Physica status solidi: Volume 20, Number 1 March 1 [Reprint 2021 ed.] 9783112502105, 9783112502099

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plxysica status solidi

V O L U M E 20

N U M B E R I . 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 0. Lattice Mechanics. Phonons 0.1 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 Dcviccs 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)

Information We should like to point out once more that a subject and author index for the volumes 1 to 15 have appeared in February 1967. This index is available at a price of MDN 35, — , independent of subscription of the journal.

physica/20/1

phys. stat. sol. 20 (1967)

Author Index I . V . ABARENKOV

643

W . DREYBRODT

337

G . B . ABDULLAEV

421

S . I . DUDKIN

427

A . A . ABDURAKHMANOVA

777

S . U . DZHALILOV

261

I . A . AICIMOV

771

M. I . ALIEV

777

S . AMELINCKX K H . I . AMIRKHANOV

613 K L 19

T . R . ANANTIIARAMAN

59

I . M . ANTONOVA

643

H . AREND

653

P . ASADI

K55,

K59,

K71,

K73

V . M . ASNIN J . AULEYTNER

755 K77

R . BABUSKOVÄ B . BARANOWSKI

K29 K37

R. R. A. V. K. M. H.

I . BASHIROV I . BAZYURA R . BEATTIE M . BENTSA BERCHTOLD BEVIS BILZ

G. BLASSE J . BLINOWSKI W.BLUM Z . BODO G . BOHN V . A . BOKOV H.P.BONZE L R . BÖTTCHER A. BRIL J . O . BRITTAIN H . BROSS M . F . BRYZHINA I . BCNGET

K119 213 K135 771 255 197 K167 551 K107 629 K45 277 745 493 121 551 443 277 745 KI63

T . N . CASSELMANN

605

N . A . CHERNOPLEKOV Y . T . CHOU

767 285

S. G. K. L. A.

Y . CHUANCI COSTE R . CRAMER CSER CZACHOR

N. A. B. R. P. L.

VAN DANG S . DAVYDOV DAYAL T . DELVES DEVATJX DOBROSAVLJEVIC

331 361 K21 581, 591 K17 557 143,153 321 693 301 K63

B . ECKSTEIN N . ECONOMOU

83 623

V . D . EGOROV

705

R . ENDERLEIN U . ESSMANN

295 95

P . FELTHAM

675

T . FIGIELSKI R . FREUD

KL K151

M . M . GADZHIALIEV

K L 19

R . R . GAL4ZKA T . GESZTI R . GEVERS R . N . GHOSHTAGORE K . GODWOD Z . GOLACKI F . T . GOLDSTEIN H . GORETZKI I . P . GREBENNIK B . A . GREENBERG N . A . GRIGORYAN M . GRYNBERG N.P.GUPTA R . P . GUPTA E . S . GUSEINOVA

113 165 613 K89 K77 KL 379 K141 213 K103 745 K107 321 291 421

W.F.HALL C. HAMANN W . HAUBENREISSER A . K . HEAD L . L . HENCH 0 . HENKEL W . HENRION 1. H E V E S I B . G . HOGG M . HÖHNE P . HUMBLE E . IGRAS B . ILSCHNER E . JÄGER M . JAROS D . A. JENKINS CH. A . JOHNSON G . JONES G. A . JONES V . M. JUDIN

KLL 481 K131 505,521 327 K127 K145 K45 331 657,667 505,521 K5 629 433 K99 327 59 K135 K95 759

792 A . A . KAMINSKII F . KUBEC

Author Index K51

K . SAKATA

K155

653

T . SAKATA

K155

G . KUNZE

179

B . N . SAMOILOV

V . I . KUTAITSEV

767

P . SCHMELING

E . P . LAUTENSCHLAGER

443

E . SCHNÜRER S. LELE S . LIBOVICKY U . LINDNER J . L I T WIN M . H . LORETTO

59 K85 K131 K77 505, 521

301

H . G . SCOTT

461

J . SEDIVI

249

A . B . SHERMAN

759

H . SICHOVÄ

249

H . SIGMUND

255 337

427

D . SILBER

W . LÜDKE

121

J . M . SILCOCK G . SMITH

H . MARKERT

K67

L . SOSNOWSKI

737

J . M . SPAETH

E . A . METZBOWER

681

S . MILOSEVIC D . D . MISHIN G . O . MÜLLER E . N . MYASNIKOV

K63 K9, K123 K173 153

V . P . NABEREZHNYKH H . NEUMANN I . NISHIDA

737 K33 K155

J . OSTANEVICH J . W . OSTROWSKI

581, 591 KL59

A . PAJ^CZKOWSKA L . PAL

K159 581,591

N . A . PANGAROV G . K H . PANOVA J . PASTRXÄK. G. S. PAWLEY PV. F . PEART H . PEIBST R . PERTHEL K . - H . PFEFFER J . R . PILBROW I . 1 . PINCHUK A . PINDOR B . Y A . PINES J . PRZYSTAWA

365, 371 767 K29 347 545 K173 433 395 225, 237 537 K17 213 451

V . RADHAKRISHNAN G . REMAUT B . REPPICH K . J . RICHARDS P.M.ROBINSON B . ROESSLER A . A . ROGACHEV M . ROSENBERG L . ROSKOVOOVA K . I . RZAEV

783 613 69 K21 461 713 755 K163 K29 261

K173

M . SCHOTT

A . F . LUBCHENKO

A . A . MABYAKHIN

767 127,597

H . N . SPECTOR J . SPYRIDELIS M . STASIW J . STOIMENOS R . STRAUBEL W . SUSKI N . SWINDELLS J . SZYMASZEK B . G . TAGIEV V. A. H. D. T. H.

V . TARASSOV N . TERENIN THIEL J . D . THOMAS C. TISONE TRÄUBLE

H . - J . ULLRICH H . P . UTECH Y. D. F. N.

725 K135 113 225, 237 605 623 657,667 623 267 451 197 K37 421 37,45 771 K173 131 443 95 K113 K41

P . VARSHNI VAUGHAN I . VILESOV A . VLASENKO

9 725 771 311

R . E . DEWAMES T . WABMINSKI R . K . WEHNER W . WINDSCH R . DE W I T Z . WOUDSZYN F . J . WORZALA

KLL K5 K167 121 567, 575 K77 K81

L . ZDANOVICZ K . ZDÄNSKY M . G . ZEMLYANOV R . ZEYHER J . ZFTKOVI. G . V . ZOZULYA L . ZSOLDOS S . A . ZYNIO

473 653 767 K167 K99 213 K25 311

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, 0 . 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 A S , Praha, H. D. M E G A W , Cambridge, T. S. M O S S , 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. Y. R O L L I N , Oxford, H. M. R O S E N B E R G , Oxford, R . V A U T I E R , Bellevue/Seine

Volume 20 • Number 1 • Pages 1 to 412, K1 to K70, and A l to A38 March 1, 1967

AKADEMIE-VERLAG•BERLIN

Subscriptions a n d orders for single copies should b e addressed to A K A D E M I E - V E R L A G G m b H , 108 B e r l i n , L e i p z i g e r S t r a ß e 3 - 4 or t o B u c h h a n d l u n g K U N S T U N D W I S S E N , E r i c h B i e b e r , 7 S t u t t g a r t 1, W i l h e l m s t r . 4 — 6 or t o D e u t s c h e B u c h - E x p o r t

und-Import

G m b H , 7 0 1 L e i p z i g , P o s t s c h l i e ß f a c h 160

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Schriftleiter und verantwortlich für den Inhalt: Professor Dr. Dr. h. c. P. G ö r l i c h , 102 Berlin, Neue Schönhauser Str. 20 bzw. 69 J e n a , Humboldtstr. 26. Redaktionskollegium: Dr. S. O b e r l ä n d e r , Dr. E. G u t s c h e , Dr. W. B o r c h a r d t . Anschrift der Schriftleitung: 102 Berlin, Neue Schönhauser Str. 20. Fernruf: 42 6788. Verlag: Akademie-Verlag GmbH, 108 Berlin, Leipziger Str. 3—4, Fernruf: 220441, Telex-Nr. 011773, Postscheckkonto: Berlin 35021. — Die Zeitschrift „physica status solidi" erscheint jeweils am 1. des Monats. Bezugspreis eines Bandes MDN 72,— (Sonderpreis für die DDR MDN 60,—). Bestellnummer dieses Bandes 1068/20. Jeder Band enthält zwei Hefte. Gesamtherstellung: VEB Druckerei „Thomas Müntzer 14 Bad Langensalza. — Veröffentlicht unter der Lizenznummer 1310 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik.

Contents Page

Review Article Y.

P.

VARSHNI

Band-to-Band Radiative Recombination in Groups I I I - V Semiconductors (II)

IV,

VI,

and 9

Original Papers V. V.

V. V.

TARASSOV

TAKASSOV

Heat Capacity and Structure of Vitreous Silica and Diamond-Like Lattices (I)

37

H e a t Capacity and Structure of Vitreous Silica and Diamond-Like Lattices (II)

45

S . L E L E , T . R . ANANTHARAMAN, a n d CH. A . JOHNSON

X - R a y Diffraction by Hexagonal Close-Packed Crystals Containing Extrinsic Stacking Faults

59

B.

REPPICH

Plastische Verformung von Eisen-II-Oxid

69

B.

ECKSTEIN

A Disorder Model of Melting and Melts

83

H . TRÄUBLE u n d U . ESSMANN

Die Beobachtung magnetischer Strukturen von Supraleitern zweiter Art

95

R . R . G A L ^ Z K A a n d L . SOSNOWSKI

Conduction Band Structure of Cd0 jHg,, 9 Te R . BÖTTCHER, W . WINDSCH, a n d W .

P.

SCHMELING

D.

J.

A. S.

D.

113

LÜDKE

E P R Investigations on Mn 2 + -Doped LiF Single Crystals

121

Diffusion of Xenon in Barium Fluoride

127

The Effect of Heat Treatment on Silicon Nitride Layers on Silicon . 131

THOMAS

On the Operator of Exciton-Phonon Interaction

DAVYDOV

143

A . S . DAVYDOV a n d E . N . MYASNIKOV

Absorption and Dispersion of Light by Molecular Excitons . . . .

153

T. GESZTI

On the Theory of Thermally Activated Processes

165

G. KUNZE

Refraktionsmodifizierte R ü c k s t r a h l - R e f l e x i o n Braggsche Beugung bei

M. BEVIS a n d H .

a s y m m e t r i s c h e r 179

SWINDELLS

The Determination of the Orientation of Micro-Crystals Using a Back-Reflection Kossel Technique and an Electron P r o b e Microanalysen 197

Contents

4

Page B . Y A . P I N E S , I . P . G R E B E N N I K , R . I . BAZYURA, a n d G . V . ZOZULYA

Surface a n d Volume Diffusion in Thin Films of the System Ag-Se . . 213 J . R . PILBROW a n d J . M .

SPAETH

E S R Studies of Cu2+ in NH„C1 Single Crystals between 4.2 and 453 °K (I ) 225 J . R . PILBROW a n d J . M .

SPAETH

E S R Studies of Cu 2 + in NH 4 C1 Single Crystals between 4.2 and 453 °K (II ) 237 J . § E D i v i a n d H . SfcHOVA

Röntgenographische Messung der charakteristischen Debye-Temperat u r bei vielkristallinem Silber mit Anwendung der Korrektur auf primäre Extinktion 249 H . SIGMUND a n d K .

BERCHTOLD

Electrical and Photovoltaic Properties of PbS-Si Heterodiodes . . 255 S . U . DZHALILOV a n d K . I .

RZAEV

On the Phenomenon of Selenium Vitrification R.

STRAUBEL

H . BROSS a n d G .

Anhysteretic Magnetization of a Collective Domain Particles

261 of Magnetic Single-

BOHN

On the Theory of Electrical Resistivity of Polyvalent Metals . . . . Y. T.

OHOTJ

R. P. GUPTA R. E N D E R L E I N

267

277

On Stacked Screw Dislocation Arrays in an Anisotropic Medium . . 285 Lattice Specific Heats of Thallium and Y t t r i u m The Influence of Collisions on the Franz-Keldysh Effect

291 295

P . D E V A U X a n d M . SCHOTT

Thermally Stimulated Currents without Optical Excitation. Application to Copper Phtalocyanine 301 N . A . VLASENKO a n d S. A .

ZYNIO

Role of Carrier Polarization in t h e Mechanism of t h e Electroluminescence of ZnS-Mn Films 311 N . P . GUPTA a n d B .

DAYAL

Equation of State of Solid Argon and K r y p t o n L. L. HENCH a n d D . A.

JENKINS

AC Conductivity of a Glass Semiconductor S . Y . CHUANG a n d B . G .

321

327

HOGG

Positron Annihilation in Alnico and (La 0 7 Pb 0 3 )Mn0 3

331

Contents

5 Page

W . DREYBRODT a n d D . SILBER

Electron Spin Resonance of T l + + Centres in KCl Crystals G. S. G.

PAWLEY

COSTE

337

A Model for the Lattice Dynamics of Naphthalene and Anthracene . 347 Oscillatory Thermomagnetic Effects in Lead Telluride

361

N. A.

PANGAROV

Calculation of the Specific Surface Energies of Crystal Planes by Investigating the Orientation of Crystal Nuclei on Inert Substrates . . 365

N. A.

PANGAROV

Twinning Processes in the Electrocrystallization of Pace-Centred Cubic Metals 371

F. T.

GOLDSTEIN

K . - H . PFEFFER

F-Center Formation by Fundamental Absorption in K I

379

Mikromagnetische Behandlung der Wechselwirkung zwischen Versetzungen und ebenen Blochwänden (I) 395

Short Notes

Z . GOLACKI a n d T . F I G I E L S K I

Extrinsic Photoconductivity in Ge Caused by Dislocations

Kl

E . IGRAS a n d T . W A R M I N S K I

Electron Mirror Microscopic Investigation of Surface Diffusion of Lithium in Silicon K5 D. D. MISHIN

On the Theory of Magnetic Hardness of Platinum Cobalt

R. E. DEWAMES a n d W . F.

K9

HALL

A Relation between the Energy Loss Spectrum of Incoherently Scattered Electrons and the Phonon Density of States in Crystals . . . K l l A . CZACHOR a n d A . P I N D O R

Dependence of Calculated Elastic Constants on the c/a Ratio for Hexagonal Close-Packed Metals K17 K . J . R I C H A R D S a n d K . R . CRAMER

Diffusion Measurements Involving Discontinuous Interfacial Contact K21 L.

ZSOLDOS

Lattice Parameter Change of F e R h Alloys due to AntiferromagneticFerromagnetic Transformation K25

6

Contents Page

L . ROSKOVCOVA, J . P A S T R N A K , a n d R . BABTJSKOVA

The Dispersion of the Refractive Index and the Birefringence of A1N . K29 H.

NEUMANN

Photosensitive Field Electron Emission from In 2 S 3

K33

B . BARANOWSKI a n d J . SZYMASZEK

The Electrical Resistance Anomaly of Nickel Hydride at Low Temperatures K37 H. P. UTECH

Interpretation of de Grindberg's Experiments on Lateral Surface Fringes in Sodium Chloride K41

Z . BODO a n d I . H E V E S I

Optical Absorption near the Absorption Edge in V 2 0 6 Single Crystals K45 A. A. P.

KAMINSKII

The Nature of "Aging" of CaF 2 -YF 3 -Nd 3 + Crystals (Type Stimulated Emission Conditions

I)

under K51

ASADI

On the Existence of a Spontaneous Polarization in Fluorite Single Crystals Doped with Lime K55

P . ASADI

On the Electrical Conductivity of Fluorite Single Crystals Doped with Uranium Dioxyde K59

S . M I L O S E V I C , a n d L . DOBROSAVLJEVIC

Evaluation of the One-Electron Potential for A1 Crystals H . MARKEBT

. . . .

K63

Decrease of Number and Size of Barkhausen Jumps during Ultrasonic Irradiation K67

Prc-printed Titles and Abstracts of papers to be published in this or in the Soviet journal , ,H3Mi;a TBepnoro T e j i a " (Fizika Tverdogo Tela) A1

Systematic List Subject classification: 1 1.2 1.3 1.4 2 3 3.1 3.2 4 5 6

Corresponding papers begin on the following pages (pages given in italics refer to the principal subject classification): 179 K25 K5, K41 131 261,327 197 365, 371 365 59, 179, 197, 371 K25 37,45, 143, 291, 347, Kll

Contents 7 8 9 10 10.1 10.2 11 12 12.1 13 13.1 13.2 13.4 14 14.1 14.2 14.3 14.3.2 14.4 14.4.1 15 16 17 18 18.2 18.3 19 20 20.1 20.2 20.3 21 21.1 21.1.1 21.3 21.6 22 22.1 22.1.1 22.1.2 22.1.3 22. 2 22.2. 1 22.2. 2 22.2. 3 22.4. 1 22.4.3 22.5.2 22.5.3 22.6 22.8 22.9 23

K67 37, 45, 249, 291, 321 121, 127, 165, 213, K5, K21 69, 165, 285, K55, K59 83, 395 379 331 165 K17 331 9, 113, 295, 361, K45, K63 9, 143, 153 9, 121, 301, 337, K1 K37, K59 277 95 327 255 165 301, 295, 311, K55 361 255, K1 K33 95, K25 267, 395, K9, K67 K67 121, 225, 237, 337 295,331 9, 153, K29, K45 9, K51 9, 311, 379 95, 277, 291, 371, K17, K21 395, K9, K17 197, 395, K25 K17 213, 249, 365 37, 45, 131, 213, 225, 237, K33 9, K9 9, 331, K l , K67 9, 255, K5 9, 213, 261 K29 9 9 9 255, 311 113, 361 121, 337, 379, K l l , K41 127, K55,K59 69, 327, K45 331, K51 301, 347 321

Contents of Volume 20 Continued on Page 415

7

Review

Article

phys. stat. sol. 20, 9 (1967) Subject classification: 20.3; 13.1; 13.2; 13.4; 20.1; 2 0 . 2 ; 22.1; 22.1.1; 22.1.2; 22.1.3; 22.2.1; 22.2.2; 22.2.3 Department of Physics,

University of Ottawa

Band-to-Band Radiative Recombination in Groups IY, YI, and III-Y Semiconductors (II) By Y . P . VARSHNI

Contents compounds 8. Results for 8.1 Gallium phosphide 8.2 Gallium arsenide 8.3 Gallium antimonide 8.4 Indium phosphide 8.5 Indium arsenide 8.6 Indium antimonide 9. Comparative

discussion

References

II)

(Part

8. Results for Compounds 8.1 Gallium

phosphide

8.1.1 Optical

properties

Transmission and reflectivity measurements on GaP in the wavelength region 1 to 40 ¡jim have been carried out by Kleinman and Spitzer [288, 289]. They observed a number of absorptions between 12 and 25 ¡j.m which they interpreted as two-phonon combination bands. Analysis of these combination bands led to the following values of the various phonons: LO = 378, TO = 361, LA = 197, TAj = 115, and TA 2 = 66 (all in cm- 1 ). Cherry [290] has observed a number of weak absorption bands in the range 8 to 12 [xm. These are believed to be due to three-phonon processes. The five characteristic phonon frequencies assigned by Kleinman and Spitzer were used to calculate the expected absorptions due to three phonon processes and the calculated values are in reasonable agreement with the experimentally observed bands. More accurate values of phonon frequencies have been resently obtained by Hobden and Russell [290a, 290b] from measurements of the Raman Spectra. Early absorption measurements on GaP were those of Folbert and Oswald [291, 292] and of Welker [293]. The absorption coefficient of GaP at room temperature in the range 2.1 to 2.8 eV was measured by Spitzer et al. [294]. They interpreted their results in

10

Y . P . VAKSHNI

terms of indirect transitions near the threshold (2.20 eV) followed by direct transitions at higher energies. Absorption measurements at room temperature were also carried out by Loebner and Poor [295]. They found that their results for values of the absorption coefficient, a, between 10 and 150 cm - 1 can be fitted to the MacfarlaneRoberts expression. The energy gap was found to be between 2.20 and 2.21 eV, and the energy of the phonon (k d) assisting in the optical transitions was estimated to be 0.026 eV. But they have not given the value of the constant A in equation (13). The energy gap found by Spitzer et al. and by Loebner and Poor is practically identical. We assumed that equation (13) can be applied to the measurements of Spitzer et al. between 2.3 and 2.4 eV and determined A = 3760 when Eg = 2.205 eV and k 0 = 0.026 eV. Gershenzon, Thomas, and Dietz [296] found that the low-temperature absorption spectra of crystals which are not highly strained contain structure in the region of low absorption coefficient (a < 10 cm - 1 ) near the band edge. A series of humps is clearly discernible when the square root of the absorption coefficient is plotted as a function of photon energy. Each hump is believed to correspond to exciton absorption with the simultaneous emission of a specific phonon. A detailed examination of the absorption spectrum of GaP single crystals at the temperature of liquid nitrogen has been carried out by Gross and coworkers [297 to 301]. They find it consists of a number of absorption steps. The steps have sharp edges, while the edge of the main absorption line is not sharp. Therefore the steps are interpreted as indirect transitions from the valence band to the levels of the free exciton with the participation of phonons (indirect excitons). Gross et al. have also obtained phonon energies from their observations, but these are different from those given by Kleinman and Spitzer [288, 289], More recently, Abagyan and Subashiev [302] have conducted a careful measurement and analysis of the fundamental absorption edge in the region 2 to 2.75 eV. They find that in the region from the edge to » 2.51 eV a l O C (hv - Eglf

(102)

which is characteristic of indirect allowed transitions. For energies greater than 2.55 eV, experimental absorption coefficients are greater than those given by the above relation. These deviations are postulated to be due to electron transitions to the minimum of the higher branch of the conduction band. The values of (a — o^), in the region 2.55 to 2.7 eV, depend on the photon energy in the form (a - a 2 ) oc (h v - E^f (103) which points to forbidden indirect transitions. The energy Eg2 corresponding to the position of the second minimum was found to be 2.51 eV. The (a, E) graph also indicates that the sharp, approximately exponential increase of a begins at energies of 2.7 eV and is presumably due to direct transitions. The effect of hydrostatic pressure on the transmission of GaP in the region 2.2 to 2.7 eV has been measured by Zallen and Paul [303]. Data adopted here: Absorption coefficient:

Results obtained from the absorption coefficients

Band-to-Band Radiative Recombination in Semiconductors (II)

