NON-ALKALI-METAL BEHAVIOR FOR LITHIUM

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NON-ALKALI-METAL BEHAVIOR FOR LITHIUM

M. Danino, M. Kaveh, N. Wiser

To cite this version:

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JOURNAL DE PHYSIQUE

Colloque C6, suppliment au

no

8,

Tome 39, aofit 1978, page

C6-1046

ION-ALKALI-METAL

BEHAVI

OR FOR

LITHI

UMf

M. Danino, M. Kaveh, and N. Wiser

Department of Physics, Bar-IZan University, Ramat-Gan, Israel

Rdsum6.- I1 est montr6 que le comportement non-alcalin de la rgsistivit6 du lithium dgcoule des valeurs anormalement grandes des dlhents de matrice du pseudopotentiel. L'approximation de l'dlectron libre, gendralement utilisle, entafne, 5 basse temp6ra- ture, des erreurs sensibles non seulement en ce qui concerne la valeur de la r6sisti- vit6 du lithiummais aussi pour sa d6pendance en tempdrature, la correction due au phonon-drag, et les 6carts 5 la loi de Matthiessen.

Abstract.- It is shown that the non-alkali-metal behavior of the electrical resistivity of lithium stems from the unusually large values of the screened electron-ion pseudopotential matrix elements. The standard free-electron resistivity calculation is significantly in error for lithium at low temperatures, not only for the magnitude of the resistivity, but also for its temperature dependence, the phonon-drag contribution, and the deviations from Matthiessen's rule.

In the study of the low-temperature transport coefficients of the simple metals, it is traditional to group the alkali metals separately from the polyvalent and noble metals (non-alkali metals). These two groups of metals differ significantly in behavior, especially regarding the electron-phonon contribution to the electrical resistivity p(T)

.

For example, for the alkali metals the maximum value for the umklapp-to-normal scattering ratio pU(T)/pN(T) lies /l/ in the range 2-10 whereas for the non-alkali metals, one obtains 121 p (~)/p (~)r10~-10~. For the alkali metals, the low-

U N

temperature deviations from Matthiessen's rule

(DMR) are quite modest /3/, say about 30 %, whereas for the non-alkali metals, the DMR may be as large /4/ as 10. In any such characterization of metals, one finds that the properties of Li are much closer to+those of the non-alkalis than to those of the alkalis. For example, the maximum value /5/ of pU(T)JpN(T) for Li is nearly 60 and the low temperature LMR for Li exceed /6,7/a factorof 2.

The non-alkali metal behavior for Li is basically due to the unusually large value of the

screened electron-ion pseudopotential matrix ele- m3nt v(G1), where G1 is the length of the shortest non-zero reciprocal lattice vector. We report here some fo the results of a detailed study of p(T) for Li that accounts for the observed non-alkali- metal behavior. Our results are based on a calcu-

- --

tResearch supported by the Israel Commission for Basic Research.

lation that includes explicitly all contributions to lowest non-vanishing order of v(G1). For the other alkali metals, it is justified /1,8/ to cal- culate p(T) using single-plane-wave pseudo-wave functions, a spherical Fermi surface, the free electron velocity, and the relaxation-time solution to the Boltzmann equation. However, these approximations are inappropriate to Li. Indeed, it is surprising how large an error is introduced in

'talc

(T) for Li if one makes the traditional alkali-metal assumptions. Not only is the magnitude of Pcalc(T) seriously in error at low temperatures, but also its temperature dependence, the magnitude of the phonon-drag corrections /8/ to P(T), and the magnitude of the low-temperature DMR 191.

For Li, one finds /l01 v(G1)' 0.1 Ry, a value that is a full order of magnitude larger than that for the other alkalis /10/. In fact, the

1

value of v(G~)/E~

'

-for Li is much more reminis- 3

cent of the noble metals than of the alkali metals. Therefore, there is no a priori justification for ignoring v(G1) and assuming free-electron-like behavior for the electrons of Li.

