Nonequilibrium transport theory of the singlet-triplet transition: Perturbative approach

We use a simple iterative perturbation theory to study the singlet-triplet (ST) transition in lateral and vertical quantum dots, modeled by the nonequilibrium two-level Anderson model. To a great surprise, the region of stable perturbation theory extends to relatively strong interactions, and this simple approach is able to reproduce all experimentally observed features of the ST transition, including the formation of a dip in the differential conductance of a lateral dot indicative of the two-stage Kondo effect, or the maximum in the linear conductance around the transition point. Choosing the right starting point to the perturbation theory is, however, crucial to obtain reliable and meaningful results.

Figure 1

The antisymmetrized vertex function containing the Coulomb and the Hund’s rule coupling.

Figure 2

(a) Counterterm, (b) first-order, and (c) second-order contributions to the self-energy. The lines denote noninteracting Keldysh Green’s functions.

Figure 3

(Color online) Hysteresis of the occupation number as a function of level splitting $\left(\Delta \right)$ for $J/U=0.0$ and $\Gamma /U=0.4$.

Figure 4

Stability region of perturbation theory (PT) for $\Gamma ={\Gamma }_{\pm }$. Empty circles mark the region of hysteresis while filled circles indicate the stable region, where PT gives meaningful results.

Figure 5

(Color online) Evolution of the singlet and triplet states as a function of $\Delta$ for an isolated dot with $U⪢J,\left|\Delta \right|$ and fixed $J>0$. Energies are measured from the energy $E_t$ of the triplet state. The singlet-triplet transition point is also indicated (S-T).

Figure 6

Sketch of the singlet, triplet, and transition regimes, as a function of $\Delta /U$ for a fixed $J>0$.

Figure 7

(Color online) Total equilibrium spectral functions, ${\varrho }_T\left(\omega \right)$, for $J/U=0$ and $\Gamma /U=0.565$, and different values of the level splitting, $\Delta /U$. Inset: evolution of the peak positions in ${\varrho }_{\pm }\left(\omega \right)$ as a function of $\Delta /U$. The dashed lines indicate $\omega =\pm \Delta$.

Figure 8

(Color online) Total equilibrium spectral functions, ${\varrho }_T\left(\omega \right)$, for $J/U=0.2$, $\Gamma /U=0.565$, for different values of level splitting, $\Delta /U$. Inset: peak positions of ${\varrho }_{\pm }\left(\omega \right)$ as a function of $\Delta /U$. The dashed lines indicate $\omega =\pm \left(\Delta -2J\right)$.

Figure 9

(Color online) Linear conductance of lateral and vertical dots for $J/U=0.2$, from numerical derivation (ND) of the current at zero bias, and from the Fermi-liquid (FL) relations. Inset: comparison of occupation numbers and phase shifts, $\delta /\pi$. The conductance of a symmetrical lateral dot shows a maximum of height $2e^2/h$ around the ST transition.

Figure 10

(Color online) Differential conductance, $G\left(V\right)$, for $J/U=0.2$ and $\Gamma /U=0.565$ for different level splittings, $\Delta /U$ both (a) for lateral and (b) for vertical quantum dots. For vertical dots the suppression and the splitting of the Kondo resonance are clearly visible in $G\left(V\right)$. For lateral dots, interference effects mask the signal on the triplet side.

Figure 11

Range of applicability of perturbation theory for ${\Gamma }_{+}\ne {\Gamma }_{-}$, as extracted from the hysteresis of occupation numbers. Filled circles indicate the region of stable perturbation theory. Couplings are measured in units of $\Gamma \equiv \sqrt{{\Gamma }_{+}{\Gamma }_{-}}$.

Figure 12

(Color online) Spectral functions for $J/U=0.2$, ${\Gamma }_{+}/U=0.4$, and ${\Gamma }_{-}/U=0.8$, for different values of level splitting, $\Delta /U$. The upper panel shows the level-projected spectral functions, ${\varrho }_{\pm }\left(\omega \right)$, while in the lower panel the full spectral functions, ${\varrho }_T\left(\omega \right)$, are presented. Notice the breaking of electron-hole symmetry for any $\Delta \ne 0$.

Figure 13

(Color online) Linear conductance for $J/U=0.2$, ${\Gamma }_{+}/U=0.4$, and ${\Gamma }_{-}/U=0.8$, as obtained from numerical differentiation of the current (ND), and from the Fermi liquid relations (FL). The inset shows the phase shifts ${\delta }_{\pm }/\pi$ extracted from the Green’s functions at $\omega =0$ and the occupation numbers, $n_{\pm }$.

Figure 14

Differential conductance of a lateral quantum dot for $J/U=0.2$, ${\Gamma }_{+}/U=0.4$, and ${\Gamma }_{-}/U=0.8$, for level splitting $\Delta /U=0$. The side peak can be attributed to the singlet excitation energy, $2J$, while the central resonance-antiresonance structure is associated with the two consecutive Kondo effects on the triplet side.

Figure 15

(Color online) Differential conductance, $G\left(V\right)$, for $J/U=0.2$, ${\Gamma }_{+}/U=0.4$, and ${\Gamma }_{-}/U=0.8$ for different level splittings, $\Delta /U$ for (a) a lateral and (b) a vertical quantum dot. The two consecutive Kondo resonances on the triplet side are clearly visible in the lateral dot (middle resonance-antiresonance structure) while the two Kondo resonances merge into a single peak in a vertical dot.