THE DIELECTRIC PROPERTIES OF THE CUBIC IV-VI COMPOUND SEMICONDUCTORS

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THE DIELECTRIC PROPERTIES OF THE CUBIC IV-VI COMPOUND SEMICONDUCTORS

E. Burstein, S. Perkowitz, M. Brodsky

To cite this version:

E. Burstein, S. Perkowitz, M. Brodsky. THE DIELECTRIC PROPERTIES OF THE CUBIC IV-VI COMPOUND SEMICONDUCTORS. Journal de Physique Colloques, 1968, 29 (C4), pp.C4-78-C4-83.

�10.1051/jphyscol:1968411�. �jpa-00213615�

THE DIELECTRIC PROPERTIES

OF THE CUBIC IV-VI COMPOUND SEMICONDUCTORS (*)

E. BURSTEIN, S . PERKOWITZ (**) and M. H. BRODSKY (***)

Laboratory for Research on the Structure of Matter and

Physics Department, University of Pennsylvania, Philadelphia, Pennsylvania

R6sum6.

-

Les proprietes dielectriques et infrarouge (dynamique de reseau) des composes semiconducteurs IV-VI cubiques sont rCsum6es ici. Les valeurs de la charge ionique dynamique macroscopique, e: = ( ~ M T / ~ u T ) E se trouvent iitre nettement plus fortes que les valeurs corres- pondantes pour les halogenures alcalins et les cristaux de type blende de zinc. Les fortes valeurs de e; qui proviennent de la redistribution de la charge klectronique, donnent une preuve de plus que le champ de Lorentz dipolaire atomique est le facteur ⁽⁽ dkstabilisateur ⁾⁾ responsable de la faible valeur de ^w~ dans les composks IV-VI.

Abstract.

-

The dielectric and infrared (lattice dynamic) properties of the cubic IV-VI compound semiconductors are reviewed. The values of macroscopic dynamic ionic, e; = ( ~ M T / ~ u T ) E are found to be appreciably larger than the corresponding values for alkali halide and zincblende type crystals. The large values of eg, which arise from the redistribution of electronic charge, provide further indication that the atomic dipole Lorentz field is the ⁽⁽ destabilizing ⁾⁾ factor res- ponsible for the low values of w~ in the IV-VI compounds.

1. Introduction. - On the basis of the ionic character and the large values of the high frequency optical dielectric constants, E,, of the lead compounds, PbS, PbSe and PbTe, it was conjectured that the low frequency (static) dielectric constant, E,, of these substances are relatively large, and that the high mobility of the free carriers in these substances at low temperatures, even in highly degenerate samples, is due to the large values of the low frequency dielectric constant [I]. This conjecture proved to be correct.

The experimentally determined values of E, of PbS, PbSe and PbTe have been found to range from 190 to 450. The large values of E , and the corresponding low values of the long wavelength transverse optical (TO) phonon frequency, a,, of the lead compounds led Cochran [2] to pose the question as to whether a diatomic crystal may have a ferroelectric phase, and he and his collaborators undertook an experimental

(*) Research Supported in part by the U.S. Office of Naval Research.

(**) Present address : General Telephone and Electronic Research Laboratory, Bayside, New York.

(***) Present address : IBM Watson Research Center, York- town Heights, New York.

(inelastic neutron scattering) and theoretical (shell model) investigation of the IV-VI compounds with this in mind. Their investigations [3] have shown that the TO phonon branch of the phonon dispersion curves of SnTe exhibits a strong temperature depen- dence similar to that of SrTiO, and that SnTe can therefore be characterized as a ⁽⁽ paraelectric ^D.

