ANION SHIFT INFLUENCE ON BAND STRUCTURE OF CRYSTALS WITH CHALCOPYRITE LATTICE

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ANION SHIFT INFLUENCE ON BAND STRUCTURE OF CRYSTALS WITH CHALCOPYRITE LATTICE

Ju. Polygalov, A. Poplavnoi, A. Ratner

To cite this version:

Ju. Polygalov, A. Poplavnoi, A. Ratner. ANION SHIFT INFLUENCE ON BAND STRUCTURE OF CRYSTALS WITH CHALCOPYRITE LATTICE. Journal de Physique Colloques, 1975, 36 (C3), pp.C3-129-C3-135. �10.1051/jphyscol:1975324�. �jpa-00216294�

JOURNAL DE PHYSIQUE Colloque C3, supple'ment au no 9, Tome 36, Septembre 1975, page C3-129

ANION SHUT INFLUENCE ON BAND STRUCTURE OF CRYSTALS WITH CHALCOPYRITE LATTICE

Ju. I. POLYGALOV, A. S. POPLAVNOI Kemerovo State University, Kemerovo, USSR

A. M. RATNER

Siberian Physical Technical Institute, Tomsk, USSR

Rhumb. - Dans cet article, nous calculons la structure de bandes d'energie de ZnSiAs2, ZnSnAs2, AgGaSz, AgGaSe2, AgGaTe2 par la methode du pseudo-potentiel avec et sans tenir compte du deplacement anionique.

On montre que ce dernier apporte des changements considerables la structure de bande.

Les constantes de deformation optiques ont et6 calculees dans I'approximation des ions rigides pour le sommet de la bande de valence et le bas de la bande de conduction pour les m&mes cristaux.

Abstract. - In this paper we calculate the energy band structure of ZnSiAs2, ZnSnAsz, AgGaSz, AgGaSez, AgGaTe2 with and no account of anion displacements by the pseudopotential method.

It is shown that account of the anion displacements leads to considerable changes in the energy band structure.

In the approximation of rigid ions the constants of optical strain were calculated for the valency band top and for the bottom of conductivity band of crystals under consideration.

Results of a wide investigation of the band structure of ternary A " B ' ~ c ~ and A'B~~'c;' semiconductors with chalcopyrite lattice were given in studies [ l , 21 and several reviews (e. g. [3] to 151). The most inte- resting results on fine structure of band edges and some additional band extrema were obtained lately by the modulation spectroscopy method [4, 51. This investi- gations turned out to be in a rather good agreement with theoretical calculations made by the pseudo- potential method [3, 6, 71.

At the same time the precise investigations of the chalcopyrite crystal structure showed a number of peculiarities which distinguish this lattice from that of sphalerite. The differences are as follows (for the sake of brevity we shall speak of the compounds

A I I B I V C ~

2) : 1) the substitution of atoms of groups I11 by those of groups II and IV of the Mendeleev periodic system, 2) the lattice compression in the direction of a tetragonal crystal axis, 3) the difference on the bond length of A"-CV and B''-CV [2, 8 to 111. On account of point, 3) the anion displacement from the lattice points of the face-centred sublattice takes place which is described by the parameter

The structure difference leads to the appearance of some disturbing terms in the band theory equation

of the undisturbed structure A"'BV. The compatibility relations for T: and 3:; group representations [l21 enable us to determine a possible symmetry of these terms. They can be transformed by the representations

r , and r,, (the line x2 + y2 - 2 z2),

of the group T:. In the approximation when the crystal potential is built up as a superposition of spherically symmetrical atomic potentials it is not difficult to determine the symmetry of disturbing terms due to 1) t o 3). Because of point 1) disturbing terms of T , and W, symmetry arise 2) gives the disturbance with the symmetry T,, _T2, X, and W,.

In the theoretical calculations of the energy band structure of A"B'"c; [3] and A'B"'c;' crystals [6, 71 the parameter X, was supposed to be equal to 0.25 due to the lack of reliable measurements. However, as the measurements [8 to 1 l ] showed the parameter 6 is not small in many cases. E. g., for CdSiP, the para- meter 6 = 0.052 [10], and the difference between the Cd-P and Si-P bond lengths is

-

differencies can effect on the spectrum structure, and for this reason while calculating one should take into account the real atom arrangement in a unit cell (Table I).

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1975324

C3-130 Ju. 1. POLYGALOV, A. S. POPLAVNOI and A. M. RATNER

TABLE I bn(W) associated with the vectors

The atomic coordinates in the chalcopyrite unit cell.

The z,ahtes are gi1;en in the units of lattice constants a.

y = cla Q 1, 114 + 6 = X,-.