11

Fig. 31. x, P(i')e(v), and U for gallium phosphide at 300 The continuous curves were obtained from the absorption coefficients of [294], the dotted ones from equation (13)

measured by Spitzer et al. [294] are shown by continuous curves in Fig. 31. Results obtained from (13) with A = = 3760, Eg = 2.205 eV, and k 0 = = 0.026eV are shown by dotted curves. Refractive index: n = 2.91 [102]. Intrinsic carrier concentration: Calculated from (48) with the values of me and mh as given in Table 1 (see Part I of this article, phys. stat. sol. 19, 459 (1967)). Energy gap: Eg = 2.205 eV at 300° K . 8.1.2 Discussion

of radiative

transitions

I t would be noticed in Fig. 31 that the P(v) o(v) curves obtained from the data of Spitzer et al. and by equation (13) merge into one at high energies, as they should. But the Spitzer curve does not show a maximum while the dotted curve does. In this region the absorption coefficient is rather small and it is usually difficult to make accurate measurements. Presumably there is some impurity absorption superimposed on the intrinsic absorption in Spitzer's measurement which is masking the expected maximum in the P(v) o(v) curve. We can expect another maximum at about 2.7 eV corresponding to the onset of direct transitions; but its contribution to the value of B would be relatively small. Luminescence resulting from the flow of current through GaP diodes has been studied by a number of investigators [304, 305, 306, 295, 307 to 314], However, the results have not been easy to analyse and the origin of the emitted radiation not unambiguously identified. Recently a series of careful experiments have been carried out by Gershenzon and co-workers [315, 316, 296, 317 to 321] at Bell Telephone Laboratories. They have found that the spectrum observed in electroluminescence and photoluminescence is fairly complicated. Here we shall consider only those results which have a bearing on radiative recombination. Gershenzon and Mikulyak [315, 316] classify the observed luminescence peaks into four groups: band-toband recombination, deep-donor recombination, the "sharp green line", and impurity bands. The situation is complicated by the fact that a "sharp green line" also occurs at about the same energy as the band-to-band recombination radiation. However, it was possible to differentiate between the two by their different characteristics : a) Recombination radiation: A weak peak (of quantum efficiency » 10~ 8 at 1 A/cm 2 at 300 °K) lying just below the band gap at three different temperatures, 80, 200, and 300 °K, was observed in the diffused Zn-Si diodes (seeFig. 32). The peak changes its position with temperature in the direction qualitatively

12

Y . P . VARSHNI Fig. 32. Forward-bias electroluminescence of a zinc-diffused G a P diode prepared from an n-type (silicon-doped) crystal grown in a graphite container. The deep-donor emission and the bandto-band recombination peak are evident. Photoluminescence spectra of uniformly zinc-doped samples and of undoped samples containing the deep donor are shown for comparison. The band gap, Eg, is indicated a t each temperature (Gershenzon and Mikulyak [316])

300°K

1

i! \

i

200°K \

1

i i i i In Deepdonor

1»

1.7

1.8

1.9

20 11 ¿2 Photon energy

2.3 2.4 (eV)—-

| | Fig. 33. Electroluminescence of zinc-diffused GaP diode prepared on n-type (sulphur-doped) floating zone crystal (Gershenzon and Mikulyak [316])

1 16

1.7

1.8

1.9

|

| £11 \ | | \ 1i

2.0 21 22 2.3 Photon energy fe/J

300°K

W0°K

80°K

2.4

expected. Also its efficiency increases with temperature as is frequently the ease with radiative recombination. b) "The sharp green line": A sharp green line is observed at 2.18 eV (see Fig. 33). Its nature was different from all the other bands observed by Gershenzon and Mikulyak. It is far narrower than any other peak in electroluminescence or photoluminescence; its width is roughly two to four times kT in contrast with « 0.2 eV widths of other bands. The band center does not shift with temperature from 20 to 500 ° K . The peak is not observed at all in the diffused Zn-Si diodes prepared directly from crystals grown in graphite boats. In fact, this is the reason that the band-to-band recombination could be observed in these diodes. Since this line is narrow and it does not shift in frequency with temperature, and its intensity does not decrease abruptly as the band gap approaches its photon energy, the emission probably arises from a fairly localized level which is well shielded from the host lattice. The efficiency of recombination radiation in GaP has been investigated by Grimmeiss and Scholz [322J. Recombination radiation emitted from point-contact GaP diodes under conditions of reverse bias has been observed by Gorton et al. [323]. The energy of the radiation was observed to be between 1.96 and 2.19 eV, with a peak density at 2.12 eV. The maximum energy 2.19 eV corresponds closely to the minimum energy gap between the conduction and valence bands.

Band-to-Band Radiative Recombination in Semiconductors (II)

8.2 Gallium

13

arsenide

Gallium arsenide is a semiconductor w i t h m a n y i n t e r e s t i n g p r o p e r t i e s a n d it h a s been s t u d i e d q u i t e extensively since t h e I I I - V c o m p o u n d s were f i r s t described b y W e l k e r [324]. A recent article b y Oliver [325] gives a brief a c c o u n t of t h e p r e p a r a t i o n , physical properties, a n d t h e m o s t i m p o r t a n t applications of gallium arsenide. A m o r e complete review h a s been given b y H i l s u m [326]. 8.2.1 Optical

properties

T h e reflection s p e c t r u m of GaAs has been well i n v e s t i g a t e d . M e a s u r e m e n t s in t h e i n f r a - r e d h a v e been m a d e b y Picus et al. [327] a n d b y H a s s a n d H e n v i s [106] (liquid H e t e m p e r a t u r e ) . Piriou a n d C a b a n n e s [328] h a v e a p p r o x i m a t e d t h e e x p e r i m e n t a l reflection f a c t o r curve b y f i t t i n g t h e p a r a m e t e r s in t h e L o r e n t z t h e o r y e q u a t i o n s . O b s e r v a t i o n s between 1.7 t o 5.1 eV h a v e been p r e s e n t e d b y T a u c a n d A b r a h a m [209]. Morrison [329] h a s m e a s u r e d t h e r e f l e c t i v i t y a t n e a r l y n o r m a l incidence f r o m 2050 A t o 15 ¡xm a t r o o m t e m p e r a t u r e , a n d h a s c o m p u t e d n a n d 1c f r o m t h e s e d a t a in t h e r a n g e 0 t o 6.0 eV using t h e dispersion r e l a t i o n between t h e p h a s e a n d t h e m a g n i t u d e of t h e reflectivity. R e f l e c t a n c e d a t a a t higher energies (1.5 t o 25 eV) h a v e been r e p o r t e d b y Phillipp a n d E h r e n r e i c h [114]. Using t h i s d a t a , D a v e y a n d P a n k e y [330] h a v e calculated values of n a n d a b s o r p t i o n i n d e x b y dispersion analysis. E a r l y m e a s u r e m e n t s of a b s o r p t i o n in t h e n e i g h b o u r h o o d of t h e edge were m a d e b y Oswald a n d Schade [331] b u t t h e y r e a c h e d a n a b s o r p t i o n level of only 100 c m - 1 . R e c e n t l y Moss a n d H a w k i n s [332] h a v e carried o u t m e a s u r e m e n t s of a b s o r p t i o n coefficient in t h e r a n g e 1.3 t o 1.6 eV on single c r y s t a l GaAs. T h e y h a v e also c o m p a r e d t h e i r results w i t h t h o s e calculated using K a n e ' s t h e o r y [333]. I n a succeeding publication, Moss [334] has also i n v e s t i g a t e d t h e s h i f t of t h e optical edge b y a p p l y i n g a n electric field. S t u r g e [335, 336] h a s m e a s u r e d t h e a b s o r p t i o n coefficient over t h e r a n g e of p h o t o n e n e r g y 0.6 t o 2.75 eV, a t t e m p e r a t u r e s f r o m 10 t o 294 °K. T h e m a i n a b s o r p t i o n edge was f o u n d t o show a s h a r p p e a k d u e t o t h e f o r m a t i o n of excitons. S t u r g e f i n d s t h e b a n d g a p t o be somewhat larger t h a n t h a t f o u n d b y p r e v i o u s workers. A n o t h e r r e c e n t i n v e s t i g a t i o n on t h e a b s o r p t i o n coefficient of GaAs is t h a t of Hill [337] who h a s studied t h e b e h a v i o u r of t h e a b s o r p t i o n coefficient, in t h e region of t h e f u n d a m e n t a l a b s o r p t i o n edge, w i t h v a r i o u s t y p e s a n d levels of d o p i n g . T u r n e r a n d Reese [338] h a v e d e t e r m i n e d optical a b s o r p t i o n coefficients a t 300 a n d 77 °K for samples of GaAs t h a t were doped t o c o n c e n t r a t i o n s c o m p a r a b l e t o t h o s e present in t h e n, p a n d p + regions of t h e GaAs i n j e c t i o n laser. T h e r e h a v e also been a n u m b e r of i n v e s t i g a t i o n s on t h e a b s o r p t i o n s p e c t r a in t h e i n f r a - r e d region. Absolute a b s o r p t i o n coefficients in t h e r a n g e of rests t r a h l e n b a n d (30 t o 40 |i.m) h a v e been m e a s u r e d b y F r a y et al. [339]. T h e i n f r a - r e d a b s o r p t i o n d u e t o free holes in p - t y p e G a A s exhibits t h r e e a b s o r p t i o n b a n d s on t h e low-energy side of t h e intrinsic a b s o r p t i o n edge [340, 341]; n - t y p e gallium arsenide does n o t show t h i s c h a r a c t e r i s t i c s t r u c t u r e , b u t i n s t e a d exhibits a m o n o t o n i c a l l y increasing a b s o r p t i o n w i t h w a v e l e n g t h . T h e s a m e s t r u c t u r e is f o u n d in all p - t y p e samples a n d c o n s e q u e n t l y t h e evidence is t h a t these b a n d s are n o t due t o t h e presence of u n k n o w n i m p u r i t i e s or lattice

14

Y. P. Vakshni

defects. Braunstein and Magid [340] interpret this spectrum in terms of hole transitions between various branches of the valence band of gallium arsenide. Spitzer and Whelan [342] have measured the infra-red absorption between 0.85 and 25 ¡i,m as a function of carrier absorption for n-type single-crystal GaAs. Infra-red absorption for p-type degenerate GaAs at room temperature for various hole concentrations has been studied by Kudman and Seidel [343]. The temperature dependence of the refractive index between 5 and 20 [xm for GaAs has been determined by Cardona [344] by measuring the interference patterns of thin single-crystal films at several temperatures. His results show that the change is only about 0.45% for a 100° change of temperature. As the accuracy in reading the absorption coefficients from the curves is much less than this, we have not taken into account the variation of n with temperature. The refractive index of GaAs has been measured by Marple [345] by the prism refraction method for photon energies from 0.7 eV to the band gap at three temperatures. The temperature and pressure dependence of the index of refraction near the absorption edge has been estimated theoretically by Stern [346] and agrees well with experimental results. Optical properties of thin films have been investigated by a number of workers [347, 348, 115, 330]. Data adopted here: Absorption coefficient: The curves given by Sturge [336] at 21, 90, 185, and 294 °K were read. Refractive index: n = 3.63 (Stern [346]). Intrinsic carrier concentration: Calculated from (48) using data given in Table 1 (see Part I, phys. stat. sol. 19, 459 (1967)). Energy gap: Sturge [336], The results at 21, 90, 185, and 294 °K are shown in Fig. 34 to 37 and in Table 2 (see Part I, phys. stat. sol. 19, 459 (1967)). 8.2.2 Discussion of radiative

transitions

Braunstein [349] observed a radiation from GaAs diodes which had a peak at 1.10 eV at room temperature and at 1.19 eV at 77 °K. The origin of this radiation was not clear and Braunstein pointed out the difficulty in attributing this radiation to the recombination of holes and electrons as the energy gap is considerably larger. The recombination radiation in GaAs has been observed by Nathan and Burns [350]. Optical injection of carriers was used. A typical spectrum observed by them at 4.2 °K is reproduced in Fig. 38. Four lines are observed. The most intense line Bx occurs at a photon energy of 1.4919 + 0.0005 eV (22.4 meV below line A). Line B 2 is at a photon energy 36.4 + 0.5 meV below line B r This energy separation is very close to the value of the LO phonon energy obtained from reststrahlen band measurements [339] (36.4meV) and from tunnel diode measurements [351] (34.9 meV). The energy separation between B 2 and B 3 is 37 + 2 meV and line B 3 appears to arise with the emission of two LO phonons. As the sample temperature is increased above 4.2 °K the intensity pattern of Fig. 38 changes rapidly with decrease in the intensity of the series B, and by 40 °K they are too weak to observe, only line A remains. Fig. 39 shows line A at 77 °K of a typical crystal grown in an 0 2 atmosphere. The line occurs at a photon energy of 1.5143 + 0.0005. (The uncertainty is

Band-to-Band Radiative Recombination in Semiconductors (II)

15

Me/; Fig. 36. x, P(v)

and V for gallium arsenide at 185 °K

Fig. 37. y, I'(i') o(v), and U for gallium arsenide at 294 °K

16

Y . P . VARSHNI Fig. 3$. Emission spectrum of a gallium arsenide crystal at T = 4.2 °K. The scale is linear in wavelength (Nathan and Burns [350])

m

m

m w Photon energy

Fig. 39. Recombination radiation in gallium arsenide at 77 °K. The solid curve represents the observations of Nathan and Burns [350], the dashed curve was calculated as explained in the text

isz (eV)—-

152 Photon energy (eVI -

due to a small variation of the position of the line from sample to sample.) The dashed curve is the calculated one and was obtained as follows: P(v) o(v) was obtained from Sturge's [336] data at 90 °K and this was shifted to the higher energy side by 0.0025 eV corresponding to the difference in the energy gaps at 90 and 77 °K. The maxima of the two curves are seen to differ by a small amount ( « 0.0007 eV) which is of the same order as the experimental uncertainty, and it would be reasonable to disregard this small difference. But there is a marked difference in the shapes of the two curves. Even if the two maxima are made to coincide, the experimental curve is much steeper than the theoretical one. Recently Sarace et al. [352] have measured the spontaneous injection luminescence of forward biased GaAs p-n junctions with special emphasis on the reduction of spectral distortion. Principal sources of distortion are self-absorption and internal reflection within the GaAs. The radiation observed by them together with the calculated one is shown in Fig. 40. The agreement is seen to be satisfactory. Recombination radiation of GaAs on excitation with fast electrons has been observed by 1M

H9

150 151 152 153 1M Photon energyleVI

155

Fig. 40. Comparison of the calculated and observed radiation from gallium arsenide at 77 °K by Saraee et al. (352]

Band-to-Band Radiative Recombination in Semiconductors (II)

17

OaAsff'K

S 'SO

I

1.3

H

hviefh Fig. 41a. Recombination radiation from a gallium arsenide p-n junction at 77 °K observed by Nasledov et al. [353]

1.1 12 13 Photon energy (eVlFig. 41b. Recombination radiation from a gallium arsenide p - n junction at 78 °K observed by Pankove and Massoulie [354]

1.1 12 13 Photon energy leV)

14

Fig. 41c. Recombination radiation from a gallium arsenide p-n junction at 77 JK observed by Keyes and Quist [356]

Basov and Bogdankevich [96], n-type GaAs was irradiated with a beam of electrons of approximately 0.6 MeV energy, obtained from a linear accelerator. In recent years the electroluminescence and photoluminescence spectrum of GaAs has been studied by a number of workers. I n the following we discuss the important features in the spectrum and their possible origin. Nasledov et al. [353] produced excess carriers by pulse injection through a p - n junction. The spectrum observed by them at 77 °K is shown in Fig. 41 a. 2

physica/20/1

18

Y. P. Vabshni

The spectrum has three maxima, one at 1.47 eV and two others at smaller energies (as 1 eV and » 1 . 3 eV). Pankove and Massoulie [354, 355] studied the injection luminescence from a forward biased p-n junction at three temperatures (4.2, 78, and 300 °K). Their spectrum at 77 ° K is shown in Fig. 41 b. Observations on an appropriately diffused GaAs diode biased in the forward direction made by Keyes and Quist [356] are reproduced in Fig. 41 c. From absolute measurements of the emitted radiation intensity, these investigators also found that at 77 ° K these diodes may be as high as 85% efficient in the conversion of injected holes into photons of the gap energy. It would be noticed from Fig. 41a, b, and c that three maxima occur in all of them but their relative intensities are very much different in the three cases. This difference shows that the relative probability of the processes giving rise to the three maxima is a sensitive function of the conditions of the experiment. Similar results have been obtained by other workers [357, 358, 359]. At first the 1.47 eV emission was thought to be the intrinsic recombination radiation of GaAs [353], but later evidence appears to indicate that it is not so. We may note that the radiation emitted at or near the band gap, as reported by different workers, is 1.44 to 1.50 eV at liquid nitrogen temperature. Assuming that these differences are not due to measurement error, they could be explained [360] in several ways: a) The energy gap varies slightly from diode to diode because of crystal strains caused by impurities, dislocations, or stresses produced by junction formation. b) Certain impurities in the crystals produce states near the band edges which tend to smear the band edges and reduce the gap. c) Exciton formation may be favoured in some crystals but not in others. The above processes may be important in explaining some of the differences in the energy and line width of the radiation observed by different workers. In 1962 stimulated emission from forward biased GaAs diodes was observed for the 1.47 eV line at 77 ° K by Hall etal. [63],Nathan et al. [64], andQuist et al. [361]. Investigations, both experimental and theoretical, have been made to determine the transition which gives rise to the 1.47 eV (77 °K) emission. This strong line occurs at an energy slightly less than the energy gap in pure material (Eg = 1.51 eV at 77 °K). Theoretical investigations of Lasher and Stern [11] show that the peak of the stimulated radiation would fall at a lower photon energy than does the peak of the spontaneous radiation, except when T = 0 ° K . Nasledov et al. [362] have carried out experiments on GaAs photocells with a thin diffused p-region (the thickness of the p-region was of the order of a few ¡xm). The excess carriers were produced by electrical injection and the radiation was observed in a direction perpendicular to the plane of the p-n junction from the p-type side. The carrier recombination occurred in a region whose width was of the order of a diffusion length which was approximately equal to 1 ¡zm, Fig. 42 shows the spectrum observed by them at three temperatures. At room temperature the recombination spectrum has two maxima; these will be called the long-wavelength and short-wavelength maxima. Only the long-wavelength maximum ( ^ 1 . 4 6 e V ) is resolved in the 77 ° K spectrum. Nasledov et al. [362, 363] believe that the radiation in the long-wavelength peak is due to transitions involving impurity centers whereas that in the short-wavelength peak is associated with direct optical transitions.

19

Band-to-Band Radiative Recombination in Semiconductors (II) Fig. 42. Spectral distribution of intrinsic recombination radiation emitted by gallium arsenide a t various temperatures. The intensity of the long-wavelength peak was assumed to be 100 relative units. The dashed curve is the spectrum taken from the n-type side (Nasledov et al. [362))

H5 Me/JMcWhorter, Zeiger, and Lax [364] believe that the emitted radiation is not due to band-to-band transitions, but probably involves transitions from donor states just below the conduction band to acceptor states just above the valence band. This is consistent with the wavelength of the emission, which is longer than that of the gap at low temperatures, and the absence of an observable shift of the spontaneous emission line with magnetic field up to 90 kG. Nathan et al. [365, 357, 358] have observed the electroluminescence and photoluminescence of GaAs at 77 °K under controlled doping conditions. The evidence from these experiments strongly indicates that the emission seen in the diodes is due to recombination involving an acceptor center. Some of the available evidence has been reviewed by Nathan [357] from whom we quote : "Now if we ask what is the detailed mechanism of the transition the answer is very difficult to give. One of the problems is the uncertainty as to the correct way to describe the states involved in the transition. However, we can say a few things. At low concentrations « 10 17 c m - 3 ) band-to-band recombination is undoubtedly excluded because the energy of the transition is well below the band gap. The transition is probably between the conduction band modified by the impurities and an acceptor center. For the higher concentrations ( > 10 18 c m - 3 ) the impurity bands are merged with the main bands, and it is probably most correct to speak of band to band recombination. At higher concentrations the difficulty in describing the states involved is most important. The conduction and valence band scates cannot be the same as in a perfect crystal. The presence of the impurities mixes states of different k, so that the k selection rule for direct transitions does not hold. The kind of band to band transitions that occur are thus different from those ordinarily encountered. In addition the impurities affect the density of states so that the bands are not parabolic near the edge. The fact that the impurities modify the energy band edges has been demonstrated from measurements of the electro-luminescent edge emission as a function of injection current. The photon energy of the peak of the emission shifts toward higher energy with increasing current [366]. The magnitude of the shift increases with increasing impurity density in the substrate [367]. Rather than actually being a shift, what is actually occurring, is a saturation of the low-energy side of the line and an increased raté of growth of the high-energy side of the line [367, 368] as shown in Fig. 43. These effects are consistent with injection into states tailing from the band edge." 2'

20

Y . P . VAKSHNI

-—Photon energyteVl 1.48 1.46 144

8400

—

142

—I

Fig. 43. Detail of the spectrum of the edge emission of a •¡^q diode showing peak shift and saturation of the low-energy side of the line (from [357])

i i x 8600 8800 Waveiength(A)

