Electronic Heat Capacity of Metals

The most important example of an assembly obeying FD statistics is the free electrons in metals. There is good reason to believe that each atom in metallic crystalline lattice parts with one or more of its outer electrons and that these free, or conducting, electrons behave essentially as a gas.

Consequently, it is common to speak of them collectively as an electron gas.

If these particles are free to move about, their behavior is similar to that of the translational motion of an ideal monatomic gas and the thermal and electrical conductivities of a metallic solid are the results of this motion within the solid. That is, this electron gas has translational degrees of freedom, which is quite independent of the metallic ions forming the crystal lattice.

Since free electrons have to obey the Pauli Exclusion Principle, a proper description of their behavior requires the use of FD statistics. That is, any energy state can be associated with no more than one electron. Nevertheless, the allowed ε_i, and g_i of electrons will still be those associated with transla-tional motion such as Eqs. (2.32) and (2.33). From them, we can derive the differential relationships between the n_i, ε_i, and g_i,

FD distribution for these electrons can also be expressed in a differential form,

dN = dg

exp[(ε− μ)/kT ] + 1. (3.22) With the help of Eqs. (3.20) and (3.21),

dg

76 Chapter 3 Heat Capacity, Entropy, and Nanothermodynamics

Equation (3.23) is actually low by a factor of two. Electrons have a spin de-generacy of two because they can be spinning in either of two directions. Each mode of storage of translational energy can occur in combination with either mode of rotation. Equation (3.23) should accordingly be replaced, in this case, by dg = 4πV (2m/h²P)^3/2ε^1/2dε. If for brevity we set A≡ 4πV (2m/h²P)^3/2, dg = Aε^1/2dε. The degeneracy therefore increases with the increasing square root of the energy. From FD distribution function of Eq. (3.22), replacing the sum with an integral, we have N = A

 _∞

0

ε^1/2

exp[(ε− μ)/kT ] + 1dε. The integral cannot be evaluated in closed form and the result can be expressed only as an infinite series. The result is, first obtained by Sommerfeld,

μ = εF a function of N/V , the number of electrons per unit volume. Thus, μ in Eq.

(3.24) is a function of T and N/V . When T = 0, μ⁰= εF. The distribution function at T = 0 is then

N_i⁰= g_i

exp[(ε_i− εF)/kT ] + 1. (3.25) The significance of εF can be seen as follows: In all levels for which ε_i <

εF, ε_i − εF is a negative quantity, and at T = 0, (ε_i− εF)/(kT ) → −∞.

Hence, N_i⁰ = g_i. That is, the average number of electrons in a level equals the number of states in the level, and all levels with energy smaller than εF

are fully occupied with their quota of one electron in each state.

In all levels for which ε_i > εF, the term (ε_i− ε^F) is positive. Hence, the exponential term equals +∞ and N_i⁰ = 0 at T = 0. There are thus no electrons in these levels and εF is the maximum energy of an electron at T = 0 K. The corresponding level is called the Fermi level.

An expression for εF can now be obtained from the requirement that dN⁰= N . Replacing the sum with an integral, introducing the distribution function at T = 0, and integrating over all levels from zero to εF, we have N = A

 _εF

0

ε^1/2dε = 2

3Aε^3/2_F , or, after inserting the expression for A,

εF= h²_P

Thus, as stated earlier, εFis a function of N/V , but is independent of T . The solid curve in Fig. 3.4 is a graph of dN⁰/dε = Aε^1/2 at T = 0. The curve extends from ε = 0 to ε = εF, and is zero at all energetic levels being greater than εF.

3.1 Heat Capacity 77

Fig. 3.4 Graphs of the distribution function of free electrons in a metal at T = 0 and at two higher temperatures T1 and T2.

As a numerical example, let the metal be Ag. Since Ag is monovalent, we assume one free electron per atom. The density of Ag is 10.5×10³kg·m⁻³ with atomic weight of 107, N/V = 5.86× 10²⁸m⁻³. The mass of an electron is 9.11× 10⁻³¹ kg and hP = 6.62× 10⁻³⁴ J·s. Then εF = 9.1× 10⁻¹⁹ J = 5.6 eV. U =

ε_iN_ifor the electrons or, replacing the sum with an integral, U = A

 _∞

0

ε^3/2

exp[(ε− μ)/kT ] + 1dε. Again, the integral cannot be evaluated in a closed form and must be expressed as an infinite series. The result is

U =3 5N εF

 1 +5π²

12

kT εF

2

−π⁴ 16

kT εF

4

+· · ·



. (3.27)

When T = 0, U⁰ = (3/5)N εF where ε⁰ = U⁰/N = 3εF/5 ≈ 3.5 eV for a single electron of Ag. The mean kinetic energy of a gas molecule at room temperature is only about 0.03 eV. If this value reaches 3.5 eV, T ≈ 28000 K.

Hence the mean kinetic energy of the electrons in a metal, even at absolute zero, is much greater than that of molecules of an ordinary gas at T ∼ 10³K.

At T = 300 K, kT /εF = (1.38× 10⁻²³× 300)/(9.1 × 10⁻¹⁹) = 4.58× 10⁻³ for Ag, which is very small. To a good approximation, one can consider that μ = εF at any T .

The dotted curves in Fig. 3.4 are dN/dε graphs at higher temperatures T1 and T2 with T2> T1. The occupation numbers change appreciably with increasing T only in those levels near εF. The reason for this is the following.

Suppose U of metals gradually increases from U⁰ at T = 0, as T raises. In order to accept a small amount of energy, an electron must move from its energy level at T = 0 to a level of slightly higher energy. However, except for those electrons near εF, all states of higher energy are fully occupied. Thus, only those electrons near εF can move to a higher level. With increasing T , those levels just below εFbecome gradually depleted, electrons at still lower

78 Chapter 3 Heat Capacity, Entropy, and Nanothermodynamics

levels can move to those that have been vacated, and so on.

For the particular level at which ε = μ, and at T > 0, the exponential term in the distribution function is equal to 1, and the occupation number is N_i= g_i/2. If T is low, as a good approximation, μ = εFand the Fermi level the other hand, C_V,m= 3R/2 for a monatomic ideal gas.

We therefore see that as a direct consequence of the Pauli Exclusion Prin-ciple, the mean kinetic energy of the electrons in metals is much smaller than that of the molecules of an ideal gas at the same T . But the excitation energy required to jar electrons into the unoccupied higher energy states is too big, except that near εF. As a result, only very few of electrons actually contribute to C_V,m. This result illustrates what had long been a puzzle in the electron theory of metallic conduction: the observed C_V,m of metals is similar to that of isolators of about 3R, as given by the Dulong-Petit law. The free electrons however, if they behave like the molecules of an ideal gas, should make an additional contribution of 3R/2 to C_V,m, resulting in a value being larger than experimental results. The fact that only those electrons having energy near εF can increase their energy with increasing T let us understand why the electrons make only negligible contribution to C_V,m.