Physica status solidi: Volume 19, Number 2 February 1 [Reprint 2021 ed.] 9783112498309, 9783112498293

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plrysica status solidi

V O L U M E 19 • N U M B E R 2 • 1967

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 F o r c e s 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 0. L a t t i c e Mechanics. P h o n o n s 0.1 Mossbauer Investigations 7. Acoustic Properties of Solids 8. T h e r m a l Properties of Solids S). Diffusion in Solids 10. D e f e c t Properties of Solids (Irradiation Defects see 11) 10.1 D e f e c t 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 ill Solids 12. Mechanical Properties of Solids (Plastic Deformations see 10)see 10.1) 12.1 Mechanical Properties of Metals (Plastic Deformations 13. E l e c t r o n S t a t e s in Solids 13.1 B a n d Structure. 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 Properties 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 Superconductivity. Superconducting 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 I o n s f r o m Solids 18. Magnetic Properties of Solids 18.1 P a r a m a g n e t i c Properties 18.2 F e r r o m a g n e t i c Properties 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 Properties (Continued

on cover three)

physica status solidi Board 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. TAUC, 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. COCHRAN, 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. MATYAS, Praha, H. D. M E G A W , 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. 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 19 • N u m b e r 2 • Pages 453 to 874, K67 to K114, and A 3 1 to A62 February 1, 1967

A K A D E M I E - V E R L A G

•

B E R L I N

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S c h r i f t l e i t e r u n d v e r a n t w o r t l i c h f ü r d e n I n h a l t : P r o f e s s o r D r . D r . h . c. P . G ö r l i c h , 102 B e r l i n , N e u e S c h ö n h a u s e r S t r . 20 b z w . 69 J e n a , H u m b o l d t s t r . 26. R e d a k t i o n s k o l l e g i u m : D r . S. O b e r l ä n d e r , D r . E . G u t s c h e , D r . W . B o r c h a r d t . A n s c h r i f t d e r S c h r i f t l e i t u n g : 102 B e r l i n , N e u e S c h ö n h a u s e r S t r . 20. Fernruf: 426788. Verlag: Akademie-Verlag G m b H , 108 B e r l i n , L e i p z i g e r S t r . 3—4, F e r n r u f : 2 2 0 4 4 1 , T e l e x - N r . 011773, P o s t s c h e c k k o n t o : B e r l i n 35021. — D i e Z e i t s c h r i f t „ p h y s i c a s t a t u s s o l i d i " e r s c h e i n t jeweils a m 1. des M o n a t s . B e z u g s p r e i s e i n e s B a n d e s M D N 72,— (Sonderp r e i s f ü r die D D R M D N 60,—). B e s t e l l n u m m e r dieses B a n d e s 1068/19. J e d e r B a n d e n t h ä l t zwei H e f t e . G e s a m t h e r s t e l l u n g : V E B D r u c k e r e i , , T h o m a s M ü n t z e r " B a d L a n g e n s a l z a . — V e r ö f f e n t l i c h t u n t e r d e r L i z e n z n u m m e r 1310 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik.

Contents Review Article Y. P.

VARSHNI

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

IV,

VI,

and 459

Original Papers W . BARTH a n d B . FRITZ

V. B.

PARIISKII

Local Mode Absorption by U-Centres in Alkali-Halide Mixed Crystals

515

Effect of Stress State on the Shape of the Dislocation Rosette near an Indentation in Some Alkali Halide Crystals

525

S. N . KOMNIK, V . Z. BENGUS, a n d E . D .

LYAK

On the Kinetics of Plastic Deformation of Some Alkali Halide Crystals

533

J . MAN, M . HOLZMANN, a n d B . VLACH

G.

DÖHLER

Microstrain Region and Transition to Macrostrain in 99.9% Polycrystalline Copper

543

On t h e Field Emission of Minority Carriers in Photoconductors.

. .

555

.

565

Recombination in Semiconductors by a Light Hole Auger Transition

577

O . E . F A C E Y a n d P . W . M . JACOBS

Photochemical Properties of SH~ Ions in KCl and K B r Crystals. A . R . BEATTIE a n d G . SMITH

R . D . L E V I N E a n d A . T . AMOS

On the Theory of Surface and Impurity States

587

W . FRANZ u n d P . MIOSGA

J . TREUSCH

Verteilungsfunktion und Driftgeschwindigkeit warmer Elektronen in Germanium

597

Green's Function Method for Energy Band Calculations Including Spin-Orbit Coupling

603

E . I . GAVRILITSA a n d S . I . R A D A U T S A N

Influence of Ordering on the Properties of Solid Solutions of t h e System ( H g S e k s - i l n j S e s h - * B . F . ROTHENSTEIN, A . POLICEC, C. ANGHEL, a n d M .

LUPULESCU

Contributions to the Mechanism of the Procopiu and Inverse Wiedemann Effects J . KOI-ODZIEJCZAK a n d E .

609

613

KIERZEK-PECOLD

Free Carrier Electro-Magneto-Optical Phenomena in Semiconductors

623

A . C . M C L A R E N , J . A . RETCHFORD, D . T . GRIGGS, a n d J . M . CHRISTIE

Transmission Electron Microscope Study of Brazil Twins and Dislocations Experimentally Produced in Natural Quartz

631

R . C. R A U a n d G . A . CHASE

Clustering of Defects in Neutron-Irradiated Beryllium Oxide . . . 30*

645

Contents

456

Page E . KÖSTER

Verformungs- und Temperaturabhängigkeit der Magnetisierungsprozesse in Nickeleinkristallen (II)

R . EXDERLEIN a n d R .

655

KEIPER

On the Theory of Electro-Absorption and Electro-Reflectance in Semiconductors

673

J . SPYRIDELIS, P . DELAVIGNETTE, a n d S. AMELINCKX

On the Superstructures of Ta 2 0 5 and Nb 2 0 5

683

B . I . B O L T A K S a n d T . D . DZHAFAROV

R.

LABUSCH

The Effect of Applied Electric Field on Diffusion of Impurities in Gallium Arsenide

705

Elastische Konstanten des Flußfadengitters in Supraleitern zweiter Art

715

A . AXMANN u n d W .

GISSLEE

Streuung langsamer Neutronen an polykristallinem Selen und Tellur

721

M . M . SHUKLA a n d B . D A Y A L

K . - H . PFEFFER

The Lattice Vibrations of Nickel in Krebs's Model

729

Wechselwirkung zwischen Versetzungen und ebenen Blochwänden mit starrem Magnetisierungsverlauf

735

M . P . VEBMA a n d B . D A Y A L

R . FISCHER

J.

VILLAIN

Lattice Dynamics of MgO

751

Zur Theorie der Erzeugung der zweiten Harmonischen an der Oberfläche eines unmagnetischen, quadratischen, optisch einachsigen Kristalls

757

Oscillations collectives de spins et ordre à courte distance

767

P . T . LANDSBERG

The Condition for Negative Absorption in Semiconductors . . . .

777

R . STOCKMEYER a n d H . H . STILLER

Lattice Dynamics of Solid Adamantane

781

Interference Effects in Spin Lattice Relaxation by Two-Phonon Processes

787

A . M . STONEHAM

O . VÖHRINGER a n d E . MACHERAUCH

The Yield Point of a-Copper-Tin Alloys

793

B . JJ-MENEZ, J . M E N D I O L A , a n d E . M A U R E R

G. A.

JONES

Orientation Dependence of the Ferroeleetricitiy of T.G.S

805

Preferred Orientation and Domain Structure of Iron Films Deposited on (100) Copper Substrates

811

E . GI'TSCIIE a n d E . J A H N E

Spin-Orbit Splitting of t h e Valence Band of Wurtzite Type Crystals

823

M . HKNZLER

Leitfähigkeit und Feldeffekt reiner Siliziumspaltflächen

833

R. H.

Electron Diffraction from a Magnetic Phase Grating

847

WADE

Contents

457 Page

L. I.

VAN T O R N E

Electron Charge Distribution about Zinc Ions in Aluminum . . . .

855

G . B J Ö R K M A N a n d G . GRIMVALL

F . BROUEKS

Polarization Vectors for Lattice Vibrations in Sodium

863

The Pair Distribution Function in the Approximation

867

R.P.A.

and in the Hubbard

Short Notes E . IGRAS a n d T . W A E M I Ä S K I

Electron Mirror Observation of Lithium Precipitation at Scratch-Induced Dislocations in Silicon K67 K . WETZIG

L. JAHN

Investigations of Surfaces of the Binary System T a - F e by Means of an Emission Electron Microscope

K71

Crystal Anisotropy and Saturation Magnetization of Hard Ferrites with a Composition near the M-Compound Prepared by Means of the Verneuil Method K75

M . ARNDT a n d A . F . RUDOLFH

Density Changes due to Dislocations in Silicon Single Crystals

. . . K79

B . SPRUSIL a n d P . VOSTRY

Resistivity Changes in Air-Quenched Gold Wires

K83

R . E . N E W N H A M a n d R . P . SANTORO

Magnetic and Optical Properties of Dioptase

K87

C . C H E R K I a n d R . COELHO

P.

GUYOT

On Charge Storage in Anodic Tantal Oxide Layers

K9I

Camel-Humps Peierls Hills in Iron

K95

H . M . OTTE, H . A . LIPSITT, P . F . LINDQUIST, M . L . HAMMOND, a n d R . H . BRAGG

R . FISCHER

Further Comments on the Paper "On the Interpretation of Electron Diffraction Patterns from 'Amorphous' Boron" K99 Ein Beispiel für die Existenz eines „Brewster-Winkels" in der nichtlinearen Optik anisotroper Kristalle K103

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

Test of the Read Model Concerning the Electron Mobility in Plastically Deformed n-Type Ge K107 M . A . N I Z AM ET DT NO VA

The Reflection Spectrum of GaSe and GaS Single Crystals near the Fundamental Absorption Edge Kill

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

A31

458

Contents

Systematic List Principal subject classification:

Corresponding papers begin on the following pages:

1 1.2 1.3 1.4 6 9 10 10.1 10.2 13 13.1 13.3 13.4 14.1 14.2 14.3 . . . 14.4.1 14.4. 2 18 18.2 18.3 20 20.1 20.2 20.3

683, K99 609 K67, K71 811 721, 729, 751, 781, 863 705 525, 533, 631, 645, K79 543, 793, K95 515, 565 597, 787, 867, K87 577, 603, 823 587, 833 855 K83 715 K107 555, K91 805 767 613, 655, 735, 847 K75 623, 673, 757, K103 Kill 777 459

The Author Index of Volume 19 Begins on Page 875 (It will be delivered together with Volume 20, Number 1.)

Review

Article

phys. stat, sol. 19, 459 (1967) Subject classification: 20.3; 13.1; 13.2; 13.4; 20.1; 20.2; 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, VI, and III-V Semiconductors (I) By Y . P . VABSHNI

Contents

1. Introduction 2. Optical properties

of

semiconductors

2.1 N o t a t i o n for optical constants 2.2 Absorption of p h o t o n s 2.3 Empirical expressions for t h e absorption coefficient 3. Theory

3.1 3.2 3.3 3.4

of radiative

4. Experimental 5. General 6. General 7. Results

7.1 7.2 7.3 7.4 7.5 7.6

recombination

rate

v a n Roosbroeck-Shockley t h e o r y More refined calculations Semiconductor masers Q u a n t u m yield or q u a n t u m efficiency methods

of excitation

and study

of radiative

recombination

references remarks for

on results

presented

here

elements

Diamond Silicon Germanium Amorphous selenium Hexagonal selenium Tellurium

References

(Part

I)

List of Principal Symbols Note: The subscripts " i " and " d " refer t o indirect and direct transitions respectively. Sometimes, when it is clear f r o m t h e context which transition is m e a n t , these subscripts h a v e been d r o p p e d . B c e

Probability for radiative recombination (cm 3 s - 1 ) Velocity of light Electronic charge

S u p p o r t e d b y t h e Defence Research B o a r d u n d e r c o n t r a c t no. PW68-200027 w i t h R . C. A. Victor R e s e a r c h L a b o r a t o r i e s , Montreal.

460 E Eex Eg Egi Egi h hv k k

m me mh mT n n* W; R T u a X

I V

a r

Y. P.

VARSHNI

Energy of photon Binding energy of exciton Energy gap Energy gap, direct Energy gap, indirect Planck's constant Energy of photon Boltzmann constant Wave vector Free electron mass Effective mass in the conduction band Effective mass in the valence band = mc mhl{mh + me) Real part of the refractive index Complex refractive index Intrinsic carrier concentration Radiative recombination rate (cm - 3 s _ 1 Temperature = hvjkT Optical absorption coefficient Absorption index Wavelength Frequency Recombination cross section (cm2) Lifetime 1. Introduction

A number of mechanisms have been suggested to account for the recombination of electrons and holes in semiconductors. General reviews of the various recombination processes have been given by Bemski [1], Smith [2], Hall [3], Bube [4], and Blakemore [5], Accounts of selected topics in this field have been provided by Shockley [6J, Vavilov [7, 8], and Guro [9]. These processes can be classified according to the answers to the following questions: 1. Is the recombination direct or through an imperfection ? 2. How is the energy of the excited carriers released in the recombination ? The answer to the first question leads to three possibilities: (i) The electron and hole combine through an electron dropping from a state in the conduction band to an empty state in the valence band in a single transition. (ii) I t is known that foreign atoms of a number of metals (e.g. Ni, Fe, Co, Cu, Au, etc.) form in semiconductors recombination and trapping centers for electrons and holes — such centers having levels in the forbidden band. The centers formed by these impurities are usually (depending on the temperature) both trapping centers (the center interacting only with one carrier type) and recombination centers (interaction takes place with both carrier types). Usually at a given temperature only one of the levels in such centers is effective. Quite often the deepest level is recombinative. Besides impurities, dislocations, and

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

461

other mechanical imperfections of the crystalline lattice can act as recombination centers. (iii) Surface recombination. This process is very similar t o t h e second one, t h e t r a p s being now a t t h e surface of t h e semiconductor. F o r the dissipation of t h e excess energy of these carriers, t h e following mechanisms have been consideied: a) R a d i a t i v e recombination [10, 11]. Most of t h e energy is carried away by a photon. b) Multiphonon recombination [12, 13, 6]. The energy given out in t h e recombination process is transformed into t h e r m a l energy of lattice vibrations. c) Auger recombination, also called " i m p a c t " or three-body recombination [14 to 19], E n e r g y released is given t o a free electron or hole t o help satisfy t h e laws of energy a n d m o m e n t u m conservation. d) Plasma oscillations [20]. The energy can be given t o t h e entire assembly of free carriers leading to the excitation of plasma oscillations. Two or more of these processes m a y be involved simultaneously. T h u s in all we h a v e 12 possibilities. Recombination radiation is usually classified in two broad categories: 1. "intrinsic recombination r a d i a t i o n " , (i) above; 2. "extrinsic recombination r a d i a t i o n " , (ii) and (iii) above. I n t h e period since 1952, the s t u d y of t h e recombination radiation spectra has given i m p o r t a n t information on the band structure, t h e defect and i m p u r i t y energy levels, and t h e lattice vibrations of groups I V and I I I — V semiconductors, a n d has led to t h e development of semiconductor lasers. The purpose of t h e present paper is to survey t h e field of intrinsic radiative recombination in semiconductors, in particular for diamond, Si, Ge, Se, Te, G a P , GaAs, GaSb, I n P , InAs, and InSb. F o r all these semiconductors we have carried out calculations of the radiative recombination rate, intrinsic lifetime, a n d radiative recombination cross section and have compared t h e m with experim e n t a l d a t a where available. Such cases where t h e energy of t h e " e x t r i n s i c " recombination radiation lies very near t h e energy gap, h a v e also been discussed. A proper analysis of t h e observations on t h e recombination radiation of a n y semiconductor requires a knowledge, for some detailed and for some elem e n t a r y , of its absorption coefficient, refractive index, phonon energies, band s t r u c t u r e , and q u a n t u m efficiency. F o r some of these properties we have a t t e m p t e d t o summarize t h e experim e n t a l and theoretical results, while for others adequate references are given so t h a t t h e interested reader m a y trace t h e sources for more detailed information. 2. Optical Properties of Semiconductors The optical properties of semiconductors have been discussed in recent years b y F a n [21], Moss [22], Smith [2], McLean [23], Parkinson [24], and Tauc [25]. I n t h e following we give a brief resume of the essential features of t h e absorption in semiconductors, relevant to t h e later discussion of radiative recombination. More detailed discussion m a y be found in t h e above mentioned articles and books.

462

Y . P . VARSHNI

2.1 Notation

for optical

constants

I t is customary to write the complex refractive index in the form or

n* = n (1 — i x)

(1)

n* = n — i k .

(2)

x is called the absorption index and k (= n x) the extinction coefficient. For a plane electromagnetic wave of frequency v and wavelength A propagating through an absorbing medium, the absorption coefficient

where

and Eex is the binding energy of an exciton. For h v pa E , equation (15) yields

so t h a t the component rises from its threshold as the three-halves power of the energy above this threshold. For h v ^ > E g , equations (15) is not valid physically, but it has some analytical significance as this corresponds to the region when the mutual interaction of the electron and hole becomes negligible. For this region, equation (15) gives

which is similar to equation (14b). For many approximate purposes, equations (13) may be used instead of the more accurate equation (15). Hartman [33] has obtained formulas for the indirect absorption coefficient as a function of photon energy, for both degenerate and non-degenerate semiconductors, taking into account the dependence of the energy denominator on the energy of the intermediate states, and has carried out calculations for germanium. 2.3 Empirical

expressions

for the absorption

coefficient

There have also been attempts to express the variation of a with h v and T by empirical or semi-empirical expressions. Urbach [34, 35] in 1953 found that for AgBr crystals, in the range of absorption coefficient 10~2 c m - 1 to 10 cm - 1 , log a is linear with h v, also t h a t the slope is given by 1 jkT at corresponding temperatures. In more general terms we may write

r a n2 u-

du .

(33)

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

469

W e note f r o m (31) t h a t P(v) o(v) is a p r o d u c t of t h r e e factors, U, n3, and x, where V = 2.2094 x 10 12 J 74 — M

?/3

e

-.

(34)

—1

If t h e dispersion is neglected [56] and n is assumed t o be constant t h e n (32) m a y be p u t as R = 2.2094 x l O 1 2 Tl n3 f ^ d u .

(35)

I n a steady-state deviation f r o m t h e r m a l equilibrium, t h e r a t e of radiative recombinations in a non-degenerate m a t e r i a l is given b y (36)

since the n u m b e r of direct recombinations m u s t be proportional t o t h e product of t h e concentrations of electrons n and holes p, and m u s t coincide with R when n p = rif, as occurs under equilibrium conditions. From t h e above expression, t h e decay time for a small disturbance in t h e carrier concentration m a y be calculated. Suppose t h e equilibrium carrier concentrations are n0, p0 and t h a t these are increased slightly in some m a n n e r to n = n0 -j- Sn, p = p0 + t h e n t h e recombination r a t e becomes R If 8n = 8p,

+

ZR = R(n0

+

Zn)P°+-8* n0p0

^ = R

W

or

+

Pol

R

= ^ + ^ . n0 p0

(37) (38)

The radiation lifetime r is given b y 8n SR

n0 p0 R (n0 + p0)

(39)

F o r t h e intrinsic case n0 = p0 = nv and n; T = 2i" The recombination probability, B, is given b y [3] (41)

5 = nf4 a n d the recombination cross section a b y »o Po v

n

i

(42)

where v is a relative velocity of t h e r m a l motion and m a y be obtained f r o m y TO v2 = kT . A general review on t h e lifetime of excess carriers in semiconductors m a y be found in Many and B r a y [57]. The effect of degeneracy on radiative recombinations has been considered b y Landsberg [58]. 31

pliysica

470

Y . P . VARSHNI

3.1.1 Intrinsic carrier concentration For some semiconductors the intrinsic carrier concentration has been determined over a range of temperatures by conductivity and Hall constant measurements. The results are usually expressed in the form n? = const T3 exp ( ^ J ^ ) ,

(43)

where Eg0 is the extrapolated energy gap at 0 ° K assuming a linear variation of Eg with T. In other cases, w, may be calculated theoretically from the well known expression ,

n! = NcNvexp

(44)

where

and

Nv = 2 m f Mv

—

j

(45)

in which M c is the number of equivalent minima in the conduction band, M v is the number of equivalent maxima in the valence band, me is the 'density of states' effective mass for electrons, and mh is the effective hole mass obtained from ml' 2 = mfjf + m'Hl , (46) wihi and mh2 being the masses of 'heavy' and 'light' holes. When M y = 1, equation (44) may be written as

= 4 (t) 3M° ^mh3)'2 6XP (ii8 ) =

= 2.332 x 10» Mk T3 i ^ Y

exp

.

(48)

3.1.2 Analytical expressions Using the theoretical expressions for the absorption coefficient a, one can obtain analytical expressions for R and other associated quantities. These expressions are specially useful in those cases where detailed absorption-coefficient measurements are lacking but data on me, mh, Eg, etc., are available. 3.1.2.1 Direct transitions We shall assume a d = b [(h v -

Et)W

+ P] ,

(49)

where b is given by (6b) and P is a constant. Substituting (49) in (33) and assuming n to be constant we obtain oo oo R, = J jlkT)V* I where

^

du + (kT)» P I

«0

«0

j _ 32 7r3e2 (2mr)3/2re/ e3 hb m

{

d ,

(50)

(47)

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

and ti0 = EJkT. we derive = J{kTf + If u0

471

Assuming that m > 1, to a good degree of approximation,

e — {(kTyi2 +

+ 3 « 0 + «;) +

+

/15 . 3 ( lie ' - y M o + M«

-p

e

+

(51)

1, the above expression may be approximated by SA = J(kT)3 e — K { ( f c r ) 1 / 2 2 " 1 / 2 + P } .

(52)

Further if the electron-hole interaction is neglected, (52) may be simplified to D

^

=

16 n3 e2 (2 mr)3/2 n (¿y)7/2 ^1/2

«„ „2

/

*

(53)

For most of the semiconductors at ordinary temperatures, u0 is much greater than 1, e.g. for Ge at 300 °K, uQ « 27. In such cases expression (53) is quite adequate. But for semiconductors with low Eg and at high temperatures, the full expression (51) should be employed. Expressions (51), (52), or (53) can be employed to obtain the recombination probability B and the recombination cross sections a. Here we give expressions for B and r corresponding to (53): R

-Sd (2w)il*he*n / m Mcc*mW (jfcT)3/2 \me + mhj

/ \

+

m ,«i\ me + m J

g d 6Xp

\

(Egl - Egd \ kT )'

{0

)

when E g i < if the case is otherwise, the exponential term drops out. Expression (54) is similar to the one given by Hall [3]. The lifetime r may be expressed in either of the following forms: n Td = 2 R c3 h? m 8ji2

TOh)3/4 e nE\d 2

/mh + me\sl2 /

(for direct gap semiconductors, Eg(\ becomes exp (Egd/2 kT)) Td = =

31*

R K + Po) C»TOW2mc(fcr)3/2/me+TOh\3/2 (27t)i/2 he*nEld\ m )

/ \

m

m

Egi, and the exponential factor in (55)

TO \ ~ 1 1 ¡Egd - Egj \ me h mj »o+^o^l j '

m

(

'

472

Y . P . VARSHNI

3.1.2.2

Indirect

transitions

We can obtain an expression for E b y using t h e analytical expressions for indirect transitions given in equation (13) oo 8nn2 k3 T3 f u2 a _ D r2k3 j e«—ldU> ^ o where u = h vjkT. The lower limit of t h e integral corresponds to t h e point where a = 0. On substituting equation (13) in t h e above expression and p u t t i n g Eg 6 Eg B = Sl + = 82 (58) k f ~ ¥ ' kf ¥ we obtain A

=

[2

12

+

+

24

1 +

At ordinary t e m p e r a t u r e s we m a y assume u0 •Sj = .s'2 = u 0 a n d = /h>$T

T

-l)

A

8/T, t h e n we h a v e

G U

' °

*

The corresponding expression for t h e recombination probability B is _ -"indirect -

2

h3AE%in2 e®/T + 1 n2 c2 M~(m(, mh)W e ^ - 1 •

(61)

E q u a t i o n (61) was obtained b y Hall [3]. The lifetime r m a y be p u t in either of t h e two f o r m s ni c2 M^ 2 > = 2B = 2 nn*AEh

/ n \s/2 \Yw)

(67)

Baryshev [58 a] has obtained t h e form of analytical expressions for a b y using other expressions for t h e absorption coefficients.

