Mesoscopics with Superconductivity. Philippe Jacquod. U of Arizona. R. Whitney (ILL, Grenoble)

Transcription

1 Mesoscopics with Superconductivity Philippe Jacquod U of Arizona R. Whitney (ILL, Grenoble)

2 Mesoscopics without superconductivity Mesoscopic = between «microscopic» and «macroscopic»; N. van Kampen 81

3 I. Weak localization Interference correction to conductance G Reduces G for TRS, SRS Enhances G for TRS, but broken SRS Not there for broken TRS -> signature in magnetoresistance Amplitude ~ e 2 /h! Th. : Gor kov, Larkin, Khmel nitskii 79 Abrahams, Anderson, Ramakrishnan 79 Hikami, Larkin, Nagaoka 80

4 II. Aharonov-Bohm oscillations Interference correction to conductance Same sign as weak localization Amplitude ~ e 2 /h! Th. : Altshuler, Aronov, Spivak 81 ; Aronov and Sharvin 87

5 III. Universal Conductance Fluctuations Parameter-dependent interference correction to conductance Reproducible noise Magnitude of fluctuations does not scale with conductance Upon application of B-field, changes in chemical potential, sample shape Magnitude ~ e 2 /h! Th. : Altshuler 85; Lee and Stone 85

6 Two-terminal (Onsager) G(H)=G(-H) Four-terminal G ij;kl (H)= G kl;ij (-H) Th. : Buttiker 86 IV. Reciprocity relations in multiterminal transport Quantum nonlocality effects Apparent violation of macroscopic Onsager symmetry in multiterminal transport measurement Magnitude ~ e 2 /h!

7 Two-terminal (Onsager) G(H)=G(-H) Four-terminal G ij;kl (H)= G kl;ij (-H) Th. : Buttiker 86 IV. Reciprocity relations in multiterminal transport Quantum nonlocality effects Apparent violation of macroscopic Onsager symmetry in multiterminal transport measurement Magnitude ~ e 2 /h!

8 V. Coherent thermal transport Thermopower and Wiedemann-Franz law Mesoscopic fluctuations of TP No average TP Symmetric in B-field Magnitude halved when breaking TRS Mesoscopic fluctuations violate Wiedemann-Franz with var(δ) ~ 1/G 2 Th.: thermopower: Esposito et al. 87; van Langen et al. 97 W-Franz: Vavilov and Stone 05.

9 Mesoscopics with superconductivity The rules of the new game: (i) Experimental findings

10 I. Aharonov-Bohm oscillations Adding superconductivity to the mesoscopic AB effect Giant enhancement of amplitude of oscillations!

11 II. Reciprocity relations in multiterminal transport 2-term G ~ 10 e 2 /h δg<e 2 /h R 3010 R 1030

12 II. Reciprocity relations in multiterminal transport and magnetoresistance 4-terminal resistance, yet even in phase Same sign as weak-loc/ab with SOI

13 III. Coherent thermal transport Thermopower Large average TP Symmetric or antisymmetric in B-field, depending on geometry parallelogram house

14 III. Coherent thermal transport Thermopower Large average TP Symmetric or antisymmetric in B-field, depending on geometry - BUT NOT ON T-GRADIENT! Nonlocal effects? Crossed Andreev reflection? Elastic cotunneling?

15 III. Coherent thermal transport : oscillations of the Wiedemann-Franz ratio Large AB oscillations of Ξ/GT Ξ and G are out of phase

16 The rules of the new game: (ii) Analytical derivation The rule of the game is simple, it s kill the quarterback. -Joe Namath

17 I. Ray optics for the 21st century Scattering approach Entrance / exit points Transport modes in leads Classical trajectories, stability and action R. Whitney and PJ, PRL/PRB 05/06; precursor theory: Richter and Sieber, PRL 02.

18 I. Ray optics for the 21st century Semiclassical expression x Semiclassical approximation (i) because N=W/λ F >> 1 (ii) Stationary phase approximation diagonal approx weaklocalization

19 II. New ingredients in presence of SC Andreev reflection (e,e F +ε) (h, E F -ε) Reflection phase : (fig taken from Wikipedia) Angle mismatch (neglect it) S phase + : h->e - : e->h

20 II. New ingredients in presence of SC Lambert 93 formula for a SC island Charging of SC Average conductance for N L =N R >>1 With even/odd # of Andreev reflections both terms give large interference contributions

21 III. Ray optics for the 21st century in presence of superconductivity 1. Contribution to T ee and T he have even and odd # of Andreev reflections resp. For T 21 ee For T 11 he Diagonal contribution to T without SC Good news : all has already been calculated! (shot-noise:pj and Whitney - FCS: Brouwer and Rahav; Berkolaiko et al.)

