weakâlocalization dip. Double-dot with barrier in ..... path in 2DEG. & ÏD â¼ time electron spends in double-dot .... function of lead-displacement. â(1âw)/(1+w).
Mirror symmetries in quantum chaos Robert S. Whitney Collaborators: Paolo Marconcini, Massimo Macucci (Universita` di Pisa) Henning Schomerus, Marten Kopp (University of Lancaster) 1
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W L Quantum dot Marcus Group, Harvard (2003)
GDR Physique Quantique Mesoscopique ´ - Aussois, Dec 2008
Mirror-symmetric dots - expectations and surprises Expectations: Baranger-Mello (1996) – symmetric dot without barrier
Double-dot with barrier in middle 1
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pa
conductance
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2
0 lead & dot size λF
B−field flux quantum
weak−localization dip
Mirror-symmetric dots - expectations and surprises Expectations: Baranger-Mello (1996) – symmetric dot without barrier
Double-dot with barrier in middle 1
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pa
conductance
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2
0 lead & dot size λF
Universal conductance fluctuations
B−field flux quantum
weak−localization dip
Mirror-symmetric dots - expectations and surprises Expectations: Baranger-Mello (1996) – symmetric dot without barrier
Double-dot with barrier in middle 1
th pa
pa
conductance
!??
th
2
0 lead & dot size λF
Universal conductance fluctuations
B−field flux quantum
weak−localization dip
Mirror-symmetric dots - expectations and surprises Expectations: Baranger-Mello (1996) – symmetric dot without barrier
Double-dot with barrier in middle 1
th pa
pa
conductance
!??
th
2
0 lead & dot size λF
Universal conductance fluctuations
B−field flux quantum
weak−localization dip
Outline of talk: • Interference between mirror-symetric paths makes barrier almost “invisible” ⇒ huge peak • symmetry breaking (B-field, disorder, etc) • Guessing experimental numbers
Semiclassics : ray optics in 21st century wavelength other scales
saddle-point of Feynman path integral
=⇒ classical paths plus interference ∝ cos (Sγ1 − Sγ2 )/¯ h
Classical paths: lots of chaos =⇒ many bounces before escape
Semiclassics : ray optics in 21st century wavelength other scales
saddle-point of Feynman path integral
=⇒ classical paths plus interference ∝ cos (Sγ1 − Sγ2 )/¯ h
Classical paths: lots of chaos =⇒ many bounces before escape Transmission probability 2 Feynman integrals → 2 paths
γ1,γ2∈ classical paths
LL
γ2= γ1
A
γ2= γ1
Pγ1 Pγ2 ei(Sγ1 −Sγ2 )/¯h
INTERFERENCE
SM
CLASSICAL
=⇒
X p
weak−localization conductance fluctuations
... but with symmetry??
“Butterfly” double-dot
Handwaving: take only 2 paths
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2
Wb W
θ’ θ
iS /¯h 2 iS /¯ h Quantum probability = rte 1 + tre 2 2|rt|2 asymmetric (S2 6=S1 ) −→ 4|rt|2 symmetric (S2 =S1 ) Symmetric = 2 × asymmetric
⇐ if θ-independent
“Butterfly” double-dot
pa
1 th pa
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2
Wb
Handwaving: take only 2 paths
W
θ’ θ
iS /¯h 2 iS /¯ h Quantum probability = rte 1 + tre 2 2|rt|2 asymmetric (S2 6=S1 ) −→ 4|rt|2 symmetric (S2 =S1 ) L lead
Double-dot: beyond handwaving ... keeping phases at barrier |rθ | iφ Sθ = e eiπ/2 |tθ |
⇒ Destructive interference between 2 paths
iπ/2
e
|tθ | |rθ |
if one tunnels 2(2j−1) times more than the other
i θ
θ
i
Double-dot: beyond handwaving ... keeping phases at barrier |rθ | iφ Sθ = e eiπ/2 |tθ |
iπ/2
e
|tθ | |rθ |
⇒ Destructive interference between 2 paths
if one tunnels 2(2j−1) times more than the other
Quantum prob. ∝ (1 − P )
∞ X
n=0
L lead n
n P SSb 41
– P = prob. to hit barrier – S Sb = double-scattering matrix (4×4) acts BOTH paths — acts on (LL,LR,RL,RR)
• put asymmetry in SSb :
put exp[iδS/¯ h] in non-corner terms
⇒ diagonalize SSb and sum geometric series in n
R lead
P =Wb /(Wb +W )
Size of conductance-peak
Tb =h|tθ |2 iθ
Wb W
P
Tb
1−P
Conductance (two parameters: P & Tb ) P (1+P )Tb e2 • sym. Gsym = h N (1−P )2 +4P Tb N modes
• asym. Gasym =
e2 h
N
P Tb 1−P +2P Tb
0.00 0.01 0.02
Ttb
optimal P
Conductance ratio: Fpeak = Gsym /Gasym
• Fpeak 1 ⇒ big peak • Fpeak =1 ⇒ no peak
10 8 6 4 2 0.4
0 0.6
P
0.8 1.0
Gsym Gasym
P =Wb /(Wb +W )
Maximizing the peak
Tb =h|tθ |2 iθ
0.00
Wb W
P 1−P
0.01 0.02
Ttb
optimal P
Tb
10 8 6
Gsym
4
Gasym
N modes 2 0.4
0 0.6
P 0.8 To see a BIG peak: 1.0 (1) Tunnelling rate Tb → 0 (2) Probability to hit barrier P → 1 ...BUT Fpeak really maximized by P = Popt ≡
1−2Tb 1/2 1−4Tb
P =Wb /(Wb +W )
Maximizing the peak
Tb =h|tθ |2 iθ
0.00
Wb W
P
0.01 0.02
Ttb
optimal P
Tb
1−P
10 8 6
Gsym
4
Gasym
N modes 2 0.4
0 0.6
P 0.8 To see a BIG peak: 1.0 (1) Tunnelling rate Tb → 0 (2) Probability to hit barrier P → 1 ...BUT Fpeak really maximized by P = Popt ≡
Then for Tb 1 & P = Popt
Gsym = Gasym =
e2 h e2 h
× 14 N × 12 N
1/2 Tb
)
1−2Tb 1/2 1−4Tb
Fpeak
1 ∝√ Tb
Family-tree of similar effects INTERFERENCE (open system)
INTERFERENCE (nearly closed system)
BIG effects
BIG effect CHAOTIC or REGULAR
SMALL effects CHAOTIC
REGULAR (integrable)
Fabry−Perot
super− conductor
EXPT (1991): Kastalsky et al THEORY (1993−4): Volkov et al Beenakker et al Nazarov−Hekking INTEGRABILITY Kosztin et al (1995)
discrete levels in dots
e
h
Resonant tunnelling
weak−localization universal conductance fluct.
