Physica status solidi: Volume 29, Number 1 September 1 [Reprint 2021 ed.] 9783112496329, 9783112496312

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plxysica status solidi

VOLUME 29 • N U M B E R 1 . 1968

Classification Scheme 1. S t r u c t u r e of Solids 1.1 Alloys. Metallurgy 1.2 Solid-State P h a s e T r a n s f o r m a t i o n s 1.3 Surfaces 1.4 F i l m s 2. Non-Crystalline S t a t e 3. Crystallography 3.1 Crystal G r o w t h 3.2 I n t e r a t o m i c Forces 4. Microstructure of Solids 5. P e r f e c t l y Periodic S t r u c t u r e s 6. L a t t i c e Mechanics. P h o n o n s 6.1 Mossbauer I n v e s t i g a t i o n s 7. Acoustic Properties of Solids 8. T h e r m a l Properties of Solids 9. Diffusion in Solids 10. D e f e c t P r o p e r t i e s of Solids ( I r r a d i a t i o n Defects see 11) 10.1 Defect Properties of Metals 10.2 P h o t o c h e m i c a l Reactions. Colour Centres 11. I r r a d i a t i o n E f f e c t s in Solids 12. Mechanical P r o p ePr rt ioepse rof (Plastic Deform 12.1 Mechanical t i eSolids s of Metals (Plastic D ae tf ioornmsasee t i o n10) s see 10.1) 13. E l e c t r o n S t a t e s in Solids 13.1 B a n d S t r u c t u r e . F e r m i Surfaces 13.2 E x c i t o n s 13.3 Surface S t a t e s 13.4 I m p u r i t y a n d Defect S t a t e s 14. Electrical P r o p e r t i e s of Solids. T r a n s p o r t P h e n o m e n a 14.1 Metals. Conductors 14.2 S u p e r c o n d u c t i v i t y . S u p e r c o n d u c t i n g Materials a n d Devices 14.3 Semiconductors 14.3.1 Semiconducting Films 14.3.2 Semiconducting Devices. J u n c t i o n s (Contact P r o b l e m s see 14.4.1) 14.4 Dielectrics 14.4.1 H i g h Field P h e n o m e n a , Space Charge Effects, Inhomogeneities, I n j e c t e d Carriers (Electroluminescence see 20.3; J u n c t i o n s see 14.3.2) 14.4.2 Ferroelectric Materials a n d P h e n o m e n a 15. Thermoelectric a n d T h e r m o m a g n e t i c P r o p e r t i e s of Solids 16. P h o t o c o n d u c t i v i t y . P h o t o v o l t a i c E f f e c t s 17. Emission of Electrons a n d Ions f r o m Solids 18. Magnetic Properties of Solids 18.1 P a r a m a g n e t i c P r o p e r t i e s 18.2 F e r r o m a g n e t i c P r o p e r t i e s 18.3 F e r r i m a g n e t i c Properties. F e r r i t e s 18.4 A n t i f e r r o m a g n e t i c P r o p e r t i e s (Continued

on cover three)

Classification Scheme — Continued 19. Magnetic Resonance in Solids 20. Optical Properties of Solids 20.1 Spectra, Optical Constants (Colour Centres see 10.2; X-Ray Spectra see 20) 20.2 Lasers 20.3 Luminescence of Solids (X-Ray Spectra see 20) 21. Metals. Alloys 21.1 Transition Metals (Group Sc to Zn) and their Alloys 21.1.1 Fe and Fe Alloys 21.2 Alkali Metals 21.3 Alkali-Earth Metals 21.4 Rare-Earth Metals 21.5 Actinides 21.6 Noble Metals 21.7 Semi-Metals (Semiconducting Alloys see 22.7) 22. Semiconductors and Ionic Crystals 22.1 Elements 22.1.1 Germanium 22.1.2 Silicon 22.1.3 Group VI Elements 22.2 I I I - V Compounds 22.2.1 Arsenides 22.2.2 Phosphides 22.2.3 Antimonides 22.3 Carbides 22.4 I I - V I Compounds 22.4.1 Sulphides 22.4.2 Selenides 22.4.3 Tellurides 22.5 Halides22.5.1 Silver Halides 22.5.2 Alkali Halides 22.5.3 Alkali-Earth Halides 22.6 Oxides (CdO, ZnO see 22.4; Spinels see 18.3 or 14.4.2) 22.7 Semiconducting Intermetallic Compounds 22.8 Tertiary and Higher Compounds (Carbonates, Phosphates, Silicates, and Mica see 22) 22.9 Organic Semiconductors 23. Solidified Gases

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

physica status solidi B o a r d of E d i t o r s P. A I G R A I N , Paris, S. A M E L I N C K X , Mol-Donk, V. L. B O N C H - B R U E V I C H , Moskva, W. D E K E Y S E R , Gent, W. F R A N Z , Münster, P. GÖRLICH, Jena, E. G R I L L O T , Paris, R. K A I S C H E W , Sofia, P. T. L A N D S B E R G , Cardiff, L. N É E L , Grenoble, A. P I E K A R A , Warszawa, A. S E E G E R , Stuttgart, F. SEITZ, Urbana, 0 . S T A S I W , Berlin, M. S T E E N B E C K , Jena, F. STÖCKMANN, Karlsruhe, G. S Z I G E T I , Budapest, J . TAUC, Praha Editor-in-Chief P. GÖRLICH Advisory Board M. B A L K A N S K I , Paris, P. C. B A N B U R Y , Reading, M. B E R N A R D , Paris, W. B R A U E R , Berlin, W. COCHRAN, Edinburgh, R. COELHO, Fontenay-aux-Roses, H.-D. DIETZE, Saarbrücken, J.D. E S H E L B Y , Cambridge, P. P. F E 0 F I L 0 V, Leningrad, J. H O P F I E L D , Princeton, G. J A C 0 B S, Gent, J . J A U M A N N , Köln, E. K L I E R , Praha, E. K R O E N E R , Clausthal-Zellerfeld, R. KUBO, Tokyo, M. M A T Y A S , Praha, H. D. MEGAW, Cambridge, T. S. MOSS, Camberley, E. NAGY, Budapest, E. A. N I E K I S C H , Jülich, L. P A L , Budapest, M. RODOT, Bellevue/Seine, B. V. R O L L I N , Oxford, H. M. ROSENBERG, Oxford, R. V A U T I E R , Bellevue/Seine

Volume 29 • Number 1 • Pages 1 to 446, K 1 to K84, and A l to A4 September 1, 1968

AKADEMIE-VERLAG

• BERLIN

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Schriftleiter und verantwortlich für den Inhalt: Professor Dr. Dr. h. c. P. G ö r 1 i c h , 102 Berlin, Neue Schönhauser Str. 20 bzw. 69 Jena, 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: 426788. Verlag: Akademie-Verlag GmbH, 108 Berlin, Leipziger Str. 3 - 4 . Fernruf: 220441, Telex-Nr. 112020, Postscheckkonto: Berlin 35021. Die Zeitschrift „physica status solidi" erscheint jeweils am 1. des Monats. Bezugspreis eines Bandes M 90,— (Sonderpreis für die DDR M 60, —). Bestellnummer dieses Bandes 1068/29. Jeder Band enthält zwei Hefte. Gesaratherstellung: VEB Druckerei „Thomas Müntzer** Bad Langensalza.—Veröffentlicht unter der Lizenznummer 1310 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik.

phys. stat. so . 2 9 ( 1 9 6 8 )