In accordance with the symmetry classification of disturbing terms it is reasonable to divide the chal- copyrite reciprocal lattice vectors into three groups :

a) the sphalerite reciprocal lattice vectors b,,(T), 6) the vectors bn(X) corresponding the vector (0 0

$1

2 n n and

(OF E)

of the star W. Making use of 3 integeres n, m, l these can be written down :

b,,(n ⁼ 2" a (n, nf,

i)

n, nr, 1 are either all even or odd

b.,(x) ⁼ a (n, m,

i)

n, m are even, l is odd and vice versa (1)

I is odd, n, m are even and odd respectively.

The Fourier coefficients of the crystal potential in the chalcopyrite lattice on the vectors (1) can be written in the form

V(b.(T)) = 2

(

[-

l

l

(

[

(

:)l

^[

⁽

:)l 1

+ ^{2 Vv(bn) [(- sin 2 nmd} + i sin 2 xn6] .exp i - n t- m + ^-

:)I

Here VII(bn), VIv(bn), Vv(bn) are Fourier coefficients of the atomic potentials normalized on the chalco- pyrite unit cell volume

in (3) the sign ⁽⁽ + ^>> refers to the case when n and m are even and the sign (c - )) to the case when, n, m are odd ; in (4) ⁽⁽ + ⁾⁾ refers to the case when n is odd and m is even, the sign cc - )) refers to the opposite case.

If in ((2) to (4)) we set 6 = 0, then V(bn(X)) will vanish. Eq. (2) will give an usual formula for the Fourier coefficients of the sphalerite crystalline poten- tial, and (4) will give disturbance on account of the substitution of atoms belonging to the group I11 by ones of I1 and IV groups.

The disturbance due to point 3) is localized on the anion sublattice and the Fourier coefficients produces nonvanishing on the vectors b,(X) which are pro- portional to atomic formfactors Vv(bn) (the corres- ponding disturbing potential transforms by represen-

. Besides, the term proportional to V,@,) is added to the Fourier coeffi-

cients on vectors bn(W). Consequently it is necessary to know the atomic formfactors Vv(bn) on much greater number of the chalcopyrite reciprocal lattice vectors than in case of calculation with ^X/ = 0.25 [3, 6 , 71. This imposes more severe requirements on the pseudopotential choice for cation sublattice atoms.

In this paper we calculate the energy band struc- ture and study the spectrum dependence on the parameter x f for two A " B ' ~ C ; - Z ~ S ~ A ~ , , ZnSnAs, compounds and for three A ~ B ' " c ~ ' - A ~ G ~ s , , AgGaSe2, AgGaTe, compounds. The crystal potential of these compounds was constructed as a sum of spherically synlmetrical atomic potentials through the lattice.

The formfactors for different simple and binary diamond-type scmiconductors were determined in a number of works from the experimental data [14, 151.

Since ternary compounds are the closest crystallo- chemical analogs of the semiconductors AIV, A1"BV, A"BV' in calculation of their crystal potential we used the atomic formfactors determined in [l4 to 181.

The recalculation of the formfactors in terms of the chalcopyrite reciprocal lattice vectors and their renormalization was carried out according to the

ANION SHIFT INFLUENCE O N BAND STRUCTURE OF CRYSTALS WITH CHALCOPYRITE LATTICE _C3-131

The atonzic form factors of pseudopotentials in ZnSiAs,, ZnSnAs,, AgGaS,, AgGaSe,, AgGaTe,.

The values are given in R, Sphalerite

star ZnSiAsz ZnSnAs2 AgGaSz AgGaSe2 AgGaTez

bn Vzn Vsi VAS VS" Vng Vc;a v s V I ; ~ VS, V,re

- - - - - - - - - - -

procedure described in [13, 191. The atomic form- factors were taken from the following substances : for zink and arsenic in ZnSiAs2 and ZnSnAs, from ZnSe and GaAs [14, 151, for tin in ZnSnAs, from a-Sn [14, 151. Silicon formfactors in ZnSiAs, were taken from paper [20], whese they have been deter- mined from experimental data for different silicon polytypes. In this paper the empirical formfactors Vsi(l K I) were determined for a great number of the vector lengths K (about 70 values) and in a considera- bly wide range of values of K. Therefore, while cal- culating Vsi(] b,Z"SiAS2~) the usage of graphic interpo- lation becomes possible. G a formfactors were taken from G a p for AgGaS, and from GaAs for AgGaSe, and AgGaTe,. The pseudopotentials of S, Se and Te for AgGaS,, AgGaSe,, AgGaTe, were determined from the compounds SnS,, SnSe, [l71 ; ZnTe [18], respectively. The pseudopotential of Ag was taken from [16]. Since in [l61 the formfactors were deter- mined for the value 1 K I < 2 K, we carried out the extrapolation according to [7] in the longer.

