Quantum Dot Source-Drain Transport Response at Microwave Frequencies

Quantum dots are frequently used as charge sensitive devices in low temperature experiments to probe electric charge in mesoscopic conductors where the current running through the quantum dot is modulated by the nearby charge environment. Recent experiments have been operating these detectors using reflectometry measurements up to GHz frequencies rather than probing the low frequency current through the dot. In this work, we use an on-chip coplanar waveguide resonator to measure the source-drain transport response of two quantum dots at a frequency of 6 GHz, further increasing the bandwidth limit for charge detection. Similar to the low frequency domain, the response is here predominantly dissipative. For large tunnel coupling, the response is still governed by the low frequency conductance, in line with Landauer-B\"uttiker theory. For smaller couplings, our devices showcase two regimes where the high frequency response deviates from the low frequency limit and Landauer-B\"uttiker theory: When the photon energy exceeds the quantum dot resonance linewidth, degeneracy dependent plateaus emerge. These are reproduced by sequential tunneling calculations. In the other case with large asymmetry in the tunnel couplings, the high frequency response is two orders of magnitude larger than the low frequency conductance G, favoring the high frequency readout.


INTRODUCTION
The ability to detect single electrons in the solid state is useful for a variety of applications, including spin qubit readout [1][2][3][4], electrical current and capacitance standards [5,6], studying cooper pair breaking [7][8][9], single-shot photodetection [10][11][12][13], and nanothermodynamics and fluctuations [14][15][16][17][18][19].While many methods exist to detect charge, one of the main ways are by utilizing quantum dots (QD).These systems make excellent charge detectors due to their high sensitivity and well-established transport theory [20,21], allowing detectors to be made predictable and with a well-understood operation principle.Originally, measurements were performed at DC, relying on a difference in current for the readout resulting in a bandwidth up to some kHz [6,22].In the last two decades, the readout methods have moved towards measuring the reflected power in a high-frequency tank circuit with resonant frequency in the 100 MHz -1 GHz range.This results in bandwidths in the MHz range allowing for µs time resolution [23][24][25].The response of the system in these studies is still governed by the low frequency response of the system, i.e. the admittance Y (ω) is equal to the DC conductance G of the system.In this article, we increase the QD sensor frequency to the 4 -8 GHz frequency range where the cavity photon energy hω is greater than the thermal energy kT [26].This opens up the avenue to increase the bandwidth correspondingly by an order of magnitude, yielding possibly a time resolution sufficient to probe the electron position in DQD systems within the recently achieved coherence times [27,28].The pioneering works have considered the dispersive response of the QD at these frequencies motivated mostly by quantum capacitance effects [29].In this article, we focus on the dissipative part that yields a stronger response, making it use-ful for charge readout [26].We present experimental results for two devices and show that for both of them at sufficiently large tunnel couplings that we are lifetime broadened, Γ > kT , the low frequency result of Y (ω) = G still applies.However, when the device is tuned to the thermally broadened limit where the tunnel couplings Γ < kT , the measured admittance is qualitatively different from the DC conductance, displaying a linewidth of 2hω in the QD level tuning and a factor two difference in admittance depending on the direction of the level shift of the quantum dot relative to the leads ε, attributed to spin degeneracy.These results are well captured by sequential tunneling theory, directly evaluating the admittance for a QD subjected to a time-periodic drive [30], or using P(E) theory in which the admittance is inferred from the absorption in the cavity [31,32].Lastly, we show in the other device which exhibits asymmetric tunnel couplings where the DC transport is suppressed while remaining lifetime broadened, the AC response in this device remains large, in line with Ref. 26, indicating a potentially useful consequence of probing QD devices at high frequencies.This response falls in a regime where neither non-interacting scattering theory nor sequential tunneling models are applicable.