Reflectionless tunnelling
Mirror−symmetry enhanced tunnelling 1 th
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Two Cousins:- resonant tunnel. and reflectionless tunnel. Resonant tunnelling
Mirror−symmetry enhanced tunnelling
INTEGRABILITY Kosztin et al (1995)
THEORY (1993−4): Volkov et al Beenakker et al Nazarov−Hekking
EXPT (1991): Kastalsky et al
super− conductor
discrete levels in dots
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2
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Reflectionless tunnelling
Peak-shape with B-field or deformation conductance
Lorentzian B
B
0
Lorentzian width B = Bc
q
τ0 τD
approx same width as weak-localization
B−field flux quantum
Bc ≡ one flux quantum 0 ∼ time in double-dot τD
Peak-shape with B-field or deformation conductance
Lorentzian B
B
0
Lorentzian width B = Bc
q
τ0 τD
approx same width as weak-localization deformation
δL
B−field flux quantum
Bc ≡ one flux quantum τD ∼ time in double-dot
Deformation: Lorentzian— width δL ∼ λF
q
τ0 τD
BAD NEWS no peak if deform small fraction of λF
More BAD NEWS : disorder and dephasing e−e interactions External noise phonons microwaves, etc
disorder
−1 τD τD Suppression of peak = 1 + + τφ τmf τφ = dephasing time (as in weak-localization) τmf = mean-free path in 2DEG & τD ∼ time electron spends in double-dot
We choose: • each dot’s diameter L= 4 µm • barrier tunnelling prob Tb =1.5×10−3
& barrier width ∼dot diameter (maximum)
⇒ Maximise peak: P =0.93 so lead’s W =310 nm (12 modes) e2 e2 ⇒ Gsym ' 3.2 h Gasym ' 0.22 h
Peak is 14×background; Fpeak '14
Suppression: 1) asymmetry irrelevant if δL < λF /20 ∼ 2nm 2) realistic disorder/dephasing: reduces peak to 10×background Detect B-fields: conductance drops by order of magnitude if fifth of a flux-quantum in each dot.
Normalized conductance (G/G 0 )
Numerics 3.5 L /2
3
L /2
L /2
Use numbers from experiment ⇒ beyond regime of theory??
L /2
(a)
2.5 W
2
W tb B
B
1.5
Fully solve for waves in double-dot W tb = L = 4 µ m; W = 310 nm
1
Recursive Green funct.
0.5 0
G tb
0
-many vertical slices 50
100
150
200
250
300
350
Normalized conductance (G/G 0 )
Applied magnetic field ( µT)
no disorder/dephasing
3.5 3
Gsym = 14 × Gasym
(b)
2.5 2
B dependence ∼ Lorentzian peak widths ∼ theory
δL
1.5 1 0.5
G tb
0 0
0.01
0.02
0.03
Barrier displacement/wavelength
0.04
Conclusions Huge new interference effect: barrier can become “invisible” in symmetric double-dot
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1
2
Wb W
θ’ θ
Perfect device (perfect symmetry & no disorder/dephasing) • peak arbitrarily large Best “available” device (cleanest 2DEGs & lowest temperatures) • peak 10× background it is > 10× weak-localization dip • detect less than quantum of flux
Conclusions Huge new interference effect: barrier can become “invisible” in symmetric double-dot
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1
2
Wb W
θ’ θ
Perfect device (perfect symmetry & no disorder/dephasing) • peak arbitrarily large Best “available” device (cleanest 2DEGs & lowest temperatures) • peak 10× background it is > 10× weak-localization dip • detect less than quantum of flux Question for expt: Make two dots symmetric on scale of 2nm ?? Question for theory: Weak-localization, UCFs, shot noise Spin-Orbit ??
Post-script on weak-loc with mirror-sym. Baranger-Mello (1996) (a) Enhancement of transmission due to left−right symmetry
1 Conductance peak = 4 G0 destroyed by sym.-breaking
γ
L γ’
... but B-field independent
Encounter near lead (b) Reduction of transmission due to left−right symmetry γ
W γ’
L Encounter in dot
(c) Reduction of reflection due to left−right symmetry γ
γ’
Encounter in dot
Post-script on weak-loc with mirror-sym. Baranger-Mello (1996) (a) Enhancement of transmission due to left−right symmetry
1 Conductance peak = 4 G0 destroyed by sym.-breaking
γ
L γ’
... but B-field independent
Encounter near lead (b) Reduction of transmission due to left−right symmetry γ
W γ’
see “shape of conductance peak” L
U
W
U
W
WL
Encounter in dot
WR
(c) Reduction of reflection due to left−right symmetry γ
