Physica status solidi: Volume 21, Number 2 June 1 [Reprint 2021 ed.] 9783112497944, 9783112497937

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pliysica status solidi

V O L U M E 21 • N U M B E R 2 • 1967

Classification Scheme 1. Structure of Solids 1.1 Alloys. Metallurgy 1.2 Solid-State Phase Transformations 1.3 Surfaces 1.4 Films 2. Non-Crystalline State 3. Crystallography 3.1 Crystal Growth 3.2 Interatomic Forces 4. Microstructure of Solids 5. Perfectly Periodic Structures 6. Lattice Mechanics. Phonons 6.1 Mossbauer Investigations 7. Acoustic Properties of Solids 8. Thermal Properties of Solids 9. Diffusion in Solids 10. Defect Properties of Solids (Irradiation Defects see 11) 10.1 Defect Properties of Metals 10.2 Photochemical Reactions. Colour Centres 11. Irradiation Effects in Solids 12. Mechanical Properties of Solids (Plastic Deformations see 10)see 10.1) 12.1 Mechanical Properties of Metals (Plastic Deformations 13. Electron States in Solids 13.1 Band Structure. Fermi Surfaces 13.2 Excitons 13.3 Surface States 13.4 Impurity and Defect States 14. Electrical Properties of Solids. Transport Phenomena 14.1 Metals. Conductors 14.2 Superconductivity. Superconducting Materials and Devices 14.3^Semiconductors 14.3.1 Semiconducting Films 14.3.2 Semiconducting Devices. Junctions (Contact Problems see 14.4.1) 14.4 Dielectrics 14.4.1 High Field Phenomena, Space Charge Effects, Inhomogeneities, Injected Carriers (Electroluminescence see 20.3; Junctions see 14.3.2) 14.4.2 Ferroelectric Materials and Phenomena 15. Thermoelectric and Thermomagnetic Properties of Solids 16. Photoconductivity. Photovoltaic Effects 17. Emission of Electrons and Ions from Solids 18. Magnetic Properties of Solids 18.1 Paramagnetic Properties 18.2 Ferromagnetic Properties 18.3 Ferrimagnetic Properties. Ferrites 18.4 Antiferromagnetic Properties (Continued

on cover three)

physica status solidi B o a r d of E d i t o r s P. A I G R A I N , Paris, S. A M E L I N C K X , Mol-Donk, V. L. B O N C H - B R U E V I C H , Moskva, W. D E K E Y S E R , Gent, W. F R A N Z , Münster, P. GÖRLICH, Jena, E. G R I L L O T , Paris, R. K A I S C H E W , Sofia, P.T. L A N D S B E R G , Cardiff, L. N É E L , Grenoble, A. P I E K A R A , Warszawa, A. S E E G E R , Stuttgart, F. SEITZ, Urbana, O. S T A S I W , Berlin, M. S T E E N B E C K , Jena, F. STÖCKMANN, Karlsruhe, G. SZIGETI, Budapest, J . TAUC, Praha Editor-in-Chief P. GÖRLICH Advisory Board M. B A L K A N S K I , Paris, P. C. B A N B U R Y , Reading, M. B E R N A R D , Paris, W. B R A U E R , Berlin, W. COCHRAN, Edinburgh, R. COELHO, Fontenay-aux-Roses, H.-D. DIETZE, Saarbrücken, J.D. E S H E L B Y, Cambridge, P . P . F E O F I L O V , Leningrad, J. H O P F I E L D , Princeton, G. J A C O B S, Gent, J. J A U M A N N , Köln, E. K L I E R , Praha, E. K R O E N E R , Clausthal-Zellerfeld, R. KUBO, Tokyo, M. M A T Y A S , Praha, H. D. MEGAW, Cambridge, T. S. MOSS, Camberley, E. NAGY, Budapest, E. A. N I E K I S C H , Jülich, L. P A L , Budapest, M. RODOT, Bellevue/Seine, B. V. R O L L I N , Oxford, H. M. R O S E N B E R G , Oxford, R. Y A U T I E R , Bellevue/Seine

Volume 21 • Number 2 • Pages 443 to 872, K97 to K187, and A33 to A62 June 1, 1967

AKADEMIE-VERLAG•BERLIN

Subscriptions and orders for single copies should be addressed to AKADEMIE-VERLAG GmbH, 108 Berlin, Leipziger Straße 3 - 4 or to Buchhandlung K U N S T U N D WISSEN, Erich Bieber, 7 Stuttgart 1, Wilhelmstr. 4—6 or to Deutsche Buch-Export und-Import GmbH, 701 Leipzig, Postschließfach 160

Editorial Note: "physica status solidi" undertakes that an original paper accepted for publication before the 8 t h of any month will be published within 50 days of this date unless the author requests a postponement. In special cases there may be some delay between receipt and acceptance of a paper due to the review and, if necessary, revision of the paper.

S c h r i f t l e i t e r u n d v e r a n t w o r t l i c h f ü r d e n I n h a l t : Professor D r . D r . h. c. P . G ö r l i c h , 102 B e r l i n , N e u e S c h ö n h a u s e r S t r . 20 b z w . 69 J e n a , H u m b o l d t s t r . 26. R e d a k t i o n s k o l l e g i u m : D r . S. O b e r l ä n d e r , D r . E . G u t s c h e , D r . W . B o r c h a r d t . A n s c h r i f t d e r S c h r i f t l e i t u n g : 102 B e r l i n , N e u e S c h ö n h a u s e r S t r . 20. Fernruf: 426788. Verlag: Akademie-Verlag G m b H , 108 B e r l i n , L e i p z i g e r S t r . 3 - 4 , F e r n r u f : 2 2 0 4 4 1 , T e l e x - N r . 0 1 1 7 7 3 , P o s t s c h e c k k o n t o : B e r l i n 3 5 0 2 1 . D i e Z e i t s c h r i f t „ p h y s i c a s t a t u s s o l i d i " e r s c h e i n t jeweils a m 1. des M o n a t s . B e z u g s p r e i s eines B a n d e s M D N 7 2 , — (Sond e r p r e i s f ü r d i e D D R M D N 60,—). B e s t e l l n u m m e r dieses B a n d e s 1068/21. J e d e r B a n d e n t h ä l t z w e i H e f t e . G e s a m t h e r s t e l l u n g : V E B D r u c k e r e i „ T h o m a s M ü n t z e r " B a d L a n g e n s a l z a . — V e r ö f f e n t l i c h t u n t e r d e r L i z e n z n u m m e r 1310 d e s P r e s s e a m t e s b e i m V o r s i t z e n d e n des M i n i s t e r r a t e s d e r D e u t s c h e n D e m o k r a t i s c h e n R e p u b l i k .

Contents Original Papers R.

KELLY

page Bubble Diffusion and the Motion of Point Defects near Surfaces .

45

G . B . A B D U L L A E V , Z . A . A L I Y A R O V A , a n d G . A . ASADOV

Preparation of Cu2Se Single Crystals and Investigation of their Electrical Properties

461

V . I . SYTJTKINA a n d E . S . YAKOVLEVA

Ò. J E C H

The Mechanism of Deformation of the Ordered CuAu A l l o y . . . .

465

Ion-Bombardment Enhanced Solubility in Solids

481

M . KARRAS a n d E . SUONINEN

J . HESSE

Energy Distribution of Photoelectrons Created by Al K a X-Rays in Alkali Halide Targets

487

FlieBspannung und Versetzungsdichte Kalzium-dotierter NaCl-Einkristalle

495

K . M . KOLIWAD, P . B . GHATE, a n d A . L . RUOFF

J . - P . ROPÉ

A.

BONNOT

Pressure Derivatives of the Elastic Constants of NaBr and K F . .

507

Etude expérimentale de la dynamique des domaines de haut champ électrique dans le couplage électron-phonon

517

Relaxation des domaines de haut champ et émission de lumière associées aux oscillations de courant dans GaAs

525

W . A. JESSER a n d D .

KUHLMANN-WILSDORF

The Geometry and Energy of a Twist Boundary between Crystals with Unequal Lattice Parameters J . A . SIGLER a n d D .

533

KUHLMANN-WILSDORF

Calculations on the Mechanical Energy of Vacancy Condensation Loops, Stacking Fault Tetrahedra, and Voids

545

F . K E L E M E N , A . N É D A , D . NICTTLESCU, a n d E . CRUCEANTJ

P.

SCHNUPP

On the Thermal Conductivity of CdTe and Some CdTei-^Sr^ and CdTei-jSe^ Solid Solutions

557

Kronig-Penney-Type Calculations for Electron Tunneling through Thin Dielectric Films

567

K . H . GUNDLACH a n d G . HELDMANN

J . POLÂK

The Effect of the i?(£)-Relation on Tunneling through Asymmetric Barriers

575

Kinetics of Quenched-in Vacancies in Pure Platinum

581

K . H . J . BUSCHOW a n d J . F . FAST

Crystal Structure and Magnetic Properties of Some Rare Earth Germanides

593

B . F . ROTHENSTEIN, A . POLICEC, M . LUPULESCU, a n d C. ANGHEL

The Electromagnetic Aspect of Magnetostrictively Induced Torsional Oscillations (the Procopiu Effect with a Cylindrical Ferromagnetic Specimen Fixed at one End) 601 M. 29«

HULIN

Selection Rules for Two-Phonon Absorption Processes

607

Contents

446

Page L . GOUSKOV, G . LECOY, a n d C . LLINARES

Carrier Recombination in Nickel Doped Germanium from Lifetime and Noise Measurements

619

L . FIERMAN'S a n d J . VEMNIK

Microprobe Investigations of Copper Precipitates in Silicon Single Crystals

627

M . POLCAROVÄ a n d J . KACZER

X - R a y Diffraction Contrast on Ferromagnetic Domain Walls in Fe-Si Single Crystals

635

D . G . ARASLI a n d M . I . ALIEV

Influence of Defects and of the Interaction between them on Phonon Scattering in Heavily Doped Ge and Si Crystals

643

K . C . A . BLASDALE

Arrays of Dislocation Dipoles in Cadmium

649

N . P . GUPTA a n d B . D A Y A L

Thermal Expansion and Compressibility of Solid Neon and Xenon .

661

M . AVEROUS a n d G . BOUGNOT

Effect of Donor I m p u r i t y Concentration of t h e Piezoresistance of n-Type GaSb

665

R . LÜCK u n d K . E . SAEGER

Über die Messung der Anisotropie der galvanomagnetischen Transversalspannung an Einkristallfolien

671

J . P. AGRAWAL Infrared Fluorescence in Manganese Activated Zinc Sulphide Phosphors

679

S . TOSCHEV a n d I . GUTZOW

Time Lag in Heterogeneous Nucleation due to Nonstationary Effects

683

P. MÜLLER

Zur Berechnung der Fehlordnung in Silberhalogeniden

693

R . HERRMANN

Oszillationen der Oberflächenimpedanz von Wolfram in schwachen

S. TAKACS

Magnetfeldern

703

Der Flußeintritt in Supraleiter I I . Art

709

P . L U K A S a n d M . Fatigue K L E S N IHardening L

P. HUMBLE

of Cu-Zn Alloys

A Comparison of Electron Microscope Images of Dislocations Based on Isotropic and Anisotropic Elasticity Theory

717 733

F . DWORSCHAK, H . SCHUSTER, H . WOLLENBERGER, a n d J . W U R M

The Influence of the Size Effect on Electrical Resistivity Measurements in Irradiated Metals S. I. MASHAROV Electrical Resistivity in Antiferromagnetic Alloys

741 747

V . HIZHNYAKOV a n d I . TEHVER

Theory of Resonant Secondary Radiation due to I m p u r i t y Centres in Crystals

755

D . M . BERCHA, A . N . BORETS, I . M . STAKHYRA, a n d K . D . TOVSTYUK

The B a n d Edge and the Energy Spectrum of In 2 Se

769

Contents

447 Page

V . P . KALASHNIKOV

Theory of Spin-Lattice Relaxation of Conduction Electrons of Ionic Semiconductors at Extremely High Magnetic Field

775

A . F . LUBCHENKO

Light Absorption and Dispersion by Impurity Centres at High Concentrations

785

I . L . SOKOLSKAYA, V . G . IVANOV, a n d G . N . F U R S E Y

Study of Barium Adsorption on Germanium by Field-Emission Microscopy

789

Y u . A . BOGOD a n d V . V . E R E M E N K O

Peculiarities of the Magnetoresistance in Bi in High Transverse and Longitudinal Magnetic Fields L . P . KHIZNICHENKO, P . F . KROMER, D . K . KAIPNAZAROV, E .

797

OTENYAZOV,

D . Y U S U P O V A , a n d L . G . ZOTOVA

The Properties of the Low-Temperature Internal Friction Peak in Silicon P . F . KROMER a n d L . P .

805

KHIZNICHENKO

On the Low-Temperature Internal Friction in Silicon

811

M . D A M O D A R - P A I a n d K . M . VAN V L I E T

K.-H.

PFEFFER

K . - H . PPEFFER

Generation-Recombination and Diffusion Noise in Cadmium Sulphide Photoconductors

819

Mikromagnetische Behandlung der Wechselwirkung zwischen Versetzungen und ebenen Blochwänden (II)

837

Zur Theorie der Koerzitivfeldstärke und Anfangssuszeptibilität . .

857

Short Notes A . C L E M E N T , N . C L E M E N T e t P . COULOMB

Paires de défauts intrinsèque et extrinsèque dans un acier inoxydable et dans un alliage cuivre-silicium K97 J . E . KNAPPETT a n d S. J . T . OWEN

Observation of the Growth Process of Gallium Arsenide on Tungsten M . PERINOVÂ

K99

Origin of Perfect Dislocations and Glide Elements in N a N 0 3 Single Crystals K103

V . K . K O N I U K H O V , L . A . KTTLEVSKII, a n d A . M . PROKHOROV

Two-Photon Absorption Spectrum in CdS near the Fundamental Absorption Edge K107 H.

PINK

H. MÜHE

The Determination of Chromium Distribution in Laser-Rubies with Neutron Activation Kill Temperature Dependence of Magnetocrystalline an Fe-Al Alloy near the Concentration Fe 3 Al

Anisotropy

of K115

I. DÉZSI, G Y . ERLAKI, a n d L . KESZTHELYI

Mössbauer Effect Studies on the a?(Rh 2 0 3 )l — a:(Fe 2 0 3 ) System

. . . K121

Contents

448

Page N . S. NATARAJAN a n d M . S. R . CHARI

The Anomalous Thermal Conductivity of Dilute Ag-Mn Alloys at Helium Temperatures K127 1. B U N G E T a n d M . ROSENBERG

A.

ZAREBA

P-Type Conduction in Barium Ferrite Single Crystals

K131

Photoconductivity in Boron at Low Temperature

K135

G . L A U T Z u n d M . SCHULZ

Schraubenförmige Dichtewellen und Oszillistor-Effekt im ElektronLoch-Plasma des n-Germaniums K139 K . W . BÖER a n d G. A. DUSSEL

Slow Moving Field Domains in CdS in the Prange of Negative Differential Conductivity K145 D . DEMCO a n d F . CONSTANTINESCU

Dispersion Relations between the Relaxation Times and the Shift of the Resonance Frequency K151 B . M . ASKEROV a n d F . M . GASHIMZADE

Determination of Parameters of Degenerate Semiconductors . . . . K155 Vu DINH KY

Susceptibility of Ferromagnetic Films in Paraprocess

2 . D I M I T R I J E V I C , S . K R A S N I C K I , H . R Z A N Y , J . TODORQVIC, A . W A N I C , H . C U R I E N , MILOJEVIC Anisotropy of Magnon Dispersion Relation in Haematite

A.

K159 and

K163

J . MERTSCHING a n d H . W . STREITWOLF

On the Derivation of the Boltzmann Equation in the Presence of a Magnetic Field K167 H.

KUHNERT

Polarization of F-Center Luminescence in KCl due to Electric Fields . K171

R.

SITTIG

Stark Effect Measurements of the O H " Absorption Band in KCl. . . K175

A.

R.

Some Remarks on the Paper „Zur Theorie des Ferromagnetismus dünner Schichten" by H. Hemschik Kl79

F.

J . KEDVES

FERCHMIN

On Quenching Experiments in Gold

KL

83

Pre-printed Titles and Abstracts of papers to be published in this or in the Soviet journal ,,®H3HKa TBepnoro T e . u a " (Fizika Tverdogo Tela)

A33

449

Contents Systematic List Subject classification:

1.1

Corresponding papers begin on the following pages (pages given in italics refer to the principal subject classification): 465

1.2

683

1.3

789

1.4

K99

2.1

733

3.1

461, 533, 683,

4

465, 627, 635

K99

5

593

6

607, 643, 661, 775

6.1

K121

7

517, 525

8

557, 643, 661,

9

451

K127

10

495, 643, 693, 805, 811, K103,

10.1

465, 533, 545, 581, 649, 717, 733, 837, 857, K97,

Kill

10.2

K171

11

481, 741, 805

12

507, 661, 805,

12.1

717

811

13

775, K139,

13.1

567, 575, 665, 769, K155,

K163

13.3

789

13.4

619, 665, 679, 755, 785, 819, K 1 3 5

14

747, 797,

14.1

671, 703, 741, K 1 8 3

K179

K167

14.2

709

14.3

461, 619, 665, K131,

14.3.1

567, 575,

K155

K99

14.3. 2

575

14.4.1

517, 525 , K 1 3 9 , K145,

14.4. 2

K131

15

461, K 1 3 1 , K 1 5 5

16

619, 819,

17

487, 789

18

593, 775,

18.2

601, 635, 837, 857, K115,

18.4

747, K 1 2 1

K175

K135 K163

19

703,

20

K139

20.1

607, 755, 769, 785, K107,

K159,

K151 K175

K171

K183

450 20.2 20.3 21 21.1 21.1.1 21.6 21.7 22 22.1.1 22.1.2 22.1. 3 22.2.1 22.2.3 22.4.1 22.5.1 22.5.2 22.6 22.8 23

Contents Kill 525, 679, 755, K171 545,649,703,733,741 465, 545, 671, 679, 717, K97, K151 601, 635, K97, K115 545, 581, K127, K183 797 461, 481, 593, 769, 775, K155 619, 635, 643, 789, K139 627,643,805,811 K135 517, 525, K99 665 679, 819, K107, K145 693 487, 495, 507, K171, K175 481, 575, K103, K i l l , K121, K163 557 661

The Author Index of Volume 21 Begins on Page 873 (It will be delivered together with Volume 22, Number 1.)

