Physica status solidi: Volume 20, Number 2 April 1 [Reprint 2021 ed.] 9783112502129, 9783112502112

Citation preview

plxysica status solidi

V O L U M E 20 • N U M B E R 2 • 1 9 6 7

Classification Scheine 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 G. 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, W. D E K E Y S E R , Gent, W. F R A N Z , Münster, P. G Ö R L I C H , 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. S E I T Z , Urbana, O. S T A S I W , Berlin, M. S T E E N B E C K , Jena, F. S T Ö C K M A N N , Karlsruhe, G. S Z I G E T I , Budapest, J. T A U C , Praha Editor-in-Chief P. G Ö R L I C H 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. C O C H R A N , Edinburgh, R. C O E L H O , Fontenay-aux-Roses, H.-D. D I E T Z E , Aachen, J. D. E S H E L B Y , Cambridge, 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, M. M A T Y Ä S , Praha, H. D. M E G A W , Cambridge, T. S. MOSS, Camberley, E. N A G Y , Budapest, E. A. N I E K I S C H , Jülich, L. P A L , Budapest, M. R O D O T , 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 20 • Number 2 • Pages 413 to 790, K71 to K178, and A39 to A78 April 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 l11 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 : P r o f e s s o r 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 . 2 0 . 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 j e w e i l s a m 1. d e s 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 72,— ( S o n d 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 1068/20. 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 ß D r u c k e r e i „ T h o m a s M ü n t z e r 4 1 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 Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik.

Contents Tage

Original Papers G . B . ABDULLAEV, E . S . GUSEINOVA, a n d B . G . TAGIEV

Electric Conductivity Studies of p-GaTe Polycrystals and Single Crystals in Strong Electric Fields

421

A . P . L I T B C H E N K O a n d S . I . DTTDKIN

Effect of Concentration on Light Absorption and Dispersion by Impurity Centres in the Case of Weak Elektron-Phonon Coupling (II) .

E . JÀGER and R .

PERTHEL

Some Remarks to the Crystal Field Theory of Spinel L a t t i c e s . . . .

E . P . LAUTENSCHLAGER, T . C. TISONE, a n d J . O.

BRITTAIN

Electron Transmission Microscopy of NiAl

J . PRZYSTAWA a n d W .

427 433 443

SUSKI

Magnetic Ordering in Uranium Compounds with Crystal Structures of Anti-Cu 2 Sb and PbFCl Types. Molecular Field Approximation . .

451

P . M . ROBINSON a n d H . G . SCOTT

Tilt Boundaries in Deformed Anthracene Single Crystals

461

L.

¿DANOWICZ

Some Optical Properties of Thin Evaporated Cd3As2 Films

473

C.

HAMANN

On the Electric and Thermoelectric Properties of Copper Phthalocyanine Single Crystals

481

Diffusion of Nickel in Silicon

493

H . P . BONZEL A. K . HEAD,

M . H . LORETTO, a n d P .

HUMBLE

The Influence of Large Elastic Anisotropy on the Determination of Burgers Vectors of Dislocations in (3-Brass by Electron Microscopy.

505

A. K . H E A D , M . H . LORETTO, a n d P . H U M B L E

The Identification of Burgers Vectors and Unstable Directions of Dislocations in p-Brass

521

I . I . PINCHUK

Thermomagnetic Effects in Anisotropic Media

537

R. F.

Effect of Pressure on the Diffusion of Fe in Ti and Ti + 1 0 % F e . . .

545

PEART

G. BLASSE and A. BRIL

Some Observations on the Cr 3+ Fluorescence in the Huntite Structure

551

Effets de la désaimantation thermique sur les cycles d'hystérésis. . .

557

R. DE WIT

Some Relations for Straight Dislocations

567

R. DE WIT

The Self-Energy of Dislocation Configurations Made up of Straight Segments

575

N.

VAN D A N G

L . C S E R , J . OSTANEVICH, a n d L . P A L

Môssbauer Effect in Iron-Aluminium Alloys (I)

L . C S E R , J . OSTANEVICH, a n d L . P Â L

P.

SCHMELING

581

Môssbauer Effect in Iron-Aluminium Alloys (II)

59 L

Non-Ideal Diffusion of Argon in Potassium Chloride

597

416

Contents Page

T . N . CASSELMAN a n d H . N . SPECTOR

Interaction of Optical Phonons with Spin Waves in Ionic Crystals. .

605

G. REMAUT, R . GEVERS, a n d S. AMELINCKX

Wavy Fringes at Domain Boundaries in Barium Titanate Observed in the Electron Microscope

J . S P Y R I D E L I S , J . S T O I M E N O S , a n d N . ECONOMOU

Optical and Photoconductive Phenomena in Cuprous Oxide . . . .

W . BLUM und B .

ILSCHNER

Uber das Kriechverhalten von NaCl-Einkristallen

I . V . A B A R E N K O V a n d I . M . ANTONOVA

The Model Potential for Systems with Closed Shells

K . 2DÄNSKY, H . AREND, and F . K U B E C

Electron Spin Resonance of Co 2+ Ions in Barium Titanate

M. HÖHNE a n d M . STASIW

T e l - and Sei" Centres in AgBr (I)

613 623 629 643 653 657

M . HÖHNE a n d M . STASIW

P.

FELTHAM

T e l " and Se|- Centres in AgBr (II)

667

F-Centres in Alkaline Earth Fluorides

675

E . A. METZBOWER

Noncentral Force Model for Hexagonal Close-Packed Crystal Lattices

681

R. T.

DELVES

The Theory of Stability during Temperature Gradient Zone Melting.

693

V. D.

EGOROV

Luminescence of CdS Excited by Electrons of Various Energies. . .

705

Observations of Ferromagnetic Domains by Anomalous Transmission of X-Rays. The Effects of Variation in the Reflecting Plane. . . .

713

B . ROESSLJER

D . VAUGHAN a n d J . M . SILCOCK

The Orientation and Shape of 0 Precipitates Formed in an Al-Cu Alloy

725

V . P . NABEEEZHNYKH a n d A. A. MARYAKHIN

Radio-Frequency Size Effects in Cadmium in Magnetic Fields Inclined to the Sample Surface

V . A . BOKOV, N . A . GRIGORYAN, a n d M . F .

737

BRYZHINA

X - R a y Diffraction and Magnetic Studies of Solid Solutions B i i - z Ca,. Mn0 3

V . M . ASNIN a n d A . A . ROGACHEV

Exciton Absorption in Doped Germanium

V . M . Y U D I N a n d A . B . SHERMAN

Magnetic Properties of NaNiF 3

745 755 759

N . A . C H E R N O P L E K O V , G . K H . P A N O V A , M . G . Z E M L Y A N O V , B . N . SAMOILOV, a n d V . I . KUTAITSEV

Investigation of the Quasi-Local Level in the Vibrational Spectrum of the Ti Lattice with Heavy Impurity Atoms

I . A . AKIMOV, V . M . B E N T S A , F . I . VILESOV, a n d A . N .

767

TERENIN

External Photoeffect from Sensitizing Dyes Adsorbed on Semiconductors

771

417

Contents

Page A . A . ABDURAKHMANOVA a n d M . I . A L I E V

On the Mechanism of Thermal Conductivity in (GaSb) 3:c —(Ga 2 Te 3 ) 1 _ a: and (GaDSb), — (GaTe)i Solid Solutions

777

V . RADHAKRISHNAN

The Penetration Depth in Two Band Model S u p e r c o n d u c t o r s . . . .

783

Short Notes P.

ASADI

P. ASADI

X - R a y Topography in Coloured and Uncoloured Zones of AntozoniteFluorite K71 On the Existence of Piezoelectric Textures in Single Crystals of Antozonite-Fluorite K73

J . A U L E Y T N E R , K . GODWOD, J . L I T W I N , a n d Z . WOLOSZYN

Application of X-Ray Methods to the Study of the Real Structure of Monocrystalline Epitaxial Layers K77 F. J . S.

WORZALA

LIBOVICKY

Hardening in Irradiated Copper Crystals

KSl

Antiphase Boundaries in Silicon Iron Single Crystals Revealed by Etching K85

R . N . GHOSHTAGORE

On the Mechanism of Substitutional Diffusion in Silicon G. A.

