Dynamical Properties of IV VI compounds [Professor Dr. Heinz Bilz, Dr. Anette Bussmann Hol(BookZZ.org)

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Springer Tracts in Modern Physics 99 
Editor: G. H5hler 
Associate Editor: E.A. Niekisch 
Editorial Board: S.Fl(~gge H.Haken J.Hamilton 
H. Lehmann W. Paul 
Springer Tracts in Modern Physics 
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78 Solid-State Physics With contributions by R. Dornhaus, G. Nimtz, W. Richter 
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97 Elements of Flow and Diffusion Processes in Separation Nozzles By W. Ehrfeld 
98 Narrow-Gap Semiconductors With contributions by R. Dornhaus, G. Nimtz, and 
B. Schlicht 
99 Dynamical Properties of IV-VI Compounds With contributions by H. Bilz, A. Bussmann- 
Holder, W. Jantsch, and P. Vogl 
100" Quarks and Nuclear Forces Edited by D. C. Fries and B. Zeitnitz 
101 Neutron Scattering and Muon Spin Rotation With contributions by R. E. Lechner, 
D. Richter, and C. Riekel 
* denotes a volume which contains a Classified Index starting from Volume 36. 
Dynamical Properties 
of IV-Vl Compounds 
Contributions by 
H. Bilz A. Bussmann-Hoider W. Jantsch P. Vogl 
With 47 Figures 
Springer-Verlag 
Berlin Heidelberg New York Tokyo 1983 
Dr. Wolfgang Jantsch 
Max-Planck-lnstitut fQr Festk0rperforschung, Heisenbergstrasse 1 
D-7000 Stuttgart 80, Fed. Rep. of Germany 
Permanent Address: Johannes Kepler Universit~tt Linz, Institut for Experimentalphysik, 
A-4045 Linz-Auhof, Austria 
Dr. Anette Bussmann-Holder 
Professor Dr. Heinz Bilz 
Max-Planck-lnstitut fQr FestkSrperforschung, Heisenbergstrasse 1 
D-7000 Stuttgart 80, Fed. Rep. of Germany 
Dr. Peter Vogl 
Universit&t Graz, Institut fQr Theoretische Physik, 
A-8010 Graz, Austria 
Manuscripts for publication should be addressed to: 
Gerhard HOhler 
Institut for Theoretische Kernphysik der Universit&t Karlsruhe 
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ISBN 3-540-12092-0 Springer-Verlag Berlin Heidelberg New York Tokyo 
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Preface 
The IV-Vl compounds such as PbS and SnTe form an unusual class of materials. They 
often show a soft-mode behaviour and exhibit characteristic d ie lect r ic anomalies. 
These properties ref lect the fact that many IV-VI compounds are real or incipient 
ferroelectr ics. Simultaneously, they are narrow-gap semiconductors with quite unique 
electronic properties. 
This volume contains a review of the present experimental situation and of the 
theoretical understanding of the dynamical aspects of IV-VI compounds with emphasis 
on the ferroelectr ic properties. 
In part icular , the d ie lectr ic and optical measurements of several IV-Vl compounds 
as a function of temperature, al loy concentration and carr ier concentration provide 
an important insight into the driving mechanisms for the structural ins tab i l i ty of 
these materials. This aspect is reviewed in the contribution by W. Jantsch. 
The second contribution by A. Bussmann-Holder, P. Vogl and H. Bi lz deals with the 
present theoretical description and interpretation of electronic and lattice-dynami- 
cal properties of IV-VI compounds. Three di f ferent microscopic models are discussed, 
which are used to explain the complex temperature dependence of soft modes and rela- 
ted properties. At present, i t seems that the recently developed po lar i zab i l i ty model 
is most successful in explaining a large variety of experimental data. 
The present volume supplements the paral lel review, by G. Nimtz and B. Schlicht, 
on IV-VI compounds which focuses on crystal preparation as well as transport and 
optical properties (Vol. 98 in this series). 
I t is hoped that our investigations turn out to be useful for both researchers 
and graduate students. They may be intrigued by the model character of these com- 
pounds which exhibit both structural s impl ic i ty and complex physical properties. 
Linz, 
Stuttgart, Graz, April 1983 
W. Jan tsch 
A. Bussmann-Holder 
H, Bilz 
P, Vogl 
Contents 
Dielectric Properties and Soft Modes in Semiconducting 
(Pb, Sn, Ge)Te 
By W. Jantsch (With 25 Figures) 
I. Introduct ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
2. Experimental Determination of the Soft-Mode Frequency . . . . . . . . . . . . . . . . . 
2.1 Neutron Scatter ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
2.2 Ine las t i c Tunneling of Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
2.3 Raman Scatter ing . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
2.4 Far - ln f rared Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
3. Experimental Determination of the Static Dielectric Constant . . . . . . . . . . . . 
3.1 D i f fe rent ia l Capacitance Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
3.1 Microwave Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
4. Effects Related to the Phase Transit ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
4.1 Changes of Band Structure and Related Phenomena . . . . . . . . . . . . . . . . . . 
4.2 Resistance Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
4.3 Acoustic and Speci f ic -Heat Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
5. Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
5.1 Temperature Dependence of the Soft Mode and Phase Trans i t ion . . . . . 
5.2 Microscopic Origin of Structura l Ins tab i l i ty and Chemical Trends 
of Cr i t i ca l Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
5.3 Inf luence of Lat t i ce Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
5.4 Inf luence of Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
6. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
Combined Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
5 
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39 
44 
45 
46 
99 
VII 
Electronic and Dynamical Properties of IV-Vl Compounds 
By A. Bussmann-Ho lder , H. B i l z , and P. Vogl (With 22 F igures ) 
I . In t roduct ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
1.1 H is to ry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
1.2 Landau Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
2. Chemical S t ruc ture and E lectron ic Bands of IV -V l Compounds . . . . . . . . . . . 
2.1 S t ruc ture and Fer roe lec t r i c i ty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
2.2 Ion ic i ty and Covalency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
2.3 E lec t ron ic Band St ruc ture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
2.4 The Zero-Gap S i tuat ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
3. Lattice Dynamics and Phase Trans i t ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
3.1 General Aspects: The Soft-Mode Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . 
3.2 E lec t ron ic Theory o f the Soft-Mode Ins tab i l i ty . . . . . . . . . . . . . . . . . . . 
3.3 The Anharmonic Lat t i ce Mode] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
3 .3 .1 Quasi-Harmonic Approx imat ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
3 .3 .2 The Mo lecu la r -F ie ld Treatment of a S ing le Mode Model . . . . . . 
3.4 The V ibron ic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
3.5 The Po la r i zab i l i ty Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
3 .5 . i Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
3 .5 .2 Three-Dimensional Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
3.6 Comparison of Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
3.7 Non l inear Exc i ta t ions in IV-VI Semiconductors . . . . . . . . . . . . . . . . . . . . 
4. Summary and Conclus ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
Combined Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
51 
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ViII 
Dielectric Properties and Soft Modes in Semiconducting 
(Pb, Sn, Ge)Te 
By W. Jantsch 
I. Introduction 
Binary compounds of Groups IV and Vl of the periodic table have been of consider- 
able theoretical, experimental and technological interest in recent years (Ravich 
et a l . , 1970; Rabii, 1974; Rauluskiewicz et a l . , 1978). For the lead chalcogenides 
PbTe, PbSe, PbS) in particular, the band structure (Dalven, 1973), ohmic (Ravich 
et a l . , 1971) and high-field transport (Jantsch et a l . , 1978a), optical (Zemel et 
a l . , 1965) and magnetooptical (Gerlach and Grosse, 1977; Bauer, 1978,1980) proper- 
t ies, and electronic states of defects (Heinrich, 1980; Lischka, 1982) have been 
studied extensively. 
The IV-VI compounds are narrow-gap semiconductors with a generally direct 
minimum energy gap of typical ly 0.2 eV. Some alloy systems, l ike Pbl_xSnxTe, ex- 
h ib i t a zero-gap transition depending on temperature and composition (Dimmock et 
a l . , 1966). This property makes IV-VI compounds especially suitable for technolo- 
gical applications as infrared sources (Preier, 1979) and detectors (Holloway, 
1980; Haas et a l . , 1975). Considerable effort has been devoted to the development 
of crystal-growth techniques. Among these, the hot-wall epitaxy has proved to be 
very successful for high-quality epitaxial material (Lopez Otero, 1978). In addi- 
tion, this method enables a great deal of variation in growth conditions. 
The IV-VI compounds exhibit a number of outstanding properties. Especially the 
dielectric properties and structural transitions of the tellurides to a low-temper- 
ature ferroelectric phase have attracted considerable attention (Lines and Glass, 
1977; Kawamura, 1978,1980). Less work has been done on the IV-VI compounds with 
lower molecular weight (GeS, GeSe, SnS, SnSe) which crystal l ize in an orthorhombic 
layer structure. Although phase transition may also occur for these compounds, the 
present review concentrates on the tellurides because of their outstandingly simple 
crystal structure and the comprehensive work published in the past few years. 
First indications for an anomalously high dielectric constant of PbTe were de- 
rived from the observation of unusually high values of the carrier mobilities at 
low temperatures (Allgaier and Scanlon, 1958; Allgaier and Houston, 1962). A dif fer 
ent interpretat ion for the small contr ibution of ionized impurity scattering was 
given later (Pratt, 1973,1974). However, from measurements of the voltage-dependent 
capacity C(V) of Schottky barriers on PbTe, Kanai and Shohno (1963) obtained a sta- 
t i c d ie lec t r i c constant of 400 for temperatures below 130 K. In view of the con-nection between the d ie lec t r i c properties and the la t t i ce dynamics of ferroelec- 
t r i c crystals (Cochran, 1960), the high value of the stat ic d ie lec t r i c constant 
led Cochran (1964) to suggest a soft-mode behavior and the poss ib i l i ty of a dis- 
placive fer roe lectr ic phase t rans i t ion in PbTe. 
