How Valley-Orbit States in Silicon Quantum Dots Probe Quantum Well Interfaces

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How Valley-Orbit States in Silicon Quantum Dots Probe Quantum Well Interfaces

J. P. Dodson, H. Ekmel Ercan, J. Corrigan, Merritt P. Losert, Nathan Holman, Thomas McJunkin, L. F. Edge, Mark Friesen, S. N. Coppersmith, and M. A. Eriksson

Phys. Rev. Lett. 128, 146802 – Published 6 April 2022

Abstract

The energies of valley-orbit states in silicon quantum dots are determined by an as yet poorly understood interplay between interface roughness, orbital confinement, and electron interactions. Here, we report measurements of one- and two-electron valley-orbit state energies as the dot potential is modified by changing gate voltages, and we calculate these same energies using full configuration interaction calculations. The results enable an understanding of the interplay between the physical contributions and enable a new probe of the quantum well interface.

Received 22 April 2021
Revised 24 December 2021
Accepted 24 February 2022

DOI:https://doi.org/10.1103/PhysRevLett.128.146802

Physics Subject Headings (PhySH)

Mesoscopics

0-dimensional systems Heterostructures Quantum dots

Spectroscopy

Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

Authors & Affiliations

J. P. Dodson ¹, H. Ekmel Ercan ¹, J. Corrigan¹, Merritt P. Losert¹, Nathan Holman¹, Thomas McJunkin ¹, L. F. Edge², Mark Friesen ¹, S. N. Coppersmith^1,3, and M. A. Eriksson¹

¹Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA
²HRL Laboratories, LLC, 3011 Malibu Canyon Road, Malibu, California 90265, USA
³University of New South Wales, Sydney, New South Wales 2052, Australia

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Vol. 128, Iss. 14 — 8 April 2022

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Images

Figure 1
Device layout and experimental setup. (a) A scanning electron microscope (SEM) image of a device lithographically identical to the one measured shows the gate electrode layout in the active region. A comsol multiphysics Thomas-Fermi simulation of the electron density is overlaid on the SEM image. The sensor dot under gate $M$ measures the average charge occupation $⟨ n ⟩$ of the $P 2$ quantum dot via lock-in amplifier detection at frequency $f_{lock-in}$ . (b) The comsol simulation shows the charge density of the $P 2$ dot, where the tunnel barrier to the left (right) reservoir is opaque (transparent). (c) Valley-orbit state splittings are measured using pulsed-gate spectroscopy. A 50% duty cycle square pulse with amplitude $V_{pulse}$ at frequency $f_{pulse}$ is applied to gate $P 2$ , rapidly pulsing the chemical potential of the $P 2$ dot between $E_{Lg}$ and $E_{Ug}$ . The change in chemical potential induces detectable shifts in the tunnel rate into and out of the $P 2$ quantum dot, allowing for measurement of excited state energies.
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Figure 2
Experimental methods. (a),(b) Measurement of orbital splitting using pulsed-gate spectroscopy. The pulse amplitude is increased until the lowest-lying orbital state is visible, shown in (a). A line cut (dashed line) is taken when the pulse amplitude exceeds the orbital splitting, allowing detection of the ground (Lg) and excited orbital ( $L_{orb}$ ) states, shown as the peaks in (b). The orbital splitting is given by $α Δ V_{orb}$ . (c),(d) Using the same method, the valley ( $E_{val}$ ) and singlet-triplet splittings ( $E_{ST}$ ) are measured. The red data points clearly show two distinct peaks in (d). The data are fit using Eq. (1), shown as the black line, giving $E_{val} = α Δ V_{val} = 53.2 μ eV$ . (e)–(g) Experimental magnetospectroscopy data (e) are reproduced in (f) by treating the dot-reservoir system as a grand canonical ensemble. The experimental peak locations in voltage space are extracted from (e) and plotted in (g) as red circles, which are then fit to Eq. (2), yielding $E_{ST} = 33.4 μ eV$ .
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Figure 3
Quantum dot orbital shape and position. The top left inset shows a device schematic where the $P 2$ quantum dot is located within the filled blue region and the $S 3$ gate is shown as the filled gray region. The minimum orbital splitting is plotted as a function of $V_{S 3}$ . At low $V_{S 3}$ , the dot has strong $y$ confinement and weak $x$ confinement, leading to a drop in the minimum orbital splitting. As $V_{S 3}$ is increased, the dot becomes isotropic, increasing the minimum orbital splitting. At high $V_{S 3}$ , the dot has weak $y$ confinement and strong $x$ confinement, reducing the minimum orbital splitting again. Thomas-Fermi electrostatic simulations are used to calculate the quantum dot shape and position within the blue region shown in the device schematic. Four points are simulated, shown as the shaded purple circles in the experimental data. The change in shape of the electron density is found to be in agreement with the experimental data. Additionally, the position of the quantum dot for each simulation is illustrated as the small solid circle in the inset, showing that the quantum dot slides down and to the left 23.8 nm.
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Figure 4
(a) The measured valley splitting ( $E_{val}$ ) and singlet-triplet splitting ( $E_{ST}$ ) are plotted as a function of $V_{S 3}$ as filled blue squares and red circles, respectively, where the measurement uncertainty is indicated by the width of the shaded regions. FCI simulations for $E_{val}$ and $E_{ST}$ , shown as open blue squares and red circles, respectively, quantitatively reproduce the experimental data once the effects of disorder in the form of atomic steps at the quantum well interface are included. (b) The nonmonotonic behavior observed in both $E_{ST}$ and $E_{val}$ as a function $V_{S 3}$ is well explained by the quantum dot position changing with respect to distinct atomic steps at the quantum well interface. The wave function and position of the $P 2$ dot singlet state is plotted with respect to atomic steps (dashed lines) at the interface, shown as the colored gradients and points. A best fit to the data produces a smooth change in position of the dot across steps spaced approximately 35 nm apart. The change in position from the Thomas-Fermi simulation, plotted as the purple line and open circles, agrees well with the FCI simulations. The inset plots the ratio $E_{ST} / E_{val}$ as a function of the orbital splitting ( $E_{orb}$ ), where the colored points map the magnitude of the orbital splitting to the position of the electron wave function with respect to atomic steps in (b). As seen in (b), the suppression of the ratio $E_{ST} / E_{val}$ below unity is a consequence of strong electron-electron interactions dominated by the magnitude of the orbital splitting.
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