Reciprocal Space Magnetic Field: Physical Implications

Reciprocal Space Magnetic Field: Physical Implications Junren Shi ddd Institute of Physics Chinese Academy of Sciences November 30, 2005

Outline Introduction Implications Conclusion 1 Introduction 2 Physical implications of the reciprocal space magnetic field Design novel transport devices Inhomogeneous phase space Effective quantum mechanics 3 Conclusion

Effective dynamics of Bloch electron Motivation: For a large class of crystalline materials, the equations of motion of Bloch electrons are different: ẋ = 1 E n (k) k k Ω(k) k = ee eẋ B Ω(k) reciprocal space magnetic field. It presents in systems: breaking time-reversal symmetry: magnetic materials breaking spatial inversion symmetry: surfaces, interfaces, nanotubes Sundaram and Niu, PRB 59, 14915(1999); Marder, Condensed Matter Physics. Physical consequences?

Aharonov-Bohm effect γ B A ds = ( A) ds = S ϕ 1 = e A ds h γ 1 ϕ 2 = e A ds h γ 2 ϕ = ϕ 2 ϕ 1 = e A ds h S B ds φ B γ

Berry phase Considering a quantum system controlled by a set of parameters that are slowly varying with time: H H[λ(t)] At t 1 : ψ(t 1 ) = ψ[λ(t 1 )] At t 2 : ψ(t 2 ) exp (iϕ) ψ[λ(t 2 )] t2 λ 2 γ λ 2 ϕ = 1 Berry Phase: t 1 E[λ(t)]dt + ϕ B ϕ B = i dλ ψ(λ) λ ψ(λ) γ λ 1 λ 1

Berry phase and k-space magnetic field k y k 2 ψ(x) = e ik x u nk (x) k 1 γ k x Ĥ(k) = Ĥ(k)u nk = ǫ n (k)u nk ( i + k)2 2m + V (x) Berry phase: A-B phase in real space: u nk ϕ B = i u nk dk ϕ AB = e A(r) ds γ k γ u k-space magnetic field: A k (k) = i u nk nk k unk Ω(k) = k A k (k) = i k u nk k

Symmetry consideration unk Ω(k) = k A k (k) = i k With time-reversal symmetry: With spatial inversion symmetry: Ω(k) = Ω( k) Ω(k) = Ω( k) u nk k With both symmetry: Ω(k) = 0

Implication Theory of everything: Ĥ = X i [ˆp i + ea(r i )] 2 2m e + X µ [ ˆP µ Z µea(r µ)] 2 2M µ X iµ Z µe 2 r i R µ + X i<j e 2 r i r j More realistic approach Effective Hamiltonian: Ĥ = i [ˆp i + ea(r i )] 2 2m e + i<j V ee (r i r j ) + Ĥe ph + Ĥph Examples: BCS theory; Quantum Hall effect... With the k-space magnetic field:?

Directions for exploration Applications the presence of an extra reciprocal field provides new freedom for designing novel devices. Fundamentals concept of phase space; effective quantum mechanics Physical effects Luttinger s theorem; orbital magnetization; magnetoresistance; superconductivity...

Outline Introduction Implications Conclusion 1 Introduction 2 Physical implications of the reciprocal space magnetic field Design novel transport devices Inhomogeneous phase space Effective quantum mechanics 3 Conclusion

Anomalous Hall effect M Hall effect in ferromagnetic metal: E ρ xy = R 0 B + (4πM)R s Intrinsic contribution: ẋ = 1 E n (k) k k Ω(k) k = ee I AH e ẋ = e2 E I AH F dk (2π) dω(k) Im[ωσ xy ](10 29 sec -2 ) Re[σ xy ](Ωcm) -1 15 10 5 0-5 1000 500 0 Karplus, Luttinger (1954); Jungwirth, Niu, MacDonald (2002); Fang et al. (2003); Yao et al. (2004). 0 1 2 3 4 5 6 7 hω (ev)

Anomalous Hall insulator σ AH = e2 F dk (2π) dω(k) For a fully occupied 2D band (band insulator): σ AH = e2 dk BZ (2π) 2 Ω z(k) d 2 kω z (k) = 2πC n BZ Quantized anomalous Hall effect: σ AH = e2 h C n Protected by the band gap, extraordinarily robust!

