INDIRECT EXCHANGE INTERACTION IN Hg1-xMnxTe AND Cd1-xMnxTe ALLOYS

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INDIRECT EXCHANGE INTERACTION IN Hg1-xMnxTe AND Cd1-xMnxTe ALLOYS

C. Lewiner, J. Gaj, Gérald Bastard

To cite this version:

C. Lewiner, J. Gaj, Gérald Bastard. INDIRECT EXCHANGE INTERACTION IN Hg1-xMnxTe AND Cd1-xMnxTe ALLOYS. Journal de Physique Colloques, 1980, 41 (C5), pp.C5-289-C5-292.

�10.1051/jphyscol:1980549�. �jpa-00219984�

INDIRECT EXCHANGE INTERACTION IN Hg;L_xMnxTe AND Cd!_xMnxTe ALLOYS,

C. Lewiner , J . A . Gaj ' and G. B a s t a r d

Groupe de Physique des Solides de I'E.N.S.jLaboratoix'e Associe au C.N.R.S. Universite Paris 7, tour 23,2, Place Jussieu 75221 Paris Cedex OS - 24 rue Lhomond, 7S231 Paris Cedex 05,Fvanae

+ Institute of Experimental Physios, Umversity of Warsaw - Warsaw - Pologne

Résumé.- Nous étudions dans les alliages Hg-^ MnxTe et Cd xMnxTe l'interaction d'échange indirecte, entre deux moments localisés, qui provient des transitions virtuelles entre la bande de valence'remplie et la bande de conduction vide(méca- nisme de Bloembergen•et Rowland). Nous examinons à la fois le cas de matériaux

zérogap (les alliages Hg-, MnTe de faible x) et de matériaux semiconducteurs ( les alliages Hg. Mn Te de grana x et les alliages Cd. Mn Te). Nous calculons la

J.""X X J.™ X X

température de Curie Weiss et nous la comparons aux résultats expérimentaux exis- tants .

Abstract.- We investigate in Hg^^MnxTe and Cd-, Kn Te alloys, the indirect exchange interaction between two localized moments, wHx&h Xarises from the virtual transi- tions between the filled valence band and the empty conduction band (Bloembergen and Rowland mechanism) . Both cases of zerogap materials (Hg-^xMnxTe alloys of low x) and semiconducting materials (Hgx-xMnxTe alloys of large x and Cdi-x M nxTe alloys) are examined. The Curie Weiss temperature is calculated and compared to the avai- lable experimental results.

1. Introduction.- Magnetic susceptibility measurements performed on Hgj_xMnxTe and Cd-._jJXlnj.Te/1 -3/ alloys revealed an anti- ferromagnetic interaction between Mn ions 2+

: X"1 (T) behaves likeT + e,9> 0,at high tem- perature. Low temperature oscillatory in- terband magnetoabsorption /4,5/, magneto- transport phenomena / 6 / and exciton magne- toreflectivity/7,8/lead to similar results.

These proved impossible to interpret by accounting only for direct exchange bet- ween closely spaced moments. In this paper we investigate the Bloembergen and Rowland

(B.R) mechanism/9/, in Hg-^xMnj/Fe and

c<3i_„Mn„Te alloys and show that it leads

x x x

to a long range interaction between spins with the right order of magnitude and the right sign. 'To get this agreement it pro- ves crucial to properly account for the symmetry of the electron Bloch functions : in fact the symmetry imposes the sign of the spin-spin indirect interaction.

2 . Background.- I n Hgx_xMnxTe /l°/andCd1_K

MnxTe alloys t h e r e i s n o t enough f r e e c a r r i e r s to g i v e a s i g n i f i c a n t i n d i r e c t i n t r a b a n d e f f e c t

(RKK50. For t h e s e compounds, B.R mechanism i s r e l e v a n t : two l o c a l i z e d s p i n s i n t e r a c t

via virtual excitation of an electron from the filled valence band to the empty con- duction band. Previous calculations used oversimplified description of the electronic band structure ; in particular no attention was paid to the symmetry of band edges Bloch functions /9,11/. The band structure of these alloys is reported on figure 1 ; for more details see ref./]-4/.

zero-gap ZG-SC transition semiconductor

\ u ^u -

"-Ho'-;.v-^7V ^s ""'"7Y»

F i g . l : Band s t r u c t u r e of Hg. Mn Te a l l o y s

1-x x J

We assume that the mobile electrons interact with the localized electrons of Mn d- levels by an Heisenberg type

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980549

JOURNAL DE PHYSIQUE

hamiltonian :

I

+ are the localized spins located where S.

l i

at

6 .

and s the bare electron spin.

+=

J (r)is the exchange integral supposed to be contact like. Due to spin-orbit coupling /13/ the second order energy shift may be rewritten as the sum of an isotropic and pseudodipolar spin-pin Hamiltonian. In

what follows we have neglected all aniso- tropic effects and calculated the Iij z z exchange integral

(31 s.

_{1 1}

^.

^{) :}

where

~ ( 2 )

are the periodic part of the Bloch states(v labels the bands and -their Kramers degeneracy). Note the crucial part played by the symmetry of the bands edges in the evaluation of the matrix element.

