Experimental realization of a quantum dot energy harvester

TL;DR

This paper reports the experimental creation of a nanoscale quantum dot energy harvester that converts heat into electrical power using resonant tunnelling, demonstrating a proof-of-concept device at millikelvin temperatures.

Contribution

It presents the first experimental realization of a quantum dot-based energy harvester utilizing resonant tunnelling for heat-to-electricity conversion.

Findings

Generates 0.13 fW of power at 75 mK.

Uses quantum dots as energy filters for heat conversion.

Operates with a 67 mK temperature difference.

Abstract

We demonstrate experimentally an autonomous nanoscale energy harvester that utilises the physics of resonant tunnelling quantum dots. Gate defined quantum dots on GaAs/AlGaAs high-electron-mobility transistors are placed on either side of a hot electron reservoir. The discrete energy levels of the quantum dots are tuned to be aligned with low energy electrons on one side and high energy electrons on the other side of the hot reservoir. The quantum dots thus act as energy filters and allow for the conversion of heat from the cavity into electrical power. This energy harvester device, measured at an estimated base temperature of 75 mK in a He3/He4 dilution refrigerator, can generate a thermal power of 0.13 fW when the temperature difference across each dot is about 67 mK.

Figures

Equations

$$S={\frac{V_{th}}{T_C-T_0}|}_{\text{I=0}}=-\frac{{\pi }^2{k}_B^2}{3e}(T_C-T_0)\frac{\partial \ln G}{\partial \mu }.$$

$${\eta }_{\text{w-f}}=\frac{V_{th}^2}{\kappa \Delta T{R}_{\mathrm{Load}}}=\frac{V_{th}^2}{GL\overline{T}\Delta T{R}_{\mathrm{Load}}},$$

$${\mathcal{I}}_{l,n}=\frac{2}{h}\int dE{E}^n{\mathcal{T}}_l(E)[f_l(E)-f_{\mathrm{C}}(E)],$$

$${\mathcal{T}}_l(E)=A_l\frac{{\Gamma }_l^2/4}{(E-{\epsilon }_l{)}^2+{\Gamma }_l^2/4},$$

$$I_{\mathrm{L}}+I_{\mathrm{R}}=0,J_{\mathrm{L}}+J_{\mathrm{R}}+\dot{Q}=0.$$

$$\eta =V_{\mathrm{th}}^2/(\dot{Q}{R}_{\mathrm{Load}}).$$

$$G=\frac{e^2}{h}\frac{1}{4k_B{T}_C}$$

$$\frac{(\Gamma /2)\pi }{(\Gamma /2{)}^2+[(e\alpha V_g-E_{res})-E{]}^2}dE.$$

$$m\text{textsubscript}S=\tan (\theta \text{textsubscript}S)={\left(\frac{\Delta V\text{textsubscript}g}{\Delta V\text{textsubscript}SD}\right)}^{(S)},$$

$$m\text{textsubscript}D=\tan (\theta \text{textsubscript}D)={\left(\frac{\Delta V\text{textsubscript}g}{\Delta V\text{textsubscript}SD}\right)}^{(D)}.$$

$$\alpha \text{textsubscript}G=\frac{1}{m\text{textsubscript}S-m\text{textsubscript}D}.$$

$$\alpha \text{textsubscript}D=\frac{1}{1-m\text{textsubscript}S/m\text{textsubscript}D}.$$

Full text

In recent years there has been an increased interest in devices which can convert waste heat into useful work White (2008). Thermoelectric generators where a temperature bias applied to an electric conductor gives rise to a charge current flow are good candidates Shakouri (2011); Benenti et al. (2017). Unfortunately, current thermoelectric devices have relatively small efficiencies Rowe (2006). This issue can be overcome by nanoscale thermoelectrics where engineered bandstructures and quantum mechanical effects can give rise to an increased efficiency Hicks and Dresselhaus (1993a, b); Mahan and Sofo (1996). Quantum dots constitute an important element in designing highly efficient thermoelectrics Beenakker and Staring (1992); Humphrey et al. (2002); Nakpathomkun et al. (2010); Esposito et al. (2009) because their discrete resonant levels provide excellent energy filters. Thermoelectric effects have been investigated in various quantum-dot setups Staring et al. (1993); Dzurak et al. (1993, 1997); Godijn et al. (1999); Small et al. (2003); Llaguno et al. (2003); Scheibner et al. (2005, 2007); Svensson et al. (2012, 2013); Thierschmann et al. (2013); Josefsson et al. (2018).

