The role of metallic leads and electronic degeneracies in thermoelectric power generation in quantum dots

TL;DR

This study investigates how metallic leads and electronic degeneracies affect thermoelectric power generation in graphene quantum dots, revealing the impact of spin degeneracy and heat exchange on device efficiency.

Contribution

It provides a detailed analysis of the influence of electronic degeneracies and lead properties on thermoelectric performance in quantum dot devices.

Findings

Spin degeneracy increases the power factor.

Non-ideal heat exchange suppresses power factor.

Temperature dependence of power factor is affected by lead properties.

Abstract

The power factor of a thermoelectric device is a measure of the heat-to-energy conversion efficiency in nanoscopic devices. Yet, even as interest in low-dimensional thermoelectric materials has increased, experimental research on what influences the power factor in these systems is scarce. Here, we present a detailed thermoelectric study of graphene quantum dot devices. We show that spin-degeneracy of the quantum dot states has a significant impact on the zero-bias conductance of the device and leads to an increase of the power factor. Conversely, we demonstrate that non-ideal heat exchange within the leads can suppress the power factor near the charge degeneracy point and non-trivially influences its temperature dependence.

Equations

$$I=\frac{e}{\hslash }{n}_N{n}_{N+1}\frac{{\gamma }_{\mathrm{L}}{\bar{\gamma }}_{\mathrm{R}}-{\gamma }_{\mathrm{R}}{\bar{\gamma }}_{\mathrm{L}}}{n_{N+1}({\gamma }_{\mathrm{L}}+{\gamma }_{\mathrm{R}})+n_N({\bar{\gamma }}_{\mathrm{L}}+{\bar{\gamma }}_{\mathrm{R}})},$$

$$\dot{Q}=\frac{n_N{n}_{N+1}}{\hslash }\frac{{\zeta }_{\mathrm{L}}{\bar{\gamma }}_{\mathrm{R}}-{\bar{\zeta }}_{\mathrm{L}}{\gamma }_{\mathrm{R}}}{n_{N+1}({\gamma }_{\mathrm{L}}+{\gamma }_{\mathrm{R}})+n_N({\bar{\gamma }}_{\mathrm{L}}+{\bar{\gamma }}_{\mathrm{R}})},$$

$${\stackrel{(-)}{\gamma }}_l$$

$${\stackrel{(-)}{\zeta }}_l$$

$$\left[\begin{array}{c}I\\ \dot{Q}\end{array}\right]=\left[\begin{array}{cc}{L}_{11}& L_{12}\\ L_{21}& L_{22}\end{array}\right]\cdot \left[\begin{array}{c}\Delta V\\ \Delta T\end{array}\right].$$

$$L_{ij}\approx -2\frac{e^{2-i}}{\hslash k_{\mathrm{B}}T}\frac{2{\Gamma }_{\mathrm{L}}{\Gamma }_{\mathrm{R}}}{({\Gamma }_{\mathrm{L}}+{\Gamma }_{\mathrm{R}})}\frac{({ϵ}_0/T{)}^{j-1}{ϵ}_0^{i-1}}{3+3\cosh (\frac{{ϵ}_0}{k_{\mathrm{B}}{T}_0})-\sinh (\frac{{ϵ}_0}{k_{\mathrm{B}}{T}_0})}$$

$$T_{\mathrm{L}}-T_{\mathrm{R}}=r_{\kappa }(T_{\mathrm{H}}-T_{\mathrm{C}}),$$

$$T_{\mathrm{L}}-T_{\mathrm{R}}=r_{\kappa }\frac{\kappa }{\kappa +2L_{22}+2L_{21}{L}_{12}\psi }(T_{\mathrm{H}}-T_{\mathrm{C}}),$$

$$V_{\mathrm{th}}=\frac{I_{\mathrm{th}}{R}_{\mathrm{\Omega }}}{\sqrt{(1+R_{\mathrm{\Omega }}G{)}^2+(C{\omega }_2{R}_{\mathrm{\Omega }}{)}^2}}.$$

