conduction electrons that a r e compressed. At the same
time, in metallic antiferromagnets the average polarization of the conduction electrons is zero, and the r e sults of ['', where the field a t the "Fe nuclei in the
antiferromagnetic alloy Pt,Fe was practically independent of the pressure, were explained naturally from
this point of view. It appears that FeSn, is not a "good"
metal and the strong dependence of the hyperfine field
at the "Fe nuclei is connected with partial localization
of the external magnetic electrons of the F e atoms in
directed chemical bonds.
In conclusion, the authors thank V. P. Mar'in for help
with a number of experiments.
'N. N. Delyagin and k. N. Kornienko, Zh. Eksp. Teor. Fiz. 61,
1946 (1971)[Sov. Phys. J E T P 34, 1036 0972)).
21.N. Nikolaev, V. P. Potapov, and V. P. Marsin, Zh. Eksp.
Teor. Fiz. 67, 1190 (1974)[Sov. Phys. JETP 40, 591 (197511
3.N. Nikolaev, L. S. Pavlyukov, and V. P. Mar'in, Zh. Eksp.
Teor. Fiz. 69, 1844 (1975)ISov. Phys. J E T P 42, 936 (1975)l.
N. Nikolaev, V. P. Potapov, and N. N. Delyagin, Zh.
Eksp. Teor. Fiz. 70, 241 (1976) [Sov. Phys. JETP 43, 125
.
".
(1976)f.
Nikolaev, L. S. Pavlyukov, V. P. Potapov, and V. P.
Marsin, Fiz. Tverd. Tela (Leningrad) 18, 1198 (1976)[Sov.
Phys. Solid State 18, 692 (19761.
61. N. Nikolaev and V. P. Potapov, Zh. Eksp. Teor. Fiz. 72,
1600 (1977)[Sov. Phys. J E T P 45, 840 (197711.
'v. I. Nikolaev, Yu. I. Shcherbina, and S. S. Yakimov, Zh.
Eksp. Teor. Fiz. 45, 1277 (1963) Eov. Phys. JETP 18, 878
(19~411
P. I. Nikolaev, Yu. I. Scherbina, and A. I. Karchevskii, Zh. Eksp. Teor. Fiz. 44, 775 (1963)[Sov. Phys. J E T P
17, 524 (196311.
8 ~ Fabri,
.
E. Germagnoli, M. Musei, and G. C. Locati, Nuovo
Cimento B 40, 178 (1965).
9 ~ Trumpy,
.
E. Both, C. Djega-Mariadassou, and P. Lecocq,
Phys. Rev. B 2, 3477 (1970).
'OW. Nicholson and E. Friedman, Bull. Am. Phys. Soc. 8, 43
(1963).
"P. K. Iyengar, B. A. Dasannacharya, P. R. Vijayaraghavan,
and A. P. Roy, J. Phys. Soc. Jpn. 17, 247 (1962).
1 2 ~B.. Steams, Phys. Rev. B 4, 4081 (1971);V. Niculescu,
K. Raj, J. I. Budnick, T. J. Burch, W. A. Hines, and A. H.
Menotti, Phys. Rev. B 14, 4160 (1976).
13v. N. Panyushkin, Fiz. Tverd. Tela (Leningrad) 10, 1915
(1968) [Sov. Phys. Solid State 10, 1515 (196811.
5 ~ N.
.
Translated by J. G. Adashko
Rearrangement of the energy spectrum of Pb,-,Sn,Se
result of band inversion under pressure
as a
N. B. Brandt, 0.N. Belousova, V. P. Zlomanov, Ya. G. Ponomarev, E. P. Skipetrov, and
V. I. Shtanov
Moscow State University
(Submitted 16 August 1977)
Zh. Eksp. Teor. Fiz. 74, 646-657 (February 1978)
An investigation was made of the oscillatory and galvanomagnetic effects in single-crystal samples of n type Pb,-,Sn,Se (x = 0.125) at pressures up to 16 kbar. The dependences of the cyclotron masses at the
Fermi level on the band gap E, were asymmetric relative to E, = 0, as predicted in the theories of
Dimmock and Martinez. The electron Fermi surface became spherical for negative values of E,. The
experimental results obtained were combined with the data of other authors in a calculation of the
parameters of the Dimmock dispersion law of carriers at the point L in the Brillouin zone.
