^{1}, Tetsuo Kodera

^{1,2,3}, Tomohiro Kambara

^{1}, Ken Uchida

^{4}and Shunri Oda

^{1}

### Abstract

Single-electron transistors (SETs) are efficient charge sensors for reading out spin or chargequbits confined in quantum dots(QDs). To investigate their capacitive parameters, which are related to the signal-to-noise ratio (SNR) during qubit readout, twin silicon single QDs were fabricated using a lithographic process on a silicon-on-insulator substrate. Since the configuration and dimensions of the QDs could be determined by direct imaging, the theoretical capacitive parameters could be compared to the measured values. Good agreement was found between the calculated and measured values, which confirms the validity of the calculation method. The results indicated that decreasing the SET diameter reduces the capacitive coupling between qubits but increases the signal-to-noise ratio for both dc and radio frequency single-shot measurements. Since these results are independent of the device materials, they are useful for establishing guidelines for the design of SET charge sensors in lateral QD-SET structures based on a two-dimensional electron gas.

This work was financially supported by JSPS KAKENHI (22246040), JST-PRESTO, and the Project for Developing Innovation Systems of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. We thank R. Suzuki, K. Usami, and T. Hiramoto for experimental assistance.

I. INTRODUCTION

II. DEVICE STRUCTURE

III. RESULTS AND DISCUSSION

IV. CONCLUSIONS

### Key Topics

- Quantum dots
- 127.0
- Charge coupled devices
- 9.0
- Qubits
- 9.0
- Electron gas
- 7.0
- Electrostatics
- 5.0

##### H01L29/00

## Figures

(a) Scanning electron microscopy image of the device and with the measurement setup indicated. (b) Schematic cross section of the device structure along the line XY in (a). (c) and (d) Stability diagrams for QD1 and QD2, respectively. (c) *V _{SG} *

_{1}= 0 V and

*V*

_{SG}_{2}= −4 V, (d)

*V*

_{SG}_{1}= 0 V and

*V*

_{SG}_{2}= −4 V. From the observed Coulomb diamonds, a charging energy of

*E*

_{QD}_{1}≈ 14.0 meV for QD1 and

*E*

_{ QD }

_{2}≈ 6.2 meV for QD2 was obtained.

(a) Scanning electron microscopy image of the device and with the measurement setup indicated. (b) Schematic cross section of the device structure along the line XY in (a). (c) and (d) Stability diagrams for QD1 and QD2, respectively. (c) *V _{SG} *

_{1}= 0 V and

*V*

_{SG}_{2}= −4 V, (d)

*V*

_{SG}_{1}= 0 V and

*V*

_{SG}_{2}= −4 V. From the observed Coulomb diamonds, a charging energy of

*E*

_{QD}_{1}≈ 14.0 meV for QD1 and

*E*

_{ QD }

_{2}≈ 6.2 meV for QD2 was obtained.

Plots of (a) *I* _{ QD } _{1} and (b) *I* _{ QD } _{2} as functions of *V* _{ TG } and *V* _{ SG } _{1}. *V _{D} *

_{1}= 1 mV,

*V*

_{D}_{2}= 1 mV, and

*V*

_{SG}_{2}= −4 V. The two red dashed lines in (b) show the positions of two Coulomb peaks in (a). Inset: Higher magnification view of a shift of a Coulomb peak. The solid white lines are visual guides.

Plots of (a) *I* _{ QD } _{1} and (b) *I* _{ QD } _{2} as functions of *V* _{ TG } and *V* _{ SG } _{1}. *V _{D} *

_{1}= 1 mV,

*V*

_{D}_{2}= 1 mV, and

*V*

_{SG}_{2}= −4 V. The two red dashed lines in (b) show the positions of two Coulomb peaks in (a). Inset: Higher magnification view of a shift of a Coulomb peak. The solid white lines are visual guides.

(a) Schematic diagram of QD1 and QD2 and various electrical parameters. (b) Schematic diagram of twin QDs for numerical calculations. (c) Calculation results for *k* _{ QD } _{1} and *k* _{ QD } _{2} as a function of *L* _{ QD } _{2} (in units of *L* _{ QD } _{1}) using the three-dimensional Poisson equation. *L* _{ QD } _{1}and the distance *d* between the edges of QD1 and QD2 are fixed. (d) Calculated E_{m}/E_{m0} as a function of *L _{QD} *

_{2}(in units of

*L*

_{ QD }

_{1}) by the same method as for panel (c).

