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Chap 17 Transport phenomena and Fermi liquid theory • Boltzmann equation • Onsager reciprocal relation • Thermal electric phenomena • Seebeck, Peltier, Thomson… • Classical Hall effect, anomalous Hall effect • Theory of Fermi liquid • e-e interaction and Pauli exclusion principle • specific heat, effective mass • 1st sound and zero sound Dept of Phys M.C. Chang Boltzmann equation • Distribution function: f(r,k,t) (“g” in Marder’s) d 3k Number of electrons within d3r and d3k f (r , k , t )d r 2 around (r,k) at time t (2 )3 3 For example, d 3k j ( r , t ) e 2 rf (r , k , t ) (2 )3 • Evolution of the distribution function t t dt f (r , k , t ) f (r dr , k dk , t dt ) =f ( r , k , t ) f f f dr k dt r t k without collision the phase-space density does not change in the comoving frame k df f f f r k 0 dt t r k Phase space is incompressible (Liouville’s Theorem) Larger △r smaller △k r supplementary With collision (due to disorder… etc): f f f f r k t r k t coll Boltzmann eq. ~ Source/drain • Transition rate for an electron at k → k’: Wk, k’ (calculated by Fermi golden rule) In a crowded space, one needs to consider occupancy and summation Wk ,k ' f k (1 f k ' )Wk ,k ' k' • On the other hand, the transition rate for an electron to be scattered into k (1 f k ) f k 'Wk ',k Grosso, SSP, p.404 k' f (1 f k ) f k 'Wk ',k f k (1 f k ' )Wk ,k ' t coll k ' = ( f k ' f k )Wk ',k if Wk ,k ' Wk ',k (for elastic scattering) k' see Marder, Sec 18.2 for more supplementary f ( f k ' f k )Wk ',k t coll k ' (valid for uniform E, B, T fields) if f k f k0 C k , then f C (k k ')Wk ',k t coll k' = C k (1 kˆ kˆ ')W (k k ') k ', k k' = where 1 f k f k0 Only the component of k’ // k contribute to the integral (if W depends only on θ) (1 kˆ kˆ ')Wk ',k Transport relaxation time k' Note: for inelastic scatterings, detailed balance requires (1 f k0 ) f k0'Wk ',k f k0 (1 f k0' )Wk ,k ' f 0 (k ) 1 ( k ) / k BT e ( k ) e k BT 1 ( k ') Wk ',k e k BT Wk ,k ' k θ k’ • Relaxation time approximation f (r , k , t ) f 0 (r , k ) f t coll (k ) f 0 (r , k ) 1 (k ) (r ) exp 1 k T ( r ) B Unperturbed equilibrium dist Relaxation (allows energy dependence) Density and temperature gradients are allowed • Boltzmann eq. f f f df f f0 r k t r k dt f rk (t ) 1 t dt ' f rk0 (t ')e (t t ')/ 0 0 ( t t ')/ df rk (t ') or = f rk dt 'e dt ' t T f f rk (t ) f rk0 dt 'e (t t ')/ vk k 0 T r r = f0 f Let’s call this t “Chamber’s formulation” df 0 f f r 0 k 0 dt r k f 0 f 0 f 0 r T r T T r T f 0 = r T r f 0 f r 0 k 1st, consider a system with electric field and temperature gradient, but no magnetic field d 3k • Electric current j (r , t ) e [dk ] rf (r , k , t ), [dk ] 2 (2 )3 density = e [dk ]vk f 0 e [dk ]vk f r 1 k k eE T f 0 ( t t ')/ dt ' e v eE r T r T f0 , G E 1 Electrochemical = v eG “force” T r e r For steady f perturbations t T f 0 j (r , t ) e [dk ] v v eG T r T σ is conductivity tensor =L11G L12 L11 is σ , T • Thermal current j Q (r , t ) [dk ]( )r f density T f0 [dk ] ( )v v eG T r T L 21G L 22 L 22 is κT κ is thermal , T conductivity tensor Coefficients of transport (matrices) f L11 e 2 [dk ] v v 0 f L 21 e [dk ] ( k )v v 0 f L12 e [ dk ] ( k )v v 0 f L 22 [dk ] ( k ) 2 v v 0 They are of the form Define energy-resolved conductivity f ij( ) e 2 [dk ] ( k ) vi v j 0 L11 Λ (0) 1 L12 L 21 Λ (1) ( e) 1 L 22 Λ (2) 2 ( e) ij ( ) e 2 [dk ] vi v j ( k ) then f ij( ) d ( ) 0 ij ( ) One example of Onsager relation T=0, Λ (0) σ ( F ) Λ (1) Λ (2) 2 3 2 3 (k BT ) 2 σ '( F ) H.W. (k BT ) 2 σ ( F ) More on the conductivity consider ▽T=0. Ohm’s law: j σG Fick’s law for “diffusion current”: j D 2 3 n * 2m F 2 F n 2 F r 3 n r 3 r 3 F j D 2 F r F 2/3 1 e r f • At T=0, 0 ( F ) • For electron gas at low T, 2 GE en σ F j e r 1 2 F 1 D σ 2 σ ( e) 3 e g ( F ) ij e2 [dk ] vi v j ( F ) =e 2 F g ( F ) vi v j vi v j FS FS is an integral over the FS Dij F vi v j FS For isotropic diffusion, Dxx F Einstein relation (for degenerate electron gas) 1 2 v 3 FS 1 vF2 F 3 • Alternative form of conductivity f 0 ij e2 [dk ] vi v j f vi v j 0 vi k j f 0 f 0 v i k j ( f 0 vi ) vi ij e [dk ] f0 k k j j If τ a 2 const. =e [dk ] f 0 m 2 ne 2 op ij m *1 ij ~ free electron gas f0 f 0 v v i j * k mij (k ) k N mijop Optical effective mass = m* if the carriers are near the band bottom • Fourier’s law on thermal conduction T Note: this would induce electric current. j Q κT L 22 To remedy this, see Marder, p.496. T L 22 2 k B2T κ σ (Wiedemann-Franz law for metals) 2 T 3 e supplementary More on the Onsager reciprocal relation (1931) 互易關係 • Transport processes near equilibrium (linear transport regime) ji ij x j (Ohm’s Law) ji Dij x j (Fick’s Law) T j ij x j ... (Fourier’s Law) Q i • They are of the form J i Lij X j Thermodynamic “flow” Thermodynamic “force” Thermodynamic conjugate variables • “kinetic coefficient” L is symmetric: For example, if Ex drives a current Jy, then Ey will drive a current Jx. • The Onsager relation is a result of fluctuation-dissipation theorem, plus time reversal symmetry. • A specific example: the conductivity tensor of a crystal is symmetric, whatever the crystal symmetry is. supplementary Simultaneous irreversible processes Note: For example, For many transport processes near equilibrium, the entropy production is a product of flow and force j L11 L12 G Q L L j T / T 21 22 also of the form: Y J i Lij x j TSirr J i X i Sirr T Xi X j 0 ∴Entropy production ~ a thermodynamic potential Same symmetry relation applies to this larger α matrix → LT12=L21 • if force 1 (e.g., a temperature gradient) drives a flow 2 (diffusion current), then force 2 (density gradient) will drive a flow 1 (heat current) ! Precursors of the symmetry relation Lij Nature likes to stay at the lowest potential → minimum entropy production (only in the linear regime) 1968 (D.G. Miller, J Stat Phys 1995) • Stokes (1851), anisotropic heat conduction • Kelvin (1854), thermoelectric effect •… http://www.ntnu.no/ub/spesialsamlingene/tekhist/tek5/eindex.htm Thermoelectric coupling (1) Seebeck effect (1821) Seebeck found that a compass needle would be deflected by a closed loop formed by two metals joined in two places, with a temperature difference between the junctions. or • In the absence of electric current Longitudinal T gradient → electrochemical potential (in metals or semiconductors) 0 L11 L12 G Q L L j T / T 21 22 G T 溫差導致電位差 • Seebeck coefficient (aka thermoelectric power) 熱電功率 L11 1 L12 2 kB2T ' T 3 e typical values observed: a few μV / K (Bi: ~ 100 μV / K) (2) Peltier effect (1834) Peltier found that the junctions of dissimilar metals were heated or cooled, depending upon the direction of electrical current. • In a bi-metallic circuit without T gradient, a current flow would induce a heat