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

3.2 More refined

473

calculations

Dumke [59] has derived expressions for the radiative-recombination lifetime based upon a microscopic analysis of both direct and indirect transitions. His results are as follows. Indirect recombination: = T

Z m+ W Z m +W

J_

«

4nh*

V

«

AEhn*

"

°8/r -

1

,fim

1

n0 + p0 e el T + 1 '

(

'

Here the sums over v and c are over the extrema in the valence and conduction bands, and Egi is the indirect energy gap. A is the factor involved in the MacFarlane-Roberts expression (13) and is obtained from an analysis of the absorption data. It may be noted that equation (63) is almost identical with the above expression. For intrinsic Ge at room temperature, Dumke calculated Tjnljirect = = 1.98 s. Direct recombination: 1_

(mV2 c3 fc3/2 \ I

2(2 n)W \ x

ex

TW

\

) U S d » " l^ifl / £ a -3/2 £ a - 3 / 2 1

p

X

•

TO

Here m/a v and m/a c are the valence- and conduction-band effective masses, m/a c0 is the effective mass of conduction-band electrons at ft = 0, Eg(i is the direct energy gap, P a is the optical matrix element, the value of which can be estimated. From this expression, Dumke found the lifetime in intrinsic Ge at 300 ° K to be r = 0.29 s. Equation (70) may be compared with equation (56). Recently Lasher and Stern [11] have calculated the spectral line shapes of the radiation produced by band-to-band recombination under the assumption that the momentum matrix element is the same for all initial and final states, i.e. that there is no momentum selection rule. The peak of the stimulated radiation falls at a lower photon energy than does the peak of the spontaneous radiation, except when T = 0 ° K . The limitations of the van Roosbroeck-Shockley approach have been pointed out. These authors have also considered the effect of a tail in the density of states on the spontaneous line shape. 3.3 Semiconductor

masers

The possibility of achieving maser operation in semiconductors was considered theoretically by several workers [60, 61, 62], The first successful observations on a semiconductor laser were made by Hall et al. [63] and by Nathan et al. [64] in 1962. Since that time, this field has developed tremendously. Recent reviews may be found in L a x [65], Hilsum [66], Geusic and Scovil [67], Birnbaum [68], and Rediker [69]. When photons of an appropriate energy fall on a semiconductor, they are absorbed with excitation of electrons from the valence band to the conduction band. This process is one we are familiar with; but there is a small but finite possibility of a second process occurring. Here one of the incident photons would interact with an electron in the conduction band and induce it to combine with a hole in the valence band. The recombination energy of the electron

474

Y. P.

VARSHNI

would be emitted as a photon, which would have to be identical to the photon which initiated the recombination. B y this process, stimulated emission can only occur with high probability when there are present in the semiconductor a large number of free carriers and many holes. In thermal equilibrium these conditions are not satisfied and absorption is then more probable than emission, but when a p-n junction is operated at extremely high-current densities a large number of holes and electrons are injected into the transition region and the conditions become more favourable for emission than for absorption. A photon travelling into this region is more likely to stimulate the emission of another photon — in other words the first photon will be amplified. The stimulated photon is identical in every way with the original photon and is emitted so that it travels in the same direction. The recombination region near the junction is only a few [im thick, but if the junction is very flat and the original photon travelled exactly in the junction plane the stimulated photon will also be in this plane, and it too may be amplified. An incident light wave can thus grow as it travels across the crystal. Initiation of amplification by light incident from outside the crystal is not necessary. The spontaneous recombination radiation can itself be amplified, provided it is travelling in the junction plane. At low current densities the junction will emit radiation isotropically but as the current increases the light in the junction plane increases rapidly and the polar diagram of the emission becomes anisotropic. Possible radiative transitions associated with laser action in semiconductors are illustrated schematically in Fig. 2. The four mechanisms are: 1 conduction to valence band, 2 donor to valence band, 3 conduction band to acceptor, and 4 donor to acceptor. Transitions between acceptor levels and the conduction band were studied by Eagles [69 a], Possibility of maser action from bandto-band transitions was considered by Dumke [62], Lasher and Stern [11], on the basis of a simple model, have calculated the spectral line shape of the spontaneous and stimulated radiation emitted in band-to-band recombination. In the case of isolated atoms it is usually easy to determine the modes in which generation of stimulated radiation occurs (the "singular" modes). For interband transitions, as a result of the discontinuity in the energy spectrum, the determination of the singular modes represents a more difficult problem. An attempt to solve this problem by determining the parameters of singular modes has been made by Vinetskii et al. [70], Callaway [71] has calculated the transition probabilities and absorption constants for the band-to-acceptor and donor-to-acceptor transitions. Recently Dumke [72] has derived expressions for the optical absorption and radiative

Ec

©

© 1

2

©

3

Fig. 2. Four mechanisms for radiative transitions: 1 2 3 4

— — — —

conduction to valence band; donor to valence band; conduction band to acceptor; donor to acceptor (after [69J)

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

475

lifetime for transitions between donor levels and the valence band or between acceptor levels and the conduction band. The available evidence appears to indicate that in some cases the radiation is due to band-to-band transitions while in others "band-to-acceptor" transition (mechanism 3) is operative. Thus the mechanism of the radiative transition is not unique; indeed it is thought that even in the same laser diode, it may be possible, by changing the junction current, to change the recombination mechanism. 3.4 Quantum

yield or quantum

efficiency

The quantum yield is defined as the number of electron-hole pairs produced per incident quantum. The van Roosbroeck-Shockley theory assumes unit quantum efficiency. However, if energy of the incident photon is high enough, it is no longer true. Investigations bearing on this topic are summarized below. Goucher's [73] photoconductivity measurements, made on a single crystal of n-type germanium indicated a yield of one electron-hole pair per photon absorbed, for wavelengths in the vicinity of, but less than, the long-wave limit. This finding was further supported by the experiments of Goucher et al. [74] on the photo-response of a germanium p-n junction. If the photon has a sufficient excess energy above threshold, additional electrons can be liberated in the crystal as a result of impact ionization. The possibility of a quantum yield of photo luminescence in excess of unity, upon excitation by quanta with energies more than double the energy of the luminescence quantum was indicated by Vavilov [75]. This phenomenon was observed experimentally by Butaeva and Fabrikant [76, 77]. To measure the quantum yield, either the electrons or the holes are collected at a p-n junction. Such measurements have been made on germanium by Koc [78] and by Vavilov and Britsyn [79], on silicon by Vavilov and Britsyn [80, 81], and on InSb by Tauc and Abraham [82]. In Fig. 3, the quantum yield curve obtained for germanium by Vavilov and Britsyn [79] is shown. I t will be seen that below a certain threshold energy of the incident quanta, the quantum yield is unity. Above this threshold the quantum yield rises steadily with increasing energy. Similar behaviour is shown by Si and InSb. Britsyn and Vavilov [83] have also examined the temperature dependence of the quantum yield in silicon. Their results are shown in Fig. 4. I t is seen that 2.0 3 15 2 10

-pittHf 1

1

3

4 5 hvteV)—-

1-ig. 3. Quantum yield versus photon energy for germanium (Vavilov and liritsyn 179])

0.

3

4 5 hv(eV)—-

.rig. 4. Quantum yield versus photon energy for silicon at different temperatures: (1) 100 ° K , (2) 300 ° K , (3) 400 °K (Britsyn and Vavilov [83])

476

Y. P.

VARSHNI

the increase in t h e q u a n t u m yield ( > 1 ) begins a t different p h o t o n energies, depending on t h e t e m p e r a t u r e ; also t h e shape of the curve for t h e region where Q 1 is different a t different t e m p e r a t u r e s . Recently Tuzzolino [84] has carried out a measurement of t h e q u a n t u m efficiency of silicon over t h e p h o t o n energy range from 4.9 t o 21 eV by measuring the photo-response of silicon surface-barrier photo-diodes. The q u a n t u m efficiency increases f r o m 2.0 a t h v = 4.9 eV to approximately 3 a t h v = 6 eV; between h v = 6 eV a n d h v m 10 eV t h e q u a n t u m efficiency is a p p r o x i m a t e l y c o n s t a n t . The t h e o r y of impact ionization has been developed b y Chuenko [85], Antoncik [86, 87], Shockley [88], a n d Hodgkinson [89, 90]. All explanations of q u a n t u m yield curves h a v e been founded on t h e following basic mechanism. The excess energy (hv — E g ) of a q u a n t u m of energy h v falling on a semiconductor of lowest energy gap Eg is divided in some w a y between t h e electron and hole produced. If one of these has sufficient energy, it can create another electron-hole pair b y collision ionization, a n d t h u s enhance the q u a n t u m yield. Antoncik assumes t h a t these processes occur in limited regions of phase space which an electron or hole must occupy if it is to t a k e p a r t in an ionizing collision. Those which do not occupy such a position are supposed not t o ionize. Antoncik's t h e o r y does not predict a t e m p e r a t u r e dependence of t h e q u a n t u m yield. Shockley [88] uses an explanation in which some of t h e more energetic carriers lose their energy in phonon collisions before they t a k e p a r t in a n ionizing collision. Both of these t r e a t m e n t s were based on specific assumptions conccrning t h e way in which t h e excess energy of t h e absorbed q u a n t u m was divided between t h e electron and hole produced. Antoncik assumed t h a t all of t h e excess energy went t o t h e electrons which were then distributed uniformly in fc-space a r o u n d the conduction band minimum. Shockley assumed t h a t t h e excess energy was divided equally between t h e electron and hole produced. Recently Hodgkinson [89] has applied optical absorption and photoemission d a t a t o analyse q u a n t u m yield in silicon and has shown t h a t most of t h e electrons and holes with energy above a certain threshold will t a k e p a r t in an ionizing collision before losing their energy. This is t a k e n to mean t h a t t h e y are scattered into a region of phase space in which t h e y can cause ionization, before their energy falls below t h e threshold. I n a succeeding paper, Hodgkinson [90] has investigated t h e conditions governing impact ionization in semiconductors and has applied t h e m t o determine t h e position in energy and fc-space of t h e various impact ionization thresholds in silicon and germanium. Combining these results with the band s t r u c t u r e and with t h e known mechanisms of optical absorption, he has succeeded in explaining t h e enhanced q u a n t u m yield thresholds in silicon and germanium. The q u a n t u m yield curve for I n S b has been interpreted b y Beattie [91]. 4. Experimental Methods of Excitation and Study oi Radiative Recombination I n studies of t h e spectra of recombination radiation, the following methods of generating t h e necessary excess concentration of carriers have been u s e d : a) Excitation b y p h o t o n s having energy greater t h a n the energy gap (photoluminescence) [92]. Laser has also been used recently b y Basov et al. [93].

477

Band-to-Band Radiative Recombination in Semiconductors (I) Fig. 5. Schematic arrangement used by Haynes [92] to measure recombination radiation in germanium

Spectrometer

Tungsten lamp

Ge Sample

PbS Detector

lb) Injection of carriers by passing a current through a n-p junction (electroluminescence) [94, 95]. c) Excitation with a beam of fast electrons (cathodoluminescence) [96], Kopylovskii et al. [97] have described electronic equipment which is used for generating and studying recombination radiation in semiconductors. A generator circuit is presented for high-power pulses having a duration of 1 to 5 [is with a continuous current adjustment from 0.5 to 400 A. A registration system for the radiation is described that makes it possible to observe on an oscillograph and to record on an automatic recording instrument the spectra of recombination radiation. The essential principles behind the experiments may be understood from the arrangement used for germanium by Haynes [92] which we describe. The technique is shown schematically in Fig. 5. The light from a ribbon filament tungsten lamp was passed through a water absorption cell 10 cm long and focused on a thin slice of germanium (1.2 X 10~2 cm thick). The radiation from the opposite side of the slice was analysed with a spectrometer and detected by a PbS cell. The water cell transmits less than 10 10 of the incident radiation at wavelengths longer than 1.4 ¡xm and the germanium transmits less than 10~ 10 of the radiation shorter than 1.4 [j.m; thus no measurable light from the tungsten source will enter the spectrometer. Any detectable radiation must arise from the recombination of electrons and holes produced by light. The results obtained by this arrangement are discussed in Section 7.3. We may note that the spectral distribution of the emergent radiation from the experimental sample may be different from the originally produced one due to self absorption in the material and other effects. Corrections for these have been discussed by Haynes [92] and Moss [98]. When injection across a p-n junction is used, an important problem is that of bringing about the emergence of the radiation from the specimen. In fact, for the simplest experimental geometry (Fig. 6a) a considerable part of the light flux undergoes total internal reflection and does not reach the entrance slit of the spectrometer. This is due to the very high index of refraction of groups I V and I I I - V semiconductors. Aigrain and Benoit a la Guillaume [99, 100] overcame this difficulty by taking the specimen in the form of a Weierstrass sphere (Fig. 6b) and having injecting contact at the Weierstrass point of the sphere. In this case the radiative volume is approximately a hemisphere with a radius of the order of magnitude of the diffusion distance of the minority carriers. However, such a specimen geometry is best only for the study of extrinsic radiation [7]; the spectrum of the intrinsic radiation is strongly distorted because of absorption. With a complicated geometry this distortion is difficult to take into account.

478

Y . P . VARSHNI

Air

a

b

6e

/

s

A •

m

• •

Fig. 6a. Total reflection at the surface of germanium

Fig. 6b. Weierstrass sphere, principle

C

Fig. 6c. Plane paraboloid combination (Aigrain [99])

Aigrain [99] has also suggested another shape which can be used but is much more critical, namely the plane-paraboloid combination shown in Fig. 6 c in which light is first totally reflected in the paraboloid face, then passes normally through the plane face. A good review of the various means of measuring the lifetime of carriers in semiconductors has been given by Suryan and Susila [101]. I t has been pointed out by Dumke [59] that even in those semiconductors where radiative recombination is the piimary mode of recombination, the observed lifetime may or may not be equal to that theoretically calculated. I f denotes the energy of the radiation peak we can distinguish between two cases. I f Eft lies in the region where the absorption coefficient is fairly low ( 1 0 c m - 1 ) , an emitted photon has got a chance of escaping the crystal before being reabsorbed with the resulting production of an electron-hole pair. B u t if Eft lies in a region where the absorption coefficient is very high ( 1 0 3 c m - 1 ) there would be heavy reabsorption and consequently no net decrease in the number of hole-electron pairs due to radiative recombination observed. The observed lifetime of these carriers due to radiative recombination would be very large. However, in very thin films, the reabsorption may be avoided. 5. General References The investigations on the properties of III—V compounds up to 1960 have been summarized in the book by Hilsum and Rose-Innes [102]. Recently an excellent and up to date monograph on the physics of I I I - V compounds by Madelung [103] has come out. Under each semiconductor we have given a brief resume of its optical properties and other necessary data. There are a number of papers and review articles in which theory or experimental data pertaining to three or more of the semiconductors considered here is presented. I t is convenient to summarize them here.

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

479

Reviews of optical properties of semiconductors are t o be found in t h e books of Moss [104] and Smith [2], A brief survey of the optical properties of I I I - V semiconductors has been given by Stern [105]. I n t h e s u m m a r y given with each substance, we have confined ourselves to giving references t o those works which are relevant t o radiative recombination calculations and those which have been publishing during t h e last few years. The infra-red lattice reflection spectra of AlSb, GaAs, GaSb, I n P , InAs, a n d I n S b were measured a t liquid helium t e m p e r a t u r e b y H a s s and Henvis [106]. B y comparing t h e observed spectrum with t h a t calculated using a single classical dispersion oscillator, t h e values of TO and LO phonon frequencies for longwavelength vibrations are obtained. Mitra and Marshall [107, 108] have noted a n u m b e r of correlations among t h e characteristic phonon frequencies of I I I - V and I l - V I compounds. J o h n s o n and Loudon [109] have presented a critical point analysis of t h e phonon spectra of diamond, silicon, and germanium. I n a recent excellent review, Johnson [110] summarises t h e results and presents the i n t e r p r e t a t i o n s of t h e infra-red lattice-band spectra for diamond, silicon, and germanium and also for AlSb, GaP, GaAs, and InSb. The f u n d a m e n t a l reflectivity spectrum of a n u m b e r of semiconductors with zinc-blende structure has been discussed b y Cardona [111] and b y Lukes a n d Schmidt [112]. Philipp and Ehrenreich [113, 114] have presented reflectance d a t a for Si, Ge, GaP, GaAs, InAs, a n d I n S b in the range of photon energies between 1.5 and 25 eV. The real and imaginary p a r t s of t h e dielectric constant are deduced f r o m t h e Kramers-Kronig relations. The absorption spectrum of thin films of Ge and several I I I - V semiconductors has been studied b y Cardona and H a r b e k e [115]. Besides t h e f u n d a m e n t a l absorption edge and its spin-orbit splitting, additional structure a t higher energies was found. T h e subject has been reviewed by Cardona [116]. Reviews on t h e band structure of groups I V and I I I - V semiconductors h a v e been given b y P a u l [117], Long [118], Hilsum [119], and H e r m a n et al. [120]. Ehrenreich [121] has discussed t h e band structure of I I I - V compounds with reference t o their t r a n s p o r t properties. Progress in t h e b a n d t h e o r y of semiconductors has been reviewed b y Pincherle [122], A simple equation for t h e variation of t h e energy gap with t e m p e r a t u r e has been proposed b y Varshni [123]. 6. General Remarks on Results Presented here Calculations of t h e radiative recombination r a t e and other related quantities were carried out within t h e framework of t h e v a n Roosbroeck-Shockley t h e o r y f o r t h e following substances: diamond, Si, Ge, Se, Te, G a P , GaAs, GaSb, I n P , InAs, a n d InSb, in some cases a t two or more t e m p e r a t u r e s . T h e values of absorption coefficients are usually given b y investigators in figures. These figures for various semiconductors were enlarged b y a p r o j e c t o r and traced on a large graph, wherefrom t h e values could be read with reasonable accuracy. I n two cases, Ge [124] and Si [125], the original large size g r a p h s were kindly supplied b y Mr. R o b e r t s and the values were read directly f r o m there. W e consider each semiconductor individually and display t h e results in figures. I n most of the figures we show the radiative recombination r a t e P(v) o(v), t h e absorption index and the q u a n t i t y U defined in equation (34). I n all figures.

Y. P. Varshni

480

Table la Data for me, mhi, my, 2, and — Most of the values are from the compilation of Cardona [127]. Values followed by a " T " are theoretical, estimated by Cardona. All raj, values, except that of diamond, were calculated from equation (46) Substance Diamond Si Ge GaP GaAs GaSb InP InAs InSb

mc/m

Ref.

»»hi/m

Ref.

mhz/m

0.2 0.13 T 0.041 0.13 T 0.07 0.047 0.073 0.026 0.0155

[144] [127] [127] [127] [127] [127] [127] [127] [127]

0.52 0.35 0.56 T 0.68 0.3 0.4 0.41 0.25

[127] [127] [127] [127] [127] [127] [127] [127]

0.16 0.045 0.13 T 0.12 0.06 0.086 T 0.025 0.012

Ref.

mj|/m

[127] [127] [127] [127] [127] [127] [127] [127]

0.4 [148]*) 0.578 0.361 0.601 0.713 0.318 0.426 0.414 0.252

*) Clegg and Mitchell [445] (see Part I I of this article, to be published in Volume 20 of phys. stat. sol.) have used the band constants deduced from the results of cyclotron resonance measurements to calculate the hole concentrations. The calculated hole concentrations may be expressed in terms of a density-of-states mass m^lm = 3.6 to 4.8. Reasons for this discrepancy have been discussed by these authors.

Table lb Data for refractive indices and energy gaps Substance Diamond Si

n 2.75 3.53

Ge

4.15

GaP GaAs

2.91 3.63

GaSb

3.72

InP

3.37

InAs

3.54

InSb

4.01

T CK) 295 20 90 195 249 290 333 77 300 300 21 90 185 294 80 300 77 298 77 298 77 295

*) See Part I I of this article.

(eV) 5.4 1.137 1.132 1.110 1.098 1.086 1.074 0.74 0.665 2.205

Reference [144] [31]

[124] text

Egd (eV)

Reference

«3

[23]

0.883 0.802

[212]

1.521 1.511 1.479 1.435 0.80 0.72 1.4135 1.3511 0.416 0.355 0.23 0.18

[336]*)

[382]*) [390]*) [404]*) [103]

B a n d - t o - B a n d Radiative Recombination in Semiconductors (I)

481

Table lc Data for Mand constants A and 6 (occurring in equation (13)). For diamond, GaAs, GaSb, I n P , InAs, and InSb, 0 has been given a value equal to the Debye temperature. Substance

Mc

Diamond Si Ge

6 ? 6 4

GaP GaAs GaSb InP InAs InSb

6 ? 1 1 1 1

A (eV" 2 cm" 1 ) «s 16000 2682 2231 (77 °K) 2722 (300 °K) 3760 3500 ss 3500 «a 3500 ss 3500 «=! 3500

Source

e (°K)

estimated see text [32]

2240 600 260

see text estimated estimated estimated estimated estimated

302 344 266 301 248 202

Table 2 Results for R, n-v B, r , and a. Values of R were obtained by numerical integration and those of H\ b y methods described under individual semiconductors. Figures in parentheses give powers of 10 b y which the figure is multipled. Thus 4.0 ( — 66) stands for 4.0 X 10~66 Substance Diamond Si

Ge Se (hex.) Te GaP GaAs

GaSb InP InAs InSb

T (°K) 295 20 90 195 249 290 333 77 300 298 100 300 300 21 90 185 294 80 300 77 298 77 298 77 295

R (cm " 3 s- 1 ) 4.0 4.8 5.9 3.4 2.3 9.2 6.1 2.4 2.85 »7 1.6 3.0 4.0 1.1 8.0 1.6 1.2 2.4 2.2 2.2 6.0 ssl 5.8 4.5 1.03

-66) -267) -42) -6) + 1) +4) + 7) -26) + 13) -4) +8) +20) -13) -336) -58) -13) + 3) -24) + 14) -65) +4) -1) + 19) +9) +22)

»i (cm - 3 )

B (cm 3 S"1) -12) -15) -15) -15) -15) -15) -15) -13) -14)

r (8)

a (cm

8.35(+37)

9.48 ( - 19)

5.77( + 27) 7.18 ( + 9 ) 2.61 ( + 6 ) 3.89(+4) 1.48( + 3) 5.04 ( + 18) 4.09 ( - 1 )

2.44 (— 22) 1.86( —22) 1.84( —22) 1.91 (— 22) 1.87 (— 22) 8.48 (— 20) 5.5 ( - 21)

)

6.68 ( - 2 8 ) ss 1 (—126) 6.81 ( - 1 4 ) 4.88 ( + 4 ) 1.2 ( + 8) 7.16(+9) 1.8 ( + 11) 2.42 ( - 7 ) 2.33 ( + 1 3 )

8.96 i=a 1 1.27 1.43 1.6 1.79 1.88 4.1 5.25

1.24( + 9) 5.93( + 15) 2.73(0)

1.04 - 1 0 ) 8.53 - 1 2 ) 5.37 - 1 4 )

3.87(0) 9.88 (— 6) 3.41 ( + 12)

1.9 ( - 17) 8.95( — 19) 5.63 ( - 21)

2.14( —25) 9.15( —3) 1.29( + 6) 9.3 ( - 9 ) 9.6 ( + 11) 1.32 ( - 2 9 ) 6.9 ( + 6 ) 2.65(+3) 8.26 ( + 14) 1.36 ( + 9 ) 1.50 ( + 1 6 )

1.75 1.91 7.21 2.77 2.39 1.26 1.26 1.42 8.5 2.43 4.58

1.34( + 32) 2.86( + 10) 5.37 ( + 2) 1.94( + 15) 2.18 (—3) 3.0 ( + 3 5 ) 5.75( + l) 1.32 ( + 4 ) 7.12(—6) 1.51 ( — 1) 7.28( —7)

3.34 ( - 15) 2.55 ( - 16) 7 . 6 4 ( - 17) 5 . 6 4 ( - 15) 2.50( — 17) 2.61 ( —14) 1.33( — 16) 2.95 ( - 15) 8.94 ( - 18) 5.04 ( —16) 4.84( — 18)

-8) -9) -10) -8) -10) -V) -9) -8) -11) -9) -11)

482

Y. P.

VARSIINI

Table 3 Calculated values of B, B, and r for indirect and direct radiative recombination obtained from equations (53), (54), (55), (60), (61), and (62). The "indirect" values for GaAs, GaSb, InP, InAs, and InSb were obtained on the assumption that the observed energy gap is "indirect". Figures in parentheses give powers of 10 by which the figure is multiplied. Thus 2.5( —67) stands for 2.5 x 10" 67 Substance

Temp.

Indirect

Bi

Bi

Direct

(°K)

(cm - 3 s - 1 )

(cm3 s - 1 )

(s)

Diamond 295 20 Si 90 195 249 290 333 77 Ge 300

2.5 (— 67)

5.6( —13) 7 . 5 ( - 15) 7 . 5 ( - 15) 7.9 ( - 15) 8.5 ( - 15) 8.9 ( - 15) 9.4 (— 15) 6.8 (— 14) 1-5 (— 13)

1.3 ( + 39)

GaP GaAs

GaSb InP InAs InSb

300 21 90 185 294 80 300 77 298 77 298 77 295

6.4(—42) 3.5( —6) 2.2( + l) 8.4(+4) 5.6( + 7) 1.9(—27) 4.7 ( + 12) 4.1( —13)

2.3(+27) 3.0(+9) 1.1 ( + 6) 1-8 ( + 4 ) 6.9 ( + 2) 4.4( + 19) 5.9(-l)

Ri

s ')

(cm3 s - 1 )

1.9( —50) 1.8( —33) 8.6 ( - 2 5 ) 5.7( —18) 1.5( —32) 1.9( + 13)

4.3( —59) 6.7 ( - 4 9 ) 9.2 ( - 4 4 ) 9.7 ( - 4 0 ) 5.2( —19) 6.4 ( - 1 3 )

1-1 ( + 5 4 ) 2.9 ( + 4 0 ) 3.5(+33) 1.3(+28) 1-1 ( + 25) 2.8(-l)

(cm

3

9.9( —63) 2.3 (—17) 5.8(-l) 3.4( —29) 6.6 ( + 11) 5.8( —71) 2.9( + l) l.l(-6) 1-9 ( + 17) 7.0(+5)

5.5 ( —14) 2 . 1 ( - 13) 2-1 (—13) 2.7 (— 13) 3.5 (— 13) 4.0 ( - 13) 7.2 ( - 13) 3.3( — 13) 6.2 ( - 13) 1 . 6 ( - 13) 2.7 (— 13) 3-1 (—13)

3.3( + 12) 1-1 ( + 3 7 ) 2.0 ( + 14) l - l ( + 6) 1.4(+20) 7.3(-l) 1-1 ( + 4 1 ) 1.2(+5) l-2(+9) 2.2( —3) l-M+3)

9.7( —59) 5.7 ( - 1 4 ) 5.3 ( + 2) 3.0 ( - 2 5 ) 3.5 ( + 14) 7.7( —67) 2.5(+4) 8.2 ( - 3 ) 7.6( + 19) 3.2(+9)

1.9( —8) 2.1(-9) 6.8( —10) 3.2 ( - 1 0 ) 3.4(-9) 3.8( —10) 4.3( —9) 5.2 ( - 1 0 ) 1.2(-9) 1.1 ( — 10) 1.4(-9)

2.2(+33) 1.6( + 11) 2.4( + 3) 3.1( + 16) 2.7( —3) 1.7( + 37) 2.8( + 2) 3.2 ( + 5 ) l.l(-5) 4.7(-l)

6.0( + 19)

4.9 (— 13)

9.2 (— 5)

1.4 ( + 22)

1.2 ( - 1 0 )

7.8( —7)

P(y) q(v) and U are in c m - 3 s - 1 . The energy shown on the abscissa is in eV. The total radiative recombination rate R was obtained by numerically integrating the P(v) Q(V) curve over u = H vjkT. In many cases, for calculating the intrinsic carrier density n { , data for m e and m h is required. In some cases experimental values of m e and m h are available. Braunstein and Kane [126] and Cardona [127] have given methods of calculating the conduction- and valence-band effective masses from the k • p theory. Where experimental values of these masses were not available, Cardona's [127] theoretical values were used. The data on me, mhl, m h2 are summarized in Table 1 . The final results for R, nv r, B, and a are recorded in Table 2. The quantities R, r, and B were also obtained from the analytical expressions (53), (54), (55), (60), (61), and (62). The necessary data employed in these calculations are shown in Tables l a , b, and c and the results in Table 3. GaAs, GaSb, InP, InAs, and InSb are usually considered as direct gap semiconductors. However, it was of some interest to see what difference in numerical magnitudes of R, B, and T is introduced if the transitions are assumed to be indirect. The constant A in equation (13) was assumed to be fa 3500 eV~ 2 c m - 1 for these five semiconductors; and 6 was put equal to the Debye temperature [128].

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

483

7. Results for Elements 7.1

Diamond

D i a m o n d , in i t s p u r e s t a t e , is a n insulator h a v i n g a resistivity a t r o o m t e m p e r a t u r e of t h e order of 10 8 £2cm. E x p e r i m e n t s show t h a t t h e r e are t w o t y p e s of d i a m o n d s , t y p e I a n d t y p e I I [129]. B o t h t y p e s of c r y s t a l h a v e a b s o r p t i o n b a n d s n e a r 3, 4.0, a n d 4.8 ¡xm a n d t h e R a m a n s h i f t is observed in b o t h t y p e s a n d is 1332 c m - 1 (7.51 ¡j.m). B u t t y p e I possesses a n a b s o r p t i o n b a n d a t 8 fim while t y p e I I does n o t . T h e t y p e of d i a m o n d m o s t s t u d i e d as a s e m i c o n d u c t o r is t h a t k n o w n as t y p e l i b ; t h i s t y p e w a s shown b y Custers [130, 131] t o b e h a v e as a s e m i c o n d u c t o r , h a v i n g a q u i t e low v a l u e for its resistivity (10 t o 100 i i c m a t r o o m t e m p e r a t u r e ) . 7.1.1 Optical

properties

Optical p r o p e r t i e s of d i a m o n d s h a v e b e e n reviewed b y Moss [132, 104]. I n r e c e n t y e a r s t h e i n f r a r e d s p e c t r a of d i a m o n d s has been investigated b y C h a r e t t e [133, 134, 135] a n d b y S m i t h a n d co-workers [136, 137]. T h e dispersion curves along [100] a n d [111] directions h a v e been d e d u c e d b y Y a r n e l l e t a l . [138, 139] f r o m inelastic s c a t t e r i n g of slow n e u t r o n s . Clark, D i t c h b u r n , a n d D y e r [140] m e a s u r e d t h e a b s o r p t i o n s p e c t r a of 52 n a t u r a l d i a m o n d s a t 80 a n d 290 °K. T y p e I d i a m o n d s were shown t o h a v e t w o k i n d s of a b s o r p t i o n centres which are n o t d e t e c t e d in t y p e I I d i a m o n d s . Differences in t h e i n f r a r e d a b s o r p t i o n s p e c t r a b e t w e e n t y p e I I a a n d t y p e l i b were r e p o r t e d . T h e y defined t h e f u n d a m e n t a l a b s o r p t i o n - e d g e c h a r a c t e r i s t i c of t y p e I I d i a m o n d s as t h a t energy E{ a t which t h e r e is first observed a c h a n g e of slope of t h e a b s o r p t i o n s p e c t r u m d u e t o f u n d a m e n t a l a b s o r p t i o n . Et w a s f o u n d t o be 5.40 + 0.03 eV. Champion a n d H u m p h r e y s [141] h a v e also r e p o r t e d s p e c t r a for a large n u m b e r of d i a m o n d s . A b s o r p t i o n coefficients u p t o a b o u t 700 c m - 1 o b t a i n e d f r o m t r a n s m i s s i o n e x p e r i m e n t s h a v e been r e p o r t e d b y Custers a n d R a a l [142], T h e changes in t h e optical a b s o r p t i o n of a relatively p u r e t y p e I I d i a m o n d (called D 1 0 0 in t h e p a p e r ) were m e a s u r e d as a f u n c t i o n of t e m p e r a t u r e b e t w e e n 80 a n d 522 °K b y Clark [143]. T h e e x p e r i m e n t a l curves, a f t e r c o r r e c t i o n f o r i m p u r i t y a b s o r p t i o n , could be f i t t e d t o t w o a b s o r p t i o n c o n t r i b u t i o n s w r i t t e n in the form

w i t h 6 = 1050 a n d 2900 ° K . Ex is t h e t h r e s h o l d p h o t o n energy of t h e a b s o r p t i o n in t h e b r a n c h a n d c is a c o n s t a n t . T h e a b o v e results are a t t r i b u t e d t o indirect t r a n s i t i o n s i n t o c o n d u c t i n g a n d exciton s t a t e s of t h e crystal. Theoretically if t h e b a n d e x t r e m a a r e a t d i f f e r e n t values of k, indirect t r a n s i t i o n s involving t h e a b s o r p t i o n or creation of p h o n o n s occur giving a b s o r p t i o n p r o p o r t i o n a l t o (h v — Ej)1'2 for each e x e i t o n b a n d , a n d (hv — E-^f for t h e c o n t i n u u m . T h u s t h e f i r s t t e r m in (h v — E¡) in t h e a b o v e expression r e p r e s e n t s t r a n s i t i o n s i n t o t h e g r o u n d s t a t e of t h e exciton, t h e n e x t t e r m t r a n s i t i o n s i n t o t h e first excited exciton b a n d . T h e last t e r m represents transitions into the continuum. T h e a b o v e work was e x t e n d e d b y Clark, D e a n , a n d H a r r i s [144], T h e y studied t h e optical t r a n s m i s s i o n s p e c t r a of several d i a m o n d s over t h e r a n g e 5.0 t o

484

Y . P . VAKSHNI Fig. 7. The absorption index x, the radiative recombination rate P(v) Q(V), and the associated quantity U (defined by equation (34)) versus photon energy for diamond at 295 °K.

