The properties of an ideal Fermi gas are strongly determined by the Pauli principle. We shall consider the limit:

Transcription

1 Chapter 13 Ideal Fermi gas The properties of an ideal Fermi gas are strongly determined by the Pauli principle. We shall consider the limit: k B T µ, βµ 1, 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 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 three 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 ). Spin-independent Hamiltonians. We will consider only spin independent Hamiltonian operator of the type Ĥ = ɛ k c k c k + d 3 r r) c rc r, k where the first and the second terms are respectively the kinetic and th potential energy. The summation over the states r whenever it has to be performed) can then be reduced to the summation over states with different wavevector k p = hk):... 2s + 1) r k where the summation over the spin quantum number m s = s, s + 1,..., s has been taken into account by the prefactor 2s + 1)...., 159

2 16 CHAPTER 13. IDEAL FERMI GAS Wavefunctions in a box. We assume that the electrons are in a volume defined by a cube with sides L x, L y, L z and volume = L x L y L z. For the one-particle wavefunction r k = ψ k r) = 1 e ik r we use periodicity condition, here along the x-direction, at the cube s walls, e ikxx = e ikxx+ikxlx, which is then translated into a condition for the allowed k-values: e ikxlx = e i2πnx, k x = 2π L x n x, n x Z. Analogously for the y- and for the z direction. Summation over wavevectors. Each state has in k-space an average volume of Δk = 2π)3 L x L y L z = 2π) ) For large we can then replace the sum r over all quantum number by 2s + 1) 1 d 3 k = 2s + 1) d 3 k Δk 2π) 3 r = 2s + 1) d 3 p, 13.2) h 3 where k = p/ h has been used. The factor 1 1 for N particles) introduced ad hoc in classical statistical physics in Sect. 8.2 appears naturally in quan- h h 3N tum statistical physics. It is a direct consequence of the fact that particles correspond to wavefunctions Grand canonical potential We consider now the expression 12.36) for the fermionic grand canonical potential ΩT,, µ) that we derived in Sect. 12.5, ΩT,, µ) = k B T r ln [ 1 + e βɛr µ)].

3 13.1. EQUATION OF STATE 161 Using the substitution 13.2) and d 3 k = 4π k 2 dk we write the grand canonical potential as [ ] β ΩT,, µ) = 2s + 1) 2π) 4π dk k 2 ln 1 + z e β h2 k 2 /2m), 13.3) 3 where we used the usual expressions z = e βµ, ɛ r ɛ k = h2 k 2 for the fugacity z and for the one-particle dispersion and used an explicit expression for the one-particle energies for free electrons ɛ k. Dimensionless variables. Expression 13.3) is transformed further by introducing with 2m β x = hk 2m, k2 dk = ) 3/2 2m β h 2 x 2 dx a dimensionless variable x. One obtains β ΩT,, µ) = 2s + 1) 4 ) 3/2 m π 2πβ h 2 x 2 dx ln 1 + z e x2). De Broglie wavelengths. By making use of the definition of the thermal de Broglie wavelength λ, 2πβ h 2 λ = m, we then get β ΩT,, µ) = 2s + 1) λ 3 4 π dx x 2 ln 1 + z e x2). 13.4) Term by term integration. We use the Taylor expansion of the logarithm, ln1 + y) = 1) n=1 n+1 yn n, in order to evaluate the integral x 2 dx ln 1 + z e x2) = = = 1) n=1 1) n=1 1) n=1 n+1 zn n n+1 zn n n+1 zn n d dn d dn dx x 2 e nx2 1 2 dx e nx2 ) π 1 n )

4 162 CHAPTER 13. IDEAL FERMI GAS term by term. The result is x 2 dx ln 1 + z e x2) = π 4 1) n+1 zn. n 5/2 n=1 Grand canonical potential. Defining we obtain f 5/2 z) = 1) n+1 zn n = 4 5/2 π n=1 for the grand canonical potential for an ideal Fermi gas. dx x 2 ln 1 + z e x2) 13.5) β ΩT,, µ) = 2s + 1 λ 3 f 5/2 z) 13.6) Pressure. Our result 13.6) reduces with Ω = P to P k B T = 2s + 1 2π h 2 f λ 3 5/2 z), λ = k B T m, z = eµ/k BT ), 13.7) which yields the pressure P = P T, µ). Density. With we find, compare Eq. 1.14), Ω = k B T ln Z = P ˆN = z ) z ln Z T, P k B T = ln Z ) P = z z k B T T, 13.8) for the number of particles N. The density nt, µ) = ˆN / is then given by n = ˆN = 2s + 1 λ 3 T ) f 3/2z). 13.9) where we have defined f 3/2 z) = z d dz f 5/2z) = 1) n+1 zn 13.1) n 3/2 n=1 Thermal equation of state. As a matter of principle one could solve 13.9) for the fugacity z = zt, n), which could then be used to substitute the fugacity in 13.7) for T and n, yielding such the thermal equation of state. This procedure can however not be performed in closed form.

