031009 Juelich 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” 3

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

Fast variable and slow variable H+2 molecule e－ electron; {nuclei} nuclei move thousands of times slower than the electron Instead of solving time-dependent Schroedinger eq. , one uses Born-Oppenheimer approximation • “Slow variables Ri” are treated as parameters λ(t) (Kinetic energies from Pi are neglected) • solve time-independent Schroedinger eq. “snapshot” solution 5

Adiabatic evolution of a quantum system • Energy spectrum: E(λ(t)) • After a cyclic evolution n+1 x n-1 0 Dynamical phase λ(t) • Phases of the snapshot states at different λ’s are independent and can be arbitrarily assigned • Do we need to worry about this phase? 6

No! Pf : • Fock, Z. Phys 1928 • Schiff, Quantum Mechanics (3 rd ed. ) p. 290 Consider the n-th level, Stationary, snapshot state ≡An(λ) Redefine the phase, An’(λ) An(λ) Choose a (λ) such that, An’(λ)=0 Thus removing the extra phase 7

One problem: does not always have a well-defined (global) solution Vector flow Contour of is not defined here Contour of C C 8

M. Berry, 1984 : Parameter-dependent phase NOT always removable! • Berry phase (path dependent) Index n neglected Berry’s face • Interference due to the Berry phase Phase difference a C 2 a b －2 interference 1 1 9

Some terminology • Berry connection (or Berry potential) • Stokes theorem (3 -dim here, can be higher) S • Berry curvature (or Berry field) C • Gauge transformation (Nonsingular gauge, of course) Redefine the phases of the snapshot states Berry curvature nd Berry phase not changed 10

Analogy with magnetic monopole Berry potential (in parameter Vector potential (in real space) Berry field (in 3 D) Magnetic field Berry phase Magnetic flux Chern number Dirac monopole 11

Example: spin-1/2 particle in slowly changing B field • Real space • Parameter space C C Level crossing at B=0 (a monopole at the origin) Berry curvature E(B) B Berry phase spin × solid angle 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? base space fiber space Fiber bundle Examples: • Trivial fiber bundle (= a product space) • Nontrivial fiber bundle Simplest example: Möbius band R 1 x R 1 fiber R 1 base 15

Fiber bundle and quantum state evolution (Wu and Yang, PRD 1975) Fiber space: inner DOF, eg. , U(1) phase Base space: parameter space • Berry phase = Vertical shift along fiber (U(1) anholonomy) • Chern number n For fiber bundle χ=2 χ=0 ～ Euler characteristic χ For 2 -dim closed surface χ =－216

u Introduction u Berry phase in solid state physics 19

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Berry phase in condensed matter physics, a partial list: v 1982 Quantized Hall conductance (Thouless et al) v 1983 Quantized charge transport (Thouless) v 1984 Anyon in fractional quantum Hall effect (Arovas et al) v 1989 Berry phase in one-dimensional lattice (Zak) v 1990 Persistent spin current in one-dimensional ring (Loss et al) v 1992 Quantum tunneling in magnetic cluster (Loss et al) v 1993 Modern theory of electric polarization (King-Smith et al) v 1996 Semiclassical dynamics in Bloch band (Chang et al) v 1998 Spin wave dynamics (Niu et al) v 2001 Anomalous Hall effect (Taguchi et al) v 2003 Spin Hall effect (Murakami et al) v 2004 Optical Hall effect (Onoda et al) v 2006 Orbital magnetization in solid (Xiao et al) v… 21

Berry phase in condensed matter physics, a partial list: v 1982 Quantized Hall conductance (Thouless et al) v 1983 Quantized charge transport (Thouless) v 1984 Anyon in fractional quantum Hall effect (Arovas et al) v 1989 Berry phase in one-dimensional lattice (Zak) v 1990 Persistent spin current in one-dimensional ring (Loss et al) v 1992 Quantum tunneling in magnetic cluster (Loss et al) v 1993 Modern theory of electric polarization (King-Smith et al) v 1996 Semiclassical dynamics in Bloch band (Chang et al) v 1998 Spin wave dynamics (Niu et al) v 2001 Anomalous Hall effect (Taguchi et al) v 2003 Spin Hall effect (Murakami et al) v 2004 Optical Hall effect (Onoda et al) v 2006 Orbital magnetization in solid (Xiao et al) v… 22