Furthermore, GeTe exhibits a ferroelectric type transition at 670 OK [4] from a NaCl structure to a rhombohedral structure similar to that of the Group V elements As, Sb and Bi, which is related to the NaCl structure by a small relative displacement of the Ge and Te fcc sub-lattices (accompanied by an elastic deformation) in the [ I l l ] direction. Cohen and co- workers, [ 5 ] on the basis of energy band calculations (using a pseudo-potential model) for the Group V elements and the reIated IV-VI coumpounds, have suggested that the cubic structure is stabilized when the difference in the pseudo-potentials of the two elements in the IV-VI compounds is large, and that the rhombohedral structure occurs when the diiTe- rence is small. On the other hand, Cochran [6] has pointed out that the phonon dispersion .curves of PbS, PbTe and SnTe are similar to those for ionic

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1968411

THE DIELECTRIC PROPERTIES C 4 - 7 9

crystals, and that, as in the case of the BaTiO, ferroe- lectrics, the Lorentz field may be the cc destabilizing ^D factor which is responsible for the low values of o, in the Pb compounds and SnTe, and for the phase transition in GeTe.

In the present paper we discuss the dielectric proper- ties of the cubic IV-VI compound semiconductors with particular emphasis on the values of the macros- copic dynamic ionic charge, e;, in these substances, which are appreciably larger than the corresponding values for alkali halide-and zinc blende-type crystals.

Since the effective electric field which acts on the electronic dipoles is essentially equal to the macros- copic electric field in the IV-VI compounds, the large values of ez arise from the redistribution of electronic charge which accompanies the relative (optical) displacement of the atoms. They provide a further indication that the atomic dipole Lorentz field is the cc destabilizing D factor which leads to low values of o, in the IV-VI compounds.

2. The Dielectric Properties of Cubic Diatomic Crystals.

-

When one neglects the frequency depen- dence of the damping constant of the TO phonons, the expression for the dielectric constant of a polar cubic diatomic crystal takes the simple form [7].

4 n ~ e : ~

E(O) = E, f

- -

m(o+

-

o2 - iyo)

a2 -

o2

-

iyo

where ^{E ,} is the high frequency (or optical) dielectric constant ; e: is the macroscopic dynamic (or effective) ionic charge ; N is the number of unit cell per unit volume ; Ei is the reduced mass of the two in the unit cell ; o, and y, are the frequency and damping cons- tant of the q

=

0 TO phonons ; and

is the plasma frequency of the atoms. In the limit y + 0, 522 is equal to ^oi - a;, where o, is the frequency of the q

=

0 LO phonons.

The macroscopic dynamic ionic charge is defined by

181

:

where u, = u,,

-

u,, is the relative transverse dis- placement of the two atoms in the unit cell ; MT is the electric moment resulting from the relative displace-

ment of the atoms ; and E is the macroscopic electric field. M T includes the contributions from the redistri- bution and transfer of charge as well as the contribution from the static ionic charge. As was first pointed out by Cochran [9], it is possible for a crystal to have a non-zero dynamic ionic charge even through the static charge of the atoms is zero.

In crystals which are predominantly ionic, such as the alkali halides, the values of the dynamic ionic charge do not, in general, differ appreciably from the static ionic charge. In such crystals the effective (or local) field

E,,,

= E

+

(4 7113) P, is assumed to act on the electronic as well as the atomic electric dipoles, and Szigeti [lo] has shown that the dynamic ionic charge can be expressed as

where

e: is the so-called Szigeti charge which includes the effects of charge redistribution resulting from short range interactions between neighboring ions, but does not include the effects of the effective electric field. The factor ^{( E ,}

+

^2)/3, arises from the coupling of the electronic and vibrational excitations via the Lorentz field, EL = (4 7113) P. The values of e,*/e for the alkali halides, which are derived from experimental data by means of eqs. (2.1) and (2.3), are found to range from 0.67 (CsI) to 0.95 (RbF) (table I). In the case of LiH and LiD which can be considered as alkali halides with H- (and its isotope D-) treated as the first in the series of halogen ions, the erle values are found to be 0.53 and 0.56 respectively (1 11.

The deviations of e:/e from unity in the alkali halides are fairly well accounted for on the basis of the shell model [I21 (or the equivalent deformation dipole model [13]) which takes into account the displacement of the outer electrons relative to the cc ion cores ^D arising from the short range repulsive interactions of the ions. The relative displacement of the electrons shell and the ion core yields a positive contribution to e: for the alkali ions and a negative contribution to e: for the halogen ions. Since the halogen ions have larger electronic polarizability (weaker spring cons- tants between the electron shell and ion core) than the alkali ions, the net contribution of the relative displacement of the electron shells and ion cores for

the positive and negative ions to e,* is negative and e:/e is less than unity. This is clearly demonstrated by the decreasing values of e,*/e in the sequence AF-ACI- ABr-Al-AH (where A is a given alkali metal) in which there is an increasing ratio of anion to cation polariza- bilities (table I).

