EXCHANGE INTERACTIONS BETWEEN LOCALIZED AND DELOCALIZED ELECTRONS IN SEMIMAGNETIC COMPOUNDS

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EXCHANGE INTERACTIONS BETWEEN

LOCALIZED AND DELOCALIZED ELECTRONS IN SEMIMAGNETIC COMPOUNDS

Gérald Bastard, J. Gaj, R. Planel, C. Rigaux

To cite this version:

Gérald Bastard, J. Gaj, R. Planel, C. Rigaux. EXCHANGE INTERACTIONS BETWEEN LO-

CALIZED AND DELOCALIZED ELECTRONS IN SEMIMAGNETIC COMPOUNDS. Journal de

Physique Colloques, 1980, 41 (C5), pp.C5-247-C5-254. �10.1051/jphyscol:1980542�. �jpa-00219977�

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EXCHANGE INTERACTIONS BETWEEN LOCALIZED AND DELOCALIZED ELECTRONS IN SEMIMAGNET1C COMPOUNDS.

G. Bastard""", J.A. Gaj** R. Planel+ and C. Rigaux +

+ Groupe de Physique des Solides de I'Eoole Normale Superieure. Lahoratoire associe au Centre National de la Recherche Saientifique, 24, rue Lhomond 75231 Paris Cedex 05, Franee. Universite Paris VII, 2, Place Jussieu 75221 Paris Cedex 06 France.

£ On leave from Institut of Experimental Physics, Warsaw University Hoza 69, Warsaw, Poland.

Résumé.- Nous reportons dans cet article les résultats d'études magnétooptiques effectuées sur les alliages ternaires Hg1_kMn,Te (k<0,15) et Cd1_ Mn Te (x<0,5).

Dans les composés Cd!_xiMnxTe les spectres Se niagnétoréf lectivitë préientent des structures bien définies attribuables à des transitions d'excitons. Les alliages Hg jjyLn^Te de bande interdite nulle ou faible sont caractérisés par des niveaux de Landau très bien définis. Les spectres de magnëtotransmission présentent une

structure oscillatoire due à l'absorption optique entre niveaux de Landau. Les deux séries d'alliages présentent un très fort accroissement des propriétés de spin des électrons délocalisés qui dépend très sensiblement de la température (T = 4.2K et T = 2K). Tous les spectres sont quantitativement interprétés en tenant compte de l'interaction d'échange entre les électrons délocalisés et les électrons localisés des ions M n2 + (effet du premier ordre). Les intégrales d'échange dans les bandes s et p sont déterminées et l'aimantation des électrons d peut être mesurée opti- quement. Dans les deux séries d'alliages les spins localisés Mnz + présentent une interaction antiferromagnétique.

Abstract.- In this paper we report magneto-optical measurements performed on the semi-magnetic ternary random alloys Hgj_Jta, Te (k<0.15) and Cdl_xMn Te (x<0.5).

In Cdi.jjMn Te compounds sharp magneto-rexlectivity structures are" associated with excitonic transitions. Narrow or zero gap Hgj_jtMn. Te alloys exhibit a pronounced Landau quantization leading to oscillatory magneto-transmission patterns connected with inter Landau levels transitions. In both kinds of compounds a dramatic increase of the spin properties of mobile carriers is observed which is strongly temperature dependent at low temperatures (T = 4.2Kand T = 2K). All the spectra are quantita- tively interpreted by including the s-d and p-d exchange interaction between the mobile carriers and the localized d electrons of Mn2 + ions (first order effect).

The exchange integrals in s and p bands are obtained and the magnetization of the d electrons can be optically measured. An antiferromagnetic interaction between Mn2 + localized spins is clearly evidenced in both series of alloys.

1. Introduction.- The semimagnetic ternary random alloys Hg._. Mn, Te, Cd1_xMnxTe have recently received increasing attention.

Their main interest is to gradually incorpo- rate the exchange phenomena, characteristic of magnetic semiconductors, to the well- known scheme of classical cubic semiconductors. Moreover, Hg, .Mn.Te and Cd, Mn Te compounds provide a wide variety of one- electron band structures : the former alloys enable the study of s - d and p - d exchange interactions when the delocalized carriers eigenstates are Landau levels ; whereas the Cd, Mn Te alloys, large gap materials, exemplify the influence of s - d and p - d exchange effects on exciton states.

