Metal - Insulator transitions: overview, classification, descriptions

Transcription

1 Metal - Insulator transitions: overview, classification, descriptions Krzysztof Byczuk Institute of Physics, Augsburg University January 19th, 2006

2 Collaboration: Walter Hofstetter - RTW, Aachen Martin Ulmke - FGAN - FKIE, Wachtberg Dieter Vollhardt - Augsburg University Plan of the talk: 1. Introduction Basic definitions Gap in insulators Phase transitions from conductors to insulators 2. Single particle insulators 3. Many body insulators 4. Mott - Hubbard MIT at integer filling 5. Mott - Hubbard MIT at noninteger fillings 6. Mott - Anderson MIT at integer filling 7. Conclusions

3 Conductors and Insulators definitions: Basic physical property of a system: how good/bad the charges (masses) are transported through it. School knowledge V A R= U I R = ρ L h A [ρ] = Ω m d 2i Transport occurs in a nonequilibrium processes. Transport can be disturbed by: ions, electron electron interactions,external fields, etc.

4 Conductors and Insulators definitions: Exact definitions of a conductor or an insulator possible only at T = 0 within linear response theory. weak external field Ohm s law j α (q, ω) = X β σ α,β (q, ω)e β (q, ω) σ α,β (q, ω) conductivity tensor Definition: Insulator is a system where σ DC α,β (T = 0) = lim T 0 + lim ω 0 lim q 0 R[σ α,β(q, ω)] = 0

5 Conductors and Insulators definitions: Drude law for typical metal τ R[σ α,β (T = 0, ω 0)] = (D c ) α,β π(1 + ω 2 τ 2 ) (D c ) α,β = πe2 n m δ α,β Drude weight τ relaxation time for electron scattering, i.g. with ions Definition: Ideal conductor is a system where R[σ α,β (T = 0, ω 0)] = (D c ) α,β δ(ω) For a translationally invariant system τ 1 0 Warning: superconductor ideal conductor + ideal diamagnet

6 Gap in the Insulator To get a charge transport in a conductor: There are low energy excitations (electron hole) above the ground state Excited states must be extended empty E F µ + (λ) = E 0 (N + 1, λ) E 0 (N, λ) µ (λ) = E 0 (N, λ) E 0 (N 1, λ) metal occupied insulator Gap: (λ) = ˆµ + (λ) µ (λ) extended λ = λ(p, x, n) control parameter There is a gap (λ) > 0 in the single particle spectrum in an insulator

7 kt<< T Insulator at finite T Experiment (λ) k B T > 0 good bad conductor obscure meaning E.g. semiconductor: ρ semi cond Ωcm semimetal: ρ semi metal Ωcm However, ρ semi cond = and ρ semi metal = 0 at T = 0! activation energy (λ): R[σ α,β (k B T (λ), ω 0)] e (λ) k B T ρ(t) insulator metal

8 Gap at finite T robust gap exists for all temperature soft gap vanishes for T > T c Roots to form a gap quantum phase transition competition between E kin and E pot thermodynamic phase transitions competition between U and S T crossover regime semimetal semiconductor H kin delocalized H pot localized robust gap without LRO T disordered Tc ordered U ordered S disordered soft gap with LRO good conductor bad conductor H kin /H pot = λ good conductor bad conductor (λ, T < T c ) 0 metal λ c insulator λ metal λ c insulator λ H = H 0 + H 1, [H 0, H 1 ] 0 SSB with LRO below T < T c

9 Types of insulators single particle: due to electron ion interactions Bloch Wilson (band) insulators Peierls (lattice deformation) insulators Anderson (lattice randomness) insulators (!) many particle: due to electron electron interactions Slater (SDW) insulators Mott Hubbard (PM) insulators (!!) Mott Heisenberg (localized AF) insulators

10 Band insulators ideal lattice Ψ k,n (r), E k,n Bloch states, 2N states in a band, completely filled bands do not participate in transport, robust gap in a single particle spectrum E (a) wide gap (5 10eV ) insulator (diamond, noble atom crystals) (b) narrow gap N e = 2N (0, 1 1eV ) insulator (Si) (c) two band metal N e = 2N (As, Sb, Bi) (d) one band metal N e = N (Na, K, Ca,...) E F a b c d quantum MIT in Yb (iterb) at p c = 13kbar

11 Peierls insulators Coupling electron phonon leads to formation of lattice deformation T>Tc a soft gap in SSB at T < T c charge density wave (CDW) n(x) cos(2k q F x) E(k) = E F ± ɛ 2 k + 2, (T) T c T T<Tc E(k) E(k) 2a E F 1/a 1/a k 1/2a 1/2a k thermodynamic MITs

12 Waves in random system: propagation of waves in a randomly inhomogeneous medium random conservative linear wave equation 2 w t 2 = c(x)2 2 w x 2 i w t = 2 w x 2 + ν(x)w self averaging diffusive motion, memory of V (0) lost, random walk over long distances, friction imposed by averaging

13 Anderson localization: propagation of waves in a randomly inhomogeneous medium random conservative linear wave equation 2 w t 2 = c(x)2 2 w x 2 i w t = 2 w x 2 + ν(x)w Anderson 1958: (no averaging) strong scattering forms standing waves, sloshing back and forth in a bounded region of space Localization is a destruction of coherent superposition of spatially separated states

