Canyon of current suppression in an interacting two-level quantum dot

Motivated by the recent discovery of a canyon of conductance suppression in a two-level equal-spin quantum dot system [Phys. Rev. Lett. 104, 186804 (2010)], the transport through this system is studied in detail. At low bias and low temperature a strong current suppression is found around the electron-hole symmetry point independent of the couplings, in agreement with previous results. By means of a Schrieffer–Wolff transformation we are able to give an intuitive explanation to this suppression in the low-energy regime. In the general situation, numerical simulations are carried out using quantum rate equations. The simulations allow for the prediction of how the suppression is affected by the couplings, the charging energy, the position of the energy levels, the applied bias, and the temperature. We find that, away from electron-hole symmetry, the parity of the couplings is essential for the current suppression. It is also shown how broadening, interference, and a finite interaction energy cause a shift of the current minimum away from degeneracy. Finally we see how an increased population of the upper level leads to current peaks on each side of the suppression line. At sufficiently high bias we discover a coherence-induced population inversion.

Figure 1

Level configuration at (a) low bias, (b) high bias, and (c) ultra-high bias. Because of the Coulomb interaction the energies of the levels are increased by an amount $U$ when the other state is filled. The applied bias is given by ${\mu }_L-{\mu }_R$, where ${\mu }_L$ and ${\mu }_R$ are the chemical potentials in the left and right contact, respectively.

Figure 2

Conductance calculated with the 2vN method for the interacting system, $U=25\hspace{0.5em}{k}_{B}T$ to the left, and the noninteracting Green’s function result to the right. The white areas in the right figures are the regions where the noninteracting formalism is not valid. The upper four figures correspond to the low-bias limit $V_{\mathrm{bias}}=k_{B}T$, while the lower four show the results for high bias $V_{\mathrm{bias}}=15\hspace{0.5em}{k}_{B}T$. The couplings are given by Eqs. (6) with ${\Gamma }_1=k_{B}T$ and $a=0.5$ for figures (a), (b), (e), and (f), while ${\Gamma }_1=4\hspace{0.5em}{k}_{B}T$ and $a=0.5$ for figures (c), (d), (g), and (h). The dashed boxes in the noninteracting figures mark the studied region in the interacting case.

Figure 3

Conductance calculated with the 2vN method. The two levels have equal parity with couplings given by $t_{L1}=t_{R1}=t$, $t_{L2}=0.4t$ and $t_{R2}=0.6t$ with ${\Gamma }_{L1}={\Gamma }_{R1}=2\pi {\rho }_0{t}^2=5\hspace{0.5em}{k}_{B}T$. Additional parameters are $V_{\mathrm{bias}}=15\hspace{0.5em}{k}_{B}T$, $U=25\hspace{0.5em}{k}_{B}T$.

Figure 4

$Q$ as a function of $V_{\mathrm{bias}}$ at electron-hole symmetry for different coupling strengths. All curves correspond to $a=0.5$ and $U=25\hspace{0.5em}{k}_{B}T$.

Figure 5

$Q$ as a function of $V_{\mathrm{bias}}$ at electron-hole symmetry for different temperatures. All curves correspond to $a=0.5$ and $U=5{\Gamma }_1$.

Figure 6

$Q$ as a function of the difference in coupling strength $a$, see Eqs. (6), at electron-hole symmetry for different temperatures. The high-temperature limit is given by Eq. (24). All curves correspond to $V_{\mathrm{bias}}=1{\Gamma }_1$ and $U=5{\Gamma }_1$.

Figure 7

(a) The current through the two-level quantum dot calculated with 2vN and 1vN formalisms, with and without renormalization of the energy levels, at ${\Gamma }_1=4\hspace{0.5em}{k}_{B}T$, $U=25\hspace{0.5em}{k}_{B}T$, $E_g=-U/2$, $a=0.5$, and $V_{\mathrm{bias}}=2.5\hspace{0.5em}{k}_{B}T$. As the broadening is not included the first-order method overestimates the peak height but underestimates the shift away from degeneracy of the conductance dip. (b) and (c) show the corresponding occupations and real part of the coherence calculated with the 2vN formalism. The vertical dashed line marks the location of the conductance minimum.

Figure 8

The conductance of the two-level quantum dot at ${\Gamma }_1=5\hspace{0.5em}{k}_{B}T$, $a=0.5$, $V_{\mathrm{bias}}=2.5\hspace{0.5em}{k}_{B}T$, and $E_g=-U/2$. Increasing $U$ leads to a shift of the conductance minimum toward positive $\Delta E$, where the less coupled state is lower in energy.

Figure 9

The conductance of the two-level quantum dot at ${\Gamma }_1=5\hspace{0.5em}{k}_{B}T$, $a=0.5$, $V_{\mathrm{bias}}=30\hspace{0.5em}{k}_{B}T$, and $E_g=-U/2$. As predicted, increasing $U$ leads to reduced conductance at level degeneracy.

Figure 10

Conductance at $E_g=0$ of the two-level quantum dot as a function of the level splitting for different values of $V_{\mathrm{bias}}$. The insert shows the occupation of the two single-particle levels for $V_{\mathrm{bias}}=2\hspace{0.5em}{k}_{B}T$, with the vertical dashed lines marking the location of the current peaks. The parameters are given by $a=1$, ${\Gamma }_1=20\hspace{0.5em}{k}_{B}T$, and $U=100\hspace{0.5em}{k}_{B}T$.

Figure 11

Occupation of the two single-particle levels for different values of $V_{\mathrm{bias}}$. Around degeneracy a region of population inversion can be found. The parameters are given by $a=1$, ${\Gamma }_1=20\hspace{0.5em}{k}_{B}T$, and $U=100\hspace{0.5em}{k}_{B}T$. The population inversion in the diagonal basis ($|\alpha \rangle$, $|\beta \rangle$) is calculated using $\Omega$ from Eq. (27).

Figure 12

The finite bias induces an anticrossing $\Omega$ between the two original states. Diagonalizing the dot Hamiltonian results in the upper state being strongly coupled to the lead with high chemical potential, while the lower state is strongly coupled to the lead with low chemical potential.

Figure 13

Populations calculated with the 1vN and 2vN methods. Couplings and charging energy are given by ${\Gamma }_1=5\hspace{0.5em}{k}_{B}T$ and $U=100\hspace{0.5em}{k}_{B}T$, respectively.