6.0 eV and at many temperatures between 90 and 600 °K. The intrinsic features of the absorptionedge spectrum could be interpreted in terms of allowed indirect electronic transitions into exciton and free-carrier states of the crystal. Analysis of the absorption data yielded six phonon energies. Three of these appear to represent combinations of two or more phonons. The reflection spectrum of diamond has been investigated by Philipp and Taft [145] and by Clark et al. [144], Walker and Osantowski [146] have measured the absolute reflection spectrum of type I I a diamond at room temperature from 4 to 30 eV and analysed it by dispersion techniques to obtain the curves for the optical parameters. These curves disagree substantially in the energy range 5.5 to 12 eV with the previous results of Philipp and Taft [145], This discrepancy has been discussed by Philipp and Taft [147] who conclude that the results of Walker and Osantowski are not entirely correct. The analysis of [146] yields large negative values of the absorption coefficient in the region where diamond is transparent [142, 143], This non-physical situation affects the computed values of the optical constants at higher energy and causes the disagreement with the previous work [145]. Data adopted here: Absorption coefficient: At 295 °K, cf. Fig. 3 of Clark et al. [144], Refractive index: n = 2.75 [146], Intrinsic carrier concentration: Obtained from equation (48) using m c = 0.2 m [144] and mh = 0.4 m [148, 149, 150], Calculated results are given in Fig. 7 and Table 2 (see Section 6). 7.1.2 Discussion

of radiative

transitions

The P(v) Q(V) curve in Fig. 7 shows two maxima at 5.27 and 5.54 eV respectively. Below 5.25 eV, the P(v) Q(V) curve shows an increasing behaviour instead of decreasing. This appears to indicate that there is still some impurity absorption left over. The absorption coefficient in this region is very small ( ss 0.3 c m - 1 ) and accurate subtraction of the impurity absorption is difficult. Intrinsic recombination radiation in diamond was first observed by Male and Prior [151]. Their intensity curve is reproduced in Fig. 8. The radiation emitted has a maximum energy of about 5.6 eV, and intensity maxima occur at energies 5.278 and 5.127 eV. The 5.278 eV maximum is in satisfactory accord with the expectations from Fig. 7. Dean and Male [152] have interpreted this peak as

485

Band-to-Band Radiative Recombination in Semiconductors ( I ) F i g . 8.

S p e c t r u m o f intrinsic r e c o m b i n a t i o n r a d i a t i o n f r o m diamond (Male and Prior [151])

5.2

5.4 Energy(e/)-

5.6

due to recombination occuring from exciton states having a Maxwell-Boltzmann kinetic energy distribution, since the low-energy threshold of the peak falls near the main exciton-phonon absorption threshold of the absorption spectrum [144], The low-energy dispersion in /.'-space of the associated optical phonons means that the shape of the emission band should be governed mainly by the kinetic energy distribution of the excitons. Dean and Male initially calculated the points for a single Maxwell-Boltzmann distribution at a temperature of 350 ° K (since it was anticipated that the crystal was subject to some Joule heating by the injected current). This distribution was corrected for the finite energy resolution of the spectrometer and was compared with the experimental curve. The experimental curve was observed to be steeper than the calculated one on the low-energy side, although the initial part of the high-energy side fitted reasonably well. The most likely explanation for this discrepancy is that there are, in fact, two unresolved components in the experimental peak. This is reasonable since two thresholds are found to occur in the absorption spectrum near 5.259 and 5.271 eV at 290 °K. The solid dots in the Fig. 9 are calculated on the assumption that each component follows a Maxwell-Boltzmann distribution for T = 350 °K. The energy separation and relative intensity of the two components were given the values observed for the corresponding absorption components, but the position of the pair of thresholds was adjusted to give the best fit to the experimental spectrum. Fig. 9 shows that the agreement obtained in this way is quite good. More recently, Dean and Jones [153] have measured the recombination radiation spectrum over the photon energy range 4.9 to 5.5 eV at 90, 160, 207, and 320 °K. The optical emission was stimulated by bombardment with an electron

ft F i g . 9. T h e d o m i n a n t p e a k o f the intrinsic r e c o m b i n a t i o n radiat i o n s p e c t r u m o f d i a m o n d . Solid d o t s are p o i n t s calculated o n the a s s u m p t i o n t h a t r e c o m b i n a t i o n occurs f r o m e x c i t o n states h a v i n g a M a x w e l l - B o l t z m a n n kinetic e n e r g y d i s t r i b u t i o n corresponding t o 350 ° K . P o i n t s A and B i n d i c a t e the l o w - e n e r g y thresholds o f the t w o assumed c o m p o n e n t s ( u n r e s o l v e d e x p e r i m e n t a l l y ) o b t a i n e d b y f i t t i n g the c a l c u l a t e d p o i n t s t o the e x p e r i m e n t a l c u r v e ( D e a n and M a l e [1521)

32

physica

vv A 5.2

'i'B

i

i

i

5.3 5.4 Quantum energy (eV) -

486

Y. P.

VARSHNI Fig. 10. R e c o m b i n a t i o n radiation spectra from a selection of natural d i a m o n d s a) Specimen Co72 ( t y p e H a ) ; 7 0 k e V , 220(¿A electron b e a m , 0.0063 eV resolution. b) Specimen K 8 0 ( t y p e I l a ) ; 7 8 k e V , 240¡¿A electron beam, 0.0063 eV resolution. c) Specimen X 6 0 (intermediate t y p e ) ; 80 keV, 250 u.A electron beam, 0.0063 eV resolution. d) Specimen W X R 1 ( t y p e I); 8 0 k e V , 2 5 0 | i A electron beam, 0.0063 eV resolution. e) Specimen C " 1 1 6 (intermediate t y p e ) ; 80 k e V , 220 [JLA electron beam, 0.0063 e V resolution. f ) Specimen D10 ( t y p e I l a ) ; 72 keV, 240 ¡¿A electron b e a m , 0.0063 eV resolution. g) Specimen E 2 ( t y p e l i b ) ; 30 keV, 220 |xA electron beam, 0.0063 eV resolution ( D e a n and Jones [153])

b e a m of energy b e t w e e n 10 a n d 80 keV. A f e w of t h e emission s p e c t r a o b t a i n e d a t 90 ° K are shown in F i g . 10. I t is seen t h a t t h e relatively b r o a d b a n d s A a n d B are t h e most p e r s i s t e n t f e a t u r e s of t h e s p e c t r a . The s h a r p b a n d s C a n d D are p r e s e n t only in t h e s p e c t r u m f r o m t h e semiconducting, t y p e l i b , specimen 5.0 5.1 5.2 53 (Fig. lOg), which also c o n t a i n s a n u m b e r Photon energy/efi—of o t h e r s h a r p a n d b r o a d b a n d s a t lower energies (bands E t o K ) . T h e effect of t e m p e r a t u r e on 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 s p e c t r a is shown in Fig. 11; t h e b a n d B is seen t o b r o a d e n a p p r e c i a b l y w i t h t h e increase in t e m p e r a t u r e . B a n d B corresponds t o t h e m a i n f e a t u r e in t h e s p e c t r u m recorded b y Male a n d P r i o r [151], B a n d s A a n d B are i n t e r p r e t a b l e in t e r m s of exciton a n n i h i l a t i o n w i t h t h e emission i n t o t h e lattice of one or m o r e p h o n o n s . 7.2

Silicon

Silicon is a simple m o n a t o m i c s e m i c o n d u c t o r a n d its p r o p e r t i e s h a v e been intensively i n v e s t i g a t e d . Reviews of its p r o p e r t i e s m a y be f o u n d in F a n [154] Moss [22], a n d S m i t h [2], 7.2.1 Optical

properties

T h e o p t i c a l p r o p e r t i e s of silicon h a v e been extensively i n v e s t i g a t e d . Collins a n d F a n [155] a n d J o h n s o n [156] d e t e r m i n e d t h e a b s o r p t i o n s p e c t r a of silicon b y m e a n s of o p t i c a l transmission studies in t h e i n f r a r e d region. J o h n son in his s t u d y correlated t h e lattice a b s o r p t i o n b a n d s w i t h multiple p h o n o n

Fig. 11. R e c o m b i n a t i o n radiation spectra from d i a m o n d a t various specimen temperatures ( t y p e l i b , specimen A 100)

5.3 5A5.Z 5.3 5A Photon energy (et'!—-

a) 90 ° K ; 80 keV, 220 |iA electron b e a m , resolution 0.0063 eV. b) 160 ° K ; 80 keV, 250 nA electron b e a m , resolution 0.0063 eV. c) 207 ° K ; 80 keV, 250 (¿A electron beam, resolution 0.0063 cV. d) 320 ° K ; 80 keV, 250 [¿A electron b e a m , resolution 0.0063 eV ( D e a n a n d J o n e s [153])

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

487

processes up to and including three phonons and covered the spectral region from 6.5 to 30 jjim. His analysis of the lattice vibration spectra is in good agreement with that as determined by Brockhouse [157] from neutron scattering experiments on Si. Dolling [158] has reported more recent results from neutron scattering experiments. Two phonon bands in the infrared absorption spectra have been observed by Balkanski and Nusimovici [159], Spectral emissivity measurements at wavelengths between 3 and 15 jjun have been made on n- and p-type silicon by Stierwalt and Potter [160]. Infrared absorption of silicon at 7.5 °K has been measured by Aronson et al. [161]. Reflectance studies have been carried out by Vavilov et al. [162], Robin-Kandare and co-workers [163, 164], Philipp and Taft [165], Sasaki and Eshiguro [166] (10 to 19.2 eV), Philipp and Ehrenreich [114] (1.5 to 25 eV), and by Subashiev et al. [167] and in some cases the values of n and x have also been derived. Very precise measurements of the ultra-violet reflectance of silicon by Lukes and Schmidt [112] have shown a hyperfine splitting of the Mj absorption edge. Phillips [168] has pointed out that this structure is exactly what one must expect when multiphonon transitions are taken into account. Investigations in the wavelength range from 2 to 7 [tm have shown that the observed coefficient in n-type is greater than is predicted by theory. This anomalous behaviour has been discussed by Yakovlev [169], The transmission of silicon between 40 and 100 [i.m has been measured by Walles [170]. The first detailed measurements on the absorption coefficient of Si in the vicinity of the absorption edge were those of Fan and co-workers [171, 30] who reported results at a number of temperatures between 90 and 602 °K. Soon after, Dash and Newman [172] measured the absorption coefficients of highpurity single crystal silicon at 77 and 300 °K in the energy range 1 to 3.4 eV. The most exhaustive work on absorption at low values of the absorption coefficient is that of Macfarlane and collaborators. In their early work, Macfar lane and Roberts [31] measured the absorption of a polycrystalline specimen of silicon in the neighbourhood of the absorption edge at temperatures from 20 to 333 °K. I t was found that in the region a < 100 c m - 1 the results can be well represented by an equation of the form (13). 6 was found to be 600 °K. This was followed by measurements of the absorption spectrum under higher resolution by Macfarlane, McLean, Quarrington, and Roberts [125, 173] near the main absorption edge at various temperatures between 4.2 and 415 °K. Fine structure was found in the absorption on the long-wavelength side of the edge. The structure was analysed and interpreted in terms of indirect transitions involving, in general, phonons with energies corresponding to temperatures of 212, 670, 1050, and 1420 °K. The original interpretation of the two components corresponding to the temperatures of 1050 and 1420 °K is now considered to be incorrect and instead these are now believed to be due to two-phonon processes [23]. Braunstein, Moore, and Herman [174] have reported measurements on the absorption coefficients at 5, 78, 197, 234, and 296 °K. At low energies they confirm the findings of Macfarlane and Roberts that the variation of >) for tellurium at 100 " K

0.35 0.W hv(eV) l'ig. 29. X and P(v) g(r) for tellurium at 300 (Read at the dotted l i n e : Theor. a used)

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

507

Fig. 30. Lifetime versus 1 / T for three very pure and structurally perfect samples, compared with the theoretical expectations of radiative (TR) and Auger (TA) recombination. Effective lifetime r e = ( 1 / T A 4 - 1 / T K ) " (Blakemore [286]) 1

7.6.2 Discussion of radiative transitions Fig. 28 and 29 show that, except at very low energies, there is fair agreement between the P(v) o{v) curves obtained from the experimental data and those from the analytical expressions for the absorption coefficient. The recombination processes in telluiium weie investigated by Redfield [285] who also attempted to observe the recombination radiation but was unsuccessful, possibly because of experimental limitations. Blakemore [286] has presented carrier lifetime data for tellui ium samples with varying degrees of purity. In extensively purified monocrystals the lifetime behaviour above 300 ° K suggests control by band-to-band transitions. Calculations were made of the radiative and Auger recombination rates, and experimental data between 300 and 420 ° K fit the sum of these two rates. Blakemore's results are shown in Fig. 30. A curve in this figure marked r e shows the calculated lifetime limited by the two band-to-band mechanisms. Recently Benoit à la Guillaume and Debever [287] have observed the recombination radiation from telluiium at 4 and 20 °K. The excitation was made by a 15 keV electron beam. The emission is polarized E c. Stimulated emission has been obtained at current above 200 ¡i.A at 4 ° K . References Note : The following abbreviations have been used : Prague Conf. = Proceedings of the International Conference on Semiconductor Physics. Prague 1960, Academic Press, New York 1961. Exeter Conf. = Report of the International Conference on the Physics of Semiconductors held at Exeter, July 1962, The Institute of Physics and the Physical Society, London 1962. Paris Conf. = Comptes Rendus du 7 e Congrès International Physique des Semiconducteurs, Paris 1964, Dunod, Paris 1964. Paris Symp. = Symposium on Radiative Recombination, Paris 1964, Dunod, Paris 1964. [1] G. BEMSKI, Proc. I R E 46, 990 (1958). [2] R. A. SMITH, Semiconductors, Cambridge University Press, 1959. [3] R, N. HALL, Proc. I E E B 106, Suppl. 17, 923 (1959) [4] R, H. BUBE, Photoconductivity of Solids, Wiley, 1960. [5] J. S. BLAKEMORE, Semiconductor Statistics, Pergamon Press. New York 1962. [6] W. SHOCKLEY, Proc. I R E 46. 973 (1958). [7] V. S. VAYILOV, Uspekhi fiz. Nauk 68, 247 (1959); Soviet Phys. - Uspekhi 2, 455 (1959).

Y . P . VARSHNI

508

[8] V. S. YAVILOV, Effects of R a d i a t i o n on Semiconductors, Consultants Bureau, New Y o r k 1965. [9] G. M. GURO, Uspekhi fiz. N a u k 72, 711 (1960); Soviet P h y s . - Uspekhi 3, 895 (1961). [ 1 0 ] W . VAN ROOSBROECK a n d W . SHOCKLEY, P h y s . R e v . 9 4 , 1 5 5 8 ( 1 9 5 4 ) .

[11] G. LASHER a n d F . STERN, P h y s . Rev. 138, A 553 (1964). [ 1 2 ] W . SHOCKLEY a n d W . T . R E A D , P h y s . R e v . 8 7 , 8 3 5 ( 1 9 5 2 ) . [13] R . N . HALL, P h y s . R e v . 87, 387 (1952). [14] L . PINCHERLE, P r o c . P h y s . S o c . B 67, 3 1 9 (1955).

[15] H . GUMMEL a n d M. LAX, P h y s . R e v . 97, 1469 (1955). [ 1 6 ] N . SCLAR a n d E . BURSTEIN, P h y s . R e v . 9 8 , 1 7 5 7 ( 1 9 5 5 ) . [ 1 7 ] A . R . B E A T T I E a n d P . T . LANDSBERG, P r o c . R o y . S o c . A 2 4 9 , 1 6 ( 1 9 5 8 ) . [ 1 8 ] A . R . B E A T T I E a n d P . T . LANDSBERG, P r o c . R o y . S o c . A 2 5 8 , 4 8 6 ( 1 9 6 0 ) .

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[219] C. D. SALZBERG and J . J . VILLA, J . Opt. Soc. Amer. 47, 244 (1957). [220] C. D. SALZBERG and J . J . VILLA, J . Opt. Soc. Amer. 48, 579 (1958).

[221] M.I. KORNFELD, Fiz. tverd. Tela 2,48 (1960); Soviet Phys. - Solid State 2, 42 (1960).

[222] F . LUKES, Czech. J . Phys. 8, 253 (1958). [223] F . LUKES, Czech. J . Phys. 10, 742 (1960). [224] F . LUKES and E . SCHMIDT, Prague Conf. (p. 371).

[225] E. ANTONCIK, Czech. J . Phys. 6, 209 (1956). [226] F. LUKES, Czech. J . Phys. 8, 423 (1958).

[227] W. H. BRATTAIN and H. B . BRIGGS, Phys. Rev. 75, 1705 (1949).

[228] M. P. LISITSA and N. G. TSVELYKH, Optika i Spektroskopiya 5, 622 (1958). [ 2 2 9 ] L . HULDT a n d T . STAFLIN, O p t i c a A c t a 6 , 27 (1959).

[230] P. P. KONOROV and O. V. ROMANOV. Fiz. tverd. Tela 2, 1869 (1960); Soviet Phys. Solid State 2, 1688 (1961). [231] F. LUKES, Czech. J . Phys. 10, 59 (1960). [232] J . RICHARD, C. R. Acad. Sci. (France) 256, 1093 (1963).

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

513

[233] P . G. BORZYAK a n d R . D. FEDOROVICH, Fiz. t v e r d . Tela 2, 3020 (1960); Soviet P h y s .

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[236a] R . NEWMAN, Phys. R e v . 105, 1715 (1957). [237] J . R . HAYNES a n d N . G. NILSSON, Paris. Symp., P a p e r R R - 0 2 . [238] N. G. NILSSON and J . R . HAYNES, Paris Symp., P a p e r R R - 0 3 . [239] J . I . PANKOVE, P h y s . R e v . L e t t e r s 4, 20 (1960).

[240] Y. M. ASNIN and A. A. ROGACHEV, Fiz. tverd. Tela 5, 1730 (1963); Soviet Phys. Solid State 5, 1257 (1963).

-

[ 2 4 1 ] J . T . NELSON a n d J . C. IRVIN, J . a p p l . P h y s . 3 0 , 1 8 4 7 ( 1 9 5 9 ) .

[242] A. G. CHYNOWETH and H. K. GUMMEL, J . Phys. Chem. Solids 16, 191 (1960). [243] P. A. WOLFF, J. Phys. Chem. Solids 16, 184 (1960). [ 2 4 4 ] P . H . BRILL a n d R . F . SCHWARZ, P h y s . R e v . 1 1 2 , 3 3 0 ( 1 9 5 8 ) .

[245] P. H. BRILL and R. F. SCHWARZ, J. Phys. Chem. Solids 8, 75 (1959). [246] A. A. ROGACHEV and S. M. RYVKIN, Fiz. tverd. Tela 4, 1676 (1962); Soviet Phvs. Solid State 4, 1233 (1962). [247] J. J . DOWD, Proc. Phys. Soc. B 64, 783 (1951). [248] M. A. GILLEO, J. chem. Phys. 19, 1291 (1951). [249] E. W. SAKER, Proc. Phys. Soc. B 65, 785 (1952). [250] J . STUKE, Z. Phys. 134, 194 (1953).

-

[251] C. HILSUM, Proc. P h y s . Soc. B 69, 506 (1956). [252] N . N . PRIBYTKOYA, O p t i k a i Spektroskopiya 2, 623 (1957). [ 2 5 3 ] W . F . KOEHLER, F . K . ODENCRANTZ, a n d W . C. W H I T E , J . O p t . S o c . A m e r . 4 9 , 1 0 9

(1959). [254] A. VASKO, Czech. J. Phys. 15, 170 (1965). [ 2 5 5 ] R . S . CALDWELL a n d H . Y . F A N , P h y s . R e v . 1 1 4 , 6 6 4 ( 1 9 5 9 ) .

[256] S. ROBIN-KANDARE, J. Phys. Radium 21, 31 (1960). [ 2 5 7 ] H . GOBRECHT a n d A . TAUSEND, Z . P h y s . 1 6 1 , 2 0 5 ( 1 9 6 1 ) .

[258] [259] [260] [261]

L. MÜLLER a n d M. MÜLLER, Studii Cercetäri Fiz. (Bucuresti) 8, 857 (1962). W . L. GOFFE a n d M. P . GIVENS, J . Opt. Soc. Amer. 63, 804 (1963). I . SRB a n d A. VASKO, Czech. J . P h y s . B 13, 827 (1963). F . R . KESSLER a n d E . SUTTER, Z. P h y s . 173, 54 (1963).

[ 2 6 2 ] H . P . D . LANYON, P h y s . R e v . 1 3 0 , 1 3 4 ( 1 9 6 3 ) .

[263] F. M. GASHIMZADE and A. I. GUBANOV, Fiz. tverd. Tela 6, 1030 (1964); Soviet Phys. - Solid State 6, 795 (1964). [264] A. I. GUBANOV, Soviet Phys. - J. exper. theor. Phys. 1, 364 (1954). [265] A. I. GUBANOV, Fiz. tverd. Tela 2, 651 (1960); Soviet Phys. - Solid State 2, 605 (1960). [266] A. I. GUBANOV, Fiz. tverd. Tela 3, 2154 (1961); Soviet Phys. - Solid State 3, 1564 (1961). [267] A. I. GUBANOV, Fiz. tverd. Tela 4, 1510 (1962); Soviet Phys. - Solid State 4, 1109 (1962).

[268] A. I. GUBANOV, Quantum Electron Theory of Amorphous Conductors, Consultants Bureau, New York 1965. [269] K . MOORJANI a n d C. FELDMAN, Rev. mod. P h y s . 36, 1042 (1964). [ 2 7 0 ] W . J . CHOYKE a n d L . PATRICK, P h y s . R e v . 1 0 8 , 2 5 ( 1 9 5 7 ) .

[271] V. PROSSER, Prague Conf. (p. 993). [272] V. PROSSER, Czech. J . P h y s . 10, 306 (1960). [ 2 7 3 ] F . ECKART a n d W . HENRION, p h y s . s t a t . sol. 2, 8 4 1 ( 1 9 6 2 ) .

[273a] F. ECKART and W. HENRION, Monatsber. Deutschen Akad. Wiss. 4, 446 (1962). [273b] F. ECKART and W. HENRION, Monatsber. Deutschen Akad. Wiss. 4, 449 (1962). [ 2 7 4 ] J . S . BLAKEMORE, D . LONG, K . C. NOMURA, a n d A . NUSSBAUM, P r o g r . S e m i c o n d . 6 , 37 (1962).

[275] J . J . LOFERSKI. Phys. Rev. 93, 707 (1954).

514

Y. P. VARSHNI : Band-to-Band Radiative Recombination (I)

[276] K . C. NOMURA and J . S. BLAKEMORE, Bull. Amer. Phys. Soc. 5, 62 (1960). [ 2 7 7 ] J . S. BLAKEMORE a n d K . C. NOMURA, P h y s . R e v . 1 2 7 , 1 0 2 4 ( 1 9 6 2 ) .

[278] V. M. KORSUNSKII and M. P. LISITSA, Fiz. tverd. Tela 2", 1619 (1960); Soviet Phys. Solid State 2, 1466 (1961). [279] J . N. HODGSON, J . Phys. Chem. Solids 23, 1743 (1962).

[ 2 8 0 ] M . P . LISITSA and N . G. TSVELYKH, O p t i k a i Spektroskopiya 4 , 3 7 3 ( 1 9 5 8 ) . [ 2 8 1 ] O. P . RUSTGI, W . C. WALKER, and G. L . WEISSLER, J . Opt. Soc. Amer. 6 1 , 1357

(1961).

[ 2 8 2 ] H . MERDY, S. ROBIN-KANDARE, a n d J . ROBIN", C. R . Acad. S c i . ( F r a n c e ) 2 5 7 , 1526 (1963). [ 2 8 3 ] H . MERDY, S. ROBIN-KANDARE, and J . ROBIN, J . P h y s . R a d i u m 2 5 , 2 2 3 (1964).

[284] P . A. HARTIG and J . J . LOFERSKI, J . Opt. Soc. Amer. 44, 17 (1954).

[285] D. REDFIELD, Phys. Rev. 100, 1094 (1955). [286] J . S. BLAKEMORE, Prague Conf. (p. 981).