flow (in metals or semiconductors) j L11 L12 G Q L L j 21 22 0 jQ j L 21 L11 T 1 (If L11 and L21 commute) • In practical applications of Seebeck/Peltier, the figure of merit (dimensionless) is (Prob 7) 2 ZT T High electric conductivity and low thermal conductivity is good. (>< Wiedemann-Franz law) • Bi2Te3 ZT ~0.6 at room temperature If ZT~4, then thermoelectric refrigerators will be competitive with traditional refrigerators. Thermoelectric cooling Peltier element: Bismuth Telluride (p/n type connected in series) advantages • Solid state heating and cooling – no liquids. CFC free. • Compact instrument • Fast response time for good temperature control Seebeck vs Peltier Hall effect (1879): classical approach dv v v eE e B m* dt c B Bzˆ; dv / dt 0 at steady state m* Ex m* / eB / c vx v e E * eB / c m / y y j env m* Ex ne 2 Ey B nec B nec m* ne 2 j 1 x 0 jy c 1 0 σρ 2 1 c c 1 c 1 1 c c 1 0 1 c nec / B 0 c 1 0 nec / B c jx 1 jy ne2 0 * m m* eB 0 2 , c * ne mc ρxy B dk e eE vk B dt c Hall effect: semiclassical approach Recall “Chamber’s formulation” (without density and T gradients) f f dt 'e (t t ')/ vk k 0 t 1 f e dt 'e (t t ')/ vk E vk B 0 c t We can now only count on vk for magnetic effect e k eE v B c B v Bk E B c e B2 B2 ExB drift Assume E⊥B v Bk EB E E 2 k c e B e B2 e (t t ')/ k (t ') EB f c 2 B d (t t ')/ 1 e k (t ') e (t t ')/ k (t ') dt ' 1 t f0 ( t t ')/ k (t ) k (t ) k (t ) , where k (t ) dt 'e Q1:How do we get the next order terms? j e [dk ]vk f 0 f f 0 EB =ce 2 [dk ] k k B k EB =ce 2 [dk ] k k B k f0 f0 0 nec σ B 0 nec B 0 0 1 c 1 c 0 2 0 0 1 If the open orbit is along the x-direction (in real space), then EB EB [ dk ] f nec 0 B2 B2 0 0 1 0 0 c 1 0 Q2: What if the orbit is not closed? • For ωcτ>>1, the first term is zero j ec 1 2 c 1 σ 0 c 0 1 c 0 0 0 0 1 1 σ 0 c 0 1 c 1 c 0 2 0 0 1 See Kitel, QTS, p.244 Q3: What about ωcτ<<1? supplementary Anomalous Hall effect (Edwin Hall, 1881): Hall effect in ferromagnetic (FM) materials FM material ρH saturation slope=RN RAHMS H The usual Lorentz force term H RN H AH ( H ), Anomalous term AH ( H ) RAH M ( H ) Ingredients required for a successful theory: • magnetization (majority spin) • spin-orbit coupling (to couple the majority-spin direction to transverse orbital direction) supplementary “Intrinsic” AHE due to the Berry curvature Mn5Ge3 Zeng et al PRL 2006 dk eE dt dr 1 k ( k ) dt k 1 e = E ( k ) k A transverse current To leading order, j e [dk ] vk f 0 e2 = [dk ] f 0 [dk ] (k ) f 0 E k e AH e2 filled After averaging over long-wavelength spin fluctuations, the calculated anomalous Hall conductivity is roughly linear in M. The S.S. refers to skew scattering. • Skew scattering from an impurity [dk ] z (k ) transition rate Wks k 's' SO s's k ' k • classical Hall effect charge B EF Lorentz force +++++++ ↑↓ y 0 L • anomalous Hall effect ↑ ↑↑↑↑↑↑↑ ↑↑↑↑ EF Berry curvature (int) ↓ B charge spin Skew scattering (ext) y ↑↑ ↑↑↑↑ 0 L • spin Hall effect spin EF ↑ Berry curvature (int) ↓ ↑↑↑↑↑↑↑ ↑↑↑↑ ↑↑↑↑ Skew scattering (ext) ↑↑↑↑↑↑↑ No magnetic field required ! y 0 L Thermo-galvano-magnetic phenomena j e σ ( B) E α ( B) T -yT j Q β( B) E κ ( B) T Bz σT ( B ) σ ( B) Onsager relations Jx β ( B) α ( B) T κ T ( B) κ ( B) • Expand to first order in B, Ohm Hall Nernst 1826 1879 1886 The effect of B on thermo-induced electric current j e E RE B T N T B Onsager relations j Q E NE B