14 Anderson MIT - cont.: Returning probability P j j (t ; V )? Ψ e R/ ξ R P j j (t ; V ) = 0 for extended states P j j (t ; V ) > 0 for localized states

15 Anderson MIT - cont.: According to one-parameter scaling theory [g = g(l)] (noninteracting system) If dim= 1 or 2 all states are localized If dim=3 there is a critical disorder above which the states are localized N(E) c band width Extended Localized E

16 Characterization of Anderson localization: Decaying of wavefunction Ψ n (r i ) e r r i /ξ(e n) metal ξ insulator ξ < Inverse participation ratio P 1 [inverse number of sites that contribute to Ψ n (r i )] metal P 1 1/N insulator P 1 const Conductance G metal G > 0 insulator G = 0 Local Density of States (LDOS) ρ i (E) = P N n=1 Ψ n(r i ) 2 δ(e E n )

17 Local DOS Heuristic arguments: P j j (t) = G j (t) 2 G j (t) e i(ɛ j +Σ j )t Σ j t e τesc t Fermi Golden Rule 1 τ esc t ji 2 ρ j (E F ) ρ j (E) ρ j (E) E F E E F E metal insulator

18 Statistics of LDOS: ρ j (E) is different at different R j! Random quantity! Statistical description P[ρ j (E)]! Exact diagonalization Schubert et al. cond-mat/ p[ρ i (E=0)] W = 18t W = 3t P[ρ i (E=0)] ρ i (E=0) ρ i (E=0) 1 Broadly distributed P[ρ j (E F )] Multifractality - M (k) L f(k) Typical escape rate is determined by the typical LDOS

19 Anderson MIT - cont.: Near Anderson localization typical LDOS is approximated by geometrical mean ρ av, ρ ty W = 12t ρ typ (E) ρ geom (E) = e ln ρ i (E) 0.12 W = 3t W = 6t W = 9t E/t W = 16t E/t Schubert et al. cond-mat/ W = 18t E/t Theorem (F.Wegner 1981): ρ(e) av = ρ i (E) > 0 within a band for any finite

20 Many body insulators; when interaction becomes important Kinetic energy t t ij = R d 3 r Φ i (r) h 2 2m + V (r) i Φ j (r) Interaction energy U = R d 3 rd 3 r Φ i (r)φ i (r ) e r r Φ i(r )Φ i (r) When U t ij 1?

21 Canonical example: V 2 O 3 V ([Ar]3d 2 4s 2 ) gives V +3 valence band partially filled should be metal? True Mott insulator persists above T N Mott Hubbard Insulator, Mott Heisenberg Insulator, and Slater Insulator

22 Slater insulators Weak coupling insulator due to LRO a soft gap in SSB at T < T c spin density wave (SDW) S z (x) cos(2k q F x) E(k) = E F ± ɛ 2 k + 2, (T) T c T E(k) E(k) 2a E F 1/a 1/a k 1/2a 1/2a k thermodynamic MITs

23 Mott-Hubbard metal-insulator transition at n = 1 U t ij, p = 0 U t ij, r = 0 Antiferromagnetic Mott insulator typical intermediate coupling problem U c t ij

24 Hubbard model to capture right physics ɛ 0 + U t U t ɛ 0 H = X iσ ɛ i n iσ + X ijσ t ij a iσ a jσ + U X i n i n i long history, many contradictions exactly solvable in d = 1 exactly solvable in d = how to approximate in 1 < d <?

25 Physical picture, n = 1 E+U atomic levels t =0 t >0 UHB E+U E U LHB E at U = U c resonance disapears gaped insulator spin flip on central site dynamical processes with spin-flips inject states into correlation gap giving a quasiparticle resonance

26 Mott MIT in binary alloy Byczuk et al., PRL03, PRB04 Disordered alloy A x B 1 x P(ɛ i ) = xδ(ɛ i + 2 ) + (1 x)δ(ɛ i 2 ) When t ij the spectral function splits into lower and upper alloy subbands E E Is there Mott MIT at n 1? DOS DMFT + G(ω) = R dɛ i P(ɛ i )G(ω, ɛ i )

27 Mott MIT in binary alloy at n 1 Byczuk et al., PRL03, PRB04 U< U> ε +U UHB ε+ UAB ε + UAB A( ω) µ 2N L n = x or n = 1 + x LAB 2xNL ω µ UAB 2(1 x)n L ε+u UHB ~ U ε LHB ε LHB alloy Mott insulator ~ alloy charge transfer insulator U µ µ LHB UHB xn L µ µ U METAL xn L INSULATOR U A(ω=0) met U=5 met U=3 met U=2 ins U=5 ins U= PI PM ν= U c = 6t x

28 Anderson and Mott transitions: Byczuk et al. PRL05, PhysicaB05 N 0 (ɛ) = 2 p D2 ɛ 2 ; η(ω) = D2 πd 4 G(ω) T = 0, n = 1, W = 2D = 1, continous disorder, geometrical averaging Anderson insulator c A crossover regime line of vanishing Hubbard subbands Metal MH c1 MH c2 coexistence regime Mott insulator U U - interaction, - disorder

29 Summary Conductors and insulators, classification Transitions between conductors and insulators Mott Hubbard MIT at n = 1 Mott Hubbard MIT at n 1 alloy band splitting Mott Hubbard MIT in alloy subband Optical lattices possible realization Anderson and Mott MIT at n = 1