DEVICE CONFIGURATION
The main device used to perform measurements is illustrated schematically in Fig 1 a).The device builds on a transmission line resonator, shown in orange, which for the fundamental mode is equivalent to the LC circuit of panel b).The right end of the resonator is connected to an input line via a coupling capacitor C C , which allows the measurement of the amplitude and phase of a reflected signal.The left end of the resonator on the other hand couples to a QD via the right junction capacitance.This configuration makes the QD source-drain transport admittance Y (ω) appear directly on the LC resonator.At low drive frequency ω = 2π f , this admittance is given just by the DC conductance G, i.e.Y (ω) = G, and the QD gives rise to dissipation in the resonator.The reflection coefficient of the input port is given by (see Appendix A) where ω r = 1/ √ LC is the resonance frequency, κ = κ QD + κ i + κ C the sum of all the couplings defining the linewidth of the resonance, κ i the internal losses, κ C = Z 0 ω 2 C 2 C /C the input coupling [33] and κ QD = Re(Y (ω))/C the QD coupling strength.The term κ QD is directly proportional to the admittance Re(Y (ω)), determining κ QD from a change in the measured reflection coefficient R thus allows us to determine the dissipative part of the QD response Re(Y (ω)).On the other hand, Im(Y (ω)) gives rise to a disperse shift δ ω QD = Im(Y (ω))/2C of Eq. ( 1), which results in a change in the resonance frequency.These dispersive shifts are typi-cally small, of the order of 10 −3 ω r as is also the case for our devices, and have been studied in detail for QDs coupled capacitively via a gate electrode [29].
To measure the DC conductance G of the QD at the same operation point as Y (ω), we apply a DC bias voltage V SD to the voltage node point in the middle of the λ /2 resonator such that it does not disturb the resonance, but appears at the source contact of the QD [12,29].The current I SD , measured from the drain contact, yields then the conductance G = dI SD /dV SD and enables the comparison of this low frequency transport result to the high frequency admittance Y (ω).These DC lines, in addition to a gate line with applied gate voltage V G to change the electron number in the QD, are shunted with a large capacitor to ground to prevent microwaves leaking out from the lines.
The physical realization of the device is presented in Fig. 1  c).The coplanar waveguide, highlighted in orange, is a 9.86 mm long metallic strip of width 10 µm with a gap of 5 µm to the ground plane.Based on Ref. 33, we estimate the lumpedelement capacitance C = 765 fF and inductance L = 871 pH, giving a characteristic impedance Z 0 = π/2 L/C = 53 Ω [33].An RF port connects to the resonator with a 400 µm long two-finger geometry which defines the input coupling κ C .The QD forms in an epitaxially grown InAs nanowire, see Fig 1 d), by altering the growth between zincblende (ZB) and wurzite (WZ) crystal phase [34].The WZ segments have a conduction band offset of 135 meV compared to the ZB segments [35], forming tunnel barriers and a ZB segment between the barriers defines the QD with length 130 nm and diameter of 80 nm [36].The location of these barriers is discerned by selectively growing GaSb on the ZB segments which highlights the features of the QD [37].The offset between ZB and WZ allow the atomically sharp definition of barriers leading to a well-defined QD.The DC lines are capacitively shunted by growing a 30 nm thick aluminium oxide layer with atomic layer deposition and evaporating 50 nm thick aluminium film to the light-gray area in Fig 1 c).Additional inductive filtering is added to all the DC pads as well as the midpoint connections to reduce RF leakage [38].The device is bonded to a printed circuit board and measured in a dilution refrigerator at the electronic temperature of T = 50 mK at base temperature.Figure 1 e) shows the measured Coulomb diamonds exhibiting a charging energy E C = 3.5 meV and excited states with energies around 300 µeV.The lever arm to the gate, α = 0.03 eV/V, is also determined.
Figure 1 f) presents the resonator response with the QD in Coulomb blockade (CB) and in the conduction resonance at V G = 5.9 V which is attained with an input power P = -130 dBm to the resonator.With the QD transport suppressed in CB (open symbols), we determine the bare resonator properties by fitting these data to Eq. ( 1) with κ QD = 0 and δ ω QD = 0. We obtain the resonance frequency f r = 6.318GHz, as well as determine the coupling strengths κ C /2π = 22.3 MHz, and κ i /2π = 0.7 MHz.The phase response (cyan rings) shows a 2π winding, characteristic for an overcoupled resonator.Next, the QD is tuned to resonance and the measurements are repeated.The corresponding data (solid markers) demonstrate that the linewidth of the resonance increases due to additional absorption in the QD.The reduction in amplitude also reflects the increase of total dissipation in the resonator by κ QD .Keeping the resonator parameters acquired from the previous data set fixed, the fit to Eq. ( 1) is now repeated to provide κ QD /2π = 4.7 MHz and vanishing δ ω QD .