JONES

Structural and Magnetic Properties of Epitaxial Films of Iron

K89 . . . K95

M . J A R O S a n d J . ZITKOVÄ

Remark on the Theory of the ESR Spectrum of the d 1 -Configuration in Rutile (Ti0 2 ) K99

B . A . GREENBERO

Effect of Segregation at Dislocations on the Shape of the NMR Line K103 J . BLINOWSKI a n d M . GRYNBERG

On the Direct Interband Magnetoabsorption in Uniaxially Stressed Germanium K107 H.-J.

ULLRICH

Precision Lattice Parameter Measurements by Interferences from Lattice Sources (Kossel Lines) and Divergent Beam X-Ray Diffraction (Pseudo-Kossel Lines) in Back Reflection K113

K H . I . AMIRKHANOV, R . I . BASHIROV, a n d M . M . G A D Z H I A L I E V

Quantum Oscillations of the Nernst-Ettingshausen Effect in Indium Arsenide K119 D. D. O.

MISHIN

HENKEL

On the Theory of Susceptibility and Coercive Force of Magnetic Materials K123 On Temperature Dependence of Interaction Effects in Ferromagnetic Fine Particle Assemblies K127

W . HAUBENREISSER a n d U . LINDNER

Some Symmetry Aspects in the Theory of Localized Magnetic States in Metals K131 G . JONES, G . SMITH, a n d A . R . BEATTIE

The Drifted Maxwellian Distribution Function in InSb H.

GORETZKI

K135

Neutron Diffraction Stiudes on Titanium-Carbon and ZirconiumCarbon Alloys K141

Contents

418

Page

W. HENRION

On an Exciton Structure in the Reflection Spectrum of Trigonal Selenium Single Crystals K145

R. FREUD

The Influence of Joule's Heat upon Characteristic Magnitudes in a Cylinder with Current in the Intermediate State K15I

T . S A K A T A , K . SAKATA, a n d I . N I S H I D A

Study of Phase Transition in N b 0 2

K155

J . W . OSTROWSKI a n d A . PAJ^CZKOWSKA

Spectral Response of Photosensitivity (SRPS) in Boron

K159

I . B U N G E T a n d M . ROSENBERG

Influence of Changes of t h e Magnetocrystalline Anisotropy on the Electrical Resistivity of Ba 3 Co 2 Fe 21 0 41 K163 H . BILZ, R . ZEYHER, a n d R . K . WEIINER

Anharmonic Sideband Effects of the U-Center Local Mode . . . .

K167

G . 0 . MÜLLER, H . P E I B S T , E . SCHNÜRER, a n d H . T H I E L

Influence of High Phonon Flux Densities on the X-Ray Reflectivity of Nearly Ideal CdS Crystals K173

Prc-printed Titles and Abstracts of papers to be published in this or in the Soviet journal ,,Oli3Hna TBepjjoro T e j i a " (Fizika Tverdogo Tela) A 39

Contents

419

Systematic List Subject classification: 1 1.2 1.4 3 3.1 3.2 4 5 6 6.1 8 9 10 10.1 10.2 11 12.1 13 13.1 13.2 13.3 13.4 14.2 14.3 14.4 14.4.1 14.4. 2 15 16 17 18 18.1 18.2 18.3 18.4 19 20 20.1 20.2 20.3 21 21.1 21.1.1 21.3 21.7 22 22.1 22.1.1 22.1.2 22.2.1 22.2. 3

Corresponding papers begin on the following pages (pages given in italics refer to the principal subject classification): 745 745, K155 K77, K95 537 693 433, 451, 681 443, 461, 613, 713, 725, K71, K73, K85 K113, K141 605, 681, 767, 777, K167 581, 591 681, 767, 777 493, 545, 597, 643, K89 461, 493, 567, 575, 629, K 8 9 , K 1 0 3 , K 1 0 7 443, 505, 521, K81 675, K 7 1 597, K81 505, 521, 545, 681 433,605, K135 643, 737, K 1 4 5 623, 705, 755, K145 771 427, 481, 551, 657, 667, 675, K 9 9 , K131 783, K151 421, 481, K 1 3 5 , K173 K73, K163 421,481 613 481, 537, K119 623, 771, K159 771 581, 591, K107 K131 451, 557, 713, 745, K 9 5 , K123, K127 433, 745, K163 451, 759, 745 605, 653, 657, 667, K99, K103 K173 427, 473, 551, 623, 755, K145, K167 705 551, 705 545, 725, 737, 783, K 1 1 3 , K 1 4 1 , K 1 5 1 443, 505, 521, 681, 767, K 8 1 545, 557, 581, 591, 713 681 537 4 2 1 , 4 7 3 , 605, K 7 1 , K 7 3 K145, K159 755, K 7 7 , K 8 5 , K 9 5 , K 1 0 7 , K 1 2 3 , K 1 2 7 493, K 8 9 K77, K119 K135

420 22.4 22.4.1 22.5 22.5.1 22.5.2 22.5.3 22.6 22.8 22.9

Contents 771 705, 771 657, 597, 675 623, 551, 481

K173 667, 771 629, 643, K167 K99, K155 613, 653, 745, 759, 777

Contents of Volume 19 Continued on Page 783

International Conference on II-VI Semiconducting Compounds 6 to 8 September, 1967, Providence, Rhode Island An "International Conference on I I - Y I Semiconducting Compounds" will be held at Brown University, Providence, Rhode Island, from on 6 to 8 September, 1967. The Conference, sponsored as a Topical Conference by the American Physical Society, is organized by a committee broadly representative of academic and industrial investigators in this field, which include R. E. Halsted, Chairman, J . L. Birman, R. H. Bube, M. Cardona, A. R. Hutson, D. W. Langer, D. C. Reynolds, and D. G. Thomas, in conjunction with an international advisory group. The Conference invites the submission of papers concerned with the basic phenomena and processes associated with the preparation and behavior of I I - V I compounds. Relevant topics include: Experimental determination of band structure parameters, band structure calculations, lattice vibrations, Raman and nonlinear optical effects, transport properties, electron-phonon interaction, magnetic properties, electrical and thermal conductivity, thermodynamic properties, crystal growth, solubility and diffusion of defects, preparation and properties of p - n junctions, surface effects, luminescence, photoconductivity, radiation damage, and E P R . I t is the intention of the committee to limit the papers accepted to a number that can be readily accommodated in a three-day conference of single sessions with ample time for presentation and discussion. I t is anticipated t h a t invited and contributed papers and subsequent discussions will be published after the Conference. The deadline for the submission of abstracts is 15 May, 1967. Full manuscripts of contributed papers will be required about two weeks prior to the Conference. Accommodation can be provided in University Dormitories and at the local hotel. Advance registration will be necessary. Inquiries should be directed to D. W. Langer, Conference Secretary, Aerospace Research Laboratories, WrightPatterson AFB, Ohio 45433.

phyaica 20/2

Original

Papers

phys. stat. sol. 20, 421 (1967) Subject classification: 14.3; 14.4.1; 22 Institute

of Physics,

Academy

of Sciences

of the Azerbaidzhán

SSR,

Baku

Electric Conductivity Studies of p-GaTe Polycrystals and Single Crystals in Strong Electric Fields By G . B . A B D U L L A E V , E . S . GTJSEINOVA, a n d B . G . T A G I E V

I t is shown t h a t in the temperature range 77 to 220 °K and under fields increasing to 3 X 104 V/cm, conduction both of polycrystalline p-GaTe and of the single crystal increases — v'e® according to Frenkel's formula (a = a 0 ). From the formula B = —-——.- and by deterIcTye

mining the slope of the isotherms lg a = fi^E 'j, the dielectric constants of p-GaTe, pertaining to the electric part of polarization, are found to be equal to six and four both along and across the layers, respectively. By extrapolating with respect to the cross-over point of isopotentials lg a = f(l/T), the metallization temperature T M = 20000 °K is determined, from which the value of kTM = 1.74 eV is found. This agrees with the value for the gapwidth of p-GaTe at absolute zero. By extrapolating the isotherms and finding a consistent solution to their equation for the electric strength, a value is obtained which is approximately equal to 6.5 X 105 V/cm. n o K a 3 a n o , HTO B HHTepBajie 77 no 220 °K c pocroM nojiH no 3 x l 0 4 B/CM n p o BOjjHMOCTb nojiH- H MOHOKpHCTajuia p-GaTe B03pacTaeT corjiacHO opMyjie — l/e® (DpeHKejiH (a = cr0 ). H 3 (jiopMyjibi /? = — — = , onpeflejiHioiueft naKJion H30kT\s

