Nonequilibrium Plasmons in a Quantum Wire Single-Electron Transistor

We analyze a single-electron transistor composed of two semi-infinite one-dimensional quantum wires and a relatively short segment between them. We describe each wire section by a Luttinger model, and treat tunneling events in the sequential approximation when the system’s dynamics can be described by a master equation. We show that the steady-state occupation probabilities in the strongly interacting regime depend only on the energies of the states and follow a universal form that depends on the source-drain voltage and the interaction strength.

Figure 1

Model system. Two long wires are adiabatically connected to reservoirs and a short wire is connected to the two by weak coupling.

Figure 2

(A) The occupation probabilities $P(\{n\})\equiv P(N=0,\{n\})$ as a function of the energies $E(N=0,\{n\})$ of the many-body states on the dot for $g=0.24$ (SWNT) and for $g=1$ (noninteracting) in the inset. The degeneracy of the energy state $E(n)$ is $D(n)\simeq \exp (\pi \sqrt{2n/3})/(4\sqrt{3}n)$, where $n=\sum _{m}m·n_m$ is the dimensionless energy [circles (○) represent individual many-body states]. (B) Averaged probabilities $P(n)=⟨P(\{n\}){⟩}_n$ as a function of the energy levels $E(n)$. Solid lines are analytic approximations $P(n)\propto n^{-(3/2)[(\alpha +1)/n_{\mathrm{s}\mathrm{d}}]n}$, where $n_{\mathrm{s}\mathrm{d}}=e{V}_{\mathrm{s}\mathrm{d}}·(\pi \hslash v_F/g{L}_D{)}^{-1}$. The inset shows the relative standard deviations of the probabilities, $⟨\delta P(\{n\}){⟩}_n/P(n)$, as a function of the dimensionless energy for $g$ values ranging from $g=1$ to $g=0.12$. Implicit parameters are $T=50\mathrm{m}\mathrm{K}$, $V_{\mathrm{s}\mathrm{d}}=g^{-1}\times 0.22\mathrm{V}$, and $N_G=0.5$.

Figure 3

Current $I$ and differential conductance $dI/dV$ as a function of source-drain voltage $V_{\mathrm{s}\mathrm{d}}$ for $g=0.24$. (A) Off-Coulomb blockade with $N_G=0.5$ and (B) a Coulomb blockade with $N_G=0.35$, for which the conductance vanishes up to $e{V}_{\mathrm{s}\mathrm{d}}=0.3\times \pi \hslash v_F/g^2{L}_D$ at zero temperature. $I$ is in units proportional to $|t_{L/R}{|}^2$. The inset in (A) shows the power law of conductance on temperature at zero bias voltage where the solid line is $G(T)=G(T_0)(T/T_0{)}^{\alpha -1}$ for $k_{B}T<\Delta E$. Implicit parameters are the same as in Fig. 2.