The connection between the stat ic d ie lec t r i c constant c 0 and the optical mode 
frequencies is evident from the LYDDANE-SACHS-TELLER re lat ion (1941), given here 
in i ts extended version (Cochran and Cowley, 1962): 
2 
~__00 = ~-[ ~_~_ (1. I ) 
~ J ~Tj 
Here E stands for the high-frequency d ie lec t r i c constant and m. , m . for the Lj T j ' 
longitudinal and transverse optical frequencies of the j th phonon branch at q = O. 
I f one of the transverse modes becomes soft at a c r i t i ca l temperature T c, 
~Tj(q : O,Tc) = 0 , (1.2) 
the crystal becomes unstable against this type of v ibrat ion. Asaresu l t , a la t t i ce dis- 
tor t ion occurs. For a diatomic cubic la t t i ce of NaCI structure (the structure of 
PbTe), there is only one type of optical v ibrat ion, sp l i t into a doubly degenerate 
transverse and a s inglet longitudinal mode. For q = O, the two sublatt ices vibrate 
against each other. On cooling below T c, the two sublatt ices are displaced with 
respect to each other along the direct ion of vibrat ion of the soft mode: the soft 
mode "freezes". Due to the lowering of symmetry a permanent dipole moment occurs 
with eight possible orientations along the cubic <111> axes. The corresponding spon- 
taneous polar izat ion can be reoriented by an external e lectr ic f ie ld , which is the 
c r i te r ion of fe r roe lec t r i c i ty . (This de f in i t ion is s t r i c t ly applicable only to in- 
sulators. In the present case, free carriers impede the application of su f f i c ient ly 
high e lect r ic f ie lds . ) 
In SnTe (Pawley et a l . , 1966) and in PbTe (Alperin et a l . , 1972), ~TO(q = O) de- 
creases with decreasing temperature. Measurements of the stat ic d ie lec t r i c constant 
using the C(V) method on graded p-n junctions (Bate et a l . , 1970) show that the 
stat ic d ie lec t r i c constant of PbTe obeys a Curie-Weiss law 
-1 
~0 = Co(T - Tc) ' (1 .3 ) 
where C O is the temperature-independent Curie constant. For PbTe only negative 
I 
values of T c have been found by extrapolat ion of ~0 ~ O, indicat ing that PbTe re- 
mains paraelectr ic for a l l temperatures. 
I 0 I 
Fig. l.1. NaCI structure and rhombohedral distor-, 
tion of the low-temperature phase of binary Group- 
IV tellurides. The arrows indicate the sublattice 
shifts 
Structural changes have been observed in GeTe (Schubert and Fricke, 1951,1953) 
and SnTe (Muldawer, 1973; lizumi et a l . , 1975; Valassiades and Economou, 1975). 
At room temperature GeTe crystallizes in a rhombohedral (C3v) structure, which may 
be viewed as a distorted NaCl structure: the angle ~ between the axes of the unit 
cell differs only s l ight ly from 90 ~ (Fig.l .1). With increasing temperature, the 
interaxis rhombohedral angle ~ of GeTe increases gradually from its room-tempera- 
ture value of 88.35 o to the cubic value of 90 ~ (NaCl structure) at about 720 K 
(Schubert and Fricke, 1951,1953). For SnTe the same kind of structural transition 
has been observed (Muldawer, 1973; lizumi et a l . , 1975). Samples with low carrier 
concentration (p < ~ �9 1020 cm -3) transform at about 100 K. With increasing car- 
r ier concentration the cr i t ical temperature decreases: the cubic structure is sta- 
bil ized. 
Alloy systems like Snl_xGexTe (Bierly et a l . , 1963), Pbl_xSnxTe (Shimada et a l . , 
1977) and Pbl_xGexTe (Kawamura, 1978; Jantsch et a l . , 1979) exhibit an increasing 
cr i t ical temperature with increasing x. X-ray analysis (Bierly et a l . , 1963; 
Schubert and Fricke, 1953) gives no indication of a discontinuity of the rhombohe- 
dral angle, which is an order parameter for this type of phase transition, at the 
cr i t ical temperature of SnTe or GeTe. The phase transition of Snl_xGexTe, however, 
has recently been shown to become f i r s t order for x > 0.27 (Clarke, 1978). Results 
for the deviation of the rhombohedral angle from 90 ~ (A~ = 90 ~ - ~) are given in 
Fig.l.2 as a function of temperature. For x > 0.27, As goes to zero discontinuously 
at the cr i t ica l temperature. 
Investigations of the Raman effect (Steigmeier and Harbecke, 1970; Sugai et a l . , 
1977a,b), inelastic neutron scattering (Pawley et a l . , 1966; Alperin et a l . , 1972) 
and measurements of far-infrared ref lect iv i ty (Jantsch et a l . , 1978b) have shown 
that the structure change coincides with a softening of the transverse optical 
phonon mode, in agreement with the soft-mode concept (Cochran, 1960; Anderson, 1960). 
In summary, the binary tellurides of Pb, Sn and Ge and their alloys form a class 
of semiconductors witha tendency to structural phase transitions caused bya soften- 
ing of the zone-center TO mode. The phase transition in GeTe and SnTe is of the 
simplest conceivable type: i t consists in a relative shift of the cation and anion 
sublattices. Compared to other "classical" ferroelectrics, the crystal structure 
is par t i cu la r ly simple: the uni t cel l contains only two atoms. In a l loy systems, 
the c r i t i ca l temperature varies wi th in an exceedingly wide range: any value between 
700 K and negative values can be attained by choice of the a l loy composition. Some 
of these compounds seem to provide rare examples of the existence of displacive 
fer roe lect r ic phase t rans i t ions of 8eoond order. These materials therefore of fer a 
rare opportunity to test the soft-mode concept in the c r i t i ca l regime close to the 
phase t rans i t ion . Such c r i t i ca l phenomena are of fundamental importance in under- 
standing phase transi t ions (Lines and Glass, 1977; Bruce and Cowley, 1980). 
0.30 
w ,,, 02. c 
,", 0.20 
0.L c 
01C 
0.0. c 
Snt.xGexTe [o ~ 
x:0,13 0 ? 
100 110 120 130' 2t0 220 230 2/.0 300 310 
TEMPERATURE ['K} 
F ig . l .2 . Deviation of the rhombo- 
hedral angle As from the cubic (90 ~ ) 
value of Sn I vGe~Te as a function 
of temperature, ~fter Clarke (1978) 
The physical properties mentioned above, together with the technological impor- 
tance of the IV-Vl compounds, make these materials interest ing both in research 
and material science. Presumably they w i l l also become standard materials in the 
f ie ld of fe r roe lec t r i c i ty . 
Complications arise, however, because IV-Vl compounds are narrow-gap semiconduc- 
tors with high concentrations of highly mobile carr iers. Therefore most of the 
standard techniques used to investigate soft modes and d ie lec t r i c properties have 
to be modified for appl icat ion to this class of materials. 
In Chaps.2 and 3, experimental methods for the invest igat ion of soft modes and the 
stat ic d ie lec t r i c constant, respectively, are reviewed. Chapter 4 discusses effects 
related to the phase t rans i t ion , some of which, especial ly an anomaly of the elec- 
t r i ca l res i s t iv i ty , have been used as simple tools to determine the c r i t i ca l 
temperature. 
F ina l ly , in Chap.5 experimental results for thed ie lec t r i c behavior and soft modes 
are discussed in the l ight of ex ist ing theoretical models. The emphasis is not on 
completeness or a detailed description of theoretical aspects, but rather on a sur- 
vey of th is f ie ld . 
2. Experimental Determination of the Soft-Mode Frequency 
According to the Cochran-Anderson (1960) theory, fe r roe lect r icphase transit ions 
are caused by "softening" of the zone-center TO-phonon mode. Four methods have been 
applied to investigate experimentally mTO(q = 0): 
i ) ine last ic neutron scattering 
i i ) ine last ic tunneling of electrons 
i i i ) Raman spectroscopy 
iv) far - in f rared spectroscopy. 
These methods are discussed below in the context of the i r appl icat ion to IV-VI com- 
pounds. 
I Neutron Scattering 
In contrast to other methods, ine last ic neutron scattering experiments enable a 
determination of the complete, wave-vector-dependent dispersion curves, thus allow- 
ing a comparison with la t t i ce dynamical models (Cochran et a l . , 1966). However, this 
method requires large crystals of good qual i ty and homogeneity, which, especial ly in 
the case of al loys of IV-VI compounds, are d i f f i cu l t to grow. Therefore, only a 
l imited amount of data is available in the l i te ra ture . The f i r s t evidence for mode 
softening on cooling in the paraelectric phase was established by Pawley et al . 
(1966) for SnTe (Figs.2.1,2). With decreasing temperature, ~TO(qSO) decreases. At 
low temperatures, mTO(q~O) saturates and no phase t rans i t ion occurs for th is par- 
t i cu la r sample. Complete dispersion curves for PbTe at 296 K and for SnTe at 100 K 
were measured by Cochran et al . (1966) and Cowley et a l . (1969), respectively. For 
SnTe, which exhib i ts carr ier concentrations as high as 1020 cm -3 due to in t r ins ic 
defects, the LO branch is expected to approach the TO branch for q ~ 0 since the 
macroscopic e lect r ic f ie ld of the LO mode is screened by free carr iers (R.A. Cowley 
and Doll ing, 1965; E.R. Cowley et a l . , 1969), (Fig.2.1). Table 2.1 summarizes ref - 
erences of available data on ine last ic neutron scattering. 
