Modelling of recent charge pumping experiments

1. Modelling of recent charge pumping experiments Vyacheslavs (Slava) Kashcheyevs Mark Buitelaar (Cambridge, UK)Bernd Kästner (PTB, Germany) Seminar at University of Geneva (Switzerland)April22st, 2008

2. Pumping = dc response to (local) ac perturbation I f 1st part

3. Part I: adiabatic pumping in CNT arXiv:0804.3219

4. Experimental data • Peak-and-dip structure • Correlated with Coulomb blockade peaks • Reverse wave direction => reverse polarity

5. 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

6. Experiment and theory

7. Interpretation: several dots

8. Interpretation and a model

9. Two “triple points” One “quadruple point” 0.3 Γ/Δ 1 3 (Static) transmission probability • If Δ is less than ΓL or ΓR (or both), the two dots are not resolved in a conductance measurement Δ

10. Adiabatic pumping (weak + strong) Charge per period Q Brouwer / PTB formula is easy to obtain analytically Q is an integral over the area enclosed by the pumping contour

11. (0,0) (1,0) (0,1) (1,1) Theory results for pumping

12. Effects of assymetry

13. Reduce frequency 5-fold

14. 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)

15. 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. SchumacherPRB 77, 153301 (2008) +arXiv:0803.0869

16. Quantization conditions !!! Conflicting mechanisms, not enough just to tune the frequency

17. (0,0) (1,0) (0,1) (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 loadand unload a full electron

18. “Roll-over-the-hill”

19. V1 V2(mV) V2 Experimental results • Fix V1and V2 • Apply Vacon top of V1 • Measure the current I(V2) V1 V2

20. ΓL ΓR Theory: step 1 • Assume a simple real-spacedouble-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) ε0

21. 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)

22. FixU1(t) and U2 Solve the scattering problem forε0(t), ΓL (t) and ΓR (t) Fix the frequency and solve the master equation ε0 Results

23. A: Too slow (almost adiabatic) Adiabatic limit – always enough time to equilibrate, unloading all we got from loading to the same leads ω<<Γ Charge re-fluxesback to where it came from →I ≈ 0

24. B: Balanced for quantization Non-adiabatic blockade of tunneling allows for left/rightsymmetry switch between loading and unloading! ω>>Γ Loading from the left, unloading to the right→I ≈ ef

25. C: Too fast ω Tunneling is too slow to catch up with energy level switching: non-adiabaticicty kills quantization as expected The charge is “stuck” →I ≈ 0

26. Frequency and gate dependence I / (ef)

27. 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)

28. Thank you!