Energy harvesting devices require that the energy source is separated from the electrical circuit, so no charge is extracted from it Sothmann et al. (2015). This can be accomplished in three-terminal devices where a hot terminal injects heat but no charge into the setup, thus driving a charge current between two cold reservoirs. There have been a number of proposals for these kinds of energy harvesters Entin-Wohlman et al. (2010); Sánchez and Büttiker (2011); Sothmann et al. (2012); Sothmann and Büttiker (2012); Ruokola and Ojanen (2012); Jiang et al. (2012, 2013); Machon et al. (2013); Jordan et al. (2013); Sothmann et al. (2013); Bergenfeldt et al. (2014); Donsa et al. (2014); Mazza et al. (2014, 2015); Sánchez et al. (2015a, b); Hofer and Sothmann (2015); Bosisio et al. (2016); Szukiewicz et al. (2016); Jiang and Imry (2018). Three-terminal heat engines based on Coulomb-coupled quantum dots Sánchez and Büttiker (2011); Sothmann et al. (2012) have been realized experimentally recently Thierschmann et al. (2015); Roche et al. (2015); Hartmann et al. (2015). Due to their design they are however limited to low power. A three-terminal energy harvester based on two resonant-tunneling quantum dots with different energy levels overcomes this issue. It can in principle reach Carnot efficiency and can be optimized to achieve a large power in combination with a high efficiency at maximum power Jordan et al. (2013).

A similar device has also been proposed Edwards et al. (1993, 1995), and later demonstrated Prance et al. (2009), as a building block of a nanoscale refrigerator. In this manuscript, we experimentally realize a resonant-tunneling energy harvester and demonstrate its ability to generate electrical power in an external load arising from energy exchanges between a hot and a cold reservoir. Importantly, no external drive or cycling is required; that is, the system is entirely autonomous and begins producing power as soon as a thermal gradient is present.

The system we have investigated, shown in Fig. 1, is comprised of two quantum dots that connect a hot cavity to two cold reservoirs Jordan et al. (2013). By putting two quantum dots in series with a hot cavity, electrons that enter via the left dot are forced to gain a prescribed energy in order to exit through the right dot, transporting a single electron charge from left to right, cf. Fig. 1(a). Constrained by the conservation of global charge and energy in the device, this thermal energy gain of electrons will be converted into electrical current Jordan et al. (2013).

Fig. 1(c) shows a false-colored scanning electron micrograph (SEM) image of a typical device we tested, along with the electrical circuit used in the experiments. \ceTi/Au gates were patterned on the surface of \ceGaAs/AlGaAs heterostructure material using electron-beam lithography. The 2DEG was 110 nm below the surface, and was contacted by annealing \ceAuGeNi ohmic contacts. The mobility $\mu$ and carrier concentration $n$ of the 2DEG were measured to be $\mu\approx 3.38\times 10^{6}\,\mathrm{cm^{2}V^{-1}s^{-1}}$ and $n\approx 1.35\times 10^{11}\,\mathrm{cm^{-2}}$ at 1.5 K. The surface gates define a cavity of $90\,\mathrm{\mu m^{2}}$ area at the central 2DEG region with two quantum dots respectively on the left and the right sides, and a heating channel on the top. The quantum dots, of 310 nm diameter, as shown in 1(d), are constructed of three barrier gates (colored red in Fig. 1(c)), one detector gate (colored green), and one plunger gate (colored blue). Both dots were found to have charging energies of approximately 1.5 meV and first excited states always at least $250\,\mathrm{\mu eV}$ above the ground state. The top heating channel (gates colored yellow in Fig. 1(c)) is connected to the central cavity via a gap of $1.26\,\mathrm{\mu m}$, which allows hot electrons to traverse into the cavity and form different temperature profiles. Measurements were performed in a He3/He4 dilution refrigerator at an estimated base temperature $T_{0}$ of 75 mK. The experiment was repeated with two samples, which are similar in design but have different resonances for the quantum dots.