[ 2 8 7 ] C. BENOIT À LA GUILLAUME and J . M. DEBEVER, Solid S t a t e C o m m u n . 3 , 19 ( 1 9 6 5 ) . (Received

June

28,

1966)

Original

Papers

p h y s . s t a t . sol. 19, 515 (1967) Subject classification: 10.2; 22.5.2 2. Physikalisches

Institut

der Technischen

Hochschule

Stuttgart

Local Mode Absorption by U-Centres in Alkali-Halide Mixed Crystals By 1

W . B A B T H ) a n d B . FEITZ

The local mode absorption b a n d s of U-centres are affected b y t h e presence of alkali or halogen foreign ions in t h e host lattice in t h e following w a y s : a) new b a n d s appear, which are displaced to higher a n d lower frequencies b) t h e entire spectrum undergoes a concent r a t i o n - d e p e n d e n t shift a n d broadening. These effects are studied for U-centres in KCl :Rb+, K C l : Na+, KCl :Br-, R b C l : K+, K B r : Rb+ a n d R b B r : K+ with doping concentrations between 10~ 3 a n d 10~'. Groups of new lines can be a t t r i b u t e d t o U-centres having doping ions in t h e n e x t three shells of neighbouring ions. The intensities of these lines are slightly greater t h a n expected f r o m t h e statistical probabilities for such configurations. The splitting observed in a characteristic pair of lines compares favourably with a simple calculation for t h e case of a U-centre p e r t u r b e d by a foreign cation in a nearest neigbour (100) position. T h e relation between t h e lattice constant a n d local mode frequency obtained in mixed crystals m a y be used t o discuss lattice relaxation near U-centres a n d "quasih a r m o n i c " t e m p e r a t u r e - d e p e n d e n t b a n d shifts. D u r c h Alkali- oder Halogenfremdionen in Alkalihalogenid-Einkristallen wird das Abs o r p t i o n s s p e k t r u m der lokalisierten U-Zentren-Schwingung folgendermaßen v e r ä n d e r t : a) E s t r e t e n neue B a n d e n mit erhöhter u n d erniedrigter Frequenz auf. b) Alle B a n d e n werden, in Abhängigkeit von der K o n z e n t r a t i o n der Fremdionen, verschoben u n d verbreitert. Diese Beobachtungen wurden an U-Zentren in den Kristallen KCl: Rb+, K C l : Na+, K C l : B r " , RbCl: K+, K B r : R b + u n d R b B r : K + gemacht. Die Fremdionenkonzentrationen lagen zwischen 10~ 3 u n d 10" 1 . Die neuen Linien können auf U-Zentren z u r ü c k g e f ü h r t werden, die von Fremdionen in den drei nächsten Nachbarschalen gestört werden. Die b e o b a c h t e t e n I n t e n sitäten sind etwas größer, als auf Grund der Wahrscheinlichkeit der entsprechenden Konfigurationen im Fall statistischen E i n b a u s zu erwarten wäre. Eine f ü r den Fall der Störung durch ein F r e m d i o n u n t e r den n ä c h s t e n N a c h b a r n d u r c h g e f ü h r t e Modellrechnung ergibt g u t e Übereinstimmung mit der beobachteten Aufspaltung des entsprechenden Linienpaars. Der gefundene Z u s a m m e n h a n g zwischen Frequenzverschiebung u n d G i t t e r k o n s t a n t e in Mischkristallen g e s t a t t e t eine Abschätzung t e m p e r a t u r a b h ä n g i g e r „ q u a s i h a r m o n i s c h e r " Frequenzverschiebungen u n d der Gitterrelaxation in der N ä h e des U-Zentrums.

Substitutional H~ ions at anion sites in alkali halides introduce an infrared active local mode vibration, which is threefold degenerated in otherwise unperturbed crystals. In this paper we report on the perturbation of this local mode absorption by foreign cations and anions. Such effects were observed for the fiist time by Schäfer [1], who found sharp new peaks, displaced to higher and lower frequencies from the strong local mode resonance peak near 500 c m - 1 in KCl: Rb + . 2

) P r e s e n t address: I n s t i t u t f ü r Höchstfrequenztechnik, Technische Hochschule S t u t t gart.

516

W . B A B T H a n d B . FRITZ

This observation leads t o t h e question of t h e t y p e s of U - c e n t r e foreign-ion aggregates t h a t m a y be f o r m e d in such crystals, a n d of t h e d i s p l a c e m e n t s f r o m t h e i r original positions of t h e neighbouring ions of such aggregates. These displacem e n t s p r o d u c e shift a n d splittings of t h e s h a r p local mode resonance line, similar t o t h e effects which are seen using e x t e r n a l u n i a x i a l stress [2, 3]. One m a y expect some of these results t o bear on discussions of residual line w i d t h s a n d splittings occuring in t h e s p e c t r a of crystals of r e a g e n t grade p u r i t y [4]. 1. Experiments T h e m i x e d crystal s y s t e m s i n v e s t i g a t e d b y us are listed in T a b l e 1. T h e crystals were air-grown f r o m t h e m e l t b y t h e K y r o p o u l o s m e t h o d . T h e r a n g e of d o p i n g p e r c e n t a g e a n d t h e m e t h o d of d e t e r m i n a t i o n thereof are also given. U - c e n t r e s were i n t r o d u c e d i n t o t h e crystals using c o n v e n t i o n a l m e t h o d s [5], T h e c o n c e n t r a t i o n of U - c e n t r e s was d e t e r m i n e d b y m e a s u r i n g t h e well-known u l t r a v i o l e t b a n d [5]. A p a r t f r o m a small shift of t h i s b a n d , t h e U V a b s o r p t i o n above 200 n m a p p e a r e d u n c h a n g e d b y t h e doping. T h e s h i f t a t r o o m t e m p e r a t u r e is typically —0.05 eV in K C l : R b + (8%) a n d is t h u s larger t h a n p r e d i c t e d b y t h e I v e y relation for t h i s b a n d in d i f f e r e n t crystals [6]. Table 1 Crystal

KCl KBr RbCl RbBr

II

Admixture Rb+ Na+ Br" Rb +

Concentration range (mol%) 0.8-8 1-5 1-50 1-10

from flame photometry in the melt

0.05 —ca. 1

(different commercial purity grades)

T h e i n f r a r e d spectra were m e a sured on a p r i s m s p e c t r o m e t e r allowing a m a x i m u m resolution of 2 c m - 1 . T h e o p t i c a l a r r a n g e m e n t a n d t h e necessary corrections were described in a r e c e n t p a p e r [4]. T h e e x p e r i m e n t a l r e s u l t s obt a i n e d are as follows. F i g . 1 demonstrates the main features of U - c e n t r e s p e c t r a a s f o u n d in KCl:Rb+: a) T h e s t r o n g e s t line d e n o t e d b y v0 h a s n e a r l y t h e same posit i o n as t h a t in p u r e KC1 w i t h U - c e n t r e s . T h i s position is indicated by the dashed curve.

• Wavelength (p.m) 21 10

550 Wavenumbers (crrr1] —

Fig. 1. Infrared absorption spectrum of Uccntres in K C l : l l b + , at T = 21 °K. v„ is a U-band of nearly unperturbed frequency. v a , etc. are new bands not seen in pure KC1

Local Mode Absorption by U-Centres in Alkali-Halide Mixed Crystals Fig. 2. Kelative intensities versus concentration c. The straight lines approximate (small c) relative probabilities of configurations with one and two foreign cations among the nearest neighbours of a l l - c e n t r e (random distribution of U-centres and foreign ions assumed!)

517

KCi.Rb*

I

U-centres

/

IVo

7

2 4 Rb* concentration in the crystal (.mot%)—

b) Five new bands, denoted b y va, Vp, vy, vg, ve are seen in the neighbourhood of vn for R b + dopings between 1 and 8 % . c) All bands, including t h a t at v0, are shifted and broadened b y like a m o u n t s , depending on t h e doping concentration (curves a, b, and c of Fig. 1). Besides these sharp bands, the usual side band structure (near 560 c m - 1 ) is seen to be present in Fig. 1, in agreement with observations on undoped KC1 :H~ crystals. This underlying side-band absorption was s u b t r a c t e d in order t o determine relative intensities of t h e mixed crystal bands reported here. The dependence on R b + concentration c of the band a t va relative t o t h a t a t v0 is given in Fig. 2. This relative intensity is found t o be proportional t o c. The bands a t Vp a n d vY follow the same dependence, which is shown in p a r t i c u l a r b y t h e linear relationship between t h e absorption a t va and (Fig. 3). The b a n d a t vp is weaker b y 1:2.16 t h a n v„ (Fig. 3) and roughly equally intense as vy. Quite a different behavior is exhibited by t h e bands vs and ve, whose relative intensities v a r y as c 2 (Fig. 2). These much weaker bands are again roughly equally strong. I n most systems listed in Table 1 groups of three lines are f o u n d , resembling those found in K C l : R b + regarding relative intensities and a m o u n t of splitting.

34

pliysica

518

W . BARTH a n d B . FRITZ

I

-UO -10

0

10 W 60 -60 -w -10 0 Band displacements I'cm •'/ —»-

10

W

Fig. 4. Band positions in the different mixed crystal systems containing U-centres. T 20 °K (schematic representation). Bands whose connection with the doping is not safely established are indicated by primes

The line with twofold intensity, va, is seen on the low-frequecny side of v0 in K C l : R b + and KCl:Br~, whereas two other lines Vp and vY are on the high-frequency side of v0. In these cases an ion of the host lattice has been replaced by a larger doping ion (see Fig. 4). In the other systems, where the foreign doping ions have smaller ionic radii than the ions they substitute for, the strongest line va is on the high-frequency side of v0 and the weaker ones on the low-frequency side, so that the splitting pattern is mirror symmetric to that above. In R b B r : K + and KC1 :Na + the connection of the and y-bands with the doping material could not be safely established, because of the lower doping concentration employed. These bands are denoted by primed letters.

6.25

626 1

1

1

1

HCl: F b*

Lattice parameter (A) 627 1

1

1

1

\

1 it 6 8 Rb* concentration in the crystal (mot%j-

Fig. 5. Positions of »•„ ( K C l : R b + ) versus doping concentration (21 °K). The lattice parameter is interpolated using a Vegard relation known to hold at room temperature. The bands va . . . vy show the same dependence

Local Mode Absorption by U-Centres in Alkali-Halide Mixed Crystals

519

Measurements on D~-U-centres in K C l : R b + gave the result that the mixed crystal bands as described here undergo the same isotopic shift as the local mode frequency v0, namely — = 1.39 » 1/2 . VD~

This behaviour is clearly different from that of the sidebands due to the twophonon absorption [4]. The concentration-dependent shift and broadening of all lines including vg were also investigated, and the results are given in Fig. 5 and 6. According to Fig. 5 the frequency of the band peak at v0, measured at 20 ° K , is a linear function of the doping concentration in KC1: Rb + . On top of this figure a lattice parameter scale is also given, which is based on the findings of Gnaedinger [7] who showed that the lattice constant in this system at room temperature follows closely a Vegard relation. We used this to extrapolate for the 20 ° K lattice parameter values at a given R b + concentration in the crystal. From these results we deduce the following relation between the frequency shift Ar and the average lattice constant change A a: "c

= 3.3— • a o

(1)

The half widths which turn out to be the same for v0, v^ and vp bands, are plotted in Fig. 6 versus temperature with different concentrations of R b + a s an additional parameter. At low temperatures the band widths reach saturation values, which increase with the Rb + doping concentration. At higher temperatures the bands broaden according to a temperature dependence almost as strong as found in undoped KC1. This dependence for the undoped case, which is roughly T 2 above 100 ° K and stronger below, is plotted for comparison.

Kig. 6. Halfwidtli versus doping concentration and temperature 34

32 Halfwidth (crn-V-

520

W . BARTH a n d B .

FRITZ

2. Discussion 2.1 Additional

lines

The additional lines appearing in mixed crystals may be interpreted by considering the influence of strain caused by nearby foreign ions on the frequency of the local mode of the U-centre. This frequency is determined to a large extent by the repulsive overlap potential between the hydrogen ion and the surrounding ions; because of the Oh symmetry it is threefold degenerate in the unperturbed lattice. The strongest perturbation is expected from one foreign cation on a nearest neighbour site (100). The symmetry is then CiV and leads to a splitting of the unperturbed local mode transition (v0) into two lines having an intensity ratio of 2 : 1 . We interpret the va and lines as arising from this configuration. The different positions of these lines, which are observed in the presence of larger and smaller foreign cations (for instance K C l : R b + and KCl:Na + ) are in good agreement with this model (see discussion, Section 2.2). An equally strong perturbation may arise from two or more foreign cations in the nearest neighbour shell. The overall probability for such configurations is lower by a factor of 10 at c = 4 % , and varies as c 2 (see Fig. 2). This agrees with the behavior of v6 and vc bands in K C l : R b + . These bands therefore may be interpreted as being due to U-centres with two R b + ions adjacent to each other within the nearest neighbour shell. The symmetry is C%v, and the splitting may be expected to be into three equally intense bands, one of them probably being hidden under the rest of the spectrum. The other possible configuration of two R b + ions having Dih symmetry occurs only in one out of five cases and is therefore unlikely to appear in the spectra. Less strong perturbations are to be expected from cations in the third shell. The probability is proportional to the doping concentration c (at low concentrations), and the intensity should be comparable to that of the and bands. The vY band, which is seen very close to v0, is likely to be the most strongly shifted line of the two expected in this case (C3r symmetry). In the case of foreign anion doping (KOI:Br"), the strongest effect should arise from a B r - ion in a second neighbour position. In fact we find bands in intensities which are close to the expected values as estimated from the doping to the melt and Vp). Because of the difficulties in resolving the vK band from the v0 line (Fig. 4) this identification is not so well established as in case of cation doping. 2.2 Calculation

of the splitting

of the

lines

The splitting of a series of lines associated with one of the configurations discussed above may be calculated from the displacements of lattice ions and the different repulsive interaction introduced by a foreign ion. In Fig. 7 we give results of a rough calculation for the and v¡¡ bands in KC1: RV" and RbCl: K+, wherein the local mode frequency was approximated by that of a H~ ion vibrating in the rigid potential well arising from nearest neighbour interactions with the H~ only. According to Fig. 7, a foreign alkali ion is assumed to occupy one of the nearest neighbour sites 1 adjacent to the H~ ion in site 0. Such a substitutional ion generally causes its own nearest neighbours to shift their positions by about one half the difference between the radii of lattice and impurity ions, as shown by recent calculations [8, 9]. This should also be true in the present case where

Local Mode Absorption by U-Centres in Alkali-Halide Mixed Crystals

J'

o h' §> ô

550 cm-1

521

calc. etp. a b KCllRb*

500

o u(0! +6.5 +6.1 RbCl:K* -6.5

H~

®

( urn + 1.6 +22 -1.6

)

ç

ul3)...u(6! (w*X) +3.3 +1.6 -33

(a) lb) (a)

m

L

500 cm'1

exp. •

va

• Va

±50 -

Fig. 7. Proposed model for a U-ceutre having local mode frequencies at v a and v^. The displacements occurring in the neighbourhood of a l i b + ion in K CI and a K + ion in RbCI are given in the table, according to different assumptions a) and b) (see text). These values are used to calculate the relative splitting from equations (2a) and (2b)

one of these neighbours is an ion, if this ion has a radius not too different f r o m t h a t of a halogen ion (e.g. CI") and has a similar form for its repulsive potential. We make this assumption about nearest neighbour displacements in our calculation and let the other displacements follow a r - 2 dependence a r o u n d t h e foreign ion as the origin (continuum approximation [10].) The values of t h e displacements are given in Fig. 7, row a). The ionic radii of the R b + and K + ion are assumed t o differ b y 0.130 A according to calculations b y F u m i and Tosi [ l l j . U n d e r b) we quote for comparison displacements which have been calculated b y Dick [9] for the case of K J : R b + , t o show the preferential strain in the [100] direction brought about by considering t h e lattice structure near the defect in detail. The relative frequency shift m a y be calculated b y fitting parameters of t h e overlap potential and the polarizability of t h e H~ ion t o t h e observed frequency shift which results from a known variation of the lattice constant. We have noted an isotropic band shift to occur in mixed crystals of known average lattice p a r a m e t e r (equation (1)); we must assume, in t h e present application of this result, t h a t da/« 0 a t the defect is t h e same as the average change in the lattice. We use a potential of the Huggins-Mayer t y p e ^ „ ^ c + ^ e x p ^ * ^ ,

(2)

where r 0 ,i is the distance between the H~ ion a t site 0 a n d its nearest neighbour ion i, and r H , r K are the crystal radii of t h e ion and the nearest neighbour cations, respectively. I t has been shown (e.g. [13]) t h a t taking t h e values of t h e parameters in equation (2) from alkali halide d a t a a n d considering t h e ions as rigid leads t o a local mode frequency which is 5 0 % too high. One way of dealing with this situation is t o consider a simple shell model for the vibrating H~ ion, which m a y be justified from the large polarizability of this ion. The

522

W . BABTH and B. FRITZ

local mode frequency o>L is then given by mHft>i=/(l + — ) '

1

'

(3)

where the overlap force constant / may be derived from (2), whereas gu is an internal shell-core spring constant introduced to take account of the deformation of the H in the (odd) local mode vibration (see also [12]). If we follow Fieschi et al. [13] in setting o = 0.34 A (equation (2)), which is the average alkali halide value, then we obtain //grH = 0.37 on our model (equation (3)), by fitting the result of equation (1). The ratio of force constants /„ changed by strain to unchanged ones /„ is obtained by taking the second derivatives of (2) and summing over six nearest neighbours at sites 1 to 6: A = 12 - i i T f exp

fx

\

-

«o / L

e

+ exp

H-sri'M)exp

w(0)

~

e

w(2)

- *JL. exp a

e

(2 a)

-u/3)

e

A r K — u(0)

Q

— • exp

u(0) -

w(2)

(2b)

A r K is the difference of impurity and lattice cation radii, and the displacements are defined in Fig. 7. These expressions do not contain the parameters and b{ of equation (2), since these are assumed to be equal for H ~ - K + and H - R b + interactions and thus drop out. This assumption may be justified from the fact that OJl differs by 5 % only between K C 1 : H - and RbOl: H~. Equations (2 a) and (2b) together with the value of grH from above give the calculated frequencies va and Vp in Fig. 7, after inserting the sets of values a) and b) for the displacements as specified in the figure. The fair agreement with the observed amount of splitting strongly supports our explanation of the and Vp bands and arising from the proposed configuration. The fact that a relatively larger displacement in site 2 than in sites 3 to 6 brought about by Dick's [9] theory diminishes the frequency shift for «-polarization probably is a hint that an H " ion is more compressible than a halogen ion and hence doe? not transmit elastic forces to the same extent. From a more accurate treatment of both elastic strain and vibrational properties of this defect we should expect predictions about band intensities as well. We have observed the va and Vp bands to be in a ratio 2.2 instead of the expected value 2.0 (Fig. 3); furthermore, their total relative intensity (/„ + Ip)IIo exceeds the statistical expectation by 1.5 (Fig. 2). I t cannot be decided at present whether the latter ratio indicates a slightly preferential formation of the defect type of Fig. 7 or is due to a change in local field and effective charge of the localised oscillator. 2.3 Dependence

of the line position

on the lattice constant

The empirical relation — Avlv0 = 3.3 Aa/a0 links the frequency change experienced by all bands on doping of the host crystal to the average lattice constant. This can only be understood if the lattice cells containing U-centres participate in a general expansion of the lattice, which is isotropic on the average. The numerical result above could also be obtained using external pressure on a crystal

Local Mode Absorption by U-Centres in Alkali-Halide Mixed Crystals

523

containing U-centres, e.g. uniaxial pressure. From an experiment with stress applied in the [100] direction the isotropic compression effect was calculated, starting from the band splitting observed with polarized light in the [100] and [010] directions [3]. The result was in satisfying agreement with our relation baove, giving a factor 3.6 instead of 3.3. The dependence on lattice constant of the local ijiode frequency is found to be somewhat weaker when different alkali halides are compared, e.g. the potassium salts with different anions (see Schäfer [1]). For the latter one finds —Av/r0 = 2.4 Aa/a0, a clear indication that an increasing relaxation of the lattice near the U-centre takes place on going from KCl to KBr or KJ. Assuming unperturbed positions of the nearest neighbours of a U-centre in KCl, one estimates that an inward displacement of these neighbours by 1.9% in KBr would be necessary to reconcile the observed frequency value and the prediction of equation (1). From lattice parameter measurements on KBr: H~ by Hilsch and Pohl [14] one deduces a relaxation of 1.6% using Eshelby's theory [10], which may be compared with our estimate. — The temperature dependent shift of the infrared U-band [4] has been discussed in a recent paper [12], It is assumed that the thermal lattice expansion contributes to the total observed shift a "quasi-harmonic" part which may also be described by equation (1). 2.4 Broadening

of lines and remnant line width

The concept of an average lattice constant change produced by statistically distributed centres necessarily comprises a spectrum of local distortions, scattering around the average dimension of a lattice cell. These local deviations are sensed by U-centres and thus contribute to the broadening of their local mode absorption. The remnant line width in Fig. 6 which is concentration dependent, is thus explained. We use the results concerning the line width in mixed crystals with known doping concentrations to extrapolate for the situation to be expected for crystals of standard as-grown impurity level. The concentration dependence of the remnant line width w0 from Fig. 6 is (W 0 ) imp «

i

C1'2

with A = 3.1 cm-1 mol % " 1 ' 2 . This gives (w0)imp = 1 cm"1 for c = 0.1% and 0.3 cm" 1 for c = 0.01%. This may be compared with the major impurity concentration in alkali halides, which for reagent grade material is in the order of 0.01% (e.g. Na in KCl) and thus contributes a few tenths of a cm -1 to the low temperature line width. The very different temperature dependence of the line width of H - and D - centers in KCl and other crystals, reported and discussed in [4] and [12], shows, however, that phonon broadening is the predominant effect at least for D - lines. After completion of this work we learned about measurements by Mirlin and Reshina [15], who obtained similar results on U-centres in KCl doped with various anions and cations. These authors give a qualitative interpretation of the various mixed crystal bands in agreement with our discussion under 2.1. Acknowledgement

We are indebted to Professor H. Pick for his interest and support of this work. One of us (W. B.) wishes to thank the Deutsche Forschungsgemeinschaft for financial support.

W. BARTH and B. FRITZ: Local Mode Absorption by U-Centres

524

References [1] G. SCHÄFER, J . Phys. Chem. Solids 12, 233 (1960). [2] W. BARTH, B. FRITZ, and U. GROSS, Internat. Symp. Colour Centers in Alkali Halides, Urbana, 1965 (p. 12). [3] W . HAYES, H . F . MACDONALD, a n d R . J . ELLIOTT, P h y s . R e v . L e t t e r s 1 5 , 9 6 1 ( 1 9 6 5 ) . [4] B . FRITZ, U . GROSS, a n d D . BÄUERLE, p h y s . s t a t . sol. 1 1 , 2 3 1 (1965).

[5] H. PICK, Struktur von Störstellen in Alkalihalogenidkristallen, in: Tracts in Modern Physics, Vol. 38, Springer-Verlag, Berlin 1965. [6] B. S. GOURARY and F. J . ADRIAN, Solid State Phys. 10, 128 (1960). [7] R. GNAEDINGER, J . chem. Phys. 21, 323 (1953). [8] Y. FUKAI, J . Phys. Soc. Japan 18, 1413 (1963). [9] B . G. DICK, P h y s . R e v . 1 4 5 , 6 0 9 (1966).

[10] D. ESHELBY, Solid State Phys. 3, 79 (1956). [11] F. G. FUMI and M. P. Tosi, J . Phys. Chem. Solids 25, 31 (1964). [12] [13] [14] [15]

H. R. R. D.

BILZ, B . FRITZ, a n d D . STRAUCH, J . P h y s . R a d i u m , t o b e p u b l i s h e d . FIESCHI, G . F . NARDELLI, a n d N . TERZI, P h y s . R e v . 138, 2 0 3 (1965). HILSCH a n d R . W . POHL, T r a n s . F a r a d a y S o c . 3 4 , 8 8 3 (1938). N . MIRLIN a n d N . N . RESHINA, F i z . t v e r d . T e l a 8, 152 (1966). (Received

September

9,

1966)

V. B. PARIISKII: Shape of t h e Dislocation Rosette near an I n d e n t a t i o n

525

phys. stat. sol. 19, 525 (1967) Subject classification: 10; 22.5.2 Physico- Technical Institute of Low Academy of Sciences of the Ukrainian

Temperatures, SSR, Kharkov

Effect of Stress State on the Shape of the Dislocation Rosette near an Indentation in Some Alkali Halide Crystals By V. B.

PABIISKII

The s t r u c t u r e of a dislocation rosette near an indentation in K B r and KCl crystals is studied. The motion of dislocations out of the rays of the dislocation rosette due to external impulse stress is shown to depend on the amplitude and duration of the pulse. The effect of stress on the shape of the dislocation rosette near the indentation is shown for KBr, KCl, and NaCl single crystals. During indentation of the stressed crystal the uniaxial surface contraction elongates the rosette rays t h a t adjoin the direction of contraction, whereas the uniaxial dilation elongates the rays t h a t adjoin the direction normal to t h e direction of extension. The effect is enhanced by increasing the preliminary stresses on t h e surface to be indented. The observed phenomenon is explained qualitatively b y considering the structure of the rosette near the indentation. This effect may be used to determine t h e magnitude and sign of the local residual elastic stresses at the surface. I n K B r und KCl Kristallen wird die Struktur der Versetzungsrosette in der Nähe des Eindrucks untersucht. Es wird gezeigt, daß die Bewegung der Versetzungen aus den Strahlen der Versetzungsrosette heraus von der Amplitude und Dauer der äußeren Impulsbeanspruchung abhängt. Der E f f e k t einer Spannung auf die Form der Versetzungsrossette in der Nähe des Eindrucks wird f ü r K B r , KCl u n d NaCl E i n k r i s t a l l e angegeben. Während des Eindrückens verlängert eine einachsige Oberflächenkontraktion die Rosettenstrahlen, die an der Richtung der Kontraktion anschließen; eine einachsige Streckung verlängert die Strahlen, die an die Richtung senkrecht zur Streckrichtung anschließen. Der Effekt wächst mit wachsender Vorspannung an der Fläche, die eingedrückt wird. Die beobachtete Erscheinung wird qualitativ erklärt unter Berücksichtigung der Rosettenstruktur in der Nähe des Eindrucks. Dieser Effekt kann zur Bestimmung der Größe und des Vorzeichens der elastischen Restspannungen an der Oberfläche benutzt werden.

Studies of the dislocation structure of rosettes appearing on the surface of alkali halide single crystals under a localized force have been reported by many authors [1 to 9]. Distortions of this type are often present in these materials. The influence of the stress state on the dislocation structure of the rosette near the indentation in KBr, KCl, and NaCl crystals is studied in this work. 1. Rosette Structure near an Indentation on an Unstressed Crystal The dislocation structure of the rosette near an indentation on NaCl [2, 5] single crystals has been studied in detail. Fig. 1 and 2 give a schematical representation of the arrangement of dislocation loops around an indentation of a diamond indentor on a {100} plane; the direction of dislocation motion out of the rosette under external stress is also shown [5],

526

V. B. PARIISKII

Fig. 1. Traces of {110} glide planes on {100} planes near the indentation and the corresponding Burgers vectors in edge planes. The intersection of the dislocation-loop edge component with the surface is marked by X

Fig. 2. Diagram of dislocation motion out of the rosette near the indentation under external stress. The external stress is marked by arrows,

Fig. 3. Dislocation motion out of the rosette near the indentation under stress pulses of different sign in a K B r crystal, a) dilation, b) contraction. The external stress is shown by dashed arrows

The investigation of dislocation rosettes near indentations on K B r and KC1 crystals by means of selective etching [10, 11] and the observation of the dislocation motion out of the rosette rays (Fig. 3) have shown that the arrangement of dislocation loops and their Burgers vector in principle agree with the above scheme. The use of stress pulses of different sign, length and amplitude allowed to study the effect of amplitude and length on the character of the dislocation motion out of the rays of the rosette. Qualitatively the result is the following: When the external stress is low or a high-amplitude but short pulse is applied, only "correct" 1 ) motion of dislocations out of the rosette is observed. This is in accord with Fig. 2. An increase of amplitude or time of stress application leads to a "mixed" 1 ) motion when an elongation of all the rays of the dislocaHere the terminology of the papers [2 and 8] is used.

Shape of the Dislocation Rosette near an Indentation in Alkali Halides

527

tion rosette takes place. This result is illustrated for NaCl crystals. The experimental procedure was carried out in the following way: The indentations were made on the surface of a freshly cleaved crystal 5 x 5 x 20 mm 3 in size along its length at a distance of 250 ¡xm from each other by means of PMT-3. Then the crystal was etched and subjected to pulses of bending stress (Fig. 4a) [11]. The stress pulses were measured by means of a tensiometer and photographed from the screen of oscillograph. The distribution of the shear stress acting on the edge dislocations in a thin surface layer calculated in the elastic approximation [13] is shown in Fig. 4b.