T L T B The effect of B on electric-induced thermo current Ettingshausen Fourier Leduc-Righi 1886 1807 (Thermal Hall effect) 1887 Landau, ED of continuous media, p.101 Beyond thermo-galvano-magnetic phenomena Optical (O) Thermal (T) • E-T: Thomson effect, Peltier/Seebeck effect Mechanical (M) • E-B: Hall effect, magneto-electric material • E-B-T: Nernst/Ettingshausen effect, Leduc-Righi effect • E-O, B-O: Kerr effect, Faraday effect, photovoltaic effect, photoelectric effect Electric (E,P) Magnetic (B,M) • E-M, B-M: piezoelectric effect/electrostriction, piezomagnetic effect/magnetostriction • M-O: photoelasticity N E B • ... solid state refrigerator solid state sensor solid state motor, artificial muscle ... Landau and Lifshitz, Electrodynamics of continuous media Scheibner, 4 review articles in IRE Transations on component parts, 1961, 1962 TABLE 1-3 Physical and Chemical Transduction Principles. (from “Expanding the vision of sensor materials” 1995) Input (Primary) Signal Output (Secondary) Signals Mechanical Thermal Electrical Magnetic Radiant Mechanical (Fluid) Mechanical effects; e.g., diaphragm, gravity balance. Acoustic effects; e.g., echo sounder. Friction effects; e.g., friction calorimeter. Cooling effects; e.g., thermal flow meter. Piezoelectricity. Piezoresistivity. Resistive. Capacitive. Induced effect. Magnetomechanical effects; e.g., piezomagnetic effect. Photoelastic systems (stressinduced birefringence). Interferometer. Sagnac effect. Doppler effect. Thermal Thermal expansion; e.g., bimetallic strip, liquid-in-glass and gas thermometers. Resonant frequency. Radiometer effect; e.g., light mill. Electrical Electrokinetic and electromechanical effects; e.g., piezoelectricity, electrometer, and Ampere's Law. Joule (resistive) heating. Peltier effect. Charge collectors. Langmuir probe. Biot-Savart's Law. Magnetic Magnetomechanical effects; e.g., magnetostriction, and magnetometer. Thermo-magnetic effects; e.g., Righi-Leduc effect. Galvano-magnetic effects; e.g., Ettingshausen effect. Thermo-magnetic effects; e.g., EttingshausenNernst effect. Galvano-magnetic effects; e.g., Hall effect, and magnetoresistance. Magneto-optical effects; e.g., Faraday effect, and CottonMouton effect. Seebeck effect. Thermoresistance. Pyroelectricity. Thermal (Johnson) noise. Chemical Thermo-optical effects; e.g., liquid crystals. Radiant emission. Reaction activation; e.g., thermal dissociation. Electro-optical effects; e.g., Kerr effect, Pockels effect. Electroluminescence. Electrolysis. Electro-migration. Landau’s theory of Fermi liquid Why e-e interaction can usually be ignored in metals? 2 • K U K 1 e2 , U m r2 r me 2 r r 2 aB Typically, 2 < U/K < 5 • Average e-e separation in a metal is about 2 A Experiments find e mean free path about 10000 A (at 300K) At 1 K, it can move 10 cm without being scattered! Why? • A collision event: k1 k3 k2 k4 • Calculate the e-e scattering rate using Fermi’s golden rule: 1 2 f | Vee | i 2 ( Ei E f ) 2 k3 , k4 | Vee | k1 , k2 i, f Scattering amplitude f | Vee | i Ei E1 E2 ; E f E3 E4 2 The summation is over all possible initial and final states that obey energy and momentum conservation Pauli principle reduces available states for the following reasons: If the scattering amplitude |Vee|2 is roughly of the same order for all k’s, then 1 Vee 2 1 k1 , k2 k3 , k4 E1+E2=E3+E4; k1+k2=k3+k4 • 2 e’s inside the FS cannot scatter with each other (energy conservation + Pauli principle), at least one of them must be outside of the FS. Let electron 1 be outside the FS: • One e is “shallow” outside, the other is “deep” inside also cannot scatter with each other, since the “deep” e has nowhere to go. • If |E2| < E1, then E3+E4 > 0 (let EF=0) But since E1+E2 = E3+E4, 3 and 4 cannot be very far from the FS if 1 is close to the FS. Let’s fix E1, and study possible initial and final states. 