The thermal power generated by the energy harvester was measured with the set-up in Fig. 1(c). An AC current $I_{\text{Heat}}$ which heats electrons at frequency $f=33\,\mathrm{Hz}$ was applied to the heating channel using a lock-in amplifier, and the thermal voltage $V_{\text{th}}$ was measured across A - B, with another amplifier locking in at frequency 2f, whilst stepping the voltage $V_{\text{LD}}$ on the left dot plunger gate and sweeping the voltage $V_{\text{RD}}$ on the right dot plunger gate through their respective Coulomb resonance. Since the heating power varies as $I^{2}$, the electron temperature in the cavity oscillates at twice the frequency of the current $I_{\text{Heat}}$. Thus it was necessary to phase lock to the 2f component of $V\textsubscript{th}$ Dzurak et al. (1997). The temperatures of the central cavity for different AC currents may be estimated by fitting the differential conductance of the quantum dots with a thermally broadened resonance, see Supplementary Information for details. The cold reservoirs, which we assume to be at base temperature $T_{0}$, are connected externally by a load resistor, $R\textsubscript{Load}$. The thermal power is then extracted by $P=V_{\rm th}^{2}/R\textsubscript{Load}$. In potential applications the heating channel would be replaced by the heat source we wish to harvest energy from and the $R\textsubscript{Load}$ represents an external device where useful work is done Jordan et al. (2013).

Figure 2(a) shows the thermal voltage between A and B (in Fig.1(c)), $V_{\text{th}}$, measured with a heating current $I_{\text{Heat}}=100\,\mathrm{nA}$ and a load resistor $R_{\text{Load}}=500\,\mathrm{k\Omega}$ in the circuit. The negative thermal voltage appears at $V_{\text{LD}}\approx-1.924\,\mathrm{V}$ for the left dot, as shown in Fig. 2(c), and $V_{\text{RD}}\approx-0.805\,\mathrm{V}$ for the right dot, as in Fig. 2(b). Fig. 2(d) is the thermal power extracted from the thermal voltage of Fig.2(a), through $P=V_{\text{th}}^{2}/R\textsubscript{Load}$. The maximum thermal power is found at $(-1.924,-0.805)\,\mathrm{V}$, followed by the second largest thermal power point at $(-1.907,-0.829)\,\mathrm{V}$, in Fig. 2(d). The maxima appear in the vicinity of the electrical conductance peaks shown in Fig. 2(e) and Fig. 2(f) respectively. This is because when both charge and heat are exclusively carried by electrons, for both diffusive and ballistic transport, the Seebeck coefficient (thermopower) $S$ is related to the energy derivative of the conductance $G$ Appleyard et al. (1998),

[TABLE]

Here $T_{C}$ is the electron temperature of the cavity, $T_{0}$ is the temperature of cold reservoirs, and $\mu$ is the chemical potential of the contacts. Meanwhile, the thermal voltage peaks in Fig. 2(a) are also related to the energy derivative of the conductance of the Fig. 2(e) and (f), as shown Eq. (1). Some thermopower is detected while only one dot is open on resonance, such as the area marked by the star $\star$ in Fig. 2(d). This arises from the influence of the external circuit impedance.

Thermopower measurements were carried out using resistance values ($R\textsubscript{Load}$) from $50\,\mathrm{k\Omega}$ to $3.9\,\mathrm{M\Omega}$ in the circuit, whilst an AC current of $60\,\mathrm{nA}$, $80\,\mathrm{nA}$ and $100\,\mathrm{nA}$ is applied on the heating channel. The heating currents of $60\,\mathrm{nA}$, $80\,\mathrm{nA}$ and $100\,\mathrm{nA}$ correspond to $122\,\mathrm{mK}$, $130\,\mathrm{mK}$ and $140\,\mathrm{mK}$ respectively, as discussed in Supplementary Material. Figure 3 depicts the maximal generated power for each measurement as a function of the load resistance and the relative thermal voltages respectively, where black circles represent the experimental data while a current of $60\,\mathrm{nA}$ is applied on the heating channel, red stars for $80\,\mathrm{nA}$, and blue triangles for $100\,\mathrm{nA}$. (Solid lines represent results from theoretical modeling and will be discussed later.) For increasing resistance $R_{\rm Load}$, the power increases, reaches a maximum and then drops down, as shown in Fig. 3(a). As the heating current in the channel is increased, the power also rises. This is because the cavity temperature increases with the heating current, resulting in more electrons tunneling through the two dots and converting more energy to electrical current efficiently, as predicted in the theoretical proposal (Jordan et al., 2013). Interestingly, the maximum power always appears around $R_{\rm Load}\approx 500\,\mathrm{k\Omega}$ for all heating currents, corresponding to impedance matching between the heat engine and the resistor. The power vs. thermal voltage in Fig. 3(b) gives an estimation of the open-circuit stall voltage of our device in each configuration. In the linear regime one expects the maximal power to occur at half the stall voltage Sothmann et al. (2015). The asymmetric dependence of the measurements suggests the presence of non-linear effects.

