The ideal fermion gas 423ef (e)Fig. 11.1The Fermi–Dirac distribution forT= 0 in eqn. (11.5.32) (solid line) and finite
temperature using eqn. (11.5.31) (dashed line).
The implication of eqn. (11.5.33) is that atT= 0, the particles fill all of the available
energy levels up to an energy valueεF, above which all energy levels are unoccupied.
Thus,εFrepresents a natural cutoff between occupied and unoccupied subspaces of
energy levels. The highest occupied energy level must satisfy the conditionεn=εF,
which implies
2 π^2 ̄h^2
mL^2|n|^2 =
2 π^2 ̄h^2
mL^2(n^2 x+n^2 y+n^2 z) =εF. (11.5.34)Eqn. (11.5.34) defines a spherical surface innspace, which is known as theFermi
surface. Although the Fermi surface is a simple sphere for the ideal gas, for interacting
systems the geometry of the Fermi surface will be considerably more complicated. In
fact, characterizing the shape of a Fermi surface is an importantcomponent in the
understanding of a wide variety of properties (thermal, electrical,optical, magnetic)
of solid-state systems.
AsTis increased, the probability of an excitation above the Fermi energy becomes
nonzero, and on average, some of the energy levels above the Fermi energy will be oc-
cupied, leaving some of the energy levels below the Fermi energy vacant. This situation
is represented with the dashed line in Fig. 11.1, which shows eqn. (11.5.31) forT >0.
The combination of a particle excitation to an energy level aboveεFand a depletion
of an energy level belowεFconstitutes an “exciton-hole” pair. In real materials such
as metals, an exciton-hole pair can also be created by bombarding the material with
photons. The familiar concept of awork function—the energy needed to just remove