1 2 3 (let the state of electron 1 be fixed) • number of initial states = (volume of E2 shell)/Δ3k number of final states = (volume of E3 shell)/Δ3k (E4 is uniquely determined) • τ-1 ~ V(E2)/Δ3k x V(E3)/Δ3k ← number of states for scatterings V ( E2 ) 4 kF2 | k2 kF | V ( E3 ) 4 kF2 | k3 kF | ∴τ-1 ~ (4π/Δ3k)2 kF2|k2-kF|×kF2|k3-kF| Total number of states for particle 2 and 3 = [(4/3)πkF3/ Δ3k]2 • The fraction of states that “can” participate in the scatterings = (9/kF2) |k2-kF|× |k3-kF| ~(E1/EF)2 (1951, V. Wessikopf) In general 1 2 2 k BT Finite temperature: ~ (kT/EF)2 ~ 10-4 at room temperature → e-e scattering rate T2 • need very low T (a few K) and very pure sample to eliminate thermal and impurity scatterings before the effect of e-e scattering can be observed. 2 Landau’s theory of the Fermi liquid (1956) assumptions • Strongly interacting fermion system → weakly interacting quasi-particle (QP) system • 1-1 correspondence between fermions and QPs (fermion, spin-1/2, charge -e). ~ a particle plus its surrounding, finite life-time Q: Is this trivial? • adiabatic continuity: As we turn off the interaction, the QPs smoothly change back to noninteracting fermions. 1962 • The following analysis applies to a neutral, isotropic FL, such as He-3. Another application: He-3 TF=7 K Similarity and difference with free electron gas • QP distribution (at eq.) at T 0, For a justification, see Marder 1 f k e( k )/ kBT 1 ( F k ) ( = F at T 0) f k f k0 ( F k0 ) ← if no other ext perturbations kF is not changed by interaction! • Due to external perturbations the distribution will deviate from the manybody ground state (no perturbation) at T=0 • Thermal f k f k ( k ) ( F k0 ) • Non-thermal (T=0) f k ( k ) ( F k0 ) (density perturbation, magnetic field… etc) • In general • QP energy f k f k ( k ) ( F k0 ) k ukk ' f k ' ' 0 k ' k ' ' interaction between QPs near FS In absence of other QPs • Total number ukk' 'is an effective N f k k N kF3 2 V 3 11 u ' kk ' uk' 'k E[ f ] E[0] k f k • Total energy k E[0] k0 f k k 1 ukk' ' f k f k ' ' O( f 3 ) 2 k k ' ' This form is not good for charged FL (with longrange interaction) • If there is no magnetic field, nor magnetic order, then k is independent of σ, and ' ukk' depends only on the relative spin directions. uk1k2' k1 , k2 | Vee | k1 , k2 For example, ukk' ' (forward scattering amplitude) 1 4 e2 2 V (k k ', 0) k k ' 0 if ' if ' Quinn, p.384 (recall the Fock interaction in ch 9) • Fermi velocity k0 vF k • Effective mass m* k kF kF vF Note: The use of k follows Coleman’s note, Baym and Pethick etc, but not Marder’s. We don’t want these quantities to depend on perturbation 0 • DOS D* ( F ) See ch 7 1 V ( 0 k F ) k 1 4 3 dS m* k F 2 2 k k0 • Specific heat dE k f k f k , f k k k 0 k f ( k0 ) T T (to lowest order) 0 E 0 f ( k ) CV k T V k T d ( k0 ) k f ( ) T f ( k0 ) =V d D ( ) See ch 6 T same as non-interacting f ( k0 ) 2 