Pr

1
Sensitivity of ac linear and non-linear
mesoscopic transport to time reversal symmetry
breaking.
Groupe de physique mésoscopique LPS Orsay
A.Chepelianskii, F.Chiodi, M.Ferrier, A.Kasumov,
R.Deblock, S. Guéron, B. Reulet, P.Billangeon,
L.Angers, Y. Noat, A.Rowe, H. Bouchiat
Coll G. Montambaux, C.Texier, Y.Gefen, A.Kamenev
D. Mailly, U.Gennser (LPN) Discussions
O.Bohigas, J.L.Pichard, D.Ulmo Random
matrices and semiclassics
2

• AC Linear mesoscopic response in isolated systems
• level spacing
• Orbital magnetism and electrical polarisability
• Magnetic field dependence and RMT
• Mesoscopic rectification at high frequency
• w1/tD
• Photovoltaic effect and photomagnetism
• Time dependent Electric field induces breaking
  of time reversal symmetry..

3
Different ways to measure conductance
Measure the conductance of a Molecule or Q dot

• Isolated

Discrete spectrum preserved

• Connected

Continuous Spectrum
?
?i
w
lt
gt
?j
ConductanceTransmissionLandauer
Conductance Absorption Kubo
Re (G)0 at w0
Non dissipative component Im G0
Non zero If delocalized states
4
Measuring the conductance of an isolated system
V
Depends on the coupling with reservoirs!
I
Electric Coupling
V(t)
connected
E(t)
Magnetic Coupling
F(t)
I(t)
I(t) GeV(t)
I(t)
How to measure Ge and Gm ? Coupling to a
resonator
V(t) -dF(t)/dt
I(t) GmV(t)
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Aharonov Bohm on a ring
Phase factor on periodic boundary conditions
F0 h/e flux quantum
F
Equilibrium Persistent current I(F ) - Sn
fn dEn (F) /d F e/ t D
Magnetic moment M -dE/dB I S
100 spins measurable!
6
Spectrum of a disordered ring
Correlated Spectrum On the Thouless energy h
D/L2 h /tD tDL2/D diffusion time around the
ring Described by random matrix theory
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Sensitivity of the energy spectrum to
fundamental symmetries
Random matrix theory and level repulsion
Poisson
GOE
GUE
R(e) Probability to find 2 levels at e
F 0, nF0/2 Time reversal symmetry F 0 No
time reversal symmetry
Orbital magnetism sensitive to level repulsion
ltIgt D/ F0 without interactions ltIgt l Ec / F0
with interactions lt 0 (Coulomb) F0/2 periodic
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Superconducting Niobium Micro-resonators
2 mm
Ag rings aligned with e-beam (D.Mailly LPN)
9
Multimode Resonator
Nb superconducting
coax
Reflected power
20cm
L nln/2
Resonances Detected between 300 MHz and 8 GHz
Impossible to separate electric and magnetic
response
10
Contactless Conductance measurements
Pabs
Perturbation of the resonances By samples
coupled to the resonator
Im G(fn) Re G(fn)
fn
Q 10 5
Sensitivity 10-10 at T10mK But no absolute
measurement Low field Magnetoconductance
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GaAs mesoscopic rings coupled to a
superconducting resonator
d1/Q Dissipative response
F0/2
10-7
df/f Non-dissipative response

• Dominant contribution Non-dissipative response
  
• Periodic in F0/2
• Low field negative dissipative
  magnetoconductance!

12
Magnetoresistance of a connected network of rings
Ferrier et al. 2003
Time reversed trajectories Weak
localization correction Positive low field
Magnetoconductance
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Electromagnetic Absorption
0.2
106 d 1/Q dG/G
_
0.0
_
-0.2
connected
x100
-1.0
-0.5
0.0
0.5
1.0
F
F
/
0

• Low field negative magnetoconductance
• Sign opposite to the magnetoconductance
• of connected rings (weak localisation)
• Amplitude much larger

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Negative Magnetoconductance
Electric Absorption inter level transitions
F0, nF0/2
Level repulsion more important in GUE than in GOE
w lt D
Experiment corresponds to pw/D0.3
15
GaAS/GaAlAs Samples
D. Mailly (LPN)
Rings

• Squares
• Nearly ballistic

D 38mK
Possibility to increase the number of electrons
N by illumination of the samples
(electroluminescent diode )
mK
20
40
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32
52
16
Ac dissipative magnetoconductance of the square
billiards
Negative Magnetoconductance Positive Magnetocondu
ctance
N600
N2400
N5900
Positive Magnetoconductance
T20mK lt D
Noat et al. PRL 99
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Electromagnetic Response of isolated systems
Magnetic Response
Electric response
Electric dipole d a E Polarisability a e0
R3 (1- ls /L) Only detectable in GaAs ls 20nm
Orbital magnetic moment
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Comparison between electric and magnetic
responses
Ratio between electric and magnetic responses
determined by Jab2/Fab2
Magnetic response
Electric response
I
I
r
(asf g) 2
Ag r10000 GaAs r0.03
asf e2/h
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Inductive measurements on isolated rings
1 mm
Superconducting resonator
f0 200-400 MHz Q2.105 Tlt1K
12 mm
inductance
capacitive
I
Magnetic Susceptibility Related to
persistent currents
Fw
I
20
Capacitive measurements on isolated rings
1 mm
Superconducting resonator
f0 200-400 MHz Q2.105 Tlt1K
12 mm
inductance
capacitive
I
- - -
Electric Polarisability

Related to screening
I
21
Silver Rings Continuous spectrum D10nK

• N109 electrons
• Disordered Diffusive Transport
• R 10W

Magnetic dominant Response
Anneaux GaAs
Gaz 2d
Discrete Spectrum D100mK

• Small number of electrons
• N1000
• weak disorder nearly ballistic transport
• R 3kW