Original

Papers

p h y s . s t a t . sol. 21, 4 5 1 (1967) Subject classification: 9 Solid State Physics

Section, Euratom,

Ispra

(Varese)

Bubble Diffusion and the Motion of Point Defects near Surfaces (Diffusion Theory for Discrete Media, Part IV 1 )) By R . KELLY

T h e theory of diffusion of inert-gas bubbles in solids is i m p o r t a n t in a v a r i e t y of fields, including reactor technology, ion b o m b a r d m e n t , powder metallurgy, a n d t h e s t u d y of large defects. B u b b l e diffusion also provides a convenient f r a m e of reference for discussing t h e motion near surfaces of point defects, i.e. of surface defects, vacancies, a n d vapourized a t o m s for a surface-diffusion or molecules. I t is shown t h a t t h e bubble diffusion coefficient, mechanism is equal t o (3/2 n a1) Ds, where a is t h e bubble radius in units of X, t h e m e a n a t o m i c spacing, a n d t h a t this expression is independent of t h e surface-diffusion j u m p distance. T h e f o r m a t i o n of vacancies a t a surface should occur a t a f r e q u e n c y given b y / f v = Dvol b, d. h. der A u s d r u c k f ü r einen Verdampfungs-Kondensations-Mechanismus, ist v o n der F o r m (3/4 7i a3) (Dg Cg A3), wobei Dg der Diffusionskoeffizient des D a m p f e s u n d C g die Gleichgewichtskonzentration des D a m p f e s in Atome/cm 3 ist.

1. Introduction There has been considerable interest recently in t h e t h e o r y of diffusion of inertgas bubbles in solids. One reason is t h a t during reactor operation, t h e fission process leads t o extensive bubble formation in the fuel a n d possibly cladding [ 1 , 2 , 3 ] . r ) P a r t I t o I I I are published in Acta metall. 12, 123 (1964), N u c l e a r I n s t r u m . a n d Methods 38, 181 (1965), a n d J . n u c l e a r Mat. 20, 171 (1966), respectively.

452

R.

KELLY

Subsequent motion of these bubbles, first demonstrated experimentally by Barnes and Mazey [4] and also observed in [1, 2, 5], enables them to coalesce, with the result that the fuel or cladding may swell. Likewise, the study of inertgas diffusion in ion-bombarded solids can be used as a probe for lattice defects. Bubble motion then manifests itself as one of a series of readily distinguishable release components [5 to 8]. Connections with powder metallurgy should be equally direct: the sintering process would be expected, in its later stages, to involve the motion of bubbles filled with the ambient gas. Finally, it is worth mentioning that most of the ideas concerning bubble motion will apply rather closely to the motion of cavities, precipitates, perforations [9], dislocation loops (i.e. conservative climb [10]), and stacking-fault tetrahedra. The first theoretical treatment relevant to bubble motion appears to have been that of Chernov [11], who considered biased motion, i.e. motion in a gradient, under the assumption of a vapourization-condensation mechanism. In the work by Greenwood and Speight [12] the important topic of random motion, i.e. motion without a gradient, was discussed for a surface-diffusion mechanism and the concept of a bubble diffusion coefficient, Z>b, was introduced. The latter authors, together with Mikhlin [13], Shewmon [14], and Speight [15], have subsequently given further treatments of biased motion. The main object of the present work will be to treat random bubble motion, thence Dh, for all three mechanisms of motion, namely the surface-diffusion, volume-diffusion, and vapourization-condensation mechanisms. In so doing, however, various aspects of point-defect motion near surfaces will be treated, and it is hoped that many of the results will have an applicability beyond the more limited topic of bubble motion. I n the concluding section, a brief discussion will be made of the problem of applying the results of the present work to bubble motion in ion-bombarded solids. 2. Surface-Diffusion Mechanism Greenwood and Speight [12], in their treatment of random bubble motion due to a surface-diffusion mechanism, point out that individual atomic jumps on a bubble surface will lead to small displacements of the centre-of-gravity of the bubble. This enables a diffusion coefficient to be defined using the general expression for three-dimensional motion, D = - ^ - / 1 i i 2 A 2 , where .T is the jump frequency, A is the mean atomic spacing, and E (in units of A) is the R.M.S. ("root mean square") jump distance. We will now briefly restate the derivation of [12] with the object of expressing D b in terms of more accessible parameters than were used in [12]. One approach is to relate the frequency _Tb at which bubble jumps occur to _TS, the frequency, per area A2 of surface, of the individual surface jumps: rb = rs(4jra2),

(la)

where a (in units of A) is the bubble radius. 1\ can, in turn, be obtained by rearranging the general expression for the surface diffusion coefficient, Ds —

=

db)

Bubble Diffusion and the Motion of Point Defects near Surfaces

453

Fig. 1. Representation of a (100) surface of a simple-cubic solid showing Its and SB, the basic distances involved in surface diffusion. i? 8 , the R . M . S . length of the individual j u m p s , will be equal to or greater than the a t o m spacing, though is shown arbitrarily a s being equal. Sa, the II.-M.S. distance between defect formation a n d condensation, is shown arbitrarily as involving a superficial ledge

where Ra (in units of X) is the R . M . S . length of the individual surface j u m p s (Fig. 1). The value of Ra has been deliberately left unclarified in ( l b ) , for estim a t e s range from unity [13, 16] to ^ 1 [17] and it is difficult to decide at present what is correct. We note also t h a t the factor 4 it a 2 , rather t h a n 4 jt a 2 S as proposed b y Shewmon [14], has been used in (1 a), where and we conclude that vacancy motion near a surface with a v unrestricted also requires a radiation boundary condition. The parameter h (in units of A -1 ) is given by h=

(1

-*v)Pv

.

(4)

The R.M.S. bubble jump distance, R h , will obviously depend on the values of the condensation efficiency, )« + i r» + i (h a + n +

1)J

"

\r < a

'' +

p],

where r (in units of A) and 6 are explained in Fig. 3, and Pn(z) is the Legendre polynomial. We now form (C) r = a and note that the mean square distance between the formation and condensation of a vacancy, will be given by the sum of two terms. The first is for those vacancies which return to the original bubble, of which the fraction is (F)

r = a

=

J D

0

_ —

• 2 n a2

s i n 6 d 6 • h(C)r=a

ha2 (a+p)

( h a + 1 )

(5)

456

R.

KELLY

and the contribution to S* is = / D • 2 Jt a2 sin 6 dd • h(C)r=a • 2 a 2 (1 o _ 2 i a ' ( 1 a 1 ~ a + p {¿a + 1 ~~ (a + p) (ha + 2)} '

cos 6»)

(6)

The second term is for the vacancies which escape from the bubble of their origin but are replaced by a like number coming from random vacancy sources and therefore taking up a random position at R.M.S. distance 2 1 ' 2 a. The fraction is { 1 — (F)r-_a} and the contribution to S* is { 1 — (F)r=a) 2 a2. The overall value of SI is thus the sum. ($y) r = a { 1 — (-iT)r=a} 2 though, for the most usual case in which L L Auger electrons from the fluorine atoms can be evaluated as the areas under the corresponding partial curves : -^-«1.16. "Au

(1)

I t is to be noted that the method used basically does not require any information on the spectrum of the secondary electrons except the assumption that their contribution is negligible in the region of the lowest peak of the spectrum. I n order to get a rough check on the quality of the model used, the contribution of the secondary electrons to the total spectrum was estimated by subtracting the calculated contributions of the K photo- and Auger electrons from the spectrum and plotting the remainder on log-log paper as a function of E. The result was, within a credible degree of accuracy, a straight line with a slope —5.9. This value differs considerably from the slope —1 assumed in [9] for the spectrum of the secondary electrons in metallic photocathodes. The authors feel that the basis of the —1 law, viz. that the secondary electrons dominate the spectrum up to 500 eV, is rather ambiguous. I n any case, the situation can be expected to be different in ionic photocathodes. The ratio of the areas under the two partial curves is rather insensitive to even fairly crude assumptions of the curve forms, as long as the same form is assumed for both curves. The estimate of the ratio is therefore probably accurate to 10 per cent, although it is unrealistic to claim the same accuracy for both contributions separately.

Fig. 3. Analysis of the photoelectric spectrum of a L i F target. T h e solid curve is a smoothed-out copy of the corresponding curve in F i g . 2. T h e relative contributions xi of the various partial processes to the t o t a l yield x = £ Xi are obtained from the areas i indicated by the corresponding symbols

Energy Distribution of Photoelectrons Created by Al K a X-Rays

491

I n this ease, when only K ionizations of F ions contribute appreciably to the spectrum of fast electrons, the formula developed by Rumsh, Shchemelev et al. [4] for the yield x can be written as 2 sin 6

rs K - 1 S*

i a

K

¿>K

a

Au.

"Au >

(2)

where ,ma is the contribution of the F atoms to the linear absorption coefficient of LiF, S K is the K absorption jump for F and 6 is the angle of incidence of the radiation. We is the Auger yield of the ionization, i.e. We = 1 — a>e, where coe is the fluorescent yield. a K and a A u are the absorption coefficients, defined by the well known Lenard's law, of the K photoelectrons and K -»• LL photoelectrons, respectively. According to Makhov [10] they are related to the electron energies by the formula a ~ E~s, where the penetration parameter s depends on the target material. Introducing this assumption into the above formula and dropping the constants common to both terms we get 1 iVk dagegen r B ß N sein. Die „kritische" Versetzungsdichte Nk ergibt sich zu Nk =

504 = A2jß2 B2.

J.

HESSE

Mit A = G 6/2 n, B = G b dj2 n (1 -

v) und iVk = 5 X 10 e /cm 2 aus

NA-

[3, 6] und Fig. 9 kann daraus ßk = — ^ für N = Nk abgeschätzt werden, wenn A

k

der Abstand d der Teilversetzungen des Dipols bekannt ist. Für NaCl ist d nicht gemessen. Für MgO ist nach Ogawa [19] d ä> 400 A. Überträgt man dieses Resultat versuchsweise auf NaCl, so erhält man ßk « 100. Diese hohe Dipoldichte ist nur schwer zu verstehen. Vermutlich ist aber auch Gleichung (2 a) in dieser Form nicht richtig, da sie den scharfen "Übergang von t r n l bevorzugt auftritt. Das fühlt jedoch zu einer besonderen Schwierigkeit: Bei einer

Dabei bedeuten Z eine Zahl der Größenordnung 1, N die Zahl der im Kristall liegengebliebenen und iVb die Zahl der beweglichen Versetzungen mit einem mittleren Laufweg L. Da für T = 294 ° K mit dem Ca-Gehalt um 2 0 % zunimmt (Fig. 6) und andererseits r/N annähernd auf 5 % konstant bleibt (Fig. 9 b und Tabelle 1), muß L mit zunehmendem Ca-Gehalt abnehmen oder aber ein größerer Bruchteil von Versetzungen liegenbleiben, was ebenfalls einer starken Behinderung entspricht. Mit der Verunreinigungskonzentration sollte die Zahl der Quergleitprozesse an Verunreinigungen und damit die Dipolbildung (ß) zunehmen. Zunächst bedeutet verstärkte Dipolbildung nicht notwenig stärkere Verfestigung ($ x ), sondern unter Umständen Entfestigung. Da aber zugleich die Laufwege verkürzt werden (Gleichung (3)), kann das dennoch zu einer zunehmenden Verfestigung führen. E s sollen hier weitere Versuche mit dem Ziel durchgeführt werden, die Hypo-

Fließspannung und Versetzungsdichte von Ca-dotiertem NaCl

505

these der begünstigten Quergleitung an Verunreinigungen experimentell zu prüfen. Dazu soll T(N) im Bereich I der VK für verschiedene Verunreinigungszusätze bestimmt werden, die sich in der Stärke ihrer Verzerrungsfelder unterscheiden. Herrn Prof. Dr. P. Haasen bin ich für die freundliche Erlaubnis, die Verformungsexperimente am Institut für Metallphysik der Universität Göttingen durchführen zu dürfen, und für seine kritische Durchsicht des Manuskripts sehr dankbar. Herr Dipl.-Phys. R. Bucksch übernahm die Programmierung der r/a-Kurven; auch ihm bin ich zu Dank verpflichtet. Literatur [1] W. IN DEE SCHMITTEN und P. HAASEN, J . appl. Phys. 8-2, 1790 (1961). [2] P. HAASEN und J . HESSE, in: The Relation between the Structure and Mechanical Properties of Metals, Her Majesty's Stationary Office, London 1963 (p. 137). [3] R . W . DAVIDGE u n d P . L . PRATT, p h y s . s t a t . sol. 6 , 7 5 9 ( 1 9 6 4 ) .

[4] [5] [6] [7] [8]

J . HESSE, phys. stat. sol. 9, 209 (1965). W. G. JOHNSTON und J . J . OILMAN, J . appl. Phys. 30, 129 (1959). B. J. SMIRNOV, in: Reinststoffprobleme, Bd. 3, Akademie-Verlag, Berlin 1967. J. DEPUTAT und Z. PAWLOWSKI, Bull. Acad, polon. Sei., Ser. Sei. techn. 14, 419 (1966). J . J. GILMAN, J . appl. Phys. 33, 2703 (1962).

[9] W . G. JOHNSTON u n d J . J . GILMAN, J . a p p l . P h y s . 3 1 , 6 3 2 ( 1 9 6 0 ) .

[10] K.-H. MATUCHA, Dissertation Universität Göttingen, demnächst. [11] J . HESSE, in: Reinststoffprobleme, Bd. 3, Akademie-Verlag, Berlin 1967. [12] R. MORAN, J. appl. Phys. 29, 1768 (1958). [13] R . P . HARRISON, P . L . PRATT u n d C. W . A . N E W E Y , P r o c . B r i t . C e r a m . S o c . 1 , 179

(1964). [14] R. L. FLEISCHER, Acta metall. 10, 835 (1962). [15] P . L . PRATT, R . CHANG u n d C. W . A . NEWEY, A p p l . P h y s . L e t t e r s 3 , 8 3 (1963).

[16] G. SCHOECK und A. SEEGER, Acta metall. 7, 469 (1959). [17] W . G . JOHNSTON, J . a p p l . P h y s . 3 3 , 2 0 5 0 (1962).

[18] S. MENDELSON, J . appl. Phys. 33, 2175 (1962). [19] K . OGAWA, P h i l . M a g . 1 4 , 6 1 9 (1966). [ 2 0 ] J . DIEIIL, S. MADER u n d A . SEEGER, Z . M e t a l l k . 4 6 , 6 5 0 ( 1 9 5 5 ) . [21] Z . S. BASSINSKI, z i t . b e i J . W . CHRISTIAN u n d P . R . SWANN, i n : M e t a l l . S o c . C o n f . ,

Vol. 29, Gordon and Breach Sc. Publ., New York 1965 (p. 105). [22] M. AHLERS, Z. Metallk. 56, 741 (1965). (Received

February

13,

1967)

K. M. KOLIWAD et al.: Pressure Derivatives of the Elastic Constants

507

phys. stat. sol. 21, 507 (1967) Subject classification: 12; 22.5.2 Department of Materials Science and Engineering, Cornell University, Ithaca, New York

Pressure Derivatives of the Elastic Constants of NaBr and K F By K . M. K O L I W A D , P. B. G H A T E 1 ) , a n d A . L. R U O F F The isothermal pressure derivatives of the adiabatic second order elastic constants at 25 °C of NaBr and K P have been measured by the ultrasonic interferometric technique. The results are:

NaBr KP

Sen dp

3 ( c ^ - ci a )/2 dp

6c'4i dp

11.50 11.74

4.91 5.04

0.423 -0.452

Here c'xß = o V%ß where Q is the density and Vxß is the velocity at the given pressure and temperature. These pressure derivatives are analyzed in the framework of the Born model of ionic solids. The pressure derivatives of the second order elastic constants can be expressed in terms of the second and third order elastic constants. The theoretical values of the second and third order elastic constants used in this article were obtained by considering the coulombic interactions of point ions, short range repulsive interactions up to second nearest neighbors, and van der Waals terms. Even though there is qualitative agreement between theory and experiment quantitative agreement is less satisfactory than desired. Die isothermen Druckableitungen der adiabatischen elastischen Konstanten zweiter Ordnung von NaBr und K P werden bei 25 °C mit der interferometrischen Ultraschalltechnik gemessen. Es werden folgende Ergebnisse erhalten: 9Cn NaBr KP

9 {c'^-c'^ß

öc'4i

dp

dp

dp

11,50 11,74

4,91 5,04

0,423 -0,452

Hierbei ist c'aß = QV%ß, wobei g die Dichte und V^ß die Geschwindigkeit bei gegebenem Druck und Temperatur sind. Diese Druckableitungen werden innerhalb des Rahmens des Bornschen Modells des Ionenfestkörpers diskutiert. Die Druckableitungen der elastischen Konstanten zweiter Ordnung lassen sich durch die elastischen Konstanten zweiter und dritter Ordnung ausdrücken. Die theoretischen Werte der elastischen Konstanten zweiter und dritter Ordnung, die in dieser Arbeit benutzt werden, werden unter Berücksichtigung der Coulombwechselwirkung der Punkt-Ionen, der kurzreichweitigen Abstoßungswechselwirkung bis zu den zweitnächsten Nachbarn und der van der Waals-Terme erhalten. Obwohl qualitative Übereinstimmung zwischen Theorie und Experiment vorhanden ist, ist die quantitative Übereinstimmung weniger befriedigend als gewünscht. Present address: Texas Instruments Inc., Dallas, Texas. 33

physica/21/2

K. M. KOLIWAD et al.: Pressure Derivatives of the Elastic Constants

507

phys. stat. sol. 21, 507 (1967) Subject classification: 12; 22.5.2 Department of Materials Science and Engineering, Cornell University, Ithaca, New York

Pressure Derivatives of the Elastic Constants of NaBr and K F By K . M. K O L I W A D , P. B. G H A T E 1 ) , a n d A . L. R U O F F The isothermal pressure derivatives of the adiabatic second order elastic constants at 25 °C of NaBr and K P have been measured by the ultrasonic interferometric technique. The results are:

NaBr KP

Sen dp

3 ( c ^ - ci a )/2 dp

6c'4i dp

11.50 11.74

4.91 5.04

0.423 -0.452

Here c'xß = o V%ß where Q is the density and Vxß is the velocity at the given pressure and temperature. These pressure derivatives are analyzed in the framework of the Born model of ionic solids. The pressure derivatives of the second order elastic constants can be expressed in terms of the second and third order elastic constants. The theoretical values of the second and third order elastic constants used in this article were obtained by considering the coulombic interactions of point ions, short range repulsive interactions up to second nearest neighbors, and van der Waals terms. Even though there is qualitative agreement between theory and experiment quantitative agreement is less satisfactory than desired. Die isothermen Druckableitungen der adiabatischen elastischen Konstanten zweiter Ordnung von NaBr und K P werden bei 25 °C mit der interferometrischen Ultraschalltechnik gemessen. Es werden folgende Ergebnisse erhalten: 9Cn NaBr KP