5 13.2. CLASSICAL LIMIT 163 Rewriting the particle density. The function f 3/2 z) entering the expression 13.9) for the particle density may be cast into a different form. Helping is here the result 13.7) for P T, µ): P k B T = 2s + 1 f λ 3 5/2 z), f 5/2 z) = λ3 ln Z 2s + 1, P k B T = ln Z, which leads to With and λ = f 3/2 z) = z d dz f 5/2z) = z d dz d dz = d dβ 2πβ h 2 )/m we then find ) dβ dz = 1 d µz dβ, λ 3 2s + 1 ) ln Z β = ln z µ 13.11) µ 2s + 1)f 3/2 z) = d λ 3 ln Z ) = 3λ3 d ln Z + λ3 dβ 2β dβ ln Z. For the particle density 13.9) we finally obtain n = 2s + 1 λ 3 T ) f 3/2z), µn = 3 2 P + d ln Z, n = N, 13.12) dβ where we have used ln Z/β = P. Caloric equation of state. The expression for µn = µn in 13.12) leads with to the caloric equation of state U = d ln Z + µ ˆN, Z dβ = r e βɛr µ). U = 3 2 P ) The equation U = 3P /2 is also valid for the classical ideal gas, as discussed in Sect. 8.2, but it is not anymore valid for relativistic fermions Classical limit Starting from the general formulas 13.7) for P T, µ) and 13.9) for nt, µ), we first investigate the classical limit i.e. the non-degenerate Fermi gas), which corresponds, as discussed in Chap. 11, to nλ 3 1, z = e βµ 1.

6 164 CHAPTER 13. IDEAL FERMI GAS Under this condition, the Fermi-Dirac distribution function reduces to the Maxwell- Boltzmann distribution function: ˆn r = 1 z 1 e βɛr + 1 ze βɛr. Expansion in the fugacity. For a small fugacity z we may retain in the series expansion for f 5/2 z) and f 3/2 z), compare 13.5) and 13.1), the first terms: f 5/2 z) z z2 βp λ 3 2s + 1)z ) 1 z 2 5/2 2 5/2 f 3/2 z) z z2 nλ 3 2s + 1)z ) 13.14) 1 z 2 3/2 2 3/2 where we have used 13.7) and 13.9) respectively. High-temperature limit. The expression for nλ 3 in 13.14) reduces in lowest approximation to z ) nλ3 2s + 1, nλ ) The number of particles nλ 3 in the volume spanned by the Broglie wavelength λ 1/ T is hence small are small. This is the case at elevated temperatures. Fugacity expansion. Expanding in the fugacity z = expβµ) z nλ3 1 } 2s {{ + 1 } 1 z2 z ) 3/2 1 z ) 2, 3/2 z1) z ) 1 + z ) 2 3/2), z ) where 1/1 x) 1 + x for x 1 was used. The equation of state 13.14) for the pressure, namely βp λ 3 2s + 1)z z 2 2 5/2 ), is then βp λ 3 2s + 1) [ z ) 1 + z ) 2 3/2) 2 5/2 z ) ) 2] ) = nλ 3 5/2 nλ , 2s + 1 when = = is used. Altogether we then find P = ˆN k B T 1 + nλ s + 1) ) ) 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.

7 13.3. DEGENERATED FERMI GAS Degenerated Fermi gas In the low temperature limit, T, the Fermi distribution function behaves like a step function: { 1 T if ɛk > µ n k = e βɛ k µ) if ɛ k < µ i.e., lim T n k = θµ ɛ k ). Fermi energy. This means that all the states with energy below the Fermi energy ɛ F, are occupied and all those above are empty. ɛ F = µn, T = ), Fermi sphere. In momentum space the occupied states lie within the Fermi sphere of radius p F. The system is then deep in the quantum regime. The Fermi energy is then be determined by the condition that the the Fermi sphere contains the correct number of states: N = states r with ɛ r < ɛ F 1, which can be written for the case of free fermions, and with 13.1), d 3 k/δk 3 = [/2π) 3 ]d 3 k, as N = 2s + 1) 2π) 3 k < k F d 3 k = 2s + 1) 2π) πk3 F ) Here, k F = p F h The Fermi energy is then is the Fermi wave number. We have n = N = 2s + 1 6π 2 k 3 F, k F = ) 6π 2 1/3 n. 2s + 1 ɛ F ) = h2 kf 2 2m, ɛ F = h2 6π 2 2/3 n. 2m 2s + 1

