Physica status solidi: Volume 9, Number 3 June 1 [Reprint 2021 ed.] 9783112497661, 9783112497654

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plxysica status solidi

V O L U M E 9 • N U M B E R . . 1965

Contents

Review Article

Page

D . C . R E Y N O L D S , C . W . LITTON, a n d T . C . COLLINS

Some Optical Properties of Group IX—VI Semiconductors (I). . .

645

B . H . K a m e e B K TepMOHHHaMHKe rattaeHÔeproBCKoro $eppoMarHeTHKa (III)

685

Original Papers

J . D . D H E E B a n d B . SHARAN

Phonon Spectrum of Caesium Bromide

701

Electronic Properties of Surface Layers of Silver Chloride

709

J . Z . DAMM

New Anisotropy Effect in Plastically Deformed Potassium Chloride Coloured by Ionizing Radiation

721

M.

Optical and Dielectric Properties of KC1 Grown from Aqueous Solution 727

B. A.

SNAVELY

GIUDICI

E . GRÜNBAUM a n d J . W . MATTHEWS

P . PODINI

Influence of Substrate Steps on the Orientation of Nuclei in Thin Deposits of Gold on Rocksalt

731

Luminescence of M-Centres in NaF Crystals

737

P . RAMA R A O a n d T . R . ANANTHABAMAN

Imperfections in Copper-Silicon Filings

743

W . M A E N H O T J T - V A N D E B VORST a n d F . V A N C B A E Y N E S T

The Green Luminescence of ZnO

749

B . SCHAEFFER, C . DUPTJY e t H . SAUCIER

Étude photoélastique des dislocations dans LiF coloré

753

A . FROVA a n d C . M . P E N C H I N A

Energy Gap Determination in Semiconductors by Electric-FieldModulated Optical Absorption

767

F . VAN D E R W O U D E a n d A . J . D E K K E R

The Relation between Magnetic Properties and the Shape of Môssbauer Spectra

775

K . S . MENDELSON a n d H . N . SPECTOR

P. H.

FANG

Interaction of Magnetoplasma Waves with Spin Waves in a Transverse Magnetic Field

787

On the Analysis of the Dielectric Dispersion of Ferrites

793

(Continued on cover three)

physica status solidi B o a r d of E d i t o r s P. A I G R A I N , Paris, S. A M E L I N C K X , Mol-Donk, W. D E K E Y S E R , Gent, W. F R A N Z , Münster, P. G Ö R L I C H , Jena, E. G R I L L O T , Paris, R. K A I S C H E W , Sofia, P. T. L A N D S B E R G , Cardiff, L. N É E L , Grenoble, A. P I E K A R A , Poznan, A. S E E G E R , Stuttgart, 0. S T A S I W , Berlin, M. S T E E N B E C K , Jena, F. S T Ö C K M A N N , Karlsruhe, G. S Z I G E T I , Budapest, J. T A U C , Praha Editor-in-Chief P. G Ö R L I C H

Advisory Board M. B A L K A N S K I , Paris, P. C. B A N B U R Y , Reading, M. B E R N A R D , Paris, W. B R A U E R , Berlin, W. C O C H R A N , Edinburgh, R. C O E L H O , Fontenav-aux-Rosea, H.-D. D I E T Z E , Aachen, J . D. É 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. N A G Y , Budapest, E. A. N I E K I S C H , Jülich, L. P A L , Budapest, M. R O D O T , Bellevue/Seine, B. V. R O L L I N , Oxford, H. M. R O S E N B E R G , Oxford, R. V A U T I E R , Bellevue/Seine

Volume 9 • Number 3 • Pages 643 to 912 and K 161 to K 224 June 1,1965

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 £ür 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 , 1 0 2 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 . 6 9 J e n a , H u m b o l d t s t r . 26. R e d a k t i o n s k o l l e g i u m : D r . S. O b e r l ä n d e r , D r . E . G u t s c h e , D r . W . B o r c h a r d t . A n s c h r i f t d e r S c h r i f t l e i t u n g : 102 B e r l i n , N e u e S c h ö n h a u s e r S t r . 20, F e r n r u f : 4 2 6 7 8 8 . Verlag: Akademie-Verlag G m b H , 108 B e r l i n , L e i p z i g e r S t r . 3 — 4 , F e r n r u f : 2 2 0 4 4 1 , T e l e x - N r . 0 1 1 7 7 3 . P o s t s c h e c k k o n t o : B e r l i n 3 5 0 2 1 . D i e Z e i t s c h r i f t „ p h y s i c a s t a t u s s o l i d i " e r s c h e i n t j e w e i l s a m 1. d e s M o n a t s . B e z u g s p r e i s e i n e s B a n d e s M D N 60,-. Bestelln u m m e r dieses B a n d e s 1068/9. 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 d e s P r e s s e a m t e s b e i m V o r s i t z e n d e n d e s M i n i s t e r r a t e s der Deutschen Demokratischen Republik.

Review Article phys. stat. sol. 9, 645 (1965) Aerospace Research Laboratories,

Wright-Patterson Air Force Base, Ohio

Some Optical Properties of Group II—VI Semiconductors (I) By D . C. REYNOLDS,

1.

C . W . L I T T O N , a n d T . C . COLLINS

Contents

Introduction

2. Exciton theory 2.1 The intrinsic exciton 2.2 Effects of external magnetic and electric fields S. Experimental observations of intrinsic excitons 3.1 Intrinsic exciton spectrum and band structure 3.2 Exciton spectrum of CdS 3.3 Magneto optical effects in the exciton spectrum of CdS 3.4 Exciton structure and Zeeman effects in CdSe 3.5 Exciton spectrum of ZnO 3.6 Exciton spectra of ZnS, ZnSe, and CdTe 3.7 Exciton structure in photoconductivity of CdS, CdSe, and CdS:Se single crystals 4. Spatial resonance 4.1 Theory

dispersion

4.2 Experimental observations References

(Part

I)

1. Introduction The optical properties of I I - V I compounds have been the subject of considerable study for many years; indeed, in the older published literature, there is a wealth of optical data, some of which is related to the fundamental properties of these materials. I t has only been in recent years, however, that even the older data has been associated with the exciton structure. The more recent (mostly since 1959) magneto-optical data (spectral data obtained at high magnetic fields and low temperatures) is very closely related to the fundamental properties of the materials; this data has come to be very well understood in terms of excitons in these compounds. In fact, the isolation and subsequent elucidation of many properties fundamental to these materials has frequently been possible because of a better understanding of the intrinsic exciton structure. We owe much of this better understanding to Thomas and Hopfield whose outstanding theoretical and experimental work on the exciton structure of CdS has served as a model for many of the other materials; also, the higher-quality single crystals (platelet form, available in recent years) have been essential to much of this work. At the outset, we chose to limit this review to the exciton 42«