Infrared (lattice vibration) parameters of alkali Halides [8]

Crys- tal

-

LiH LID LiF LiCl LiBr NaF NaCl NaBr NaI K F KC1 KBr KI RbF RbCl RbBr RbI CsCl CsBr CsI

W L

(cm - ')

-

1 120 880 662 398 325 414 264 209 209 326 214 165 139 286 173 127 103 165 112 85

(a) BRODSKY (M. H.) and BURSTEIN (E.), J. Phys. Chem.

Solids, 1967, 28, 1655.

(b) GOTTLICH (M.), J. Opt. SOC. Am., 1960,50, 343.

(.) HASS (M.), J. Phys. Chem. Solids, 1963, 24, 1159.

HOHLS (H. W.), Ann. Phys. Lpz., 1937, 29, 433.

(9 JONES (G. O . ) , MARTIN (D. H.), MAWER (P. A.) and PERRY (C. H.), Proc. Roy. SOC. (London), Ser., A261, 1961, 10.

(f) LOWNDES (R. P.) and MARTIN (D. H.), to be published.

In the case of the zinc blende type crystals, Slater [12]

has suggested from considerations of atomic and ionic radii that the covalent contributions to the binding dominate the ionic contributions and that the atoms have essentially zero static ionic charge. This suggestion has been confirmed by Herman [15] whose calculations of the charge distribution in the iso-elec- tronic series Agl, CdTe, InSb and Sn indicate that the atoms in the compounds have nearly zero static charges, even in the case of AgI which would ⁽⁽ a priori ^D be

expected to have an appreciable static ionic character.

On the other hand, ZnS type crystals, without excep- tion, exhibit strong infrared (lattice vibration) absorp- tion and have a dynamic ionic charge e; greater than unity (table 11). Since the static ionic charge is pre- sumably very small, the observed values of e;/e may be largely attributed to charge redistribution effects.

The homopolar trigonal crystals Se and Te are further, even more striking, examples of substance having sizeable dynamic ionic charges and identically zero static ionic charges 1161. In these crystals the dynamic ionic charge must be attributed entirely to charge redistribution effects.

Infrared (lattice vibration) parameters for ZnS type crj~stals [8]

0, 0,

Crystal ^{E ,} c0 (cm-l) (cm-I) e;/e Ref.

-

S i c AlSb G a p GaAs GaSb InP InAs InSb ZnS ZnSe ZnTe CuCl

(a) SPITZER (W. G.), KLEINMAN (D. A.) and FROSCH (C. J.), Phys. Rev., 1959, 113, 133.

(b) TURNER (W. J.) and REESE (W. E.), Phys. Rev., 1962, 127, 126.

(C) KLEINMAN (D. A.) and SPITZER (W. G.), Phys. Rev., 1960, 118, 110.

(&) HASS (M.) and HENVIS (B. W.), J. Phys. Chem. Solids, 1962,23, 1099.

( e ) CZYZAK (S. J.), BAKER (W. M.),, CRANE (R. C) and

HOWE (J. B.), J. Opt. Soc. Am., 1957, 47,240.

(f) MANABE (A.), MITSUISHI (A.) and YOSHINAGA (A.), Japan.

J. Appl. Phys., to be published.

(8) MARPLE (D. T. F.), J. Appl. Phys., 1964,35, 539.

( h ) IWASA (S.), Thesis (Physics Department, University of

Pennsylvania, 1965).

