper lowers 0 so that G increases when this texture is removed. On the other hand, a < l l l ) - t e x t u r e in copper raises E so that E decreases if the texture is removed. This is opposite to the results shown in Fig. 2 so that we conclude that the texture effect is not decisive in our problem. In order to specify the internal situation of the specimens somewhat better we have added measurements of the Yickers hardness (VH) and of the electrical conductivity. The decrease of VH is a measure of the disappearance of the sources of the hardness, i.e. of the structural defects. The electrical conductivity x, on the other hand, is sensitive not only to these defects but also to impurity atoms of another species. Griineisen reports y. = 53 and 32 m/iimm 2 for the specimens I and I I , respectively, whereas our results on rather pure copper lie within 56.7 and 58.2 m/Qmni 2 . They provide evidence that Griineisen's specimens were not of very pure copper. Griineisen himself reports an impurity content of 0.15% As and 0.03% Fe for specimen I. The effect of these impurities on the moduli, however, is certainly much smaller than the effect under discussion in this paper so that we need not go into more details. 4 . Conclusions

All theories known to us which calculate effective polycrystal moduli from the single crystal moduli (C,A ) assume that the latter moduli are those of the in situ crystallites. Thus if one wants to derive the cold-work (e.g. Griineisen) moduli one has to use the moduli of single crystals which have the defect structure of the crystallite in situ. These moduli, however, are not available. Cold-working of a single crystal do;\s not help because the defect structure developing in a plastically deformed single crystal is different from that obtained in a crystallite of a deformed polycrystal. On the other hand, both well-annealed single crystals and the crystallites of a well-annealed polycrystal are practically defect-free. This means that the quota of lattice defects (point defects and dislocations) influencing the elastic parameters in both species is negligible so that they have the same elastic moduli. This allows us to use the experimental moduli of well-annealed single crystals as if they were measured crystallite moduli of well-annealed polycrystals. Clearly the theoretical polycrystal moduli then obtained are to be compared with experimental moduli of well-annealed polycrystals, for instance those of Bradfield. This discussion allows us to draw the following conclusion: Good agreement between theoretical predictions based upon the moduli of annealed single crystals and Griineisen's polycrystal moduli (e.g. for copper G = 4.552, E = 12.263) shows that this theory is quantitatively not correct. As examples which stand for many others we quote the theories of Laurent and Eudier [5], Huber and Schmid [15], and Shibuya [16]. Good theories based on defectfree single crystals give results close to those of Bradfield (e.g. for copper G = 4.83, E = 12.96). In order to calculate such values one needs the moduli of well-annealed single crystals. Such moduli have been provided in large numbers by the precise 3 ) The term "practically" indicates that the number of defects is not zero but so small that they have only a negligible influence on the elastic moduli.

Role of Grüneisen's Elastic Parameters in the Theory of Elasticity

295

measurements of the National Physical L a b o r a t o r y in Teddington [17]. F o r copper t h e values are Cu = 16.905, C 12 = 12.193, C 44 = 7.55. I f the comparison between theory and experiment is performed along t h e indicated lines one finds t h a t the so-called self-consistent elastic moduli, first discussed b y Hershey [18] and K r ö n e r [19], are very close to t h e experimental values of t h e annealed specimens in almost all cases. There are exceptions, for instance for graphite, where the texture- and defect-free state cannot be easily reached. T h e self-consistent moduli always lie between the second-order bounds of Hashin and S h t r i k m a n [20] and the third-order bounds of K r o n e r [11]. F o r instance, the self-consistent shear modulus of an annealed polycrystal of copper is 4 . 8 2 6 when calculated with the single crystal moduli given above. T h e third-order bounds are 4.77 and 4 . 8 6 . T h e Voigt-Reuss-Hill ( V R H ) arithmetic average is 4.72, i.e. outside t h e thirdorder bounds. I t has recently been explained b y K r ö n e r [12] t h a t the self-consistent moduli are the true effective moduli of a macro-homogeneous and macro-isotropic polycrystal provided the arrangement of the grains is as irregular as possible, i.e. the polycrystal is in a state of " p e r f e c t disorder". T h e l a t t e r term has been given a quant i t a t i v e meaning in K r o n e r ' s work. Apparently, t h e annealed polycrystal comes very close to t h e condition of perfect disorder. On t h e other hand, there is no theoretical basis at all that the V R H prescription has anything to do with this condition. Our considerations certainly do not devaluate the great merit of Grüneisen's work whose aim was to provide the numerical values of the elastic moduli of materials of practical importance. Cold-worked materials are very important for m a n y applications. T h e study of Grüneisen's work shows that his specimens were produced b y cold-work after which "tiefgreifende Änderungen durch thermische oder mechanische B e h a n d lung vermieden wurden" ([1], p. 846). At t h a t time, nobody knew b e t t e r t h a n Griineisen about the complicated nature of the internal s t a t e of solid materials. Hei writes: " W e d e r der Einfluß mechanischer oder thermischer Behandlung noch derjenge der chemischen Zusammensetzung scheint mir bisher hinreichend festgestellt zu s e i n " ([1], p. 802). T h e point is that thermal and mechanical t r e a t m e n t change the internal state of a material thereby also changing its elastic moduli, sometimes b y a considerable amount.