Full text

A thermoelectric device converts a temperature difference between two metallic reservoirs, $\Delta T$, into a thermovoltage, $V_{\mathrm{th}}$. The extent of this conversion is quantified by the Seebeck coefficient (thermopower) defined as: $S=-\frac{\Delta V_{\mathrm{th}}}{\Delta T}$. However, the Seebeck coefficient alone does not provide a good measure of the heat-to-energy conversion efficiency. In the linear response regime (i.e. for an operating temperature $T$ such that $\Delta T\ll T$) one should instead consider the power factor: $\mathcal{P}=S^{2}G$ (where $G$ is the electrical conductance), which is proportional to the maximum power output density of the thermoelectric generator Liu2016 . Since the seminal work of Dresselhaus and Hicks Hicks1993_1D ; Hicks1993_0D , who predicted an increase of the thermoelectric efficiency with decreasing dimensionality, numerous experimental realizations of low-dimensional thermoelectric generators have been achieved Harman2002 ; Ibanez2016 ; Lee2016 ; Venkatasubramanian2001 ; Heremans2002 . Particularly promising here are quantum-dot (QD) devices as they allow for precise control of heat and charge flows. Recently, the role of the Kondo effect, quantum-interference phenomena and coupling asymmetry in the thermoelectric energy conversion in these systems has been examined theoretically Sanchez2011 ; Wierzbicki2011 ; Krawiec2006 ; Daroca2018 ; Gehring2017 . Additionally, recent experimental studies on QD thermoelectric devices demonstrated novel ways of reaching high thermoelectric efficiencies Thierschmann2015 , even approaching the Carnot efficiency limit Josefsson2018 . However, the role of intrinsic device features, such as metallic leads, has not yet received much experimental attention.

In this work, we experimentally study the thermoelectric properties and power factor of electrostatically-controlled graphene quantum dots (GQD). Our devices comprise a few-nanometer sized GQD located between two graphene leads which are connected to Au leads deposited on a Si/SiO2 chip, see Fig. 1a. The GQDs are fabricated using feedback-controlled electroburning of a bow-tie shaped graphene constriction Gehring2016 , which reliably produces quantum dots with relatively high addition energies Gehring2017_1 . As shown in Fig. 1a, to create a temperature gradient across the GQD, we pass an electric current (to induce Joule heating) through a micro-heater located next to the source electrode. We simultaneously measure both the gate dependent conductance, $G$, and thermovoltage, $V_{\mathrm{th}}$, to avoid artificial offsets between the two quantities (which can lead to wrong estimates of the power factor Zuev2009 ). To this end, we apply an AC voltage to the source through a resistor in series at a frequency of $\omega_{1}=91$ Hz. The heating current is applied at a frequency $\omega_{2}=17$ Hz while the thermovoltage drop over the quantum dot is measured at the second harmonic, i.e. at $2\times\omega_{2}$. The temperature difference between the gold contacts is obtained by a calibration using a four point resistance measurement Gehring2017 .

When heating up one side of the device, the Fermi distribution of the “hot” contact gets broader than the one of the “cold” contact, inducing a thermocurrent between the two leads. In an open circuit configuration, a thermovoltage $V_{\mathrm{th}}$ will be established so as to nullify the thermocurrent, as schematically shown in Fig. 1b.

