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Home , Condensed matter physics, Density of states, Fermi liquid theory

Phys 769 Selected Topics in Condensed Summer 2010 Lecture 3: Fermi- theory

Lecturer: Anthony J. Leggett TA: Bill Coish

1 General considerations concerning condensed matter

(NB: Ultracold atomic gasses need separate discussion) Assume for simplicity a single atomic species. Then we have a collection of N (typically 1023) nuclei (denoted α,β,...) and (usually) ZN (denoted i,j,...) interacting ∼ via a Hamiltonian Hˆ . To a first approximation, Hˆ is the nonrelativistic limit of the full Dirac Hamiltonian, namely1

~2 ~2 1 e2 1 Hˆ = 2 2 + NR −2m ∇i − 2M ∇α 2 4πǫ r r α 0 i j Xi X Xij | − | 1 (Ze)2 1 1 Ze2 1 + . (1) 2 4πǫ0 Rα Rβ − 2 4πǫ0 ri Rα Xαβ | − | Xiα | − | For an isolated , the relevant scale is the Rydberg (R) – Z2R.

In addition, there are some relativistic effects which may need to be considered. Most important is the -orbit interaction: µ Hˆ = B σ (v V (r )) (2) SO − c2 i · i × ∇ i Xi

(µB is the Bohr magneton, vi is the velocity, and V (ri) is the electrostatic potential at 2 3 2 ri as obtained from HˆNR). In an isolated atom this term is o(α R) for H and o(Z α R) for a heavy atom (inner-shell electrons) (produces fine structure). The (-electron) magnetic dipole interaction is of the same order as HˆSO. The (electron-nucleus) hyperfine interaction is down relative to Hˆ by a factor µ /µ 10−3, and the nuclear dipole-dipole SO n B ∼ interaction by a factor (µ /µ )2 10−6. In addition to the above “intrinsic” terms, there n B ∼ may be terms due to external magnetic and electric fields; with currently available fields (E . 107 V/m, B . 60 T) the maximum value of ea E is 0.5meV (i.e. 10−4 R) and 0 ∼ ∼ the maximum value of µ B is a little larger, 4 10−4 R. B ∼ × 1For the moment, ignore externally applied fields

1 of Hˆ : ,P ,T (translation), SO(3) , SU(2) . Relativistic corrections NR T orb spin break the last two, but not , P , or T (but external fields do). T When combine to form a liquid or , the interatomic spacing is of the order of the atomic size. Then it is usually a good approximation to regard the closed-shell (“valence”)2 electrons as rigidly tied to nuclei (so, in particular, we can regard rare- atoms as “fixed units” even in the liquid/solid state). However, open-shell (“conduction”) electrons (Zc/atom) may be distributed over the whole space. Then, in principle, we can implement the Born-Oppenheimer approximation: that is, we solve the time-independent Schr¨odinger equation (TISE) for the conduction electrons for fixed ionic positions, then feed back the resultant into the ionic Hamiltonian. Since the time scale of ionic motion is much greater than that of the conduction electrons, this generally gives good results (but use caution for ). In solving the TISE for the conduction electrons, we must in principle use the Coulomb potential of the ions and require orthogonalization with core states; (“pseudo-potential” method). However, in practice it is often adequate to replace the core electrons by a modification of the Coulomb potential to some phenomenological potential U(r (possibly also spin-dependent).

Thus, if for the moment we ignore ionic motion, the new problem is that of NZc conduction electrons described to lowest order by the Hamiltonian3 ~2 2 ˆ ′ 2 1 e 1 HNR = i U(ri)+ (3) −2m ∇ − 2 4πǫ0 ri rj Xi Xi Xij | − | where U(r) is the phenomenological potential of the (supposed fixed) ionic cores; this may ˆ ′ be either periodic () or aperiodic (liquid or ). Note that in general HNR is not invariant under either T or SO(3)orb (but is still invariant under SU(2)spin, and more importantly, under P and ). The relativistic terms break the SU(2) invariance but (in T spin zero external field) still preserve P and . T Energy scales: Since U(r) is at least of the order of magnitude of the Coulomb potential of the ions, and the atomic size ( interatomic spacing) is determined by the competition ∼ of kinetic and potential energies, the order of magnitude of both the first two terms in Eq.

