Physica status solidi: Volume 27, Number 2 June 1 [Reprint 2021 ed.] 9783112500620, 9783112500613

Citation preview

pliysica status solidi

VOLUME 17 • NUMBER 2

196S

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

physica status solidi B o a r d of E d i t o r s P. A I G R A I N , Paris, S. A M E L I N C K X , Mol-Donk, V. L. B O N C H - B R U E V I C H , Moskva, W. D E K E Y S E R , Gent, W. F R A N Z , Münster, P. 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 E 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 , Saarbrücken, J . D . E S H E L B Y, Cambridge, P. P. F E O F I L 0 V, Leningrad, J. H O P F I E L D , Princeton, G. J A C O B S, Gent, J. J A U M A N N , Köln, E. K L I E R , Praha, E. K R O E N E R , Clausthal-Zellerfeld, R. K U B O , Tokyo, M. M A T Y À S , Praha, H. D. M E G A W , Cambridge, T. S. MOSS, Camberley, E. N A G Y , Budapest, E. A. N I E K I S C H , Jülich, L. P A L , Budapest, M. R O D O T , Bellevue/Seine, B. V. R O L L I N , Oxford, H . M . R O S E N B E R G . Oxford, R. V A U T I E R , Bellevue/Seine

Volume 27 • Number 2 • Pages 465 to 764, K85 to K172, and A5 to A8 June 1,1968

AKADAMIE-V ERLAG • BERLIN

Subscriptions and orders for single copies should be addressed to AKADEMIE-VERLAG G m b H , 108 Berlin, Leipziger Straße 3 - 4 or to Buchhandlung K U N S T UND WISSEN, Erich Bieber, 7 Stuttgart 1, Wilhelmstr. 4 - 6 or to Deutsche Buch-Export und - I m p o r t GmbH, 701 Leipzig, Postschließfach 160

Editorial Note: 14

physic a status solidi" undertakes t h a t an original paper accepted for publication before t h e of any month will be published within 50 days of this date unless t h e author requests a postponement. I n special cases there m a y be some delay between receipt and acceptance of a paper due to t h e review and, if necessary, revision of the paper.

S c h r i f t l e i t e r u n d v e r a n t w o r t l i c h f ü r d e n I n h a l t : Professor D r . D r . h . c. P . G ö r l i c h , 102 B e r l i n , N e u e S c h ö n h a u s e r S t r . 20 b z w . 69 J e n a , H u m b o l d t s t r . 26. R e d a k t i o n s k o l l e g i u m : D r . S. O b e r l ä n d e r , D r . E . G u t s c h e , D r . W . B o r c h a r d t . A n s c h r i f t d e r S c h r i f t l e i t u n g : 102 B e r l i n , N e u e S c h ö n h a u s e r S t r . 20. F e r n r u f : 4 2 6 7 8 8 . Verlag: Akademie-Verlag G m b H , 108 B e r l i n , L e i p z i g e r S t r . 3 — 4 , F e r n r u f : 2 2 0 4 4 1 , T e l e x - N r . 1 1 2 0 2 0 , P o s t s c h e c k k o n t o : B e r l i n 3 5 0 2 1 . D i e Z e i t s c h r i f t „ p h y s i c a s t a t u s solidi 44 e r s c h e i n t jeweils a m 1. des M o n a t s . B e z u g s p r e i s eines B a n d e s M 90,— ( S o n d e r p r e i s f ü r die D D R M 60,—). B e s t e l l n u m m e r dieses B a n d e s 1068/27. J e d e r B a n d e n t h ä l t z w e i H e f t e . G e s a m t h e r s t e l l u n g : V E B D r u c k e r e i „ T h o m a s M ü n t z e r 4 4 B a d L a n g e n s a l z a . — V e r ö f f e n t l i c h t u n t e r d e r L i z e n z n u m m e r 1310 des P r e s s e a m t e s b e i m V o r s i t z e n d e n d e s M i n i s t e r r a t e s der D e u t s c h e n D e m o k r a t i s c h e n R e p u b l i k .