Author Index F . AGULLÖ-LÖPEZ L . AHMAD G . M . ALIEV Z . I . ALIZADE M . ALTWEIN T . ANAGNOSTOPOULOS H . AREND G . E . ARKHANGELSKII I . ARTZNER CH. M . ASKEROV V . M . ASNIN M . AVEROUS H . BACHERT D . J . BACON G . R . BAESCH A . N . BASTJ D . BÄUERLE N . P . BAUM V . BECKMANN E . P . BERDNIKOV C . B . VAN DEN BERG H . BILGER I . A . BLECH H . BLUME O . V . BOGDANKEVICH V . L . BONCH-BRUEVICH D . BONNET H . BORIK N . A . BORISOV G . BOUGNOT A . BOUKRET D . I . BOWER R . BOYN V . A . M . BRABERS S . E . BRONISZ W . BRUCKNER W . BRÜCKNER R . A . BURLEY K . H . J . BUSCHOW J . CALAS B . CHALUPA G . CHANUSSOT M . S . R . CHARI K . P . CHIK V . S . CHINCHOLKAR E . CLARIDGE W . COCHRAN L . N . COJOCARU P . CRACIUN A . CZACHOR

217 179 K47 K19 203 K179 K149 831 KL 17 K47 443 807 K175 543 121 367 639 107 781 K145 837 K179 653 159 715 9 K65 133 715 807 . 283 617 307 K73 K95 781 211 551 189, 825 807 K51 K149 K69 455 669 617 K81 K L 19 761 423

S. D. P. J. L. B. A. J. D. V. F. E.

DÄBRITZ DAUTREPPE DEKKER DELAPLACE J . DEMER G . DICK M . VAN DIEPEN DILLINGER L . DOUGLASS A . DROZDOV DWORSCHAK Z . DZIUBA

G . L . ERISTAVI J . F . FAST H . FINKENBATH R . J . FLEMING P . FLÖGEL M . P . FONTANA E . FORTIN C . T . FORWOOD W.FRANK B . FRITZ W . FUCHS

K89 283 K73 819 K141 587 189 707 K95 K145 75, 81 813 443 825 203 K77 K89 159 K153 99 391, 767 639 781

I . GAAL F . B . GADZHIEV E . E . GALLONI H . GAMARI-SEALE B . GARBEN D . GEIST H . GENGNAGEL H . GERMER J . H . GIESKE F . GIUSIANO H . J . GLÄSER Y U . A . GOLDBERG A . S . VAN DER GOOT V . V . GORSKII N . A . GORYUNOVA B . VON GUERARD J . P . GUIGAY S . E . GUKASYAN J . GYULAI

K163 K47 K91 323 K27 167 91 K65 121 341 167 K103 825 45 435 K59 799 49 K85

P . HAEN P . H . HANDEL B . HARRIS E . HARTMANN S . HAUSSÜHL H . - P . HENNIG

743 299 383 KLL K159 K167

894 K . H . HERRMANN U. HOFMANN R . G. HOWELL I . HRIANCA P . HUMBLE P . HUTH K . IBEL R . VON JAN A . JOSTSONS J. C. JOUSSET R . KERSTEN E. KIERZEK-PECOLD J. KINEL K . KLEINHENZ K . KLEINSTÜCK L . I . KLESHCHINSKII B. R . KNOTTNERUS J. KO&ODZIEJCZAK C. KONÄK W . KRESS M. A . KRIVOGLAZ I . V . KRUKOVA A . B. KUNZ S. KUPCA L . B. KVASHNINA

Author Index 193 91 697 761 99 K35 403 755 873 K127 575 K183 K39 627 211 435 K43 645, K183 203, 707 133 53, 61 715 115 K15 53, 61

E. G. LANDSBERG F . P . LAVBENTEV B. M. LAVRUSHIN J. L . LEVEQUE M. E. LEVINSHTEIN J. E. LEWIS S. LIGENZA G. V . LOSHAKOVA M. LUUKKALA

9 569 715 K179 K103 743 K99 435 377

T . G. MAGARRAMOV G. MAMBRIANI P . G. MCDOUGALL I . S. MCLINTOCK A . MEHRA H . MEHRER E. S. MEIERAN R . MICHALEC E. MÖHLER Z. L . MORGENSHTERN J. W . MORON P . MOSER L . A . MOZGOVAYA G. M. DE'MUNARI

K19 341 873 K157 847 231 653 K51 K55 831 K39 K179 K145 341

V . P . NABEREZHNYKH D. N . NASLEDOV

K23 K103

N . S. NATARAJAN V . NAUNDORF E. NEBAUER V . V . NEMOSHKALENKO V . B. NEUSTRUEV C. W . A . NE WE Y D. J. NEWMAN J. C. NICOUD D. NICULESCU N . NICULESCU J. C. ORR E. O. OSMANOV

K69 K123 269, K 8 9 45 831 357 697 819 813 813 K157 435

P . PAPAMANTELLOS Y . S. PARK R . T . PASCOE B . PEGEL H . PEISL C. M. PENCHINA R . PERTHEL P . PETRESCU V . PETRZÎLKA K . PFEUFFER H . PFLEIDERER P . B . PICKAR J. PIEKOSZEWSKI Y. PLAKHTY K . PLATZÖDER I . PRACKA V . PROSSER J. PRZYBYLA J. PUHLMANN

323 K7 357 K133 K59 K7 211 333, K L , K 3 K51 171 597 153 K99 K81 K63 K183 707 K39 193

S. RAAB H . RABENHORST R . RAMJI RAO 0 . N . RASUMOV P . REIMERS G. RITTER H . RODOT A . A . ROGACHEV G. ROTH B. P . ROTHENSTEIN D. K . ROY A . R . RUFFA P . RUNOW W . RUPPEL

K175 K65 865 45 K31 781 743 443 K123 K117 KILL 605 627 K31

J. SAK P . P . SALHOTRA O . P . SALITA C. SÂNCHEZ D. H . SASTRY E. V . R . SASTRY

707 859 569 217 K137 K107

Author Index M . SAUVAGE W . SCHILZ K . SCHRÖDER G . E . R . SCHULZE D . SCHUMACHER H . SCHUSTER W . J . VAN SCIVER A . SEEGER H . SELIGER S . SENGUPTA D . SHAW V . S . SHPINEL K . N . SHRIVASTAVA C . SIMOI S. J . SIVONEN R . SIZMANN R . SRINIVASAN T . M . SRINIVASAN J . STUKE E . C. SUBBARAO A . SUKIENNICKI E . J . SUONINEN J . SUWALSKI E . TAGLAUER R . C. T H I E L J . TICH* H . D . TILLER I . S . T . TSONG B . TUCK

725 559 107 211 819 75, 81 159 231, 455 K27 367 145 49 737 761 K171 403 865 K107 203 859 417 K171 K99 259 837 K51 153 617 793

H . - J . ULLRICH H . - G . UNRUH L . URAY A. K. P. D. V. G. H.

A . VAIPOLIN I . VASU VENKATESWARLU YESELY L . VLADIMIROVA VOGL R . VYDYANATH

R . H . WADE W . WAIDELICH W . W . WALKER P . WALZ H . WEGENER R . K . WEHNER D. T. Y. WEI H . WEIJMA R . C . WHELAN H . W . DE WUSR W . WINDSCH H . WOLLENBERGER J . WURM J . ZELENKA P . ZIMMER G . ZIMMERER R . ZIMMERMAN M . ZVARA

895 K89 669 K163 435 K137 859 675, 685 569 819 K137 799 K59 K141 245 781 133 K7 K43 145 189 KLL 75, 81 75,81 K51 K89 203 K91 707

Contents Page

Review Article V . L . BONCH-BRUEVICH a n d E . G . LANDSBERG

Recombination Mechanisms. Orginal Papers V . V . NEMOSHKALENKO, O . N . RASUMOV, a n d V . V . GORSKII

Investigation of the Mössbauer Effect in Some Fe-Al A l l o y s . . . .