TepMOB lg or = fi^E),

naiiflena jiH3Jiei{TpmiecKaH npoHiinaeMOCTb p-GaTe, OTHOCH-

MAHCH K AJIEKTPOHHOIL NACTM NOJIHPH3AUHH, KOTOPAN 0KA3AJIACB PABHOÜ 6 H 4

Bjiojib h nonepeK cJioeB, cooTBeTCTBeHHO. ripw 3KCTpanojiauHH no TOHKC n e p e C6iIGHHH H3onoTeHqnaJiOB lg a = f(l/T) onpeaejieHa TeMnepaTypa MeTajuiH3auHn Tu = 2 0 0 0 0 ° K , n o KOTopoií HaiifleHO kTM= 1,74 3B, HTO cooTBeTCTByeT iimpHHe 3anpemenH0H 3 0 h h p-GaTe n p n aScojiiOTHOM Hyjie TeMnepaTypbi. BKCTpanojiHUHeft H30TepM0B H COBMeCTHOM peiUeHHH HX ypaBHeiIHH JIJ1H 3JieKTpHHeCKOii npoHHocTH HañaeHO 3HaqeHHe npHMepHO paBHoe 6 , 5 x l 0 5 B/CM. F r o m electric c o n d u c t i v i t y m e a s u r e m e n t s in strong electric fields o n e can determine t h e w i d t h of t h e forbidden e n e r g y gap, t h e energetic positions of t h e i m p u r i t y levels, t h e electric strength, t h e electronic c o m p o n e n t of t h e dielectric c o n s t a n t , t h e m e t a l l i z a t i o n temperature, etc. [1, 2], I t is of great i m p o r t a n c e t h a t t h e a b o v e m e n t i o n e d m e t h o d m a y be used t o determine t h e w i d t h of t h e forbidden energy gap e v e n i n t h e case of o p a q u e s e m i c o n d u c t o r s h a v i n g small sizes and a n y shapes. B y t h i s m e t h o d t h e afore-mentioned parameters for monocrystalline p-GaSe a n d n - I n S e h a v e been determined [1, 2], Single crystals of p-GaTe also belong t o t h e A m B V I - t y p e s e m i c o n d u c t o r s and h a v e a l a m i n a t e d structure similar t o t h a t of gallium a n d i n d i u m selenides.

422

G. B. A b d u l l a e v , E. S. G u s e i n o v a , and B. G. T a g i e v

The electric conductivity of p-GaTe polycrystals and single crystals has been measured till now only in weak electric fields [3, 4], and there are no data on the mechanism of the electric conductivity growth in strong electric fields, on electric strength, "metallization" temperature, and the electronic component of the dielectric constant. For this reason, we are interested in studying the electric conductivity of p-GaTe polycrystals and single crystals in strong electric fields. To avoid heating of the specimens the measurements were carried out under pulsed operation conditions. The pulses were of rectangular shape, they had a duration of 1 (xs and a repetition rate of 4 Hz [5], The single crystals of p-GaTe were prepared by slow cooling at a constant temperature gradient [6]. The electric conductivity of twenty monocrystalline and four polycrystalline p-GaTe specimens were measured between 77 and 220°K in electric fields up to 3 X 104 V/cm. The ohmic contacts of the specimens consisted of indium. The electric conductivity of monocrystalline specimens has been measured both along and across the layers. It has been shown [7] that the structure of p-GaTe consists of four-layer TeGaGaTe-packets, packed up according to the law of the double-layer hexagonal packing. This phase was metastable, and a short time annealing at a temperature of 100 to 105 °C caused its conversion into a monoclinic modification. As shown by our measurements, annealing at this temperature for 30 min does not change the conductivity of the p-GaTe single crystals and its dependence on the electric field strength. This gives us good reason to believe that our specimens must have been stable. The studied specimens has a length of 4 to 5 mm, a width of 2 to 4 mm, and a thickness of 0.1 to 0.4 mm. The contact area ranged from 0.3 to 0.8 mm 2 . In order to measure the electric conductivity along the layers, dumbbell-like specimens were used which ruled out the injection of carriers from the metal electrodes. In this case the resistance of the specimens mainly originates in the narrow middle part. The electric conductivity of the specimens at room temperature was of the order of 1 0 - 1 Q _ 1 c m - 1 . A rise of conductivity with an increasing electric field was observed in all polycrystalline and monocrystalline p-GaTe specimens (both along and across

150 200 VEi/'Wj—

Fig. 1. Logarithm of the electric conductivity a ( Q - ' c m - 1 ) va. fE (E in V/cm) for p-GaTe a t different temperatures T: 1 - 208 °K, 2 - 179 "K, 3 - 166 °K, 4 - 151 °K, 5 - 125 °K

Electric Conductivity Studies of p-GaTe in Strong Electric Fields

423

t h e layers). I n both directions the electric conductivity of the specimens increased with growing electric field strength according to Frenkel's theory [8], i.e. there is a linear relationship between lg a and ^ E : a

(1)

=

where a 0 is the electric conductivity in weak fields, ¡1 is the slope of t h e lg a = == i{l/E) straight lines, and E is the electric field strength. Similar results were obtained from polycrystalline p-GaTe specimens. I n Fig. 1 and in Table 1 the dependence of a\\ on t h e electric field is shown at different temperatures for a monocrystalline and a polycrystalline specimen, respectively. Table 1 Electric conductivity crxlO 4 ( i l _ 1 cm - 1 ) of the polycrystalline GaTe specimen No. 2 vs. electric field E at different temperatures T T(0K)

E (kV/cm) 5 10 14 22

215

187

178

12 20 31 56

8 15 23 46

5 10 18 35

As it is known, the Frenkel mechanism of growth of current-carrier concentration leading to an increased electric conductivity consists in a field-induced increase of thermal release of carriers. This is referred t o as thermoelectric ionization. According t o this theory the slope of lg a = f(^E) straight lines is determined b y the expression kT |/c {e ist the electronic charge, k the Boltzmann constant, T the absolute temperature, and e the electronic component of the dielectric constant). This conclusion made is well confirmed by measurements on monocrystalline p-GaTe. The temperature dependence of the slopes of the lg a = f(^E) isothermes (see Fig. 1) is presented in Fig. 2. I t is shown there t h a t for GaTe single crystals t h e dependence ß 12

I 3

I 4

I 5

I 1 " 6 7 fW-',—

Fig. 3. Logarithm of the electric conductivity vs. reciprocal temperature for p-GaTe. Curves 1, 2, 3, and 4 have been obtained at 40000, 22500, 10000, and 2500 V/cm, respectively

100

200 YElV1/2lcm1/2J-~-

Fig. 4. Activation energy A E of majority carriers vs. electric field strength E (V/cm)

Electric Conductivity Studies of p-GaTe in Strong Electric Fields

425

B y extrapolating the lg a = /(1/7 7 ) straight lines the "metallization" temperature of monocrystalline p-GaTe was determined to be 2 0 0 0 0 ° K . To this temperature corresponds an energy of 1.74 eV which is the width of the forbidden energy gap for p-GaTe single crystals at absolute zero. The electric strength of monocrystalline p-GaTe in the direction perpendicular to the c-axis determined by extrapolating the isotherms, lg a = approximately amounts to 6 to 7 X 10 5 V/cm. The electric strength also can be determined from the curves presented in Fig. 1. In this case a value for the electric strength of approximately 6.5 X 10 6 V/cm has been obtained. This value is in agreement with the theoretical conclusions of other authors [10]. According to the calculations presented in [10], the electric strength of semiconductors depends on the width of the forbidden energy gap Eg and corresponds to breakdown field strengths between 5 X 10 4 and 8 x 10 5 V/cm for Eg between 0.5 and 2 eV, respectively. References [1] G. B . ABDULLAEV, E . S . GUSEINOVA, and B . G. TAGIEV, phys. s t a t . sol. 1 6 , 2 0 5 (1966).