*E*

_{m}_{0}is the value of

*E*when

_{m}*L*

_{QD}_{2}=

*L*

_{QD}_{1}and

*d*=

*L*

_{QD}_{1}. In panels (c) and (d), the calculation results are shown for

*d*=

*L*

_{QD}_{1}and

*d*= 0.5

*L*

_{QD}_{1}.

(a) Schematic diagram of QD1 and QD2 and various electrical parameters. (b) Schematic diagram of twin QDs for numerical calculations. (c) Calculation results for *k* _{ QD } _{1} and *k* _{ QD } _{2} as a function of *L* _{ QD } _{2} (in units of *L* _{ QD } _{1}) using the three-dimensional Poisson equation. *L* _{ QD } _{1}and the distance *d* between the edges of QD1 and QD2 are fixed. (d) Calculated E_{m}/E_{m0} as a function of *L _{QD} *

_{2}(in units of

*L*

_{ QD }

_{1}) by the same method as for panel (c).

*E*

_{m}_{0}is the value of

*E*when

_{m}*L*

_{QD}_{2}=

*L*

_{QD}_{1}and

*d*=

*L*

_{QD}_{1}. In panels (c) and (d), the calculation results are shown for

*d*=

*L*

_{QD}_{1}and

*d*= 0.5

*L*

_{QD}_{1}.

(a) Plots of *I* _{ QD } _{2} as a function of *V* _{ TG } and *V* _{ SG } _{1} for another device. *V _{D} *

_{1}= 1 mV,

*V*

_{D}_{2}= 5 mV, and

*V*

_{SG}_{2}= −5.8 V. (b) Schematic diagram of RF measurement setup. (c) Numerical calculation results for the SNR as a function of

*L*

_{QD}_{2}(in units of

*L*

_{ QD }

_{1}) for optimized gate bias conditions. Using y-axis value

*Y*= 10

^{SNR/20}

*e*(

*R*

*Δf*

*/*

*E*

_{SET}_{0})

^{1/2}in (c), the SNR can be expressed as SNR = 20Log

_{10}[

*Y*/{

*e*(

*R*

*Δf*

*/*

*E*

_{SET}_{0})

^{1/2}}]. Here,

*R*=

_{L}*R*=

_{R}*R*,

*k*

_{B}

*T*= 0.01

*E*

_{SET}_{0}(

*E*

_{SET}_{0}is the charging energy of QD2 when

*L*

_{QD}_{2}=

*L*

_{QD}_{1}and

*d*=

*L*

_{QD}_{1}),

*A*= 0.2

*E*

_{SET}_{0}/

*e*,

*V*

_{0}= 0, and

*L*

_{ QD }

_{1}is fixed. Calculations were performed for

*C*/

_{L}*C*

_{QD}_{2}= 0.1, 0.2, and 0.3. Calculation results for

*d*=

*L*

_{QD}_{1}and

*d*= 0.5

*L*

_{QD}_{1}are shown.

(a) Plots of *I* _{ QD } _{2} as a function of *V* _{ TG } and *V* _{ SG } _{1} for another device. *V _{D} *

_{1}= 1 mV,

*V*

_{D}_{2}= 5 mV, and

*V*

_{SG}_{2}= −5.8 V. (b) Schematic diagram of RF measurement setup. (c) Numerical calculation results for the SNR as a function of

*L*

_{QD}_{2}(in units of

*L*

_{ QD }

_{1}) for optimized gate bias conditions. Using y-axis value

*Y*= 10

^{SNR/20}

*e*(

*R*

*Δf*

*/*

*E*

_{SET}_{0})

^{1/2}in (c), the SNR can be expressed as SNR = 20Log

_{10}[

*Y*/{

*e*(

*R*

*Δf*

*/*

*E*

_{SET}_{0})

^{1/2}}]. Here,

*R*=

_{L}*R*=

_{R}*R*,

*k*

_{B}

*T*= 0.01

*E*

_{SET}_{0}(

*E*

_{SET}_{0}is the charging energy of QD2 when

*L*

_{QD}_{2}=

*L*

_{QD}_{1}and

*d*=

*L*

_{QD}_{1}),

*A*= 0.2

*E*

_{SET}_{0}/

*e*,

*V*

_{0}= 0, and

*L*

_{ QD }

_{1}is fixed. Calculations were performed for

*C*/

_{L}*C*

_{QD}_{2}= 0.1, 0.2, and 0.3. Calculation results for

*d*=

*L*

_{QD}_{1}and

*d*= 0.5

*L*

_{QD}_{1}are shown.

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