B r a u n s t e i n , P a n k o v e , a n d Nelson [369] h a v e shown t h a t t h e r e c o m b i n a t i o n s p e c t r u m d e p e n d s also on t h e d o n o r c o n c e n t r a t i o n , a n d t h a t t h i s d e p e n d e n c e m i g h t be a t t r i b u t e d t o a c o n c e n t r a t i o n d e p e n d e n t shrinkage of t h e e n e r g y g a p . L u c o v s k y a n d R e p p e r [370] h a v e s t u d i e d t h e c h a r a c t e r i s t i c s of t h e s p o n t a n e o u s r a d i a t i o n as a f u n c t i o n of d o p i n g d e n s i t y in order t o i d e n t i f y t h e e n e r g y levels associated w i t h t h e r a d i a t i v e t r a n s i t i o n s . Laser action in GaAs excited b y a b e a m of 50 k e V electrons a t liquid helium t e m p e r a t u r e h a s been r e p o r t e d b y H u r w i t z a n d K e y e s [371]. B a g a e v et al. [372] h a v e s t u d i e d t h e s p o n t a n e o u s a n d induced emission s p e c t r a associated w i t h t h e i n j e c t i o n of c a r r i e r s t h r o u g h p - n j u n c t i o n s p r e p a r e d b y d i f f u s i n g zinc i n t o gallium arsenide which h a d a t e l l u r i u m c o n c e n t r a t i o n r a n g i n g f r o m 1 X 10 17 t o 2 x 10 18 c m - 3 . T h e y f i n d t h a t t h e i r r e s u l t s are satisf a c t o r i l y e x p l a i n e d b y r a d i a t i v e t r a n s i t i o n s of electrons f r o m t h e c o n d u c t i o n b a n d which h a s m e r g e d w i t h t h e d o n o r levels t o a n i m p u r i t y b a n d f o r m e d b y zinc. T h e r e c o m b i n a t i o n r a d i a t i o n u n d e r conditions of low i n j e c t i o n i n t o a p - n j u n c t i o n h a s also been i n v e s t i g a t e d b y B a g a e v et al. [373]. I t was f o u n d t h a t for a n i m p u r i t y c o n c e n t r a t i o n of a b o u t 10 18 c m - 3 t h e m a x i m u m of t h e recombin a t i o n r a d i a t i o n s h i f t s t o w a r d lower energies w h e n t h e c u r r e n t is decreased. Y u n o v i c h et al. h a v e s t u d i e d r e c o m b i n a t i o n r a d i a t i o n in gallium arsenide p - n j u n c t i o n s f o r m e d b y beryllium d i f f u s i o n [374] a n d b y zinc d i f f u s i o n [375]. Cusano [376] h a s d e t e r m i n e d t h e salient f e a t u r e s of r a d i a t i v e r e c o m b i n a t i o n near t h e b a n d edge in GaAs b y electron b e a m e x c i t a t i o n of d o n o r a n d a c c e p t o r doped c r y s t a l s r a n g i n g in i m p u r i t y c o n t e n t f r o m 10 16 t o 10 20 c m - 3 . T h e laser a c t i o n in GaAs is i n f e r r e d f r o m t h e s p e c t r a l n a r r o w i n g of t h e light e m i t t e d f r o m p - n j u n c t i o n s w i t h i n c r e a s i n g applied c u r r e n t in t h e f o r w a r d direction a t 77 ° K . I t is r e a s o n a b l e t o ask w h a t role excitons p l a y i n t h e recomb i n a t i o n process. Casella [377] h a s discussed t h i s q u e s t i o n . H e f i n d s t h a t excitons (either f r e e or b o u n d t o impurities) do n o t p l a y a role in r e c o m b i n a t i o n process a t h i g h - c u r r e n t levels. 8.3 Gallium 8.3.1 Optical

antimonide properties

T h e r e f l e c t i o n s p e c t r u m of G a S b h a s b e e n i n v e s t i g a t e d b y a n u m b e r of workers. I n t h e i n f r a - r e d , studies h a v e b e e n m a d e b y several workers [327, 106, 344]. Cardona [378] h a s m a d e m e a s u r e m e n t s in t h e e n e r g y r a n g e 0.5 t o 5 eV a n d T a u c

Band-to-Band Radiative Recombination in Semiconductors (II)

21

and Abraham [209] in the energy range 1.6 to 5 eV. Fine structure in the reflection spectrum in the energy range 3 to 5.5 eV has been observed by Lukes and Schmidt [211]. The first measurements on absorption coefficients were those of Welker [324] and of Oswald and Schade [331], who reported results up to about 125 c m - 1 . Roberts and Quarrington [379] carried out measurements at a number of temperatures (20, 77, 195, 249, and 289 °K). Their data goes up to ), and U for gallium antimonide at 80 °K

Fig. 45. x, P(v) q(v), and 17 for gallium antimonide at 300 °K

22

Y . P . VABSHNI

8.3.2 Discussion of radiative transitions I t would be noticed from Fig. 44 and 45 that, at both temperatures, the P(v) Q(V) curve does not fall continuously for energies below the energy gap. Indeed at 80 °K, below 0.79 eV it shows a marked increase with decrease in energy. This appears to indicate that the absorption data includes some contribution due to other causes. An early attempt to observe the recombination radiation in GaSb at room temperature was made by Braunstein [349] who found a peak at 0.625 eV. The peak in Fig. 45 occurs at » 0.73 eV and we conclude t h a t the radiation observed by Braunstein was not due to intrinsic recombination. Calawa [384] has recently observed the radiation emitted from forwardbiased GaSb p - n junctions: At 77 °K the radiation from the diode consists of a single peak at 0.775 eV while at 298 °K it consists of a single but broader peak at 0.679 e'V. At both temperatures the emission occurs at an energy below the optical energy gap of GaSb, which is 0.725 eV at 298 °K and 0.80 eV at 77 °K. Calawa found no significant line shift in both absorption measurements and measurements in magnetic fields up to 90000 G; further the radiation spectrum was found to change by the addition of a different impurity. All this evidence indicates t h a t the observed radiation is not produced by band-to-band transitions but is probably connected with impurities. Experiments on several GaSb diodes at high-current densities have been carried out by Deutsch, Ellis, and Warschauer [385]. They find t h a t at 77 °K there are two emission lines, at about 0.72 eV and 0.78 eV, either or both of which may occur in the same diode. They have presented a model to explain the observed effects in terms of the band structure of GaSb. Recently Benoit a la Guillaume and Debever [386] have observed laser action in GaSb excited by electron bombardment. 8.4 Indium

phosphide

8.4.1 Optical properties The optical properties of indium phosphide were first investigated by Oswald and Schade [331] and Oswald [292, 293]. Reflectance studies in the infra-red region have been made by several workers [387, 327, 106], Cardona [388] has measured the room-temperature reflectivity of a polished n-type sample before and after etching. The spectral emittance of indium phosphide has been measured by Stierwalt and Potter [389] in the 3 to 44 |im region at several temperatures from 77 to 473 °K. Thirteen features were observed which were attributed to two phonon processes. Using the allowed dipole-transition selection rules, values were derived for the phonon energies at the critical points. Newman [387] has reported absorption coefficient in the vicinity of the absorption edge of n-type I n P at 77 and 300 °K. Differences were found in the spectra of samples of differing origin. The effects are believed to be due to impurities. Recently Turner, Reese, and Pettit [390] have presented very accurate optical-absorption data obtained on undoped samples of n-type I n P in the region of the intrinsic edge at 6, 20, 77, and 298 °K. From this data they were able to derive the exciton energy, exciton binding energy, and intrinsic optical gap.

Band-to-Band Radiative Recombination in Semiconductors (II)

23

U2 IU hvleVI Fig. 46. x, P(v) Q(V), and U for indium phosphide at 77.4 °K

Fig. 47. x, P(v) q(v), and V for indium phosphide at 208 °K

The temperature dependence of the refractive index has been investigated by Cardona [344], Data adopted here: Absorption coefficient: At 77.4 and 298 °K from Fig. 1 and 2 of Turner, Reese, and Pettit [390], Refractive index: n = 3.37 [391]. Intrinsic carrier concentration: Calculated from equation (48) using data given in Table 1 (see Part I, phys. stat. sol. 19, 459 (1967)). Energy gap: Eg77 = 1.4135 eV, E% 2 9 8 oK = 1.3511 eV [390]. The results are shown in Fig. 46 and 47 and Table 2 (see Part I , phys. stat. sol. 19, 459 (1967)). 8.4.2 Discussion of radiative transitions An attempt to observe the recombination radiation in I n P was made by Braunstein [349]. Recently Turner and co-workers [392, 390] have observed the photo-induced recombination radiation in InP. Photo-excitation of hole and electron pairs was accomplished by light of photon energies greater than 2 eV. The emitted radiation was observed from that side of the sample which received the exciting radiation. Measurements were made at 2.2, 6, and 77 °K. Their results at 77 °K are shown in Fig. 48. Line 1 is believed to be due to free exciton (and possibly some free electron-hole) recombination. The observed energy of the peak 1.409 eV is in good agreement with that expected from absorption data calculations (Fig. 47), 1.4075 eV. The small difference of 0.0015 eV is almost equal to the accuracy of measurement of the exciton peak energy, given as + 0 . 0 0 1 3 by Turner et al. [390]. Lines 2 and 3 result from recombination of a hole and electron at a shallow impurity with the emission of 0 and 1 optical phonons. The spacing of these

24

Y . P . VARSHNI Fig. 48. Photoluminescence of I n P at 6 °K observed by Turner and P e t t i t [392]. Net electron concentration for sample shown is 3 x 10 1 4 c m " 3

lines is 0.043 eV which is exactly the LO phonon energy reported by Newman [387] from reststrahlen data. Stimulated light emission from forwardbiased I n P diodes has been observed by Weiser and Levitt [393]. 8.5 Indium

8.5.1 Optical

arsenide

properties

Reflectivity measurements on InAs have been made by several workers: 0.70 0.80 0.90 1.00 1.10 1.20 1.30 Spitzer and Fan [394], Tauc and AbraPhoton energy!eV}— ham [209] (1.7 to 5.2 eV), Lukes and Schmidt [112] (2.3 to 2.9 eV), Greenaway and Cardona [395], and Philipp and Ehrenreich [114] (1.5 to 25 eV). Morrison [329] has measured the reflectivity of InAs at nearly normal incidence from 2050 A to 15 [Am at room temperature. The optical constants were computed from these data in the range 0 to 6 eV using the dispersion relation. Studies on the infra-red reflection band have been made by Picus et al. [327] and by Hass and Henvis [106]. The observed spectrum was interpreted in terms of a single classical dispersion oscillator and values of the transverse and longitudinal optical frequencies for long-wavelength vibrations were obtained. Absorption studies in the fundamental absorption-edge region were made by Hrostowski and Tanenbaum [396], Oswald [292, 397], Stern and Talley [398, 399], Spitzer and Fan [394], Matossi and Stern [400, 401], and Nasledov et al. [402], However, significant discrepancies exist in these results. For example, the values reported for the width of the forbidden energy gap at room temperature range from 0.31 to 0.36 eV. Recently Dixon and Ellis [403, 404] have measured the absorption coefficient of InAs in the fundamental absorption-edge region as a function of the impurity content. Their experiments were carried out on single crystal material of higher purity than used by most of the previous workers. Trie et al. [405] have observed the transmission spectrum of a thin specimen of InAs at liquid hydrogen temperatures using monochromatic light modulated at 250 Hz. The simultaneous irradiation by light modulated at 175 Hz caused a slight reduction in the absorption coefficient in the region of 3.1 [i.m; this was attributed to laser action. Recently Melngailis [406] has observed laser action in InAs diodes. Data adopted here: Adsorption coefficient: a) Room temperature data from Dixon and Ellis [404]. b) For absorption coefficient at 77 °K it was assumed that the shape of the (a, E) curve remains the same as that at room temperature but it is shifted by an amount equal to the change in the energy gap. This change is 0.061 eV [404]. Refractive index: n = 3.54 (Morrison [329]).

Band-to-Band Radiative Recombination in Semiconductors (II)

MeV!

at

25

" °K

Intrinsic carrier concentration: Calculated from equation (48) using data given in Table 1 (see Part I, phys. stat. sol. 19, 459 (1967)). The results are shown in Fig. 49 and 50 and Table 2 (see also Part I). 8.5.2 Discussion of radiative transitions Forward-biased InAs diodes emit an intense line of radiation at an energy slightly below the optical band-gap energy. Recently Melngailis [407] has obtained detailed results on the spontaneous emission of various InAs diodes in an effort to identify the transitions involved.

Photon energy (eV!—-

Fig. 5 1 b . Comparison of the temperature dependence of the band gap and of the peak energy of diode emission of Fig. 5 1 a

2W 280

T(°K) :—-

26

Y . P . VABSHKI

Fig. 51a shows the spectrum [407] at 2, 77, and 300 °K for a diode fabricated by diffusing Zn into an n-type base with a donor concentration of 2 x 10 16 c m - 3 . The intensity scale is different for each temperature. The emission line at 300 °K occurs at 335 meV or 3.70 ¡xm and has a halfwidth of 25 meV. At 2 °K the line has shifted to 400 meV and narrowed to 12 meV. The photon energy of this line agrees closely with the energy of a line obtained in optical excitation experiments with a similar material [405, 408, 409]. Fig. 51b compares the temperature variation of the photon energy at the diode emission peak with the temperature variation of the optical band gap published by Dixon and Ellis [404] for n-type InAs of comparable purity. On the basis of two points, the diode emission line follows closely the band gap variation between 298 and 77 °K at an energy about 20 meV below the gap energy. At lower temperatures the shift of the diode emission is less than the change in the gap. A shift of the spontaneous emission spectrum to higher energies has been observed in InAs diodes in magnetic fields up to 90 kG [410]. For the diode of Fig. 51 a the energy of the peak shifts linearly with the magnetic field with a slope that is in approximate agreement with the slope expected from the lowest Landau level of the conduction band. This points to conduction band states as the origin of the transitions. The terminal states then should be acceptor levels 20 meV above the valence band, in agreement with Trie et al. [405]. This is further supported by a linear variation of radiation intensity with current. Spectrum of the same diode at 2 °K with details of the low-energy tail is shown in Fig. 52. An additional peak occurs at 372 meV and a knee is observed at 340 meV. The first satellite 28 meV below the main peak can be explained by the recombination of an electron with the simultaneous emission of an optical phonon and a photon. The energy difference of 28 meV agrees within experimental error with the energy of 29.5 + 0.5 meV measured for a LO phonon in InAs tunnel diodes by Hall et al. [411]. Picus et al. [327] have determined a value of 28.8 meV for the phonon from optical reflectivity data. The second weaker satellite 60 meV below the main peak corresponds to the energy of two phonons, also observed in tunnel diodes at 59 + 0.5 meV. The emission of these phonons can be expected in InAs, since it is a strongly polar compound and has a strong coupling of electrons to the lattice. Benoit a la Guillaume and Debever [412] have observed laser action in InAs by electron bombardment. T-2°K

-

»/

'30

0.32

i

i

0.3b

i 0.36

i

i

i

0.38 Photon energy lev I

i

010 — -

F i g . 52.

R a d i a t i o n f r o m indium arsenide a t 2 c Jv (Melngailis [ 4 0 7 J )

Band-to-Band Radiative Recombination in Semiconductors (II)

8.6 Indium

27

antimonide

A m o n g all t h e I I I - V c o m p o u n d s , i n d i u m a n t i m o n i d e h a s p r o b a b l y received the most attention. T h e p r o p e r t i e s of I n S b h a v e been reviewed b y Moss [413] a n d H u l m e a n d Mullin [414], T h e f o r m e r review deals p r i m a r i l y w i t h t h e f u n d a m e n t a l p r o p e r ties of I n S b while t h e l a t t e r one c o n c e n t r a t e s on m e t a l l u r g y , d i f f u s i o n , dislocations, a n d device applications. 8.6.1 Optical

properties

T h e lattice a b s o r p t i o n b a n d s of I n S b were i n v e s t i g a t e d b y Spitzer a n d F a n [415] in t h e w a v e l e n g t h r a n g e 5 t o 115 |im. T h e y f o u n d f o u r b a n d s a t 28.2, 30, 52, a n d 95 ¡Am respectively. Y o s h i n a g a a n d O e t j e n [416] m e a s u r e d t h e reflectivity a n d t r a n s m i s s i o n of I n S b crystals (n-type) b e t w e e n 20 a n d 200 [Am. A series of detailed m e a s u r e m e n t s of t h e a b s o r p t i o n s p e c t r u m of h i g h - q u a l i t y single crystals of n - t y p e I n S b has been carried o u t b y F r a y , J o h n s o n , a n d J o n e s [417, 418] in t h e w a v e l e n g t h r a n g e 15 t o 130 ¡xm a n d over t h e t e m p e r a t u r e r a n g e 4.2 t o 90 ° K . T h e positions a n d t e m p e r a t u r e d e p e n d e n c e of all t h e p r i n c i p a l a b s o r p t i o n p e a k s could be i n t e r p r e t e d in t e r m s of multiple p h o n o n i n t e r a c t i o n s involving p h o n o n s w i t h energies of 180, 156, 118, a n d 43 c m - 1 . T h e f u n d a m e n t a l lattice reflection b a n d of I n S b has been m e a s u r e d a t liquid helium t e m p e r a t u r e b y H a s s a n d H e n v i s [106], B y c o m p a r i n g t h e o b s e r v e d s p e c t r u m w i t h t h a t calculated using a single classical dispersion oscillator, t h e following optical p h o n o n frequencies were o b t a i n e d : Vtransv

= 184.7 + 3 c m " 1 ,

»>iong = 197.2 + 2 c m - 1 .

O t h e r r e f l e c t i v i t y m e a s u r e m e n t s h a v e been m a d e b y t h e following w o r k e r s : T a u c a n d A b r a h a m [209] (2 t o 5 eV), Morrison [329] (2050 A t o 15 ¡xm a t r o o m t e m p e r a t u r e ) , G r e e n a w a y a n d Cardona [395] (1 t o 5.5 eV), P h i l i p p a n d E h r e n reich [114] (1.5 t o 2 5 eV), K u r d i a n i [419] (0.22 t o 17 [Am), S a n d e r s o n [420] (20 t o 200 cm" 1 ). Moss, S m i t h , a n d H a w k i n s [421] h a v e used t r a n s m i s s i o n m e a s u r e m e n t s t o d e t e r m i n e a b s o r p t i o n coefficients over t h e w a v e l e n g t h r a n g e 1.5 t o 7.5 ¡Am a n d m a d e i n t e r f e r o m e t r i c d e t e r m i n a t i o n s of t h e r e f r a c t i v e i n d e x f r o m 7 t o 20 ¡Am. A b s o r p t i o n coefficients in t h e vicinity of t h e a b s o r p t i o n edge in p u r e a n d n - t y p e d e g e n e r a t e samples a t r o o m t e m p e r a t u r e a n d liquid n i t r o g e n t e m p e r a t u r e h a v e been m e a s u r e d b y Gobeli a n d F a n [422]. T h e v a r i a t i o n of t h e r e f r a c t i v e i n d e x w i t h t h e t e m p e r a t u r e h a s been s t u d i e d b y Cardona [344]. Kessler a n d S u t t e r [423] h a v e r e p o r t e d m e a s u r e m e n t s of t h e free-carrier a b s o r p t i o n . T h e effect of d o p i n g on t h e reflection s p e c t r u m of I n S b in t h e e n e r g y region 0.35 t o 1.6 eV has been i n v e s t i g a t e d b y K u r d i a n i et al. [424]. Data adopted here: Absorption coefficient: 77 ° K a f t e r Gobeli a n d F a n [422], 295 °K a f t e r Moss et al. [421], Refractive index: n = 4.01 ( e x t r a p o l a t e d f r o m Fig. 4 of reference [421]). Intrinsic carrier concentration: I n v e s t i g a t i o n s of H a l l coefficient a n d conduct i v i t y over a wide t e m p e r a t u r e r a n g e h a v e been m a d e b y H o w a r t h , J o n e s ,

28

Y. P. Vabshni

0.22 02S W Fig. 53.

0.25 0.26 0.27 0.28 0.23

hvieVl

P{v) o(v), a n d V for indium a n t i m o n i d e a t 77 ° K

F i g . 54.

x, P(v) e(r),

and U f c r indium a n t i m o n i d e at 295 °K

and Putley [425] who give

(

0 255\

cm~6» (104) where kT is in eV. Calculated values of P(V) Q(V), X, and U are shown in Fig. 53 and 54 and other results are recorded in Table 2 (see Part I, phys. stat. sol. 19, 459 (1967)).

8.6.2 Discussion

of radiative

transitions

The radiative recombination rate and the lifetime of carriers in InSb at room temperature have been calculated by earlier workers. Goodwin and McLean [426] obtained R = 2.25 X 10 22 cm" 3 s" 1 and r = 0.36 ¡as. Calculations of Moss et al. [421] gave R = 1.02 X 10 22 cm" 3 s" 1 and r = 0.79 ¡xs. Recombination radiation in InSb was observed by Moss and Hawkins [427, 428]. I t was pointed out by Moss [98] that the spectral distribution of the radiation which is observed experimentally may differ considerably from that generated, depending on the experimental conditions. For example, in the experiments of Moss and Hawkins, the excess carriers were generated optically by highly absorbed wavelength radiation so that all the excess carriers can be considered to the generated at the illuminated surface of a thin slab and the recombination radiation was observed from the dark surface. The emitted radiation would suffer attenuation in its passage through the thin slab and would also undergo multiple reflections at the two surfaces. After making allowances for these effects, the calculated spectral distribution is shown as curve D in Fig. 55. Curve A is the measured emission curve. Allowing for the fact that the experimental curve was broadened by the low resolution which was used, the agreement between the two curves is reasonable.

29

Band-to-Band Radiative Recombination in Semiconductors (II) Fig. 55. Measured and expected recombination radiation distribution from indium antimonide according to Moss [98]. (A) Measured emission curve, (D) expected emission curve after taking into account the corrections

9 W Wavelength (pm) —

F r o m t h e absolute i n t e n s i t y of t h e r e c o m b i n a t i o n r a d i a t i o n , Moss [98] e s t i m a t e d t h e p e r c e n t a g e of r e c o m b i n a t i o n s which are r a d i a t i v e , which equals t h e r a t i o t b / t r of t h e b u l k lifetime t o t h e r a d i a t i v e lifetime, a n d f o u n d it t o b e nearly 20%. W e h a v e m e n t i o n e d a b o v e t h e calculated r R = 0.79 [xs. T y p i c a l m e a s u r e d values of r in good q u a l i t y crystals are « 4 X 10~ 8 s, which indicates t h a t o n l y œ 5 % of t h e r e c o m b i n a t i o n s are r a d i a t i v e . T h i s is a t v a r i a n c e w i t h Moss' e s t i m a t e . L a n d s b e r g [429] has suggested t h a t t h i s discrepancy could be exp l a i n e d if t h e s h o r t - w a v e l e n g t h light used t o g e n e r a t e t h e excess carriers p r o d u ced m o r e t h a n one electron-hole p a i r p e r p h o t o n . Such an e n h a n c e d q u a n t u m efficiency has been f o u n d in I n S b b y T a u c a n d A b r a h a m [82] a n d b y I v a k h n o a n d N a s l e d o v [430], R e c e n t l y B e n o i t à la Guillaume a n d L a v a l l a r d [431, 432] h a v e o b s e r v e d r e c o m b i n a t i o n r a d i a t i o n in I n S b a t 77 a n d 20 ° K . Their o b s e r v a t i o n s are rep r o d u c e d in Fig. 56. T h e s h a p e of t h e s p e c t r u m a t 77 °K is u n s y m m e t r i c a l , being steeper on t h e low-energy side which is similar t o t h a t in F i g . 54. T h e m a x i m u m in t h e e x p e r i m e n t a l c u r v e occurs a t 230.5 m e V as is t h e case w i t h t h e t h e o r e t i c a l c u r v e . T h e observed s p e c t r a m a y be considered t o be in good agreement with the absorption data. Recently Pehek and Levinstein \ j I T-20'K T-TTK / [433] h a v e carried o u t extensive i n v e s t i g a t i o n s on t h e recombinat i o n r a d i a t i o n s p e c t r a of a v a r i e t y of n- a n d p - t y p e I n S b single cryst a l s a t 300, 200, 100, a n d 12 °K. .¡3 § T h e emission s p e c t r a included conMiO t r i b u t i o n s f r o m direct b a n d - t o - b a n d t r a n s i t i o n s , indirect phonon-assisted band-to-band transitions, transitions t o flaws, a n d t r a n s i t i o n s bet-

A

\

\i^ N \ I*

' \\

j

/

25

/

/ r i g . 56. Recombination radiation spectra from indium antimonide at 77 and 20 °K as measured by lienoit a la Guillaume and Lavallard [432]

225

A i\

230

i

i

\\ \

i

235 Photon energy ImeVI-

30

Y . P . VARSHNI

Fig. 57. Theoretical and experimental recombination emission at 300 °K (Pehek and Levinstein [433]). (a) Experimental emission adjusted for constant wavelength resolution, (b) van ltoosbroeck-Shockley theory taking into account self absorption

ween localized states not thermally coupled to the lattice, the transition energies of which were larger t h a n the band gap of InSb. Their results on the band-to-band emission at 300 °K are shown in Fig. 57. Alferov et al. [434] have observed recombination radiation from p - n - n + structures in InSb at 77 °K. The spectrum was found to have an almost symmetrical shape and the maximum was located at an energy of about 0.215eV, which does not agree with the energy ( 0 . 2 3 0 5 eV) at which the maximum occurs in Fig. 54. This inconsistency between the absorption data and the recombination radiation data appears to indicate that the radiation observed by Alferov et al. [434] was not the "intrinsic radiation". Vul et al. [435] have observed the recombination radiation in degenerate InSb on injection of carriers through a p - n junction. Recombination radiation from InSb under avalanche breakdown has been observed by Basov et al. [436, 437]. Investigations on the carrier lifetime have been made by Wertheim [438], Baev [439], and Gulyaeva, Iglitsyn, and Petrova [440]. Gulyaeva et al. measured the lifetime in p- and n-type InSb using steady-state photomagnetic and photoconductivity methods. They conclude t h a t the main recombination process at high temperatures is Auger recombination. Laser action in InSb has been observed by Phelan et al. [441] and by Benoit a la Guillaume and Lavallard [442]. Several important magnetic effects have been observed in InSb. The first of these is shift in the maser frequency which can be explained by the shift of the energy gap with magnetic field. Thus as the field is increased the excitation is transferred from one axial mode to the adjacent mode at a shorter wavelength. Another effect is the lowering of the threshold current density. The reasons for this are discussed by Lax [443]. Ivanov-Omskii et al. [444] have observed an increase in recombination radiation from single-crystal samples of p-type InSb when placed in combined electric and magnetic fields. The increase was interpreted as due to a nonequilibrium increase in charge carrier density at the sample surface resulting from the combined action of the electric and magnetic fields. 9. Comparative Discussion The results are summarized in Table 2 (see P a r t I of this article, phys. stat. sol. 19, 459 (1967)). It would be here appropriate to consider the accuracy of these quantities.