No data has been published for the low-temperature phase of any member of the 
Group IV te l lu r ide family as yet. However, from measurements of the temperature de- 
pendence of e last ic neutron scattering on Pb1_xSnxTe and SnTe with low defect con- 
centration, evidence for a displacive phase t rans i t ion was reported by Komatsubara 
et al. (1974) and lizumi et a l . (1975), respectively. In this method, the Bragg 
peak intens i t ies of odd-index ref lect ions, which are very weak in the cubic NaCI 
structure, are measured as a function of temperature. The phase t rans i t ion manifests 
i t se l f by an increase of the "forbidden" Bragg intens i t ies due to the lowering of 
symmetry. The sublatt ice sh i f t , which is an order parameter, can be evaluated from 
the normalized intens i ty . The results indicate that the fer roe lect r ic phase transi - 
tions of SnTe and Pbl_xSnxTe are of second order. 
Table 2.1. References for ine last ic neutron-scattering investigations 
Material Temp. [K] Measured phonon branch Reference 
SnTe 6-300 T0,LO(qlI[001]) 
SnTe 100 complete 
PbTe 4,293 TO,LO(qlI[111]) 
4-293 TO(q ~ O) 
PbTe 296 LO([O01], [110], [111]) 
PbTe 296 complete 
Pbo.2Sno.8Te 94 TO, A 
Pbo.63Sno.37Te 4.2 TO, A 
Pbo.63Sno.37Te 4-220 TO(q ~ O) 
Pbo.87Sno.13Te 19-293 TO(q ~ 0), ] inewidth 
Pbo.8Sno.2Te 5-239 TO(q ~ 0), l inewidth 
Pb0.8Sn0.2Se 80 T0,L0,LA,TA(q[I[O01]) 
Pb0.8Sn0.2Se 5-250 T0(q ~ O) 
Pb0.93Sn0.o7Se 5-250 TO(q ~0), linewidth 
Pawley et al . (1966) 
E.R. Cowley et al . (1969) 
Alperin et al . (1972) 
R.A. Cowley and Doll ing (1965) 
Cochran et al . (1966) 
Do]ling and Buyers (1973) 
Daughton et al . (1978) 
Vodopyanov et a l . (1978) 
2 / /A j 
z | j~ / , , TO 
[>;f 
u_1~ t/ 
qU [ 001 ) 
o I i I 
0 0.2 0/~ 
~-. SnTe 
i? LO \ o \ 
xo 
.300K 
~210K 
ot00K 
. 6K 
100 
REDUCED WAVE VECTOR aq/2~ 
- - 1.C 
~ 0.~ 
u_ 
Y 
SnTe / / 
/ 
/ 
/ 
/ 
,[ 
/ 
/ 
~'~/ q "-> 0 
/ I " 
I ~ 0 0 ~ ~ J 
0.6 08 1.0 Fi g. 2.2 * 0 100 2O0 3OO 
�9 ~ Fig.2.1 TEMPERATURE [K] 
2000 
t500 
-T- 
1000 
o 
3 
500 
Fi9.2.1. Dispersion of the LO and TO modes of SnTe at various temperatures ob- 
tained from ine last ic neutron scattering, af ter Pawley et a l . (1966) 
Fig.2.2. Squared soft-mode frequency [~TO(q ~ 0)] from Fig.2.1, after Pawley et 
al . (1966) 
2. 2 Inelastic Tunneling of Electrons 
Phonon-assisted tunneling was investigated for p-type Pb1_xSnxTe at l iqu id helium 
temperatures by Takasaki and Tanaka (1977). Tunneling junctions were prepared as 
metal-insulator-semiconductor (MIS) structures of the type Pb-SiO 2 (or ZnS)-Pbl_ x 
SnxTe. Peaks observed in the second derivat ive of the current-voltage characteris- 
t i cs , ~21/~V2, versus bias voltage were attr ibuted to the TO and LO modes at the r 
and L points of the Br i l l ou in zone. The energy gap can be determined also from this 
kind of experiment. With increasing x, a decrease of the energy gap and a decrease 
of mTO(q ~ O) occurs. With increasing carr ier concentration, mTO(q ~ O) increases. 
These results were interpreted in terms of the interband electron-phonon coupling 
model by Kawamura et al . (1975). For PbTe the carr ier concentration was not found 
to influence the LO-phonon energy. 
2. 3 Roman Scattering 
First-order Raman scattering in Group IV te l lur ides is restr icted to the low-tem- 
perature rhombohedral phase, since for the NaCl structure (T > Tc) a l l the atoms 
are situated at centers of inversion. In the la t te r s i tuat ion the optical phonon 
modes are IR active but Raman inactive. 
The optical phononsin the cubicNaCl phase (O h structure) belong to a 3-fold de- 
generate Flu mode, whose degeneracy is l i f ted by the macroscopic e lectr ic f ie ld 
associated with the longitudinal mode. In the rhombohedral phase (C3v structure), 
the Flu mode is sp l i t into a doubly degenerate E- and a single A I mode. The eigen- 
vector of the A I mode corresponds to the stat ic deformation due to the rhombohedral 
d is tor t ion . I t is this mode which is responsible for the phase t rans i t ion (Steig- 
meier and Harbeke, 1970). 
Both the E- and the A 1 modes are IR- and Raman active in the C3v structure. The 
Raman tensors are given by (Loudon, 1964): 
I ~ I;i c i) T(Alz) = \0 oa , T(Ey) = -Cd ' T(E-x) = O0 ' (2.1) 
where the extraordinary A 1 mode is polarized in the rhombohedral c-d irect ion 
(~ z-axis) and the ordinary E modes in the x-y plane. For "diagonal" polar izat ion 
[same polar izat ion of incident and scattered beam, e.g., z(xx)z using the notation 
of Damen et a l . , 1966] both the A 1- and E modes can be observed, whereas for 
crossed polar izat ion [e.g. , z(yx)z] only the E modes give a nonvanishing contr i - 
bution. 
The interpretat ion of Raman spectra of fer roe lectr ic IV-VI compounds may be com- 
plicated by several effects: 
( i ) Due to the high absorption coef f ic ient of these materials in the v is ib le 
range, wave vector conservation is not s t r i c t ly fu l f i l l ed . Therefore the result ing 
l ine shape is expected to contain contributions from a f in i te volume in q space 
7 
rather than the q ~ 0 component of the dispersion curves (Abstreiter et a l . , 1979; 
Katayama and Mi l l s , 1979). The observed peak posit ion in th is case does not coin- 
cide with the phonon frequency and the l ine width is enhanced with respect to the 
pure phonon l ine width at q = O. 
( i i ) In the presence of high concentrations of free carr iers , coupled LO-phonon- 
plasmon modes instead of pure phonon l ines and s ing le-part ic le excitat ion spectra 
may occur. In both cases the Raman spectra depend on the carr ier concentration and 
on the scattering wave vector. Both effects are sensit ive to surface states,since 
surface states may cause a band bending and hence gradients in the carr ier con- 
centration. 
( i i i ) In the rhombohedral phase the crystal divides into fer roe lect r ic domains 
(Snykers et a l . , 1972; Jantsch et a l . , 1979,1981b; Lewis et a l . , 1980): the di - 
pole moment of an individual domain is oriented along one of the four possible d i - 
rections corresponding to the cubic < I I I> axes. The domain size is of the order of 
I um (Snykers et a l . , 1972; Jantsch et a l . , 1981b). Therefore the usual selection 
rules cannot be used to ident i fy the various modes. 
F irst-order Raman scattering has been observed in GeTe (Steigmeier and Harbeke, 
1970), SnTe (Br i l l son et a l . , 1974; Murase et a l . , 1976; Sugai et a l . , 1977a,b; 
Murase and Sugai, 1979) and in Pbl_xGexTe (Sugai et a l . , 1979). The experiments 
were performed in backscattering geometry in the v is ib le range. Due to the high ab- 
sorption coef f ic ient , Raman scattering is restr icted to a surface region extending 
to a depth of about I00 ~. Consequently, the Raman intens i t ies are very low 
(30-100 counts/s for typical experimental conditions) and the spectra are suscep- 
t ib le to the surface preparation method (Br i l l son and Burstein, 1971; Shimada et 
a l . , 1977; Cape et a l . , 1977). Frequently an iodine f i l te r is used with the 
5145 X argon-ion laser l ine to suppress the strong elast ic component, which other- 
wise obscures the low-frequency phonon l ines (Devlin et a l . , 1971). 
In SnTe and Pbo.95Geo.o5Te two l ines have been observed below 50 cm - I whose fre- 
quencies decrease on approaching the c r i t i ca l temperature from below (Figs.2.3,4). 
These l ines do not occur above the c r i t i ca l temperature. In Pbo.95Geo.o5Te a direc- 
t ional dispersion of the upper l ine was also found (Fig.2.3). Therefore the upper 
l ine has been assigned to the extraordinary A I mode. For k lL [ l l l ] , (using pseudo- 
cubic indices) the A I mode is observable only due to the presence of fe r roe lect r ic 
domains, whose c-axis is not paral le l to the laser beam. In later experiments on 
Pbl_xGexTe, the A1-E mode sp l i t t ing could not be resolved (Jantsch and Stolz, 1980). 
