Fermi Statistics

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6.3 Fermi Statistics

We now move on to consider the Fermi gas at finite temperatures. We shall need to derive the distribution function or occupation probability f…E;T†for non-zero temperatures. This is a thermodynamical problem, since we are in- quiring about the distribution that arises when various quantum mechanical states are in equilibrium with one another. To derive the distribution f…E;T†

we must therefore apply some basic concepts of thermodynamics.

We consider an atomic system with single-particle energy levels Ej. We assume that the energy levels Ej lie very close to one another as in a solid.

We can then consider new energy levels Ei each of which consists of many E_j. The degeneracy of these new levels is denoted byg_iand their occupation number by n_i where both g_i and ni are large numbers. On account of the Pauli principle we must have nigi. From thermodynamics we know the conditions that the system must fulfill in order for all energy levels to be in equilibrium: the free energyFof the total system must be stationary with re- spect to a variation in the relative occupation numbers of the levels. In other words we must have

dFˆX

i

@F

@n_idniˆ0 …6:18†

with the subsidiary condition of conservation of particle number X

i

dniˆ0: …6:19†

For the specific case of exchange of electrons between two arbitrary levels k andlthe equilibrium conditions read

@F

@nkdnk‡@F

@nldnlˆ0; …6:20†

dnk‡dnlˆ0: …6:21†

From this it follows immediately that the derivatives of the free energy with respect to the occupation numbers must be equal

@F

@n_kˆ@F

@n_l : …6:22†

Since the two levels were selected at random, at equilibrium all@F/@ni must be equal and we denote this quantity by a new constant, l, defined as the

``chemical potential'' of the electrons.

We will now calculate the free energy of the system of electrons. From thermodynamics we have the relation

FˆU TS …6:23†

with the internal energyU UˆX

i

niEi …6:24†

and the entropyS.The entropy is given by

Sˆ lnP; …6:25†

wherePrepresents the number of possible ways of distributing the electrons among the states. The number of ways of accommodating one electron in the levelEiisgi. For a second electron, also in levelEi, the number of possi- bilities isgi±1, and so on. There would therefore be

g_i…g_i 1†…g_i 2†. . .…g_i n_i‡1† ˆ g_i!

…gi ni†! …6:26† possible ways of accommodatingnielectrons at definite positions within the energy levelEi.However, arrangements which di€er only in the exchange of electrons within the energy level are not distinguishable. Since there are ni! such possibilities, the total number of distinguishable ways of accommodat- ingnielectrons in the levelEiis given by

gi!

ni!…gi ni†! : …6:27†

The number of waysP of realizing the total system is then the product over all possibilities for occupying each level:

PˆY

i

g_i!

ni!…gi ni†! : …6:28†

Thus the entropy can be expressed as

k

Sˆ X

i

‰lngi! lnni! ln…gi ni†!Š; …6:29†

where the factorials can be replaced by using Stirling's approximate formu- la

lnn!nlnn n …for largen†: …6:30†

It now becomes a straightforward matter to calculate the chemical potential, i.e., the derivative of the free energy F with respect to the occupation num- ber of an arbitrary leveli:

lˆ@F

@niˆE_i‡ Tln ni

gi ni : …6:31†

We can rearrange this expression to obtain the occupation numberni: n_iˆg_i…e^{Ei l†= T}‡1† ¹: …6:32† The probability that a quantum mechanical state (degenerate states are also considered distinct here) is occupied is given by the distribution function f…E;T†, which, from (6.32), can be seen to be (Fig. 6.6)

f…E;T† ˆ 1 e^{E l†= T}‡1

: …6:33†

This function is also known as the Fermi distribution. It is the distribution function for particles, only one of which may occupy each quantum state.

For electrons and all particles with half-integral spin, i.e., fermions, this dis- tribution function guarantees that the Pauli principle is obeyed. However, it

k

k k

k

Fig. 6.6.The Fermi distribution function at various temperatures. The Fermi temperature TF=E⁰F/khas been taken as 510⁴K. The tangent at the point of inflection (± .. ±) intersects the energy axis at 2k^TaboveE0Fat all temperatures

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…Tˆ0 K† ˆE⁰_F: …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;T†can be approximated by the classical Boltzmann distribution,f…E;T† exp…EF E†= T; see (12.5).