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6.4 The Specific Heat Capacity of Electrons in Metals
would be wrong to claim that the Fermi distribution is only valid for particles with spin 12; it is equally valid for atoms or molecules that are dis- tributed among predetermined fixed positions, whenever only one atom or molecule can occupy such a position. Corresponding situations arise in the thermodynamics of defects (Sect. 2.7), the solubility of gases in solids, and adsorption processes.
The significance of the chemical potential l in the Fermi distribution is most readily seen in the limiting case of T= 0 K. At zero temperature the Fermi function becomes identical with the step function introduced pre- viously. It has a value of 1 for E <l and of 0 for E >l. Thus at T= 0 K the chemical potential of the electrons is equal to the Fermi energy:
l T0 K E0F: 6:34
Because of this equality, one often speaks of the ``Fermi level'' in place of the chemical potential and uses the symbolEF. This Fermi level, however, is then a temperature-dependent quantity!
At higher temperatures the sharp edge of the Fermi distribution be- comes more rounded; states below EF have a finite probability of being un- occupied, while those slightly aboveEF may be occupied (Fig. 6.6). The size of the region, over which the Fermi function deviates significantly from the step function, is of the order of 2 T to each side, as shown by the tangent to f T;E at EF drawn in Fig. 6.6. One sees that as the temperature is raised, only a small fraction of the electrons is able to gain energy. This has important consequences, e.g., for the specific heat capacity of the electron gas (Sect. 6.4).
If one is interested in the occupation probability for energies or tem- peratures in the range |E±EF|2 T, it is possible to use approximations to the Fermi function (6.33).
The conditionE±EF2 Tis often fulfilled, for example for the conduc- tion electrons in semiconductors (Sect. 12.2). In this regime, with energies E far above the Fermi edge, the Fermi functionf E;Tcan be approximated by the classical Boltzmann distribution,f E;T exp EF E= T; see (12.5).
value. The reason is simple: electrons, in contrast to a classical gas, can only gain energy if they can move into free states in their energetic neighbor- hood. The number of such electrons, expressed as a fraction of the total density n, is only of the order of 1/100, as demonstrated by the following simple estimate:
The ``smeared out'' region of the Fermi function has a width on the or- der of 4 T, i.e., according to Fig. 6.7, the Pauli principle dictates that only a fraction of about 4 T/EF of all ``free'' electrons (density n) can absorb thermal energy. The energy per electron is around T and so the total en- ergy of these thermally excited electrons is on the order of
U4 T2n=EF: 6:35
With TF=EF/ as the Fermi temperature, one obtains the following order of magnitude estimate for the specific heat of the electrons
cv@U=@T8 nT=TF: 6:36
As seen in Table 6.1, the Fermi temperatures are typically on the order of 105K and this, on account of the factorT/TF in (6.36), explains the vanish- ingly small contribution of the conduction electrons to the specific heat ca- pacity.
The exact calculation to determine the specific heat capacity of the gas of free electrons is as follows:
Upon heating the Fermi electron gas from 0 K to a temperature T, the internal energy per unit volume is increased by an amountUgiven by
U T
1
0
dEED Ef E;T
E0F
0
dEED E: 6:37
k k
k k
k k
Fig. 6.7.Explanation of the specific heat capacity of quasi-free metal electrons. The eect of raising the temperature from 0 K to T is to allow elec- trons from 2kT below the Fermi energy to be promoted to 2kT aboveEF. The tangent (± ± ±) intersects the energy axis atEF+ 2kT
We also have the relation EFnEF
1
0
dE D Ef E;T; 6:38 where n is the total concentration of free electrons. Dierentiating (6.37) and (6.38) yields
cv@U=@T
1
0
E D E @f=@TdE; 6:39
0EF @n=@T
1
0
EFD E @f=@TdE: 6:40
Subtracting (6.40) from (6.39), the specific heat capacity cvof the electrons is obtained as
cv@U=@T
1
0
dE E EFD E @f=@T: 6:41
The derivative @f/@T (Fig. 6.7) has significant values only in the ``smeared out'' region of 2 TaboutEF. The density of states D(E) does not vary a great deal in this region and may be approximated byD(EF), i.e.,
cvD EF
1
0
dE E EF @f=@T; 6:42
where the temperature derivative of the Fermi function (6.33) is given by
@f
@TE EF
T2
exp E EF= T
fexp E EF= T 1g2 : 6:43
With the abbreviationx E EF= Twe then have cv 2TD EF
1
EF= T
dx x2expx expx1 2 : 6:44
Since the factor of expxin the integrand is negligible forx±EF/ T, the low- er integration limit can be extended to minus infinity. The resulting integral
1 1
dx x2expx expx1 2p2=3 6:45
is a standard integral whose value can be found in tables.
k
k k
k k k
k
k
One thus has the following general result for the specific heat capacity of the ``free'' electrons in metals:
cvp2
3 D EF 2T: 6:46
In deriving (6.46) no use has been made of the explicit form of the density of states D(E). Equation (6.46) is therefore also valid for cases where the density of states deviates from that of the free electron gas, as indeed is ex- pected in the majority of cases. For metals, measurements of the electronic heat capacity are therefore used as a method of determining the density of states D EFat the Fermi level.
In the free electron gas model D EF can be very simply expressed in terms of the electron concentration. In the case of metals, the validity of the relation TTF implies
n
EF
0
D EdE; 6:47
and the density of states for this case can be written
D E D EF E=EF1=2: 6:48
It follows that n2
3D EFEF; 6:49
and from (6.46) we have cvp2
2 n T
EFp2 2 n T
TF : 6:50
Thus the only dierence between this exact calculation and the previous rough estimate (6.36) is the appearance of the factor p2/2 in place of the factor 8.
The predicted linear dependence of the electronic specific heat capacity on temperature is well confirmed experimentally. For low temperatures, where the phonon contribution to cv displays the Debye T3 dependence, one expects to observe
cvcTbT3; c;bconst: 6:51
The experimental results in Fig. 6.8 show the linear dependence expected from (6.51) for a plot ofcv/TagainstT2.
For the transition metals in particular, the experimentally determined values of cbear little resemblance to those calculated from the electron gas model, as is seen in Table 6.2.
k
k k k
Fig. 6.8.Plot of cv/T against T2 for copper. The experimental points (* and `) stem from two separate measurements [6.3]
Table 6.2.Comparison of experimentally determined values of the coecientcof electro- nic specific heat with values calculated using the free-electron-gas model. At low tempera- tures one has cv=cT+bT3for the combined electronic (!T) and (!T3) contributions to the specific heat
Metal cexp(10±3J/Mol K2) cexp=ctheo
Li 1.7 2.3
Na 1.7 1.5
K 2.0 1.1
Cu 0.69 1.37
Ag 0.66 1.02
Al 1.35 1.6
Fe 4.98 10.0
Co 4.98 10.3
Ni 7.02 15.3
Fig. 6.9. Qualitative behavior of the density of states D E for the con- duction band of a transition metal.
The strong contribution of thed-elec- trons in the vicinity of the Fermi le- vel lies on top of that of the s-band (partially dashed)
The large deviations for Fe, Co, and Ni are attributed to the partially filled d-shells of these transition metals, whosed-bands thus lie at the Fermi energy. Because of the strong localization of the d-electrons at the atoms, the overlap of their wavefunctions is small. This means that the correspond- ing energy band is relatively narrow and therefore yields a large contribu- tion to the density of states as shown in Fig. 6.9.