[2] G. B . ABDULLAEV, E . S. GUSEINOVA, and B . G. TAGIEV, phys. stat. sol. 17, 593 (1966). [3] G. HARBELE and G. LAUTZ, Z. N a t u r f . 1 1 a , 1015 ( 1 9 5 6 ) . [4] G. FISCHER and J . Z. BREBNER, J . P h y s . Chem. Solids 2 3 , 1363 ( 1 9 6 2 ) .

[5] A. R. REGEL and B. G. TAGIEV, Fiz. tverd. Tela 5, 1419 (1963). [6] R . F . MEKHTIEV,

G. B . ABDULLAEV, and

Azerb. SSR 18, 11 (1962).

G. A. AKHUNDOV,

Dokl.

Akad.

Nauk

[7] S . A . SEMILETOV and V . A . VLASOV, K r i s t a l l o g r a f i y a 8, 8 7 6 (1963).

[8] Ya. I. FRENKEL, Zh. eksper. teor. Fiz. 8, 1292 (1938).

[9] J . Z. BREBNER, G. FISCHER, a n d E . MOOSER, J . P h y s . Chem. Solids 2 3 , 1417 (1962). [10] A. E . GLAUBERMAN a n d Y a . I . STETSIV, Lvovskii fiz. S b o r n i k 3 3 , 4 2 ( 1 9 5 5 ) . (Received

December

28,

1966)

A. F . LUBCHENKO and S. I . DUDKIN : Ligth Absorption and Dispersion (II)

427

phys. stat. sol. 20, 427 (1967) Subject classification: 20.1; 13.4

Institute of Physics, Academy of Sciences of the Ukrainian

SSR,

Kiev

Effect of Concentration on Light Absorption and Dispersion by Impurity Centres in the Case of Weak Electron-Phonon Coupling (II) By A. F . LUBCHENKO a n d S. I.

DTTDKIN

The two-particle Green's function method is used to investigate the dependence of light absorption and dispersion due to substitutional impurity centres upon their concentration in the case of weak electron-phonon coupling, taking into account the change of the quasi-elastic constants due to the substitution. At low impurity concentrations the halfwidth of the zero-phonon line and the corresponding dispersion curve as well as their position in the spectrum are shown to vary linearly with concentration; at higher concentration the absorption coefficient may deviate from linearity.

MeTonoM nByxiacTHqHbix $yHKUHii rpiraa HccjiejioBaHa KOHueHTpauHOHHaH 3aB0CHMOCTt> cncKTpoB norjiomeHHH H HHcnepcHH CBeTa npHMecHbiMii ueHTpaMH 3aMemeHHH npn cJia6oii 3JieKTp0H-Ke HX nojio>KeHHe B cneKTpe H3MeHHIOTCH C KOHUeHTpaUHeli JIHHeilHO; K03(j)(j)HUHeHT nOrjlOmeHHH C pOCTOM KOHqeHTpaijHH MOJKGT OTKJIOHHTBCH OT jiHHeftHoro pocTa K S K B CTopony yBejiHneHHH, TaK H yMeHLIIieHHH. 1. Polarization of the Solid Solution Consider a cubic crystal with one atom per unit cell which can be substituted by another kind of atom, thus forming a disordered substitutional solid solution. I t is evident that upon such a substitution both the mass of the crystal and its quasi-elastic constants change. In the previous paper [1] the absorption and dispersion of light by such impurity centres in the case of weak electron-phonon coupling was investigated taking into account only the ratio of atomic masses M

of host and impurity centres. In the present work we assume that ¡x = — — 1, where M and m are atomic masses of the host and the impurity centre, respectively, be negligibly small in comparison with the relative change of the quasielastic constants of the solid solution. We further assume that the impurity absorption region is far from that of the host crystal and the concentration of impurities is so small that the interaction between the centres can be neglected. Then the stationary states of strongly coupled electrons of such a system can be determined in the adiabatic approximation [2]. However, if the spacings between the energy levels of the optical electrons of the impurity atom are of the order of the phonon energy of the ideal host crystal the adiabatic approximation cannot be used to determine their stationary states. Let us assume that in the case under consideration we have to deal with weak electron-phonon coupling. Then writing the potential energy of the nuclear sub-system [3] in

428

A . F . LTJBCIIENKO a n d S . I . D U D K I N

the form

U(R) = V0(R) + VC(R) ,

where F 0 (R) is the potential energy of the ideal host nuclear sub-system, Ve(R) is a correction to F 0 (ii) due to the impurity substitutional atoms with the concentration c, and R are the nucleus coordinates, the Hamiltonian of the whole system will be H = hZ vm{l) a+i aml + 2 h cos (b+ b, + + 2 Bs (bs + bts) ml s \ ^f s

+

+ 2 A « , n (bSi + btSi) + V L.{m I, m V) a+i amV (b, + 6 t . ) . •s,' i -- 1 smll'

(1)

Here m is the number of the lattice point, I the set of quantum numbers characterizing the states of the impurity optical electrons with the energy h v(l), ojg are the normal vibrational frequencies of the ideal host lattice, s = (q, a), q is the phonon wave vector, o the branch number, a^i and ami are the electron creation and annihilation operators for the m-th impurity centre, bj and bs the phonon creation and annihilation operators, Ls(m I, m V) is the matrix element of electron-phonon coupling energy, B„ and AgiSt are the coefficients of the VC(R) expansion in powers of the nuclear displacements una from the equilibrium positions R0 determined by 8F 0 (R) = 0 . In particular, A S i S i is of the form

where M mass of the host atom, N the number of atoms in the system; c„ = 1, when the n-th site is occupied by an impurity atom, and cn = 0, when this site is occupied by a host crystal atom; v^s) is the component of the polarization vector of the .s-th normal vibration, Vrn are the VC(R) expansion coefficients in powers of c„: VC(R) = Z cn + £ cn < - 0 «*> Pfi Pff\ Gfr> lr(-v) « v a jcff IV Here Gjf,iv(—v) is the Fourier transform of the Green's function:

.

(4) ~

G/r.ir = iO (t - t') is the dyadic product of matrix elements of dipole moments belonging to transitions between the states |Z>, |Z'} and |/>, |/'> for the fc-th impurity centre, k numbers the impurity centres in the volume d3, where d < X, P1 is the part of polarization caused by the transitions of strongly coupled electrons of the system and of the impurity optical electrons with large I, /, i.e. upon excitation of states far from the investigated set of closely lying levels. For the disordered solid solutions considered here only values averaged over all possible configurations of impurity centres have a physical meaning. Therefore, denoting by the symbol (()) the configurational average, the calculation of the polarization will be reduced to evaluating ( ( < ? / / - , v ) ) ) T o calculate the function w{v) we differentiate the function (3) with respect to t : i

= +

à(t I

H {nkv

{L,(kf,

-

nkl)

ôfV ôri

kp)

+ {vk(f')

+ vk(f)} G j f j v +

Ls(k p, k /) G ^ r , »•} ,

-

(6)

where G%,iv

nkl — (tiki akly J n , , att(t')

akr(t')]}

(7)

.

In (7) we have omitted the argument t in all operators left t o the comma. Calculating further the ¿-derivative of the functions (7) and pairing the boson operators containing identical times, we obtain 8G(i) fp_ r„. .. u\\ rM = {n(p) — vk(t)} &»fPt IV + ft)s GsfPt ; r + 81 +

2" LSi{k p, k I") GSSi(0) G.pr,iv sll"

-

, (8)

Z LSi(kr,kf)GSSi(Q)Gri>,,r I" Si

where G'sfp, ir = i6 {t — t') ([akf akp (bs — bts),

afi{t')

akV(t')]>

,

GssX0) = . The quantity C?SSi(0) may be calculated with the help of the Matsubara Green's function [4], Using the technique developed in [1], one can easily show that G I f l (0) = (2 n, + 1) + A , (9) where A is a contribution to G'SfP> n> which is small of higher order than the first term of (9), therefore in the following the quantity A will be neglected. On the other hand, we obtain for the ¿-derivative of the function G' s j Pt w in the same approximation t i

= in(p)

+ 4 Z Asi(_s) «i

G(SX

lv

-

"*(/)} G'sfPi

+ Z 'i

tr

+ 2 B_s GfPt

p, k I,) Gflulv

lv

+ coGi%,lv

+ 2 L-.it h

+

h> * /) Gt,P,ir

• (10)

Proceeding now in (8) and (10) to the Fourier representation, one can readily obtain G%, w(v) = ti^Gff, 28

physica/20/2

lt.(v)

+ 2 Dpf(si) Gi%, iv{v) + Rx{v) , Sl

(11)

430

A . F . L u b c i h e n k o a n d S. I.

where

= L_s(k p, k /') (

+

D u d k i n

1

+

—1.

