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.

2

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

5

Two-slit interference--a quintessential QM example:

““Two slit formula””When is it valid???

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A. Tonomura: Electron phase microscopy

 

Each electron produces a seemingly random spot, but:Single electron events build up to from an interference pattern in the double-slit experiments.

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h/e osc. –mesoscopic fluctuation. Compare:

h/2e osc. – impurity-ensemble average,

Altshuler, Aronov, Spivak, Sharvin2

scatterer

scatterer

Closed system!

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The AB interferometer

Use 2-slit formula:

AB phase shift

Measure a- begof a QD) from dependence of I?

9

Semiconducting Quantum Dots

Blue=metal

White=insulating

Red=semiconducting

2D electron gas

10

Model for Quantum Dot:

Transmission:QD

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 D

11

Transmission through a “QD”

Landauer conductance:

How to measure the “intrinsic” phase ?

???

13

Solid-State Aharonov-Bohm interferometers

)interference effects in electronic conduction(

2|| tI Landauer formula

14

?Higher harmonics?

16

The Onsager (Casimir) (1931) relations:

Time reversal symmetry +Unitarity (conservation of

Electron number(

Phase rigidity holds for CLOSEDSystems!

2-slit formula no good??

(e.g. M. Buttiker and Y.I., J. Phys.C18, L467 (1985),for 2-terminal Landauer)

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For 2-slit formula, must use (HOW?) OPEN (non-unitary) interferometer!

Nature 385, 417 (1997)

See: Hackenbroich and Weidenmuller

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-0.58 -0.56

7.0

7.5

8.0

8.5

Col

lect

or V

olta

ge (

a.u.

)

Plunger Gate Voltage [V]

AB-oscillations along a resonance peak

Col

lect

or V

olta

ge (

a.u)

-15 -10 -5 0 5 10 15

Magnetic Field [mT]

( )tQD

e Adl 2

0

C

E

IC

B

B

V

P

A

VE

19

A

B

What is

G

20

What is the difference between 2-slit and the AB interferometer?

D

S

2-slit: NO reflectionsFrom S or D:

Waves MUST beReflected from S and D

K real

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Theory, Three results:

*“Intrinsic” QD transmission: can deduce

*Closed AB interferometer: one can measure the intrinsic phase , without violating

Onsager!

*Open AB interferometer: the phase shift depends on how one opens the system,

but there exist openings that give

PRL 88, 166801 (2002); PRB 66, 115311 (2002); PRL 90, 156802 (2003); cond-mat/0308382

33

0

2 p

4 p

6 p

8 p

PHI

-5

0

5

V

0

0.5

1

T

0

2 p

4 p

6 p

8 p

PHI

0 5 10 15 20 25-10

-5

0

5

10

V

Example:No interactions

34

0

2 p

4 p

6 p

8 p

PHI

-5

0

5

V

0

0.5

1

T

0

2 p

4 p

6 p

8 p

PHI

44

Phase increases by around the Kondo

resonance, sticks at /2 on the resonance

SCIENCE 290, 79 2000

46

47

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).

48

Early history of normal persistent currents

L. Pauling: “The diamagnetic Anisotropy of Aromatic molecules”, J. Chem. Phys. 4, 673 (1936);

Induced currents in anthracene

F. London: “Theorie Quantique des Courants Interatomiques dans les Combinaisons aromatiques”, J. Phys. Radium 8, 397 (1937);

49

2

02

2

2

mRE ,....2,1,0

Thermodynamic persistent current in one-dimensional ring

gpc

EI zero temperature

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`normal’ thermodynamic currents in response to a phase

I. O. Kulik: “Flux Quantization in Normal Metals”, JETP Lett. 11, 275 (1970);

M. Buttiker, Y. Imry, and R. Landauer: “Josephson Behavior in Small Normal One-dimensional Rings”, Phys. Lett. 96A, 365 (1983): ELASTIC SCATTERING IS OK!

weak-disorder

persistent currents in impure mesoscopic systems

(BUT: coherence)!!!