Design an anomalous Hall insulator Objective: A 2D ferromagnetic band insulator A ferromagnetic 2DEG confined in an asymmetric quantum well Patterned with non-magnetic rings arranged as a triangular lattice Electron density tunable by the gate voltage Ferromagnetic 2DEG Parameters: Strength of Rashba spin-orbit coupling; Exchange interaction between electron and magnet; Lattice constant; ring radius

ε 25 20 15 10 5 0-5 -10-4 -3-2 -1 0 1 2 3 4 k x σ AH 3 2 1 0-10 -5 0 5 10 15 ε Reversed magnetization ε 30 25 20 15 10 5 0-5 -10-4 -3-2 -1 0 1 2 3 4 k x σ AH 2 1 0-1 -2-3 -10-5 0 5 10 15 ε

Outline Introduction Implications Conclusion 1 Introduction 2 Physical implications of the reciprocal space magnetic field Design novel transport devices Inhomogeneous phase space Effective quantum mechanics 3 Conclusion

Breakdown of Liouville s theorem Liouville s theorem: Ensemble density in phase space is conserved. k V = r k V (t 2 ) 1 V d V (t) dt = x ẋ + k k = 0 V (t 1 ) r With k-space magnetic field: 1 V d V (t) dt 0 V (0) V (t) = 1 + (e/ )B Ω(k) V (t 1 ) k V (t 2 ) V (t 1 ) r

Non-uniform phase space Phase volume of a quantum state: (2π) d Statistical Physics Phase space measure: (2π) d 1 + (e/ )B Ω(k) D(r,k) = 1 (2π) d D(r,k) = 1 (2π) d ( 1 + e B Ω(k) ) Physical quantity: Ô = drdkd(r, k)o(r, k)f(r, k) f(r,k) Distribution function

Physical effects Introduction Implications Conclusion Fermi sea volume: Z n e = V F +δv F δv F = e dk (2π) d 1 + e B Ω Z V F B Ω B F. S. F. S. V F V F + δv F Orbital magnetization: Z E(B) = V F +δv F M = E B = e Z 2 V F Luttinger s theorem breaks down! dk (2π) d 1 + e B Ω [ǫ 0 (k) B m(k)] fi dk (2π) d i unk k h 2µ ǫ(k) Ĥi fl u nk k Confirmed by: Thonhauser, Ceresoli, Vanderbilt, Resta, cond-mat/0505518

More physical effects Density of states at Fermi surface and specific heat: dk ( ρ(µ, B) = (2π) d 1 + e )δ B Ω (ǫ(k) B m(k) µ) C e = π2 3 k2 BTρ(µ,B) Magnetoresistivity in linear B: σ xx = e 2 ρ(µ,b)υ 2 F (µ,b)τ(µ,b) υ F(µ, B) = υ0 F(µ) B [ m(k F)/ k F] 1 + (e/ )B Ω Z 1 dk = 1 + e W τ(µ,b) (2π) d B Ω kk 1 k «k k 2 µ

Outline Introduction Implications Conclusion 1 Introduction 2 Physical implications of the reciprocal space magnetic field Design novel transport devices Inhomogeneous phase space Effective quantum mechanics 3 Conclusion

Effective quantum mechanics Conventional: With k-space magnetic field: Ĥ = E n (ˆk) eφ(ˆr) Ω = 0 Pierles substitution: ˆk i r + e A(r) i ψ [ ( t = E n i + e ) ] A eφ(r) ψ ǫ µνγ Ω γ [ˆx µ, ˆx ν ] = i 1 + (e/ )B Ω [ˆk µ, ˆk ν ] = i (e/ )ǫ µνγb γ 1 + (e/ )B Ω [ˆx µ, ˆk ν ] = i δ µν + (e/ )B µ Ω ν 1 + (e/ )B Ω B = 0 : ˆr i k + A k(k) General case: Quantum mechanics in non-commutative geometry

Many-body Hamiltonian Theory of everything: Ĥ = X i [ˆp i + ea(r i )] 2 2m e + X µ [ ˆP µ Z µea(r µ)] 2 2M µ X iµ Z µe 2 r i R µ + X i<j e 2 r i r j More realistic approach: Ĥ = [ˆp i + ea(r i )] 2 2m + V ee (r i r j ) + Ĥe ph + Ĥph i e i<j With the k-space magnetic field: Ĥ = i 2ˆk2 i 2m e + i<j V ee (ˆr i ˆr j ) + Ĥe ph(ˆr) + Ĥph [ˆr µ i, ˆrν i ] 0

Superconductivity Boson exchange mechanism Theory of superconductivity: Electrons develop attractive interaction through exchanging bosons (collective excitation). boson Cooper pairs e e Bosonic excitations responsible for the superconductivity: Phonon conventional BCS superconductivity Charge density wave Kohn and Luttinger, PRL, 1965 Spin fluctuation 3 He; heavy-fermion superconductors; high-t c superconductors?...

