Noise spectra of an interacting quantum dot

We study the noise spectra of a many-level quantum dot coupled to two electron reservoirs, when interactions are taken into account only on the dot within the Hartree-Fock approximation. The dependence of the noise spectra on the interaction strength, the coupling to the leads, and the chemical potential are derived. For zero bias and zero temperature, we find that as a function of the (external) frequency, the noise exhibits steps and dips at frequencies reflecting the internal structure of the energy levels on the dot. Modifications due to a finite bias and finite temperatures are investigated for a noninteracting two-level dot. Possible relations to experiments are pointed out.

Figure 1

The top panel depicts the averages, computed self-consistently from Eq. (21), as a function of the interaction strength $U$. The horizontal curves are the diagonal averages, $\langle d_n^{†}{d}_n\rangle$; the “cloud” around the lowest ones represents all the off-diagonal averages $\langle d_n^{†}{d}_m\rangle$ with $n\ne m$. The red (thin-dashed) steplike curves represent the diagonal average corresponding to the energy levels crossing the Fermi energy ${\epsilon }_{\mathrm{F}}$, computed within the Hartree-Fock approximation; the purple (thick-dotted) curves show the same average computed within the Hartree approximation alone. Lower panel: The effective energies ${\tilde{\epsilon }}_n$, Eq. (19), are presented by the blue (dashed) and red (dashed-dotted) curves. The green (continuous) curves are ${\tilde{J}}_{nm}$, Eq. (20), and the dotted (black) horizontal curve marks the Fermi energy. Parameters: ${\epsilon }_{\mathrm{F}}=0.5$, ${\epsilon }_L={\epsilon }_R=0.598$, ${\omega }_L={\omega }_R=0.8$, ${\mathcal{N}}_g^{}=30$. The level widths [see Eqs. (30) and (31)] are then equal, ${\Gamma }_L={\Gamma }_R=0.05$. (All energies are in units of the level spacing $\Delta$.)

Figure 2

Same as Fig. 1, with ${\mathcal{N}}_g^{}=33$.

Figure 3

The dc conductance as a function of the Fermi energy. The continuous (blue) curve is the conductance of a noninteracting dot, Eq. (38); the dashed (red) curve corresponds to $U=0.5$. Parameters used in Eq. (31): ${\omega }_L={\omega }_R=0.8$, ${\epsilon }_L={\epsilon }_R=0$. (All energies are in units of the level spacing $\Delta$.)

Figure 4

The noise spectrum for a dot coupled symmetrically to the leads, as a function of the frequency, for various values of the interaction strength ($U=0,$ 0.1, $...$, 1, from left to right) computed within the Hartree approximation. Inset: The effective single-particle energy levels ${\tilde{\epsilon }}_n/\Delta$ on the dot, as a function of the interaction strength for the same parameters. Parameters: ${\epsilon }_F=-0.8$, ${\mathcal{N}}_g^{}=30$, and ${\Gamma }_L={\Gamma }_R=0.025$. These widths are given by Eqs. (30) and (31) with ${\epsilon }_L={\epsilon }_R=0$ and ${\omega }_L={\omega }_R=0.8$. (All energies are in units of the level spacing $\Delta$.)

Figure 5

Same as Fig. 4 for a dot coupled asymmetrically to the leads. The black dashed line is for $U=0$ while the solid blue is for a noninteracting case. The vertical red line in the inset indicates the chosen interaction strength, $U=0.5$, and the dashed black line indicates the Fermi energy, ${\epsilon }_F=-0.9$. The dot-lead coupling parameters are ${\Gamma }_L=0.022$ and ${\Gamma }_R=0.014$. The widths are found from Eqs. (30) and (31) with ${\epsilon }_L=0$, ${\epsilon }_R=1$, and ${\omega }_L={\omega }_R=0.8$.

Figure 6

Same as Fig. 4 for an interaction-free dot, shown by the solid (black) curve, and for $U=0.7$ computed within the Hartree approximation, shown by the dotted-dashed (blue) curve, and within the Hartree-Fock approximation, shown by the dashed (red) curve. The Fermi energy is ${\epsilon }_F=-0.7$.

Figure 7

The noise spectrum of a noninteracting dot as a function of the frequency, for various values of the Fermi energy, ${\epsilon }_F=-0.7$, $-1.3$, $-1.7$, and $-2.3$ (from left to right). Parameters: ${\Gamma }_L={\Gamma }_R=0.028$, $0.014$, $0.009$, $0.004$ [from left to right, obtained from Eqs. (30) and (31) with ${\omega }_L={\omega }_R=0.8$ and ${\epsilon }_L={\epsilon }_R=0$]. (All energies are in units of the level spacing $\Delta$.)

Figure 8

The correlations $C^{(+)}$ (upper panel) and $C^{(-)}$ (lower panel) for the two-level case, as a function of the frequency at zero temperature and zero bias. The different curves in the lower panel correspond to ${\Gamma }_L=0.1$ and ${\Gamma }_R=0$ (brown continuous curve), ${\Gamma }_L=0.0745$ and ${\Gamma }_R=0.0255$ (blue dashed curve), and ${\Gamma }_L={\Gamma }_R=0.05$ (black dotted curve). Parameters: ${\epsilon }_1=1$, ${\epsilon }_2=-2$, and ${\epsilon }_F=0.5$. All energies are in units of $\Delta$.

Figure 9

The noise spectra $C^{(+)}\left(\omega \right)$ (top figures) and $C^{(-)}\left(\omega \right)$ (bottom figures) of the biased dot at zero temperature with ${\epsilon }_1=\Delta$ and ${\epsilon }_2=-2\Delta$. The Fermi energy is set to $0.5$ in all figures. The different figures are for for $\mathrm{eV}=3$ (first and third figures from the top) and 6 (second and fourth figures from the top). The five curves correspond to ${\Gamma }_L=0.1$ and ${\Gamma }_R=0$ (brown continuous curves), ${\Gamma }_L=0.0745$ and ${\Gamma }_R=0.0255$ (blue dashed curves), ${\Gamma }_L={\Gamma }_R=0.05$ (black dotted curves), ${\Gamma }_L=0.0255$ and ${\Gamma }_R=0.0745$ (gray long-dashed curves), and ${\Gamma }_L=0$ and ${\Gamma }_R=0.1$ (green dotted-dashed curves). All energies are in units of $\Delta$.

Figure 10

Same as Fig. 9, with $T=1.5\Delta$ and $\mathrm{eV}=6\Delta$.

Figure 11

The peak in the differential conductance as a function of the bias voltage, for various couplings $J$: $J=0$ (black continuous curve), $J=0.1$ (red dotted curve), and $J=0.2$ (blue dashed curve). Parameters: ${\epsilon }_{\mathrm{F}}=0.5$, ${\epsilon }_1=1$, ${\epsilon }_2=2$, ${\Gamma }_L={\Gamma }_R=0.05$. (All energies are in units of the level spacing $\Delta$.)