45

S. E . GUKASYAN a n d V . S. SHPINEL

Mössbauer Effect in Antimony-Organic Compounds

49

M . A . KRIVOGLAZ a n d L . B . K V A S H N I N A

The Correlation Function 8ZS( co) for Weakly Coupled I m p u r i t y Atom Spin in a Ferromagnetic

53

M . A . KRIVOGLAZ a n d L . B . K V A S H N I N A

On the Spin Broadening of Mössbauer Lines of Impurities in Ferromagnetics and of Atoms in Antiferromagnetics

61

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

Analysis of Point Defect States in Copper (I)

75

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

Analysis of Point Defect States in Copper (II)

81

H . GENGNAGEL a n d U . HOFMANN

Temperature Dependence of the Magnetocrystalline Energy Constants Kv K2, and K3 of Iron

91

P . H U M B L E a n d C . T . FORWOOD

A Dynamical Treatment of the Stress-Induced Dissociation of Triangular F r a n k Dislocation Loops in F.C.C. Metals K . SCHRÖDER a n d N . P .

A. B. KUNZ

99

BAUM

Absolute Thermoelectric Power and Electrical Resistivity of Chromium-Rich Chromium—Nickel Alloys

107

Electronic Energy Bands for Rubidium Chloride and the FaceCentered Cubic Alkali Bromides

115

J . H . GIESKE a n d G. R .

BARSCH

Pressure Dependence of the Elastic Constants of Single Crystalline Aluminum Oxide W . KRESS, H . BORIK, a n d R . K .

121

WEHNER

Infrared Lattice Absorption of Silicon and Germanium

133

R . C. W H E L A N a n d D . SHAW

Evidence of a Doubly Ionized Native Donor in CdTe

145

P . B . PICKAR a n d H . D . TILLER

Optical Energy Gap in TISe

153

M . P . F O N T A N A , H . B L U M E , a n d W . J . VAN SCIVER

Properties of Exciton States in N a l (I)

159

H . J . GLÄSER u n d D . GEIST

Elektrische Leitfähigkeit und Halleffekt von mangandotiertem Germanium und der Einfluß von Wärmebehandlungen auf den Einbauzustand des Mangans l

167

4

Contents

K . PFEUFFER

L.

AHMAD

Determination of the Electron Spin-Lattice Relaxation Time T1 in the Direct Process Change in Electron Affinity of a Semiconductor due to Different Forms of Chemisorption

Page

171 179

A . M . VAN D I E P E N , H . W . D E W I J N , a n d K . H . J . B U S C H O W

Nuclear Magnetic Resonance and Susceptibility of Equiatomic RareEarth-Aluminum Compounds

189

K . H . HERRMANN a n d J . PUHLMANN

Piezoresistance of Tellurium (II)

193

M . A L T W E I N , H . F I N K E N R A T H , Ö. K O N A K , J . S T U K E , a n d G . ZIMMERER

The Electronic Structure of CdO (II)

203

W . BRÜCKNER, R . P E R T H E L , K . KLEINSTÜCK, a n d G . E . R . SCHULZE

Magnetic Properties of ZrFe 2 and TiFe 2 within Their Homogeneity Range

211

C. SANCHEZ a n d F . AGULLÖ-LÖPEZ

Transient Effects in the Room-Temperature F-Colouring of NaCl Irradiated with X- or y-Rays

217

A . SEEQER a n d H . MEHRER

F. E.

WALZ

TAGLAUER

E . NEBAUER

Interpretation of Self-Diffusion and Vacancy Properties in Gold . .

231

Untersuchung der magnetischen Nachwirkung in Nickel nach Neutronenbestrahlung und plastischer Verformung

245

Small Angle Scattering of Subthermal Neutrons from Deformed Polycrystalline Copper

259

Vapour-Pressure Investigations of Impurity Diffusion and Solubility in A n - B I V Compounds Demonstrated for the System CdS:Au:S 2 . .

269

A . BOURRET a n d D . DAUTREPPE

P.

H . HANDEL

R. Bo YN

Study of Agglomerated Defects in Irradiated Pure Nickel by Electron Microscopy

283

Instabilities and Turbulence in Semiconductors

299

Optical Absorption due to Intrinsic Defects in CdS Single Crystals. .

307

H . GAMARI-SEALE a n d P . PAPAMANTELLOS

P.

PETRESCU

A Neutron Diffraction Study an Gallium-Substituted Magnetite . .

323

On the L-Bands and the First Exciton Bands in the Photoemission Spectra of Alkali Iodides

333

G . M . DE'MUNAEI, F . GIUSIANO, a n d G . MAMBRIANI

A Phase Transition and the Photoemissive Quantum Yield at Low Temperatures in Cs3Sb Films

341

R . T . PASCOE a n d C. W . A . N E W S Y

Deformation Modes of the Intermediate Phase NiAl

357

A . N . BASU a n d S. SENGUPTA

M.

LUUKKALA

A Deformable Shell Model for the Alkali Halides

367

On Coherent Phonon Fusion in Quartz Transducers

377

Contents

5 Page

B. Habbis

Solution-Hardening and Deformation of Niobium-Tungsten Alloys .

383

W. F r a n k

Die kritische Schubspannung mit zweiwertigen Kationen dotierter Alkalihalogenide (I)

391

K. I b e l and R. Sizmann Energy Loss of Channelled a-Recoil Atoms in Gold

403

A. Sukiennicki On a New Type of Stripe Domains

417

A. Czachor

423

On the Lattice Dynamics of Hexagonal Structure Metals

A. A. Vaipolin, N. A. Goryunova, L. I. K l e s h c h i n s k i i , G. V. Loshakova, and E. O. Osmanov The Structure and Properties of the Semiconducting Compound ZnSnP 2

435

V. M. Asntn, G. L. Eristavi, and A. A. Rogachev Effect of Weak Electric Fields on the Absorption Edge in Doped Germanium

443

Short Notes P. P e t e e s c u

On the Photoemission Spectrum of LiF: Mg

K1

P. P e t e e s c u

Evidence for L-Bands in Photoemission Spectra of K F and KCl. . .

K3

D. T. Y. W e i C. M. Penchina, and Y. S. P a e k Observation of Oscillatory Lifetime in CdS

K7

W. Windsch and E. Hartmann Switching Effect in Vanadyl-Doped Ferroelectric Triglycine Sulphate Observed by Electron Paramagnetic Resonance Kll 3. Kupöa

Hardness of NaCl Single Crystals Prepared with Various Concentrations of BaCl 2 in the Melt K15

Z. I. Alizade and T. G. Magabramov Galvanomagnetic Effect in Ferromagnetic Alloys of Nickel-Tantalum and Nickel-Niobium Systems K19 V. P. Nabebezhstykh Radio-Frequency Size Effect on Electrons in Boundary Cross-Sections Having Extremum Displacement during a Cyclotron Period . K23 B. Garben und H. S e l i g e r Faraday-Effekt an amorphem Selen

K27

P. Reimebs and W. Ruppel F. H u t h

The Preparation of CdS-CdSe Graded Single Crystals

K31

Hall Effect and Mobility in n-Type GaAs

K35

J . K i n e l , J . W. Moron, and J . P b z y b y l a A Diffusional Magnetic Viscosity Effect in Iron-Carbon Martensite . K39 H. Weijma and B. R. K n o t t n e r u s Ionic Conductivity of Pure and Ba 2+ -Doped CsCl

K43

F. B. Gadzhiev, Ch. M. Askebov, and G. M. A l i e v Effect of Heat Treatment and Natrium Admixtures on the Temperature Dependence of Electric Conductivity of Amorphous Selenium. . K47

Contents

6

Page B . CHALTTPA, R . M I C H A L E C , Y . P E T R Z Î L K A , J . T I C H Y , a n d J . Z E L E N K A

E.