Compared Activities of the Main Abstracting and Indexing Services Covering Physics, Chemistry, and Biology during the Year 1965 ICSU Abstracting Board, 17 rue Mirabeau, Paris 16e December 1967 — 83 pages — U.S. $ 5.00 plus mailing charges As indicated by its title, this report gives detailed information about the Main Abstracting and Indexing Services which are members of the ICSU Abstracting Board: Referativnyi Zhurnal, Bulletin Signaletique, Chemical Abstracts Service, Biological Abstracts, Physikalische Berichte, Chemisches Zentralblatt, Astronomische Jahresberichte, Physics Abstracts. The number of periodicals scrutinized, non-periodical literature covered, number of abstracts published, Abstracting and Indexing practices, use of computers, etc., are described, compared, and commented.

physica status solidi, Bd. 27, Heft 2

Contents

Page

Original Papers T . M . FITZGERALD a n d B . D . SILVERMAN

M. I. W.

KLINGER

ULRICI

Ultrasonic Attenuation in Unirradiated and Neutron Irradiated Quartz

473

On the Experimental Observation of Small Polarons in Semiconductors

479

Untersuchung dreiwertiger Übergangsmetallionen in Silberhalogeniden (II)

489

F . P . B U L L E N a n d S . M c K . COUSLAND

The Temperature Dependence of the Flow Stress of Copper Single Crystals

501

O . BRUMMER u n d G . DRÄGER

S.

CERESARA

J . SAK F.

W.

FELIX

Untersuchungen zur Deutung der kantenfernen Feinstruktur der Röntgen-K-Absorptionsspektren

513

A Step Annealing Procedure for the Determination of Diffusion Coefficients in Metals by the Resistometric Method — Application to the Diffusion of Cu in Al

517

On the Theory of Shallow Impurity States

521

Rare-Gas Diffusion in Neutron-Irradiated Rubidium Halides . . .

529

B . K . D A N I E L S a n d D . B . MEADOWCROFT

Twist Boundaries and Electroluminescence

535

G . F . A L F R E Y a n d D . B . MEADOWCROFT

The Influence of Structure on the Electroluminescenec of Zinc Sulphide P . LUKAS,

M. KLESNIL, a n d J .

541

KREJM

Dislocations and Persistent Slip Bands in Copper Single Crystals Fatigued at Low Stress Amplitude 545 R . J . M U R P H Y a n d J . L . CRAWFORD

C. J E C H

T.

JOSSANG

Electron Microscope Image Profiles of Paired and Triple Dislocations

559

Depth Distribution Measurements Using Bombardment-Enhanced Solubility

573

The Self-Energy of a Dislocation and the Elastic Energy of Piecewise Straight Dislocation Configurations

579

H . HORA a n d G . B . KABIERSCH

Combined Infrared Photoemission from Cs3Sb

593

H . - D . DIETZE u n d K . SCHRÖDER

Magnetisierung in der Umgebung unmagnetischer Einschlüsse in Ferromagnetika (I) K . SCHRÖDER u n d H . - D . D I E T Z E

601

W

Magnetisierung in der Umgebung unmagnetischer Einschlüsse in Ferromagnetika (II)

611

B. S. Tosiö and R. B. MAKULA Kinematic and Dynamic Interaction between Elementary Excitations in Ferromagnetics with Spin S = 1/2

623

468

Contents Page

P . SUPTITZ a n d R . W E I D M A N N

Diffusion of Mn 2+ in AgCl and AgBr Crystals

B . H . ZIMMEBMANN,

H . JENA,

G . ISCHENKO,

H . KILIAN, a n d D . SEYBOTH

Mossbauer Experiments with Coulomb-Excited Recoil Implantation J.

631 73

Ge after Coulomb 639

+

AUTH

On the Reverse-Biased Capacitance of Step p - n Junctions with Traps

653

G. APPELT

Fine Structure Measurements in the Energy Angular Distribution of Secondary Electrons from a (110) Pace of Copper

657

P . R . HERCZFELD,

K . M . VAN V L I E T , a n d M . D . P A I

Experiments on Fluctuations in Optically Quenched CdS and CdSe Crystals

671

P . R . H E R C Z F E L D a n d K . M . VAN V L I E T

Electronic Processes and Fluctuations in Optically Quenched CdS and CdSe Crystals

681

S. D . MCLAUGHLAN a n d H . W . EVANS

Production of Colloidal Calcium by Electron Irradiation of CaF 2 Crystals M. WUTTIG,

695

J . T . STANLEY, a n d H . K . BIRNBAUM

Interstitial Solute Trapping in Irradiated and Quenched Iron . . .

