Cotunneling-Mediated Transport through Excited States in the Coulomb-Blockade Regime

We present finite-bias transport measurements on a few-electron quantum dot. In the Coulomb-blockade regime, strong signatures of inelastic cotunneling occur which can directly be assigned to excited states observed in the nonblockaded regime. In addition, we observe structures related to sequential tunneling through the dot, occurring after it has been excited by an inelastic cotunneling process. We explain our findings using transport calculations within the real-time Green’s function approach, including diagrams up to fourth order in the tunneling matrix elements.

Figure 1

(a) Atomic force microscope micrograph of structure with designations of gates: source (S) and drain (D) of the quantum dot, lateral gates G1 and G2 to control the coupling of the dots to the reservoirs, and plunger gates P1 and P2 to tune the number of electrons on the dot. (b) Finite-bias measurement of the dot’s differential conductance $dI/dV$ at $B=0.1\mathrm{T}$. (c)-(e) Finite-bias transport measurement at $B=0.2\mathrm{T}$, $B=0.3\mathrm{T}$, and $B=0.5\mathrm{T}$, respectively. Note the different vertical distances between the diamond edge and the diagonal structures inside the Coulomb-blockaded region. (f) Single trace of the bias dependence of the current in the blockaded regime, obtained by averaging over the gate voltage range marked in (f) by two vertical dashed lines. (g) Color scale used in (b)-(e): To show all features inside and outside the Coulomb-blockaded region, our data are presented in a highly nonlinear color scale. The plotted quantity is calculated as $\sqrt[10]{|x|}\mathrm{sgn}(x)$ with $x/1\Omega =(dI/dV)$.

Figure 2

(a) Schematic picture of the various tunneling regimes within the Coulomb blockade. $A$ and $B$ label the marked intersection points. Labels (b) and (c) mark the gate voltage values of the corresponding diagrams. E is for elastic and I for inelastic cotunneling, S for sequential tunneling. (b), (c) Relative position of transport channels and contact electrochemical potentials for different bias voltages.

Figure 3

(a) Calculated charging diagram within the Coulomb-blockade regime including the cotunneling contribution. The peak conductances due to resonant sequential tunneling [also responsible for the increasing background in (b)] have been “cut” from the color scale (outer white regions). Solid, dotted, and dashed lines correspond to traces in (b). (b) Cotunneling signatures within the Coulomb-blockade regime. Shown is the differential conductance versus the applied transport voltage for three different gate voltages. The arrow marks the onset of inelastic cotunneling for the dashed trace. The parameters used for this calculation are: $\Gamma =0.1k_{B}T$, $\delta =14.67k_{B}T$, and $U=52k_{B}T$. Bottom: color scale for (a).