Thermal rectification properties of multiple-quantum-dot junctions
It is illustrated that semiconductor quantum dots (QDs) embedded into an insulating matrix connected with metallic electrodes and some vacuum space can lead to significant thermal rectification effect. A multilevel Anderson model is used to investigate the thermal rectification properties of the multiple-QD junction. The charge and heat currents in the tunneling process are calculated via the Keldysh Green’s function technique. We show that pronounced thermal rectification and negative differential thermal conductance (NDTC) behaviors can be observed for the multiple-QD junction with asymmetrical tunneling rates and strong interdot Coulomb interactions.
Records of thermal rectification date back to 1935 when Starr discovered that copper oxide/copper junctions can display a thermal diode behavior. Recently, thermal rectification effects have been predicted to occur in one dimensional phonon junction systems. Such a thermal rectification effect is crucial for heat storage. Scheibner and coworkers have experimentally observed the asymmetrical thermal power of the two-dimensional electron gas in QD under high magnetic fields. So far the rectification mechanism of a single QD is still ambiguous owing to the unclear relation between the thermal power and the thermal rectification effect. This inspires us to investigate whether the QD junction system can act as a thermal rectifier. A useful thermal diode to store solar heating energy not only requires a high rectification efficiency but also high heat flow. The later requires a high QD density in the QD junction thermal diode. The main goal of this study is to illustrate that the multiple QDs embedded into an insulator connected with metallic electrodes and with a vacuum layer insert can give rise to significant thermal rectification and negative differential thermal conductance (NDTC) effects in the nonlinear response regime. We also clarify the relation between the thermal power and the rectification effect.
The proposed isulator/quantum dots/vacuum (IQV) double barrier tunnel junction system (as illustrated in Fig. 1) can be adequately described by a multi-level Anderson model. Here, the vacuum layer serves as a blocking layer for phonon contributions to thermal conduction, while allowing electrons to tunnel through. We assume that the energy level separation between the ground state and the first excited state within each QD is much larger than , where is the temperature of concern. Therefore, there are only one energy level for each QD. We have ignored the interdot hopping terms due to the high potential barrier separating QDs. The key effects included are the intradot and interdot Coulomb interactions and the coupling between the QDs with the metallic leads. Using the Keldysh-Green’s function technique, the charge and heat currents through the junction can be expressed as
where is the transmission factor. and is the Fermi distribution function for the left (right) electrode. The chemical potential difference between these two electrodes is related to the bias difference created by the temperature gradient. denotes the temperature maintained at the left (right) lead. denotes the average Fermi energy of the electrodes. and [ denote the tunneling rates from the QDs to the left and right electrodes, respectively. and denote the electron charge and Plank’s constant, respectively. For simplicity, these tunneling rates are assumed to be energy- and bias-independent. Eqs. (1) and (2) have been employed to study the thermal properties of single-level QD in the Kondo regime. Here, our analysis is devoted to the multiple-QD system in the Coulomb blockade regime. The expression of the retarded Green function for dot of a multi-QD system, can be found in Ref. 
To study the direction-dependent heat current, we let and , where is the equilibrium temperature of two side electrodes and is the temperature difference. Because the electrochemical potential difference, yielded by the thermal gradient could be significant, it is important to keep track the shift of the energy level of each dot according to , where is the ratio of the distance between dot and the mid plane of the QD junction to the junction width. Here we set . A functional thermal rectifier requires a good thermal conductance for , but a poor thermal conductance for . Based on Eqs. (1) and (2), the asymmetrical behavior of heat current with respect to requires not only highly asymmetric coupling strengthes between the QDs and the electrodes but also strong electron Coulomb interactions between dots. To investigate the thermal rectification behavior, we have numerically solved Eqs. (1) and (2) for multiple-QD junctions involving two QDs and three QDs for various system parameters. We first determine by solving Eq. (1) with (the open circuit condition) for a given , and an initial guess of the average one-particle and two-particle occupancy numbers, and for each QD. Those numbers are then updated according to Eqs. (5) and (6) in Ref.  until self-consistency is established. For the open circuit, the electrochemical potential will be formed due to charge transfer generated by the temperature gradient. This electrochemical potential is known as the Seebeck voltage (Seebeck effect). Once is solved, we then use Eq. (2) to compute the heat current.