We next turn to the efficiency of heat to work conversion which is defined as the ratio of the generated electrical power $P$ to the heat current from the hot reservoir ${\dot{Q}}$. The heat current is given by $\dot{Q}=\kappa\Delta T=\kappa(T_{\rm C}-T_{0})$ where the thermal conductance $\kappa$ can be estimated from the electrical conductance $G$ via the Wiedemann-Franz law, $\kappa=GL\overline{T}$, where $L$ is the Lorenz number, $\overline{T}=(T_{\rm C}+T_{0})/2$ and $G$ is the combination of the conductance at $V_{\rm RD}=-0.805\,\mathrm{V}$ in Fig. 2(e) and that at $V_{\rm LD}=-1.924\,\mathrm{V}$ in Fig. 2(f). We remark that the Wiedemann-Franz law in general is violated for mesoscopic conductors with strongly energy dependent transmissions such as quantum dots Kubala et al. (2008); Tsaousidou and Triberis (2010); Erdman et al. (2017); Dutta et al. (2017). As the thermal conductance cannot be measured directly in our setup, we still use it to obtain an lower bound on the thermoelectric efficiency given by

[TABLE]

which can be compared with the theoretical efficiency calculated below. Figure 3(c) depicts the ratio of the estimated efficiency from Equation (2) to the Carnot efficiency ($\eta_{C}=1-T_{\rm 0}/T_{\rm C}$) for $60\,\mathrm{nA}$, $80\,\mathrm{nA}$ and $100\,\mathrm{nA}$ on the heating channel respectively.

The experimental data in Fig. 2 and Fig. 3 is reproduced by the model of Ref. Jordan et al. (2013) which we generalize to incorporate the effect of the external circuit. The thermoelectric transport through each dot can be described by the Landauer-Büttiker formalism, with the expression:

[TABLE]

giving the charge $I_{l}=e\mathcal{I}_{l,0}$ and energy $J_{l}=\mathcal{I}_{l,1}$ currents at lead $l$=L,R. The quantum dot resonances are defined by a transmission coefficient

[TABLE]

where the parameter $A_{l}$ depends on the asymmetry of the quantum dot barriers Büttiker (1988). The quantum dot resonant energies are tuned with gate voltages, $\varepsilon_{l}=\varepsilon_{l,0}-e\alpha_{l}V_{{\rm g}l}$. In our experiment, the width $\Gamma_{l}$ is thermally broadened beyond the natural line width of the level. As no charge is injected from the heating channel into the conductor, the conservation laws for charge and energy read:

[TABLE]

where ${\dot{Q}}$ is the heat current injected into the central cavity. For a closed circuit where the energy harvester powers an impedance $R_{\rm Load}$, the voltages are set via Ohm’s law, producing the thermovoltage, $V_{\rm th}=I_{\rm L}R_{\rm Load}$ and power of Fig. 4. Accounting for the external resistance in the circuit gives rise to additional features not present in an open-circuit model Jordan et al. (2013), such as the vertical and horizontal lines in Fig. 4. Our simple model based on resonant tunneling captures all the features of the experiment, seen by the comparison of experimental data (points of black circles, red stars and blue triangles) and theoretical modeling results (solid lines in black, red and blue) in Fig. 3. The theoretical efficiency, shown as solid lines in Fig. 3(c), is computed with the general expression of the heat current evaluated at the obtained thermovoltage,

[TABLE]

Fig. 3(c) suggests the top theoretical efficiency of the device is $\sim 0.5\eta_{C}$, for the considered parameters. Given its overall good agreement with experimental results, this theoretical model provides a more realistic estimate of the efficiency with its direct access to the heat current, $\dot{Q}$. The experimental estimates in Fig. 3(c), extracted by Eq. (2), are only the lower bound of the efficiency, where the thermal conductance is overestimated because quantum dots have a smaller Lorenz ratio than $L$ due to the violation of the Wiedemann-Franz law as discussed earlier Kubala et al. (2008); Tsaousidou and Triberis (2010); Erdman et al. (2017); Dutta et al. (2017).