References [1] [2] [3] L4] [5] [(>] ¿7J

¡8] [9| I'lOj [Ill [12 j [13]

[14] [15] [16]

E. G R Ü N E I S E N , Ann. Phys. 22. 801 (1907). E. G R Ü N E I S E N , Ann. Phys. 25, 825 (1908). D. A. G . B R U G G E J I A N N . Z. Phys. 92, 570 (1934). H. RÖHL. Ann. Phys. Hi. 904 (1933). P . L A U R E N T and M. E U D I E R , Rev. Metall. 4 7 . 5 8 2 ( 1 9 5 0 ) . P . S . R U D J I A N . Trans. M S A 1 M E 2 3 3 , 8 6 4 ( 1 9 6 5 ) . \Y. K Ö S T E R and H . F R A N Z . Metallurg. Rev. (>, 1 ( 1 9 6 1 ) . E. VOGT. Physikalische Eigenschaften der Metalle, Vol.I. Akademische Verlagsgesellschaft Geest & Portig, Leipzig 1958 (p. 125). J . K. V A R M A and M. D. A G G A R V A L , J . appl. Phys. 411. 2841 (1975). H. M. L E D B E T T E R and E. R. N A I M O X , J . appl. Phys. 45. 66 (1974). E. K R Ö N E R . J . Mech. Phys. Solids 25. 137 (1977). E. K R Ö N E R . J . Phys. V 8, 2261 (1978). G . B R A D F I E L D . Use in Industry of Elasticity Measurements in Metals with the Help of Mechanical Vibrations, Kotes on Applied Science No. 30, Nat. Phvs. Lab. London, 1960 (p. 4). H. H. WAWRA, Z. Metallk. (»5, 584 (1974). A. H U B E R and E. S C H M I D . Helv. phys. Acta 7, 620 (1934). Y. S H J B U Y A . Sei. Rep. Res. Inst. Töhoku Univ. Al, 161 (1949).

H . H . WARWA : Role of Griineisen's Elastic P a r a m e t e r s in t h e T h e o r y of E l a s t i c i t y

296

[17] H . PURSEY, P r i v a t e Communication, National Physical L a b o r a t o r y T e d d i n g t o n (Middlesex) 1974.

[18] A. V. HERSHEY, Trans. ASME, Ser. E (J. appl. Mech.) 21, 127 (1963). [19] E. KRONER, Z. P h v s . 151, 504 (1958). [20] Z. H A S H I Ï T a n d S. S H T R I K M A N , J . Mech. P h y s . Solids 11, 127 (1963). (Received

January

3, 1979)

Obituary Dr. H . H . W a w r a lost his life in a car accident in J u n e 1978. W i t h Dr. W a w r a t h e scientific world lost an e m i n e n t researcher who was always full of ideas; he has significantly a d v a n c e d t h e field of t h e elasticity a n d other properties of polycrystalline materials. 1 personally lost a good friend a n d coworker. E. Kroner

V. A. SOLOVEV: The '"Superdislocation" Model of a Plastic Zone near a Crack

297

phys. stat. sol. (a) 54, 297 (1979) Subject classification: 10 Institute of Metallography and Metal Central Scientific Institute for Ferrous

