Mapping the chemical potential landscape of a triple quantum dot

We investigate the nonequilibrium charge dynamics of a triple quantum dot and demonstrate how electron transport through these systems can give rise to nontrivial tunneling paths. Using a real-time charge sensing method, we establish tunneling pathways taken by particular electrons under well-defined electrostatic configurations. We show how these measurements map to the chemical potentials for different charge states across the system. We use a modified Hubbard Hamiltonian to describes the system dynamics and show is reproduces all experimental observations.

Figure 1

A triple quantum dot in Si:P with an adjacent single-electron-transistor used as a charge sensor. (a) A scanning tunneling micrograph of three small Si:P QDs incoherently tunnel coupled to a larger QD which itself is coupled to source $\left(S\right)$ and drain $\left(D\right)$ leads and acts as a sensing single electron transistor (SET). A gate $G_{\text{SET}}$ is used to control the SET while gates $\{G_1,G_2,G_3\}$ are used to control the QDs $\{1,2,3\}$. Inset shows a close up image of the three QDs. From the lithographic area it is estimated that they contain $\sim 5$ donors each. The distances between the QDs as well as their individual distances to the SET are shown in nanometers. (b) The conductance of the SET as a function of the gates $\{G_{\text{1}},G_{\text{3}}$} for $G_{\text{2}}=0.9$ V and $G_{\text{SET}}=0.5$ V. Breaks in the Coulomb blockade peaks (lines running at $-{45}^{\circ }$) of the SET show where a charge transition from one of the QDs to the SET occurs. The inset provides a guide for where the charge transitions occur. The equivalent charge numbers on the QDs are given by $(n_1,n_2,n_3)$. (c) A schematic representation of the modified Hubbard model showing the energies (not to scale) of different charge configurations at a detuning given by the star, also shown in (b). The incoherent tunnel rate ${\mathrm{\Gamma }}_{s,s^{\prime }}$ couples two charge configurations $s$ and $s^{\prime }$.

Figure 2

Comparison of theoretical and experimental tunnel rate maps. (a) The theoretically predicted tunnel rate map for the TQD device scaled by the highest dot-reservoir tunnel rate, ${\gamma }_2$. Solid magenta lines show the charge transitions visible in standard charge stability maps [see Fig. 1], while the dashed and dotted black lines correspond to transitions that can only be obtained from the tunnel rate map. The inset shows a zoom-in around the (0,1,0) charge region. (b) The points (i), (ii), (iii), and (iv) corresponding to those shown in (a) are examined in terms of the chemical potentials of the QDs and the SET Fermi level (shown by the grey shaded region on the right of each panel). The levels shown in red and green correspond to the chemical potentials of those charge states at the points shown by the red and green circles in (a). The dashed lines in (i) and (ii) represent the SET Fermi level and corresponds to the dashed equivalent line shown in (a). Likewise, the dotted lines in (iii) and (iv) represent the relevant chemical potentials of charge states also shown in (a). (c) The experimental procedure used to produce a tunnel rate map involves a two-level pulse from the (1,1,1) charge region (red square) to the detuning position $\mathrm{\Lambda }$ (green square). The position marked by the red square is stepped across the $V_{G1}-0.6V_{G3}$ direction, while the final detuning position shown by the green grid is stepped across the charge transitions (magenta lines). (d) The SET current is monitored in time to detect any electron tunneling during the two level pulse. All data (blue circles) are comprised of 200 repetitions of the two level pulse (black line). A measure of the tunnel rate $\mathrm{\Gamma }\left(\mathrm{\Lambda }\right)$ is obtained from an exponential fit (green line) during the first phase of the two level pulse. The second phase of the pulse indicates the reloading of the electrons into the (1,1,1) charge region. (e) A tunnel rate is obtained for different final detuning positions [green marker in (c)] $\mathrm{\Lambda }$. There is a good qualitative agreement between the theoretical tunnel rate map shown in (a), which predicts all of the features obtained from the measurement. The areas enclosed by the white lines indicate the regions in gate space where an electron initially on $\text{QD-}1$ can only tunnel directly to the SET. The dashed white line indicates the cut used to calculate $\beta$ for Eq. (3).

Figure 3

Time-resolved charge stability maps for a triple quantum dot. Maps showing the conductance of the SET charge sensor at different times from $t=0.32$ ms to $t=16.9$ ms. The solid magenta lines show those transitions that are apparent from a standard stability map, whereas the dashed lines are only visible using time-resolved charge sensing. (a) The conductance through the SET charge sensor at 0.32 ms before any electrons have tunnelled off the QDs. There are no SET breaks corresponding to QD transitions. (b) At 2.43 ms, both the electrons from $\text{QD-}2$ and 3 have tunnelled to the SET creating breaks in the Coulomb blockade where ${\mu }_2$ and ${\mu }_3$ are equal to ${\mu }_{\text{RES}}$. Point (i) shows where ${\mu }_2\text{(1,1,1)}={\mu }_{\text{RES}}$; whereas point (ii) indicates the ${\mu }_1\text{(1,0,0)}={\mu }_2\text{(0,1,0)}$ condition. (c) At 7.49 ms, the electron on $\text{QD-}1$ begins to tunnel to the SET, thus two transitions in the upper right corner of the map are visible at points (iii) and (iv). At point (iii), the electron is tunneling straight from $\text{QD-}1$ to the SET. The electron on $\text{QD-}1$ is also able to tunnel from the (1,0,0) charge state to the (0,0,0) on the bottom left of the map at point (iv). (d) Finally, after a time 16.9 ms the map is equivalent to the standard charge stability map from Fig. 1, as it shows all three of the QD transitions at equilibrium.