In this review we shall concentrate on magnetooptical experiments, but we want to stress that very interesting phenomena were also observed in the Shubnikov-de Haas effect of HgMnTe alloys /4/.

2. Model.- In semimagnetic compounds, HgMnTe, CdMnTe ..., the electron states can be divided into :

- localized d states, originating from the half-filled d shells of the Mn atoms. These are characterized by spatially localized wavefunctions centered around the Mn sites. The d electrons form localized magnetic moments. According to Hund's rule, the five d electrons group into a s.,, multiplet whose degeneracy is not lifted by

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980542

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JOURNAL DE PHYSIQUE

the cubic ligand field,

-

extended states giving rise to wide bands. These are built from the hybridiza-

2 2 2

tion of 6 s (Hg) or 5 s (Cd) or 4 s ( M n ) with 5 p (Te) orbitals. 6

The hamiltonian describing the electronic states of Hgl-kMnkTe, Cdl-xMnxTe alloys can then be written as :

H = H + H

c d

'

^Hcd ⁽¹⁾

Hc is the one electron band structure hamiltonian, Hd the hamiltonian of the d levels and Hcd the exchange term between localized and delocalized Carriers. It is possible to account for this exchange interaction /1/ by setting :

where

zi

^is

+ 5

Ri (Si = Z) carrier and

the localized spin at the site

; -+ o the spin of the mobile J

(z

-Si) an exchange integral rapidly varying over a unit cell.

In the following, we shall only account for Hcd by a first order perturbation calculus and replace H by its thermo- dynamical average <Hcd> (2-6'. This molecular field approach is justified since the eigenfunctions of Hc are spatially extended over the whole crystal and average the fluctuations Hcd

-

<Hcd> to zero.Hence :

If the localized moments were not interacting <S z > would reduce to a Brillouin function corresponding to a spin 5/2. In both Hgl-klfinkTe and Cdl-xMnxTe alloys, there exist significant Mn

-

^{M n}interactions which are antiferromagnetic /3,4,7/. Since there is practically no free carriers in HgMnTe and CdbmTe alloys, all the intraband

processes may be discarded. The actual antiferromagnetic interaction arises from

virtual interband processes (Bloembergen and Rowland mechanism) /8/. At low temperature, no explicit form of < S z > may be guessed a priori.One of the most striking results of magnetooptical experiments

involving delocalized electrons will be a precise determination of the d electrons magnetization. However, to extract this

information from the magnetooptical spectra it is necessary to know the very details of the symmetry and band parameters of the eigenstates of Hc.

2.1. Eigenstates of Hc

. -

VThereas CdmTe alloys are "classical" semiconductors with a rather large gap, the HgMnTe alloys exhibit narrow or zerogap properties.

2.1.1 _Ze~o-m_ag$e_t&c_~ie&~.- At H = 0, the CdMnTe alloys display the conventional ban2 structure of cubic I1 VI semiconductors. At the center of the Brillouin zone

( r

point), the

r6

conducti.on band (two s-type levels) is separated by an energy gap co from the

t w o upper

r8

valence bands (four p-type

levels with total angular momentum J=3/2).

A lower valence band (r7 symmetry, L=l, J=1/2) is split by the spin orbit coupling from the I' 8 bands ( A = E ~ ~ - E ~ ~ ~ U ~ ~ V ) 'b (Fig. 1)

.

Fig.1 : Band structures and interaction gap

E~ versus composition in Cdl-xMnxTe and Hgl_pnkTe alloys (T=4.2 K)

.

s o increases linearly from 1.6eV (CdTe) to

3.2eV (hypothetic cubic MnTe). Away from k=O, the I'g levels give rise to light

(r81h) and heavy hole T8 v bands.

Hgi-kMnkTe alloys may show either the CdMnTe band ordering (k>7% at T=4.2 K) or the zerogap (Z.G.) structure. The latter is

-

characterized by an inversion of the relative positions of T6 and T8 states : E~ =

- E < 0 (Fig.l). At k # 0,

r8

^levels

'I'6 _{r 8}

(4)

vanishing near k = 7% /2,4/ where a zerogap

+ semiconductor transition takes place.

+ +

From k.p pertubation calculus one obtains the effective masses and gyromagnetic factors at the band edges :

where mo is the free electron mass. In eq.

(4) K., yl, y, F are small parameters (of the order of one) : the dominant contributions arise from the

2.6

coupling between s and p levels (Ep

*

^{19 eV)}

^.