Physics, Metallurgy,

Moscow

The "Superdislocation" Model of a Plastic Zone near a Wedge-Shaped Crack By V . A . SOLOVEV

A simple anisotropic dislocation theory of the plastic zone near a nonsymmetric wedge-shaped crack is developed. The zone is described by some superdislocations with multiple Burgers vectors. The main parameters of the equilibrium zone (the strength and spread of the relaxation shears) are determined. Simple expressions for a symmetrical quasi-brittle crack are derived both, in anisotropic and isotropic elasticity. Capabilities of the theory developed to describe the interaction of different shear-type planar defects are discussed. The critical pressure is calculated necessary for crack plastic closing as well as the parameters of cracks arising during deformation under pressure.

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OKOJIO IICCHM-

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1. Introduction A dislocation model of a plastic zone arising near cracks in crystal materials was first proposed in f l , 2], According to this model the zone forms on t h e crack continuation in the crack plane. There is a " d r y f r i c t i o n " force acting on the dislocations of t h e zone. T h e formation of such a zone may relax plane and antiplane shears due to t h e c r a c k ; however, in the model of [1, 2] there is no opportunity to relax strains induced by the crack opening in the direction of the normal to the crack plane. As a consequence, the geometry of the zone formation accepted in [1] is too simplified (that is most clearly seen from t h e T E M observations [3]). T h e simple model of a plastic zone consisting of four noncoplanar symmetric "superdislocations" near the tips of symmetric, symmetrically loaded cracks has been developed in [4], using t h e solution for the stress field due to the interaction of a dislocation and a crack given in [5]. Nevertheless, in [4, 5] there was not taken into account the Burgers vector of t h e crack itself (i.e. wedging) which is quite important because crack nucleation in t h e internal parts of the crystal appear to be a result of t h e coalescence of dislocations piling up against obstacles. Moreover, only such cracks (Stroh cracks) are able t o provide nucleation, storage, and development of microcracks followed b y critical crack formation [6]. Contrary to the cracks considered in [4, 5] (Griffith cracks) t h e Stroh cracks are characterized b y not only the critical configuration but also b y t h e steady-state one which can pass into the critical one under proper conditions [7].

298

V . A. S o l o v e v

I n [8] we had found stresses near a nonsymmetric Stroh crack in the presence of outer dislocations. I n the present work the "superdislocation" model of a plastic zone is developed for the case of such a crack. T h e case of a quasi-brittle crack is analysed in more detail. T h e analysis is carried out in anisotropic elasticity theory, and then for symmetric loading the isotropic formulae are found by the limiting procedure developed in [9]. T h e important applications of the model worked out are discussed in t h e last part of the article. 2. The Model oi the Stroh Crack with the Plastic Zone T h e model under consideration of the plastic zone near the general slit-like crack is shown in Fig. 1. T h e crack whose wedging vector is h is influenced by the homogeneous stress cr'j0> and by the stress field of the zone which is described by a set of effective superdislocations in the vicinity of the crack termination. T h e superdislocations lie parallel to each of the crack edges in planes passing through these edges. Only t h e planes most favourable for plastic deformation are taken into account. T h e zone parameters are as follows: the magnitudes of effective dislocation Burgers vectors m s b ( s ) and their distances r ^ ' from the crack ends. T h e y can all be simultaneously determined by the following conditions: 1. T h e shear stresses in slip planes (t) in the vicinity of the crack tips are equal to t h e dislocation emission stresses cr^. 2. T h e total forces acting on superdislocations (s) in their slip planes are equal to " d r y f r i c t i o n " forces cr^'W") in the material investigated multiplied b y the magnitudes m s of the Burgers vectors of the effective superdislocations with respect to the Burgers vectors of the elementary dislocations in corresponding slip planes. T h e stress field of the s-th effective dislocation lying parallel to Oz is given by js) Okj

=

v

isv) « ntmsTT Ukj

L

V =

1 V

S SV

.

