HAL Id: jpa-00217941
https://hal.archives-ouvertes.fr/jpa-00217941
Submitted on 1 Jan 1978
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
ENERGY DEPENDENCE OF THE ELECTRON RELAXATION TIME
A. Bergmann, M. Kaveh, N. Wiser
To cite this version:
A. Bergmann, M. Kaveh, N. Wiser. ENERGY DEPENDENCE OF THE ELECTRON RELAXATION TIME. Journal de Physique Colloques, 1978, 39 (C6), pp.C6-1044-C6-1045.
�10.1051/jphyscol:19786461�. �jpa-00217941�
JOURNAL DE PHYSIQUE Colloque C6, supplPment au no 8, Tome 39, aoiit 1978, page ~ 6 - 1 0 4 4
ENERGY DEPENDENCE OF THE ELECTRON RELAXATION T I M E X
A. Bergmann, M. Kaveh, and N. Wiser
Deparhnent of Physics, Bar-IZan University, Ramat-Gun, Israel
Rdsum6.- La rdsistivit6 dlectrique des m6taux polyvalents, nobles et alcalins a dt6 calculde en te- nant compte de la aspendance en 6nergie du temps de relaxation des Electrons. I1 est montrE que l'introduction de la ddpendance en Energie ne conduit 2 une rdduction notable de la rdsistivit6 que si deux critPres sont remplis. Le comportement particulier B chacune des classes de mdtaux est ainsi expliqu6.
Abstract.- The energy dependence of the electron relaxation time for electron-phonon scattering is included in the calculation of the electrical resistivity of the polyvalent, noble, and alkali metals.
The analysis yields that a sizeable reduction of the resistivity is obtained only if two criteria are met. The applicability of these two criteria explains the different behavior obtained for dif- ferent classes of metals.
The electron relaxation time for electron- phonon scattering contains, in general, both an an- gular dependence and an energy dependence. Whereas there is a considerable body of work dealing with the angular dependence, the energy dependence has received very little attention. Only within the last year has the energy dependence of the electron re- laxation time T(E) been included 11-51 in the calcu- lation resistivity p(T), and then only for the alkali metals. We here report the results of a ge- neral analysis of the reduction of the calculated resistivity due to the energy dependence of T(E), where E is the energy of the electron. The relaxa- tion-time approximation consists of replacing T(E) by a constant T ~ , and leads to an approximate value for the resistivity, denoted by pRT(T). Including the energy dependence of T(E) leads to a lower va- lue 161 for the calculated electron-phonon resisti- vity, and this reduction is denoted by bp(T).
The present analysis of Ap(T) is applicable to the polyvalent and the noble metals, as well as to the alkali metals, and includes the magnitude of Ap(T), its temperature dependence, and its varia- tion from metal to metal. The analysis reveals that the magnitude of Ap(T) for any particular metal is essentially determined by two general criteria which depend only on geometrical properties of the Fermi surface of the metal. The usefulness of this result lies in the fact that by applying these two crite- ria to any particular metal, one may obtain a relia-
ble estimate of both the magnitude and the tempera- ture dependence of Ap(T) without the necessity of performing a detailed calculation. These results have been verified by comparing our prediction for Ap(T) d t h explicit calculations for several metals, including the alkalis, At, and Cu.
It is found that in order to obtain a large value for the ratio Ap(T)IpRT(T), the following two criteria must both be met. (i) The weighted average change in the velocity of the scattered electrons,
<AV>, must be comparable with the Fermi velocity, i.e., AV(T) E <Av>/vF I , and (ii) the weighted a- verage phonon energy, H<u>, must be large compared with kgT, i.e., Z(T) Z H<w>/kgT >> 1 .
The general result of the present analysis can be summarized by stating that the value of Ap(T)lpRT(T) is closely correlated with the value of the product AV(T)Z(T). For large (small) values of AV(T)Z(T), one obtains large (small) values for Ap(~)lp~~(T), and for hV(T)Z(T) = 1, one obtains Ap(T)/pRT(T) " 0.1.
