Thermal Gating of Charge Currents with Coulomb Coupled Quantum Dots

We have observed thermal gating, i.e. electrostatic gating induced by hot electrons. The effect occurs in a device consisting of two capacitively coupled quantum dots. The double dot system is coupled to a hot electron reservoir on one side (QD1), whilst the conductance of the second dot (QD2) is monitored. When a bias across QD2 is applied we observe a current which is strongly dependent on the temperature of the heat reservoir. This current can be either enhanced or suppressed, depending on the relative energetic alignment of the QD levels. Thus, the system can be used to control a charge current by hot electrons.

and QD2, respectively [15,17]. Reservoir H defined by gates 1, 2, 3 and 4 forms a channel of 20 µm length and 2 µm width. Opposite QD1, a constriction is created by gates 1 and 2 which can be used as a voltage probe in the channel. Its conductance is set to G = 10 e 2 /h by adjusting the gate voltages V 1 and V 2 thus ensuring that no thermovoltage is created across this junction when the temperature in reservoir H is increased [18].
The sample is mounted in a top-loading dilution refrigerator with base temperature T base = 80 mK. For a conductance characterization of QD2 with all reservoirs at T base , an excitation voltage V ac = 5 µV (f = 113 Hz) is applied between reservoirs S and D. The current is measured with a lock-in, using a current amplifier that connects D to a virtual ground potential. By varying V P1 and V P2 one obtains the so-called stability diagram of the QD-system [17], shown in Fig. 1 (b) where the conductance G of QD2 is displayed in a gray scale as a function of the voltages V P1 and V P2 .
Along the horizontal axis V P2 , we observe two conductance resonances which identify those gate voltage configurations for which µ (2) is aligned with µ S and µ D . They are separated by the Coulomb charging energy of QD2. Due to the mutual capacitive coupling the energetic position of µ (2) is affected by the energy of QD1. This leads to a continuous shift of the conductance resonances for larger V P1 towards smaller V P2 [dashed, red lines in Fig. 1 When µ (1) aligns with µ H , N changes by one [solid, red lines in Fig. 1 (b)]. This causes discrete jumps for the conductance resonances of QD2, indicated by red arrows in Fig. 1 (b).
These jumps are a result of the capacitive inter dot coupling which leads to the transfer of the energy E C : µ (2) [15,17]. Hence, the charge occupation numbers of both QD1 and QD2 are stable only in the regions enclosed by solid and dashed lines in Fig. 1 (b). The energy E C can be calculated from the displacement of the conductance resonance along the V P2 -direction, ∆V C , indicated by yellow dotted lines in the figure. Using the gate efficiency α 2 = 0.032 obtained from dI/dV characterization of QD2 yields E C ≈ 90 µV.
In order to subject the QD-system to a temperature difference, we make use of a current heating technique [18]: An ac-current I h = 150 nA with frequency f = 113 Hz is applied to the heating channel (reservoir H). Because of the strongly reduced electron-lattice interaction in GaAs/AlGaAs 2DEGs at low temperature, the energy is dissipated into the lattice only in the wide contact reservoirs. On a length scale of a few µm, however, electronelectron scattering dominates electron-phonon scattering, resulting in a thermalized hot Fermi distribution of the electrons in the channel only. Based on QPC-thermometry [18] we estimate that for a current of I h = 150 nA, T H increases by ∆T ≈ 100 mK. The ac-heating causes the temperature in the heat reservoir to oscillate at 2f = 226 Hz between T base and T max = T base +∆T . This ensures that all temperature-driven effects also oscillate at frequency 2f , enabling straight forward lock-in detection. Next, a dc-voltage source is connected to S which applies V S,GND while the current amplifier (input impedance R imp = 2 kΩ) connecting D to ground potential is read out by a lock-in amplifier detecting at 2f = 226 Hz. This allows us to determine the change of the current in the drain contact ∆I D due to variation of T H .
With V S,GND ≈ −30 µV we obtain the data shown in Fig. 1  . In a next step the dc-voltage applied to S is reversed, so that V S,GND = 100 µV. The result is given in Fig. 2 (b). Clearly, the clover leaf pattern is reproduced, however, with all signs inverted.
We now discuss qualitatively how we can understand this behavior. As is evident from  Fig. 2 (a) and (c)]. Furthermore, the 2f -detection of the signal indicates that these current signals are triggered by a temperature change in reservoir H. In the vertex region, the occupation numbers of both QDs can fluctuate while the occupation number becomes fixed when moving away from this region. It is thus apparent that the current changes which give rise to the clover-leaf structure originate from fluctuating occupation numbers of QD1 and QD2. As an example, Fig. 3 (a) shows the alignment of µ (1) and µ (2) with V SD < 0 for section 1 with N +1 electrons on QD1 and M electrons on QD2. Due to the ac-character of the heating current T H oscillates between the two values T H = T base and T H = T max > T base .
The first case is shown on the left side of Fig. 3 (a): QD1 is occupied with N +1 electrons, i.e. µ (1) is below µ H and therefore the electron number of QD1 is fixed at N +1. QD2 is occupied with M electrons and the chemical potential µ (2) (N +1, M +1) which is required to add the (M +1) th electron lies outside the bias window V SD . Thus, transport across QD2 is blocked. When the temperature in reservoir H is increased such that T H = T max [right hand side in Fig. 3 (a)], empty states are created below µ H in this reservoir. This increases the charge fluctuation rate on QD1. However, when due to these fluctuations QD1 relaxes to the N -state, the energy required to add an electron to QD2 is reduced by E C . The corresponding µ (2) (N, M +1) is below µ S and the current across QD2 increases [indicated by red arrows in Fig. 3 (a)]. Since T H oscillates between T base and T max , this effectively leads to a temperature driven modulation of the conductance of QD2: If T H increases, the current across QD2 increases as well. For T H at a minimum, transport is blocked. The resulting current modulation at the drain contact is then detected by the lock-in amplifier as a positive signal.