MOHLER

D i f f r a c t i o n of N e u t r o n s on a Vibrating Quartz Crystal

K51

Linear Electro-Optic E f f e c t of Excitons in CuCl

K55

B . VON G U É R A R D , H . P E I S L , a n d W .

WAIDELICH

Equilibrium Vacancy Concentration in KC1

K . PLATZODER

T e m p e r a t u r e Effects on t h e Vacuum Ultraviolet Reflectance of Quartz

K59 oc-

K63

D . BONNET, H . GERMER, a n d H . RABENHORST

Photoconduction in Cd(S x Sei_z) Films with Graded Composition . . K 6 5 M . S. R . CHARI a n d N . S . NATARAJAN

The Phonon-Electron Scattering Coefficients in Dilute Silver-Manganese Alloys K69 V . A . M . BRABERS a n d P . D E K K E R

I n f r a r e d Spectra a n d Cation Distributions of Manganese Ferrites . . K 7 3 R. J.

FLEMING

E v a l u a t i o n of 1 s —2 p Optical Transition E n e r g y for F-Centres in K77 CaF 2

V . P L A K H T Y a n d W . COCHRAN

X - R a y S t u d y of t h e Phase Transition a n d Lattice Vibrations of L a n t h a n u m Aluminate

K8l

Pre-printed Titles of p a p e r s to be published in t h e n e x t issue

A1

7

Contents Systematic List

Subject classification :

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

1.2

341,

3

341

3.1

K81

K31

4

259, 283, 417, 435

5

153, 323, 435, K31, K51

6

9, 133, 171, 367, 377, 423, 435, K 6 9 , K81

6.1

45, 49, 61

7

377 K69

8 9

231, 269

10 10.1

145, 307, 391, 435, K15, K43, . . '

10.2

217, 333, K l , K3,

11

75, 81, 217, 245, 283, 403

12 13

K59

75, 81, 99, 231, 245, 259, 283, 357, 383 K77

121 9, 45, 53, 61, 189, K 2 7 , K 6 9

13.1

107, 115, 153, 193, 203, 341, K23,

13.2

9, 159, 333, 443, K 5 5 , K 6 3

13.3

179

13.4

9, 145, 167, 307, K 3 5 , K 7 7

K63

14.1

107,

14.3

9, 145, 167, 193, 435, K 1 5 , K35, K47,

14.3.1

203

14.4. 1

299

14.4. 2 15

K19

Kll 107

16

153, K7,

17

333, 341, Kl,

18

K55

K65 K3

K27

18.1

171, 189

18.2

53, 61, 91, 211, 245, 417, K19,

18.3

323, K 7 3

18.4

61, 107, 211

19

53, 167, 171, 189,

20

9, K27,

K39

Kll

K55

20.1

133, 153, 203, 307, 443, K 3 1 , K63,

20.3

9, 159

K73

21

189, 383, K 2 3

21.1

75, 81, 99, 107, 245, 259, 283, 357, K 1 9

21.1.1

45, 91, 211, 417, K 3 9

8 21.3 21.6 22 22.1.1 22.1. 2 22.1.3 22.2.1 22.4 22.4.1 . 22.4. 2 22.4. 3 22.5 22.5.2 22.5. 3 22.6 22.7 22.8 22.9

Contents 423 99, 231, 403, K69 153, 299 9, 133, 167, 443 9, 133 193, K27, K47 K35 203 269, 307, K7, K31, K65 K31, K65 145 K55 115, 159, 217, 333, 367, 391, K l , K3, K15, K43, K59 K77 121, 377, K51, K63 341 171, 179,^435, K81 449

Contents of Volume 29 Continued on Page 449

Review

Article

phys. stat. sol. 29, 9 (1968) Subject classification: 13.4 and 14.3; 6; 13; 13.2; 20; 20.3; 22.1.1; 22.1.2 Faculty of Physics, Moscow State University, and Institute of Radioengineering and Electronics, Academy of Sciences of the USSR, Moscow

Recombination Mechanisms1) By V . L . BONCH-BRUEVICH a n d E . G . LANDSBERG

Contents 1.

Introduction

2. Capture

cross-sections

3. Recombination

and

lifetimes

mechanisms

3.1 Phonon mechanism 3.2 Impact mechanism 3.3 Plasmon, exciton, and spin mechanisms 4. Capture 5. Radiative

by charged recombination

centres as a tool of studying

the energy

spectrum

of a

solid

Appendix References

A review is given of various mechanisms of the charge carriers recombination in semiconductors. A theoretical estimate of the relative importance of various mechanisms is given as well as the relevant temperature dependences of the capture cross-sections. Influence of the static electric field upon the recombination rate is considered in connection with the electrical instability problem. 1. Introduction Recombination of the charge carriers is one of the processes t h a t establish the thermodynamic equilibrium in semiconductors. I t is one of the specific features of these materials as contrasted to metals t h a t the charge carrier concentration is not a constant but a thermodynamic variable. I t may be varied by means of some exterior perturbations (illumination, injection through the p - n junction, etc.) and thus rather significant deviations from the thermodynamic equilibrium may be produced (see, for example, references [1] and [2]). The J ) An enlarged version of the report given by one of the authors (V.L.B.-B.) at the Tagung der Physikalischen Gesellschaft in der DDR, April 1967. The hospitality of the Physikalische Gesellschaft in der DDR and of the Humboldt-Universität in Berlin is gratefully acknowledged.

10

V . L . BONCH-BRUEVICH a n d E . G. LANDSBERG

process of establishing the thermodynamic equilibrium with respect to the charge carrier concentration is called a recombination (or a generation if the concentration is increased during the process). The study of the various aspects of the recombination was one of the main lines of development of the semiconductor physics during the last 15 years. This is rather an old age for the semiconductor physics; nevertheless, the problem is still attracting serious attention both of experimental and of theoretical physicists. This seems to be due to at least three reasons. Firstly, the lifetime of the non-equilibrium charge carriers determined by the recombination processes is one of the most important characteristics of the device materials. According to the purpose of the devices both materials with very high and very low lifetimes may be needed. Thus governing the recombination rate is governing the parameters of the semiconductor devices. Secondly, the study of the recombination processes is of serious interest from a purely scientific point of view since these are the processes of the energy exchange in a solid. I t will be seen from what follows t h a t the microscopic recombination theory is tied up with almost all of the most difficult principle problems of the semiconductor theory (and of the theory of solids in general). Thirdly, the experimental study of some recombination transitions (mostly optical) provides us with a unique way of the direct study of some subtle peculiarities of the energy spectrum of a semiconductor (see Section 5). By definition a single recombination act consists of the disappearance of the two oppositely charged carriers: a conduction electron and a hole. This process is usually described in the one-electron language as the conduction band-valence band transition. We shall also use this language in what follows though this is not necessary in principle: all the phenomena to be considered may be described as well using exact many-body concepts. Experiment shows t h a t there are two different ways of the recombination process: The first one is just a direct 2 ) band-to-band transition with no intermediary steps in between. The second one includes intermediary stages of electron and hole transitions to some discrete levels produced in the forbidden band by various imperfections. Such band-to-level transitions are usually described as the "capture processes" while the relevant imperfections are called the recombination centres or "traps". The role of the latter may be played by impurities, vacancies, dislocations, etc. This way of "recombination via centres" is often the dominant one, the most popular examples being those of germanium and silicon. One of the reasons is probably the smallness of the (non-radiative) transition probability if the energy transfer is too large. Under such conditions the two- (or three-) step process may turn out more profitable. Another reason is t h a t in presence of the imperfections the electron crystal momentum must not necessarily be conserved, this leading to an increase of the phase space volume where the direct band-to-band transitions take place. Recombination via centres is of particular interest in connection with the problem of preparing the materials with the prescribed properties: in this case the recombination rate may be purposely varied to a large extent by changing the nature and concentration of the imperfections. 2 ) Not to be mixed with the direct optical transitions. In our case both phonons and impurities may take part in the process. The only point of importance is that the electron passes directly from one region of the continuous spectrum to the other.

