Detecting weak coupling in mesoscopic systems with a nonequilibrium Fano resonance

S. Xiao, Y. Yoon, Y.-H. Lee, J. P. Bird, Y. Ochiai, N. Aoki, J. L. Reno, and J. Fransson

Phys. Rev. B 93, 165435 – Published 25 April 2016

Abstract

A critical aspect of quantum mechanics is the nonlocal nature of the wave function, a characteristic that may yield unexpected coupling of nominally isolated systems. The capacity to detect this coupling can be vital in many situations, especially those in which its strength is weak. In this work, we address this problem in the context of mesoscopic physics, by implementing an electron-wave realization of a Fano interferometer using pairs of coupled quantum point contacts (QPCs). Within this scheme, the discrete level required for a Fano resonance is provided by pinching off one of the QPCs, thereby inducing the formation of a quasibound state at the center of its self-consistent potential barrier. Using this system, we demonstrate a form of nonequilibrium Fano resonance (NEFR), in which nonlinear electrical biasing of the interferometer gives rise to pronounced distortions of its Fano resonance. Our experimental results are captured well by a quantitative theoretical model, which considers a system in which a standard two-path Fano interferometer is coupled to an additional, intruder, continuum. According to this theory, the observed distortions in the Fano resonance arise only in the presence of coupling to the intruder, indicating that the NEFR provides a sensitive means to infer the presence of weak coupling between mesoscopic systems.

Received 12 November 2015
Revised 1 March 2016

DOI:https://doi.org/10.1103/PhysRevB.93.165435

Physics Subject Headings (PhySH)

Nonlocality

Two-dimensional electron gas

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

Authors & Affiliations

S. Xiao¹, Y. Yoon¹, Y.-H. Lee¹, J. P. Bird^1,2,*, Y. Ochiai², N. Aoki², J. L. Reno³, and J. Fransson^4,†

¹Department of Electrical Engineering, University at Buffalo, the State University of New York, Buffalo, New York 14260-1900, USA
²Graduate School of Advanced Integration Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan
³CINT/Sandia National Laboratories, Department 1131, MS 1303, Albuquerque, New Mexico 87185, USA
⁴Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 21, Uppsala, Sweden

^*jbird@buffalo.edu
^†jonas.fransson@physics.uu.se

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Vol. 93, Iss. 16 — 15 April 2016

