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
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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
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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
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the double-slit experiments.
Closed system!
scatterer
scatterer
h/e osc. –mesoscopic fluctuation.
Compare:
h/2e osc. – impurity-ensemble average,
Altshuler, Aronov, Spivak, Sharvin2
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The AB interferometer
Use 2-slit formula:
AB phase shift
2
Measure aa- ab (e.g. of a QD) from f dependence of I?
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Semiconducting Quantum Dots
Red=semiconducting
2D electron gas
White=insulating
Blue=metal
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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:
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Transmission through a “QD”
Landauer conductance:
How to measure the
“intrinsic” phase a?
???
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Solid-State Aharonov-Bohm interferometers
(interference effects in electronic conduction)
Landauer
formula
I | t |
2
f
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?
Higher harmonics?
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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!
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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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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
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VE
G(f)
A
B

What is ??
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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
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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
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Example:
No interactions
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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
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6p
8p
8
-5
Phase increases by 
around the Kondo
resonance, sticks at /2
on the resonance
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SCIENCE 290, 79 2000
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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).
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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
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Thermodynamic persistent current in
one-dimensional ring
 
 
 

E 
2 
2mR   0 
2
I pc 
E g

2
  0,1,2,....
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);
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!!!)
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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!
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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);
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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
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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 '
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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.
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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
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phonon-assisted transition probabilities
charge conservation on the triadthe difference is odd in the AB flux
(phonon-assisted) persistent current-
Pij  Pi  Pji  Pi
Pij  Pji
Pij  Pji
does not violate the Onsager-Casimir relations!
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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  bq
 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
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


)
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
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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)
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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.
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`open’ interferometers
f
What is left of the Holstein mechanism?
Can the current be manipulated by controlling the radiation?
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`open’ interferometers-the model
circulating current:
I cir
1
1
 ( I1  I 2 )   ( I1  I 2 )  
2
2
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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  dteit d  d (t )
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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
iq t
 (1  N q )e
iq t
K
factor
]
nQD
aq
Nq
q
dot occupation
elec.-ph. coupling
Bose occupations
phonon frequency
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L. I. Glazman and R. I. Shekhter , JETP 67, 163 (‘88)
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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.
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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.
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the end
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