9 {c'^-c'^ß

öc'4i

dp

dp

dp

11,50 11,74

4,91 5,04

0,423 -0,452

Hierbei ist c'aß = QV%ß, wobei g die Dichte und V^ß die Geschwindigkeit bei gegebenem Druck und Temperatur sind. Diese Druckableitungen werden innerhalb des Rahmens des Bornschen Modells des Ionenfestkörpers diskutiert. Die Druckableitungen der elastischen Konstanten zweiter Ordnung lassen sich durch die elastischen Konstanten zweiter und dritter Ordnung ausdrücken. Die theoretischen Werte der elastischen Konstanten zweiter und dritter Ordnung, die in dieser Arbeit benutzt werden, werden unter Berücksichtigung der Coulombwechselwirkung der Punkt-Ionen, der kurzreichweitigen Abstoßungswechselwirkung bis zu den zweitnächsten Nachbarn und der van der Waals-Terme erhalten. Obwohl qualitative Übereinstimmung zwischen Theorie und Experiment vorhanden ist, ist die quantitative Übereinstimmung weniger befriedigend als gewünscht. Present address: Texas Instruments Inc., Dallas, Texas. 33

physica/21/2

508

K . M . K O L I W A D , P . B . GHATE, a n d A . L . R U O F F

1. Introduction I n recent years there have been a number of theoretical treatments of t h e cohesion of ionic solids with a view to elaborate and extend the original and rather quite successful Born model [1, 2], However, as would be expected, t h e Born model is only a rather good approximation when a t t e m p t s are made to calculate such physical quantities as elastic constants. Since it involves only central forces in a centrosymmetric lattice, it must predict t h a t t h e Cauchy relations hold at T = 0. Careful measurements of elastic constants indicate t h a t these relations fail and t h a t the failure is often marked at low temperatures. The quasiharmonic approximation developed b y Leibfried and H a h n [3] can be used to calculate the temperature dependence of the elastic constants. I n this approximation the failure of the Cauchy relations at finite temperatures follows as a consequence of the vibrational free energy of the crystal. N r a n y a n [4] and Ghate [5] have used such a quasiharmonic model for t h e calculation of t h e third order elastic constants and their temperature variation. Recently Lincoln, Koliwad, and Ghate [6] have included the v a n der Waals interaction in their calculation of the second and third order elastic constants of some NaCl t y p e alkali halides. The second order elastic constants and their pressure dependence provide convenient tests of t h e theories of cohesion. The strain derivatives of t h e second elastic constants are related to t h e higher order elastic constants which in t u r n are related to the anharmonicity of the lattice. A cubic crystal with 4-fold symmetry has three and six independent second and third order elastic constants, respectively. The second order elastic constants, c u , c12, and c44, can be determined by measuring t h e velocities of t h e acoustic waves propagating along certain crystallographic directions. A cubic crystal subjected to hydrostatic pressure maintains its cubic symmetry and lends itself amenable t o t h e measurement of t h e pressure derivatives of its second order elastic constants. I t is interesting to note t h a t t h e magnitude and the sign of the pressure derivatives differ from one system to another. A number of such measurements on several alkali halides are now available [7 to 13]. I n p a r t , t h e present work on NaBr and K F was undertaken to assess t h e theoretical conclusions reached by Lincoln et al. [6] regarding the sign of dc^ 4 /dp. I n the present article, we shall first describe t h e measurements of t h e elastic constants of NaBr and K F at room pressure and also as a function of pressure. The experimental values of the pressure derivatives of the elastic constants will be presented. The calculations of the elastic constants in the framework of t h e Born model will be reviewed briefly. The short range interactions have been included u p to second nearest neighbors. The v a n der Waals terms, neglected in earlier calculations, have also been included. The theoretical values of (dc^/dp) computed by using the calculated values of t h e second and T.O.E. constants will be compared with the experimental values. 2. Experimental Procedure 2.1 Sample

preparation

Approximately 1/2" cubes of NaBr and K F single crystals were obtained from Harshaw Chemical Company. The faces were oriented in t h e [110] and [100] crystallographic directions. The accuracy of the orientations of t h e crystals as obtained was about + 2 ° . The deviation from the exact orientation was

509

Pressure Derivatives of the Elastic Constants of NaBr and K F l c44. The shear constant c 44 increases with pressure in the case of NaBr and decreases in the case of K F . However, the Cauchy-Love relation (c'12 — c'i4) = 2 is nearly satisfied for K F and fails in the case of Sp NaBr. The second order elastic constants and their pressure derivatives are calculated on the basis of the Born model. These values are taken from the work of Lincoln et al. [6], The derivatives dc^/dp can be evaluated for a given pairwise interaction using equations (8) and (9) of [6] and equations (74), (75), and (76) of [5]. The latter set of equations relate the pressure derivatives of the effective elastic constants to the second and third order elastic constants of the stress free crystals. I t should be noted that the theoretical expressions for dc'zpjdp used by Lincoln et al. [6] and used in our analysis can be shown to be identical to those given by Miller and Smith [9], The interaction potential used in the theoretical calculations contains the Coulomb term, short range cation-anion interaction, short range second nearest neighbor interaction and van der Waals interactions. The short range interactions are expressed in the form A exp (—r/b). For most of the alkali halides b is of the order of 0.3333 A. Tosi [2] has shown that different values of b result if one used different equations of state. Lincoln, Koliwad, and Ghate [6] have used b = 0.340 A and 0.338 A for NaBr and K F , respectively, which are the values obtained by Tosi making use of the Hilderbrand equation of state. The various parameters needed for the calculation are listed in Table 1 of [6]. Comparison of our experimental results with the calculations is made in Tables 2 and 3. We note that the theory fails to predict the correct sign of d c ' t J d p for K F . Tables 2 and 3 also list the results of the calculations based on the quasiharmonic approach. In this approximation, if one starts with the correct potential energy, the coupling parameters of the second order and therefore the oscillator or eigenfrequencies a>t are functions of the lattice constant. This part of the anharmonic effects are already described by the dependence of

K. M. Koliwab, P. B. Ghatb, and A. L. Ruoff

514

Table 2 Second order elastic constants of N a B r and K F in units of 1011 dyn/cm 2 *) Room temperature

Experiment

Quasi-harmonic

with vW

no v W

with vW

no v W

3.063 3.437 1.003 0.789 1.003 0.789

3.373 3.685 1.370 1.191 1.370 1.191

2.668 3.002 0.777 0.576 0.974 0.781

3.168 3.459 0.990 1.362 1.362 1.194

2.445 2.860 1.978 1.721 1.978 1.721

4.087 4.429 2.157 1.945 2.157 1.945

1.951 2.335 1.707 1.462 1.926 1.686

3.833 4.161 1.921 1.713 2.146 1.942

r? W •j

NaBr Cu Cj 2 C

44

4.037

3.817

1.013

0.958

1.015

1.015

6.485

6.212

1.427

1.367

1.281

1.281

KF

C

12

*) The values in the first row in each case correspond to b = 0.340 A for N a B r and t o b = 0.338 A for K F while the values in t h e second row correspond to b = 0.3333 A . **) The experimental d a t a are adiabatic a t 25 °C while the calculated values are isothermal. The experimental values have been converted to isothermal ones b y using thermal expansion d a t a [20] a t 0 °C. Table 3 Pressure derivatives of the second order elastic constants of N a B r and K F *) Room temperature

NaBr Scii ap 8C 12 dp Ki dp KF 9 Cn dp dc'12 dp

Quasi-harmonic

Experiment

with vW

no vW

with vW

no v W

11.87 13.94 2.60 2.05 0.60 0.05

8.26 9.88 2.46 2.11 0.46 0.11

12.72 14.94 2.80 2.16 0.73 0.14

9.15 10.03 2.54 2.15 0.52 0.14

11.50

11.76 12.54 3.15 2.70 1.15 0.70

8.07 8.10 2.60 2.32 0.60 0.32

12.43 12.54 3.40 2.90 1.55 0.90

8.10 8.53 2.68 2.38 0.68 0.38

11.74

1.68 0.423

1.66 -0.452

dp *) The experimental values are adiabatic and t h e calculated values are isothermal. Because of the lack of d a t a on the variation of thermal expansion with temperature conversion t o completely isothermal derivatives has not been made.

Pressure Derivatives of the Elastic Constants of NaBr and KP

515

a>i(a). With these assumptions one can calculate the free energy as a function of the lattice parameter and temperature. This is essentially a high temperature approximation. Following the suggestion of Born, each frequency o>i is then substituted by []1''2- This simplification is equivalent to using Einstein's model for frequencies. A more exact procedure would be to evaluate each partition function knowing the individual a>i(a). The inadequacies of this approach can be seen in the case of c12 and c 44 for K F . Acknowledgements

We wish to thank Richard Lincoln for his assistance in some of the calculations. We acknowledge the support of the U.S. Atomic Energy Commission. We also thank the Advanced Research Projects Agency for support through the Cornell Materials Science Center Facilities. Appendix I t is worth mentioning here that one must not use the expression for cap given by equation (8) of [6] (which has already made use of the equilibrium condition d^/dr = 0) to obtain dlnc„p/dr. For a further discussion of this point the reader is referred to the article by Blackman [17]. In this same regard we note that some authors choose to discuss the calculated values of ca/3 in terms of the relative amount of the coulomb and the non-coulomb contribution. Thus, c,p = clp + Cp .

For example, Kellerman [18] does this and obtains different answers for the individual contributions than do Miller and Smith [9] although the expressions for c„p are the same. Huntington has noted this in his review paper [19]. W e feel that this point needs further clarification. The expressions given by Miller and Smith [9] for clp give the total coulomb contribution. Upon application of the equilibrium condition to their equation we note that e.fi = dp

+ A dp + CTp + A cTp ,

where the equilibrium condition states that Aca% =

-

Ac?p .

I t is then mathematically correct to write a p = CI p + CTp

c

and physically correct to call CI p the net coulomb contribution which in fact corresponds to the expression given by Kellerman [18]. However, if one is interested in comparing the actual coulomb contribution to the actual noncoulomb contribution then the ratio dp/c™p (and not ClpjC^p) should be used. References [1] M. BORN and K. HUANG, Dynamical Theory of Crystal Lattices, Oxford University Press 1954. [2] M. P. Tosi, Solid State Phys. 16, 1 (1964). [ 3 ] G. L E I B F R I E D a n d H . HAHN, Z . P h y s . 1 5 0 , 4 9 7 ( 1 9 5 8 ) .

W. LTJDWIG, Solid State Phys. 12, 275 (1961).

Also see G. L E I B E R I E D

and

516

K. M. KOLIWAD et al.: Pressure Derivatives of the Elastic Constants

[4] A. A. NRANYAN, Fiz. tverd. Tela 5, 177 (1963); Soviet Phys. - Solid State 5, 129 (1963). Fiz. tverd. Tela 6, 1865 (1963); Soviet Phys. - Solid State 5, 1361 (1964). [5] P . B . GHATE, P h y s . R e v . 1 3 9 , A 1 6 6 6 (1965). [6] R . C. LINCOLN, K . M . KOLIWAD, a n d P . B . GHATE, p h y s . s t a t . sol. 1 8 , 2 6 5 (1966). ¿7] D . LAZARUS, P h y s . R e v . 7 6 , 5 4 5 ( 1 9 4 9 ) .

[8] W. B. DANIELS and C. S. SMITH, The Physios and Chemistry of High Pressures, Gordon and Breach Science Publishers, Inc., New York 1963 (p. 50). [9] R. A. MILLER and C. S. SMITH, Bull. Amer. Phys. Soc. 9, 687 (1964); J . Phys. Chem. S o l i d s 2 5 , 1 2 7 9 (1964).

[10] P . J . REDDY and A. L. RUOFF, in: Physics of Solids at High Pressures, Ed. C. T. TOMIZUKA a n d R . M . EMRICK, A c a d e m i c P r e s s , 1965.

[11] Z. P. CHANG, Phys. Rev. 140, A1788 (1965). [12] K. M. KOLIWAD and A. L. RUOFF, Bull. Amer. Phys. Soc. 10, 1113 (1965). [13] K. M. KOLIWAD, P. B. GHATE, and A. L. RUOFF, Bull. Amer. Phys. Soc. 11, 47 (1966). [14] J . WILLIAMS a n d J . LAMB, J . A c o u s t . S o c . A m e r . BO, 3 0 9 ( 1 9 5 8 ) .

[15] R. H. MARTINSON, Ph. D. Thesis, Cornell University, 1966. [16] K. M. KOLIWAD, Ph. D. Thesis, Cornell University, 1967. [17] M . BLACKMAN, P r o c . P h y s . S o e . 8 4 , 3 7 1 (1964). [18] E . W . KELLERMAN, P h i l . T r a n s . R o y . S o c . ( L o n d o n ) A 2 3 8 , 5 1 3 (1940).

[19] H. B. HUNTINGTON, Solid State Phys. 7, 213 (1958). [ 2 0 ] F . A . HENGLEIN, Z. E l e c t r o c h e m . 8 1 , 4 2 4 ( 1 9 2 5 ) ; Z . p h y s . C h e m . 1 1 5 , 9 1 ( 1 9 2 5 ) . (Received

March

14,

1967)

517

J.-P. ROPÉ: Dynamique des domaines de haut champ électrique phys. stat. sol. 21, 517 (1967) Subject classification: 14.4.1; 7; 22.2.1 Laboratoires

E.R.6., La Badiotechnique — Coprim Suresnes (Hauts-de-Seine)

R.T.C.,

Étude expérimentale de la dynamique des domaines de haut champ électrique dans le couplage électron-phonon Par J . - P . ROPÉ Nous étudions la répartition de potentiel à l'intérieur des domaines de haut champ électrique qui se propagent dans les échantillons de GaAs présentant des oscillations de courant dues au couplage électron-phonon. Nous montrons en particulier que le champ électrique dans le domaine augmente lorsque le domaine disparaît. A study is made of the voltage distribution inside high electric field domains which propagate in GaAs samples exhibiting current oscillations due to the electron-phonon interaction. In particular, the electric field inside the domain is seen to increase as the domain disappears.

1. Introduction Des oscillations de courant, dues à l'effet acoustoélectrique, ont été observées dans différents semiconducteurs lorsque le champ électrique appliqué dans une direction piézoélectriquement active dépasse une valeur critique E s (champ de seuil) [1], Ces oscillations de courant peuvent être attribuées à la diminution du courant due à l'amplification du flux thermique ultrasonique par les porteurs. La diminution du courant présente un temps r d'incubation dépendant du champ électrique appliqué. Il est maintenant bien établi [2] que: 1. La distribution de potentiel dans l'échantillon devient non linéaire à t = r (Fig. 1). Il existe trois parties dans l'échantillon: une région de fort champ électrique ou „domaine" et deux parties de faible champ. 2. Le domaine se déplace à l'intérieur de l'échantillon avec la vitesse du son, de la cathode C à l'anode A pour un échantillon du type n. Nous avons utilisé une sonde capacitive pour étudier le champ électrique

Fig. 1. Distribution de potentiel à l'intérieur d'un échantillon a) pour I < T, b) pour t > t

0

p

L

517

J.-P. ROPÉ: Dynamique des domaines de haut champ électrique phys. stat. sol. 21, 517 (1967) Subject classification: 14.4.1; 7; 22.2.1 Laboratoires

E.R.6., La Badiotechnique — Coprim Suresnes (Hauts-de-Seine)

R.T.C.,

Étude expérimentale de la dynamique des domaines de haut champ électrique dans le couplage électron-phonon Par J . - P . ROPÉ Nous étudions la répartition de potentiel à l'intérieur des domaines de haut champ électrique qui se propagent dans les échantillons de GaAs présentant des oscillations de courant dues au couplage électron-phonon. Nous montrons en particulier que le champ électrique dans le domaine augmente lorsque le domaine disparaît. A study is made of the voltage distribution inside high electric field domains which propagate in GaAs samples exhibiting current oscillations due to the electron-phonon interaction. In particular, the electric field inside the domain is seen to increase as the domain disappears.

1. Introduction Des oscillations de courant, dues à l'effet acoustoélectrique, ont été observées dans différents semiconducteurs lorsque le champ électrique appliqué dans une direction piézoélectriquement active dépasse une valeur critique E s (champ de seuil) [1], Ces oscillations de courant peuvent être attribuées à la diminution du courant due à l'amplification du flux thermique ultrasonique par les porteurs. La diminution du courant présente un temps r d'incubation dépendant du champ électrique appliqué. Il est maintenant bien établi [2] que: 1. La distribution de potentiel dans l'échantillon devient non linéaire à t = r (Fig. 1). Il existe trois parties dans l'échantillon: une région de fort champ électrique ou „domaine" et deux parties de faible champ. 2. Le domaine se déplace à l'intérieur de l'échantillon avec la vitesse du son, de la cathode C à l'anode A pour un échantillon du type n. Nous avons utilisé une sonde capacitive pour étudier le champ électrique

Fig. 1. Distribution de potentiel à l'intérieur d'un échantillon a) pour I < T, b) pour t > t

0

p

L

518

J.-P. ROPÉ

à l'intérieur de plusieurs échantillons de GaAs présentant ces oscillations de courant. Cette sonde diffère de celle utilisée par Gunn [3] essentiellement parce que les fréquences à étudier sont beaucoup plus faibles que celles de l'effet Gunn, ce qui a permis une technique plus simple. Nous avons plus particulièrement étudié la répartition du champ électrique à l'intérieur du domaine lorsque ce dernier se trouve loin des extrémités d'une part et près de l'anode d'autre part. 2. Utilisation du signal recueilli par la sonde Etant donné la périodicité du phénomène, la durée 6 de l'impulsion de tension appliquée sur l'échantillon sera toujours limitée à une valeur légèrement supérieure à T (période des oscillations de courant). Considérons la courbe F p ( i ) obtenue sur une sonde située à une distance x de la cathode (Fig. 1 et 2). Pour t < r l'échantillon est ohmique et F p = Vx. Lorsque x — A/«s > i > r (A étant la largeur du domaine et vs la vitesse du son dans l'échantillon), la répartition du potentiel est non linéaire et nous avons F p = Vz (à condition que le champ appliqué E& soit suffisamment élevé pour que T

(i)

2 ax a 2 a

(2)

a

Consider now an arbitrary reference vector, r, situated in the twist-misfit boundary. Crystallographically, r corresponds to the vector in the overgrowth and r 2 in the substrate ,where r x and r z have the magnitudes Arctan

r a,

•¡-a-i cos0/

ri

and §1 ri I t

=

a

(3a) (3 b)

while r = T(r1

+

(3c)

If these three vectors are imagined to have the same origin, as in Fig. 1, a Burgers circuit begin-

m

Fig. 1. Diagram of the relative orientations of the reference vector r in the twist-misfit interface, the two vectors and rs which crystallographically correspond to r in overgrowth and substrate, respectively, as well as the closure failure, D, and the direction of one set of the square grid of dislocations which accomodates the twist misfit, for the case that r is paralîel to a cube axis of the reference lattice

Twist Boundary between Crystals with Unequal Lattice Parameters

535

ning, say, at the tip of rlt passing the origin of the three vectors, and ending at the tip of r 2 , will have a closure failure D, given by D =

r

i

- r

1

.