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Images

Figure 1
(a) Shown left is a schematic representation of the standard two-path FR, while on the right we represent the possible variation of transmission as a function of energy ( $E$ ) for such a system. The line shape of the resonance reflects the relative values of the matrix elements $w$ and $v$ , which in this case are shown to yield a weakly asymmetric peak. (b) The schematic on the left indicates the situation in which the Fano system of (a) is coupled to an additional, intruder, continuum ( $I$ ). On the right, we sketch the expected variation of $T (E)$ for this system. The dotted line represents the FR in the absence of the intruder, while the solid line indicates the line shape distortion that can be induced by the coupling to $I$ . (c) The left panel is a schematic representation of the energy alignment in the standard FR. The host continuum connects points $A$ and $B$ and is coupled to the discrete level ( $D$ ) by the matrix element $v$ . An FR results when monoenergetic waves are injected into the system with an energy close to $E_{o}$ . The right panel shows the corresponding alignment relevant to discussions of the NEFR. Here, the host is coupled to the discrete level and the intruder ( $I$ ), with respective matrix elements $v$ and $t$ . The NEFR occurs when waves with a spread of energies, ranging from $E_{o}$ to $E_{o} + Δ$ , are simultaneously injected into the system to access both $D$ and $I$ . The dotted line in the figure indicates the energy alignment of the intruder.
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Figure 2
(a) False color micrograph of a coupled-QPC device, in which the lighter-colored gates are held at ground potential ( $G$ ), while the gold-colored gates are used to form the coupled QPCs. $V_{s}$ and $V_{d}$ represent the (dc) voltages applied the gates of the swept- and detector-QPCs, respectively. The white circle represents the natural quantum dot formed in the swept-QPC at pinch off, which is tunnel coupled (white dotted line) to electrons injected (solid white line with arrow) from the source of the detector. In measurements of the NEFR, the detector conductance is measured by superimposing a nonzero dc voltage ( $V_{D}$ ) on top of a smaller ac component ( $v_{D}$ ). (b) A schematic illustration of the density of states (DoS) in a QPC. This structure is only expected to be valid near pinch-off, where a spontaneously-formed discrete level ( $D$ ) is present inside the QPC. The 1D subbands of the QPC represent the intruder and are separated from $D$ by an energy detuning $Δ$ . (c) Concept of the QPC-based intruder scheme. The Fano interferometer is created by coupling the DoS in (b) to the different reservoirs of the device (the source, drain, and floating regions, at electrochemical potentials $μ_{S}, μ_{D}$ , and $μ_{F}$ , respectively).
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Figure 3
The main panel shows measurements of the linear conductance ( $V_{D} = 0$ ) of the detector QPC for the two coupled-QPC geometries identified as configurations 1 and 2 in the false-color electron micrographs that form the upper insets to the figure. Red data correspond to the geometry identified as configuration 1, while blue data correspond to configuration 2. Dotted lines through the data represent the background subtracted from the raw conductance to obtain the resonant component. The schematic at the bottom of the panel represents the realization of the intruder scenario in the coupled QPCs. The intruder and the discrete level are formed within the same QPC, as indicated by the red dotted line enclosing these two entities.
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Figure 4
(a) Measurements of the NEFR in the two different configurations. The red curves were measured in Device 1 in Configuration 1, while the blue curves were obtained in configuration 2 for Device 2. (i) Red data: $V_{D}$ = 0 mV. Blue data: $V_{D}$ = 0 mV. (ii) Red data: $V_{D}$ = 3 mV. Blue data: $V_{D}$ = 2 mV. (iii) Red data: $V_{D}$ = 4 mV. Blue data: $V_{D}$ = 3 mV. (iv) Red data: $V_{D}$ = 5 mV. Blue data: $V_{D}$ = 4 mV. (v) Red data: $V_{D}$ = 6 mV. Blue data: $V_{D}$ = 5 mV. (b) Contour plot revealing the full variation of detector conductance as a function of $V_{s}$ and $V_{D}$ , for device 2 in configuration 2. A monotonic background has been subtracted from $G_{d}$ to construct the contour. The white dashed line shows the evolution of the original antiresonance, present at $V_{D} = 0$ , while the white dotted line represents the signature of the intruder that emerges for nonzero $V_{D}$ .
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Figure 5
Measurements of the floating-reservoir voltage ( $v_{F}$ ) as a function of $V_{s}$ , in the configuration indicated in the inset to the figure. The different curves were obtained by applying various ac voltages ( $v_{D}$ , RMS values indicated in the figure) across the detector, and then measuring the voltage of the floating reservoir relative to ground. Also shown is the corresponding variation of $G_{s} (V_{s})$ .
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Figure 6
(a) Model calculations of the NEFR for different system parameters. The top panels assume $t = 30$ meV and $v = 30$ meV, and are intended to reproduce the top set of panels of Fig. 4. The lower panels, on the other hand, assume $t = v = 40$ meV, and capture the behavior for the bottom set of panels of Fig. 4. (b) Schematic illustrations showing the level alignments in the system under different conditions. The left and center panels are for thermal equilibrium ( $v_{D}$ = $V_{D}$ = 0), where either the localized state ( $D$ , left) or the edge of the 1D subbands (center) is aligned with the reservoir chemical potentials. Shown right, however, is the nonequilibrium situation, where the applied voltage ( $V_{D}$ ) opens up an effective energy window that may be used to simultaneously couple to both $D$ and $I$ . (c) NEFR computed for $t = 0$ , corresponding to the usual two-path FR shown in the upper schematic of Fig. 1. The calculations assume $v = 30$ meV and are performed for different $V_{D}$ (values indicated). (d) Similar NEFR ( $v = 30$ meV, values of $V_{D}$ again indicated), but for $t = 30$ meV.
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