Detection of the relaxation rates of an interacting quantum dot by a capacitively coupled sensor dot

Jens Schulenborg, Janine Splettstoesser, Michele Governale, and L. Debora Contreras-Pulido

Phys. Rev. B 89, 195305 – Published 16 May 2014

Abstract

We present a theoretical study of the detection of the decay time scales for a single-level quantum dot by means of a capacitively coupled sensor dot, which acts as an electrometer. We investigate the measurement back-action on the quantum-dot decay rates and elucidate its mechanism. We explicitly show that the setup can be used to measure the bare quantum-dot relaxation rates by choosing gate pulses that minimize the back-action. Interestingly, we find that besides the charge relaxation rate, also the rate associated to the fermion parity in the dot can be accessed with this setup.

Received 5 March 2014

DOI:https://doi.org/10.1103/PhysRevB.89.195305

Authors & Affiliations

Jens Schulenborg^1,2, Janine Splettstoesser^1,2, Michele Governale³, and L. Debora Contreras-Pulido^1,4

¹Institut für Theorie der Statistischen Physik, RWTH Aachen University, D-52056 Aachen, & JARA - Future Information Technologies, Germany
²Department of Microtechnology and Nanoscience (MC2), Chalmers University of Technology, SE-41298 Göteborg, Sweden
³School of Physical and Chemical Sciences and MacDiarmid Institute for Advanced Materials and Nanotechnology, Victoria University of Wellington, P.O. Box 600, Wellington 6140, New Zealand
⁴Institut für Theoretische Physik, Albert-Einstein Allee 11, Universität Ulm, D-89069 Ulm, Germany

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 89, Iss. 19 — 15 May 2014

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Sketch of the system. The upper dot (d) is driven out of equilibrium at time $t_{0}$ ; the lower dot (SQD) is used for detection of the relaxation process. The two single-level dots are coupled capacitively to each other and they are tunnel-coupled to different electronic reservoirs. Since the SQD electron spin is not important for the measurement, we neglect it for simplicity.
Reuse & Permissions
Figure 2
(a) Schematic representation of the dot (d) coupled to a reservoir (C). The dot is driven out of equilibrium at time $t_{0}$ and then relaxes to a new equilibrium state. (b) Relaxation rates that govern the exponential decay of the charge $(λ_{c})$ , the spin $(λ_{σ}),$ and the fermion-parity mode $(λ_{fp})$ of the dot as a function of the level position $ε_{d}$ after the switch. The temperature is set to $k_{B} T = U_{d} / 10$ .
Reuse & Permissions
Figure 3
(a) and (b) Deviation of the spin relaxation rates $λ_{σ}^{\pm}$ from their values in the noninteracting limit $U, U_{d} \to 0$ . (c) and (d) The coefficients $a_{\pm}$ entering the time-dependent spin decay expressed in Eq. (22), for a switch applied to a system in which the quantum dot is initially occupied with a spin-up electron and the SQD is initially empty. All quantities are plotted as a function of the final level positions $ε_{SQD}$ and $ε_{d}$ . The other system parameters are $μ_{L} = μ_{R} = μ_{C} = 0$ , $Γ_{L} = Γ_{R} = Γ_{C} = Γ / 2$ , $U_{d} = 8 U / 5$ , $k_{B} T = U_{d} / 10 = 0.16 U$ , and $Γ / k_{B} T ≪ 1$ .
Reuse & Permissions
Figure 4
(a) Deviation of the dot related relaxation rates $λ_{\tilde{c}}, λ_{\tilde{fp}}$ from their respective values in the noninteracting limit $U, U_{d} \to 0$ as a function of the final level positions $ε_{SQD}$ and $ε_{d}$ . We set $μ_{L} = μ_{R} = μ_{C} = E_{F} = 0$ , $Γ_{L} = Γ_{R} = Γ / 2$ , $U_{d} = 8 U / 5$ , and $k_{B} T = U_{d} / 10 = 0.16 U$ . With $Γ_{C} = Γ / 10$ , we must also assume $Γ / k_{B} T ≪ 0.1$ , even though this ratio does not enter explicitly in our leading order calculations. (b) Typical gate pulse into the back-action regime $- U < ε_{d} < ε_{SQD} < 0$ for which the dot charge relaxation is strongly suppressed. The initially filled SQD induces Coulomb blockade in the dot, and the latter therefore cannot reach its stationary limit, which would be the singly occupied state. Note that for simplicity, the graph does not indicate the dot on-site repulsion strength $U_{d}$ . (c) An initially empty dot gate-pulsed into the regime $- U_{d} - U < ε_{d} < - U_{d}$ . The dot is driven into double occupation only if the SQD is empty, but at least one electron can always enter the dot as long as $U_{d} > U$ , even if the SQD is occupied.
Reuse & Permissions
Figure 5
Logarithm of (a) the current signal $Δ I (t)$ and (b) the dot charge deviation $Δ n_{d} (t)$ , as a function of time for two different gate pulses, both for a spinless and spin-degenerate SQD. The quantities are normalized by their respective initial values $Δ I (0)$ , $Δ n_{d} (0)$ . The red dash-dotted line represents in each case the decay of $Δ m (t) / Δ m (0)$ . (c) and (d) The two gate pulses are visualized as arrows in a plot of the effective charge- and dot fermion-parity rate $λ_{\tilde{c}}$ , $λ_{\tilde{fp}}$ as a function of $ε_{SQD}, ε_{d}$ . The starting/end point of each arrow indicates the level positions before/after the switch, with the initial state $P^{in} = P^{st} (ε_{SQD}^{in}, ε_{d}^{in}, ...)$ . The system parameters are: $μ_{L} = μ_{C} = 0$ , $μ_{R} = - e V_{sd} = - U$ , $Γ_{C} = Γ / 10$ , $U_{d} = 8 U / 5$ , and $k_{B} T = U_{d} / 10 = 0.16 U$ . The SQD on-site interaction strength is $U_{SQD} = 2 U$ . As before, we must require $Γ / k_{B} T ≪ 0.1$ , even though not explicit in the calculation.
Reuse & Permissions
Figure 6
Relative difference between the slopes $s$ of the logarithmic current signal and dot charge curves as obtained for a spinless SQD, see Fig. 5, and a spin-degenerate SQD. System parameters are: $μ_{L} = μ_{C} = 0$ , $μ_{R} = - e V_{sd} = - U$ , $Γ_{C} = Γ / 10$ , $U_{d} = 8 U / 5$ , $U_{SQD} = 2 U$ , and $k_{B} T = U_{d} / 10 = 0.16 U$ . We assume $Γ / k_{B} T ≪ 0.1$ .
Reuse & Permissions

Physical Review B

covering condensed matter and materials physics