Berry phase in solid state physics

03/10/09 @ Juelich Berry phase in solid state physics - a selected overview Ming-Che Chang Department of Physics National Taiwan Normal University Qian Niu Department of Physics The University of Texas at Austin 1

Taiwan 2

Paper/year with the title Berry phase or geometric phase 80 70 60 50 40 30 20 10 0 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 3

Introduction (30-40 mins) Quantum adiabatic evolution and Berry phase Electromagnetic analogy Geometric analogy Berry phase in solid state physics 4

Fast variable and slow variable H + 2 molecule H ( r, p;, ) { R } i Pi e - electron; {nuclei} Born-Oppenheimer approximation Slow variables R i are treated as parameters λ(t) (Kinetic energies from P i are neglected) solve time-independent Schroedinger eq. H ( r, p; λψ ) ( x) = E ψ ( x) nuclei move thousands of times slower than the electron Instead of solving time-dependent Schroedinger eq., one uses n, λ n, λ n, λ snapshot solution 5

Adiabatic evolution of a quantum system Hrpλ (, ; ) Energy spectrum: E(λ(t)) x x 0 λ(t) n+1 n n-1 After a cyclic evolution λ( T ) = λ(0) ψ T i dt ' En ( t') 0 e = ψ n, λ( T ) n, λ( 0) Dynamical phase Phases of the snapshot states at different λ s are independent and can be arbitrarily assigned ψ iγn ( λ) e ψ n, λ( t) n, λ( t) Do we need to worry about this phase? 6

No! Fock, Z. Phys 1928 Schiff, Quantum Mechanics (3rd ed.) p.290 Pf : Consider the n-th level, Redefine the phase, iγ ( ) i dt' En ( t') n λ 0 t e e λ Ψ () = HΨ () t = i Ψ () t λ λ t γ ψ ψ λ λ t ψ n, λ n = i 0 n, λ n, λ ψ ' A n (λ) i n ( ) e φ = λ ψ n, λ n, λ φ = A n n (λ) λ A n (λ) Stationary, snapshot state Hψ = Eψ n, λ n n, λ Choose a φ (λ) such that, A n (λ)=0 Thus removing the extra phase 7

Contour of φ One problem: = A( λ ) λ φ Vector flow A does not always have a well-defined (global) solution Vector flow A φ is not defined here Contour of φ C C A dλ = 0 A dλ 0 C C 8

M. Berry, 1984 : Parameter-dependent phase NOT always removable! ψ iγ i dt' E( t') C 0 e e = ψ λ( T ) λ(0) Berry phase (path dependent) γ ψ C i ψ = d λ C λ λ λ T Index n neglected 0 Berry s face Interference due to the Berry phase ( ) C 1 2 if = + 0, then = + 0 1 2 1 2 Phase difference 1 1 a a b C 2-2 interference 9

Some terminology Berry connection (or Berry potential) A( λ ) i ψ ψ λ γ C C λ Stokes theorem (3-dim here, can be higher) = A dλ = A da S λ λ λ() t λ 3 S λ 2 Berry curvature (or Berry field) F( ) A( ) = i λ λ ψ ψ λ λ λ λ λ λ 1 C Gauge transformation (Nonsingular gauge, of course) i i i i iφ( λ) ψ e ψ λ λ A( λ) A( λ) λφ F( λ) F( λ) γ C γ C Redefine the phases of the snapshot states Berry curvature nd Berry phase not changed 10

Analogy with magnetic monopole Berry potential (in parameter space) A( λ ) i ψ ψ λ λ Berry field (in 3D) F( λ) A( λ) λ λ Vector potential (in real space) A( r ) Magnetic field Br ( ) Ar ( ) Berry phase A C ( ) d γ = λ λ C = F da S Chern number 1 ( ) integer 2 F π λ da = S Magnetic flux Φ = = S A( r) dr C B da Dirac monopole 1 ( ) integer 4 B π λ da = S 11

Example: spin-1/2 particle in slowly changing B field Real space z Parameter space B z x B() t H S C = μ B BB σ λ = y B x B() t C B y (a monopole at the origin) Level crossing at B=0 E(B) B + Berry curvature 1 F± ( B) = i Bψ, Bψ = ± B ±, B 2 Berry phase 1 γ ± = F da ( C) S ± = Ω 2 spin solid angle Bˆ B 2 12