Note that mrR > 0 , gri < 0, describes the light electron band in 'z.G. 8 materials

(&o<O) whereas m R<O, g R > O accounts for the r8

light hole levels

r8

^r8in open gap compounds (CdPln~e and Hgl-kMnkTe, k> 0.0 7)

.

2.1.2. Effect ~f_ag-externs&lgggneti~ fie&+.- The basic parameter which governs the behaviour of HgMnTe and CdMnTe alloys in a magnetic field is the order of magnitude of the band gap (co) separating s and p levels.

In HgMnTe alloys, a small gap

1 ~

( I E ~ ~ <

~ 1

0.3 eV) leads to small effective masses (mr6, m ^< 3~10-~m,,) i.e. large cyclotron

8

frequencies and large g factors (g > 40).

An external magnetic field

8

strongly affects the delocalized electron states by quanti- zing the electronic motion into well defined one-dimensional Landau subbands (w - r > > l )

C

even at low fields.

On the other hand, in CdMnTe alloys, the Landau quantization is almost negli- geable since E is large (E % 2eV) and

o a

cyclotron splittings rather small.

The effective mass approximation

functions at the

r

point (1 = 1, 2,

...

⁸⁾^;

FNl (r) are slowly varying envelope functions and C1(N) are normalization coefficients.

If .Q is the volume of the crystal one has : 1 -+ +

F ~ l

( :

) = exp i k.r for CdMnTe alloys (6 whereas the Landau quantization leads to :

in HgMnTe alloys.

In eq(7), h2 = Kc/eH is the magnetic length and 4; is the nth oscillator function centered at

-

^~~k_Y'

2.2, Pertubation treatment.- The effect of

<Hcd> is treated by first order pertubation calculus taking into account the quasi degeneracy of some unperturbed states /3,4, 6/

-

2.2.1. IlqklnTe alloys-C~,+~.- Due to the smallness of the 8 bands are conside- red as'being quasi degenerate. When projec- ted on the 8 ulo Bloch functions, <Hcd> is expressed by a quasi diagonal matrix The F ~ l

are now the eigenstates of : (HI + (F) = E (F)

-

⁽⁸

where H (H) is the 8x8 unperturbed hamilto- a

nian in the presence of an external magnetic field which exactly diagonalizes the

g.5

interaction inside the ulo basis /lo/. Vll,

= < u lo has only diagonal terms

except for two off-diagonal

r7-r8

components Vll, depends on H through <S

z

> and involves only two parameters :

a =

<s/J(~)/s>

and

B=<X/J

/x>

= <Y/J

(g)

^/Y>= <Z/J (jr) /z> ( 9 ) which represent the s

-

d and p

-

d exchange integrals respectively.

2

-

²

-

²

- E~?!?!TS-~~~~YS-LPL.-

As Eo is large, H is taken independent of the magneticfield

6

^The

^2.;

interaction is taken into account up to k2. This enables one to rewrite H in the form of three independent block matrices

a

17

(5)

representing the effective hamiltonians in

r 6

bands (2x2) ;

r 8

p bands (4x4) and T,p bands (2x2). Neglecting the

r

7 8

-r

coupling, V does not couple these effective hamiltonians : the exchange perturbation <Hcd> is diagonalized separately in the

r 6

and

r 8

subspaces.

As a result, we obtain k-dependent eigenstates whose Kramers degeneracy is completely lifted owing to the exchange pertubation.

The conduction S type levels retain very simple dispersion relations :

whereas the T8 valence energy levels cannot be obtqined in closed forms. In Eq.(lO)

*&~n is the number of Mn ions per unit

-

R volume

.

In CdMnTe alloys electron-hole interaction gives rise to significant exciton effects. To build the exciton levels, the simplest approach is to assign an exciton state to each interband transition taking place between exchange-perturbed k-dependent eigenstates.

2.2.3. Pertubed _q_ factors.- The previous pertubation treatments enable one to derive the effective masses and gyromagnetic factors of tt!e exchange pertubed eigenstates. Near the band edges (parabolic limit), one finds:

As a conclusion, the exchange perturbation <HCd >leaves the effective masses unchanged but modifies the g factors. Such an effect was expected since < H > displays

c?

the same analytical form as an extra Zeeman term.

2.2.4. "Tep2ggggre depe~aegt band structure':- At low temperature (T 5 4.2K) band

----

parameters of non magnetic semiconductors hardly show an appreciable variation. Such

is not the case for HgblnTe and CdMnTe alloys, since the s

-

^{d, p}

-

d exchange couplings lead to eigenenergies which depend on the d electron magnetization.