_ 4 i < l ( - , ) ws V! ~ Q„2 2 ' pf ~ Qpj = v — vk(p) + !**(/)> and all terms not contributing to t h e zero-phonon line are denoted by R^v). The Fourier transform of the function Crspfjv is calculated quite similarly: U

Gf'r, ,v{v) = rf? Qfr, iv{v) + 2 Dfp(Sl)

G[f> r , lv(v) + Rt(v) .

(12)

Here

Q

f p - ms Qfp = v — Vidf) + v/c{p). R^iv) denotes all terms not contributing to t h e zero-phonon line. The relations (11) and (12) are used to obtain t h e iteration series connecting G^fp, w{v), G^spf, ii'{v), and O f f , w(v). However, t h e configurational averaging of these series and their summation fail in the general case. Therefore, we / e2 f c \ approximate the function I by a simpler one. Let us put r ^ \Suniaidun^JR=Rll J 2 °n ) ~ 2 cn in -^n,«! > n \ OMn,,,, OM^«, lR=R, n where the dynamic coefficients are the same as those for the ideal crystal. Such approximation is in general used in the dynamic theory of crystal lattice with imperfections, though it can ignore the local vibrations, taking into account only some changes of all normal vibrational frequencies of the system due to the change of its quasi-elastic constants. I n this approximation 2

D

8

= - r f - 1 2 2 in Cn j E Cn (n s *r =

2

/

i + t Up ^ j- —t OJg 2 =

the Fourier transform

'-

=

2 , 2s \ Ufp — cos n' of Gyy iv will be written in the form - ^

Gfi lv{v)

+

iVlkv -

}

\

^ki)Sridfi'

v - vk(f>) + vk(f) - Mn(v)

11 - I ^ i

"

( U )

Ligth Absorption and Dispersion by Impurity Centres (II)

431

Here .

/ «.. -i-i

i

A„ = Z {L,(t sp

r

hp)

«..

\

+

tr„ - L,(kr,i;/)«•}[»-

K = a>! VPS. [ £ p f - a > * ( 1 + in)]- 1 ,

M/1

+ ».(/)

% = fl>; r? c y; l [

„ M

Vl> I)

Hence, it follows that if y?; > 0, there is a tendency of x to increase less than linearly with increasing concentration, while the halfwidth of the zero-phonon line increases linearly. If y n < 0, the deviations of x from linearity and the behaviour of the halfwidth of the zero-phonon line are expected to be opposite. In the general case the variations of x and halfwidth will be determined by the value and sign of ^ as well as by ¿a. n

References [1] A. P.

and S. I. D U D K I N , phys. stat. sol. 18, 853 (1966). and H U A N G K U N , Dynamical Theory of Crystal Lattices, Oxford University Press 1954. [3] A. P. L U B C H E N K O and S. I. D U D K I N , phys. stat. sol. 14, 227 (1966). [ 4 ] F . M A T S U B A K A , Progr. theor. Phys. 1 4 , 3 5 1 ( 1 9 5 5 ) . [2]

LUBCHENKO

M . BORN

(Received

November

23,

1966)

E. JAGER and R. PERTHEL: Crystal Field Theory of Spinel Lattices

433

phys. stat. sol. 20, 433 (1967) Subject classification: 13; 3.2; 18.3 Institut für Magnetische Werkstoffe der Deutsehen Akademie der Wissenschaften zu Berlin,

Jena

Some Remarks to the Crystal Field Theory of Spinel Lattices By E . JÄGER and R .

PEBTHEL

The crystal field potential of an octahedral site in the lattice of a normal, and an inverse spinel is calculated on the basis of the point charge model. The ten nearest neighbouring environments are taken into account. Finally, the tables of the matrix elements of the operator equivalents 0\ and 0\ for J = 1/2, 1, 3/2, . . . , 8 are given. Auf der Grundlage des Punktladungsmodells wird das Kristallfeldpotential für einen Oktaederplatz im Gitter eines normalen und eines inversen Spinells berechnet. Dabei werden die zehn nächsten Umgebungen berücksichtigt. Schließlich werden die Tabellen für die Matrixelemente der äquivalenten Operatoren 0\ und 0\ für J = 1/2, 1, 3/2, . . ., 8 angegeben.

1. Introduction Within the last ten years some authors (e.g. [1, 2]) have shown that it is possible to explain certain properties of spinel ferrites by means of the crystal field theory (one-ion model). In order to estimate the magnitudes of the parameters the calculations are often based on the point charge model taking into account only the nearest neighbours. The purpose of this paper is to determine the electrostatic potential of the neighbourhood of an octahedral site in a spinel lattice taking into account not only the first sphere of neighbouring atoms but the first ten spheres. The calculations of the matrix elements of the potential are simplified by passing over from the spherical harmonics Y„ into which the potential is expanded to the operator equivalents 0 „ because most of the matrix elements needed are tabulated in [3]. Only in the case of an inverse spinel these tables must be completed by the matrix elements of the operators 0\ and 0\ which are given in Table 2 and 3 of this paper. 2, Environment of an Octahedral Site in a Spinel Lattice Although it is well known that in the point charge model such important effects as the finite dimensions of the ions, the overlap of the wave functions, screening effects, etc., are neglected, it seems reasonable to study the limits of this theory by comparing the theoretical results with the experimental data (e.g. E P R measurements). B y numerical calculations one can easily see that the contributions of the second, third, fourth etc. sphere to the crystal field potential, which mostly are neglected, are not small compared with that of the first one. In order to calculate the crystal field it is necessary to known which sites belong to the corresponding spheres of the environment of an octahedral site. The first sphere consists of 6 0 2 ~ ions with the distance A^, given in Table 1. We use a spherical coordinate system with the origin in one octahedral site with

S (M

1

CD O

O

1

1

1

+

m

j^Jw

11 CO FH 1 CO

i-H

r

a o lO < ©N

«H O «M

II e

leo e CO

©

e ©

II 8

-r

e m m

o II e

II a 13 |«

«e

0

I.o

[gj

is

CO CO

l-H Ico I ? CO

|j3 [ 13 •p.

"3. I 23 —B L ^ ! ^ [co Uo Ico (MM

s

[g< (M CD

—I

(I?)

but • cos ( 2 ^ +

Wx

+ y2) + r\ ^ e - 2 « - ^

,

(1) v

where A

=

(«? + fcf) "2 [(1 + nxf +~*f] [(„, +

+ jfc»] •

'

Here are rx and r2 complex Fresnel coefficients for the air-film and film-glass interfaces, respectively [11 to 13], d1 phase change of a light wave traversing the film, Wj the refractive index of the film, the extinction index of the film, aj = ^ ^

1

the absorption coefficient of the film, nz the refractive index of

the glass substrate, the film thickness, A the wavelength of the radiation in air, yij and ipz phase angles arising at the interfaces of the film [10, 12]. The refractive index nx and absorption coefficient of the film cannot be eliminated analytically from equation (1) [13, 14]. However, in the spectral region where these equations may be reduced. With occurrence of interference fringes cos (2 dil> a fi o a

•a S g o Ss

fi O

he a •si cô 5h OO 'ö tí O cS fi 3 O M

so

o o w . fi S .2 ® to 'S -ta O o S fi o ¡sfi O a O $ 0)» S 31 ft1 ®

S .-S .S í í g 5 o-

T3 fi s 2 § S £ 'S MH » £

50 fi

.fi ft-

-

-ta 'e fi .35 ®

'S =9 -ta C > fi fi s i o CS BO ^ ° fi $ g O O

S.g § s 2® > .s O

cT

S

® ft

«

JS Oh

s i i S I ' fi "O -S © O O cö£C '-3 • S eôh s ° *tH -t^ .2 '-S O g CS -»a S fi 2 o %fi ''S .E e o ® 'S 3 g o 3 fi ® 8 'S 5 © fi 13 cS © G fi ï ^ S s o os

c® .2 o ®S 0 "S o fi -ta «fiS O -ta fi m O fi '-ta ,9 Is O tj

60 fi

« o

»

