ANHARMONIC EFFECTS
THERMODYNAMIC PROPERTIES OF CONDUCTION ELECTRONS
2. Thermodynamic functions
2.1. Fermi-Dime function and the chemical potential
The electrons in a solid can be divided into two groups; those that form the closed electron shells of the constituent atoms (the core electrons), and the remaining electrons of higher energy (the valence electrons). In metals the valence electrons can move more or less freely through the lattice. One usually refers to them as the conduction electrons.
We first consider the Sommerfeld model (Sommerfeld and Bethe 1933) in which the conduction electrons are assumed to form a gas of fermions with energies E^ and a density of states N(E). The Fermi-Dirac distribution function is
f
{E)= —^ T7TTTT7'
( i a 2>
exp[(£ - fi)/kBT] + 1
where [i = fi(T) is the chemical potential. We shall frequently encounter df/dE or df/dT. Some useful expressions are
9 / 1 3E kBT
1
4kBT cosh2[(£ - ii)/2k*T]' (10.3) and
ff-*(^rii-i- m
The function —df/dE is symmetrically peaked around /x(T), with an approximate width of a few kBT (fig. 10.1).
The chemical potential /x(T) is determined by the condition that the total number of conduction electrons, Afei, is conserved, i.e.
••r
J - c
N(E)f(E) &E = Neh (10.5)
The integration limits in eq. (10.5) mean that the integration is over all energies for which the integrand is nonvanishing. Often one takes E = 0 to be the bottom of the conduction band, i.e. N(E) = 0 for E < 0. The
Thermodynamic functions 169
Fig. 10.1. The peaked shape of —(df/dE) around the Fermi level.
lower integration limit then is 0. The Fermi energy (Fermi level) EF is defined to be the chemical potential at zero temperature, EF = /x(0).
Sometimes the energies are counted relative to EF instead, i.e. E = 0 at the Fermi level. It even happens, in theoretical calculations, that still another reference level is chosen for the energies, e.g, the zero-level for the so-called muffin-tin potential. It is common to express EF through the Fermi temperature TF\
kBTF = EF. (10.6)
From what has just been said about the various conventions for the reference level of EF, it is obvious that the Fermi temperature TF is not a unique quantity. However, in free-electron-like systems, one almost invariably takes E = 0 at the bottom of the conduction electron band, and textbook values of TF then refer to this case.
It is a standard technique (Ashcroft and Mermin 1976) to evaluate integrals such as eq. (10.5) in the form of a series expansion in powers of T. Then, to lowest order in T2,
/ x ( D - / x ( 0 ) - fi(T)-EF = -(n2/6)(kBT)2
x(dN/dE)E==E¥/N(EF), (10.7)
where dN/dE is evaluated at E = EF. Because df/dE is peaked around
JJL(T) with an approximate width ~ AkBT (fig. 10.1), it is of interest to know how large is the shift ii(T) — /x(0) expressed in units of kBT.
Fromeq. (10.7)
kB
T = ~ *
B 7T yw) * -r
gn(N}-
(ia8)170 Ch. 10. Thermodynamic properties of conduction electrons Table 10.1
The Fermi temperature Tp in a free-electron description Element charac-teristic temperature such that N{E) varies significantly (e.g. a variation comparable to N(E) itself) when E is altered by an amount k^Ty. For free electrons, Tv ~ TF - 105 K (table 10.1). With T lower than the melting temperature Tfus, the shift IJL{T) — /x(0) therefore is negligible in free-electron-like metals. In transition metals, however, N(E) may vary considerably over energies ^Tfus around the Fermi level. Then one cannot neglect the temperature dependence of JJL{T) in the calculation of high temperature properties. See also the related discussion of the heat capacity (eqs. (10.13) and (10.14)).