COMPARISON OF CONDUCTANCE AND HIGH FREQUENCY ADMITTANCE
Now we turn to comparing the low frequency conductance G and the high-frequency response Y (ω) presented in Figs. 2  a) and b) for a lifetime-broadened resonance at V G = 6.7 V.The QD conductance G has a peak width of 60 µeV > 4 kT, hence, a fit (solid line) to Landauer Büttiker theory [21] yields the tunnel couplings Γ L = 6 µeV and Γ R = 55 µeV.In Fig. 2 b) the measured admittance Re(Y (ω)) = κ QD C is shown for the same resonance with an input power P = -120 dBm.This admittance response is identical to the DC conductance G within 30 %, as expected from the low-frequency prediction of Y (ω) ≈ G.A numerical calculation based on Landauer-Büttiker theory (orange line) for this system, see appendix B and Ref. [39], also predicts the equivalence Re(Y (ω)) = G for this configuration.
The energy of a single microwave photon is hω r = 26 µeV, hence in the configuration of Fig. 2 a)-b) the lifetime broadening exceeds the photon energy.By reducing the gate voltage, the tunnel coupling reduces, decreasing the lifetime broadening.This allows us to make the linewidth of the DC conductance peak smaller than the photon energy.Tuning from V G = 6.78V to V G = 4.72 V results in a thermally broadened peak, shown in Fig. 2 c).The Landauer Büttiker theory fit again reproduces the results with Γ L = 0.25 µeV.As the linewidth is now set by temperature and not the tunnel couplings, the larger coupling may vary from Γ R = 0.5 to 6 µeV without disrupting the fit to the data.Now the measured Y (ω), presented in Fig. 2 d), shows a broader peak with a qualitatively different peak shape than the conductance has, hence, the equivalence Y (ω) = G is broken.The response has two plateaus at 2.3 and 4.6 µS, extending out by 30 µeV to either direction from the midpoint, and matching the photon energy in line with the DC current response studied in Ref. 40.The broadening of Y (ω) arises since with the energy of the photon, the system overcomes an additional charging energy cost up to hω r as depicted in Figs. 2 e) and f).The factor of two difference arises from the spin degeneracy in the QD.The photon absorption rate is twice for tunneling out of the QD (applies for ε > 0) as compared to tunneling into the QD (applies for ε < 0).A sequential tunneling model with either time dependent voltage drive [30] or P(E) theory [31,32] describes the full response.The two theories agree at low resonator -QD couplings, but as the coupling increases (e.g. by increasing cavity impedance), the P(E) theory predicts spontaneous emission events which further change the transport.See Appendices C and D for details.Here we have fitted the value of the larger tunneling rate to Γ R = 1.65 µeV which sets the overall height of the response.The spin degeneracy shifts the resonance point and the total height of the resonance peak by a small amount, see Eq. A.43, thus the DC fitting parameter values were adjusted to Γ L = 0.25 µeV and α∆V G0 = −1.5 µeV.
-300 0 300 0 0.5  3 c).As the voltage amplitude V MW enters the high power regime, the broadening of the RF and DC response both arise from the amplitude of the microwave signal.For the microwave response, the boundary point P ≈ −100 dBm between the high and low power regime is set by the condition eV MW = hω r , i.e. whether the energy related to the amplitude or single photon is dominant.However, the width of the DC feature continues to be defined by the drive amplitude until the power P = −120 dBm, at which point the energy corresponding to the drive amplitude becomes smaller than the thermal energy i.e. eV MW < kT .The measurements of Y (ω) performed in the measurements of Fig. 2 were performed at P ≤ −120 dBm, allowing highpower effects to be ignored in the analysis.Note also that the DC measurements of Fig. 2 were performed without applied microwave drive, though applying the drive does not change the DC response at this power level.Figures 4 a) and b) repeat the study in the lifetime broadened case for a second device.Now we have a much more asymmetric device with fitted values of Γ L = 1.4 µeV and Γ R = 357.5 µeV catching again the equivalence Y (ω) = G, valid in both experiment and theory.Tuning the QD to a lower gate voltage V G 0 = 6.73 V, Figs. 4 c) and d), has again the effect of reducing the tunnel barriers such that the conductance G is suppressed by two orders of mangitude.This results in a correspondingly smaller Γ L = 6.7 neV while the right barrier Γ R = 80 µeV still provides a lifetime broadening to the system.In the measured high-frequency response, we observe a peak with the same linewidth as the DC feature but with an amplitude value of Y (ω)| ε=0 = 3.6 µS, which is two orders of magnitude greater than the peak value of the DC conductance of 30 nS.In this case, Landauer-Büttiker theory still predicts Y (ω) = G.Therefore the linewidth of Y (ω) of fig 4 d) is re-produced correctly but the predicted overall magnitude is two orders of magnitude lower than the measured response.The sequential tunneling calculations miss lifetime broadening effects, thus not replicating the linewidth.The predicted peak admittance of Y (ω) = 30 µS on the other hand predicts qualitatively correct that the RF response is stronger, though the predicted value is an order of magnitude larger than the measured.With these arguments and findings, we interpret that the correct picture to describe is closer to the sequential tunneling case where the microwave drive is divided between the junction capacitances and then a considerable fraction of the drive arises across the transparent junctions and leads to large dissipation as observed before for lower frequencies [45].The Landauer-Büttiker theory differs from this as the total admittance of the system determine the voltage division, which in our case would lead to the same dissipation as at DC.To describe the response quantitatively, a more advance theory combining the above aspects would be needed [46].