In conclusion, we have experimentally realized an energy harvester based on resonant-tunneling quantum dots Jordan et al. (2013) which can generate a power of $0.13\,\mathrm{fW}$ in an estimated efficiency with a lower bound around $0.1\,\mathrm{\eta_{\rm C}}$. Our theoretical model (not affected by limitations of the Wiedemann-Franz law) suggests the actual efficiency to be about $0.5\,\mathrm{\eta_{\rm C}}$. Experimental observations of thermal power, voltage and efficiency at different values of $I_{\rm Heat}$ and $R_{\rm Load}$ have also been reproduced by this model. There are small quantitative differences between experimental results and theoretical modeling in terms of parameters, such as electrical temperatures and energy level difference. This may be explained by asymmetric barriers, accidental degeneracies or the broadened lifetime width of the quantum dots, as well as charging effects in the non-linear regime. Also, the oscillation brought with the AC heating and AC measurements can increase thermal broadening in the cavity, and therefore cause inaccuracy in the measurement results. Overall, this proof-of-principle experiment demonstrates the basic soundness of the theory of mesoscopic energy harvesting with energy filtering techniques at the quantum level, realizing a heat engine.

We propose several possible improvements for future work. First, we can improve the power and efficiency by optimizing the resonance width $\Gamma_{l}$ as well as the symmetry of the quantum dots. Second, DC heating and measurement techniques can be used to avoid unnecessary oscillations of voltages and temperatures in the device. Finally, the performance of the energy harvester may be enhanced by scaling it up in size with resonant-tunneling quantum wells, which may increase the maximum power up to fractions of W/cm2 at 300 K Sothmann et al. (2013), or by using smaller dots or molecules, whose large level spacing allows the system to operate at higher temperatures Leobandung et al. (1995); Park et al. (2002); Jordan et al. (2013).

We are grateful to Dr. J. Waldie for technical assistance. This work was funded by EPSRC(UK). GJ acknowledges financial support from China Scholarship Council and GBCET. RS acknowledges financial support from the Spanish MINECO via grant FIS2015-74472-JIN (AEI/FEDER/UE), the Ramón y Cajal program RYC-2016-20778 and through the “María de Maeztu” Programme for Units of Excellence in R&D (MDM-2014-0377). Work by ANJ was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES) under Award #DE-SC0017890. BS acknowledges financial support from the Ministry of Innovation NRW via the “Programm zur Förderung der Rückkehr des hochqualifizierten Forschungsnachwuchses aus dem Ausland”. This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958.

I Supplementary Material: Experimental realisation of a quantum dot energy harvester

I.1 Estimating temperature differences

In this section, we discuss how the temperatures in Fig. 1 (b) are estimated. Figure 5 shows the circuit used for temperature measurements. Two potential dividers of 1/10th are set on either side of the heating channel, followed by a 1 M$\Omega$ resistor, to provide constant currents in the channel. DC voltages of the same magnitude but opposite sign are applied spontaneously to either side of the heating channel via NicDAQ-9178, to avoid a bias voltage being applied across the quantum dots. Two capacitors of 22 $\mu$F and 47 $\mu$F are set on either side of the cavity, to block heating currents from leaking through the dots. Therefore, heating currents only go to the ground of the potential divider on the other side the heating channel. The differential conductance, $G=dI/dV$, are measured through the left quantum dot while different DC heating currents are applied in the channel, and the cavity is defined. To extract the temperature of the cavity at each DC heating current, $G$ is fitted to a thermally broadened Lorentzian, parameterised by a full width at half-maximum $\Gamma$ Foxman et al. (1993),

[TABLE]

Where $e$ is the electron charge, $k_{B}$ is the Boltzmann Constant, $T_{\text{C}}$ is the electron temperature of the cavity, $B$ is the temperature-independent energy-integrated strength of the resonance, $E$ and $E_{res}$ are the energy of the dot level and the energy of resonance respectively. Here $E=e\alpha V_{LD}$, with $\alpha=0.025$ the lever arm of the plunger gate, ascertained via measurement of the Coulomb Diamonds for each dot, which is discussed in the next section. For DC currents from $0\,\mathrm{}$ to $100\,\mathrm{nA}$, Eq. (7) gives a temperature range from $75\,\mathrm{mK}$ to $150\,\mathrm{mK}$, as shown in Fig. 6(b).