(4)

B y the reasoning first given by Frank [5], this closure failure must be equal to the sum of the Burgers vectors of all misfit dislocations intersected by the vector r . In order to gain a better understanding of the general twist-misfit boundary, its limiting cases shall first be briefly considered: We know that the misfit dislocations in the case of parallel alignment, i.e. in a pure misfit boundary, form a square grid of two sets of uniformly spaced edge dislocations, parallel to the two cube axes. Thus if r is oriented parallel to one of these axes, only dislocations belonging to one set are intersected, and we find P M , the spacing of the dislocations in a pure misfit boundary, as = M

D

b_ I ax + aa\ 2 \ a2 - aj

«i a2 , a2 - %

=

W

In the case of a pure twist interface, i.e. for r1 = ra but with the crystals in non-parallel alignment, rt and r2 are rotated in opposite directions relative to r, each through the angle 0/2 if the angle of twist equals 0. Again a closure failure D = r2 — r1 results, but now D is normal to r, while it was parallel to r for the case of pure misfit. The dislocation arrangement in this case is well known to be a square grid of two sets of uniformly spaced screw dislocations parallel to the two cube axes of the reference lattice. Again, as in the case of the pure misfit boundary, one set of the dislocations is normal to D if r is aligned parallel to one of the cube axes of the reference lattice. Consider now the general case of a twist-misfit boundary, outlined in Fig. 1 The closure failure D = r2 — r , is now neither parallel nor perpendicular to r, but includes an angle W = arc tan (

\ a2 —

^

)

c o s CP/

(6)

with r2, as shown. This closure failure could, in principle, be accomodated by misfit dislocations in a variety of ways. However, in both, the pure misfit as well as the pure twist boundary, the Burgers vectors remained the same, namely parallel to the cube axes of the reference lattice, and equal in magnitude to the lattice constant of the reference lattice. Since these Burgers vectors are given geometrically, we expect the Burgers vectors to be the same also in the general case. Furthermore, the dislocations in both of the limiting cases formed a uniform square grid of dislocations. This, too, is a feature which one must expect to be retained, — definitely for reasons of symmetry. Finally, we recall a third common feature of the two limiting cases, namely that if r is oriented parallel to one of the cube axes of the reference lattice, then one set of dislocations is parallel to D, while the other set is normal to D. Whe shall therefore assume that this feature is also retained as indicated in Fig. 1. Correspondingly we shall proceed to evaluate dislocation orientations and spacing in a square grid of dislocations, consistent with the twist-misfit, in which the dislocations have the two Burgers vectors described (equation (2)), and the axes of which are normal and parallel to D if r is oriented parallel to one of the cube axes of the reference lattice.

W. A. Jesser and D. Kuhlman n-Wilsdorf

536

In order to find the dislocation spacing — the same in each of the two grids —, orient r parallel to one of the cube axes of the reference lattice as in Fig. 1. The component of D parallel to r must be the sum of the Burgers vectors parallel to r which belong to dislocations intersected by the vector r. These belong only to the dislocations whose axis direction is indicated in Fig. 1, while the set of dislocations normal to these have the other of the two Burgers vectors. Thus, then, the number of dislocations contributing to the parallel component of D equals (D sin a)/b. With the spacing of the dislocations designated by P , the number of dislocations which are intersected by r and belong to the set considered, is given by (r sin (x)/P. Hence, D sin a r sin a S = (?) or P = r b/D as before, except that now D =

yielding

(rl

+

r\ — 2 rx r2

cos &) 1 ! 2

(8)

rb 1)

(al

I - 2 a t a2 cos

The angle a is found from Fig. 1 as 71

a = — À'

0 &

— arc tan

2

(9)

'

I sin 0 \ \a2 — a-L cos 0 /

(10)

assuming that a 2 is larger than a1. The equations derived above for the dislocations in twist-misfit interfaces are the same that also govern the spacing and orientation of moiré fringes generated by substrate and overgrowth when viewed normal to their interface. The same considerations with minor appropriate modifications may be made with respect to interfaces of hexagonal symmetry. In that case, three similar uniform sets of dislocations at 120° angles contribute, which, through reactions at their nodes, form the well-known hexagonal dislocation networks. These networks, consisting of edge dislocations in pure misfit boundaries, rotate,

16 :12

ft?

8

16

24

32

40 48 „ = b cos a (16) while P according to equation (9) is substituted for PT in equation (12) and for P M in equation (14). Obviously, the energy of the boundary is a complicated function of 0 and misfit, and therefore the computation of the boundary energy as a function of these two parameters as well as all subsequent numerical computations were performed with a B 5500 computer. The boundary energy, or interfacial energy, is the sum of the dislocation network energy and a minimum energy, E0, which has been evaluated elsewhere [4], and which is independent of misfit or misalignment. The interfacial energy, E\, per unit area is then El =

sEd

+

eE&

(17)

+ E0 .

In all numerical examples E0 has been arbitrarily set at zero, which affects the absolute value of the interfacial energy but not its functional dependence on any of the parameters involved. 4. Elastically Strained Overgrowth 4.1 Variation

of interfacial

energy

with misfit and

misalignment

I n order to investigate the behavior of the interfacial energy with superimposed elastic strains modifying the misfit, the lattice parameter of the overgrowth is varied by introducing an elastic strain, ev while e2, the elastic strain of the substrate, is held in a fixed ratio to that of the overgrowth. If a prime is used to denote a parameter which is changed by a superimposed homogeneous elastic strain, then a[ = ax (1 + ex) (18a) 35 physica 21/2

540

W . A. JESSER a n d D .

i

•

i

•

i—

\ 105 I s

\0-r

4 90

-

-

^ 75 60

0-oy\

45

"

Eo-0

-

i

.

i

.

i

20 15 lin e^lO3

J^mar 25 -

KUHLMANN-WILSDOBF

Fig. 5. The energy of a twist-misfit boundary (Ei according to equation (17) in conjunction with equations (9) to (16) and (24), substituting «,' = (1 + «,) for a, and a 2 ' = a a (1 - «2) for a2), as a function of the elastic strain in the overgrowth. Three different values of 4>, the angle of twist between substrate and overgrowth, were considered as indicated. The present example refers to the interfacial energy between nuclei of platinum on gold. In this example the radius of the nuclei is 46 A, under which circumstances the pseudomorphic strain, «max, becomes 0.0225. 0 is chosen as 8.5 x 1 0 " dyn/cm 2 , since previous experiments indicated this as the probable value. The intercepts of the curves with the ordinate («: = 0) indicate the values of the interfacial energy if the overgrowth is unstrained. The superimposed elastic strain partly relieves the misfit between substrate and overgrowth, and the interfacial energy drops with increasing strain, reaching a minimum at the pseudomorphic strain «max, i.e. t h a t elastic strain in the overgrowth which (together with the strain in the substrate beneath the nuclei or continuous overgrowth) just compensates the misfit. If the strain becomes still larger, misfit of a sign opposite to the strain-free misfit is introduced and the interfacial energy rises again. Note in particular t h a t for 0 — 0, i.e. in the case t h a t substrate and overgrowth are in parallel alignment, the energy minimum is cusped, while it is rounded for all finite values of

542 j

W. A. Jesser and D. Kuhlmanm-Wilsdorf m

1= 136 fi-464 /

^ 128 /

120-

/

1 1 1

1

0-7"/

/ X

112 104^

/1 1 / ' 1 / 1 1 1 1

/

/

| i 1 i 1

/

ill IIill I I/I 111 i

— s

i1

/ M '

F i g . 8 . T h e e q u i l i b r i u m e l a s t i c s t r a i n in the o v e r g r o w t h , elt a s a function of the angle of twist, f o r t h e c a s e o f 9 % m i s f i t , corr e s p o n d i n g t o nuclei o f gold with r a d i i o f a b o u t 1 0 A o n m o l y b d e n i t e (MOS 2 ). T h e p a r a m e t e r s e m p l o y e d in t h i s e x a m p l e a r e

\ l / \ 11

6, = a,lY2 = 2.884 A, h = 3.150 A, i> = 2 6, bjib, + bz) = 3.010 A,

Bmax. f ,

15 20 strain e^lO3-

12 % t(W'ral)

V

25

G, = 2.84 x 10 1 1 d y n / c m 2 , G„ = 2 x 1 0 1 1 d v n / c m 2 , r = 0 . 4 , a n d G = 8 X 1 0 " dyn/cm2

F i g . 7. T h e d e p e n d e n c e of t h e s u m o f i n t e r f a c i a l e n e r g y a n d e l a s t i c s t r a i n e n e r g y , in s u b s t r a t e a n d o v e r g r o w t h n u c l e u s , a s a f u n c t i o n o f et, t h e e l a s t i c s t r a i n in t h e n u c l e u s . T h e e x a m p l e chosen is the s a m e a s t h a t in F i g . 5 a n d 6, n a m e l y p l a t i n u m nuclei o f 46 A r a d i u s on a n i n f i n i t e l y thick g o l d s u b s t r a t e . P s e u d o m o r p h i s m e x i s t s a t e, = «max, which is close t o 2 . 2 5 % . T h e c a s e of p a r a l l e l a l i g n m e n t = 0) is c o m p a r e d with t w o d i f f e r e n t m i s a l i g n m e n t s , 0 = 0.2° and = 1 ° , a s i n d i c a t e d . N o t e t h a t t h e p s e u d o m o r p h i c c a s e c e a s e s to r e p r e s e n t a n energy m i n i m u m when the a n g l e o f t w i s t e x c e e d s s o m e s m a l l , critical v a l u e . H o w e v e r , a n e n e r g y m i n i m u m for s o m e f i n i t e s t r a i n p e r s i s t s , which is n e a r 0 . 5 % in the p r e s e n t e x a m p l e

which has been derived previously for the case of a pure misfit boundary [3, 4], By substituting ez from (24) into (23), the problem has been reduced to one of finding the equilibrium elastic strain e1. The results are shown in Fig. 7 which is a graph of equation (23) as a function of overgrowth strain for various values of angular misalignment. Here it is seen that the equilibrium elastic strain is a function of 0. In a preceding paper it was shown for the case of perfect alignment that a relative minimum and a relative maximum in the energy curve as a function of superimposed elastic strain were always either both present or both absent [4]. As is seen now from Fig. 7, beyond a critical value of the angle of twist, the relative energy maximum is eliminated. Also, it is seen that the cusped energy minimum, occurring when a[ = «2 and 0 = 0, becomes rounded and is displaced toward smaller values of e1 by increasing 0, until this minimum disappears entirely for sufficiently large angles of twist. As has been discussed elsewhere [2 to 4], if the overgrowth is sufficiently thin, it has the lowest energy when it is pseudomorphic, i.e. is elastically strained to match the substrate lattice exactly. As the overgrowth becomes thicker, a relative minimum in the energy curve occurs, making it energetically favorable for the overgrowth to reduce its strain so as to have an atomic spacing closer to that of a bulk overgrowth crystal. In order to transform from the pseudomorphic form into the less strained form, the overgrowth must go through the relative maximum in the energy curves as shown in Fig. 7. It is apparent, however, that such a transition from pseudomorphism will be facilitated by introducing an angular misalignment, since for increasing values of 0 the height

Twist Boundary between Crystals with Unequal Lattice Parameters

543

of t h e r e l a t i v e m a x i m u m is f i r s t reduced a n d t h e n eliminated, as is t h e energy m i n i m u m which c o r r e s p o n d s t o p s e u d o m o r p h i s m . T h e position of t h e a b s o l u t e m i n i m u m in t h e i n t e r f a c i a l energy, i.e. t h e equilibrium elastic s t r a i n of t h e o v e r g r o w t h , d e p e n d s on b o t h , t h e size of t h e overg r o w t h nucleus a n d t h e slope of t h e interfacial energy curve. T h e l a t t e r varies w i t h 0 in a complicated m a n n e r . F o r a given r a d i u s of t h e hemispherical nucleus, t h e v a l u e of t h e equilibrium elastic s t r a i n as a f u n c t i o n of 0 is s h o w n in Fig. 8, for t h e p a r t i c u l a r case of v e r y small gold nuclei on m o l y b d e n i t e . — App a r e n t l y t h e c h a r a c t e r i s t i c s of t h e c u r v e a r e n o t sensitive t o t h e p a r t i c u l a r values of t h e p a r a m e t e r s chosen. I t is seen f r o m Fig. 8 t h a t w i t h increasing values of 0 t h e equilibrium s t r a i n initially s h i f t s t o smaller values, a f a c t which is also e v i d e n t f r o m Fig. 7. Therea f t e r t h e equilibrium elastic s t r a i n rises quite s t r o n g l y a n d t h e n d r o p s slowly according t o Fig. 8. This result, t h a t t h e equilibrium elastic s t r a i n is a s t r o n g a n d complicated f u n c t i o n of r o t a t i o n a l misalignment, would n o t follow f r o m t h e simpler p r e v i o u s a p p r o a c h according t o which t h e i n t e r f a c i a l e n e r g y is f o u n d b y simply a d d i n g t h e energy of a t w i s t b o u n d a r y t o t h a t of t h e m i s f i t dislocation n e t w o r k . I n t h a t case, t h e screw dislocation n e t w o r k would be only a f u n c t i o n of 0, its e n e r g y would n o t d e p e n d on t h e misfit, a n d t h u s t h e equilibrium s t r a i n would n o t be a f f e c t e d b y misalignments, while according t o t h e t h e o r y p r e s e n t ed in t h i s p a p e r t h e equilibrium s t r a i n is q u i t e s t r o n g l y d e p e n d e n t on 0. 5. Discussion F o r p r a c t i c a l p u r p o s e s , p e r h a p s t h e t w o m o s t i m p o r t a n t f e a t u r e s which h a v e emerged f r o m t h i s investigation, a r e : i) The equilibrium elastic s t r a i n d e p e n d s s t r o n g l y on t h e angle of m i s a l i g n m e n t , a n d ii) t h e e n e r g y g r a d i e n t d r i v i n g t h e o v e r g r o w t h t o w a r d s parallel a l i g n m e n t rises w i t h increasing s t r a i n e 1; r e d u c i n g t h e misfit, i.e. rises as t h e dislocation n e t w o r k is r o t a t e d t o w a r d s screw orient a t i o n — as m a y b e clearly seen f r o m Fig. 5 t o 7. T h e implications of i) a r e discussed elsewhere [6, 7] w h e r e it is shown t h a t c e r t a i n discrepancies b e t w e e n t h e simple t h e o r y [2 t o 4] a n d e x p e r i m e n t a l evidence m a y b e d u e t o t h e presence of r o t a t i o n a l misalignments. T h e significance of p o i n t ii) a b o v e is t h a t t h e smaller t h e m i s f i t , t h e larger t h e t e n d e n c y f o r t h e o v e r g r o w t h nuclei t o g r o w in parallel alignment. I n o t h e r words, t h e larger t h e s t r a i n in a given nucleus so as t o r e d u c e t h e m i s f i t , t h e larger t h e t e n d e n c y for t h a t nucleus t o be in p a r a l l e l a l i g n m e n t . Since t h e elastic s t r a i n in nuclei h a s b e e n s h o w n t o increase w i t h decreasing size of t h e nuclei [3, 4], one would expect f r o m t h e a b o v e considerat i o n s t h a t t h e r e d u c e d misfit m a y result in an i m p r o v e m e n t in t h e a l i g n m e n t of t h e small nuclei as c o m p a r e d t o t h e larger nuclei, a t least u n d e r s u i t a b l e conditions. Acknowledgements T h i s r e s e a r c h was s u p p o r t e d b y t h e U.S. A t o m i c E n e r g y Commission u n d e r C o n t r a c t N o . AT-(40-l)-3108. T h e a u t h o r s wish t o t h a n k D r . J . W . M a t t h e w s , U n i v e r s i t y of t h e W i t w a t e r s r a n d , J o h a n n e s b u r g , S o u t h A f r i c a f o r p o i n t i n g o u t t h e correlation b e t w e e n moiré fringes a n d i n t e r f a c i a l dislocations a t a t w i s t misfit boundary.

544

W . A. JESSER

a n d D. K U H L M A N N - W I L S D O R F : Twist B o u n d a r y

References [1] M. B E T T M A N , Single Crystal Films, E d . M . H . F R A N C O M B E a n d H . SATO, MacMillan Co., N e w Y o r k 1964 (p. 177). [2] J . H . VAN DER MERWE, Proo. P h y s . Soc. (London) A63, 616 (1950); J . appi. P h y s . 84, 117 (1963); 34, 123 (1963). J . H . VAN D E R M E R W E , Single Crystal Films, E d . M . H . F R A N C O M B E a n d H . SATO, Macmillan Co., New Y o r k 1964 (p. 139). [3] N . CABRERA, Surface Sci. 2, 320 (1964); Mém. sci. R e v . Métall. 62, 205 (1965). [ 4 ] W . A. J E S S E R a n d D . K U H L M A N N - W I L S D O R F , p h y s . s t a t . sol. 1 9 , 95 (1967). [5] F . C. FRANK, P i t t s b u r g h S y m p . Plastic D é f o r m a t i o n of Crystalline Solids, 1950 (p. 150). [6] W . A. JESSER, P h . D . Dissertation, U n i v . of Virginia, 1966. [ 7 ] W . A. J E S S E R , J . W . M A T T H E W S , a n d D . K U H L M A N N - W I L S D O R F , Appi. P h y s . L e t t e r s 9, 176 (1966).

(Received

February

28, 1967)

J . A . SIGLER a n d D . KUHLMAOTT-WILSDORF : M e c h a n i c a l E n e r g y

545

phys. stat. sol. 21, 545 (1967) Subject classification: 10.1; 21; 21.1, 21.6 Department of Physics and Department of Materials Science, University of Virginia, Charlottesville, Virginia

Calculations on the Mechanical Energy of Vacancy Condensation Loops, Stacking Fault Tetrahedra, and Voids1) By J . A . SIGLER a n d D .

KUHLJIANN-WILSDORF

Employing the most accurate theoretical expressions known for the energy of five different types of vacancy condensation products, the relative stabilities of these are evaluated by means of a B 5500 computer. The calculations are performed for the cases of aluminum, nickel, platinum, copper, gold, and silver, while the defects considered are t h e hexagonal P r a n k loop on {111} with 1/3

¡Zi

O

o ¡2! 0 ¡5

¡Zi

O

o ¡z;

O £

¡Zi

o

m

œ

o ¡z;

CO 0) ¡H 04)9 >H

04)9 ¡x

o ¡2i

¡Zi

o

¡Zi

o

o

O Ä

¡Zi

o

04)9 ¡>H 049>

¡Zi

O

e«n O Q

ai a

¡Zi

O 04)3

0O 9

¡Zi

O Yes

Diamond loop

ai » ¡Zi

Yes

ai

o

Silver T\y = 55

>H ¡Z¡

Gold rjy = 25

Hexagonal loop

o £

Copper r/y = 22

Frank loop

ai ®

Platinum r¡y = 20

Tetrahedron

>H

Nickel r/y = 5

Void

Basic agreement between theory and experiment

Frank loops are only rarely seen

Frank loops and hexagonal perfect loops should not be stable

Basic agreement between theory and experiment

Voids are probably present. S.F. Energy must be quite low if faulted defects are seen

Frank loops and hexagonal perfect loops should not be stable

Remarks

Calculations on the Mechanical Energy of Vacancy Condensation Loops 551

o

ai

O

C O 10) are stable initially, and remain stable up to about 103 vacancies. Beyond t h a t size, the perfect loop becomes the most stable defect. Due to the activation energy necessary for conversion, the initially stable form may continue to grow to sizes much larger than predicted on the basis of minimum energy considerations.

555

Calculations on the Mechanical Energy of Vacancy Condensation Loops Acknowledgement

The authors acknowledge the support of the Atomic Energy Commission for this work under AEC Contract No. AT-(40-l)-3108. The computer calculations presented were made possible by a grant from the University of Virginia Computer Science Center. References [1] J . SILCOX and P. B. H I R S C H , Phil. Mag. 4, 72 (1959). [ 2 ] M. H. L O R E T T O , L . M. CLAREBROUGH, and R . L . S E G A L L , Phil. Mag.