Berry phase in solid state physics v Persistent spin current v Quantum tunneling in a magnetic cluster v Modern theory of electric polarization v Semiclassical electron dynamics ü Quantum Hall effect (QHE) ü Anomalous Hall effect (AHE) ü Spin Hall effect (SHE) Spin • Persistent spin current • Quantum tunneling Bloch state • AHE • Electric polarization • SHE • QHE 23

Electric polarization of a periodic solid ü well defined only for finite system (sensitive to boundary) ü 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 … + － … P Choice 2 － … + • However, the change of P is well-defined ΔP … … Experimentally, it’s ΔP that’s measured … 33

Modern theory of polarization One-dimensional lattice (λ=atomic displacement in a unit cell) Iℓℓ-defined Resta, Ferroelectrics 1992 However, d. P/dλ is well-defined, even for an infinite system ! King-Smith and Vanderbilt, PRB 1993 Berry potential • For a one-dimensional lattice with inversion symmetry (if the origin is a symmetric point) (Zak, PRL 1989) • Other values are possible without inversion symmetry 34

Berry phase and electric polarization g 1=5 g 2=4 Dirac comb model … … 0 b a Lowest energy band: γ 1 Rave and Kerr, EPJ B 2005 ← 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, R 107 (2000) 35

Berry phase in solid state physics v Persistent spin current v Quantum tunneling in a magnetic cluster v Modern theory of electric polarization v Semiclassical electron dynamics ü Quantum Hall effect ü Anomalous Hall effect ü Spin Hall effect 36

Semiclassical dynamics in solid Limits of validity: one band approximation E(k) n+1 x Negligible inter-band transition. n x “never close to being violated in a metal” n-1 0 2π • Lattice effect hidden in En (k) • Derivation is harder than expected 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 … 37

Semiclassical dynamics - wavepacket approach 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 3. Minimize the action Seff[rc(t), kc(t)] and determine the trajectory (rc(t), kc(t)) → Euler-Lagrange equations Wavepacket in Bloch band: Berry potential (Chang and Niu, PRL 1995, PRB 1996) 38

Semiclassical dynamics with Berry curvature “Anomalous” velocity Berry curvature Cell-periodic Bloch state Wavepacket energy Bloch energy Zeeman energy due to spinning wavepacket If B=0, then dk/dt // electric field Simple and Unified → Anomalous velocity ⊥ electric field • (integer) Quantum Hall effect • (intrinsic) Anomalous Hall effect • (intrinsic) Spin Hall effect 39

u Why the anomalous velocity is not found earlier? In fact, it had been found by • Adams, Blount, in the 50’s u Why it seems OK not to be aware of it? For scalar Bloch state (non-degenerate band): • Space inversion symmetry both symmetries • Time reversal symmetry u When do we expect to see it? • SI symmetry is broken ← electric polarization • TR symmetry is broken ← QHE • spinor Bloch state (degenerate band) ← SHE Also, • band crossing ← monopole 40

Berry phase in solid state physics v Persistent spin current v Quantum tunneling in a magnetic cluster v Modern theory of electric polarization v Semiclassical electron dynamics ü Quantum Hall effect ü Anomalous Hall effect ü Spin Hall effect 41

Quantum Hall effect (von Klitzing, PRL 1980) classical 2 DEG σH (in e 2/h) 3 quantum 2 1 Increasing B 1/B Each LL contributes one e 2/h z Al. Ga. As EF Increasing B LLs EF energy CB B=0 2 DEG Density of states VB 42

Semiclassical formulation Equations of motion (In one Landau subband) Magnetic field effect is hidden here Hall conductance Quantization of Hall conductance (Thouless et al 1982) Remains quantized even with disorder, e-e interaction (Niu, Thouless, Wu, PRB, 1985) 43

Quantization of Hall conductance (II) For a filled Landau subband Brillouin zone 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) 44