701

K . MAIER a n d W . GEBHARDT

H. K.

MULLER

H. K.

MULLER

B . A . STRUKOV,

Stress-Induced Inhomogeneous Effects of the K-Absorption in RbCl

713

Electrical and Optical Properties of Sputtered l n 2 0 3 Films (I) . . .

723

Electrical and Optical Properties of Sputtered l n 2 0 3 Films (II) . . .

733

M . AMIN,

a n d V . A . KOPCHIK

Comparative Investigation of the Specific Heat of K H 2 P 0 4 (KDP) and K D 2 P 0 4 (DKDP) Single Crystals

741

V . FEDOSEEV a n d V . HIZHNYAKOV

On the Theory of Low-Temperature Exciton Absorption Spectra

751

J . K . POZHELA a n d K . K . REPSHAS

Thermoelectric Force of Hot Carriers

757

A . I . M I T S E K a n d N . P . KOLMAKOVA

Erratum

763

Short Notes S . A . MIRONOV,

K . V . SHEVLYAGIN,

A . G . GTJREVICH, a n d B . M . L E B E D

Excitation of Magnetostatic, Magnetoelastic, and Acoustic Waves in Ca-Bi-V-Fe Garnet K85 J . L . BRIMHALL a n d B . MASTEL

Defect Clusters in Neutron Irradiated Rhenium A.

GRAJA

O. G.

VENDIK

K89

Production of the Second Harmonic of Light in Ammonium Pentaborate and Other Powdered Piezoelectric Crystals K93 On the Nonlocal Connection between Polarization and Field in Ferroelectrics

Electric K99

Contents

469 Page

W . SzYMANSKA a n d J . GINTER

On the Scattering of Electrons by Ionised Impurities in t h e Case of Strong Degeneracy K103 E . KIERZEK-PECOLD,

J . KOLODZIEJCZAK, a n d I . PRACKA

Anisotropy in the Reflection Coefficient of ß-Rhombohedral Boron in the Region 1 to 6 eV K107 E . W . KREUTZ,

H . PAGNIA, a n d W . WAIDELICH

X - R a y Induced Conductivity Changes in InSb Study of Iron Ions in

K . HENNIG

57

Kill

Fe-Doped AgCl Using t h e Mössbauer Effect

K115

R . K O E P P a n d K . MARINOVA

High Field Domains in CdS Crystals Moving in the Anode-Cathode Direction K117 YA. 0 . DOVHYJ

P . BYSZEWSKI,

Connection between the Circular Dichroism and Rotatory Dispersion of Gyrotropic Crystals and its Experimental Verification in the Region of the Excitation of Circular Excitons K121 J . HOLZMAN, a n d J . KOLODZIEJCZAK

Oscillatory Magneto-Photoconductivity Effect in Germanium . . . K125 A . P . ZHUKOV,

I. S. REZ,

V . I . PAKHOMOV, a n d G . K . S E M I N

Temperature Dependences of the Nuclear Quadrupole Resonance Spectra of As 75 in KH 2 As0 4 , RbH 2 As0 4 , CsH 2 As0 4 , NH 4 H 2 As0 4 , and of their Deuterated Analogues K129 D . HAHN a n d J . MIMKES

On the Frequency Dependence of Electroluminescence of ZnS :Cu Phosphors K133 I . LICEA

Warm Electron Temperature and Mobility in Polar Semiconductors K137

I . LICEA

H o t Electron Temperature and Mobility in Polar Semiconductors

K143

W . HAUBENREISSER

Nonlinear Two-Magnon-One-Phonon Garnet

Confluence- Process in BVC KJ45

P . GROSSE a n d B . K R A H L - U R B A N

The Contribution of Free Carriers to the Complex Dielectric Constant of Tellurium a t 9 GHz K149 R . P . HARRISON

The Yield Strength of Sodium Chloride Containing Manganese

M. ZVARA

I n t e r b a n d F a r a d a y Rotation in Gallium Arsenide

K . HENNIG,

.

