Applications of Fermi-Dirac statistics

Transcription

1 Statistical Mechanics Phys504 Fall 2006 Lecture #12 Professor Anthony J. Leggett Department of Physics, UIUC Applications of Fermi-Dirac statistics 1. Electrons in metals The electron in metals are not much lie a system of free fermions: they interact strongly with the ions and with one another via the Coulomb interaction. Nevertheless, considerable progress has been achieved in the theory of metals by starting with a free-electron model and gradually adding the complications. In increasing order of sophistication, we have the Sommerfeld model, the Bloch model, and the Landau-Silin (Fermi-liquid) model. I discuss these briefly in turn, assuming that the metal in question remains normal (i.e., not superconducting) in the limit T 0. Sommerfeld model This is the simplest possible picture: it just regards the N conduction electrons (of spin 1/2) of the metal as moving freely in a volume V, without interaction, so that the states are characterized by wave vector and spin σ, Ψ σ = V 1/2 exp i r σ >, and the single-particle spectrum is just ɛ(p) = p 2 /2m h 2 2 /2m (1) where the allowed values of are 2πn x /L x, 2πn y /L y, 2πn z /L z, n i integral, V = L x L y L z. According to the results of l.11, the Fermi energy is then given by (11), i.e., (since g s = 2 in this case) ɛ F = (3π 2 n) 2/3 h 2 /2m h 2 F 2 /2m (2) and all states with energy < ɛ F are occupied (with one spin- and one spin- electron). In wave vector () space, those states occupy the interior of a sphere of radius (3π 2 n) 1/2 F, called the Fermi sea ; the surface of this sphere is called the Fermi surface. For a typical metal F is 1Å 1, so ɛ F a few ev. The thermodynamics of this system is exactly as described in l.11. However, it is very important to note that the Fermi temperature T F ɛ F / s is almost invariably much greater than the melting temperature of the metal in question, so that in the solid state (and actually usually also in the liquid state of a metal) the electrons are always strongly degenerate (T T F ). Thus we can always use (in this model) the formulae of l.11, in particular eqns. (30), 1

2 (31) and (40) 1 with ρ(ɛ F ) given by 3n/2ɛ F. The Sommerfeld model gives a surprisingly good qualitative account of the static properties 2 of most simple metals, but the experimental numbers often differ from the predicted ones. To an extent this discrepancy is remedied by the Bloch model In this model one imagines the ionic cores to be fixed and to provide a static periodic potential for the conduction electrons; apart from this the latter are regarded as noninteracting. The one-electron energy eigenfunctions are now no longer simple plane waves, but have the general form Ψ n (r) = exp(i r)u n (r) (3) where n is a band index and runs over the so-called first Brillouin zone of the crystal lattice in question; the volume of this zone is (2π) 3 /Ω where Ω is the volume of a lattice cell, but it may have various shapes depending on the lattice structure. The allowed values of within the FBZ is the same as in the Sommerfeld model (2π/L x, etc.), so the number of allowed values is just the number of unit cells in the crystal lattice. The energy is a function of both the wave vector and the band index n, giving a band structure : ɛ = ɛ(, n) (4) Note that even within a single band, ɛ is in general a function of the direction of as well as its magnitude. Just as in the Sommerfeld model, the electrons are mutually noninteracting, so the standard FD statistics are applicable and the single-electron states are occupied with 2 electrons each (one for each spin) up to the Fermi energy ɛ F. However, the Fermi surface so generated (the locus of ɛ(, n) = ɛ F ) is no longer the surface of a simple sphere; indeed in general it may intersect several energy bands. I will confine myself to simplicity to the case when the FS lies entirely within a single band and is simply connected i.e., it is a single surface enclosing N/2 allowed values of (the more general case simply complicates the notation without introducing any qualitatively new aspects). The Fermi sea thus occupies a fraction p/2 of the FBZ, where p is the number of conduction electrons per primitive unit cell of the crystal lattice. The DOS at the FS, ρ(ɛ F ), is now no longer given by the Sommerfeldmodel formula 3n/2ɛ F (though this is usually a good order-of-magnitude estimate); in fact, 1 Eqn. (40), of course, gives only the electronic contribution to the specific heat; in addition there will be a contribution from the lattice vibrations (phonons). 2 And also of the transport properties (cf. second half of course). 2

