Antisite defects in Sb2−xInxTe3 mixed crystals

J. Phys. Chem. Solids Vol. 49, No. 2. pp. Printed in Great Britain.

ANTISITE

191-198,

0022-3697/M $3.00 + 0.00 Pergamon Press plc

1988

DEFECTS

IN Sb,_,In,Te,

J. HORAK, Technical

University

MIXED

CRYSTALS

Z. STAR+, P. LOW&K

of Chemical

Engineering,

Pardubice,

Czechoslovakia,

and J. PAN& J. Heyrovsky Institute of Physical Chemistry and Electrochemistry, Czechoslovak Academy of Sciences, Prague, Czechoslovakia (Received

20 November

1986; accepted

in revised form 9 June 1987)

Abstract--Changes in the values of the plasma resonance frequency and some transport coefficients (electrical conductivity, Hall coefficient, Seebeck coefficient) of Sb, _xIn,Te, crystals show that an increase in the content of incorporated In atoms causes a decrease in the free carrier concentration. This result is interpreted in the following way. In atoms are incorporated in the Sb-sublattice and form uncharged substitutional defects In&, which, of the higher electropositivity of In than Sb, cause an increase in bond polarity and also in crystal ionicity. This effect leads to an increase in the energy of formation of antisite defects, such that the concentration of antisite defects is decreased with increasing values of X. The energy of formation of antisite defects in Sb, _,In,Te, crystals has been determined for x = O.fLO.4, these values lying in the range 0.354.44 eV. The conclusions about the bond polarity and the changes in the energy of formation of antisite defects are supported by quantum chemical calculations, which also provide an explanation of the changes in the lattice parameters of Sb,_XIn,Te, crystals with increasing values of x.

Keywords: Antisite defects, Sb,_,In,Te, crystals. 1. INTRODUCTION

Sb,_,In,Te, (x = 0.0-0.4) belong to the family of layered compounds having the tetradymite structure space group Did. The properties of polycrystals Sbz _xInxTe3 compounds were first described by Rosenberg and Strauss [l], who proved that the structure of the mixed crystals corresponds to the Dzd space group for In concentrations up to x = 0.40. Increasing concentration of In built into the Sb,Te3 crystal lattice results in a marked change of the free carrier concentration [ 1,2]; we explained this effect in our earlier papers by the interaction that takes place between the In atoms and the antisite defects. This interaction brings about a decrease in the concentration of Sb;, point defects and, at the same time, a decrease in the concentration of holes. In the present paper we gave the results of the measurements of several transport coefficients and optical properties that enable us to determine the concentration of free carriers. We derive the relations between the concentration of free carriers and that of the antisite defects, and we present an explanation of the relations between the concentration of antisite defects and the change in bond polarity as well as in other bonding parameters due to building In atoms into the crystal lattice; the changes in the bonding character are related to the parameters obtained by quantum-chemical calculations. 2. EXPERIMENTAL

The mixed compounds of Sb,_,In,Te, were prepared from 5 N purity elements by the method de-

elsewhere [2,5]. The single crystals were g:lowu from the synthesized mixture by means of a modified Bridgman technique and the In content in the crystals varied over the range x = O&0.4. The orientation of the cleavage planes was checked by means of the Laue back-reflection technique. The cleavage planes were identified as the (0001) planes. The lattice parameters of the crystals were determined from X-ray powder data obtained with an HZG4BX-ray diffractometer (Freiberger Prazisionsmechanik, G.D.R.). The diffraction maxima were measured using a stepwise procedure with a step of 0.01”. The measurements were carried out with Cu-Ko! radiation in the range of 20 = 545” and with Cu-Ka, radiation in the range of 20 = 45-loo”, using a nickel filter to reduce the K/3 radiation. Calibration of the equipment was carried out on polycrystalline Si. The data published by Dijnges [6] were used for indexing the maxima of the powder patterns and the lattice parameters a and c (Fig. 9) were calculated from the diffraction pattern by a least-squares method. The concentration of In in the prepared single crystals was determined by an electron microprobe. The reflectivity R of the crystals was measured in

the plasma resonance frequency region at 294 K. The measurement was carried out with the incident radiation nearly perpendicular to the natural cleavage face (OOOl), i.e. for the electromagnetic field intensity E perpendicular

to the crystal c-axis. The Hall coefficient R, was measured at room temperature by an a.c. method with a frequency of 170 Hz for B 1)c (B is the magnetic induction vector)

191

192

J.

HOCK

et

al.

Table 1. Transport coefficients and plasma resonance frequency of Sb,_,In,Te, No.

