Calculations of electron-phonon mass correction

Volume 27A, number 2

PHYSICS

An asymptotic expression of E at J

>>

J~is given

by

Pf(~~TJc) E ePf( ~ ~ c)/~ ‘6 2ii and e [cf~(1 k)/2B}sin29. where A critical pf current ‘p0B/c density Jc in eq. (3) does not diverge at 0 = 0 but shows a weak dependence on 9, H0

~

-

‘

-

while the angular dependence of E~is approximately the same as the one derived by the guided motion [1]. To confirm these results, parallel and transverse voltages were measured on the specimens of Pb-25wt.%Tl, each of which was cut out from a single cold-rolled sheet into the dimensions of 3 . . 3 x 0.3 x 0.01 cm . In fig. 1, the experimental data were compared with a theoretical curve of Jc(0)/Jc(4r) with k = 1.1 which was determined

CALCULATIONS

LETTERS

at 9

=

~7T.

3 June 1968

The observed data for E

1(O) show fairly scattered values; e.g.,0.83
References 1. A.K.Njessen, J.Van Suchtelen, F.A.Staas and W.F. Druyvestyn, Philips Res. Rept. 20 (1965) 226. 2. F. Irie and K. Yamafuji, submitted to Phys. Letters. 3. F. Irie and K. Yamafuji, J. Phys. Soc. Japan 23 (1967) 255.

OF ELECTRON-PHONON

MASS CORRECTION

J. F. JANAK IBM Watson Research Center, Yorktown Heights, New York, USA Received 6 May 1968

The Animalu-Heine pseudopotential is used to compute the mass correction for several metals. Good agreement is obtained with experimental specific-heat and cyclotron masses when Rice’s values for Coulomb corrections are used.

The effects of the electron -phonon interaction are measured by the parameter A, whose average over the Fermi surface is given by [1] dS -

dS

~ k ~ 3 ~ v~EJ~ Vk,EI ~

I

M

i dS

to the free-electron sphere, one finds that the er-

kk’,ji / Wq,I.L V~E 1k

/

~‘ ‘~ / 4ir In this expression, E(k) is the one-electron energy computed in the absence of electron-phonon interactions (but including both band-structure and electron-electron effects), Mkk~~ is the matrix element between states k and k’ of the elec-

tron-phonon interaction for phonons on branch

I.L,

Wqj~is

the phonon frequency (q

=

k’

-

k

-

I

Fermi sphere, into and replacing VfrE~by kF/meff, ror introduced eq. (1) by integrating over the where meff includes the energy dependence of the pseudopotential and the electron-electron corrections, is now larger than the error introduced into the density of states by such approximations (typically [6] <2%). The phonon frequencies to be used in eq. (1) were obtained from spherical averages of neu-

K

3 is obtained by reducing k’ - k into the central Brilbum zone), and all integrations extend over the area of the Fermi surface E(k) = EF. Computations by Ashcroft and Wilkins [2] and Pytte [3] based on eq. (1) replace V~E by kp/(m), where
I

actual, distorted Fermi surface. Actually, when one considers distorted multiple-OPW Fermi surfaces on which no Bragg plane is nearly tangent

data. The phonon Brillouin zone was replaced by the Debye sphere, and the polar ization vectors appropriate to an isotropic solid were used (somewhat more complicated approximations are necessary for hcp metals). Computations were performed using both 1- OPW and 2-OPW [31expansions for the matrix element, Rice’s [4] values for Coulomb corrections and tron-diffraction

105

Volume 27A, number 2

PHYSICS LETTERS

3 June 1968

Table 1 Computed values of A, C/C

0. and (mcyc/m)theor. using Heine-Animalu [71 pseudopotential, and experimental specific heats and cyclotron masses.

Metal

_______

~

I (~~theor= ~~exp[61 ~i~oPw ~2~OPW

meff m0

m1jmo

mb OmCOU1 =~1’i÷

\(mcyc\ m )exp

meff

[4]

[5]

Na

0.06

0.13

‘~0.1 —0.01

1.00 0.99 1.04

1.06

K Al

1.09 1.03

0.12 0.50

Pb

0.0

0.86

0.86

1.67 1.14

Mg Cd

0.01 0.01

1.01 0.87

1.02

m0\

,.mcyc\ ~~theor

=

mb /

0.25

0.13 a 0.12 0•53b

1.20 1.22 1.6

1.3 1.1 1.6

1.20 1.22 1.51

1•55C

l.26~

2.2 1.95

2.1

2.55 2.26

0.39 1.1

1.41 0.75

1.33 1.29

1.40

1.24 1.21 1.55

±

0.02 [91 [9]

±0.02 —

1.6

[10]

2.2 [8J 1.2

-

1.3

[11]

a. Compare to A = 0.18, computed by Darby [2]. b. Compare to X= 0.49 [2], and X= 0.46 [3]. c. Upper row for Pb: computations with Animalu-Heine pseudopotential; lower row, computations with Harrison pseudopotential. Compare to A = 1.05 [2], and experimental value of A = 1.33 ±0.02 [12].

Weaire’s [5] values for band mass arising from the energy dependence of the Heine-Animalu [7]

I am indebted to P. E. Seiden, N. Wiser, T. D. Schultz, and P. M. Marcus for a number of helpful

pseudopotential were used. The results of these computations are given in table 1, along with experimental specific-heat data (taken from table 3-4 of ref. 6), and values for the ratio of experimental [8-11] to band-theoretical cyclotron masses. The latter ratio is equal to the many-body correction to the cyclo-

discussions.

tron mass:

~~)theor m

(1

+

)

Om~o~ m0 (1 + A)

.

(2)

This quantity is also given in table 1. The relatively good agreement in table 1 between theory and experiment for cubic metals suggests that approximate computations of A made by sphericalization of phonons are reasonably accurate in these metals. This is apparently not true for hcp metals, and it is evidently important in such metals to treat the anisotropy of the phonon spectrum more carefully.

106

1. 5. Nakajima and M.Watabe, Progr. Theor. Phys.

29 (1962) 341. 2. N.W.Ashcroft and J.W.Wilkins, Phys. Letters 14 (1965) 285. 3. E. Pytte, J. Phys. Chem. Solids 28 (1967) 93. 4. T.M.Rice, Ann. Phys. (USA) 31 (1965) 100. 5. D.Weaire, Proc. Phys. Soc. 92 (1967) 956. 6. A.O.E.Animalu 7. Inc., W.A.Harrison, New York, Theory and 1966). V.Heine. of metals Phil. (W.A.Benjamin, Mag. 12 (1965) 1249; (see also table 8-4 in ref. 9). 8. J.R.Anderson and A.V.Gold, Phys. Rev. 139 (1965) A1459. 9. C. C. Grimes and A. F. Kip, Phys. Rev. 132 (1963) 1991. 10. F. W. Spong and A. F. Kip, Phys. Rev. 137 (1965)

A431.

11. J.B.Ketterson andR.W.Stark, Phys. Rev. 156 (1967) 748. 12. W. L. McMillan and J. M.Rowell, Phys. Rev. Letters 14 (1965) 108.