. K153

K157

K I M Y U N G , a n d B . S . SKORCHEV

On States of Iron I m p u r i t y Ions in AgCl and NaCl Investigated by the Mössbauer Method K161 R . BALZER,

H . PEISL, a n d W . WAIDELICH

Evidence for Frenkel Defect Recombination during Thermal Annealing of Low-Temperature X-Irradiated KCl K165 I . IVANOV-OMSKII,

V. A.

SKRIPKIN

F . P . KESAMANLY,

B . T . KOLOMIETS,

A . SH. MEKHTIEV,

Density of States Effective Mass of Holes in HgTe

and

K169

470

Contents

Pre-printed Titles of papers to be published in the next issue

471

Contents

Systematic List Subject classification:

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

1.2

741

1.3

545

1.4

559, 573

4

541

6

473, 521, 751, K145

6.1

639, K115, K161

7

473, K85

8

741

9

517, 529, 631

10

489, 529, 535, 541, 559, 579, 631, K115, K153, K161,

10.1

501, 545, 559, 701, K89

10.2

695, 713

11

473, 529, 573, 695, K89, K i l l , K165

13

479, 623, K137, K143, K145, K157

13.1

521, 657, K169

13.2

751, K121

13.3

Kill

13.4

489, 521, 653, 681, 713, 723, K103

14.3

479, K103, K i l l , K137, K143,

14.3. 1

723

14.3. 2

653

14.4

489

14.4.1

757, K117, K143

14.4. 2

741, K99, K129

15

757, K169

16

671, 681, K117, K125

17

593, 657

18

K125, K157

18.2

601, 611, 623

18.3

K85,

K149

K145

19

695, K129

20

513, K121,

20.1

479, 489, 657, 695, 723, 733, 751, K107

K157

20.2

K93

20.3

535, 541, K133

K165

Contents

472 21

517, K89

21.1

501, 545, 657

21.1.1

701

22

513, 521, 573, 593, 751, K93, K103, K121

22.1

K107

22.1.1

513, 639, 757, K125

22.1.2

573, 653

22.1. 3

K149

22.2.1

K157

22.2.3

K i l l , K137, K143

22.4.1

535, 541, 671, 681, K117, K133

22.4.2

671, 681

22.4. 3

K169

22.5.1

489, 631, K115, K161

22.5.2

529, 713, K153, K161, K165

22.5.3

695

22.6

473, 479, 513, 723, 733

The Author Index of Volume 27 Begins on Page 765 (It will be delivered together with Volume 28, Number 1.)

Original

Papers

phys. stat. sol. 27, 473 (1968) Subject classification: 7; 6; 11; 22.6 NASA ¡Electronics

Research Center, Cambridge, Massachusetts

Ultrasonic Attenuation in Unirradiated and Neutron Irradiated Quartz By T . M . FITZGEKALD and B . D . SILVEBMAN The ultrasonic attenuation of unirradiated and neutron irradiated quartz is discussed in terms of recent theoretical formulations of the three-phonon scattering mechanism. I t is shown that the temperature dependence of the attenuation in the unirradiated quartz crystal can be understood by invoking this scattering mechanism. I t is also shown that upon neutron irradiation the observed changes in the attenuation can be qualitatively understood by assuming that neutron irradiation decreases the thermal phonon lifetimes in the crystal. Die Schwächung von Ultraschallwellen in mit Neutronen bestrahltem und unbestrahltem Quarz wird im Zusammenhang mit kürzlich erhaltenen theoretischen Formulierungen des Drei-Phonon-Streumechanismus diskutiert. Es wird gezeigt, daß die Temperaturabhängigkeit der Schwächung im unbestrahlten Quarzkristall mit diesem Streumechanismus verstanden werden kann. Es wird ferner gezeigt, daß die bei Neutronenbestrahlung beobachteten Änderungen der Schwächung qualitativ verstanden werden können, indem man annimmt, daß Neutronenbeschuß die Lebensdauer thermischer Phononen im Kristall verringert.

1. Introduction For several insulating crystals, it appears that a longitudinal sound wave decays via a third order anharmonic interaction with the longitudinal thermal phonons of the crystals [1, 2], This process dominates the attenuation even though it is not allowed if the participating longitudinal thermal phonons have infinite lifetime. Several theoretical investigations [3] have been devoted to obtaining an expression for the ultrasonic attenuation resulting from this third order process. The expression obtained by Maris is 8.68 h y A (q s + q1 — q2) \0(qa g,; -g 2 )| 2 6n(w)

_

OC —

N x

s

Qi q±

—

w2

(A + A) (n + u>1 + Ax - a>2 - A2f +

___ + r2f

•

X

m

This expression involves a sum over the wave vectors of the two thermal phonons. To generate a simpler and more useful expression for the attenuation, it is assumed that the thermal phonon frequencies are given approximately by the "dominant thermal phonon frequency" at a given temperature [4], i.e., h o)T « kT. Conservation of wave vector, however, requires an angular integration to be performed over the angle between one thermal phonon wave vector and the sound wave vector. One might expect the result of this angular integration to be significant for the following reason. For a nondispersive longitudinal branch, only collinear third order processes are allowed for infinite thermal phonon lifetime. For finite thermal phonon lifetime, the wave vectors of the ther-

>

474

T . M . FITZGERALD a n d B . D .

