Single-electron pumping in common-gate triple-dot devices with arbitrary asymmetric gate capacitance distributions


Single-electron pumping in common-gate triple-dot devices is examined for arbitrary asymmetric gate capacitance distributions. Here, the ratio of the three gate capacitances is assumed to be a + d/2:1:ad/2, where a and d are arbitrary positive real numbers. To ensure that pumping occurs, the gate voltage should be swung such that the set of excess electron numbers in the three dots is (ns, nc, ns) at low gate voltage and (ns, nc + 1, ns) or (ns + 1, nc − 1, ns + 1) at high gate voltage, where ns and nc are arbitrary integers. Moreover, (a, d) should be within a region that depends on the set of excess electron numbers at high and low gate voltages. Thus, the regions for pumping in the ad plane are revealed. The absolute value of the bias voltage for pumping is limited to a maximum value, called the critical voltage. The critical voltage is also demonstrated as a function of a and d. The largest critical voltage obtained in this study is 2.5 times that reported in previous work for common-gate triple-dot devices.

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  1. 1.

    Geerligs, L.J., Anderegg, V.F., Holweg, P.A.M., Mooij, J.E., Pothier, H., Esteve, D., Urbina, C., Devoret, M.H.: Frequency-locked turnstile device for single electrons. Phys. Rev. Lett. 64, 2691–2694 (1990)

    Article  Google Scholar 

  2. 2.

    Pothier, H., Lafarge, P., Urbina, C., Esteve, D., Devoret, M.H.: Single-electron pump based on charging effects. Europhys. Lett. 17, 249–254 (1992)

    Article  Google Scholar 

  3. 3.

    Ono, Y., Takahashi, Y.: Electron pump by a combined single-electron/field-effect-transistor structure. Appl. Phys. Lett. 82, 1221–1223 (2003)

    Article  Google Scholar 

  4. 4.

    Kouwenhoven, L.P., Johnson, A.T., van der Vaart, N.C., Harmans, C.J.P.M., Foxon, C.T.: Quantized current in a quantum-dot turnstile using oscillating tunnel barriers. Phys. Rev. Lett. 67, 1626–1629 (1991)

    Article  Google Scholar 

  5. 5.

    Fujiwara, A., Zimmerman, N.M., Ono, Y., Takahashi, Y.: Current quantization due to single-electron transfer in Si-wire charge-coupled devices. Appl. Phys. Lett. 84, 1323–1325 (2004)

    Article  Google Scholar 

  6. 6.

    Fujiwara, A., Nishiguchi, K., Ono, Y.: Nanoampere charge pump by single-electron ratchet using silicon nanowire metal-oxide-semiconductor field-effect transistor. Appl. Phys. Lett. 92, 042102 (2008)

    Article  Google Scholar 

  7. 7.

    Nakazato, K., Blaikle, R.J., Ahmed, H.: Single-electron memory. J. Appl. Phys. 75, 5123–5134 (1994)

    Article  Google Scholar 

  8. 8.

    Waugh, F.R., Berry, M.J., Crouch, C.H., Livermore, C., Mar, D.J., Westervelt, R.M., Campman, K.L., Gossard, A.C.: Measuring interactions between tunnel-coupled quantum dots. Phys. Rev. B 53, 1413–1420 (1996)

    Article  Google Scholar 

  9. 9.

    Nuryadi, R., Ikeda, H., Ishikawa, Y., Tabe, M.: Ambipolar Coulomb blockade characteristics in a two-dimensional Si multidot device. IEEE Trans. Nanotechnol. 2, 231–235 (2003)

    Article  Google Scholar 

  10. 10.

    Nuryadi, R., Ikeda, H., Ishikawa, Y., Tabe, M.: Current fluctuation in single-hole transport through a two-dimensional Si multidot. Appl. Phys. Lett. 86, 133106 (2005)

    Article  Google Scholar 

  11. 11.

    Ikeda, H., Tabe, M.: Numerical study of turnstile operation in random-multidot-channel field-effect transistor. J. Appl. Phys. 99, 073705 (2006)

    Article  Google Scholar 

  12. 12.

    Moraru, D., Ono, Y., Inokawa, H., Tabe, M.: Quantized electron transfer through random multiple tunnel junctions in phosphorus-doped silicon nanowires. Phys. Rev. B 76, 075332 (2007)

    Article  Google Scholar 

  13. 13.

    Yokoi, K., Moraru, D., Ligowski, M., Tabe, M.: Single-gated single-electron transfer in nonuniform arrays of quantum dots. Jpn. J. Appl. Phys. 48, 024503 (2009)

    Article  Google Scholar 

  14. 14.

    Jalil, M.B.A., Ahmed, H., Wagner, M.: Analysis of multiple-tunnel junctions and their application to bidirectional electron pumps. J. Appl. Phys. 84, 4617 (1998)

    Article  Google Scholar 

  15. 15.