646

D . C. R E Y N O L D S , C. W . LITTON, a n d T . C . COLLINS

problem in the I I B - V I B compound semiconductors. One could cite many reasons for making such a choice but perhaps the most important of these are the following: 1. Many of the fundamental properties of these compounds have now been explained in terms of intrinsic exciton structure; hence, excitons were a natural choice. 2. There has been a recent upsurge of interest in bound-exciton-complex spectra in semiconductors, yet this work, a subject unto itself, has not been collected and summarized elsewhere. 3. As usual, particularly in a case like this where the subject is so diverse, it is necessary to limit the scope of the review. Obviously, this is necessary in order to treat at least some of the subject in sufficient depth. It is the purpose of the present review to summarize some of the recent theoretical and experimental work on intrinsic and extrinsic excitons in the II-VI's, and to discuss certain cooperative phenomena (such as phonon interactions) that are related to the exciton problem. In the present review, most of the emphasis has been placed on CdS, a fact which is not surprising since most of the exciton studies have been performed on this material. 2. Exciton Theory 2.1 The intrinsic

exciton

In order to understand some of the details of the spectra which are observed in I I - V I compounds, it is first necessary to recall some of the fundamentals of exciton theory, i.e., the exciton structure and the coupling of the exciton field with other fields in the crystal. For this reason we shall briefly review the historical development of the exciton before we proceed to its application in the I I - V I compounds. The introduction of this quasi-particle, the exciton, was made by Frenkel [1] in his attempts to gain insight into the transformation of light into heat in solids. He was able to explain the transformation by first order perturbation of a system of N atoms with one electron per atom which had the following properties: 1. The coupling between different atoms in a crystal is small compared with the forces holding the electron within the separate atoms. 2. The Born-Oppenheimer approximation is valid. 3. The wavefunction is a product of one-electron functions. If one represents the ground state atomic wavefunction by Wi and the excited atomic wavefunction by Wn, the ground state of the crystal will be (neglecting that the wavefunction should be antisymmetric) 0o = nwI(i). i

(2.i)

The states of the crystal in which one electron is excited and the rest in the normal state are formed from linear combinations of Qi^VvWnWM).

(2.2)

There are N sets of coefficients { C ( } n corresponding to the splitting of the undisturbed states of N isolated atoms with the total energy equal to Wii + (N — 1) Wi into N different states denoted as the "excitation multiplet"

Some Optical Properties of Group I I - V I Semiconductors (I)

647

of the crystal. The corresponding value of the coefficients are determined by the equations N Z U p t Ct = W ' C P , (2.3) where the Uri's are matrix elements of the mutual potential energy of all the atoms, U, with respect to the functions of equation (2.2). One finds U is of the form u = z U(oc, p, S.p) , (2.4) a

(2 32

(2.33) 2< ' where m* and m* are the effective masses of the electron and hole, respectively. Further, transforming to the center of mass coordinates plus the use of U'(ft) = eiaKP

U(P)

(2.34)

to eliminate the cross terms of K V^g, U(fi) must satisfy the following equation: h2 h2 k2 (2.35) E — E0 — Eq

2«

651

Some Optical Properties of Group II-VI Semiconductors (I)

where Eg = Ee — Ev and ¡x is the reduced mass of the exciton. Since the operator on the left of equation (2.35) is the hydrogenic operator, the eigenvalues have the form fi ei h K2 \ W (2.36) &vK — -&0 + &G 2 2 2 (to* + 2h n where v represents the quantum numbers associated with the hydrogenic problem, n is the principle quantum number, and the last term of equation (2.36) is the kinetic energy of the exciton. The exciton wavefunctions thus have the form TO*

XcwK = ¿ " e x p i « Kp Uv{fi) &ve(fi, K) . (2.37) yj L me T mh Let us return to the terms of the Hamiltonian which have been neglected; i.e., those of equation (2.24). The matrix elements with respect to the wavefunctions in (2.37) are

„*

3€„' = E e x P fi fi" X E é K

m* + m%

K R

K(0

¡V O ah, cpo,

-

fi') U*(P) U,.(P) X -(^2-1)

vRa'h,c

(R + 0')

+

(2.38)

fi t In examining the above equation, we will use a definition found in Knox's [3]1) book on exciton theory. Namely, if one considers the hole described by Wannier function, a v j{ a h being localized on a particular lattice point, the excited electron has the form of a "molecular orbital" consisting of a linear combination of Wannier functions. The coefficients of this linear combination are the hydrogenlike ones. The electron wavefunction is -K(i\uAP)ac{n+fi)o,{r). = E exp nh J fi Using equation (2.39), the matrix element X' may be written as

(2.39)

S W )

3€vv' — (vo(P)

=

I

¥mfs{r)

z

(2.102)

dr

and m and m label the bands or atomic states used in constructing the exciton states i and j. If the hole is common t o i and j, then m and ra' label electron states; if the electron is common to i and j, then m and m' label hole states. The two-center m a t r i x elements of z have been ignored in the above. I n the case, where t h e excitons connected by t h e Stark perturbation are built from the same bands, one m a y change variables of integration equation (2.102) from r to r' = r — (i and obtain Z

m m

( P ) =

/

(z'

+

ft)

y

m /

» ( r ' ) dr'

.

(2.103)

For the moment consider t h e crystal point group t o contain inversion. t h e n finds there is no contribution from z' and z(ft) = /?z . The Stark matrix element becomes X '

=

- e E

Z

U*mnvK (p)

pz

U

m n v

.

K

( p ) .

One

(2.104)

P

Substituting the hydrogenic coefficients and using the quasi-continuous variable equation (2.104) becomes

X' = - e E f U*(r) z U} (r) dr .

(2.105)

Let us now return to equation (2.58). This t e r m m a y be written in t h e f o r m (neglecting the anisotropy parts and the constants)

K A = KH

x

( F x H) r .