Although the assumption of an effective field for the electronic dipoles E,,, = E -I- (4 7113) P is gene- rally believed to be valid for the alkali halides [17], it is not believed to be valid for ZnS type crystals such as the I-VII, 11-VI, 111-V and IV-IV (Sic) compound semiconductors [l8]. For these substances the electrons

THE DIELECTRIC PROPERTIES C 4 - 8 1 and holes have extended wavefunctions and, conse- ponding to the excitation of q

x

0 LO phonons. In quently, the effective field acting on the electronic the case of the tunneling curves for p-n junctions of dipoles may be expected to be essentially equal to the 111-V compound semiconductors, such structure is macroscopic field, i-e., E,, x E. Under these circums- found to correspond closely in energy to that of the tances the electronic and vibrational electric dipole q z 0 LO phonons as determined from infrared excitations are not coupled by the Lorentz field and reflectivity spectra [21] [22]. Since the LO phonons

*

⁹

e~ x e,

.

are coupled to plasmons in the n-and p- regions of the

3. The Dielectric Properties of the IV-VI Com- pounds. - The low frequency dielectric constant of semiconductors can be determined by capacitance measurements on samples (at low temperatures) in which the free carriers are frozen out or by capaci- tance measurements on p-n junctions.. Alternatively, it can be determined from values of the long wave- length (q

=

0) TO and LO phonon frequencies and the high frequency dielectric constant by means of the Lyddane-Sachs-Teller (LST) relation

However, the relatively high residual carrier densities which are present in the IV-VI compounds, and the coupling of the LO phonons with the plasma oscilla- tions of the free carriers [19] prevents one from unam- biguously determining w, and w, from infrared reflec- tivity data.

Kanai and Shohno [20] have determined the low frequency dielectric constant of PbTe by measuring the barrier-capacitance of abrupt p-n junctions at low temperatures. Their data yield a value of E , z 400 and indicate that E , is relatively independent of temperature in the range from 4O to 130 OK.

Values of w, for PbS, PbSe and PbTe have been obtained by Hall and Racette [21] from structure in the electron tunneling curves of p-n junctions corres-

samples, the experimental data imply further that the inelastic scattering of the tunneling electrons take place in the ^<( carrier free ⁾⁾ transition region of the junctions.

Values of a, for PbS, PbSe and PbTe have been established by various investigators from room tem- perature and low temperature infrared transmission data on thin films [23]. Cochran and co-workers [3]

have obtained values of w, and w, for PbS, PbTe and SnTe from inelastic neutron scattering data. Their values for PbS and PbTe are in good agreement with the a, values determined from thin film transmission data and with the values of w, determined from elec- tron tunneling data.

Application of the LST relation to the data for E,,

w, and w, for the cubic IV-VI compounds (table 111) leads to E , values of 190, 280, 450 and 1770 for PbS, PbSe, PbTe and SnTe respectively. We noted that the

E , value for PbTe calculated by means of the LST

relation is in agreement with the value which Kanai and Shohno have obtained from p-n junction capa- citance measurements.

It is of interest to note that an effort to obtain an independent experimental estimate of E , for PbTe from magneto-optical data at microwave frequencies has been carried out by Sawada and co-workers [24].

On the basis of data on Fabry-Perot resonances of helicon waves, similar to that obtained by Libchaber

Infrared (Lattice Dynamics) Parameters for Cubic IV-VI Compounds

PbS PbSe PbTe SnTe

-

18.5 (") (77 OK) 25.2 (") (77 OK) 36.9 (") (77 OK) 45 (1 (300 OK)

-

66.7 ^(b) (77 OK) 44

("1

(4 OK) 31.5 (*)

(4 OK) 22.3 (3 (100 OK)

- ^{- -}

200 (f) 190 10 4.5

(4 OK)

155

(3

^{280 11 5.2}

(4 OK)

110 (f) 450 12 6.2

(4 OK)

140 (g) 1 770 49 7.8

(1 00 OK)

and Veilex for InSb [25], and data on the cut-off magne- tic field, i.e., the magnetic field a t which the real part of the dielectric constant for the cyclotron inactive EM mode, goes to zero, they derived an E, value of 3 000 for a p-type sample having p ⁼ 1 x l o t 7 cm-3 and a value of 1 x lo4 for a sample with

Perkowitz [26] has recently carried out similar studies on n-type PbTe. He obtains a value of E,

x

_{1 x lo4} for a sample with n = 5 x 1017 ~ m - ~ , essentially the same resultats for p-type PbTe. The nature of the discrepancy between the value of ^E, for PbTe obtained from magneto-optical measurements and that obtained by the other methods is still unresolved.