3 by considering each small piece of the FS separately and noting that the number of added states is 3 2V/(2π) 3 S and ɛ = ɛ d we find for the DOS (per unit volume) ρ(ɛ F ) = 2 (2π) 3 ds ɛ() (5) (where the integral runs over the FS and ɛ() is ɛ n for the relevant (conduction) band). Once we have the expression for ρ(ɛ F ) we can use all the results of the l.11, in particular eqns. (30), (31) and (40). However, it is worth noting one qualitative difference from the Sommerfeld model: the quantity ρ (ɛ F ) is not necessarily positive (in particular if the FS is close to the top of the band) and as a result the first temperature corrections to the zerotemperature value of µ(t = 0)( ɛ F ) may be positive. (It is clear that this complication does not effect the formula for the specific heat, which involves ρ(ɛ F ) not ρ (ɛ F )). What about the temperature-dependence? Unless the band structure is very pathological, the order of magnitude of ɛ F is the same as in the Sommerfeld model, and thus the strong degeneracy condition T T F is satisfied at all temperatures up to melting. Hence, unless p is very close to 0 or 2 (in which case special care is necessary) the low-temperature expressions (30), (31), and (40) may be safely used at all temperatures. The Bloch model gives a remarably good semi-quantitative account of the behavior of many normal metals. This is at first sight surprising, since it ignores both the electronelectron interactions and the motion of the ions. A partial explanation is given by the Landau-Silin model This is an adaptation of Landau s Fermi-liquid theory (originally devised for liquid 3 He (see below) to metals. As it will be discussed in detail later in the course, I just describe the bare essentials here. It is argued that provided perturbation theory in the interactions starting from the Bloch picture converges, then the effect of the interactions is simply (1) to renormalize the band structure ɛ(, n), and (2) introduce a sort of molecular field analogous to the Weiss molecular field in the theory of ferromagnetism, which are generated by macroscopic polarizations. The most important such field is the long-range Coulomb one (generated by electric charge buildup), but in addition, there are short-range fields associated with spin and other types of polarization. As a result, the response to external fields is modified; for example, the expressiion (31) for the spin susceptibility is multiplied by a Stoner-lie factor (1 + Fo a ) 1, where Fo a parametrizes the relevant molecular field. However, a mere rise in temperature does not correspond to any ind of polarization, and hence the 3 The factor of 2 is for the spin. 3

4 specific heat is still given by eqn. (11.40), where, however, in the expression (12.5) for the DOS ρ(ε F ) we must use the renormalized band spectrum ε(, n). Generally speaing, the effect of this replacement is only a factor 1; however, there is one class of materials, the so-called heavy fermion metals, where the experimental specific heat can be of the order of 1,000 times the Sommerfeld value, and this enormous increase is usually attributed to the effect of interactions (rather than to the Bloch-model band structure). Correspondingly, it is usually believed that the estimate ρ(ɛ F ) 3n/2ɛ F still holds to an order of magnitude in those materials, so that T F is typically reduced by a factor 1, 000 to a few K. Thus, one would not expect the standard low-temperature formulae to remain valid above these temperatures, and indeed this seems to be the case experimentally. However, the theory of these materials is still quite controversial. If one excepts the class of heavy-fermion materials, the statement that the strong degeneracy condition T T F is always satisfied in metals generally remains valid at the Landau-Silin level. However, since this model tries to handle inter alia the electron-phonon interaction, it introduces a new energy (or temperature) scale associated with the phonons, the Debye energy ɛ D hω D (or Debye temperature T D ɛ D / B ) which is typically T F and is in the accessible regime (typically T D room temperature). The usual belief is that a model of this Landau-Silin type applies both for T T D, and for T T D, but with different parameters; the behavior in the crossover origin T T D is expected to be complicated. It should be stressed that the importance of the LS model of metals is primarily conceptual, in that it shows that the interactions mostly do not change the results calculated from the Bloch model qualitatively. In fact, it is rather difficult, in most metals, to detect unambiguously the effects specific to the LS model (such as that of the molecular fields), and in many cases these effects are quite small and not at all spectacular. 2. Liquid 3 He Liquid 3 He is a set of effectively structureless particles of (nuclear) spin 1/2, and thus forms a translationally invariant Fermi system. If one could neglect the interatomic interactions, it should be described as a free Fermi gas obeying simple FD statistics and thus by the formulae developed in l.11. However, the calculated Fermi temperature is a few K, comparable to the temperature of the experimental critical point, so the condition T T F is fulfilled only at the lowest temperatures. In fact, experiment shows that the liquid does behave qualitatively lie a degenerate Fermi gas at T 100 < mk, but there are substantial quantitative discrepancies (e.g. the spin susceptibility χ is something lie 20 times the value 4