51

53

54

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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!

57

Persistent currents in Aharonov-Bohm interferometers:

Coupling to an incoherent sonic/em source

does the electron-phonon interaction have necessarily a detrimental effect on coherence-related phenomena?

)as long as the sonic/em source does not destroy coherence(

T. Holstein: “Hall Effect in Impurity Conduction”, Phys. Rev. 124, 1329 (1961);

64

The Holstein process-invoking coupling to phonons

'', ''

''''')(

''

,||,,||,

qn qi

qqqqji

qi

njVnnVnit

coupling with a continuum, with exact energy conservation>-

the required imaginary (finite!) term

)(11

xix

Pix

)energy conservation with intermediate state!(

0

''''',''

'' ,||,,||,)(''

qqqqnq

qi njVnnVniiq

65

66

the Holstein process--doubly-resonant transitions

For DISCRETE I and j

ji

requires two phonons (at least)

The transition probability jiP

qi

'qji

through the intermediate site

67

The Holstein mechanism-consequences

The transition probability—

dependence on the magnetic flux

evenij

oddijijij PPPP 0

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.

result from interference!

68

Violation of detailed balance

evenij

oddijijij PPPP 0

Persistent current at thermal equilibrium

)( ji

evenij

oddijijji PPPP 0 )( ij

oddji

oddij PP

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ijiiij PPPP charge conservation on the triad-

jiij PP )phonon-assisted (persistent current-

jiij PP the difference is odd in the AB flux

phonon-assisted transition probabilities

does not violate the Onsager-Casimir relations!

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Detailed calculation

phononelectorntunnelingphononsiteon HHHHH

polaron transformation

effphononsiteon HHHH jiiji

ijijeff ccQetH ij

q

qqq

ijq

ij bbv

Q

exp

ijQDebye-Wallerfactor

O. Entin-Wohlman, Y. I, and A. Aronov, and Y. Levinson (‘95)

jiiji

ijij ccQetI ij Im2

the current:

71

persistent currents and electron-phonon coupling

in isolatedisolated rings-summary

-reduction due to Debye-Waller factor;

-counter-current due to doubly-resonant (energy-conserving) transitions, which exist only at T>0.

][ 0counterpc

Kpc IIeI

Ke

counterI

non-monotonic dependence on temperature

72

manipulating the orbital magnetic moment

by an external radiation

O. Entin-Wohlman, YI, and A. Aronov, and Y. Levinson, (‘95)

][ 0counterpc

Kpc IIeI

all phonon modes phonon modesof doubly-resonant transitions

73

Using boson-assisted processesbetween 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.

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What is left of the Holstein mechanism?

Can the current be manipulated by controlling the radiation?

`open’ interferometers

75

`open’ interferometers-the model

)(

2

1)(

2

12121 IIIIIcircirculating current:

76

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.

QD

AQD

RQD GGG ,,

)()( tdddteiG tiQD

84

Coupling to a phonon source

)]()()1[()( 21 GnGnieG QDQDK

QD

)(

0

)(2,1 )( tti edteG extQD

Bose occupations

elec.-ph. coupling

phonon frequency

qqNq

Debye-Waller

factorKe

])1([||

)(2

2ti

qti

qq q

q qq eNeNt

L. I. Glazman and R. I. Shekhter , JETP 67, 163 (‘88)

QDn dot occupation

85

90

Acousto-magnetic effect in open interferometers

(as compared to the Holstein process in closed interferometers)

One virtual and one real phonon

-reduction due to Debye-Waller factor;

-counter-current due to doubly-resonant (energy-conserving) transitions, which exist only at T>0.

Original Holstein process:

operative at a specific frequency-band

open ring:

operative in a wide frequency-band

-reduction due to Debye-Waller factor;

-no need for exact resonance conditions, exists also at T=0.

-no need for 2nd “real” phonon.

single (virtual) phonon

Both controllable by boson intensity

92

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.

93

the end