(3) is R (or perhaps better, ZcR). The last term might also be expected to be of the same order of magnitude; thus, prima facie, the conduction electrons in any (non-rare-gas) liquid or solid are strongly interacting (More technically, U V ǫ ). ∼ ∼ F 2I am using the terms “valence” and “conduction” somewhat differently from the standard ways, e.g., in physics. 3Strictly speaking, this ignores the possible effect of polarization of the core shells by the conduction 2 2 electrons. This may be handled phenomenologically by e → e /ǫc.

2 Let’s now take into account the possibility of ionic motion. Then we must include in the Hamiltonian an ionic kinetic energy, a “direct” ion-ion interaction (which will not in general be pure Coulomb, because of, e.g., exclusion-principle effects) and an interaction between the (displaced) ions and the conduction electrons. Most of the last can be eliminated by the Born-Oppenheimer technique in favor of an extra effective ion-ion interaction. Thus we ′ get extra terms in HNR of the form ~2 2 ′ H = + V (R , R , R ,...)+ H − (4) ion −2M ∇α ion α β γ ion el α ··· X αβγX where M is the ionic ( nuclear mass) and V (R , R , R ,...) is in general very ≃ ion α β γ complicated; however, a crucial point is that its order of magnitude will still be R or ∼ ZcR. The last term is any part of the ion-electron interaction which we have been unable to eliminate by the Born-Oppenheimer technique. Since the order of magnitude of Vion is the same as that of U or V in Eq. (3), while the mass in the kinetic-energy term is much larger, we see that the characteristic frequencies (vibrational energies) of the ionic system, which are (V ′′/M)1/2, are down by a factor (m/M)1/2 10−2 10−3 relative to the ∼ ∼ − characteristic electronic energies. This is a quite general conclusion, and independent of whether the system is crystalline, amorphous, or even liquid. (In particular, in a crystalline solid Debye energies are typically room , which is about 1/300 of R). ∼

2 Sommerfeld model

Free of spin 1/2 (electrons) moving in volume Ω. (No periodic/other potential, no interaction). Plane wave states are specified by k,σ: (periodic boundary conditions)

1 ik·r ψkσ = e σ ; σ = 1. (5) √Ω | i ± ~2k2 ǫ = (6) k 2m In thermal equilibrium at temperature T ( = 0): H 1 nkσ = (7) e(ǫk−µ)/kBT + 1 2 µ(T )= µ(0) + o((kBT ) /µ(0)). Hence at zero temperature

nk = θ(µ(0) ǫ ) put µ(0) ǫ . (8) σ − k ≡ F so fills sphere of radius

k = (3π2n)1/3 (n N/Ω) ǫ = ~2k2 /2m = (~2/2m)(3π2n)2/3 (9) F ≡ F F

3 T ǫ /k typically 104 105 K at all solid/liquid k T ǫ ⇒ F ≡ F B ∼ − ⇒ B ≪ F → µ(T ) µ(0), and all the “action” is close to the . ≃ The (DOS) (both spins) is dn/dǫ = 3n/2ǫ [N(0) 1 (dn/dǫ)], we also ≡ F ≡ 2 define p ~k , v p /m = Fermi velocity. F ≡ F F ≡ F As such, the model implies that σ (no ). If we introduce a phenomenological → ∞ mean-free path l, we have a time τ l/v , then the d.c. conductivity σ is ≡ F (Drude): ne2τ 1 dn σ = = v2 τ (10) m 3 F dǫ (the second form refers only to the Fermi surface). More generally, σ(ω) is complex and is given by 2 2 2 (ne /m)τ ǫ0ωpτ 2 ne σ(ω)= , ωp (11) 1+ iωτ ≡ 1+ iωτ ≡ mǫ0 If we combine this result with Maxwell’s equations, we find that electromagnetic radiation is absorbed (a) for ωτ . 1 (“Drude peak”), and also (b) in a δ-function peak at ωp (“plas- mon”). In most “textbook” metals, ωp & ǫF (visible/UV). Most predictions agree well with experiments on “conventional” (textbook) metals.

3 Bloch model

This model takes into account the periodic crystalline potential (but still no interactions). −1/2 Now ψk Ω uk exp ik r. ǫ( k ) ǫ (k). If the number of electrons per unit cell is ∼ n · | | → n even, then (usually) filled bands imply an . If odd, then the Fermi surface intersects one (or more) band(s). Properties are qualitatively similar to the Sommerfeld model, except for σ(ω) (interband absorption) (also, σ σ ) (anisotropic crystal)) → ij

4 Landau Fermi-liquid theory

Originally done for (normal) liquid 3He (no crystalline potential, interactions short-ranged), later generalized to electrons in metals (crystalline potential, long-ranged Coulomb interac- tion).