2 * * VD ( F ) d = k BTVD ( F ) result except for the T 3 effective mass. * Heavy fermion material (CeAl3) C is proportional to T, but the slope gives an effective mass 103 times larger! He-3 ρ is proportional to T2, also a FL behavior Specific is linear in T below 20 mK Giamarchi’s note, p.88 (30 bar) Effective mass of a QP (I) (total) “Particle” current JN k k k f k f k m k m (1) Z.Qian et al, PRL 93, 106601 (2004) Effective mass of a QP (II) JN On the other hand, give particles an active boost (with p-h excitations) dE dk (see next page) k k dk f k f k dk , f k dk f k k f f k0' dE k dk k k k k ' J N = k f k vk ukk' 'vk ' ' ( k0' F ) f k k k k k ' ' (1)=(2) → (δf is arbitrary) (2) k k ' k ' * ukk ' * ( k0' F ) m m k ' ' m ' If m* is spin-indep, m* ' k F k F ' 0 1 u ( kk ' k ' F ) 2 (nonmagnetic FL) m kF k ' ' then (an integral over the FS) (Only for QPs near the Fermi surface) see Fradkin’s note, Pathria p.296 (a passive “boost”) Nozieres and Pines, p.37 Introducing Fermi liquid parameters ' • Moments of ukk ' over the FS provide the most important information about interactions (e.g., see the previous m* formula ) ukk' ukk' ukks ' ukka ' let ukk' ukk' ukks ' ukka ' and decompose u k θ k’ s kk ' ukka ' For spherical FS, ukk’ depends only on θ s 2 1 1 u u kk ' kk ' u P (cos ) u d cos P (cos ) 1 0 2 2 1 u u 2 1 a a kk ' kk ' u P (cos ) u d cos P (cos ) 2 1 2 0 s • Dimensionless parameters F VD ( F )u , F VD ( F )u s For example, * s a * a A small set of parameters for various phenomena m* k '2 dk ' 0 1V d ' u u cos ' ( kk ' kk ' k' F ) 3 m 2 1 =1+ F1s 3 H.W. determined from specific heat. m*/m~3 for He-3 More on the effective mass • recall m* 1 1+ F1s m 3 pF pF F1s pF * m m 3 F1s m Backflow correction (to ensure current conservation) • m* k F m* 1 * s * 1+ VD ( F )u1 , D ( F ) 2 2 m 3 m m* V 1 D ( F )u1s 3 diverges when V D( F )u1s 1 (~ Mott transition) 3 Compressibility of Fermi liquid 1 V V P 1 n T 2 n T Note: dP SdT nd At fixed S or T (little difference near T=0) T n At T=0, f k k F k0 = 1 P T n Before compression 1/ n P N ,T 1 n n T 1 n 2 P T n T k F f k f k ' ' F k0 1 ukk ' ' k ' ' Note: Slightly different from Marder’s (see Baym and Pethick, p.11) Both k and k’ lie on FS, and Ukk’ depends only on cosθ, ∴ Ak is indep. of k. ≡ Ak (indep. of σ if not magnetized) A ukk' ' k ' ' F0s A 1 F0s f k ' ' ukk' ' F k0' 1 A k ' ' = F0s Dependence of various quantities on δμ f k F 0 k 1 A = F k0 1 1 F0s k ukk' ' f k ' ' k ' ' Note: • For attractive interaction, Fs 0 If F s 1 , then κT diverges, 0 and FS will become unstable to deformation (spontaneous breaking of rotational symmetry). F0s = 1 F0s also, N V 1 1 * f D ( ) k F s V k 1 F0 n 1 D* ( F ) V 1 F0s D* ( F ) 1 1 n T 2 n V ,T n 2 1 F0s • This is called Pomeranchuk instability (1958). For example, nematic FL. 向列型 T m* / m 0 T 1 F0s 0910.4166 Deformation of Fermi sphere and the FL parameters ' 0 ukk ' k ' F k ' ' 1 ' k F k F ' u ( k0' F ) kk ' 2 3 k ' ' kF From Coleman’s note summary For He-3, F1s 6 (larger effective mass) F0a 0.5 (more spin polarizable) compressibility κ F0s 10 (less compressible) From Coleman’s note Travelling wave: firstly, 1st sound (i.e., the usual pressure wave) Velocity of the 1st sound c1 so P P , s s,N mn V P 1 n V s s 1/2 n s = 1 F 0 * mD ( ) F m*k F k F3 * D ( F ) 2 2 , n 2 3 1/2 m* c1 vF 1 F0s 3m F0S=10.8 for He-3, determined by measuring C1 vF 