11, 459

(1965).

[ 3 ] L . M . CLAREBROUGH, R . S E G A L L , M . H . LORETTO, a n d M . E . H A R G R E A V E S , P h i l .

Mag.

9, 372 (1964). [4] L. M. CLAREBROUGH, Conf. on Deformation of Crystalline Solids, Ottawa, August 1966 (in press). [ 5 ] L . M. CLAREBROUGH, R . L . S E G A L L , and M. H . L O R E T T O , Phil. Mag. 1 3 , 1 2 8 5 ( 1 9 6 6 ) . [6] R. M. J . COTTERILL, Lattice Defects in Quenched Metals, Academic Press, New York 1965 (p. 97). [7] R . E . SMALLMAN, K . H . WESTMACOTT, a n d J . COILEY, J . I n s t . M e t a l s 8 8 , 127 ( 1 9 5 9 ) . [8] [9] 10]

R. M. J . COTTERILL and R. L . S E G A L L , Phil. Mag. 8 , 1 1 0 5 ( 1 9 6 3 ) . R. M. J . C O T T E R I L L , Phil. Mag. 6 , 1 3 5 1 ( 1 9 6 1 ) . S. Y O S H I D A , Y . SHIMOMURA, and M . K I R I T A N I , J . Phys. Soc. J a p a n

17,

1196

(1962).

1 1 ] P . B . H I R S C H , J . SILCOX, R . E . SMALLMAN, a n d K . H . W E S T M A C O T T , P h i l . M a g . 3 , 8 9 7 12] 13]

(1958). M. H . LORETTO, L . M. V . C . K A N N A N and G .

14] D .

and P. H U M B L E , Phil. Mag. appi. Phys. 3 7 , 2 3 6 3 ( 1 9 6 6 ) .

CLAREBROUGH, THOMAS, J .

KUHLMANN-WILSDORF,

R.

MADDIN,

a n d H . G. F .

13, 953 (1966).

WILSDORF, S t r e n g t h e n i n g

Me-

chanisms in Solids, ASM 1962 (p. 737). 15] S . MADER, A . SEEGER, a n d E . SIMSCH, Z . M e t a l l k . 5 2 , 7 8 5 ( 1 9 6 1 ) .

and I. G R E E N F I E L D , Proc. Internat. Conf. Crystal Lattice Defects, J . Phys. Soc. J a p a n 1 8 , Suppl. I l l , 2 0 ( 1 9 6 3 ) . 17] K. CHIK, A. SEEGER, and M. RÜHLE, Proc. 5th Internat. Congr. Electron Microscopy, Vol. I, J - l l , Academic Press, New York 1962. 1 8 ] M. J . M A K I N and B . H U D S O N , Phil. Mag. 8 , 4 4 7 ( 1 9 6 3 ) . 1 9 ] M . K I R I T A N I , J . Phys. Soc. J a p a n 1 9 , 6 1 8 ( 1 9 6 4 ) . 20] M . K I R I T A N I , Y . SHIMOMURA, and S. Y O S H I D A , J . Phys. Soc. J a p a n 1 9 , 1624 (1964). 21] G. DAS and J . W. WASHBURN, Phil. Mag. 11, 955 (1965). 2 2 ] K . H . W E S T M A C O T T , Phil. Mag. 1 4 , 2 3 9 ( 1 9 6 6 ) . 2 3 ] S. Y O S H I D A , M . K I R I T A N I , Y . SHIMOMURA, and A . Y O S H I N A K A , J . Phys. Soc. J a p a n 2 0 , 16]

H.

G . F . WILSDORF

1962;

628 (1965).

24] S. YOSHIDA, M. KIRITANI, a n d Y . SHIMOMURA, L a t t i c e D e f e c t s in Q u e n c h e d M e t a l s ,

Academic Press, New York 1965 (p. 713). 25] D. K U H L M A N N - W I L S D O R F and H. G . F. W I L S D O R F , J . appi. Phys. 31, 516 (1960). 2 6 ] K . A . J A C K S O N , Phil. Mag. 7 , 1 1 1 7 ( 1 9 6 2 ) . 27] G . THOMAS and J . W . W A S H B U R N , Rev. mod. Phys. 35, 992 (1963). 2 8 ] J . A . S I G L E R and D. K U H L M A N N - W I L S D O R F , Harwell Conf. on the Nature of Small Defect Clusters, Vol. 1, H.M.S.O., London 1966 (p. 125). 29] R. M. J . COTTERILL, Harwell Conf. on the Nature of Small Defect Clusters, Vol. 1, H.M.S.O., London 1966 (p. 144). 30] D. KUHLMANN-WILSDORF, Lattice Defects in Quenched Metals, Academic Press, New York 1965 (p. 269). 31] D. J . BACON and A. G. CROCKER, Lattice Defects in Quenched Metals, Academic Press, New York 1965 (p. 667). 3 2 ] T . JOSSANG and J . P. H I R T H , Phil. Mag. 1 3 , 6 5 7 ( 1 9 6 6 ) . 36

physica 21/2

556

J . A . SIGLER a n d D . KUHLMANN-WILSDORF : M e c h a n i c a l

Energy

[33] R. A. JOHNSON, Phil. Mag., to be published. [ 3 4 ] G . CZJZEK, A . SEEGER, a n d S . MADER, p h y s . s t a t . s o l . 2 , 5 5 8 ( 1 9 6 2 ) . [ 3 5 ] R . C. FABINIAK a n d D . KUHLMANN-WILSDORF, C o n f . R e p t . o n t h e

Environment-Sen-

sitive Mechanical Behavior of Materials, Baltimore 1965, Gordon and Breach, in the press. [36] H . SUZUKI, Sci. Rep. Res. Inst. Tohoku Univ. 4, 455 (1952). [37] H . SUZUKI, J . Phys. Soc. J a p a n 17, 332 (1962). [38] H . SUZUKI, Dislocations and Mechanical Properties of Crystals, Wilev, New York 1962 (p. 3 6 1 ) . [ 3 9 ] D . KUHLMANN-WILSDORF, P h i l . M a g . 1 1 , 6 3 3 ( 1 9 6 5 ) .

[40] F . W. C. BOSWELL and G. E. RUDDLE, Canad. J . Phys. 43, 2096 (1965). [ 4 1 ] S . YOSHIDA, M . KIRITANI, a n d Y . SHIMOMURA, J . E l e c t r o n M i c r o s c o p y 1 2 , 1 4 8 ( 1 9 6 3 ) . [ 4 2 ] R . BULLOUGH a n d A . J . E . FOREMAN, P h i l . M a g . 9 , 3 1 5 ( 1 9 6 4 ) . [ 4 3 ] D . J . BACON a n d A . G . CROCKER, P h i l . M a g . 1 3 , 2 1 7 ( 1 9 6 6 ) .

[44] D. J . BACON, Phil. Mag. 14, 715 (1966). [ 4 5 ] T . R . D U N C A N a n d D . KUHLMANN-WILSDORF, J . a p p l . P h y s . 1 9 6 7 ( i n p r e s s ) . (Received

February

27,

1967)

557

F. KELEMEN et al.: Thermal Conductivity of CdTe phys. stat. sol. 21, 557 (1967) Subject classification: 8; 22.8 Faculty of Physics, "Babes-Bolyai" University, Cluj (a), and Institute of Physics, Academy of Sciences of the R.8.R., Bucharest O n

t h e

T h e r m a l

C o n d u c t i v i t y CdTe^a.Se,,.

of

C d T e

Solid

a n d

S o m e

(b)

C d T e ^ S *

a n d

S o l u t i o n s

By F . K E L E M E N ( a ) , A . N E D A ( a ) , D . N I C U L E S C U ( b ) , a n d E . CRTJCEANU ( b )

The thermal conductivity of CdTe and some solid solutions of CdTe l _ 3 . S x and CdTe^^. Sej. (x = 0.1 and 0.2) is studied. Measurements are made a t temperatures ranging f r o m 90 t o 700 °K, on two kinds of monocrystal specimens, prepared from elements of different purity. I n specimens synthetized from less pure elements, the thermal conductivity starts to increase with temperature between 300 and 400 °K, whereas in specimens obtained f r o m elements of 99.9999 per cent purity this increase only occurs a t higher temperatures (about 500 °K). The additional thermal conductivity in purer specimens may be ascribed t o internal electromagnetic radiation, b u t in less pure specimens this phenomenon is probably due to ionization of impurities. F o r impurities in CdTe, t h e values of ionization energy calculated from t h e temperature dependence of the thermal conductivity are comparable with those determined b y other methods. Es wird die Wärmeleitfähigkeit in CdTe und in einigen C d T e ^ ^ S x - u n d CdTe 1 _ a . Se x Mischkristallen (x = 0,1 und 0,2) untersucht. Die Messungen werden im Temperaturbereich von 90 bis 700 °K an zwei verschiedenen Sorten von Einkristallproben, die aus Elementen von verschiedenem Reinheitsgrad hergestellt worden sind, ausgeführt. An den aus weniger reinen Elementen hergestellten Proben wird schon im Bereich von 300 bis 400 °K eine Zunahme der Wärmeleitfähigkeit beobachtet. An den Proben, die aus Elementen mit einem Reinheitsgrad von 99.9999% bestehen, zeigte sich diese Zunahme n u r bei höheren Temperaturen um 500 °K. I n reineren Proben kann m a n die beobachtete zusätzliche Wärmeleitfähigkeit der inneren elektromagnetischen Strahlung zuschreiben, während in weniger reinen Proben die Erscheinung wahrscheinlich von der Ionisierung der Verunreinigungen herrührt. F ü r die Verunreinigungen im CdTe sind die Werte der Ionisierungsenergien, die auf Grund der temperaturbedingten Variationen der Wärmeleitfähigkeit bestimmt werden, mit den durch andere Methoden erhaltenen Werten vergleichbar.

1. Introduction The optical [1 to 5], electrical [2, 5], and photoelectrical [1, 2, 4, 5] properties of CdTe have thorougly been studied. In the last years attention has been paid also to its thermal properties. Thermal conductivity measurements have been carried out by Chasmar et al. [6, 7] in a temperature range from 300 to 500 °K, and by Devyatkova and Smirnov [8] from 100 to 500 °K. At low temperatures, from 3 to 300 °K, Slack and Galginaitis [9] have investigated in detail the processes of thermal conduction in pure CdTe as well as in CdTe doped with Mn, Fe, or Zn. Horch and Nieke [10] have recently studied the electrical and thermal properties of CdTe specimens with stoichiometric composition and of specimens with an excess of tellurium and cadmium between 36«

557

F. KELEMEN et al.: Thermal Conductivity of CdTe phys. stat. sol. 21, 557 (1967) Subject classification: 8; 22.8 Faculty of Physics, "Babes-Bolyai" University, Cluj (a), and Institute of Physics, Academy of Sciences of the R.8.R., Bucharest O n

t h e

T h e r m a l

C o n d u c t i v i t y CdTe^a.Se,,.

of

C d T e

Solid

a n d

S o m e

(b)

C d T e ^ S *

a n d

S o l u t i o n s

By F . K E L E M E N ( a ) , A . N E D A ( a ) , D . N I C U L E S C U ( b ) , a n d E . CRTJCEANU ( b )

The thermal conductivity of CdTe and some solid solutions of CdTe l _ 3 . S x and CdTe^^. Sej. (x = 0.1 and 0.2) is studied. Measurements are made a t temperatures ranging f r o m 90 t o 700 °K, on two kinds of monocrystal specimens, prepared from elements of different purity. I n specimens synthetized from less pure elements, the thermal conductivity starts to increase with temperature between 300 and 400 °K, whereas in specimens obtained f r o m elements of 99.9999 per cent purity this increase only occurs a t higher temperatures (about 500 °K). The additional thermal conductivity in purer specimens may be ascribed t o internal electromagnetic radiation, b u t in less pure specimens this phenomenon is probably due to ionization of impurities. F o r impurities in CdTe, t h e values of ionization energy calculated from t h e temperature dependence of the thermal conductivity are comparable with those determined b y other methods. Es wird die Wärmeleitfähigkeit in CdTe und in einigen C d T e ^ ^ S x - u n d CdTe 1 _ a . Se x Mischkristallen (x = 0,1 und 0,2) untersucht. Die Messungen werden im Temperaturbereich von 90 bis 700 °K an zwei verschiedenen Sorten von Einkristallproben, die aus Elementen von verschiedenem Reinheitsgrad hergestellt worden sind, ausgeführt. An den aus weniger reinen Elementen hergestellten Proben wird schon im Bereich von 300 bis 400 °K eine Zunahme der Wärmeleitfähigkeit beobachtet. An den Proben, die aus Elementen mit einem Reinheitsgrad von 99.9999% bestehen, zeigte sich diese Zunahme n u r bei höheren Temperaturen um 500 °K. I n reineren Proben kann m a n die beobachtete zusätzliche Wärmeleitfähigkeit der inneren elektromagnetischen Strahlung zuschreiben, während in weniger reinen Proben die Erscheinung wahrscheinlich von der Ionisierung der Verunreinigungen herrührt. F ü r die Verunreinigungen im CdTe sind die Werte der Ionisierungsenergien, die auf Grund der temperaturbedingten Variationen der Wärmeleitfähigkeit bestimmt werden, mit den durch andere Methoden erhaltenen Werten vergleichbar.

1. Introduction The optical [1 to 5], electrical [2, 5], and photoelectrical [1, 2, 4, 5] properties of CdTe have thorougly been studied. In the last years attention has been paid also to its thermal properties. Thermal conductivity measurements have been carried out by Chasmar et al. [6, 7] in a temperature range from 300 to 500 °K, and by Devyatkova and Smirnov [8] from 100 to 500 °K. At low temperatures, from 3 to 300 °K, Slack and Galginaitis [9] have investigated in detail the processes of thermal conduction in pure CdTe as well as in CdTe doped with Mn, Fe, or Zn. Horch and Nieke [10] have recently studied the electrical and thermal properties of CdTe specimens with stoichiometric composition and of specimens with an excess of tellurium and cadmium between 36«

558

F . KBLEMEÏT, A . N É D A , D . NICULESCU, a n d E .

CRUCEANU

150 a n d 450 °K. T h e y have observed an increase of t h e r m a l conductivity a t t e m p e r a t u r e higher t h a n about 350 °K. The electrical and t h e r m a l properties of a CdTe-CdSe system of some different compositions have been investigated b y Stuckes and Farrel [11]. B u t for t h e CdTe-CdS system — according t o t h e a u t h o r s ' knowledge — no thermal conductivity measurements have been performed so f a r . I n this paper we present t h e results obtained f r o m t h e investigation of thermal conductivity of CdTe and some of t h e solid solutions mentioned above. The measurements were carried out at t e m p e r a t u r e s ranging f r o m 90 t o 700 °K on two kinds of monocrystalline specimens prepared f r o m elements of different p u r i t y . W i t h these experiments we aimed a t obtaining some information about heat conduction mechanisms in these semiconductors a t high temperatures. 2. Specimen Preparation The p r e p a r a t i o n of b o t h CdTe and its solid solutions was carried out according t o B r i d g m a n ' s horizontal method in evacuated q u a r t z ampoules under h e a t t r e a t m e n t conditions described in [12], as well as b y means of a technique similar t o t h a t used in [11] for CdTe 1 _ ;c Se 3: . The specimens denoted in this paper with I t o I I I were prepared f r o m elements of 99.9999% p u r i t y , while for t h e synthesis of specimens denoted with 1 t o 5 we used cadmium pro analysi. F o r obtaining solid solutions, CdTe and CdS or CdTe and CdSe were t a k e n in proper amounts. Crystal s t r u c t u r e determinations carried out b y t h e X - r a y diffraction method in a 114 m m diameter Debye-Scherrer camera showed t h a t b o t h CdTe and its solid solutions had a sphalerite s t r u c t u r e whose lattice parameter decreased as t h e a m o u n t of CdS or CdSe increased in t h e solid solutions. 3. Measuring Methods 3.1 Thermal

conductivity

The t h e r m a l conductivity was determined by measuring t h e t h e r m a l diffusivity. I n this case t h e coefficient of t h e r m a l conductivity (K) was calculated f r o m t h e expression K = qc a , (1) where c specific heat, q density, and a t h e r m a l diffusivity. Thermal diffusivity measurements were performed b y means of t h e heat pulse m e t h o d described in [13] and [14] for a single length of t h e specimen. The junction of t h e thermocouple was soldered t o a t h i n silver plate, t o which t h e specimen was fixed a t one of its ends with " D e g u s s a " silver paste. Then t h e specimen was introduced into a glass t u b e and k e p t in horizontal position b y t h e wires of t h e thermocouple. The measurements were carried out a t atmospheric pressure. T h e heat pulse was produced b y illuminating t h e f r o n t surface of t h e specimen for ti = 0.5 or 1 s. Recording t h e variation of t e m p e r a t u r e on t h e rear surface of t h e specimen as a function of time, t h e thermal diffusivity was calculated f r o m t h e expression L2 a

= 2tm(l

+ 2btm)

'

(2)

where L length of specimen, tm t i m e in which the t e m p e r a t u r e reaches its maxi-

Thermal Conductivity of CdTe and Some CdTe 1 _ x S a . Solid Solutions Fig. 1. Variation of temperature on the rear surface of a specimen as a function of time after transmission of a heat pulse

559

A

0

tm

Time —»-

mum value, b a coefficient characterizing the heat transmission between the specimen and the ambient atmosphere. This coefficient is determined from the branch A B of the curve T = f(t) of Fig. 1. The method gives accurate results when ti < ¿ m /10. If this condition is not satisfied, a correction is made according to the procedure indicated in [13]. The accuracy of measurements was about 4% at room temperature and about 7% near 100 ° K . The values of specific heat for the compounds studied were taken from [15] and for solid solutions they were determined from the specific heat of the component compounds on the basis of the Neumann-Kopp law. The density of the specimens was measured at room temperature, and its variation with temperature was determined by calculating the coefficient of linear expansion from the expression given by [16]. 3.2 Electrical

conductivity

To interpret the experimental data it was necessary to know also the variation of the electrical conductivity of the specimens in the studied temperature range. The electrical resistivity of specimens I to I I I was high, their conductivity was determined by means of an electrometer with high input resistance (Ri > 1015 ß). Ohmic contacts were prepared according to the method described by Kroger and de Nobel [17]; their quality was checked by recording V-I curves at room temperature. The electrical conductivity measurements were carried out subsequently to the thermal conductivity measurements on the same specimens. The size of specimens I to I I I was 1 0 x 3 x 2 . 5 mm 3 and that of specimens 1 to 5 was about 8 x 4 x 3 mm 3 . 4. Experimental Results Fig. 2 shows the variation of thermal resistance 1/iiT versus temperature for specimens I to III. On this figure the variation of I j K for CdTe according to the data obtained by [9] (curve A), [8] (curve B), [7] (curve C), and [10] (curve D) is also presented. The data of curve A refer to a specimen of high purity (marked by the authors with R-96) for which the concentration of impurities was lower than 1018 cm - 3 . (The value of K obtained by us for specimen I at 300 ° K is lower by approximately 15%, and at 100 ° K by 40% than that of curve A.) From the shape of curves I to I I I on can see that the thermal conductivity shows an increase beginning from 530 ° K for specimen I, and from about 480 ° K for the other two specimens.

