Thermal rectification properties of multiple-quantum-dot junctions
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Thermal rectification properties of multiple-quantum-dot junctions
David M.-T. Kuo and Yia-chung Chang Department of Electrical Engineering and Department of Physics,National Central University, Chungli, 320 Taiwan and 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 rectificationeffect. A multilevel Anderson model is used to investigate the thermal rectification properties of themultiple-QD junction. The charge and heat currents in the tunneling process are calculated via theKeldysh Green’s function technique. We show that pronounced thermal rectification and negativedifferential thermal conductance (NDTC) behaviors can be observed for the multiple-QD junctionwith asymmetrical tunneling rates and strong interdot Coulomb interactions.
Records of thermal rectification date back to 1935when Starr discovered that copper oxide/copper junc-tions can display a thermal diode behavior. Recently,thermal rectification effects have been predicted to oc-cur in one dimensional phonon junction systems. − Sucha thermal rectification effect is crucial for heat storage.Scheibner and coworkers have experimentally observedthe asymmetrical thermal power of the two-dimensionalelectron gas in QD under high magnetic fields. So far therectification mechanism of a single QD is still ambiguousowing to the unclear relation between the thermal powerand the thermal rectification effect. This inspires us toinvestigate whether the QD junction system can act asa thermal rectifier. A useful thermal diode to store so-lar heating energy not only requires a high rectificationefficiency but also high heat flow. The later requires ahigh QD density in the QD junction thermal diode. Themain goal of this study is to illustrate that the multipleQDs embedded into an insulator connected with metallicelectrodes and with a vacuum layer insert can give rise tosignificant thermal rectification and negative differentialthermal conductance (NDTC) effects in the nonlinear re-sponse regime. We also clarify the relation between thethermal power and the rectification effect.The proposed isulator/quantum dots/vacuum (IQV)double barrier tunnel junction system (as illustrated inFig. 1) can be adequately described by a multi-level An-derson model. Here, the vacuum layer serves as a block-ing layer for phonon contributions to thermal conduction,while allowing electrons to tunnel through. We assumethat the energy level separation between the ground stateand the first excited state within each QD is much largerthan k B T , where T is the temperature of concern. There-fore, there are only one energy level for each QD. We haveignored the interdot hopping terms due to the high poten-tial barrier separating QDs. The key effects included arethe intradot and interdot Coulomb interactions and thecoupling between the QDs with the metallic leads. Usingthe Keldysh-Green’s function technique, the charge andheat currents through the junction can be expressed as J e = − eh X ℓ Z dǫγ ℓ ( ǫ ) ImG rℓ,σ ( ǫ ) f LR ( ǫ ) , (1) Q = − h X ℓ Z dǫγ ℓ ( ǫ ) ImG rℓ,σ ( ǫ )( ǫ − E F − e ∆ V ) f LR ( ǫ ) , (2) where γ ℓ ( ǫ ) = Γ ℓ,L ( ǫ )Γ ℓ,R ( ǫ )Γ ℓ,L ( ǫ )+Γ ℓ,R ( ǫ ) is the transmissionfactor. f LR ( ǫ ) = f L ( ǫ ) − f R ( ǫ ) and f L ( R ) ( ǫ ) =1 / ( exp ( ǫ − µ L ( R ) ) / ( k B T L ( R ) ) + 1) is the Fermi distributionfunction for the left (right) electrode. The chemical po-tential difference between these two electrodes is relatedto the bias difference µ L − µ R = e ∆ V created by thetemperature gradient. T L ( T R ) denotes the temperaturemaintained at the left (right) lead. E F = ( µ L + µ R ) / ℓ,L ( ǫ )and Γ ℓ,R ( ǫ ) [Γ ℓ,β = 2 π P k | V ℓ,β, k | δ ( ǫ − ǫ k )] denote thetunneling rates from the QDs to the left and right elec-trodes, respectively. e and h denote the electron chargeand Plank’s constant, respectively. For simplicity, thesetunneling rates are assumed to be energy- and bias-independent. Eqs. (1) and (2) have been employed tostudy the thermal properties of single-level QD in theKondo regime. Here, our analysis is devoted to themultiple-QD system in the Coulomb blockade regime.The expression of the retarded Green function for dot ℓ of a multi-QD system, G rℓ,σ ( ǫ ) can be found in Ref. [7]To study the direction-dependent heat current, welet T L = T + ∆ T / T R = T − ∆ T /