Experimental realizations : Bitter and Dubbers, PRL 1987 Tomita and Chiao, PRL 1986 13

Geometry behind the Berry phase Why Berry phase is often called geometric phase? Examples: base space Trivial fiber bundle (= a product space) Fiber bundle fiber space Nontrivial fiber bundle Simplest example: Möbius band R 1 x R 1 fiber R 1 R 1 base 14

Fiber bundle and quantum state evolution (Wu and Yang, PRD 1975) Fiber space: inner DOF, eg., U(1) phase Base space: parameter space φ Berry phase = Vertical shift along fiber (U(1) anholonomy) Chern number n n 1 = da π F 2 S For fiber bundle χ = 2 χ = 0 ~ Euler characteristic χ χ 1 = 2π da G S For 2-dim closed surface χ =-2 15

Introduction Berry phase in solid state physics 16

Berry phase in condensed matter physics, a partial list: 1982 Quantized Hall conductance (Thouless et al) 1983 Quantized charge transport (Thouless) 1984 Anyon in fractional quantum Hall effect (Arovas et al) 1989 Berry phase in one-dimensional lattice (Zak) 1990 Persistent spin current in one-dimensional ring (Loss et al) 1992 Quantum tunneling in magnetic cluster (Loss et al) 1993 Modern theory of electric polarization (King-Smith et al) 1996 Semiclassical dynamics in Bloch band (Chang et al) 1998 Spin wave dynamics (Niu et al) 2001 Anomalous Hall effect (Taguchi et al) 2003 Spin Hall effect (Murakami et al) 2004 Optical Hall effect (Onoda et al) 2006 Orbital magnetization in solid (Xiao et al) 18

Berry phase in solid state physics Persistent spin current Quantum tunneling in a magnetic cluster Modern theory of electric polarization Semiclassical electron dynamics Quantum Hall effect (QHE) Anomalous Hall effect (AHE) Spin Hall effect (SHE) Persistent spin current Quantum tunneling Spin AHE SHE Bloch state Electric polarization QHE 20

Electric polarization of a periodic solid 1 = V r ρ( ) 3 P d r r well defined only for finite system (sensitive to boundary) or, for crystal with well-localized dipoles (Claussius-Mossotti theory) P is not well defined in, e.g., covalent crystal: P Unit cell Choice 1 + - Choice 2 - + P However, the change of P is well-defined ΔP Experimentally, it s ΔP that s measured 21

Modern theory of polarization One-dimensional lattice (λ=atomic displacement in a unit cell) P q λ λ = ψnk r ψnk L nk Ill-defined λ ir ( ) k λ ψ r = e u () r nk nk Resta, Ferroelectrics 1992 However, dp/dλ is well-defined, even for an infinite system! where dp Δ P = dλ = P( λ2) P( λ1) dλ P ( λ ) = q = q n For a one-dimensional lattice with inversion symmetry (if the origin is a symmetric point) γ n n γ n 2π BZ dk u 2π = 0 or π (Zak, PRL 1989) Other values are possible without inversion symmetry λ nk i k u λ nk Berry potential King-Smith and Vanderbilt, PRB 1993 22

Berry phase and electric polarization g 1 =5 g 2 =4 Dirac comb model 0 b a Rave and Kerr, EPJ B 2005 Lowest energy band: γ 1 1 P1 q γ 2π Δ = Δ g 2 =0 γ 1 =π similar formulation in 3-dim using Kohn-Sham orbitals r =b/a Review: Resta, J. Phys.: Condens. Matter 12, R107 (2000) 23

Semiclassical dynamics in solid Limits of validity: one band approximation Negligible inter-band transition. never close to being violated in a metal dk = ee er B dt dr 1 En = dt k E(k) Lattice effect hidden in E n (k) Derivation is harder than expected x x 0 2π n+1 n n-1 Explains (Ashcroft and Mermin, Chap 12) Bloch oscillation in a DC electric field, quantization Wannier-Stark ladders cyclotron motion in a magnetic field, quantization LLs, de Haas - van Alphen effect 25

