modelling of recent charge pumping experiments

Vyacheslavs (Slava) Kashcheyevs

Mark Buitelaar (Cambridge, UK)Bernd Kästner (PTB, Germany)

Seminar at University of Geneva (Switzerland)April 22st, 2008

Modelling of recent charge pumping experiments

Pumping = dc response to (local) ac perturbation

f

I

weak strong

adiabatic (linear in ω)

carbon nanotubewith acoustic waves

carbon nanotube with acoustic waves

non-adiabatic (all orders in ω)

GaAs nanowirewith direct gating

1st part

Part I: adiabatic pumping in CNT

arXiv:0804.3219

Experimental data• Peak-and-dip structure• Correlated with Coulomb blockade

peaks• Reverse wave direction => reverse

polarity

Experimental findings

• At small powers of applied acoustic waves the features grow with power and become more symmetric

• For stronger pumping the maximal current saturates and opposite sign peaks move aparpt

Experimentand theory

Interpretation: several dots

Interpretation and a model

(Static) transmission probability

• If Δ is less than ΓL or ΓR (or both), the two dots are not resolved in a conductance measurement

Δ

Γ/Δ310.3

Two “triple points” One “quadruple point”

Adiabatic pumping (weak + strong)

Charge per period Q

Q is an integral over the area enclosed by the pumping contour

is easy to obtain analytically

Brouwer / PTB formula

(0,0)

(0,1)

(1,0)

(1,1)

Theory results for pumping

Effects of assymetry

Reduce frequency 5-fold

Conclusions of part I• Simple single-particle model

describes many experimental features (robust)

• Most detailed experimental test of the adiabatic pumping theory to-date?

• Alternative mechanisms Barrier modulation + level renormalization? Rectification?

• Work in progress: Connect wih the overlapping peak regime

(moving quantum dot picture, no sign change)

Single-parameter non-adiabatic quantized charge 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

PRB 77, 153301 (2008) + arXiv:0803.0869

weak strong

adiabatic (linear in ω)

carbon nanotubewith acoustic waves

carbon nanotube with acoustic waves

non-adiabatic (all orders in ω)

GaAs nanowirewith direct gating

Quantization conditions

!!! Conflicting mechanisms, not enough just to tune the frequency

weak strong

adiabatic (linear in ω)

quantization: NO quantization: YESwith ≥ 2 paramters

non-adiabatic (all orders in ω)

quantization: YES with ≥ 1 paramters

(0,0)

(0,1)

(1,0)

(1,1)

Single-parameter quantization?• Quantization = loading form

the left + unloading to the right

• One-parameter kills quantization because the symmetry (i.e. ΓL / ΓR)

at loading and the symmetryat unloading are the same

• Non-adiabaticity kills quantization because not enough time to load

and unload a full electron

“Roll-over-the-hill”

V2(mV)

• Fix V1 and V2

• Apply Vac on top of V1

• Measure the current I(V2)

V1

V2

Experimental results

V1 V2

Theory: step 1• Assume a simple real-space

double-hill potential:

• For every t, solve the “frozen-time” scattering problem

• Fit the lowest resonance with Breit-Wigner formula and obtain ε0(t) , ΓL (t) and ΓR

(t) ΓRΓL

ε0

Theory: step 2

• Write (an exact) equation-of-motion for P(t)

• If max(ΓL,ΓR, ω) << kT one gets a Markovian master equation

Flensberg, Pustilnik & Niu PRB (1999)

For the adiabatic case, see Kashcheyevs, Aharony, Entin, cond-mat/0308382v1 (section lacking in PRB version)

• Fix U1(t) and U2

• Solve the scattering problem forε0(t) , ΓL (t) and ΓR (t)

• Fix the frequency and solve the master equation

Results

ε0

A: Too slow (almost adiabatic)

Adiabatic limit – always enough time to equilibrate, unloading all we got from loading to the same leads

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

ω<<Γ

B: Balanced for quantization

Non-adiabatic blockade of tunneling allows for left/rightsymmetry switch between loading and unloading!

Loading from the left, unloading to the right

→ I ≈ e f

ω>>Γ

C: Too fast

Tunneling is too slow to catch up with energy level switching: non-adiabaticicty kills quantization as expected

The charge is “stuck” → I ≈ 0

ω

Frequency and gate dependence

I / (ef)

Outlook for part II

• Single-parameter dc pumping possible due to non-adiabatic blockade of tunneling

• In progress: two-parameter “bare-bones” model

for quantitative fitting same type of pump with carbon nanotubes

• In the same device there exists a range of qunatized ac current! not measured (yet)