Typical Raman spectra obtained in backscattering geometry (z(xx)z) are given in 
Fig.2.5. The l inewidth is comparable to that of each of the two l ines in Fig.2.3 
but much larger than the l inewidth obtained from far - inf rared spectroscopy (Sect. 
2.4), which can be attr ibuted to effects of the extremely small penetration depth 
(see above). Mode sp l i t t ing is discussed in Sect.2.4. 
With increasing carr ier concentration, a decrease in frequency of a single l ine 
at about the E-mode frequency has been reported by Sugai et a l . (1977b) (Fig.2.6). 
8 
2000 
1500 
1000 
500 
. oo 'oo> I I 
�9 , 
- ~ 50 cm" 
0 50 100 200 
Temperature [K] 
Fig.2.3. Temperature dependence of squared 
Raman line frequencies observed for (111) 
and (100) surfaces, respectively, of 
Pb1_xGexTe (x = 0.05) in the rhombohedral 
phase. The inset shows a typical Raman 
spectrum, after Murase and Sugai (1979) 
1500~ Sn Te 
p : lj, x102o rn. 3 
~176176 I 
~3 01 , , ~ i , , , ~ , 
o 50 1oo 
TEMPERATURE [ K) 
Fig.2.4. Temperature dependence of 
squared Raman line frequency observed 
on a SnTe (100) surface in the rhom- 
bohedral phase, after Murase and 
Sugai (1979) 
'7- 
o 
o 
c 
oJ 
c 
" Pbl-x Gex Te 
x:0.06 
�9 : ' "=" ; 1OK p:ixlOm~ ~ 
' , . .4S" " ":.m" . . . �9 : : . . . : : .~ : ~. : : . ; . , : . . . . 
"....::"'~.% 20K Z(XX)Z 
'J-" ..... ~" (111) surface "~., ..r.~.. . 
":. " �9 :~" . 'C : "~ 
't ":..-~',:'.*.. 30K . .....~." �9 =....,..> .. .'I. 
.~.:.;~-/~" & 0 K 
;. "-:;~. . 
,:':..-..- 7OK " " ....... ~',:": .... . ..~- ...: 
�9 " . . . . 
�9 =.--;,(,':,.~:.....o 
".~:. 100 K 
20 40 60 80 
Romon shift [cm "l] 
Sn Te 
sO(: o P"~JOm'3 
~oc ",,.,,,,. �9 p,.,.,o~o~, 
-- 30C " " o p=4.2xlO2Ocm "3 
200 " " �9 
100 
0 50 100 
TEMPERATURE IKI 
Fig.2.6. Temperature dependence of 
squared-Raman line frequencies ob- 
served for p-SnTe with different hole 
concentrations. Scattering is prob- 
ably due to the ETO mode, after 
Sugai et al. (19776) 
F ig .2 .5 . Raman spect ra ( z (xx)z ) o f a Pb n qaGe~ n6Te f i lm grown ep i tax ia l l y on a 
(111) o r iented BaF 2 subst ra te . For c ros~-po l&~zat ion (~(xy)z ) the spect ra a re 
essent ia l l y unchanged, a f te r Jantsch and Sto lz (1980) 
Softening of a mode in the low-temperature phase corresponds to decreasing crit ical 
temperature, i .e . , a stabilization of the cubic phase due to free carriers or intrin- 
sic lattice point defects, which are the origin of free carriers in undoped material. 
For a coupled plasmon-phonon polariton, one would expect an increase in frequency 
9 
E 15 u 
% 
[ a 
t �9 OeTe ,present work 
2Sx103~ ~ GeTe ~ Steigme=er and Horbeke 
| * Te . Pine ond Dresselhaus 
~ 
200 400 600 800 
TEMPERATURE [K} 
.d 
=; 
Ge Te 295 K 
Z(xx)Z 
k0=4762 
I I I I 
50 100 150 200 
RAMAN SHIFT [cm ~1] 
Fig.2.8. Typical Raman spectrum of GeTe 
at 295 K in backscattering geometry 
(Jantsch and Stolz, 1980) 
Fig.2.7. Squared Raman l ine sh i f ts as observed in GeTe [from Steigmeier and 
Harbeke (1970) and Jantsch and Stolz (1980)] and in Te [from Pine and Dresselhaus 
(1971) and Richter (1973)]. The arrow indicates the c r i t i ca l temperature of GeTe 
obtained from d i f fe rent ia l thermal analysis and X-ray investigations (Steigmeier 
and Harbeke, 1970) 
with increasing carr ier concentration, in contrast to the experimental result . 
Therefore th is effect (Sugai et a l . , 1977b) is e i ther overcompensated by the sta- 
b i l i za t ion of the phonon mode or only the pure unscreened phonon modes are observed. 
For GeTe two Raman l ines have been observed (Steigmeier and Harbeke, 1970), 
about a factor of 3 higher in frequency than in the case of SnTe or Pbo.95Geo.o5Te 
(Fig.2.7). In contrast to the results for SnTe and Pbl_xGexTe, selection rules 
depending on the re lat ive polar izat ion of laser and Raman l ight allowed an assign- 
ment of these l ines to the A 1-andE modes. The observation of selection rules ind i - 
cates that only a single domain was involved in this experiment. The conditions for 
preparing large or even single domain crystals have not been discussed. For kI Ic 
and k_Lc the same results with respect to the l ine posit ion and i t s temperature de- 
pendence were obtained (Harbeke and Steigmeier, 1980) thus indicat ing that the ob- 
served TO- and LO modes are degenerate. The sp l i t t ing due to the macroscopic f ie ld 
of the LO mode is perfect ly screened by the free carr iers, whose concentration is 
of the order of 1020 cm -3 in SnTe and GeTe. 
In Fig. 2.7 recent results obtained on GeTe are given together with results for 
pure Te. The close s imi la r i ty suggests that a Te surface-layer was present in the 
experiment on GeTe (Jantsch and Stolz, 1980): Te surface-layers have also been 
observed on CdTe (Z i t te r , 1971) and Pbl_xSnxTe (Cape et a l . , 1977) by Raman spectro- 
scopy. The observed Raman spectra are very s imi lar to those obtained for GeTe given 
2 ~ 0 y ie lds a c r i t i ca l temperature far in Fig.2.8. A l inear extrapolation of ~TO 
above the value obtained from X-ray investigations (Schubert and Fricke, 1951; 
10 
Table 2.2. References for available Raman data 
Material Investigated Carrier concen- References 
Temp. range [K] tration [cm -3] 
GeTe 55-480 
SnTe 3-80 1.5. 1020 
SnTe 20-80 1.1�9 1020 
SnTe 20-80 1.1. 1020 
SnTe 15-80 1.4-4.2 �9 1020 cm -3 
SnTe 15-80 1.4-4.2. 1020 cm -3 
SnTe 10-300 1.9-7.8 �9 1020 
Pbl_xSnxTe 10-300 
(0 < x < 0.54) 
Pbl_xGexTe 5-130 4.2 �9 1018 
(x : 0.05) 
Pbl_xGexTe 5-130 4.2. 1018 
(x = o .o5) 
Steigmeier and Harbeke (1970) 
Brillson et al. (1974) 
Murase et al. (1976) 
Sugai et al. (1977a) 
Sugai et al. (1977b) 
Kawamura et al. (1978) 
Shimada et al. (1977) 
Shimada et al. (1977) 
Murase et al. (1979) 
Sugai et al. (1979) 
Steigmeier and Harbeke, 1970). This discrepancy may also indicate that the Raman 
spectra observed on GeTe surfaces are actually due to Te. 
A summary of available references on Raman scattering in Group IV tellurides is 
given in Table 2.2. So far, a detailed analysis of line shapes and scattering cross 
sections has not been performed, probably because of the experimental di f f icult ies 
associated with this kind of material. 
2. 4 Far-lnfrared Spectroscopy 
Far-infrared (FIR) spectroscopy as a method for investigating the soft-mode fre- 
quency of IV-Vl compounds has some advantages compared to the Raman effect. First 
of a l l , the optical phonons are also infrared (IR) active in the cubic phase, where- 
as first-order Raman scattering is restricted to the low-temperature rhombohedral 
phase. Surface effects are less important, since the l ight penetration depth in 
FIR is much larger than in Raman experiments, the latter being carried out in the 
visible range far above the fundamental absorption edge. Sample inhomogeneities 
and space charge layers can be recognized and ruled out (Tennant and Cape, 1976). 
Evaluation of phonon-mode parameters from the FIR optical properties is some- 
what complicated, since a model for the dielectric function is required. For typical 
samples, the plasma frequency, Up, is above the Reststrahlen regime. Thus the trans- 
mission is very low and only the ref lect iv i ty can be measured with sufficient accur- 
acy. The ref lect iv i ty is governed by the complex dielectric function in the frequency 
range considered. In principle, the dielectric function could be evaluated from the 
11 
re f lec t iv i ty using the Kramers-Kronig relat ions. This method requires, however, a 
high photometric accuracy, which is d i f f i cu l t to achieve in the FIR. Therefore 
the re f lec t iv i ty is calculated from a model d ie lec t r i c function, which contains 
phenomenological phonon- and plasmon-parameters. These are determined by f i t t ing 
the model re f lec t iv i ty to the experimental data. 
The accuracy of th is method can be improved s ign i f i cant ly by using epitaxia l 
f i lms of several um thickness, comparable to the l ight penetration depth in the FIR. 
Due to the high d ie lec t r i c constant and the high carr ier concentrations, bulk 
samples exh ib i t nearly total re f lec t iv i ty in th is spectral regime. The character- 
i s t i c structures, caused by phonons, etc. , are rather small. In thin f i lm samples, 
however, mult ip le internal re f lect ion and interference effects occur, which tend 
to enhance small dips in the re f lec t iv i ty spectra. In the model calculat ion, mul- 
t ip le interference wi th in the f i lm and the substrate must be taken into account. 