(2.106)

This t e r m represents the quasi-electric field which an observer riding with t h e center of mass of the exciton would experience because of the magnetic field in the laboratory. The quasi-field would produce a Stark effect linear in H, and this would give rise to a maximum splitting interpretable as a "gr-value". To detect this effect Thomas and Hopfield [10] made use of an electric balancing technique. The Stark effect on excitons is dominated by the simple "hydrogenic" Stark effect (see equation (2.105) above). For an exciton of zero wavevector, all Stark effect energy shifts should t h u s be a function only of the absolute value of the electric 43 physica

660

D . C . R E Y N O L D S , C . W . LITTON, a n d T . C. COLLINS

field. All energy levels will be the same for an applied laboratory field E and — E, and the exciton energy level spectrum will be symmetric about E = 0. The same argument holds even in the presence of a uniform magnetic field, and follows from the inversion symmetry of the effective-mass Hamiltonian. An exciton of finite wavevector k feels in addition to the applied external electric field E the quasi-field Eq. If Eq and E are arranged to be collinear, Eg can be directly measured as follows. The Stark shift is plotted against the applied electric field E ; in the absence of a magnetic field the plane of symmetry is at E = 0. I n the presence of a magnetic field the plane of symmetry occurs not at zero applied field, but at a value Es, and Es = — Eq. The quasi-field Et can thus be measured without the need for an analysis of the energy level spectrum. 3. Experimental Observations of Intrinsic Excitons 3.1 Intrinsic

exciton

spectrum

and band

structure

A detailed study of the absorption edge of CdS was made by Dutton [11] in the temperature range 90 to 340 °K. The dichroism reported by Gobrecht and Bartschat [12] and also by Furlong and Ravilious [13] was clearly demonstrated. The reflection spectra observed at 81 °K is shown in Fig. 2. Dutton was unable to account for the source of the dichroism but felt that the absorption near the edge was due to exciton transitions. Gross and co-workers [14 to 16] studied the absorption near the edge in CdS at 4.2 °K. They attributed the short wavelength intrinsic absorption lines to exciton transitions. Birman [5] attacked the problem from a theoretical standpoint. Assuming a tight binding approximation in conjunction with group theory, he arrived at the irreducible representations, band symmetries, and selection rules for the zincblende and wurtzite structures. The interaction between the magnetic moment of the electron spin and orbital angular momentum may result in the splitting of energy states in crystals as it does in isolated atoms. Since the outer electrons in isolated atoms are chiefly responsible for the splitting, the splitting in the solid state is not unlike that in atoms. Application of group theory to the problem of spin orbit interaction in crystals yields information describing the symmetries of the electronic wavefunctions and the way in which states degenerate in the absence of the interaction split when the perturbation is turned on. Any calculation of the spin-orbit splitting in crystals must begin with some type approximation for the electronic wavef unctions. I n the tight-binding approximation, the wavef unctions at the center of the Brillouin Zone are taken Fig. 2. Reflection Spectrum. Reproductions of original data records giving reflected light intensity as a function of wavelength, (a) for light E|[c, (b) for E J _ c . The upturn at the left of each curve marked by heavy arrows is due to additional reflection from the second surface, since the crystal becomes transparent at these points. The vertical pips are wavelength markers. Note the similarity of anomoly B in the two polarizations (Dutton)

S o m e O p t i c a l P r o p e r t i e s of G r o u p I I - V I S e m i e o n d u e t o r s

661

(I)

to be linear combinations of atomic wavefunctions. In CdS it is assumed that the bottom of the conduction band is formed from the 5 S levels of Cd, and the upper valence bands are formed from the 3 P levels of sulfur. The upper valence band states are constructed out of appropriate linear combinations of products of P x , P y , and P z hydrogen-like orbitals with spin function. In the absence of both spin-orbit and crystalline field effects, these states are degenerate. The crystalline field of the wurtzite lattice partially removes this degeneracy, separating the P z form the P x and Pw orbitals. The J?x, P y band is further split by spin-orbit coupling. This structure at K = 0 along with the band symmetries and selection rules is shown in Fig. 1. This band structure could readily account for the dichroism observed in CdS and other wurtzite type structures, as described in Section 2. 3.2

Exciton

spectrum

of

CdS

Thomas andHopfield [17] studied the reflection spectra from CdS crystals at 77 °K and 4.2 °K. Their data at 77 °K were in good agreement with that obtained by Dutton. They observed one reflection peak at 2.544 eV (4873 A) active only for E_j_c and another at 2.559 eV (4844.5 A) active in both modes of polarization. These correspond to transitions involving the two top valence „ FLUOkWENT A PEAKS

B

H I -

\

' 4 : \

He

-t

A—

B'

^

NOISE LEVEL

1k

j

%

260

156 -—PHOTON

¿52 ENERGY (eV)

He

1—

te

— j —

v

n>

^ OA

\\

I Fig. 3. Reflection of CdS at 4.2 °K for light polarized with (a) E J _ e and (b) E||e. These are traces taken from microphotometer recordings of Kodak 103—0 plates. The noise level is as indicated in Fig. 3 a. The helium calibration lines indicate the resolution used (Thomas and Hopfield) 43»

\ L

^02

I 0 w

I 260

256

2.52 PHOTON ENERGr (ef)

662

D . C. R E Y N O L D S , C. W . LITTON, a n d T . C. COLLINS

CONDUCTION BAND

Fig. 4. Energy band structure of CdS at K = 0. The levels A, B, and C refer to the hole bands from which the exciton causing reflections A, B, and C arise. The band symmetry is given at the right (Thomas and Hopfield)

" h

bands. They also observed a third and broader peak at 2.616 eV that Dutton did not observe. This peak was active for both modes of polarization and is fff associated with transitions from the third valence band. They repeated the measurements at 4.2 °K and found a more complex reflection spectra. Densitometer traces are shown in Fig. 3 a and b. In addition to the parent transitions A, B, and C involt ving the three valence bands, additional Erte structure A' and B' was observed. The polarization properties of A' and B' VALENCE indicate that they are associated with 4 -Ec BANDS the transitions A and B respectively. £b S The energy separations between A — A' and B — B' are identical and equal _ to 0.021 eV. It is reasonable to assume that A and B correspond to the ground state excitons associated with the first and second valence bands respectively. The weaker transitions A' and B' result from the n = 2 states of the parent transitions. If the excitons are hydrogen-like, an estimate of the binding energy can be made (4/3x0.021 = 0.028 eV). Using this value for the binding energy and the expression [18] G

(3.1)

2 h2 e2

the reduced exciton mass was calculated giving a value of 0.18. From the analyses of their reflection data Thomas and Hopfield deduced many of the parameters relating to the band structure (Fig. 4) of CdS as follows: — -^exciton A

Eg

Eg — (Eb Eg - (Ec

2.554 eV , (3.2)

E A ) - E e x c i t o n * = 2.570 eV , EÄ)

-

-Eexciton

c = 2.632 eV

a represents the binding energy of exciton Assuming ^ e x c i t o n A = -^exciton B then

^exciton

E

a

—

E

b

=

A

and similarly for

0.016 eV ,

B

and

C.

