Shot noise of inelastic tunneling through quantum dot systems

We present a theoretical analysis of the effect of inelastic electron scattering on current and its fluctuations in a mesoscopic quantum dot (QD) connected to two leads, based on a recently developed nonperturbative technique involving the approximate mapping of the many-body electron-phonon coupling problem onto a multichannel single-electron scattering problem. In this, we apply the Büttiker scattering theory of shot noise for a two-terminal mesoscopic device to the multichannel case with differing weight factors and examine zero-frequency shot noise for two special cases: (i) a single-molecule QD and (ii) coupled semiconductor QDs. The nonequilibrium Green’s function method facilitates calculation of single-electron transmission and reflection amplitudes for inelastic processes under nonequilibrium conditions in the mapping model. For the single-molecule QD we find that, in the presence of the electron-phonon interaction, both differential conductance and differential shot noise display additional peaks as bias-voltage increases due to phonon-assisted processes. In the case of coupled QDs, our nonperturbative calculations account for the electron-phonon interaction on an equal footing with couplings to the leads, as well as the coupling between the two dots. Our results exhibit oscillations in both the current and shot noise as functions of the energy difference between the two QDs, resulting from the spontaneous emission of phonons in the nonlinear transport process. In the “zero-phonon” resonant tunneling regime, the shot noise exhibits a double peak, while in the “one-phonon” region, only a single peak appears.

Figure 1

Schematic description of the inelastic scattering problem for (a) a single site and (b) two sites in series, both with on-site EPI. Each phonon state of the site, taken jointly with the Bloch state of the electron in the lead, can be visualized as a pseudochannel labeled by $n$, which is connected to the two leads with hopping parameters ${\Gamma }_{L∕R}$. These channels are connected vertically by the EPI, ${\lambda }_j$.

Figure 2

(a) The calculated total current $I$ (thick line), elastic current $I^{\mathrm{el}}$ (thin line); (b) the zero-frequency shot noise $S$ (thick line), its elastic part $S^{\mathrm{el}}$ (thin line); and (c) the corresponding differential conductance $dI∕dV$ (dashed line) and differential shot noise $dS∕dV$ (solid line) as functions of applied bias voltage for a single-molecular QD with bare level ${ϵ}_d=1.0$. For comparison, we also plot the current and shot noise in the case without the phonon bath. The other parameters used in the calculation are $\Gamma =0.04$, ${\omega }_{ph}=1.0$, and $\lambda =0.5$. The temperature is set to $T=0.04$.

Figure 3

Fano factor vs bias-voltage $V$. Solid line denotes results with EPI, while the dashed line denotes results without EPI. The parameters are the same as in Fig. 2.

Figure 4

Predicted tunneling current as a function of the energy difference, $\epsilon$. (a) The total current $I_{\mathrm{tot}}$, its elastic component $I_{\mathrm{el}}$, its inelastic part $I_{\mathrm{inel}}$, and the current without phonon bath (thin line) vs $\epsilon$ (with $\Gamma =T_c=0.25$, EPI constants ${\lambda }_1=-{\lambda }_2=0.1875$, and temperature $T=0.125$); (b) the total currents (thick lines) and currents without EPI (thin lines) for various couplings, $T_c$, between the two QDs.

Figure 5

Zero-frequency shot noise vs $\epsilon$. (a) Total shot noise, $S_{\mathrm{tot}}$; its elastic part, $S_{\mathrm{el}}$; its inelastic part, $S_{\mathrm{inel}}$; and shot noise with no EPI (thin line). The parameters are the same as in Fig. 4a; (b) shot noise with (thick lines) and without (thin lines) EPI for several different values of $T_c$.

Figure 6

Fano factor vs $\epsilon$. (a) Fano factors with (thick curve) and without (thin curve) EPI for the same system as in Fig. 4a; (b) Fano factors for several different values of $T_c$.