Andreev bound states in a few-electron quantum dot coupled to superconductors

Yulin Ma, Tianqi Cai, Xiyue Han, Yaowen Hu, Hongyi Zhang, Haiyan Wang, Luyan Sun, Yipu Song, and Luming Duan

Phys. Rev. B 99, 035413 – Published 8 January 2019

Abstract

A direct measurement of the density of Andreev bound states (ABSs) is experimentally investigated in a superconductor–quantum dot–superconductor hybrid nanowire system. A hard proximity-induced superconducting gap is observed, arising from superconducting correlations, in the transport spectrum of the hybrid system. Conductance peaks observed inside the superconducting gap reveal that the subgap states participate in the transport of the hybrid junction. We explore the evolution of low-energy Andreev bound states in a few-electron quantum dot (QD) coupled to superconductors, by probing the magnetic field dependence of exquisite detailed conductance spectra with a small bias voltage applied on the superconducting lead. In the presence of low magnetic fields, the resonance current is enhanced and broadened, attributed to the transport through Andreev bound states in QD, as the energy of ABSs reaches the threshold set by the applied bias voltage with the increase of the magnetic field. The simulated transport spectrum matches the experimentally observed evolution patterns of conductance, further implying the superconducting correlation nature of the observed electron transport. At high magnetic fields, the conductance maxes as the Fermi level reaches the degeneration point of Landau levels, leading to conductance peaks shown in the alternating narrow and wide patterns of Coulomb blockade oscillations.

Received 19 September 2018
Revised 15 November 2018

DOI:https://doi.org/10.1103/PhysRevB.99.035413

Physics Subject Headings (PhySH)

Andreev reflection Hybrid quantum systems Proximity effect

Quantum dots Superconducting devices

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Yulin Ma¹, Tianqi Cai¹, Xiyue Han¹, Yaowen Hu², Hongyi Zhang¹, Haiyan Wang¹, Luyan Sun¹, Yipu Song^1,*, and Luming Duan^1,†

¹Center for Quantum Information, IIIS, Tsinghua University, Beijing 100084, China
²Department of Physics, Tsinghua University, Beijing 100084, China

^*ypsong@mail.tsinghua.edu.cn
^†lmduan@tsinghua.edu.cn

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Issue

Vol. 99, Iss. 3 — 15 January 2019

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Images

Figure 1
(a) Schematic cross section of the S-QD-S hybrid nanowire device. (b) SEM image of a measured hybrid device. The scale bar is 500 nm. (c) Schematics of a S-QD-S device with tunnel couplings to the superconducting probe ( $Γ_{l}$ ) and reservoir ( $Γ_{r}$ ), respectively. $Δ_{0}$ is the superconducting energy gap, $μ_{s}$ is the chemical potential of the superconducting lead, $ɛ_{0}$ is the orbital energy of the dot, $E_{c}$ is the charging energy of the dot, $\pm ɛ$ represents the energy levels of Andreev bound states in the subgap, $V_{sd}$ is the bias voltage applied on the superconducting lead. In tunnel spectroscopy measurements, an electric current measured at $| V_{sd} | < Δ_{0} / e$ is carried by Andreev bound states, transferring quasiparticles from the superconducting probe, to a Cooper pair in the superconducting reservoir. The alignment of $μ_{s}$ to an energy level of ABSs yields a conductance peak.
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Figure 2
(a) Typical conductance curves versus back gate for a device measured at 10 mK with a source-drain voltage $V_{sd} = 1.0 mV$ , in the presence of a perpendicular magnetic field 0 and 5 T for the black and red line, respectively. The curve in black is shifted up 0.8 $y$ -axis unit for clarity. (b) Differential conductance as a function of source-drain voltage $V_{sd}$ with different back gate voltages $V_{g}$ at a zero magnetic field. Eight conductance curves from bottom to top are measured at $V_{g} = - 3.009$ , −3.008, −3.007, −3.006, −3.005, −3.004, −2.986, and −2.985 V, respectively. For clarity, the conductance curves in dark yellow, magenta, cyan, blue, green, red, and black are shifted up 5, 10, 15, 20, 25, 30, and 35 $y$ -axis units, respectively. Conductance peaks inside the superconducting gap, marked by red arrows, reveal that the subgap states participate in the transport of the S-QD-S hybrid junction. (c) Differential conductance as a function of source-drain voltage $V_{sd}$ with different back gate voltages $V_{g}$ at $B = 3.078 T$ . Six conductance curves from bottom to top are measured at $V_{g} = - 2.9994$ , −3.0046, −3.011, −3.0144, −3.0238, and −3.0324 V, respectively. For clarity, the curves in red, green, blue, dark yellow, and magenta are shifted up 20, 40, 70, 95, and 115 $y$ -axis units, respectively.
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Figure 3
Differential conductance of the measured device as a function of the back gate voltage $V_{g}$ and bias voltage $V_{sd}$ with a perpendicular external magnetic field $B = 0 T$ (a) and $B = 3.078 T$ (b), respectively. Coulomb diamond patterns in (b) are significantly modulated at a zero magnetic field. Resonance patterns of Andreev bound states, marked by yellow arrows, emerge inside the superconducting gap arising from superconducting correlations. (c) The spectrum of transport current as a function of the back gate voltage and magnetic field with $V_{sd} = 1.0 mV$ . The resonance current peak is enhanced and broadened at a particular magnetic field. The overall trend of the transition value of magnetic field increases as sweeping the back gate voltage to more negative. (d) A simulation of conductance resonant patterns corresponding to the odd occupation valley at $V_{g} = - 2.982 V$ in (c). The simulated conductance peak is broadened as the magnetic field goes up to 0.38 T. The dimensionless gate voltage $x = 1 + \frac{2 ɛ_{0}}{E_{c}}$ . $x = 0$ marks the center of the odd valley. The parameters used for the calculation are taken as $E_{c} = 3.1 meV$ , $Δ_{0} = 1.3 meV$ , $V_{sd} = 1.0 mV$ , $ϕ = π / 3$ , $\frac{Γ_{α}}{E c} = 0.135$ , $g_{ce} = 0$ , $g_{d} = 10$ . (e) Energy levels of Andreev bound states (the energy difference between $| D 〉$ and $| S 〉$ ) at different magnetic fields $B = 0, 0.4$ , 0.8, 1.2, and 1.6 T for the black, red, green, blue, and magenta curve, respectively. The parameters used for simulation are $E_{c} = 3.1 meV$ , $Δ_{0} = 1.3 meV$ , $V_{sd} = 1.0 mV$ , $ϕ = π / 3$ , $\frac{Γ_{α}}{E c} = 0.135$ , $g_{ce} = 0$ , $g_{d} = 10$ . The black dashed line at 1.0 meV marks the bias window in the measurement.
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Figure 4
(a) Energy spectroscopy of the S-QD-S hybrid system in a high magnetic field up to 5 T in a range of back gate voltage $- 3.2 V < V_{g} < - 2.8 V$ with $V_{sd} = 1.0 mV$ . (b) The simulated magnetic field dependence of the energy spectrum of Landau levels for $n$ from one to ten with $l = 1 \sim 10$ . A parabolic confinement of 4.85 meV is used for calculation.
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