The atom formfactors determined by the above method are given in table 11.

The calculation of the energy band structure was carried out at the symmetrical points T, T and N of the Brillouin zone, and about 170 plane waves were taken into account in expanding of the pseudo-wave function. In calculations the parameters of crystalline lattices [l, 2, I l ] given in table 111 were used.

To study the energy band structure dependence on the value of the anion shift the calculations at different values of 6 were performed. In figure 1 the width of forbidden band the value of crystalline splitting and some most important energy levels in ZnSiAs, and ZnSnAs, as functions of 6 are represented. In the figure 1 the levels belonging to the valency band are

The crystal lattice parameters

X, in the units of Compounds a, A 2 c, A 2 c/a a lattice const.

- - ^-

ZnSi As2 5.606 10.890 1.94 0.269 ZnSnAsz 5.852 11.703 2.00 0.231 AgGaS2 5.743 10.26 1.78 0.291 AgGaSez 5.973 10.88 1.82 0.250 AgGaTez 6.283 11.94 1.80 0.250

labelled by ⁽⁽ v ^D and those to the conduction band by

(< c D. The zero of energy scale is chosen at the valency

band top E,(T4) at the experimental values of S.

The energy spectrum dependence upon ( 6 1 for AgGaS,, AgGaSe,, AgGaTe, in many respects is similar to the one which is observed for ZnSnAs,, as all these sen~iconductors have a direct energy gap.

In all semiconductors under consideration the value A cr. decreases with the growth of the parameter 1 6 1.

In ZnSnAs,, AgGaS,, AgGaSe,, AgGaTe, this is due to the rise of the levels r4 and r, at valency band top. The level r, turns out to rise slower than that of r,. The bottom of the conductivity band (T,) practically does not change in these crystals as 6 changes. This can be easily explained by the fact that the level T,(c) is placed at considerable distance from both the valency and conductivity band levels, and it weakly interacts with these after switching on the disturbing terms of l), 2) and 3).

The 6 dependence of the energy band edges in ZnSiAs, turns out to be somewhat different (see Fig. 1). This is the result of the differences in the nature of the conductivity band bottom in ZnSiAs, (crystal with pseudo-direct energy gap [4]) and

C3-132 Ju. I. POLYGALOV, A. S. POPLAVNOI and A. M. RATNER

Fra. l. - The dependence of the most important parameters of the energy structure of ZnSiAsz. ZnSnAsz versus the value

of anion shift.

l

ZnSnAs,, AgGaS,, AgGaSe,, AgGaTe, (crystals with direct energy gap [4, S]).

For the same reason the value E, in ZnSiAs, increases and in ZnSnAs,, AgGaS,, AgGaSe,, AgGaTe, it discreases with the growth of 1 6 1.

The dependence of the additional maxima of the valency band top at the points T and N of the Bril- louin zone upon the value of anion displacement is of interest. The additional maximum at the point T grows in all the semiconductors under consideration and, in particular, in ZnSiAs, a t the experimental value of 6, and the maximum T, @ T, turns out to be competing with the valency band top. The addi- tional maximum at the point N in ZnSiAs, essentially does not change with 8, but it slowly grows in ZnSiAs, (Fig. 1). In AgGaS, with increasing of 6 to the experimental value 0.041 the level NI rises on 0.20 eV, while the level f4 rises on 0.50 eV. In AgGaSe, the level N, varies very weakly (within 0.01 eV) and in AgGaTe, it falls with 6 at approximately the same rate at which this1 eve1 grows in AgGaS,. The calculat- ed energy band structure of ZnSiAs,, ZnSnAs,,

FIG. 2 . - The energy band structure of ZnSiAsz, xf = 0.269 (101.

AgGaS, at the experimental values of 6 is illustrated in figures 2, 3, 4 respectively. Since no reliable experi- mental values of 6(x,) for AgGaSe, and AgGaTe, are known, figures 5 and 6 give the calculated energy structures for these crystals at 6 = 0 ( x f = 0.25).

It is interesting to note that in all the crystals except AgGaTe, at both 6 = 0 and 6 Z 0 (in the range of the experimental values of this value [8 to 111) the valency band top is placed in the centre of the Brillouin zone (level r,). Tn AgGaTe, at 6 = 0 the top of the valency band is at the point N and the additional maximum at r .