SILVERMAN

mal phonons that interact with the sound wave are found in a cone about the sound wave vector [5]. How this cone opens up with increasing temperature or decreasing thermal phonon lifetime, one might expect, indicates in a quantitative manner how the ultrasonic attenuation will depend upon the thermal phonon lifetime and hence the temperature. In the expression for the attenuation obtained by Silverman ([7], eq. 2), the angular integration has been performed by neglecting only terms of order t the thermal phonon frequency.) Hence, one might expect this expression to reflect a more accurate behaviour of the ultrasonic attenuation with temperature than obtained by approximating the angular integration more crudely. At high temperatures (Q T 1) equation (1) yields, for an isotropic material, a result independent of the dispersion of the longitudinal branch. The result of the angular integration leading to equation (2) has this property. Even though {1 — y i T ) } at higher temperatures is significantly different from 1 and violates the assumption that only slight deviations from a nondispersive branch at the thermal phonon frequencies will be considered, we believe that equation (2) is sufficiently meaningful to be evaluated into the Q x 1 regime. One is in a sense not using the linear chain dispersion relation but a parabolic form for the dispersion of the longitudinal branch. Anyhow we emphasize that it is important to retain the { 1 — y{T)} in the denominator of equation (2) otherwise an anomalous high temperature behavior will result. I t should also be pointed out that we have used the thermal conductivity data of deHass and Biermasz for X-cut quartz to obtain thermal phonon lifetimes. At low temperatures, these are the appropriate values to use if we are to take this anisotropic effect into account within the context of our isotropic theory, i.e. we have chosen the thermal conductivity data measured in the same direction as the attenuation since the thermal phonons that damp the sound wave propagate predominantly in this direction. I t has previously been suggested [1, 6, 7] that the most significant modification of the ultrasonic attenuation arising from neutron irradiation, resulted from a decrease in thermal phonon lifetimes. At high temperatures, (Q r 1), the decrease in the thermal phonon lifetime allows the thermal phonons to relax more rapidly to the instantaneous phonon equilibrium distribution and the attenuation is therefore decreased. At low temperatures, (Q 1), the reduction in thermal phonon lifetime relaxes energy conservation, and hence the attenuation is increased. For slight neutron irradiation, the unirradiated and irradiated attenuation curves cross in the vicinity of Q r = 1. In section 4, these ideas are illustrated. 2. Ultrasonic Attenuation The equation that will be used to fit the temperature dependence of the ultrasonic attenuation in quartz is [7] a

"

8.68 8

Si T e

CV(T)

* {1 -

r

\

2

y ( T ) }

a r

°

t

a

n

\l

+

2(1 - y ( J ) ) A t (2 - y ( T ) ) (Q

y ( T )

) r)»J

( Z

>

with y { T ) = 0.32 d k T j s h . This equation is derived assuming the sound wave to decay via a third order anharmonic interaction with the higher frequency thermal phonons in the crystal. To obtain this relatively simple expression for the attenuation, a number of simplifying assumptions have been made. First, 2