Cl

1

0

2 3 4 5 6

0.050 0.098 0.22 0.29 0.41

[cm’ A-’ SK’]

Rn (B IIcl

P/yA, lo-l9 [cm-‘]

0.058 0.067 0.090 0.14, 0.24 0.19

10.78 9.33 6.94 4.22 2.60 3.29

o,(Elc)~W4 [&-1, 74 80 85 114 172 215

at a B-value of 1.1 T. The values of R, (B 11 c) are shown in Table I. The Seebeck coefficient S, was measured with the temperature gradient VT perpendicular to the c-axis (VT N 7 Kcm-‘).

3. RESULTS

The dependence of the reflectivity R(E I c) on the wavenumber v of the incident radiation for samples of Sb,_,In,Te, is shown in Fig. 1. The reflectivity curve was fitted by means of the expression resulting from the Drude-Zener theory [7]; in this way one can obtain the values of the plasma resonance frequency or (E I c), the optical relaxation time z,r, (E I c) and the high-frequency permittivity. The values of or (E 1 c) and z,rt (E I c) are given in Table 1. The plasma resonance frequency is given by the relation

crystals

W’l

(P/m?). IO-20 [cme3]

1.95 1.86 1.70 1.26 1.19 0.82

5.85 5.33 4.36 2.34 2.00 0.94,

s ‘““‘r’ 2.5 2.4 1.8 0.9, 0.8, 0.8,

the concentration of the built-in In atoms; the ratio y(O)/y(x) is taken as constant in the subsequent calculation of the polarization energy. The values of P/[A, y (x)] characterizing the concentration of holes in the crystals of varying composition are given in Table 1. The reduced Fermi level can be estimated from the value of the Seebeck coefficient S,,; with increasing In content in the mixed crystals the Seebeck coefficient increases, the value of the reduced Fermi level decreases, and hence the hole concentration also . decreases. The value of the Hall factor A, is also band structure dependent; for lack of any relevant data we take A, = 1 in all the samples investigated. The three experimentally dependent quantities w&E I c), RH(B l/c) and S,, are plotted in Fig. 2 as a function of the concentration of the In atoms built into the crystal lattice.

(1)

r-

R

where P is the free carrier concentration, e the electron charge, 6. the permittivity of free space and m, the effective mass in the direction perpendicular to the trigonal c-axis. Using this relation, we determined the value of P/(m,/m,) characterizing the change in the concentration of the free carriers in Sb2 _ .In,Te, crystals with increasing content of the In atoms built into the lattice. The Hall coefficient RH (B I/c) for layered crystals derived from Sb,Te, is given by the expression

where A, is the Hall factor and y the structure factor whose value depends on the band structure model. According to Stordeur and Simon [8], the valence band of p-Sb,Te, is well described by means of the model of a non-parabolic six-valley one valence band structure, which leads to the value of y = 0.77. This value has been used to determine the concentration of the free carriers in undoped Sb,Te,. The band structure of Sbz_ .In,Te, is not known and hence there are no data on the value of y. Therefore, we adopted the simplification that the structure factor of the undoped crystal, y (0) does not change significantly with

k

I

*P

I

1 2

i

3

I

\‘I__--i,

1s

0.:

jt

L

Fig. 1. The positions of the plasma resonance frequency.

Antisite defects in Sb,,,_,In,Te,

193

mixed crystals

Fig. 2. Dependences of the Hall and Seebeck coefficients and the plasma resonance frequency on the composition of Sb,_,In,Te, crystals.

Calculation crystals

of the bonding character in the

In order to be able to evaluate the relations between the concentration of antisite defects and the changes in bonding conditions as a function of In content, we carried out a quantum-chemical calculation of the bonding conditions in Sb,_,In,Te, crystals for x E (0.0, 0.66) using the computer program TOPOLOGY [9]. Data are shown in Table 2. The quantum chemical procedure used in this paper is merely a version of the HMO method [lo] extended on the basis of all valence orbitals. Unlike the Del Re method [1 11, the calculation is performed in the basis of equivalent orbitals and interactions of all equivalent orbitals between the nearest neighbours are taken into account. Coulomb integrals were identified with scaled orbital electronegativities and the resonance integrals were calibrated on a set of homonuclear diatomics. The Coulomb integrals are presented in Table 3. The details of the method and the calibrations can be found in Ref. [12]. The calculation was carried out for a section of the crystal layer between the van der Waals gaps. This

section was formed by 31 atoms that had altogether 174 valence orbitals. The arrangement of the crystal section used in the calculations is represented in Fig. 3. For clarity, only three atomic layers are depicted: the Te’, Sb and Ten layers. The next Sb layer has the same arrangement as the Te’ layer and the following Te” layer is of the same type as the Sb layer shown. The atomic planes forming one five-layer stack are thus arranged as ABCAB. The bonds shown in the figure are drawn with respect to the Sb atoms. The results obtained, in particular the distribution of electrons, were used for a qualitative interpretation of notions such as bond ionicity and