The dielectric parameters for the cubic IV-VI compound semiconductors are summarized in Table 111.

We note that the values of ^E, and ^E, increase and that of o, decreases in the sequence PbS, PbSe, PbTe and SnTe. On the other hand the value of o, decreases in the sequence PbS, PbSe and PbTe, but increases on going from PbTe to SnTe. We note also that

for the Pb compounds and

x

50 for SnTe, and that therefore, w, 2 ⁼ o;

+

^{52' w}

a',

i.e., the frequency of the q w 0 phonons is determined largely by the plasma frequency of the ions, rather than by o,, the spring constant frequency of the optical phonons.

4. The Dynamic Ionic Charge of the Cubic N - V I Compounds.

-

The macroscopic dynamic ionic charge e; can be calculated from values of o,, o, and E ,

either by using the relation

-

-

me,

a2

rns,(o2 - o$)

e:=--

-

4 nN 4 EN (4.1)

or from values of E,, E, and o,. by using the relation

The two procedures are of course equivalent since the four parameters are macroscopically related to one another through the LST relation. The values of e:/e are found to be 4.5 (for PbS), 5.2 (for PbSe), 6.2 (for PbTe) and 7.8 (for SnTe). These values are considera- bly larger than the e:/e values for the alkali halides (e;/e

w

1) and for the ZnS type compound semicon- ductors (e:/e w 2).

As in the case of the ZnS type semiconductors, we do not believe that the assumption of an effective

field, E,, = E

+

(4 4 3 ) P, acting on both the elec- tronic and atomic dipoles is valid for the IV-VI compound semiconductors, but rather that the eKec- tive field acting on the electronic dipoles is essentially equal to the macroscopic field, i.e., E,,,

w

E. On this basis, the electronic and vibrational (electric dipole) excitations are not coupled and e: w e,. In the absence of the (E,

+

2)/3 enhancement factor, the large values of e: imply that the relative displacement of atoms in the optical vibration modes is accompanied by a rather Iarge redistribution of electronic charge. A Lorentz field does act on the ⁽⁽ local ⁾⁾ part of the atomic displacement dipoles [27] and presumably, does lead to a decrease in the frequency of the q x 0 TO phonons of the form

where e:, corresponds to the ⁽⁽ local >> dynamic ionic charge and w, is the spring constant frequency in the absence of the Lorentz field. A microscopic model would be needed to establish the relation between e:, and e;. In the absence of such a model, it is not even possible to guess the relative magnitudes of e;, and e:, nor the trend in the values of e;, among the IV-VI compounds. However, it appears likely that an appreciable shift in frequency of the q

=

⁰ TO phonons by the atomic dipole Lorentz field, is the

(( destabilizing ^)> factor which is responsible for the relatively low values of o~ among the IV-VI com- pound semiconductors and for the paraelectric beha- vior of SnTe.

Acknowledgements.

-

We wish to aclcnowledge valuable discussions with

J.

_{N. Zemel.}

References

[I] BURSTEIN (E.) and EGLI (P.), Advances in Electronics and Electron Physics, 1955, 7, 56. See also SCAN-

LON (W. W.), Solid State Physics, 1959, 9, 83.

Edited by S ~ r r z (F.) and TURNBULL (D.) (Acade- mic Press, Inc., New York 1958).

[2] COCHRAN (W.), Phys. Letters, 1964, 13, 193.

[3] COCHRAN (W.), COWLEY (R.), DOLLING (G.) and ELCOMBE (M. M.), PYOC. Roy. Soc. (London),

1966, A293,433.

[4] BIERLY (J. N.), MULDAWER (L.) and BECKMAN (O.), Acta Met., 1963, 11, 447.

[5] COHEN (M. H.), FALICOV (L. M.) and GOLIN (S.), IBM J. Res. Develop., 1964, 8, 215.

[6] COCHRAN (W.), Proceedings of the General Motors Symposium on Ferroedectricity, edited by E. Weller, p. 62 (Elsevier Publishing Co., 1967).

