Current noise in a vibrating quantum dot array

We develop methods for calculating the zero-frequency noise for quantum shuttles, i.e., nanoelectromechanical devices where the mechanical motion is quantized. As a model system we consider a three-dot array, where the internal electronic coherence both complicates and enriches the physics. Two different formulations are presented: (i) quantum regression theorem and (ii) the counting variable approach. It is demonstrated, both analytically and numerically, that the two formulations yield identical results, when the conditions of their respective applicability are fulfilled. We describe the results of extensive numerical calculations for current and current noise (Fano factor), based on a solution of a Markovian generalized master equation. The results for the current and noise are further analyzed in terms of Wigner functions, which help to distinguish different transport regimes (in particular, shuttling versus cotunneling). In the case of weak interdot coupling, the electron transport proceeds via sequential tunneling between neighboring dots. A simple rate equation with the rates calculated analytically from the $P\left(E\right)$ theory is developed and shown to agree with the full numerics.

Figure 1

Schematic picture of the three dot system. The outer dots are fixed—the left one (L) at the position $-x_0$ and the right one (R) at $x_0$, while the central one (C) can move (position $\widehat{x}$) in a harmonic confining potential. It also interacts with a heat bath causing damping and thermal noise. The outer dots whose respective energy levels are dealigned by the device bias $\left({\epsilon }_b\right)$ are coupled to the full or empty electronic reservoirs (leads), respectively. The current flows within the system due to tunneling between the left and central dot and the central and right dot and is described by the corresponding current operators ${\widehat{I}}_{CL}$, ${\widehat{I}}_{RC}$.

Figure 2

(Color online) The mean current $I$, zero-frequency noise $S\left(0\right)$, and the Fano factor $F$ as functions of the device bias ${\epsilon }_b$ for the static three dot array (dotted line) and the vibrating array at different temperatures given by the mean oscillator occupation number $\overline{n}$. The other parameters are $V_0=0.5\hslash {\omega }_0$, $\alpha =0.2\sqrt{2}\sqrt{m{\omega }_0∕\hslash }$, $x_0=(5∕\sqrt{2})\sqrt{\hslash ∕m{\omega }_0}$, $\gamma =0.025{\omega }_0$, $\Gamma =0.05{\omega }_0$ which corresponds to the case studied in Ref. 16, Fig. 6.

Figure 3

(Color online) The mean current and Fano factor for $V_0=0.76\hslash {\omega }_0$, $\alpha =0.28\sqrt{m{\omega }_0∕\hslash }$, $x_0=5\sqrt{\hslash ∕m{\omega }_0}$, $\Gamma =0.2{\omega }_0$, $T=0$ and different values of the damping coefficient (in units of ${\omega }_0$) corresponding to shuttling around ${\epsilon }_b\approx \hslash {\omega }_0$, $2\hslash {\omega }_0$.

Figure 4

(Color online) The mean current and Fano factor for $V_0=0.76\hslash {\omega }_0$, $\alpha =0.28\sqrt{m{\omega }_0∕\hslash }$, $x_0=5\sqrt{\hslash ∕m{\omega }_0}$, $\Gamma =0.2{\omega }_0$, $T=0$ and different values of the damping coefficient (in units of ${\omega }_0$) in the strong inelastic cotunneling/shuttling regime. The dots on the curves corresponding to $\gamma =0.0125$ denote the points for which the Wigner functions in Fig. 5 are plotted.

Figure 5

(Color online) Phase space representation of the oscillator around the transition from the shuttling to the strong inelastic cotunneling regime at ${\epsilon }_b∕\hslash {\omega }_0=3.96$, 4.04, 4.12, respectively (columns from the left to the right). The respective rows show the Wigner distribution functions for the empty $\left(W_{UU}\right)$ or occupied $\left(W_{CC}\right)$ central dot, and the sum of the two $(W_{\mathrm{tot}}=W_{UU}+W_{CC})$ in the oscillator phase space (horizontal axis—coordinate in units of $\sqrt{\hslash ∕m{\omega }_0}$, vertical axis—momentum in $\sqrt{\hslash m{\omega }_0}$, the grid is at 2.5 in the dimensionless units). The other parameters are: $V_0=0.76\hslash {\omega }_0$, $\alpha =0.28\sqrt{m{\omega }_0∕\hslash }$, $x_0=5\sqrt{\hslash ∕m{\omega }_0}$, $\gamma =0.0125{\omega }_0$, $\Gamma =0.2{\omega }_0$, $T=0$. The parameters correspond to the dots in Fig. 4. The Wigner functions are normalized within each column.

Figure 6

The four states 0 (device empty), $L$ (left dot occupied), $C$ (central dot occupied), $R$ (right dot occupied) and the transition rates as described by the Markov process given by the transition matrix (46).

Figure 7

(Color online) The Fano factor in the two-state sequential tunneling limit (zero temperature, large $\Gamma$). The thick line is the computed Fano factor while the thin lines with circles are given by the formula $n_C^2+{(1-n_C)}^2$ where $n_C$ are the occupation of the central dot. The collapse of the two curves marks the sequential tunneling region. The values of the other parameters are $V_0=0.1\hslash {\omega }_0,x_0=5\sqrt{\hslash ∕m{\omega }_0},\gamma =0.1{\omega }_0,\Gamma =0.1{\omega }_0,T=0$.

Figure 8

(Color online) Comparison between the numerical rates and the ones calculated by the $P\left(E\right)$ theory for $V_0=0.1\hslash {\omega }_0,\alpha =0.2\sqrt{m{\omega }_0∕\hslash },x_0=5\sqrt{\hslash ∕m{\omega }_0},\gamma =0.1{\omega }_0,\Gamma =0.1{\omega }_0,T=0$. The numerical rates are calculated assuming that the two-state sequential tunneling picture holds which is only true for ${\epsilon }_b\gtrsim 1.5\hslash {\omega }_0$, see Fig. 7. In that region the two results match almost perfectly.

Figure 9

(Color online) Comparison between the numerics and the $P\left(E\right)$ theory based sequential tunneling picture for $V_0=0.05\hslash {\omega }_0$, $\alpha =0.2\sqrt{m{\omega }_0∕\hslash }$, $x_0=5\sqrt{\hslash ∕m{\omega }_0}$, $\gamma =0.1{\omega }_0$, $\Gamma =0.1{\omega }_0$, $\overline{n}=1$. Due to the nonzero temperature the two-state model considered in Figs. 7, 8, had to be extended and there is a sequential tunneling region also for a negative bias. We observe a nearly perfect match between the two approaches for $\mid {\epsilon }_b\mid \gtrsim 1.5\hslash {\omega }_0$. The behavior around ${\epsilon }_b=0$ is clearly governed by the physics of the static array since the oscillator is largely decoupled from the electronic degrees of freedom.