Table 3. Diagonal members of H core matrix (scaled electronegativity)

Sb Te In

S

P

d

sp’d’

1.245 1.409 0.926

0.669 0.772 0.479

0.132 0.167 0.072

0.7702 0.7883 0.7366

Table 2. Some data obtained bv the oroeram TOPOLOGY Bond polarity Sb,_,In,Te, x 0.000

0.165 0.330 0.495 0.660

P

I 0.404, 0.453, 0.492, 0.536, 0.579,

lel II 0.226, 0.267, 0.307, 0.353, 0.400,

Electron density inside the gap pp

lel

3.904, 3.910, 3.912, 3.915, 3.9186

Electron density inside the layers

Bond order BO

I

II

I

0.665, 0.675, 0.683, 0.685, 0.687,

0.521, 0.520, 0.522, 0.523, 0.526,

7.130, 7.160, 7.180, 7.207, 7.234,

III 4.105, 3.884, 3.672, 3.440, 3.211,

5.542, 5.593, 5.612, 5.631, 5.666,

J. HORAK et al. 4. DISCUSSION

0

Td"

Sb

Fig. 3. A part of the crystal section used for the calculation. The bonds are drawn relative to the Sb atomic plane.

polarity. AS the cluster which replaces the infinite crystal does not exhibit its translational symmetry, average values were calculated and presented in Tables for all the quantities defined below. The results obtained are summarized in Tables 3 and 4 and in Figures 41.

According to the results of [13], undoped Sb,Te, crystals exhibit a pronounced superstoichiometric content of Sb atoms (approx. 8 x lOI cm-‘) that are built into the Te sublattice and thus form antisite defects Sbr,; the charge on these defects is compensated by holes. The results of the measurements of the Hall coefficient and the values of the plasma resonance frequency wp (E I c) obtained on a series of Sb, I In,Te, crystals show that increasing the content of the built-in In atoms gives rise to a decrease in the free carrier concentration. In agreement with this finding, the Seebeck coefficient increases with x and the value of the Fermi level decreases. Hence, it has been shown by independent methods that the built-in indium atoms lower the concentration of the holes. These results enable one to evaluate the nature of the point defects formed by the incorporation of

Dejinitions of the notions 1. Polarity of the bond c between atoms A and B (AL-B) is defined as the difference in the electron density on the orbitals in the direction of bond c on atoms A and B. Its numerical value is averaged over all bonds of the same kind existing in the crystal model. It is given in units of the elementary charge and denoted by P. 2. Electron density between the stacks is the density of electrons on the orbitals pointing to the van der Waals gap. It is given in units of the elementary charge and referred to one atom. Its symbol is pg. 3. Electron density in the layers is also referred to one atom and given in units of the elementary charge. The corresponding symbol is p,. 4. Bond order: average order of the bonds of the same type. It expresses the bond multiplicity. This corresponds to the electron density in the space of the bond. It is a dimensionless quantity and is denoted by BO. Whenever electrons are mentioned, valence electrons are to be understood. Table 4. Energy of formation of AS defects and polarization energy Sb,_,In,Te, x 0.000 0.050 0.098 0.220 0.290 0.410

E+AE [eV 0.350 0.361 0.384 0.422 0.459 0.441

BE

0.21

,

am Fig. 4. Dependence Sb,_,In,Te,

,

OS65

0.330

,

0x95 -x

,

0.66'0

of bond polarity on the composition crystals. (I, ?&Tel; II, SbTe”.)

of

0.528 0.524 -

lmevl 0.520t\ 11 34 12 109 91

Qocl Fig. 5. Dependence Sb,_,In,Te,

0 0.165 --I a330

0.495 -x

0.660

of bond order on the composition crystals. (I, Sb-Te’; II, Sb-Tel’.)

of the

Antisite defects in Sb,,,_,In,Te,

L-

L

A-.-.

