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>
HAL Id: jpa-00213615
https://hal.archives-ouvertes.fr/jpa-00213615
Submitted on 1 Jan 1968
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JOURNAL DE PHYSIQUE
ColjOque C 4, supplkment au no 11-12, Tome 29, Nouembre-Dkcembre 1968, page C 4 - 78
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
se) E
trouvent iitre nettement plus fortes que les valeurs corresmacroscopique, e: = ( ~ M T / ~ u T
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
)E
semiconductors are reviewed. The values of macroscopic dynamic ionic, e; = ( ~ M T / ~ u Tare
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 responsible 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, Yorktown 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 dependence 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 coworkers, [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 diiTerence 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
crystals, and that, as in the case of the BaTiO, ferroelectrics, 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 properties of the cubic IV-VI compound semiconductors
with particular emphasis on the values of the macroscopic 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 macroscopic 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 dependence 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].
E(O) = E, f
4n ~ e : ~ 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 constant 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 displacement of the two atoms in the unit cell ; MTis the
electric moment resulting from the relative displace-
C4-79
ment of the atoms ; and E is the macroscopic electric
field. M T includes the contributions from the redistribution 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 (111.
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 constants 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
C4-80
E. BURSTEIN, S. PERKOWITZ AND M. H. BRODSKY
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,*/ein the sequence AF-ACIABr-Al-AH (where A is a given alkali metal) in which
there is an increasing ratio of anion to cation polarizabilities (table I).
Infrared (lattice vibration) parameters
of alkali Halides [8]
Crystal
WL
(cm - ')
-
-
LiH
LID
LiF
LiCl
LiBr
NaF
NaCl
NaBr
NaI
KF
KC1
KBr
KI
RbF
RbCl
RbBr
RbI
CsCl
CsBr
CsI
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-electronic 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 exception, exhibit strong infrared (lattice vibration) absorption and have a dynamic ionic charge e; greater than
unity (table 11). Since the static ionic charge is presumably 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,
Crystal
E,
c0
0,
(cm-l) (cm-I)
e;/e
Ref.
-
Sic
AlSb
Gap
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 generally 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
and holes have extended wavefunctions and, consequently, the effective field acting on the electronic
dipoles may be expected to be essentially equal to the
macroscopic field, i-e., E,, x E. Under these circumstances the electronic and vibrational electric dipole
excitations are not coupled by the Lorentz field and
* 9
e~ x e, .
3. The Dielectric Properties of the IV-VI Compounds. - 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 capacitance measurements on p-n junctions.. Alternatively,
it can be determined from values of the long wavelength (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 oscillations of the free carriers [19] prevents one from unambiguously determining w, and w, from infrared reflectivity 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-
C4-81
ponding to the excitation of q x 0 LO phonons. In
the case of the tunneling curves for p-n junctions of
111-V compound semiconductors, such structure is
found to correspond closely in energy to that of the
q z 0 LO phonons as determined from infrared
reflectivity spectra [21] [22]. Since the LO phonons
are coupled to plasmons in the n-and p- regions of the
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 temperature 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 electron 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 capacitance 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)
(4 OK)
155 (3
(4 OK)
110 (f)
(4 OK)
140 (g)
(100 OK)
190
10
4.5
280
11
5.2
450
12
6.2
1 770
49
7.8
E. BURSTEIN, S. PERKOWITZ AND M. H. BRODSKY
C4-82
and Veilex for InSb [25], and data on the cut-off magnetic field, i.e., the magnetic field at 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 lot7 cm-3
and a value of 1 x lo4 for a sample with
+
field, E,, = E (4 4 3 ) P, acting on both the electronic and atomic dipoles is valid for the IV-VI
compound semiconductors, but rather that the eKective 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
2)/3 enhancement factor, the large values
of the (E,
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
+
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
-
-
e:=--
me, a2- rns,(o2 - o$)
4 nN
4 EN
or from values of
E,,
E,
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 compound semiconductors and for the paraelectric behavior of SnTe.
Acknowledgements. - We wish to aclcnowledge
valuable discussions with J. N. Zemel.
(4.1)
and o,. by using the relation
References
[I] BURSTEIN
(E.) and EGLI(P.), Advances in Electronics
and Electron Physics, 1955, 7, 56. See also SCANLON (W. W.), Solid State Physics, 1959, 9, 83.
Edited by S ~ r r (F.)
z and TURNBULL
(D.) (Acade-
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 considerably larger than the e:/e values for the alkali halides
(e;/e w 1) and for the ZnS type compound semiconductors (e:/e w 2).
As in the case of the ZnS type semiconductors, we
do not believe that the assumption of an effective
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.
(W.), Proceedings of the General Motors
[6] COCHRAN
Symposium on Ferroedectricity,edited by E. Weller,
p. 62 (Elsevier Publishing Co., 1967).
C4-83
THE DIELECTRIC PROPERTIES
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.
[ll] 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 OVERHAUSER (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 Semiconductors, p. 1061 (Dunod, Paris, 1964).
(PbSe) BURSTEIN(E.), WHEELER(R.) and ZEMEL
(J. N.), Proc. Int. Conf. on Physics of Semiconductors, p. 1065 (Dunod, Paris, 1964).
(PbTe) BYLANDER
(E. G.) and HASS(M.), Solid State
Commun., 1966, 4, 51.
[24] SAWADA,
BURSTEIN,CARTERand TESTARDI,
Proc.
Symp. on Plasma Effects in Solids, p. 71 Dunod,
Paris (1964).
[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 communication) 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 on position,
also the electrons are distributed (diffuse) so that in
the equation of motion, one should have diffuse electrons 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 oversimplification that fixed charges move rigidly when the
lattice deforms. Obviously a new oversimplification is
required, and work is now underway to develop it.