PACS numbers: 71.25.Pi, 72.15.Gd, 62.50.+p
INTRODUCTION
Lead selenide (PbSe) and its alloys Pbl-,SqSe (x
< 0.43) crystallize in the NaCl lattice and a r e semicon-
ductors with a narrow direct gap E, a t the point L in the
Brillouin zone. In the case of PbSe and Pbl-,SqSe with
x < x , (x, = 0.15 a t T = 4.2 OK) the terms L,' and L l , which
correspond to the bottom of the conduction band and the
top of the valence band a t atmospheric pressure>1 become inverted a t higher pressures.c2-51 The transition
to the zero-gap state [E,= E(L;) - E ( L ~=) 0] at a pressure p =Po, which depends on the composition x and
temperature T, is accompanied by vanishing of the effective c a r r i e r masses m*(O) a t the bottom of the conduction band and a t the top of the valence band. The
340
Sov. Phys. JETP 47(2), Feb. 1978
effective masses a t the Fermi level m *(E,) a r e affected
much less by transition to the zero-gap state and the
relative reduction in these masses decreases with increasing c a r r i e r density in the bands.[']
The nature of the rearrangement of the c a r r i e r spectrum a s a result of inversion of the terms a t L depends
on the strength of the interaction of the inverted terms
with the more distant bands at the point L.['-~*~]If this
interaction can be ignored, the c a r r i e r dispersion law
is of the Kane typeC71and the sign of the band gap E, i s
unimportant. Then, all the €,-dependent quantities vary
symmetrically relative to the value E, = 0 [the effective
masses m *(E,) pass through a minimum and the Fermi
energy E, passes through a maximum if the c a r r i e r
0038-564617814702-0340$02.40
01978 American Institute of Physics
density remains constant]. It should be added that in
the case of the two-band Kane model the electron and
hole dispersion laws a r e m i r r o r images of one another,
the constant-energy surfaces a r e strictly ellipsoidal,
and the anisotropy of these surfaces is independent of
the Fermi energy E , and of the magnitude and sign of
4.
'
The influence exerted on the electron and hole dispersion laws of the four more distant bands a t the point L,
. arranged in pairs above and below the Fermi
complicates greatly the nature of the changes in the
spectrum a s a result of band inversion a t ~ . 1 2 - ~ " * ~ * ~ ]
Theoretical calculations carried out in the six-band
predict a reduction in the anisotropy
of the constant-energy surfaces at L on transition to
negative values of E, and transformation of the Fermi
surface to the perfectly spherical shape a t some value
of E:< 0. This effect has been observed in studies of
the anisotropy of the Fermi surface of Pb,-,SqSe alloys of variable composition (0 s x G 0.2) a t atmospheric
pressure[8*9]and is clearly the most convincing proof
of the validity of the models of the energy spectrum
based on the six-band approximation.C4s61A further increase in the absolute value of e, should split the extremum a t L into three equivalent extrema located in
the hexagonal plane of the Brillouin zoneL4](this r e presents a transition to a spectrum of the SnTe type).
It follows from the
that the interaction with
the more distant bands results in an asymmetric, r e lative to E, = 0 , dependence of the effective masses a t
the Fermi level m *(E,) (for N = const) on the energy
gap: the average r a t e of r i s e of m*(~,) with increasing
( ~ beyond
~ 1 the point of inversion should be considerably
greater than the average rate of change of m*(e,) before inversion. Moreover, the minima in the band-gap
dependences of m * ( E , ) should shift toward the positive
values of eg on increase in the c a r r i e r density N . [ ~ * ~ ]
To detect experimentally the theoretically predictedCzs*'l asymmetry of the band-gap dependences of the
effective masses and thus confirm anew the validity of
the six-band DimmockCG1
and ~ a r t i n e z models
~ ~ ] of the
carrier spectrum, it is necessary to c a r r y out detailed investigations of the band-gap dependences of
the cyclotron masses in Pb,-,SqSe in a wide range of
values of leg[ before and after passing through the zerogap state. Such investigations have not yet been carried
out. In an earlier paperL5]we reported the results of a
study of the Shubnikov-de Haas effect in n-type PbSe
on transition to the zero-gap state under pressure. The
dependences of the cyclotron masses mclo0on eg obtainedC5]in the range 145 meV2 e, 3 0 were insufficient
to make an assured selection between the Kane model
and the six-band Dimmock model.