Semiclassical dynamics - wavepacket approach rw r () c t kw k () c t 1. Construct a wavepacket that is localized in both r-space and k-space (parameterized by its c.m.) 2. Using the time-dependent variational principle to get the effective Lagrangian for the c.m. variables Leff ( rc, kc; r c, kc) = W i H W t 3. Minimize the action S eff [r c (t),k c (t)] and determine the trajectory (r c (t), k c (t)) Euler-Lagrange equations Wavepacket in Bloch band: L = k ea r + k E r k ( ) Rn (, ) eff c c c n c c Berry potential (Chang and Niu, PRL 1995, PRB 1996) 26

Simple and Unified Semiclassical dynamics with Berry curvature dk dt dr dt = ee er B 1 E n = k Ω k Wavepacket energy n ( k) (integer) Quantum Hall effect (intrinsic) Anomalous Hall effect (intrinsic) Spin Hall effect Anomalous velocity 0 e En( rc, kc) = En( kc) + n( kc) B 2m L Bloch energy If B=0, then dk/dt // electric field Berry curvature Anomalous velocity electric field ( k Ω ) = i u u n k nk k Cell-periodic Bloch state nk Zeeman energy due to spinning wavepacket Ln ( k c ) = mw( r rc ) vw 27

Why the anomalous velocity is not found earlier? In fact, it had been found by Adams, Blount, in the 50 s Also, Why it seems OK not to be aware of it? For scalar Bloch state (non-degenerate band): Space inversion symmetry Time reversal symmetry Ω ( k) = Ω ( k) n Ω ( k) = Ω ( k) When do we expect to see it? SI symmetry is broken TR symmetry is broken spinor Bloch state (degenerate band) band crossing n n n Ω n both symmetries Ω n( k ) = 0, k ( k ) 0 electric polarization QHE SHE monopole 28

Quantum Hall effect (von Klitzing, PRL 1980) 2 DEG σ H (in e 2 /h) 3 2 1 quantum Increasing B classical 1/B Each LL contributes one e 2 /h CB AlGaAs z GaAs E F E F B=0 Increasing B LLs 2 DEG energy VB Density of states 30

Semiclassical formulation Equations of motion 2 dk J = e r (In one Landau subband) 2 (2 π ) filled dk 2 2 e d k = ee =0 E Ω( k ) 2 dt (2π ) filled dr 1 E = k Ω( k) 2 e 1 2 dt k Jx = d k Ωz ( k ) E h 2π filled Magnetic field effect is hidden here Hall conductance σ H y Quantization of Hall conductance (Thouless et al 1982) 1 d 2 z 2π B Z σ = H k Ω ( k ) = integer n 2 e n h Remains quantized even with disorder, e-e interaction (Niu, Thouless, Wu, PRB, 1985) 31

Quantization of Hall conductance (II) Brillouin zone For a filled Landau subband 2 dkω z ( k) = dk A BZ BZ Counts the amount of vorticity in the BZ due to zeros of Bloch state (Kohmoto, Ann. Phys, 1985) In the language of differential geometry, this n is the (first) Chern number that characterizes the topology of a fiber bundle (base space: BZ; fiber space: U(1) phase) 32

Berry curvature and Hofstadter spectrum 2DEG in a square lattice + a perpendicular B field tight-binding model: (Hofstadter, PRB 1976) Width of a Bloch band when B=0 energy LLs Landau subband Φ 1 = Φ 3 0 Magnetic flux (in Φ 0 ) / plaquette 33

Bloch energy E(k) Berry curvature Ω(k) C 1 = 1 C 2 = 2 C 3 = 1 34

Re-quantization of semiclassical theory L = k ea r + k R E r k eff ( ) (, ) Bohr-Sommerfeld quantization 1 1 γ ( Cm) eb ( ) ˆ 2 2 k dk dz = π m+ 2 2 C m π Berry phase γ ( C ) = R dk m C m Would shift quantized cyclotron energies (LLs) Bloch oscillation in a DC electric field, re-quantization Wannier-Stark ladders cyclotron motion in a magnetic field, re-quantization LLs, dhva effect Now with Berry phase effect! 35

cyclotron orbits (LLs) in graphene QHE in graphene E Dirac cone B Cyclotron orbits k Ω ( k) = πδ( k) γc = π ρ L σ H Ek ( ) = ν Fk π k γ C eb = 2π m+ 2 2π 2 1 Novoselov et al, Nature 2005 1 γ C En = vf 2eB n+ 2 2π 36