    Weiss, D.N., Brokmann, X., Calvet, L.E., Kastner, M.A., Bawendi, M.G.: Multi-island single-electron devices from self-assembled colloidal nanocrystal chains. Appl. Phys. Lett. 88, 143507 (2006)

    Article  Google Scholar 

  16. 16.

    Kang, Y.B., Hu, G.Y., O’Connell, R.F., Ryu, J.Y.: Effect of stray capacitances on single electron tunneling in a turnstile. J. Appl. Phys. 80, 1526–1531 (1996)

    Article  Google Scholar 

  17. 17.

    Mizuta, A., Moriya, M., Usami, K., Kobayashi, T., Shimada, H., Mizugaki, Y.: Coulomb blockade conditions for detailed model of single-electron turnstile device including finite self-capacitances of island electrodes. Jpn. J. Appl. Phys. 46, 3144–3148 (2007)

    Article  Google Scholar 

  18. 18.

    Danilov, V., Golubev, D.S., Kubatkin, S.E.: Tunneling through a multigrain system: deducing sample topology from nonlinear conductance. Phys. Rev. B 65, 125312 (2002)

    Article  Google Scholar 

  19. 19.

    Imai, S., Kawamura, D.: Analytical study on a single electron device with two islands connected to one gate electrode. Jpn. J. Appl. Phys. 47, 9003–9009 (2008)

    Article  Google Scholar 

  20. 20.

    Imai, S., Kato, H., Hiraoka, Y.: Stability diagrams of single-common-gate double-dot single-electron transistors with arbitrary junction and gate capacitances. Jpn. J. Appl. Phys. 51, 124301 (2012)

    Article  Google Scholar 

  21. 21.

    Imai, S., Kawamura, D.: Analytical study on a single-electron device with three islands connected to one gate electrode. Jpn. J. Appl. Phys. 48, 124502 (2009)

    Article  Google Scholar 

  22. 22.

    Imai, S.: Stability diagrams of triple-dot single-electron device with single common gate. Jpn. J. Appl. Phys. 50, 034302 (2011)

    Article  Google Scholar 

  23. 23.

    Imai, S., Moriguchi, S.: Single-common-gate triple-dot single-electron devices with side gate capacitances larger than the central one. Jpn. J. Appl. Phys. 53, 094002 (2014)

    Article  Google Scholar 

  24. 24.

    Imai, S., Iwasa, N.: Stability diagrams and turnstile operations of single-common-gate triple-dot single-electron devices with outer junction capacitances different from inner ones. Jpn. J. Appl. Phys. 54, 064001 (2015)

    Article  Google Scholar 

  25. 25.

    Imai, S., Ito, M.: Anomalous single-electron transfer in common-gate quadruple-dot single-electron devices with asymmetric junction capacitances. Jpn. J. Appl. Phys. 57, 064001 (2018)

    Article  Google Scholar 

  26. 26.

    Imai, S., Nakajima, A., Kobata, T.: Single-electron pumping in single-common-gate triple-dot devices with asymmetric gate capacitances. Jpn. J. Appl. Phys. 54, 104001 (2015)

    Article  Google Scholar 

  27. 27.

    Imai, S., Ito, Y.: Single-electron pumping in single-common-gate quadruple-dot devices with asymmetric gate capacitances. Jpn. J. Appl. Phys. 58, 034001 (2019)

    Article  Google Scholar 

  28. 28.

    Azuma, Y., Yasutake, Y., Kono, K., Kanehara, M., Teranishi, T., Majima, Y.: Single-electron transistor fabricated by two bottom-up processes of electroless Au plating and chemisorption of Au nanoparticle. Jpn. J. Appl. Phys. 49, 090206 (2010)

    Article  Google Scholar 

  29. 29.

    Okabayashi, N., Maeda, K., Muraki, T., Tanaka, D., Sakamoto, M., Teranishi, T., Majima, Y.: Uniform charging energy of single-electron transistors by using size-controlled Au nanoparticles. Appl. Phys. Lett. 100, 033101 (2012)

    Article  Google Scholar 

  30. 30.

    Kano, S., Maeda, K., Tanaka, D., Sakamoto, M., Haranishi, T., Majima, Y.: Chemically assembled double-dot single-electron transistor analyzed by the orthodox model considering offset charge. J. Appl. Phys. 118, 134304 (2015)

    Article  Google Scholar 

  31. 31.

    Katz, E., Shipway, A.N., Willner, I.: Chap. 6. In: Schmid, G. (ed.) Nanoparticles—from Theory to Application, pp. 368–421. Wiley-VCH, Weinheim (2004)

    Google Scholar 

  32. 32.