The d ie lec t r i c function of the substrate can be determined from the same kind of 
experiment. 
The d ie lec t r i c funct ion, E(u), of PbTe can be described by a conventional dis- 
persion osc i l l a to r mode] including a Drude term for the free carr iers (Perkowitz, 
1975; Burkhard et a l . , 1976): 
2 2 
s(~) = ~ + Xph + xFC = s + 2 2 imr ~(~+i~ ) (2.2) 
wTO - m - 
Here u denotes the photon frequency, uTO the TO-phonon frequency and As = ~0 - s 
the difference between the stat ic d ie lec t r i c constant and the high-frequency dielec- 
t r i c constant. The la t te r takes into account contributions from short wavelength 
exci tat ions; ? and ~ are la t t i ce - and free carrier-damping parameters, respectively, 
T 
and Up stands for the unscreened plasma frequency 
2 4~Ne 2 (2.3) Up : . - 
m 
P 
is 3K/(2K+I) : plasma ef fect ive mass, where m t (N: carr ier concentration; m~n = m t �9 
f - 
the transverse ef fect ive mass and K = mJm t the effective-mass anisotropy). The 
phonon contr ibut ion is frequently wr i t ten also in the fol lowing form: 
2 2 
~LO - uTO (2.4) ~ 
Xph = ~ " --2 2 ' 
~TO - ~ i~F 
or 
2 2 
~LO - ~ - i~F ~ ~ 
Sph = Xp h+s =~ �9 2 2 ' (2.5) 
~TO - ~ - iur 
where the LYDDANE-SACHS-TELLER (1946) re lat ion has been used 
2 2 
~0/%o = ~LO/mTO (2.6) 
12 
From the model dielectric function the ref lectivity and transmission are calculated 
for the film-substrate sandwich: 
R123 = {R12e~ + R23e-~ + 2(Re{r12}-Re{r23} + Im{r12}.Im{r23})cos~ + 
+ 2(Re{r12}.Im{r23} - Re{r23}.Im{r12})sina}/D (2.7) 
and 
T 123 
where 
= T12T23T31e-n/D , (2.8) 
D = e B + RI2R23 e-B + 2(Re{r12}-Re{r23} - Im{r12}.Im{r23})cosa + 
and 
Rjk 
+ 2(Re{r12}-Im{r23} + Re{r23}-Im{r12})sina (2.9) 
13 
= irjki2 rj k nj - nknk , - ~jj u , nj = /~j(m) = nj - iKj 
with j = I (vacuum), 2 (film), 3 (substrate), and Tjk = (rjk - 1)-(rjk - 1); 
6 = 2n2md2/c; B = 2K2md2/c; and q = 2~3md3/c. Here the film and the substrate are 
characterized by their complex refractive indices n2 and n3 and their thicknesses 
d 2 and d 3, respectively. In (2.7) multiple interference within the substrate is not 
taken into account. The optical path length within the substrate (thickness 
d 3 ~ 1 mm) is large compared to that in the film. Interference effects due to a 
perfectly parallel substrate thus introduce a modulation of much higher periodicity 
in the frequency domain. The complete expression for this case has been given by 
Burkhard et al. (1977). Using wedged substrates or lower wavelength resolution, this 
additional complication is avoided. 
Figure 2.9a illustrates this method with the aid of results from the dispersion 
oscillator model for the real part of the dielectric function Re{E(m)}. The para- 
meters used (given in Fig.2.9b) are typical for PbTe. The dashed curve gives the 
free-carrier contribution, the dotted one the contribution from the phonon oscilla- 
tor. The solid curve represents the total dielectric function, that is, the sum of 
both susceptibility contributions plus the high-frequency dielectric constant. Ar- 
* and mC, where rows indicate the mode frequencies mTO, mLO' mp 
= ~pl ~r (2.10) m R 
is the plasma frequency screened by the valence electrons and 
+ m~L20 *2 m L : + mp (2.11) 
is the upper branch frequency of the coupled plasmon - phonon mode at q : O. 
0-= 
z 
0 
5000 
Z 
2000 
LL 
1000 
so~ 
n,,, 
uJ 
N -lo 
-IOC 
U_ 
O -50C 
-1000 m~ 
-200C 
_J -5000 <( 
uJ 
m _ _ 
Y 
~Re ~)) 
I I~ 
0 
10 
W TO : 
N = 5,6x10 ~crn'~ 
W~ : 2cm- 1 
20 ~0 60 
WAVE 
1oo 200 ~o0 6oo 1ooo 
NUMBER I crn-~l 
Fig.2.9. (a) Model calculat ion for 
the real part of the d ie lec t r i c 
function of PbTe due to the optical 
phonons Re{cph } = Re{~ph } + 
(dots), the free carr iers Re{EFC } 
= Re{xFC} + c (dashed l ine) , and 
the total d ie lec t r i c function 
Re{E} = Re{~ + ~ _} + ~ (solid 
. r~ pn ~ 1/3 
l lne). A nonlinear scale (c ) has 
been chosen to emphasize the zeros 
of the various contributions at m~, 
+ P 
~LO and mL" (b) Reflectivity for a 
film(PbTe)-substrate (BaF2) sandwich 
taking into account multiple inter- 
ference in the PbTe film. The para- 
meters used are given 
In Fig.2.9b the ref lect iv i ty resulting from (2.5) is given as a function of 
frequency. For the substrate the parametersof BaF 2, 
s = 5.04, c 2.16 , = 185 cm - I r = 3 cm -1 
= ~TO ' ' 
have been used. As can be seen from Fig.2.9b, the re f lec t iv i ty is very high below 
the plasma edge (m<~C). Close to the TO-phonon resonance, there f lec t iv i ty drops due 
to the large peak in the phonon contr ibut ion to the real part of the d ie lec t r i c func- 
t ion. The high-frequency onset of th is structure approximately coincides with the 
TO-mode frequency, mTO(q~O), indicated by an arrow in Fig.2.9a. The additional 
structure at lower frequencies is due to interference effects wi th in the f i lm. 
At ~ m E, the real part of ~(m) becomes posit ive and the th in the plasma edge, 
f i lm sample becomes nearly transparent. The re f lec t iv i ty exhibits osc i l la t ions due 
to Fabry-Perot-type interference effects. Additional structure in this range or i - 
ginates from the Reststrahlen band of the BaF 2 substrate. 
To f i t the osc i l l a to r model to the observed re f lec t iv i ty spectra in the cubic 
phase, the fo l lowing parameters are required: 
~ : high-frequency d ie lec t r i c constant 
~ = c O - ~ : phonon osc i l l a to r strength 
mTO : soft-mode frequency 
14 
F 
Up 
CO 
T 
: phonon damping parameter 
: plasma frequency 
: f ree-carr ier damping. 
The high-frequency d ie lectr ic constant is obtained from the extrapolation of the 
of the refract ive index towards low frequencies (co~ _ < m << Eg/h). The refrac- square 
t ive index can be evaluated from the per iodic i ty of the interference fringes be- 
low the fundamental absorption edge (~co < Eg) (Jantsch, 1980). Literature values 
for Eooare given in Table 2.3. An anomaly of the temperature dependence of e at 
the phase transit ion is discussed in Sect.4.1. 
Table 2.3. Optical d ie lectr ic constant ~ of (PbSnGe)Te 
x Carrier conc. [cm -3] TEK] c Remarks References 
(<0 for n type) 
Pbl_xSnxTe 0 5 �9 1017 373 32.6 Zemel et al . 
300 33.4 (1965) 
0.06 4.66 1017 77 38.5 �9 300 34.5 Dependence on Dionne and 
82 38.8 temperature Woolley 
(84-300 K) and (1972) 
0.208 2.56 ~ 1018 300 40.5 carr ier concen- 
82 43.3 trat ion 
(5 LIO 17 - 1020 
cm - j ) invest i - 
gated 
0 -8.2 �9 1016 300 33 
80 37 
6 39 
0.24 9.8 �9 1017 300 45 
80 53 
6 55 
0.36 4.8 �9 1018 300 50 
6 59 
0.39 8.0 �9 1017 300 50 
80 60 
6 60 
0.4 7 �9 1018 77 59 
6 59 
1 5 �9 1019 300 50 
1 3.5 �9 1020 300 42 
Lowney and 
Senturia 
(1976) 
Dependence on Riedl et al. 
carrier con- (1967) 
centration 
(5-1019 - 
5. 1020 cm -3) 
Dependence on Finkenrath 
carrier con- and K~hler 
centration (1966) 
(3.5. 1020- 
1.4. 1021 ) 
15 
Table 2.3 (cont.) 
x Carrier conc. [cm -3] T[K] ~ Remarks References 
(<0 for n type) 
Pbl_xSnxTe i 2 �9 1019 300 52 Dependence on Ota and 
carr ier con- Rabii (1971) 
centration 
(2.1- I019 - 
4.5. 1020 ) 
I 1.3 �9 1020 300 53 Dependence on Murase and 
temperature Sugai (1979) 
(4.2-300 K) 
0 5.6. 1016 300 34.2 Dependence on Murase and 
temperature Sugai (1979) 
(4.2-300 K) 
Pbl_xGexTe 0.07 4.7 �9 1018 300 39 Dependence on Murase and 
temperature Sugai (1979) 
(4.2-300 K) 
1 - 300 36 E polarization Burstein 
et al . (1971) 
The amplitude of the interference fringes is very sensit ive to the carr ier damp- 
ing parameter m . For an optimum f i t , m turns out to be frequency dependent. A de- 
T T 
ta i led analysis of the frequency dependence of the f ree-carr ier damping parameter 
of PbTe has been carried out by Burkhard et al . (1978). At low temperatures a strong 
resonance-like increase of the electron damping parameter was observed in the region 
of the coupled LO-phonon-plasmon excitations. This behavior can be explained quanti- 
ta t ive ly by the Mycielski (1978) model, which considers absorption of electromagne- 
t i c radiation from interaction of free carriers and col lect ive plasma osci l lat ions 
in a nonperfect, polar crystal . At higher temperatures the peak in m disappears due 
T 
to d i f ferent screening and thermal broadening and only a smooth variat ion of ~ with 
T 
frequency is required to f i t the re f lec t iv i ty spectra (Burkhard et a l . , 1976,1978). 