Ferromagnetic superconductors 60 UGe 2 30 ZrZn 2 T C UGe2 Temperature (K) 40 20 Ferromagnetism Superconductivity T (K) 20 10 T FM 10 T SC T (K) 10 T SC 0 0 1 2 Pressure (GPa) 0 0 5 10 15 20 25 P (kbar) Sexena et al., Nature, 2000 Pfleiderer et al., Nature, 2001 Prevailing picture: Attractive e-e interaction induced by the enhanced spin fluctuation near a quantum critical point.

Spin fluctuation? Introduction Implications Conclusion Puzzle: The superconducting state is only observed in the ferromagnetic phase, not in the paramagnetic phase. T Theory T Experiment PM PM FM FM SC P c SC P SC P c P Alternative theory?

Effective quantum mechanics Quantum mechanics in non-commutative geometry: ǫ µνγ Ω γ [ˆx µ, ˆx ν ] = i 1 + (e/ )B Ω iǫ µνγω γ [ˆk µ, ˆk ν ] = i (e/ )ǫ µνγb γ 1 + (e/ )B Ω 0 [ˆx µ, ˆk ν ] = i δ µν + (e/ )B µ Ω ν 1 + (e/ )B Ω iδ µν Zero B field limit: ˆx µ i k µ + A µ (k) Electron-electron interaction: V ee (x i x j ) V ee (ˆx i ˆx j )

Electron-electron interaction Classical picture Scattering between two electrons: ẋ = 1 E n (k) F ee Ω(k) k k = F ee (r r ) υ r υ θ υ r F F υ θ = F Ω Screw scattering transient pairing of electrons with finite angular momentum Superconductivity with unconventional pairing?

Quantum approach Second quantization V = 1 2V kk K V ee (x i x j ) υ(ˆx i ˆx j ) ˆx µ i k µ + A µ (k) u(k,k ;K)c K 2 +k c K 2 k c K 2 kc K 2 +k, u(k,k ;K) = υ(k k)e iγ( K 2 +k, K 2 +k)+iγ( K 2 k, K 2 k) Aharanov-Bohm phase under the k-space magnetic field k y Ω γ k γ(k,k ) = k k A(k) dk k k x

Application to a simple ferromagnetic metal Model: An isotropic two-dimensional ferromagnetic metal with a constant topological field Ω z (k) = Ω 0 Bare electron-electron interaction: υ(r) = V 0 exp[ κ TF ( r 2 + a 2 a)] (r/a) 2 + 1 Berry phase: γ(k,k ) = 1 2 Ω 0kk sin θ Effective e-e interaction: u(k,k ;K = 0) = υ( k k )exp [ iω 0 kk sin θ ] = m u m (k,k )e imθ

Effective electron-electron interaction For channel m: κ u κ TF / 1 + κ TF a, φ Ω Ω 0 κ 2 u ( ) I kk u m (k,k m 1 φ κ 2 ) 2 u Ω, φ Ω 1 ( ) ( 1) m J kk m sgn(φ κ 2 Ω ) φ 2Ω 1, φ Ω > 1 u Attractive interaction: Ω 0 κ 2 u > 1 m is odd The angular momentum is υparallel to the Ω-field. More realistic model: u m (k,k)/υ 0 0.4 (a) m=-1 0.2 m=0 0.0 m=3 m=1-0.2 0 5 10 15 20 (k/κ 5.0 B ) 2 Ω(k) = Ω 0κ 2 B (k 2 + κ 2 B )2

Superconductivity Introduction Implications Conclusion (k) = 1 u(k,k ) tanh 2 E(k ) k βe(k ) 2 (k ) Pairing is determined by the most attractive m-channel. 6 5 (k) k x + ik y kt/ε B (10-3 ) 4 3 2 1 p-wave Superconductor Normal Metal 0 0 1 2 3 4 5 ε F /ε B F (0)/1.76ε B F (0) kt c 1.76 F (0) (ǫ F +ǫ B ) [ ] 1 exp ρ F u(ǫ F,ǫ F )

Conclusion Reciprocal space magnetic field: Design novel transport devices anomalous Hall insulator Inhomogeneous phase space Breakdown of Luttinger theorem; Fermi-surface related phenomena; orbital magnetization... Quantum mechanics in non-commutative geometry Unconventional superconductivity in ferromagnetic metals... Collaborators: Q. Niu and D. Xiao (UT-Austin) Publication: Phys. Rev. Lett. 95, 137204 (2005); More is coming

Conclusion Reciprocal space magnetic field: Design novel transport devices anomalous Hall insulator Inhomogeneous phase space Breakdown of Luttinger theorem; Fermi-surface related phenomena; orbital magnetization... Quantum mechanics in non-commutative geometry Unconventional superconductivity in ferromagnetic metals... Collaborators: Q. Niu and D. Xiao (UT-Austin) Publication: Phys. Rev. Lett. 95, 137204 (2005); More is coming Thank You For Your Attention!