Conductance of a spin-1 quantum dot: The two-stage Kondo effect

We discuss the physics of a spin-1 quantum dot, coupled to two metallic leads, and develop a simple model for the temperature dependence of its conductance. Such quantum dots are described by a two-channel Kondo model with asymmetric coupling constants, and the spin screening of the dot by the leads is expected to proceed via a two-stage process. When the Kondo temperatures $T_{K1}$ and $T_{K2}$ of each channel are widely separated on cooling, the dot passes through a broad crossover regime $T_{K2}⪡T⪡T_{K1}$ dominated by underscreened Kondo physics. A singular or non-Fermi-liquid correction to the conductance develops in this regime. At the lowest temperatures, destructive interference between resonant scattering in both channels leads to the eventual suppression of the conductance of the dot. We develop a model to describe the growth, and ultimate suppression, of the conductance in the two channel Kondo model as it is screened successively by its two channels. Our model is based upon large-$N$ approximation in which the localized spin degrees of freedom are described using the Schwinger boson formalism.

Figure 1

(Color online) Energy diagram of a spin-1 quantum dot with strong Hund’s coupling.

Figure 2

The noncrossing approximation for the self-energies of conduction electrons and $\chi$ fermions. The solid line denotes the Larkin Ovchinnikov matrix propagator for the conduction electrons. The dashed line denotes the corresponding Green’s function of the auxiliary $\left(\chi \right)$ fermions and the wavy line is the bosonic propagator. Thin lines denote the bare propagator and full lines the dressed propagator. Each vertex corresponds to the factor $i\frac{\tilde{t}}{\sqrt{N}}$.

Figure 3

Imaginary part of the $t$ matrix for a variety of voltages for $K∕N=0.4$, $S=1$. As the voltage is increased, the singular central peak splits into two components.

Figure 4

Temperature dependence of the differential conductance, normalized with respect to $g_U=N\frac{e^2}{h}{\sin }^2(\pi K∕N)$ for the representative case $K∕N=0.4$, $S=1$. The inset shows the $1∕T$ divergence of the derivative $dg∕dT$.

Figure 5

Voltage-dependent conductance $G\left(V\right)=I\left(V\right)∕V$ for the case $K∕N=0.4$, $S=1$. Inset: derivative of $G\left(V\right)$ showing $1∕V$ divergence.

Figure 6

Dyson equations for the conduction electron propagators (a) and the slave fermion propagators (b) for the two-channel quantum dot.

Figure 7

(Color online) Linear conductance of the two-channel quantum dot normalized with respect to $g_U=N\frac{e^2}{h}{\sin }^2(\pi K∕N)$, calculated for $K∕N=0.4$, $S=1$, and $\alpha =\beta =1∕\sqrt{2}$, using Eqs. (74, 75), for three ratios of the two Kondo temperatures $\alpha =T_{K1}∕T_{K2}=10$ (short-dashed curve), $\alpha =T_{K1}∕T_{K2}=100$ (long-dashed curve), and $\alpha =T_{K1}∕T_{K2}=1000$ (solid curve). Arrows show size of $T_{K2}$ in each case.

Figure 8

NCA contributions to the self energies of (a) lead electrons, (b) auxiliary $\chi$ holes, and (c) Schwinger bosons. The solid line denotes the Larkin Ovchinnikov matrix propagator for the conduction electrons. The dashed line denotes the corresponding Green’s function of the auxiliary $\left(\chi \right)$ fermions and the wavy line is the bosonic propagator. Each vertex corresponds to the factor $i\frac{\tilde{t}}{\sqrt{N}}$.

Figure 9

The Dyson equation for ${\widehat{G}}_{1\phi }$ Green’s function (${\widehat{G}}_{1\phi }$ is a matrix in Keldysh space).

Figure 10

Diagrammatic contributions to the off-diagonal Green’s function $D_{12}\left(\omega \right)$, describing interchannel contributions to the conduction electron propagator.