(i)

where = x + pvy, = xs + prys\ x, y, z are rectangular coordinates of the observation point; xs, ys, —oo z + c o designate the dislocation (s) position; pv are the complex roots of the six-order characteristic equation of anisotropic elasticity t h e o r y ; V ^ P are defined b y pv, the Burgers vectors 5 ( s ) , the dislocation direction, and the elastic moduli of the media [10, 11], T h e slit-like crack parallel to Oz and passing through the interval [ — /J, A] of Ox is adequately described b y the three dislocation arrays distributed with densities Qx(x) (a = 1, 2, 3) and having linearly independent shear vectors the choice of which is governed b y reasons of convenience (in particular we assume hereafter ft'3) = n M , where ii ( 1 ) is the unit vector perpendicular to the crack edge and lying in the crack plane, the unit vector perpendicular to the crack plane, }J ) +

(9a) dj = ± A, + S ws6M . 1 «il a n d [Sj] is a set of systems emitted b y the left tip only. Similarly, for the dislocation of the i-th system of the set [s2] of systems emitted from the right-hand crack tip (t),(t) f (0) . »• Oj I a,, +

{

d2 = i

v

2. Is)

S mW

(«) .

+

ij

y

.,

= 0 their effects e n h a n c e each o t h e r n e a r t h e s h a r p t i p a n d relax each o t h e r n e a r t h e base of t h e crack. So we a s s u m e t h e crack h a s a zone only n e a r t h e r i g h t - h a n d (sharp) tip a n d it consists of t w o edge dislocations w i t h B u r g e r s vectors which lie in planes inclined t o t h e crack p l a n e a t a n angles + I n such a case t h e p a r a m e t e r s m1 a n d rx of t h e equilibrium zone can be expressed in a closed f o r m : m

rx

A ( k

+

1

_ _

A*b*(h

°}A ~ 2Aql(avb +

2 m A l 2 / A ) 2

(glgo

+

ga)

(16)

- oflnfi,)

a>A-

2mAJAf

2Aql(avb—a^)nlbj)

(

M o

+

\2

I n these formulae A = 2 Qgp >

A

12 = 2

(18)

a n d all t h e a n i s o t r o p y is localized in t h e nondimensional f a c t o r s q0, qlt a n d q2 all d e p e n d i n g on t h e m u t u a l o r i e n t a t i o n of t h e dislocations a n d t h e crack a n d being of the form 6 2rjl>

N

and the ratio of the lengths of these defects is A 21

(qtqf + qtf ~ (?o*)2

_(ofnl?%

-

ajb)*_

(26)

{pb-ofn^btf

This is another proof of the estimation carried out in [12]. Formulae (25), (26) give also a way of evaluating the pressure necessary for a plastic closing of the crack. W e have pointed out that under tension a plastic zone is mainly concentrated near the sharp tip of the crack. Application of pressure after unloading reverses the picture and leads to the zone formation near the crack base and therefore to a plastic closing of the crack. I f we take into consideration that instead of ( Exe ( + oo, qVm] ,

(22)

y e [ylt 1] =» Ex e [qVm, 0] .

(23)

This means t h a t / T E , equation (18), expresses t h e flux of electrons passing over t h e barrier top, e.g. thermionic emission, while / E E , equation (19), refers to t h e flux of those passing t h r o u g h t h e barrier, e.g. field emission. I n m a t h e m a t i c a l terms, the value yx is chosen so t h a t t h e second t e r m s in t h e brackets of (18) and (19) are less t h a n u n i t y in t h e whole integration intervals except just a t y = i/j. Therefore, we m a y t r e a t the expression in the brackets as geometrical series and write 2/i-e

J T E = lim / £-•0 o

- B~hf + (B'hf)2 ...] dy ,

/;. E = lim / Jy-\ 1 - By-" + (By-)2 e -> 0 }/1 r£

(24)

...] dy ,

(25)

where e ¡5: 0. B y integrating the series term by term a n d taking the limit, we obtain + oo I 7 t e = #i/0-i 2 (_l)» , n=0 nc + 1 -r OO

1

»=0

1 — (n + 1) C

(26) +00 n—0

1 1 — (w + 1) c

T h e sequences of terms belonging to these series converge to zero. I n t h e cases of t h e first a n d the t h i r d series, it is obvious; in the second series, this depends upon t h e value of t h e p a r a m e t e r B. B u t b y definition (15), B 1. Hence, t h e corresponding sequence of terms converges, too. I n addition to t h a t , t h e series are alternating, so according to Leibniz' criterion, t h e y must be convergent. The expressions (26) a n d (27) provide an easy way of evaluating t h e numerical value of t h e integral (14) or (11). Because of t h e properties of alternating series, t h e m a x i m u m value a n d sign of the t r u n c a t i o n error can be easily established. Moreover, it turned out t h a t it was possible to find simple analytical expressions a p p r o x i m a t i n g the sums of these series.

