Cotunneling Spectroscopy in Few-Electron Quantum Dots

Few-electron quantum dots are investigated in the regime of strong tunneling to the leads. Inelastic cotunneling is used to measure the two-electron singlet-triplet splitting above and below a magnetic field driven singlet-triplet transition. Evidence for a nonequilibrium two-electron singlet-triplet Kondo effect is presented. Cotunneling allows orbital correlations and parameters characterizing entanglement of the two-electron singlet ground state to be extracted from dc transport.

Figure 1

(a) Differential conductance $g$ (log color scale) as a function of source-drain bias $V_{\mathrm{S}\mathrm{D}}$ and gate voltage $V_G$ at $B_{\perp }=0$, at base electron temperature $T_{\mathrm{e}\mathrm{l}}=45\mathrm{m}\mathrm{K}$. Numbers zero through four are the number of electrons in the dot. White vertical lines identify the locations for data shown in (c) and (d). (b) Same as (a), at $B_{\perp }=1\mathrm{T}$. (c) Differential conductance through the $N=2$ diamond showing step with overshoot at $V_{\mathrm{S}\mathrm{D}}=J(B_{\perp })/e$ at $B_{\perp }=0$ and 1 T. (d) Differential conductance through the $N=3$ diamond showing Kondo peak at $V_{\mathrm{S}\mathrm{D}}=0$ for $B=0$, split by $B_{\perp }=1\mathrm{T}$.

Figure 2

(a) Differential conductance $g$ as a function of $V_{\mathrm{S}\mathrm{D}}$ in the $N=1$ diamond ($V_G=0.1\mathrm{V}$) for in-plane fields $B_X=0$, 0.4, 0.6, 0.8. 1 T, (top to bottom, curves offset). Dashed gray lines are guides to the eye showing the cotunneling gap. (b) $g(V_{\mathrm{S}\mathrm{D}})$ shows a zero-bias peak in the $N=3$ valley ($V_G=0.42\mathrm{V}$) that splits in an in-plane field $B_Y=0$, 0,25, 0.45, 0.7, 0.95 T (top to bottom, curves offset). (c) and (d) splitting energies (see text) versus magnetic field as in (a) and (b) with linear fits. Insets: angular dependence of the $g$ factor in the plane of the 2DEG indicating isotropic behavior. Dashed circles show direction-averaged $g$ factors. Directions $X$ and $Y$ in the plane are arbitrary.

Figure 3

(a) Differential conductance $g$ (log color scale) as a function of $V_{\mathrm{S}\mathrm{D}}$ and $V_G$ for $B=0\mathrm{T}$ in the vicinity of the $N=1\to 2$ transition. (b) $g(V_{\mathrm{S}\mathrm{D}},B_{\perp })$ at $V_G=0.164\mathrm{V}$ [vertical white line in (a)] [33] shows the perpendicular field dependence of the singlet-triplet gap. (c) Cuts showing $g(V_{\mathrm{S}\mathrm{D}})$ at the positions of the vertical lines in (b), marked A, B, C, and D. (d) Horizontal cut [marked as E in (b)] showing $g(B_{\perp })$ at zero bias. Note the asymmetric peak in $g$ at the singlet-triplet transition.

Figure 4

(a) First and second one-electron excited-state energies ${\Delta }_1$ and ${\Delta }_2$, measured from sequential tunneling along with fits to a 2D anisotropic harmonic oscillator model with $\hslash {\omega }_a=1.2\mathrm{m}\mathrm{e}\mathrm{V}$ and $\hslash {\omega }_b=3.3\mathrm{m}\mathrm{e}\mathrm{V}$ (see text). (b)  Singlet-triplet splitting $J$ as a function of magnetic field $B_{\perp }$. (c) Dependence of $J$ on gate voltage $V_G$ for various $B_{\perp }$ as indicated. (d) Average slopes $\Delta J(V_G)/\Delta V_G$ from (c) as a function of magnetic field $B_{\perp }$, showing strong reduction of gate voltage dependence of $J$ at large $B_{\perp }$.

Figure 5

Differential conductance $g$ as a function of $V_{\mathrm{S}\mathrm{D}}$ for temperatures $T_{\mathrm{e}\mathrm{l}}=45$, 140, 170, 240, 280, 330, 380, 420, 570 mK (top to bottom) showing overshoot at $V_{\mathrm{S}\mathrm{D}}\sim J/e$. Inset: peak conductance as a function of temperature with best-fit $log(T)$ dependence (solid line).