2.2. Heat capacity
The total conduction-electron energy is (in the single-particle descrip-tion, i.e. with the neglect of explicit many-body corrections)
/ oo
EN(E)f(E) dE, (10.9)
-OO
which gives the heat capacity
= ^ / ( ^ ) V £ > ( - ^ ) d
£. (KM0)
In the integrand, a term 0 = d{2EFNQ\)/dT was subtracted, and a higher-order correction from the temperature dependence of JJL(T)
was ignored. The integration is over all energies where N(E) is non-vanishing, but as we shall see below only a narrow energy interval around £F gives a significant contribution. If N(E) is slowly varying
Thermodynamic functions 171
with E near the Fermi level, N(E) ~ N(E?) can be taken outside the integral. Then
Cel = ^-N(EF)klT. (10.11)
It is common to write Cei = yT. This form allows for electron-phonon many-body corrections etc. in the parameter y. For the Sommerfeld result (eq. (10.11)) we use the notation
Cei = yj = (mb/m)Cfe. (10.12)
The subscript b means that electron band structure effects are included.
Cfe is the heat capacity calculated in the free-electron model. The band mass rab is defined in Appendix B.
If the density of states around the Fermi level is expanded in a series in E - E¥, one obtains (AT = dN(E)/dE\ Nff = d2N(E)/dE2)
lit2 9
i-(*
Br)
2^r
N'\2 1 N"
N 5 N (10.13)
For a free-electron density of states, N(E) ~ El/2, and (Nf/N)2 = l/(4E2) and N"/N = -l/(4E2). Then eq. (10.13) becomes the well-known result from textbooks (e.g. Wilson 1965)
2n2 2 ( 3n2/T\2}
Cd = — W , ) » i r { ! - — ( - ) j . (.0.14)
It is not an unusual mistake that the (r/rF)2-term in Cei is neglected in transition metals, invoking an argument that TF = £ F / & B is of the order of 100,000 K (cf. table 10.1). A correct treatment has to consider what is the energy interval AE around E? over which N(E) varies signifi-cantly, and replace 7p in the low temperature expansion (eq. (10.14)) by AE/kB, a quantity which may be <1000 K when E¥ falls at a sharp structure in N(E).
Example: CQ\for a realistic density of states N(E). This example serves to give an idea about corrections to eq. (10.11) in a real metal. The inset
Ch. 10. Thermodynamic properties of conduction electrons
T (K)
Fig. 10.2. The inset shows a density of states N(E) for ^-electrons characteristic of Pd.
The main part of the figure shows the corresponding heat capacity Ce\(T) in three mod-els, as described in the text. The arrow marks the energy width A £ that corresponds to
AE/kB= 20,000 K.
in fig. 10.2 shows the gross features of N(E) for Pd. The main figure shows the heat capacity Ce\(T) calculated from this N(E) in three mod-els. The two straight lines refer to a constant N(E) = N(E¥) for all E\
the full-drawn line with a tentative constant many-body enhancement parameter X = 0.7 (allowance for electron-phonon and other correc-tions, see below), and the dashed line with X = 0. The full-drawn curve is based on the full N(E) and therefore reflects the true behaviour at high T (where electron-phonon effects are small).
2.3. Entropy
A useful expression for the entropy is
/
oo
{/(£)[ln/(£)]
-OO
+[l - f(E)]\n[l - f(E)]}N(E) dE. (10.15) The function inside the braces { . . . } in eq. (10.15) is an even function of E — ii{T) and is sharply peaked at fi(T). When the density of states
Electronic entropy and heat capacity in real metals 173
varies slowly with E near the Fermi level, N(E) can be taken outside the integral, as a constant N(Eft). We get
5d = ^-N(EF)klT. (10.16)
Thus 5ei = Cei, a result which also follows immediately from the general thermodynamic relation
S*(T) = [ ^ P dT
f(10.17)
Jo I
when Ce\(Tf) is linear in T'. However, we noted above that structure in the energy dependence of N(E) near E? may be important. Then Sei ^ Cei at high T (cf. Chapter 19, §12).
3. Electronic entropy and heat capacity in real metals