2

2

2

2

Ultrasonic Attenuation in Q u a r t z

475

it is assumed that the major contribution to the attenuation results from a third order R a m a n type process in which the sound phonon and a thermal phonon are destroyed with the creation of another thermal phonon. This assumption should be valid for temperatures such that k kT h Q. Next, approximations involved in evaluating certain integrals over energy are made, which are equivalent to assuming exponential decay of the two participating thermal phonons. These approximations lead to the Lorentzian linewidths that replace ¿-function energy conservation. Third, due to the relatively low acoustic frequency, approximate conservation of energy requires that the two thermal phonons involved in the process differ in frequency by approximately the acoustic frequency and hence can be considered to have equal density of states at the dominant phonon frequency. In making the dominant phonon approximation, it is assumed that the lifetime of the thermal phonon should be evaluated at the dominant phonon frequency, and hence should be temperature dependent. If phonon lifetimes obtained from thermal conductivity measurements are substituted for the thermal phonon lifetime r in equation (2) the temperature dependents of the lifetime is taken into account. In section 4, we will investigate the effect on the ultrasonic attenuation arising from reducing the thermal phonon lifetime by neutron irradiation of the sample. Here again, if values of r are obtained from thermal conductivity measurements on neutron irradiated samples, the temperature and hence frequency dependence of the imperfection scattering is implicitly taken into account. In section 4, we will also investigate the implications of frequency independent imperfection scattering, since it reveals in a simple manner the qualitative features of experiment; namely, with neutron irradiation, the attenuation increases at low temperatures and decreases at higher temperatures. 3. Unirradiated Quartz

Si02 X-cut 1000 MHz Experiment— Theory •

where x(T) is determined from the thermal conductivity data of deHass and Biermasz [8] for heat flow parallel to the optic axis. The results of these calculations depend to some extent on the "dominant thermal phonon" frequency', mr = 1.6 B kTjh where B is consi-

'0.001,

0.0001 10

A theoretical curve calculated using equation (2) is compared in Fig. 1 with the experimental data of Bommel and Dransfeld [1] for the attenuation of a 1000 MHz longitudinal wave in X-cut quartz. The lifetime of the thermal phonons r appearing in this equation is calculated using the expression: 1 x ( T ) = - C v ( T ) s*r (3)

SO

50

70 Tl'Kl-

Fig. 1. Comparison of theory and experiment for the temperature 110 dependence of the attenuation of a 1000 MHz longitudinal wave in X-cut quartz

476

T . M . FITZGERALD a n d B . D . SILVERMAN

dered a n a d j u s t a b l e c o n s t a n t . T h e high t e m p e r a t u r e (Q x of e q u a t i o n (2) reduces t o 4

1) limiting f o r m

Q s3

which is essentially t h e expression o b t a i n e d b y Akhiezer [9]. F o r t h e case of a non-dispersive longitudinal b r a n c h where d = 0, e q u a t i o n (2) reduces t o 8.68 TCy(T) 2 r Q tan-1 2 Ü x (5) which is t h e result o b t a i n e d by Woodruff a n d Ehrenreich [3], This result reduces t o e q u a t i o n (4) for Q x 1. Thus, in t h e high t e m p e r a t u r e limit, e q u a t i o n (2) reduces t o a well-behaved expression t h a t is i n d e p e n d e n t of b o t h t h e d o m i n a n t p h o n o n f r e q u e n c y a n d dispersion. I n other words, a t high t e m p e r a t u r e where t h e t h e r m a l phonons h a v e a v e r y s h o r t lifetime, each p h o n o n m a k e s a n equal c o n t r i b u t i o n t o t h e a t t e n u a t i o n ; a n d t h e concept of a d o m i n a n t p h o n o n no longer applies. H o w e v e r , a t low t e m p e r a t u r e s , where t h e t h e r m a l p h o n o n lifetime is long, b o t h t h e m a g n i t u d e a n d t h e t e m p e r a t u r e dependence of t h e a t t e n u a t i o n depends on t h e d o m i n a n t p h o n o n f r e q u e n c y . These concepts are illust r a t e d in Fig. 2 where we h a v e chosen d i f f e r e n t values of B which is e q u i v a l e n t t o v a r y i n g t h e f r e q u e n c y of t h e d o m i n a n t p h o n o n . The e x p e r i m e n t a l d a t a for longitudinal waves in p u r e X - c u t q u a r t z exhibits a T 7 dependence which suggest t h a t (or 6 kTjh. Similar calculations [10] for A1 2 0 3 also suggest t h a t wT «

6

kTjh.

