The properties of an ideal Fermi gas are strongly determined by the Pauli principle. We shall consider the limit: µ >> k B T βµ >> 1,

Transcription

1 Chapter 3 Ideal Fermi gas The properties of an ideal Fermi gas are strongly determined by the Pauli principle. We shall consider the limit: µ >> k B T βµ >>, which defines the degenerate Fermi gas. In this limit, the quantum mechanical nature of the system becomes especially important, and the system has little to do with the classical ideal gas. Since this chapter is devoted to fermions, we shall omit in the following the subscript ) that we used for the fermionic statistical quantities in the previous chapter. 3. Equation of state Consider a gas of N non-interacting fermions, e.g., electrons, whose one-particle wavefunctions ϕ r r) are plane-waves. In this case, a complete set of quantum numbers r is given, for instance, by the 3 cartesian components of the wave vector k and the z spin projection m s of an electron: r k x,k y,k z,m s ). Since the Hamiltonian operator of the system, Ĥ = N Hi) = i= N i= p 2 i 2m +V r i), is spin independent, the summation over the states r whenever it has to be performed) can be reduced to the summation over states with different k p = h k) as... 2s+) r k where we have already summed over m s = s, s+,...,s to get the prefactor 2s+) ,

2 64 CHAPTER 3. IDEAL FERMI GAS We assume that the electrons are in a volume defined by a cube with sides L x, L y, L z and volume V = L x L y L z. This imposes on the one-particle wavefunction ϕ k r) = V e i k r the following periodicity condition at the cube s walls: e ikxx = e ikxx+ikxlx e ikxlx = e i2πnx, which is then translated into a condition for the allowed k-values: k x,y,z = 2π L x,y,z n x,y,z, n x,y,z Z. Each state has in k-space an average volume of In the thermodynamical limit, i.e., when k = 2π)3 L x L y L z = 2π)3 V. V, N and n = N V = const, k so that the sum r can be substituted by an integral over k-space as 2s+) d 3 V k = 2s+) d 3 k k 2π) 3 r = 2s+) V d 3 p, 3.) h 3 where the equality k = p/ h has been used. Note that the factor h h 3N for N particles), which we had to introduce ad hoc in classical statistical physics, in quantum statistical physics appears naturally. Grand canonical potential We consider now the expression for the grand canonical potential ΩT,V,µ) that we derived in Chapter 2: ΩT,V,µ) = k B T r ln [ +e βεr µ)]. Through the substitution 3.), it can be rewritten as V βωt,v,µ) = 2s+) 2π) 34π dk k 2 ln [+ze β h2 k 2 2m ], 3.2)

3 3.. EQUATION OF STATE 65 where we also introduced z = e βµ and used an explicit expression for the one-particle energies for free electrons ε r given by ε r ε k) = h2 k 2 2m. Expression 3.2) is then transformed further by introducing a dimensionless variable x, into β x = hk 2m k2 dk = ) 3/2 2m β h 2 x 2 dx, βωt,v,µ) = 2s+) 4V ) 3/2 m π 2πβ h 2 x 2 dxln +ze x2). By making use of the definition of the thermal de Broglie wavelength λ, we get λ = βωt,v,µ) = 2s+) λ 3 2πβ h 2 m, 4V π dx x 2 ln +ze x2). In order to calculate the integral, we perform a Taylor expansion of the logarithm: ln+y) = ) n+yn n for y. Then, x 2 dx ln +ze x2) = = = = π 4 ) n+zn n ) n+zn n ) n+zn n ) d dn d dn n+ zn dx x 2 e nx2 2 n 5/2. dxe nx2 ) π n ) We define the following functions: f 5/2 z) = 4 π dx x 2 ln +ze x2) = ) n+ zn n 5/2

4 66 CHAPTER 3. IDEAL FERMI GAS and f 3/2 z) = z d dz f 5/2z) = ) n+ zn n 3/2 so that the grand canonical potential for an ideal Fermi gas can now be written as βωt,v,µ) = 2s+ λ 3 Vf 5/2 z). Since Ω = PV, βp = 2s+ λ 3 f 5/2 z) with λ = λt). 3.3) In order to find the thermal equation of state, we substitute the fugacity z by the particle density n =< ˆN > /V. Thus, ) < ˆN >= z z lnz T,V ) = Vz z βp T,V n = < ˆN > V = 2s+ λ 3 T) f 3/2z). 3.4) Substituting eq. 3.4) into eq. 3.3) one can obtain, in principle, the thermal equation of state. The caloric equation of state can be obtained from One first finds µ < ˆN >: and then, using eq. 3.3), U: U = β lnz +µ < ˆN >. µ < ˆN > = µv 2s+ z d λ 3 dz f 5/2z) = V 2s+ ) λ 3 λ 3 β V2s+) lnz = β lnz + 3 2β lnz = β lnz PV U = 3 2 k BTV 2s+ λ 3 f 5/2 z) = 3 2 PV. The equation U = 3 2 PV is also valid for the classical ideal gas, but it is not anymore valid for relativistic fermions.