5b Ci

Cl Ss Cl

fi 'S O fi C^L O ts So O .ce

cT

tì ta S.3 ~ o .s o £ « fi s II 1 c* _® •S ó >

CS C .S ».H -fi •-® 3S3 S'-S 3S b S fr-S ft fi a g a

£

s £ s'g -S §c a

o h ft M

-S »

bo fi 'fi fi «

Ph

CT 3 g W o

•tí o fi £1 tS _ h tí O ° /S 02 M

Diffusion of Nickel in Silicon

497

2. Experimental Techniques The radioactive source for all experiments was a solution of NiCl2 which contained 4 mC Ni 63 . The specific activity of the nickel was 5188 mC/g (Oak Ridge National Laboratories). The absorption coefficient (linear) for the ^-radiation of Ni 63 (E& = 67 keV) in silicon is 3.30 X10 3 cm" 1 [18, 20]. Because of the strong absorption small amounts of nickel penetrating into the crystal can be detected. A brief collection of technical details is now given for completeness. The Czochalski-grown silicon single crystals (Mallinkrodt) had a resistivity of 80 to 100 Qcm and p-type conductivity. Wafers of orientation near (111) with a diameter of 14.3 cm and a thickness of 0.25 cm were cut from the rods with a diamond saw. After lapping and mechanical polishing (SiC powder) they were chemically polished in a mixture of 3 parts H N 0 3 and 1 p a r t H F (48%). For determination of the etch-pit density the polished surfaces were etched in a solution of Cr0 3 in H F [50 g Cr0 3 in 100 ml H 2 0 , 2 parts of this solution + 1 part HF], The triangular pits were counted under the microscope, the distribution of the pits being inhomogeneous (more pits in the edge region). The density varied between 400 and 1000 cm - 2 . A layer of nickel was deposited on one surface of each crystal by electrodeposition (65 V, 0.3 to 2.0 mA/cm 2 , room temperature). The amount of radioactivity deposited by this process varied between 4 x 103 and 105 counts/min. The ^-radiation of Ni 63 was detected with a gas-flow counter (2 Tc-geometry) with pre-amplifier and scaler (Nuclear, Chicago). The counter operated in the proportional region at 2.15 kV (counting gas: 90% argon, 10% methane) for which the counting efficiency is about 100%. The heat treatment of the samples was carried out in a vacuum of iS5 X X 10~5 Torr. A preheated furnace which was held at constant temperature ( + 1.5 °C, Leeds & Northrup "Speedomax H " ) could be moved over t h e quartz tube which contained the sample. Therefore the time required to heat the specimen to the diffusion temperature was kept within 5 to 7 minutes. At the end of a diffusion anneal, the furnace was removed so t h a t the specimen could cool down within 10 to 15 minutes. The temperature was measured separately near the sample location with a second thermocouple. Other experiments were carried out with quartz-encapsulated samples. After heat treatment, these were quenched in ice water (time of the order of 1 second) by breaking the quartz capsule as it entered the quenching medium. The capsules were evacuated under fore-pump vacuum (10~2 Torr). 3. Results Diffusion experiments were carried out in the temperature range 450 to 800 °C. The surface activity of the Ni 63 -radiation was measured before the diffusion annealing (A0) and also for several different diffusion times at constant temperature by interrupting the heat treatment and taking the sample out of the system for counting (activity A). The originally inactive backside of each sample was also investigated for radioactivity. 3.1 Results

for the air-cooled

samples

The absolute counts of surface radioactivity of the front and backside of several samples are shown in Fig. 1. The important result of these experiments is the finding of a considerable amount (depending on initial activity as well as on

498

H. P.

50r

1 4

front side activity Filled symbolsback side activity •• Run at 739 X " Run at 788 X

BONZEL Fig. 1. Surface radioactivity o f front and backside o f diffusion samples as a function o f annealing time

temperature) of radioactivity on the backside of the sample after the first heating period. In the following periods 10a decrease of activity on both surfaces generally could be observed, while the 0 20 40 60 3 gradient at the backside was less steep Kl0 s)than that at the front side. In some exD periments the backside activity remained Front side activity • Back side activity nearly constant throughout the period Run at 707 X t of measurement (Fig. l b ) . For one experiment only a single spot b of Ni 63 was deposited in the center of 20 40 tWs)the sample surface. This sample was heat-treated at 650 and 770 °C, and, in addition to the usual surface counting, radiographs of front and backside were taken after each run in order to estimate the extent of surface diffusion. The originally sharp contours of the spot remained unaltered even after heating for a long time, although, after two hours annealing, uniform concentration of nickel was detected on both sides of the crystal. This indicates that the nickel spot did not serve as a source for surface diffusion because in that case a diffusionlike concentration distribution around the spot would be expected as a function of time. Therefore, the possibility of surface diffusion as an explanation for the result of this experiment is reasonably excluded. Although some nickel had migrated (mainly during the first period of heat treatment) from the front to the backside, which is a disturbance of the boundary condition for the diffusion process, an attempt was made to treat the data within the framework of simple diffusion theory. Applying the method of decrease in surface activity, the measured values of activity ratio, A/A0, and time, t, 20 yielded the data fi2D t by the help of the function .6 30\ -it § 20

•

fsi

/

i Run at 788 X 5 Run at 700 °C

0

=

e x p (fx2Dt)

[1 -

e r f (^Dt)]

,

(10)

where /x absorption coefficient of (irradiation of Ni 63 and D effective diffusion coefficient. The plot of ¡i?Dt vs. t should be a straight line through the origin in an ideal diffusion experiment. In most cases, this line could be drawn and the diffusion coefficient was calculated from its slope (Fig. 2). The effective diffusion coefficients have been determined in this manner for the 50 SO t(103s)-

F i g . 2.

P l o t of fi'Dt vs. t for several experiments at different temperatures

499

Diffusion of Nickel in Silicon

TÌXÌ 300 800 700

Fig. 3. Semi-log plot of a p p a r e n t diffusion coefficient vs. reciprocal temperature. Open points result from single measurements with one s a m p l e a t three different temperatures

500 450 WO

temperature range 450 to 800 °C. They were plotted in an Arrhenius diagram, Fig. 3, resulting in a straight line (best fit) which can be expressed by i) =

0 . 1 e x p ( -

3.2 Results

4 4 0 k

; y

m o l

) c

m

for the quenched

V s .

(11)

samples

Two samples with initial surface activities of 5.5 X 10 4 and 2 X 10 s counts/min were annealed at 700 °C for about 4 hours and then quenched. The fast quen ching of the silicon caused high internal stresses which resulted in cracks and fracture. Therefore, only a piece of the original sample could be used for further investigation of the concentration profile in the interior. The measured surface activities and the residual activities after having polished off part of the sample, are plotted in Fig. 4. The thickness of the lapped silicon was calculated from the weight loss of the sample. The concentration profiles show a plateau region inside the crystal after a sharp initial decrease near the surface. The activity of such a concentration plateau can be used to calculate the concentration of nickel per unit volume. The rest-activity on the surface of the sample is equal to A

1

c

=

=

1.87x10» J

C I c(x, t) v

'

1.87 x IO9 fi Ar

da; =

'

4 X 10

35

12

40

x(W 3cm)—-

F i g . 4. R e s i d u a l surface activity AT as a function of thickness x of removed material for water-quenched samples. Arrow indicates another point which is o f f scale

F e (1 1.87 x 10» fi y

e~

f t

"),

(12)

Ar atoms/cm , 3

x(W' 3cm)Fig. 5. R e s i d u a l surface a c t i v i t y .1 r as a function of thickness x of removed material for air-cooled sample (annealed at 700 °C for 20 hours). Arrow indicates another point off scale, r s o i s the extrapolated equilibrium concentration measured in reference [17]