4. Irradiated Quartz T h e t e m p e r a t u r e dependence of t h e ultrasonic a t t e n u a t i o n is a sensitive f u n c t i o n of t h e lifetime of t h e t h e r m a l phonons. If t h e lifetime of t h e t h e r m a l phonons is c h a n g e d b y i n t r o d u c i n g scattering centers i n t o t h e crystal, t h e r e should be a corresponding change in t h e a t t e n u a t i o n . These changes are shown in Fig. 3 for f a s t n e u t r o n i r r a d i a t e d q u a r t z . As a result of irradiation, t h e a t t e n u a tion is reduced a t high t e m p e r a t u r e s a n d increased a t low t e m p e r a t u r e s . The theoretical curves were calcul a t e d using e q u a t i o n (2) (as in sect i o n 3) a n d t h e t h e r m a l conduct i v i t y d a t a of B e r m a n [11] for neutron irradiated quartz. For the slightly i r r a d i a t e d sample, t h e agreem e n t is quite reasonable. F o r t h e heavily i r r a d i a t e d sample, t h e calculated a n d e x p e r i m e n t a l curves

60 70 T!°K1 —

Fig. 2. E f f e c t s of changing t h e d o m i n a n t p h o n o n frequency on the temperature d e p e n d e n c e of the ultrasonic a t t e n u a t i o n (read SiO E instead of SiO)

477

Ultrasonic Attenuation in Quartz

Fig. 3. Effects of fast neutron irradiation on the temperature dependence of the ultrasonic attenuation in X-cut quartz

Unirradiated

agree at high temperatures while at low temperatures the calculated curve falls away more rapidly than the experimental one. The results of systematically decreasing the thermal phonon lifetime which simulates increasing the defect concentration are shown in Fig. 4. Here it is assumed that the total relaxation time r can be written as

10"n/cm

Si02 X-cut 16Hz Neutron irradiated

1

T„

+

.

(6)

where r a is the anharmonic contributions of a pure crystal and r-t is a finite contribution arising from the presence of imperfections. Strictly speaking, r i should have its own 20 30 W 50 60 70 80 frequency and temperature depenTi'K)dence depending on the type of imperfection. However, for low imperfection densities in neutron irradiated quartz the assumption rl = constant in equation (6) is reasonably consistent with thermal conductivity data. As can be seen from Fig. 4, there are a number of features that result from changing the lifetimes of the thermal phonons by imperfection scattering. At high temperatures, the attenuation of the irradiated sample decreases with increasing irradiation as the thermal phonons can relax more rapidly to the instantaneous phonon equilibrium distribution. At low temperatures, the attenuation of the irradiated sample is higher than the pure sample as energy conservation is relaxed and more thermal phonons take part in the interaction. The cross-over region, where the attenuation in the unirradiated and irradiated samples is equal, moves to lower temperatures with increasing irradiation : The net result of lowering the cross-over region is to increase the temperature range in which the "Akhiezer limit" is appropriate. 0.001

60

/Y7l7

70

80

Fig. 4. Effects of decreasing the thermal phonon lifetime on the temperature dependence of the ultrasonic attenuation (dashed line represents experimental curve for unirradiated quartz)

478

T . M . FITZGERALD

a n d B . D . S I L V E R M A N : Ultrasonic A t t e n u a t i o n in S i 0 2

References and K . DRANSFELD, Phys. Rev. 1 1 7 , 1 2 4 5 (1960). E . H . JACOBSEN, i n : Q u a n t u m Electronics, Columbia U n i v e r s i t y Press 1960 (p. 468). R . N A V A , R . A Z R T , I . CICCARELLO, a n d K . D R A N S F E L D , P h y s . R e v . 134, A 5 8 1 ( 1 9 6 4 ) . I . S. CICCARELLO a n d K . D R A N S F E L D , P h y s . R e v . 134, A 1 5 1 7 ( 1 9 6 4 ) . M. POMERANTZ, P h y s . R e v . 139, A501 (1965). D. W . O L I V E R a n d G. A. S L A C K , J . appl. P h y s . 37, 1542 (1966). J . D E K L E R K a n d P . G. K L E M E N S , P h y s . R e v . 147, 5 8 5 ( 1 9 6 6 ) . T. D. W O O D R U F F a n d H . E H R E N R E I C H , P h y s . R e v . 123, 1553 (1961). L . D. L A N D A U a n d G. R U M E R , P h y s . Z. SU 11, 18 (1937). K . KAWASAKI, Progr. theor. Phys. 2 6 , 7 9 5 ( 1 9 6 1 ) . H . J . MARIS, Phil. Mag. 12, 89 (1965). S. SIMONS, Proc. P h y s . Soc. 82, 401 (1963). A . J . L E G G E T T a n d D . TER H A R R , P h y s . R e v . 1 3 9 , A 7 7 9 ( 1 9 6 5 ) . P . C. K W O K , P . C. M A R T I N , and P . B . M I L L E R , Solid S t a t e C o m m u n . 3 , 1 8 1 ( 1 9 6 5 ) . R . ORBACH, U n p u b l i s h e d Thesis, 1960, U n i v e r s i t y of California, Berkeley. P . G. KLEMENS, i n : Physical Acoustics, Vol 3 b , Chap. 5, Academic Press 1965. P . B . MILLER, Phys. Rev. 1 3 7 , A 1 9 3 7 (1965). N. S . S H I R E N , P h y s . L e t t e r s (Netherlands) 2 0 , 1 0 ( 1 9 6 6 ) . J . M. ZIMAN, Electrons a n d Phonons, Oxford Univ. Press 1960. H . J . MARIS, Phil. Mag. 9, 901 (1964).