31

Band-to-Band Radiative Recombination in Semiconductors (II)

Moss, Smith, a n d H a w k i n s [421] have calculated R a t 295 °K, using t h e i r own absorption d a t a . The value t h e y obtain is R = 1.02 X 10 22 which m a y be compared with t h e value calculated here independently, viz. 1.03 X l O 2 2 using t h e same absorption d a t a . The good agreement shows t h a t t h e absorption coefficients could be read f r o m enlarged diagrams with resonable accuracy. The values of R depend on t h e accuracy of t h e absorption coefficients. F o r most entries of R, we believe the accuracy is 5 % or better. Values for Si a t temperatures other t h a n 290 °K m a y be in error u p t o 15% a t medium t e m p e r a t u r e s a n d u p t o 5 0 % a t low t e m p e r a t u r e s as t h e constant A in equation (16) appears t o v a r y with t e m p e r a t u r e . F o r InAs (77 °K) t h e value is obtained t h r o u g h an approximation a n d could be in error b y a f a c t o r of 5. The Se (hexagonal) value gives only an order of magnitude a n d m a y be in error b y a factor of 10. As regards n-v for Si, Ge, a n d InSb, t h e values have been obtained f r o m equations whose p a r a m e t e r s were derived f r o m certain experimental d a t a and these should be of reasonable accuracy. On t h e other hand, for most of t h e other substances, ni values have been obtained f r o m t h e effective masses a n d are subject t o appreciable uncertainties. The results show t h a t R a n d r v a r y v e r y rapidly with change in t e m p e r a t u r e . On t h e other h a n d , t h e recombination probability B shows only a slow change with t e m p e r a t u r e a n d is a convenient q u a n t i t y for interpolation or extrapolation. I n Fig. 58 a n d 59, we show log B versus log T for t h e various substances. W h e r e only two points are available, these have been joined b y a straight line. At least for direct gap semiconductors, there is some justification for doing so, as t h e t h e o r y shows t h a t B oc T~3I2. However, t h e slope of t h e lines in Fig. 59 is nearer ( — 3) r a t h e r t h a n t h e theoretically expected (—3/2). P e r h a p s p a r t of

• 6aAs * GaSb a inP T InAs n InSb

r

V

\

\

\—

\

-

•

N.

-

SO 90 100

i ii rm-

Fig. 58. Dependence of the radiative recombination probability B oil temperature for Oe, Si, a n d T e

i

i

N

i

TfV Fig. 59. Dependence of the radiative recombination probability B on temperature for the I I I - V semiconductors

32

Y . P . VAKSHNI

t h i s d i s c r e p a n c y could be a t t r i b u t e d t o t h e v a r i a t i o n of effective masses w i t h t e m p e r a t u r e . A t i n t e r m e d i a t e t e m p e r a t u r e s t h e s e curves m a y be u s e d f o r e s t i m a t i n g values w i t h o u t too g r e a t a n e r r o r . F o r Ge a n d G a S b a t T > 400 ° K a n d for I n S b a t T > 350 ° K , r a d i a t i v e r e c o m b i n a t i o n lifetimes would be v e r y s h o r t . B u t w h e t h e r or n o t t h i s would be t h e d o m i n a n t m o d e of r e c o m b i n a t i o n , d e p e n d s on o t h e r c o m p e t i n g processes. Allowing for t h e a p p r o x i m a t i o n s m a d e in deriving t h e t h e o r e t i c a l expressions for R etc., a n d t h e u n c e r t a i n t i e s in t h e values of t h e i n p u t quantities, t h e results given in T a b l e 3 for t h e a p p r o p r i a t e t y p e of t r a n s i t i o n are in r e a s o n a b l e agreem e n t w i t h t h e c o r r e s p o n d i n g results in T a b l e 2 (Tables 2 a n d 3 see P a r t I of t h i s article). T h e R-values for GaAs, GaSb, I n P , I n A s , a n d I n S b (Table 3) show t h a t f o r a given v a l u e of t h e e n e r g y gap, direct r e c o m b i n a t i o n is m o r e p r o b a b l e t h a n t h e i n d i r e c t one. Acknowledgements P a r t of t h e w o r k w a s carried o u t d u r i n g a s u m m e r s p e n t a t t h e R.C.A. V i c t o r R e s e a r c h L a b o r a t o r i e s , Montreal. T h e a u t h o r wishes t o express his t h a n k s t o D r . R . W . J a c k s o n for his s u p p o r t a n d e n c o u r a g e m e n t , a n d D r . R . J . M c l n t y r e for m a n y h e l p f u l discussions a n d for r e a d i n g a n d c o m m e n t i n g on t h e m a n u s c r i p t . T h e w o r k was s u p p o r t e d b y t h e D e f e n c e R e s e a r c h B o a r d (Canada) u n d e r c o n t r a c t No. PW69-200027 w i t h t h e R.C.A. V i c t o r R e s e a r c h L a b o r a t o r i e s ( E C R D C P r o j e c t N o . T68). H e l p f u l c o m m u n i c a t i o n f r o m Mr. V. R o b e r t s (Royal R a d a r E s t a b l i s h m e n t , Gt. Malvern) is also g r a t e f u l l y acknowledged. References (The references [1] to [287] have been published in Part I of this article, phys. stat. sol. 19, 459 (1967). For explanation of the abbreviations "Prague Conf.", "Exeter Conf.", "Paris Conf.", and "Paris Symp." see also the references of Part I.) [288] D. A. KLEINMAN and W. G. SPITZER, Phys. Rev. 118, 110 (1960).

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28,

1966)

Original Papers phys. stat. sol. 20, 37 (1967) Subject classification: 6; 8; 22 D. I. Mendeleev Chemico-Technological

Institute, Department of Physics, Moscow

Heat Capacity and Structure of Vitreous Silica and Diamond-Like Lattices (I) By V . V . TARASSOV

The phonon spectra of lattices with crystalline and glassy structures possessing skeletons of the purely covalent or ionic-covalent type are discussed on the basis of the "one-dimensional-three-dimensional model". The upper frequency band, from co' to co max (co' co max ), corresponds to a one-dimensional frequency distribution gi((o) in which elastic waves may propagate only in certain principal directions. In the lower band from 0 to co', the frequency distribution g3(co) is three-dimensional, as in the usual Debye-Born-von Karman theory. For the one-dimensional distribution ¡^(co) the exact solution is used and taken from papers by Pirenne and Renson, Deltour and Kartheuser for monoatomic and diatomic chains. The phonon spectrum for vitreous silica is discussed in terms of the present model, since its structure resembles that of ß-cristobalite. It is shown that the "chain model" applies to the thermal capacity of vitreous silica up to 900 °K. Die Phononenspektren von Gittern mit kristalliner und glasartiger Struktur mit einem Gerüst von rein kovalentem oder ionisch-kovalentem Typ werden mit dem „eindimensionalen-dreidimensionalen Modell" diskutiert. Das obere Frequenzband von co' bis co max (co' ) is also three-dimensional. Thus, the phonon spectrum of diamond-like lattices (the application of chain dynamics to diamond-like structures has been treated by the author at the 7th International Crystallography Congress in Moscow in J u l y 1966), according to our new hypothesis, consists of two main bands: an upper band corresponding to a one-dimensional law of frequency distribution, and a lower band with a three-dimensional distribution. This gives a very good description of the entire heat capacity curve for Ge, Si, Sn (grey), GaSb, GaAs, AlSb, InP, InAs, InSb, etc., as well as Si0 2 ((3-cristobalite and glass) and GeOa in the crystalline and vitreous states. I n the case of Ge one need not fit a single constant empirically; one simply uses the dispersion curves co = f(k) for the optical and acoustic longitudinal and transverse branches of the lattice vibrations obtained by Brockhouse and Iyengar [14] by the method of inelastic scattering of cold monochromatic neutrons. I t was by this method that we computed the heat capacities of germanium, given in Part I I of the present paper. On the other hand, in computing the heat capacities of diamond-like I I I —V(b) compounds the data for the acoustical and optical frequencies were again not fitted empirically, but obtained from the results of a study of multiphonon processes (Marshall and Mitra [15]). As is well known, in the lattice of high-temperature cristobalite the silicon atoms occupy the sites of the diamond lattice, whereas the oxygen atoms lie midway between the silicon atoms. (According to [16] and [17] the angle Si-O-Si in p-cristobalite is less than 180°. This, however, does not essentially affect the general scheme of lattice vibrations described here.) Hence, the laws of the dynamics of diamond-like lattices mentioned above and fully presented in Part I I are applicable to the vibrations of (3-cristobalite, as well as to the skeletons of vitreous Si0 2 and GeOa and to glasses based on them. I t should be observed t h a t the exact theory of monatomic and diatomic chains taking into consideration the effect of the mass ratio (M/m) of the heavy and light atoms alternating in the chain, and a corresponding computation of the thermodynamic functions, including the heat capacity, were given by the Belgian scientists Pirenne and Renson [18] (for a monatomic chain) in 1962. The exact theory for diatomic chains with different mass ratios [1,19 to 22] was published later by Deltour and Kartheuser [19, 20] in 1964 and 1965. Pirenne and Renson's heat capacity function (PR) for a monatomic chain is slightly different from our function tabulated up to the fourth decimal at small intervals by Wunderlich (to be denoted in the following by (TW)] [21] on an electronic computer. The (PR) function differs, too, from the results of the well-known paper of Blackman [22]; although it approaches absolute zero according to a T 5 law like (TW), it has a point of inflexion, which Blackman's results do not show. Thus, the strict (PR) theory of the lattice heat capacity confirms the existence of a region between T = 0 and the point of inflexion where the T1 law holds. The same can be said of the heat capacity functions of diatomic chains of Delthour and Kartheuser (DK) which also have a T1 region below the point of inflexion. For not too large mass ratios, when M/m < 2, the (DK) function is very close to the (TW) function, but for M\m ^ 4 the differences between (DK) and (TW) become quite considerable. I n view of this, in the present paper, especially for

40

V . V . TARASSOV

Ge0 2 , I n P and AlSb, for which the ratio M\m is large, we shall apply the exact dynamics of chain lattices according to Tarassov, Pirenne, Renson, Deltour, and Kartheuser (to he denoted in the following by ( T P R D K ) ) . Although the present paper deals mainly with the dynamics of vibration of vitreous silicon dioxide, nevertheless, the examples of the structures G e 0 2 , I n P , and AlSb with large mass ratios, for which the curve of the heat capacity diffeis essentially from the (TW) curve are very important as a proof of the validity of the ( T P R D K ) dynamics of vibrations in the skeletons of glass-forming compounds and glasses. 2. Results of Measurements of the Heat Capacity of Vitreous Silica over a Wide Temperature Range Westrum measured the heat capacity of vitreous silica over a wide temperature range (from 10 to 900 °K). The resulting values are given in Table 1; they represent the arithmetic mean values for samples of glass stabilized at 1300 and 1070 °C. These figures are taken from [5] (see Table 3 in [5]). I n Table 1 the second column contains Westrum's data, while the third column gives C(x) the data of Turdakin obtained in our laboratory in 1965 on samples of superpure quartz glass. The experimental part of Turdakin's work with a detailed description of the calorimetric apparatus ( K Y 300 M) and samples of vitreous silica is to be published. The results of Turdakin's measurements are given in Table 2. The experimental values of the heat capacities of vitreous silica according to Westrum and Turdakin, are plotted in Fig. 1. In order to convince the reader that the ordinary three-dimensional Debye function is not applicable we have drawn a dashed curve in the Figure, representing the Debye function for which the characteristic temperature was chosen so as to give coincidence with experiment at 300 ° K . I t is evident from the figure that the Debye curve for Cv decreases to zero far more steeply than follows from experiment, whereas our one-parameter function for a chain (continuous curve) agrees well with the experimental points of

Fig. 1. Heat capacity of vitreous silica. A after "Westrum, O after Turdakin, solid curve -- our one-parameter function for a "chain structure" with 6t = 1550°K, dashed curve — Debye function, 0 3 chosen so that C',.,. = C ^ '

^

Heat Capacity and Structure of Vitreous Silica (I)

41

Table 1 C in (cal/mole deg) T °K

C[ 5]

10 20 30 40 50 60 70 80

0.064 0.350 0.745 1.161 1.590 2.027 2.468 2.930

C(x)

T °K

C[ 5]

C(x)

T° K

Ci 5]

C(x)

3.852 4.753 5.607 6.405 7.140 7.819 8.448 9.036

3.814 4.723 5.568 6.358 7.117 7.820 8.458 9.018

260 280 300 400 500 600 700 800 900

9.589 10.114 10.605 12.404 13.879 14.845 15.550 16.084 16.423

9.545 10.05 10.51

1.525 2.008 2.452 2.911

100 120 140 160 180 200 220 240

Table 2 Smoothed values of the heat capacities of super-pure vitreous silica T °K

C (molar)

T °K

C (molar)

T °K

50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130

1.525 1.787 2.008 2.226 2.452 2.688 2.911 3.144 3.365 3.588 3.814 4.043 4.275 4.501 4.723 4.940 5.157

135 140 145 150 155 160 165 170 175 180 185 190 195 200 205 210 215

5.362 5.568 5.773 5.973 6.169 6.358 6.551 6.739 6.920 7.117 7.292 7.480 7.650 7.820 7.991 8.155 8.305

220 225 230 235 240 245 250 255 260 265 270 275 280 285 290 295 300

G (molar) 8.458 8.603 8.746 8.882 9.018 9.153 9.290 9.418 9.545 9.675 9.793 9.920 10.05 10.16 10.29 10.39 10.51

Turdakin and Westrum throughout the entire range up to 900 °K, even without correction for the dispersion of the phase velocity of the phonons and the mass ratio MSJM0 of the atoms according to (TPRDK). In the region 120 to 250 °K the experimental data are somewhat higher than the computed curve which represents our simple one-parameter function of the heat capacity of a chain structure [6, 8]. For three moles of Si0 2 it has the form n

n e>

T

f

e

"

dx

h v

a

^"max

0 In the calculations we made use of the most exact five-decimal tables of this function, given in Wunderlich's paper [21]. The computed curve in Fig. 1 has

42

V. V. TABASSOV Table 3

T (°K) 18.980 16.062 8.982 6.769 6.226 5.504 4.565 2.344

CEXP 0.3057 0.2097 0.04136 0.01564 0.01155 0.007318 0.003525 0.000335

£COMP 0.2770 0.1850 —

0.01390 0.01076 0.00743 0.00424 0.000547

been plotted for d1 = 1550 ° K , which corresponds to a frequency « 1080 cm " 1 , very close to the main peak of infrared absorption of vitreous silica. (This circumstance, however, is not the reason for our choice of 6 1 — 1550 ° K . ) Below 80 ° K the experimental points lie below the computed curve, plotted according to (1). I n order to represent the heat capacity of vitreous silica as a function of temperatur, over the entire temperature range including the low temperatures, one must apply our two-parameter formula [7, 9]

The second characteristic temperature d3 < dv which determines the "lateral interaction of the chains", is insignificant in the given case, and corresponds to a ratio d3ld1 = 0.0916. Since (2) goes over to the T3 law at T < 0.1 0 3 , it follows that in the given case, i.e. at Q1 = 1550 ° K and 0 3 /0 1 = 0.09, one can expect the T3 law to hold at temperatures below 0.009 0 t , i.e., below 15 ° K . Table 3 shows the results of our computations, carried out according to (2) for 0! = 1550 ° K and Osjd1 = 0.09, for temperatures below 20 ° K . The experimental data are taken from [2], The results of the calculations given in Table 3 show that the parameters of (2) chosen at high temperatures enable one to extrapolate this formula to low temperatures, i.e., over a range in which the heat capacity changes by a factor of one hundred. More detailed results of the one-dimensional and three-dimensional description of the phonon spectra according to ( T P R D K ) applied to the case of vitreous silica are given in Part I I of this paper. One of the main points of the present paper is illustrated by Fig. 1, namely the agreement between experimental data on the heat capacities of vitreous silica and the theoretical curve, obtained on the basis of the one-dimensional description of the phonon spectrum. This means, apparently, that in the case of vitreous silica as well as in other glasses [9] a "memory" exists not merely of the crystalline short-range order, but also of the polymeric long-range order of the crystal which has "generated" the glass. This " m e m o r y " of the long-range order is evident in the similarity of the phonon spectra of the glass to that of the crystalline phase which is nearest to its structure.

Heat Capacity and Structure of Vitreous Silica (I)

43

References [1] E. F. WESTRUM, IV Congres Intern, du Verre, Paris 1956 (p. 396). [2] P . FLUBACHER,

A . J . LEADBETTER,

J . A . MORRISON, a n d B . P . STOICHEFF, J .

Phys.

Chem. Solids 12, 41 (1954). [3] G. K . WHITE a n d J . A. BIRCH, P h y s . a n d Chem. Glasses 6, 85 (1965).

[4] V. V. TARASSOV, Bulletin of the Moscow Chemico-Technological Institute 11, 5 (1947). [5] R . C. LORD a n d J . C. MORROW, J . c h e m . P h y s . 2 6 , 2 3 0 (1957).

[6] V. V. TARASSOV, Dokl. Akad. Nauk SSSR 46, 117 (1945). [7] V. V. TARASSOV, Dokl. Akad. Nauk SSSR 58, 577 (1947). [8] V . V . TARASSOV, Z h . f i z . K h i m . 2 4 , 111 ( 1 9 5 0 ) .

[9] V. V. TARASSOV, New Problems of the Physics of Glass, published for the US Department of Commerce and the National Science Foundation, Washingotn D.C. by the Israel Program for Scientific Translations, Jerusalem 1963. [10] W. DE SORBO, J . chem. Phys. 21, 1144 (1953). [11] W. DE SORBO, J . chem. Phys. 21, 764 (1953). [12] L . S. KOTHARI a n d V . K . TEWARY, J . c h e m . P h y s . 3 8 , 4 1 7 ( 1 9 6 3 ) .

[13] V. V. TARASSOV, Dokl. Akad. Nauk SSSR 84, 321 (1952). V. V. TARASSOV a n d YA. S. SAVITZKAYA, Zh. fiz. K h i m . 27, 744 (1953). [14] B . N . BROCKHOITSE a n d P . K . JYENGAR, P h y s . R e v . I l l , 7 4 7 (1958).

[15] R . MARSHALL a n d S. S. MITRA, P h y s . R e v . 134, A 1019 (1964).

[16] T. F. W. BARTH, Amer. J . Sei. 23, 350 (1932). [17] W. NIEUWENKAMP, Z. Krist. 96, 454 (1937). [ 1 8 ] J . PIRENNE a n d P . RENSON, P h y s i c a 2 8 , 2 3 3 (1962). [ 1 9 ] J . DELTOUR a n d E . KARTHEUSER, B u l l . S o c . R o y . Sei. L i è g e 3 3 , 8 0 1 ( 1 9 6 4 ) . [ 2 0 ] J . DELTOUR a n d E . KARTHEUSER, P h y s i c a 3 1 , 2 6 9 ( 1 9 6 5 ) .

[21] B. WUNDERLICH, J . chem. Phys. 37, 1207 (1962). [ 2 2 ] M . BLACKMAX, P r o c . R o y . S o c . A 1 4 8 , 3 6 5 ( 1 9 3 5 ) . (Received

August

15,

1966)

V. V.

TARASSOV

: Heat Capacity and Structure of Victreous Silica (II)

45

phys. stat. sol. 20, 45 (1967) Subject classification: 6; 8; 22 D. I. Mendeleev Chemico-Technological

Institute, Department of Physics,

Moscow

Heat Capacity and Structure of Yitreous Silica and Diamond-Like Lattices (II) By V. V.