It should be pointed out that a t present there a r e f a r
too few experimental data to determine reliably the
validity of the six-band models.c4161Some of these data
a r e only in qualitative agreement with the models o r
a r e even in conflict with them.C91101For example, the
theory predicts an increase in the anisotropy of the
Fermi surface of Pb,-,SqSe on increase in the c a r r i e r
density,L6] whereas it is reported[g] that the anisotropy
34 1
Sov. Phys. JETP 47(2), Feb. 1978
of the hole Fermi surface decreases on increase in the
hole density P in the alloy with x = 0.06. Deviations of
the shape of the Fermi surface from an ellipsoid, due
to the presence of the terms proportional to p4 in the
dispersion law, a r e observed only a t c a r r i e r densities
] six-band models of the
exceeding 1019~ m - ~ . [ ' ~The
c a r r i e r spectrum predict saturation of the c a r r i e r density dependences of the effective masses when the
density reaches -loz0 cm-'. Kucherenko et a ~ . [ ~de]
tected this saturation experimentally but found only a
qualitative agreement between the experimental and
theoretical dependences. The amount of experimental
data on the degree of departure of the electron and hole
dispersion laws at L from mirror-image symmetry is
also insuff i ~ i e n t . [ ' - ~ * ~ - ~ ~ 1
We investigated the galvanomagnetic effects using
weak magnetic fields in the temperature range 4.2 OK
s T s 300°K, and also the Shubnikov-de Haas effect a t
helium temperatures in n-type Pb,-,SqSe (x = 0.125
0.01) a t pressures of 1 bar < p s 16 kbar. A comparison of the results obtained in the present study with
those deduced by u s earlier from a study of the Shubnikov-de Haas effect in n-type pbseC5] made it possible
to deduce the band-gap dependences of the cyclotron
mass mcl0, in a wide range of eg: 145 meVz E, 2 - 110
meV. These dependences were strongly asymmetric
relative to eg = 0, as predicted by DimmockCS1and Martine^.[^] Moreover, the electron energy surfaces became spherical a t a pressure corresponding to e f
= 100 meV. The results obtained were used to determine the parameters of the Dimmock dispersion law.
*
-
1. MEASUREMENT METHOD. SAMPLES
The electrical resistivity p and the Hall coefficient
R, were determined in weak magnetic fields ( p H << 1)
f o r single crystals of n-type Pb,_,SqSe (x=0.125) in the
temperature range 4.2 OK c T c 300°K. The temperature was controlled by a thermal regulation unit.["]
The galvanomagnetic effects and temperature were recorded with a two-channel digital measuring system.D41
Quantum oscillations of the magnetoresistance (Shubnikov-de Haas effect) were investigated a t temperatures
2.1 OKs T c 4.2 OK in the field of a superconducting solenoid H < 55 kOe. In investigations of the angular dependences of the period of the Shubnikov-de Haas oscillations A ( ~ / H the
) measurements were carried out
in the field of a superconducting Helmholtz coil system
(H s 20 kOe).
The precision of the measurements of the cyclotron
masses and oscillation period was improved by suppressing the monotonic field dependence of the resistivity p ( ~ with
)
the aid of a computing device producing
a signal proportional to CY H * pH2.
P r e s s u r e s p c 16 kbar were produced in a chamber
made of heat-treated beryllium bronze.C151 The pressure-transmitting medium was a mixture of 50% pentane, 25% kerosene, and 25% transformer oil. At
helium temperatures the pressure in the working channel was deduced from the shift of the superconducting
transition temperature Tc of a tin sensor.D61 When the
Brandt et a/.
34 1
w ,cm'
. -
V1 sec-'
4
7
FIG. 1. Pressure dependences of the Hall mobility
in samples 4-1 (curve 1)
and 4-2 (curve 2) at T
=4.2 OK.
p, kbar
temperature was varied in the range 4.2"Kc T a 100°K
the pressure in the chamber remained practically constant and for T- 300°K, i t increased by 2-3 kbar.[17]
Rectangular single-crystal Pb,-,SqSe blanks were
split from an ingot a t liquid nitrogen temperature.
Samples of 0.5x 0.7% 3.0 mm dimensions were then cut
by spark machining. Current leads made of tinned copper wire were soldered with indium to the ends of the
samples. Potential and Hall contacts were sparkwelded to the samples. The dimensions of the samples
and the distances between the potential contacts were
determined with an MBS-1 microscope.