Rate-equation model.—Charge transport through a quantum dot weakly coupled to the metallic leads can be described using a rate-equation (RE) approach. It models the overall transport as a sequence of electron hopping events, assigning a rate for hopping on ($\gamma_{l}$) and off ($\bar{\gamma}_{l}$) the QD (where $l=\mathrm{L,R}$ and L (R) stands for the left (right) reservoir), see Fig. 1b Sowa2018 . In what follows, each lead is found at a temperature $T_{l}$ which determines the shape of the Fermi distributions: $f_{l}=[\exp((\epsilon-\mu_{l})/k_{\mathrm{B}}T_{l})+1]^{-1}$, where $k_{\mathrm{B}}$ is the Boltzmann constant, and $\mu_{l}$ is the electrochemical potential of the lead $l$. The quantum-dot energy level (located at $\epsilon_{0}$) is coupled to the left and right reservoirs with the tunnel coupling strength of $\Gamma_{l}$. $\epsilon_{0}$ can be tuned using the gate electrode: $\epsilon_{0}=\epsilon_{00}-\alpha\lvert e\rvert V_{\mathrm{g}}$ where $\epsilon_{00}$ is the position of $\epsilon_{0}$ at zero gate, $\alpha$ is the lever arm and $V_{\mathrm{g}}$ is the gate voltage Hanson2007 . We assume that (due to strong electron-electron interactions) only one additional (transport) electron can be found on the QD at any given time. The electrical current across the QD, $I$, is then given by Sowa2019 ; SI :

[TABLE]

where $n_{N}$ and $n_{N+1}$ are the degeneracies of the electronic ground states of $N$ and $N+1$ charge states, respectively. We assume that in the considered QDs only the spin degeneracy plays a role, meaning that $n_{N}=1$ and $n_{N+1}=2$, or vice versa. As we will demonstrate, these factors can be extracted from the temperature-dependent shift of the conductance peak Beenakker1991 .

Similarly, the heat current through the dot, $\dot{Q}$, is SI :

[TABLE]

where the charge and energy transfer rates in Eqs. (1) and (2) are given by:

[TABLE]

In the above equations, lifetime broadening is introduced via the density of states of the QD, which has a Lorentzian line shape: $K(\epsilon)=\Gamma/[\Gamma^{2}+(\epsilon-\epsilon_{0})^{2}]$ where $\Gamma=(\Gamma_{\mathrm{L}}+\Gamma_{\mathrm{R}})/2$. The Fermi-Dirac distributions are $f_{+l}(\epsilon)\equiv f_{l}(\epsilon)$ and $f_{-l}(\epsilon)\equiv 1-f_{l}(\epsilon)$.

Electronic conductance.—Fig. 1c shows the stability diagram of device A from which we extract the lever arm of $\alpha=7.9$ meV. We next consider the zero-bias differential conductance G=\frac{\mathrm{d}I}{\mathrm{d}V}\Bigr{|}_{V=0} as a function of $V_{\mathrm{g}}$. We begin by fitting the experimental conductance trace at $T=3.1$K using the rate-equation model described above, see Fig. 2a. The conductance trace has an approximately Lorentzian lineshape, in agreement with our model and as expected for a single-electron transistor Kouwenhoven1991 . From the fit (black dashed line in Fig. 2a), we obtain tunnel couplings of $\Gamma_{\mathrm{L}}\approx 0.2$ meV and $\Gamma_{\mathrm{R}}\approx 1.1$ meV.

Fig. 2b shows the zero-bias conductance as we increase the temperature from 3.1K to 32K. Two separate effects can be observed. First, the conductance peak thermally broadens and its maximum decreases accordingly. Secondly, we observe that the position of the conductance peak shifts with temperature. This effect has been theoretically predicted by Beenakker Beenakker1991 and can be understood as originating due to the changes in entropy of the system, as has been discussed elsewhere Hartman2018 . The semi-classical theory of Ref. Beenakker1991 predicts that (for a doubly degenerate energy level as considered here) the position of the peak should shift by: $\Delta E_{\mathrm{shift}}(G_{\mathrm{max}})\approx\mathrm{ln}(2)k_{\mathrm{B}}T/2$, so that $\mathrm{ln}(2)k_{\mathrm{B}}/2$ can be regarded as the entropy associated with populating a doubly-degenerate electronic level. This is also in agreement with our theory. The expected position of the conductance peak is shown by the white dashed line in Fig. 2b, which has a slope of $\mathrm{ln}(2)k_{\mathrm{B}}T/2$.

