ANHARMONIC EFFECTS
THERMODYNAMIC PROPERTIES OF CONDUCTION ELECTRONS
3. Electronic entropy and heat capacity in real metals 1. Introduction
The Sommerfeld electron theory of metals, leading to expressions such as eq. (10.11) for the heat capacity CQ\ and eq. (10.16) for the entropy Sei, is not in quantitatively good agreement with experiment, even if one uses an electron density of states N(E^) determined from accurate elec-tron band calculations. The main reason is that the Sommerfeld model neglects important electron-phonon many-body corrections. Their ex-istence was realised by Buckingham (1951) and Buckingham and Schafroth (1954) but their magnitude was unknown until much later (e.g. Ashcroft and Wilkins 1965, Allen and Cohen 1969). The electron-phonon many-body correction to the thermal properties of conduction electrons has been reviewed by Grimvall (1976, 1981).
The complications caused by a rapidly varying density of states N(E) around the Fermi level have already been discussed above. It may be necessary to allow the chemical potential /x to vary with T.
In numerical calulations of the electron band-structure term (i.e. without many-body corrections) one may use eq. (10.5) to determine fi(T), then eq. (10.15) to obtain the entropy, and finally calculate the heat capacity as C = T(dS/dT). Examples showing the non-linear T-dependence of
Ch. 10. Thermodynamic properties of conduction electrons
Cei at high temperatures are given in Eriksson et al. (1992) and in fig.
10.2.
3.2. Effects of electron scattering
In the treatment of electron states with energies Ek, and a correspond-ing density of states N(E), it is assumed that the electrons move in a perfect periodic lattice, with sharp energy eigenvalues Ek. In a real solid, scattering against phonons or impurities gives these electron states a finite life time r which may be described by an imaginary part
—2/T = —ih/x in Ek. We can still use the expression (10.15) for the entropy if we replace the density of states N(E) by a "smeared"
function N*(E) defined as
Ar(£)
-£/ w ( B - q £U d '- <iai8)
When r -> 0, the Lorentz function in the integrand becomes a 8-function, 8(s - E), and then N*(E) = N(E). If we expand N(E) around E¥ as N(s) = N(EF) + (s - EF)N'9 (eq. (10.18)) formally di-verges, but the contribution from (s — E¥)Nf vanishes over a symmetric interval around £F. Therefore, in real cases, it is only when N(E) has a strong non-linear energy dependence near £F that the finite lifetime of the electrons affects the entropy. Electron states within several &B T from
£F contribute to the entropy (eq. (10.15)), so the effect of a finite T is negligible if T <S kBT. For the phonon-limited life time one has (Chap-ter 15 and Grimvall 1981) T = 7rAei-Ph^B7" at high T, where Xei_Ph is the electron-phonon parameter introduced in §3.3. Since A.ei_ph is of the order of unity, the phonon-limited T could have a significant influence on Sei. However, calculations with realistic N(E) (Thiessen 1986) show that the effect is usually small (< 10%, any sign). In the case of impurity scattering it is convenient to relate F = h/2r to the electron lifetime as it enters the electrical conductivity, a = ne2r/m\y (eq. (15.9)). It follows that when the electrical resistance due to impurities is less than that due to electron-phonon scattering, as is the case with dilute impurities, the influence of impurity scattering is negligible. Finally, note that even if N(E) is a 5-function, N*(E) decays as \/E for large E and therefore has an infinite width. Integrals for thermal properties are taken over a finite energy range, and N*(E) must be multiplied by a renormalisation factor so that eq. (10.5) is still fulfilled.
Electronic entropy and heat capacity in real metals 175
3.3. Electron-phonon many-body corrections to the electronic entropy The electron-phonon many-body corrections are significant only at low temperatures (T < #D) and may usually be neglected when T is so high that one has to consider the variations in N(E) dealt with above.
Therefore, in this section we let the electron density of states be a constant, N(E¥). We also assume that the electron-phonon interaction, expressed through a so-called electron self energy Mei_ph, is isotropic.