The concentration and homogeneity of the distribution
of Sn in the alloy was determined with a JXA-50A
x-ray microprobe analyzer. The Sn content in the
alloy was determined to within 1 at. %,
-
2. RESULTS OF MEASUREMENTS
At helium temperatures the Hall coefficient R, and
the Shubnikov-de Haas oscillation period A,,(~/H) of
the investigated Pb,,SqSe ( x =0.125) n-type samples
remained independent of pressure (within the limits of
the experimental error). The 4.2 OK electrical resistivity p passed through a minimum and the Hall mobility pH=R,/p passed through a maximum a t the press u r e p * = 1.5* 0.5 kbar (Fig. 1). The maximum of the
pressure dependence of the c a r r i e r mobility pH was due
to the passage of the cyclotron mass m,(~,) through a
FIG. 3. Temperature dependences of the Hall coefficient:
a) sample 4-1; b) sample 4-2. hessure p (kbar): 0) 2.5;
X) 12.5.
minimum a s a result of band inversion under pressure.
The temperature dependences of the electrical r e s i s tivity p(T) were "metallic" throughout the investigated
range of pressures because of the high c a r r i e r density
in our samples (Fig. 2). After band inversion under
pressure a t T = 4.2 OK, i.e., a t pressures p >Po, there
was a return transition to the zero-gap state when the
temperature was increased. Therefore, a t helium
temperatures the electrical resistivity rose with increasing pressure (eg< 0) but a t room temperature it
fell with rising pressure (6, > 0) (Fig. 2).
The temperature dependences of the Hall coefficient
R,(T) exhibited a small maximum at about 200°K (Fig.
3). An increase in the pressure p made this maximum
more pronounced and shifted it toward higher temperatures.
Oscillations of the transverse magnetoresistance p ( ~ )
were usually investigated in a magnetic field H 11 (100).
The extremal sections of the four electron Fermi s u r faces were identical and the oscillations were monochromatic. The cyclotron masses a t the Fermi level
m,,o(c,) were deduced from the temperature dependences of the amplitudes of the Shubnikov-de Haas
I
0
5
I
I
10
15
1
20
p, kbar
FIG. 2. Temperature dependences of the electrical resistivity
of sample 4-2 at pressures p (kbar): 1)6.0; 2) 12.5.
342
Sov. Phys. JETP 47(2), Feb. 1978
FIG. 4. Pressure dependences of the cyclotron mass at the
Fermi level in H 11 (100) for n -type Pbl,Sn,Se ( x =0.125):
0 ) sample 4-1; X ) sample 4-2. The dashed curves are plotted
in accordance with the Kane model using the parameters given
in the text.
Brandt e t a/.
342
cating constancy of the electron density in our samples. However, the anisotropy of the Fermi surface a t
L decreased monotonically with rising pressure until
the Fermi surface became spherical when the band gap
reached E;= - 100 meV (Fig. 5). The extremal section
of the F e r m i surface S, corresponding to the H(((100)
orientation did not change greatly under pressure. This
was due to theproximity of S,,, to the average section of
the Fermi surface.
The nonparabolicity of the dispersion law of c a r r i e r s
a t the point L resulted in a strong r i s e of the effective
masses on increase of the absolute value of the band gap
E, (Fig. 4), which reduced the Hall mobility p H in the
range p > p, (Fig. 1).
5
10
1s
s
I0
H, kOe
IS
20
H, kOe
FIG. 5. Oscillations of the magnetoresistance p ( H ) on rotation
of the field H in the (100) plane, obtained for sample 4-2 at
T=4.2 "K: a) at pressure of 6.0 kbar; b) at pressure of 15.5
kbar (the angle was measured from the direction of H 11 (loo)).
oscillations. At pressures p > P o the cyclotron masses
rose with rising pressure (Fig. 4), and this happened
much more rapidly than predicted by the two-band
Kane model.L5]
One of the samples (4-2) was used to investigate the
angular dependences of the Shubnikov-de Haas oscillation period A ( ~ / H when
)
the field was rotated in the
(100) plane and this was done throughout the selected
pressure range. At low pressures there were beats of
two oscillation periods from pairwise coincident extremal sections of the four electron ellipsoids and this
was observed over a wide pressure range (Fig. 5a).
An increase in the pressure altered considerably the
beat pattern and at pz 15 kbar the oscillations became
completely monochromatic (Fig. 5b). The oscillation
period was independent of the angle, which was evidence
of the spherical shape of the Fermi surface in this
case.