22 III. Ray optics for the 21st century in presence of superconductivity 1. Contribution to T ee and T he have even and odd # of Andreev reflections resp. For T 21 ee For T 11 he Diagonal contribution to T without SC BAD NEWS!!! Generate new contributions by adding crossings with new legs touching SC terminal?! infinite number of terms to resum!?

23 III. Ray optics for the 21st century in presence of superconductivity 2. Consider N s << N n -> consider only minimal # of Andreev reflections For T 21 ee Diagonal contribution to T without SC For T 11 he Note : unitarity is preserved at order (N s /N n ) 2

24 III. Ray optics for the 21st century in presence of superconductivity Large oscillations of transmissions probs. if N T 1/2 <<N S (Compatible with N S <<N T ) φ : phase-difference between the two SC contacts Note: RMT for any N S : Beenakker, Melsen and Brouwer 95; but backscattering??

25 We have the tools, let s construct the rules!

26 Charge and thermal conductance single-dot model Charge and Heat currents Vs. 2-term V and T differences Negative magnetoconductance Positive magnetothermoconductance Note: T 11 he = N 2 N s /N T 2 i.e. not giant backscattering!

27 Charge and thermal conductance single-dot model with SOI Spin rotation along path γ unchanged =1/4 =1/4 =-1/2

28 Charge and thermal conductance single-dot model with SOI Spin rotation along path γ No change in sign of magnetoconductance Amplitude of oscillations reduced by factor of 4

29 Charge and thermal conductance single-dot model Wiedemann-Franz ratio Signs of magnetoconductance and of magneto-thermoconductance agree with exps. by Chandrasekhar Note : circuit theory results show that this remains valid at N s ~ N n for amplitude; with more harmonics

30 Conductance fluctuations To calculate var(g), take any two contributions we considered for G and pair them in a non-connected way. This requires to add at least two encounters - the first one so that the diagram is connected, the second one to bring back together trajectories splitted by the first one. Conductance fluctuations remain universal, O(e 2 /h)

31 Thermopower T L,V L,I L T R,V R,I R Heat up one reservoir Set V L and V R such that I L =I R =0 Thermopower S= (V L -V R )/(T L -T R ) S=- Semiclassical contributions to ~ δt: difference in duration of two e-h path-pair segments No contribution to S, unless L-R symmetry is broken

32 Thermopower - breaking LR symmetry Double-cavity model ~ parallelogram or Asymmetric single dot Paths hitting R sc from L take more time ~δt correlated with φ Contributions to proportional to φ-antisymmetric contributions to S Vanish as T->0 on the scale of E T =1/δt

33 Thermopower - breaking LR symmetry Triple-cavity model Paths hitting R sc from L take more time ~δt correlated with φ Contributions to proportional to φ-antisymmetric contributions to S Vanish as T->0 on the scale of E T =1/t

34 CONCLUSIONS - new rules 1)large, positive magnetoresistance only magnitude depends on SOI 2) large, negative magneto-thermoresistance hence large oscillations of Wiedemann-Franz ratio 3) UCF remain O(1) 4) same symmetry for multiterminal transport as without SC - but apparent symmetry at large G 5) large average thermopower (broken LR symmetry) antisymmetric in φ no need to invoke T-gradients across sample (see: Volkov; Heikkila and Virtanen; Titov) Open question: even TP in house interferometers? vs. mesoscopic fluctuations?? vs. charge imbalance (Titov)??

35 III. Ray optics for the 21st century in presence of superconductivity Feynman rules for calculating contributions : (i) Factors of N i and N T : N T -1 for each path-pair -N T -1 for each in-cavity encounter N i for each on-lead encounter on lead i N i for each leg terminating on lead i (ii) Phase factors : exp[-i arccos(ε/δ)] for each e->h or h->e exp[-i φ Sa ] for each e->h at sc contact a exp[2iεt] for each eh path pair

36 Our apologies for the upcoming slide

37 Symmetry of multiterminal transport 1) Define conductance matrix including SC leads 2) Define generalized transmission coefficient (SC island) 3) Define Onsager coefficients

38 Symmetry of multiterminal transport then the 4-terminal conductance reads and has the same symmetry as without SC (Buttiker 86) BUT : take e.g. leads 1 and 2 close, 3 and 4 close and treat G 14 ~G 13 +dg 14 aso 1 3 Then, large, φ-symmetric contributions that exist for G ii, but not G ij render the 4-terminal conductance symmetric 2 4 when N n >>1 Agreement with numerics