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Converging theoretical perspectives on charge pumping

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Slava Kashcheyevs

Colloquium at Physikalisch-Technische Bundesanstalt (Braunschweig, Germany)

November 13th, 2007

I

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

f

Interested in “small”pumps to witness:

Pumping overlaps with:

- quantum interference
- single-electron charging

- 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

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

a phase-coherentconductor

Pump by deforming

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

Brouwer formula gives I in terms of

Vary shape via X1(t), X2(t), ..

Solve for “frozen time”

scattering matrix

Brouwer formula gives I in terms of

Brouwer formula gives I =

Brouwer formula gives I in terms of

- Depends on a phase
- Allows for a geometric interpretation
- Need 2 parameters!

“B”

- 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

- 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)

X1X2

- 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)

- 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

- 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)

h f

- 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)

h f

- 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

ΓL

ΓR

- A different starting point
- Consider states of an isolated, finite device
- Tunneling to/from leads as a perturbation!

ε0- i (ΓL +ΓR)

- 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

- 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

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”

V1

V2(mV)

V2

- Fix V1and V2
- Apply Vacon top of V1
- Measure the current I(V2)

V1

V2

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

Enough time to equilibrate

ω<<Γ

Charge re-fluxesback to where it came from →I ≈ 0

Tunneling is blocked, while the left-right symmetry switches to opposite

ω>>Γ

Loading from the left, unloading to the right→I ≈ ef

ω

Tunneling is too slow to catch up with energy level switching

The chrage is “stuck” →I ≈ 0

I / (ef)