2 1/3 Recap: Sommerfeld model at T = 0, Fermi sea filled up to kF = (3π n) . Formally, n(p,σ)= θ(ǫ ǫ(p)). F − Excited states (N-conserving): take a particle from below the Fermi surface with

4 p, place it above the Fermi surface at a state with momentum p′. δE = ǫ(p) ǫ(p′). More − generally,

E E0 = pσ ǫ(p)δn(pσ), − δn(pσ) = 0 or 1,p>pF , p with (12) Sz = σ σδnP ( σ), ( δn(pσ)=0or 1,p

It is convenient toP measure ǫp from ǫ , and consider not E but E µN (recall that at low F − T µ(T ) const. = ǫ ) With this definition of E and ǫp ≃ F ∆E = ǫ(p)δn(pσ) (13) pσ X independent of whether pσ δn(pσ) = 0. Energy eigenstates are completely specified by δn(pσ) . { } P Now, turn on the interaction adiabatically: assume the interaction is of the form 1 V ( r 2 ij | i− r ) (true to a good approximation in 3He) conserves linear and angular momentum as j| ⇒ P well as spin.

** Fundamental assumption: Adiabatic evolution of low-lying states.

If true, we can label each (low-lying) state of the interacting system by the set δn(pσ) { } which described the original free system by construction the Fermi momentum is un- ⇒ changed. Terminology: δn(pσ)=+1( p >p ) “” in state pσ, δn(pσ)= | | F → 1 ( p < p ) “quasihole” in state pσ (Idea of “dressing”). Note that since [V, S] = − | | F → [V, P]=0,

Sz = σδn(pσ) P = pδn(pσ) (14) pσ pσ X X and, trivially, δN = δn(pσ) (15) pσ ! X However, note that while for the noninteracting system, the spin current Jspin is given by

Jspin = σpδn(pσ), (16) σ X this is not true for Landau Fermi-liquid theory (since in general [J , V ] = 0). (in partic- spin 6 ular, Jspin is not even diagonal in the basis).

Taylor expansion of energy E E δn(pσ) : ≡ { } 2 δE 1 δ E ′ ′ E E = δn(pσ)+ δn(pσ)δn(p σ )+ (17) − 0 δn(pσ) 2 δn(pσ)δn(p′σ′) ··· pσ p p′ ′ X ,X,σ,σ 5 Definition: δE ǫ(pσ) 1 (18) δn(pσ) ≡ ∼ 2 δ E ′ ′ − f(pp ,σσ ) Ω 1 (19) δn(pσ)δn(p′σ) ≡ ∼ Why can we stop at the second term? Suppose N δn(pσ) N. Then the second ex ≡ pσ | | ≪ term is N 2 f N 2 /Ω N (N /N), while the third term is N 3 Ω−2 N (N /N)2, ∼ ex ∼ ex ∼ ex ex P ∼ ex ∼ ex ex so second term. On the other hand, we often (in fact, almost always) find that in the ≪ first term, the contribution linear in Nex vanishes from , so the second and first are of the same order of magnitude. Thus, terms explicitly kept in Eq. (17) are enough.

Symmetry: For a rotationally invariant system, we must have

(a) ǫ(pσ)= ǫ( p ). | | Expand around ǫ (recall ǫ measured relative to ǫ µ): F F ≡ dǫ ǫ ǫ = (p p )+ o (p p )2 (20) − F dp − F − F  pF  Definition: dǫ ∗ v , p /v m (“effective mass”) (21) dp ≡ F F F ≡  p=pF so the single-particle energy spectrum is completely parametrized by m∗. In particular, the density of states of the Fermi liquid is given by dn p2 m∗p m∗ dn = F = F (22) dǫ π2~2v π2~2 ≡ m dǫ F    free gas (b) Landau interaction function: rotational invariance ⇒ ′ ′ ′ ′ ′ f(pp σσ )= f( p , p , ∠pˆ pˆ ,σσ σ ) (23) | | | | · · (σ σ′ is a generalization of σσ′) but can set p p′ p , so: · | |≃| |≃ F ′ ′ ′ ′ ′ f(pp σσ )= f(cos θ,σσ σ )= f (cos θ)+ f (cos θ)σ σ θ ∠pˆ pˆ (24) · s a · ≡ ·