Electric dominant Response
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Magnetic Response of silver rings
resonator
Silver rings
5 mm
T 40mK
Deblock et al. PRL 2002

• Magnetic Susceptibility cmdI/df
• periodic in F0/2 (ensemble average)
• amplitude I 0.3 nA e/tD
• diamagnetic at low field!
• . Small BCS attractive interactions?
• But similar results with GaAS rings

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Capacitive coupling
Resonance frequency
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Mesoscopic Magnetopolarisability
Phase contribution induced by an electric
potential
Dj1 Dj2
Static potential
Zero average static Magnetopolarisability
But not true any more for a time dependent
electric field
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Magnetopolarisability of GaAs Rings
F0/2
100 Hz
Deblock et al. PRL 2001
Electric response larger than magnetic one!

• periodic in F0/2
• positive at low field field enhanced screening
• amplitude da/a 7 10-4

Finite frequency hw D
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Frequency dependent magnetopolarisability
F screened potential, g level width, g
dimensionless Drude conductance
Efetov PRL(1996), Noat et al. EPL(1996),PRB(2002)
Blanter et Mirlin PRB(1998,2001)
Sensitivity of matrix elements Fab to time
reversal symmetry

f (w) 1 w gtgt D depends on level
occupation Isolated Rings Canonical Ensemble
Tltlt D fa0 or 1 Flux dependence of energy
denominators and matrix elements cancel each
other!
27
f
statistics of levels
Energy denominators
from Blanter et Mirlin PRB(2001)
x
Experiment agrees with theory Isolated rings at
finite frequency.
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High frequency Mesoscopic conductance of
Isolated systems

• System Connected     Conductance transmission
• Isolated System
• AC Dissipative Conductance absorption
• Importance of spectral correlations (Level
  statistics)
• Dominant Response non-dissipative
• Magnetic orbital magnetism and persistent
  currents
• Same order of magnitude 1nA in Ag and GaAs
  rings
• Diamagnetic sign not fully understood
• Electric Response mesoscopic
  magnetopolarisability
• GaAs rings 1000 electrons (discrete
  spectrum)
• Exist only at finite frequency !!

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Mesoscopic rectification
What kind of diodes are mesoscopic systems? At
low frequency w lt 1/ t D Field Asymmetry
related to electron-electron interactions
Amplitude of electron-electron interactions can
be measured! At high frequency w gt 1/ t D
Photovoltaic effect and photomagnetism Time
dependent Electric field induces breaking of
time reversal symmetry.. Frequency dependent
Asymmetry of AB oscillations Microwave induced
DC Orbital magnetism
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Symmetry in magnetic field of G2 and G1
Casimir Onsager Symmetry rules G1(B)G1(-B)
Do not apply for G2(B)
G1
S
G1
AS
G1AS 0 G2AS 0
Low frequency measurements
31
Asymmetry of the AB oscillations on G2
Same behavior than UCF at large field
But AB oscillations have a rigid phase close to
B0!
Can be understood semiclassically
Amplitude modulation with UCFs
32
High frequency non-linear transport Photovoltaic
effect
High frequency excitation
DC current amplifier
DC Induced current by high frequency voltage
excitation V(t) Vw cos wt f
w/2p 1_18 GHz
I DC G2 (w, f) ltV2(t) gt
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High frequency rectification
Asymmetric function of magnetic field
1

lt
GHz
f
4
2ptD
Phase of the AB oscillations still rigid Strong
2nd harmonics
Angers, Chepelianskii et al 2007
34
High frequency rectification
1

gt
GHz
f
4
2ptD
Phase of the AB oscillations is not rigid
35

-
36
Photovoltaic effect at high magnetic field
Hall bar Geometry
A.Chepelianskii 2008
dn RF induced modification of the charge
distribution on the edge states
Completely antisymmetric function of field
37
Experiment Hall magnetometry
Copper shield
Side gate
Bare probe
Hall probe coupled to the stadium
I (courant)
38
Hall probe response to a microwave field
Hall probe coupled to the stadium sign changes
Amplitude a Erf2
Empty Hall probe Heating effect a Erf
dRXX
Empty
dRXy
With stadium
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Hall probe response to a microwave field
Hall probe coupled to the stadium sign changes
Amplitude a Erf2
dRH
Empty Hall probe Heating effect a Erf
dR34,12
I
dRXX
Empty
dRXy
With stadium
dRH
dR12,34
Signal detected dR34,12 - dR12,34 Exhibits
the symmetry of a magnetic field But amplitude
larger than theoretical predictions by a factor
100!!!
V
40
Non-Homogeneous RF Field
Asymmetry also detected on the empty probe
! Billiard generates inhomogeneties in the RF
field! No Onsager relations in time dependent
transport
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• AC mesoscopic response
• Sensitive to spectral statistics (discrete
  spectrum)
• Yields information on Orbital magnetism
  (persistent currents)
• but also on flux dependent electric
  Polarisability
• Finite frequency effect!
• Mesoscopic rectification
• w gt 1/ t D
• Electric field induces breaking of time reversal
  symmetry.
• Phase of the AB oscillations takes any value
  between 0 and p
• Detection of photo induced magnetism in closed
  sytems?
• Inhomogeneous time dependent electric field on a
  Hall bar
• mimics a dc magnetic field!
• Thanks to Oriol and his collaborators in
  LPTMS!!!!

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La méso au labo
Lionel Angers
Francesca Chiodi
Bertrand Reulet
Sophie Gueron
Richard Deblock
Alexei Chepelianskii
Meydi Ferrier
Yves Noat
Collaborations Gilles Montambaux Christophe
Texier Dominique Mailly (LPN)