TABASSOV

In a diamond-like covalent lattice, and in vitreous silica, the phonon spectrum for the main high-frequency band (containing 80 to 90% of all the modes of vibration) is one dimensional. This leads to the proposed chain model of Tarassov, Pirenne and Renson, and Deltour and Kartheuser. This model is used here to calculate the specific heat of germanium by using the data of Brockhouse and Iyengar. The same method is also applied to InP, AlSb, and vitreous silica. In einem kovalenten Gitter vom Diamant-Typ und im glasigen Siliziumdioxyd ist das Phononenspektrum für das hochfrequente Hauptband (das 80 bis 90% aller Schwingungsmoden enthält) eindimensional. Dies führt zu dem vorgeschlagenen Kettenmodell von Tarassov, Pirenne und Renson und Deltour und Kartheuser. Dies Modell wird benutzt, um die spezifische Wärme von Germanium mit den experimentellen Ergebnissen von Brockhouse and Iyengar zu berechnen. Dieselbe Methode wird auch für InP, AlSb und glasiges Siliziumdioxyd verwendet. On the Chain Heterodynamism 1 ) of Vitreous Silica As early as 1952 t o 1953 the present author published papers on the heterodynamism of silicate glasses [1, 2]. In these papers it was shown t h a t the temperature dependence of the heat capacity of crystals and glasses, in which the main structural elements are metasilicate chains, is well approximated by a function of the type of equation (1) which describes the heat capacity of a chain structure : (1) J ) The term "heterodynamism" is suggested by the author (see [1 to 4]) for polymeric structures possessing two types of bonds: stronger bonds within the chains and weaker bonds between the chains. In the present paper this term is applied in a wider sense (see farther on this paper). 2 ) In (1) D1 and D3 are the one-dimensional and three-dimensional Debye-funcitons, respect ively:

o

e/T o

where It is the gas constant and x

h co

x T

46

y . v . takassov

Still earlier the author showed [3, 4] t h a t the phonon spectrum of such chain structures consists of two b a n d s : an upper band corresponding t o large values of the wave number, from kj t o ¿ m a x , and to the m a j o r p a r t of all the lattice vibrations, and a d j a c e n t t o it a band of lower frequencies, from to k = 0 (ifc = 2 n f i ) . The chain heterodynamism of t h e phonon spectrum manifests itself in the circumstance t h a t t h e frequency distribution law in t h e upper band is onedimensional. This corresponds to a one-dimensional frequency density «) = =

mal

and to the propagation of t h e phonon waves only along the chains.

The lower band corresponds t o a three-dimensional frequency density g3{a>) = diVVi I and the propagation of phonon waves in all directions, not only [da) Jo along the chains. Equation (1) expresses this two-band character of the phonon spectrum. We believe t h a t the main result of P a r t I, Section 2 is the proof of t h e fact t h a t vitreous silica, too, possesses chain heterodynamism, i.e., it has a onedimensional frequency distribution almost in the entire vibrational spectrum of the skeleton (from 0.1i»max t o v max ) (since ^ =

= 0.0916; see P a r t i ,

Section 21, while only in a small range of frequencies in the lower band (from 0.1 1'max t o 0) it has a three-dimensional frequency distribution. I t follows from the above statements t h a t the chain heterodynamism (agreement of the experimental d a t a with (1)), if inferred only from the form of the heat capacity curves, does not always indicate uniquely t h e existence of only linear polymer chains. The case of vitreous silica, (3-cristobalite, and several other substances illustrated in the present paper shows t h a t in structures with intersecting chains, such as p-cristobalite, we find the same clearly pronounced chain heterodynamism of the heat capacity as in structures with isolated chains. I t is very important to stress, as was done in p a r t I of this paper, t h a t t h e same situation exists in the case of diamcnd-like elements and compounds, i.e., here, too, the structure consists of intersecting zigzag chains with twelve, and only twelve, directions perpendicular t o the faces of a rhombic dodecahedron {110}. All the high-frequency phonons (from fcmax to kj) of t h e upper band of frequencies propagate in these and only these directions, i.e., perpendicular t o the faces of a rhombic dodecahedron (see Fig. 1). I n t h e range of wave numbers from &max to kj there are no phonons propagating in other directions. I n this range the diamond-like structure is one-dimensional with respect t o its phonon spectrum, which is here described much more exactly b y a one-dimensional frequency distribution function g^co) t h a n by a three-dimensional function grs(co). I t should be stressed, in addition, t h a t over the entire frequency range from 0 to comax the spectrum of diamond-like structures and of (3-cristobalite is described much more exactly b y our one-dimensional-three-dimensional model t h a n in any of the theoretical papers based on various force models of the lattice or on other assumptions We shall not quote these papers here, since this is done in a special paper on the vibration dynamics of diamond-like lattices, prepared for publication. We have to make three more essential remarks. The first refers t o the real

Heat Capacity and Structure of Vitreous Silica (II)

Fig. 1. Zigzag chains in the diamond lattice which are directed perpendicular to the faces of a rhombic dodecahedron. The greater part of the phonon waves propagate in these directions

47

Fig. 2. Structure of cristobalite. Each oxygen and silicon atom belong to three and six chains — S i - 0 — Si — O —, respectively

law of propagation of phonons with large wave numbers. Since such phonons can propagate only in the twelve directions indicated above, and since at higher temperatures they determine the main heat content of the lattice, we can predict the existence of an essentially new phenomenon: selective heat conductivity in Ge, Si, [3-cristobalite, and several diamond-like structures. The second remark refers to our coefficient of heterodynamism of the diamond lattice (y). I t will be shown that diamond-like structures, for which the ratio of the limiting frequency of transverse acoustical vibrations COTA* to the frequency of longitudinal acoustical vibrations COLA* is less than 0.5, i.e., ...max rA

possess the greatest chain heterodynamism, i.e., the greatest fraction of modes in the upper "one-dimensional" band. As will be shown below (see Table 3), for diamond itself y = 0.75; hence, the diamond lattice does not show pronounced heterodynamism. I t resembles more closely homodynamic structures. On the other hand, in the case of germanium y = 0.3, and the heat capacity can be well represented b y (1), where d1 and 6J0 3 are calculated from the data of Brockhouse and Iyengar [5] on the inelastic scattering of cold neutrons. The third remark refers to the chain heterodynamism of vitreous silica and glasses in general. In passing from ^-cristobalite to Si0 2 -glass the long-range order of the cristobalite lattice, and any long-range order in general, disappears, of course. However, the polymer long-range order, i.e., the character of the intertwining of the - S i - O - S i - O - S i - chains remains the same as in cristobalite, although the chains are distorted in the three dimensions of space. This means that the frequency distribution function g(co) does not change essentially until considerable depolymerization of the skeleton takes place. Let us turn to a description of the chain structure of (3-cristobalite, of which vitreous silica is a polymer analogue. Fig. 2 shows five tetrahedra of the structure of (3-cristobalite. We have marked three zigzag chains passing

48

V . Y . TARASSOV

Fig. 3. In the 3_cristobalite structure the chains are parallel to the face diagonals of a cube, i.e., they are directed along the edges of a regular tetrahedron

Fig. 4. Six directions of the Si-0 chains and twelve main directions of propagation of phonon waves in the lattice of (3-cristobalite

Fig. 5. Model of the diamond lattice

49

Heat Capacity and Structure of Vitreous Silica (II)

through the oxygen atom " 0 " :

Si/

cr

o

N

Nsi^

o

x

o

Nsi'

I n the "catenogenic" (catenogenic — from the Latin " c a t e n a " = chain) approach to the (3-cristobalite lattice it must be stressed that each oxygen atom belongs to three chains - S i - O - S i - O - S i - O - S i - in three directions. I n Fig. 2 these chains are denoted by A B , CD, and E F . Correspondingly each silicon atom belongs to six chains in six directions (each of the six chains is perpendicular to two opposite and mutually parallel faces of a rhombic dodecahedron). Thus, each diatomic chain (in the sense of the dynamics of Born and Karman) —Si—0—Si-0— contains one third of each oxygen atom (which always belongs simultaneously to three chains) and one sixth of each silicon atom (which always belongs simultaneously to six chains). In the catenogenic approach tne stoichiometric composition of the chains of [i-criatobalitc corresponds to O1/3 Sil/6

which corresponds to stoichiometric composition S i 0 2 . Fig. 3 shows the direction of the chains in (3-cristobalite parallel to the face diagonals of a cube, i.e., in the direction of the edges of a regular tetrahedron. Fig. 4 presents the same for the real structure of [3-cristobalite. We presume that Fig. 2, 3, and 4 give a clear indication of the directions of the chains chosen, i.e., of the directions of propagation of the phonons. A similar catenogenic picture of the diamond lattice with the same " c h o s e n " directions of the chains and phonons can be seen in the photographs of the model of a diamond structure (Fig. 5). To illustrate the chain heterodynamism of diamond-like structures we shall now discuss the investigation of the optical and acoustical branches of the germanium lattice by means of the inelastic scattering of monochromatic cold neutrons. We refer to results repeatedly cited in the text [5]. Fig. 6 shows the form of the dispersion curves for germanium obtained by this method. The lower branches, TA and LA, are the transverse and longitudinal acoustical branches; the upper ones, LO and TO, are the longitudinal and transverse optical branches. They express the relation v ~ f(k) for a beam of neutrons incident on the { 1 0 0 } plane, i.e. perpendicular to the Ge chains. The maximum optical frequency obtained by Brockhouse and Iyengar is (9.0 + 0.3) X 10 1 2 Hz. This limiting frequency corresponds to a characteristic temperature oGe ?disp

= — 9.0 X l 0 1 2 =

431.9 °K .

Fig. 6. Dispersion relations in [100] directions: (1) transverse acoustic, (2) longitudinal acoustic, (3) longitudinal optical, (4) transverse optical branch 4

physica/20/1

02

OA 0.6 0.8 1.0

k/kmax

60

V. Y.

TARASSOV

Assuming that in the case of germanium we are dealing with a chain or quasione-dimensional case, with a law of dispersion corresponding to the monatomic chain of Born and Karman, we set the zero-point energies of the non-dispersive chain of Debye and of the chain of Born and Karman equal to one another : Vmax

(rmax)disp

hv

2

0

or

3

f

3 N vmax

d

h t'max

3

==

_— N 71

dv

1 r - N . r - . 2 n j/vïax

J

0

7V7h i.Vmux

— N N 4

" =

%

m a

x)disp

/ox

_

.

•

(2)

(2')

This brings us to a characteristic temperature d1 equivalent to $disp for the BornKarman chain. Hence, from the results of neutron scattering we can find the equivalent temperature d1 for the Debye chain. This idea is illustrated by Fig. 7 where the upper hatched area corresponds to the zero-point energy, while the lower area corresponds to the heat content: H

= / b „ dT o

.

The determination of the equivalent value d1 by means of formula (2') leads to the equi-enthalpy value of the Debye characteristic temperature with respect to the Born-Karman characteristic temperature. Integrating (2) we find "disp

_ _^max— (i'max/disp

=

A

^

=

i .2732 .

(3)

Taking for 0 d i sp the value 431.9 °K obtained from neutron scattering data, we find for the equivalent Debye chain (06 e ) e i = — 4 3 1 . 9 ° K = 550.0 °K. 71

(4)

Since the equivalent value of the characteristic temperature for germanium, (gGe)eq __ ggQ w a g obtained on the basis of neutron scattering data we shall label it by the subscript " n e u t r " (fl?e)SSatr = 550 ° K .

Fig. 7.

and 0eq- H a t c h e d area : zero-point energy. Area under the c u r v e : h e a t content

Fig. 8. Simplest distribution function for a chain structure

51

Heat Capacity and Structure of Vitreous Silica (II)

We draw attention to two remarkable features in the acoustical and optical curves of germanium according to Brockhouse and Iyengar (Fig. 6). The first one is the anomalously big difference between the maximum frequencies of the transverse and longitudinal acoustical vibrations of the germanium lattice. It corresponds to the value (see Table 3) roe

=

=

"'ta

0.302

.

The second one is the form of the transverse acoustical branch with its extended "plateau", where the curve co = f(k) is parallel to the ¿-axis (for k > 0.6 &max). As is well known (see [6], p. 54), the frequency density function g(u>) becomes infinite at dco/dfc -> 0. Hence, starting from

and utilizing both the acoustic (A) and theoptical (0) branches, we can represent g(a>) as schematically drawn in Fig. 8. The resulting picture fully corresponds to our two-band structure of the phonon spectrum. A similar curve was obtained by Wunderlich [8] for polyethylene (see Fig. 9). It is very interesting to observe that Cole and Keneke [7] from a study of the thermal scattering of X-rays in the silicon lattice obtained a very similar shape of the frequency density function g(co) (Fig. 10). All this, in our opinion, clearly indicates that chain heterodynamism is present in the Ge and Si lattices and that the frequency distribution in the upper frequency band of the lattice is one-dimensional: Here the phonons propagate only perpendicular to the faces of the dodecahedron {110}. The correspondence between our formula (1)

t

t

3

(after H.Cole and E. Keneke f 7] j Si

^ 2

ii

h

I I I I I

0

500 k(cm' 1l—•

Fig. 9. Frequency spectrum of polyethylene, determined from the heat capacity curve of Wunderlich [8] 4'

2U

1.2

12.0

16.9

uxl0~ u(s'V— Fig. 10. Frequency-density function for the vibrations of the silicon lattice, obtained from thermal scattering of X-rays (Cole and Keneke [7])

52

V . Y . TARASSOV Fig. 11. Heat capacity of Ge. Experimental results according to Piesbergen [12], Solid theoretical curve according to formula (6). Upper dotted line: one-parameter function (TW) for 0, = 550 ° K . Lower dotted line: function ( P R )

and the curves for the acoustical and optical branches of Ge is very good if we put 6»! = (fljCutr = 550 ° K and = 0.302 . 3 )

6i

(5)

I n other words, all the phonons in the lattice of germanium whose frequencies exceed COTA* are "captured" in the chains, i.e., they propagate only in the directions

2

2 til

exp

= — oo

2

n

i

mh*

(i)

where ip 2 is a function of h1 and h2 which vanishes except when hx = H and h2 = K, and & m is the phase difference between X-rays scattered through H B1 + K B2 by two layers, m3 and ml, which are m layers apart. For a perfect crystal +

l ) " i - ( - l H .

(2)

In general, & m can be expressed as the sum of the individual phase difference, (pk , across successive layers: m3 o

( _ 1 )m+k

(11a)

2

while if it is t y p e B the phase change is "1 - (-1)»»+*' (^ffl)oo = — M o o ] e x p [ i ( 0 M A ! ) O o ] + P[(2)oo] + k

+ P[(tf£).o]exp[i(o) = Pa Pb + PbPc

+ Pc Pa =

+

=

p(

where T = Ts — T1; and h is the thickness of the nitride layer. As the thickness of the silicon is very much greater than t h a t of the nitride layer it is taken as equal to the total thickness, t, of the composite. Combining equations (1) and (2) and remembering that the curvature is found to be o n J Es t2 '

^

By inserting this value for the curvature into equation (1) the force may be determined: p _ E$ [ j ^ Y

' +

6 - and k and one may write M(k,

)f

(8)

co) ,

w) -

+

)

co

[CO

i e'yy{Q,

a,)]' + y2(Q,

A(Q,

in a small region of frequencies, the

eyy

3. Choice of the Crystal Model I t is convenient to calculate in cylindrical coordinates with the cylinder axis along the vector d (the basis vector ay). Let us assume that the first Brillouin zone is determined by the values of wave vectors fc satisfying the conditions O ^ e ^ l ,

(10)

where n y = k ay ,

7i q = a \k\ +

£

,

tp = arctg ~ . Kz

We shall consider molecular crystals having the structure of exciton bands determined by the function E(k)

=

L { e

+

S(y,

Q

)},

(11)

156

A. S . DAVYDOV a n d E . N . MYASNIKOV

where 2

2 a ( j ! + g2)

V

'

The transversal excitons are characterized in this band by the value y = 0. As £ 0 we pass to the point corresponding to the bottom of the exciton band with energy i? m i n = e L. The effective mass of transversal excitons is positive and equal to a 7r2/a2X. The width of the exciton band is equal to L/2 y)] d{> d y

.

The results of numerical integration of the functions and £>¿(0 are plotted in Fig. 1 and 2. If Die L is the effective frequency of acoustic phonons with which excitons interact most strongly at low temperature when kT < Qlc L, the real and the

159

Absorption and Dispersion of Light by Molecular Excitons

imaginary parts of the mass operator (7) are determined by the expressions F ¿Qicl ¿¡¡c(C) = KH0CTb oSblHHOrO KBa3H-paBH0BeCH0r0 OnHCaHHH. IIoKa3aHo, MTO MaKpocKonHiecKan nH cu" 1 , cf. [11]). I n both sub-cases a) and b) the intra-well thermalization of the defects is dominated b y a q u a n t i t y slowly varying with respect t o tp (say, t h e energy of the local or quasi-local vibration). I t is for this case t h a t an equation like t h a t of Prigogine and Bak may hold (actually their derivation is applicable t o our sub-case a) only). As shown below in Section 5, in case I rate theory is always applicable, t h u s deviations from the predictions of rate theory are connected with case I I behaviour, if only the jumping picture itself is valid (Section 2, condition (1)). 4. The Separation of Time Scales Different components of non-equilibrium distributions relax at different rates. Any kinetic description of t r a n s p o r t phenomena is based on the simplification obtained b y averaging out f a s t processes and describing events on a slow time scale. The classic formulation of this idea is t h a t of Bogolyubov [12] (cf. also [13]). We analyse the situation in thermally activated processes in terms of t h e characteristic times tp, tE, t*, and tt. a) The basic fast process in this framework is the intra-well directional thermalization, i.e. t h e disappearance of a directed component from the m o m e n t u m distribution. I t is characterized b y t h e time tp (cf. Section 2). The kinetic description is based on averaging over times tp. b) As stated in Section 3, in some cases there are intra-well relaxation processes which are very slow with respect to tp. These are connected with t h e existence of a slow variable t h a t we denote hereafter b y E, t h e corresponding relaxation time being tE. c) I t is appropriate to introduce i*, the characteristic time of t h e slowest intrawell thermalization process. F o r the cases treated in Section 3 t* w tp t* » tE > tp t* > tp

(case I) , (case II) , (intermediate case) .

(2a) (2b) (2c)

I n the intermediate case t* has no simple meaning, but it is always a finite time determined by the intra-well transitions only and independent of the distribution over different potential wells. d) The index i labeling t h e potential wells may be regarded as a discrete coordinate of t h e particle. The corresponding relaxation time t t characterizes the disappearance of spatial inhomogeneities. Obviously f, Si 1 ¡ r (cf. Sections 6 and 7), thus (1) implies tt > tp

(3)

t h a t is, i is a slow variable in hopping processes. 5. The Classical Kinetic Equations F r o m now on we restrict ourselves to classical mechanics. The dynamical s t a t e of the defects is characterized by some coordinates and momenta symbolized by t h e single letter x.

169

On the Theory of Thermally Activated Processes

In order to handle kinetic phenomena we take advantage of the separation of time scales expressed in (2a), (2b), and (3), by applying Bogolyubov's ideas. In the spirit of his theory, on averaging over times tp (i.e. averaging out fluctuations on the fast time scale) the complete one-particle distribution function F(x) will depend on time only as a functional of the distribution function of the slow variables c,-(£) (case I) or fi(E, t) (case II) (in the intermediate case generally such reduction is not possible; but see Section 6!): F = F[x\cl(t)) F =

0)

F{x\fi(E,

(case I) ,

(4a)

(case II) .

(4b)

Moreover, the distribution functions for the slow variables satisfy kinetic equations of the type ^ = Z r

t

j

— = Z f d E '

c,(t) Kti(E,

E') f,(E',

t)

(case I) ,

(5a)

(case II) .

(5b)

Equations (5) are the most general Markovian (single-time) linear equations of the first order in the time variable. 3 ) Their linearity means that we restrict ourselves to a low concentration of defects, when interaction between defects can be neglected. The Markovity is a common feature of slow relaxation processes [22, 23], The quantities r{} and Kij(E, E') are transition rates governing the time change of the slow variables. In case I I an important reduction can be achieved by noting that the very act of a passage over the barrier has a very short duration (w (coD)_1, negligible on the slow time scale); therefore during a passage i j the value of the slow variable E does not change. This implies 4 ) Kti(E,

E')

=

kt(E, E')

t*. (11) _

1

In the common diffusion experiments (10) always holds (actually, (10) is a necessary condition for applying the macroscopic diffusion equation), thus in any case intra-well relaxation processes are fast with respect to macroscopic diffusion. Hence (4a) and all the subsequent case I formulas are always applicable to macroscopic diffusion. Consequently macroscopic diffusion is always correctly described by rate theory, whenever the jumping picture itself is valid (condition (1)), irrespective of the detailed mechanism of intra-well thermalization. (This is a special case of the general statement that memory effects do not appear in steady-state transport coefficients [26].) The same conclusion can be reached by the following argument due to Lifshits [9], The macroscopic diffusion equation describes the time change of a smoothly varying probability distribution function for a single particle that together with the surrounding crystal forms a system in thermal equilibrium. The motion of the particle is part of the equilibrium fluctuations of the system. The number of passages of the particle through given surfaces (namely the barriers) can be calculated by equilibrium statistical mechanics [2]. The resulting quantities in (8 a)) completely characterize the diffusion whenever each successive passage over the barrier is a step of a random walk in the lattice, directionally independent of the former one (condition (1)). Our conclusions are contrary to theories predicting "non-equilibrium" deviations from the rate theory expression for the diffusion constant. The theories of Prigogine and Bak [4] and of Rockmore and Turner [5] investigate the relaxation of an initial distribution in which one potential well is occupied and all the surrounding ones are empty. In such a distribution the density varies strongly within a lattice constant, so condition (10) clearly does not hold. Thus deviations from the rate theory really appear, however the relaxation time obtained has nothing to do with macroscopic diffusion experiments. Besides this, we emphasize again the importance of a clear distinction between loss of momentum and thermalization (this question is clearly connected with the three-dimensional saddle-point character of the barrier and the essential non-separability of the lattice potential that makes any one-dimensional treatment unrealistic) a neglecting of which causes several theories of diffusion to fail. The only effect in diffusion due to the existence of a slow variable E is a change of the distribution in time of hopping events: this will be no longer a Poissondistribution (as it is in the pure fast-thermalization case), but there will be long successions of frequent jumps (when E is high) followed by a long time with no jump, the average number of jumps per unit time being the same as it would be according to rate theory. 7. Relaxation Effects Genuine "non-equilibrium effects", if any, may be found studying the response of defect systems to external disturbances, the most typical cases being dielectric and anelastic relaxation. In these phenomena usually one has to do with

172

T . GESZTI

unequal occupations of a few neighbouring potential wells (e.g. the possible orientations of a dipole), thus the relaxation time tt is of the order of 1 ¡T, the mean time between two successive jumps. l / T is a well-defined microscopic time, and in some cases (namely under case I I behaviour) tt fa t* may occur, i.e. spatial relaxation may interfere with intra-well relaxation, which may give observable effects in resonance absorption. In this section we deal with a crystal containing a low concentration of defects, distributed over nd classes of lattice sites (potential wells). nd is usually a small number. The sites belonging to different classes differ in orientation only and they are dynamically equivalent when the crystal is unperturbed. Lifting the symmetry by external forces gives rise to the relaxation processes in question [17]. Dielectric or anelastic linear response of the defects is characterized by thermal equilibrium time correlation functions of the electric dipole moment fi a and the elastic dipole moment Xap of the defect [17] (see the Appendix: a and /? are vectorial indices). In the limiting cases I and I I we may suppose that the polarization processes contributing to the dipole moments follow the slow variables without retardation, so the average value of the dipole moments on the slow (with respect to tp ) time scale will be unique functions of the slow variables and p (easel) or f i \ ( E ) and X^^E) (case I I ) . As a further simplification we may neglect the ^-dependence in case I I and put fi\ and A® p; then these quantities are good even for the intermediate case (i is always a slow variable). This is the meaning of the common term "dipole moments in the ¿-th configuration" [17]. The time variation of i, that is the random walk of the defect from one potential well into another in thermal equilibrium, may be regarded as a stationary stochastic process. The information about this process needed to calculate linear response functions is contained in the quantity Ga> (t — t'). This is the conditional probability of finding the defect in the «-th well at time t if it had been in the ¿'-th one at t': «„,,_,, - = ^ ¡ 2 ;

, > , .