To get a volume-independent quantity, multiply by the total density of states Ω(dn/dǫ), thus F (cos θ) Ω(dn/dǫ)f (cos θ), F (cos θ) Ω(dn/dǫ)f (cos θ) (25) s ≡ s a ≡ a F , F dimensionless. Finally, expand in Legendre polynomials: ⇒ s a s Fs(cos θ)= Fl Pl(cos θ) (etc.) (26) Xl 6 The interquasiparticle interaction is completely parametrized by an infinite set of dimen- ⇒ s a sionless numbers Fl , Fl . (but for most purposes, we only need l . 2). NB: In the most general case, δn(pσ) has to be a matrix. (hence σ σ′ not σσ′). ·

5 Generalized molecular fields

a ex. only F0 nonzero, i.e. (for δnpσ diagonal in the Sz-basis) ′ ′ − − ′ f(pp σσ ) = Ω 1(dn/dǫ) 1F a σσ (27) 0 · (2) −1 −1 a ′ δE = Ω (dn/dǫ) F σσ δnp δnp′ ′ (28) ⇒ 0 σ σ p p′ ′ σ,Xσ 1 − − = Ω 1(dn/dǫ) 1F aS2 (29) 2 0 equivalent molecular field: − dn 1 = F aS (30) Hmol − dǫ 0   so calculate the response of the spin density to an external field by Hext S(kω)= χ0(kω) (kω) [response of free with effective mass m∗] s Htot  tot(kω)= ext(kω)+ mol(kω) (31)  H H H  (kω)= (dn/dǫ)−1F aS(kω) Hmol − 0 the true susceptibility χ (kω) δS(kω)δ (kω) is given by ⇒ s ≡ Hext χ0(kω) k s χs( ω)= −1 a 0 (32) 1 + (dn/dǫ) F0 χs(kω) e.g. for ω 0 then k 0, χ0 is the static response = dn/dǫ, so → → s 0 a −1 χs/χs = (1+ F0 ) (“Wilson ratio”) (33)

General principle: for no net polarization of the Fermi surface, all molecular field effects vanish. Ex.: no effect on specific heat, de Haas van Alphen, ....

(When do molecular fields have an effect? e.g., zero sound, high-field NMR...).

Lifetime of quasiparticles: due to e.g.

qp qp + qp + qh (34) → Result of calculation: − ǫ τ 1(ǫ) F ǫ2 + π2k2 T 2 /ǫ2 (35) ∼ ~ B F so for ǫ, k T ǫ , τ −1(ǫ) ǫ. (Justifies the original Landau argument). B ≪ F ≪

7 6 Generalization to metals

1. Crystalline lattice: can handle by adiabatic technique, but [P, V ] = 0. So we can 6 “label” the Bloch states k by the original states from which they evolved, but ~k is no longer “momentum of quasiparticle with wavevector k”: (rather, quasimomentum). Also, ǫ(k) has gaps at the boundaries.

K = kσ knkσ not conserved because of Umklapp processes Lack of rotational invariance we must classify ǫ(k), f(k, k′) by symmetry operations P ⇒ of the crystal group (actually sometimes simplifies things).

2. Coulomb interaction: must handle as a special type of molecular field. Result:

χ0(q, ω) χ(q, ω)= 2 2 (36) 1 + (e /ǫ0q )χ0(q, ω)

where χ0(qω) is the “bare” Fermi-liquid density response.

3. Impurity scattering: (quasi)-momentum not conserved. However, we can reformu- late Fermi-liquid theory in terms of exact s,p eigenstates of the noninteracting gas in the presence of impurities (so we no longer have plane waves). NB: Subtle interaction of impurity and interaction effects (Altshuler, Aronov).

4. Electron- interactions two different Fermi-liquid theories (ω ω : ⇒ ≫ D ineffective, ω ω : all parameters renormalized). ≪ D

“Topological” interpretation of Fermi-liquid theory.

7 Some references

D. Pines and P. Nozi`eres, Theory of Quantum , Addison-Wesley, 1989 W. Harrison, Solid State Theory, McGraw-Hill 1970 (concentrates on band theory) P. Nozi`eres, Theory of Interacting Fermi Systems, Benjamin 1964 (very technical) A. J. Leggett, Quantum Liquids, Oxford 2006, appendix 5A. (closest to notes) Ashcroft and Mermin, Solid state physics, Hold Reinehart and Winston; 1976 (good on Sommerfeld model, less detail on Fermi-liquid theory)

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