Magnetoresistance in parallel fields J.S. Meyer, University of - PDF document

Magnetoresistance in parallel fields J.S. Meyer, University of Minnesota A kk’ W(z) E x z B JSM, A. Altland, and B.L. Altshuler, PRL 89 , 206601 (2002); JSM, V.I. Fal’ko, and B.L. Altshuler, cond-mat/0206024. ISSP – August 18, 2003

Motivation I main focus of previous works: • orbital effect → perpendicular fields • parallel fields → spin effects F B purely 2 d system: no magnetoresistance due to the orbital effect in parallel fields ↔ 2DEGs A kk’ W(z) E x (subband structure) z B [Falko, J. Phys.: Cond. Matt. 2 , 3797 (’90); Mathur, Baranger, Phys. Rev. B 64 , 235325 (’01).] B B 2

Motivation II AlGaAs/GaAs heterostructures: - clean interfaces and - weak z -dependence of impurity scattering z W(z) y x → z -inversion ( P z ) symmetry? magnetoresistance extremely sensitive to P z ⇒ probe properties of 2DEG special case: 1 occupied subband 3

Outline Weak localization & magnetoresistance ⊲ ⊲ Spin effects ⊲ Berry-Robnik symmetry effect ⊲ Subband structure & the Cooperon matrix Results: ⊲ M > 1 ⊲ ⊲ M = 1: virtual processes ⊲ Conclusions 4

REMINDER: Weak localization & magnetoresistance I consider return probability P ( r , r ) → sum over paths k with amplitude A j = a j e iϕ j 2 � � � � � P ( r , r ) = A j � � � � j multiple scattering due to the random phases ϕ j of different paths, interference terms in general average out 2 � � ⇒ classical result: P cl ( r , r ) = � � A j � � j � however: additional averaging-insensitive contribution due to (constructive) interference of time-reversed paths k | 2 | A k + A ¯ | A k | 2 + | A ¯ k | 2 + A ∗ k + A ∗ = k A ¯ k A k ¯ 4 | A k | 2 = thus, P ( r , r ) = 2 P cl ( r , r ) 5

REMINDER: Weak localization & magnetoresistance II enhanced return probability ↔ reduced conductance corrections to classical Drude conductivity: e i φ r f -i φ e r i interference of time-reversed paths – Cooperon = + + + ... = + 1 � d 2 q C ( q , 0) C ( q , ω ) ∼ ⇒ δσ ∼ D q 2 − iω + 1 τ φ ∼ ln( τ/τ φ ) D diffusion constant τ φ phase coherence time elastic scattering time τ temperature-dependence of τ φ ∝ T − p ⇒ temperature-dependence of δσ ( T ) ∼ p ln T 6

REMINDER: Weak localization & magnetoresistance III magnetic field breaks time-reversal invariance � A d r ]): (electrons pick up a phase factor exp[ ± i phase difference proportional to enclosed flux � � 2 A d r = 2 H d F = 2Φ ⇒ limits phase coherent propagation and suppresses interference phenomena ⇒ Cooperons short-ranged (‘massive’): 1 /τ φ → 1 /τ φ ( H ) = 1 /τ φ + 1 /τ H ⇒ negative magnetoresistance 7

Parallel vs perpendicular fields relevant field scale: 1 flux quantum through typical area → ‘magnetic length’ l H perpendicular magnetic field: B φ 0 l l � H ⊥ l 2 ⊥ = φ 0 ⇒ l ⊥ = φ 0 /H ⊥ ⇒ 1 /τ H ⊥ ∝ H ⊥ parallel magnetic field: l φ 0 d B H � l � d = φ 0 l � = φ 0 / ( H � d ) ⇒ • different from conventional magnetic length • geometry dependent 1 /τ H � ∝ H 2 ⇒ � if 1 /τ H > 1 /τ φ : δσ ( H ) ∼ ln( τ/τ H ) ∼ α ln H ( α =1 , 2) 8

Spin effects [REVIEW: cond-mat/0206024] • interplay of Zeeman splitting and spin-orbit coupling ( → weak antilocalization) −π π Aleiner & Fal’ko, PRL 87 , 256801 (2001) • spin-flip scattering at magnetic impurities → decoherence → suppression of WL magnetic field: polarization of impurities → no spin-flip → restoration of WL Vavilov & Glazman, PRB 67 , 115310 (2003) 9

Berry-Robnik symmetry effect (quasi-)2 d system with disorder potential U and confining potential W in parallel magnetic field 1 2 m ( p − e A ) 2 + U ( x, y, z ) + W ( z ) H = gauge choice: A = − Hz e y + W + E F AlGaAs GaAs symmetry transformations: • time reversal ( T ) • z -inversion ( P z ) T P : z z T P z z p p - A e - A e iff U = U ( x, y ) and W ( z ) = W ( − z ), Hamiltonian invariant under T P z : T P z H = H ! Berry-Robnik (’86): unitary ↔ orthogonal spectral statistics Berry-Robnik symmetry effect = additional discrete symmetry compensates for time-reversal symmetry breaking 10