Fig. 4. JsaCI crystal stressed by pulses of different length a) t>) c) d) e)

Scheme of pulse application; 1 — knives, 2 — sample, 3 — indentations on the sample surface. Epure of the distribution of resolved shear stress acting on edge dislocations in a thin surface layer. No dislocation motion in the regions AC and A'C'. Correct dislocation motion in the regions CD and C D ' . Mixed motion of dislocations in the region DD'.

528

V. B.

PARIISKII

When the amplitude of the pulse was high enough repeated etching revealed some areas differing by the character of dislocation motion out of the rosette near the indentation (Fig. 4b). In the area of low stress (AC and A'C') there was no dislocation motion out of the rosette near the indentation at all (Fig. 4c). As stress increased at first the "correct" (CD and C'D') (Fig. 4d) motion of dislocations began and then the "mixed" one (DD' area) (Fig. 4e). The stress at which the mixed motion of dislocations started decreased with increasing pulse length. In these experiments the pulse length varied approximately by a factor of 100 from 2.6 X 10~ 2 to 2.4 s. The relation between the stress r ' , at which the mixed motion of dislocations begins, and the pulse length appears to be linear in semilogarithmic coordinates, and can be represented in the form t = A exp (— a r') ,

(1)

where A = 1150 s and a = 8.25 X 10~7 d y n - 1 cm 2 . I t should be noted that such a relation between the time of load action and the stress is typical of thermally activated processes [12], The mixed motion of dislocations is due to the generation of dislocation loops of the opposite Burgers vector (opposite direction of by-pass [5]) in the rays of the rosette which, according to Fig. 2, should shorten for the chosen sign of stress. One of the mechanisms for the formation of these loops is double crossslip, which was considered in connection with the motion of rosette rays in [5, 8]. The observed dependence of the character of dislocation motion out of the rosette on the amplitude and time of application of stress does not contradict the mechanism of double cross-slip. 2. Eifcct of Stress State on the Shape of the Dislocation Rosette near the Indentation This study was carried out on K B r , KC1, and NaCl single crystals having different yield points. The samples were annealed for 50 to 100 h at temperatures about 50 ° K below the melting point. The cooling and heating rates were 1 to 2°/min. The dislocation density in annealed samples was 10 3 to 10 4 c m - 2 . The samples having a size of about 4 x 4 X 20 mm 3 were placed in a special device where they were subjected to pure bending stress (Fig. 5a). The device was fixed on the plate of PMT-3 in such a way as to enable indentations on stressed crystals at different loads on the diamond pyramid indentor. The indentations were made across the sample as shown in Fig. 5b, c on the side of stress gradient, the external stress being chosen so that no plastic deformation occurred before the indentation. Then the sample was unloaded and etched. A change of the symmetry of the rosette near the indentation on stressed samples was observed for all substances; schematically this is shown in Fig. 5b. In the contracted area the rays close to the direction of contraction are elongated, whereas in the dilated area the rays close to the direction normal to the direction of dilation increase in length. On the neutral axis all the rays of the rosette have the same length. Fig. 6a and b show the corresponding pictures for a K B r crystal. A similar change of the shape of the rosette near the indentation has been observed for KC1 and NaCl crystals. If the length difference of adjacent edge rays of a rosette is taken to be a measure of the effect the following is noticed:

Shape of the Dislocation Rosette near an Indentation in Alkali Halides

529

Fig. 5. a) Attachment to PMT-3 for four-point bend crystal loading; 1 base, 2 — lever, 3 — bearing, 4 — scale for load3, 5 — block, 6 — string. 7 — crystal, 8 — knives, 9 — cuvette, b) Scheme of intendation arrangement on the crystal. A and B : dilated and contracted surfaces, respectively, c) Epure of normal stress distribution

y -^/f 'W * «

• * f

/

m

- P / m

>

N

20fjw *

j

*

Fig. fi. Micropliotojiraphs of dislocation rosettes near the indentation on a stressed ICBr crystal, a) Dilation, b) contraction. External stress is shown by arrows

1. The length difference is the larger the larger the magnitude of the stress acting in this region of the crystal. 2. At given stress the length difference of the rays decreases with increasing rigidity of the sample (increasing yield point). Fig. 7 shows the dependence of the length difference A of dislocation-rosette rays, obtained for each rosette by averaging over all four rays, on the shear stress r applied to sample surface before the indentation. This dependence is shown for a KC1 single crystal with the critical shear stress r c = 270 p/mm 2 . Curve 1 was obtained for a dilation of the part adjoining the surface A and con-

530

V. B.

PARIISKII

Fig. 7. Dependence of the length differences of the rays of a dislocation rosette (A) on the shear stress (r) applied to sample surface before indentation for a KC1 crystal. A and B: sample surfaces (see Fig. 5b); (1) dilation of the surface A, contraction of the surface B; (2) dilation of the surface B, contraction of the surface A a, b, c: loads of 1, 2, and 5 p, respectively

rn

Fig. 8. Dependence of the length differences of the rays of a dislocation rosette (A) on shear stress (T) applied to the sample surface before indentation for a KBr crystal

traction of t h e p a r t adjoining t h e surface B (see Fig. 5b). Curve 2 was obtained when t h e signs of stress were reversed. The experimental points correspond to indentor loads of 1, 2, and 5 p. F r o m the diagram it is evident t h a t t h e magnitude of t h e indentor load dees not affect this dependence in t h e limits of experimental error. I n this case a linear relation is observed between t h e length difference of t h e r a y s of t h e dislocation rosette A and t h e local shear stress r in t h e range of dilational and small contractional stress. I n t h e range of larger contractional stress this dependence becomes stronger. The similarity of curves 1 and 2 obtained on t h e same sample for different signs of stress shows t h a t the above mentioned effect is not due t o t h e presence of residual elastic stress. To verify this condition t h e sample was examined in t r a n s m i t t e d polarized light before loading. An analogous dependence was found for a K B r crystal, whose yield stress was determined f r o m t h e appearance of t h e first slip-bands (r c = 250 p/mm 2 ) for t h e area of dilation (Fig. 8). I t should be mentioned t h a t internal inhomogeneities in the sample cause some scatter of the experimental points, particularly in t h e presence of internal stress in t h e sample. Therefore, t o get "calibration curves" similar to those of Fig. 7 and 8 carefully annealed undeformed crystals are t o be used. I t is useful t o analyse t h e forces which act on t h e head dislocations in the rays of a dislocation rosette. W h e n t h e sample is indented elastic a n d plastic deformation occurs. Therefore the analysis can only be given in general outline. Two cases will be considered. 1. Diamond-pyramid indentor acting on an unstressed crystal. — I n this case plastic deformation takes place forming a dislocation rosette near t h e inden-

Shape of the Dislocation Rosette near an Indentation in Alkali Halides

531

tation. At the same time an elastic stress field is built up around the pyramid indentor which hinders the further intrusion into the crystal. We can write the equilibrium condition for the head dislocation of the rosette ray under load T1 + T11 = Tc ,

(2)

1

where r is the stress due to plastic deformation, i.e. the superposition of the elastic fields of all dislocations formed, t 1 1 the elastic stress balanced by the external stress, and r c the shear strength. 2 ) When the diamond pyramid is lifted the elastic stress t 1 1 disappears (elastic return of indentation). The stress r 1 is balanced by the lattice resistance that is always lower t h a n or equal to r c . We write the equality (2) in the form r 1 = t c — r 11 . Hence, in order to move the head dislocation out of the rosette after the indentor has been lifted, it is necessary to apply an external stress which exceeds or is equal to r 11 . 2. Diamond-pyramid

indentor

acting on a stressed

crystal.

— I n t h i s case t h e

external stress tends to elongate some, and to shorten other rays of the dislocation rosette as shown in Fig. 1 and 2. The equilibrium condition for the head dislocation will be written as for an elongating ray and as

T} + T? + T = Tc tI + T\1-T

= tc

(3) (4)

for a shortening one. r is the external shear stress. The stress r11 decreases with increasing r, where r is the distance from the point of stress application to a given point in the crystal. From the relations (3) and (4) it is evident t h a t r j 1 < t^ 1 (as x\ f=t! t ' ) and, consequently, r, > r2, where rx is the distance from the centre of indentation to the head dislocation in the elongating and r2 to t h a t in the shortening rays. This is just the circumstance which leads to the change of the shape of dislocation rosettes on stressed crystals. When r = t c the condition of equilibrium (3) is not fulfilled and the rays pass through the whole crystal, which is just observed. When the indentor is lifted and the crystal unloaded, r 1 1 and r go to zero and a rearrangement of stress occurs in accordance with the equalities (3) and (4). From the relations (3) and (4) it is seen that the equilibrium conditions for the head dislocation contain the value r c which in real crystals strongly depends on the quantitative, qualitative, and structural state of impurities. Hence it is evident that the length difference of dislocation-rosette rays near the indentation (A) depends not only on the value of external stress r, but also on r^. Therefore the dependences similar to those shown in Fig. 7 and 8 are different for crystals of the same substance which have different yield stresses. In order to determine the values of internal stress in a crystal from the change of the form of the dislocation rosette it is necessary to obtain a calibration curve for the given substance similar to Fig. 7 and 8. 2 ) Here the term shear strength refers to the stress which is needed to displace the dislocations in the lattice in a definite fixed time of its influence. It is necessary to introduce this limitation because the process of dislocation movement is thermally activated, and the stress that leads to some displacement decreases to the definite value, (to athermal constituent) with increasing time of load application.

532

V . B . PARIISKII

: Shape of the Dislocation Rosette near an Indentation

T h e sensitivity of t h e m e t h o d depends on t h e rigidity of t h e c r y s t a l a n d is a b o u t 10% of t h e yield stress d e t e r m i n e d f r o m t h e a p p e a r a n c e of t h e first slip b a n d s in t h e crystal. This m e t h o d is r a t h e r sensitive f o r s o f t crystals a n d p e r m i t s t o d e t e r m i n e i n t e r n a l stresses of some p / m m 2 . T h e a d v a n t a g e s of t h i s m e t h o d lie in t h e f a c t t h a t it is r a t h e r simple, r a p i d , a n d p e r m i t s t o m e a s u r e t h e local stress in t h e sample. T h e effect observed is t h o u g h t to be useful for t h e d e t e r m i n a t i o n of t h e i n t e r n a l stress in single crystals w h e r e t h e stress-birefringence m e t h o d c a n n o t be used. Acknowledgement

T h e a u t h o r is g r e a t l y i n d e b t e d t o V. I . S t a r t s e v for his continuous a t t e n t i o n t o t h e work a n d S. V. L u b e n e t s for helpful discussions. References and W . G . JOHNSTON, Dislocations and Mechanical Properties of Metals, Izd. innostr. Lit., Moscow 1960 (p. 82), (in Russian).

[ 1 ] J . J . OILMAN

[ 2 ] A . A . PREDVODITELEV, V . N .

ROZHANSKII, a n d V . M . STEPANOVA, K r i s t a l l o g r a f i y a

7,

418 (1962).

and G . F . DOBZHANSKII, Kristallografiya 7 , 1 0 3 ( 1 9 6 2 ) . [4] A. A. SHPUNT, Kristallografiya 7, 474 (1962). [5] W. M. N A D G O R N Y I and A. V. STEPANOV, Fiz. tverd. Tela 5, 1006 (1963). [6] A . A . U R U S O V S K A Y A , R . T H Y A G A R A J A N , and M . V . K L A S S E N - N E K L Y U D O V A , Kristallografiya 8, 625 (1963). [ 7 ] A . A . U R U S O V S K A Y A and R . T H Y A G A R A J A N , Kristallografiya 8 , 5 3 1 ( 1 9 6 4 ) . [ 3 ] M . P . SHASKOLSKAYA

[ 8 ] A . A . PREDVODITELEV, V . X . ROZHANSKII, V . M . STEPANOVA, a n d X . A . [9] [10]

Kristallografiya 9, 695 (1964). N . A . TOROPOV and Yu. P . U D A L O V , Izv. VUZOV SSSR, Fiz. S . V . L U B E N E T S and N . F. K O S T I N , Kristallografiya 7 , 3 2 8

TUMANOVA,

9, 187 (1966). (1962).

[11] V. B . PARIISKII, S. V . LUBENETS, a n d V. I. STARTSEV, F i z . t v e r d . Tela 8, 1227 (1966).

[12] H. G. VAN BUEREN, Defects in Crystals, Izd. innostr. Lit., Moscow 1962, (in Russian). [13] S. TIMOSHENKO, Strength of Materials, Izd. Nauka, Moscow 1965, (in Russian). (Received November 9, 1966)

S. N.

KOMNIK

et al.: Kinetics of Plastic Deformation of Alkali Halides

533

phys. stat. sol. 19, 533 (1967) Subject classification: 10; 22.5.2 Physico-Technical Institute of Low Temperatures of the Ukrainian Academy of Sciences, Kharkov

On the Kinetics of Plastic Deformation of Some Alkali Halide Crystals1) By S . N . K O M N I K , V . Z . BENGTTS, a n d E . D . L Y A K

A study is made of the kinetics of plastic deformation in LiF, NaCl, KCl, and KBr crystals. This includes the non-uniformity of the plastic deformation, the strain rate dependence of this non-uniformity, the strain rate dependence of the flow stress, and the dislocation structure of the deformed crystals. In the above crystals a regular change of the kinetics is found for a decrease of the elastic anisotropy factor A = 2 CM/(Cn — C12). The correlation of the macroscopic behavior of the crystals with their elastic anisotropy is assumed if other conditions are similar. Die Kinetik der plastischen Deformation wird in LiF-, NaCl-, KCl- und KBr-Kristallen untersucht, wobei die Ungleichförmigkeit der plastischen Deformation, ihre Abhängigkeit von der Deformationsgeschwindigkeit, die Abhängigkeit der Flußspannung von der Deformationsgeschwindigkeit und die Versetzungsstruktur der deformierten Kristalle eingeschlossen werden. In den g e n a n n t e n Kristallen wird eine regelmäßige Änderung der Kinetik mit abnehmendem elastischen Anisotropiefaktor A = 2 Cil/(C11 — C12) gefunden. Zwischen makroskopischem Verhalten der Kristalle und ihrer elastischen Anisotropie wird eine Korrelation a n g e n o m m e n , wenn andere Bedingungen ähnlich sind.

1. Introduction It is of great interest to predict the macroscopic behavior of a crystal during plastic deformation in terms of its basic physical constants (elastic, thermal, etc.). However, only properties of individual dislocations may probably be immediately correlated with physical constants of the crystal. But in macroscopic plastic deformation of the crystal a great number of dislocations takes part simultaneously. The macroscopic behavior of a crystal is not so much determined by the properties of individual dislocations (for example, by start stress) as by the behavior of the great ensemble of dislocations. The macroplasticity of a crystal is largely determined by the interaction of dislocations and their multiplication in this ensemble with statistical regularities as well. Therefore one may expect some correlation between the macroscopic behavior of the crystal and the physical constants which essentially determine the interaction of dislocations. As was shown by Chou the elastic interaction of dislocations and the shape of the dislocation elastic stress field are determined by the elastic anisotropy of the crystal [1,2], It seems therefore expedient to look for a correlation between the macroscopic behavior of the crystal during plastic deformation and its elastic anisotropy. 1

) The contents of this article was reported at the Seventh International Congress of Crystallography, Moscow, July 1966. 35 physica

534

S. N. Komnik, V. Z. Bengtjs, and E. D. L y a k

When studying the plastic behavior of different alkali halide crystals differing b y the value of the anisotropy factor A = 2 C' 44 /(C 11 —0 12 ) we found a regular change in their behavior. This is the subject of the present paper. 2. Method The investigations were m a d e with L i F , NaCl, KC1, and K B r crystals having yl-values of 1.82, 0.7, 0.375, and 0.35, respectively. Samples about 4 x 4 x 10 m m 3 in size were obtained by cleavage and deformed b y compression a t room temperature, which implied nearly identical temperature (relative to the melting point) for all the crystals studied. The deformation was carried out in an apparatus of about 2 x 10 3 kp/mm rigidity, the time dependence P(t) of the load P acting on a crystal a t a given strain rate being recorded photographically [3], The accuracy of the force measurement was about 10 p. The experiments were carried out ou fairly pure (with respect to cation impurities) crystals grown from the melt in air by the K y r o p o u l u s method. F o r example, the KC1 crystals contained less than 30 p p m M g 2 + , less than 5 p p m C a 2 + , and much less than 2 p p m B a 2 + . The grown-in dislocation density was 3 X 10 4 to 106 c m - 2 in different crystals. The non-uniformity of plastic deformation, the strain rate dependence of this non-uniformity, the strain r a t e dependence of flow stress, and the dislocation structure of the deformed crystals were studied in this work. 3. Results 3.1 Non-uniformity of plastic and the strain rate dependence

deformation of flow stress

The non-uniformity of plastic deformation during creep was studied by Klassen-Neklyudova in NaCl [4]. The non-uniformity of plastic deformation in KC1 a t constant strain r a t e manifests itself in sudden changes of load with time [5], The original " m i c r o s c o p i c " yield drops in the P(t) curve a p p e a r simultaneously with the creation of the first slip bands (before the macroscopic yield stress) and accompany the whole process of plastic deformation of KC1 crystals. Apparently each drop corresponds to the lateral broadening of existing slip bands or to the appearance of new ones. The drops in the P(t) curve are found [5] to become more pronounced with decreasing strain rate, and the rise before the drop becomes often elastic. This elastic rise indicates the absence of a storage of moving dislocations in the crystal. I t implies t h a t the condition N v is fulfilled, where n is the rate of multiplication of moving dislocations, N the density of moving dislocations, v their velocity, l 0 the sample length in the direction 0

120

2W

360

f f o ) — F i g . 1. Time dependence of load P near the yield point for L i F

535

Kinetics of Plastic Deformation of Some Alkali Halide Crystals

of the normal to the slip plane (l 0 = (I + ®)/2 here, I and a being the length and width of the sample). The appearance of moving dislocations at a certain time instant leads to an immediate elongation which ceases when the dislocations leave the crystal or are pinned in it. A similar behavior is characteristic of all alkali halide crystals. In particular for L i F crystals it is shown in Fig. 1. Considering the shape of a certain drop in the P(t) curve one can evaluate the number and average velocity of dislocations leading to the drop of load. From the equation

+

S" + Hj (2) in which H j denotes an interstitial hydrogen atom. A small band with an apparent maximum at 255 nm appears as a shoulder on the larger 235 nm absorption band [1, 2]. Fischer and Griindig suggest [7] that this band is also due to transitions within the S" ion. We have now investigated the photochemical properties of KC1 + KSH and also of KBr + K S H in greater detail. I t appears t h a t both the primary photochemical process and the subsequent thermal processes which occur on annealing are more complicated than are implied by equation (2). 2. Experimental Procedure Crystals doped with S H - or SI)" ions were prepared by three different methods. (i) Suitably sized cleaves of KC1 crystals (either from Harshaw or grown from Analar KC1 which had been melted under chlorine) were sealed off in silica ampoules under a pressure of H 2 S gas in the range 60 to 160 Torr and annealed at 700 °C for times varying from 2 hours to 5 days. All such crystals showed the S H - absorption band provided the specimen examined was cleaved from the outside of the crystal. This shows that the reaction (1) proceeds rapidly at the surface of the KC1 but that diffusion of S H - into the interior of the crystal is exceedingly slow. To obtain more uniform doping either of the following two methods was employed. (ii) Crystals were grown from the melt containing 0.01 mol% anhydrous KSH. The latter salt was prepared by passing dry H 2 S through a solution of potassium in absolute ethyl alcohol. The first-formed precipitate is a mixture of K 2 S and KSH and this was discarded. On adding absolute ether to the filtrate, a precipitate of KSH forms. This was purified by re-precipitation from alcohol with ether and kept in a desiccator over Si0 2 gel + CaCl2. All operations involved in the preparation and handling of KSH were performed in a dry box under nitrogen. The KC1 was pre-melted under Cl2 to remove OH" ions and then cooled to room temperature before adding the KSH. This is probably the least ambiguous method of doping with SH" ions. (iii) Crystals were grown from the previously chlorinated melt under 0.33 atm pressure of hydrogen (or deuterium) after adding 3 mg of sulphur to the charge. For SD" doping, the chlorinated melt was treated with deuterium at 1 atm pressure at 500 °C and allowed to cool in deuterium. This procedure minimized the possibility of contamination from hydrogen dissolved in the silica container. For KBr method (iii) was used but pre-treatment with Cl3 was omitted. Spectra were run either on a Perkin-Elmer UV137 or on a Cary 14R spectrophotometer. 3. Results 3.1 KCl +

KSH

The absorption spectra obtained with a 0.72 mm thick cleave from a KCl crystal doped with KSH are shown in Fig. 1. The large band at 186 nm (the peak of which is just off scale) is due to SH". This band always shows a long tail, on the long wavelength side, which includes a small peak in the region of

567

Photochemical Properties of SH Ions in KC1 and K B r Crystals

25

250

WO Wavelength (nm)

Fig. 1. Absorption spectra at 97 °K of KC1 + K S H : (1) before irradiation, (2) after 10 min, (3) after 20 min, (4) after 41 min irradiation

0

3

6

9

12

15 18

Time of irradiation jminj

—-

Fig. 2. Kinetics of formation of the absorption bands 193 nm (1), 235 nm (2), 212 nm (3), and 255 nm (4) in irradiated KC1 • K S H . Results from three different runs at 90 °K are included

215 nm. There is no optical evidence for OH~ ion impurity which should have been completely removed by the chlorination procedure. On irradiation with uv light from a mercury low-pressure lamp, the SH" band decreases, and bands develop at 193, 212, 235, and 255 nm. The kinetics of formation of these bands are shown in Fig. 2. For short irradiation times the intensity of the 193 and 235 nm bands are well correlated as are those of the weaker 212 and 255 nm bands. For long irradiations, however, the intensity of the 235 nm band passes through a maximum and then decreases, owing to the thermal instability of the interstitial H atoms. While the 255 and 193 nm bands saturate, the U-band at 212 nm grows steadily and eventually becomes as intense as the 193 nm band. The sensitivity of the U 2 -band at 235 nm to temperature is shown in Fig. 3. The thermal bleaching of this band may be followed in the temperature range 85 to 100 °K. At these temperatures the 212 and 255 nm bands do not bleach, but the S~ band decreases partially in intensity along with the U 2 -band. An F-band is apparent in KC1 + K S H only after very long irradiations lasting several hours. On warming the F-band intensity increases. The kinetics of the bleaching of the U 2 -band was investigated by measuring the absorbance as a function of time at constant temperature. Three possibilities seem likely, namely that the reaction should be unimolecular, bimolecular

Fig. 3. Decrease in absorption coefficient at 235 nm with time on annealing a uv irradiated KSHdoped KC1 crystal at 100 °K 37»

0

50

100

150 Time of anneal (min)-

568

O . E . F A C E Y a n d P . W . M . JACOBS Fig. 4 a . Kinetics of the thermal decay of U 2 -centres (a) plotted according to equation (3), (b) according to equation (4) b. Kinetics of the thermal decay of U 2 -centres plotted according to equation (5)

Time(min) -

Time1/2fmin7/i) -

or diffusion controlled. Let a. denote the fraction of H atoms which have reacted in time t; 1 — a is given by the ratio A(t)IA(0) where A(t) is the absorbance (optical density) at time t. Then for a unimolecular process for a bimolecular process

— In (1 — a) = kx t , (1 - a)" 1 =

kat,

(3) (4)

and for a diffusion controlled process a = k3 i 1 ' 2 . (5) Plots testing these kinetic equations are shown for a typical run in Fig. 4. Only the plot of (1 — a ) - 1 vs. t is linear so that the thermal decay of U 2 -centres is a bimolecular reaction. The rate constants k2 obey the Arrhenius equation, the activation energy being 0.2 eV. 3.2 KCl +

KSD

Absorption spectra of KCl crystals containing K S D are very similar to those for KCl + KSH, and their behaviour on irradiation is analogous. In view of the danger of hydrogen contamination from the quartz, it was necessary to check that the crystal really did contain deuterium. The half-widths of the D~ and Dj bands were therefore measured by assuming their shape to be gaussian and the resulting values are compared in Table 1 with the corresponding values for K S H doped crystals. The deuterium bands are significantly sharper. The kinetics of formation of the four bands at 193, 212, 235, and 255 nm are shown in Fig. 5. These data show (i) the good reproducibility for two runs at the same temperature, (ii) that the 193 and 235 nm bands form at the same rate until the thermal bleaching of the latter becomes important, and (iii) the importance

Photochemical Properties of SH~ Ions in KC1 and K B r Crystals

569

Irradiation time (mm) —-

Table 1 Half-widths in eV for the U- and U 2 bands in KC1

Hydrogen Deuterium

U-band

U £ -band

0.35 0.27

0.39 0.30

of the U-band, which continues to grow after the 255 and 193 nm bands have saturated. This is also true for K S H doped crystals. At the doping levels used the absorption coefficient of the 235 nm depended on thickness unless thin crystals 0.6 mm were used. This is because the intense absorption by SD~ ions causes so much attenuation of the light that the deuterium atoms are not produced at uniform concentration throughout the whole width of the crystal. The kinetics of decay of the U 2 -band again obey equation (4). The rate constants are slightly smaller than for the corresponding hydrogen band as would be expected for the heavier deuterium atoms, and the activation energy is slightly larger (m 0.25 eV). 3.3 KBr +

KSH

The K B r + K S H crystals display a large absorption band peaking at 194 nm at 80 ° K . As in KC1 there is a pronounced tail on the long wavelength side which can sometimes be resolved into a shoulder at 230 nm. Attempts to fit this SH~ band by an equation of gaussian form were not successful. The major part of the band is gaussian with a full width at half-height of 0.28 eV, but the long wavelength side cannot be fitted completely by either a single gaussian or by a superposition of gaussian bands. It was not possible to do a detailed study of the temperature variation of the half-width for above 123 ° K the SH~ band disappears into the K B r band edge. On irradiation the 194 nm SH~ band decreases and new bands form at 204 and 300 nm. The hydrogen bands also appear: the U-band at 225 nm and the U 2 -band at 270 nm. The latter is somewhat less prominent than in KC1 because the band bleaches thermally more easily than in KC1, and is hardly apparent

570

0 . E . T A C E Y a n d P . W . M . JACOBS Fig. 6. Kinetics of decomposition of SH~ ions in K B r shown by plotting the change in absorption coefficient at various wavelengths. ( 1 ) U-band formation, ( 2 ) S H " decomposition, (3) 204 nm band, ( 4 ) U 2 band, (5) F-band, (6) F-band in undoped K B r , (7) 300 nm band

Time(min)-

350 Wavelength (nm) -

F i g . 8. Changes in the absorption spectrum of K B r + K S H on irradiation and annealing. ( 1 ) Unirradiated crystal at 88 (3) after 90 min uv irradiation at 88 ° K , (2) after overnight warm-up and recooling to 90 K

Photochemical Properties of SH~ Ions in KC1 and KBr Crystals

500

571

550 Wavelength (nm) -

\/Ti W-E(eV)

150

300

350

Wavelength (nmj Fig. 9. Changes in the absorption spectrum of KC1 + KSH on irradiation and annealing. (1) Unirradiated crystal at 85 °K, (2) after 2 h uv irradiation at 85 °K, (3) after 1 day at room temperature and recooling to 85 °K