Recombination Mechanisms

11

Calculation of the rate of recombination via centres is naturally divided into two parts which may be called a dynamical and a statistical one, respectively. The first of them consists of studying the elementary capture and generation processes and of calculating the relevant transition probabilities. The second part [3 to 6, 35] has to do with calculating the occupancies of the various energy levels under the non-equilibrium conditions as well as with calculating the directly observable quantities: the lifetimes of electrons and holes, the probabilities of the intermediate elementary processes being given. We consider only the dynamical part of the problem since the statistical part is now adequately treated in the standard textbooks [6]. In doing so we limit ourselves to the most often met case when the recombination proceeds much slower t h a n the processes establishing equilibria with respect to the crystal momentum and the energy in any separate band. I n other words we assume t h a t the recombination time r r greatly exceeds the energy mean free path r. 3 ) Then the recombination may be considered to proceed under the conditions of equilibrium with respect to the energy and the crystal momentum. The fact of there being no equilibrium with respect to the carrier concentration is taken account of only by using the two separate Fermi levels — for the conduction electrons and for the holes — instead of a single one. I t is shown in the statistical part of the problem t h a t under such conditions all the relevant dynamical information is contained in the electron and hole capture coefficients, C n and C p , defined by the relations of the following type: Cn = I f

D

(p)v(p)an(p)dp.

(1.1)

Here f n ( p ) is the electron distribution function normalized to unity, v ( p ) is the absolute value of the electron velocity if the crystal momentum is equal to p, an(p) is the electron capture cross-section for the trap level in question. The integral is taken over the Brillouin zone. The same equation (with the obvious change of notation) holds for O p . The quantities C n and Op have dimensions of Ls i 1 - 1 . Multiplying them by concentrations of the vacant (occupied) centres of the type considered one obtains the transition probabilities of the electron (hole) from the conduction (valence) band to the level in question. 4 ) Every level is thus characterized by the two capture coefficients, Cn and C p . Sometimes they are written in the form Cn,p =

VT

>

(1-2)

where F T is the thermal velocity of the free electron while > 3 S A ^ © 'E © u OO -2- T) O Sb m © fl % s A - 2 3 .2 « 8 S 1 ° O " ©m ^® SK» rS B O S3 A &1 £ 'O © '-*3 « • S o "S «8 A 8 r ® fi ^ O 3 H O a

cl cs ¿3 © © A

o 2 gn £ 3 t 02 S

«3

£ O

® sC (B a I I 6

i

®

* ®

29

30

V . L . BONCH-BKITEVICH a n d E . G . L A N D S B E R G

The situation might change in case of t h e " d e e p " excitons (A — J e 0 and az 0 in cases of the attractive and repulsive fields, respectively). In doing the thermodynamical average in equation (1.1) the "thermal" values of p ~ (TO kT)1!2 are most important. Since (to IcT)1^ h¡\az\, equation (4.1) simplifies to Here a, =

N « or to N «

2

n h

P az

(Coulomb attraction)

exp j — 2 71 h \ p\az\ r [ p\az\\

(Coulomb repulsion) .

(4.2 a)

(4.2b)

17 ) In case of a neutral centre one may usually neglect the influence of its field upon the wave functions of the continuous spectrum — in any case at the present stage of recombination studies.

32

V . L . BONCH-BRUEVICH a n d E . G . LANDSBERG

In case of the shallow (hydrogen-like) traps the factor (4.2 a) may increase the capture cross-section by two orders of magnitude [130]. I n case of the deep traps the problem is quantitatively less clear, though it seems that there too the Coulomb correction to the wave functions of the continuous spectrum may substantially influence the recombination rate. The factor (4.2 b) clearly decreases the capture cross-sections especially for the slowly moving charge carriers. The reason is clear: to be captured by a negatively (positively) charged ion the electrons (holes) have to surpass a Coulomb potential barrier surrounding the region where the short-range attractive forces dominate which are responsible for the formation of the bound states. Note, however, t h a t the role of this factor cannot be described in classical terms. Indeed in the latter case the cross-section a(p) would be exactly equal to zero if the carrier energy W(p) = p2j2 m were less than some critical value (the barrier height) W0. On the other hand the right-hand side of equation (4.2b) is finite at any finite p. The reason of such a non-classical behaviour is evident: it is a manifestation of the quantum-mechanical tunneling through the potential barrier. This fact is automatically taken account of in (4.2b) and (4.1) since they were obtained using the exact wave functions of the Coulomb problem. 18 ) Experimentally interesting is the influence of the Coulomb field of the trap upon the temperature dependence of the capture coefficients. An exact calculation (especially in case of the attractive forces) is rather difficult due to the reasons discussed in Section 2. However, in case of the repulsive forces the comparatively strong energy dependence of the factor (4.2b) makes it possible to use a simple semi-phenomenological method [133, 112]. Namely experiments on the capture of the charge carriers by the neutral centres show (see Tables 3 and 4) t h a t in most cases of this kind there is no exponential temperature dependence of the cross-sections. Thus one may think t h a t the most pronounced form of the temperature dependence is due just to the Coulomb barrier effect. Consequently it seems reasonable to present the capture cross-section of a charged (repulsive) centre in the form of a product of the Sommerfeld factor N by some slowly varying function of the energy (say, of the power law type). 19 ) Using the Boltzmann form of the distribution function f(p) the integral inequation (1.1) is then easily done following paper [136] and one obtains Cn~exp{-(^-)1/3}, where Tn =

(4.3)

27 7i2 tn z2 e4

„ . .. ,— is the characteristic constant having the dimen2 £2 h2 k ° sion of a temperature. Putting e = 16 and m = 0.2 m0, where m 0 is the free electron mass, one obtains T0 = 3.26 z2 104 (°K). The additional factor which is not written down explicitly in (4.3) may as well be temperature-dependent but in a less pronounced way. (say, as some small power of T). As mentioned before to calculate this weak temperature dependence the model of the t r a p is to be specified which is very difficult to do at present in a convincing way. ,8 ) Note that the problem in question is formally similar to that of the rate of the nuclear reactions induced by the proton-nuclear collisions [131, 132]. 19 ) It is easy to take account of the eventual energy conditions limiting the available energy range and leading to an exponential temperature dependence of the cross-sections [133, 112]. The result is of course different from that given by equation (4.3).