0.00

0.165

_.---L__

0.330

0.495 -x

-J

195

sublattice (b), owing to the electron configuration of In and Sb atoms is impossible; such a process would lead to an increase in hole concentration. This is in contradiction with the experiment. Even the substitutional defect In atom in the Te sublattice (c, d) (considered in paper [1]) would cause a negatively charged defect and so lead to an increase in the hole concentration, which is contrary with the experiment. Therefore, we accept the following process for the incorporation of In atoms in the Sb*Te, lattice. The In atoms introduced into the Sb,Te, crystal form uncharged substitutional defects in the Sb sublattice; in the light of the ideas presented in [4], their formation can be described as follows

I

3.901

mixed crystals

a59

2In (s) + $Te, (g) + 2 Vsb+ 3 V,,

0.660

= 21ns, + 3Te,,

Fig. 6. Dependences of the electron density in the van der Waals gap and of the average bond order on composition.

In atoms in the Sb,Te, lattice. The incorporation of In atoms into the Sb,Te, can be realized in several ways: (a) In atoms enter as interstitials within the structural layers; (b) In atoms occupy positions in the Sb sublattice; (c) In atoms occupy Te” sites in the Te sublattice with octahedral coordination to the Sb atoms; (d) In atoms occupy Tei sites in the Te sublattice; (e) In atoms enter the van der Waals gaps in the layered structure of Sb*Te,. According to the results obtained we can exclude the occupation of interstitial sites by In atoms (a), or their ionisation in van der Waals gaps (e), because both defects would lead to the creation of free electrons, i.e. to a very strong decrease in the concentration of the holes. Also the fo~ation of In,“, defects created by the incorporation of In atoms in the Sb

because the equation 2Vsb -t 3 VT,= 0 is valid. The change in the electron configuration of the In atoms forming the Insl, point defects can be symbolically expressed as In (Ss*5~‘) -+ In,, (Ss”5p3). The formation of the uncharged In& defects is not in contradiction with the theory of chemical bonds. After Krebs [14] only p-electrons participate in the a-bond formation. Accepting this assumption we explain the decrease in the number of antisite defects in the following way: introduction of In atoms into the Sb sublattice gives rise to formafly uncharged In,, defects. In Table 3 are entered the scaled orbital electronegativities of Sb, Te and In atoms for s, p and d orbitals. The value of the scaled electronegativity for the sp3d2 orbital is given in the form of the diagonal term of the H-core matrix. It is evident from these values that indium is more electropositive than antimony, so that one can expect the In& defect to acquire [see Fig. 7(a)] a partial positive charge (Ink”). Therefore, in the Ii@-Te bonds there will be a shift of the electron density

(a) Td

x =0.66

Sb Ten f

(3)

Sb

Fig. 7. Electron density along the cross-section of one crystal layer. (-

Sb,Te,; ---

Sb,.,Ib,,Te,.)

196

J.

HORAK et al.

towards the Te’ atom. A shift of the electron density towards the Te” atoms can also be expected. The assumed changes in the bonding conditions in the crystal lead to an increase in ionicity; larger ionicity suppresses the probability of the transition of a positively polarized particle into the sublattice of negatively polarized particles, leading thus to a decrease in the concentration of antisite defects. The above-mentioned qualitative idea of the changes in bonding parameters is supported by a quantum chemical calculation which permits us to calculate the changes in the electron density in the Tel--S&Te” layers; Fig. 7(b) shows the change in the electron density on the individual layers of the Sb,_,In,Te, crystal (for x = 0.0-0.66). As is evident from the figure, the charge corresponding to the antimony sublattice becomes markedly more positive and the electron density on the outer atomic layers Te’ increases. Also the Te” layers acquire a higher electron density. The polarity of the bonds between the atoms of individual layers (see Table 2) was determined from the calculated data of electron distribution. In Table 2 are shown other parameters characterizing the bonding conditions for different quantities of In atoms introduced into the Sb, _ .In,Te, lattice. These are: density of electrons in the gap, bond order, and the electron densities corresponding to the individual atomic layers. The results of quantum-chemical calculations show unambiguously that the In atoms introduced into the Sb sublattice bring about an increase in the polarity of both the In&-Te’ and In&-Te” bonds; the average bond ionicity in the Sb, _ I In,Te, crystals will therefore increase with a number of In atoms incorporated into the lattice. The results of our earlier studies [2,4, 15-171 imply that increasing ionicity of the crystal leads to a decrease in the probability of the formation of antisite defects in the crystal lattice.