Fig. 10. Long wavelength side of the S" band in KC1 at liquid helium temperature

a t t e m p e r a t u r e s > 90 °K. T h e F - b a n d a t 610 n m is also f o r m e d in v a r i a b l e a m o u n t . T h e f o r m a t i o n of a n F - b a n d in u n d o p e d K B r was also observed (Fig. 6), b u t it is of smaller i n t e n s i t y t h a n in K B r + K S H . T h u s , F - c e n t r e f o r m a t i o n in t h e l a t t e r is a genuine effect. N o U- or U a -centre f o r m a t i o n was d e t e c t e d in t h e u n d o p e d K B r , b u t it r e m a i n s a n open question as t o w h e t h e r or n o t t h e F centre f o r m a t i o n is d u e t o a small residual O H - i m p u r i t y . T h e d e c a y of t h e S H b a n d a n d t h e g r o w t h of t h e 204, 300, a n d 610 n m b a n d s are shown in F i g . 6. T h e kinetics of decomposition of t h e SH~ centres follow a simple f i r s t order law (Fig. 7), i.e. t h e r a t e of photochemical decomposition is simply p r o p o r t i o n a l t o t h e a m o u n t u n d e c o m p o s e d . T h e r a t e c o n s t a n t A; increases slightly w i t h t e m p e r a t u r e , t h i s increase corresponding t o a t h e r m a l a c t i v a t i o n energy of 300 cal/mol. T h e r m a l bleaching in i r r a d i a t e d K B r + K S H is shown in Fig. 8. On w a r m i n g overnight t h e U 2 - b a n d bleaches a n d t h e 204 n m b a n d a l m o s t d i s a p p e a r s . T h e recovery of t h e SH~ b a n d a t 193 n m is, however, incomplete while t h e U - a n d F b a n d s show o n l y a n a p p r o x i m a t e l y 5 0 % bleach u n d e r t h e s e conditions. T h i s beh a v i o u r of t h e F - b a n d c o n t r a s t s w i t h t h a t in KC1 + K S H for t h e r e t h e F - b a n d a t first increases on w a r m i n g (Fig. 9). This figure also shows t h e slowness of t h e U - b a n d bleach c o m p a r e d t o t h a t of t h e U 2 - b a n d a n d t h e e x t e n t of t h e S " bleach which accompanies t h e d e c a y of t h e U a - b a n d . 3.4 Band

shapes

T h e 193 n m S~ b a n d a n d t h e 186 n m SH~ b a n d in KC1 overlap t o s u c h a n e x t e n t t h a t a complete analysis of t h e b a n d s is impossible. B y m a k i n g r e a s o n a b l e s u b t r a c t i o n s f o r SH~ b a c k g r o u n d , however, we h a v e been able t o a n a l y z e t h e leading edge (long w a v e l e n g t h side) of t h e S " b a n d . Fig. 10 shows a plot of [In ( A ^ J A ) ] 1 ! 2 , w h e r e A is t h e a b s o r b a n c e a t a p h o t o n e n e r g y E , a g a i n s t E . A r e a s o n a b l y good s t r a i g h t line is o b t a i n e d f r o m which one f i n d s a h a l f - w i d t h W of 0.173 eV. This a c t u a l l y represents twice t h e h a l f - w i d t h on t h e low-energy side, i.e. t h e f u l l w i d t h if t h e b a n d is s y m m e t r i c . T h e low-energy side of t h e

572

0 . E . F A C E Y a n d P . W . M . JACOBS

0.5

Fig. 11. Temperature dependence of the half-width of the 193 nm band in KCl. O and • iefer to two different runs; • helium run, + Fischer and Gründig

% iOJ 0.2 0.1 11

8

(Temperature) 1/2 CK)

,„

, 1.0

Fig. 12. Temperature dependence of halfwidth of 255 nm band. Symbols as for Fig. 11

(Temperature)" 2(°K) y2~

255 nm band was analyzed similarly and the temperature dependences of W for the two bands are shown in Fig. 11 and 12, respectively. Because of the difficulties involved in the analysis of overlapping bands high accuracy is not expected. Nevertheless, both sets of data show the anticipated T 1! 2 dependence at high temperatures i.e. > 80 °K. Both graphs show data obtained with four different crystals, namely two runs covering the temperature range 80 to 230 °K, a single point at liquid helium temperature and the liquid nitrogen point from the data of Fischer and Griindig [7], The absorbance of the two bands was measured on different cleaves above 80 ° K so as to allow increased accuracy by using longer irradiations when making measurements on the 255 nm band. The continuous lines in Fig. 11 and 12 were calculated from the equation W{T)

=

PF(0) [coth

( h v J k T ) y i

2

,

(6)

where W(T) is the half-width at temperature T and v„ the vibrational frequency in the ground state. The values of vg used were 1.98 X 10 12 s _ 1 for the S " band and 2.09 X l 0 1 2 s" 1 for the 255 nm band. The dotted line in Fig. 12 was calculated assuming no error in the helium point. The corresponding frequency is 2.99 xlO 1 2 s" 1 . 4. Discussion Irradiation of KC1 containing SH~~ ions with uv light produces bands at 193, 212, 235, 255 nm together with a weak F-band which is only observed for long irradiation times. K B r behaves similarly, the positions of the four uv bands being 204, 226, 273, and 300 nm. The F-band is more prominent in K B r than in KC1. Four of these eight bands are the well-known hydrogen bands due to interstitial hydrogen atoms (235 and 273 nm) and hydride ions on lattice sites (212

Photochemical Properties of SH~ Ions in KC1 and KBr Crystals

573

a n d 226 n m ) . T h e kinetics of f o r m a t i o n (Fig. 2, 5) a n d t h e bleaching experim e n t s show t h a t 193 a n d 235 n m b a n d s a r e well correlated in KC1. Similarly t h e 204 a n d 273 n m b a n d s are correlated in K B r . T h e decrease in t h e S H a b s o r p t i o n w h e n t h e s e b a n d s f o r m a n d t h e increase in t h e SH~ b a n d w h e n t h e y decay m a k e s t h e a s s i g n m e n t of t h e 193 n m b a n d in KC1 a n d t h e 204 n m b a n d in K B r t o S" ions on n o r m a l anion sites certain [7]. T h u s one p h o t o c h e m i c a l r e a c t i o n is S H - + h v -> SH~* S" + H, . (7) A t t e m p e r a t u r e s above 85 °K t h e hydrogen a t o m s are mobile a n d t h e U a - b a n d bleaches. T h e kinetics of t h i s process indicate a bimolecular reaction. Since t h e initial c o n c e n t r a t i o n s of S~ a n d H i are equal, e q u a t i o n (4) does n o t distinguish between H ; + Hi - Hg (8) and H , + S~ ->• S H - . (9) T h e r e is a t h i r d possible r e a c t i o n Hi + S H " -» H 8 S - ,

(10)

since H a S~ has been identified b y H a u s m a n n [8] in KC1 a n d K B r using E S R techniques. T h e H a S - molecular ion f o r m s r a p i d l y a t 110 °K b u t is u n s t a b l e a b o v e 150 °K. R e a c t i o n (10) would display unimolecular kinetics as t h e SH~ was p r e s e n t in large excess in our experiments. W e conclude t h e r e f o r e t h a t (9) is t h e m a j o r process w i t h some c o n t r i b u t i o n f r o m (10) which is, however, insufficient in a m o u n t t o u p s e t t h e p r e d o m i n a n t l y bimolecular kinetics. T h e H 2 S~ ion t h u s a c t s as a t r a p for some of the H a t o m s a n d a t higher t e m p e r a t u r e s these are given off again a n d r e a c t i o n (9) concluded. This discovery of H a S~ resolves a n a p p a r e n t a n o m a l y in t h e d a t a . W h e n H a t o m s a n n e a l o u t a t 100 °K t h e S~ b a n d only bleaches p a r t i a l l y a l t h o u g h t h e U 3 -band will be 9 0 % bleached in t h e t i m e of t h e e x p e r i m e n t . H o w e v e r , when an i r r a d i a t e d K B r + K S H c r y s t a l is allowed t o w a r m u p slowly over a period of several h o u r s t h e n t h e S " bleaches completely along w i t h t h e U 2 -band (Fig. 8). This means t h a t reaction (8) m u s t occur t o a v e r y limited e x t e n t if a t all in K B r , since t h e reaction H 2 + S SH~ + H; would require a good deal of t h e r m a l energy t o split t h e s t r o n g H - H b o n d . T h e beh a v i o u r of KC1 + K S H is a little different in t h a t t h e S - b a n d bleach is incomplete even on w a r m i n g t o r o o m t e m p e r a t u r e (Fig. 9) a l t h o u g h it is m o r e extensive t h a n a t 100 °K. T h u s t h e f o r m a t i o n of molecular hydrogen, reaction (8), occurs t o some e x t e n t in KC1. Fig. 8 shows t h a t t h e r e c o v e r y of t h e S H " is incomplete w h e n t h e S" b a n d has bleached. This f a c t , t o g e t h e r w i t h t h e absence of t h e U - b a n d in i r r a d i a t e d u n d o p e d K B r , shows t h a t t h e U-centres come f r o m t h e S H - ions a n d n o t f r o m other impurities. T h u s besides t h e reaction (7) t h e r e m u s t be a t least one m o r e photochemical r e a c t i o n : SHS H * -»• H _ + Sj . (11) There is n o direct evidence or even indirect (optical) evidence for interstitial sulphur a t o m s b u t , since lattice h y d r i d e ions are f o r m e d , t h a t is t h e only other possibility. If we assume t h a t t h e S H - ions are excited b y t h e a b s o r p t i o n of 185 n m p h o t o n s t o S H - * a n d t h a t t h e S - H b o n d is b r o k e n in t h e excited s t a t e , t h e n t h e ejection of a h y d r o g e n a t o m i n t o a n interstitial position seems m u c h

574

0 . E . FACEY a n d P . W . M .

JACOBS

more likely than the ejection of a sulphur atom because of the greater mass of the latter. We thus reject this mechanism of bond fission in the excited state implied by (7) and (11) and consider two other possibilities. Both these are similar in principle to the two main theories of F-centre formation in alkali halides. 4.1 Ionization

mechanism

If the excited SET * state lies either in or just below the conduction band, then the primary photochemical step, possibly with the assistance of a small amount of thermal energy, will be the formation of an SH radical and an electron in the conduction band. The neutral SH centre no longer has the benefit of the coulombic energy to keep it on an anion site and therefore moves easily into an interstitial position. The electron is then trapped by the resulting anion vacancy forming an F-centre with the SH radical close by. This is an unstable situation with respect to the formation of either S - o r H ~ i o n s , so bond fission in the radical and the movement back into the anion site of either the H or S atoms occurs. Clearly the activation energy for this would be much smaller for the H atom and so S - ions are formed initially at a greater rate than H - ions. This process may be summarized: -

or

SH- + h v - * S H - *

91'T

SH + e SHi + F

SH; + e + R kT

&7t

SH; + F ^ S " + Hj

H " + Si .

(12)

IcT signifies the need for thermal energy. In K B r it was possible to measure the temperature dependence of the photochemical reaction though admittedly over a rather limited range. The thermal activation energy required turned out to be slight, only « 0.013 eV or hardly more than kT at the temperatures under consideration. The annealing experiments show that the interstitial sulphur atoms move very slowly indeed even at room temperature (e.g. the slow bleaching of the U-band in Fig. 8). The annealing of the U 2 -band is orders of magnitude more rapid, but even this requires an activation energy of 0.2 eV. Thus it seems that the ionization mechanism cannot be sustained because of the thermal energy needed to move the interstitial back into the anion site. However, this argument may not be quite correct since the activation energies may refer to the migration of the species from one interstitial site to the next whereas in (12) the moving atom benefits from the recombination energy with the electron. The main weakness in (12) seems to be that the energy available from the formation of the F-centre is simply dissipated instead of being usefully employed. This is avoided in the mechanism (ii). 4.2 Radiationless

transition

mechanism

Pooley [9] has shown the energy available from radiationless electron-hole recombination may be sufficient to result in the formation of a negative ion vacancy and a negative ion interstitial. The electron is first trapped in an excited state by a V K centre and a radiationless transition from the excited state to the ground state results in a [110] replacement sequence. We consider that a similar mechanism may be operative in KC1 and K B r containing S H - ions. Photon absorption results in a bound excited state and a radiationless transition makes available nearly all the energy of the original quantum. The violent vibrations

Photochemical Properties of SH~ Ions in KC1 and KBr Crystals Fig. S + E2 In

575

13. Schematic representation of the energy of the H + e system. In KC1 E = 6.68 eV, E1 =•= 0.013 eV, and E3 are unknown, Et = 0.2 eV, and Es • • Ex. KBr E = 6.08 eV and other values are unknown. (a) SH", (b) SH"», (c) Si + H", (d) S" + Hi

of t h e S H " ion are now sufficient t o split t h e S - H bond with consequent ejection of either t h e S or t h e H a t o m into an interstitial position. Schematic curves showing t h e energy of t h e S + H + e system in various possible configurations are given in Fig. 13. Only E, Ex, and Ei are known a t present f r o m experiment a n d t h e energy diagram is t h u s highly schematic. Nevertheless, we believe it represents t h e main features of t h e system in a qualitative m a n n e r . Precise computations would be r a t h e r difficult t o achieve. I t seems unlikely t h a t F-cent r e s are formed as p a r t of t h e p r i m a r y photochemical reaction if mechanism (ii) is valid. However, once U-centres are formed t h e n absorption within t h e Ub a n d would lead t o t h e reactions H" -

H + Hr ,

(13)

H r ^ R e + H,.

(14)

The increase in t h e F - b a n d on warming irradiated K.C1 ; KSII is difficult t o explain. The same phenomenon occurs in KC1 + K O H where Kerkhoff [10] has suggested t h a t it results from t h e reaction 2 0"

02 + 2 e + 2 F j .

(15)

Neither 0~ nor S~ ions would have t h e requisite mobility for this process, however. 2 ) T h e continued growth in t h e U-band after t h e S~ b a n d s a t u r a t e s (Fig. 5, 6) requires comment. Because of t h e mobility of t h e interstitial H atoms at the irradiation t e m p e r a t u r e , a steady state is set u p in which t h e back reaction (9) just balances t h e f o r w a r d reaction (7). The II i a t o m concentration actually decreases because of t h e operation of two f u r t h e r reactions, namely (10) a n d H, + F

U .

(16)

However, t h e S; atoms are f a r less mobile t h a n H ; and so, t h e back reaction being much slower, t h e U-band continues t o grow. The assignment of t h e 255 nm band in KC1 a n d t h e 300 n m b a n d in K B r cannot be m a d e with certainty. The bands form a n d decay along with t h e respective S" bands and so could be due t o transitions in t h e S" centre. The similarity in t h e ground s t a t e local mode frequenices determined for t h e t w o bands t e n d t o s u p p o r t this conclusion. The possibility t h a t t h e 225 and 300 n m bands be due t o charge t r a n s f e r transitions from CI" t o interstitial sulphur atoms has 2 ) A possible mechanism is the movement of H atoms from hydride ions to S~ ions located on near-neighbour sites: S~ + H~ -> SH~ + F.

576

0. E. FACEY and P. W. M. JACOBS: Photochemical Properties of SH Ions

not been overlooked. However, this is disproved by the kinetics which show the growth in the U-band after the 255 and 310 nm bands saturate (Fig. 5, 6). The growth curves would rather, as stated above, tend to support Fischer and Griindig's assignment [7]. Unfortunately it was not possible to determine the ground state vibrational frequency with sufficient accuracy to decide whether or not the two transitions have a common ground state. The two continuous lines in Fig. 10 und 11 both correspond to 2 X 10 12 s - 1 , but a frequency of 3 X 10 12 s - 1 would give almost as good a fit for the 255 nm band for which the experimental errors are greater. Acknowledgement We are indebted to Dr. A. Runciman for the use of a liquid helium cryostatReferences [1] U . M. GRASSANO and P . W . M. JACOBS, J . appl. P h y s . 35, 2 3 9 1 (1964). [2] P . W . M. JACOBS and H . A. PAPAZIAN, P h y s . R e v . 1 2 7 , 1567 (1962).

[3] H. THOMAS, Ann. Phys. (Germany) (5) 38, 601 (1940).

[4] C. J . DELBECQ, B . SMALLER, and P . H . YUSTER, P h y s . R e v . 1 0 4 , 5 9 9 ( 1 9 5 6 ) . [5] F . KERKHOFF, W . MARTIENSSEN, a n d W . SANDER, Z. P h y s . 1 7 3 , 184 ( 1 9 6 3 ) .

[6] J. ROLFE, Appl. Phys. Letters 6, 66 (1965). [7] F. FISCHER and H. GRUNDIG, Phys. Letters 13, 113 (1964); Z. Phys. 184, 299 (1965). [8] A . HAUSMANN, Z. P h y s . 1 9 2 , 3 1 3 ( 1 9 6 6 ) .

[9] D. POOLEY, Solid State Commun. 8, 241 (1965). [10] F. KERKHOFF, Z. Phys. 158, 595 (1960). (Received

October 28,

1966)

A . R . BEATTIE a n d G . SMITH: R e c o m b i n a t i o n in S e m i c o n d u c t o r s

577

p h y s . stat, sol. 19, 577 (1967) Subject classification: 13.1; 16; 22.2.1; 22.2.3 Department

of Applied

Mathematics

University

and Mathematical

College,

Physics,

Cardiff

Recombination in Semiconductors by a Light Hole Auger Transition By A . R . BEATTIE a n d G. SMITH

T h e transition rates for an Auger collision process in semiconductors which involves t h e light hole b a n d is calculated using a q u a n t u m mechanical p e r t u r b a t i o n m e t h o d . Spherical energy surfaces a r e assumed although non parabolic energy b a n d s are allowed for. T h e t e m p e r a t u r e dependence of t h e lifetime of excess carriers due t o this process is investigated a n d t h e results applied t o I n S b a n d InAs in t h e t e m p e r a t u r e r a n g e 200 t o 500 °K. T h e s h a p e of t h e t e m p e r a t u r e dependence of this theoretical lifetime f o r I n S b agrees well w i t h e x p e r i m e n t a t room t e m p e r a t u r e a n d above, a n d when estimates of overlap p a r a m e t e r s which occur in t h e t h e o r y are m a d e t h e absolute m a g n i t u d e of t h e lifetime also agrees w i t h e x p e r i m e n t . T h e probability per u n i t t i m e t h a t a light hole created b y a p h o t o n of energy h v will t a k e p a r t in a n i m p a c t ionizing transition is also given as a f u n c t i o n of h v. I t is concluded t h a t t h e transition r a t e for this process is a t least comparable t o t h a t of t h e more u s u a l Auger transitions involving t h e h e a v y hole a n d conduction b a n d s only. Die Ü b e r g a n g s r a t e n f ü r einen Augerkollisionsprozeß in H a l b l e i t e r n w e r d e n u n t e r E i n beziehung der Energiebänder leichter Löcher m i t einer q u a n t e n m e c h a n i s c h e n Störungsr e c h n u n g berechnet. E s werden sphärische Energieflächen a n g e n o m m e n , obwohl a u c h nichtparabolische E n e r g i e b ä n d e r e r l a u b t sind. Die T e m p e r a t u r a b h ä n g i g k e i t der Lebensd a u e r der d u r c h diesen Prozeß erzeugten überschüssigen L a d u n g s t r ä g e r wird u n t e r s u c h t u n d die Ergebnisse w e r d e n auf I n S b u n d InAs im T e m p e r a t u r b e r e i c h 200 bis 500 ° K angewendet. Die T e m p e r a t u r a b h ä n g i g k e i t dieser theoretischen L e b e n s d a u e r s t i m m t f ü r I n S b bei Z i m m e r t e m p e r a t u r u n d d a r ü b e r g u t m i t dem E x p e r i m e n t überein. W e n n die Überlappungsp a r a m e t e r der b e n u t z t e n Theorie abgeschätzt werden, s t i m m t die absolute Größe der Lebensd a u e r ebenfalls m i t d e m E x p e r i m e n t überein. Die Wahrscheinlichkeit pro Zeiteinheit, d a ß ein leichtes Loch, d u r c h ein P h o t o n der Energie h v erzeugt, a n einem Stoßionisationsüberg a n g t e i l n i m m t , wird als F u n k t i o n von h v gegeben. E s wird geschlossen, d a ß die Übergangsr a t e f ü r diesen P r o z e ß mindestens vergleichbar ist m i t d e m gebräuchlicheren Augerübergang, a n d e m n u r d a s B a n d der schweren Löcher u n d d a s L e i t u n g s b a n d beteiligt sind.

1. Introduction If the conduction, heavy hole, and light hole bands in semiconductors are considered, there are ten Auger transitions which lead to recombination, or, if the reverse processes are considered, to impact ionization [1]. A theory for the two transitions involving the conduction and heavy hole bands only has been developed by Beattie and Landsberg [2, 3] who applied it to the problem of recombination of excess carriers. The theory for the transitions shown in Fig. 1 is now developed. In Fig. l a two electrons in the heavy hole band interact via their Coulomb repulsion so that one electron drops into a hole in the light hole band while the other goes into the conduction band, conserving energy and

578

A . R . B E A T T I E a n d G . SMITH

£

E

•H

a

b

Fig. 1. Auger transitions involving the light hole band for a) recombination and b) impact ionization are shown. Curves C, H, L represent the conduction, heavy hole, and light hole bands, respectively. The initial and final states are 1, 2 and 1', 2', respectively

momentum in the process. Fig. l b shows the reverse of Fig. l a and leads to recombination. The processes of Fig. 1 have been considered because their possible importance has been suggested by measurements of the quantum efficiency of incident photons in InSb [4, 5]. When Beattie [1] examined the threshold energies of all ten possible impact ionization transitions, he found that only two would be likely to cause a quantum efficiency greater than unity in the energy range of interest. These were the electron collision process of Beattie and Landsberg and the transition of Fig. l a . In Tauc's [4] measurement of the quantum efficiency in InSb as a function of photon energy the two threshold energies which appeared agreed well with the theoretical values and, although the experimental error was such that there was some doubt about the existence of the second threshold (Antoncik [6]), its presence has since been confirmed by Nasledov [5], Although the threshold photon energies are different for the two processes the threshold kinetic energies of the ionizing carriers are almost the same (1.069 X energy gap for the Beattie-Landsberg transition and 1.073 X energy gap for the present transition). Thus, if the transition of Fig. l a is important in determining the quantum efficiency, then, the transitions of Fig. 1 should also be important in the recombination of excess carriers. The calculation of recombination lifetimes is preferred to a direct calculation of quantum efficiency which entails not only considering the impact ionization transitions but also all other energy loss mechanisms. In calculating the transition probability in Section 2 a quantum mechanical perturbation treatment [2] is followed. The net transition rate, the difference between the recombination and impact ionization rates, is found for small departures from equilibrium by integrating over all allowed states the transition probability multiplied by the appropriate probabilities of occupancy and vacancy of the initial and final states. The overlap integrals which occur in the theory are discussed in Section 3 where their functional dependence on fc-vectors is given for small wave vector differences, and estimates of an overlap parameter are obtained by three methods which, in the case of InSb, agree within an order

579

Recombination in Semiconductors by a Light Hole Auger Transition

of magnitude. I n t h e numerical work of Section 4, where t h e t h e o r y is used to evaluate intrinsic carrier lifetimes in InSb a n d InAs a n d t o give t h e probability per unit time t h a t a light hole of energy E will impact ionize, it is assumed t h a t transitions between light and heavy holes are sufficiently rapid t o permit the use of the same quasi-Fermi level for these two bands.

2. Transition Rates and Lifetimes I n calculating t h e net transition r a t e for t h e processes of Fig. 1 Bloch wave functions are used in t h e q u a n t u m mechanical p e r t u r b a t i o n t r e a t m e n t of [2]. U m k l a p p processes are neglected a n d the assumptions of spherical energy surfaces, quasi-Fermi levels a n d non-degenerate semiconductors are made. However, t h e possibility of non-parabolic energy bands is t a k e n i n t o account. The net transition r a t e per unit volume follows directly f r o m equation (3.6) of [2] a n d is given b y

t- (8FC +8Fl-2 8FK) J J f 0(1, 1', 2, )2'X ^(1,1') F(2,2') F (1, 2') F(2,2 1') F{ 1, I') F(2, 2') F( 1, 2') F(2, 1') X A + f A +1 A + 1 A + ; 1 — cos X , X dfcj dk2 dfc2 ,

R=

e4

8 Ä2 £2

2

2

2

2

2

2

X

(1)

where e is the electronic charge, t is time, h is Dirac's constant, e is t h e dielectric constant, 8Ft is t h e d e p a r t u r e of t h e quasi-Fermi level f r o m its equilibrium value lor the i - t h band, subscripts C, H, L refer t o the conduction, heavy hole, and light hole bands, 6 (1, 1', 2, 2') is t h e appropriate probability of occupancy or vacancy of states 1, 1', 2, 2', g = kl — k[, I = fc2 — k[, where

x=±\E1 + E,-E'1-E'a\

and where and

(2a)

F(i, l')=f «£(*,, r) uc(kv, r) d r F(i, 2') = / u£(kt, r) w (fc -, *•) d r L

2

(2b)

are overlap integrals of t h e modulating p a r t s of Bloch functions. The neglect of the screening parameter 1 in (1) results in a slight overestimate of t h e t r a n sition r a t e . This effect, however, is small since t h e screening length in nondegenerate semiconductors is large and t h e correction t o t h e Auger lifetime due t o screening was found in [2] t o be no more t h a n 5 % . The overlap functions which are discussed in Section 3 are t a k e n t o be of the form 2

\F(i, j)I = cLtj I kt - WHEt - E}) ,

where J?f is t h e energy of the t-th state a n d ay is a p a r a m e t e r . Since t h e factors are slowly varying in comparison with the rest of t h e integrand t h e y may be removed from t h e integral of (1) a n d evaluated for t h e most probable transition. F o r the most probable transition k1 = lt? (see [1]), a n d hence

(3)

A. R. Beattie and G. Smith

580

IE'l\ ,

then mL = m L 0 {1 + (Ey -

EL)IEg}

.

(30)

With the use of (29) or (30) the threshold energy is calculated from (11) and the lifetime is found from (15). The effective mass TO* is obtained from (27 a) by finding the second derivative of the energy with respect to k. Since the conduction, heavy hole, light hole, and split-off bands only are included in the sum rules determination of a { j, the correction term S(E^) is also omitted in finding TO* for InAs. The material parameters which were used are summarized in Table 1, and the values of the overlap parameters corresponding to the threshold transition are given in Table 2. The total optical absorption near the band edge includes not 38*

584

A . R . B E A T T I E a n d G . SMITH

Fig. 2. Lifetimes for intrinsic I n S b based on the theory of Section 2 are plotted as functions of inverse t e m p e r a t u r e . I n a) where the lifetimes are multiplied b y t h e overlap parameters, curve (a) is based on the assumption of parabolic bands, curve (b) represents the a p p r o x i m a t i o n to the non-parabolic case, a n d curve (c) gives the result using t h e computed integral. I n b) the overlap p a r a m e t e r s have been estimated by (a) sum rules, (b) optical absorption, a n d (c) b a n d s t r u c t u r e calculations. Curve (d) shows t h e experimental lifetimes f r o m Zitter et al. [8]

Fig. 3. Theoretical lifetimes multiplied b y t h e overlap parameters for InAs are plotted as f u n c t i o n s of inverse t e m p e r a ture. The donors a n d acceptors are assumed to be completely ionized

IntrinsicND-NA-I016cm"J

Fig. 4. The probability per u n i t time t h a t a light hole of wave vector k will impact ionize is plotted as a f u n c t i o n of t h e p h o t o n energy required to produce this light hole

\

-NA-ND=1016cm3

_L 2.5

I 3

L_ 3.5

4

45

5

0.5 0.6

0.7

0.8

03

10 1.1 17 hv(eV)

Recombination in Semiconductors b y a Light Hole Auger Transition

585

Table 1 Material parameters for InSb and InAs Parameter

InAs

InSb (0.24-0.26x10" T) eV 1.073 E 0.0114 TO 0.18 m 0.0117 TO 16.8 4.1

Ef TOCO «H0 tolo E

n

(0.426-0.22 xlO" 3 T) eV 1.303 Eg 0.026 to 0.4 m 0.025 to 10.9

Table 2 Overlap parameters for the threshold transition S u m rules

Optical absorption

Direct evaluation from wave function

InSb « C E «HL 1, t h e occurrence of localized

On the Theory of Surface and Impurity States

593

states depends on the magnitude of y

2 0 [ 1 - ( 1 +*)»] These results are equivalent to the conclusions derived from the study of the limiting behaviour of the finite chain [6, 21]. For the solutions with outgoing boundary conditions (1 + xf) - y - p (a - a) (1 + a?)»]"1

= [/?(« + 5) (1 = [ 2 $ (1 -

c 2 ) cos

+ |0> . n+0

(4.15)

If \y\ > (1 — c 2 ) 2 ft, the impurity absorption is outside the band (c = 0 is the case studied by Rashba). In this case a = exp (— x) and the wave function decays rapidly in space unless a + y is just outside the band. From the relation PGP = (E-PH P ) " 1 + P W Q G Q W+ P and using the fact that the change in the density of states is Im(Tr P G P ) it is obvious that the change in the density of states due to the impurity is compensated by the same amount in the density of the remainder. The impurity density of states vanishes at the band edges, as expected (compare [22 to 24]). The lifetime for the transfer of excitation among sites is unchanged as the phase of the wave operator remains the same. (For the connection between the phase and the lifetime see [9, 25].) The change in the imaginary part of ^Oli^lO^ is thus entirely due to the further delay experienced by the excitation at the impurity which cannot permanently trap the excitation. This is in contrast with the pure chain in which there is no delay due to the occupation of the site. The phenomenon is rather similar to the delay of a plane wave at a point of discontinuity of the potential, and may explain the observation that minute quantities of impurities are enough to change the emission spectra to that of the impurity [26]. Since the phenomenon is exhibited even by molecules rather similar to that of the host crystal, it is perhaps best explained as due to the enhanced probability of the excitation to be found on the site of the impurity. 5. Surface and Chemisorption States Theoretical studies of surface and chemisorption states have generally concentrated on localized states in which the electrons of the foreign atom remain near the surface. I t seems to us that it is by no means clear that these are the most important. Indeed, for surface states it seems that the existence condition

594

R . D . LEVINE a n d A. T. AMOS

for localized states is, in practice, too restrictive to be of general significance, while in the case of chemisorption states, if the bond order between the adsorbed atom and the surface is considered as a measure of the effective surface interaction then simple arguments show that delocalized states can be just as important as the localized ones. Indeed, there are cases when only the delocalized solutions are bonding. In this section, therefore, we shall consider surface and chemisorption states corresponding to propagating solutions inside the band of the unperturbed chain. Following Koutecky we use a tight-binding Hamiltonian for the linear chain with alternating Coulomb and exchange integrals and fi as the unit of energy, i.e. = d„,m{cc

and

=]

+ y(-l)"}

(1

for even n ,

Ix

for odd n .