CONCLUSIONS
In summary, we studied the high frequency source-drain response of a quantum dot.We showed experimentally that the low frequency result of Y (ω) = G holds for quantum dots tuned to sufficiently large tunnel couplings in line with the slow-drive limit.However, when the tunnel couplings are tuned to be smaller than the photon energy, the measured linewidth of the admittance Y (ω) is set by the photon energy.This response is well-described by sequential tunneling theory.Additionally, the low-frequency limit does not hold when the drive amplitude is made sufficiently large or with large asymmetry in tunnel couplings of the junctions.For the highly asymmetric case, it is also shown that the admittance Y (ω) can be orders of magnitude larger than the conductance G, indicating a potential benefit of measuring at high frequencies, as the readout strength remains large even for weakly conducting dots.
For the reflection probability, inserting Z R and Z I into Eq.(A.3), writing the capacitive couping rate κ C = Z 0 ω 2 r C 2 C /C and neglecting terms proportional to the small parameter C C ω r Z 0 1, we arrive at where we introduced the total κ = κ QD + κ i + κ C and ω * r = ω r (1 + C C /C), the capacitive coupling renormalized resonance frequency.By noticing that C C /C 1 we can put ω * r ≈ ω r and we then arrive at Eq. ( 1) in the main text.this appendix, we calculate the QD admittance within the Landauer-Büttiker formalism.With this approach, the QD admittance Y (ω) is evaluated within a time-dependent scattering approach, neglecting Coulomb blockade effects but fully accounting for the current conservation at the QD via the flow of dynamic screening currents.Our result is an extension of the discussion presented by Pretre, Thomas and Büttiker, [39], here including QD-lead capacitive couplings.We therefore present only the main steps in the derivation.The staring point for the calculation is the energy dependent, symmetric scattering matrix S(E) of the QD, assuming effectively a single transport channel, given by where the reflection and transmission amplitudes are given by the Breit-Wigner expressions Here ε = ε d − αV G is the energy of the discrete QD level where ε d is the bare dot energy and α = eC G /(C L +C R +C G ) the lever arm for the gate potential V G .Unprimed (primed) amplitudes correspond to particles incident from the left (right) lead.We consider the case with a pure AC-voltage V (t) = V cos(ωt) at contact R, while contact L is grounded and the gate contact is kept at the constant potential V G corresponding to the experimental settings.The case with a pure DC-voltage bias is discussed below.As a result of the oscillating potential V (t), a potential U(t) is induced on the QD.The effect of the oscillating potentials is that electrons can pick up or loose quanta of energy hω when scattering at the QD.
Our focus is on the regime of weak microwave drive, where the response is linear in the potentials.In this regime V hω and only a single quantum can be picked up or lost.As a consequence the time dependent particle current at lead L/R has only a single Fourier component, The current component I L/R (ω) can be expressed in terms of the scattering amplitudes in Eq. (A.7) and the lead Fermi distribution f (E) as and we have introduced the Fourier components V (ω) and U(ω) of the potentials V (t) and U(t).We note that V (ω) = V /2, independent on ω, but the frequency dependent notation is kept for convenience.
Inserting the scattering amplitude expressions in Eq. (A.7) into the current components in Eqs.(A.9) and (A.10) we can write and For non-zero frequencies, the particle currents flowing into the QD typically do not add up to zero, i.e.I L (ω) + I R (ω) = 0.As a consequence, there is nonzero charge Q(t) on the QD dot which induces AC screening, or displacement, currents flowing between the QD and the leads L/R as well as the gate G.The total screening current into the QD is given by I sc (t) = dQ(t)/dt, with the charge determined from classical electrostatical considerations, via the potentials V (t) and U(t) and the capacitances C L , C R , C G .This gives the screening current Fourier component I sc (ω) as (A.16) The induced QD potential U(ω) can then be determined from the condition that the total current flowing into the dot is conserved, I L (ω) + I R (ω) + I sc (ω) = 0, (A.17) giving