5 3.2. CLASSICAL LIMIT Classical limit Starting from the general formulae 3.3) and 3.4), we first investigate the classical limit non-degenerate Fermi gas) corresponding to z = e βµ <<. Under this condition, the Fermi-Dirac distribution function reduces to the Maxwell- Boltzmann distribution function: < ˆn r >= z e βεr + ze βεr. Also,intheseriesexpansionforf 5/2 z)andf 3/2 z)onlythefirsttwotermscanberetained, i.e., f 5/2 z) z z2 2 5/2, f 3/2 z) z z2 2 3/2. Then, eq. 3.3) and 3.4) reduce to βpλ 3 2s+)z z ) 2 5/2 and nλ 3 2s+)z z ). 2 3/2 The simplest approximation in this limit is z ) nλ3 2s+, which is equivalent to assuming that nλ 3 <<, i.e., that the particle density and the thermal de Broglie wavelength are small and, since λ T, high temperatures. In the next approximation step we have: ) ) z ) z ) + z) = z ) z) z ) +z ) 2 3/2), 2 3/2 2 3/2 where +x, x << x was used. The equation of state for pressure is then βpλ 3 2s+) [ z ) +z ) 2 3/2) 2 5/2 z ) ) 2] ) = nλ 3 5/2 nλ3 +2 2s+

6 68 CHAPTER 3. IDEAL FERMI GAS or PV =< ˆN > k B T + ) nλ s+) In this expression, the first term corresponds to the equation of state for the classical ideal gas, while the second term is the first quantum mechanical correction. 3.3 Fermi energy In the low temperature limit, T, the Fermi distribution function behaves like a step function: { T if εk > µ, n k = e βε k µ) + if ε k < µ, i.e., n k T θµ ε k ). This means that all the states with energy below the Fermi energy ε F, ε F = µn,t = ), are occupied and all those above are empty. In momentum space the occupied states lie within the Fermi sphere of radius p F see Fig. 3.). Such a situation is denoted quantum degeneracy as we already mentioned at the beginning of this chapter. Figure 3.: Fermi surface in momentum space. The Fermi energy can be determined from the following condition: N = states r with ε r < ε F which, in the case of free fermions, can be written as N = 2s+)V d 3 k = 2s+)V 4 2π) 3 k < k r F 2π) 3 3 πk3 F = N. ε r < ε F,

7 3.4. GROUND STATE 69 Here, k F = p F h is the Fermi wave number. Since n = N V = 2s+ 6π 2 k 3 F, k F = 6π 2 n 2s+ ) /3 so that we can get the following expression for the Fermi energy: 3.4 Ground state ) ε F = h2 kf 2 2m = h2 6π 2 2/3 n. 2m 2s+ At T =, the system is in its ground state, with the internal energy U given by U = k <k F ǫ k = Using the expression for total particle number N, = kf V2s+) dk 4πk 2 ) h2 k 2 2π) 3 2m V h 2 ) 4π 2π) 3 2m 5 k5 F. N = V2s+) 4π 2π) 3 3 k3 F, one can show that the internal energy per particle at absolute zero is given as U N = 3 h 2 kf 2 5 2m = 3 5 ǫ F independent of s). Since PV = 2 3 U, one can derive now an expression for the zero-point pressure P : P P T= = 2 5 nǫ F. This zero-point pressure arises from the fact that there must be moving particles at absolute zero since the zero-momentum state can hold only one particle of a given spin state. Let us use our model of a Fermi gas of electrons contained in a box in order to describe electrons in a metal. For this case, since the density of electrons in a metal is n 22 cm 3, we find a zero-point pressure to be equal to: P erg cm 3.

