Phonon-mediated spin dynamics in a two-electron double quantum dot under a phonon temperature gradient

Kazuyuki Kuroyama, Sadashige Matsuo, Seigo Tarucha, and Yasuhiro Tokura

Phys. Rev. B 108, 115308 – Published 20 September 2023

Abstract

We have theoretically studied phonon-mediated spin-flip processes of electrons in a GaAs double quantum dot (DQD) holding two spins, under a phonon temperature gradient over the DQD. Transition rates of interdot phonon-assisted tunnel processes and intradot spin-flip processes involving spin triplet states are formalized by the electron-phonon interaction accompanied with the spin-orbit interaction. The calculations of the spin-flip rates and the occupation probabilities of the spin-states in the two-electron DQD with respect to the phonon temperature difference between the dots are quantitatively consistent with our previous experiment. This theoretical study on the temperature gradient effect onto spins in coupled QDs would be essential for understanding spin-related thermodynamic physics.

Received 9 March 2023
Revised 14 August 2023
Accepted 16 August 2023

DOI:https://doi.org/10.1103/PhysRevB.108.115308

Physics Subject Headings (PhySH)

Electron-phonon coupling Spin blockade Spin-orbit coupling

Double quantum dots

Stochastic processes & statistics

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Kazuyuki Kuroyama ^1,2,*, Sadashige Matsuo ², Seigo Tarucha ^2,3,†, and Yasuhiro Tokura ^4,‡

¹Institute of Industrial Science, The University of Tokyo, 7-3-1 Komaba, Meguro-ku, Tokyo, 153-8505, Japan
²Center for Emergent Materials Science (CEMS), RIKEN, 2-1 Hirosawa, Wako-shi, Saitama, 351-0198, Japan
³RIKEN Center for Quantum Computing (RQC), RIKEN, 2-1 Hirosawa, Wako-shi, Saitama, 351-0198, Japan
⁴Pure and Applied Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, 305-8571, Japan

^*kuroyama@iis.u-tokyo.ac.jp
^†Corresponding author: tarucha@riken.jp
^‡tokura.yasuhiro.ft@u.tsukuba.ac.jp

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Issue

Vol. 108, Iss. 11 — 15 September 2023

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Images

Figure 1
Geometrical configuration of the DQD and the crystallographic orientations. The DQD array is aligned along the [01-1] crystallographic axis, which is denoted by $x$ , and each dot is located at $x = \pm d$ . The $B$ field is applied in the [0-1-1] direction. The $x^{'} y^{'}$ plane, that is spanned by [010] and [001] crystallographic axis, is used to express the spin-orbit interaction.
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Figure 2
(a) Energy diagram which explains the phonon-induced spin-flip tunnel processes in the two-electron DQD. A phonon excitation from $| T_{+} (1, 1) 〉$ to the excited state of $| T_{+} (0, 2) 〉$ (Process I), which is followed by the intradot spin-flip relaxation processes from $| T_{+} (0, 2) 〉$ to $| S (0, 2) 〉$ (Process II), is illustrated. (b) Transition diagram of the two-electron spin states with phonon excitations.
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Figure 3
[(a)–(c)] Color-coded maps of $| Λ_{l, k, 36} |^{2}$ calculated for the different $k$ -vector orientations, $θ$ and $ϕ$ , at $Δ = 0.6$ , 0.8, and 1.0 meV, respectively. Phonons propagating in the angle marked in a bright color contribute to the electron-phonon interaction. Note that since the electron-phonon interaction is symmetric with respect to $z = 0$ , the color-coded maps are plotted only for $0 \leq θ \leq 0.5 π$ . [(d)–(f)] Color-coded maps of $| Λ_{μ, k, 56} |^{2}$ calculated for the different $k$ -vector orientations, at $Δ = 0.6$ , 0.8, and 1.0 meV, respectively.
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Figure 4
(a) Color-coded map of the spin-conserving tunnel rate $Γ_{15}$ calculated for $Δ$ and $d$ . (b) Color-coded map of the spin-flip tunnel rate $Γ_{35}$ calculated for $Δ$ and $d$ . (c) $γ_{0}$ plotted with respect to the singlet-triplet energy separation $Δ$ . (d) Color-coded map of $γ_{0}^{'}$ plotted in the plane of $Δ$ and $d$ .
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Figure 5
(a) Spin-flip tunnel rate from the left to right QD, $Γ_{R \leftarrow L}$ , plotted as a function of $T_{L}$ . (b) Color-coded map of the spin-flip tunnel rate from the right to left QD, $Γ_{L \leftarrow R}$ , plotted as functions of $T_{L}$ and $T_{R}$ . (c) Color-coded map of the spin-flip rate ratio, $Γ_{L \leftarrow R} / Γ_{R \leftarrow L}$ , plotted with respect to $T_{L}$ and $T_{R}$ . A dashed line indicates $T_{L}$ and $T_{R}$ satisfying $P_{02} \sim 0.2$ [see Fig. 6]. (d) $Γ_{L \leftarrow R} / Γ_{R \leftarrow L}$ ratio for various temperature gradients, i.e., $T_{L} = r T_{R}$ , plotted as a function of $T_{R}$ . The colors correspond to the values of $r$ as shown in the annotation in the figure. (e) $Γ_{L \leftarrow R} / Γ_{R \leftarrow L}$ ratio extracted from 5 at $P_{02} \sim 0.2$ [see a dashed line in (c) and Fig. 6] is plotted by a black line. The same ratio at the equilibrium condition, i.e., $T_{L} = T_{R}$ , is plotted by a green line, as a reference.
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Figure 6
[(a)–(c)] Color-coded maps of $P_{02}, P_{AP}$ , and $P_{P}$ , respectively, calculated for $T_{L}$ and $T_{R}$ . (d) $P_{02}$ (red), $P_{AP}$ (green), and $P_{P}$ (blue) plotted as a function of $T_{R}$ under the equilibrium condition, i.e., $T_{L} = T_{R}$ . $P_{AP}$ , and $P_{P}$ take the same value. (e) $P_{AP}$ and $P_{P}$ extracted from (b) and (c), respectively, at $P_{02} \sim 0.2$ [see a dashed line in [(a)–(c)]].
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