FIG. 1 .
FIG. 1.(a) A schematic diagram of the studied device.A microwave resonator (orange) is driven with a high-frequency signal (RF) through the coupling capacitor C C and the reflected amplitude A and phase φ is measured.A QD with tunnel couplings Γ R and Γ L , connects to the resonator via the source contact.The DC electrical current I SD is measured from the drain contact and DC voltage bias V SD is applied via the resonator and a gate voltage V G via a separate gate electrode.(b) The equivalent lumped-element LC circuit for the device with the complex admittance Y (ω) arising from the QD.(c) An optical micrograph of the device.The microwave resonator and DC lines are defined using a Nb etch-back method.The DC lines are capacitively shunted towards the resonator with a 30 nm aluminium oxide -50 nm aluminium stack (white area).The contacts nearby the QD, visible in the scanning electron micrograph of the inset, are defined using EBL and deposited using Ni/Au evaporation.(d) A zoom-in of panel c showing the InAs nanowire in which the QD is defined.(e) The measured detector current I SD as a function of bias and gate voltages V SD and V G .(f) Measured reflection coefficient R and phase φ as a function of frequency f with the QD in Coulomb blockade (open circles) and conducting at zero bias voltage (dots) at V G = 5.9 V. Solid lines are fits to Eq. (1) with f r = 6.315GHz, κ C /2π = 22.3 MHz and κ QD = 0 for Coulomb blockade and κ QD /2π = 4.7 MHz for the QD in resonance.

FIG. 2 .
FIG. 2. (a)The DC conductance G measured at zero bias (V SD = 0) as a function of the level shift ε = −α(V G − V G0 ) for the QD resonance at V G0 =6.78 V without applying the RF drive.A small gate voltage V G,2 = 0.5 V is applied to the two remaining gate contacts in order to tune the tunnel couplings of the single dot slightly.The line shows a fit to Landauer-Büttiker (LB) theory, Eq. (A.22).(b) The admittance data Re(Y (ω)) around the same resonance as in a).The solid line is the finite-frequency Landauer Büttiker theory of Eq. (A.20).(c, d) Data for another resonance at V G0 = 4.72 V.The fit in panel c) is done using Eq.(A.34), and in panel d) the dashed line is a Sequential Tunneling (ST) calculation of Eq. (A.35) based on the formalism in Ref. 30.(e, f) Band diagrams with the two energy level ε settings corresponding to the two plateaus of panel d).The orange arrows indicate tunneling processes involving photon absorption, while the grey arrows indicate the tunneling events returning the system back to the lowest energy state shown with the blue arrows.The number of orange/grey arrows specify the number of electrons which can participate in the corresponding tunneling process between the N = 1 and N = 2 electrons on the dot.

FIG. 3 .
FIG. 3. The DC conductance (a) and RF amplitude (b) are measured simultaneously as the microwave power to the resonator input is changed.The measured linewidths are plotted in (c) along with a dashed black line indicating 2hω r and a solid black line corresponding to the calculated microwave amplitude of Eq. (2).

Figure 3
extends the measurements of Figs. 2 c) and d) as a function of drive power P. At low drives P < −100 dBm, the linewidth of Y (ω) is essentially set by 2hω r while the linewidth of G remains thermally broadened for P < −120 dBm.At high power additional broadening is observed in both the DC and RF result, in line with the previous works by Refs.41, 42.The amplitude of the microwave oscillations inside the resonator is estimated by following the steps of Refs.43, 44 yielding the microwave amplitude FIG. 4. The same measurements as of Fig. 2 for a second device at V G0 = 9.4 V in (a) and (b), and V G0 = 6.4 V in (c) and (d).For this second QD, the charging energy and lever arm are E C = 2 meV, and α = 0.04 eV/V.
FIG. 5. Sketch of the quantum dot showing, in addition to Fig. 1 of the main article, the tunnel junction capacitances C L ,C R and the applied potential V (t) on the right contact and induced potential U(t) in the QD.