8 166 CHAPTER 13. IDEAL FERMI GAS Ground state properties At T =, the system is in its ground state, with the internal energy U given by U = 2s + 1) kf ɛ k = dk 4πk 2 ) h2 k 2 2π) 3 2m k <k F 2s + 1) h 2 ) 4π = 2π) 3 2m 5 k5 F. Using the expression for total particle number N, for kf 5 = k3 F k2 F, one obtains N = 2s + 1) 4π 2π) 3 3 k3 F, U N = 3 h 2 kf 2 5 2m, U N = 3 5 ɛ F independent of s). for internal energy per particle at absolute zero. Pressure. Since P = 2U/3, we obtain now an expression for the zero-point pressure P : P P T = = 2 5 nɛ F. The zero-point pressure arises from the fact that fermionic particles move even at absolute zero. This is because the zero-momentum state can hold only one particle of a given spin state. Taking a Fermi energy of typically ɛ F 1 e = J and an electron density of n m 3 we find a zero-point pressure of P J m bar, where we have used that 1 P = 1 J/m 3 = 1 5 bar Fermi temperature At low but a finite temperature, the Fermi distribution function ˆn r = nɛ) for the occupation number smooths out around the Fermi energy.

9 13.3. DEGENERATED FERMI GAS 167 Such an evolution of nɛ) with increasing temperature is due to the excitation of fermions within a layer beneath the Fermi surface to a layer above. Holes are left beneath the Fermi surface. Fermi temperature. We define the Fermi temperature T F as ɛ F = k B T F. T T F For low temperatures T T F, the Fermi distribution deviates from that at T = mainly in the neighborhood of ɛ F in a layer of thickness k B T. 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. T T F For T T F, the Fermi distribution approaches the Maxwell-Boltzmann distribution. The quantum nature of the constituent particles becomes irrelevant. Frozen vs. active electrons. The typical magnitude, ɛ F 2 e, T F K, of the Fermi temperature in metals implies that room temperature electrons are frozen mostly below the Fermi level. Only a fraction of the order of T T F.15 of the electrons contributes to thermodynamic properties involving excited states. Particles and holes. We can define that the absence of a fermion of energy ɛ, momentum p and charge e corresponds to the presence of a hole with energy = ɛ momentum = p charge = e The concept of a hole is useful only at low temperatures T T F, when there are few holes below the Fermi surface. The Fermi surface disappears when T T F, with the system approaching the Maxwell-Boltzmann distribution function. Specific heat. Since the average excitation energy per particle is k B T, the internal energy of the system is of order ) T U U + Nk B T, 13.18) T F

10 168 CHAPTER 13. IDEAL FERMI GAS where U is the ground-state energy. The specific heat capacity C is then of the order of C Nk B T T F, C = ) U T ) The electronic contribution to the specific heat vanishes linearly with T. The roomtemperature contribution of phonons lattice vibrations) to C is therefore in general dominant Low temperature expansion In this section we will derive the scaling relations 13.18) and 13.19) together with the respective prefactors Density of states We will work from now on with density of state per volume DE) = ΩE)/, which is defined as the derivative of the integrated phase space per volume, φe) = ΦE)/ : DE) = φe) E, φe) = 1 ɛ k E d 3 k Δk 3, The spin degeneracy factor 2s + 1 will be added further below. Δk3 = 2π)3 Fermi sphere. With the phase space being isotropic we may write the volume of the Fermi sphere as. Δk 3 ) φe) = 4 3 πk3 E = 4π 3 2mE h 2 ) 3/2, E = h2 k 2 E 2m. Introducing the spin degeneracy factor 2s + 1 we then obtain DE) = { A = 2s + 1 2π) 2 A E if E, otherwise, 13.2) ) 3/2 2m h 2 Since we did not use any special properties of a fermionic system, this expression is also valid for bosons. For both types of systems, DE) shows a E dependence. Energy and particle density. The energy density U/ is given by UT, µ) = + de E ne)de), ne) = 1 e βe µ) + 1, 13.21)