3 (w/o interaction) Zero sound (predicted by Landau, verified by Wheatley et al 1966) • usual sound requires ωτ<<1 (mean free path ℓ<<λ) when ωτ→1 , sound is strongly absorbed • when ωτ>>1, sound propagation is again possible • zero sound is a collisionless sound ~ plasma wave in charged FL Can exist at 0 K no thermal equilibrium in each volume element • to get the zero sound, one can increase ω or decrease T (to increase τ) Oscillation of Fermi sphere 1st sound t zero sound fk cos s cos γ t (egg-like shape) Giamarchi’s note, p.102 Boltzmann-like approach (requires f f f r k 0 t r k consider (collisionless) Instead of the semiclassical equations, one uses rk No r-dependence hidden in ε. f k f k0 f k (r , t ), F , q kF ) k ; k f k0 k0 F f k k ei ( qr t ) (indep of spin) f k f r k i ( vk q ) f k t r To order δf f k k f k0 k k vF vF k0 F r k r k k f k ' ' ukk' ' iq ukk' ' f k ' ' r k ' ' r k ' ' ( v q ) k vk q k0 F ukk' ' k ' 0 k ' ' k k0 F vk q ukk' ' k ' vk q k ' ' k k r let k k0 F k then k u vk q ukk' ' k ' (1) vk q k ' ' k ' ukk ' F k ' ' kk ' ' k ' ' k' ' = 0 k' d ' a( ')k ' 4 V * D ukk ' ukk' 2 d a ( ) F0s 4 a ( ) Assume a(θ) ~const. let (q , k ) F0s cos 1 d cos , s 2 s cos vq F0s s s 1 = 1 ln 2 2 s 1 s s 1 1 or ln 1 s 2 s 1 F0 1 F0s decompose k ( ) P (cos ) 0 d ' 4 k ( ') 0 d vk q (1) 0 F0s 0 4 v q k • only F0s > 0 (repulsion) can have a solution • for F0s 0, S 1 svF q c0 svF 3c1 • when QP velocity > C0 (s<1), the integral has a pole at cos 0 s , a QP would emit “supersonic” zero sound Analogies: • Supersonic shock wave • Cherenkov EM radiation from “superluminal” charged particles Transition from the 1st sound to the zero sound Dispersion of zero sound in He-3 (from neutron scattering exp’t) vF Superfluid transition Vollhardt and Woelfle, p.45 Aldrich et al, PRL 1976 coherent In addition to collective excitations (zero sound, plasma), there are also particle-hole excitations incoherent particle-hole excitation: k q k For charged FL only (more in ch 23) k q k 2 2k 2m F qq 2 2 2k 2m F 2k q q 2m 2 2 q q2 Q: what is the particle-hole band for 1-dim electron liquid? H. Godfrin et al, Nature 2012 From Altshuler’s slide Beyond the Fermi liquid Quasiparticle decay rate at T = 0 in a Fermi Liquid: • • 2 F 2 F log F e e d 3 d 2 d 1 For d =3,2, from it follows that / , i.e., the QPs are well ee F determined and the Fermi-liquid approach is applicable. For d =1, ee is of the order of / , i.e., the QP is not well defined and the Fermi-liquid approach is not valid. → Tomonaga-Luttinger liquid in 1D (1950,1963) For more, see Marder, Sec 18.6 Giamarchi, p.6 Features of a Luttinger liquid (cited from wiki) • Even at T=0, the particles' momentum distribution For electron, not for QP function has no sharp jump (Z=0). (in contrast to QP dist) • Charge and spin waves are the elementary excitations of Z the Luttinger liquid, unlike the QPs of the FL (which carry both spin and charge). • spin density waves propagate independently from the charge density waves (spin-charge separation). Luttinger liquids reported in literatures • electrons moving along edge states in the fractional Quantum Hall Effect (Th: Kane and Fisher 1994; Ep: 1996) 朝永 振一郎 • electrons in carbon nanotubes (McEuen group, 1998) • 'quantum wires' defined by applying gate voltages to a two-dimensional electron gas. (Auslaender et al, 2000) Non-Fermi-liquid in 2D? Luttinger