Coulomb-blocked transport through a quantum dot with spin-split level: Increase of differential conductance peaks by spin relaxation

Nonequilibrium transport through a quantum dot with one spin-split single-particle level is studied in the cotunneling regime at low temperatures. The Coulomb diamond can be subdivided into parts differing in at least one of two respects: what kind of tunneling processes (i) determine the single-particle occupations and (ii) mainly contribute to the current. No finite systematic perturbation expansion of the occupations and the current can be found that is valid within the entire Coulomb diamond. We therefore construct a nonsystematic solution, which is physically correct and perturbative in the whole cotunneling regime, while smoothly crossing over between the different regions. With this solution, the impact of an intrinsic spin-flip relaxation on the transport is investigated. We focus on peaks in the differential conductance, which mark the onset of cotunneling-mediated sequential transport. It is shown that these peaks are maximally pronounced at a relaxation roughly as fast as sequential tunneling. The approach as well as the presented results can be generalized to quantum dots with few levels.

Figure 1

(Color online) (a) Schematic picture of the diamond-shaped cotunneling regime showing its subdivision into areas with different possible tunneling processes (hatched areas) as well as into CR, SR, and IR, in which the occupations $P_g$ and $P_e$ are determined either by cotunneling, sequential tunneling, or both [colored (shaded) areas]. $E$ stands for elastic and $I$ for inelastic cotunnelings and $S$ is for sequential tunneling. (b) Calculated charging diagram [based on Eqs. (9, 11)] of the cotunneling regime with parameters $\Delta =45k_{B}T$, $U=225k_{B}T$, $\Gamma =4.5\times {10}^{-3}{k}_{B}T$, and $\theta =\Gamma /2$. High values of the differential conductance $\tilde{G}=d\tilde{I}/d{V}_{\text{bias}}$ (in units of $G_0=\beta \Gamma e^2/\hslash$) outside and at the border of the diamond are clipped by the color scale (values near the border and in the exterior, see text). The short horizontal line corresponds to the range of bias values in Fig. 2. Both (a) and (b) can be extended to regions with opposite signs of $e{V}_{\text{bias}}$ by reflection with respect to the $e{V}_{\text{bias}}=0$ axis.

Figure 2

(Color online) Current $\tilde{I}$ (in units of $I_0=e\Gamma /\hslash$) and differential conductance $\tilde{G}$ versus bias $e{V}_{\text{bias}}$ and relaxation $\theta$ with $e{\Phi }_D^{\epsilon }=\left(U+\Delta \right)/2$ and parameters $\Delta ,U$ and $\Gamma$ are as in Fig. 1. It can be seen how the height of the (a) cotunneling-mediated current step and of the (b) corresponding conductance peak as well as their positions depend on the relaxation rate. In the range of $0\le \theta \lesssim {\Gamma }^2/\Delta$ (linear scale), the system is hardly affected by the relaxation. Between ${\Gamma }^2/\Delta$ and $\Delta$ (logarithmic scale), the height of current step and conductance peak first grow to a maximum value at ${\theta }_0\approx 0.58\Gamma$ [dashed lines in (a) and (b)] for increasing $\theta$, then decrease again and vanish before $\theta =\Delta$. Even faster relaxation (reciprocal scale) has no further effect. (c) and (d) show cuts through (a) and (b), respectively, for five different values of $\theta$. Both the step in (a) and (c) and peak in (b) and (d) slightly shift toward higher absolute values of $e{V}_{\text{bias}}$ for increasing rates between ${\Gamma }^2/\Delta$ and $\Delta$.

Figure 3

(Color online) Position of the Fermi levels ${\mu }_L$ and ${\mu }_R$ of left and right reservoirs [colored (shaded) rectangles] relative to the four energy differences ${\mu }_D\left({\phi }^{\prime },\phi \right)={\epsilon }_{{\phi }^{\prime }}-{\epsilon }_{\phi }-e{\Phi }_D$ [light red (light gray) horizontal lines] between initial and finial state energies of the single-particle transitions (parameters $\Delta ,U$, and ${\Phi }_D^{\epsilon }$ as in Fig. 1). In (a) bold green (light gray) and red (dark gray), arrows represent elastic tunneling via virtual state $|d⟩$ through the dot in ground and excited states, respectively, which are indicated by filled green (light gray) and red (dark gray) points $\left(e{V}_{\text{bias}}=1.5\Delta \right)$. Double pointed arrows on the left show for both cases how much energy the tunneling electron must gain to get into the virtual state. (b) Coherent inelastic processes causing transitions $|g⟩\to |e⟩$ [green (light gray) arrows] and $|e⟩\to |g⟩$ [red (dark gray) arrows] are illustrated $\left(e{V}_{\text{bias}}=2\Delta \right)$. Electrons in the red (dark gray) colored [green (light gray) hatched] part of reservoir $R$ can tunnel into the dot, if it is in initial state $|e⟩$ $\left(|g⟩\right)$.