The plasma frequency mp can be calculated from (2.3) using d.c. Hall data for the 
carr ier concentration and effect ive mass values as determined from magneto-optical 
investigations (Bauer, 1978). Using (2.6), the phonon osc i l la tor strength in (2.2) 
can be replaced by the phonon frequencies, giving (2.4 or 2.5). 
The longitudinal mode frequency, mLO(q ~ 0), of PbTe is well known from inelast ic 
neutron scattering (Cochran et a l . , 1966; Alperin et a l . , 1972) and tunneling 
- I 
studies (Takasaki and Tanaka, 1977). A temperature-independent value of 114 cm 
is generally accepted. In a l loy systems, a s l ight variat ion of mLO with composition 
was found (Takasaki and Tanaka, 1977). 
Thus, to f i t the re f lec t iv i ty data in the neighborhood of the TO resonance we 
are le f t with two adjustable parameters: the soft-mode frequency mTO(q ~ 0), and the 
phonon damping parameter ~. No frequency dependence of r is required for an optimum 
f i t of the FIR spectra. 
18 
/ 16K 
\ . . . . . . . . . . . 
i f - - - - - ~ 
~l.xGex ~ y 
i m a~ Tc 
- - - ~t~ 
o lb ~ ~ c~ 
FREQUENCY"" i 
Fi9.2.10. Re f lec t iv i~ spectra 
of a Pbl_xGexTe f i lm (thickness 
5.4 nm) for various temperatures 
above ( ) and below ( - - - ) the 
c r i t i ca l temperature, after 
Jantsch et a l . (1978b) 
Fig.2.12. Results for m~O (e) 
and the damping constant r (+) 
of Pbl_xGexTe obtained from 
the osc i l la tor model f i t to 
FIR re f lec t iv i~ . Below T c, mTO 
corresponds to the ordina~ mode 
l i - 
1.0 
�9 , WTO 
[ x :0.0056 
�9 [ T= 205 K 
~./ ~.r : 3.5cm" 
, , / 
"~ c,,.= 34 
09 cs : 1950 
I i I i i 
0 1 20 cm -1 30 
FREQUENCY - 
i 
Fig.2.11. Osci l lator model f i t (dots) to an ex- 
perimental re f lec t iv i ty spectrum of Pbl_xGexTe, 
parameters given, after Jantsch et al . (1978b) 
>.. 
p... 
p- 
0.8 
. J 
I.L 
u.I 
n~ 
300 
2OO 
'E , j 
11}( 
\ 
t 
j.+ 
o 
o 
/ 
/ n-Pb1_xG % Te 
/ x :2 .7% 
/ 
....-,,/* 
/ 
/ 
I / 
J 
/ +--.. 
/ T + + 
/+ I 
too 200 
TEMPERATURE I K) 
w 
'3 
'2 
-I 
0 
300 
The FIR method has been used for investigations of mTO in PbTe (Buss and Kinch, 
1973; Burkhard et a l . , 1976; Bauer et a l . , 1978) and Pbl_xGexTe (Jantsch et a l . , 
1978b,1979). For GeTeandSnTe this method i s less suitable, since samples of these 
compounds have carr ier concentrations of the order of 1020 cm -3, whereas PbTe and 
Pbl_xGexTe samples with carr ier concentrations of 1016- 5. 1017 cm -3 are available. 
For high carr ier concentrations the negative contribution from free carr iers, 
Re{~FC}, dominates the FIR optical properties and the d ie lect r ic anomaly at the TO 
resonance becomes too small for experimental observation. 
Experimental results for the re f lec t iv i ty of < I I i> oriented thin f i lms of 
Pbl_xGexTe are given in Fig.2.10 for various temperatures above and below the cr i - 
t ica l temperature of T c=36Kfor this part icular sample (Jantsch et a l . , 1978b). 
Above T c, the onset of the drop in re f lec t iv i ty is shifted towards lower frequencies 
with decreasing temperature. This behavior indicates softening of the TO mode. Be- 
low T c, the soft mode is stabi l ized and the structure ~n the re f lec t iv i ty extends 
17 
to higher frequencies again. Figure 2.11 examplifies the oscil lator model f i t in 
this spectral regime. The solid line is experimental, the dots resultfrom the os- 
c i l la tor model. 
Typical results for the temperature dependence of the soft-mode frequency and 
its damping constant ?, obtained from the FIR method, are given in Fig.2.12. The 
cr i t ica l temperature T c is obtained from linear extrapolation of m~O. ~ O. Above and 
below T c, the curves rise l inearly according to a Curie-Weiss law. A low-frequency 
l imit for the determination of the soft-mode frequency of 10 cm -I is imposed ex- 
perimentally, which corresponds to an inaccessible temperature interval of 
IT - Tcl ~ 20 K. In this cr i t ica l regime the mode softening cannot be studied by 
FIR- or Raman-spectroscopy. 
In the rhombohedral phase, spl i tt ing of the phonon modes is expected due to the 
lowering of crystal symmetry: the two low-frequency Raman lines have been interpre- 
ted accordingly (Fig.2.3). The structure observed in the FIR ref lect iv i ty (m < 20 
- I 
cm ) yields a TO frequency comparable to that of the lower frequency Raman l ine, 
which was attributed to the ordinary ETO modes. No evidence was found for a second 
phonon mode in the low-frequency regime (m < 60 cm-1). The rest of this section is 
devoted to a discussion of mode spl i t t ing due to the rhombohedral distortion. 
In the rhombohedral phase, the 3-fold degeneracy of the cubic Flu mode is l i f ted 
by the lowering of symmetry at the phase transition. In the rhombohedral (C3v) phase 
a doubly degenerate, "ordinary" E mode, and a single "extraordinary" A 1 mode exist. 
For the A I mode the two sublattices oscil late along the rhombohedral axis with re- 
spect to each other, the E modes correspond to a relative motion perpendicular to 
the c-axis. For the ordinary mode transverse solutions exist: the electric f ie ld E 
and the polarization P are perpendicular to both the c-axis and the wave-propagation 
direction k. The E modes are isotropic: they do not show directional dispersion. 
For the extraordinary wave, however, the polarization in general is not perpen- 
d4cular to the c-axis or k. Therefore the A 1 mode contains both transverse- and longi- 
tudinal-components. Its frequency depends on the angle e between k and the c-axis. 
The dielectric function EE(~ ) of the E modes has the same form as that of the 
Flu modes (2.5) (Loudon, 1964; Claus et a l . , 1975): 
2 2 . 
~~ ~ ~EL 0 - ~ - Im~ 
~E (~) ~ ~ (~) = ~ 2 ~ (2.12) 
- m - imF 
VETo 
For the extraordinary mode the following expression is obtained: 
" (m,e) = ~=(m)';"(m) (2.13) 
~A 1 E~(~)sin2e + E"(m)cos2e ' 
where 
18 
1.0 
>- 
> 0.5 
W 
_J 
Ii 
W 
.~-.~.. Ooo oo . , ^ _ Pb+_ x Ge x Te . . . . . . . . . "~Z ~-~. -~- .~ 
- " ,~'~..~"~.~o-- 13 K ...... X :0 .027/ , 
q, "~ /23K-- 
c' l 83 . - 
 ,oE A I "','.\/sT 
' I %* 7 7 
o" gr ~o" 
0 
0 50 100 150 200 250 
FREQUENCY Icrn -1) 
Fig.2.13. Reflectivity of a Pbl_xGexTe film grown on a BaF2-(lll)-surface for tem- 
peratures above and below the crit ical temperature of T c=73K.The drop in ref lect i - 
vity above 100 cm -1 is due to the combined plasmon-phonon edge. Experimental data 
(lines) are compared to results from a two-phonon oscillator model f i t (circles 
and triangles). The inset shows the directional dispersion of the A 1- and E-phonons, 
corresponding to the best-f it parameters obtained (triangles), from Jantsch et al. 
(1979) 
2 _ 2 i~?" 
,, ~AILo 
~"(~) = E 2 2 
- m - imr" 
mAlT 0 
(2.14) 
In (2.12-14) quantities related to a polarization parallel and perpendicular to the 
c-axis are designated by the superscripts " and , respectively. 
The total phonon dielectric function including the high-frequency dielectric 
constants is given by the sum of (2.12,13). For kllc (0 = 0), (2.13) reduces to 
(2.12). In this case, the low-frequency dielectric anomaly is caused only by the 
ETO mode. The ref lect iv i ty data from [111] oriented samples (Fig.2.10) have been 
interpreted accordingly. However, in the neighborhood of the plasma edge, the spec- 
tra given in Fig.2.13 show an anomaly (Jantsch et a l . , 1979): below T c, the com- 
bined plasmon-phonon edge shows additional structure. I f the rhombohedral distor- 
tion occurs only in that [111] direction, which is perpendicular to the sample sur- 
face, only the E modes are observable. The additional structure can be explained 
i f we assume that ferroelectric domains with their c-axis along the other, equiva- 
lent cubic <111> directions are also present in the rhombohedral phase. 