3. DISCUSSION OF RESULTS
The band gap E, of the investigated samples of the
Pb,-,SqSe (x = 0.125) alloy was E,= +24 meV a t atmospheric pressure and helium temperature. This gap was
deduced from['']
The pressure of the transition to the zero-gap state p,
did not correspond to the maximum in the pressure dependence of the Hall mobility (Fig.1) and could be found
by using the reliably established pressure coefficient
ar,/ap = 8.5 m e ~ / k b a r . [ ~ *In~ our
~ ] case, this pressure wasp0=2.8*0.2 kbar and in the r a n g e p > p , the
negative band gap E, increased in the absolute sense
right up to E,=
110 meV a t the maximum pressure
empfoyed.
-
-
The T = 4.2 OK value of the Hall coefficient remained
constant (within the limits of the experimental e r r o r )
throughout the investigated range of pressures, indi343
Sov. Phys JETP 47(2), Feb. 1978
A characteristic feature of the Pb,-,SqSe samples
with the inverted spectrum a t T =4.2"K was a change in
the sign of ap/ap with rising temperature (Fig. 2). This
was the qualitative difference from PbSe and from
Pb,,SqSe alloys with the normal spectrum.C51 This
effect provided an additional confirmation of the band
inversion in Pb,-,SqSe (x = 0.12 5) under pressure.
It was worth noting the nontrivial nature of the temperature dependences of the Hall coefficient R, of the
investigated samples under pressure (Fig. 3). The
maximum in the dependences R,(T) was most likely due
to indirect transitions of thermally excited electrons
from the L to other extrema located higher in the conduction band.['] However, the available experimental
data were insufficient to estimate the indirect gap.
The results obtained in the present study for the
Pb,-,S q S e (x = 0.12 5) alloy may be compared with our
earlier data on the transition to the zero-gap state in
n-type PbSe under pressure.[5] Table I lists the prope r i t e s determined a t atmospheric pressure and helium
temperature for our samples and those investigated
earlierJ5] In the case of samples 4-2 (x = 0.125) and
1-1 (x = 0) the electron densities N agreed to within a
few percent, which made it possible to plot a single
band-gap dependence of the cyclotron mass a t the Fermi
level rn,,(~,) in the band-gap range + 145 meV2 E,
> - 110 meV. These band-gap dependences of the cyclotron masses a r e calculated using Eq. (1) and assuming
that B E ~ / ~=P- 8.5 meV/kbar is independent of the press u r e p and composition x
. It ~is clear
~ from
~ Fig.
~ 6~
that the dependence of m,,(~,) on E, for the N = 5.0
x 1017~ m case
- ~(samples 4-2 and 1-1) is asymmetric
relative to E, = 0. The dashed curves in Fig. 6 a r e
plotted in accordance with the Kane model f o r the parameters E, =3.8 eV and E,, =2.9 evJ5] The Kane dependences, symmetric relative to E, =0, a r e in poor agreement with the experimental points after the inversion.
TABLE I.
3&&
n,.o//o.
Sample
x
(n-type)*
I
'.1
I
m~
Oe-'
I
.y*
(
ID..
g ; x1 . 1
cm-y
cm' .v-8
.sec-*
I
I
*The results are given for T = 4 . 2 "K and p= 1bar.
B randt eta/.
343
~
m: and rn; a r e the transverse and longitudinal correction masses representing the interaction with the more
distant bands (the plus sign corresponds to holes and
the minus to electrons). If the terms containing l/m:,
l/m f , l/m;*, and l/m :* can be neglected, the dispersion law (2) becomes of the Kane type. The terms with
l/m; and l / m f on the left-hand side of Eq. (2) govern
the degree of departure of the electron and hole spectra
a t I from the m i r r o r symmetry.
I
745
I
I
o
so
700
-
I
50
I
I
- roo
.€
,
meV
FIG. 6. Band-gap dependences of the cyclotron mass at the
Fermi level in H 11 (100) obtained at helium temperatures for
n-type samples of PbSe: 0) 1-1, 0)1-2, A) 2-2, and n-type
samples of Pb-,Sn,Se (x =0.125): 0) 4-1, X ) 4-2. The
dashed curves are plotted on the basis of the Kane model and
the continuous ones on the basis of the Dimmock model. The
parameters used in the calculations are given in the text.
