#### Phase measurements and Persistent Currents in A-B interferometers Yoseph Imry The Weizmann Institute

Download Report#### Transcript Phase measurements and Persistent Currents in A-B interferometers Yoseph Imry The Weizmann Institute

Phase measurements and Persistent Currents in A-B interferometers Yoseph Imry The Weizmann Institute In collaboration with Amnon Aharony, Ora Entin-Wohlman (TAU), Bertrand I. Halperin (HU), Yehoshua Levinson (WIS) Peter Silvestrov (Leiden) and Avraham Schiller (HUJ). Inspired by results of A. Jacoby, M. Heiblum et al. Discussions with: J. Kotthaus, A. stern, J. von Delft, and The late A. Aronov. Outline • • • • • • • • The Aharonov-Bohm (AB) interferometer, with a Quantum dot (QD) Experiment: Open vs closed ABI. Theory: Intrinsic QD, (Fano) ,Closed ABI+ QD, Open ABI + QD (The sensitivity of the phase to Kondo correlations.) Mesoscopic Persistent Currents The Holstein Process Phonon/photon induced persistent current Conclusions PRL 88, 166801 (2002); PRB 66, 115311 (2002); PRL 90, 106602 , 156802 (2003), 91, 046802, (2003), cond-mat/0308382, 0311609 2 Two-slit interference--a quintessential QM example: “Two slit formula” When is it valid??? 5 A. Tonomura: Electron phase microscopy Each electron produces a seemingly random spot, but: Single electron events build up to from an interference pattern in 6 the double-slit experiments. Closed system! scatterer scatterer h/e osc. –mesoscopic fluctuation. Compare: h/2e osc. – impurity-ensemble average, Altshuler, Aronov, Spivak, Sharvin2 7 The AB interferometer Use 2-slit formula: AB phase shift 2 Measure aa- ab (e.g. of a QD) from f dependence of I? 8 Semiconducting Quantum Dots Red=semiconducting 2D electron gas White=insulating Blue=metal 9 Model for Quantum Dot: Basic model for “intrinsic” QD: (a) On QD: single electron states plus interactions. (b) QD connected to 2 reservoirs via leads. No interactions on the leads. S QD D Transmission: 10 Transmission through a “QD” Landauer conductance: How to measure the “intrinsic” phase a? ??? 11 Solid-State Aharonov-Bohm interferometers (interference effects in electronic conduction) Landauer formula I | t | 2 f 13 ? Higher harmonics? 14 The Onsager (Casimir) (1931) relations: Time reversal symmetry + Unitarity (conservation of Electron number) (e.g. M. Buttiker and Y.I., J. Phys.C18, L467 (1985), for 2-terminal Landauer) 2-slit formula no good?? Phase rigidity holds for CLOSED Systems! 16 For 2-slit formula, must use (HOW?) OPEN (non-unitary) interferometer! Nature 385, 417 (1997) See: Hackenbroich and Weidenmuller 17 AB-oscillations along a resonance peak 8.0 Collector Voltage (a.u) Collector Voltage (a.u.) 8.5 7.5 7.0 -0.58 -0.56 Plunger Gate Voltage [V] A C -15 -10 -5 0 5 10 15 Magnetic Field [mT] (t QD ) IC B VP B 2 e Adl 0 E 18 VE G(f) A B What is ?? 19 What is the difference between 2-slit and the AB interferometer? D S 2-slit: NO reflections From S or D: Waves MUST be Reflected from S and D 20 K real Theory, Three results: * “Intrinsic” QD transmission: can deduce a! * Closed AB interferometer: one can measure the intrinsic phase a, without violating Onsager! * Open AB interferometer: the phase shift depends on how one opens the system, but there exist openings that give a! PRL 88, 166801 (2002); PRB 66, 115311 (2002); PRL 90, 156802 (2003); cond-mat/0308382 21 Example: No interactions 10 V 5 1 T 5 0.5 0 0 0 2p 4p PHI f V 0 -5 6p 8p -5 -10 0 5 10 15 20 33 25 f 1 T 5 0.5 0 0 0 V 2p 4p PHI f 34 6p 8p 8 -5 Phase increases by around the Kondo resonance, sticks at /2 on the resonance 44 SCIENCE 290, 79 2000 46 A-B Flux in an isolated ring • A-B flux equivalent to boundary condition. • Physics periodic in flux, period h/e (Byers-Yang). • “Persistent currents”exist due to flux (which modifies the energy-levels). • They do not(!!!) decay by impurity scattering (BIL). 47 Early history of normal persistent currents L. Pauling: “The diamagnetic Anisotropy of Aromatic molecules”, J. Chem. Phys. 4, 673 (1936); F. London: “Theorie Quantique des Courants Interatomiques dans les Combinaisons aromatiques”, J. Phys. Radium 8, 397 (1937); Induced currents in anthracene 48 Thermodynamic persistent current in one-dimensional ring E 2 2mR 0 2 I pc E g 2 0,1,2,.... zero temperature 49 `normal’ thermodynamic currents in response to a phase I. O. Kulik: “Flux Quantization in Normal Metals”, JETP Lett. 11, 275 (1970); weak-disorder M. Buttiker, Y. Imry, and R. Landauer: “Josephson Behavior in Small Normal One-dimensional Rings”, Phys. Lett. 96A, 365 (1983): ELASTIC SCATTERING IS OK! persistent currents in impure mesoscopic systems (BUT: coherence!!!) 50 51 53 54 Persistent current induced by a flux of phonons/photons Due to Holstein 2nd order process (boson emission and absorption), generalizing previous work (discrete and equilibrium case) with Entin-Wohlman, Aronov and Levinson. boson number (if decoherence added, T, DW fixed…)! Leads make it O(2), instead of O(3) for discrete case. Sign opposite to that of electrons only. Process retains coherence! 