Ferroelectric domains in Group IV tellurides have been observed by electron 
microscopy (Snykers et a l . , 1972; Jantsch et a l . , 1981b). The interpretation of 
cyclotron resonance spectra (Lewis et a l . , 1980) gives further evidence for the 
existence of ferroelectric domains in Pbl_xGexTe. For [111] oriented samples, two 
types of domains exist: a-domains with their c-axes perpendicular to the surface, 
and b-domains, whose c-axes include an angle of about 70 ~ with the c-axis of the 
a-domains. According to Jantsch et al. (1981b), the domain size in Pb1_xGexTe is 
19 
in the micron range, wnich is small compared to the wavelength of FIR radiat ion. 
Therefore the optical properties can be described by an average d ie lec t r i c function 
containing weighted contributions from both ~E(~) and ~Al(~,e = 70~ Assuming 
equal populations of the four possible domain or ientat ions, the spectra given in 
Fig.2.13 have also been f i t ted for T < T c. Results from the osc i l l a to r model f i t 
are given in Fig.2.13 together with the corresponding direct ional dispersion. The 
level ordering obtained, mAT 0 > mELo, does not agree with that from an interpre- 
tat ion of Raman data (Sect.2.3). At low temperatures, samples of comparable compo- 
s i t ion and c r i t i ca l temperature show Raman l ines at 30-50 cm - I , which have been at- 
tr ibuted to the A I mode due to the i r d i rect ional dispersion. An extensive search in 
this spectral regime using FIR spectroscopy fa i led to show any evidence for an extra- 
ordinary mode at a frequency d i f ferent from the ETO mode. 
The level ordering depends on the re lat ive magnitude of the anisotropy of the 
force constants due to the rhombohedral d is tor t ion and, on the other hand, the 
e lectrostat ic ones, which are responsible for the difference between longitudinal 
and transverse modes (Loudon, 1964). In the present case, the rhombohedral d is tor t ion 
is rather small. Therefore, we expect the e lectrostat ic forces to be larger than 
the i r anisotropy and hence a level ordering according to: [mAITO - mETOI 
<< (mAILO - mAITO ) and (mEL 0 - mETO), in agreement with the Raman data. 
Al ternat ive interpretat ions of the addit ional structure in the neighborhood of 
the plasma edge below T c have been discussed recently (Jantsch et a l . , 1981b). In 
the rhombohedral phase the plasma ef fect ive mass depends on the angle e as wel l . 
Thus, the f ree-carr ier contr ibut ion to a- and b-domains in the d ie lec t r i c function 
w i l l be d i f ferent . Using the ef fect ive mass tensor obtained from cyclotron resonance 
invest igat ions, the re f lec t iv i ty spectra can be explained quant i tat ive ly with a 
corresponding model by assuming a negl ig ib le E-A I phonon mode sp l i t t ing (Jantsch 
et a l . , 1981b). The E-A I sp l i t t ing is not resolvable in the FIR spectra, although 
the damping parameter F (Fig.2.12) is much smaller than the l ine width observed in 
Raman experiments, since the la t te r is strongly enhanced due to the extremely small 
penetration depth (Sect.3.2). 
3. Experimental Determination of the Static Dielectric Constant 
3. 1 Differential Capacitonce Measurements 
The bestand purest samples of IV-VI compounds exhib i t concentrations of 1016-1020 
-3 cm of highly mobile carriers due to in t r ins ic defects. The standard methods to 
determine the stat ic d ie lec t r i c constant, which are based on measurements of the 
complex low-frequency impedance of capacitors f i l l ed with the sample d ie lec t r i c , 
are not applicable because of the high conductivity of these materials. The prob- 
lems result ing from the dominating real part of the conductivity are avoided by 
measuring the d ie lec t r i c constant in the space charge region of p-n junctions, 
Schottky barriers or MIS structures. 
20 
Investigations on junctions were f i rst reported by Kanai and Shohno (1963) for 
PbTe. The capacitance C of an abrupt junction with an applied bias V is given by 
(Sze, 1969) 
| ~ ~sNe ]1/2 
C/A (3.1) 
[8~(V + Vbi)] ' 
where A is the junction area, E s the static dielectric constant, (Ne) the space 
charge density and Vbi the built-in voltage of the diode. From (3.1) we obtain 
a (._~_~ 8~ (3.2) = .,--T~-2-. ~C ~ csNeA 
Thus, linear curves are obtained by plotting C "2 as a function of the bias voltage. 
The slope given in (3.2) depends on ~s and N. The space-charge density of homo- 
geneous samples equals the bulk carrier concentration determined from the d.c. 
Hall effect times the elementary charge. Thus, the static dielectric constant is ob- 
tained from the slope of C-2-V curves using (3.2). A detailed investigation of 
Schottky barriers on PbTe (Walpole and Nil l , 1971) and Pb1_xSnxTe (Nill et al., 
1971) has shown that (3.1) describes experimental C-V characteristics even when 
inversion layers exist at the surface. The name "differential capacitance method" 
derives from (3.2), although the junction capacitance is measured by conventional 
techniques using radio frequency bridges or lock-in amplifiers without differen- 
tiation. 
Kanai and Shono (1963) investigated n- and p-type samples of PbTe with carrier 
concentrations between 1.5. 1017 and 8-1018 cm -3. Abrupt junctions were obtained 
by alloying In to p-type and Ag-Te to n-type PbTe. For temperatures between 4.2 K 
and 130 K a constant value of ~s = 400 was found, independent of carrier type and 
concentration. 
For paraelectric materials we expect a linear dependence of ~ I on temperature, 
according to the Curie-Weiss law. This behavior was f i rst observed by Bate et al. 
(1970) on diffused p-n junctions of PbTe. For linearly graded junctions (Sze, 1969) 
reZ(aN/ax)e li/3 
C/A = L3~(v +Vbi)] , (3.3) 
is obtained. Here aN/@x denotes the constant doping gradient across the junction. A 
linear dependence of C -3 on bias voltage was observed on PbTe and Pbo.83Sno.17Te 
diodes (Bate et al., 1970). Only at low temperatures deviations from linearity were 
found on PbTe and Pb1_xSnxTe diodes of different doping gradients. The deviations 
became apparent essentially at the same threshold polarization of about 10 -6 C/cm 2. 
The maximum polarization Pm~x within the linear graded junction was estimated from 
21 
Equation (3.4) is obtained from the f ie ld-st rength maximum in the space charge 
region of a graded junct ion (Sze, 1969): 
E(x) = -e ~-~N1w/2)2 - x2 (3.5) 
~x 2~ s 
Here w stands for the depletion layer width: 
A~s (3.6) 
w = C 
Combining (3.5) for x = 0 with (3.6,3) and P ~ ~s E for ~s ~ i , then (3.4) is ob- 
tained. The nonl inear i t ies of C -3 versus V at low temperatures and high reverse 
bias V were attr ibuted to a nonlinear dependence of the polar izat ion P on the elec- 
t r i c f ie ld (Cochran, 1960): 
E = Cl(T - Tc)P + c3 P3 + c5P5 + . . . (3.7) 
Bate's et a l . (1970) results for ~s are qua l i ta t ive ly d i f ferent from those of Kanai 
and Shono (1963), who did not observe the temperature dependence of ~s" This discre- 
pancy was attr ibuted to a maximum polar izat ion of 40. 10 -6 C/cm 2, estimated from 
the data of Kanai and Shono. This value seems to indicate that a polar izat ion well 
in excess of the quoted threshold value of i �9 10 -6 C/cm 2 may occur in C-V experi- 
ments on abrupt junctions. In th is s i tuat ion , nonlinear terms in (3.7) would domi- 
nate and ~s and hence ~C-2/~V in (3.1) would depend on the applied bias voltage, 
in contrast to the experimental results of Kanai and Shono. 
Unfortunately Bate et al . did not explain how the maximum polar izat ion was est i - 
mated for the abrupt junction. From a treatment analogous to that described above 
for graded junctions, we obtain for abrupt junctions a maximum polar izat ion of 
pa ~ 2(V + ) C (3.8) 
max Vbi 
From this expression and the experimental data of Kanai and Shono a maximum polar iz - 
ation of 1.5 �9 10 -6 C/cm 2 is estimated, comparable to the value obtained for graded 
junctions. I t should be pointed out that the deviations observed by Bate et a l . 
(1970) on l inear graded junctions at low temperatures can be also explained in 
terms of doping pro f i le nonl inear i t ies : the width of the space-charge region w of 
l inear graded junction is obtained from (Sze, 1969): 
w 3 12~s(V + Vbi) (3.9) 
= - e(~Ni~x) " 
For a speci f ic bias voltage V, the depletion width increases with increasing d i - 
e lect r ic constant. Since the d ie lec t r i c constant increases with decreasing tempera- 
ture in paraelectr ic PbTe, w increases with decreasing temperature. An evaluation 
of Bate's et al . (1970) experimental data shows that the threshold voltage observed 
at 4.2 K corresponds to a depletion width unobtainable at the next higher tempera- 
22 
ture of 77 K within the investigated voltage range. The positive deviations of the 
C-3-V curves observed at 4.2 K indicate a flattening of the doping gradient, which 
is expected for large distances away from the junction. 