At the bottom of the conduction band the ratio of the
maximal and minimal Fermi surface sections S,,,/Smin
is equal to the ratio of the maximal and minimal cyclotron masses mc,ax/mc,in (S,, = S,,,, memi,'- mcll,) and it
is given by1)
SSrnrs
-=-=
memomrn
If the electron and hole spectra do not have the mirr o r symmetry, the band-gap dependences of Smax/S,,,
and mcmax/mc,in a t the bottom of the conduction band
It should be noted that the variation of E L and El, makes
[see Eq. (311 have a kink at E, = 0 (in the zero-gap state)
it possible to obtain agreement with the experimental
when the bottom of the conduction band and the top of
data either f o r E,> 0 o r for E,< 0 but not for all the
the valence band a t L "interchange" their correction
values of the band gap. The asymmetric band-gap demasses. In the zero-gap state the anisotropy of the
pendences of the cyclotron masses and transformation
Fermi surface sections and cyclotron masses a t the
of the Fermi surface to the spherical shape under presbottom of the band is governed solely by the parameters
sure, a similar transformation of the Fermi surface in
E, and E,I. The values of the ratios S,,, /Sminand
Pb,-,&Se alloys with rising X , C ~ ' ~the
] difference bemc,a,/mcmin, determined in the Dimmock approximatween the anisotropies of the electron and hole contion, depend not only on E, but also on the c a r r i e r denstant-energy s~rfaces,[l2~'~]
and the dependence of the
sity in the bands, and away from the bottom of the band
Fermi surface anisotropy on the c a r r i e r den~ity[lL'Z~4~l] the values of S,,,,, /S,, and m,,,, /mCmincorresponding to
all indicate the need to use models more complex[4861
a given c a r r i e r density a r e , strictly speaking, unequal.
than the Kane model for the c a r r i e r spectrum at the
When the spectra do not have the m i r r o r symmetry,
point L. The large number of parameters occurring in
the carrier-density dependences of S,,,/Smin and
the six-band Dimmock model[61 and particularly in the
mcmax/mcmin
a r e different for the conduction and valence
Martinez
a r e known to cause difficulties in
bands.
comparisons of the theory and experiment. We shall
The expression for the minimal electron cyclotron
consider the Dimmock model,[61which-as indicated by
mass
m,,
(HI I(111)) in the Dimmock approximation
estimates reported by ~ a r t i n e z [ ~ ] - g i v e spractically
has
(in
coiitrast
to m,,,,) the relatively simple form:c8]
identical results with the Martinez modelCq in the
range of band gaps of interest to us.
The Dimmock dispersion model[61can be written in a
form used by ~ b r i k o s o v : @ ~ ]
where S' = Smin/2nm
here, the energy E is measured from the midpoint of
the gap, E, is the band gap after the inversion ( E , < O),
@, and PL a r e the longitudinal and transverse matrix
elements, m, is the free electron mass,
-=m,'-(),1
1
2
1
1
m,-
-=-(1
m;.
i
2
1
1
m.++,-)'
,.
The theory predicts an asymmetric dependence of the
cyclotron mass on E, in the case of band inversion and
a constant c a r r i e r density.['] Such a dependence m,,,
( E ~is
) exhibited by our samples (Fig. 6). The theory
predicts also a shift of the minimum in the energy-gap
dependences relative to E, = 0, in the direction of positive values of the gap away from the bottom of the band.
In principle, this shift may explain the discrepancy
between the values of p, and of the pressure c o r r e s ponding to the maximum of the Hall mobility exhibited
by samples 4-1 and 4-2 of the investigated alloy (Fig. 1).
The experimental results obtained in the present
study and those reported elsewhere were used by us in
344
Sov. Phys. JETP 47(2), Feb. 1978
Brandt eta/.
344
the determination of the parameters of the Dimmock
dispersion law. The calculations were carried out on a
computer. In these calculations we used: 1) the experimental dependences of the electron cyclotron mass e s m,,, on the gap E, (Fig. 6); 2) the dependence of the
electron cyclotron mass m,,,, on the c a r r i e r density in
PbSe;831 3) the value of the band gap E; a t which the
Fermi surface became spherical in the investigated
x = 0.125 alloy with N = 5X lo1' ~ m - ~4) ; the anisotropies
of the electron and hole Fermi surfaces at the bottom
(top) of the band of PbSe; c12*211 5) the dependence of the
anisotropy of the hole Fermi surface at L on the hole
density in PbSe a t atmospheric
We recall that simple analytic dependences of the
extremal sections and cyclotron masses a r e obtained
from the dispersion law (2) only in the H I I (111) case.
