Transcript Converging theoretical perspectives on charge pumping

Converging theoretical
perspectives on charge pumping
Slava Kashcheyevs
Colloquium at Physikalisch-Technische Bundesanstalt
(Braunschweig, Germany)
November 13th, 2007
Pumping: definitions
I
f
Interested in “small”
pumps to witness:
• quantum interference
• single-electron charging
Pumping overlaps with:
•
•
•
•
rectification
photovoltaic effect
photon-assisted tunneling
ratchets
Outline
• Adiabatic Quantum Pump
 Thouless
pump
 Brouwer formula
 Resonances and quantization
• Beyond the simple picture
 Non-adiabaticity
(driving fast)
 Rate equations and Coulomb interaction
• Single-parameter, non-adiabatic, quantized
Adiabatic pump by Thouless
DJ Thouless, PRB 27, 6083 (1983)
• If the gap remains open at all times,
I = e f  (exact integer)
• Argument is exact for an infinite system
Adiabatic Quantum Pumping
Pump by deforming
a phase-coherent
conductor
Change of interference pattern can
induce “waves” traveling to infinity
Brouwer formula: “plug-and-play”
Vary shape via
X1(t), X2(t), ..
Solve for “frozen time”
scattering matrix
Brouwer formula gives
I in terms of
Brouwer formula: “plug-and-play”
Brouwer formula gives
I in terms of
Brouwer formula: “plug-and-play”
Brouwer formula gives
I in
= terms of
• Depends on a phase
• Allows for a
geometric interpretation
• Need  2 parameters!
“B”
Outline
• Adiabatic Quantum Pump
 Thouless
pump
 Brouwer formula
 Resonances and quantization
• Beyond the simple picture
 Non-adiabaticity
(driving fast)
 Rate equations and Coulomb interaction
• Single-parameter, non-adiabatic, quantized
Resonances and quantization
• Idealized double-barrier resonator
X1
X2
• Tuning X1 and X2 to match a resonance
• I  e f, if the whole resonance line encircled
Y Levinson, O Entin-Wohlman, P Wölfle Physica A 302, 335 (2001)
Resonances and quantization
• How can interference lead to quantization?
X1
X2
• Resonances correspond to quasi-bound states
• Proper loading/unloading gives quantization
V Kashcheyevs, A Aharony, O Entin-Wohlman, PRB 69, 195301 (2004)
Outline
• Adiabatic Quantum Pump
 Thouless
pump
 Brouwer formula
 Resonances and quantization
• Beyond the simple picture
 Non-adiabaticity
(driving fast)
 Rate equations and Coulomb interaction
• Single-parameter, non-adiabatic, quantized
Driving too fast: non-adiabaticity
• What is the meaning of “adiabatic”?
Thouless:
staying in the
ground state
• Can develop a series:
Brouwer:
a gapless
system!
O Entin, A Aharony, Y Levinson
PRB 65, 195411 (2002)
• Q: What is the small parameter?
Floquet scattering for pumps
• Adiabatic scattering matrix S(E; t)
is “quasi-classical”
hf
• Exact description by
• Typical matrix dimension
(# space pts)  (# side-bands)
LARGE!
M Moskalets, M Büttiker PRB 66, 205320 (2002)
Adiabaticity criteria
• Adiabatic scattering matrix S(E; t)
• Floquet matrix
hf
• Adiabatic approximation is OK as long as
≈ Fourier T.[ S(E; t)]
• For a quantized adiabatic pump,
the breakdown scale is f ~ Γ (level width)
M Moskalets, M Büttiker PRB 66, 205320 (2002)
Outline
• Adiabatic Quantum Pump
 Thouless
pump
 Brouwer formula
 Resonances and quantization
• Beyond the simple picture
 Non-adiabaticity
(driving fast)
 Rate equations and Coulomb interaction
• Single-parameter, non-adiabatic, quantized
Rate equations: concept
• A different starting point
ΓL
ΓR
• Consider states of an isolated, finite device
• Tunneling to/from leads as a perturbation!
Rate equations: an example
• Loading/unloading of a quasi-bound state
ε0- i (ΓL +ΓR)
• Rate equation for the occupation probability
• Interference in an almost closed system
just creates the discrete states!
For open systems & Thouless pump, see GM Graf, G Ortelli arXiv:0709.3033
Rate equations are useful!
• Backbone of Single Electron Transistor
theory
DV Averin, KK Likharev “Single Electronics” (1991)
CWJ Beenakker PRB 44, 1646 (1991)
• Conditions to work:
 Tunneling
is weak:
 No coherence between
multiple tunneling events:
Γ << Δε or Ec
Γ << kBT
• Systematic inclusion of charging effects!
Outline
• Adiabatic Quantum Pump
 Thouless
pump
 Brouwer formula
 Resonances and quantization
• Beyond the simple picture
 Non-adiabaticity
(driving fast)
 Rate equations and Coulomb interaction
• Single-parameter, non-adiabatic, quantized
Single-parameter
non-adiabatic
quantized
pumping
B. Kaestner, VK, S. Amakawa, L. Li, M. D. Blumenthal, T. J. B. M.
Janssen, G. Hein, K. Pierz, T. Weimann, U. Siegner, and H. W.
Schumacher,
arXiv:0707.0993
“Roll-over-the-hill”
Experimental results
V1
V2(mV)
V1
V2
V2
• Fix V1 and V2
• Apply Vac on top of V1
• Measure the current I(V2)
A simple theory
ε0
• Given V1(t) and V2 , solve
the scattering problem
• Identify the resonance
ε0(t) , ΓL (t) and ΓR (t)
• Rate equation for the occupation probability P(t)
A: Too slow (almost adiabatic)
Enough time
to equilibrate
ω<<Γ
Charge re-fluxes back to
where it came from → I ≈ 0
B: Balanced for quantization
Tunneling is blocked,
while the left-right
symmetry switches to
opposite
ω>>Γ
Loading from the left,
unloading to the right
→I≈ef
C: Too fast
ω
Tunneling is too slow
to catch up with
energy level switching
The chrage is “stuck”
→I≈0
A general outlook
I / (ef)