The conductance of small mesoscopic disordered rings Doron Cohen, Ben-Gurion University Collaborations: Tsampikos Kottos (Wesleyan) Holger Schanz (Gottingen) Swarnali Bandopadhyay (BGU) Yoav Etzioni (BGU) Tal Peer (BGU) Rangga Budoyo (Wesleyan) Alex Stotland (BGU) Discussions: Michael Wilkinson (UK) Bernhard Mehlig (Goteborg) Yuval Gefen (Weizmann) Shmuel Fishman (Technion) $ISF, $GIF, $DIP, $BSF
Driven Systems Non interacting “spinless” electrons in a ring. H ( Q, P ; Φ( t )) − ˙ Φ = electro motive force (RMS) G ˙ Φ 2 = rate of energy absorption ����������������� ����������������� Flux ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� Metallic ����������������� ����������������� ����������������� ����������������� Ring ����������������� ����������������� grain ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� ����������������� Radiation wire Flux dot
Linear Response Theory (LRT) H = { E n } − Φ( t ) {I nm } |I mn | 2 δ T ( E n − E F ) δ Γ ( E m − E n ) � G = π ¯ h n,m h ( ̺ ( E F )) 2 ��|I mn | 2 �� = π ¯ G applies if EMF driven transitions ≪ relaxation otherwise connected sequences of transitions are essential. leading to Semi Linear Response Theory (SLRT)
Semi Linear Response Theory (SLRT) H = { E n } − Φ( t ) {I nm } dp n � dt = − w nm ( p n − p m ) m const × g nm × EMF 2 w nm = Scaled transition rates: |I nm | 2 g nm = 2 ̺ − 3 ( E n − E m ) 2 δ Γ ( E n − E m ) F
Semi Linear Response Theory (cont.) −J E n J E +J |I nm | 2 g nm = 2 ̺ − 3 ( E n − E m ) 2 δ Γ ( E n − E m ) F The SLRT analog of the Kubo formula: h ( ̺ ( E F )) 2 ��|I mn | 2 �� = π ¯ G where ��|I mn | 2 �� ≡ inverse resistivity of the network ��|I mn | 2 �� harmonic ≪ ��|I mn | 2 �� ≪ ��|I mn | 2 �� algebraic
Conductance of mesoscopic rings (a) (b) (c) (d) S (e) S Naive expectation (assuming Γ > ∆): e 2 h M ℓ � ∆ � G = L + O 2 π ¯ Γ L = perimeter of the ring ℓ = mean free path ∝ W 2 ℓ ∞ = localization length ≈ M ℓ Ballistic regime: L ≪ ℓ Diffusive regime: ℓ ≪ L ≪ ℓ ∞ Anderson regime: ℓ ∞ ≪ L
Conductance versus disorder ∼(∆/Γ) 1 G G Drude Ballistic Diffusive Anderson−Mott regime regime regime disorder strength (1/l) 4 10 Drude Kubo Meso 0 10 G -4 10 -8 10 -3 -2 -1 0 1 10 10 10 10 10 w
The RMT modeling {| v nm | 2 } ≡ { X } Characterization of the perturbation matrix: • bandwidth ( b ) • sparsity ( p ) • texture 0 10 G Meso /G Kubo -2 10 numerical results for the tight binding model corresponding RMT results VRH approximation for the RMT results -4 10 -6 10 0 2 4 10 10 10 L / l Comparison between: • Actual results based on “real” matrices • RMT results based on “artificial” matrices • Semi-analytical VRH estimate
Ergodicity of the eigenstates • Weak disorder (ballistic rings): Wavefunctions are localized in mode space. • Strong disorder (Anderson localization): Wavefunctions are localized in real space. 30 25 20 PR 15 10 5 0 −2 −1 0 10 10 10 1−g T 1 0.6 PR F(X) p 0.8 0.5 mode space 0.4 0.6 position space 2 10 mode-pos. space 0.3 0.4 p W=0.05 W=0.35 0.2 W=7.50 0.2 0.1 0 0 10 0 -2 0 2 -4 -2 0 2 10 10 10 10 10 10 10 w X/<X>
Modeling of sparsity 1 − x � � | v nm | 2 M 2 v 2 X ≡ ∼ F exp l ∞ <X> = p X 1 BiModal distribution p X log(<X>) X 0 X 1 LogBox distribution X X 0 X 1 log(X) X ∈ LogBox[ X 0 , X 1 ] (ln( X 1 /X 0 )) − 1 p ˜ ≡ � � p ≡ Prob X > � X � ≈ − ˜ p ln ˜ p � X � ≈ pX 1 ˜ ∼ pX 1
The VRH estimate � e � 2 � | v mn | 2 δ T ( E n − E F ) δ Γ ( E m − E n ) G = π ¯ h L n,m � e G = 1 � 2 � ˜ ̺ F C qm ( ω ) δ Γ ( ω ) dω 2 L ˜ C qm-LRT ( ω ) ≡ 2 π̺ F � X � ˜ C qm-SLRT ( ω ) ≡ 2 π̺ F X � ω � � � where by definition: Prob X > X ∼ 1 ∆ For strong disorder we get: − ∆ ℓ � � v 2 X ≈ F exp ω � � � � − ∆ ℓ −| ω | � ∝ exp exp dω G | ω | ω c
LRT, SLRT and beyond − ˙ Φ = electro motive force (RMS) G ˙ Φ 2 = rate of energy absorption Semi linear response theory [1] D. Cohen, T. Kottos and H. Schanz, “Rate of energy absorption by a closed ballistic ring” , (JPA 2006) [2] S. Bandopadhyay, Y. Etzioni and D. Cohen, The conductance of a multi-mode ballistic ring , (EPL 2006) [3] M. Wilkinson, B. Mehlig, D. Cohen, The absorption of metallic grains , (EPL 2006) [4] D. Cohen, “From the Kubo formula to variable range hopping” , (PRB 2007) [5] T. Peer, R. Budoyo, A. Stotland, T. Kottos and D. Cohen, The conductance of disordered rings , (arXiv 2007) Beyond (semi) linear response theory [6] D. Cohen and T. Kottos, “Non-perturbative response of Driven Chaotic Mesoscopic Systems” , (PRL 2000) [7] A. Stotland and D. Cohen, ”Diffractive energy spreading and its semiclassical limit” , (JPA 2006) [8] A. Silva and V.E. Kravtsov, Beyond FGR , (PRB 2007) [9] D.M. Basko, M.A. Skvortsov and V.E. Kravtsov, Dynamical localization , (PRL 2003)
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