55 Persistent currents in Aharonov-Bohm interferometers: Coupling to an incoherent sonic/em source does the electron-phonon interaction have necessarily a detrimental effect on coherencerelated phenomena? (as long as the sonic/em source does not destroy coherence) T. Holstein: “Hall Effect in Impurity Conduction”, Phys. Rev. 124, 1329 (1961); 57 The Holstein process-invoking coupling to phonons ti ( ) j i, nq | V | , nq '' , nq '' | V | j, nq ' i q '' i nq '' , q '' 1 1 P i ( x) x i x i ( q '', nq '' i 0 q '' )i, nq | V | , nq '' , nq '' | V | j , nq ' (energy conservation with intermediate state!) coupling with a continuum, with exact energy conservation-> the required imaginary (finite!) term 64 65 the Holstein process--doubly-resonant transitions For DISCRETE I and j The transition probability Pi j through the intermediate site requires two phonons (at least) i q f i j i j q ' 66 The Holstein mechanism-consequences The transition probability— dependence on the magnetic flux Pij P P 0 ij odd ij P even ij result from interference! 1. When used in the rate equations for calculating transport coefficients yields a term odd in the flux, i.e., the Hall coefficient. 2. Coherence is retained. 67 Violation of detailed balance Pij P P P (i j ) Pji P P P ( j i) 0 ij odd ij 0 ij odd ij even ij even ij Persistent current at thermal equilibrium odd ij P P odd ji 68 phonon-assisted transition probabilities charge conservation on the triadthe difference is odd in the AB flux (phonon-assisted) persistent current- Pij Pi Pji Pi Pij Pji Pij Pji does not violate the Onsager-Casimir relations! 69 Detailed calculation H H on site H phonon H tunneling H electorn phonon polaron transformation H H on site H phonon H eff H eff tij e Qij ci c j ifij ij vqij Qij exp bq bq q q ( the current: ifij ij i I ij 2 Im tij e Q c c j Debye-Waller factor O. Entin-Wohlman, Y. I, and A. Aronov, and Y. Levinson (‘95) Qij 70 ) persistent currents and electron-phonon coupling in isolated rings-summary -reduction due to Debye-Waller factor; e K -counter-current due to doubly-resonant (energy-conserving) transitions, which exist only at T>0. counter I K I pc e [ I I counter ] 0 pc non-monotonic dependence on temperature 71 manipulating the orbital magnetic moment by an external radiation K I pc e [ I I counter ] all phonon modes 0 pc phonon modes of doubly-resonant transitions O. Entin-Wohlman, YI, and A. Aronov, and Y. Levinson, (‘95) 72 Using boson-assisted processes between two leads • Quantum analogue of “peristaltic pump”, to transfer charge between the leads. • We will discuss the flux-sensitive circulating current produced by the boson (incoherent) source. 73 `open’ interferometers f What is left of the Holstein mechanism? Can the current be manipulated by controlling the radiation? 74 `open’ interferometers-the model circulating current: I cir 1 1 ( I1 I 2 ) ( I1 I 2 ) 2 2 75 Method of calculation All interactions are confined to the QD Use Keldysh method to find all partial currents Express all partial currents in terms of the exact (generally, un-known) Green fn. on QD Use current conservation to deduce relations on the QD Green fn. R A GQD , GQD , GQD GQD ( ) i dteit d d (t ) 76 Coupling to a phonon source GQD ( ) ie [(1 nQD )G1 ( ) nQDG2 ( )] K i ( QD ext ) t ( t ) G1, 2 ( ) dte 0 (t ) q | aq | 2 q 2 [ Nqe Debye-Waller e e iq t (1 N q )e iq t K factor ] nQD aq Nq q dot occupation elec.-ph. coupling Bose occupations phonon frequency 84 L. I. Glazman and R. I. Shekhter , JETP 67, 163 (‘88) 85 Acousto-magnetic effect in open interferometers (as compared to the Holstein process in closed interferometers) Both controllable by boson intensity Original Holstein process: open ring: operative at a specific frequency-band One virtual and one real phonon operative in a wide frequency-band single (virtual) phonon -reduction due to Debye-Waller factor; -counter-current due to doubly-resonant (energy-conserving) transitions, which exist only at T>0. -reduction due to Debye-Waller factor; -no need for exact resonance conditions, exists also at T=0. -no need for 2nd “real” phonon. 90 Conclusions • Experimentalists and theorists benefit talking to each other! • THREE Ways to determine transmission phase. • Phase measured in the open AB interferometer depends on method of opening; Need experiments which vary the amount of opening; must optimize • One CAN obtain the QD phase from dot’s transmission and from closed interferometers! -- Need new fits to data. • Phase is more sensitive to Kondo correlations than transmission. • Possible to “pump” persistent currents in open and closed ABI’s by phonons/photons. Differences between the two. 92 the end 93