Berry curvature and Hofstadter spectrum 2 DEG in a square lattice + a perpendicular B field (Hofstadter, PRB 1976) Landau subband energy Width of a Bloch band when B=0 tight-binding model: LLs Magnetic flux (in Φ 0) / plaquette 45

Bloch energy E(k) Berry curvature Ω(k) C 1 = 1 C 2 = 2 C 3 = 1 46

Re-quantization of semiclassical theory Bohr-Sommerfeld quantization 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, d. Hv. A effect • … Now with Berry phase effect! 48

cyclotron orbits (LLs) in graphene ↔ QHE in graphene E Dirac cone B … ρL σH Cyclotron orbits k Novoselov et al, Nature 2005 49

Berry phase in solid state physics v Persistent spin current v Quantum tunneling in a magnetic cluster v Modern theory of electric polarization v Semiclassical electron dynamics ü Quantum Hall effect ü Anomalous Hall effect ü Spin Hall effect Mokrousov’s talks this Friday Buhmann’s next Thu QSHE) (on Poor men’s, and women’s, version of QHE, AHE, and SHE 50

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 Ingredients required for a successful theory: • magnetization (majority spin) Anomalous term • spin-orbit coupling (to couple the majority-spin direction to transverse orbital direction) 51

Intrinsic mechanism (ideal lattice without impurity) • Linear response • Spin-orbit coupling • magnetization gives correct order of magnitude of ρH for Fe, also explains that’s observed in some data 52

Smit, 1955: KL mechanism should be annihilated by (an extra effect from) impurities Alternative scenario: • Skew scattering (Smit, Physica 1955) ～ Mott scattering Spinless impurity Extrinsic mechanisms (with impurities) e－ • Side jump (Berger, PRB 1970) anomalous velocity due to electric field of impurity ～ anomalous velocity in KL (Crépieux and Bruno, PRB 2001) e－ 2 (or 3) mechanisms: In reality, it’s not so clear-cut ! Review: Sinitsyn, J. Phys: Condens. Matter 20, 023201 (2008) 53

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…” 54

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 55

Old wine in new bottle Berry curvature of fcc Fe (Yao et al, PRL 2004) Karplus-Luttinger theory (1954) = Berry curvature theory (2001) → intrinsic AHE • same as Kubo-formula result • ab initio calculation Ideal lattice without impurity 56

• classical Hall effect charge EF B ü Lorentz force +++++++ ↑↓ y 0 L • anomalous Hall effect ↑↑↑↑ ü Berry curvature EF ↑ ↑↑↑↑↑↑↑ ↓ B charge spin ü Skew scattering y ↑↑ ↑↑↑↑ 0 L • spin Hall effect ↑↑↑↑↑↑↑ No magnetic field required ! EF ↑ ↑↑↑↑ ü Berry curvature ↓ ↑↑↑↑↑↑↑ spin ü Skew scattering y 0 L 57

Murakami, Nagaosa, and Zhang, Science 2003: Intrinsic spin Hall effect in semiconductor Band structure • Spin-degenerate Bloch state due to Kramer’s degeneracy → Berry curvature becomes a 2 x 2 matrix (non-Abelian) • (from Luttinger model) curvature for HH/LH Berry 4 -band Luttinger model The crystal has both space inversion symmetry and time reversal symmetry ! Spin-dependent transverse velocity → SHE for holes 58

Only the HH/LH can have SHE? : Not really • Berry curvature for conduction electron: 8 -band Kane model spin-orbit coupling strength • Berry curvature for free electron (!): Dirac’s theory electron mc 2 positron Chang and Niu, J Phys, Cond Mat 2008 59

Observations of SHE (extrinsic) Science 2004 Nature 2006 Nature Material 2008 Observation of Intrinsic SHE? 60

• Summary Spin • Persistent spin current • Quantum tunneling Bloch state • AHE • Electric polarization • SHE • QHE • Three fundamental quantities in any crystalline solid E(k) Berry curvature Ω(k) Bloch energy L(k) Orbital moment (Not in this talk) 61

Thank you! 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? ) 63