Recombination Mechanisms

33

Note t h a t the function (4.3) is appreciably less sharp t h a n the simple exponential dependence

~ e x p ^ — ^ j j which would have been obtained in case

of the classical potential barrier of the height W0. As had to be expected tunneling weakens the effect of the barrier. There is a certain complication t o be met if one tries to compare t h e theoretical result (4.3) with experimental d a t a on germanium and silicon. Namely, equation (4.3), following from (4.1), was obtained assuming t h e parabolic and isotropic dispersion relation (W(p) = = p2j2 m). On the contrary the true electron 20 ) energy surfaces in these substances are anisotropic: they are represented by the sets of ellipsoides of t h e form

where mY ]> m t . The components of the crystal momentum px, py, and are referred here to the centre of the ellipsoid in question, which is located at one of the crystallographic axes. Strictly speaking one ought to solve the Coulomb problem using the dispersion relation (4.4). However, it is very difficult since the Schrodinger equation is not separable. Therefore, it seems desirable to keep the simple equation (4.3) making clear, however, what is to be put instead of a scalar effective mass m. I n [133] this quantity was f i t t e d so as to obtain the correct ionization energy of t h e shallow donor described by the same hydrogen-like problem. (It was in this way t h a t t h e above mentioned value of T0 was found.) Then t h e scalar mass model, however crude it might be, is at least self-consist ant. On the other hand an effective mass m t was used instead of m in the tunneling problem considered in [134]. The argument was t h a t these were just the low mass particles t h a t dominated the tunneling process. However, this procedure seems to us not very convincing since it overestimates somewhat the role of s-electrons in the continuous spectrum (see Appendix). Equation (4.3) was found to be in satisfactory agreement with experiment [24, 44, 84] using the value of T0 cited above. On the other hand several authors (see Tables 3, 4) found a weak temperature dependence of t h e cross-sections which was close to exponential with a small activation energy (of t h e order of 0.01 eV). I n principle this does not contradict the t h e o r y : according t o [133] such a possibility is realized when there are some additional limitations on t h e available energy range due to, say, energy conservation condition. Note, however, t h a t the temperature interval covered in the papers in question was r a t h e r narrow (cf. [135]). There was an a t t e m p t [136] to take account of screening in the tunneling problem. As was to be expected the screening weakens the barrier effect still further. Note, however, t h a t the usual self-consistent approach used in [136] is only valid provided e2 ml/3 I n this case the equations derived in [136] are reduced to (4.3) (weak screening). Thus the problem of the strong screening effect on t h e charge carrier capture by the repulsive centres remains still open. 20 ) In case of Ge and Si equation (4.3) is of some experimental interest in connection with electron capture by the negatively charged centres.

3 physica 29/1

34

Y . L . BOKCH-BBUEVICH a n d E . G. LANDSBERG

Up till now only the capture by point imperfections was considered in this paper. However, the same approach is valid in case of the imperfections possessing finite dimensions, such as dislocations. The conditions are rather easily met under which the recombination rate in germanium is controlled by an electron capture by negatively charged edge dislocations (see, e.g., [137, 138]). The relevant capture coefficient (at not too low temperatures) was found to be but weakly temperature-dependent — in contradiction with the theory [139] based upon the classical treatment of the potential barrier (the latter led of course to an exponential temperature dependence). More adequate seems to be the quantum-mechanical treatment given in paper [140], The tunneling, taking account of a comparatively weak temperature dependence of the capture coefficient On is obtained theoretically for the temperature range T ig 100 °K. However, at lower temperatures the theoretical situation changes substantially. The point is that it is a matter of principle [141] to take account of screening when treating the quantum-mechanical problem of the electron states in the Coulomb field of a charged dislocation (or of some other one-dimensional imperfection). Even the very weak screening is essential. Thus strictly speaking the very form of the potential barrier becomes temperature-dependent. At low temperatures this effect is appreciable and it leads to a drastic temperature dependence of the capture coefficients: On ~ where /S is a constant. No relevant experimental data are as yet known to the authors. Now turn to the problems of the second group. It is clear from what has been said above that some more or less convincing calculations may be done in the two cases only. The first of them is the case of a capture by some shallow traps described by the hydrogen model. The second one is that of a capture by repulsive centres when the tunneling probability plays the dominant role while the detailed shape of the potential near the trap is relatively unimportant. The relevant theoretical results are qualitatively clear a priori. Really the attractive centres are especially effective in capturing the slowly moving carriers. The electric field is expected to produce two effects. Firstly, it has to decrease the probability of the free carriers' transitions to the excited states of the centre, since the average velocities of electrons and holes increase in the field. Secondly, the probability of inverse transitions increases in the field due both to an impact ionization and to an autoionization, this leading to an effective decrease of the sticking probability. Both effects lead to a decrease of the capture coefficients on raising the electric field strength once the capture proceeds to attractive centres. Experimentally this is indeed the case [83, 89, 90]. In [89] data are reported on the field dependence of the electron capture coefficients, shallow donors of the group V (Sb, As, P) serving as recombination centres in germanium. At helium temperatures 4.2 °K) the capture coefficients begin to decrease at the field strength of about 0.2 Y/cm, following the relation Cn ~ E1-8. It is this fact that leads to the superlinearity of the I-V curve of n-type germanium doped with group Y donors as observed at low temperatures. Analogously the non-linear form of the I—V curve of Cu-doped p-type germanium [90] is due to the field dependence of the hole capture coefficient, the free holes making transitions to the 0.04 eV levels of the negatively charged copper ions. Convincing data are reported in [89] on the field-induced decrease of the hole capture coefficients in silicon, charged boron acceptors playing the role of the traps. In this work an essentially non-linear dependence of the current density upon the electric field strength was observed and shown to result both from the de-

35

Recombination Mechanisms ©

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53

phys. stat. sol. 29, 53 (1968) Subject classification: 13; 18.2; 19

Institute of Metal Physics,

Academy of Sciences of the Ukrainian

SSR,

Kiev

The Correlation Function for Weakly Coupled Impurity Atom Spin in a Ferromagnetic By M . A . KBIVOGLAZ a n d L . B . KVASHNIHA The closed chain of equations for the temperature Green's functions for the z-th component of the spin of an impurity atom in a ferromagnetic is obtained for the case of quasi-local spin excitations in the neighbourhood of the impurity atom. The solution of this system is obtained in explicit form for the cases of high and low temperatures and of small values of the spin of the impurity atom (8 — 1/2, 1, 3/2). For these cases the spectral representation Szz(co) of the correlation function eppoMarHeTHKe, B0JIH3H Kcrroporo HMeiOTCH KBa3HjioKajibHbie cnHHOBbie BO30y}KneHHH. PeiueHHe 3TOIT CHCT6MH n o j i y n e i i o B HBHOM BHRe HJIH cJiy^aeB BHCOKHX H HH3KHX TeMnep a T y p h HJIH HeSoJibuiHX 3HaieHHii cnHHOB npHMecHoro aTOMa ( S = 1/2, 1, 3/2). JI,JIH 3THX cjiynaeB onpeHejieHO cneKTpajibHoe npeHCTaBJieHHe Szz(a>) KoppejiHUMOHHott (JiyHKiiHH cnHHa npHMecHoro aTOMa. OTMe^eHO, MTO KBa3HJi0KajibHbie cnHHOBbie COCTOHHHH MoryT npHBonHTb K Ha HecKOJibKO IXOPHHHOB SojibineMy MOHyJiHUHOHHOMy yuiHpeHHio JIHHHH HAepHoro MarHHTHoro p e 3 0 HaHca, neM cnHHOBbie BOJIHH B HfleajibHOM KpHCTajuie.

