Integer Quantum Hall effect basics theories for the quantization - PowerPoint PPT Presentation

Integer Quantum Hall effect • basics • theories for the quantization • disorder in QHS • Berry phase in QHS • topology in QHS • effect of lattice • effect of spin and electron interaction Dept of Phys M.C. Chang

Hall effect ( 1879 ), a classical analysis � � � � � dv v v * = − − × − * m eE e B m τ dt c � � � = = ˆ; / 0 at steady state B Bz dv dt • Hall resistivity ⎛ ⎞ ⎛ ⎞ ⎛ τ ⎞ * / / v E m eB c = − x x ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ e − * τ / / v E ⎝ ⎠⎝ ⎠ ⎝ ⎠ eB c m y y � � = − ⎛ ⎞ j env * m B ⎜ ⎟ ⎛ ⎞ ⎛ ⎞ ω τ ⎛ ⎞ ⎛ ⎞ 1 * τ E 2 j j m eB ne nec ρ = τ ω = ⎜ ⎟ 2 , = = ρ x x x c ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 * c 0 − ω τ n e m c ⎜ ⎟ 1 * E j ⎝ ⎠ j ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ B m − y y y c ⎜ ⎟ ⎝ τ ⎠ 2 nec ne ρ xy • Hall conductivity − ω τ ⎛ ⎞ σ 1 τ 2 ne = − = 1 σ ρ 0 c σ = ⎜ ⎟ ( ) ω τ 0 2 1 * B + ω τ ⎝ ⎠ 1 m c c − ω τ ⎛ ⎞ 1 ω τ << ⎯⎯⎯ 1 → σ c ⎜ ⎟ c 0 ω τ 1 ⎝ ⎠ c − ⎛ 0 / ⎞ nec B ω τ >> ⎯⎯⎯ → 1 ⎜ ⎟ c / 0 ⎝ ⎠ nec B

y Resistance and conductance W x L Σ Σ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ V R R I I V = = x xx xy x x xx xy x , ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ Σ Σ V R R I I V ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ y yx yy y y yx yy y Note: So it’s possible to have R xx and Σ xx Σ = yy R simultaneously be zero (provided R xy and Σ xy Σ xx det are nonzero). V E W L W = ρ ≡ = = ρ 3 : y y D R R xx xx yx yx A I J A A = x 0 x I y E W L = ρ = = ρ 2 : y D R R xx xx yx yx W J W x ρ = ρ = 0, . Quantum Hall effect const xx yx ρ 1 1 R → Σ = = = = = σ yx yx ρ ρ yx det det yx R R yx yx

Measurement of Hall resistance 2-dim electron gas (2DEG)

GaAs/AlGaAs heterojunction Dynamics along z- direction is frozen in the ground state subband (broadened) Landau levels in a magnetic μ field Energy

Effect of disorder on σ xy ( theoretical prediction before 1980 ) Si(100) MOS inversion layer 9.8 T, 1.6 K Ando, Matsumoto, and Uemura JPSJ 1975 Kawaji et al, Supp PTP 1975

Quantum Hall effect ( von Klitzing, 1980 ) 1985 ρ xy deviates from (h/e 2 )/n by less than 3 ppm on the very first report. • This result is independent of the shape/size of sample. • Different materials lead to the same effect ( Si MOSFET, GaAs heterojunction …) → a very accurate way to measure α -1 = h/e 2 c = 137.036 (no unit) → a very convenient resistance standard.

An accurate and stable resistance standard (1990) • experiment Kinoshita, • theory Phys. Rev. Lett. 1995

Condensed matter physics is physics of dirt - Pauli clean dirty • Flux quantization h φ = 0 2 e • Quantum Hall effect • … Often protected by topology, but not vice versa.

e The triangle of quantum metrology (to be realized) QCP f I Josephson QHE effect h / e 2 e / h V

Quantum Hall effect requires • Two-dimensional electron gas • strong magnetic field < � ω ( ) k T • low temperature B c Note: Room Temp QHE in graphene ( Novoselov et al, Science 2007 ) Aoki, CMST 2011 Plateau and the importance of disorder Broadened LL due to disorder The importance of localized states Why R H has to be exactly (h/e 2 )/n ? • see Laughlin’s argument below Filling factor

Width of extended states? 256 states in the LLL. ε ( Φ ) periodic in Φ 0 Aoki 1983

• Finite-Size Scaling Exponent for correlation length Huo and Bhatt PRL 1992 − Δ = ν ~ , 1/ 2 x E N x States that can carry Hall current (with non-zero Δ E Δ E Chern number ) Ensemble average over • experiment 100-2000 disorder configurations Li et al PRL 2005

Quantization of Hall conductance , Laughlin’s gauge argument (1981) 1 � 2 ⎛ ⎞ 1 � � � e ∑ = + + + ( ) ( ) ⎜ ⎟ H p A r V r V i i i e e 2 ⎝ ⎠ m c i • Simulate a longitudinal EMF by a fictitious time-dependent flux Φ − ∂ ⎡ ⎤ 1 � � e ∑ e = + ( ) j A r ⎢ ⎥ ∂ x x i ⎣ ⎦ m L L i x c i x y ∂ ∂ c H c H Φ = − = − Φ = A L ∂ ∂Φ L L A L x x x y x y ψ >= ψ > | | H E solve Φ Φ Φ Φ By the Hellman-Feynman theorem, one has y ∂ ∂ ∂ H E x < ψ ψ >= < ψ ψ >= Φ Φ | | | | H Φ Φ Φ Φ Φ ∂Φ ∂Φ ∂Φ ∂ c E ∴ = − Φ x j ∂Φ L y • Due to gauge symmetry, the system needs to be invariant under Φ→ Φ + Φ 0 , • E F at localized states, no charge transfer whatever Φ is. − 2 ( ) V n e e E • E F at extended states, only integer charges may transfer = − = y j c n h Φ x y along y when Φ is changed by one Φ 0. L 0 y

