Anomalous resistivity in structural phase transition of IV–VI

Solid State Communications,Vol. 19, Pp. 38 1—383, 1976.

Pergamon Press

Printed in Great Britain

ANOMALOUS RESISTIVITY IN STRUCTURAL PHASE TRANSITION OF IV—VI COMPOUND: p-SnTe S. Katayama Department of Physics, Osaka University, Toyonaka, 560 Japan (Received 8 March 1976 by Y. Toyozawa) The contribution of a carrier-soft optical phonon interaction to the electrical resistivity of IV—VI compounds is discussed. The calculated resistivity based on a soft mode theory shows a large increase near the transition temperature T~.It is found that the observed increase of resistivity ofp-SnTe is reproduced by the carrier—soft TO phonon interaction. RECENT MEASUREMENTS of the neutron’ and X-ray2 diffractions revealed the cubic—rhombohedral phase transition for p-SnTe at relatively high temperature. An anomalous increase of the electrical resistivity at this transition temperature has been first observed by Kobayashi et a!.3 The relative resistivity increment ‘~P/Pois of the order of 0.04 around T~= 97.5 K where Po is the value of the background resistivity. Kobayashi et a!. have tried to explain their data assuming a carrier— soft transverse optical (TO) phonon interaction. However, their explanation was quite qualitative.

2 Im D(kFbl, w)) l)(1 e_~~i~BT)~ (1)

‘-~

X

(e~~BT—

—

0

In this equation VF, a, and D(kpj.z, w) are the Fermi velocity, lattice constant and the Fourier transform of the phonon Green function, respectively; Mis the reduced mass density per unit cell. The carrier—phonon interaction potential is assumed to be

In this paper we report the result of the theoretical calculation of the resistivity increase near T~for p-SnTe. We investigated the effect of carrier—longitudinal optical (LO) phonon interaction as well as a carrier—soft TO phonon interaction on the electrical resistivity. The key ingredient of the present calculation is that we took into account the dispersion of soft TO phonon and of screened LO phonon. We assume that both the carrier— acoustic phonon and carrier—defect interactions give rise to the background resistivity. To simplify the problem we assume that the conduction electrons in four valleys have an isotropic effective mass m*. For p-SnTe, the change of the band structure due to the sublattice displacement ~ at T< T~is assumed to be too small to affect the electrical resistivity. The basic formula of the electrical resistivity by carrier—soft optical phonon interaction is derived by the use of the lattice resistivity formula due to Ziman.4 In order to include explicitly the dispersion and damping of soft phonon mode relevant to the scattering events, we used the correlation function for the atomic displacement field, instead of the square of its amplitude, in Ziman’s expression. Using the spectral representation for the time correlation function and for

VT0

=

PkFh ~!

VLO

=

~

2

q e(q)

êTO)

a(q

.

for TO phonon,3

eLo)

)MI

\E~— ~

1/2

(2a) for phonon, (2b)

whereP k~,eTo, eLo, E~and e~are the momentum matrix element, the Fermi wave number, unit polarization vectors for TO and LO phonons, static and optical dielectric constants, respectively.! and EG in equatjon (2a) are the interband deformation potential5 and the gap energy at L-point, respectively. The dielectric function is given by e(q) = 1 + q~~/E~~q2 where ~rF is the Thomas—Fermi wave number. If the Planck distribution function is approximated by kBT/hw, the integral over win equation (l)is given simply by ~rRe D(q, 0), with the use of Kramers—Kronig relation. Let us first discuss the resistivity due to the soft TO phonon. We assume that the soft phonon is described by the following Green function: ,

Re D~o(q,0)

=

—

w~o(0)+ A(T)q2

(3)

where w~~ 0(0) = a(T— T~)

for

T> T~,

(4a)

phonon Green function we get Mi 2Mz3a2kBT

p

=

l6e

2 fi.t3 dizl V(kFp)12

= 2cr(T~—T) forT
382

ANOMALOUS RESISTIVITY OF N—VI COMPOUND: p-SnTe I

—

Vol. 19, No.4

I

186Exp.

TcIOOK

E16U ~-~~i2~j T~975~ ~LO -

30

~ V~4O

I

I

0

~56• c~:2°

0.6 0.8 REDUCED

I

0

0L i I I 70 80 90 100 110 1~130 140 150

1.0 1.2 1.4 TEMPERATURE T/Tc

TEMPERATURE (K)

Fig. 1. The function TG(T, kF)I7 in P~ocalculated accordin~to equation (6) vs TIT~.The value of 7 = 4Ak~./afor each curves is 0, 10 and 40K, respectively. sufficient to take only up to quadratic term in the dispersion of soft mode as in equation (3). Using equations

Fig. 2. Calculated resistivity PTO (solid line) and PLO (broken-dotted line) vs temperature for p-SnTe. Open circles 0 represent the observed 3 resistivity increments ~p by Kobayashi et al. important. Taking account of this aspect we get the real part of D~

