Measuring the Degeneracy of Discrete Energy Levels Using a $\mathrm{GaAs}/\mathrm{AlGaAs}$ Quantum Dot

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Measuring the Degeneracy of Discrete Energy Levels Using a $GaAs / AlGaAs$ Quantum Dot

A. Hofmann, V. F. Maisi, C. Gold, T. Krähenmann, C. Rössler, J. Basset, P. Märki, C. Reichl, W. Wegscheider, K. Ensslin, and T. Ihn

Phys. Rev. Lett. 117, 206803 – Published 11 November 2016

Abstract

We demonstrate an experimental method for measuring quantum state degeneracies in bound state energy spectra. The technique is based on the general principle of detailed balance and the ability to perform precise and efficient measurements of energy-dependent tunneling-in and -out rates from a reservoir. The method is realized using a $GaAs / AlGaAs$ quantum dot allowing for the detection of time-resolved single-electron tunneling with a precision enhanced by a feedback control. It is thoroughly tested by tuning orbital and spin degeneracies with electric and magnetic fields. The technique also lends itself to studying the connection between the ground-state degeneracy and the lifetime of the excited states.

Received 4 July 2016

DOI:https://doi.org/10.1103/PhysRevLett.117.206803

Physics Subject Headings (PhySH)

Energy levels

Quantum dots

Inelastic electron tunneling spectroscopy

Condensed Matter, Materials & Applied PhysicsStatistical Physics & Thermodynamics

Authors & Affiliations

A. Hofmann^*, V. F. Maisi, C. Gold, T. Krähenmann, C. Rössler, J. Basset, P. Märki, C. Reichl, W. Wegscheider, K. Ensslin, and T. Ihn

Solid State Physics Laboratory, ETH Zurich, CH-8093 Zurich, Switzerland

^*andrea.hofmann@phys.ethz.ch

Signatures of Hyperfine, Spin-Orbit, and Decoherence Effects in a Pauli Spin Blockade

T. Fujita, P. Stano, G. Allison, K. Morimoto, Y. Sato, M. Larsson, J.-H. Park, A. Ludwig, A. D. Wieck, A. Oiwa, and S. Tarucha

Phys. Rev. Lett. 117, 206802 (2016)

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Vol. 117, Iss. 20 — 11 November 2016

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Figure 1
Measurement setup. (a) The surface of the crystal with $Ti / Au$ top gates in light gray and the two-dimensional electron gas underneath the dark area. An electron (white circle) tunnels back and forth between the quantum dot and the reservoir (blue and green arrows). The dotted line indicates a closed barrier, and biased gates are plotted in a brighter color than grounded gates. (b) The current (red arrow) through the charge detector as a function of time measures the occupation of the quantum dot [ $(N - 1) \approx 1.46 nA$ , $N \approx 1.21 nA$ ]. The voltage, converted to energy, applied to the gate plunger gate (black) indicates the energies $E_{\pm}$ of the state $μ_{N}$ , as indicated in the energy diagram of the quantum dot-reservoir system in (c).
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Figure 2
Tunneling rates of the first eight resonances around the reservoir Fermi energy (zero energy reference). Green and blue denote the tunneling-in and tunneling-out rate, respectively. The solid lines are fits to Eq. (1), and the red lines are guides to the eye indicating the ratio $W_{0, in} ∶ W_{0, out}$ of the tunnel couplings to alternate between $2 ∶ 1$ and $1 ∶ 2$ .
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Figure 3
Tunneling rates at a finite magnetic field for the first (a) and second (b) resonance. The Zeeman energy is $Δ E_{Z} = g μ_{B} B = | E_{es} |$ with the $g$ factor $| g | = 0.44$ and the Bohr magneton $μ_{B}$ . The solid blue and green lines are fits to a modification of Eq. (1) with adding energetically offset tunneling rates. The red lines are guides to the eye and indicate the ratios $1 ∶ 1 ∶ 1$ for tunneling into the ground state, tunneling into the Zeeman-split excited state and tunneling out of the ground state. The ratios for the first and second electron are $W_{0, out} ∶ W_{0, in} ∶ W_{0, in}^{Z} = 76.7 ∶ 78.5 ∶ 75.4$ and $W_{0, out} ∶ W_{0, in} ∶ W_{0, in}^{Z} = 49.9 ∶ 54.7 ∶ 59.1$ , respectively. (c) Excited state spectroscopy of the second electron. The solid lines are fits to Eq. (1) with three (two) energetically offset tunneling-in (-out) rates. The red lines indicate the ratio $2 ∶ 1$ for tunneling into the triplet states $T_{-}$ and $T_{0}$ , $W_{0, in}^{T_{-}} ∶ W_{0, in}^{T_{0}} = 105 ∶ 54$ .
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Figure 4
Excited state spectroscopy ( $k T \approx 3.4 μ eV$ ), showing that the feedback technique allows us to infer degeneracy, excitation spectrum, and spin states from one quick measurement. The quantum dot states are driven symmetrically as shown in (a) to measure the tunneling rates $Γ_{in}$ and $Γ_{out}$ . The data are plotted as open squares (closed circles) in (b) for the one-electron (two-electron) state being driven around the Fermi energy. $N$ -electron excitations are indicated with red bars and show a short-lived excitation (no step) for the $N = 1$ state and long-lived excitation (step visible) for the $N = 2$ state. The spectroscopy around $μ_{N = 5}$ is shown in (c) for the spin-degenerate (top-left panel, $d = 2$ ) and the orbital degenerate (bottom-left panel, $d = 4$ ) ground state in open squares and closed circles, respectively. The right panel shows an enlargement around zero energy with the tunnel barrier being slightly more open.
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Physical Review Letters

Measuring the Degeneracy of Discrete Energy Levels Using a $GaAs / AlGaAs$ Quantum Dot

A. Hofmann, V. F. Maisi, C. Gold, T. Krähenmann, C. Rössler, J. Basset, P. Märki, C. Reichl, W. Wegscheider, K. Ensslin, and T. Ihn

Phys. Rev. Lett. 117, 206803 – Published 11 November 2016

Abstract

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Signatures of Hyperfine, Spin-Orbit, and Decoherence Effects in a Pauli Spin Blockade

T. Fujita, P. Stano, G. Allison, K. Morimoto, Y. Sato, M. Larsson, J.-H. Park, A. Ludwig, A. D. Wieck, A. Oiwa, and S. Tarucha

Phys. Rev. Lett. 117, 206802 (2016)

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Physical Review Letters

Measuring the Degeneracy of Discrete Energy Levels Using a GaAs/AlGaAs Quantum Dot

A. Hofmann, V. F. Maisi, C. Gold, T. Krähenmann, C. Rössler, J. Basset, P. Märki, C. Reichl, W. Wegscheider, K. Ensslin, and T. Ihn

Phys. Rev. Lett. 117, 206803 – Published 11 November 2016

Abstract

Physics Subject Headings (PhySH)

Authors & Affiliations

See Also

Signatures of Hyperfine, Spin-Orbit, and Decoherence Effects in a Pauli Spin Blockade

T. Fujita, P. Stano, G. Allison, K. Morimoto, Y. Sato, M. Larsson, J.-H. Park, A. Ludwig, A. D. Wieck, A. Oiwa, and S. Tarucha

Phys. Rev. Lett. 117, 206802 (2016)

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Measuring the Degeneracy of Discrete Energy Levels Using a $GaAs / AlGaAs$ Quantum Dot