500

H . P . BONZEL

where Ar surface activity of plateau (cpm), a length of plateau 10~2 cm), and c constant concentration within plateau region. The calculated concentrations inside the quenched samples were 3 X 1013 and 8 X 10 13 atoms/cm3, respectively. I t is believed that this difference in concentration arises from the difference in the amount of nickel initially deposited on the crystals (in addition to a slight difference of annealing time). For comparison, one of the slow-cooled samples which had been annealed at 700 °C for about 20 hours was also checked for the concentration profile. The residual concentration-« curve is shown in Fig. 5. There is also a sharp drop in the concentration near the surface, followed by a concentration valley. After that a depleted region is found for greater depth x and, beginning at a depth of 0.05 cm, a plateau region with a concentration of 12 X 10 13 atoms/cm3 which is higher but of the same order as that for the quenched samples. 4. Discussion The diffusion of nickel as an impurity in silicon has not yet been the subject of investigation although data on solubility and precipitation effects as well as on electrical properties [14 to 19] have been published. Some of these experimental results may be summarized here because they will be useful for the comparison with the present observations. The solid solubility of nickel in silicon is retrograde and shows a maximum of about 6 X 1018 atoms/cm3 ( X M 10" 4 ) between 1200 and 1300 °C [15, 17, 18]. The nickel distribution inside the crystal depends strongly on the cooling rate after heat treatment; "concentration valleys" near the surface as a consequence of a finite cooling rate indicate a fast mobility of nickel atoms (D 10~4 cm2/s at 1100 °C) [17, 18]. Precipitation of nickel in the interior and on the surface of the crystals is observed; precipitates of different shapes are reported and the kinetics of precipitation seem to follow an expression of the type N(t) -

N(oo)

N( 0) -

N( oo)

= exp

(13)

with n = 1, N(0) and N(oo) initial and final concentrations of precipitated phase [14, 16, 18]. Precipitation in the neighborhood of dislocations is generally observed; vacancies have been considered as precipitation nuclei [14]. Two acceptor levels have been detected in nickel-diffused silicon. I t is believed that the acceptor nickel is probably identical with substitutional nickel while there is also the fast moving interstitial nickel which is apparently electrically inactive [14]. The dissociative mechanism of diffusion which has been established for the diffusion of Cu in Ge [4, 7], Au in Si [9, 10, 21], Cu in Si [7], and probably Au in InSb [22] is now also proposed for the diffusion of Ni in Si. Most of the experimental results can be interpreted in terms of this model and this is evident from the following statements: First, the accumulation of nickel on the backside of the specimen after a relatively short annealing time (at all temperatures) indicates the fast mobility of nickel. Surface diffusion and transport through the vapor phase can be ruled out in this case. Therefore, only the transport through the bulk remains possible. The crystals had a low dislocation density so that pipe diffusion is also very unlikely. The diffusion of nickel via interstitial sites is the most probable

501

Diffusion of Nickel in Silicon

explanation. The accumulation occurs because of two reasons: If the temperature is lowered at the end of the heat treatment, the solubility of interstitial nickel drops rapidly. Nickel diffuses to the free surface which acts as a preferential, heterogeneous nucleation site for nickel precipitate. Also some nickel atoms react with vacancies which migrated into the crystal from the surface resulting in a certain concentration of stable nickel atoms in substitutional sites near the surface (proportional to the vacancy gradient). Second, in subsequent annealing periods, it is observed by measuring the decrease of the surface activity of Ni 63 that nickel is still absorbed by the crystal. This behavior indicates a secondary process with a slower rate constant than the interstitial diffusivity. This process is the already-mentioned reaction of interstitials with vacancies: i + v s. This reaction is slow in the interior of the crystal because of the slow supply of vacancies (mainly by dislocation climb), and faster near the surfaces. The result of the surface-region reaction is a diffusion-like concentration profile of substitutional nickel which is experimentally verified by the straight line, fi2Dt vs. t, and by the steep concentration gradient found in the polished samples near the surface. Therefore, it seems allowable to examine the diffusion data further. If the apparent diffusion coefficient of this investigation is seen in the light of the Frank-Turnbull theory, it should be proportional to the diffusion coefficient of vacancies or that of self-diffusion of silicon (equation (2)). Although the assumption of local equilibrium between vacancies and substitutional atoms is still in question, one may try to estimate Dai with the help of this equation. The solubility of nickel in silicon has been measured above 750 °C [17]. Based on the extrapolation of the solubility data to lower temperatures, and with the use of equation (11), one finds c s 0 = 5.4 X 1026 exp | —

atoms/cm 3

= 1.08X10« e x P ( - 5 3 - 5 ^ / m o 1 ) ,

•

Z)sd = c s 0 B = 1 0 3 e x p ( - 9 - ^ A n o l )

(14)

cm*/s .

This result compares with x.

, „

/

Asd = 1-8 X 103 exp ^

110 kcal/mol \ ^

J cm 2 /s,

which has been measured recently by Peart [24], and D s d = 1.81 X10< exp cm2/s which has been deduced indirectly by Wilcox et al. [9]. The diffusion coefficients of substitutionally-soluted elements (for example B , Al, P, Sb) in silicon which should lie in the same order of magnitude as the self-diffusion of silicon have pre-exponential factors up to 103 cm 2 /s and activation energies in the range 80 to 107 kcal/mol [23]. Therefore the values reported for self-diffusion are reasonable. On the other hand, if the energy of solution

502

H . P . BONZEL

(or formation) for vacancies and substitutionals can be assumed to be nearly equal, the ratio Dv/D will be constant and equal to the ratio of the solubilities, cso/Cvo- The activation energy for the diffusion of vacancies (energy of motion) in this case would be 44 kcal/mol. It should be emphasized that with the method chosen to evaluate the diffusion data, it is possible to get only approximate values for the diffusion coefficients. Especially it has been assumed that diffusion and solution of interstitial nickel should not affect the diffusion of vacancies and their subsequent reaction with each other, and that precipitated regions as a consequence of cooling should dissolve again in the following annealing period (reversibility). The same should be valid for vacancies and possible cluster formation. Third, looking at the concentration profiles of nickel in silicon it becomes obvious that for the slow-cooled sample (annealed at 700 °C), as well as the quenched sample, a certain concentration of nickel in the interior is detected. Since the slow-cooled sample had been heat-treated much longer than the quenched samples, its plateau concentration is considerably higher. Therefore one also can conclude that the nickel detected in the interior is the product of the reaction and hence located at substitutional sites. The amount of interstitial nickel is probably a small fraction of the substitutional portion because, otherwise, nearly no nickel could be detectable inside the slow-cooled sample. The extrapolated value of the equilibrium concentration of nickel at 700 °C [17] is about 5.3 X 1014 per cm3 which is still larger than the plateau value of 1.2 X 1014 per cm3 for the one specimen. This indicates that the rate of supply of vacancies inside the crystal is slow, consistent with the low dislocation density. Finally, in this context we note that while the diffusion sample is cooled from the annealing temperature in 10 to 15 minutes, a great portion of the interstitial nickel will diffuse to the free surface because its solubility is lowered. This back-diffusion should cause a broad and deep valley in the concentration profile where the width is determined by the diffusion constant and the depth by the maximum concentration of interstitial nickel. The concentration valley actually observed is rather shallow and this indicates that only a small fraction of the nickel is dissolved interstitially. Most of it is located at the more stable substitutional site. During the cooling of the sample the crystal becomes undersaturated in interstitial nickel because of its fast mobility. This causes the dissociation of substitutional atoms into vacancies and interstitials to take place in order to re-establish equilibrium. This reaction also lowers the concentration of substitutional nickel resulting in a shallow concentration valley near the surface. The depth of this valley is determined by the rate constant kx of the dissociative reaction. Thus far, the explanation of the observed experimental facts has been discussed in the framework of the dissociative diffusion process. Similar observations have been made for the diffusion of Au in Si [12], although in that work the steep concentration gradients near the free surfaces of the diffusion samples appeared to be independent of time. Also, for the diffusion of Au in InSb, steep gradients near the surface followed by long flat regions have been found [22]. These have been evaluated separately resulting in two different diffusion coefficients more consistent with the explanation given for the diffusion of Ni in Si. The diffusion of Cu in Ge also has been studied thoroughly [4], and, in this case, it was possible to measure the concentration of radioactive and electrically active copper independently. This permits a separation of all the copper into

503

Diffusion of Nickel in Silicon

substitutionals (single acceptor) and interstitials (single donor). The concentration of substitutional copper as a function of annealing time obeys nearly a pure exponential law of the kind given by equation (9) and this fact constitutes more evidence for the reaction between vacancies and interstitials in the bulk. Again, this is only true for material with high dislocation content, while for crystals with low dislocation density a much smaller saturation rate is observed. 6. Comments In conclusion it is pointed out that it might be worthwhile to measure more accurately the concentration profiles caused by diffusion of impurities in semiconductors, possibly for crystals of different dislocation content. However, an attempt must also be made to get theoretical expressions for the concentration profiles as a function of diffusion time, in solving the differential equations 0C ~

=

dc -± = t =

0:

t ^ 0:

V 2 cv +

Dv

-

k

l C s

+

cs(0) = 0 for all

h

e8 — k2 c i 0 c v ,

k2ci0cv;

C; =

ci0;

(5a) (6a)

x,y,z ,

cv(i) = cvo at free surface and dislocations.