2, where T = ( T L + T R ) / T = T L − T R is the temperature dif-ference. Because the electrochemical potential difference, e ∆ V yielded by the thermal gradient could be significant,it is important to keep track the shift of the energy levelof each dot according to ǫ ℓ = E ℓ + η ℓ ∆ V /
2, where η ℓ is the ratio of the distance between dot ℓ and the midplane of the QD junction to the junction width. Herewe set η B = η C = 0. A functional thermal rectifier re-quires a good thermal conductance for ∆ T >
0, but apoor thermal conductance for ∆
T <
0. Based on Eqs.(1) and (2), the asymmetrical behavior of heat currentwith respect to ∆ T requires not only highly asymmetriccoupling strengthes between the QDs and the electrodesbut also strong electron Coulomb interactions betweendots. 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 var-ious system parameters. We first determine ∆ V by solv-ing Eq. (1) with J e = 0 (the open circuit condition) fora given ∆ T , T and an initial guess of the average one-particle and two-particle occupancy numbers, N ℓ and c ℓ for each QD. Those numbers are then updated accord-ing to Eqs. (5) and (6) in Ref. [7] until self-consistencyis established. For the open circuit, the electrochemicalpotential will be formed due to charge transfer generatedby the temperature gradient. This electrochemical po-tential is known as the Seebeck voltage (Seebeck effect).Once ∆ V is solved, we then use Eq. (2) to compute theheat 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 E A = E F − ∆ E/ E B = E F + α B ∆ E , where α B istuned between 0 and 1. The heat currents are exporessedin units of Q = Γ / (2 h ) through out this article. Theintradot and interdot Coulomb interactions used are U ℓ =30 k B T and U AB = 15 k B T . The tunneling rates areΓ AR = 0, Γ AL = 2Γ, and Γ BR = Γ BL = Γ. k B T ischosen to be 25Γ throughout this article. Here, Γ =(Γ AL + Γ AR ) / ≪ k B T so the Lorentzianfunction of resonant channels can be replaced by a deltafunction. We have Q/γ B = π (1 − N B )[(1 − N A )( E B − E F ) f LR ( E B )(3)+ 2 N A ( E B + U AB − E F ) f LR ( E B + U AB )] , Here N A ( B ) is the average occupancy in dot A(B). There-fore, it is expected that the curve corresponding to E B = E F + 4∆ E/ E B is far away from the Fermi energy level.For cases when E B is close to E F , the approximation isnot as good, but it still gives qualitatively correct behav-ior. Thus, it is convenient to use this simple expression toillustrate the thermal rectification behavior. The asym-metrical behavior of N A with respect to ∆ T is mainlyresulted from the condition Γ AR = 0 and Γ AL = 2Γ.The heat current is contributed from the resonant chan-nel with ǫ = E B , because the resonant channel with ǫ = E B + U AB is too high in energy compared with E F .The sign of Q is determined by f LR ( E B ), which indi-rectly depends on Coulomb interactions, tunneling rateratio and QD energy levels. The rectification behavior of Q is dominated by the factor 1 − N A , which explains whythe energy level of dot-A should be chosen below E F andthe presence of interdot Coulomb interactions is crucial.The negative sign of Q in the regime of ∆ T < η Q = ( Q (∆ T = 30Γ) −| Q (∆ T = − | ) /Q (∆ T = 30Γ).We obtain η Q = 0 .
86 for E B = E F + 2∆ E/ E B = E F + 4∆ E/
5. Fig. 2(c) shows DTCin units of Q k B / Γ. It is found that the rectificationbehavior is not very sensitive to the variation of E B .DTC is roughly linearly proportional to ∆ T in the range − < k B ∆ T < E B = E F + 4∆ E/
5. Similar behavior was reported inthe phonon junction system. Fig. 3 shows the heat current, differential thermalconductance and thermal power ( S = e ∆ V /k B ∆ T ) asfunctions of temperature difference ∆ T for a three-QDcase for various values of Γ AR , while keeping Γ B ( C ) ,R =Γ B ( C ) ,L = Γ. Here, we adopt η A = | Γ AL − Γ AR | / (2Γ)instead of fixing η A at 0.3 to reflect the correlation of dotposition with the asymmetric tunneling rates. We as-sume that the three QDs are roughly aligned with dot Ain the middle. The energy levels of dots A, B and C arechosen to be E A = E F − ∆ E/ E B = E F + 2∆ E/ E C = E F + 3∆ E/ U AC = U BA = 15 k B T , U BC = 8 k B T , U C = 30 k B T , and all other parametersare kept the same as in the two-dot case. The thermalrectification effect is most pronounced when Γ AR = 0 . as seen in Fig. 4(a). (Note that the heat current isnot very sensitive to U BC ). In this case, we obtain asmall heat current Q = 0 . Q at ∆ T = − Q = 0 . Q at ∆ T = 30Γ and therectification efficiency η Q is 0.79. However, the heat cur-rent for Γ AR = 0 is small. For Γ AR = 0 . Q = 1 . Q at ∆ T = − Q = 5 . Q at ∆ T = 30Γ,and η Q = 0 .