(3.3)

with -^exciton a = 0.028 eV the band gap is Eg = 2.582 eV. In order to determine (Ec — EA ) it was necessary to compute 2?exciton c which they did using a quasi cubic model based on the similarity between the wurtzite and zincblende structures. The values obtained are -Bexciton c =

0.026 eV ,

Ea

- Ec = 0.073 eV .

(3.4)

Some Optical Properties of Group I I - V I Semiconductors (I) 3.3 Magneto

optical effects in the exciton

spectrum

663

of CdS

From the theory of section 2 it is seen that the observation of exciton spectra as a function of polarization and magnetic field and the interpretation of the spectra in the framework of the theory gives information relevant to the band structure of the material. Thomas and Hopfield [7] have made a detailed study of the exciton spectra in CdS. They observed two overlapping exciton series. The absorption lines observed for light polarized with E c and E\\c are shown in Tables 1 and 2 respectively. The lines identified as I a (a = 1, 2, . . .) are impurity lines and are discussed in the section on bound excitons. In the orientation E_\__c the ground state of the A exciton series is the transverse exciton. It is seen that the n = 2 state of the A exciton series overlaps the ground state of the B exciton series. For the orientation E\\c the lowest Table 1 A symmary of the data for some of the absorption lines seen in light polarized perpendicular to the hexagonal c-axis E J_ c, temperature 1.6 °K

Line

Ii h h An = i ( 1 S T ) h Bn = i An = 2

Posi tion

(Approximately apparent width)x10 3

(Â)

(eV)

(eV)

4888.6 4868.7 4859.0 4854.5 4837.7 4826.4 4812.9

2.5359 2.5463 2.5513 2.5537 2.5626 2.5686 2.5758

0.3 0.5 0.3 3.5 1.7 3.5 3.0

Crystal

GEB 3 GEAI GEA 1 GEA 1 GEB 3 GEA 1 GEA 1

Table 2 A summary of the data for some of the absorption lines seen in light polarized parallel to the hexagonal c-axis E\\c, temperature 1.6 °K

Line

Position

(Approximately apparent width)X10 3

J

(eV)

(eV)

2.5524 2.55455 2.5626 2.5687 2.57508 2.57575 2.57977 2.58094 2.59085

0.1

(Â) a (1 s r 6 ) AL (1 S £ ) h Bn = 1 A„ = 2 A„=s •An = 4 B„ = 2

4857.0 4852.9 4837.7 4826.1 4814.21 4812.96 4805.46 4803.28 4784.9

1.7 4.3 0.1 0.4 0.5 0.3 3.5

Crystal

GED7 GED7 GEB 3 GEA 1 GED 7 GED7 GED 7 GED 7 GEA 1

664

D . C. R E Y N O L D S , C. W . LITTON, a n d T . C . COLLINS

energy intrinsic transition is the _T6 exciton in which the hole and electron spins are parallel. This exciton splits in a magnetic field in the orientation (H||c) with a g-value of 2.93. I n the orientation E\\C t h e lowest energy A exciton is the longitudinal exciton. The absorption coefficient for this line was measured as a function of angle of incidence. As the crystal was rotated away f r o m normal incidence about an axis normal t o t h e c-axis the absorption coefficient increased rapidly. The effect was evident for rotations of less t h a n 10°. The deviation from normal incidence permits a component of the E vector in t h e direction perpendicular to c and t h u s interacts with the allowed r 5 transition. The exciton propagates in the direction of the photon which is almost in the direction of t h e polarization vector of the exciton which results in a longitudinal exciton [9]. The n = 2 state of A has two components as shown in Table 2. A densitometer trace of the two components is shown in Fig. 5 c. When viewed in a magnetic field of 31000 G, H\\e, (q is the wave vector of the exciting light) t h e line designated as P z shows a diamagnetic shift and also splits with a g-value of 0.62 + 0.06. This splitting along with t h e magnetic behavior of the other component of the n = 2 state is shown in Fig. 6. A total of seven lines are derived from the two zero field components. Absorption measurements were made at the peak magnetic field as the crystal was rotated about an axis normal to the c-axis. As the crystal was rotated away from normal incidence the line marked Sx rapidly increased in intensity. The results in Fig. 6 are shown for t h e orientation of t h e crystal c-axis 5° away from normal incidence. These rotational experiments show t h a t S^ is a longitudinal exciton derived from t h e S state active for light polarized with E_\_c. Observation of the line (ST, Px — i Pw) in the same rotational experiment described above showed t h a t it also increased in intensity with angle of rotation, but not as much as S £ . Also the intensity did not go to zero at normal incidence as was observed for SL. This intensity behavior indicates t h a t the transverse exciton is contained in this line. However, since it does not vanish a t

"J

PHOTON ENERGY (eV)

bj

-

c)

Tig. 5. Microphotometer t r a c e of Zeeman e f f e c t s of t h e n = 2 exciton s t a t e s in CdS crystal G E D 8 a t 1.6 ° K . c_|_JJ, HII®, qJ_H, a n d g j _ c . q is here t h e wave vector of t h e exciting p h o t o n (Hopfieid a n d Thomas)

Some Optical Properties of Group II-VI Semiconductors (I)

665

15770

2.S750 Fig. 6. The Zeeman effect at 1.6 ° K for H||e. At H = 0 the limits indicate the apparent line widths, otherwise they indicate the approximate uncertainty in the line positions (Hopfield and Thomas)

Fig. 7. The Zeeman effect at 1.6 °K for H J _ c and q\\H. i.e., the light is here travelling parallel to the magnetic field. In this diagram the lines represent theory; the mass values A have been used and the following values are employed: rydberg = 0.027 eV, y = 0.222, ge J _ = 1.73 and g\ _]__ = 8.3. As before, the limits on the experimental points a t H = 0 indicate the apparent Jinewidth; otherwise they indicate the approximate uncertainty in the line positions (Hopfield and Thomas)

normal incidence it must contain another state. This state must be a P state. The remaining state must be the lowest energy P state having zero angular momentum about the c-axis and denoted as P 2 . The magnetic field behavior of the two components of the A n= ,2 state for the orientation HJ_c,

0.0028 0.0017

0.0016;

The agreement between theory and experiment is quite good. The 2 P 0 state splits into two components in a magnetic field (H\ |c) as shown in Fig. 6. The _T5 P 2 state has antiparallel spins therefore the ¡/-value should be flre|l — ¡7/,11 . The _T6 P z state has parallel spins and the ¡/-value should be ffeii + 9h\\ • Since the -T6 transition is forbidden, the observed ¡/-value of g = = 0.62 i 0.06 is associated with the difference of the values. The ¡/-value of the 1 S r s state was determined to be 2.93 ± 0.03. The spin-orbit coupling in CdS is small, and the conduction band is treated as S-like; therefore, it is likely that the electron ¡7-value will be near —2.0. From the measured g- values the most reasonable electron and hole ¡7-values would be =

- 1 . 7 8 ± 0.05 ,

gh|| = - 1 . 1 5 ± 0.05 .