Therefore, in computer calculations of the Fermi surface anisotropy and c a r r i e r density a t given values of
the gap parameter E, and energy E we used the numerical integration method. The anisotropy of the cyclotron masses was determined using approximate rela~b~]
tionships obtained by Foley and ~ a n ~ e n b e rwhich
were sufficiently accurate in the range of electron densities of interest to us (N < 2 X 1018 ~ m - ~ )We
. also
assumed that in the investigated range of pressures
p and compositions x the changes in the correction
masses and in the parameters EL and El, were negligible.kl
Variation of the parameters in the dispersion law (2)
gave us the following set:
satisfying all the experimental data given above. The
degree of agreement between the experiment and theory
can be judged on the basis of Fig. 6, where the continuous curves a r e plotted in accordance with the Dimmock
model.
Figure 7 gives the theoretical dependences of the
anisotropy coefficient K = (S,,, /S,, )' of the electron
Fermi surface at L on the band E, for several electron
densities. It is worth noting the kink in the dependence
TABLE 11.
K(E,) which occurs at E, = 0 for the curve representing
the bottom of the conduction band [see also Eq. (3)].
Similar calculations were also carried out for p-type
materials. For all values of E, the c a r r i e r density dependence of K was much stronger for the valence band
than for the conduction band. Hence, we concluded that
the plotting of the dependence K(E,) on the basis of the
experimental data obtained for Pb,-,SqSe alloys with
different impurity concentration^[^] was an incorrect
procedure.
The experimental and theoretical values of the anisotropy coefficient K' of n- and p-type PbSe and of p-type
Pbo~g2Sno~o,Se
samples with different c a r r i e r densities
a t T = 4.2"K a r e compared in Table 11. The agreement
between the theory and experiment can be regarded a s
fully satisfactory.
The band gap E ;corresponding to the transformation
of the Fermi surface to the spherical shape shifts to ward higher absolute values of this gap when the electron density N is increased (Fig. 7). For N = 5X 1017
cm-3 the Fermi surface becomes spherical at ~f = - 85
meV [our experimental value i s E ; = - (loo* 20) meV].
The universality of our parameters of the Dimmock
model was checked by calculating the cyclotron masses
a t the Fermi level for different c a r r i e r densities in
n- and p-type samples of PbSe and Pbl-,SqSe alloys
(O<x < 0.2). A comparison of the calculated values of
m,,, and m,,,with those found experimentally by various
a ~ t h o r ~ [ ~
showed
* ~ ~a ~good
~ ~agreement
* ~ ~ ] between the
theory and experiment (the discrepancy between the calculated and experimental data did not, on the average,
exceed 10%).
The reported results indicate that the Dimmock model
is in satisfactory agreement with the available experimental data for PbSe and Pb,-,SqSe alloys in the band
gap range 145 meVa E, 3 - 110 meV. However, we
may expect the model to become unsatisfactory in the
case of the energy spectra of c a r r i e r s in Pb,,,SqSe
beyond the point at which the Fermi surface becomes
spherical.[41
The authors take this opportunity to thank S. D. Beneslavsk: for extremely valuable discussions.
FIG. 7 . Dependences of the anisotropy coefficient of the electron Fermi surface^=(^,,/^,^,)^ on the band gap c, at helium temperatures: l ) at the bottom of the conduction band;
2), 3) for electron densities of N = l x 10" cm-3 andN=5x id9
~ m - ~The
. calculations a r e based on the Dimmock model and
the parameters a r e given in the text.
345
Sov. Phys. JETP 47(2), Feb. 1978
' I ~ h e r ea r e e r r o r s in the expression for the anisotropy of the
Fermi surface a t L obtained in the Dimmock approximation
by Kucherenko et a1
.['
'P. J. Lin and L. Kleinman, Phys. Rev. 142, 478 (1966).
'5. Melngailis, T. C. Harman, and J. A. Kafalas, in: Physics
of IV-VI Compounds and Alloys (ed. by S. Rabii), Gordon
Brandt et a/.
345
and Breach, New York, 1974, p. 59.
in: Physics of IV-VI Compounds and Alloys (ed.
by S. Rabii), Gordon and Breach, New York, 1974. p. 215.
4 ~ Martinez,
.
Phys. Rev. B 8 , 4678 (1973).
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. Brandt, 0. N. Belousova, V. P. Zlomanov, Ya. G.
Ponomarev, and E. P. Skipetrov, Fiz. Tverd. Tela (Leningrad) 19, 437 (1977) [Sov. Phys. Solid State 19, 250 (1977)l.