11 13.4. LOW TEMPERATURE EXPANSION 169 where ne) is the Fermi distribution as a function of the energy. The analogous expression for the particle density is nt, µ) = ˆN = + de ne)de) ) For a bosonic system one substitutes the Boson distribution function 1/e βe µ) 1) for ne) Sommerfeld expansion We are interested in the thermodynamic properties of a fermionic system at small but finite temperature, viz at a low temperature expansion of expectation values like 13.21) and 13.22). Sommerfeld expansion. We denote with HE) a function depending exclusively on the one-particle energy E. We will show that H = dehe)ne) µ dehe) + π2 6 k BT ) 2 H µ) 13.23) holds terms or order T 4 or higher. The first term on the r.h.s. of 13.23) survives when T and ne) θµ E). It represents the ground-state expectation value. The second term results from expanding both HE) and ne) around the chemical potential µ. Partial integration. definition KE) = We start the derivation of the Sommerfeld expansion with the E de HE ), which allows as to perform the partial integration H = de dke) de ne) = HE) = dke) de, de KE) dne) ) ) de For the integration we have used lim E ne) =, namely that the probability to find particles at elevated energies E falls of exponentially. We have also assumed that lim E HE) =. Taylor expansion. Substituting the first two terms of the Taylor expansion KE) = Kµ) + E µ)k µ) + E µ)2 K µ) + OE µ) 3 ) 2

12 17 CHAPTER 13. IDEAL FERMI GAS of KE) around the chemical potential µ into 13.24) leads to H de [Kµ) + E µ)k E ] µ)2 µ) + K µ) dne) ) 2 de ) Ground state contribution. The first term in 13.25) is Kµ) de dne) ) = Kµ) [ n) n ) ] = de viz the T = value of H. Symmetry cancellation. The second term in 13.25) vanishes because dn de = is symmetric in E µ. µ dehe), βe βe µ) [e βe µ) + 1] 2 = βe βe µ) [1 + e βe µ) ] 2, ne) = 1 e βe µ) + 1, Finite temperature correction. The third term in 13.25) yields the first non-trivial correction K E µ)2 µ) de dne) ) = π2 2 de 6 k BT ) 2 H µ), where the scaling with k B T ) 2 = 1/β 2 follows from a transformation to dimensionless variables y = βe µ). The factor π 2 /6 results form the final dimensionless integral. This concludes our derivation of the Sommerfeld expansion 13.23) Internal energy at low temperatures We start applying the Sommerfeld expansion 13.23) to the particle density 13.22): nt, µ) = = = + µ ɛf de DE)nE) 13.26) de DE) + π2 6 k BT ) 2 D µ) { µ de DE) + de DE) ɛ } F {{} µ ɛ F )Dɛ F ) } + π2 6 k BT ) 2 D µ), where we have taken into account that µ = µt ) may be different from the Fermi energy ɛ F = lim T µt ). We have also assumed that the density of states DE) is essentially constant around the Fermi energy. Constant particle density. The terms inside the bracket in 13.26) need to cancel if the particle density n is to be constant. The chemical potential µ varies hence as µ = ɛ F π2 6 k BT ) 2 D ɛ F ) Dɛ F ), µ = ɛ F π2 k B T ) ) 12 ɛ F as a function of temperature.

13 13.4. LOW TEMPERATURE EXPANSION 171 We have approximated D µ) in 13.27) consistently by D ɛ F ). The scaling DE) E, as given by 13.2), leads to D /D = 1/2E). Internal energy. The energy density 13.21) evaluated with the Sommerfeld expansion is u = U/ = + = u + µ de E ne)de) ɛ F de E DE) + π2 6 k BT ) 2 d de EDE) ) E=µ } {{ } 3 2 Dɛ F ), where u = ɛ EDE)dE is the ground state energy. The substitution µ ɛ F performed for the argument of last term, for which we used DE) E, is correct to order T 2. The second term is µ de E DE) µ ɛ F )ɛ F Dɛ F ) = π2 k B T ) 2 ɛ F 12 when DE) E. One finds hence with 3/12 1/12 = 1/6 that Dɛ F ). U = u + π2 k B T ) 2 Dɛ F ) u u k B T ) 2 D ) 6 The internal energy increases quadratically with the temperature, being at the same time proprotional to the density of states D = Dɛ ) at the Fermi level. Specific heat. Assuming a constant density of states DE) Dɛ F ) close to the Fermi energy ɛ F we find c = C = 1 U T, c π2 k 2 3 B T ) Dɛ F ) 13.29) for the intensive specific heat. A trademark of a fermionic gas is that c temperature. The heat capacity per volume saturates however for T, where it becomes identical with the ideal gas value C / = 3nk B /2 derived in Sec is linear in the

14 172 CHAPTER 13. IDEAL FERMI GAS