(12,

Then if A and B stand for any of the /1„ andAa/3 > w e have for the classical correlation function i id < A { t ) R > = - Z A n d i i'

i

G

i

e

( t ) B

i

(13)

' ,

where the summation goes over one representative of each nonequivalent lattice site; and for the generalized defect susceptibilities ([18], see the Appendix) oo XAB(M)

= ~

=

— n d

v

( A B )

-

i c o f ( A ( t ) B }

P

A*

[ d i i ' - i ( 0 Gi

o

z ii>

o

e ~ i m t d< v (co)] B

where c is the molar concentration of the defects (c of the host crystal, = 1 j k B T , and OO G

i f

( m )

=

/

o

G w ( t )

e ~

i m t

d t .

i r

,

(14)

1), v0 is the molar volume (15)

O n t h e T h e o r y of T h e r m a l l y A c t i v a t e d P r o c e s s e s

173

The i- and ¿'-dependence of Gn>(a)) can be diagonalized by means of the so-called "normal relaxation modes" [19] (or "symmetry coordinates of the concentration" [17]) y>a?u regarded as a column vector for i. y>aj is the basis vector belonging to the a-th row of the s-th irreducible representation of the defect symmetry group. The basis vectors satisfy the orthogonality and completeness relations Mi 2

i

2 s

..

. Wa'.i =

fa'i

< W )

sin (d + yj)

w2 — q sin a 0 .

(8 a) (8b)

Bei Einsetzen in (7 b) findet man dann nach einigen trigonometrischen Umformungen als Variante des Braggschen Gesetzes g12 = 2 d (1 — sin y

1

—

Auf gleiche Weise ergibt sich mit Randbedingung (4 b) g12 = 2 d sin §1C1 = m A0

mit

(15a)

cos xp sin (2 & — y)

^

=

1

ri^

2ö -

y - ] /

1

\

- ^ ( 2 0 - y ) / '

(15b)

wobei gemäß ( l a , c) y> = &

0

- y =

-

+ (20

-

(15c)

y)

ist. Bei verschwindendem Dekrement des Brechungsindex (Ö = 0) gehen das allgemeine Beugungsgesetz und seine Varianten in das gewöhnliche Braggsche Gesetz über, so daß in diesem Fall (1 - 6) = A = C0 = Cx = 1

und

0 = 0 = 0O =

= 0B

(16)

wird. I n allen anderen Fällen bestehen zwischen diesen Winkeln bzw. Koeffizienten Differenzen, die nicht nur von Ö, sondern auch vom Inzidenzwinkel y abhängen.

186

G. KUNZE

2.5 Symmetrische

Rückstrahl-Reflexion

I n Gleichung (11), die sich auf die asymmetrische Rückstrahlbeugung in Seemann-Bohlin-Systemen [7, 8] anwenden läßt, ist auch der Spezialfall der symmetrischen Rückstrahlbeugung, wie er in Bragg-Brentano-Systemen [9, 10] gegeben ist, mit enthalten. In diesem Fall stimmen Inzidenz- und Exzidenzwinkel miteinander überein: 2 0 - y = y ,

d.h.

0 = y .

(17a)

Diese Gleichheit ist nur realisierbar, wenn die reflektierenden Netzebenen parallel zur Kristalloberfläche liegen, d. h. gemäß Fig. 2 für (17b) f = & 0 - y = 0° . Dasselbe ergeben auch die Randbedingungen (4 a, b) bei Gleichsetzung. (17 a) und (17b) liefern mit (lc) 0

= y

=

#o

=

#1=£0t

(18a)

und für die Koeffizienten ergibt sich B = A* = Cl = Cl = l - ^

t

£ .

(18b)

An die Stelle von (11) t r i t t dann als Beugungsgesetz der symmetrischen Rückstrahl-Reflexion der streng gültige Ausdruck g12 = 2 d sin 0 y1

- ^

q = 2 d |/w2 - cos2 0 = m?.0.

(19)

Eine approximative Form hiervon wurde bereits von Darwin [2] und Ewald [4] angegeben, während die bisherige Literatur ein allgemeingültiges (kinematisches) Beugungsgesetz für die asymmetrische Rückstrahl-Reflexion, wie es Gleichung (11) darstellt, nicht aufweist. 3. Approximatives Beugungsgesetz Für viele Fälle genügt eine approximative Form des strengen Gesetzes (11). Sind nämlich \2ô — ô21\ ,1 u n d , |2 < < < > > < <
< < < > < < <
< > > > < > >

Vi & & 0 0 0 & »i

b) Ungleichungsketten, abgeleitet aus a)**) (I):

&1>@>&0>#>#b

(II): & 0 > & > & 1 > & > # B (III): & < > > & > & > & ! > # * (IV): # B > # o > # > & > & ! c) Ungleichungsketten der Koeffizienten

(I): C^A

< C 0 < ( 1 - d) < 1

(II): C 0 < A gL > 2 [23, 24], Trappeniers und Hagen [ 1 0 ] report a centre in N H 4 C 1 : C U (their centre II) with ^-values, A and B for 77 °K close to the values we obtained for our centre I at the same temperature, and which they associate with a |3 z2 — r 2 ) ground state [ 1 1 ] . They observe an analogous spectrum in N D 4 C 1 : C U which shows a resolved hyperfine structure from four equivalent Cl~ ligands, which can only be explained by an admixture of \x2 — y2) into the |3 z2 — r2) ground state. Such an admixture is also evident in their ¡/-values, since g\\ > 2, but they ignore these admixtures in their hyperfine structure analysis [11]. Their observed 2+ C I " ligand hyperfine interaction in N D 4 C 1 : C U also suggests that the Cu site is interstitial. The striking temperature dependence of the spin-Hamiltonian parameters invites a detailed analysis, which we have carried out and present in the following paper. Acknowledgements

We are greatly indebted to Mr. T. D. Smith for his helpful cooperation in growing the crystals, without which this work would not have been possible. It was appreciated that he accomplished this in view of his own very busy research programme. We would also like to thank the following: Dr. C. K. Coogan and Dr. I. D. Campbell of the Chemical Physics Division, Commonwealth Scientific and Industrial Research Organisation, Clayton, Victoria, for allowing us to use their Varian X-Band ESR Spectrometer and Variable Temperature Unit; Mr. W. R. Barry for the use of his Q-Band Spectrometer; Mr. R. L. Davis for help with the X-ray measurements and Dr. Coogan and Mr. R. H. Dunhill for comments on the manuscript. One of us (J. M. S.) acknowledges a Post-Doctoral Fellowship from the Studienstiftung des deutschen Volkes, which enabled him to work at Monash University. References [ 1 ] A . ABRAGAM a n d M . H . L . PRYCE, P r o c . R o y . S o c . A 2 0 5 , 1 3 5 (1951). [2] A . ABRAGAM, J . HOROWITZ, a n d M . H . L . PRYCE, P r o c . R o y . S o c . A 2 2 8 , 1 6 6 ( 1 9 5 5 ) . [3] A . J . FREEMAN a n d R . E . WATSON, M a g n e t i s m I I A , E d . G . T . RADO a n d H . SUHL,

Academic Press, 1955 (p. 167).

E S R Studies of Cu2+ in NH 4 C1 Single Crystals (I) [4] [5] [6] [7] [8] [9]

235

V. HEINE, Phys. Rev. 107, 1002 (1957). R . E . W A T S O N and A. J . F R E E M A N , Phys. Rev. 1 2 3 , 2027 (1961). S. GESCHWIND, Bull. Amer. Phys. Soc. [II] 8, 212 (1963). M. M. ZARIPOV and G. K . C H I R K I N , Soviet Phys. - Solid State 6, 1290 (1964). M. M. ZARIPOV and G. K . C H I R K I N , Soviet Phys. - Solid State 7, 74 (1965). M . M . ZARIPOV and G. K . C H I R K I N , Soviet Phys. - Solid State 7 , 2 3 9 1 ( 1 9 6 6 ) .

[10] H . J . TRAPPENIERS a n d S. H . HAGEN, P h y s i c a 81, 122 (1965). [11] H . J . TRAPPENIERS a n d S. H . HÄGEN, P h y s i c a 81, 2 5 1 (1965).

[12] [13] [14] [15] [16]

F. E. SIMON, Ann. Phys. 68, 241 (1922). R . E X T E R M A N and J . W E I G L E , Helv. phys. Acta 15, 453 (1943). H . ABE and H . SHIRAI, J . Phys. Soc. J a p a n 15, 711 (1960). T. J . SEED, J . chem. Phys. 41, 1486 (1964). R . W. G. WYCKOFF, Crystal Structures, 2nd edition, Vol. 1, Interscience Publishers, 1963. [17] P. DINICHERT, Helv. phys. Acta 15, 462 (1943). [18] H. BUCKLEY, Crystal Growth, 4th edition, Wiley, 1958. [19] F. EHRLICH, Z. anorg. Chem. 203, 26 (1931). [20] H . A . LEVY a n d S. W . PETERSON, P h y s . R e v . 8 6 , 7 6 6 (1952).

[21] J . S. GRIFFITH, The Theory of Transition Metal Ions, Cambridge University Press, 1961.

[22] B. BLEANEY, Phil. Mag. 41, 441 (1951). [ 2 3 ] B . B L E A N E Y , K . D . B O W E R S , and M. H . L . P R Y C E , Proc. Roy. Soc. [24] W . H A Y E S and J . W I L K E N S , Proc. Roy. Soc. A 281, 340 (1964). [25] Z . S R O U B E K and K . Z D A N S K Y , J . chem. Phys. 44, 3078 (1966).

A 228, 166 (1955).

[26] R . H . DUNHILL, J . R . PILBROW, a n d T . D . SMITH, J . c h e m . P h y s . 4 5 , 1474 (1966). [27] M.

C.

M. O'BRIEN,

Proc. Roy. Soc. (Received

A 281, 323 (1964).

November

30,

1966)

J . R. PILBROW and J . M. SPAETH: ESR Studies of Cu 2+ in NH 4 C1 (II)

237

phys. stat. sol. 20, 237 (1967) Subject classification: 19; 22 Physics Department, Monash University, Clayton,

Victoria

ESR Studies of Cu2+ in NH4C1 Single Crystals between 4.2 and 453 °K II. Theoretical Analysis: Vibrational Admixtures, Spin Polarization, and for Cu2+ Ions By J . R . PILBROW a n d J . M. SPAETH 1 ) The temperature dependence of the spin-Hamiltonian parameters for two Cu 2+ centres in NH4C1 single crystals, which were reported in the previous paper (I), is analysed in detail. The results show that, apart from zero point vibrations, higher lattice modes also contribute to the admixture of the |a;2 — y2) state into the |3 z2 — r 2 > ground state for these centres. For one of the centres there is also evidence t h a t vibrations and reorientations of a single nearest neighbour NH 4 + ion contribute to the admixture. From the analysis, the temperature dependence of ~3/3z* _ r 2 , a crystal field splitting parameter and the contact contribution to the hyperfine field a t the Cu 2+ nucleus is obtained. All these quantities vary strongly with temperature and show sudden changes at the X-point ( — 30.5 °C) for NH4C1. The temperature dependence of the contact term in the hyperfine interaction can be qualitatively understood in terms of core spin polarization by taking into account the observed changes in the radial distribution of the 3d ground state. The analysis gives further support for the models proposed for the two Cu 2+ centres. Es wird eine eingehende Analyse der Temperaturabhängigkeit der Spin-Hamilton-Parameter von zwei Cu 2 + -Zentren in NH 4 C1-Einkristallen, über welche in der vorangehenden Arbeit (I) berichtet wurde, durchgeführt. Die Ergebnisse zeigen, daß abgesehen von Nullpunktsschwingungen auch höhere Gitterschwingungen zu der Zumischung des \x2 — «/2> Zustandes in den |3 z 2 — r2> Grundzustand f ü r diese Zentren beitragen. F ü r eines der Zentren wird gezeigt, daß auch die Schwingungen und Reorientierungen eines einzelnen nächstbenachbarten NH 4 + -Ions zu der Zumischung beitragen. Die Analyse ergibt die Temperaturabhängigkeit von 3Z! _ r !, eines Kristallfeld-Aufspaltungsparameters und des FermiKontaktbeitrages zum Hyperfeinfeld am Cu 2 + -Kern. All diese Größen ändern sich sehr stark mit der Temperatur und zeigen plötzliche Änderungen beim X-Punkt ( — 30.5 °C) von NH4C1. Die Temperaturabhängigkeit des Kontaktterms in der Hyperfeinwechselwirkung kann qualitativ auf Grund der „core spin polarization" verstanden werden, wenn man die beobachteten Änderungen der Radialverteilung des 3d-Grundzustandes berücksichtigt. Die Analyse gibt weitere Begründungen f ü r die für die zwei Cu 2+ -Zentren vorgeschlagenen Modelle.

1. Introduction In Part I 2 ), we reported the temperature dependence of hyperfine interactions and ^-factors for two Cu2+ centres in NH4C1. For these centres we tentatively proposed models. Centre I consists of an interstitial Cu2+ ion surrounded by four Cl~ ligands, with one nearest neighbour (n.n.) NH4 ion and a n.n. NHJ Post-Doctoral Fellow of Studienstiftung des deutschen Volkes. On leave of absence from I I . Physikalisches Institut, Technische Hochschule Stuttgart. 2 ) P a r t I see phys. s t a t . sol. 20, 225 (1967).

238

J . R . PILBEOW a n d J . M . SPAETH

ion vacancy (see Fig. 6 of Part I), whereas centre I I differs only in having two n.n. NHi ion vacancies. Here we present a detailed analysis of the temperature dependence of the measured spin-Hamiltonian parameters. The ¡/-values are discussed in terms of vibrational admixtures of the \x 2 — y 2) state into the |3 z2 — r 2 > ground state, and the hyperfine contact interaction in terms of core spin polarisation. All the experimental results quoted are given in Part I. 2. Theory of Cu2+ Spin Hamiltonian in a Tetragonal Crystalline Field The energy levels of the Cu 2+ ion (3d 9 ; 2 D configuration) in octahedral cubic and tetragonal crystalline fields are shown in Fig. 1 [1, 2]. From the symmetry of the crystalline field alone, it is impossible to decide whether |3 z2 — r 2 > or \x 2 — y 2) is lowest. From the observed gr-values, however, we will show that the ground state is [3 z2 — r 2 ). Spin-orbit coupling mixes the eg and t2g orbitals, but does not cause any mixing within the eg orbitals. The wave functions, correct to second order in X

X

X

—7, — , and — , are given explicitly elsewhere [3], where X is the spin-orbit coupling constant for Cu 2 + . We define the following parameters: 3 ) ^

=

A N D

^

=

W

The wave functions correct to second order in £ are denoted by |3 z2 — r 2 ) ' and I* 2 -

y 2)'-

The appropriate Hamiltonian for the Zeeman and hyperfine interactions for the 3d9 configuration, is in the usual notation [2], fiH(L + 2 S) + X = +

p \ l • / -

K S • I + y [4 S • I -

le> = l

3

( L • I ) { L • S ) j . (2)

x 2-y 2>

\6>=\3z z-r z> Free ion + cubic field + tetragonal field

{L • S) (L • I) -

Fig. 1. Energy level diagram for Cu 3+ in an octahedral and tetragonal crystalline field. Notation for wave functions as in [2]

) i , p, and q are related to u, v, and w of [1, 3] by v = —{, w =-= —pi,

and u = —q f.

ESR Studies of Cu2+ in NH4C1 Single Crystals (II)

239

H e r e P = 2 y (3 ~ 3 ) 4 ), w h e r e y is t h e m e a n n u c l e a r g-iactor f o r t h e t w o c o p p e r isotopes Cu 6 3 a n d Cu 6S , a n d (r~s~) t h e m e a n inverse cube r a d i u s for t h e g r o u n d s t a t e (y = 1.513). T h e t e r m in x, t h e F e r m i c o n t a c t t e r m , is i n t r o d u c e d phenomenologically t o give a g r e e m e n t w i t h t h e e x p e r i m e n t a l results, k is a numerical p a r a m e t e r , for convenience, b u t — x P h a s t h e dimensions of t h e interaction. E q u a t i o n (2) is e v a l u a t e d for t h e |3 z2 — r 2}' a n d \x 2 — y' 1/ s t a t e s a n d t h e n m a t c h e d w i t h t h e spin H a m i l t o n i a n (equation (1) of P a r t I) w i t h i n t h e m a n i f o l d of s t a t e s I, Ms, M7>, w h e r e S = 1/2, / = 3/2 a n d t h e Ms a n d Mt are respect i v e l y t h e electron spin a n d nuclear m a g n e t i c q u a n t u m n u m b e r s . T h u s *—r', i.e. ( i 1 " 3 ) ^ .

Fig. 5 shows ,. Hc is negative and of the order of magnitude predicted for 3d configurations by Freeman and Watson [21] from calculations of spin polarization. Below Tx, —H c decreases rapidly for both centres, but shows a minimum for centre I at about 100 °K, and increases again towards 4 °K. Above Tx, —H c decreases for centre I but is constant for centre II. Such a temperature dependence is not easy to explain in terms of configurational mixing of 4 s electrons, whereas the results can be understood qualitatively in terms of spin polarization [21]. The net spin polarization of closed s shells at the nucleus due to exchange interactions between the 3d electrons and the electrons in the closed Is, 2s, and 3s shells was calculated for theMn 2+ (3d 5 ), Fe 2+ (3d 6 ), and Ni 2+ (3d 8 ) free ions and for Ni 2+ ions in a cubic field. Unfortunately, calculations are not yet available for Cu 2+ ions. Watson and Freeman found for the above mentioned cases t h a t the contribution to H c is small and negative for the Is shell, large and negative for the 2s shell, and quite large but positive for the 3s shell. The net hyperfine field at the nucleus is found to be negative and this is confirmed by our results. Thus the resultant Hc arises from the competition between terms of opposite signs. The Is and 2s electrons with spin parallel to the 3d spin are "attracted" outwards leaving a net negative spin density at the nuclear site and hence a negative contribution to H c . The 3d electron distribution overlaps strongly with the 3s shell so t h a t the latter lies neither completely "inside" nor "outside" the 3d function. The 3s contribution, therefore, depends strongly on the radial distribution of the d-function. If the 3s shell were completely "inside" the 3d shell its contribution to H c would be negative and much larger numerically than the negative contribution from 2s, while if it were wholly "outside" the 3d shell its contribution would be

E S R Studies of Cu 2+ in NH4C1 Single Crystals (II)

247

positive and large. The overlapping of the 3d electrons with the 3s shell leads to competing tendencies leaving, as a result, a positive value smaller in magnitude than the negative 2s contribution. The behaviour of 32>_r! (Fig. 5) shows that between Tx and about 100 ° K the radial 3d distribution contracts for both centres, thus the positive part of the 3s contribution increases and hence —H c decreases ("3d moves more inside the 3s"). For centre I, below 100 ° K , (r~3ySzt-r% shows a further sharp increase and thus the wave function contracts still more, and parallel with it, —H c increases again. This can be explained since the 3d electrons are now close to, but still outside the 2s shell whose negative contribution to —H e must increase, whereas the positive contribution of the 3s shell will decrease, these electrons being more and more "outside" the 3d orbital. Hence, we conclude that the behaviour of Hc below Tx, for both centres, can be understood qualitatively in terms of spin polarization. For a 3d 8 Fe configuration, Watson and Freeman showed (Figs. 2 and 3 of [21]) that He is very sensitive to the value of r at which the radial wave function peaks. In these terms it is also readily understood why H c for centre I I is nearly constant above Tx, since (r~3/3zi ri (Fig. 5) is also constant in this temperature range. This observation is further experimental support for the proposed explanation of the s-contact hyperfine interaction of Cu 2 + ions. For centre I, one would expect on the same basis, since the 3d wave function still expands, that —H c would increase. That this is not the case indicates that the reorientation motion of the single NH4 ion has a dominant influence. During reorientation of the NH4 ion about the tetragonal axis, its electron distribution "passes through" the |3 z 2 — r2}' lobe and through overlap 3d spin density is "removed", thus decreasing the polarizing effect. The higher v0I the more the 3d spin density overlaps the NH4 distribution on the average. Thus with increasing temperature it can be understood why |// c | increases. Charge transfer as well as overlap may occur, but both would have the effect of removing 3d spin density. Below Tx, the experimental results for which is closely related to the peak position of the radial wave function, and Hc provide a good basis for comparison with future unrestricted Hartree-Fock (U.H.F.) calculations for Cu 2 + ions in NH4C1. So far U.H.F. calculations have only been compared with experiments for a few ions in various environments, and the agreement for Hc was only fair. This is not surprising since / / c is calculated from the differences between pairs of numbers which are large compared with the size of the resultant effect. 5. Conclusion On the basis of the two models the temperature dependence of the experimentally observed g-values and hyperfine interaction, and the parameters obtained from them can be satisfactorily understood. Whether or not the N H J ion vacancies contain water molecules we cannot say. We have found experimental evidence that as well as zero point vibrations as discussed by O'Brien [5], there are higher lattice modes contributing to the admixture of \x2 — y2/ into the |3 z2 — r2}' ground state. We have furthermore established experimentally for the first time, as far as we know, that the contact contribution to the hyperfine field at the Cu 2 + nucleus is strongly dependent on the spatial distribution of the 3d wave function.

248

J . R . PILBROW a n d J . M . SPAETH: E S R S t u d i e s of C u 2 + i n N H 4 C 1 ( I I )

A

cknowledgements

We wish to thank Dr. C. K. Coogan, Division of Chemical Physics, Commonwealth Scientific and Industrial Research Organization, Clayton, Victoria, for many helpful discussions and for comments on the manuscript, and Mr. R. H. Dunhill for reading the manuscript. One of us (J.M.S.) acknowledges a Post-Doctoral Fellowship from the Studienstiftung des deutschen Volkes. References [ 1 ] B . B L E A N E Y , K . D . BOWERS, a n d M . H . L . PRYCE, P r o c . R o y . S o c . A 2 2 8 , 1 6 6 ( 1 9 5 5 ) .