Berry phase in solid state physics Persistent spin current Quantum tunneling in a magnetic cluster Modern theory of electric polarization Semiclassical electron dynamics Quantum Hall effect Anomalous Hall effect Spin Hall effect Poor men s, and women s, version of QHE, AHE, and SHE Mokrousov s talks this Friday Buhmann s next Thu (on QSHE) 37

Anomalous Hall effect (Edwin Hall, 1881): Hall effect in ferromagnetic (FM) materials FM material ρ H saturation slope=r N R AH M S H ρ ρ The usual Lorentz force term H AH = R H + ρ N ( H) R AH AH Anomalous term ( H ), M( H) Ingredients required for a successful theory: magnetization (majority spin) spin-orbit coupling (to couple the majority-spin direction to transverse orbital direction) 38

Intrinsic mechanism (ideal lattice without impurity) Linear response Spin-orbit coupling magnetization gives correct order of magnitude of ρ H for Fe, also explains ρ AH that s observed in some data ρ 2 L 39

Smit, 1955: KL mechanism should be annihilated by (an extra effect from) impurities Alternative scenario: Extrinsic mechanisms (with impurities) Skew scattering (Smit, Physica 1955) ~ Mott scattering Spinless impurity e - Side jump (Berger, PRB 1970) ρ AH ρ L anomalous velocity due to electric field of impurity ~ anomalous velocity in KL (Crépieux and Bruno, PRB 2001) e - ρ AH δ 1A ρ 2 L 2 (or 3) mechanisms: In reality, it s not so clear-cut! ρ = am ( ) ρ + bm ( ) ρ 2 AH L L Review: Sinitsyn, J. Phys: Condens. Matter 20, 023201 (2008) 40

CM Hurd, The Hall Effect in Metals and Alloys (1972) The difference of opinion between Luttinger and Smit seems never to have been entirely resolved. 30 years later: Crepieux and Bruno, PRB 2001 It is now accepted that two mechanisms are responsible for the AHE: the skew scattering and the side-jump 41

However, Science 2001 Science 2003 And many more Karplus-Luttinger mechanism: Mired in controversy from the start, it simmered for a long time as an unsolved problem, but has now re-emerged as a topic with modern appeal. Ong @ Princeton 42

Old wine in new bottle Berry curvature of fcc Fe (Yao et al, PRL 2004) Karplus-Luttinger theory (1954) = Berry curvature theory (2001) σ AH 2 3 e dk = Ω( k ) 0 3 (2π ) filled intrinsic AHE same as Kubo-formula result ab initio calculation Ideal lattice without impurity 43

classical Hall effect B E F charge +++++++ Lorentz force 0 L y anomalous Hall effect B Berry curvature Skew scattering E F charge spin 0 L y spin Hall effect spin Berry curvature Skew scattering E F y No magnetic field required! 0 L 44

Murakami, Nagaosa, and Zhang, Science 2003: Intrinsic spin Hall effect in semiconductor Band structure 4-band Luttinger model The crystal has both space inversion symmetry and time reversal symmetry! Spin-degenerate Bloch state due to Kramer s degeneracy Berry curvature becomes a 2x2 matrix (non-abelian) (from Luttinger model) Berry curvature for HH/LH dx dt Ω HH 3 kˆ ( k ) = σ 2 2 k En ( k) e = k z E Ω Spin-dependent transverse velocity SHE for holes n 45

Only the HH/LH can have SHE? : Not really 8-band Kane model Berry curvature for conduction electron: 1 Ω= α σ+ Ok ( ) spin-orbit coupling strength Dirac s theory mc 2 electron positron Chang and Niu, J Phys, Cond Mat 2008 Berry curvature for free electron (!): Ω= + 2 λ 2 C Ok 1 σ ( ) λ C = / mc 12 10 m 46

Observations of SHE (extrinsic) Science 2004 Nature 2006 Nature Material 2008 Observation of Intrinsic SHE? 47

Summary Spin Bloch state Persistent spin current Quantum tunneling AHE SHE Electric polarization QHE Three fundamental quantities in any crystalline solid E(k) Bloch energy Berry curvature Ω(k) L(k) Orbital moment (Not in this talk) 48

g{tç~ çéâ4 Slides : http://phy.ntnu.edu.tw/~changmc/paper Reviews: Chang and Niu, J Phys Cond Matt 20, 193202 (2008) Xiao, Chang, and Niu, to be published (RMP?) 49