In order to test in a consistent manner the effects on P (T) of including terms of order v(Gl),

one must include all the following corrections. (i) The pseudo-wave function must be taken to con- sist of a linear combination of plane waves. In fact, we find that one must include 4 plane waves to reach convergence for the calculation of p(T).

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(ii) The non-sphericity of the Fermi surface must be included. Although the Fermi surface of Li dif- fers from a sphere by only 1-2 % on the average, the maximum bulge of 4 % occurs precisely in the important

E101

direction. This seemingly small deviation from sphericity plays a crucial role at low temperatures, as will be seen. (iii) The true Fermi velocity of the electrons must be used, which is significantly smaller than the free-electron Fermi velocity. (iv) the local field corrections /l]/ to the dielectric screening function must be included. (v) An improved solution to the Boltmann transport equation must be used.

To illustrate the importance of the v(G1) corrections, we present as an example the calcula- ted values of the important umklapp-scattering term pU(T). The importance of PU(T) lies in the fact that for Li, p (T) makes a negligible contribution

N

to p(T) for all temperatures. For T > 7 K, pU(T) exceeds p (T) by more than an order of magnitude.

N

For very low temperatures, phonon drag can be expected/8/ to eliminate p (T) almost completely.

N

Therefore,

£'er

all temperatures, by far the dominant contribution to p(T) arises from pU(T).

The curve plotted in Figure 1 is the ra- approx

tio pU(T)/pU (T), where p (T) includes the approx U

v(G1) corrections and p

U (T) denotes the values obtained from the simplified free-electron calculation.

Fig. 1 : Temperature dependence of the ratio pU (T) /pU approx (T)

.

For T < 20 K, there is a reversal of the relative approx

magnitudes of p

(T)

and pU

U (T), with the ra-

approx

tio pU(T)/pU (T) increasing rapidly with de- creasing temperature. The ratio reaches 10 at about 5 K and, in fact, diverges as T + 0. This effect occurs because of the non-sphericity of the true Fermi surface of Li. For low temperatures,

pU(T) is essentially proportional /8/ to exp

(-lrlw min/kBT), where ylw min/kB 40 K is the mini- mum transverse phonon frequency in the dominant

b1a

direction. 11, this direction, the 4 % bulge of the non-spherical Fermi surface toward the Brillouin zone boundary corresponds to a 30 % re- duction in w Since 30 % of Mw

min. min/kB is 12 K, we may immediately conclude that for T < 12 K, it is crucial to use the true non-spherical Fermi surface for the calculation of pU(T). Otherwise, one obtains the low-temperature errors shown in Figure 1.

In summary, the calculation of the electrical resistivity of Li is similar to that of the polyvalent and noble metals because of the unusually large value of v(G1). The free-electron calculation is inadequate and does not yield quantita- tively reliable values for p (T) for Li, espe-

calc cially at low temperatures.

References

/l/ Kaveh, M. and Wiser, N., Phys. Rev. B

5

(1972) 3648.

1.21

Lawrence, W.E. and Wilkins, J.W., Phys.Rev.B

5

(1972) 4466.

/3/ Ekin, J.W. and Maxfield, B.W., Phys.Rev. B

4

(1971) 4215.

141 Bass, J.,Advan.Phys.

2

(1972) 431.

151 Danino, M., M.Sc.thesis, Bar-Ilan University, Ramat-Gan, Israel (1977) unpublished.

161 Dugdale, J.S. and Gugan, D., Cryogenics

2

(1961) 13.

/7/ Krill, G. and Lapierre, M.F., Solid State Com- mun

9

(1971) 835.

/8/ Kaveh, M. and Wiser, N., Phys. Rev. B

2

(1974) 4042.

/9/ Bergman, Y.,Kaveh, M. and Wiser, N., Phys.Rev. Letters

32

(1974) 606.

/l01 Cohen, M.L. and Heine, V., Solid State Phys. 24 (1970) 37.