In Fig. 2a we further plot the experimental (navy dots) and theoretical (white dashed line) values of the zero-bias conductance at $T=32$K. Both of the effects discussed above are captured well by our theoretical model (using only the parameters extracted from the earlier fit at 3.1K). The excellent match between the experimental and theoretical values of electronic conductance also at higher temperature further supports the validity of our theoretical model.

Thermovoltage.—We proceed to analyse the thermovoltage measurements. As discussed, a temperature gradient across the QD is induced using the micro-heater shown in Fig. 1a. $V_{\mathrm{th}}$ measured at a base temperature of 3.1K is shown in Fig. 2c. Similarly to the electrical conductance, the thermovoltage exhibits a gate-dependent behaviour. It switches from negative to positive values as the QD energy level crosses the Fermi energy of the electrodes, indicating a change from a hole to an electron-based thermocurrent Lunde2005 . In contrast to what is predicted by the Mott relation (which connects $V_{\mathrm{th}}$ to the derivative of $G$) Dzurak1993 , however, we observe a wide region of suppressed $V_{\mathrm{th}}$ around the charge degeneracy point. As we shall discuss, we attribute this suppression to non-ideal heat exchange within the leads resulting in a different effective temperature difference across the QD, as shown schematically in Fig. 2f. Furthermore, as the temperature increases (from 3.1K to 32K), the amplitude of the thermovoltage signal decreases by an order of magnitude, also in disagreement with a single level model Gehring2017 .

To explain these effects, we consider the case where the immediate contact regions coupled to the QD are at temperatures $T_{\mathrm{L}}$ and $T_{\mathrm{R}}$, respectively, and are not perfectly thermalized with the “hot” and “cold” reservoirs (see Fig. 2e) which are at temperatures $T_{\mathrm{H}}$ and $T_{\mathrm{C}}$, respectively. Instead, the left (right) contact is coupled to the hot (cold) reservoir via a thermal conductance $\kappa(T)$ accounting for both the electrical and phononic heat transport. Additionally, the thermal conductance through the quantum dot (i.e. between left and right) consists of an electrical (gate- and temperature-dependent) contribution $\kappa_{\mathrm{el}}(\epsilon,T)$ and a (strictly temperature-dependent) phonon/substrate contribution $\kappa_{\mathrm{loss}}(T)$. The latter accounts for any phononic heat transport through the substrate as well as the QD. The resulting thermal circuit diagram is shown in Fig. 2f. Since it is very challenging to realise perfect thermal contacts between the leads and the QD Apertet2012 , we believe that similar models should be relevant to most zero-dimensional systems (although the effects discussed here may not alaways be as pronounced as in our study).

To quantify the interplay between the heat and charge flows across the QD and determine the gate-dependent heat flow, we consider the Onsager matrix given by:

[TABLE]

where $L_{ij}$ are the Onsager modes. To obtain $L_{ij}$, we expand $I$ and $\dot{Q}$ to the first order with respect to $\Delta T$ and $\Delta V$. In the linear response regime and assuming small lifetime broadening $(\Gamma\rightarrow 0)$ the modes are given by SI :

[TABLE]

where we used $n_{N}=2$ and $n_{N+1}=1$. The magnitude of the electronic heat exchange through the QD (from left to right) is given by $L_{22}$ since the contribution of the $L_{21}$ term can be neglected under open circuit conditions SI so that $\kappa_{\mathrm{el}}=L_{22}$.

Firstly, we note that the phononic thermal transport contribution, $\kappa_{\mathrm{loss}}$, dominates the thermal transport between the left and right contact off-resonance, i.e. $\kappa_{\mathrm{el}}\ll\kappa_{\mathrm{loss}}$ when $\lvert\epsilon_{0}\rvert\gg\Gamma,k_{\mathrm{B}}T$. Far away from resonance, therefore, the effective temperature gradient across the QD, ($T_{\mathrm{L}}-T_{\mathrm{R}}$), is determined only by $\kappa$ and $\kappa_{\mathrm{loss}}$ SI :

[TABLE]

where $r_{\mathrm{\kappa}}=\kappa/(2\kappa_{\mathrm{loss}}+\kappa)$.