Under a magnetic field, the latter conside- rably changes with T (Curie Weiss behaviour).

Hence, one may expect unusual modifications of the magnetooptical spectra as T. is lowered, for example from 4.2 K to 2 K. This effect is one of the most convincing proofs of the influence of H

cd'

3. Results in CdMnTe alloys.- The Cdl-xMnxTe alloys have been first grown by R.R. Galazka for various Mn concentrations, up to x = 0.7 /11/, with a good control on composition Basic measurements attest the semiconducting character of these compounds, with the same zinc blende structure as pure CdTe, and show a linear increase with x of the energy gap between the valence and conduction bands. Actually, around the composition x = 0.5, this energy becomes greater than the transition energy (fiu =2.2 eV) due to the optical excitation of the localized Mn states, and the absorption edge remains constant /12/.

As a consequence, most experiments have been, up to now, performed on smaller composition where the optical properties in zero magnetic field are very similar to those of a classical cubic semiconductor :

the absorption peak or reflectivity structure can be attributed, by continuity from x+O, to the free exciton states associated with the valence-to-conduction band transitions. One just notices a broadening of these structures with increasing alloying, which could actuallv be expected /11,/.

Under external maqnetic field, the structure associated with exciton ground state is strongly split into six components /9/ as shown on figure 2 : two in LI

+

polarization, two in u- polarization (Faraday configuration) and two in .rr polarization

(Voigt configuration). Let us point out that these splittings AE, expressed in terms of' "effective g factors" (g=AE/pB~ )

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164 Id3 162 161 PHOTON ENERGY [ r V )

unit volume, x the Mn mole fraction and

<sZ>

the mean spin value of a Mn 2+ ion.

Between the two weaker conponetts, the splittings will be :

The linear dependencesof the splittings AEl, AE2 on

<sZ>

can be checked by comparison of the optical data with the Fiq. 2 : Exciton reflection spectra of direct measurement of the magnetization

~ d - Mn 2Te at T = 1.4 K and 11 kG.

(aPtZB r&f?rence [91)

.

using classical extraction technics /7/.

This is done on figure 4 for several PJI These six components may be easily explained and magnetic field intensi-

from a simple pattern of the J53/2 valence ties. Let us point out at this stage some band (four fold degenerate) and J = 1/2 consequences of such a good agreement.

conduction band (two fold degenerate) (Fig.

3) in which all the effects of the magnetic field are neglected on the delocalized electron states, except the giant exchange splittings of the two bands due to the magnetization of the ~ n ~ + i o n s ( 6 ) .

Fig. 3 : Pattern of optical transitions Fig. 4 : Splitting of strong components of between valence (J = 3/2) and conduction free exciton line in Cd Mn Te plotted

(J = 1 / 2 ) election states. Note that 2 versus mean value per

uAir

cgll of the

transitions remain optilally forbidden in spin along the magnetic field. Composition any polarization (AM =

-

2). (after ref./9/) values indicated in %. (after ref./7/).

100-

-

a

z

_t

t 10- -I Q cD

Z 0

t

z

1

According to'this pattern, only two exchange parameters, a for the conduction band and

B

for the valence band, determined the observed energy shifts for a given rnagneti- zation. For example, the energy difference between the two stronger components in Faraday configuration (corresponding to the

(-3/2, -1/2) : a+ and to the (+3/2, +1/2) :

a transitions) will be readily

-

:

X 0.5

; ;

+ 5 0 10 - A 20

30

- /'

B

-

-/ 8 /"

/ I I , I

i) The experimental exchange parameters are reasonably independent of the concentration, magnetic field and temperature. From AEl, AE and x <sZ>, one gets

2

/7/ N a = -0.22 eV and N o B =

+

0.88 eV.

ii) The use of molecular field

approximation to describe the s

-

d exchange interaction seems justified.

iii) It seems also sufficient to

-10-2 -10-1

SPIN PER UNIT CELL

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neglect all the direct influences of the magnetic field on the delocalized band states, and to build up excitons with each pair of split bands. However, this should not be true neither for the excitonic exci- ted states, as shown by recent magnetoab- sorption experiments /13/, nor in very high magnetic fields /14/. In these cases, more sophisticated models based on the Altarelli Lipari description of excitons in cubic semiconductors could be useful /9,13/.

iv) One gets an optical access to the magnetization of the ~ n ions ~ +: as an illustration, the experimental splittings depend on temperature as well as magnetic field and Mn concentration. As a consequence, one may consider the curves of figure 5 as plots of magnetization versus magnetic field.