(3.5)

This electron (/-value agrees with that obtained by Lambe and Kikuchi [19] from spin resonance measurements. Hopfield and Thomas were further successful in analysing the higher excited states from which (/-values and effective masses of the electrons and holes both parallel and perpendicular to the c-axis were determined. I n Fig. 5 a and 5 b [20] for the orientation HJ_c, q_[_H and q_\_c it is seen that two different spectra are obtained for two directions of H. I n Fig. 5 a P j - , T?x- and T y - are more intenset han the -f- counterparts. — corresponds to the electron spin orientation parallel to the magnetic field and - f antiparallel. When the magnetic field is reversed in Fig. 5 b the intensity ratios reverse. I n crystals of the wurtzite structure, which do not have inversion symmetry, time reversal reverses the sign of the magnetic field, but leaves the selection rules for infinite-wavelength, plane-polarized light unchanged. I n zero magnetic field, the four states P x ± and V y ± (eight states including hole spin) are degenerate. Thomas and Hopfield have shown by utilizing group theory that one linear combination of these eight states (derived from i ^ ) has an optical matrix element in the limit q —> 0, and that one linear combination (derived from _T5) has an optical matrix element proportional to qy. Utilizing these linear combinations they computed the optical matrix elements for states P x ± . They found for the respective states having spins in the + and — ^-directions Matrix element for spin + x = A + qy B . Matrix element for spin — x = A — qy B .

(3.6)

The absolute magnitude of these two matrix elements are different as long as both A and B are nonzero. Reversing the direction of qy reverses the relative magnitudes of the two matrix elements. One state (say, with spin in the plus a-direction) has an energy corresponding to P x + or P ^ - , according to the direction of H. Reversing H therefore interchanges the energy levels of the two spin orientations without altering the optical matrix elements.

Some Optical Properties of Group I I - V I Semiconductors (I) 3.4 Exdton

structure

and Zeeman

667

effects in CdSe

The optical absorption and reflection spectra of CdSe were studied by Wheeler and Dimmock [8, 21], after earlier approaches by Gôrlich and Heyne Table 3 Series 1 From H\\c diamagnetic shift, /ux = 0.100 ± 0.005 From zero field positions, nx — 2 states hence:

a = 1 - fi x ej(/x x ez) = 0.32 ± 0.02, Pdfz = ° - 7 5 ± °- 0 4 > Pz = 0 1 3 ± 0 0 1 From H J_ c diamagnetic shift, ftJ/Xa = 0.77 ± 0.04, fiz = 0.13 ± 0.01, B y = 106 ± 5 cm" 1 , a 0 = 54 À, £ g a p = 14850.5 ± 2.0 cm-' Experimental

State IS 2P„ 2P±i 3P-n

14727 14818.6 14822.5 14839

± ± ± ±

1 0.3 0.2 1

Calculated

cm" 1 cm" 1 cm" 1 cm- 1

14734

± 6

cm" 1

14838.0 ± 1.5 cm" 1

Mass parameters m* x = 0.13 ± 0.01 m m*z = 0.13 ± 0.03 m

j m%x = 0.45 ± 0.09 m j m%z m Series 2

State IS 2S 2P„

Experimental

2P ± i

Calculated series limit By = 120 ± 10 cm" 1 HX = 0.11 ± 0.01 := 0.13 ± 0.01 m

14931 15032 15022 15050

± 3 ± 2 ± 3 ± 15

cm" 1 cm" 1 cm" 1 cm" 1

m%x = 0.9 ± 0.2 m

Series 3 IS Calculated series limit Crystal field splitting Spin-orbit splitting Electron ¡/-values3)

18218 18340 200 3490 \9ex\

± 10 cm" 1 ± 20 cm" 1 ± 15 cm" 1 ± 20 cm" 1 = 0 . 5 1 ±0.05;

\gez\ = 0.6

±o.i

3 ) There is some evidence that the electron g-values are negative. The Zeeman splitting between the 2 P ± i , F 1 — r 2 , and F 6 states indicates a negative gez.

668

D . C . R E Y N O L D S , C . W . LITTON, a n d T . C . COLLINS

[105] 4 ) had become known. The identification and interpretation of the spectra was aided by the Zeeman structure it displayed. The spectra were analysed in terms of the theory to obtain the band parameters at K = 0 . Three non-overlapping exciton series resulting from the three valence bands of the wurtzite structure were identified. Here one sees a significant difference between CdSe and CdS. In the case of CdS the splitting between the two top valence bands is less than the binding energy of the exciton. This leads to more valence band mixing, which is evidenced by the fact that unallowed transitions are observed in CdS. The agreement between theory and experiment is in general good for CdSe. The significant band parameters obtained by Wheeler and Dimmock are given in Table 3. 3.5 Exciton

spectrum

of ZnO

Thomas [22] has investigated the fundamental exciton structure of ZnO by absorption and reflection measurements. At 4.2 ° K he observes three reflection peaks, peaks A and B being active in the polarization mode E_\_c while peak C is active for E\\c. These peaks are identified with the ground state exciton for the three valence bands of ZnO. Three additional peaks are observed and iden-. tified as A', B', C'. These peaks are all on the short wavelength side of the respective A, B, C peaks by approximately the same energy, and are associated with the n = 2 state of the respective ground state excitons. The energy positions of these peaks with some appropriate parameters are given in Table 4. Table 4 (4.2 ° K )

Oscillator

A A' B B' C C'

Position (eV) (±0.0005 eV)

Difference

3.3768 3.4225 3.383 3.4275 3.4215 3.465

0.0457

(eV)

0.0445 0.0435

Oscillator strength

Full width at half height (eV)

13 xlO" 4 3.5 XlO" 4 45 X10~ 4 2.2 XlO" 4 60 XlO" 4 15 XlO" 4

0.0015 0.003 0.0022 0.0025 0.0017 0.009

Assuming the excitons are hydrogenic the ionization energy is calculated to be 0.059 eV, however, a reflection anomaly was not detected at the expected series limit. From the expression for the ionization energy of the exciton E

i

-

ei/xm p B2

'

(3-7)

where m is the free electron mass and using the low frequency dielectric constant 8.5 [23] the reduced exciton mass is /i = 0.31 m .

