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Real-time millikelvin thermometry in a semiconductor-qubit architecture

V. Champain, V. Schmitt, B. Bertrand, H. Niebojewski, R. Maurand, X. Jehl, C.B. Winkelmann, S. De Franceschi, and B. Brun

Phys. Rev. Applied 21, 064039 – Published 17 June 2024

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Abstract

We report local time-resolved thermometry in a silicon-nanowire quantum dot device designed to host a linear array of spin qubits. Using two alternative measurement schemes based on rf reflectometry, we are able to probe either local electron or bosonic bath temperatures with microsecond time scale resolution and a noise-equivalent temperature of $3 mK / \sqrt{Hz}$ . Following the application of short microwave pulses, causing local periodic heating, time-dependent thermometry can track the dynamics of thermal excitation and relaxation, revealing clearly different characteristic time scales. This work opens important prospects to investigate the out-of-equilibrium thermal properties of semiconductor quantum electronic devices operating at very low temperature. In particular, it may provide a powerful handle to understand heating effects recently observed in semiconductor spin-qubit systems.

Received 25 August 2023
Revised 16 February 2024
Accepted 17 May 2024

DOI:https://doi.org/10.1103/PhysRevApplied.21.064039

Physics Subject Headings (PhySH)

Temperature

Doped semiconductors Double quantum dots Quantum dots

Thermometry

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

V. Champain ^1,*, V. Schmitt ¹, B. Bertrand ², H. Niebojewski², R. Maurand ¹, X. Jehl ¹, C.B. Winkelmann ¹, S. De Franceschi ¹, and B. Brun ^1,†

¹Commissariat à l’énergie atomique et aux énergies alternatives (CEA), Grenoble INP, Interdisciplinary Research Institute of Grenoble (IRIG)–Quantum Photonics, Electronics, and Engineering Laboratory (Pħeliqs), Université Grenoble Alpes, Grenoble, France
²CEA Laboratoire d’électronique des technologies de l’information (Leti), Université Grenoble Alpes, Minatec Campus, Grenoble, France

^*Corresponding author: [email protected]
^†Corresponding author: [email protected]

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Vol. 21, Iss. 6 — June 2024

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Figure 1
The device and charge configuration for two thermometry techniques. (a) A false-color tunneling electron micrograph of a representative MOS device with a silicon-nanowire channel (blue), highly doped conductive leads (yellow), and eight parallel gates (red). Note that the gates can be considered metallic; however, the contacts on the leads and gates are made of tungsten vias, which are superconducting at base temperature. Three different samples have been measured in this work (for details, see the Supplemental Material [17]). (b) A schematic of an eight-gate device, illustrating the two thermometry methods. The source and drain (yellow) are highly $p$ doped to form metallic leads acting as hole reservoirs. Quantum dots (QDs) can be defined along the undoped silicon channel (blue) by means of gates G1–G8. In the first type of thermometer (setting I), G1 defines a single QD tunnel coupled to the source reservoir as illustrated by the hole-energy diagram just below. The hole reservoir is modeled by a Fermi distribution with electronic temperature $T_{e}$ , while the dot is modeled by a single level the line width of which is determined by the tunnel rate $Γ$ . The detuning, $ε$ , is defined here as the difference between the QD electrochemical potential and the Fermi level in the reservoir. In the second thermometer (setting II), a double QD isolated from the reservoirs is defined under G6 and G7. Both QDs are modeled by single levels with a tunnel coupling $t$ and detuning $ε$ . The corresponding energy diagram is shown just below, with $| L ⟩$ ( $| R ⟩$ ) representing the state localized in the left (right) QD at large detuning, and $| + ⟩$ ( $| - ⟩$ ) the hybridized bonding (antibonding) state at zero detuning. The coupling to the leads is completely suppressed by means of the two side gates G5 and G8.
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Figure 2
An experimental demonstration of the two thermometry techniques. (a) Setting I: the phase of the reflectometry signal from the tank circuit as a function of the detuning, for a dot-lead transition at different mixing-chamber (MC) temperatures. The peak amplitude is decreasing with temperature while the peak width increases. (b) Setting II: the phase of the reflectometry signal for an interdot transition as a function of the MC temperature. The peak amplitude is decreasing, while the width remains unchanged. (c) The $T_{MC}$ dependence of the electronic temperature, $T_{e}$ , in the source reservoir as obtained from fitting to Eq. (1). The inset shows a close-up of the low-temperature saturation. (d) The $T_{MC}$ dependence of $T_{B}$ , as obtained from fitting Eq. (2)).
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Figure 3
The thermometer calibration and optimal operation regime for setting II. (a) The peak amplitude, $φ_{0}$ , as a function of $T_{MC}$ in the case of setting II. The experimental data (open circles) are fitted to Eq. (3) (solid red line), giving a tunnel rate $t / h = 1.72 \pm 0.04 GHz$ . The dashed gray line is a linear fit to the data in the $55 -- 100$ range of highest sensitivity ( $| \partial φ_{0} / \partial T | = 1.28 mrad / mK$ ). (b) The calculated noise-equivalent temperature, NET, versus ( $T_{MC}$ , $t$ ). Iso-NET curves are drawn in blue and yellow colors. The dashed white line highlights the $t$ dependence of the NET minimum (i.e., the $T_{MC}$ at which $\partial NET / \partial T_{MC} = 0$ ).
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Figure 4
Real-time thermometry with setting II. Microwave bursts are periodically applied to gate G5, adjacent to the double-dot thermometer. The microwave of frequency 15 GHz and power $- 40$ dBm is modulated by a square signal of frequency 20 kHz, resulting in 50- $μ s$ -long bursts, with 50- $μ s$ waiting time between each burst. The phase signal is acquired with a synchronized lock-in at a sampling rate of $1.7 MHz$ , with a $400$ -ns integration time. It is averaged over $2 \times 10^{6}$ cycles and is plotted as a function of time (open circles). (a) Heating: at each cycle, the microwave burst is turned on at $Time = 0$ and the phase $φ_{0}$ decreases from $φ_{c} \sim 220$ to $φ_{on} \sim 200 mrad$ . (b) Cooling: the microwave excitation is switched off at $Time = 0$ and the phase increases back to $φ_{on}$ with a two-stage relaxation (for details, see the text).
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Physical Review Applied

Real-time millikelvin thermometry in a semiconductor-qubit architecture

V. Champain, V. Schmitt, B. Bertrand, H. Niebojewski, R. Maurand, X. Jehl, C.B. Winkelmann, S. De Franceschi, and B. Brun

Phys. Rev. Applied 21, 064039 – Published 17 June 2024

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