0 2 4 6

MAGNETIC FIELD ( T I

Fig. 5 : Splitting of the strong component of exciton spectra in CdMnTe, at T = 1.4 K.

Composition values given in mole % of Mn.

Lines added for clarity. (after ref./9/).

This allows for the future magnetic measurements in very low field, using the gigantic Faraday rotation associated with such splittings /11/. Up to now, the data for ordina- ry fields reveal an interaction between Mn 2+

ions of antiferromagnetic type /7/.

4. Results in HgMnTe alloys.- Magnetoabsorp- tion experiments /2,3/ have been performed at T = 4.2 K and T = 2 K on Hgl-kMnkTe alloys with Mn content k ranging from 0 to

0.15. The samples were grown by A. Mycielski

from P .A.N. Institute (Warsaw)

.

^{As the}

magnetic field is swept up, oscillatory

magneto transmission spectra are observed.

These oscillatory structures are associated with T6 +T8 (Z.G. configuration) or

r8 -+r6

(semiconducting configuration) inter Landau levels transitions.

For Z.G. alloys, these experiments provide a direct access to the reduced cyclotron frequencies of T6 and i"' bands

(IiR) and the spin splittings of

$

^conduc-

tion levels (sc (n) ) :

Fiq. 6 : S (1) and K Q as a function of c o

in HgCdTe ?open symbols) and HgMnTe (black symbols) at T = 4.2 K and 2 K. H = 20 KG.

(after ref ./3/).

On figure 6,KR and Sc (1) are plotted, at H = 20 kG, versus the interaction gap for Z.G. HgMnTe and (non magnetic) Z.G.

HgCdTe alloys /15/. As may be seen from eq.(4), unperturbed band structure calculations predict almost identical Sc and KR for two different alloys with the same Ep and E,. Moreover, one should get 6R % 4Sc.

~ctually, as seen of figure 6, one obtains similar KR in the two types of compounds whereas the

gc

spin splitting of HgMnTe alloys greatly exceeds the value obtained in HgCdTe alloys. These results cannot be interpreted in the framework of conventional + -+

k.p calculations and provide a striking proof of exchange-enhanced Land6 g factors in Hgl.4nTe alloys.

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exchange-perturbed quantities, such as

'L

"I;(" ^,

do depend on temperature through the

magnetization

<sZ>.

This implies a marked temperature dependence of interband T6

-+g

magnetotransmission spectra of Z.G. HgMnTe alloys. To illustrate this feature we present, on figure 7, the shifts of magne- toabsorption lines observed at fixed photon energy in Faraday a+ polarization, when T is lowered from 4.2 K to 2 X. In Faraday a- polarization, one observes an opposite behaviour : lines shift towards larger field by lowering T.

Fig. 7 : blagnetotransmission recording at f i w = 243.3 m eV for an HgMnTe alloys with k = 1.5 % in Faraday a+ polarization. T = 4.2 K and 2 K.

As a result of temperature decrease, the

: r

spin splitting is increased (Fig.5) ;

which implies @>o since g c < o (see eq. 11).

From the saturation of the magnetization,

rs

observed on a very dilute HgMnTe alloy, one obtains by a fitting procedure (3,4) :

N , a = -0.75 eV N o @ = 1.5 eV.

The d electrons magnetization was also obtained /3/ : the magnetization curves strongly depart from a non interacting spin behaviour exemplifying an antiferromagnetic interaction between localized moments.

From the knowledge of band and exchange parameters one may calculate the

try-induced

5

bands degeneracy, leading to the opening of an energy gap between TV

8 and : T Landau levels.

The exchange perturbation <Hcd> dras- tically modifies this picture /3,4/. We show on figure 8 the evolution of

rC

_{8 '}

gV

Landau levels with an increase of Mn conte&.

~ f g . 8 : Magnetic field dependence of the

r8 , kc

Landau levels energy, at kZ =

o

for several compositions. T = 4.2 K and T = 2 K.

(after ref . / 3 / )

.