    Devoret, M.H., Grabert, H.: Chap. 1. In: Grabert, H., Devoret, M.H. (eds.) Single Charge Tunneling: Coulomb Blockade Phenomena in Nanostructures, pp. 1–19. Plenum, New York (1992)

    Google Scholar 

  33. 33.

    Jo, M., Uchida, T., Tsurumaki-Fukuchi, A., Arita, M., Fujiwara, A., Ono, Y., Nishiguchi, K., Inokawa, H., Takahashi, Y.: Fabrication and single-electron-transfer operation of a triple-dot single-electron transistor. J. Appl. Phys. 118, 214305 (2015)

    Article  Google Scholar 

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Appendix 1

The derivations of the CB conditions as linear inequalities of Vsd, Vg, and n1 to n3 are similar to those for quadruple-dot SE devices described in Appendices A and B of Ref. [25]. Thus, only the results for triple-dot devices are presented here because the derivation procedure is very long.

The total effective charges on the left-hand side of Eq. (1) and the right-hand side of Eq. (2) are presented as follows.

$$C_{{{\text{T}}10}} (V_{1} - V_{ 0} ) = Q_{1} + r_{{{\kern 1pt} 21{\text{R}}}} \left( {Q_{2} + r_{{32{\text{R}}}} Q_{3} } \right) - C_{{{\text{T}}10}} V_{0} ,$$
$$C_{{{\text{T}}21}} (V_{2} - V_{ 1} ) = - r_{{{\kern 1pt} {\text{R}}21}} Q_{1} + r_{{{\kern 1pt} {\text{L}}12}} \left( {Q_{2} + r_{{{\kern 1pt} 32{\text{R}}}} Q_{3} } \right),$$
$$C_{{{\text{T3}}2}} (V_{3} - V_{ 2} ) = - r_{{{\kern 1pt} {\text{R3}}2}} \left( {r_{{{\kern 1pt} 21{\text{L}}}} Q_{1} + Q_{2} } \right) + r_{{{\kern 1pt} {\text{L}}23}} Q_{3} ,$$
$$C_{\text{T43}} (V_{ 4} - V_{ 3} ) = C_{\text{T43}} V_{ 4} - r_{{{\kern 1pt} 32{\text{L}}}} \left( {r_{{{\kern 1pt} 21{\text{L}}}} Q_{1} + Q_{2} } \right) - Q_{3} .$$

Here, Q1, Q2, and Q3 are defined as follows.

$$Q_{{{\kern 1pt} 1}} = - n_{1} e + C_{10} V_{ 0} + C_{\text{g1}} V_{\text{g}} ,$$
$$Q_{{{\kern 1pt} 2}} = - n_{2} e + C_{\text{g2}} V_{\text{g}} ,$$
$$Q_{{{\kern 1pt} 3}} = - n_{3} e + C_{43} V_{ 4} + C_{\text{g3}} V_{\text{g}} .$$

The source and drain voltages are defined as follows.

$$V_{ 0} = - \frac{{V_{\text{sd}} }}{2},$$
$$V_{ 4} = \frac{{V_{\text{sd}} }}{2}.$$

By substituting Eqs. (7)–(11) into the right-hand sides of Eqs. (3)–(6), the total effective charges are expressed as linear combinations of Vsd, Vg, and n1 to n3.

The total capacitances are as follows.

$$C_{{{\text{T}}10}} = \frac{{C_{{{\text{Lg}}1}} C_{21} + C_{{{\text{Lg}}1}} C_{{{\text{Rg}}2}} + C_{21} C_{{{\text{Rg}}2}} }}{{C_{21} + C_{{{\text{Rg}}2}} }},$$
$$C_{\text{T21}} = \frac{{C_{21} C_{{{\text{Lg}}1}} + C_{21} C_{{{\text{Rg}}2}} + C_{{{\text{Lg}}1}} C_{{{\text{Rg}}2}} }}{{C_{{{\text{Lg}}1}} + C_{\text{Rg2}} }},$$
$$C_{\text{T32}} = \frac{{C_{32} C_{\text{Lg2}} + C_{32} C_{\text{Rg3}} + C_{\text{Lg2}} C_{\text{Rg3}} }}{{C_{{{\text{Lg}}2}} + C_{\text{Rg3}} }},$$
$$C_{\text{T43}} = \frac{{C_{\text{Rg3}} C_{32} + C_{\text{Rg3}} C_{\text{Lg2}} + C_{32} C_{\text{Lg2}} }}{{C_{32} + C_{\text{Lg2}} }}.$$

Only Eqs. (12) and (15) are used as coefficients of V0 and V4 on the right-hand sides of Eqs. (3) and (6), respectively.