Then (Grimvall 1976, 1981, 1986)
s = N(EF)kBh2 f00 I" co 1
d (kBT)2 J^ |_cosh2(/to/2/:Br) J
x[hco - R e Md_p hO , T)] dco. (10.19)
This result holds for all temperatures. In the low temperature limit (T <& 0D where #D is a Debye temperature) the integral (10.19) picks up Mei_ph very close to the Fermi level, i.e. for small co. There one may expand Mei_ph as Me\_vh(co, &F; T) — —Xe\-phhco. The resulting integral has the same form as in the Sommerfeld model, apart from a factor 1 + A.ei_ph, and the final low temperature result is
Sei = (27T2/3)Ar(£F)(l + Ae l_p h)£2r. (10.20) Thus the effect of electron-phonon many-body interactions is easily
ac-counted for, at very low temperatures. The high temperature limit of eq. (10.19) agrees exactly with the Sommerfeld model, since Mei_ph
goes to zero as (0D/T)2 for T > #D- At intermediate temperatures, one has to perform the integration in eq. (10.19) numerically. It is convenient to split 5ei into two parts; 5b which is the Sommerfeld (or electron-band theory) result, and 5ei_Ph which is the correction caused by electron-phonon many-body interactions;
Sel = 5b + S e n * . (10.21)
In the low temperature limit,
Sel-ph = ^el-ph^b- (10.22) If Mei_ph is calculated with an Einstein model for the lattice vibrations
Sei-Ph can be expressed as a universal function 5ei_ph(7")/(yb^ei_phr) (fig. 10.3).
Fig. 10.3. The temperature dependence of the electron-phonon renormalisation con-tribution to the electronic heat capacity yel-ph(T)> a nd to the electronic entropy,
Sel-ph(^)> calculated in an Einstein model for the phonons. From Grimvall (1981).
The quantity Aei_Ph discussed in this section is closely related to the BCS theory of superconductivity. In fact, Aei_ph can be estimated from the critical temperature Tc (Appendix B). Numerical values of A,ei_ph
are given in Appendix I. Because A.ei-Ph is anisotropic, the listed values of Aei-ph are averages over the electron states at the Fermi level. There is also a close relation between A,ei_ph and the "transport" coupling pa-rameter A.tr that appears in the electrical resistivity of metals (Chapter
15).
3.4. Electron-phonon many-body corrections to the electronic heat capacity
The electronic heat capacity (at constant volume) is obtained from
c " = T {w) v - <m23)
Equation (10.20) gives the low temperature (i.e. T < #D) result
Cel = (27t2/3)N(E¥)(l + Ae l_p h)^r = ^ ( i + Ael_ph)7\ (10.24) We define a "thermal" effective electron mass m^ by
rnth = (Y/Yfe)m, (10.25)
Electronic entropy and heat capacity in real metals 111
where y is the measured coefficient in Cei = yT,m is the usual (free) electron mass, and /fe is the coefficient resulting from the free-electron version of the Sommerfeld model. With the electron-phonon many-body correction written explicitly we have
mth = (KbMeXl + K\-Ph)m = rab(l + Aei_ph). (10.26) In analogy to our treatment of the entropy, the electronic heat capacity
is split into two parts;
Cei = [yb + yd-phCmr, (10.27) where yei_ph(r = 0) = ybAei_ph. At high temperatures, yei-ph tends to
zero. There is no temperature dependence in y^ because we have as-sumed the approximation that N(E) is a constant, within the energy interval probed by the heat capacity. Figure 10.3 shows ye\-ph(T) in an Einstein phonon model. As a rough rule of thumb, we can take Xei-ph(^)/Kb = ^-ei-ph for T < 9^/4 and zero for T > 0D/3, where 9D is a characteristic Debye temperature. The temperature dependence of Ye\-ph(T) and Sei-ph(^) given in fig. 10.3 is not much altered if one uses the true phonon spectrum instead of the Einstein model; the peaks in fig.
10.3 will be somewhat broader and with smaller maxima. The decrease, and eventual absence, of the electron-phonon enhancement factor is not easy to see in experiments on Cp because it is difficult to separate it from the temperature dependence of the lattice part. However, there seems to be clear evidence for the effect, including a spin fluctuation part (see below) in Lu and Sc (Tsang et al. 1985, Pleschiutschnig et al. 1991, Swenson 1996).