0(q,0) as (1), (2a), (3) and (4) we obtain with

PTO

=

3ir kBT G( T, kF), 2M~4 VTO~2 al M(T)k~ l6e o.4

(

0(0) 4A(T)k~ (6) log~1± 2 G(T, kF) = 2 2A(T)k~ wTo(O) 2F
J

Kobayashi Ct a!. used the divergent behavior of resistivity near T~,to interpret their data. In Fig. 1 we plotted aTG(T, kF)/4Ak~which is the temperature dependent part of Ppo, as functions of TITC according to equation (6) for three values of ~ = 4Ak~/a.The dispersion coefficientA(T) is assumed to be independent on the temperature. It is observed that the anomalous increase of the resistivity at T~is greatly reduced by taking account of the dispersion in the soft mode. If the soft mode has no dispersion, the mode frequencies in the entire region of the Brillouin zone become zero at 1’,, simultaneously, resulting in the divergence of the resistivity. However, in the presence of dispersion, the phonon mode responsible for the enhancement of the resistivity is limited in the small region near the zone center,which is relevant to the small angle scattering, Fordevelop the casea of the carrier—LO phonon scattering we can similar analysis as for TO phonon scattering. When a carrier density becomes of the order of 1018 _1020 cm3, the coupling6 between plasmons and LO phonons as well as the screening effect7 becomes

2 —l e~q) ________ w~o~w TO

ReD~0(q,0) = w~.

(5)

,

(8)

0+ where WLO is the LO phonon frequency. Substituting equation (8) into equation (1) and integrating over p we

have

PLO

=

fh~~112kB T Po~ EF hwLo F(T,kF),

)

Po n

(

.J2 1

where =

—

(9)

1/2 ~)m*3~’2 WL0

nh3”2e~

being the carrier density. In the case of ~4

0/w~0 ~ 1, F(T, kF) in equation (9) is approximately given by F(T, kF)

i

~ + log (i + (10) 4 \ Q) with Q = 4F/e,,,k~..The function F(T, kF) in the expression of PLO [equation (9)] also has a peak at T = T,~,.However, this peak is very small compared with that of PTO• This is because of the steeper curvature of dispersion of the screened LO phonon as well as of the strong electronic screening. In Fig. 2 the resistivity due to the carrier—LO phonon scattering is shown by broken-dotted line. This is obtained by taking = 2.65 x 1013 sec~ c,,, = 45, 3 WLO and the conductivity effective n = 1.17 x 0.06m. 1020 cmIt is seen that the anomaly around mass m* = T~= 97,5 K is vanishingly small. The solid lines are the contribution from the TO phonon scattering calculated —

~-‘~

,

Vol. 19, No.4

ANOMALOUS RESISTIVITY OF IV—VI COMPOUND: p-SnTe

from equation (5) with the same values of parameters as for the LO phonon scattering. The solid curves are adjusted so that the calculated curve and the experimental points ofp-SnTe by Kobayashi eta!, agree with each other. For this fitting we chose = 2.1 eV, E0 = 0.332 K’ eV and 4A4/a If wedata8 use ain= the 0.30above x 10~ taken from =the20K. neutron sec equation, the dispersion coefficient of soft mode becomes A = 1.63 x 1010cm2 sec’2 .This value is very small compared with A ~ 2.07 x 1011cm2 sec2 esti-

383

to us that the curvature of dispersion curve changes rapidly near the zone center. In conclusion the observed resistivity increase for p-SnTe near T~is mainly caused by the increase of the atomic displacement associated with the soft TO phonon. Takingofaccount of the dispersion the soft mode, divergence the resistivity increase is of suppressed. We could determine the dispersion coefficient for soft phonon mode from the comparison between theory and experiment.

mated from the neutron data.8 However it should be

noted that, while the wave number of the soft modes responsible for the carrier scattering is smaller than 2kg, the wave number of the soft phonon observed with the neutron scattermg is much larger than kF. This suggests

Acknowledgements The author wishes to express his thanks to Dr. K.L.I. Kobayashi for valuable discussions on the experimental data. He is Brateful to Professor H. Kawamura, Professor Y. Yamada and Dr. K. Murase for helpful comments and discussions on this work. —

REFERENCES 1. 2.

IIZUMI M., HAMAGUCHI Y., KOMATSUBARA K.F. & KATO Y., J. Phys. Soc. Japan 38,443(1975). MULDAWER L.,J. Nonmetals 1, 177 (1975).

3.

KOBAYASHI K.L.I., KATO Y., KATAYAMA Y. & KOMATSUBARA K.F., Solid State Commun. 17, 875 (1975). ZIMAN J.M., Electrons and Phonons, p. 358. Oxford University Press, London (1967). MILLS D.L. & BURSTEIN E.,Phys. Rev. 188,1465(1969); KAWAMURA H., KATAYAMA S., TAKANO S. & HOTTA S., Solid State Commun. 14, 259 (1974). YOKOTA I.,J. Phys. Soc. Japan 16,2075 (1961).

4. 5. 6. 7. 8.

EHRENRICH H.,J. Phys. Chem. Solids 8, 130 (1959); DONIACH S.,Proc. Phys. Soc. 73, 849 (1959); COWLEY R.A. & DOLLING G., Phys. Rev. Lett. 14,549(1965). PAWLEY G.S., COCHRAN W., COWLEY R.A. & DOLLING G., Phys. Rev. Lett. 17, 753 (1966).