Our data shows the power dissipated per electron Appleyard et al. (1998) is best fit with $P=I^{2}R/(n_{e}C)=\beta(T_{C}^{5}-T_{0}^{5})$ with $R=500\,\mathrm{\Omega}$, $C=350\,\mathrm{\mu m^{2}}$ and $\beta=1.8\times 10^{-15}\,\mathrm{WK^{-5}}$, shown as the red line in Fig. 6(b). Here, $I$ is the heating current, $R$ is the resistance of the heating channel and $C$ is the area of the heating channel. This $T^{5}$ behavior was attributed to acoustic phonon scattering in the Grüneisen-Bloch regime in the heating channel, with coupling via a screened piezoelectric potential Proskuryakov et al. (2007). The $\beta=1.8\times 10^{-15}\,\mathrm{WK^{-5}}$ is larger than the theoretical prediction of $9.6\times 10^{-18}\,\mathrm{WK^{-5}}$ ($60\,\mathrm{eV/sK^{5}}$) Appleyard et al. (1998). This can be because some heat is leaking through ohmic contacts at each end of the heating channel. For $T\sim 500\,\mathrm{mK}$, the heated electrons relax to the lattice temperature over a distance $l\textsubscript{e-ph}\sim 200\,\mathrm{\mu m}$, and as $T$ is lowered further, $l\textsubscript{e-ph}$ can significantly exceed the size of the cavity of the device Dzurak et al. (1997). In this regime, the energy redistribution is achieved via electron diffusion to the cold reservoirs (Mittal et al., 1996). Therefore, in this device, hot electrons diffuse out from the heating channel to the central cavity and replace cold electrons, serving to redistribute energy and leading to a well-defined electron temperature profile.

We assume the central cavity and the reservoir of the heating channel share the same temperature $T_{C}$, and the left and the right reservoirs have the same temperature with the base temperature, $T_{L}=T_{R}=T_{0}$. The base temperature estimated by this analysis is $75\,\mathrm{mK}$. The quantum dot was used as the thermometer in this experiment, because it has greater accuracy than the thermometer in the mixing chamber of this dilution refrigerator, which gave a base temperature of $50\,\mathrm{mK}$.

I.2 Coulomb blockade analysis

When an isolated island of charges have a sufficiently small capacitance, the energy required to change its charge by even one electron may be large. Until this energy is supplied, no charge may move onto or off the island. This is called Coulomb blockade. During our experiment, the two quantum dots have firstly been tuned to have energy levels lying between the potentials of source and drain reservoirs. In this situation, the Coulomb blockade has been lifted and electrons with energies that match the energy level lying within the bias window can tunnel through the dots. When the potential of the dots is changed by a gate electrode, the plunger gate in Fig. 1, the current through a quantum dot shows periodic oscillations in plunger-gate voltage, which is known as the Coulomb Peak. Increasing the source-drain bias across a quantum dot will increase the window of energies over which states in the source are full and states in the drain are empty. With a large enough bias, the widened conductance peaks from adjacent charge states of dots overlap leaving the diamond shape regions, which are commonly referred to as Coulomb Diamonds Prance et al. (2009). In reality, the excited states start contributing to transport before the transport window is wide enough to include the next charge state of the dot, shown as light blue lines in Fig. 7. The boundaries of the Coulomb diamonds are labeled as ‘source resonance’ and ‘drain source’ in Fig. 7. This is because they correspond to the dot level being aligned with the source and drain potentials respectively. The gradient of these resonances can be used to calibrate the conversion factor between $\Delta V\textsubscript{g}$ and the electrochemical potential of the dot. The gradient are defined as Prance et al. (2009):

[TABLE]

[TABLE]

The superscript in the right hand expression denotes whether the gradient is of the source or drain resonance line. The gate electrode lever arm is given by the value as:

[TABLE]

The lever arm for the gating effect on the dot energy from the biased reservoir (the drain) can also be found:

[TABLE]

With the bias applied to only the drain reservoir, $\alpha\textsubscript{S}$, the lever arm of the source cannot be found.

The lever arm of a gate (or lead) quantifies the effect of the gate on the potential of the dot. It is a useful parameter to convert the experimentally measured voltage to energy. During this experiment, Coulomb diamonds from the left dot gives 0.025 lever arm, which is crucial to the analysis of electrical temperatures.