[2] J. S. GRIFFITH, The Theory of Transition Metal Ions, Cambridge University Press, London 1961. [ 3 ] W . H A Y E S a n d J . WILKENS, P r o c . R o y . S o c . A 2 8 1 , 3 4 0 ( 1 9 6 4 ) . [ 4 ] U . OPIK a n d M . H . L . PRYCE, P r o c . R o y . S o c . A 2 3 8 , 4 2 5 ( 1 9 5 7 ) . [ 5 ] M . C. M . O ' B R I E N , P r o c . R o y . S o c . A 2 8 1 , 3 2 3 ( 1 9 6 4 ) . [ 6 ] L . BELFORD, M . CALVIN, a n d G. BELFORD, J . c h e m . P h y s . 2 6 , 1 1 6 5 ( 1 9 5 7 ) . [ 7 ] D . S. MCCLURE, S o l i d S t a t e P h y s . 9 , 4 0 0 ( 1 9 5 9 ) . [ 8 ] L . P . H . BOVEY a n d G. B . B . M . SUTHERLAND, J . c h e m . P h y s . 1 7 , 8 4 3 ( 1 9 4 9 ) .

[9] L. P. H. BOVEY, J. chem. Phys. 18, 1684 (1950); J. Opt. Soc. Amer. 41, 836 (1951). [ 1 0 ] H . S . GUTOWSKY, G. E . P A K E , a n d R . BERSOHN, J . c h e m . P h y s . 2 2 , 6 4 3 ( 1 9 5 4 ) . [ 1 1 ] R . BERSOHN a n d H . S . GUTOWSKY, J . c h e m . P h y s . 2 2 , 6 5 1 ( 1 9 5 4 ) . [ 1 2 ] H . A . L E V Y a n d S . W . PETERSON, P h y s . R e v . 8 6 , 7 6 6 ( 1 9 5 2 ) . [ 1 3 ] A . H . COOKE a n d L . E . DRAIN,

Proc. P h y s . Soc. A 65, 894 (1952).

[14] C. K. COOGAN, J. chem. Phys. 43, 823 (1965). [ 1 5 ] A . ABRAGAM,

J . HOROWITZ, a n d M . H . L . PRYCE, P r o c . R o y . S o c . A 2 8 0 , 1 6 9 ( 1 9 5 5 ) .

[16] Z. SROUBEK and K. ZDANSKY, J. chem. Phys. 44, 3078 (1966). [ 1 7 ] R . H . DUNHILL,

J . R . PILBROW, a n d T . D . SMITH, J . c h e m . P h y s . 4 5 , 1 4 7 4 ( 1 9 6 6 ) .

[18] R. E. WATSON, Tech. Report No. 12, Solid State and Molecular Theory Group, M.I.T., June 15, 1959 (unpublished). [19] W. Low, Solid State Phys. Suppl. 2, 11 (1960). [ 2 0 ] A . J . FREEMAN

a n d R . E . WATSON,

Magnetism IIA,

E d . G . T . RADO

Academic Press, 1965 (p. 167). [ 2 1 ] R , E . WATSON a n d A . J . FREEMAN,

(Received

P h y s . R e v . 123, 2027 (1961).

November 30,

1966)

a n d H . SUHL,

J . SEDIVY u n d H . SICHOVA: Messung d e r D e b y e - T e m p e r a t u r

249

p h y s . s t a t . sol. 20, 249 (1967) Subject classification: 8 ; 21.6 Lehrstuhl für Festkörperphysik der Mathematisch-Physikalischen Karlsuniversität, Prag

Fakultät,

Röntgenographische Messung der charakteristischen Debye-Temperatur bei vielkristallinem Silber mit Anwendung der Korrektur auf primäre Extinktion Von J.

SEDIVY

und H. SicHovÄ

Die D e b y e - T e m p e r a t u r & von polykristallinem Silber wird aus R ö n t g e n s t r a h l e n b e u gungsmessungen b e s t i m m t , die n u r bei Z i m m e r t e m p e r a t u r d u r c h g e f ü h r t w u r d e n . Die Met h o d e (unter der Bezeichnung „single-temperature m e t h o d " b e k a n n t ) b e r u h t auf einer relativ g e n a u e n K o r r e k t u r der ursprünglichen E x t i n k t i o n . Sechs auf verschiedene Weise p r ä p a r i e r t e polykristalline Silberproben w u r d e n u n t e r s u c h t . Der Mittelwert aus diesen Messungen ergibt einen W e r t f ü r & von (220 ^ 8) °K. Die Differenzen zwischen den (9W e r t e n d e r verschiedenen P r o b e n werden diskutiert, wobei berücksichtigt wird, d a ß jede P r o b e z e h n m a l u n t e r s u c h t wurde. T h e D e b y e t e m p e r a t u r e 0 of polycrystalline silver a t room t e m p e r a t u r e is d e t e r m i n e d f r o m X - r a y d i f f r a c t o m e t e r m e a s u r e m e n t s a t room t e m p e r a t u r e only. T h e m e t h o d (which is k n o w n as t h e " s i n g l e - t e m p e r a t u r e m e t h o d " ) is based on a relatively precise correction for t h e effect of t h e p r i m a r y extinction. Six polycrystalline silver specimens p r e p a r e d in various w a y s are investigated. The average of these m e a s u r e m e n t s gives a final value for 0 of (220 ^ 8) °K. T h e difference between t h e values o b t a i n e d for 0 for p a r t i c u l a r specimens is discussed t a k i n g i n t o a c c o u n t t h e f a c t t h a t each sample was studied 10 times.

1. Einleitung und Problemstellung Die charakteristische Debye-Temperatur (91) von Silber wurde bereits von vielen Autoren bestimmt. Eine kurze Übersicht der bisherigen Ergebnisse ist in Tabelle 2 angeführt und diskutiert. Die D-T wird bei vielkristallinen Stoffen röntgenographisch meistens durch Messungen bei zwei unterschiedlichen Temperaturen ermittelt. Diese Methode werden wir weiterhin als Zweitemperaturmethode bezeichnen (siehe z. B. [1, 2 und 3]). Die vorliegende Arbeit stellt sich als Ziel, den Wert von © bei Zimmertemperatur für vielkristallines Silber mittels röntgenographischer Messung nur bei Zimmertemperatur zu bestimmen. Diese Methode werden wir weiterhin als Eintemperaturmethode bezeichnen. Die erfolgreiche Anwendung der Eintemperaturmethode bei Untersuchungen mittels des Zählrohrgoniometers fordert die Erfüllung der nachstehenden Voraussetzungen : a) genaue Kenntnis der Abhängigkeit des Atomfaktors des untersuchten Stoffes vom Braggschen Winkel, b) genaue Justierung und glatte Oberfläche der flachen Probe, I m folgenden kurz D - T oder n u r

250

J.

SEDIVY

und H. SfcirovÄ

c) Abwesenheit von Vorzugsorientierung (Textur) der Kristallite in der Probe, d) die genaue Erfassung des Einflusses der Extinktion, besonders der primären Extinktion. Die Voraussetzung a) k a n n man bei Silber als erfüllt betrachten; eine neue genauere Berechnung war nur bei den Übergangsmetallen nötig [4]. Die Korrektur des Einflusses der anomalen Dispersion ist dabei berücksichtigt worden. Die entsprechende Lage des Präparathalters wurde mittels eines Fernrohres genau justiert. Die glatte Oberfläche ist durch sorgfältiges Pressen des Silberpulvers mit einer Teilchengröße von ca. 1 [xm erreicht worden. Die Abwesenheit von Vorzugsorientierung, die Teilchengröße sowie die innere Spannung haben wir mittels Film-Rückstrahlaufnahmen kontrolliert. Bei der Auswertung der Intensitätsmessungen h a t sich eine relativ genaue Korrektur des Einflusses der primären Extinktion als notwendig erwiesen. Wir haben eine entsprechende Methode ausgearbeitet, die im weiteren behandelt wird. 2. Herstellung der Proben Für die genaue Erfassung des Einflusses der primären Extinktion hielten wir eine Kristallitgröße (d. h. die Größe der kohärenten Gebiete) im Intervall von i0 = 1 X 10" 5 bis 8 X 10~5 cm am vorteilhaftesten. Als geeignetste Methode zur Herstellung des Silberpulvers h a t sich die chemische Methode nach H u n d und Müller [5] erwiesen. Die Spektralanalyse des hergestellten Silbers f ü h r t e zu folgendem Ergebnis: 99,99% Ag, Spuren von Cd, Co, Cu und P d . Auch auf diese Weise hergestelltes Silberpulver weist manchmal nicht die optimale Größe der kohärenten Gebiete auf. Auch die Kristallitgröße variiert in einem relativ breiten Intervall. Aus diesem Grund wurde das zu untersuchende Pulver in einer Achatschale gerieben u n d eventuell einer Glühung in evakuierten und abgeschmolzenen Glasröhrchen unterzogen (siehe Tabelle 1). 3. Die Meß- und Auswertungsmethode Die Messungen wurden mittels eines sowjetischen Zählrohrgoniometers URS-50-I nur bei Zimmertemperatur ausgeführt. Als Strahlung wurde mit L i F monochromatisierte Kupferstrahlung verwendet „„ = 1.542 A). Die Ausmessung jeder vorkommenden Debye-Scherrer-Linie bei jeder Probe wurde insgesamt lOmal wiederholt. Die Intentsität der Linien ist immer durch Planimetrieren sowie auch durch Impulszählung ermittelt worden. Das Silberpulver ist in einen Präparatträger aus Messing eingepreßt. Die Abmessungen der eigentlichen Probe waren 20 X 10 X 1 mm 3 . Die gemessenen Intensitäten weisen bei der gleichen Probe eine gute Reproduzierbarkeit auf, der mittlere Fehler von 10 Messungen beträgt ca. + 2 % . Zur Auswertung der Meßergebnisse und damit auch zur Bestimmung der D-T geht man vom Logarithmus der Intensität der Debye-Scherrer-Linie aus: (1)

g ist die gemessene integrale Intensität der Linie (hkl). E ist der Extinktionsfaktor, der in unserem Fall unter Anwendung eines Monochromators nach

Röntgenographische Messung der charakteristischen Debye-Temperatur

251

Zachariasen [6] und Lang [7] durch _ tgh n g„ + cos 2 2 ot |cos 2 &\ • tgh (n q0 |cos 2 fl|) — n q0 (1 + cos 2 2 The oxide window was fabricated by planar technique. The ohmic contact on PbS was achieved by evaporated gold films in both cases. In order to grow the PbS layers, the crystals were mounted on a metallic support and submerged into a solution consisting of 2 parts 0.39 molar thiourea, 1 part 0.33 molar lead acetate and 0.003 parts 50% hydracine hydrate. After 3 min 0.3 parts 17.5 molar NaOH have been added. The reaction lasted about five minutes. Then the samples were rinsed with distilled water and methanol. 3. Structure of the PbS Films The PbS film structure was investigated by transmission electron diffraction. In order to remove the PbS films from the substrate the crystals were submerged shortly in a 1 molar HC1 solution. A typical transmission pattern is shown in Fig. 2. The apparent Debye-Scherrer rings indicate a polycrystalline structure

F i g . 2. Electron diffraction pattern of a P b S layer

257

Electrical and Photovoltaic Properties of PbS-Si Heterodiodes

of the P b S films. Therefore, the corresponding films of the P b S - S i heterodiodes will also be of polycrystalline material, provided that during the preparation process the original structure had not been modified significantly. 4. Electrical Properties The U-I characteristic of the heterodiodes in the dark and under illumination are shown in Fig. 3. I f the Si side of the heterojunction is positive biased, the reverse characteristic of the diodes is observed. The reverse current below 1 V bias is proportional to Un, with n varying between 1 and 0.5. For larger bias the current is proportional U; for bias exceeding about 30 V strong irreversible changes in the U-I characteristic are observed. Forward characteristics are shown in more detail in Fig. 4. All characteristics change their slopes at about 0.1 V. According to the theories of Riben [13] and Donnelly [14] it is supposed, that the forward current below this point is a pure recombination current, and above this point it is mainly a tunnelling current. The recombination current is proportional to exp (e UjnkT), n being a characteristic constant varying between 1.3 and 2 according to Fig. 4. On the other hand, the temper ature-dependent tunnelling current is proportional to exp (A U), where the constant A ranges between 20 and 30 V - 1 . Further measurements confirmed the independence on temperature of the constant A.

urn—perature 17

physica/20/1

Fig. 4. Forward characteristics of different P b S - S i heterodiodes at room temperature in semilogarithmic plot

258

H . SIGMUND a n d K . BEBCHTOLD

ft

SN2

° 300°K *170°K

•• ,4If ^ :

\\ X \\

V

1.0

1.5

2.0

2.5 3.0 Mpmi

MfimlFig. 6. Relative spectral response of two lieterodiodes a t room temperature. The response refers to equal irradiated power

3.5

Fig. 5. Spectral response of a typical P b S - S i heterodiode at 300 and 170 °K. The response refers to equal irradiated power

5. Photovoltaic Properties The spectral response of the heterojunctions (Fig. 5 and 6) shows the "window effect" which is caused by the different band gaps. The peak of the photovoltage at 1 ¡xm (Si absorption) depends on the temperature in contrast to the peak at 2 (jim (PbS absorption, see Fig. 7). The temperature coefficient of the Siphotopeak in the measured temperature range is approximately 1.6% per deg. At low temperatures (170 °K) the known extension of the spectral response of PbS to longer wavelengths has been observed (Fig. 5). The absolute sensitivity of several diodes at room temperature was measured. The radiant power of a black body at 1000 °C incident to the sensitive area was 1.5 X 10~5 W. The heterodiodes exhibit a sensitivity of /Shiack, 1000 "c = 200 V / W . The spectral response at 1 [JUII is ^m = 0.8 to l x l 0 4 V / W ; at 2.4 ¡xm 4 ^A=2.4nm = 0.5 to 0.7 X 10 V/W. The photovoltage is up to 20 mV proportional to the incident power.

Î3 S

1

2

7 0

220

2b0 ' .

260

280

300> Fig. 7. Temperature dependence of the photovoltage at 1 (¿m (Si »absorption) and 2 [jtm (PbS absorption)

Electrical and Photovoltaic Properties of PbS-Si Heterodiodes

259

6. Discussion According to the electron diffraction patterns, the PbS layers on the Si substrate exhibit obviously polycrystalline structure. However, due to the experimental results concerning the U—I characteristic of the diodes, the heterojunction model after Riben [13] and Donnelly [14] can be applied. Hence, the forward current below 0.1 V is a recombination current, for larger bias it is mainly a tunnelling current. The so-called "power-law", however, for the reverse current could not be confirmed. Thermoelectric measurements indicate p-conduction of the deposited P b S films. Therefore, the device can be regarded as a p - P b S - n - S i heterojunction. The measurements of spectral response and sensitivity promise an application of this device as an /i?-detector in the wavelength range from 1 to 4 [j.m. Preliminary measurements indicate a time constant of several [is. Acknowledgements We are indebted to Dr. I . Ruge and Dr. W. Harth for stimulating discussions. We also would like to thank Mrs. H. Leitermann for technical assistance. References [1] [2] [3] [4] [5] [6] [7]

R . RUTH, J . C. MARINACE, and W . DUNLAP, J . appi. P h y s . 3 1 , 9 9 5 (1960). J . C. MABINACE, I B M J . Res. Developm. 4, 2 4 8 (1960). R . L . ANDERSON, Solid S t a t e Electronics 5, 341 (1962). R . H . REDIKER, S. STOPEK, and J . H . WARD, Solid S t a t e Electronics 7, 621 (1964). L . Y . W E I and J . SHEWCHUN, P r o c . I E E E 5 1 , 9 4 6 (1963). J . SHEWCHUN and L . Y . W E I , J . Electrochem. Soc. I l l , 1 1 4 5 (1964). W . G. OLDHAM and A. G. MILNES, Solid S t a t e Electronics 7, 153 (1964).

[8] M. J. HAMPSHIRE and G. T. WRIGHT, Brit. J. appi. Phys. 15, 1331 (1964).

[9] S. YAWATA and R . L . ANDERSON, phys. stat. sol. 12, 2 9 7 (1965). [10] J . P . DONNELLY and A. G. MILNES, Solid S t a t e Electronics 9, 174 (1966). [11] J . L . DAVIS and M. K . NORR, J . appi. Phys. 37, 1 6 7 0 (1966).

[12] J . R. DALE, phys. stat. sol. 16, 351 (1966). [13] A. RIBEN, Thesis, Carnegie Institute of Technology, Febr. 1965.

[14] J . P . DONNELLY, U . S . A r m y Research, Contract D A - 3 1 - 1 2 4 - A R O (D)-131. ( Received December

17*

20,

1966)

S. U. DZHALILOV and K. I. RZAEV: Phenomenon of Selenium Vitrification

261

phys. stat. sol. 20, 261 (1967) Subject classification: 2; 22.1.3 Institute of Physics, Academy of Sciences of the Azerbaidzhan SSR,

Baku

On the Phenomenon of Selenium Vitrification By S . U . DZHALILOV a n d K . I . R Z A E V

The temperature range for the vitrification of selenium (18 to 36 °C) is determined from volumetric measurements. This is correlated with that obtained from data on the temperature dependence of the viscosity (the function aT), the specific heat, and the velocity of propagation of ultrasonic waves in selenium. It is shown for the first time that the vitrification temperature T g of selenium depends on the experimental time base. A drop in the cooling rate of two orders of magnitude (from 0.5 to 0.005 deg/min) cause the Tg value to be decreased by 9 °C. Increasing the concentration of Te in selenium lowers the vitrification temperature of the Se-Te co-polymer according to a linear relation. The vitrification temperature of the specimen increases linearly with the concentration of Sb in selenium. This is explained as being due to the binding action of Sb, which consists of the formation of chemical cross-links between the selenium molecules. Der Temperaturbereich für die Verglasung von Selen (18 bis 36 °C) wird aus volumetrischen Messungen bestimmt. Er wird in Beziehung gesetzt zu den Werten, die aus der Temperaturabhängigkeit der Viskosität (der Funktion ay), der spezifischen Wärme und der Ausbreitungsgeschwindigkeit von Ultraschallwellen in Selen erhalten wurden. Es wird gezeigt, daß die Verglasungstemperatur Tg von Selen von der experimentellen Zeitbasis abhängt. Eine Senkung der Abkühlgeschwindigkeit um zwei Größenordnungen (von 0,5 zu 0,005 grd/ min) verringert den Tg-Wert um 9 °C. Eine wachsende Te-Konzentration in Selen erniedrigt die Verglasungstemperatur des Se-Te-Kopolymers gemäß einer linearen Beziehung. Die Verglasungstemperatur der Probe steigt linear mit der Sb-Konzentration in Selen an. Dies wird durch die Bindung von Sb erklärt, die in der Bildung von chemischen Kreuz-Verbindungen zwischen den Selenmolekülen besteht. 1. Introduction The vitrification temperature, Tg, of selenium was determined previously by different methods including measurements of the temperature dependence of volumetric expansion, specific heat, dielectric constant [1], and electric conductivity [2], The values of T g found in these works are within the temperature range 2 9 . 8 ± 0.3 °C t o 30.7 + 0.6 °C. I t has been shown in a paper [3] dealing with measurements of temperaturespecific gravity relations for pure selenium annealed at different temperatures (270, 375, and 5 0 0 °C) t h a t the value of Tg is within the range 31.0 + 0.5 °C and does not depend on the temperature of annealing. However, until now not a single paper has ever analysed the question of different values of T% for selenium obtained in various experiments of diverse nature. (At any rate we do not know any publication dealing with this question.) There is a trend to consider T g of selenium as a fixed temperature equal t o approximately 31 °C. I t seems t o us t h a t the disagreement of the T g values obtained in various experiments m a y be hardly explained only by experimental errors. Taking into

262

S . U . DZHALILOV a n d K . I . RZAEV

account the polymeric nature of selenium, the above disagreement of the Tg values may be explained by the influence of the experimental time base which varies for experiments different in nature. It has been proved that the value of Tg for polymers depends on the rate of heating (or cooling) [4 to 6]. An investigation of factors causing variations in the Tg value is of great interest, as above the vitrification temperature (within the range of Tg and Tg -f+ 100 °K) all viscoelastic and viscofluid properties of polymeric materials are determined by the magnitude of Tg [7]. In addition, an investigation of the process of vitrification of selenium, which is typical of inorganic polymers, may in general throw more light on the vitrification nature that cannot be as yet looked at as being completely elucidated. 2. Experimental Procedure In this work the method of volumetric dilatometry described elsewhere [14] was used to determine T g . Specimens for the experiments were obtained by quenching a high-purity selenium melt to 0 °C. All specimens were melted in evacuated Pyrex glass ampoules during 2 hours at a temperature of 350 °C and then cooled at the same rate. The temperature of the heat bath was held constant (to within + 0.1°) during the experiments. Distilled water was used as a dilatometric fluid since it does not react with selenium at low temperatures and meets the requirements for dilatometric fluids. The level of the fluid meniscus in the capillary tube of the dilatometer was determined by a cathetometer. The diameter of the capillary tube was 2.1 mm. The weight was 60 g at a diameter of 15 mm. 3. Discussion of Results In Fig. 1 the dependence of the specific volume V on temperature T is presented at a cooling rate of W = 0.25 deg/min. The value of the non-equilibrium temperature of vitrification, Tg, was determined as a crossover point resulting from the continuation of the low- and high-temperature branches of the curve V(T), which proved to be equal to 31 °C. A good arrangement of experimental

Fig. 1. Effect of temperature on the specific volume F, the functions ar [10], the specific heat Cp [8], and the propagation velocity of longitudinal ultrasonic waves, F] 19]. The upper abscissa gives T — TB; read 10 s vi instead of ve on the right-hand ordinate

On the Phenomenon of Selenium Vitrification

263

points on the high temperature side of the straight line indicates that as the specimen is cooled, its volume has enough time to assume an equilibrium value. However, at temperatures lower than 36 °C the experimental points practically begin to deviate from the straight line corresponding to the equilibrium of the fluid. This temperature Tg corresponds to the beginning of the vitrification region. At temperatures lower than 18 °C the experimental points lie again on the straight line. The temperature of Tg = 18 °C seems to correspond to the lower limit of selenium vitrification. To compare the vitrification region obtained from dilatometric data with its behaviour in other experiments, the temperature dependence of the function aT, entering in the Williams-Landel-Ferry (WLF) equation [7], and that of the specific heat Cp [8] and the propagation velocity v of longitudinal ultrasonic waves [9] are presented in Fig. 1. The comparison of these curves with that of V(T) shows the existence of a certain correlation for the vitrification region in different experiments. In this case one should pay attention to the experimental data on the temperature dependence of the specific heat, since the time base of the experiment for evaluating specific heat is close to that of the dilatometric measurements. Due to this, one can expect a better agreement of data related to the vitrification region. The results of the dilatometric data show good agreement with the value of the function aT. At temperatures of 36 and 18 °C the experimental values of aT are intersected by a curve (solid line) evaluated on the basis of the W L F equation for Ts = 68 °C [10]. A continuation of the low- and high-temperature branches of the propagation velocity curve, vh gives a crossover point corresponding to a temperature of approximately 35 °C. This suggests the idea that the propagation velocity of ultrasonic waves in selenium responds to the beginning of the temperature region of vitrification Tg, rather than to Tg. I t should be stressed that an accurate determination of the lower edge of the vitrification region, Tg, presents some experimental difficulties and depends on the accuracy of the experiment and its time base [11]. The nature of the vitrification region deduced from the CP(T) curve also agrees with the values found by volumetric measurements. This leaves no doubt that the vitrification region of selenium extends from 18 to 36 °C. The macroscopic coefficients of thermal expansion have values of oij = 5.425 X X 10~ 4 deg - 1 and a g = 1.640 X 10~ 4 deg - 1 above and below T g , respectively. According to the iso-free-volume hypothesis of Fox-Flory [12], the fraction of free volume, f(Tg), and the thermal expansion coefficient of free volume, a f , at the temperature of vitrification must be equal to 0.025 and 4.85 X 10~ 4 deg - 1 , respectively. The values of f(Tg) and a, determined by us for selenium proved to be equal to 0.030 and 4.82 X 1 0 - 4 deg - 1 , respectively. The coefficients and a g are related to T by the Simha-Boyer relation [13] (a, - a g ) Tg = 0.11 .