Let us first apply this result to the alkali metals, which are characterized by a spherical Fermi
surface. For normal scattering, it is readily shown that at low temperatures, AV(T) + 0 as T*
,
whereas for high temperatures, Z(T) + 0 as T-l. From geome- trical considerations, it can be shown 171 that AV(T)Z(T) reaches a maximum value of about unity(see figure 1). Therefore, if only normal scattering were present, Ap(T)lpRT(T) would vanish at low and
at high temperatures and be small at all temperatu- res. These results are in precise agreement with the Research supported by the Israel Commission for
Basic Research classic calculation of Sondheimer /6/.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786461
also include in figure 2, the calculated /1,4,5/
curves for K and for Li.
Fig. 1 : Temperature dependence of AV(T)Z(T), sca- led to the Debye temperature 0, for normal scatte- ring (dashed curve) and for umklapp scattering
(solid curve) for a typical alkali metal and for AR (dash-dot curve)
We now turn to umklapp scattering for the alkalis. At low temperatures, AV(T) = 1 and
<m> -t mmin, implying that Z(T) -t as T -+ 0. There- fore, if only umklapp scattering were present, one would obtain Ap(T)/pkT(T) -t 1 for T -+ 0 and the electron-phonon resistivity would be reduced to zero by the energy dependence of -c(E). For higher tempe- ratures, AV(T)Z(T) decreases rapidly, as shown in figure 1, implying a corresponding decrease in AP (T) /pRT (T)
.
We now consider polyvalent and noble metals, which are characterized by a Fermi surface which intersects the Brillouin zone boundaries. The cru- cial difference from the alkalis concerns umklapp scattering. For the polyvalent and noble metals, Z(T) does not diverge at low T, but rather approa- ches a finite value. Moreover, AV(0) = 0, because of the distortion of the Fermi surface at the Brillouin zone boundaries. Therefore, AV(O)Z(O) = 0, rather than infinity, as was the case for the alkalis. It is precisely this zero value for AV(O)Z(O) that produces a relatively small value for Ap(T)pRT(T).
Indeed, the temperature dependence of AV(T)Z(T) for umklapp scattering for the polyvalent and noble me- tals is qualitatively similar to that for normal scattering for the alkali metals. The only diffe- rence is that whereas the maximum value of AV(T)Z(T) is about 2-3 for the polyvalent and noble metals, it is less than unity for normal scattering for the alkali metals (see figure 1 ) .
From the above analysis, one can immediately predict that for the polyvalent and noble metals, the maximum value of Ap(T)/pRT(T) is about 0.2. The calculated /7/ temperature dependence of Ap(T)lp (T)
RT for AR and Cu is plotted in figure 2. The agreement with the prediction is evident. For comparison, we
Fig. 2 : Temperature dependence of Ap(T)/p (T) for four metals, scaled to the Debye temperature 0 RT
In summary, it is shown that the values for AV(T)Z(T) determine quantitatively the temperature dependence and the magnitude of the correction Ap(T)/p (T) due to the energy dependence of the
RT
electron relaxation time. Moreover, the temperature dependence and the magnitude of both AV(T) and Z(T) can be determined from geometrical considerations alone, without the need for a detailed calculation, both for the alkali metals and for the polyvalent and noble metals.
References
/ l / Leavens, C.R., J. Phys. F
7
(1977) 1297/2/ Takegahara, K. and Wang, S., J. Phys. F
7
(1977) L293/3/ Jumper, W.D. and Lawrence, W.E., Phys. Rev. B
16
(1977) 3314
/4/ Kus, F.W. and Zaremba, E., J. Phys. F (in press) / 5 / Danino, M. (unpublished)
161 Ziman, J.M., Electrons and Phonons oxford) 1960
/7/ Bergmann, A,, Kaveh, M. and Wiser, N.
(unpublished)