560

F. Kelemen, A. Néda, D. Niculescu, and E. Cruceantt - T(°K) 500350250200 150 125 100 \

J

- A k

X

\ !

¡1

VL

I la

i

T/°K}~ Fig. 2. Temperature dependence of thermal resistance as a function of temperature for specimens I to III: I - CdTe, II - CdTe„.»S,,.,, I l l - CdTe0.,S,,.s. I a specimen I after heating to 580 A, B, and C — CdTe, according to measurements of [9], [8], and [7]; D — CdTe + 0.1% excess Cd, according to measurements of [10] (specimen 10.2); E — variation calculated by [9]

i

i

i

i

l

1

..

1 2 3 k 5 6 7 8 9 10 11

Fig. 3. Electrical conductivity as a function of reciprocal temperature for specimens I to III. The specimen composition is given in Fig. 2

Curve I a represents the variation of 1 ¡K after the first measurement when specimen I was heated to 580 °K. It can be observed that the thermal conductivity changed to a large extent after heating: at temperatures below 300 °K it becomes lower, and above 300 °K it did not show any essential variations with temperature. In the case of specimens II and III a variation due to heating was also observed, but this was much lower than for specimen I. Fig. 3 represents the variation of the electrical conductivity (log a) versus 1 \T for specimens I a, II, and III. It seems possible that the deviation of log cr from a linear relationship observed above 300 °K is due to structural changes that appeared in specimens after heating during the first thermal conductivity measurement. It should be mentioned that for CdTe similar changes were observed in its electrical [10, 18, 19], optical, and photoelectrical properties [1,2, 4], It may be assumed that after heating some defects appeared in the crystal lattice of CdTe (Frenkel disorder [2]). The phonons are scattered by these defects. This phenomenon is probably the cause of the decrease of thermal conductivity at temperatures below 300 °K. At the same time, at higher temperatures the ionization of vacancies may result in an increase of thermal conductivity (cf. discussion). For specimens 1 to 5 the variation of thermal resistance, illustrated in Fig. 4, is also linear at low temperatures. But the deviation from the linear relationship appears at lower temperatures than for specimens I to III. Specimens 1 to 5 did not show any essential changes in thermal conductivity after heating to 700 °K. In Fig. 4 the variation of \\K for three specimens (8.2, 6, and 9) according to Horch and Nieke [10] is also given for comparison. Specimens 8.2

Thermal Conductivity of CdTe and Some CdTe 1 _ I S x Solid Solutions T(°K) 700 500 m

'7 0

700 200 300 WO

300

250

561

200

170

'i

500 600 700 T(°K)

Fig. 4. Temperature dependence of thermal resistance for specimens 1 to 5: 1 — CdTe, 2 — CdTeo.jSj.j, 3 — CdTc 0 .,S(,. 2 , 4 — CdTe 0 .,Se 0 .„ 5 - CdTe„.,Se 0 . a . (Specimens 8.2 (CdTe + 1 % Cd), 6 (CdTe + 1 % Te), and 9 (CdTe + 0 . 1 % Te) according to [10])

Fig. 5. Electrical conductivity as a function of reciprocal temperature for specimens 1 to 5 (specimens 8 . 2 , 6 , and 9 according to [10]). The specimen composition is given in Tig. 4

and 6 contained a cadmium and tellurium excess of 1 % while specimen 9 contained a tellurium excess of 0.1%. (These specimens were prepared from cadmium and tellurium of 99.999% purity [10].) Fig. 5 shows the variation of electrical conductivity (log a) versus 1 ¡ T for specimens 1 to 5. The activation energies calculated on the basis of the curves of this figure show that specimens 1 to 5 have an impurity type conductivity. I t should be noted that the activation energies obtained for specimen 1 (CdTe) from the two straight-line portions of log a, are comparable to those found by [18] for a p-type CdTe specimen doped with lithium. One can observe on Fig. 4 that the maximum of IjK appears approximately at the same temperatures at which the variation of log a is very small in Fig. 5. A similar result has already been reported [10]. 5. Discussion

Slack and Galginaitis [9] have analysed in detail the variation of thermal conductivity of pure and doped CdTe at low temperatures. The results show that at temperatures higher than 30 °K the variation of K is determined by Umklapp processes. In Fig. 2 curve E represents the variation of 1 ¡K calculated by these authors, taking TD = 158 °K for the Debye temperature and y = 2 for the Griineisen constant. For temperatures T > TD the variation of 1 \K is linear. At high temperatures, beside Umklapp processes, other heat conduction mechanisms should be taken into consideration. The electrical conductivity of specimens 1 to 5 remains below 10 _ 1 O - 1 c m - 1 . Therefore, the observed variation cannot be explained by ambipolar diffusion [20], Taking into account

562

F. Kelemen, A. Néda, D. Niculescu, and E. Crtjceanu

that the increase of thermal conductivity begins in general at the same temperature at which the variation of electrical conductivity is very small, Horch and Nieke [10] have assumed that the additional thermal conductivity is due to excitons. The role of excitons in thermal conductivity has been suggested by other authors too [21], But the value of the exciton energy found from the thermal conductivity variation of CdTe [10] is higher by one order of magnitude than that found from optical measurements [22, 23]. Consequently, it does not seem probable that the excitons play any part in heat conduction in the observed phenomenon. For selenium [24], tellurium [20, 25], CdSb [26], and GaSe [27], the thermal conductivity increase above 200 to 250 °K and 400 °K may be explained by internal electromagnetic radiation. The additional thermal conductivity that results from this effect can be calculated from the expression given by Genzel [28]:

where cr0 Stefan-Boltzmann constant, n refractive index, a absorption coefficient for wavelength / m at which the radiation of a black body reaches its maximum value at temperature T. The value of Am at temperatures ranging from 300 to 700 °K varies between 10 and 4 ¡xm. According to the measurements performed by [29 to 31] on CdTe, optical transmission and absorption do not vary essentially between 1 and 10 [xm, Therefore, the value a. = 9 c m - 1 found in [32] at A = 1 fim may be taken also for the range X = 1 to 10 (i,m. The refractive index is 2.66 for CdTe and CdSe, and 2.30 for CdS [23]. By substituting the values of a and n for CdTe into equation (3) and taking T = 580 °K, we found Ktíá = 4.6 x X 10~ 3 W/cm deg which is comparable with 4.8 X 10~ 3 W/cm deg obtained experimentally for specimen I. As regards specimens I I and I I I , their experimental values at 580 °K were found to be 10 X 10" 3 and 7.3 X 10" 3 W/cm deg, respectively. But it should be noted that the calculated -K"rad value varies less slowly with temperature than the experimental one. In addition, the absorption coefficient in the infrared region near the absorption edge depends to a large extent on the purity of the specimens and generally increases with the concentration of impurities and current carriers [5, 33]. Consequently, we believe that for specimens 1 to 5 (prepared from elements of less purity) the thermal conductivity increase above 300 to 400 °K cannot be exclusively explained by internal electromagnetic radiation. Krumhansl [34] has shown that at high temperatures other heat transport mechanisms may also contribute, in which, similar to the exciton heat transfer, there is an excitation energy Ei between the ground state and the first excited states of the system. The additional heat conductivity for each of these phenomena may be expressed by =

P ( - § ; ) >

W

where k Boltzmann constant, g number of all current-carrying states above the ground state, and E¡, L¡, and v¡ excitation energy, characteristic length for

Thermal Conductivity of CdTe and Some C d T e ^ S j Solid Solutions Fig. 6. Variation of log (KiT) vs. 1 IT. K\ = K m — -KL is the additional thermal conductivity due to ionization of impurities or defects. The specimen composition is given in Table 1

-—T(°K) 700 600 500

563

WO 350

damping of temperature differences, and characteristic speed of transport, respectively. Depending on the specific case, Ei may be the band gap, the exciton, the dissociation, or ionization energy; L x and i>t are chosen in an appropriate manner [34], Exciton thermal conduction is unlikely, as already stated. Therefore, other similar heat transport mechanisms have to be taken into account. For scattering of electrons and holes by lattice vibrations the product vl Ll is proportional to T~112 [5,35], and for scattering by impurity ions % T5/'2 [35]. Consequently, as a first approximation it may be assumed that Li ~ T. I n this case, according to equation (4) the function log (Ki T) = /(1/T) must yield a straight line. Fig. 6 proves the linearity for all specimens. In Table 1 the Ei values obtained on the basis of Fig. 6 and equation (4) Table 1 Ionization energy of impurities or lattice defects in CdTei_ x Sx and CdTei _ x $ e x , determined by thermal conductivity and other measurements Ì Speci- I men 1 Composition number | I !

1

CdTe

Our thermal conductivity measurements (equation (4))

Impurities or defects

Cu, Ag, As

Ionization energv (eV)

0.32

Other thermal conductivity measurement ([10], equation (3))

Other measurements

I lonizaExcess in CdTe j t i o n energy (eV)

Impurities or defects

Ionization energy (eV)

Cu, Au, Ag

0.33

2

Cd, 1%

0.28

Li

0.27 0.32

36 18

Te, 1 %

0.22

Sb, P

0.36

36

Te, 0 . 1 %

0.31

Reference

2

CdTeo.gSo.i

Cu, Ag, As

0.22

-

-

-

-

-

3

CdTeo.8So.2

Cu, Ag, As

0.21

-

-

-

-

-

4

CdTeo.gSeo.i Cu, Ag, As

0.21

-

-

-

-

5

CdTeo.sSeo.2 Cu, Ag, As

0.217

-

-

-

-

la

CdTe

Cd-vacancies

0.145 Cd-vacancies

0.15

2,4

—

—

564

F . KELEMEN, A . NEDA, D . NICULESCU, a n d E . CRUCEANU

are given for specimens 1 to 5 and l a . This table also comprises the ionization energies, determined by other methods, for some impurities and defects in CdTe [36]. The agreement between these values and those obtained by us is satisfactory. The data obtained by [10] for Eit which are considered as exciton energies, are also given in the table. One can see that these values have the same order of magnitude as the ionization energy, but greatly differ from the exciton binding energy which is 0.012 eV in CdTe [22, 23]. For CdTej-^Ss and CdTei-^Se^ specimens with a;=0.1 and 0.2, the Rvalues are lower than those for CdTe. In the literature there are no data concerning the ionization energy of these solid solutions. It should be mentioned that in the case of CdSe the ionization energy is 0.60 eV for copper impurities and cation vacancies, and 0.14 eV for anion vacancies [36]. The ionization energy for copper impurities in CdS decreases from 0.80 to 0.20 eV when their concentration increases from 1017 to 1020 cm - 3 [37]. Taking into account that the sulfur and selenium content contributes to the increase of impurity concentration in CdTe, the Ei values obtained by us for these solid solutions are acceptable. References [ 1 ] C. Z . VAN DOORN a n d D . DE NOBEL, P h y s i c a 2 2 , 3 3 8 ( 1 9 5 6 ) .

[2] D. DE NOBEL, Philips Res. Rep. 14, 361 (1956). [3] T. S. Moss, Optical Properties of Semiconductors, Izd. IL, Moscow 1961 (p. 250) (in Russian). [4] G. B. DUBROVSKII, Fiz. tverd. Tela 3, 1305 (1961). [5] R. A. SMITH, Semiconductors, Izd. IL, Moscow 1962 (p. 410,198, and 152) (in Russian). [6] R. P. CHASMAS, E. W. DURHAM, and A. D. STUCKES, Proc. Internat. Conf. Semiconductor Physics (Prague 1960), Czechoslovak Academy of Sciences, Prague 1961 (p.1018). [7] A. D. STUCKES, Brit. J . appl. Phys. 12, 675 (1961). [8] E . D . DEVYATKOVA a n d I . A . SMIRNOV, F i z . t v e r d . T e l a 4 , 2 5 0 7 ( 1 9 6 2 ) .

[9] G. A. SLACK and S. GALGINAITAS, Phys. Rev. 133, A253 (1964). [10] R. HORCH and H. NIEKE, Ann. Phys. (Germany) 16, 289 (1965). [11] A. D . STUCKES a n d G. FARREL, J . P h y s . Chem. Solids 25, 477 (1959).

[12] E. CRUCEANU and D. NICULESCU, C. R. Acad. Sei. (France) 261, 935 (1965). [13] F . KELEMEN, F . BOTA, a n d A. NEDA, Studii Cere. Fiz. (Bucuresti) 16, 809 (1964).

[14] F. KELEMEN, Acta phys. Hungar. 23, 111 (1967). [15] LANDOLT-BÖRNSTEIN, Vol. II/4, Springer-Verlag, Berlin/Göttingen/Heidelberg 1961 (p. 4 8 3 ) . [16] K. A. GSCHNEIDER, Jr., Solid State Phys. 16, 397 (1964). [17] F . A . KRÖGER a n d D . DE NOBEL, J . E l e c t r o n i c s a n d C o n t r o l 1 , 1 9 0 ( 1 9 5 5 ) .

[18] B. M. VUL and V. A. CHAPNIN, Fiz. tverd. Tela 8, 256 (1966). [19] P . HÖSCHL, phys. s t a t . sol. 13, K 1 0 1 (1966).

[20] J . R. DRABBLE and H. J . GOLDSMID, Thermal Conduction in Semiconductors, Izd. IL, Moscow 1963 (p. 143 and 205) (in Russian). [21] A. F. IOFFE, Canad. J . Phys. 34, 1342 (1956). ¿22] W . G. SPITZER a n d C. A. MEAD. J . P h y s . Chem. Solids 25, 443 (1964). [23] R . E . HALSTED, M . R . LORENTZ, a n d B . SEGALL, J . P h y s . C h e m . S o l i d s 2 2 , 109 ( 1 9 6 1 ) . ¿24] G . B . ABDULLAEV, S. I . MEKHTIEVA, D . SH. ABDINOV, G . M . ALIEV, a n d S . G . ALIEVA,

phys. stat. sol. 13, 315 (1966). [ 2 5 ] E . D . DEVYATKOVA, (1959).

B . YA. MOIZHES, a n d I . A . SMIRNOV,

Fiz. tverd.

T e l a 1,

613

[26] I . M. PILAT, L. I . ANTICHUK, a n d A. V. LUBCHENKO, Fiz. t v e r d . Tela 4, 1649 (1962). [ 2 7 ] G . D . GUSEINOV a n d A . I . RASULOV, p h y s . s t a t . sol. 1 8 , 9 1 1 ( 1 9 6 6 ) .

[28] L. GENZEL, Z. Phys. 135, 177 (1953); Glastechn. Ber. 26, 69 (1953).

Thermal Conductivity of CdTe and Some CdTe1_I.S;c Solid Solutions

565

[29] W. D . L A W S O N , S. N I E L S E N , E. H. P U T L E Y , and A. S. Y O U N G , J . Phys. Chem. Solids 9, 325 (1959). [30] J . C. W O O L E Y and B. B A Y , J . Phys. Chem. Solids 1 5 , 27 (1960). [31] O. G . LORIMOR and W. G . S P I T Z E R , J . appl. Phys. 3 6 , 1841 (1965). [32] P. W. D A V I S and T. S . S H I L L I D A Y , Phys. Rev. 118, 1020 (1960). [ 3 3 ] W. G . S P I T Z E R and M. W H E L A N , Phys. Rev. 1 1 4 , 59 (1959). [34] J . A. KRUMHANSL, J . Phys. Chem. Solids 8, 343 (1959). [35] A. F. J O F F E , Physik der Halbleiter, Akademie-Verlag, Berlin 1958 (p. 294). [36] R. H . B U B E and E . L . L I N D , Phys. Rev. 110, 1 0 4 0 ( 1 9 5 8 ) . [37] R. H. B U B E and A. B . D R E E B E N , Phys. Rev. 1 1 5 , 1578 (1959). (Received

February

20,

1967)

P. SCHNUPF: Kronig-Penney-Type Calculations for Electron Tunneling

567

phys. stat. sol. 21, 567 (1967) Subject classification: 14.3.1; 13.1 Max-Planck-Institut für Physik und Astrophysik, Abteilung Numerische Rechenmaschinen, München

Kronig-Penney-Type Calculations for Electron Tunneling through Thin Dielectric Films By P.

SCHNUPF

Using a Kron ig-Penney model and matrix methods the probability D = exp ( — 2 x a) of an electron tunneling through a thin dielectric film can be calculated exactly for an arbitrary value of the thickness a. The relation between the damping constant x and the electron energy E in the band gap of the dielectric depends on the thickness a. For thick films containing more than 10 to 15 crystal planes the jB(>i2)-curves approximate to the usual ones obtained from band theory. However, the simple model involving potential walls without any potential wells is the limiting case for very thin films. Nevertheless, the departure of the complex band structure of the dielectric from the free electron-approximation usually used in tunneling calculations should be observable qualitatively even for very thin films. Mit Hilfe von Matrixmethoden läßt sich für ein eindimensionales Kroning-PenneyModell eines Dielektrikums die Tunnelwahrscheinlichkeit D — exp ( — 2 x a) durch Schichten beliebiger Dicke a exakt angeben. Die Beziehung zwischen der Dämpfungskonstante x und der Elektronenenergie E im verbotenen Band ist von der Schichtdicke abhängig. F ü r dicke Schichten (mehr als 10 bis 15 Atomlagen) geht sie in die aus der üblichen Bändertheorie zu erhaltende Beziehung über. Für sehr dünne Schichten verformt sich die E(x2)Kurve immer mehr zu der, die für eine einfache Potentialschwelle ohne Potentialmulden berechnet werden kann. Qualitativ bleibt jedoch die aus der komplexen Bandstruktur des Dielektrikums folgende Abweichung von der üblicherweise für Tunnelstromrechnungen benutzen „freien Elektronen-Näherung" auch bei sehr dünnen Schichten erhalten.

1. Introduction To calculate the tunneling probability D of electrons through thin dielectric films one assumes in most cases an 2?(fc)-dependence of the form E — V = = h2 P / 2 m*. V is the electron energy at the lower edge of the conduction band (Fig. lb), m* the effective mass of the electrons at the band edge; in the forbidden band one assumes a complex k. In the following this method will be designated as "free-electron approximation". Against this approximation even from the band-theoretical viewpoint one may raise the objection that it neglects the existence of the valence band which lies below the conduction band in an energetical distance Eg. At the edge of the valence band the tunneling probability should increase to unity again. Hence one expects an E(k)-dependence differing from the free-electron approximation. Therefore in [1] the tunneling current is calculated using a parabolic approximation for the E(k2)curve as suggested by Franz [2] and it is shown that this assumption really improves the consistency of the experimental with the theoretical tunneling characteristic for thin A1203 films. Moreover, it may be argued that the band model which is derived usually assuming an infinite dielectric may not be a good approximation for thin films.