Heavy fermion systems get their name from the fact that the effec-tive electron mass (quasi particle mass) at the Fermi level is very high;
several orders of magnitude larger than in conventional metals. The heat capacity parameter y = Ce\/T shows a strong temperature dependence, varying with the material. The ground state of a heavy-fermion system may be superconducting (e.g. CeCu2Si2, UPt3), magnetic (e.g. UCdn, U2Z1117) or normal metallic (e.g. CeAl3, CeCu6). We may still write
^th = /wo(l + K\-ph) where ra0 is the result in the absence of electron-phonon interactions, but in this case the effect of the electron-electron-phonon interaction is to reduce meff, i.e. A,ei__ph < 0 (Fulde et al. 1993).
3.5. Other many-body corrections
Electron-electron interactions: In a uniform electron gas, the electron-electron many-body corrections to the thermal electron-electron mass are small.
In free-electron-like metals the corrections are at most a few percent (Grimvall, 1975b). In transition metals there are important electron-electron many-body terms, but to a large extent they are folded into the single-particle density of states N(E) obtained in a band structure cal-culation. The remaining correction to the thermal mass probably is only a few percent and of uncertain sign. Lacking more detailed information it is therefore best to neglect these effects.
Electron-paramagnon interactions: In metals that are close to a mag-netic instability, there are electron-paramagnon many-body corrections.
We can write (Gladstone et al. 1969, Burnell et al. 1982, Leavens and MacDonald 1983)
™th = mh(\ + A-el-ph + A-el-sp), (10.28)
where Aei-sp refers to spin fluctuations (i.e. paramagnons). It is difficult to calculate Aei_sp accurately (Daams et al. 1981, Leavens and Mac-Donald 1983). Among several proposed expressions we quote that of Doniach and Engelsberg (1966);
Aei-sp = 3IN(EF) (1 + V / J V ( £ F ) 1. (10.29)
el sp v w
[ 12[1 - IN(EF)] J v }
Here we recognise the term 1 — IN(EF) from the Stoner model of magnetism, where the susceptiblity x = Xo/U — IN (Eft)] diverges when IN(E?) -> 1 which signals a transition to a magnetically ordered state (eq. (19.24)). The parameter v is roughly of the order of 1/2. In free-electron-like metals IN(EF) «; 1, and A.ei-sp is negligible. Also for most transition metals A,ei_sp < 0.05. However, there seem to be metals (LuCo2; Ikeda and Gschneidner 1980, MnSi; Taillefer et al. 1986) with Aei-sp of the order of 4, i.e. a larger enhancement than the highest known A,ei_ph from electron-phonon interactions.
Electron-magnon interactions: In magnetically ordered materials, there are electron-magnon many-body corrections which add a term analogous to Aei_sp in eq. (10.28). The magnitude of their influence on mth is not very well known, but it may be comparable to Aei_ph in Ni and Co (Phillips 1967, Batallan et al. 1975) and even be the dominating
Electron density of states in real metals 179
enhancement in rare earths (Cole and Turner 1967, Nakajima 1967, Kim 1968). Fulde and Jensen (1983) gave a unified theoretical treat-ment of electron mass enhancetreat-ments due to phonon, electron-paramagnon and electron-magnon interactions. The corrections may be appreciable near T = 0 K, but disappear at high temperatures (cf. fig.
10.3).
Single-particle density of states in magnetic metals: In the Stoner model of ferromagnetic metals, one considers separate density-of-states functions, N+(E) and N-(E), for the two spin directions. Relations such as eq. (10.15) for the entropy and eq. (10.1) for the low temperature heat capacity remain valid if we put
2N(E) = N+(E) + N-(E). (10.30) However, one should note that the splitting of the two spin bands,
and hence N+(E?) and N^(E^), varies with the temperature. The many-body enhancement factor need not be the same for the two spin directions, but this is of no concern if we let the enhancement factor be an average over all electron states, in analogy to the case of anisotropic enhancement referred to earlier.