View
221
Download
2
Embed Size (px)
Citation preview
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???
6
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.
7
h/e osc. –mesoscopic fluctuation. Compare:
h/2e osc. – impurity-ensemble average,
Altshuler, Aronov, Spivak, Sharvin2
scatterer
scatterer
Closed system!
8
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)
17
For 2-slit formula, must use (HOW?) OPEN (non-unitary) interferometer!
Nature 385, 417 (1997)
See: Hackenbroich and Weidenmuller
18
-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
21
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
50
`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
55
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
69
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!
70
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.
74
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