THE DIELECTRIC PROPERTIES C 4 - 8 3 171 BURSTEIN (E.), J. Phys. Chem. Solids., 1967, 28, 1655.

[8] BURSTEIN (E.), BRODSKY (M. H.) and LUCOVSKY (G.), J. Quantum Chem, IS, 1967,759.

[9] COCHRAN (W.), Nature, 1961,191,60.

[lo] SZIGETI (B.), Trans Faraday Soc., 1949, 45, 155.

[ l l ] BRODSKY (M. H.) and BURSTEIN (E.), J. Phys. Chem.

Solids., 1967,28, 1655.

[12] YAMASHITA (J.) and KURASAWA (T.), J. Phys. SOC.

Japan, 1955,10, 610. DICK (B. G.) and OVERHAU-

SER (A.), Phys. Rev., 1958, 112, 90. HAULON (J. E.) and LAWSON (A.), Phys. Rev., 1959, 133, 472.

[13] TOLPYGO (K. B.), Fiz. Tverd. Tela., 1959, 1, 211.

[14] SLATER (J. C.), Quantum Theory of Molecules and Solids I1 (McGraw-Hill Book Co., New York, 1961) Chap. 4, p. 95.

[15] HERMAN (F.), Int. J. Quantum Chem., 1967, IS, 533.

[16] LUCOVSKY (G.), KEEZER (R. C.) and BURSTEIN (E.), Solid State Commun., 1967,5,439.

[17] TESSMAN (J. R.), KAHN (A. H.) and SHOCKLEY (W.), Phys. Rev., 1953, 92, 890.

1181 BRODSKY (M. H.) and BURSTEIN (E.), Bull. Amev. Phys.

SOC., 1953, 7 11, 214.

[19] VARGA (B.), Phys. Rev., 1965,137A, 1896.

[20] KANAI (Y.) and SHOHNO (K.), Jap. J. Appl. Phys., 1968,2, 6.

(211 HALL (R. N.) and RACETTE (J. N.), J. Appl. Phys., 1961,32 (Supplement) 2078.

1221 IWASA (S.), BALSLEV (I.) and BURSTEIN (E.), Proc.

Int. Conf. on Physics of Semiconductors p. 1078 (Dunod, Paris 1964).

1231 (PbS) GEICK (R.), Phys. Letters, 1965, 10, 5. ZEMEL (J. N.), Proc. Int. Conf. on Physics of Semicon- ductors, p. 1061 (Dunod, Paris, 1964).

(PbSe) BURSTEIN (E.), WHEELER (R.) and ZEMEL (J. N.), Proc. Int. Conf. on Physics of Semi- conductors, p. 1065 (Dunod, Paris, 1964).

(PbTe) BYLANDER (E. G.) and HASS (M.), Solid State Commun., 1966, 4, 51.

[24] SAWADA, BURSTEIN, CARTER and TESTARDI, Proc.

Symp. on Plasma Effects in Solids, p. 71 Dunod, Paris (1 964).

[25] LIBCHABER (A.) and VEILEX (R.), Phys. Rev., 1962, 127, 774.

[26] PERKOWITZ (S.), Thesis, University of Pennsylvania, 1967.

[27] BURSTEIN (E.) (( Phonons and Phonon Interaction ^)), edited by T. Bak, p. 276 (W. A. Benjamin, 1964).

DISCUSSION

BIRMAN, J. L. - I wish t o emphasize that Szigeti's recent work (with Leigh and also private communi- cation) shows that the ((local field D is not a well defined object. I n fact, the field acting on the electrons in a solid is not a constant but depends o n position, also the electrons are distributed (diffuse) so that in the equation of motion, one should have diffuse elec- trons clouds acted o n by crystal fields. Which vary in position over the clouds. Then the effective charge concept is correspondingly a weakened one.

In principle one knows how t o proceed : in the adiabatic approximation one solves the Schroedinger equation in each instantaneous configuration to find electron eigenstates, electron charge densities, etc.

Clearly these quantities are diffuse. The problem we are observing, with wide discrepancies between egigeti and e: results from a breakdown of a certain oversim- plification that fixed charges move rigidly when the lattice deforms. Obviously a new oversimplification is required, and work is now underway to develop it.