5 predicted by the Fermi-gas model). This observation was the original impetus for Landau s Fermi-liquid theory, which is considerably simpler for 3 He than the LS version applicable to metals. In particular, because of the simple isotropic form of the ideal gas energy spectrum., the relevant effects on it of the interactions can be encapsulated in a simple replacement of the real 3 He mass or by an effective mass m*, so that the DOS at the FS, ρ(ɛ F ), is just (m /m)ρ free (ɛ F ). However, the molecular-field effects are much stronger in this system than in metals. [Brief discussion of nuclear matter] 3. Electrons in semiconductors. 4 If in a given crystal the total number of electrons per primitive unit cell of the crystal p is an even integer, then at zero temperature the occupied states may (though they need not) fill up exactly p/2 energy bands. In that case there will in general be a nonzero band gap call it E g to the bottom of the next band, and the Fermi energy lies within this gap (exactly where is, for the moment, unimportant), so we have the situation setched. The highest completely filled band is traditionally called the valence band and the lowest unoccupied one, the conduction band. Since there are no states at the Fermi energy, at T = 0 one can formally regard this situation as simply a special case of that treated in l.11 with ρ(ɛ F ) = 0; it is then clear that the comparability K, the Pauli spin susceptibility χ and the coefficient of the linear term in the specific heat c v all vanish. Indeed, the electrons in the valence band are completely inert and cannot respond to any ind of wea perturbation, and this inter alia means that at T = 0 the system is a perfect electrical insulator in wea fields. At nonzero temperature things are more interesting. It is clear that the minimum energy necessary to excite an electron from the valence band into the conduction band is the band gap E g, so one would thin prima facie that the number of such electrons which are normally excited at a temperature T would be proportional to exp E g / B T. (Actually, for a reason which will appear below, it is e Eg/2BT, but this will do for an order-of-magnitude estimate). Let us consider the situation at say room temperature ( B T = 0.025eV ). If E g is very large (say 5 ev as in diamond) then the exponent E g /2 p T is 100, and even for a 1 cm 2 crystal, there is little chance of even a single electron being excited. If, on the other hand, E g is < 1 ev as in pure Ge or Si, then the exponent, though still large, is small enough that an appreciable number of electrons are excited at room temperature and are freed for electrical 4 KIUH ch. 4: note differences in notation. 5

6 conduction, etc. In this latter case, the crystal in question is called a semiconductor (the distinction between a semiconductor and an insulator is only quantitative, not qualitative). Let s now try and mae these considerations more quantitative. To do so we need to now something about the energy spectrum ɛ() in both the valence and the conduction bands. Assuming for notational simplicity that the maximum of the former and minimum of the latter both occur at the BZ center 5 = 0, we can mae a Taylor expansion around this point and thus write (taing for the moment the zero of energy to lie in the middle of the band gap) ɛ v () = E g A v ij i j +... (6a) i,j=1 ɛ c ( = E g + 3 i,j=1 A c ij i j +... (6b) where the quantities A v, A c are fixed tensors, which in general define a tensor effective mass for the valence and conduction bands respectively. For simplicity I will assume that these effective masses are both isotropic (i.e. A v ij, A c ij δij), in which case we can rewrite (6a-b) in the simpler form ɛ v () = 1 2 E g h o( 4 ) 2m v ɛ c () = 1 2 E g + h o( 4 ) 2m c (7a) (7b) The quantities m v and m c may each be either larger or smaller than the electron mass m and are in general unequal; however, I will assume they do not differ in order of magnitude. Under normal conditions B T is small compared not only with E g but with the width of the V and C bands (which generally are > E g ) so the higher-order terms in can be safely neglected. We now want to consider the thermal distribution at nonzero temperature. In the following I will assume that the condition B T E g /2 is always satisfied. Quite generally, the average occupation of a (V or C) state with wave vector and spin σ(= ±1) is given by the standard FD formula n (v,c) 5 so that the semiconductor is of direct-gap type. = (exp β(ɛ v,c µ) + 1) 1 (8) 6