On the other hand, accounting for all the heat-flow contributions shown in Fig. 2f allows us to obtain the effective temperature gradient across the QD as SI :

[TABLE]

where $\psi={R_{\mathrm{\Omega}}}/{\sqrt{(R_{\mathrm{\Omega}}+G)^{2}+(C\omega_{2}R_{\mathrm{\Omega}})^{2}}}$. Here $R_{\mathrm{\Omega}}=1\mathrm{T}\Omega$ is the input impedance of our high-impedance amplifier, and the system capacitance is extracted from our low temperature fit as $C\approx 3$ nF. The zero-bias thermocurrent, $I_{\mathrm{th}}$, is calculated using Eqs. (1) and (8). Accounting for the load resistance ($R_{\Omega}$) and the inherent capacitance of the system ($C$), gives the thermovoltage as SI ; Svensson2012 :

[TABLE]

As can be inferred from Eqs. (8) and (9), the electronic heat flow across the QD, in the presence of non-ideal contacts, should result in a strong suppression of the thermovoltage around the resonance. This is indeed observed experimentally in Fig. 2b which also shows the theoretical fit of the rate-equation model (with $C$ and $\Delta T$ as fitting parameters and $r_{\mathrm{\kappa}}=1$ held constant). In order to extract $r_{\mathrm{\kappa}}$ we fit the thermovoltage at corresponding temperatures with $\kappa$ and $\kappa_{\mathrm{loss}}$ as the fitting parameters and holding all other parameters constant with respect to the low-temperature fit SI .

We next turn to the second non-trivial effect observed in our thermopower measurements: a rapid decrease of $V_{\mathrm{th}}$ with increasing temperature shown in Figs. 2c and d and the inset of Fig. 3c. This effect cannot be explained with a simple single-level model (within either the Landauer or a RE approach) and recent experimental findings of an inverse relation studied a metallic island with a continuum of energy levels which is not applicable in our case Erdman2019 . Nonetheless, we note that thermovoltage decreasing with increasing temperature is almost universally reported for zero-dimensional structures Staring1993 ; Molenkamp1994 ; Small2003 ; Scheibner2007 .

This trend can be explained by changes in the thermal conductances ($\kappa_{\mathrm{loss}}$ and $\kappa$) between the reservoirs. As shown in Fig. 3c and Eq. 8, $r_{\mathrm{\kappa}}$ obtained from our fit and therefore the effective temperature drop over the QD ($T_{\mathrm{L}}-T_{\mathrm{R}}$) decreases as the temperature increases. The thermovoltage must therefore follow a similar trend. The behaviour of $r_{\mathrm{\kappa}}$ could be attributed to a combination of the temperature dependent thermal conductance in graphene which has a non-linear behavior Seol2010 and size-dependent effects due to the geometry of the device Xu2014 . We not that the substrate used in this study, SiO2, has a negligible temperature dependence in our measurement range, therefore $r_{\mathrm{\kappa}}$ contains mostly information about our nanostructure Wingert2016 .

Independent of the effects discussed above, the thermovoltage lineshape can be also influenced by the degeneracy of the relevant charge ground states as reported recently Kleeorin2019 . Electronic degeneracy can result in a slight asymmetry between the positive and negative peaks of $V_{\mathrm{th}}$.

