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THE PHONON-ELECTRON INTERACTION IN Pb-Bi
ALLOYS
C. Thompson, Jr, C. Tsai, H. Weinstock, W. Overton
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, supplkment au no 8, Tome 39, aoiit 1978, page
C6-1024
THE
PHONON-ELECTRONINTERACTION IN
Pb-Bi ALLOYS+I C.W. Thompson, Jr.
,
C.L. Tsai, H. Weinstock, and W.C. Overton, Jr.IZZinois I n s t i t u t e of TeehnoZogy, Chicago, IL 60616, USA
T Los AZamos S c i e n t i f i c Lab. Los AZarnos,
EM 87545, USA
RQsumQ.- Le coefficient de diffusion phonon-Qlectron (C) est d6duit des mesures de conductivitd thermique entre 0,5 K et 10 K, et de r6sistivitB Blectrique B 4,2 K. Une relation est Qtablie entre les variations de C avec la concentration des allia- ges et les changements du nombre de porteurs libres.
Abstract.- The phonon-electron scattering coefficient (C) is deduced from measure- ments of the thermal conductivity between 0.5 K and 10 K, and of the electrical resistivity at 4.2 K. The variation in C with alloy content is related to changes in the number of free carriers.
1. INTRODUCTION.- The phonon-electron scattering coefficient (C) of Pb-TI alloys, determined from low temperature thermal conductivity measurements /l/ increaseswith the percentage of T1. By combi- ning Ziman's /2/ evaluation of C with the free
electron model, one can show that C nq3
,
wheren is the charge carrier concentration. Thus C may be correlated with the changing electron density at the Fermi surface of Pb. Herein we report the effect on C of alloying Pb with Bi.
2. EXPERIMENT.- Thermal conductivity and electrical resistivity measurements were made on five poly- crystalline Pb-Bi samples with from 1 % to 15 % Bi. Electrical resistivity was measured at 300 K, 77 K, and 4.2 K. The value at 4.2 K was taken as
the residualresistivity p Thermal conductivity
0.
was measured between 0.5 K and 10 K.
All the samples were superconducting below 7.2 K. Thermal conductivity measurements were made in both the superconducting and normal
states. The normal state measurementswere done in magnetic fields which ranged from 2.5 kG to 6.5 kG. Thermal conductivity was measured before and after annealing each sample apprbximately 25'~ below its melting point. Further details of the experimental apparatus are described elsewhere 131.
3. RESULTS AND ANALYSIS.- The electron and lattice contributions to the total thermal conductivity were separated for the normal and superconducting states. The normal state electronic contribution 'supported by the U.S. Department of Energy.
was taken as
Ken = (p /LT + B ~ ~ 1 - l
,
eP (1)
where B is a constant describing electronic
eP
thermal resistance due to electron-phonon scatte- ring 141. The normal state lattice contribution (Kgn) was obtained by subtracting K from Kn.
en
The superconducting state electronic contribution was taken as K es = ReKen, where Re is the reduced
thermal conductivity calculated by Kadanoff and Martin (KM) /5/. KM theory was used to evaluate R because normal state electron scattering at T contains an electron-phonon component /6/ in addi- tion to that due to electron-impurity scattering. The superconducting state lattice contribution
(Kgs) was obtained by subtracting K from Ks. The
es
lattice contribution in both states (K and K )
g S gn
were then fitted to the expression
where the total phonon scattering rate, T-l = .c -l
b + h4 + Cg(w,T)w, is due to grain boundaries, point defects and electrons, respectively. BRT theory was used to evaluate g(w,T) 171, which is proportional to the number of normal electrons. Figure 1 shows K vs. T for Pb-3.5 % Bi, with the
g
curves representing the best fits of the K ) g points to Equation (2). The scatter in the results and the closeness of fit is typical of that for all five samples. The large increase in annealed
K was founddueonly to a corresponding increase
gs
in grain size. Prior to anneal, all samples had grains about 100 p wide; after anneal, these
increased to as much as 8 0 0 ~ .
4 0 0 r
Fig. 1 : Phonon thermal conductivity vs.temperatu- re for Pb-3.5 % Bi ( A
--
Kgs annealed,unannealed, and
m--
K annealed and u n a ~ n ~ ~ l ~ % ~€9
Direct: microscopic observations confirmed this
analysis. K was found to decrease with higher Bi
g S
concentrations (e.g., K (1 %)/K (15 %) = 2.5) as
g S g S
a result of both a larger density of point defects and an enhanced phonon-electron interaction.
4. DISCUSSION.- The variation in C with the per- centage of Bi is shown in Figure 2. The Fermi sur- face of Pb falls in the second and third Brillouin zones, with holes in the second zone and electrons in the third.
Fig. 2 : Phonon-electron scattering coefficient C vs. % Bi ( &
--
from Pb-Bi best fit curves, and A--
calculated from pure Pb data181).
electrons to the second zone hole states. This in- terpretation can be extended to the variation in C deduced for Pb-TI /l/, i.e., we interpret the in- crease in C with the percentage of TI as due to the removal of electrons from the second zone, thereby increasing the number of holes.
References
/l/ Ho
,
J., Ph.D.Thesis (Illinois Institute of Technology) 1974.121 Ziman, J.M., Philos Mag. i(1956) 191. 131 Thompson, Jr., C.W., Ph. D. Thesis (Illinois
Institute of Technology) 1978.
141 Hilsch, R. and Steglich, F., Z. Phys.
226
(1969) 182.151 Kadanoff, L.P. and Martin, P., Phys. Rev.
124
(1961) 670.161 Mrstik; B.J. and Ginsberg, D.M., Phys. Rev. B5 (1972) 1817.
-
171 Bardeen, J., Rickayzen, G., and Tewordt, L. Phys. Rev.
113
(1959) 982.181 Fletcher, R. and Stinson, M.R., J. Low Temp. Phys.