7 where, as usual, µ(t ) is the chemical potential, which we will have to fix self-consistently from the condition that v,c σ n σ = N where N is the total number of electrons (twice the number of primitive unit cells) which occupied the valence band at T = 0. Let us mae the assumption (which will be subsequently checed to be self-consistent) that the chemical potential µ(t ) is never far from the center of the band gap. Thus, the quantity β(ɛ (c) Maxwell-Boltzmann form: µ) is always > E g /2 B T, and thus n (c) n (c) = exp β(ɛ (c) On the other hand, in the valence band β(ɛ (v) and, in fact, we can write can be well approximated by its µ) (9) µ) < E g /2 B T, so n (v) is always close to 1 1 n (v) = exp β(ɛ (v) µ) 1. (10) It is, therefore, very convenient to introduce the concept of a hole in the V band. By definition, a hole in the (electron) state (, σ) in the V band has momentum h, spin σ, charge e( + e ) and, most importantly in the present context, energy ɛ (v) (> 0). Also, the chemical potential for a hole is the energy necessary to remove an electron, not to add it, and is therefore µ e where µ e is the electron chemical potential. Holes obey FD statistics just lie electrons. With these prescriptions one has the exact relation (since ɛ σ = ɛ, σ ) that the equilibrium number of holes of momentum and spin σ is given by n (hole) = (exp β(ɛ (hole) σ µ (hole) ) + 1) 1 (exp β(ɛ σ µ) + 1) 1 = 1 n (v) (11) so that everything is consistent. In the limit B T E g /2 holes obey a Maxwell-Boltzmann distribution (cf. eqn. (10)) n (hole) = exp β(ɛ (hole) µ (hole) ) (12) The quantity ɛ (hole) is (taing as above the zero of energy in the center of the gap) ɛ (hole) = E g + h 2 2 /2m v (13) We can now use these formulae to calculate the total number of electrons per unit volume in the conduction band (traditionally denoted n, for negative carriers) and the total number of holes in the valence band (p. for positive carriers): in the present case n and p must evidently be equal, and this will allow us to fix the chemical potential µ(t ). By a calculation 7

8 exactly analogous to that done in l.6 for the free monatomic gas, and remembering to insert a factor of 2 for the spin states, we have n = 2(λ (c) T ) 3 exp β( 1 2 E g µ) (14) p = 2(λ (v) T ) 3 exp β( 1 2 E g µ (hole) ) (15) where the quantity λ (c,v) (2π h 2 /m (c,v) / B T ) 1/2 (16) is the thermal de Broglie wavelength for the c(v) band. Remembering that µ (hole) = µ and setting n = p, we have n = p = (np) 1/2 = 2(λ (c) T λ (v) T ) 3 exp 1 2 βe g 2((m c m v ) 1/2 B T/2π h 2 ) 3/2 exp E g /2 B T. Also, comparison of (14) and (15) gives exp 2βµ = (λ (c) T /λ (v) T ) 3 /2, i.e. (17) µ(t ) = 3 4 BT ln (m v /m c ) (18) Thus, at T = 0 µ lies exactly in the center of the gap, and provided m v and m c do not differ in order of magnitude and B T E g it never migrates far from this position, so that the approximations we made in deriving MB statistics for the electrons and holes are self-consistent. Note that e.g. for m c = m v = m and E g = 0.7eV we have n = p m 3, about 1 part in of all the electrons in the V band, but still enough to give an appreciable electrical conduction. The above results apply to the case of intrinsic semiconductors (no impurities). Analogous results can be obtained when there are a certain number of donor and/or acceptor impurities which at T = 0 are electrically neutral (e.g. P in Si). Suppose for example we have N D donor impurities per unit volume, each with a single level E D, below the C band minimum, which can bind one electron only 6 For the moment we allow only a single spin state to be associated with this level. Let us assume that E D E g and B T E g (but we mae no assumption about the ratio B T/E D ). Under these conditions it will turn out that the number of holes excited in the valence band is negligible, so that all electrons in the conduction band must have come from donor levels. Thus if n D is the number of electrons in the donor levels, we must have n = N D n D (19) 6 This is a physically reasonable assumption when the onsite Coulomb repulsion U is greater than E D, as is usually the case. 8