Power factor.—We finally turn to consider the power factor which, as we discussed, can be obtained from the electrical conductance and thermovoltage measurements: $\mathcal{P}=S^{2}G=G[V_{\mathrm{th}}/(T_{\mathrm{H}}-T_{\mathrm{C}})]^{2}$. Fig. 3a shows the experimental power factor obtained using $G$ and $V_{\mathrm{th}}$ measured at $T=3.1$K. Firstly, we note that it exhibits a clear asymmetry with respect to the charge-degeneracy point. This asymmetry stems from the discussed shift of the conductance peak (since $V_{\mathrm{th}}$ is always zero at the charge degeneracy point, irrespective of temperature). Secondly, we observe a strong reduction of the power factor around the charge degeneracy point. This can be attributed to the effects of non-ideal contacts discussed above [which induce a corresponding suppression in $V_{\mathrm{th}}$, see Eq. (8)]. Fig. 3b shows the calculated power factor for the degenerate ($n_{N}=2$ and $n_{N+1}=1$) and non-degenerate ($n_{N}=n_{N+1}=1$) QD level, keeping all other parameters constant. A good qualitative agreement between experiment and theory can be observed in the case of a degenerate electronic level.

Previously, it has been predicted that (for a non-interacting single-level model) the maximum power factor $\mathcal{P}^{\mathrm{max}}$ should exhibit a maximum as a function of temperature at $k_{\mathrm{B}}T\approx 1.1\Gamma$ Gehring2017 . This is a result of an interplay between $S$ and $G$ which are expected to increase and decrease with increasing $k_{\mathrm{B}}T/\Gamma$, respectively. For the doubly-degenerate level considered here, the maximum of the power factor should instead be expected to occur at $k_{\mathrm{B}}T\approx 1.05\Gamma$ SI . As we show in Fig. 3d, however, we find that this trend is not observed experimentally (blue dots) since, in our experiments, the thermovoltage decreases with increasing $T$ (see inset in Fig. 3c). A decreasing power factor with increasing temperature was measured in all devices included in this study, even when a less pronounced suppression of the thermovoltage signal was observed SI . This suggests that this is a persistent effect which needs to be taken into account when designing a high-efficiency thermoelectric QD generator.

However, in our calculation of the power factor we can replace the applied temperature difference ($T_{\mathrm{H}}-T_{\mathrm{C}}$) with the effective one, so that the effective power factor is $\mathcal{P}_{\mathrm{eff}}=G[V_{\mathrm{th}}/(T_{\mathrm{L}}-T_{\mathrm{R}})]^{2}$. We then recover the expected behaviour of the power factor. This is shown in Fig. 3d (orange triangles) where $\mathcal{P}_{\mathrm{eff}}^{\mathrm{max}}$ reaches a maximum for $k_{\mathrm{B}}T\approx 1.3\Gamma$, in a relatively good agreement with the theoretical predictions.

Conclusions.—We investigated the influence of non-ideal contacts and electronic degeneracy on the thermoelectric properties of graphene quantum dots by simultaneously measuring their thermovoltage and conductance. We have shown that the spin degeneracy of the QD level leads to a (temperature-dependent) shift in the electronic conductance peak (as compared to a non-degenerate or non-interacting electronic level). As discussed, this gives rise to an asymmetric enhancement of the thermoelectric power factor. Conversely, non-ideal heat exchange within the leads and heat transport through the QD were found to have a deleterious impact on the observed thermovoltage. In particular, due to an efficient electronic heat transport across the quantum dot, this effect results in a strong suppression of $V_{\mathrm{th}}$ in the vicinity of the charge-degeneracy point. Our experimental results are supported by a rate-equation model which successfully captures all the phenomena described above. We believe that the effects discussed here are ubiquitous to quantum-dot thermoelectric devices. This work, therefore, opens the door to engineering zero-dimensional devices with increased theremoelectric power factor and provides further understanding of phenomena governing heat-to-energy conversion in such systems.

This work was supported by the UK EPSRC (Grants EP/K001507/1, EP/J014753/1, EP/H035818/1, EP/K030108/1, EP/J015067/1, and EP/N017188/1). P.G. acknowledges a Marie Skłodowska-Curie Individual Fellowship under Grant TherSpinMol (ID: 748642) from the European Union’s Horizon 2020 research and innovation programme. J.A.M. acknowledges a RAEng Research Fellowship.