(5.1)

B y suitable choice of x and y we can use (5.1) to treat several different models. For example, with x = 1 we can apply the results to mixed crystals [27] or deal with correlation effects [28], while with y = 0 we can deal with alternating bond lengths [29], The recurrence relation for the matrix elements Wr = /t\ 10/ of the wave operator is W2n = ( £ + - « - yJ"1 (W2n+1 + a; J f 2 „ - i ) • (5.2) Iterating once: W2n = ( £ + - « - y)-1 (E+-a

+ y)'1 [x (W2n+2

+ J f 2 ( i _ 2 ) + (1 + x2) W2n] . (5.3)

The translational symmetry of the wave operator is obvious from (5.3), so that we can put W2»_i = k W2n

and

W2n+l

= kz W2n .

(5.4)

From (5.3) and (5.4) where

z + z-1 =

X -

1

(a2 -

2± = y t

1

1 - x2 - y2)

(say, = U)

(5.5)

+ (1 - 4 C 7 - 2 ) 1 - ' - ] .

Koutecky investigated the problem subject to V2 > 4. Here we are interested in the complementary case U2 iS 4 which corresponds to propagating solutions inside the band. We also need to know = d . (5.12) Then < — 1|JF|0> = (e — c) _ 1 d and an extra term is present in _T: = (c - c)"i d2 .

(5.13)

Proceeding as in equations (5.8), (5.9), and (5.11) this makes a further contribution to Re k. We are not able, therefore, to explain the selective properties of surfaces as consistency conditions on the parameters of the tight-binding Hamiltonian, since there is no reason to exclude the in-band solutions of the wave equation in the absence of experimental support. Using the extended Hellmann-Feynman theorem [16] it is possible to evaluate the dependence of all relevant expectation values on the matrix elements of Vs or F 3 , in a manner analogous to equation (5.11). The success of similar methods in rationalizing reactivity data in a series of similar organic molecules suggests that it may be possible to correlate data on the adsorption states of a series of similar molecules on the same surface [19]. Acknowledgements

One of us (R.D.L.) would like to thank the Friends of the Hebrew University and the Humanitarian Trust for financial support, and Professor C. A. Coulson, F.R.S., for his hospitality in the Mathematical Institute, Oxford, where most of the work was carried out.

596

R. D. LEVINE and A. T. AMOS : Theory of Surface and Impurity States References

[1] K. M. WATSON, Phys. Rev. 105, 1388 (1957). K . M . WATSON a n d M . H . MITTLEMAN, P h y s . R e v . 1 1 3 , 198 (1959).

[2] J . KOUTECKY, Adv. chem. Phys. 9, 85 (1965). [3] J . CALLAWAY, J. math. Phys. 5, 784 (1964). [4] S. TAKENO, J . chem. Phys. 44, 853 (1966). [ 5 ] Y U . A . IZYNMOV, A d v . P h y s . 1 4 , 5 6 9 ( 1 9 6 5 ) . [ 6 ] K . KATASURA a n d M . INOKUTI, J . P h y s . S o c . J a p a n 1 8 , 1 4 8 6 (1963).

[7] [8] [9] [10]

H. MATSUDA and K. OKADA, Progr. theor. Phys. 34, 539 (1965). E. I. RASHBA, Soviet Phys. - Solid State 4, 2417 (1963). R. D. LEVINE, J . chem. Phys. 44, 2035 (1966). A. MESSIAH, Quantum Mechanics, Part II, North-Holland Publ. Comp. 1962.

[11] G. F . KÖSTER a n d J . C. SLATER, P h y s . Rev. 95, 1167 (1954).

G. F. KOSTER, Phys. Rev. 95, 1436 (1954). [12] B. A. LIPPMANN a n d J . SCHWINGER, P h y s . Rev. 79, 469 (1950). [13] M. GELL-MANK a n d M. L. GOLDBERGER, P h y s . R e v . 91, 398 (1953).

[14] T. B. GRIMLEY, Adv. Catalysis 12, 1 (1960). [15] A. T. AMOS, Modern Quantum Chemistry III, Academic Press, 1965 (p. 265). [16] R. D. LEVINE, Proc. Roy. Soc. A294, 467 (1966). [17] P . O. LOWDIN, J . m a t h . P h y s . 3, 969 (1962). [ 1 8 ] C. A . COULSON a n d H . C. LONGUET-HIGGINS, P r o c . R o y . S o c . A 1 9 1 , 3 9 ( 1 9 4 7 ) .

[19] [20] [21] [22] [23] [24]

R. D. LEVINE, D. P. Thesis, University of Oxford, 1966, and to be published. R. D. LEVINE, Kinetics of Unimolecular Breakdown, VI, to be published. R. E. MERRIFIELD, J . chem. Phys. 38, 920 (1963). P. W. ANDERSON, Phys. Rev. 124, 41 (1961). P. A. WOLFF, Phys. Rev. 124, 1030 (1961). A. M. CLOGSTON, Phys. Rev. 125, 439 (1962).

[ 2 5 ] R . D . LEVINE, J . c h e m . P h y s . 4 4 , 2 0 2 9 (1966).

[26] A. S. DAVYDOV, Soviet Phys. - Uspekhi 7, 145 (1964). [ 2 7 ] A . T . AMOS a n d S . G . DAVISON, P h y s i c a 3 0 , 9 0 5 (1964).

[28] S. G. DAVISON and A. T. AMOS, J . chem. Phys. 43, 2223 (1965). M. TOMASEK, Surface Sei. 4, 471 (1966). [29] J . A. POPLE a n d S. H . WALMSLEY, Mol. P h y s . 5, 15 (1962).

[30] V. HEINE, Phys. Rev. 138, A 1689 (1965). J . D. LEVINE, Phys. Rev. 140, A 586 (1965). (Received

November

11,

1966)

W. FRANZ und P. MIOSGA: Verteilungsfunktion und Driftgeschwindigkeit

597

phys. stat. sol. 19, 597 (1967) Subject Classification: 13; 14.3; 22.1.1 Institut für Theoretische Physik der Universität

Münster

Verteilungsfunktion und Driftgeschwindigkeit warmer Elektronen in Germaniumx) Von W . FRANZ u n d P .

MIOSGA

Das in [1] und [2] beschriebene Verfahren, nach welchem die Verteilungsfunktion heißer Elektronen durch iterative Quadraturen bestimmt wurde, wird für den Fall kleiner elektrischer Felder modifiziert. Die Ergebnisse einer Rechnung für Germanium zeigen bei höheren Temperaturen befriedigende Übereinstimmung mit Messungen. The iterative method described in [1] and [2] is modified for the special case of low electric fields. Numerical results for germanium agree fairly well with measurements made at moderate temperatures.

1. Berechnung der Verteilungsfunktion 2

Das in [1] und [2] ) aufgestellte Gleichungssystem soll im folgenden verwendet werden, u m die ersten Abweichungen vom Ohmschen Verhalten bei kleinen Feldstärken F zu berechnen. I m Gleichungssystem (I, 6) bis (I, 14) ist F ausschließlich in den Parametern Bt von Gleichung (I, 8) enthalten, welche zu F2 proproportional sind. F ü r F = 0 werden die Gleichungen durch die Boltzmannverteilung gelöst, wobei die Größen St(x) und

(1)

worin (2)

die Boltzmannverteilung ist, so normiert, daß CO

/ d z )jx

o

X Q ( X )

=

1

.

Da die Größen S( und o* proportional F2 sind, k a n n man in Gleichung (I, 11) '/j durch Xo ersetzen, so daß m a n hat Si(x)

= od*) Xo(x) •

(3)

Setzt man Gleichung (1) und (3) in Gleichung (I, 6) ein und vernachlässigt ') Auszug aus der Diplomarbeit von P. Miosga, Münster 1965. ) Im folgenden bei Gleichungen zitiert als I, z. B. (I, 6).

2

39 physica

598

W . FRANZ u n d P .

MIOSGA

Größen, die in F vom vierten Grade sind, so ergibt sich (4)

o Aus Gleichung (I, 10) andererseits folgt (beachte, daß 3) - Zij 3= 1

-

& + 3 +

I dyiy

Ui(y)

(y + m3)%o(y)

- 2

(5)

Uv)

3= 1

Mittelt man (4) und (5) über sämtliche Täler, so erkennt man, daß sich die Mittelwerte der Größen t;((x) und at(x) allein durch den Mittelwert der Größen Bt bestimmen, und zwar zu ihm proportional sind. Genauso ergibt sich, daß auch die Abweichung des Wertes er, vom Mittelwert proportional zu der Abweichung der Größe Bt vom Mittelwert ist. Man kann deshalb zur Lösung des Gleichungssystems den folgenden Ansatz machen: U*)

=

(x)+retr

(8) (9)

Die beiden Gleichungssysteme für die Variabeinpaare \ a (v) sind aus (4) und (5) unmittelbar abzulesen. Sie enthalten Feldstärke und Feldrichtung nicht mehr. — Die Konstante C (1> bestimmt sich aus der Forderung, daß die Gesamtzahl der Elektronen durch das Feld nicht verändert wird: Zur Bestimmung von 0 ( 2 ) hat man Gleichung (I, 13) zu benutzen: oc / da: ix (x + «3)X0(Z)[| (x + co3) + f(*)] = 0 . Die numerischen Ergebnisse für die

(11)

zeigt für einige Temperaturen Fig. 1.

Verteilungsfunktion und Driftgeschwindigkeit warmer Elektronen in Ge

39*

599

600

W . FRANZ u n d P . MIOSGA

Aus den Gleichungen (6) und (9) ersieht man, daß sich für ein kaltes Tal (FUe,) die Beiträge von und £(2> zum großen Teil kompensieren, während sie sich für ein heißes Tal (F_L e i) verstärken. 2. Korrekturen an der Driltgeschwindigkeit

Setzt man Gleichung (6) in Gleichung (I, 45) ein und multipliziert mit der Feldstärke, so ergibt sich für die Driftgeschwindigkeit » = T1 ¿ V i

(12)

i=1

mit Vt

=

P0 F • ~ Tili

[1 +

P

± 1 _ 2 mkT °° J d E p z a

F*pm

Y iP (2)] .

+

(13)

Dabei bedeutet ¡dEE^rXo = ~

_

J p(v)

dE Em

zxo

n4)

'

U

i

X o

fiM-M-

d|«

(15)

= 2 J iE E3/2

T

Setzt man (13) in (12) ein und führt die Summation über die Täler explizit aus, so ergibt sich

» = ß0 ((1 + F 2 PW) F+ 2 ( m ' " ] W M 2 (

\ZTO]+

mtr/

F - (Fl, F 3y, J-J)l} . (16) J

Für Germanium ist \2 m x + m t r /

V

'

ß0 ist die Ohmsche Beweglichkeit. Die eigentlich interessanten Größen sind die Komponente von v parallel zum Feld, Vy, und der Winkel (p zwischen Feld und Driftgeschwindigkeit: VU

N [F(r) - 0(r) (1

(14)

the radial solutions then being spherical Bessel functions jt(x r) (with x = |/E) independent of whether A is positive or negative. 1 ) The Green's function can be expanded as G(r,

r')

=

% % [ . A l m V m . j^x r) jv{x I m V m'

r')

+

x biv

Smm.-nt(x r) jt(x

r')]

Yl

m

{r)

Yf.m.(r')

(15)

for

In order to make compatible the expansions (13) and (15), the summation over A and ju in (13) is replaced by a summation over I and m, making use of (10b) and setting ¡j, — ms = m. C/^ and R>.(r) are called C\m and R\{r) with j = 1 , 2 for A > 0, respectively, the m-summation runs over —I

-

for

l ^ m ^ l

—I

j =

< ^ m < L l - 1 for

1,

(16)

7=2.

The spherical harmonics belonging to one spinor are Yf for the upper, Yf l~ l for the lower component. Substituting (15) and (13) in the rewritten form into (7), and then using the condition (TS, equation (16)) of continuity of the wave function, one gets a system of linear equations for the Cjm, which is soluble, if the following determinant equals zero: det

\ A

{

_L

A

A

r)

m x r)

-

711

~ 3i

Lf(r)

r = gj

X

x c ( r i , ; < m s ) | = 0.

(17)

') Since 0(r) is of the character 1 /r 3 that seems impossible, but one most keep in mind that there is no sense in taking r smaller than e.g. the Compton wavelength, due to physical and computational reasons.

606

J. Treusch

Alm] ;-„,' are the structure constants of the spinless case, Lf(r) — (dli\(r)j(lr)jR\(r). V runs from 0 to N, the m'-summation obeying (16), I and m run in the usual sense with Ir^N, j takes the values 1 and 2, indicating whether X < 0 or X > 0, and ms takes the values + 1 / 2 and —1/2, denoting the upper and lower component of the wave function. Thus, by the double-valued indices j and ms, the dimension of the secular determinant is doubled as compared to the spinless case. The C\m for I > N are given by two linear equations corresponding to (TS equation (21)). They couple C}m and Cfm and can easily be set up following equations (20) and (21) of TS. 3. Conclusion

The main results of this paper are 1. that the K K R method — esteemed because of its mathematical rigor and good convergence — is able to include spin-orbit coupling from first principles, 2. that the computational effort is not much larger than that for the spinless calculation, since the same structure constants can be used. To show the connexion to the work of Onodera [6] and Takada [7], we transform our resultant secular equation (17) as follows: Multiplying the (I, m, ms) row by C (l -^-j; m, m s j and adding to it the (/, m + 2 ras, — ms) row times C {l

j; m, —wi-sJ with j = 1 , 2 for ms = + -i-, respectively, one gets an equi-

valent secular equation of the form

with

det \B>,+ I H — m& = m

x (5aa- and find E':P in the form 4 n i .eff E =

1 -

4

71 i

r eff

EW .

(41)

Ef> is the longitudinal component of the electromagnetic wave since it is parallel to the propagation vector N. Since o f f vanishes only for vanishing magnetic field the longitudinal component of the electromagnetic wave is associated with the magnetic field only. 41 physica

630

J . KOLODZIEJCZAK

and

E . KIEKZEK-PECOLD

: Free Carrier Optical Phenomena

References [1] J . KOLODZIEJCZAK and H . STRAMSKA, phys. stat. sol. [2] J. KOLODZIEJCZAK, phys. stat. sol. 19, 231 (1967). (Received

November

21,

17,

1966)

701 (1966).

A. C. MCLAREN e t a l . : T r a n s m i s s i o n E l e c t r o n Microscope S t u d y

631

phys. s t a t . sol. 19, 631 (1967) Subject classification: 10; 4 ; 22.6 j Department

of Physics, Monash University, Victoria (a), and Institute Planetary Physics, University of California, Los Angeles

of Geophysics (b)

and

Transmission Electron Microscope Study of Brazil Twins and Dislacations Experimentally Produced in Natural Quartz*) By A . C . M C L A K E N ( a ) , J . A . R E T C H F O R D ( a ) , D . T . GRIGGS ( b ) , a n d J . M . C H R I S T I E ( b )

Transmission electron microscopy has been used to examine single crystals of n a t u r a l q u a r t z which h a v e been plastically deformed in compression a t high t e m p e r a t u r e (500 t o 700 °C) a n d confining pressure (15 t o 20 kbar) w i t h high shear stress on (0001). I n specim e n s deformed a t 500 °C, t h e only features f o u n d are closely spaced pairs of p l a n a r defects parallel t o {0001}. S t u d y of t h e diffraction contrast fringes reveals t h a t each pair of defects b o u n d s a Brazil t w i n b a n d which has been produced b y t h e high shear stress. A t t h e higher t e m p e r a t u r e there are dislocations in t h e twin b a n d s a n d other regions of t h e crystals with dislocations only. T h e dislocations are almost invariably either pure screw (b = a ) or of a p a r t i c u l a r m i x e d character (predominantly edge, b u t w i t h a 30° screw component). A r r a y s of t h e mixed dislocations are observed which would produce stress fields consistent w i t h t h e stress-optical effects observed b y optical microscopy. Elektronenmikroskopische D u r c h s t r a h l u n g s m e t h o d e n werden zur U n t e r s u c h u n g von n a t ü r l i c h e n Quarzeinkristallen b e n u t z t , die in Kompression bei h o h e n T e m p e r a t u r e n (500 bis 700 °C) im Druckbereich 15 bis 20 k b a r mit hoher S c h u b s p a n n u n g auf (0001) plastisch deformiert wurden. I n P r o b e n , die bei 500 °C v e r f o r m t wurden, sind die einzigen Merkmale eng beieinander liegende P a a r e von ebenen Defekten parallel zu {0001}. E i n e U n t e r s u c h u n g der B e u g u n g s k o n t r a s t b i l d e r zeigt, d a ß jedes D e f e k t p a a r ein Zwillingsband v o m Brazil-Typ begrenzt, das d u r c h die hohe S c h u b s p a n n u n g erzeugt wurde. Bei den höheren T e m p e r a t u ren f i n d e t m a n Versetzungen in den Zwillingsbändern u n d a n d e r e Kristallbereiche, die n u r Versetzungen e n t h a l t e n . Die Versetzungen sind meist gleichbleibend von reinem Schrauben- (b = a 0. Beide Abweichungen können daher bei unseren Messungen vernachlässigt werden. 3.3.5 Die Bedeutung der Spannungsanisotropie bei der Untersuchung der Versetzungsstruktur in plastisch verformten Kristallen In ähnlicher Weise wie aus der reversiblen Suszeptibilität im Gebiet der Einmündung in die Sättigung [10] kann aus der absoluten Größe der Spannungsanisotropie Ka eine Aussage über die Versetzungsanordnung in plastisch verformten Einkristallen gewonnen werden, da die Größen c (Ai00, A m ) und C' empfindlich von der Versetzungsanordnung abhängen. Beide Methoden geben im Detail aus den folgenden zwei Gründen etwas voneinander verschiedene Aussagen. Einmal wird die Wechselwirkung der Magnetisierung mit den Versetzungen bei ein und demselben Kristall in zwei verschiedenen Richtungen gemessen; dies sind in kleinen Feldern die -Richtungen, und bei der Einmündung in die Sättigung ist es die Feldrichtung. Zum anderen ist für beide Fälle eine verschieden große Austauschlänge maßgebend, nämlich für kleine Feldstärken [2] (15a)

666

E . KÖSTER

und für die Einmündungssuszeptibilität [10]

die angibt, ab welchem gegenseitigen Abstand Versetzungen als Einzel Versetzungen gemessen werden. Durch eine Veränderung von x~x kann die Versetzungsstruktur sozusagen abgetastet und die Zahl der über l < ar 1 angeordneten Versetzungen, d. h. die in der Verfestigungstheorie [3] wichtige Zahl der aufgestauten Versetzungen pro Versetzungsgruppe, angegeben werden. In Nickel erhält man bei 273 °K und H = 2700 Oe mit der Austauschkonstanten A = = 8,6 X 10~7 erg/cm [14] und der Anisotropiekonstanten K1 = 7,7 X 104 erg/cm 3 die beiden Austauschlängen XR1 = 410 A (16a) und x ^ = 114A. (16b) Die xK entsprechende Feldstärke bei Einmündungsmessungen aus xK = y.H ist H = 210 Oe ,

(17)

bei der der Kristall jedoch bereits in Weißsche Bezirke aufgeteilt ist. Beide Arten von Messungen zusammen gestatten im Prinzip eine Untersuchung der Versetzungsstruktur über einen weiten Bereich der Austauschlänge x~x, der durch Temperaturerhöhung, d. h. Verkleinerung von Klt nach Gleichung (15a) zu noch größeren Werten ausgedehnt werden kann. Ein quantitativer Vergleich der Ergebnisse beider Untersuchungsmethoden erfordert eine eingehende theoretische Untersuchung, die zur Zeit von Holz [15] in Stuttgart ausgeführt wird. Die bisher vorliegenden Ergebnisse zeigen, daß sowohl die in Abschnitt 3.6 wiedergegebenen Messungen der reversiblen Suszeptibilität im Gebiet der Einmündung in die Sättigung als auch der Betrag von Ka nur durch weitreichende Spannungsfelder aufgestauter Versetzungsgruppen im Bereich I I der Verfestigungskurve erklärt werden kann. 3.4 Das

Rayleigh-Gebiet

Für das Rayleigh-Gesetz eines einachsigen Polykristalls leitete Neel [16] die Beziehung J =

XkH

H + 0,16 ^ H « )

+ « H* =

(18)

ab. Hierbei ist %A die Anfangssuszeptibilität, ) differs from J(m) b y an amount J0 equal t o t h e direct current set up b y t h e static electric field E0 and which does not depend on E(t): J'(to) = J0+J(m) . (4) As we are dealing with the electro-optical effects which exhibit a non-linear dependence on E0, it is necessary t o give an accurate description of the interaction of the crystal with E0. W e therefore incorporate the static electric field into the unperturbed system. The H a m i l t o n i a n of the whole system is H'(t) = H + AH(t) ,

(5)

where AH(t) = -

(6)

Z e E(t) rt i

describes t h e interaction of the crystal electrons with the light w a v e E(t) in the self-consistent-field approximation. (ri is the radius vector of the ¿-th electron.) T h e Hamiltonian H of the unperturbed system can be written, using the adiabatic approximation, in the f o r m H = H0 -

Z e K r,.

(7)

i

Here H 0 denotes the Hamiltonian of the crystal electrons in t h e absence of external fields. The second term in H describes t h e interaction of the electrons with the static field E0. The density operator o'(t) of the system satisfies the equation of motion i h g'(t) = [H'(t), e ' ( i ) ] .

(8)

W e assume t h a t both external fields, E0 and E(t), are switched on at the time t0 and t h a t the system was previously in thermodynamical equilibrium. T h e initial condition for n'(t) is therefore e'Co) = 6o .

i?o = [Sp

•

,

(9)

where ¡x a n d N are the chemical potential and the t o t a l particle-number operator, respectively.

Theory of Electro-Absorption and Electro-Reflectance in Semiconductors

675

Denoting by p(t) that part of o'(t) which is independent of E(t) and by Aq(t) the part linear in E(t) we have, neglecting terms of higher order in E(t),

e'(0 =

+

(10)

.

W e then obtain, using (8) and (9), '-II Q(t)

=

+

q)2

( 1 = 1 , - 1)

(50) (51)

lQf

From (47) one obtains approximately < ' = 1 - ^ ,

where

+ Q transmitted in both positive and negative directions along the 2-axis. The generation of "harmonics" a> + Q is quadratic in the field-induced quantity in contrast to electroabsorption and electro-reflectance which are linear in the field contribution e(0) — (1 + Ae). The relative intensity of the "harmonics" is of the order of magnitude of |e(0) — (1 + Ae)|2 and depends on the band structure. 5. Franz-Keldysh Effect and Electro-Reflectance As was shown in the previous section, the absorption and reflectivity of a crystal in a static field E0 is determined by the Fourier coefficient £ (0) (oj). At the absolute zero and for values of co in the vicinity of the threshold we obtain from (39) approximately («,) = l

e

"2

-1- ~n i" i' "e 1\(2 2 0 co

\m J

\px\ 10)!2 h co

dad k C To ,

¿/

3

2

(55)

where v = 1 and fi = 2 denote the valence band and the conduction band, respectively. In order to determine the absorption coefficient a we consider according to (53) and (55) the real part of C210- We have cosh y — e~y

Re I—wvJt;

|e-i/m _ i j R e

44*

r

1

l - i f m _

(56)

r ! = lim 4" cosh y — cos f(T)

J

1

r

= n

_>0 2 00

J ^ J { t ( T ) ~

1

2 n s ) - - .

(57)

680

R. Enderlein and R. Keiper

Taking (37) and (34) into account it follows that Re

/

die-™

2

(/(T) — 2 ns) .

(58)

-2V2

1

The second factor in (58) describes the Stark ladder in the absorption spectrum [7, 12], If we replace it according to 2 d { f ( T ) - 2 n s ) - > ~ S ^ 71

(59)

by 1/2 we find the standard expression a = K61l2 where

/

oo

da; |Ai(x)|2 ,

(60)

o>„—1. One can easily show that the formal substitution (59) is equivalent to making allowance for collisions. We assume that the frequency co21(fe) of an interband oscillator becomes a complex number if collission processes are taken into account i

W 21 (fc)^0) 21 (fe) T

.

(62)

21

The relaxation time t 2 1 must satisfy Q t 3 1 < 1 < « T21 .

(63)

(56) is not changed by (62) provided T j r 2 1 is substituted for y. Because of (63) T\t21 is a very large number and we have approximately Re and therefore Re C210 =

Yt

772

/ dte-W) -27 2

(65)

Collisions have been neglected in the integrals in (37). A comparison of (65) and (58) shows in fact that taking collisions into account in our approximation is equivalent to the substitution (59) (see also [12]). The imaginary part of O210 which determines the reflectivity in the presence of a static electric field can be found from (65) with the help of a KramersKronig relation. Acknowledgement

The authors wish to express their thanks to Dr. M. Porsch for numerous discussions and valuable suggestions.

Theory of Electro-Absorption and Electro-Reflectance in Semiconductors

681

Appendix Let 2 to the lattice specific heats in addition to the usual electronic contribution y T. The spin-wave contribution introduces complication in the analysis of the heat capacity data. An important reason for the choice of nickel for the present study is that nickel is one of the few metals whose frequency distribution curve has been observed experimentally by means of inelastic scattering of neutrons. This makes the comparison of the theoretical frequency distribution curve with the experimental one possible and thus provides an independent check to the validity of the force models.