Edge state in quantum Hall system • Classical picture • Bending of LLs Chiral edge state Gapless excitations at the edges (skipping orbit) • Robust against disorder (no back-scattering) • number of edge modes = n

Inclusion of lattice (more details later) • Bulk states: E n (k x ,k y ) (projected to k y ); Edge states: E n (k y ) Figs from Hatsugai’s ppt • when the flux is changed by 1 Φ 0 , the states should come back. → Only integer charges can be transported.

2 Streda formula ( 1982 ) Nonzero L = ∇ × ˆ c M z along edge R Degeneracy of a LL: D=BA/ Φ 0 • If ν bands are filled, then the number of electrons per unit area is n= ν eB/hc ∴ σ H = ν e 2 /h Giuliani and Vignale, Sec 10.3.3

Current response: conductivity � � ∂ 1 ( ) A t • Vector potential of = − ( ) E t ∂ an uniform electric field c t � � � � � � ω i = − ω = − ω = ( ) , then ( ) ; i t i t E t E e A t A e E A ω ω ω ω c � � 2 ⎛ ⎞ 1 � � e e = + + = + ⋅ + 2 ( ) H ⎜ p A ⎟ V H A p O A 0 latt 2 ⎝ ⎠ m c m c 0 0 � � e − ω = ⋅ ' i t H p A e ω m c 0 = σ • 1st order perturbation in E → j E α αβ β α β − 2 f f v v e ∑ Kubo-Greenwood σ ω = ( ) � � � m m m α β ω ω + ω formula � iV � � � m m m ω ≡ ω − ω α ≡ ψ α ψ , v v � � � � m m m m

3 Quantization of Hall conductance Thouless et al’s argument (1982) α β − β α 2 1 e p p p p ∑ σ = � � � � ℓ , m = (n, k ) DC m m m m f α ≠ β � ω 2 2 � im V � 0 m � m ⎛ ⎞ ∂ ∂ ∂ ∂ 2 2 u u u u e ∑ α 1 � = − ⎜ ⎟ p nk nk nk nk f ∂ + � = � ⎜ ⎟ m u k u ∂ ∂ ∂ ∂ � nk α α � i V k k k k m ⎝ ⎠ m m i α β β α nk 0 0 ∂ ε ∂ u δ + ω • Berry curvature = � � u � � ∂ ∂ m m m � k k ⎛ ⎞ α α � ∂ ∂ ∂ ∂ u u u u Ω ≡ ⎜ − ⎟ ( ) nk nk nk nk k i ⎜ ⎟ cell-periodic γ ∂ ∂ ∂ ∂ n k k k k ⎝ ⎠ α β β α function u m α β γ ( , , are cyclic) • Berry curvature (for n-th band) • Hall conductivity for the n-th band � � Ω = ∇ × ∇ � ⎡ ⎤ ( ) 2 k i � u � u 1 e ( ) ( ) ∫ n n n σ = Ω 2 k k ( ) ⎢ ⎥ � � d k k π H 2 n = ∇ × n z ( ) h ⎣ ⎦ � A k BZ n k • Berry connection � an integer for a � ≡ ∇ � ( ) A k i u u filled band n n n k

d c � 1 ∫ Ω = 2 ( ) intege r d k k n π 2 z Brillouin B Z zone g y � ∫ ∇ × 2 Pf: d k A a b BZ � � � � � � � � g x ∫ b ∫ c ∫ d ∫ a = ⋅ + ⋅ + ⋅ + ⋅ dk A dk A dk A dk A a b c d ∫ ∫ ⎡ ⎤ ⎡ ⎤ = − + − ( ,0) ( , ) ( , ) (0, ) dk A k A k g dk A g k A k ⎣ ⎦ ⎣ ⎦ x x x x x y y y x y y y → ↑ θ ( ) θ = = ( ) i k , i k 1 u � e y u � u � e 2 u � x + ˆ + ˆ k k g x k k g y x y ∫ ⎡ ⎤ − = θ − θ ( ,0) ( , ) ( ) ( ) dk A k A k g a b ⎣ ⎦ 2 2 x x x x x y → Zeros and vortices � etc θ = ( ) i a u e 1 u � a b ∫ ∇× 2 θ d k A = ( ) i b u e 2 u b c BZ − θ = ( ) i d u e 1 u = θ − θ + θ − θ ( ) ( ) ( ) ( ) a b d a c d 2 2 1 1 − θ = ( ) i a = π u e 2 u 2 n BZ d a total vorticity in the BZ [ ] θ + θ − θ − θ ( ) ( ) ( ) ( ) ∴ = i a b d a 1 2 1 2 u e u a a • Niu-Thouless-Wu generalization to system with disorder and electron interaction (PRB 1985). Czerwinski and Brown, PRS (London) 1991