(b).
Starting again with the condition T H = T base so that N +1 is fixed, we find that transport across QD2 is enabled because µ (2) (N +1, M +1) is situated within V SD . However, charge fluctuations on QD1, which increase with increasing T H [right side in Fig. 3 (b)], tend to block transport across QD2: The corresponding µ (2) Fig. 2 (b)] is also straightforward: Because a sign change of the bias voltage reverses the dc-current through QD2 this leads to an overall reversal of the observed signal.
We have performed simple model calculations to substantiate the qualitative discussion presented above. Assuming sequential transport across QD2, the current I D can be related to the applied difference in electro-chemical potential V SD = µ S −µ D by considering Fermi-Dirac occupation statistics f (µ (2) , T j ) = 1/(1 + exp(µ (2) − µ j /k B T j )), j = S, D in the source and the drain contact and a single resonant QD level µ (2) which is located at µ (2) ). The current I D across QD2 when QD1 hosts N +1 electrons can be treated likewise, with µ (2) = +E C /2. The total current I D through QD2 is now the sum of I D and I D , weighted with the appropriate probabilities of QD1 hosting N or N +1 electrons. Thus, µV and E C = 90 µV. As expected, the results strongly resemble the conductance stability diagram in the vertex region. However, major differences for different T H are not directly obvious. In order to model our experiment we subtract the calculated data sets in Fig. 4 (a) from each other such that we obtain ∆I D = I D (T max ) − I D (T base ), which corresponds to the change in current through QD2 due to a change of T H by ∆T . The result is given in Fig. 4 (b). Evidently, the clover-leaf pattern found in the experiments, is reproduced nicely.
We point out that a similar four-leafed clover pattern has been observed in connection with Coulomb coupled double QDs previously: McClure et al. [16] have reported on experiments addressing the cross-correlation of shot noise in such a system. There, the authors observed regions of positive and negative correlation at the stability region vertex arranged in a cloverleaf shaped pattern similar to the one discussed here. The underlying mechanism is actually closely related to the one active in our experiments: Negatively correlated shot noise indicates that charge fluctuations of one QD tend to suppress fluctuations on the other one (and vice versa). Correspondingly, those are the configurations for which we observe a reduced current through QD2 if the temperature in reservoir H is increased. A positive correlation implies that occupation fluctuations tend to occur simultaneously on both dots. Thus, we observe an enhancement of the current through QD2 with temperature in those regions.
In order to estimate the gating range of our device we analyze the data shown in Figs. 2 (a) and 2 (c): Using the G of QD2 at those configurations for which a maximal ∆I D = 18 pA is observed (G = 0.09 e 2 /h) we calculate the drain current for V SD = 100 µV and T H = T base , which gives I D = 360 pA. Relating this current to ∆I D then yields a gating amplitude of 5%.
Although this ratio is rather small, it can be strongly enhanced by tuning the parameters Finally, we note that the thermal gating effect presented here could be used, e.g., to monitor carrier heating in quantum circuits. Furthermore, it could be utilized to also manipulate heat flow across QD2: Since the thermal conductance κ of a QD as a function of µ usually follows the Wiedemann-Franz rule and thus has a similar line shape as the conductance [19], the mechanism presented here would allow gating of heat currents to be accomplished, thus suggesting a route to realizing a QD-based all thermal transistor.