P. SCHNUPF: Kronig-Penney-Type Calculations for Electron Tunneling

567

phys. stat. sol. 21, 567 (1967) Subject classification: 14.3.1; 13.1 Max-Planck-Institut für Physik und Astrophysik, Abteilung Numerische Rechenmaschinen, München

Kronig-Penney-Type Calculations for Electron Tunneling through Thin Dielectric Films By P.

SCHNUPF

Using a Kron ig-Penney model and matrix methods the probability D = exp ( — 2 x a) of an electron tunneling through a thin dielectric film can be calculated exactly for an arbitrary value of the thickness a. The relation between the damping constant x and the electron energy E in the band gap of the dielectric depends on the thickness a. For thick films containing more than 10 to 15 crystal planes the jB(>i2)-curves approximate to the usual ones obtained from band theory. However, the simple model involving potential walls without any potential wells is the limiting case for very thin films. Nevertheless, the departure of the complex band structure of the dielectric from the free electron-approximation usually used in tunneling calculations should be observable qualitatively even for very thin films. Mit Hilfe von Matrixmethoden läßt sich für ein eindimensionales Kroning-PenneyModell eines Dielektrikums die Tunnelwahrscheinlichkeit D — exp ( — 2 x a) durch Schichten beliebiger Dicke a exakt angeben. Die Beziehung zwischen der Dämpfungskonstante x und der Elektronenenergie E im verbotenen Band ist von der Schichtdicke abhängig. F ü r dicke Schichten (mehr als 10 bis 15 Atomlagen) geht sie in die aus der üblichen Bändertheorie zu erhaltende Beziehung über. Für sehr dünne Schichten verformt sich die E(x2)Kurve immer mehr zu der, die für eine einfache Potentialschwelle ohne Potentialmulden berechnet werden kann. Qualitativ bleibt jedoch die aus der komplexen Bandstruktur des Dielektrikums folgende Abweichung von der üblicherweise für Tunnelstromrechnungen benutzen „freien Elektronen-Näherung" auch bei sehr dünnen Schichten erhalten.

1. Introduction To calculate the tunneling probability D of electrons through thin dielectric films one assumes in most cases an 2?(fc)-dependence of the form E — V = = h2 P / 2 m*. V is the electron energy at the lower edge of the conduction band (Fig. lb), m* the effective mass of the electrons at the band edge; in the forbidden band one assumes a complex k. In the following this method will be designated as "free-electron approximation". Against this approximation even from the band-theoretical viewpoint one may raise the objection that it neglects the existence of the valence band which lies below the conduction band in an energetical distance Eg. At the edge of the valence band the tunneling probability should increase to unity again. Hence one expects an E(k)-dependence differing from the free-electron approximation. Therefore in [1] the tunneling current is calculated using a parabolic approximation for the E(k2)curve as suggested by Franz [2] and it is shown that this assumption really improves the consistency of the experimental with the theoretical tunneling characteristic for thin A1203 films. Moreover, it may be argued that the band model which is derived usually assuming an infinite dielectric may not be a good approximation for thin films.

568

P.SCHNUPP Fig. 1.

One-dimensional energy diagrams for a dielectric film, a) Actual potential, b) energy diagram as used in the band model, c) Kronig-Penney model

Metal 1

6-potentio/ wells

As shown in [3] using a KronigPenney model of a thin film with "shallow" potential wells, there may be considerable discrepancies with the usual expressions for the tunneling probability even in absence of a valence band, i.e. for the case of tunneling below the lowest allowed band. In the following we shall investigate in which manner the _E(&2)-dependence as calculated for an infinite dielectric using conventional band-theoretical methods will be modified by limiting the thickness of the dielectric to a small number of atomic layers.

2. The ¿ - W e l l Model for a Limited Dielectric

As in [3], also in this paper the real potential in the thin dielectric film (sketched one-dimensionally in Fig. 1 a) will be replaced by an one-dimensional ¿-well model as shown in Fig. l c . As in [3], the potential wells in the metal electrodes (dotted in Fig. l c ) will be ignored. Using this Kronig-Penney model one can calculate rigorously the tunneling probability for the thin film. On the other hand, for an unlimited sequence of ¿-potential-wells we can calculate the band structure and hence the E(k2)-curve in the forbidden band, not using a kp-approximation or even the semi-empirical parabolic approximation as used by Franz. Hence this model is especially well suited to compare the unlimited crystal with the thin dielectric film. From Fig. 1 c one obtains the following energy values for a tunneling electron as a function of the coordinate x : Potential energy for x

x x =

v a v a — N

E = ft2 ky2

V(x) = 0

0 and x ^ a

for 0 < x < a,

Kinetic energy

E -

V(x) = V 0 < v< N V(x) = V -

Vd d{x)

h2 «

m

V = h2 ¡fcfj/2 TO

Kronig-Penney-Type Calculations for Electron Tunneling through Thin Films

569

k n is the electron wave number in the sections of constant potential in the dielectric which are separated by the (5-wells. For V > E, kB is imaginary. kM denotes the wave number in the metal electrodes, rj measures the "depth" of the potential wells and has the dimension of a wave number. In each section of constant potential the Schrodinger equation may be solved by a pair of plane waves or exponential functions: for

for

v = 0:

0 = B0 exp ( — i kM x) ,

l ^ v ^ N :

1). I n the case of the Procopiu effect with a cylindrical ferromagnetic thin film (much thinner than the films used by us due to experimental difficulties caused by the presence of the copper wire) fixed at one end, the energy stored at a given moment in the volume unit of the specimen is given by [12] (Fig. 4) 3 W = K sin 2 (a — yi) + — a, sin 2 (45° — a) — H J cos a — H J sin a , y

F i g . 4. R e l a t i v e position between [magnetization vector J a and specimen axis in the case t h a t the easy a x i s does not coincide with the specimen axis

s

X

S

Fig. 5. Critical curve for a constant value of hp < < ./ 8 Hy, j / 2 K ihy = 0.2) in the plane (kg = 3/2 ?.B ai/2 K, hx = H z J s l 2 K )

Magnetostrictively Induced Torsional Oscillations

-2.0 -1.5 -10 -05

605

0

0.5 W

1.5 2.0

| 50 ¿r2.5-2.0 -15 -10

05 -25 -

\

1.5 2J) hx

-5.0-75Fig. 6. Ax-dependence of the magnetic flux @ff through the surface limited by the test coil, plotted on the basis of the proposed model, as well as of the e.m.f. Ei induced in the test coil obtained by graphic derivation of 0y (h x and the torsional oscillation are phase shifted by 90° and yi = 18°)

Fig. 7. /¿¿-dependence of the magnetic flux through the surface limited by the test coil plotted on the basis of the proposed model, as well as of the e.m.f. Ei induced in the test coil obtained by graphic derivation of (hx and the torsional oscillation are in phase)

where K is the anisotropy constant of the sample, ip the angle between the specimen axis and the easy axis, the saturation value of the magnetostriction and cTj the shear strain in phase with the corresponding Wiedemann twist. The QW

d2W

8W

equilibrium value of the angle a is determined by — = 0 and —2- > 0. — = 0 6 J ^ 82W 8a 8a ^ 8a and = 0 represent straight lines in the plane (ha, hx). Their crossing point separates the region of stable equilibrium from the region of unstable equilibrium. The geometric locus of the crossing points is the critical curve. As long as the specimen does not perform torsional oscillations (a i = 0) the effect is the same as in the case of the Procopiu effect with a ferromagnetic specimen fixed at both ends, which has been investigated in a previous paper [10]. For a constant value of hy > 3 X oH J \

( by

~ Y

Js H ^

I:1 the effect can be easily followed in the

c r ^ c a l c u r v e represented in Fig. 5 is given cos 2 y> cos3 a -)- by 1 . hs = 2 — ~ sin a (1 + 2 cos a) y sm 2 V > ^ _ cos 2 y> + hu cos a (1 + 2 sin2 a) * ~~ sin a (1 + 2 cos2 a) '

~

2 k)'

,

If hx and hs are phase-shifted by 90°, the point characterizing the state of the sample described an ellipse in the plane (ha, hx). The variation of the magnetic flux & y through the surface limited by the coil, plotted on the basis of the proposed model, as well as the variation of the e.m.f.

606

B. F. ROTHENSTEIN et al.: Magnetostrictively Induced Torsional Oscillations

E j induced in the test coil, obtained by graphic derivation of & y , are given in Fig. 6 as a function of hx. I n good agreement with the experiment (Fig. 2 and 3) the magnetization vector performs partial switching from region I V towards region I and from region I I I towards region I I (see Fig. 4). The switching are released at points a and b of the critical curve. The proposed model shows that the tension between the ends of the sample presents partial switching signals alternating in sign. If hx and hs are in phase, the point which characterizes the state of the sample describes a straight line in the plane (hs, h x ). The variation of the magnetic flux & y through the surface limited by the test coil, plotted on the basis of the proposed model, as well as the variation of the e.m.f. Ei induced in the test coil obtained by graphic derivation of & v , are given in Fig. 7 as a function of hx. I n good agreement with the experiment (Fig. 2 and 3) the magnetization vector performs only continuous rotations. If Hy > Hy l the partial switchings of the magnetization vector from region I V towards region I and from region I I I towards region I I are equivalent with passing from a right torsion towards a left torsion and from a left torsion towards a right torsion, respectively. If hx and the torsional oscillation are phase-shifted by 90°, energy transfer from h x towards the specimen fixed at one end becomes possible. References [1] G. WIEDEMANN, Lehre von der Elektrizität, 3, 680 (1883). [2] I. R. SMITH and K . J . OVERSHOT, Brit. J . appl. Phys. 16, 1247 (1965).

[3] I . R . SMITH, B . K . GAZEY, and J . L . BLACK, J . sei. Instrum. 4 3 , 251 (1965).

[4] B. ROTHENSTEIN and I. HRIANCA, Czech. J . Phys. 11, 179 (1961). [5] B . ROTHENSTEIN and I . HRIANCA, Czech. J . P h y s . 1 3 , 3 1 8 (1963). [6] B . ROTHENSTEIN and A. POLICEC, Czech. J . P h y s . 14, 137 (1964).

[7] B. ROTHENSTEIN and A. POLICEC, Z. Phys. 187, 6 (1965). [8] S. PROCOPIU, J . Phys. Radium 1, 306 (1930).

[9] S. PROCOPIU and G. VASILIU, C. R . Acad. Sei. (Prance) 2 0 4 , 6 7 3 (1937). [10] B . F . ROTHENSTEIN, A. POLICEC, C. ANGHEL, and M. LUPULESCTJ, phys. s t a t .

19, 613 (1967).

[11] W . B . HATFIELD, J . appl. P h y s . 3 6 , 2 6 6 2 (1965). [12] E . C. STOKER and E . P . WOLFARTH, Trana. R o y . Soc. 2 4 0 , 5 9 9 (1948).

(Received March 18, 1967)

sol.

M. HULIN: Selection Rules for Two-Phonon Absorption Processes

607

phys. stat. sol. 21, 607 (1967) Subject classification: 20.1; 6 Laboratoire

de Physique

de l'École Normale

Supérieure,

Paris1)

Selection Rules for Two-Phonon Absorption Processes By

M. H u l i n A model for the interaction between photons and phonons which includes the crystal electrons is proposed in order to investigate the possible selection rules for absorption processes involving the emission of two phonons. The main results can be rederived; special attention is given to the problem of harmonics especially in crystals possessing inversion symmetry. Um die möglichen Auswahlregeln für Absorptionsprozesse mit Emission zweier Phononen zu untersuchen, wird ein Modell für die Wechselwirkung zwischen Photonen und Phononen vorgeschlagen, das die Kristallelektronen einschließt. Mit diesem Modell können die bekannten hauptsächlichen Ergebnisse abgeleitet werden; eingehend behandelt wird das Problem der Harmonischen speziell in Kristallen mit Inversionssymmetrie.

1. Introduction I n the past few years, much attention has been paid to the problem of defining complete selection rules for processes occurring in crystals, involving several periodic perturbations (such as photons and phonons) and connected with infrared absorption, Raman or Brillouin emission, electron or phonon scattering, etc. While Birman [1, 2] derived his results using the full space-group techniques, other authors restricted their consideration to the group of operations that leave all perturbations unchanged [3, 4], Recently, L a x [5] claimed to demonstrate the equivalence of both treatments, a conclusion still subject to discussion [6], The special case of crystals endowed with inversion centers has received particular attention : the matter is not clear whether all "overtone" transitions are then forbidden, nor is it obvious that consideration of the full crystal space-group is still unnecessary to prove the result [5, 7], We propose here to re-investigate shortly some of these selection rule problems in a quite "pedestrian way", using a particular model to describe the absorption of light. We hope the resulting loss of generality and elegance may be compensated by a greater ease to visualize the demonstration mechanism, and the opportunity to introduce some examples of particular interest. The following discussion will be restricted to the problem of photon absorption with emission of two phonons. 2. Description of the Transition Model Following Gurevich and Ipatova [8], we shall suppose that the p h o t o n phonon interaction is brought in by virtual transitions of the crystal electrons. J

) Laboratoire associé à la Faculté des Sciences de Paris et au C.N.R.S.

M. HULIN: Selection Rules for Two-Phonon Absorption Processes

607

phys. stat. sol. 21, 607 (1967) Subject classification: 20.1; 6 Laboratoire

de Physique

de l'École Normale

Supérieure,

Paris1)

Selection Rules for Two-Phonon Absorption Processes By

M. H u l i n A model for the interaction between photons and phonons which includes the crystal electrons is proposed in order to investigate the possible selection rules for absorption processes involving the emission of two phonons. The main results can be rederived; special attention is given to the problem of harmonics especially in crystals possessing inversion symmetry. Um die möglichen Auswahlregeln für Absorptionsprozesse mit Emission zweier Phononen zu untersuchen, wird ein Modell für die Wechselwirkung zwischen Photonen und Phononen vorgeschlagen, das die Kristallelektronen einschließt. Mit diesem Modell können die bekannten hauptsächlichen Ergebnisse abgeleitet werden; eingehend behandelt wird das Problem der Harmonischen speziell in Kristallen mit Inversionssymmetrie.

1. Introduction I n the past few years, much attention has been paid to the problem of defining complete selection rules for processes occurring in crystals, involving several periodic perturbations (such as photons and phonons) and connected with infrared absorption, Raman or Brillouin emission, electron or phonon scattering, etc. While Birman [1, 2] derived his results using the full space-group techniques, other authors restricted their consideration to the group of operations that leave all perturbations unchanged [3, 4], Recently, L a x [5] claimed to demonstrate the equivalence of both treatments, a conclusion still subject to discussion [6], The special case of crystals endowed with inversion centers has received particular attention : the matter is not clear whether all "overtone" transitions are then forbidden, nor is it obvious that consideration of the full crystal space-group is still unnecessary to prove the result [5, 7], We propose here to re-investigate shortly some of these selection rule problems in a quite "pedestrian way", using a particular model to describe the absorption of light. We hope the resulting loss of generality and elegance may be compensated by a greater ease to visualize the demonstration mechanism, and the opportunity to introduce some examples of particular interest. The following discussion will be restricted to the problem of photon absorption with emission of two phonons. 2. Description of the Transition Model Following Gurevich and Ipatova [8], we shall suppose that the p h o t o n phonon interaction is brought in by virtual transitions of the crystal electrons. J

) Laboratoire associé à la Faculté des Sciences de Paris et au C.N.R.S.

608

M . HTJLIN

Limiting the perturbation treatment to the first relevant order, we are led to a transition probability proportional to \A\ 2, with A = Z { < 3 | t f _ 0 | l > x 1,2,3

X [(-Bj + h co - E2) (E1 + hw - E-s-

h coq )]" 1} +

+ 2 { X 1,2,3

X [(E1 + hw - E2) {E1 + ha> — Es — h «q)]" 1}

.

(1)

The numerator of each fraction is the product of three matrix elements, one related to the electron dipole moment along the photon polarization e„( V«) and two involving the perturbation potentials, as felt by the electrons, associated with the two phonons (UQ and U_Q). (As we neglect the photon wave vector, the phonon wave vectors are opposite: Q and — Q; hence our notation for the potentials.) These matrix elements are between electronic states which we call, for short, |2), |3>, and |3). |1> and |2) have the same wave vector, say k; |3> and ¡3) have wave vectors (k + Q) and (k — Q) respectively. The denominator of each fraction is an "energy denominator". Elt E2, E3, E% are the electron energies in states |1), |2>, |3>, and |3>; OO, Q, COQ are the photon and phonon frequencies. The conservation of energy requires CO =

COQ

+

COQ

.

(2)

The first series of fractions in (1) represents virtual transitions of electrons, into state [2) with absorption of the photon. The electron initially in state then returns to the initial state |1> with successive emission of both phonons (Fig. 1). The second series of fractions in (1) corresponds to an inverted order of phonon emission (Fig. 2). At the end of both processes, the initial electron configuration has been restored, one photon absorbed and two phonons emitted. Our description of the electronic coupling between phonons and photons thus plays the role of a model, since no overall modification of the electron assembly appears.

\2>

^ j f

Î

Phonon UQ

\

Photon \

I

XL

\ Phonon BQ

¡I7> 1 1

k

k+Q Electron wave vector —

F i g . 1. The absorption process with successive emission of phonons with wave vectors Q and — Q

k-Q

k Electron wave vector

F i g . 2. The absorption process with reverse order of phonon emission

Selection Rules for Two-Phonon Absorption Processes

609

(Truly, A should also include processes where the photon is absorbed after the emission of one or both phonons. This addition does not modify our study which will be based upon the simpler expression (1).) 3. Introducing and Using Symmetry Properties Our task is to state when A is bound to be zero because of the symmetry properties of our problem: the transition probability is then zero and we get a selection rule. The symmetry properties we can advocate are the following: 1. The crystal is invariant under the operations R of some space group G. If Wn> k(r) is an electron wave function (in band n, with wave vector k and energy En, k), R XVn,fc(i")= xVn, fc(-K_1 r ) is also a wave function in the same band, with the same energy, and with wave vector R k. 2. In the same band, with the same energy and opposite wave vector, there also appears the wave function K Wni k where K is the time reversal operator. If we do not take the electron spin into account — a justified approximation here, since the electrons only have an intermediary role in our model — K W reduces to W*, the complex conjugate of W. In the same way, if some phonon of wave vector Q produces the perturbation potential UQ for the electrons, there exists a phonon with the same frequency and opposite wave vector which produces the potential UQ. (Depending on the space group structure, these time-reversed potentials may or may not be related to the space-symmetry transformed potentials: we shall discuss this point later.) In the expression of A, as given by (1), the photon and both phonons are specified: V„, UQ, and U_Q are the same for all terms in the sum. Relations may appear only between terms for which the matrix elements for all three operators V«, UQ, and U_Q are themselves related, which is the case only if the wave functions |1>, . . ., |3> are somehow connected. So we will tentatively associate

with

,

with

, etc.,

S being any symmetry operator R € G or K. To calculate A, we proceed as follows: start with some set of wave functions |1>, |2>, |3>, yielding some term in (1); determine all other terms related to this first one by relevant symmetry properties, and add all the terms of this group. As no simple relation can appear between the contributions from distinct groups, A can be zero only if each contribution separately is compelled to zero by symmetry arguments. Let |1>, |2> (with wave vectors fc), |3> (with wave vector k Q) be the starting set; the corresponding term in (1) is < 3 | f f _ 0 | l > [ ( ^ + h w - E2) (E, + h co - Ea - h cog)]" 1 . (3 a) We shall first look for symmetry-connected terms corresponding to the same

610

M. Hulin

order of phonon emission: < 3 | J R l 7 _ 0 | l > + rzqq

+ } . (11) Now, what are the connections between the degenerate phonons UQ and J U_Q ? Using the well-known Herring's criterion we may state that a) if /J[(J i?)2] = , |2>, |3>) =

2

R J) ,2)

X

jS, v, v'

X

2

{mR-i)t[D,(R)7p[Dl(J-iRJYp.+

R E G

Q

+ [DV((J R)-1)]P [A( w

m j

R

.