Subband structure I consider system of finite width d : system splits into M subbands ( p z quantized) A kk’ W(z) E x z B 1 2 m ( p − A ) 2 + U ( x, y ) + W ( z ) H = • diagonalize z -dependent problem − ∂ 2 � � z 2 m + W ( z ) − ǫ k φ k ( z ) = 0 • rewrite H in subband basis � 1 � δ kk ′ + 1 2 m ( p y − A ) 2 2 mp 2 H kk ′ = x + U ( x, y ) + ǫ k kk ′ � dz φ k ( z ) z φ k ′ ( z ) e y ⇒ vector potential A kk ′ = − H no H � : subbands decoupled ⇒ each subband contributes separately to the WL correction:   M d 2 q C kk ( q ) ∝ M ln τ �  δσ ∼ �  τ φ k =1 11

Subband structure II parallel magnetic field 1.) couples the subbands & 2.) breaks T -invariance coupling leads to a splitting of formerly degenerate Cooperon modes   M − 1 τ  δσ ∼ � ⇒ ln  τ k φ ( H ) k =0 H and 1 /τ k � =0 where 1 /τ k φ ( H ) = 1 /τ φ + 1 /τ k � = 0 H 1 /τ 0 H = . . . ? 2 X kk ′′ 2 X kk ′ ( C − 1 � ( q − 2 A kk ) 2 + � � q , 0 ) kk ′ = δ kk ′ + � D k D k ′ D k k ′′ 2 (1 − δ kk ′ ) D k + D k ′ where X kk ′ = 1 1+( ǫ kk ′ τ ) 2 A kk ′ A k ′ k ( D k diffusion constant of subband k , ǫ kk ′ = ǫ k − ǫ k ′ ) k k’ k k’’ k k k’ k 12

Results ( M � = 1) 2 X kk ′′ 2 X kk ′ � ( C − 1 ( q − 2 A kk ) 2 + q , 0 ) kk ′ = � � δ kk ′ + √ D k D k ′ D k k ′′ a) P z -symmetry: 1 /τ 0 H = 0 one Cooperon mode is unaffected by the field! ⇒ δσ s ( H, T ) ∼ p ln T + 2( M − 1) ln H T P z ���� ���� ���� ���� ������ ������ ������� ������� ���� ���� ���� ���� ������ ������ ������� ������� ���� ���� ���� ���� ������ ������ ������� ������� ���� ���� ���� ���� ������ ������ ������� ������� ���� ���� ���� ���� ������ ������ ������� ������� ���� ���� ���� ���� ������ ������ ������� ������� ���� ���� ���� ���� ������ ������ ������� ������� ���� ���� ���� ���� ������ ������ ������� ������� P T ���� ���� ���� ���� ������ ������ ������� ������� z ���� ���� ���� ���� ������ ������ ������� ������� ���� ���� ���� ���� ������ ������ ������� ������� ���� ���� ���� ���� ������ ������ ������� ������� b) no P z -symmetry: 1 /τ 0 H � = 0 • asymmetric confining potential W ( z ) � = W ( − z ) 1 /τ 0 (as) ∝ H 2 H • z -dependent disorder δU ( x, y ; z )  H 2 H < H c 1 /τ 0 (imp)  ∝ H 1 /τ ′ H > H c  where 1 /τ ′ inter-subband scattering rate 13 δσ as ( H, T ) ∼ 2 M ln H ⇒

Example: M = 2 for simplicity: D 0 = D 1 ≡ D � � D ( q − A ) 2 +2 X 01 + 1 2 X 01 C − 1 = 1 τ φ D ( q + A ) 2 +2 X 01 + 1 2 X 01 D τ φ where A = A 00 − A 11 if A = 0: 1 /τ (0) = 1 /τ φ and 1 /τ (1) = 1 /τ φ + 4 X 01 H H 1 /τ ( k ) � � = D A 2 if A � = 0: δ H at small H : δσ ( H ) − δσ (0) ∼ 2 X 01 τ φ (independent of A ) T - and H -dependence of the conductivity: H ∆σ (T) saturates H=H ❋ ❍ φ ∆σ ln (T) ~ T ● ● ■ ❍ ■ ∆σ 2 (H) ~ H ❒ ∆σ (H) ~ ln H ■ H=H φ ● ❒ asymmetry definition of characteristic fields H φ and H φ ∗ : D A 2 ( H φ ∗ ) = 1 /τ φ 4 X 01 ( H φ ) = 1 /τ φ note: W ( z ) = W ( − z ) ⇒ A = 0 14

M = 1 occupied subband I A 00 pure gauge field ⇒ no effect ? residual magnetoresistance due to virtual processes → modifies electron dispersion p 2 + α p 2 p 2 y + β p 3 → y where α ∼ H 2 and β ∼ H 3 o u1 u2 o o u1’ u2’ o → effective vector potential 1 /τ H ∝ H 6 ⇒ (note that β = 0 in the P z -symmetric case!) z -dependent impurities: U ( x, y ) + δU ( x, y, z ) [Falko, J. Phys.: Cond. Matt. 2 , 3797 (’90).] → coupling to unoccupied subbands even in the absence of a magnetic field + inter−subband scattering unoccupied 2 H unoccupied 1 H H H δ U occupied 1 /τ H ∝ H 2 ⇒ ( H 2 → H 6 crossover at H M =1 ∝ τ ′− 1 / 4 ) c 15

Experiments quantum dots: Zumbuhl et al., PRL 89 , 276803 (2002) → fit 1 /τ H ∼ a H 2 + b H 6 16