It is necessary for some problems (for example for the investigation of spin broadening of a Mossbauer line or of lines of the nuclear magnetic resonance in ferromagnetics) to know the spectral representation S2Z(m) of the time dependent correlation function ( S ^ ) — Sz, Sz(t2) — Sz) for the z-component of the spin operator S of the impurity atom in ferromagnetics (z is the direction of magnetization, ° lies in the region of low state density of the spin waves. To determine Szz(m) in this case we shall consider the impurity atom in a ferromagnetic crystal. Assume that the exchange energy I of interaction of this atom with crystal atoms has the same sign as the exchange energy I 0 between the crystal atoms, but the value of I is much smaller than that of / 0 . Then the inequalities I and I 3 10 S0 are satisfied (S and S0 are the spins of impurity and crystal atoms, respectively). We shall assume that the temperature is low compared to the Curie temperature Tc, but may be comparable with (or greater than) oj°/i B (kB is the Boltzmann constant). Then in the Hamilto-

54

M. A. Krivoglaz and L. B. Kvashnina

nian of the exchange interaction one may go from the spin operators of crystal atoms (but not of the impurity atom) to the operators of spin waves (distorted by the presence of the defect) at, ax and can neglect in the "harmonic" approximation terms of 4-th order with respect to these operators. The spin deviations of the crystal atoms from the z-axis at T +

+ y*

- 2 Z yt' ((Sz a+. ax\ +

+ H y*xx'

Sz

+ Z y*'*" x' y-"

(5)

(«£ a*." —

a

*") a«\ •

(6)

I n the considered case of weakly coupled spin the last two terms on the right hand side of equations (5) and (6) may be neglected as compared to the third 1 terms (since Z j21. After this, taking into account t h a t S+ S- = 8 (S + 1) + Sz -

si,

8~ s+ = 8 (S + 1) -

Sz -

SI

(7)

it is possible to express the functions (() Gn _k+2

-

R(a>) Gn _k

+ 1]

,

(11)

k\ (n — k)\ '

Here lc in the sums

and

k

k

is only odd or even, respectively.

Since the Green's function G2 s+i may be expressed by functions with smaller values of n with the help of the well-known operator equality (8, -

S) (8, -

8 + 1) . . . (8, + 8) =

0,

the equation system (11) for Gn is closed on the equation for G 2 s and is reduced to the system of 2 8 equations. Having solved this system one may find all the functions Gn and among them the function G1 in which we are interested. I n particular, the expressions for Gt at the values of impurity spins S = 1/2; $ = ! ; $ = 3/2 are found to be G1

Gi

.

A to - R'

A, = -— cosh" 4 71

-Ai (co — 3 R') + BA2 ((a — R') (to - 3 R') + R* '

2 2 + cosh x n (1 + 2 cosh xf '

(S = 1) ,

.

A0

2 sinh x = — n ( 1 + 2 cosh x)2 '

(13)

The Correlation Function for Weakly Coupled Impurity Atom Spin

(co - 3 R') (co - 6 R')

A Oi =

1 =

A =

-

-L ii 2

-

A2

(CO

- 6 R')

R

+ 2 A3

57

R2

(co - 3 ii') [(co - i?') (co - 6 -R') + 6 £ 2 ]

1 cosh 2 a; + 4 cosh a; + 5 1 sinh 2 a; + 2 sinh a; / q ' =— 5 -TV a; ' 71 cosh — x 4- cosh —\ /cosh —- a: 4- cosh I 2 2/ \ 2 1 13 cosh 2 x + 28 cosh x + 41

(14)

(

(cosh — x + cosh —\ 2

2/

At arbitrary values of the spin 8 simple expressions for G1 may be obtained in the limiting cases of high and low temperatures (compared with S a>°lkB). In the first case, when kBT^> 8 a>° (but T Tc), for the region of small co which is important in applications, R may be neglected with respect to R', and the value

A2

A

= ^ (4 S2 + 4 8

—

3)

oj°/kBT

is small compared with

A1

=

= S (8 + l)/3 n. Thus, according to equations (8) and (10) (taking into account that Gs in (10) is not larger by an order of magnitude than S2 G,) the last term in equation (8) may be neglected. Then =

To-^W

> ^ > 8 ^ ) .

(15)

At low temperatures, when kBT it is convenient to substitute G2 by the function G'2 = {{(iS'2 — 8) (Sz — S + 1); Sz — S2)). Since at low temperatures those levels are mainly filled which correspond to quantum numbers m = 8 and m = S — 1 (co° > 0), the function G'2 is exponentially small as compared to G1 (or G2). Thus substituting G2 by (2 S — 1) Gl + G!2 in equation (8), omitting the term with G'2 and having in mind that at low temperatures A1 & — exp (— co°/&BT), and R'm R, one may find the following expression 71

for ( T J :

Using the relationship between the spectral representations of the correlation functions and Green's functions (for anticommutator) = i ^exp

+ 1

[Gj (co + i d) — Gx (co — i ABr>jSo»o) and at low temperatures

8„(o>) = | exp ( - ^

[exp ¿ L + 1

co2 + r 2

,

(¿BT°, for which it was deduced, b u t in the whole temperature range T Tc. The above expressions for the correlation function Slz(a>) were used in paper [5] for the analysis of the spin modulational broadening of the Mossbauer spect r u m of impurity nuclei in ferromagnetics. They m a y be used as well for the investigation of line broadening of the nuclear magnetic resonance on such nuclei. The spin modulational broadening 8 co of lines of the nuclear magnetic resonance on impurity atoms in ferromagnetics which is due to quasi-local excitations is determined by (27) where ¡x is the magnetic moment of the nucleus; dt% is the effective magnetic field at T = 0. The case is considered here when only superfine (but not dipolar) interaction between the nuclear and atom spins is essential. I n the presence of quasi-local excitation Szz(0) Xe). Since, according to equations (20) to (24), Szz(0) AJT is inversely proportional to a small quantity this spin broadening m a y happen to be some orders of magnitude greater t h a n in the perfect crystal, where Szz(0) has the order of magnitude «¡a A

1 2

/T\2

(—) In

B C \-»C/

kriT O)o

(Te is the Curie temperature, a>0 is the gap in the spin wave spectrum, see e.g. [6, 7]). References [1] T . WOLFRAM a n d J . CALLAWAY, P h y s . R e v . 1 3 0 , 2 2 0 7 ( 1 9 6 3 ) . T . WOLFRAM a n d W . HALL, P h y s . R e v . 1 4 3 , 2 8 4 ( 1 9 6 6 ) .

[2] Yu. IZYUMOV and M. V. MEDVEDEV, Zh. eksp. teor. Fiz. 48, 574 (1965); Soviet Phys. — J. exp. theor. Phys. 21, 381 (1965). Yu. IZYUMOV, Adv. Phys. 14, 569 (1965). [3] D . HONE, H . CALLEN, a n d L . R . WALKER, P h y s . R e v . 1 4 4 , 2 8 3 ( 1 9 6 6 ) .

[4] N. N. BOGOLYUBOV and S. V. TYABLIKOV, Dokl. Akad. Nauk SSSR 126, 53 (1959); Soviet Phys. - Doklady 4, 589 (1959). D. N. ZUBAREV, Uspekhi fiz. Nauk 71, 71 (1960); Soviet Phys. — Uspekhi 3, 320 (1960). [5] M. A . KRIVOGLAZ a n d L . B . KVASHNINA, p h y s . s t a t . sol. 2 9 , 61 ( 1 9 6 8 ) .

