At T = 0 the Fermi-Dirac distribution has every quantum state with ε<µ occupied, f F ( ε ) = 1 for ε < µ , and every state with ε > µ unoccupied, f F ( ε ) = 0 for ε > µ . ε = µ = ε F is the Fermi surface. 12 ε ε F D( ) ε f ( ) ε F T=0 At T = 0 the product f F ( ε ) D ( ε ) cuts off at ε = ε F and drops to zero. The distribution of quantum sates in k -space at T = 0 is the interior of a solid ball of radius k F : 12 Actually the plot of f F ( ε ) above is produced assuming µ is independent of T . As we have seen this is not a good assumption for a classical ideal gas (the classical case is best visualised by the high T curves in f B ), but is often a good assumption for low T , in particular when k B T<<µ . As we shall see for a real Fermi gas µ has a slight T dependence near ε F and the curves for f F ( ε ) at different T do not all cross at exactly the same point. 62
k k k k F y z x For T > 0 some electrons are thermally excited above ε , leaving unoccupied states below ε . The excited and unoccupied states lie in a band of width k B T around ε F . ε ε F ε D( ) ε Equal areas k T B f ( ) F T>0 The above graph of f F ( ε ) D ( ε ) as a function of ε shows the number of electrons with energy ε : the total area under the curve is N , the number of electrons in the crystal. The chemical potential can be determined as a function of T and N by the condition N = integraldisplay ∞ 0 f F ( ε ) D ( ε ) dε = V 2 π 2 parenleftbigg 2 m ¯ h 2 parenrightbigg 3 2 integraldisplay ∞ 0 ε 1 2 parenleftBig e ε − μ k B T + 1 parenrightBig dε. (33) 63
Heat capacity of a free electron gas Classically, an ideal gas of N particles has heat capacity 13 C V = 3 2 Nk B , (34) and specific heat c V = C V V = 3 2 N V k B = 3 2 n e k B , (35) where n e = N V is the number of electrons per unit volume. 14 The observed c V in metals is only about ∼ 1% of (35), a phenomenon which perplexed nineteenth century physicists but which, from a more modern perspective, can be qualitatively be understood as being due to the Pauli exclusion principle. Only those electrons within a distance ∼ k B T of the Fermi surface are free to contribute to the specific heat, electrons any deeper than k B T below the Fermi surface have their dynamics ‘frozen’ by the exclusion principle: there are no unoccupied energy states nearby and so these electrons have no degrees of freedom to contribute to the specific heat. To get a quantitative expression for the specific heat due to free electrons in a metal we first need the internal energy U ( T ) = integraldisplay ∞ 0 εf F ( ε ) D ( ε ) dε (36) . The heat capacity is then ∂U ∂T vextendsingle vextendsingle V but one subtlety is that we need to calculate U ( T ) at constant N , while the right hand side of (36) is a function of µ which in turn depends on T and N through (33). It is not possible to perform the integral (36) analytically so we resort to an approximation, but it is a very good approximation. Define the Fermi temperature , T F by k B T F = ε F .
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