The coefficients on the right-hand sides of Eqs. (3)–(6) are as follows.

$$r_{{{\kern 1pt} 21{\text{R}}}} = \frac{{C_{21} }}{{C_{21} + C_{\text{Rg2}} }},\quad r_{{{\kern 1pt} 32{\text{R}}}} = \frac{{C_{32} }}{{C_{32} + C_{\text{Rg3}} }},\quad r_{{{\kern 1pt} 21{\text{L}}}} = \frac{{C_{21} }}{{C_{21} + C_{\text{Lg1}} }},\quad r_{{{\kern 1pt} 32{\text{L}}}} = \frac{{C_{32} }}{{C_{32} + C_{\text{Lg2}} }},$$
$$r_{{{\kern 1pt} {\text{R}}21}} = \frac{{C_{\text{Rg2}} }}{{C_{{{\text{Lg}}1}} + C_{\text{Rg2}} }},\quad r_{{{\kern 1pt} {\text{R}}32}} = \frac{{C_{\text{Rg3}} }}{{C_{{{\text{Lg}}2}} + C_{\text{Rg3}} }},\quad r_{{{\kern 1pt} {\text{L}}12}} = \frac{{C_{{{\text{Lg}}1}} }}{{C_{{{\text{Lg}}1}} + C_{\text{Rg2}} }},\quad r_{{{\kern 1pt} {\text{L}}23}} = \frac{{C_{\text{Lg2}} }}{{C_{\text{Lg2}} + C_{\text{Rg3}} }}.$$

The parallel capacitances CLgi and CRgi in the coefficients and the series capacitances CLi and CRi are obtained inductively as follows.

$$C_{{{\text{L}}1}} = C_{10} ,C_{{{\text{Lg}}1}} = C_{{{\text{g}}1}} + C_{\text{L1}} ,$$
$$C_{\text{L2}} = \frac{{C_{21} C_{\text{Lg1}} }}{{C_{21} + C_{{{\text{Lg}}1}} }},C_{{{\text{Lg}}2}} = C_{{{\text{g}}2}} + C_{\text{L2}} ,$$
$$C_{\text{L3}} = \frac{{C_{32} C_{\text{Lg2}} }}{{C_{32} + C_{\text{Lg2}} }},C_{{{\text{Lg}}3}} = C_{{{\text{g}}3}} + C_{\text{L3}} .$$
$$C_{{{\text{R}}3}} = C_{43} ,C_{{{\text{Rg}}3}} = C_{{{\text{g}}3}} + C_{\text{R3}} ,$$
$$C_{{{\text{R}}2}} = \frac{{C_{32} C_{{{\text{Rg}}3}} }}{{C_{32} + C_{\text{Rg3}} }},C_{{{\text{Rg}}2}} = C_{{{\text{g}}2}} + C_{\text{R2}} ,$$
$$C_{{{\text{R}}1}} = \frac{{C_{21} C_{\text{Rg2}} }}{{C_{21} + C_{\text{Rg2}} }},C_{{{\text{Rg}}1}} = C_{{{\text{g}}1}} + C_{\text{R1}} .$$

In this paper, the junction capacitances C(k+1)k are uniform and equal to Cj.

Appendix 2

Tunneling event Tkk′ can occur when CTk′k (Vk′ − Vk) is greater than e/2, where CTk′k is equal to CTkk′ if k = k′ + 1. Here, (Vk′ − Vk) is explained to be the dominant factor that determines which of Tkk′ and T(4–k′)(4–k) is preferable when the operating point leaves an SR along the V′g axis.

If d > 0, that is, Cg1 > Cg3, then V1 is always greater than V3 for Vsd = 0, Vg > 0, n1 = n3 ≥ 0 and n2 ≥ 0. The reason is as follows. If Vg is positive and no excess electrons exist at the three islands, then V1 is increased by Vg more strongly than V3 because island I1 is capacitively coupled with the gate electrode more strongly than island I3 is. If excess electrons exist in the three islands symmetrically (n1 = n3) and Vg is zero, then V1 is decreased from zero by excess electrons more weakly than V3 because island I1 is capacitively coupled with the zero-biased gate electrode more strongly than island I3 is. Thus, if Vg is positive and excess electrons exist, then V1 is always greater than V3 because of the principle of superposition.

Thus, V1 − V2 > V3 − V2, V2 − V3 > V2 − V1, V1 − V0 > V3 − V4, and V4 − V3 > V0 − V1 because V0 = V4 = 0. If (Vk′ − Vk) is the dominant factor, then these inequalities are consistent with the fact that T21, T32, T01, and T34 are preferred over T23, T12, T43, and T10, respectively. Consequently, b21, b32, b01 and b34 are chosen as the boundary segments of an SR.

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Imai, S., Takanoya, R. Single-electron pumping in common-gate triple-dot devices with arbitrary asymmetric gate capacitance distributions. J Comput Electron (2020).

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  • Single-electron pumping
  • Common-gate triple-dot structure
  • Stability regions
  • Coulomb blockade