8 7 CHAPTER 3. IDEAL FERMI GAS 3.5 Fermi temperature At a finite temperature, the Fermi distribution function occupation number) < n r >= nε) has the shape as shown in Fig Such an evolution of nε) with increasing Figure 3.2: Fermi distribution function. temperature can be interpreted in terms of fermions within a layer beneath the Fermi surface being excited to a layer above and leaving holes beneath the Fermi surface Fig. 3.3). Figure 3.3: Momentum space. We define the Fermi temperature T F as ε F = k B T F. ForlowtemperaturesT << T F, thefermidistributiondeviatesfromthatatt = mainly in the neighborhood of ε F in a layer of thickness k B T, i.e., particles at energies of order k B T below the Fermi energy are excited to energies of order k B T above the Fermi energy. For T >> T F, the Fermi distribution approaches the Maxwell-Boltzmann distribution. For electrons in a metal, ε F 2eV T F 2 4 K, which means that at room temperatures the electrons are frozen below the Fermi level, except for a fraction T T F.5 of them. Since the average excitation energy per particle

9 3.6. LOW TEMPERATURE PROPERTIES 7 is k B T, the internal energy of the system is of order ) T Nk B T and the specific heat capacity C V, T F C V Nk B T T F, tends to zero so that the electronic specific heat contribution can be neglected at room temperature. T >> T F T << T F classical limit degenerate limit 3.6 Low temperature properties In section 3., we found that nλ 3 2π h 2 = f 3/2 z)2s+), with λ = k B Tm. 3.5) The low temperature limit, T << T F, at a fixed density corresponds to the condition nλ 3 >>. In the low temperature limit, z is large and one cannot use power series expansion for deriving an asymptotic formula for f 3/2 z). Instead, such a formula is obtained by the Sommerfeld expansion: f 3/2 z) 4 ] [lnz) 3 3/2 + π ) π 8 lnz We substitute eq. 3.6) into eq. 3.5) keeping only the first term: ) 3 π lnz ) nλ 3 2/3 = µ 4 2s+) k B T = ε F k B T = T F T, which gives the correct limiting value for the chemical potential ε F. Including also the next order in the expression 3.6), we get nλ 3 4 2s+) 3 π [lnz) 3/2 + π2 8 lnz ],

10 72 CHAPTER 3. IDEAL FERMI GAS which is then rewritten as lnz) 3/2 3 π 4 nλ 3 2s+ π2 8 lnz. We now substitute lnz on the right-hand side of this equation by lnz ) = T F /T: ) 3/2 ) ] 2 lnz) 3/2 TF [ π2 T T 8 T F [ lnz T ) ] 2 F π2 T, 3.7) T 2 T F where we used x) 2/3 2 3 x+... From 3.7) follows the asymptotic formula for the chemical potential: µ ε F [ π2 T 2 T F ) 2 ]. Figure 3.4: Chemical potential of the ideal Fewrmi gas at a fixed density. In Fig. 3.4, the Maxwell-Boltzmann curve refers to the high-temperature limit ) nλ 3 µ = k B T ln. 2s+ Now, we can calculate the internal energy U from U = k ε k n k = V 2π) 3 4π h 2 2m 2s+) dk k 4 n k

11 3.7. PARTICLES AND HOLES 73 by partial integration: U = V 32s+)4π h2 2π) 2m = = V 2π) 32s+)4π h2 2m β V 2π 2 m 2β h4 2s+) dk k5 5 dk k5 5 n k k z e βε ε [z e βε +] 2 k dk k6 e βε µ) [e βε µ) +] 2, where ε = h2 k 2 2m. At low temperatures, the integrand is peaked at k = k F. By expanding k 6 around this point and using the expression for µ obtained above, we get [ U = 3 ) 2 5 Nε F + 5π2 T +...], 2 T F [ P = 2 U 3V = 2 ) 2 5 mε F + 5π2 T +...] 2 and the heat capacity with C V Nk B = π2 2 T T F +..., N =< ˆN >. T F 3.7 Particles and holes We can define that the absence of a fermion of energy ε, momentum p and charge e corresponds to the presence of a hole with

12 74 CHAPTER 3. IDEAL FERMI GAS energy = ε momentum = p charge = e The average occupation number for a fermion is and for a hole with n k = n k = e βε k µ) + e βµ ε k) +, ε k = k2 h 2 2m. The concept of a hole is useful only at low temperatures T << T F, when there are few holes below the Fermi surface. When T >> T F, the Fermi surface disappears and the system approaches the Maxwell-Boltzmann distribution function. Only holes and excited particles participate in the conduction; the rest of fermions remain inactive beneath the Fermi surface.