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OutlineOutline

Converging theoretical perspectives on charge pumping

Slava Kashcheyevs

Colloquium at Physikalisch-Technische Bundesanstalt (Braunschweig, Germany)

November 13th, 2007

Pumping: definitions

I

- rectification
- photovoltaic effect
- photon-assisted tunneling
- ratchets

f

Interested in “small”pumps to witness:

Pumping overlaps with:

- quantum interference
- single-electron charging

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 = ef (exact integer)

- Argument is exact for an infinite system

Adiabatic Quantum Pumping

a phase-coherentconductor

Pump by deforming

Change of interference pattern can induce “waves” traveling to infinity

Brouwer formula gives I in terms of

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”- 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
- Tuning X1 and X2 to match a resonance
- I e f, if the whole resonance line encircled

X1X2

Y Levinson, O Entin-Wohlman, P Wölfle Physica A 302, 335 (2001)

Resonances and quantization

- How can interference lead to quantization?
- 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”?
- Can develop a series:
- Q: What is the small parameter?

Thouless:

staying in the ground state

Brouwer:

a gapless system!

O Entin, A Aharony, Y LevinsonPRB 65, 195411 (2002)

Floquet scattering for pumps

- Adiabatic scattering matrix S(E; t) is “quasi-classical”
- 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
- Adiabatic approximation is OK as long as ≈ FourierT.[ S(E; t)]
- For a quantized adiabatic pump, the breakdown scale is f ~ Γ (level width)

M Moskalets, M Büttiker PRB 66, 205320 (2002)

- 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

ΓR

Rate equations: concept- A different starting point
- 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
- Rate equation for the occupation probability
- Interference in an almost closed systemjust 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
- Conditions to work:
- Tunneling is weak: Γ << Δε or Ec
- No coherence between multiple tunneling events: Γ << kBT
- Systematic inclusion of charging effects!

DV Averin, KK Likharev “Single Electronics” (1991)CWJ Beenakker PRB 44, 1646 (1991)

- 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

GivenV1(t) and V2, solve the scattering problem

Identify the resonanceε0(t), ΓL (t) and ΓR (t)

Rate equation for the occupation probability P(t)

ε0

A simple theoryA: Too slow (almost adiabatic)

Enough time to equilibrate

ω<<Γ

Charge re-fluxesback 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)

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