(16)

(16) has the same structure as (8): possible selection rules depend on the bracket which, as it should, is independent of electronic terms. The bracket in (16) is a sum over the elements of the symmetry group (GQ + J GQ) of products of matrix elements of the irreducible representation of the photon and the product representation constructed on the functions Up Up'. The bracket is different from zero only if this second (generally reducible) representation contains the first one. This can be checked using character tables. The character associated with R € GQ in the product representation is — using (15) — s

m

m

[Dr(B)]>:

=

(R)

Xl

v{R) = %i{R) xiiJ- 1 R J) •

X

The character associated with J R is zv m

m

m j

R m

=

m

•

The overtone band will appear only if R

S €

GQ

(XviR-1) Xi(R) XiiJ-1

Xvl(J m1]

Xi[(J tf)2]} = 2 ngQ

(17)

with n some positive integer (Q and —Q not equivalent). If the band is symmetry allowed, the possible associated phonon polarizations ¡x and n' for a phonon polarization e„, must satisfy 2

R ^ G Q

{[DAR'1)]?

[AWEtW"1

R J)V

X [D,(J R J)]p'} = 2 » .

+ Wv (J

[ A W x (18)

The foregoing analysis may be considerably simplified when Q and —Q are equivalent. The distinction between cases a), b x ), and b a ) is still valid. No selection rule can be brought in by time-reversal alone and we shall restrict attention to cases a) and b x ). Any element of GQ could be taken as J but we can keep (12) 2 ) This relation is valid for both cases a) and bj). In case a), however, we might simply write

DV(R) = Dt(B)* .

M. Hulin

614

and all the calculations t h a t followed only if J is p u t equal to E, t h e identity. All t h e derivation is then straightforward and leads to the simple result , |2>, |3>) = 2 X x

z [Dv(R-i)t ReGg

{[A(«)is

+

m s m

.

There appears the symmetrized square of the phonon representation, already introduced by Lax and Birman, with characters

[»(*)]. = y

+

•

The overtone absorption is possible if this representation contains RY: 2 X,(R~ 1 )[Xi{R)\ = n g Q ReGg (Q and —Q

(19)

equivalent).

6. Crystals Possessing Inversion Symmetry Let I be the inversion (possibly associated with some — generally understated — non-primitive translation). Whatever Q, we may t a k e J = I and apply our formulae (17), (18), and (19). W e then have [Z>v(Z?-l)]f = ~[DV{(J , and therefore R)-i) = -XviR'1) • Xy((J We shall distinguish four main cases described in t h e following. 6.1 Q is an ordinary wave vector If Q is an ordinary wave vector in t h e Brillouin zone GQ reduces to the identity E, with %i(E) = 1. The left-hand side of (16) is always zero, and overtones are forbidden. This general result has been given by a number of authors. However, it was missed by Lax and Burstein in their study [9] of the "one-dimensional rock-salt lattice", as a result of the incorrect assumption t h a t the displacements of the two sublattices are in phase (equation (7.13) in their paper): re-establishing the correct phase factor (c®0''2) leads to the vanishing of all transition matrix elements Hft,. A continuity argument has been p u t forward [5, 7] to extend this selection rule for wave vectors Q with higher symmetry. This assumption is only valid when t h e considered phonons are non-degenerate: overtones being forbidden everywhere in the vicinity of Q are also forbidden for their limit, the phonon at Q.3) 3

) This can readily be checked using (18) or (19). D\ being one-dimensional, yj(R) is identical to Dj. If Q is not equivalent to —Q we calculate Z XviR-1) {Di{R) D,(J~i RJ)Di(J RJ R)} = 0 , RzGq since J'1 = I_1 = J. If Q is equivalent to —Q, the left-hand side in (19) is Z XviR'1) Df • ReGg Df is the same for R and I R, while XviR1) h a s opposite signs. The total sum is zero again.

Selection Rules for Two-Phonon Absorption Processes

615

This continuity argument no more applies if several phonon branches cross at Q (i.e., if Ri is pluri-dimensional): then, overtones at Q are the limiting processes not only of the forbidden overtones for neighbouring wave vectors, but also of — possibly permitted — summation processes. We shall now present some situations where it seems t h a t overtones become possible — in opposition to what may have been stated —, and we shall define some general cases for which they are effectively excluded. We wish to stress here t h a t the problem is not purely academic: symmetry points of course make up a set of measure zero as compared to the whole of the Brillouin zone. However, they often correspond to critical points for the combined density of states which appears in the summation bands. This makes it important to know whether overtones with degenerate phonons may or may not appear at symmetry points, since it will profoundly affect the shape of the summation band. 6.2 Q is a symmetry

point

not equivalent

to —Q

We shall first take the example of the space group DS D which is symmorphic and contains I; we consider point K at the edge of t h e Brillouin zone (cf. Roster's article [10], p. 210): Q at K is not equivalent t o —Q. GQ is D3, a degenerate representation of which is B: E

2 C3

*«: 2

-1

3 C'2 , 0.

I commutes with all R in GQ, SO t h a t (17) reads Z X^R-1)

{[*.(*)]" - *(*')}

=%ngQ.

(20)

R

For a photon polarization along the trigonal axis, t h e characters for %Y are: 1, 1, —1, for the three above classes. The left-hand side is then equal to 12 (i.e. n = 1); which indicates t h e possible presence of overtones, a result we can check with Birman's full space-group techniques. Direct examination of all particular cases is thus necessary before drawing conclusions as t o the reality of selection rules. Let us give some general indications t h a t can help such an investigation: 6.2.1 Q inside the Brillouin zone, whatever the group GQ Q is then rigorously invariant in all B e GQ (and not only sent into some equivalent wave vector). If t h e photon polarization e„ is t a k e n parallel t o Q it is conserved in all the elements of GQ and DV is the unity representation. Let us suppose I is associated with the non-primitive translation r , and B e GQ has the general form (a/a). ( J - 1 R J) and (J B J) are both equal to (a/a x — x — a). Since Q is inside the Brillouin zone, Dt for (a/a) is equal to ei q • a w h e r e Dl is associated to 3 x 10~17 cm2 at 100 ° K , ffp «

3.5 X 10-16 cm2 at 200 ° K .

Within this range, free electron fluctuations due to transitions between Ec and Ei can account for the experimental noise measurements. The spectral density of electron fluctuations associated with this two level process is given by

»« + y + - g with r =

and V the sample volume.

For cor

(1)

M. Averous and G. Bougnot

666 where II

piezoresistance tensor of order 4,

m

elastoresistance tensor of order 4,

yki

stress tensor (order 2),

/.iki

strain tensor (order 2),

where Q like % and fi are symmetric tensors with six components which we number 1 to 6, so that equation (1) becomes i n

j

i i X j

=

£miJ(Mi.

j

(2)

The symmetry of the point groups 43 m, 432, and 4/m 3 2/m reduces the set of independent piezoresistance and elastoresistance coefficients to 77 n , 77i2, 7744 and m n , m12, m44. The connecting relations between 77^ and m(i are [4] ( m n + 2m12)/3 = (77n + 2/712) (6

the C, j being the elasticity tensor of the material. W e need three kinds of measurements under uniaxial stress to determine the three coefficients l l l l t 7712, and 7744: a) two longitudinal measurements with j \ | F 11 (100)

and j\\F\\ (110) , where j

is the current density,

F is the applied force, (100) and (110) are the chosen sample axis; b) one transverse measurement with j || F J _ (100) or (110). W e remark that case b) is generally less precise than the others [4, 5] because of the very low transfer resistance (due to geometry and to the low resistivity) and because of geometrical difficulty in defining the 4 terminal transfer resistance. W e deduce the piezoresistance coefficients from the longitudinal measurements by the relation

~~ = nn -2P y. Q„

7 7 ' + k',

(4)

where k' is a correction term due to sample dimension changes, and P = l\ mi\ -p m\ n\ -f- n\ l\ a function of sample orientation. l l t m1, Wj being the direction cosines of the specimen axis along which the stress is applied in the crystal axis system, and finally 77' = 77n - 77J2 - 77 44 .

Effect of Donor Impurity Concentration on the Piezoresistance of GaSb

667

3. Experimental Technique The samples employed are cut from single crystals obtained in our laboratory (Czochralski method). The parent single crystals are oriented t o + 0.5 0 by X-rays. The sample parallelepipeds are cut by ultrasound on a cross-movement turntable whose precision is + 10 ¡xm and + 3 min. Four integral side arms permit t h e attachment of potential leads, t h e current passing by the ends. The potential and current contacts are made by unilateral micro-spot-welding and are coated with conducting silver paint; with a probability of 0.8, they are ohmic to 4 °K. The current source is a 135 V battery in series with a resistance of 3.5 k O or greater. The voltage measurements are made by a "solartron" integrating digital voltmeter; the temperature is measured by a Cu-AuCo thermocouple. Variable temperature measurements are made with the aid of a Sulfrian cryostat modified so as to permit the introduction of the mechanical load. The sample is loaded in compression between its support and a bracket in t h e form of a C-clamp which transmits a compression to the top of t h e sample when one pulls down on the bracket. Thus the load rod is in tension rather t h a n compression. Each measurement is made at constant temperature and gives isothermal coefficients directly. 4. Results Measurements have been made on two samples of n-type GaSb and compared with one of Sagar's samples. Table 1 lists the resistivity and Hall coefficient d a t a taken at 300 °K. From g and ii H , we have calculated the Hall mobility and impurity concentration which are also shown in Table 1. Table 1 Sample A

q (iîcm)

Ä H (cm 3 /As)

^H (cm2/Vs)

Nd (cm - 3 )

Orientation axis

0.007

23.2

3300

5.7 x 1017

(100)

18

(100) (100)

B

0.003

9.2

3060

1.6 XlO

S (Sagar)

0.004

10.5

2730

1.4 xlO 18

The electron densities for the samples are deduced from a two-band model with the two carrier types of same signs and different mobility (fi0 and The hole contribution is neglected. This two-band model was fully studied by Sagar [1], Aukerman and Willardson [6], Strauss [7], and more recently by Allgaier [8]. The expressions for conductivity and Hall coefficient for this model are given by R

r_ ,«2 + e 0uo n0 + /¿I

O = e (fiQ nQ + 43

physics 21/2

Wj) ,

(6)

M . AVEROUS a n d G . BOUGNOT

668

and the formula relating the Fermi level to the total carrier concentration

(

m hT\3/2

- V " )

where

N„ [F1I2(Vf)

+ p Fll2 fo -

Vg)]

,

(7)

oo ^ ^ / { [ l + e x p Î , - , ) ] } ^ ' o

and

p 1

=

m \Nj

m312. \m0)

The subscripts 0 and 1 denote the properties of the (000) and (111) hands. For sample B we have taken account of the degeneracy effect. The Hall coefficient is shown in Fig. 1. i? H is a decreasing function of 1/T for the three samples whose slope

is increasingly negative for increasing

sample purity. The resistivity is shown in Fig. 2. q is also a decreasing function of 1/T. For a given T, as evident, o increases as Nd decreases. The piezoresistance coefficient is shown in Fig. 3. For the three samples studied, the sample axis is (100) and thus the coefficient obtained is 77n. W e give the 1/T dependence of 77n, in the cases observed 77u becomes negative and goes through a minimum. The minimum moves toward higher temperatures as the donor concentration increases. The sample with the smallest Nd gives distinctly the most strongly negative 77n (minimum). 23 r

: 20-

\

+ Sample A Nd-5.7>J0 ?cm3 ° Sample B Nd-W'Wncm3 A SampleS Nd-U>101scms

V

15 + Sample A Nd-5.7-1017cm'3 ° Sample B Nd • I6'10wcm3 Û Sample S Nd-1A-W^cm'3

7

8

10

F h l Fig. 1. Hall coefficient as a function of 1 IT for n-type GaSb samples of different carrier concentrations

"3,

4 - 5

6

7

8

9

10

F / R ? — Fig. 2. Resistivity as a function of 1/T for n-type GaSb samples with different carrier concentrations

669

E f f e c t of Donor Impurity Concentration on the Piezoresistance of G a S b F i g . 3.

l'iezoresistance v s . 1/T GaSb

for ri-rvpe

20

0

° f\o to^

o\ J-

.St

I P

O

* Sample A fl/d-5.7-W^cnf 3 o Sample B Nd-W'10' scm' 3 ASampleS Nd'lk'10 Kar>- 3

o o

/

P

-20

/ /+

/

f4

/

9

70

5. Interpretation of Results 5.1 Hail effect

The increase of jK h with T is due to the existence of both the (000) and (111) bands. The population of the latter increases with T, while the total number of carriers Nd is constant so that n0 falls as T increases. The (111) electrons have the lower mobility [1], The experimental results are in agreement with the two-band model since equation (5) predicts the increasing of R n with i.e. with temperature. Sagar and the others [6 to 10] have shown that d(i? H )/d T depends on a) the energy separation AE, b) scattering mechanism and mobilities in the two bands, c) the densities of states in the two bands. 5.2

Piezoresistance

Fig. 3 shows the minimum position of IIu for the three samples A, B, and S. If we have no good agreement for the value of IIn minimum between our samples A and B and Sagar's sample S, on the other hand, it appears very clearly that this minimum moves toward higher temperatures as Nd increases. The existence of this minimum can be explained by mixed conduction between the two subbands, as for the variation of i? H with T. If we apply for mixed conduction between bands of the same sign the analysis given by Potter [11] and by Tuzzolino f 12] for bands of different sign 43*

670

M. AVEROUS and G. BOUGNOT: Effect of Donor Impurity Concentration

in InSb, we obtain

7 7 il „

= A

6A E

a —1

8 log V 2 kT a

where a =

1 +

4 n1

(a - 1) (a + 1)

'

(8)

nx

n011/2

k Boltzmann's constant, V volume, A is a constant which depends on Ci; (elastic constants of GaSb) and % (compressibility). We observe t h a t even with two bands having the same sign we must find the displacement [11, 12] of the minimum toward high temperatures when N d is increased. 6. Conclusion The experiment here reported shows t h a t the piezoresistance coefficient n u > in the case of a stress along the (100) axis, passes through a minimum when the temperature is varied, and t h a t the position of this minimum is correlated with the donor concentration. This effect can be interpreted qualitatively if we assume mixed conduction subbands (000) and (111) and if we admit t h e validity of equation (8). Acknowledgement

We wish to express our appreciation to Rector C. Chalin for his continued support and assistance in this investigation. Also the m a n y helpful discussions with Dr. Rouzeyre are appreciated. We wish to t h a n k J . L. Robert and M. Marques for their assistance in making Hall measurements. References [ 1 ] A . SAGAS,

Phys. Rev. 117,

93 (1960).

[ 2 ] S. ZWERDLING, B . LAX, K . BUTTON, a n d L . M. ROTH, J . P h y s . Chem. S o l i d s 9 ,

320

(1959). [3] HERMANN-MAUGUIN, System of Point and Space Group Notation. [ 4 ] C . SMITH, Phys. Rev. 94, 4 2 ( 1 9 5 4 ) . [5] R. W. KEYES, Solid State Phys. 11, 158 (1960). [6] L. W . AUKERMAN a n d R . K . WILLARDSON, J . appl. P h y s . 3 1 , 939 (1960).

[7] A. J. STRAUSS, Phys. Rev. 121, 1087 (1961). [8] R. S. ALLGAIER, J. appl. Phys. 36, 2429 (1965). [9] C. HERRING, Bell Syst. tech. J. 34, 237 (1955). [ 1 0 ] O . M A D E L U N G , Phys. of III—V Compounds J .

WILEY

L o n d o n (1964).

[11] R. F. POTTER, Phys. Rev. 108, 652 (1957). [12] A. J . TUZZOLINO, Phys. Rev. 109, 1 9 8 0 ( 1 9 5 8 ) . (Received March 28,

1967)

and sons, Inc., N e w York,

R. LÜCK und K. E. SAEGER: Anisotropie der galvanomagnetischen Spannung

671

phys. stat. sol. 21, 671 (1967) Subject classification: 14.1; 21.1 Max-Planck-Institut für Metallforschung, Institut für Metallkunde, Stuttgart

Über die Messung der Anisotropie der galyano magnetischen Transversalspannung an Einkristallfolien Von R . LÜCK u n d K . E . SAEGEB

Herrn Professor Dr. W. Köster zum 70. Geburtstag gewidmet Die Grundlagen der Meßmethode werden theoretisch begründet, und die Anwendbarkeit wird abgegrenzt. Über Meßergebnisse eines zusätzlich zum Halleffekt auftretenden Nebeneffektes (transverse-even voltage) an Kupfereinkristallen wird berichtet. Die Messungen wurden bei 4,2 °K in Abhängigkeit von Orientierung und magnetischer Feldstärke durchgeführt. Mit Hilfe einer neuen allgemeingültigen Beziehung zwischen diesem Effekt und dem Widerstand im Magnetfeld, die auch den sättigenden Anteil des Widerstandes berücksichtigt, wird der Winkel zwischen Stromrichtung und Richtung der offenen Bahnen berechnet und die Übereinstimmung mit röntgenographisch gewonnenen Werten überprüft. Die Feldabhängigkeit dieses Effektes wird benutzt, den Widerstand im Magnetfeld in einen sättigenden und einen mit der magnetischen Feldstärke quadratisch ansteigenden Anteil zu zerlegen. The theoretical basis is given for the experimental method used and the range of applicability is defined. Results are given for transverse-even voltage measurements on copper single crystals. The measurements are made at 4.2 °K as functions of crystal orientation and magnetic field intensity. Using a relation between this effect and magnetoresistance, which is generally valid and also takes into account the saturating part of the magnetoresistance, the angle between current and open orbit direction is calculated and compared with data obtained from Laue diagrams. Using the field dependence of the transverseeven voltage the magnetoresistance curve is found to be composed of a saturating part and another part which increases quadratically with magnetic field. 1. Einleitung D i e orientierungsabhängige B e s t i m m u n g des H a l l k o e f f i z i e n t e n a n Einkristallen l ä ß t sich r e l a t i v g ü n s t i g durchführen, w e n n Einkristallfolien m i t f e s t anges c h w e i ß t e n P o t e n t i a l a b g r i f f e n i m M a g n e t f e l d g e d r e h t w e r d e n [1, 2], H i e r b e i ist j e d o c h die Orientierung des Magnetfeldes zur P r o b e n o b e r f l ä c h e g e s o n d e r t in R e c h n u n g z u ziehen. E s ist E{