For the alloy with k = 0.14 % , the level orderingis still the same as the one prevailing in non magnetic Z.G. alloys ;

one just notices an increase of spin splitting and a narrowing of the forbidden gap between

5

bands. This situation holds up to k % 0.4 %, For more concentrated HgMnTe alloys, the exchanged-enhanced

%c spin splitting becomes larger than fiw c leading to a rebarkable inversion of \he

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JOURNAL DE PHYSIQUE

relative positions of n-c and (n+l)p.

+

However, the most striking ef r8 Hect induced by <Hcd> consists of adisappearance of the T;

-

^r8V forbidden gap which is replaced by a finite overlap between

5

C ^and

r:

bands for k > 0.4 %. This overlap is due to the exchange shift of the bv(-1) heavy hole level which at kZ = 0 stays above the 0

+

conduction level.

When existing, the overlap A ^{= E} (-1) bv

- E first increases with H (<S >O=H). Then

0

+

^z

saturates because the exchange contribution increases sublinearly with H as <sZ>

does, whereas the unperturbed contribution (leading to A<O) raises linearly with H.

A then decreases, vanishing at some criticd field Hcrit where a semimetal -+ semiconductor transition takes place. For H>Hcrit, A<o : the normal opening of an energy gap between

gv

^and is. recovered. Note that

?. 37 kG at T = 2 K for k % 0.4 %

'crit?.

whereas A<o for all H at T = 4.2 K on the same sample. For k 2 0.8 % the semimetallic structure exists in the whole investigate field range.

Acknowledgements.- The results reported in this paper were obtained thanks to a constant cooperation between the E.N.S.

Laboratory and the Polish Institutes of Physics : P.A.N. (Polish Academy of Sciences) and Warsaw University. It is then a pleasule to express our gratitude to Pr. C. Benoit

a

la Guillaume, Drs. G. Fishman ; Y. Guldner and C. Lewiner from Paris as well as to Drs. R.R. Galazka, J. Kossut, A. Mycielski, Pr. J. Mycielski and Dr. M. Nawrocki from Warsaw. Pr. J.K. Furdyna from Purdue University kindly supplied us with some HgMnTe crystals.

References

Kossut, J., Phys. Status Solidi (b) 78 (1976) 537.

-

Bastard, G., Rigaux, C. and Mycielski,

A . , Phys. Status Solidi (b)

2

⁽¹⁹⁷⁷⁾

585.

Bastard, G., Rigaux, C., Guldner, Y., tiycielski, J. and Mycielski, A., J.

Physique

2

(1978) 8 7 .

Bastard, G., Proc. 3d international conference on the Physics of Narrow Gap semiconductors, Warsaw 1977

(Polish Scientific Publishers).

Jaczynski, M., Kossut, J. and Gazazka, R.R.; phy&. Status Solidi (b)

88

(1978) 73.

Dobrowolska, M., Dobrowolski, W., Gazazka, R.R.,and Kossut, J., Solid State Commun

30

(1979) 25.

Gazazka, R.R., Proc. 14th Internatio- nal Conference on the Physics of semiconductors, Edinburgh (1978), edited by B.L.H. Wilson. The Institute of Physics, Bristol and London.

Gaj, J.A., Ginter, J.,and Galazka, R.

R . , Phys. Status Solidi (b)

89

(19781655.

Gaj, J.A., Planel, R.,and Fishman, G., Solid State Commun

9

(1979) 435.

Lewiner, C., Gaj, J.A.,and BastarQG., This Conference.

Gaj, J.A., Byszewski, P., Cieplak, M.

Z., Fishman, G., Gapazka, R.R., Gin&, J., Nawrocki, K., Nguyen The Khoi, Planel, R., Ranvaud, R . , and Twardow-

ski, A., in ref. /5/.

Pidqeon, C.R., and Brown, R., Phys.

Rev.

186

(1966) 575.

Gaj, J.A., Gapazka, R.R. and Nawrocki, M., Solid State Commun

15

(1978) 193.

Khoi, N.T. and Gaj, J.A., Phys. Status Solidi (b)

83

(1977) K 133.

Twardowski. A , . Nawrocki.M., and

Ginter, J., Phys.Status Solidi(inpress) Cieplak, M.Z. and Byszewski, P.,

Solid State Commun

2

(1979) 81.

Guldner, Y., Rigaux, C., Mycielski, A. and Couder, Y., Phys. Status Solidi

(b)

81

(1977) 615.

Pastor, K., Grynberg, M. and Gapazka, R.R., Solid State Commun

29

(1979) 739.