342

R. S. Popovic

First of all, note t h a t parameter B is practically always in the range 0 < B < 1.

(28) 200

10

I t s typical value lies between 10~ and 10" for an unbiased metal-silicon contact with impurity concentration between 1015 and 1018 c m - 3 and & b n = 0.7 eV. Therefore, for c ]> 1 the sum of the second series in (26) and (27) is negligible compared to the other two: 1 « /TE + 7 t f e ,

(29)

^ ^ " I H ' h i i c - I -

(3o)

where

The relative error is of the order of _B2 (i — i/^). Since we are dealing with higher temperatures, we may assume that the last series (30) refers to the thermal-field emission fraction of the total current. Next, by adding the corresponding terms of series (26) and (30), we have I « £ 1, the quality of approximation depends again upon the magnitude of B. The approximation by our expression (36) is slightly better than that obtained by the use of expression (32) of Ridout and Crowell's type. However, the application of the latter is fully justified for higher values of c. It is significant to note that expression (32) evidently represents the total current of the contact, rather than only its thermionic pait, as it was claimed in [9]. In addition and connection to that, expression (32) is still valid if 1.5 < c < 2. (iii) Neither of these appioximations is valid in the close proximity of the point c = 1. However, we can see from Fig. 3 that each of the three curves 1(c) can be approximated very well in the vicinity of this point by a straight line (in log-log coordinates). Therefore, since we know the exact values of I for c = 1 (equation (51) or (52)), and for any other (close) c (equations (26) and (27)), we can always write the equation of this line. This equation will be a good approximation of the curve 1(c) if c ~ 1.

Non-Degenerate Schottkv Diode Current-Voltage Characteristics

347

Appendix A Consider expression (26) for / T E . W e can rewrite it in the form 1

TE

=

1

h

£(i/0-i

(Al)

S. T B

where »Ste =

2 n=i

(A2)

n + 1/c

If c 1, t h e latter series does not differ so much from the numerical series expansion of In 2 [17], + 0O 1 In 2 = 2 ( - l ) » + i — . (A3) n »=i T o make use of this similarity, we shall form the difference (A4)

i?TE = $ t e — In 2 , + oo 1 ^te= 2 (—i)n-,—¡-tt—• (nc

n = 1

+

1

(A5)

) n

Note the series (A5) converges much faster t h a n t h a t in (A2). I t is alternating and therefore the magnitude of its sum is smaller t h a n its first term, the sign of the sum being the same as t h a t of the first term. T h u s we can write

.

-Rte =

(A6)

, , . c + 1

(A7)

0 < 0, < 1 . Now, from (A4) and (A6) Srv. = In 2

ft

-

(A8)

c + 1

Substituting (A8) into ( A l ) , we obtain JTE =

5(1/0-1 1 -

— In 2 c

c(c + 1)

(A9)

Likewise, starting from 7 t f e = JSd/«)-i c

+ n =

£ ( - 1 )« + i i

n

1 — — 1/c

(A10)

we come to / T F E = .BdM"1 — [ l n 2 c o
0 ,

(2.12)

ym < o .

T h e parameter a m in (2.12) is determined by (2.6). I t arises in the expression for the vacuum wave vector because of the dispersion law ltlm = ko• Formula (2.10) is valid for the Bragg-reflected beams in the space outside the entrance crystal surface and for the Laue-diffracted beams in that outside the exit surface. T h e coefficients of transmission R0 and reflection Rm, m =j= 0, are defined simply in terms of the amplitude Let the incident radiation be plane-polarized, for example synchrotron radiation. I f qj is the angle between the polarization plane and the vector e0n then Rm{