One usually assumes it to be constant which is only valid for parabolic spherical s- type bands. For p symmetry levels their

-b -+

matrix elements depend on k/l kland k l / lk'l 3. Analytic calculations ( A = - ) . - 3.1 Para- bolic zerogap semiconductors (Hgl-xMnxTe alloys x <l%) .-The absence of forbidden gap leads to a power law decline for Iij /lo/ :

where B = < X J(r) X> is the exchange in-

I I

tegral in the p bands, mv the effective massof the heavy hole valence band and

f (s) has been evaluated for two cases : actual symmetry induced band (f si) and accidental banddegeneracy (fa)i.e.

a constant matrix element

.

^{f si (s)} and fa (s) are presented on f i ~ u r e 2 which is a stri- king illustration of-the art played Ly the symmetry : fa>O for all s leading to a ferromagnetic interaction whereas f

s i shows a sign reversal and is negative for all existing Z.G. materials (s>3).

Fig. 2 : Dependence of the effective exchange integral for parabolic ZG materials in the cases of accidental band degeneracy (f ) and symmetry induced degeneracy (f

sif '

Note that fa and fsi vanish in the limit of infinite s (mv finite,~~=m~=O). In this limit however, it is impossible to use a parabolic dispersion relation for the conduction band which forcO= 0 is linear ink.

3.2 Extreme non parabolic limit (E,=O)

V

There are two types of virtual interband transitions :

The former is analogous to case a) and involves only the matrix elements of J$) between p symmetry states since<~I..J (r)k>=~ + /:5/. The latter is induced by s and p mixing and involves two matrix elements :

B = < X .

I

J

(g) I

X > and _hh+c < S C J ^I I;)

I

S>=a. _lh-tc

Then /15/ : Iij

-

-Iij +Iij ⁽⁵⁾

--- -

with 6 = and A = 43 mo 1

x

2J2-<

(<sIvzIp,>

I

m, is the bare electron mass.

reversal for 6 % 0.4 corresponding to I over many neighbours is performed these

0 j

R 20.44

;

which is much less than the nea- oscillatory effects will compensate; however rest neighbourg distance (4.53 A) .We may if only one or two neighbours are involved thus conclude that the electron heavy hole it will he necessary to smoothen the Iij(R) contribution is in all realistic cases curves.

antiferromagnetic with R-5 dependence

Superimposed on this behaviour eq(3) at large R : shows that as long as R<< (-- 2mcc h2 1 1/2 the

lh c-- - m v X l o o ? $ 43 $ 2 extreme non parabolic limit results may be Iij 64sjaLRk ( a Z -

-

₃ ⁺

-

_{9 ) ( 7 )} used and a power law I~.,LXR'~ with 4<n< 5 for the experimental values a and 6 the Prevails. For large R an exponential decay electron light hole contribution is ferro- /9/ will be recovered.

magnetic and behaves like R-'; however the 5. Results and conclusion.- In order to total effect remains antiferromagnetic compare our theory with susceptibility for all distances. measurements /1-2/ (high temperature) it

is necessary to calculate the Curie-Weiss

F i g . 3 : Plot of q(6) versus 6

temperature /18/: kgi3 = 35x C I (8) 12 aRRj Oj

In low temperature experiments/5,8/

the first and second neighbouring pairs are frozen and the long range interaction temperature involved is To given by /5/:

kB To=

;;

^{x (1-X)}

^I

^IOj

Roj>a ⁽⁹⁾ (a is the lattice parameter).

' A s pointed out the osciLlatry beha- viour of Iii

-

prevents us of obtaining any reliable values of To. For Cdl-xMnxTe we get 8 = 2.2' for x = 1% and the results 4. General calculation.- For non parabolic for HgMnTe are displayed in table I and Z.G. alloys (Hgl-xMnxTe l%<x<7%) or for show a qualitative agreement with experi- opengap semiconductors (Hgl_,M.nxTe x > 7% or mental measurements.

Table 1

.-

Curie VJeiss temperature 8 as a Cdl-XMnXTe of it is use function of x in Hgl-xMnxTe alloys. The the correct dispersion relation since s and numerical values of a B are from ref, p band mixing are then responsible for part

/14/ and mv=ov6mo.

of the affect ( 2 . G ) or for the whole effect

(S.G.)

.

For finite spin-orbit coupling the In conclusion, the B.R. mechanism latter is no longer analytic and numerical used together with a Kane band structure

provides an interpretation of the anti- calculations were perfomed using a very

simplified band model (the Kane four band ferromagnetic long range interaction approximation /16/)and limiting k and k r observed in Hgl-xMnxTe and Cdl-xMnxTe to a spherical Brillouin zone. This sharp alloys.

ow ever

for opengaps the wave vec- cut off leads to an oscillatory behaviour tors near the Brillouin zone edge contri- of Iij with the distance exactly as a sharp bute significantly showing clearly the Fermi level is needed to get the RKKY limitations of the Kane model. Moreover

one should take properly into account oscillations. These oscillations are spurious

and will be smeared out by several factors anisotropic effects.

among them the use of a non spherical Bril- louin zone /17/ and of a finite spatial

JOURNAL DE PHYSIQUE

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