The levels T4 and T, at the point r rise and the level at N falls with 6 so that at d = 0.027 they coin- cide. At further growth of d the level r4 turns out to be upper and becomes, an additional maximum.

Thus, in AgGaTe, the anion displacement can pro- duce the rearrangement of the valency band top structure.

ANION SHIFT INFLUENCE ON BAND STRUCTURE OF CRYSTALS WITH CHALCOPYRITE LATTICE C3-133

FIG. 3. - The energy band structure of ZnSnAsz.

.rf : 0.231 1101.

The possibility of rearrangement of the valency band top in A'B"'c~' due to the anion displacement is of great interest. The calculations of the band struc- ture for several crystals we carried out with x f = 0.25 showed the valency band top in these semiconductors to have a complicated structul-e. In CuGaS,, Se,, Te,, AglnS,, AgGaS,, Se,, with calculated value E, 2 2 eV the valency band top turns out to be at the point r and the nearest additional maximum at the point N. In CuInS,, Se,, Te,, AgInS,, Se,, Te,, AgGaTe, where the calculated value E, at the point l- is of

-

The valency band top is at the point N. In the compounds where the valency band top is at N the edge of the fundamental absorption is to be non- direct. Besides, the value of Eg obtained from electrical and photoelectrical properties and from electro- reflection experiment are to bc different. This conclu- sion, however, is not confirmed by experimental data [ 5 ] . In addition, as was noted in [5] the calculated values E, for A'B"'c~' are consistently higher than the experimental data t o explain these discrepancies of theory and experiment.

FIG. 4. - The energy band structure of AgGaS2, xf = 0.291 [23].

A supposition is made in [5], that the band struc- ture of the A'B"'c~' crystals is very sensitive to the d-electrons of noble metals. Small value of the spin- orbit splitting in these compounds as compared with that of their analogs is also explained by the influence of d-electrons on the valency band top [5]. Our results enable to put forward another possible expla- nation of the above discrepancies : the rearrangement of the valency band top and the decrease of E, are effected by displacements of anions from the lattice points of the face-centred cubic sublattice.

The anion shifts which do not vary the crystal lattice symmetry correspond to the optical vibrations with symmetry f , of group 3:; [21].

The intra-valley scattering of charge carriers at the zeroth order in the wave vector K takes place on this mode. The intravalley scattering on the optical phonons with symmetry different from that of T , occurs in higher orders in K [21]. Since the mode of symmetry f, is nonpolar the relaxation time can

C3-134 Ju. I. POLYGALOV, A. S. POPLAVNOI and A. M. RATNER

FIG. 5. - The energy band structure of AgGaSe2, xf

-

T r

be evaluated by the deformation potential method [22].

The optical deformation potential for the oscillation

r , can be written as B.t, where t is a relative dis- placement of an anion atom, ^{f i )} is the shift value of the corresponding energy level per unit displacement.

The calculation carried out by the authors allows to estimate the constant 'Q in the rigid ion model. The results of these estimations are listed in table IV.

U p to now no experimental data on the contribu- tion of the nonpolar optical scattering to the relaxation times of charge carriers in A~'B'"c; and A'B"'c;' are available. Therefore, we can compare our estima-

FIG. 6 . - The energy band structure of AgGaTcn, xy = 0.25.

tions of the constant 9) with the data available for germanium of n- and p-type [22].

The value of 9) for n-Ge obtained from various experimental data is in the range (4-9) 108 eV/cm ; in p-Ge that is one order higher. Hence, one can conclude that the scattering of the nonpolar optical oscillation in n-ZnSnAs, and ~-A'B"'c;' is two orders of magnitude lower than in n-Ge, and in n-ZnSiAs,, one order of magnitude lower. Scattering

The constants of the optical deformation potentiel a) in the tmits 10' eV/cm

Compounds ZnSiAs, ZnSnAsz AgGaSz AgGaSe,

- ^{- - - -}

9 (conductivity band) 0.3 + 0.5 0.05 0.5 0.9

6 (valency band) 1 + 2 2 + 4 4 2 + 3

ANION SHIFT INFLUENCE ON BAND STRUCTURE O F CRYSTALS WITH CHALCOPY RITE LATTICE C3-135

in p-ZnSiAs,, p-ZnSnAs,, p - ~ ' ~ " ' ~ ; ' is one order fact that in the vibrations of symmetry F , the atoms weaker than in p-Ge. The smaller value of the defor- of one sublattice take part, while in A" semiconduc- mation potential constant for nonpolar optical tors the atoms of both sublattices take part in the

and A ' B ~ ' ' C ~ ' / ~ is due to the optical vibrations.

scattering in A ~ ' B I ' ' C ~

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