Modelling of recent charge pumping experiments
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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
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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
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Experimental data• Peak-and-dip structure• Correlated with Coulomb blockade
peaks• Reverse wave direction => reverse
polarity
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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
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Experimentand theory
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Interpretation: several dots
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Interpretation and a model
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(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”
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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
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(0,0)
(0,1)
(1,0)
(1,1)
Theory results for pumping
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Effects of assymetry
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Reduce frequency 5-fold
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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)
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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
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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
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(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
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“Roll-over-the-hill”
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V2(mV)
• Fix V1 and V2
• Apply Vac on top of V1
• Measure the current I(V2)
V1
V2
Experimental results
V1 V2
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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
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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)
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• 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
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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
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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
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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
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Frequency and gate dependence
I / (ef)
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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)
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