730

M. M. SHUKLA a n d B .

DAYAL

Recent studies of Krebs [2, 3], Mahesh and Dayal [4, 5], Shukla [6, 7], and Shukla and Dayal [8, 9J have shown that Krebs's model yields very good results in the case of some body-centered and face-centered cubic metals, the agreement with the experimental data being almost exact in a few cases. Although this model is phenomenological in its nature like those of de Launay [10], Bhatia [11], and Sharma and Joshi [12], it is more satisfactory inasmuch as it satisfies the requirements of the lattice periodicity in the reciprocal space. We have, therefore, thought it proper to study the lattice dynamics of nickel also with the help of this model. The results are reported in the present paper. 2. Numerical Computation The secular determinant for calculating the frequencies and the method of evaluating the frequency distribution have been described earlier b y us [9] in a paper dealing with the lattice vibrations in aluminium. The same procedure is followed in this paper. As in that paper the force constants have been calculated from the experimental values of the elastic constants and one observed phonon frequency. The elastic constants have been taken from the work of Alers et al. [13] and refer to a temperature of 296 °K. The experimental phonon frequency is taken to be the one corresponding to the zone boundary for the longitudinal branch of [100], This is taken from the work of Birgeneau et al. [1]. These data together with the other input data are given below: Cn = 2.46 x 10 12 dyn/cm 2 , C 12 = 1.50 X 10 12 dyn/cm 2 , Cu = 1.22 x l O 1 2 dyn/cm 2 , v = 7 . 3 0 X l O 1 2 Hz, a = 3.5239 A. The force constants calculated from the above data are found to have the following values: «! =

36.347 x 10 3 dyn/cm,

a 2 = - 2 . 2 9 1 X 10 3 dyn/cm, a3 =

1.007 X l O 3 dyn/cm.

In the evaluation of force constants we have to take into account the screening parameter (A) and the number of free electrons per atom (weff). In the case of transition metals weff is rather uncertain. In our present study we have used weff = 2 along with the Bohm and Pines value of the screening parameter A. The specific heat data allow us to choose the value of (w^f/A)1'6. Any uncertainty in weff is thus balanced by the corresponding uncertainty in A. The computed dispersion curves along the three symmetry directions [100], [110], and [111] are shown in Fig. 1, 2, and 3 respectively. The experimental data of Birgenau et al. [1] have also been plotted there to facilitate comparisons. The specific heats have been calculated by the same method as reported in the previous work from this laboratory (see, e.g., reference [7]) by solving the secular determinant for only 48 non-equivalent points distributed uniformly within the l/48th part of the first Brillouin zone. The whole spectrum was

731

The Lattice Vibrations of Nickel in Krebs's Model

0.4 Q6 aq/ZrcVIF i g . 1.

Dispersion curve for nickel in [100] direction calculated b y u s ,

* * * J experimental points of Birgeneau et al. [1]

as ay/2nV3—F i g . 3. Dispersion curve for nickel in [111] direction calculated b y us, experimental points of Birgeneau et al. [1]

0.75

F i g . 2. Dispersion curve for nickel in [110] direction calculated b y us,

x x x 1 • • • f

ooo'

experimental points of Birgeneau et al. [1]

7 8 912 1 10 11 v(10 s )— F i g . 4. Frequency distribution curve for nickel calculated b y us, } C h e r n o p l e k o v et al. [14] experimental of B r u g g e r [15] Mozer et al. [16]

divided into intervals of Av = 4 X10 1 2 . The frequency distribution curve obtained in this way is shown in Fig. 4. The experimental frequency distribution curves of Chernoplekov et al. [14], Brugger [15], and Mozer et al. [16] are also shown there. The results of Brugger [15] and Mozer et al. [16] have not yet been published. Birgeneau et al. [1] have, however, plotted them in their paper. The present authors have, therefore, reproduced these results from the paper of Birgeneau et al. [1]. The specific heats Cv were not calculated below 15 ° K because the mesh becomes too coarse to give accurate results. The experimental data available on the heat capacities of nickel are those of Keesom and Clark [17], Busey and Giauque [18], and Rayne and Kemp [19], Keesom and Clark [17] have carried out their measurements from 1 to 19 ° K and Busy and Giauque [18] from 13 to 300 ° K . Rayne and Kemp [19] confined themselves to liquid helium temperatures. Since nickel is ferromagnetic below 631 °K, there is also a spin-wave contribution to the lattice specific heat in addition to the usual electronic contribution. These two contributions have to be subtracted from the experimental heat capacities in order to obtain the

732

M . M . SHUKLA a n d B . DAYAL Fig. 5. (0 — T) curve for nickel calculated by us, experimental of Rayne and Kemp [19], calculated by Birgeneau et al. [1]

Theoretical of BMBENEAU et a! [11

0

20

W

120

150

T(°K)~

lattice contribution. Rayne and K e m p [19] have analysed t h e data of Busey and Giauque [18] and have given a (6 — T) curve after subtracting t h e electronic and spin-wave contributions from the experimental heat capacities. In this way they have found the value of electronic heat capacity coefficient y to be 7.05 X 10~3 J / m o l deg 2 . The value of Cm , the spin-wave contribution to the heat capacities, was found to be Cm = 8.8 x 10~5 T 31' 2 J / m o l deg 3 ' 2 . The experimental (0 — T) curve presented by Rayne and K e m p [19] has been reproduced in Fig. 5 together with the theoretical curve obtained by us using Krebs's model. We have also given the theoretical curve calculated by Birgeneau et al. [1] from t h e general Born-Karman force model including fourth neighbours. 3. Discussion A study of Fig. 1 to 3 shows t h a t Krebs's model yields a very good description of the experimental dispersion curves in the three symmetry directions. At low vectors the calculated frequencies almost coincide with the experimental results. The maximum divergence of less t h a n 3 % is found only at the zone boundaries. Since the experimental phonon frequencies were determined with an accuracy of about 2 % the agreement must be considered excellent. Birgeneau et al. [1] have also attempted to fit their experimental results by the use of general interatomic forces up to fourth nearest neighbours, and also with an axially symmetric model in which the interactions are extended u p to f i f t h nearest neighbours. They have also reported an excellent agreement in both cases. This is, however, not surprising in view of the fact t h a t their general force model and the atomic-shell model both employ a considerably large number of parameters (twelve and ten respectively), all of which have been determined entirely from neutron scattering results by the method of least square fit. The agreement with the experimental results, therefore, depicts nothing but the self-consistency of the experimental results. In spite of this self-consistency the elastic constants and the low-frequency region of t h e disperison curves are not described correctly rs these autors have pointed out themselves. This appears to be due to the fact t h a t they did not use the elastic constants to determine their force constants. The great superiority of Krebs's model over the models employed b y Birgeneau et al. [1] lies in the fact t h a t it gives excellent agreements with elastic data as well as with the dispersion curves with a very small number (four only) of parameters. These are the central force constants with the three neighbours and the electronic force constant Jc„.

733

The Lattice Vibrations of Nickel in Krebs's Model

A study of Fig. 5 reveals that our calculated (d — T) curves show an excellent agreement with the experimental one. We have also compared our results with the theoretical curve of Birgeneau et al. [1] calculated by them by means of the general interatomic force model extended up to fourth neighbours. In the low-temperature region our curve almost resembles to the curve of Birgeneau et al. [1], but at higher temperatures it reproduces the experimental results much better than is given by the latter. Fig. 4 gives the frequency distribution of nickel calculated on the basis of Krebs's model. Since there are three different sets of experimental g(v) curves available we have compared our theoretical curve with all of them. The real difficulties about these experimental curves are that they differ from each other considerably. Our calculated curve is also not found to describe fully the course of anyone of these experimental curves but, on the whole, it makes a good compromise amongst them in a wide range of spectrum. Acknowledgement

The authors are thankful to Dr. R . P. Gupta for useful discussions. References [ 1 ] R . J . BIRGENEAU, 1359 (1964).

J . CORDES,

G. DOLLING, a n d A . D . B . WOODS, P h y s .

Rev.

136,

[2] K. KREBS, Phys. Letters (Netherlands) 10, 12 (1964). [3] K. KREBS, Phys. Rev. 138, 143 (1965). [4] [5] [6] [7]

P. P. M. M.

S. MAHESH a n d B . S. MAHESH a n d B . M. SHUKLA, p h y s . M. SHUKLA, p h y s .

DAYAL, phys. s t a t . sol. 9, 3 5 1 ( 1 9 6 5 ) . DAYAL, P h y s . R e v . 1 4 8 , 4 4 3 ( 1 9 6 6 ) . s t a t . sol. 7, K L L ( 1 9 6 4 ) . s t a t . sol. 8 , 4 7 5 ( 1 9 6 5 ) .

[8] M. M. SHUKLA and B. DAYAL, J . Phys. Chem. Solids 26, 1343 (1965). [ 9 ] M. M. SHUKLA a n d B . DAYAL, p h y s . s t a t . sol. 1 6 , 5 1 3 ( 1 9 6 6 ) .

[10] [11] [12] [13]

J . DE LAUNAY, J . chem. Phys. 21, 1979 (1953). A. B. BHATIA, Phys. Rev. 97, 363 (1955). P. K. SHARMA and S. K. JOSHI, J . chem. Phys. 39, 2633 (1963). G. A. ALERS, J . R. NEIGHBOURS, and H. SATO, Bull. Amer. Phys. Soc. 4, 131 (1959).

[ 1 4 ] N . A . CIIERNOPLEKOV, M. G. ZEMLYANOV, A . G. CHETSEHIN, a n d B . G. LYASHCHENKO,

Inelastic Scattering of Neutrons in Solids and Liquids, Vol. 2, International Atomic Energy Agency, Vienna 1962 (p. 159). [15] R. M. BRUGGER, A.E.R.E. (Harwell) Rep. R4562 (1964) (unpublished). [16] B. MOZER, K. OTNES, and H. PALEVSKY, J . Phys. Chem. Solids, to be published. [ 1 7 ] W . H . KEESOM a n d C. W . CLARK, P h y s i c a 2, 5 1 3 ( 1 9 3 5 ) . [ 1 8 ] R . H . B U S E Y a n d W . F . GIAUQUE, J . A m e r . Chem. S o c . 7 4 , 3 1 5 7 ( 1 9 5 2 ) . [ 1 9 ] J . A . RAYNE a n d W . R . G. KEMP, Phil. Mag. 1 , 9 1 8 ( 1 9 5 6 ) . (Received

November

20,

1966)

K.-H. PFEFFER: Wechselwirkung zwischen Versetzungen und Blochwänden

735

phys. stat. sol. 19, 735 (1967) Subject classification: 18.2; 10.1; 21.1; 21.1.1 Institut für Physik am Max-Planck-Institut für Metallforschung, Stuttgart, und Institut für theoretische und angewandte Physik der Technischen Hochschule Stuttgart

Wechselwirkung zwischen Versetzungen und ebenen Blochwänden mit starrem Magnetisierungsverlaufl) Von K . - H . PFEFFER Mit Hilfe der Peach-Koehlerschen Formel werden die Wechselwirkungskräfte zwischen 180°-Blochwänden, deren Magnetisierungsverlauf als starr angenommen wird, und verschiedenen Versetzungskonfigurationen berechnet. Numerische Ergebnisse werden für die (110)- und (112)-180°-Blochwände in Nickel und für die (001)- und (110)-180°-Blochwände in Eisen angegeben. Die Temperaturabhängigkeit der für die verschiedenen Versetzungskonfigurationen charakteristischen effektiven Wechselwirkungslängen und die in der statistischen Analyse der Blochwandbewegung zu beachtenden geometrisch-statistischen Parameter werden diskutiert. The interaction forces between 180°-Bloch walls, the magnetization of which is assumed to be f i x e d and different dislocation configurations are calculated by the Peach-Koehler method. Numerical results are given for the (110)- and (112)-180°-Bloch walls in nickel and for the (001)- and (110)-180°-Bloch walls in iron. The temperaturedependence of the effective interaction lengths of the different dislocation configurations a n d t h e geometrical parameters relating to a statistical analysis of the Blochwall movement are discussed.

1. Einleitung Die ausgeprägte Abhängigkeit der Kenngrößen der Magnetisierungskurven ferromagnetischer Kristalle von deren Verformungszustand beruht vorwiegend auf der magnetoelastischen Wechselwirkung zwischen der Magnetisierung und den durch Versetzungen hervorgerufenen inneren Spannungen. Durch Arbeiten von Dietrich und Kneller [1], Krause [2], Rieger [3], Träuble und Bilger [4] und Träuble [5] ist experimentell gesichert, daß in den magnetisch mehrachsigen Kristallen Eisen und Nickel die Koerzitivfeldstärke Hc und die Anfangssuszeptibilität Xu hei hinreichend tiefen Temperaturen (bei Fe unterhalb 500 °C, bei Ni unterhalb Raumtemperatur) durch Verschiebungen von 180°Blochwänden bestimmt werden. Die Wechselwirkung der 180°-Blochwände mit inneren Spannungen ist bisher in verschiedenen Näherungen und unter verschiedenen Annahmen untersucht worden. In der von Kondorskii [6] und Kersten [7] entwickelten Spannungstheorie der Koerzitivfeldstärke wird ein mittlerer Eigenspannungsbetrag d i eingeführt, ohne die Art und die Anordnung der Spannungsquellen zu spezifizieren. Vicena [8], Rieder [9] und Träuble [5] berücksichtigten die spezielle Natur der Eigenspannungsquellen und berechneten die Wechselwirkungskraft zwischen einer Blochwand und einer Versetzung unter folgenden Annahmen: 1. Der Magnetisierungsverlauf innerhalb und außerhalb der Blochwand wird durch die inneren Spannungen der Versetzung nicht beeinflußt. 2. Die Versetzungen verlaufen parallel zur Blochwand. J

) Dissertation, Teil I, Technische Hochschule Stuttgart, 1966.

736

K . - H . PFEFFER

Zur Begründung der Annahme 1. schätzte Vicena [8] den Einfluß der durch die inhomogene MagnetisierungsVerteilung in der Umgebung einer Versetzung [10] bedingten Streufelder auf die Wechselwirkung zwischen Blochwand und Versetzung ab und fand, daß dieser Streufeldeffekt [5] eine um zwei bis drei Größenordnungen kleinere Wechselwirkungskraft liefert, als die Näherung der starren Blochwand ergibt. Bei dieser Abschätzung bleibt jedoch die entscheidende Tatsache unberücksichtigt, daß sich die Magnetisierungsverteilungen in der Umgebung einer Versetzung und im Innern einer Blochwand gegenseitig beeinflussen, wodurch eine zusätzliche Energieänderung im System Bloch w a n d VerSetzung a u f t r i t t . Bevor wir eine mikromagnetische Behandlung dieses Sachverhaltes durchführen, untersuchen wir in der vorliegenden Arbeit unter Beibehaltung der Annahme 1. die Wechselwirkung von Blochwänden mit gekrümmten Versetzungen und schaffen damit die Grundlage f ü r eine genauere Berücksichtigung der in einem Kristall vorliegenden Versetzungsanordnung bei der Berechnung von Hc und xa2. Methoden zur Berechnung der Wechselwirkungskraft I n allen Verfahren zur Berechnung der Wechselwirkungskraft zwischen Versetzungen und Blochwänden wird der Magnetisierungsverlauf innerhalb einer isolierten Blochwand und damit der Tensor e M der magnetostriktiven E x t r a dehnungen derselben als gegeben betrachtet. Bei Kenntnis des Spannungstensors a v der Versetzung läßt sich die magnetoelastische Kopplungsenergie —cM • o v berechnen und hieraus durch Differentiation die Wechselwirkungskraft bestimmen (Methode von Vicena [8]). Rieder [11] dagegen ermittelt den Tensor o M der mit dem inhomogenen Magnetisierungsverlauf in der Blochwand verknüpften Extraspannungen und berechnet mit Hilfe der Peach-Koehlerschen Formel [12] die K r a f t , die die Blochwand auf die Versetzung ausübt. Durch Anwendung des Gesetzes „Actio = Reactio" ergibt sich hieraus die K r a f t , die die Versetzung auf die Blochwand ausübt. Letztere Methode ist die bequemere, da nur der Verlauf und der Burgersvektor, nicht aber der Spannungshof der Versetzung in die Rechnung eingeht. Wir werden im folgenden diese Methode benutzen. Die K r a f t , die eine geschlossene Versetzungsschleife auf eine Blochwand ausübt, ist als Flächenintegral [13] zu berechnen: = - f (dS X V) X a M • b . (1) s b ist der Burgersvektor der Versetzungslinie, 8 eine beliebige, von der Versetzungslinie berandete Fläche. o M ist der Extraspannungstensor der Blochwand. Weist die Blochwandnormale in z-Richtung, so sind nur die Spannungskomponenten a\j mit {i, j) = (1,2) von Null verschieden. Diese berechnen sich f ü r eine isolierte 180°-Blochwand 2 ) nach Rieder [11] zu P^

afj = Aij sin 2 0

0

+ Bij sin 2 & 0 ;

Die Konstanten A { j und B ( j enthalten Konstanten des betreffenden Kristalls. Blochwand in Ni bzw. die (001)-180"sind die A-ij und Bij aus Tabelle 1 bzw.

0 ^ &0 ^ n .

(2)

die magnetostriktiven und elastischen Für die (110)-180°- und die (112)-180°und die (110)-180°-Blochwand in Fe 2 zu entnehmen.

2 ) Die «/-Richtung fällt mit einer leichten Magnetisierungsrichtung zusammen; bei Ni ist dies eine -Richtung, bei Fe eine -Richtung.

737

Wechselwirkung zwischen Versetzungen und ebenen Blochwänden

\ \

\

v

v(

»

h/ /

/—*

r~im°K

'

A fao°K \y' i i i \W/ Y 1 2 3 4 5

F i g . 2. Verlauf der E x t r a s p a n n u n g e n in 180°-Bloch\vänden in Nickel, a) Ol2)-180°-Blochwand, b ) (110)-180°-Blochwand

738

K . - H . PFEFFER

739

Wechselwirkung zwischen Versetzungen und ebenen Blochwänden

o o o o o

o

o o o o o

o

J—1 PS 00 Iii © ©

t and wl corresponding to zero wave vector and the equilibrium condition. The longitudinal optical frequency col is itself determined from a>0 and the high- and low-frequency dielectric constants e^ and e0 in accordance with the Lyddane-Sachs and Teller relation (1)

The six equations used for determining the six parameters are best satisfied at 0 °K. The values of the elastic constants of MgO at 0 °K have been taken from measurements of Durand [10] and are cu = 29.89, c1S! = 8.57, and c 44 = = 15.679 in units of 10 12 dyn c m - 2 . Because of lack of measurements of e 0 and £oo at 0 °K we have used the values e0 = 9.8 and £jo = 2.95 as given by Born and Huang [11] (Table 17, p. 85). Calculations were made for two different values 10.9 and 7.462 X 10 13 s - 1 of the infra-red absorption frequency a»0 corresponding to / 0 = 17.3 and 25.26 ¡im respectively. We are, however, reporting the results only for a> 0 =7.462 x 10 1 3 s _ 1 because the other value does not give even an approximate agreement with the observed heat capacity except at very low temperatures. The values of the six parameters obtained by solving equations (10), (12), and (13) of I are as follows: A1 = 9.364, B1 = - 0 . 6 9 7 , At = 5.104, Bz = - 0 . 4 6 8 , «„ =

-1.264,

and s = 1.377. The lattice vibration frequencies have been calculated for a 1000 point division of the first Brillouin zone on an I.B.M. 1620 computer at I.I.T., Kanpur. The Debye temperatures 6 n in the range 0 to 300 °K have then been determined in the usual way. The plot of 0 D versus T together with the corresponding experimental graph obtained from the measurements of Barron et al. [12] has been shown in Fig. 1. The agreement is good and proves that Saksena and Vishwanathan's value of the principal infra-red absorption frequency is correct.

0

50

100

150

200

250 300 T(°K)

Fig. 1. 0 d - T curve for MgO theoretical, experimental

0

10 20 30 40 ivii' ^¡(lO12 s'1) —

Fig. 2. The combined density-of-states curve for MgO. Arrows indicate positions of infra-red absorption peaks observed by Willmott [3]

754

M . P . VERMA a n d B . D A Y A L

The combined density of states can be obtained from the lattice vibration frequencies in accordance with the two selection rules. Since the vibrational frequencies at wave vector —q a must be the same as those at +)] -

2

E0(2

(23)

co) +

[ » . ( 2 co) x p f

co)

L

[ n a o ( 2 co) X e a o ( 2 co)] E a 0 ( 2 co) ( 2 co)] .

(24)

OL

Die Polarisationsvektoren e 0 (2 co) und e ao (2 co) in (23), (24) können als b e k a n n t angesehen werden, denn sie sind aus und

e 0 (2

co)

= [n 0 (2

co) X

c] (25)

e,0(2 co) = (e0(2 co) 1 - n a o (2 co) n a o (2 co)) • c zu berechnen [12], wenn man die Refraktionsvektoren kennt. Die Refraktions-

R . FISCHEB

762

vektoren nG(2 co) und n a o (2 co) sind jedoch bei bekannter kristallographischer Orientierung wegen (20) und (21) ebenso eindeutig festgelegt wie n r (2 co). Zur Bestimmung der Refraktionsvektoren wird der Ansatz «¡(2 co) = &(co) +

mit

i = r, o, ao

r]i q;

(26)

b(co) = [q x [ne(w) x g]]

gemacht. Die zu q parallelen Komponenten errechnet man aus den Gleichungen für die Normalflächen [12] n%{2 co) =

£ „ ( 2 co) ,

co): n a o ( 2 co) n a o ( 2 co) =

e(2

e0(2

co) e a o ( 2 co)

(27)

zu %

^°(2C0)

e(2o!):gg

=

-e(2co)

( 2 co) =

co) -

(e0(2

^ ( 2 co) =

-

{~£(2ct>)

: qb{CO)

: q q

b » ) V 2

co) -

[nl(2

+

,(28)

b2(co))1/2 ,

[H2ft>) :

co) : b ( c o ) b ( c o ) -

(e(2

,

(29)

^&(ft,))2

-

£ 0 ( 2 co) e a o ( 2 « ) ) ] ! / « } .

(30)

Bei Totalreflexion sind f]0 oder oder beide rein imaginär. In (23), (24) sind deshalb nur E0{2 co), E&0{2 co) und ET(2 co) unbekannt, sie können durch geeignete Multiplikation von (23), (24) mit linear unabhängigen Vektoren eliminiert werden. Wählt man als linear unabhängige Vektoren n 0 (2 co), n ao (2 co), e 0 (2 co) und e ao (2 co), ergibt sich Z? ( 2 co) rv

=

'

nr(2 cu) • [Mj x m2] ¿ 5 , ( 2 co) • [ n r ( 2 t o ) x n

^o(2

( 2 co)] -

Z

* » a o ( 2 eo) • K ( 2

co) X p ?

L

(2

co)]

e0(2 »).[» 0 (2a»)x»„(2a.)] ' Er(2 co) • [n0(2 co) xn r (2 co)] + 2 n0(2 co) • [n„(2 co) xpNL(2 co)]

0>) =

2 ? aaoo((22 cCO) 0)

a o

v

=

^(»i.wMxigä,.)]

mit

'

« i = e 0 • [ « 0 x n J [ e , 0 x g ] - tf • [ e a o x e 0 ] [ n a o xVni ,r 1 «2 = eao • K X n a o ] [e 0 x i>i = [ e a o X e 0 ] = [ea„xe0] ^

[n. x p f ^

1 : n«ao &0q

: n0q

q] -

q

[e ao x e 0 ] [nQ x n•], r]

[ n 0 x n a o ] fq x

2

(33)

(34)

J

P« L| : e ao

(32)

% >

— [w0 x n a o ] ^qr x 2 ' l » ? I , j = e o e a o

(35) (36)

Durch (31) bis (36) ist das Problem der Bestimmung der Amplituden der Harmonischen gelöst. Diese Gleichungen sind nicht brauchbar bei senkrechtem Einfall (n0(2 co) und n a o (2 co) sind parallel) und falls die optische Achse in der Einfallsebene liegt (n0(2 co), » a o (2 co) und e a o (2 co) sind linear abhängig). Unter Verwendung von (25) errechnet man für senkrechten Einfall, wenn man den be-

'

Theorie der Erzeugung der zweiten Harmonischen an der Oberfläche

763

kannten Fall qf = c [6] ausschließt, 2 E,(2

co)

(»« -

=

»o) P ? (»r +

2" K

a

-

• [ q x c ] [qr X c ]

L

o)

[gxc]

n

»ao)p? K

E0(2

w)

ov

'

L

2

- [gx[gxc]]

+

% o ) [g x c]

[gx[gxc]]

2

„NL/.(2 tu)j • [ g x c ] ^ r (2ö>) + Z J», = -i , 2 ra0(2

(38)

co) [ g x c ]

( E r ( 2 co) +

E U 2 a>) = ^

Z

p*

L

(37)

'

v

( 2 co) j • [ g x [ g X e ] ]

,„/,,,

•

(39)

In dem Fall, daß c in der Einfallsebene liegt, folgt aus (23), (24) E,{2

co)

=

v 3 [ n r ( 2 co) X [ w 3 X n r ( 2 t o ) ] ] [nr(2 co)Xw3]2 Z

n 0 ( 2 co) • [ » 1 , ( 2 co) X p ? L ( 2 co)] [ n 0 ( 2 co) x n r ( 2 co)] [ n 0 ( 2 CO) X N r ( 2 co)] 2

n 0 ( 2 co) • T g X ( E r ( 2 co) +

*«< «> =

^

2

e

Z

„NL/ 2

co))]

^

o ( 2 co) • I g X ( £ r ( 2 co) +

(40)

•

(41)

2 " p N L ( 2 co))~|

•®ao(2 co) =

,

(42)

mit v*

=

1(2" [ » . X f » ? L ] ) [ ? x e a 0 ] -

«3 = { K x g ] [ n a o X e a o ] -

[ n a o x e a o ] J : e0 e0 ,

[ e 0 X n r ] [ q x e a o ] | • e0 .

(43) (44)

Die Beziehungen (31) bis (44) erlauben auch die Bestimmung der Amplituden der Grundwellen, wenn in ihnen 2 co durch w und die spezielle Lösung der inhomogenen Gleichung durch die einfallende Grundwelle ersetzt werden. Sie sind sehr allgemein, denn sie gelten für alle optisch einachsigen Kristalle ohne Inversionssymmetrie, und werden wesentlich durch die konkrete Form des Tensors dritter Stufe in (2) bestimmt. Wir werden nun begründen, weshalb sie auch für absorbierende Kristalle gültig sind. 3. Berücksichtigung der Absorption Es wird nun der Fall näher untersucht, in dem das quadratische Medium absorbierend ist, das lineare Medium sei weiterhin nichtabsorbierend. Dieser Fall ist vor allem deshalb von Interesse, weil man durch Messung der von einem absorbierenden Kristall reflektierten Harmonischen Aussagen gewinnen kann, die man in Transmission nicht erhält. Die Ausbreitung des Lichtes in linearen, unmagnetischen, absorbierenden, anisotropen Kristallen wurde ausführlich in [12] diskutiert. In absorbierenden Kristallen breiten sich inhomogene, elliptisch polarisierte Wellen aus. Die Phasen- und Amplitudennormalen dieser Wellen stimmen (angesehen vom Fall des senkrechten Einfalls) nicht überein. Bei optisch einachsigen Kristallen ist

764

R.

FISCHER

es durch dieselbe orthogonale Transformation möglich, sowohl den Real- als auch den Imaginärteil der tensoriellen, komplexen, dielektrischen Permeabilität auf Hauptachsen zu transformieren [12], man kann in Analogie zu (13)

e'(2 w) = £¿(2 co) 1 + (ßao(2 co) - e'0(2 co)) c c

(45)

schreiben. In (45) ist c wieder ein reeller Einheitsvektor in Richtung der optischen Achse. e0{2 co) und £ao(2 co) lassen sich mit Hilfe der Eigenwerte der reellen dielektrischen Permeabilität e(2 co) und des Leitfähigkeitstensors a ausdrücken £¿(2 co) = £0(2 CO) — ^ FF0(2 co) , Z CO

£^(2 co) =

e ao (2

co) — y^