The concentration of antisite defects NAs in undoped crystal can be expressed by the equation

NAs= k, ev( - J%lkaT) where E,, is the formation energy of the antisite defects, k, the Boltzmann constant and T the absolute temperature. The introduction of foreign atoms which alter the bond polarity results in a change in the formation energy of the antisite defects. In doped or mixed crystals the formation energy of the antisite defects is given by the sum E, + AE, where AE is the polarization energy; the sign of AE can be positive in the case of the impurity atoms enhancing the bond polarity, or negative in the case of the impurity atoms lowering the bond polarity. The concentration of the antisite defects is then given by the equation

Nis=k,exp(

Further, we adopt the assumption that the constant of the crystal and that its value does not change with the foreign atoms built into the lattice [4]. In the case of Sb, _,In,Te, mixed crystals, E,, = 0.35 eV at x = 0, according to [4]; the polarization energy BE will increase with increasing concen-

k, is characteristic

tration of In atoms because In is more electropositive than Sb. Figure 8 shows the dependence of E, + AE on the concentration of the In atoms in the lattice. In calculating AE we adopted the assumption that only one type of charged defect exists in the mixed crystal lattice, i.e. the antisite defects in the sense of In,,. Therefore, the concentration of the antisite defects is equal to the concentration of holes and, using relation (2), we can determine AE from the equation

& -= NM

Fig. 8. Dependence of average bond polarity, free carrier concentration and formation energy of the antisite defect on the composition of Sb, .In,Te, crystals.

-F).

Y(X)~I(XM”(O) y(O)A,(O)R,(x)

= exp

.

(6)

This equation was used by us to calculate the AE values for individual crystal compositions. The results are given in Table 4. Increasing In content results in increasing bond polarity. Along with bond polarity the AE value also increases (Table 4, Fig. 8). Both quantities increase approximately linearly with the In content. Thus, one can assume close agreement between the bond polarity and the formation energy of the antisite defects and, consequently, their concentration. The incorporation of foreign atoms that are more electropositive than Sb into the cation sublattice leads, therefore, to an increase in bond polarity, an increase in bond ionicity and to a decrease in the concentration of antisite defects. Hand in hand with the effect of bond polarization goes the dependence of the lattice parameters u and c on the In content (see Fig. 9). These dependences are in good agreement with the experimental results

Antisite defects in Sb,,,_.In,‘I$

mixed crystals

197

However, the value of the c parameter grows only until another effect, which is the shortening of the bonds, starts to dominate. Figure 5 from which the increase in the bond order is evident, depicts schematically the relation between the growth of the bond order and the change in the lattice parameters. All the described processes are represented in parts (a), (b) and (c) of Fig. 10. 5, CONCLUSION

T= 294 K

0.990ao

0.1

0.2 I

0.4 1 -x

0.3

0.5

Fig. 9. Lattice parameters as a function of X.

published in [l] as well as with our earlier data [2]. In view of the fact that In and Sb atoms have comparable radii, we cannot expect any pronounced change in the a and c parameters due to the difference in the size of the atoms, It is evident from Fig. 9 that the Q value decreases monotonically with increasing In& content whereas the c value passes through a maximum. This fact can be explained in the follawing way: it is clear that the electron density in the van der Waals gap increases with In concentration (see Fig. 6) Therefore, an electrostatic repulsion between the individual layer stacks takes place. One can further assume that the value of the angle (b (see Fig. 10) increases due to the removal of electrons from the atoms in the antimony sublattice, i.e. due to the increase of the partial positive charge on this sublattice. Both these circumstances are responsibIe for the initial growth of the c value. The increase in the angle # results also in the decrease in the a value.

The formation of antisite defects is a consequence of the very weak polarity of the bonds between the atoms in the crystal lattice. If, by introducing suitabie impurity atoms, we provoke an increase in bond polarity, the formation energy of the antisite defects increases and their concentration decreases; conversely, by suppressing the bond polarity-which is rather low in Sb,Te, crystals-we create the conditions favourable to the formation of antisite defects. The experimentally determined changes of the properties of the crystals in the Sbz_,InXTe3 system, as well as the observed decrease in the concentration of the antisite defects with increasing In content, corroborate the correctness of the ideas presented. This qualitative view of the relations between the concentration of antisite defects and bond polarity is supported by quant~~hemieal calculations carried out on the Sb,_.In,Te, system; they enabie us to interpret the polarity changes and to derive the relations between bond polarity and the change in the formation energy of the antisite defects. In our opinion, the idea of the relation between the concentration of antisite defects and the bond polarity has a general validity and can also be applied to other crystals containing antisite defects

Acknowledgements--We are indebted to professor Richter, I. Physikalisches Institut der Rheinis~h-W~tf~~sc~e Hochschule, Aachen, for providing the measurement of the reflectivity spectra.One of us (Hi.) thanks Professor P. Grosse for the invitation and the Deutsche Forschungsgemeinschaft for financial support.

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b)

Cl

Fig. IO. Schematic graphic description of the chartges of the lattie parameters and bond angles.

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