[ 1 ] H . E . BOMMEL

[2]

[3]

[4] [5]

[6] T . M. FITZGERALD, B . B . CHICK, a n d R . TRUELL, J . a p p l . P h y s . 3 6 , 1639 (1964).

[7] B. D. SILVERMAN, P r o g r . theor. P h y s . , t o b e published. [8] J . W . DEHASS a n d T . BIERMASZ, P h y s i c a 2, 6 7 3 (1935).

[9] A. AKHIESER, J . P h y s . (SU) 1, 277 (1939). [10] T. M. F I T Z G E R A L D a n d B. D. SILVERMAN, P h y s . L e t t e r s (Netherlands) 2 5 A , 245 (1967). (Received January

16,

1968)

M. I. K l t n g e r : Observation of Small Polarons in Semiconductors

479

phys. stat. sol. 27, 479 (1968) Subject classiification: 13; 14.3; 20.1; 22.6 Institute

of Semiconductors,

Academy

of Sciences of the USSR,

Leningrad

On the Experimental Observation of Small Polarons in Semiconductors By M. I . K l i n g e r An analysis is made of experimental methods for observing small-polaron conductivity in order to study the properties of small polarons. Special attention is paid to infrared absorption measurements. Some recent data on TiO s are discussed. AHajIH3HpyiOTCH 3(J)$eKTHBHMe 3KCnepHMeHTajIbHbie CII0C06M, n03B0JIHK)IHHe O0Hapy>KHTb HOMHHHpyiOmyiO npOBOHHMOCTb MaJlblX nOJIHpOHOB H HCCJieflOBaTb h x CBoiicTBa. CymecTBeHHoe BHMMaHiie ynejieHO nepcneKTHBHOMy MeTojiy H3yqeHHH HH(j)paKpacHoro norjiomeHHH. B btom acneKTe, aHajiH3HpyioTCH HegaBHiie onbiTHbie jiamibie, KacaiomHecH T i 0 2 .

1. Introduction 1. The problem of the experimental observation of small-polaron conductivity (SPC), especially of hopping SPC in semiconductors, i.e. the problem of the experimental verification of the concept of small polarons, is the subject of numerous though controversial discussions [1 to 5]. The model low-mobility materials are generally transition-metal compounds of ionic-crystal type (e.g. NiO, LaCoOj, etc.). The problem is rather complex and not clear because of both its inherent complexity and some inadequate theoretical treatments. Therefore the conclusions are often contradictory and the situation is rather uncertain. This is particularly the case for NiO and TiO a (see Section 3). Under such circumstances, for a further elucidation of the problem it seems useful to discuss here (Section 2) 1 ) the theoretical conclusions which, if verified experimentally, can furnish actually effective evidence whether or not SPC occurs (and then permit estimates of SPC parameters). The attention is focused on such comments and conclusions which seem to be important and previously treated either not adequately or not at all (some unpublished results are also described — their derivation will be given elsewhere). Below only transport (and optical) phenomena in weak (ohmic) electric fields are treated which are mainly not associated with magnetic effects. Section 3 presents, from this standpoint, critical comments on some recent conclusions about the transport in TiO a . 2. Further, extrinsic, impurity-doped, semiconductors with a compensation degree K, 0 5S K 5S 1, are considered. Let us refer to the crystal as " p u r e " or "doped" when the concentration Nx of the majority (effectively univalent) impurity is low, N-, < N? = 3/(4 n Rf0)