Antcl i f fe et al. (1973) reported capacitance measurements on Pbl_xGexTe junction 
diodes at various fixed dc biases as a function of temperature. At the phase transi- 
tion temperature, a maximum of the diode capacitance was observed, indicating the 
highest latt ice polarizabi l i ty at this point. The transition temperature derived 
from this method is in good agreement with values obtained by Hohnke et al. (1972) 
from X-ray analysis. Antcl i f fe et al. also observed additional structure in the tem- 
perature dependence of the diode capacitance and a tentative interpretation was given 
in terms of an electric-field-induced phase transition at temperatures well above the 
phase transition temperature. However, there is no other indication in the l iterature 
of the appearance of this effect, and other interpretations of the reported anoma- 
lies are possible in the absence of a detailed investigation of the diode character- 
ist ics. 
An influence of latt ice defects on the dielectric behavior of PbTe (Jantsch and 
Lopez-Otero, 1976) and Pbl_xGexTe (Jantsch et a l . , 1981a,1982) was postulated from 
investigations of the capacitance of Schottky diodes. Differential capacitance 
measurements on Schottky diodes allowa determination of absolute values of E s in 
contrast to measurements on graded junctions, since in the latter case the doping 
gradient is experimentally inaccessible. 
-1 
In Fig.3.1 results for the inverse quasi-static dielectric constant E s are 
given as a function of temperature for two p-type samples of PbTe with different 
carrier concentrations. Here ~s is designated as quasi-static, since measurements 
are performed in the MHz regime. True static measurements are not possible because 
of leakage currents in Schottky diodes on IV-VI compounds. The maximum polarization 
involved here was estimated from (3.8) to be less than 10 -6 C/cm 2. At high tempera-tures, ~s varies l inearly with temperature according to the Curie-Weiss law, whereas 
at low temperatures the inverse dielectric constant saturates due to zero-point 
fluctuations (Barrett, 1952; MUller and Burkard, 1979). The inverse dielectric 
constant depends strongly on the carrier concentration of the sample. In the C-V 
method, however, the dielectric constant is determined in a space-charge region in 
the absence of free carriers. In undoped PbTe, free carriers originate from doubly 
ionized Pb and Te vacancies (Pratt, 1973,1974; Heinrich et a l . , 1975,1980). There- 
fore the apparent influence of the carrier concentration on ~s is attributed to vacan- 
cies rather than to the free carriers. Additional evidence for this conclusion is 
derived from a variation of the crystal growth conditions (Jantsch and Lopez-Otero, 
1976) and recently also from investigations of electron-irradiated Pb1_xSnxTe 
samples (Suski et a l . , 1982). 
Samples of equal carrier concentration grown at different temperatures also ex- 
h ib i t different dielectric behavior. The carrier concentration of compensated 
samples is given by 
23 
1000 9 T 
-1oo o 
PbTe / / 
p= 7,.d~3 
p= 2x1~'r ~ 
I I �9 
lOO 200 
TEMPERAIURE [KI 
Fi9.3.1. ' Temperature dependence of the 
inverse quasi-static dielectric constant 
of two p-PbTe samples of different hole 
concentration, after Jantsch and Lopez- 
Otero (1976) 
N = 2[VTe] - 2[Vpb] + [Ipb] , (3.10) 
where [Vpb] and [VTe] stand for the concentration of Pb and T~ vacancies, respec- 
tively, and [Ipb], for the concentration of interstit ial Pb azoms (Heinrich, 1980). 
Obviously, the total number of point defects and hence the dielectric properties 
depend on the growth conditions, whereas because of compensation the carrier concen- 
tration need not change. Unfortunately, the usual methods for determining the compen- 
sation or the total number of defects are not applicable for IV-Vl compounds because 
of the special nature of the vacancy states. Ionized impurity scattering is very 
weak because of the high dielectric constant (Allgaier and Scanlon, 1958) and/or 
the short-range nature of the vacancy potential (Pratt, 1974). Defect spectroscopy 
methods do not work either, since the native donors and acceptors in PbTe form re- 
sonant states (Pratt, 1974; Gresslehner et al. , 1978). Therefore, at present a 
quantitative evaluation of the influence of defects is not possible. 
Differential capacitance measurements have also been performed on Pb1_xGexTe 
(Grishechkina et al. , 1978; Jantsch et al. , 1981a,1982). The alloy systemPbl_xGexTe 
undergoes a phase transition for x > 0.005 (Jantsch et al. , 1979). Results for 
the temperature dependence of ~;I obtained from differential capacitance measure- 
ments on Schottky barriers are given in Fig.3.2 (Jantsch et al., 1981a). For com- 
parison, data for the static dielectric constant, ~0' derived by the FIR method 
(Sect.2.4) via the Lyddane-Sachs-Teller relation (1.1) are also given. The results 
obtained with the quasi-static method show substantial deviations from those ob- 
tained from the phonon data. The inverse quasi-static dielectric constant ~;1 ex- 
hibits a minimum at a temperature, which coincides with the critical temperature ob- 
tained with the FIR methods. However, extrapolations of the more or less linear parts 
I - i above and below T c intersect at ~s �9 O. The ~s curves are shifted towards higher 
24 
";•' 
2x10-~ 
7 
ffl 
z 
o 
mr 
,,el, lxm-3 
UJ 
if) 
W 
0 
I I | 
': Pbl x Ge Te 
i 
~, ~s (c-v) ,., j .: / 
�9 ~ a : 0,0035 .~' ,o 
/~X : v: 0.0022 ~ q / 
~'~o~ 7" / o: 0 .0032 / o 
P ", f q? 
'" i i 9" (FIR) 
- , . , ! ! z 
-,--., \ / , . . .zS /,/I" 
| , , 
0 100 20O 300 
TEMPERATURE I K ! 
Fi9.3.2. Comparison of results for the 
inverse guasi-static d ie lectr ic con- 
stant E~ I obtained from C-V measure- 
ments, ~nd the stat ic d ie lectr ic con- 
stant cO 1 calculated from FIR phonon 
data using the LST relat ion, after 
Jantsch et al . (1981a) 
with respect to the ~01- data, which were obtained from the phonon frequen- values 
cies. This discrepancy indicates that additional dispersion occurs in the frequency 
range between the Reststrahlenband in the THz range and the low-frequency MHz 
regime, where ~s is determined. The same effect obviously causes the variation of E; I 
of PbTe with defect concentration. Since PbTe is paraelectric, no minimum of 
- i ~s occurs at f in i te temperatures and a negative Curie temperature is obtained by 
extrapolating ~;I ~ O. This Curie temperature is just a parameter which describes 
the d ie lectr ic properties rather than the crystal s tab i l i ty . The la t ter is charac- 
terized by a c r i t i ca l temperature derived from phonon data (Fig.5.3). In Fig.3.1 
the apparent variat ion of T c with defect concentration is mainly caused by the ad- 
dit ional po lar i zab i l i ty in the low-frequency range below the Reststrahlen regime 
(Sect.5.3). 
2 Microwave Techniques 
A different approach to the experimental determination of the static dielectric 
constant of IV-VI compounds is based on investigations of the optical properties in 
the microwave regime in the presence of high magnetic fields. Without magnetic field, 
electromagnetic waves with frequencies below the Reststrahlen-plasma regime cannot 
propagate into the material, since the real part of the dielectric function (2.2) 
is negative. In the presence of a magnetic field B the free-carrier part of the 
dielectric function, xFC' becomesasecond-rank tensor even for materials with iso- 
tropic band structure. A general introduction to the field of magnetoplasma effects 
in semiconductors was given by Palik and Furdyna (1970). 
25 
In the cubic phase of PbTe, SnTe and GeTe, the constant energy surfaces of the 
valence- and conduction band are described by e l l ipso ids oriented along the <111> 
direct ions. A discussion including a l l possible configurations of the magnetic 
f ie ld B and the direct ion of l ight propagation re lat ive to the crystal axes would 
be tedious. A number of configurations has been investigated by St i les et a l . (1962) 
Nii (1964), Perkowitz (1969),Takanoet al . (1974) and Foley and Langenberg (1977). 
In the case of highest symmetry, B11<lOO>,and Faraday configuration klIB, the two 
normal magnetoplasma modes are c i rcu la r ly polarized waves. The dispersion of the 
magnetoplasma waves is described by the d ie lec t r i c function 
k 2• : (#)2 (Exx • iLxy ) ~ (#)2 ~,(m,B) (3.11) 
wi th 
~• 
~• = ~ + Xph(~) + XFC(~,B) , (3.12) 
where the plus and minus signs correspond to r ight- and le f t -c i rcu lar polarized 
waves. In the local approximation, X~C is given by (Wallace, 1965) 
2 
= Up ~3 • (~ + i~) 
, (3 .13) 
(~ + i~) 2 - ~ 
~+ 
xFC 
where 
2 4~Ne 2 2K + 1 m 
* T ' K=-& 
m t m t 
2 eB 2 K + 2 eB K + 2 
~c =~- -T~- - ' ~3 =--~-K-+--r 
m C m t 
* and * stand for the longi tudina l - and transverse-effective mass, respec- Here m s m t 
t i ve ly , and N is the f ree-carr ier concentration. 
The main structures in the optical properties resul t from two types of s ingu lar i - 
t ies in the d ie lec t r i c function: poles, which are caused by cyclotron- or TO-reso- 
nances, and zeros associated with longitudinal excitat ions. Points where Re{e} = 0 
are referred to as "d ie lec t r i c anomalies." 
A discussion of optical properties is s impl i f ied by considering contour maps of 
the zeros and poles of the d ie lec t r i c function in the ~ versus ~c plane in the l im i t 
of negl ig ib le damping r and ~ (Palik and Furdyna, 1970). A contour