9 It is convenient now to measure energies, including the chemical potential µ from the bottom of the conduction band (so the donor level lies at E D ). We then have (since the donor levels are distinguishable) n = N D n D (exp β( E D µ) + 1) 1 (20) and for the number of conduction electrons in states (, σ) n (c) σ = (exp β(ɛ(c) µ) + 1) 1 (21) Let us now suppose that B T is not only E g but also E D. Thus it will turn out that it is self-consistent to treat the conduction electrons as nondegenerate, i.e. to approximate eqn. (21) by its Maxwell-Boltzmann limit and, performing the usual sum over, we get Rewriting (20) in the form we find n = 2(λ (c) T ) 2 exp βµ N c exp βµ (22) n D N D n D = exp β(µ + E D ) (23) n D (n/n c ) = exp βe D ( ) (24) N D n D Eliminating n D from (19) and (24), we finally find for n the implicit equation If B T E D as assumed, this has the approximate solution n 2 = N c (N D n) exp βe D (25) n = (N c N D ) 1/2 exp βe D /2 (26) so that provided N D is not very large compared to N c, the nondegeneracy condition n N C is fulfilled. Note that under these conditions µ lies approximately at E D /2 as we might intuitively expect. If the condition B T E D is not fulfilled, then it is not a priori obvious that the nondegeneracy condition holds and thus that we can use the above results. However, we can see that since the maximum possible number of electrons in the conduction band is N D, a sufficient condition for it to hold is N c (T ) N D (i.e. much less than one impurity per λ 3 T ). If this condition is not fulfilled, and B T E > D then the effects of degeneracy in the conduction band cannot be neglected and the problem is more complicated. 9

10 In the real-life problem of an impurity semiconductor there is a complication we have neglected: namely, although we allow only one electron per donor, this electron can actually be in either of 2 spin states. Because of the double occupancy prohibition, this complication cannot be handled simply by replacing N D by 2N D. Rather, we now have for each donor 2 occupied levels and 1 unoccupied one, so the single-level grand PF is Z (1) G = i e β(e i µn i ) = 2e β(e D+µ) + 1 (27) so that the probability of occupation is now ( 1 exp(ε 2 D µ) + 1) 1. The effect is to multiply the RHS of eqn. (24), and hence of (25), by 1/2. 4. The degenerate Fermi gas in an external potential: Thomas-Fermi approximation. Consider a gas of noninteracting Fermi particles in some external potential V (r) (a typical example might be a single hyperfine species of 40 K atoms in a laser or magnetic trap). What can we say about the density distribution? In the classical limit, i.e. where the condition B T n 2/3 h 2 /m is satisfied everywhere, this is given by the classical formula. n(r) = Zq 1 exp βv (r) (Z q dr exp βv (r)) (28) What happens at low temperatures? The fundamental principle we have to use is that where particles of a given type can be exchanged between two different volumes 1 and 2 which are at the same temperature, then the chemical potentials must be equal: µ 1 = µ 2. Suppose that the two regions are held at a different potential, V 2 V 1 ; then the chemical potential µ j is the intrinsic contribution µ 2 (n j, T ) which is a function only of n j and T, plus V j, and thus we have µ(n 1, T ) + V 1 = µ(n 2, T ) + V 2 (29) and this, together with the conservation of the total particle number N N 1 + N 2, is sufficient to fix n 1 and n 2. It is now plausible that when a system of particles (Fermi or other) moves in a continuously varying potential, V (r), the continuum version of eqn. (21) should apply; that is, µ {n(r) : T } + V (r) = const. (30) 10

11 This is essentially the Thomas-Fermi approximation (it was originally applied to fermions (electrons), but is more generally applicable). Note that for a classical gas we have according to the results of l. 6 µ(n, T ) = T ln(g s /nλ 3 T ) (31) so with an appropriate choice of the constant (cf. below) (30) agrees with (28). What is the condition for the validity of the TF approximation, eqn. (27)? A sufficient condition is certainly that the relative change of V (r) over the thermal de Broglie wavelength is small. However, it turns out that this condition (which will certainly fail at sufficiently low temperatures) is not necessary; it is enough that δv/v is small over a distance of the order of the interparticle spacing. At intermediate temperatures we do not usually have an analytic expression for µ as a function of n, so it is necessary to resort to numerical computation. However, in the strongly degenerate regime things simplify again, since now µ is now to a good approximation the Fermi energy ɛ F, which is given (in 3D) by formula (11.27), namely ɛ F = ( h 2 /2m)(6πn/g s ) 2/3 An 2/3 (A ( h 2 /2m)(6π 2 /g s ) 2/3 ) (32) where g s is the spin degeneracy. Thus (30) reduces to An 2/3 (r) + V (r) = const. (33) The constant must be fixed from the consideration that the integral of n(r) over all space is the total number of particles N. that As an example, consider a 3D isotropic harmonic trap, where V (r) = 1 2 mω2 o r 2. We see and the constant λ is given by n(r) = ( (λ 1 2 mω2 or 2 )/A) 3/2 (34) λ = const N 1/3 hω o (35) so the extent r o of the level is N 1/3 the zero-point length ( h/mω o ) 1/2. This agrees with an argument based on phase-space considerations (see Problem). 11