arXiv:1001.1204v1 [cond-mat.mes-hall] 8 Jan 2010 Thermal rectification properties of multiple-quantum-dot junctions David M.-T. Kuo 1 and Yia-chung Chang 2 $^{1}$Department of Electrical Engineering and Department of Physics, National Central University, Chungli, 320 Taiwan and $^{2}$Research Center for Applied Sciences, Academic Sinica, Taipei, 115 Taiwan (Dated: November 21, 2018) It is illustrated that semiconductor quantum dots (QDs) embedded into an insulating matrix con- nected 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. 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 An- derson model. 7 Here, the vacuum layer serves as a block- ing 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 k$_{B}$T , where T is the temperature of concern. There- fore, there are only one energy level for each QD. We have ignored the interdot hopping terms due to the high poten- tial 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, 8 the charge and heat currents through the junction can be expressed as To study the direction-dependent heat current, we let T$_{L}$ = T$_{0}$ + Δ T/ 2 and T$_{R}$ = T$_{0}$ − Δ T/ 2, where T$_{0}$ = ( T$_{L}$ + T$_{R}$ ) / 2 is the equilibrium temperature of two side electrodes and Δ T = T$_{L}$ − T$_{R}$ is the temperature dif- ference. Because the electrochemical potential difference, e Δ V 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 ϵ$_{ℓ}$ = E$_{ℓ}$ + η$_{ℓ}$ Δ V/ 2, where η$_{ℓ}$ is the ratio of the distance between dot ℓ and the mid plane of the QD junction to the junction width. Here we set η$_{B}$ = η$_{C}$ = 0. A functional thermal rectifier re- quires a good thermal conductance for Δ T > 0, but a poor thermal conductance for Δ T < 0. Based on Eqs. (1) and (2), the asymmetrical behavior of heat current with respect to Δ T 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- Records of thermal rectification date back to 1935 when Starr discovered that copper oxide/copper junc- tions can display a thermal diode behavior. 1 Recently, thermal rectification effects have been predicted to oc- cur in one dimensional phonon junction systems. 2 − 5 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. 6 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 so- lar 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 re- sponse regime. We also clarify the relation between the thermal power and the rectification effect. where γ$_{ℓ}$ ( ϵ ) = Γ$_{ℓ,L}$( ϵ )Γ$_{ℓ,R}$( ϵ ) Γ$_{ℓ,L}$( ϵ )+Γ$_{ℓ,R}$( ϵ ) is the transmission factor. f$_{LR}$ ( ϵ ) = f$_{L}$ ( ϵ ) − f$_{R}$ ( ϵ ) and f$_{L}$$_{(}$$_{R}$$_{)}$ ( ϵ ) = 1 / ( exp ( ϵ − µ$_{L}$$_{(}$$_{R}$$_{)}$ ) / ( k$_{B}$T$_{L}$$_{(}$$_{R}$$_{)}$ ) + 1) is the Fermi distribution function for the left (right) electrode. The chemical po- tential difference between these two electrodes is related to the bias difference µ$_{L}$ − µ$_{R}$ = e Δ V created by the temperature gradient. T$_{L}$ ( T$_{R}$ ) denotes the temperature maintained at the left (right) lead. E$_{F}$ = ( µ$_{L}$ + µ$_{R}$ ) / 2 de- notes the average Fermi energy of the electrodes. Γ$_{ℓ,L}$( ϵ ) and Γ$_{ℓ,R}$( ϵ ) [Γ$_{ℓ,β}$ = 2 π ∑ $_{k}$| V$_{ℓ,β,}$$_{k}$ | $^{2}$δ ( ϵ − ϵ$_{k}$ )] denote the tunneling rates from the QDs to the left and right elec- trodes, respectively. e and h 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. 9 Here, our analysis is devoted to the multiple-QD system in the Coulomb blockade regime. The expression of the retarded Green function for dot ℓ of a multi-QD system, G r $_{ℓ,σ}$( ϵ ) can be found in Ref. [7] Q = − 2 h ∑ ℓ ∫ dϵγ$_{ℓ}$ ( ϵ ) ImG r $_{ℓ,σ}$( ϵ )( ϵ$_{−}$ E$_{F}$ $_{−}$e Δ V ) f$_{LR}$ ( ϵ ) , (2) J$_{e}$ = − 2 e h ∑ ℓ ∫ dϵγ$_{ℓ}$ ( ϵ ) ImG r $_{ℓ,σ}$( ϵ ) f$_{LR}$ ( ϵ ) , (1)