(3.8)

4 ) P . GÖHLICH and I. HEYNE, Optik 4, 206 (1948); for a summarizing presentation cf. also P. GÖRLICH, Photoeffekte, vol. 2, Akad. Verlagsges. Geest u. Portig K G , Leipzig 1963 (p. 1 0 7 ) .

Some Optical Properties of Group II-VI Semiconductors (I)

669

The exciton Bohr radius is given by A ^ J ^ n * .

(3.9)

jimer

For the first Bohr orbit

A = 14 A . (3.10) In absorption spectra a line is observed in the E\\c orientation at the energy of the A exciton line. This indicates a mixing of the A and C bands in ZnO. Here the crystal field splitting dominates, the spin orbit splitting being only one tenth the binding energy of the exciton. I n the case of CdS the spin orbit splitting dominates so that mixing occurs in the B and C bands so that the intensity of these two peaks is essentially equal in both modes of polarization. The interpretation of the experimental results of polarization in ZnO places the ri valence band above the I\ valence band. These are the PXiV valence bands whose degeneracy is lifted primarily by spin orbit interaction. This aspect of the experiment is not in agreement with the theory of Hopfield [6]. Placing the _T7 band above the f 9 band places the wrong sign on the spin orbit energy. I n the case of atoms an inverted multiplet can occur only from configuration interactions between different spatial one-electron wavefunctions. I t may be that a model which does not allow for such interactions is not adequate in the case of ZnO. 3.6 Exciton

spectra

of ZnS, ZnSe, and

CdTe

Birman, Samelson and Lempicki [24] have studied the reflectivity spectra of structurally pure, cubic and hexagonal ZnS single crystals (platelets) at 77 °K and 14 °K. The cubic crystals were doped with 1 mol% CI. They have examined the spectra near the fundamental absorption edge and find two reflection anomalies in the cubic {A' and B') and three (A, B and C) in the hexagonal ZnS. Fig. 8 shows a cubic ZnS spectrum at 14 °K, while Fig. 9 shows a spectrum at 77 °K for crystals of the hexagonal modification. Although the peaks are not too sharp for the cubic crystal at 14 °K, the reflection anomalies in both the cubic and hexagonal crystals undoubtedly derive from the ground state exciton transition, at least at 14 °K. Excited states of the exiton were not obser-

Fig. 8. Reflection coefficients of cubic ZnS at 14 °K (Birman, Samelson, and Lempicki)

Fig. 9. Reflection coefficients of hexagonal zinc sulfide at 77 °K (Birman, Samelson, and Lempicki)

670

D . C. R E Y N O L D S , C. W . L I T T O N , a n d T . C. COLLINS

ved. With regard to spectral resolution, the spectra were automatically recorded -with a Hilger D121 double monochromator; a spectral band width of about 2.5 A was achieved. As illustrated in Fig. 9, they have also observed the absorption dichroism in the wurtzite ZnS, i.e., the ^4-peak was active only for the E_|_c mode of polarization, while the B and C peaks were active for both EJ_c and E\\c. As in CdS, they have explained the dichroism in terms of the group theoretic arguments of symmetry-allowed and symmetryforbidden transitions; this may best be visualized in terms of the energy band models for the zincblende and wurtzite structures shown in Fig. 1. The average line positions (resonant frequencies or energies associated with the reflection peaks), together with the energy band separations, are given in Table 2. A Kramers-Kronig inversion analysis was not performed on the data to determine the resonant frequencies associated with the reflection anomalies. The resonance frequencies were determined approximately by locating the inflection points (inflection between maxima and minima) on the reflectivity curves. The error introduced by such an approximation is probably not greater than 10 3 eV. Aven, Marple and Segall [25] (AMS) have measured the absorption and reflectivity spectra of cubic ZnSe single crystals at 300 and 23 ° K . Shown in Fig. 10 is a plot of the absorption coefficient, a, as a function of h v in the absorption edge region. The curve at 23 ° K (black points) was obtained from both absorption and reflectivity measurements. The curve up to about 2.795 eV is calculated from absorption values while the cur-ve from about 2.785 eV toward higher energies is calculated from reflectivity spectra using a Kramers-Kronig inversion analysis to calculate a spectrum of 2 n k vs. h v and hence i, and this may be seen from (4.20), (4.21) and (4.22) in the above Section. In the development of their model of spatial resonance dispersion, Hopfield and Thomas [37] have considered in detail the problem of exciton reflection by an infinite potential barrier near a crystal surface. Shown in Fig. 20 is a graphical illustration of the effect of exciton reflection from potential barriers at arbitrary depths, I, below the crystal surface. Here they have calculated a reflection anomaly from the SRD (spatial resonance dispersion) model using CdS parameters for the A-band (the SRD model predicts the extra peak in the reflectivity minimum). One can see that as the barrier depth increases, the subsidiary maximum in the reflection anomaly minimum increases in intensity; at the same time, one can also observe that the principal maximum decreases in intensity with increasing I until it is almost flattened out at 1 = 154 A. As can be seen from Fig. 20 d, I can be increased to the point where the subsidiary maximum becomes more intense than the principal maximum in the reflection anomaly. At this point the anomaly appears to have suffered a reversal in shape (intensity profile) i.e., the reflection maximum appears to be at higher energy than the reflection minimum, rather than vice versa; also, the principal reflection maximum (the low-energy maximum) is flattened not unlike one would expect for an increased damping factor in classical dispersion theory. The effective barrier depth for the CdS ^4-band is in the range of 80 to 100 A, as can be seen by comparison of Fig. 20a to 20d with Fig. 19a [fc_|_c]. Since the

b%(Xo-0.125 co-8.0 Z0.80

H

0.60

-

3.60

1.80—f 0.40

Fig. 21. The calculated reflectivity taking s p a t i a l dispersion into account (but without damping) for the case I = 0 as a function of the exciton effective m a s s (in units of free electron mass). For an infinite exciton m a s s , the classical result of total reflection would occur between the indicated l i m i t s ( H o p f i e l d a n d T h o m a s )