9.0.Dimmock, in: Proc. Intern. Conf. on Physics of Semimetals and Narrow-Gap Semiconductors, Dallas, 1970 (ed.
by D. L. Carter and R. T. Bate), in: J. Phys. Chem. Solids
32, Suppl. 1 , 319 (1971).
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T. C. Harman, and W. C. Kernan, Phys. Rev.
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'OV.
N. Ivanov, I. V. Kucherenko, V. N. Moiseenko, M. S.
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690 (1975) [Sov. Phys. Semicond. 9, 453 (1975)).
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55, 1215 (1968) [Sov. Phys. JETP 28, 635 (1969)l.
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.
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Skipetrov, Prib. Tekh. Eksp. No. 6, 160 (1973).
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. Harman, in: Proc. Intern. Conf. on Physics of Semimetals and Narrow-Gap Semiconductors, Dallas, 1970 (ed.
by D. L. Carter and R. T. Bate), in: J. Phys. Chem. Solids
32, Suppl. 1, 363 (1971).
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i tverdykh rastvorov na ikh
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Translated by A. Tybulewicz
Oscillations of the impedance of thin tungsten plates in a
magnetic field
0.A. Panchenko, V. V. Vladimirov, P. P. Lutsishin, and M. A. Mukhtarov
Institute of Physics, Ukrainian Academy of Sciences
(Submitted 16 July 1977)
Zh. Eksp. Teor. Fi. 74, 658-664 (February 1978)
The derivative with respect to the magnetic field of the surface impedance of thin tungsten plates (the skin
layer depth being comparable with the thickness) in circularly polarized radio-frequency fields Cf = 5
MHz) is investigated experimentally and theoretically. The magnetic field is perpendicular to the (001)
face of the sample. It is shown that in both polarizations the impedance oscillations are due to a group of
holes lying at the inflection of the Fermi surface octahedron. The "-" polarization oscillation series can
be attributed entirely to the Gantmakher-Kaner radio-frequency effect [Sov. Phys. JETP 21, 1053
(1965)l. The impedance oscillations in the "+" polarization are due to both the Gantmakher-Kaner effect
and to doppleron excitation [McGraddy et a l . , Phys. Rev. 141, 437(1966); Fisher et a l . , Sov. Phys.
JETP 33, 410 (1971)l.
PACS numbers: 72.15.Gd
1. It is known that weakly attenuating e l e c t r o m a g netic waves, calleddopplerons, can e x i s t i n m e t a l s
n e a r the threshold of Doppler-shifted cyclotron resona n ~ e . [ ' ~ T~ h' e dopplerons are m o s t c l e a r l y observed in
compensated m e t a l s , w h e r e the helicons cannot propagate. Tungsten is one s u c h metal. T h e excitation of
dopplerons c a n lead to a n o s c i l l a t o r y dependence of the
impedance of the p l a t e o n the magnetic field.
In the p r e s e n t work, we have investigated the s u r f a c e
r e s i s t a n c e of thin s i n g l e - c r y s t a l s a m p l e s of tungsten
(D 6, D is the thickness of the plate, 6 is the skin depth)
i n the radiofrequency region of the s p e c t r u m . A c i r c u l a r l y polarized exciting field w a s u s e d in the experiments.
A s h a r p l y delineated s e r i e s of oscillations of the d e r i v a -
-
346
Sov. Phys. JETP 47(2), Feb. 1978
tive of the impedance with r e s p e c t to the magnetic field
w a s o b s e r v e d in s t r o n g magnetic f i e l d s in both p o l a r izations. T h e p e r i o d of the oscillations in the "+" p o l a r ization (direction of rotation of the radiofrequency field
c o r r e s p o n d s to the cyclotron rotation of the holes)
changes by about 5% with the magnetic field, and in the
polarization it i s p r a c t i c a l l y independent of the field.
In v e r y s t r o n g fields, the two p e r i o d s coincide and correspond t o the cyclotron d i s p l a c e m e n t s of the holes located a t the inflection points of the octahedron of the
F e r m i s u r f a c e . T h e conclusion h a s b e e n drawn, on the
b a s i s of t h e s e r e s u l t s , that a l l the o b s e r v e d phenomena
are connected with a s i n g l e g r o u p of c a r r i e r s . It will b e
shown that the s e r i e s of oscillations with constant period
("-" polarization) i s due to the Gantmakher-Kaner ef-
"-"
0038-564617814702-0346$02.40
01978 American Institute of Physics
346