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

Abstract

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:a − d/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 (n_s, n_c, n_s) at low gate voltage and (n_s, n_c + 1, n_s) or (n_s + 1, n_c − 1, n_s + 1) at high gate voltage, where n_s and n_c 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 a–d 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.

Appendices

Appendix 1

The derivations of the CB conditions as linear inequalities of V_sd, V_g, 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.

$$C_{{{\text{T}}10}} (V_{1} - V_{ 0} ) = Q_{1} + r_{{{\kern 1pt} 21{\text{R}}}} \left( {Q_{2} + r_{{32{\text{R}}}} Q_{3} } \right) - C_{{{\text{T}}10}} V_{0} ,$$

(3)

$$C_{{{\text{T}}21}} (V_{2} - V_{ 1} ) = - r_{{{\kern 1pt} {\text{R}}21}} Q_{1} + r_{{{\kern 1pt} {\text{L}}12}} \left( {Q_{2} + r_{{{\kern 1pt} 32{\text{R}}}} Q_{3} } \right),$$

(4)

$$C_{{{\text{T3}}2}} (V_{3} - V_{ 2} ) = - r_{{{\kern 1pt} {\text{R3}}2}} \left( {r_{{{\kern 1pt} 21{\text{L}}}} Q_{1} + Q_{2} } \right) + r_{{{\kern 1pt} {\text{L}}23}} Q_{3} ,$$

(5)

$$C_{\text{T43}} (V_{ 4} - V_{ 3} ) = C_{\text{T43}} V_{ 4} - r_{{{\kern 1pt} 32{\text{L}}}} \left( {r_{{{\kern 1pt} 21{\text{L}}}} Q_{1} + Q_{2} } \right) - Q_{3} .$$

(6)

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

$$Q_{{{\kern 1pt} 1}} = - n_{1} e + C_{10} V_{ 0} + C_{\text{g1}} V_{\text{g}} ,$$

(7)

$$Q_{{{\kern 1pt} 2}} = - n_{2} e + C_{\text{g2}} V_{\text{g}} ,$$

(8)

$$Q_{{{\kern 1pt} 3}} = - n_{3} e + C_{43} V_{ 4} + C_{\text{g3}} V_{\text{g}} .$$

(9)

The source and drain voltages are defined as follows.

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

(10)

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

(11)

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

The total capacitances are as follows.

$$C_{{{\text{T}}10}} = \frac{{C_{{{\text{Lg}}1}} C_{21} + C_{{{\text{Lg}}1}} C_{{{\text{Rg}}2}} + C_{21} C_{{{\text{Rg}}2}} }}{{C_{21} + C_{{{\text{Rg}}2}} }},$$

(12)

$$C_{\text{T21}} = \frac{{C_{21} C_{{{\text{Lg}}1}} + C_{21} C_{{{\text{Rg}}2}} + C_{{{\text{Lg}}1}} C_{{{\text{Rg}}2}} }}{{C_{{{\text{Lg}}1}} + C_{\text{Rg2}} }},$$

(13)

$$C_{\text{T32}} = \frac{{C_{32} C_{\text{Lg2}} + C_{32} C_{\text{Rg3}} + C_{\text{Lg2}} C_{\text{Rg3}} }}{{C_{{{\text{Lg}}2}} + C_{\text{Rg3}} }},$$

(14)

$$C_{\text{T43}} = \frac{{C_{\text{Rg3}} C_{32} + C_{\text{Rg3}} C_{\text{Lg2}} + C_{32} C_{\text{Lg2}} }}{{C_{32} + C_{\text{Lg2}} }}.$$

(15)

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 C_Lgi and C_Rgi in the coefficients and the series capacitances C_Li and C_Ri 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 C_j.

Appendix 2

Tunneling event T_kk′ can occur when C_Tk′k (V_k′ − V_k) is greater than e/2, where C_Tk′k is equal to C_Tkk′ if k = k′ + 1. Here, (V_k′ − V_k) is explained to be the dominant factor that determines which of T_kk′ 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, C_g1 > C_g3, then V₁ is always greater than V₃ for V_sd = 0, V_g > 0, n₁ = n₃ ≥ 0 and n₂ ≥ 0. The reason is as follows. If V_g is positive and no excess electrons exist at the three islands, then V₁ is increased by V_g 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 V_g 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 V_g 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 (V_k′ − V_k) 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.