\ 1

% cy

—j

0.90-J/\ 0M5 S

0.20

l

2M5

I—CLASSICAL TOTAL I REFLECTION REGION 1

jl

111

| 1 1 1 1 1 1 1

i

i III 2.550

i 2.555

Some Optical Properties of Group II-VI Semiconductors (I)

683

Bohr radius of the >4-exciton in CdS is approximately 27 A, we may conclude that the 4-exciton is reflected from an infinite potential barrier whose depth, I, below the crystal surface is of the order of a few Bohr radii. In terms of the SRD model, Hopfield and Thomas have also shown how the exciton effective mass, ¡x, effects the shape of a calculated reflection anomaly in CdS. For example, a finite increase in ¡jl causes an increase in the principal maximum, while a decrease in ¡1 gives rise to a decrease in the peak height, rather like one would expect for an increased damping. This is shown in Fig. 21 where they have calculated the reflectivity for the case 1 = 0, using the effective masses shown and the other CdS parameters. The effects of varying ¡x and I can now be compared in Fig. 20 and 21. As can be seen from Fig. 18, the unusual shapes and subsidiary structures are not confined to the 4-band reflection anomaly; in fact, every reflectivity peak in CdS is characterized by an unusual shape. In the B band anomaly, e.g., there is an extra peak in thefc_|_cdirection (EJ_c) which is not analogous to the extra peak observed in the A anomaly: The extra peak in B can be explained in terms of SRD effects but is of different origin than that of the extra peak in A. This will be discussed below. Also observed in Fig. 18 are the unusual shapes of the A-exciton exited states — the excited states of the _B-exciton are similar in appearance. In the k_|_c case, note that the reflection anomalies for the A n = 2 and A n = s excited states appear to be reversed, i.e., the reflection minimum appears at lower energy than the maximum. Actually, the maxima in the An=2 and An=3 anomalies are the extra peaks that arise from SRD effects, namely exciton reflection from a potential barrier, as in the ground state of exciton A. It is interesting to make a qualitative comparison of the An=n and An=s peaks of Fig. 18 with the calculated peaks of Fig. 20. From such a comparison, it appears that excitons in these excited states are reflected from potential barriers that are, when compared to the barriers for the ground state, at a somewhat greater distance, I, below the crystal surface. It has been suggested that I is probably on the order of the exciton Bohr radius for all semiconductors [38]. Since the exciton Bohr radius increases as the square of the principal quantum number for the excited states, one would expect the subsidiary maximum to become more pronounced in the excited-state reflection anomalies. Such arguments provide a natural explanation for the apparent "reversal" of the excited-state anomalies. In a later paper, Mahan and Hopfield [38] have treated spatial resonance dispersion effects in still further detail; in this treatment, they have considered the origin of the peak in the 5 - b a n d anomaly (See Fig. 18b). They have attributed this subsidiary peak in the -B-anomaly, observed only for the casefcJ_C, E±c, to energy terms linear in wavevector for the second and third valence bands. Such energy terms (called linear crossing terms) will lift a degeneracy between two states at fc = 0. The energy term contains the factor 0 , called the exciton splitting factor; it is this factor in the exciton Hamiltonian (off-diagonal element) that gives rise to a mixing of exciton states. The mixing of exciton states gives rise, in turn, to a splitting of the r s exciton state into its longitudinal and transverse parts, _T6 L and T^y at 1c = 0. The known energy value for the t exciton state places the _T5 T energetically near the extra peak in the Banomaly; also SRD theory predicts an extra peak at the frequency of the longitudinal exciton, r & L . The L peak is not observed experimentally, probably because the spectral line width is too great.

684

D. C. REYNOLDS et al.: Optical Properties of I I - V I Semiconductors FL) References

[1] J . FRENKEL, Phys. Rev. 37, 17 (1931). [2] G. H. WANNIER, Phys. Rev. 52, 191 (1937). [3] R. S. KNOX, in: Solid State Physics, edited b y F. SEITZ and D. TURNBULL, Academic Press, Inc., New York 1963 (Suppl. 5). [4] J . J . HOPFIELD and D. G. THOMAS, J . Phys. Chem. Solids 12, 276 (1960). [5] For Example: G . DRESSELHAUS, P h y s . R e v . 1 0 5 , 1 3 5 ( 1 9 5 7 ) ;

J . L. BIRMAN, Phys. Rev. Letters 2, 157 (1959); J . Phys. Chem. Solids 8, 35 (1959); Phys. Rev. 114, 1490 (1959); R. C. CASELLA, Phys. Rev. 114, 1514 (1959); M. BALKANSKI and J . DES CLOIZEAUX, J . Phys. Radium 21, 825 (1960). [6] J . J . HOPFIELD, J . Phys. Chem. Solids 16, 97 (1960). [7] J . J . HOPFIELD a n d D. G. THOMAS, P h y s . R e v . 122, 35 (1961). [8] R . G . WHEELER a n d J . 0 . DIMMOCK, P h y s . R e v . 1 2 5 , 1 8 0 5 ( 1 9 6 2 ) .

[9] J . J . HOPFIELD and D. G. THOMAS, J . Phys. Chem. Solids 12, 276 (1960). [ 1 0 ] D . G . THOMAS a n d J . J . HOPFIELD, P h y s . R e v . 1 2 4 , 6 5 7 ( 1 9 6 1 ) . [11] D . DUTTON, P h y s . R e v . 1 1 2 , 7 8 5 (1958).

[12] H . GOBRECHT a n d A. BARTSCHAT, Z. P h y s . 136, 224 (1953). [13] L. R . FURLONG a n d C. F . RAVILIOUS, P h y s . R e v . 98, 954 (1955).

[14] E. F. GROSS, Nuovo Cimento Suppl. 3, 672 (1956). [ 1 5 ] E . F . GROSS, B . S . RAZBIRIN, a n d M . JAKOBSON, Z h . t e h k . F i z . 2 7 , 1 1 4 9 ( 1 9 5 7 ) ; ( T r a n s -

lation: Soviet Physics J . tech. Phys. 2, 1043 (1957)). [16] E. F. GROSS and B. S. RAZBIRIN, Zh. tekh. Fiz. 27, 2173 (1957); (Translation: Soviet Physics J . tech. Phys. 2, 2014 (1957)). [17] D . G . THOMAS a n d J . J . HOPFIELD, P h y s

R e v . 1 1 6 , 5 7 3 (1959).

