to Δ V can lead to significant change in the heat current. It is found that NDTC is accompanied with low heat current for the case of E$_{C}$ = E$_{F}$ + Δ E/ 5 [see Fig. 4(b)]. Even though the heat currents exhibits rectification ef- fect for E$_{C}$ = E$_{F}$ + Δ E/ 5 and E$_{C}$ = E$_{F}$ + 3Δ E/ 5, the thermal powers have very different behaviors. From Figs. 3(c) and 4(c), we see that the heat current is a highly non- linear function of electrochemical potential Δ V . Conse- quently, 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 signif- icant. We envision a thermal rectification device made of an array of multiple QDs (e.g. three-QD cells) with a 2D density N$_{2}$$_{d}$ = 10 $^{11}$cm − $^{2}$. For this device, the heat 

1 C. Starr, J. Appl. Phys. 7, 15 (1936). 2 

M. Terraneo, M. Peyrard, G. Casati, Phys. Rev. Lett. 88 , 094302 (2002). 3 

Baowen Li, L. Wang and G. Casati, Phys. Rev. Lett. 93 , 184301 (2004). 4 

B. Hu, L. Yang and Y. Zhang, Phys. Rev. Lett. 97 , 124302 (2006). 5 

N. Yang, G. Zhang and B. Li, Appl. Phys. Lett. 95, 033107 (2009). 6 

R. Scheibner, M. Konig, D. Reuter, A. D. Wieck, C. Gould, H. Buhmann and L. W. Molenkamp, New. J. Phys. 10 , 083016 (2008). 7 

D. M. T. Kuo and Y. C. Chang, Phys. Rev. Lett. 99 , 086803 (2007). 8 

H. Haug and A. P. Jauho, Quantum Kinetics in Trans- port and Optics of Semiconductors (Springer, Heidelberg, 1996). 9 

M. Krawiec and K. I. Wysokinski, Phys. Rev. B 75 , 155330 (2007). 

current density versus Δ T is given by Figs. 3 and 4 with the units Q$_{0}$ replaced by N$_{2}$$_{d}$Q$_{0}$ , which is approxi- mately 965 W/m 2 if we assume Γ = 0 . 5 meV . Similarly, the units for DTC becomes N$_{2}$$_{d}$k$_{B}$Q$_{0}$/ Γ, which is approx- imately 34 W/ $^{0}$Km$^{2}$. Since the phonon contribution can be blocked by the vacuum layer in our design, this de- vice should have practical applications near 140 $^{0}$K with ( k$_{B}$T$_{0}$ ≈ 12 . 5 meV ). If we choose a higher tunneling rate Γ > 1 meV and Coulomb energy > 300 meV (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 dif- ferences between the dots. 

## Acknowledgments 

This work was supported by Academia Sinica, Taiwan. Email-address: mtkuo@ee.ncu.edu.tw; yi- achang@gate.sinica.edu.tw 

10 D. Segal, Phys. Rev. B 73 , 205415 (2006). 

## Figure Captions 

Fig. 1. Schematic diagram of the isulator/quantum dots/vacuum tunnel junction device. 

Fig. 2. (a) Heat current (b) average occupation num- ber, and (c) differential thermal conductance as a func- tion of Δ T for various values of E$_{B}$ for a two-QD junc- tion. Γ$_{AR}$ = 0, η$_{A}$ = 0 . 3 and Δ E = 200Γ. 

Fig. 3. (a) Heat current, (b) differential thermal con- ductance and (c) thermal power as a function of Δ T for various values of Γ$_{AR}$ for a three-QD junction. 

Fig. 4. (a) Heat current, (b) differential thermal conductance and (c) thermal power as functions of Δ T for various values of E$_{C}$ for a three-QD junction with Γ$_{AR}$ = 0 and η$_{A}$ = 0 . 3. 