By sectioning the diffused samples and measuring the concentration of (substitutional) impurities as a function of depth it would be possible to separate the contribution of vacancies due to diffusion from the free surface and due to dislocation climb. Consequently, this would give more information about the climb rate of dislocations in semiconductors, e.g. whether vacancy migration is the rate-limiting factor for the occurrence of climb. Another question of interest which might be solved by knowing the exact solution of equations (5a) and (6a): Are the assumed values of the constant source concentration of vacancies, c v 0 , at the free surface and at the dislocation core identical with the "equilibrium" concentration of vacancies in the bulk ? The steep concentration gradient near the surface of a diffusion sample seems to indicate that in this region the concentration of substitutional nickel and therefore probably also the vacancy concentration are considerably higher than the presently assumed "equilibrium" values for the bulk of this material. Finally, it would be possible to derive the rate constants ^ and k2 of the defect reaction, and to prove whether the assumption of local equilibrium between these defects — made by Frank and Turnbull — is justified or not. Acknotvledgements

I would like to express my gratitude to Professor J . P. Hirth and Professor G. P. Powell for providing the opportunity for this work and helpful discussions. I want to thank Dr. J . Patterson for valuable help and Mr. H. Pagean (Department of Electrical Engineering) for preparing the silicon wafers. I am indebted to Dr. G. F. Boiling for his invaluable comments on the manuscript. This work was made possible by a fellowship of the Graduate School of the Ohio State University and by support of the National Science Foundation.

504

H. P. BONZEL : Diffusion of Nickel in Silicon

References [1] B. I. BOLTAKS, Diffusion in Semiconductors, Academic Press, New York 1963. [2] C. S. PULLEB,

J . D . STRUTHERS,

J . A . EITZENBERG ER, a n d K . B . WOLFSTIRN, P h y s .

Rev. 93, 1182 (1954). [3] A. G. TWEET a n d C. J . GALLAGHER, P h y s . R e v . 103, 828 (1956). [4] C. S . FULLER a n d J . A . DITZENBERGER, J . a p p i . P h y s . 2 8 , 4 0 ( 1 9 5 7 ) . [5] P . VAN DER MAESEN a n d J . A . BRENKMAN, J . E l e c t r o c h e m . S o c . 1 0 2 , 2 2 9 ( 1 9 5 5 ) . [6] P . C. FRANK a n d D . TURNBULL, P h y s . R e v . 1 0 4 , 6 1 7 (1956).

[7] R. N. HALL and J. H. RACETTE, J . appi. Phys. 35, 379 (1964). [8] A. HIRAKI, J . P h y s . Soc. J a p a n 21, 34 (1966). [9] W . R . WILCOX a n d T . J . LACHAPELLE, J . a p p i . P h y s . 3 5 , 2 4 0 ( 1 9 6 4 ) .

[10] G. J . SPROKEL, J . Electrochem. Soc. 112, 807 (1965). [11] T. I. KUCHER, Fiz. tverd. Tela 6, 801 (1964); Soviet Phys.-Solid State Phya. 6, 623 (1964). [12] G. J . SPROKEL a n d J . M. FAIRFIELD, J . Electrochem. Soc. 112, 200 (1965).

[13] P. PENNING, Philips Res. Rep. 14, 337 (1959). [14] Y. TOKUMARU, Japan. J . appi. Phys. 2, 542 (1963). [15] J . H . AALBERTS a n d M. L . VERHEIJKE, Appi. P h y s . L e t t e r s 1, 19 (1962).

[16] T. IIZUKA, M. KIKUCHI, and K. KANASAKI, Japan. J. appi. Phys. 2, 309 (1963). [17] M. YOSHIDA a n d K . FURUSHO, J a p a n . J . appi. P h y s . 3, 521 (1964). [18] Y . YAMAGUCHI, M . YOSHIDA, a n d H . AOKI, R e v . E l e c t r . C o m m u n . L a b . 1 2 , 4 0 6 ( 1 9 6 4 ) .

[19] [20] [21] [22]

M. YOSHIDA and K. FURUSHO, Japan. J . appi. Phys. 4, 74 (1965). H. P. BONZEL, Ber. Bunsenges. phys. Chem. 70, 73 (1966). W. C. DASH, J . appi. Phys. 31, 2275 (1960). B. I. BOLTAKS and V. 1. SOKOLOV, Fiz. tverd. Tela 6, 771 (1964); Soviet Phys.-Solid State Phys. 6, 600 (1964). [23] N. B. HANNAY (Ed.), Semiconductors, Reinhold Pubi. Comp., New York 1958. [24] R . F . PEART, phys. s t a t . sol. 15, K119 (1966). [25] D . G . HURST, A E C L - 1 5 5 0 ( N o v . 1962). [26] H . GAUS, Z . N a t u r i . 2 0 a , 1 2 9 8 (1965).

[27] A. SOSIN a n d W . BAUER, P h y s . R e v . 147, 478 (1966). (Received

January

12,

1967)

A. K. HEAD et al.: Influence of Large Elastic Anisotropy

505

phys. stat. sol. 20, 505 (1967) Subject classification: 10.1; 12.1; 21.1 Commonwealth Scientific and Industrial Research Organization, Division of Tribophysics, University of Melbourne

The Influence of Large Elastic Anisotropy on the Determination of Burgers Vectors of Dislocations in ß-Brass by Electron Microscopy By A . K . H E A D , M. H . LOBETTO1), a n d P . HUMBLE

The theory of electron diffraction images of dislocations based on isotropic elasticity is found to be inadequate for ß-brass because of its large elastic anisotropy. I n particular, the isotropic invisibility criteria (g • b — 0 and g • b X u = 0) are not valid in general in anisotropy except for pure screw or pure edge dislocations lying perpendicular to a symmetry plane of the crystal. Theoretical image profiles have been computed using the twobeam dynamical approximation and the full anisotropic strain fields of dislocations. Comparison of the character of theoretical and experimental images has established t h a t the majority of dislocations in ß-brass are screw dislocations having a Burgers vector of a whole and gliding on {110}. Es wird gefunden, daß die Theorie der Elektronenbeugungsbilder von Versetzungen, die auf einer isotropen Elastizität beruht, für ß-Messing wegen dessen großer elastischer Anisotropie nicht anwendbar ist. Insbesondere sind die isotropen Nichtbeobachtbarkeitskriterien (g • b = 0 und g • b x u = 0) nicht mehr allgemein bei Anisotropie gültig, außer f ü r reine Schrauben- oder reine Stufenversetzungen, die senkrecht zu einer Symmetrieebene des Kristalls liegen. Theoretische Beugungsbilderprofile wurden mit der dynamischen Zweistrahlnäherung und den vollständigen anisotropen Spannungsfeldern der Versetzungen berechnet. Ein Vergleich des Charakters der theoretischen und experimentell beobachteten Beugungsbilder ergibt, daß die Mehrzahl der Versetzungen in ß-Messing Schraubenversetzungen mit Burgersvektoren in und Gleitebene {110} sind.

1. Introduction In a recent paper, Head f l ] has analysed the stability of dislocations with Burgers vectors in