[18] R. J . ELLIOTT, Phys. Rev. 108, 1384 (1957). [19] J . LAMBE a n d C. KIKUCHI, J . P h y s . Chem. Solids 8, 492 (1959). [ 2 0 ] J . J . HOPFIELD a n d D . G . THOMAS, P h y s . R e v . L e t t e r s 4 , 3 5 7 ( 1 9 6 0 ) . [21] J . O. DIMMOCK a n d R . G . WHEELER, J . a p p l . P h y s . 3 2 , 2 2 7 1 ( 1 9 6 1 ) .

[22] D. G. THOMAS, J . Phys. Chem. Solids 15, 86 (1960). [23] A . R . HUTSON, P h y s . R e v . 1 0 8 , 2 2 2 (1957). [ 2 4 ] J . L . BIRMAN, H . SAMELSON, a n d A . LEMPICKI, G . T . & E . R e s . D e v e l o p m . J . 1 , 2 ( 1 9 6 1 ) .

[25] M. AVEN, D. MARPLE, and B. SEGALL, J . appl. Phys. Suppl. 32, 226 (1961). [26] D. G. THOMAS, J . appl. Phys. Suppl. 32, 2298 (1961) [27] K . W . BOER a n d H . GUTJAHR, Z. P h y s . 152, 203 (1958).

[28] E. F. GROSS and B. V. NOVIKOV, Soviet Physics - Solid State Physics 1, 321 (1959). [29] V. V. EREMENKO, Soviet Physics - Solid State Physics 2, 2315 (1961). [30] E. F. GROSS, K. F. LIDER, and B. V. NOVIKOV, Soviet Physics — Solid State Physics 4 , 8 3 6 (1962).

[31] Y. S. PARK and D. C. REYNOLDS, Phys. Rev. 132, 2450 (1963). [32] Y. S. PARK and D. L. LANGER, Phys. Rev. Letters 13, 392 (1964). [33] H . J . STOCKER, C. STANNARD J R . , H . KAPLAN, a n d H . LEVINSTEIN, P h y s . R e v . L e t t e r s 12, 163 (1964). [ 3 4 ] M . H . HABEGGER a n d H . Y . FAN, P h y s . R e v . L e t t e r s 1 2 , 9 9 ( 1 9 6 4 ) . [ 3 5 ] R . MARSHALL a n d S. S . MITRA, P h y s . R e v . 1 3 4 , A 1 0 1 9 ( 1 9 6 4 ) .

[36] S. I. PEKAR, Soviet Physics -

J E T P 6, 785 (1958); Soviet Physics - Solid State

P h y s i c s 4, 953 (1962).

[37] J . J . HOPFIELD a n d D. G. THOMAS, P h y s . R e v . 132, 563 (1963). [38] G . D . MAHAN a n d J . J . HOPFIELD, P h y s . R e v . 1 3 5 , A 4 2 8 ( 1 9 6 4 ) .

(Received January 6, 1965)

Original

Papers

phys. stat. sol. 9, 685 (1965)

Hmmumym 0U3UKU AnadeMuu. Hayn JIameuucKou CCP, Puea K TepMOAHHaMHKe

raii3eH6eproBCKoro

4>eppoMarHeTHKa

(III)

B. H. KameeB C noMombio pacuenjieHHH ToMHTbi-TaHaKH [10] nojiyneHO BbipaweHHe hjih KoppejiHTopa npoflojibHbix K0Mn0HeHT ciihhob H30TponHoro rail3eHSeproBCKoro $eppoMarHeTHKa. IIpH h h 3 k h x T e M n e p a r y p a x h MajiHX |g| oho cobnanaeT c pe3yjibTaTOM Teopmi cnniiOBtix bojih. TepMommaMHKa MaraeTHKa npw b h c o k h x TeMnepaTypax, nocTpoeHHan c Hcn0Jib30BaHHeM aToro KoppejiHTopa, npHBO«HT k cJienyromHM pe3yjibTaTaM: TeMnepaTypa KiopH Tc, HecKOjibKO 6ojibrnaH, MeM b n p y r n x Teopwax, Hcnojib3yiomHx opMajiH3M yHKUHft TpHHa; nojiioc BToporo nopfWKa b TOK^e K i o p n hjih npoAOJibHoii napaMarHHTHoii bociiphhmih-

bocth: oSbWHan j/r c — T 0C06eHH0CTb cnoHTaHHOil HaMarHHHeHHOCTH B Tc;

KOHe^HWii CHa^OK enHHOBOH TCIIJIOCMKOCTH B TOHKe K r o p n .

On the basis of the Tanaka-Tomita splitting [10], an expression is derived for the correlation fc -

co,) (JVfc + 1)] = -

Ä 2 « ( - co) • ( 2 . 1 9 )

3aMeTHM TyT m e , »ito HeneTHocTb (JiyHKijHñ i?¿,(co) y c T a H a B H H B a e T c a c p a 3 y , ecJiH b ¿ " b ( 2 . 1 8 ) h ( 2 . 1 9 ) coBepniHTb c n B i i r a p r y M e H T a —k—q h ynecTb ( 2 . 1 6h) . I I p H noACTaHOBKe ( 2 . 1 7 ) b (2.7) Hcnojib3yeM n3BecTHoe TownecTBo (x ± i e)" 1 = P a r 1 =F i n ô(x)

;

e ^ + 0

(2.20)

( P — CHMBon r j i a B H o r o 3HaHeHHH), KOTopoe npHMeHHTejibHO K ( 2 . 2 8 ) , ( 2 . 1 9 ) npHMeT BHJJ: R i q (co + i e ) = P¿,(co)

i riq(œ)

;

rlq(a>) = ~E (u)k — co, + f c ) (Nq+u — Nk) ô (to + iv

k

Ag( - N(Szy2]

(3.7)

(fieppoMarHeTHKa.

IIoaTOMy BCioay z z B H a j i b H e n m e M Mbi 6 y « e M p a c c M a T p H B a T b B e j i H H H H y (S (q) S (— q)y n p n q^O. HMCH TOHHoe (B n p e n e j i a x npHMeHHMocTH TeopHH cnHHOBbix BOJIH) B b i p a î K e H H e ( 3 . 6 ) , MH c p a B H H M e r o c n o j i y n a i o m H M C H n o $ o p M y j i e n P H / S > / S c , Ä = 0.

(2.24)

CpaBHeHHe BbipaHieHHH ( 2 . 2 4 ) h ( 3 . 6 ) 6 y n e T n p 0 H 3 B e n e H 0 n p H T a n n a n o 6 a 3TH B b i p a w e H H H p a c x o n H T c n , TO H a n ö o j i e e H H T e p e c H O n o B e z z HEHHE