[6] A. H. MITCHELL, J. chem. Phys. 27, 17 (1957). [7] V . G. BARYAKHTAR, S. V . PELETMINSKII, a n d E . G. PETROV, F i z . t v e r d . T e l a 1 0 , 7 8 5 (1968).

(Received May 22, 1968)

M. A. KRIVOGLAZ and L. B. KVASHNINA: Spin Broadening of Mossbauer Lines

61

phys. stat. sol. 29, 61 (1968) Subject classification: 6.1; 13; 18.2; 18.4 Institute

of Metal Physics,

Academy of Sciences of the Ukrainian

SSR,

Kiev

On the Spin Broadening of Mossbauer Lines of Impurities in Ferromagnetics and of Atoms in Antiferromagnetics By M . A . KRIVOGLAZ a n d L . B .

KVASHNINA

The broadening and shift of Mossbauer lines for the nuclei of impurity atoms in ferromagnetics due to the interaction between the nucleus and atomic spins are discussed. Detailed examination is made of the effects associated with quasi-local and local spin excitations. It is shown that low-frequency quasi-local excitations can lead to a value of broadening some orders greater as compared with the case of ideal ferromagnetics. The broadening of Mossbauer lines is also determined for the ideal antiferromagnetic. P a c c M O T p e H b i y i i m p e H H e H c j j B H r M e c c S a y s p o B C K H X J I H H H H HJIH H n e p n p m v i e c H b i x aTOMOB B ( f i e p p o M a r H e T H K a x , o S y c j i o B j i e H H L i e B 3 a i i M o ; i e i i c T B H e M H j i p a c o c n n H a M n a T O M O B . B HaCTHOCTH, H C C J i e n O B a H B I 3eKTM, C B H 3 a H H h i e C K B a S H J I O K a j I b H b l M H H J I O K a j l b H B I M H c n H H O B b l M H B03f>y>KiI,eHHHMH. I l 0 K a 3 a H 0 , HTO H H 3 K O J i e > K a i H H e K B a 3 H J I 0 K a J l b H b i e B 0 3 6 y H t H e H H H M O r y T H a HeCKOJIbKO n O p H H K O B y B e J I H H H T b B e j l H i H H y y n i H p e H H H no c p a B H e H H i o co c j i y q a e M n n e a j i b H o r o ( J ) e p p o M a r n e T H i ; a . OnpeH e j i e H o T a K > K e y m n p e H H e MECCSAYSPOBCKOII J I H H H H B n n e a J i b H O M aHTH$eppoMarHeTHKe.

1. Introduction The fluctuations of magnetic moment in the vicinity of a Mossbauer nucleus in a ferromagnetic or antiferromagnetic crystal and connected with them fluctuations of the effective magnetic field result in the modulation of energetic levels of the nucleus. These fluctuations cause the modulational broadening of the lines of the fine structure of Mossbauer spectrum. This broadening as well as the broadening and the shift which are connected with transitions between multiplet levels have been considered by Kagan and Afanasiev [1] for nuclei in the perfect ferromagnetic by the method of diagrams which have been developed in [2], In the present work we consider the Mossbauer spectra of impurity nuclei in ferromagnetics. In particular, effects are investigated which are due to quasilocal spin excitations (QSE). I t is shown that thisQSE may cause the broadening of Mossbauer lines by some orders of magnitude greater than the spin waves in perfect crystals. The broadening and the shift of Mossbauer lines in perfect antiferromagnetic crystals will be considered, too. We shall use the method apparently rather convenient for such problems which has been applied earlier for the investigation of a narrow electronic line in optical spectra of impurity absorption [3 to 5]. I t is based on the application of time dependent correlation functions (CF) for the interaction Hamiltonian (see also [6]).

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M. A. Krivoglaz and L. B. Kvashnesta 2. General Formulae for the Cross Section of Mossbauer Absorption by Impurity Nuclei in Magnetically Ordered Crystals

Let us consider a magnetically ordered crystal in which the average spins of all atoms and effective magnetic fields are parallel to a definite axis, say to the •z-axis. We shall limit ourselves to the case where the dipolar interaction of nuclear spin with orbital magnetic moments of electrons and with spins of other atoms is not large; t h a t is one may take into account only the interaction of the nuclear spin with its own atom spin. Then the Hamiltonian of the system is of the form H = H0 + Hi + H,, s. Jtyp t e — J in yT defines the modulational broadening. 1 ) 3. Broadening of Mossbauer Line in Ferromagnetics due to Quasi-Local Spin Excitations At some relationships between the spins and the exchange integrals of impurity atoms and crystal atoms QSE are produced in the vicinity of impurity atoms [7 to 10]. Their energy lies in the band of energies of the spin waves but are found in the region of low density of states. The energy levels with a finite b u t not large width correspond to these excitations. If there are such QSE the Fourier component in CF for the spin of an impurity S_+(a>) must have a sharp peak a t the frequency of QSE. The presence of such a peak should have significant effect on the value of the function Szi(a>) and on the modulational broadening of Mossbauer line for an impurity. To calculate CF Szz(co) in a "harmonic" approximation one must have in mind t h a t in the fourfold sum (16) in this approximation are considered only terms with xy = xt, x2 = x3. Those terms which have the same four indices x2, x3, x 4 in the absence of "true local" states in the limit of the infinite crystal give an infinitesimal contribution. Hence, taking into account the fact t h a t for a ferromagnetic S++{co)

= S__(a>) = 0

(21)

and employing equation (15), the expression (16) m a y be rewritten as oo oo >s'zz(co) =

f S_+(a>') S+_ (CO — co') don' = f m ' (a a + > m _ m ' d co0 the expressions for the shape of spectral distribution and for its width are somewhat altered (see analogous expressions which describe the optical spectra in [5]).

Spin Broadening of Mossbauer Lines of Impurities in Ferromagnetics

67

C F ( a a + ) ( 0 a n d valid in the whole temperature range kBT

/o i\ (« = D •

/or^ (30)

The quantity r , entering into equations (25) and (27) to (30), was determined in the approximation of nearest neighbours in [8] and may be written as r = 2 7r-£(co0)2(?(co0) , (31) «0 where G(a>°) is the state density for the spin waves in the perfect crystal. I n the discussed "harmonic" approximation, according to (31), r does not depend on temperature. Therefore, as one may see from equations (27) to (30), the modulational broadening y'ss> due to QSE at first exponentially increases with rising temperature, then at kBT « S OJ° reaches the maximum and at high temperatures decreases proportional to T'1. The plots of the temperature dependence y'SS' for the spins 8 = 1/2, S = 1, and S = 3/2 are shown in Fig. 1. Let us compare y'SS' with the width y°s' in an ideal ferromagnetic formed by crystal atoms. In the approximation of the nearest neighbours one may put in equation (19) /3 = 1 /4 JV I0 80 Z v2'3 where for s.c., b.c.c., and f.c.c. lattices we

Spin Broadening of Mossbauer Lines of Impurities in Ferromagnetics

69

have JV = 2/3, JV = 2*/3/3 and JV = 2" 1 / 3 , respectively. Noting t h a t G(OJ) = 1}

= — -2 -^r^r2 , from equations

4 n y'ss'l'/L'

p**/

at the temperature 0

kBT

(19), (28), and (31) we find t h a t the ratio &

S

oj° is of the order

+ 1) / 1 ,