Quantum dot admittance probed at microwave frequencies with an on-chip resonator

We present microwave frequency measurements of the dynamic admittance of a quantum dot tunnel-coupled to a two-dimensional electron gas. The measurements are made via a high-quality 6.75 GHz on-chip resonator capacitively coupled to the dot. The resonator frequency is found to shift both down and up close to conductance resonance of the dot corresponding to a change of sign of the reactance of the system from capacitive to inductive. The observations are consistent with a scattering matrix model. The sign of the reactance depends on the detuning of the dot from conductance resonance and on the magnitude of the tunnel rate to the lead with respect to the resonator frequency. Inductive response is observed on a conductance resonance when tunnel coupling and temperature are sufficiently small compared to the resonator frequency.

Figure 1

(a) Optical image of the microwave resonator–double quantum dot sample. Ohmic contacts (M), top gates (C), resonator (R), ground plane (GND), and on-chip inductor (inset, I). (b) Scanning electron micrograph of the double quantum dot structure (LD, RD) defined by the gates $V_{\mathrm{L}}$, $V_{\mathrm{LP}}$, $V_{\mathrm{C}}$, $V_{\mathrm{RP}}$, and $V_{\mathrm{R}}$. RG marks the gate connected to the resonator. The mesa edge is highlighted by a dashed line. The double quantum dot is connected to the 2DEG source (S) and drain (D) contacts. (c),(d) Dependence of the transmission amplitude (c) and phase (d) of the resonator on the frequency ${\nu }_{\mathrm{R}}$ of the applied microwave signal in the Coulomb blockade (1) and on a conductance resonance (2).

Figure 2

(a) Relative resonator transmission amplitude $A/A_{\max }$ at fixed measurement frequency ${\nu }_{\mathrm{M}}$ as a function of $V_{\mathrm{L}}$ and $V_{\mathrm{R}}$. Points labeled (1) and (2) mark gate voltage positions where the full resonance spectra shown in Figs. 1c and 1d were measured. Green roman numerals label four different measurement regions separated by dash-dotted green lines; for details, see the text. (b) Phase change of the microwave resonator for the same gate voltage ranges as in (a). (c) Direct current measurement through the double quantum dot for gate voltage settings $V_{\mathrm{L}}$ and $V_{\mathrm{R}}$ within the dashed region highlighted in (a). (d) Schematic of the charging diagram of a double quantum dot for (N,M) electrons close to the two triple points as a function of $V_{\mathrm{L}}$ and $V_{\mathrm{R}}$. (e) Zoom of the direct current measurement in region II. (f) Magnified view of the transmission amplitude change measurement in the same gate voltage range as in (e). (g),(h) Schematic of the double quantum dot for the direct current measurement (g) and for the microwave measurement (h), respectively.

Figure 3

(a) Relative resonator transmission amplitude $A/A_{\max }$ at fixed measurement frequency as a function of $V_{\mathrm{L}}$ and $V_{\mathrm{R}}$. (b) Relative amplitude change vs. $V_{\mathrm{R}}$ along the dashed line shown in (a). (c) Phase change $\Delta \phi$ for the same gate voltage range as shown in (a). (d) Change of transmission phase for a fixed measurement frequency ${\nu }_{\mathrm{M}}$ along $V_{\mathrm{R}}$ as indicated in (c). (e) Detail of the transmission phase in a selected region of $V_{\mathrm{R}}$ as indicated by a dash-dotted line in (c). Dashed circles refer to gate voltage settings investigated in more detail in Fig. 4. (f) Change of transmission phase $\Delta \phi$ at fixed ${\nu }_{\mathrm{M}}$ as a function of $V_{\mathrm{R}}$ and $V_{\mathrm{L}}$ for a dataset acquired in a similar regime, but with slightly different gate voltages necessary after a charge rearrangement in the sample.

Figure 4

(a) Schematic of the right quantum dot tunnel-coupled to the drain lead (D). The dot is capacitively coupled to the resonator gate (RG) and to a capacitor ($C_{\mathrm{Gs}}$) formed by the capacitive coupling of the center gate ($C_{\mathrm{C}}$) and the right plunger gate ($C_{\mathrm{P}}$). The displayed model circuit is used to calculate the dynamic admittance $g^{\mathrm{QD}}\left(\omega \right)$ of the dot coupled to the lead at finite frequencies. The $V_{\mathrm{x}}$ refer to the voltages applied by external voltage sources, with $\mathrm{x}=\mathrm{dot}$, $\mathrm{Gs}$, and $\mathrm{RG}$. $U_{\mathrm{dot}}$ describes the locally induced potential in the quantum dot. (b) Circuit representation of the schematic in (a); see Eq. (1) in the text. (c) Lumped element model of the microwave resonator with the dynamic admittance of the quantum dot connected in parallel. (d) Measured change of resonator frequency $\Delta {\nu }_0$ for the three resonant dot lead states (1–3) indicated by dashed circles in Fig. 3e. (e) Calculated imaginary part of complex external admittance $g_{\mathrm{dot}}^{\mathrm{ext}}$ of the quantum dot [Eq. (2)] for tunnel couplings ${\gamma }_1/h=20\mathrm{MHz}$, ${\gamma }_2/h=58\mathrm{MHz}$, and ${\gamma }_3/h=125\mathrm{MHz}$. Inset: Imaginary part of the external admittance for tunnel couplings ${\gamma }_4/h=2\mathrm{GHz}$, ${\gamma }_5/h=1.5\mathrm{GHz}$, and ${\gamma }_6/h=1\mathrm{GHz}$ from top to bottom. (f) Results of the model calculations [Eq. (3)] for three different tunnel couplings from top to bottom as in (e).