Fig. 2 shows the heat currents, occupation numbers, and differential thermal conductance (DTC) for the two-QD case, in which the energy levels of dot A and dot B are and , where is tuned between 0 and 1. The heat currents are exporessed in units of through out this article. The intradot and interdot Coulomb interactions used are and . The tunneling rates are , , and . is chosen to be throughout this article. Here, is the average tunneling rate in energy units, whose typical values of interest are between 0.1 and 0.5 meV. The dashed curves are obtained by using a simplified expression of Eq. (2) in which we set the average two particle occupation in dots A and B to zero (resulting from the large intradot Coulomb interactions) and taking the limit that so the Lorentzian function of resonant channels can be replaced by a delta function. We have
Here is the average occupancy in dot A(B). Therefore, it is expected that the curve corresponding to obtained with this delta function approximation is in good agreement with the full solution, since is far away from the Fermi energy level. For cases when is close to , the approximation is not as good, but it still gives qualitatively correct behavior. Thus, it is convenient to use this simple expression to illustrate the thermal rectification behavior. The asymmetrical behavior of with respect to is mainly resulted from the condition and . The heat current is contributed from the resonant channel with , because the resonant channel with is too high in energy compared with . The sign of is determined by , which indirectly depends on Coulomb interactions, tunneling rate ratio and QD energy levels. The rectification behavior of is dominated by the factor , which explains why the energy level of dot-A should be chosen below and the presence of interdot Coulomb interactions is crucial. The negative sign of Q in the regime of indicates that the heat current is from the right electrode to the left electrode. We define the rectification efficiency as . We obtain for and 0.88 for . Fig. 2(c) shows DTC in units of . It is found that the rectification behavior is not very sensitive to the variation of . DTC is roughly linearly proportional to in the range . In addition, we also find a small negative differential thermal conductance (NDTC) for . Similar behavior was reported in the phonon junction system.
Fig. 3 shows the heat current, differential thermal conductance and thermal power () as functions of temperature difference for a three-QD case for various values of , while keeping . Here, we adopt instead of fixing at 0.3 to reflect the correlation of dot position with the asymmetric tunneling rates. We assume that the three QDs are roughly aligned with dot A in the middle. The energy levels of dots A, B and C are chosen to be , and . , , , and all other parameters are kept the same as in the two-dot case. The thermal rectification effect is most pronounced when as seen in Fig. 4(a). (Note that the heat current is not very sensitive to ). In this case, we obtain a small heat current at , but a large heat current at and the rectification efficiency is 0.79. However, the heat current for is small. For , we obtain at , at , and . We see that the heat current is suppressed for with decreasing . This implies that it is important to blockade the heat current through dot A to observe the rectification effect. Very clear NDTC is observed in Fig. 3(b) for the case, while DTC is symmetric with respect to for the case.
From the experimental point of view, it is easier to measure the thermal power than the direction-dependent heat current. The thermal power as a function of is shown in Fig. 3(c). All curves except the dash-dotted line (which is for the symmetrical tunneling case) show highly asymmetrical behavior with respect to , yet it is not easy at all to judge the efficiency of the rectification effect from for small (). Thus, it is not sufficient to determine whether a single QD can act as an efficient thermal rectifier based on results obtained in the linear response regime of . According to the thermal power values, the electrochemical potential can be very large. Consequently, the shift of QD energy levels caused by is quite important. To illustrate the importance of this effect, we plot in Fig. 4 the heat current for various values of for the case with , and . Other parameters are kept the same as those for Fig. 3. The solid (dashed) curves are obtained by including (excluding) the energy shift . It is seen that the shift of QD energy levels due to can lead to significant change in the heat current. It is found that NDTC is accompanied with low heat current for the case of [see Fig. 4(b)]. Even though the heat currents exhibits rectification effect for and , the thermal powers have very different behaviors. From Figs. 3(c) and 4(c), we see that the heat current is a highly nonlinear function of electrochemical potential . Consequently, the rectification effect is not straightforwardly related to the thermal power in this system.
Comparing the heat currents of the three-dot case (shown in Figs. 3 and 4) to the two-dot case (shown in Fig. 2), we find that the rectification efficiency is about the same for both cases, while the magnitude of the heat current can be significantly enhanced in the three-dot case. For practical applications, we need to estimate the magnitude of the heat current density and DTC of the IQV junction device in order to see if the effect is significant. We envision a thermal rectification device made of an array of multiple QDs (e.g. three-QD cells) with a 2D density . For this device, the heat current density versus is given by Figs. 3 and 4 with the units replaced by , which is approximately if we assume . Similarly, the units for DTC becomes , which is approximately . Since the phonon contribution can be blocked by the vacuum layer in our design, this device should have practical applications near with (). If we choose a higher tunneling rate and Coulomb energy (possible for QDs with diameter less than 1 nm), then it is possible to chieve room-temperature operation.
In summary, we have reported a design of multiple-QD junction which can have significant thermal rectification effect. The thermal rectification behavior is sensitive to the coupling between the QDs and the electrodes, the electron Coulomb interactions and the energy level differences between the dots.
This work was supported by Academia Sinica, Taiwan.
Email-address: firstname.lastname@example.org; email@example.com
Fig. 1. Schematic diagram of the isulator/quantum dots/vacuum tunnel junction device.
Fig. 2. (a) Heat current (b) average occupation number, and (c) differential thermal conductance as a function of for various values of for a two-QD junction. , and .
Fig. 3. (a) Heat current, (b) differential thermal conductance and (c) thermal power as a function of for various values of for a three-QD junction.
Fig. 4. (a) Heat current, (b) differential thermal conductance and (c) thermal power as functions of for various values of for a three-QD junction with and .
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