Finite-bias charge detection in a quantum dot

We present measurements on a quantum dot coupled capacitively to a quantum point contact used as a charge detector. The transport current through the dot and the time-averaged charge on the dot are measured simultaneously. At finite bias voltage through the dot, the differential charge signal coincides with some of the Coulomb diamond boundaries and with the signatures from excited states in the dot current. Combining the resulting integrated charge data with the simultaneous measurements of the dot current allows us to estimate the coupling of different energy levels inside the dot to both leads individually.

Figure 1

(a) Atomic force microscopy (AFM) micrograph of the structure with designations of gates: source $\left(S\right)$ and drain $\left(D\right)$ of the QD and the QPC used as a charge detector; lateral gates G1 and G2 to control the coupling of the dots to the reservoirs; plunger $\left(P\right)$ gate to tune the QPC detector. Only the part of the circuit related to the readout functionality is included. $V_{\mathrm{qpc}}$ consists of a dc and a small ac component $(V_{\mathrm{qpc},\mathrm{ac}}⩽5\mu \mathrm{V})$. (b) Example measurement of the current through the dot. For these low bias measurements, the voltage across the dot was $V_{\text{bias},\text{dot}}=10\mu \mathrm{V}$. (c) Simultaneous measurement of the conductance through the QPC, where each step corresponds to a change of the dot’s charge by one electron. (d) Simultaneous measurement of the transconductance $d{I}_{\mathrm{qpc}}∕d{V}_{\mathrm{qpc}}$.

Figure 2

(Color online) (a) Finite bias transport measurement through the dot at a magnetic field of $B=0.1\mathrm{T}$. Dark regions represent larger positive (blue) or negative (red) differential conductance. (b) Corresponding measurement of the transconductance. Dark regions represent more negative values. (c) Single transconductance trace for low bias [$V_{\text{bias}}=20\mu \mathrm{V}$, see the upper horizontal line in (b)]. (d) Single transconductance trace for $V_{\text{bias}}=-0.2\mathrm{mV}$ [see the dashed horizontal line in (b)]. (e) Averaged transconductance. Averaging was performed over a set of bias voltages $-1.6\mathrm{mV}<V_{\text{bias}}<-0.6\mathrm{mV}$. Individual traces were laterally shifted with respect to each other so that the two lines visible in the lower left part of (b) [parallel to the diamond edges in (a)] yield two distinct sharp peaks. The region delimited by dotted lines in (b) marks the range over which averaging took place. As a consequence, the step corresponding to the second Coulomb peak is broadened, because for this peak the slope of the corresponding lines in (b) is different. A constant and a linear background were subtracted. (f) Averaged transconductance integrated with respect to $V_{\mathrm{G}2}$.

Figure 3

(a) Detail from Fig. 2b, showing a broadened line with negative slope at zero bias and a splitting at finite bias. (b) Illustration of the connection between the coupling of a single state $\left(G\right)$ to source $\left(S\right)$ and drain $\left(D\right)$ and the mean charge on the dot. This representation was generated using a simulation based on a rate equation approach (Ref. 9), and is compatible with our experimental observations (with the exception of line broadening due to strong coupling, which is not included in the theory). Our model for this case includes two single-particle levels: an asymmetrically coupled state accounting for the imbalance of line weight observed at positive and low negative bias (where the line with positive slope, corresponding to alignment of $G$ with $D$, is practically absent), and a higher energy level H with more symmetric coupling, responsible for the branching at finite negative bias. We used coupling asymmetries ${\Gamma }_{\mathrm{D},\mathrm{G}}∕{\Gamma }_{\mathrm{S},\mathrm{G}}=1∕50$ for the level G, and ${\Gamma }_{\mathrm{D},\mathrm{H}}∕{\Gamma }_{\mathrm{S},\mathrm{H}}=1∕4$ for the level H. In additon, we set ${\Gamma }_{\mathrm{D},\mathrm{G}}+{\Gamma }_{\mathrm{S},\mathrm{G}}={\Gamma }_{\mathrm{D},\mathrm{H}}+{\Gamma }_{\mathrm{S},\mathrm{H}}$. The diagonal line $\left(G\text{−}S\right)$ with negative slope corresponds to the alignment of a single energy level (G) with the source (S) contact, and the line (H-D) with positive slope in the negative bias range marks an alignment of an excited state (H) with the drain (D) contact. Letters in parentheses [(c)–(f) located in the blockaded (left and right) and nonblockaded (top and bottom) regions] refer to the corresponding figures. These illustrate how the time-averaged electron number depends on the bias and the chemical potential of the dot for the case of an asymmetric (here $P^{+}>1∕2$) coupling.

Figure 4

(a) Detail from 2b, showing discontinuity (splitting) of lines in the transconductance signal at zero bias. (b) Schematic representation of (a) generated by a numerical simulations. To explain the observed line splitting, a simplified model spectrum with four states is used, consisting of two pairs of (quasi)degenerate levels. Small letters in parentheses refer to (c)–(e), where the spectrum is depicted for different values of gate and bias voltage, illustrating how the interplay of several levels with different couplings to the leads influences the time-averaged occupation of the dot. The labeled lines in (b) refer to the alignment of a level [$L,G$, or $H∕H^{\prime }$, see (c)] with either source $\left(S\right)$ or drain $\left(D\right)$. (c) For low positive bias, charge passes through the degenerate levels $G$ and $L$. Due to the asymmetric coupling, the mean charge remains low until alignment with the drain contact. (d) For small negative bias, the asymmetry in the coupling of levels $L$ and $G$ leads to a finite mean charge on the dot as soon as they align with the drain’s chemical potential [see line $G\text{−}D$ in (b)]. (e) In this parameter range, the mean charge reaches a high level, due to the asymmetric coupling of the levels $H$ and $H^{\prime }$ that trap charge entering through the source contact.