_

(1)

A check of the relation (1) for selenium shows that it is met adequately: At Tg = 304 °K we get -

= i «>( + , n) = C (1 + x y zf , »=0

= 2 «*">( + , n) = C (z x + yf n=0

.

The probability to find a negative magnetization in the centre of the above chain is given by oo

p^

= 2

«=0

2

w(-,

n) = C y {x + y zf ,

2)L"

= 2

>

n=0

«>(-,

n) = C y (z + x yf

.

With the relations n i6' ( + ) ( + , n) = z^~w \ . / « + » — wAl sin v + sin u — sin I 1 — sin I 1 +

(

u — vw\

(u -(- k)2 — W2

. ¡U — V — wW 1 — sin I

+ four additional cyclicly permuted terms j ,

ax a

setting u = —^r— , v =

qv a

q, a

,w=

,.

1 + (10)

,

(a lattice constant).

The corresponding expression with a b.c.c. crystal is 2 9

^

=

i

u

(m* - v*) (m - w ) l 2

2

~2

WSin

^COS U

+

C0S V +

* + (usinu + vsmv +

C0S W

u — u sin -^-j3 u \+ two additional cyclicly permuted terms. + w sin w) cos — (ii) I t is easy to prove the desired behaviour f0;

q =

U;

q = 0

0,

of these expressions. Also the cubic symmetry, resulting from taking into account the shape of the atomic polyhedron, is easily seen. The important difference from the factor g ^ i q ) , proposed by Srivastava and Dayal [3], is the fact that this factor is zero for \q\ = \K\. The electron scattering seems to be underestimated by these authors. We would also like to emphasize the considerable difference between our expression V1 (ft + q, ft) for the case q «s K and that proposed by Pfennig [17].

On the Theory of Electrical Resistivity of Polyvalent Metals

281

3. The Temperature Dependence of the Electrical Resistance of Aluminium and the Mass Shift of the Conduction Electrons due to the Electron-Phonon Interaction

The calculation of electrical resistivity of aluminium is essentially analogous to that of lithium, described earlier [15], so that we can restrict ourselves to considerations specific for the present problem. The frequencies co and the polarisation vectors e are determined by lattice dynamics: M mz e =

(13)

T • e .

The characteristic form of the matrix T(] for a f.c.c. crystal is given in an earlier paper [18]. Numerical values for the coupling parameters of neighbours of first, second, and third order are taken from Walker who determined these magnitudes from the non-elastic scattering of X-rays [12]. In contrast to Pytte [4] their temperature dependence is neglected. Furthermore, the values for the energies E0 of the state fe = 0 and for the potential on the surface of the atomic polyhedron are needed. Their order of magnitude can be taken from the band calculations by Segall [9,10] who gets for the difference E0 — F(r s ) = —0.335 Rydberg or —0.130 Rydberg, respectively, depending on the potentials used. Complete agreement of theoretical and experimental values of the electrical resistivity at the temperature T = 300 °K is achieved with E0 — F(rJ = —0.172 Rydberg which is very well compatible with Segall's. For the sake of time economy the dependence of electrical resistivity on the temperature was only investigated for this special value. By the approximation of free electrons an essential simplification is attained by the fact that the matrix elements V ! ( k ' , fe) depend only on the difference vector fe' — fe = q. Like in an earlier paper [18] we expand N Q

^ I Vi(k

2

+ q, fc)| 2

sinh2 -

de

=

Z L

F*(q)

K*(q°)

(14)

2KT

in terms of cubic harmonics which is done by least-squares fit. The expansion coefficients F L only depend on the angle 6 between fe and fe' and may be expanded in series of Legendre polynomials by means of the Fehlberg procedure [20]. The resulting expressions are of the form Pn (cos 6) K L. As they have the special property of not changing their numerical value if both fe and fe' are submitted to a cubic symmetry operation of the space group, they may be expanded in terms of the complete set of bicubic spherical harmonics Bff (k°, k'[21]. In the present problem we have confined ourselves to L = 0, 4, and 6. The proper transport problem is again solved by means of Kohler's variation method by taking into account the trial functions K«0-1», K

0

1, R2( h , we have

Equations (23) and (25) indicate that in the stacked pile-ups the number of dislocations in each of the arrays decreases as the separation between the pileups decreases. The stress at a point (x, y) produced by the stacked arrays is * ( f > = j — a' 19

physica/20/1

sech a' tanh t' ^ r . . r 2 tanh (C - t') y*tanh a': " -" tanh.2 t'. . . . •

290

Y. T. CHOIX: Stacked Screw Dislocation Arrays in an Anisotropic Medium

After integration, we have /

,

,

*(C> = V * , . + > a x z = , c^sech a = V*(,

^ ^

tanh f

-

= - l ) .

\ysinh 2 C - sinh 2 a'

, \

l) = (27)

J

Along t h e plane y = 0, t h e t o t a l stress due t o t h e stacked arrays a n d t h e applied stress a is therefore Oxt

=

0

and =

f

, l/sinh 2 x — s i n r a,

°r

W>l«l-

(28)

E q u a t i o n s (27) and (28) indicate t h a t t h e stress concentration r o u n d t h e tips decreases as t h e separation between t h e arrays decreases. I t is n o t e w o r t h y t h a t , if t h e applied stress a is superimposed in equation (27), t h e equation will represent t h e stress field produced b y a n infinite sequence of stacked shear cracks subjected t o t h e applied stress in t h e anisotropic medium. I n t h e case r\ = 1, the expression becomes t h e same as given b y B a r e n b l a t t a n d Cherepanov [9] for an isotropic medium. 3. Summary A direct method is given for solving problems of screw dislocation a r r a y s uniformity stacked one above t h e other in an infinite sequence in a n anisotropic medium. Using this method, one can readily find the distribution a n d t h e n u m ber of dislocations in each array, a n d t h e stress function produced b y t h e stacked sequence. The simplicity of t h e present approach is illustrated b y an example of double-ended pile-ups. I t was shown t h a t both t h e n u m b e r of dislocation pairs in each pile-up a n d t h e stress concentration around t h e tips decrease as t h e separation between pile-ups decreases. Such a system is elastically equivalent t o an infinite stack of parallel shear cracks. The present method can be applied t o stacked screw arrays in heterogeneous materials. A detailed analysis of t h e problem will be reported elsewhere. References [ 1 ] N . LOUAT, P h i l . M a g . 8 , 1 2 1 9 ( 1 9 6 3 ) . [ 2 ] L . D . WEBSTER a n d H . H . JOHNSON, J . a p p i . P h y s . 3 6 , 1 9 2 7 ( 1 9 6 5 ) .

[3] N. I. MUSKHELISHVILI, Singular Integral Equations, P. Noordhoff, Ltd., Groningen 1953 (p. 249). [ 4 ] A . K . H E A D a n d N . LOUAT,

Austral. J. P h y s . 8, 1 (1955).

[5] A. E. GREEN and W. ZERNA, Theoretical Elasticity, Oxford University Press, London 1954 (p. 159). [6] Y. T. CHOU, J. appi. Phys. 34, 429 (1963). [ 7 ] S . AMELINCKX a n d W . DEKEYSER, S o l i d S t a t e P h y s . 8 , 4 0 9 ( 1 9 5 9 ) .

[8] J. C. M. Li and C. D. NEEDHAM, J. appi. Phys. 31, 1318 (1960). [9] G. I. BAEENBLATT and G. P. CHEREPANOV, Prikl. Mat. Mekh. 25, 1110 (1961); Translation: J. appi. Math. Mech. 25, 1654 (1961). (Received

November

17,

1966)

R. P. GUPTA: Lattice Specific Heats of Thallium and Yttrium

291

phys. stat. sol. 2«, 291 (1967) Subject classification: 8; 6; 21 Department

of Physics,

Banaras

Hindu

University,

Varanasi

Lattice Specific Heats of Thallium and Yttrium By R . P . GUPTA

The lattice specific heats of thallium and yttrium are calculated using the electron gas model of Gupta and Dayal for hexagonal metals. The interactions are restricted to fourth neighbours. The force constants are determined from experimental values of the elastic constants. The d-T curves calculated from the computed specific heats show a fairly satisfactory agreement with the experimental values, i.e. the difference is not greater than 3 % for thallium and 6 % for yttrium. Mit dem Modell für ein Elektronengas von Gupta und Dayal für hexagonale Metalle wird die spezifische Wärme des Gitters für Thallium und Yttrium berechnet. Die Wechselwirkungen werden auf die vierten Nachbarn beschränkt. Die Kraftkonstanten werden aus experimentellen Werten der elastischen Konstanten bestimmt. Die © - T - K u r v e n , die aus den berechneten spezifischen Wärmen bestimmt werden, zeigen befriedigende Übereinstimmung mit experimentellen Werten, d. h. ihre Abweichungen sind nicht größer als 3 % für Thallium und 6% für Yttrium.

1. Introduction Thallium and yttrium are metals of the hexagonal class belonging to the third group of the periodic table. Yttrium is transitional also and has the electronic configuration 4d 5s2. Clusius and Vaughen [1], Keesom and Kok [2], Hicks [3], Snider and Nicol [4], and Van Der Hoeven and Keesom [5] determined the heat capacities of thallium, and Jennings et al. [6], Montgomery and Pells [7], and Morin and Maita [8] those of yttrium in various temperature regions. No attempt has, however, been made so far to discuss the available experimental data for them theoretically. This lack of theoretical attention seems to be due only to the fact that the elastic constants for these metals were not available until a few years back when they were determined by Ferris et al. [9] for thallium (4.2 to 300 °K) and by Smith and Gjevre [10] for yttrium (4.2 to 400 °K). Gupta and Dayal [11] have recently discussed a model for hexagonal metals which takes an explicit account of the role of conduction electrons on lattice vibrations. This model has given a successful description of the existing experimental data on elastic constants, phonon dispersion curves, and specific heats of beryllium [11, 12], magnesium [13], titanium [14], zirconium [14], and hafnium [14]. This has tempted the present author to study the lattice specific heats of thallium and yttrium also with this model. The results of this study have been reported in this paper. 2. Numerical Computations The method of calculation and the notation used in the present paper remains the same as in our previous papers [11 to 14], The interactions have been con19*

292

R. P.

GUPTA

sidered to extend only out to fourth neighbours. Thus there are only five force constants including the electronic bulk modulus K e . These force constants have been determined exclusively from the experimental elastic constants [9, 10] corresponding to the temperature 4.2 °K. There are some minor errors in the values of the elastic constants of yttrium reported by Smith and Gjevre [10]. We have, however, used the corrected elastic constants which have been obtained from them privately. The corrected value of C13 at 4.2 °K is (1.8 + 0.4) X X 1011 dyn/cm 2 . Thus the uncertainty in this constant is quite large. I n the present work the value of C13 has been taken to be 2.2 x 1011 dyn/cm 2 because this value is found to yield the best results and also lies within the limits of experimental accuracy. The elastic constants [9, 10] and the other constants [15] which have been used to determine the force constants are given in Table 1. The calculated force constants are given in Table 2. Table 1 Experimental data for thallium and yttrium Elastic constants (10 1 1 dyn/cm2) Constant Thallium c

Yttrium

4.44 3.76 6.02 0.88 3.00

u

C12 C33 C44 G13

Lattice constants (À) and atomic mass (a.m.u.)

8.340 2.960 8.010 2.715 2.200

Constant Thallium

Yttrium

3.4566 5.5248 204.37

3.6451 5.7305 88.919

a c m

Table 2 Force constants (103 dyn/cm) and Ke (10 u dyn/cm2) for thallium and yttrium Constant

Metal Thallium Yttrium

a

ß

V

Ò

Kc

6.5937 18.060

9.3542 20.987

-1.4942 -0.125

-0.5948 -1.636

2.12 -0.515

250

'i

— Theoretical • Experimental

7 100-

— •

SO-

h

121

[240

—Theoretical • Experimental

\

230

220

*

210 10

20

30

40 T(°K)

Fig. 1. 6-T curve for thallium

0

20

40 T(°K)~

Fig. 2. d-T curve for y t t r i u m

293

Lattice Specific Heats of Thallium and Yttrium

The specific h e a t s h a v e been calculated b y t h e same m e t h o d as employed in t h e case of other m e t a l s [12 t o 14]. The f r e q u e n c y s p e c t r u m h a s been divided in steps of Ai> = 0.1 X 10 12 s _ 1 in t h e case of t h a l l i u m a n d Av = 0.2 X 10 12 s" 1 in t h e case of y t t r i u m . The 0-T curves obtained f r o m t h e calculated specific h e a t s are shown in Fig. 1 a n d 2 respectively. The values calculated f r o m t h e e x p e r i m e n t a l lattice specific h e a t s h a v e also been p l o t t e d t h e r e . The calculation of specific h e a t s has n o t been carried out below 5 ° K f o r t h a l l i u m a n d 10 ° K for yttrium. 3. Results and Discussion 3.1

Thallium

The experimental d a t a on t h e specific heats of t h a l l i u m h a v e been t a k e n f r o m t h e work of Hicks [3], it being t h e latest available. H i c k s h a s given only t h e Cp values a n d h a s not converted t h e m t o C„ . Cv f r o m t h e e x p e r i m e n t a l Cp have, therefore, been calculated b y t h e use of t h e well k n o w n relation C,-CV

= AC'T.

(1) -5

I n t h i s relation t h e c o n s t a n t A h a s been t a k e n t o be 2.85 X 1 0 g a t o m / c a l f r o m t h e earlier work of Clusius a n d Vaughen [1], The electronic specific h e a t coefficient y h a s been t a k e n t o be 3.51 X 10~4 cal/g a t o m deg 2 f r o m t h e most recent work of Van Der Hoeven a n d Keesom [5]. The electronic specific h e a t has been s u b t r a c t e d f r o m t h e e x p e r i m e n t a l Cv in order t o o b t a i n t h e c o n t r i b u t i o n of t h e lattice specific h e a t . 3.2

Yttrium

The e x p e r i m e n t a l specific h e a t d a t a for y t t r i u m h a v e been t a k e n f r o m t h e work of J e n n i n g s et al. [6]. These workers also h a v e given only t h e Cp values a n d h a v e not converted t h e m t o Cv. The l a t t e r h a v e t h e r e f o r e been calculated b y t h e present a u t h o r himself f r o m relation (1). The c o n s t a n t A was d e t e r m i n e d f r o m t h e relation A = 0 . 2 1 4 / T m , where Tm is t h e melting t e m p e r a t u r e . F o r y t t r i u m Tm = 1782 ° K [15] so t h a t A = 12 x 10" 6 g a t o m / c a l . The electronic specific h e a t s h a v e been calculated b y t a k i n g y = 24.4 X 10~ 4 cal/g a t o m deg 2 f r o m t h e work of M o n t g o m e r y a n d Pells [7]. T h e agreement between t h e theoretical a n d t h e e x p e r i m e n t a l characteristic t e m p e r a t u r e s for b o t h t h e m e t a l s is fair. Acknowledgements

The a u t h o r is deeply g r a t e f u l t o Professor B. D a y a l f o r his c o n s t a n t interest and guidance during t h e course of t h e present investigations. H e is also t h a n k f u l t o t h e authorities of t h e C o m p u t e r Centre, I n d i a n I n s t i t u t e of Technology, K a n p u r , for providing t h e facilities of their I B M 1620 c o m p u t e r , a n d t o t h e University G r a n t s Commission for t h e a w a r d of a Senior Research Fellowship. References [1] K. CLUSIUS and J. V. VAUGHEN, J. Amer. Chem. Soc. 52, 4686 (1930). [2] W. H. KEESOM and J. A. KOK, Physica 1, 175 (1934). [3] J. P . G. HICKS JR., J . A m e r . Chem. Soc. 60, 1000 (1938). [ 4 ] J . L . SNIDER a n d J . NICOL,

P h y s . R e v . 105, 1242 (1957).

[ 5 ] B . J . C. V A N D E R HOEVEN J R . a n d P . H . KEESOM,

P h y s . R e v . 135, A631 (1964).

294

R. P. GUPTA: Lattice Specific Heats of Thallium and Yttrium

[ 6 ] L . D . JENNINGS, R . E . MILLER, a n d P . H . SPEDDING, J . c h e m . P h y s . 8 8 , 1 8 4 9 ( 1 9 6 0 ) . [ 7 ] H . MONTGOMERY a n d G . P . PELLS, P r o c . P h y s . S o c . 78, 6 2 2 ( 1 9 6 1 ) . [8] F . J . MORIN a n d J . P . MAITA, P h y s . R e v . 1 2 9 , 1 1 1 5 (1963).

[9] R. W. FERRIS, M. L. SHEPARD, and J . F. SMITH, J . appi. Phys. 34, 768 (1963). [10] J . F. SMITH and J. A. GJEVRE, J. appi. Phys. 31, 645 (1960). [ 1 1 ] R . P . GUPTA a n d B . DAYAL, p h y s . s t a t . sol. 8, 1 1 5 ( 1 9 6 5 ) . [ 1 2 ] R . P . GUPTA a n d B . DAYAL, p h y s . s t a t . sol. 9 , 87 ( 1 9 6 5 ) .

[13] R. P. GUPTA and B. DAYAL, phys. stat. sol. 9, 379 (1965). [14] R. P. GUPTA and B. DAYAL, phys. stat. sol. 13, 257 (1966). [15] International Tables for X-Ray Crystallography, Vol. I l l , 1962 ( Received

December

23,

1966)

R . ENDERLEIN: Influence of Collisions on t h e F r a n z - K e l d y s h E f f e c t

295

phys. s t a t . sol. 20, 295 (1967) Subject classification 20; 13.1; 14.4.1 Department

of Physics,

State University,

Moscow

The Influence of Collisions on the Franz-Keldysh Effect By R. E n d e b l e i n

A s t u d y is m a d e of t h e effect fo e l e c t r o n - p h o n o n a n d e l e c t r o n - i m p u r i t y scattering o n t h e field-induced change in t h e optical i n t e r b a n d a b s o r p t i o n for p h o t o n energies n e a r t h e d i r e c t b a n d g a p h to0. I t is f o u n d t h a t in t h e presence of collisions t h e exponential tail of t h e a b s o r p t i o n coefficient a t co < cu0 oscillates a n d t h a t t h e oscillations a t ai > cu0 a r e d a m p e d . This d a m p i n g seems t o b e in q u a n t i t a t i v e a g r e e m e n t w i t h t h e e x p e r i m e n t a l results of H a m a k a w a , Germano a n d H a n d l e r [4]. E s wird der E f f e k t der E l e k t r o n - P h o n o n - u n d der E l e k t r o n - S t ö r s t e l l e n s t r e u u n g auf die feldinduzierte Ä n d e r u n g in der optischen I n t e r b a n d a b s o r p t i o n f ü r P h o t o n e n e n e r g i e n in der N ä h e der direkten B a n d k a n t e h a>0 u n t e r s u c h t . E s wird g e f u n d e n , d a ß beim Vorhandensein v o n S t ö ß e n der exponentielle Ausläufer des Absorptionkoeffizienten bei co < co0 oszilliert u n d d a ß die Oszillationen bei eo > tu0 g e d ä m p f t sind. Diese D ä m p f u n g scheint m i t d e n experimentellen Ergebnissen von H a m a k a w a , G e r m a n o u n d H a n d l e r [4] q u a n t i t a t i v übereinzustimmen.

1. Introduction The purpose of this paper is to calculate approximately the field-induced change in the optical absorption coefficient of a semiconductor near the direct band gap, taking into account impurity and phonon scattering of the electrons. It was shown [1] that the relation between D(a>) and E(o>) in the presence of a static electric field F has the form1) 00

D{w)

= / d o / e(w', co -

co') E{(o')

(1)

— OO

with E(CO', co —

o/)

=

—

D (co — c o ' )

oo +

J r .

^

/ ¿ —oo

+

t e i ( m

~

a , ) t

f t„

d r

elo/(i

"°

V -

*o). i S ' -

W]>0 •

(2)

We express the current density correlator in (2) in terms of the correlation functions (CF) t') =

C^k.(t')

CvK{t')\

,

(3)

where C^u and Cvjt are the creation and annihilation operators for electrons in Bloch states \v k). ') T h e n o t a t i o n s here are t h e s a m e as in [1],

296

R . ENDERLEIN

Because intraband transitions give a negligible contribution to the absorption in the optical frequency range, we can restrict our considerations to the inter band transitions. The effect of collisions on the interband transitions is, generally, to " d a m p " the direct transitions and to introduce indirect transitions. However, near the direct band gap the field-induced change in the direct absorption is much greater than that in the indirect absorption [2]. Furthermore, as we will show, the effect of collisions on the field-induced change in the direct absorption is of the same order as the field-induced change itself. Therefore, in the following calculations we can neglect the contribution of the indirect transitions. This implies, in particular, that we have to consider the interbandCF (¡u = v', fi' = v) only. 2. Interband Correlation Functions

We have Fv/vUt,

n

=