69. We see that the heat current is sup-pressed for ∆
T < AR . This im-plies that it is important to blockade the heat currentthrough dot A to observe the rectification effect. Veryclear NDTC is observed in Fig. 3(b) for the Γ AR = 0 . T for theΓ AR = Γ AL case.From the experimental point of view, it is easier tomeasure the thermal power than the direction-dependentheat current. The thermal power as a function of ∆ T isshown in Fig. 3(c). All curves except the dash-dotted line(which is for the symmetrical tunneling case) show highlyasymmetrical behavior with respect to ∆ T , yet it is noteasy at all to judge the efficiency of the rectification effectfrom S for small | ∆ T | ( k B | ∆ T | / Γ < T /T ≪ According to thethermal power values, the electrochemical potential e ∆ V can be very large. Consequently, the shift of QD energylevels caused by ∆ V is quite important. To illustrate theimportance of this effect, we plot in Fig. 4 the heat cur-rent for various values of E C for the case with Γ AR = 0, U BC = 10 k B T and η A = 0 .
3. Other parameters are keptthe same as those for Fig. 3. The solid (dashed) curvesare obtained by including (excluding) the energy shift η A ∆ V /
2. It is seen that the shift of QD energy levels dueto ∆ V can lead to significant change in the heat current.It is found that NDTC is accompanied with low heatcurrent for the case of E C = E F + ∆ E/ E C = E F + ∆ E/ E C = E F + 3∆ E/
5, thethermal 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 ∆ V . Conse-quently, the rectification effect is not straightforwardlyrelated 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 inFig. 2), we find that the rectification efficiency is aboutthe same for both cases, while the magnitude of the heatcurrent can be significantly enhanced in the three-dotcase. For practical applications, we need to estimate themagnitude of the heat current density and DTC of theIQV junction device in order to see if the effect is signif-icant. We envision a thermal rectification device madeof an array of multiple QDs (e.g. three-QD cells) with a2D density N d = 10 cm − . For this device, the heat current density versus ∆ T is given by Figs. 3 and 4with the units Q replaced by N d Q , which is approxi-mately 965 W/m if we assume Γ = 0 . meV . Similarly,the units for DTC becomes N d k B Q / Γ, which is approx-imately 34 W/ Km . Since the phonon contribution canbe blocked by the vacuum layer in our design, this de-vice should have practical applications near 140 K with( k B T ≈ . meV ). If we choose a higher tunneling rateΓ > meV and Coulomb energy > meV (possible forQDs with diameter less than 1 nm), then it is possible tochieve room-temperature operation.In summary, we have reported a design of multiple-QDjunction which can have significant thermal rectificationeffect. The thermal rectification behavior is sensitive tothe coupling between the QDs and the electrodes, theelectron Coulomb interactions and the energy level dif-ferences between the dots. Acknowledgments
This work was supported by Academia Sinica, Taiwan.Email-address: [email protected]; [email protected] C. Starr, J. Appl. Phys. 7, (1936). M. Terraneo, M. Peyrard, G. Casati, Phys. Rev. Lett. ,094302 (2002). Baowen Li, L. Wang and G. Casati, Phys. Rev. Lett. ,184301 (2004). B. Hu, L. Yang and Y. Zhang, Phys. Rev. Lett. , 124302(2006). N. Yang, G. Zhang and B. Li, Appl. Phys. Lett. 95, 033107(2009). R. Scheibner, M. Konig, D. Reuter, A. D. Wieck, C. Gould,H. Buhmann and L. W. Molenkamp, New. J. Phys. ,083016 (2008). D. M. T. Kuo and Y. C. Chang, Phys. Rev. Lett. ,086803 (2007). H. Haug and A. P. Jauho,
Quantum Kinetics in Trans-port and Optics of Semiconductors (Springer, Heidelberg,1996). M. Krawiec and K. I. Wysokinski, Phys. Rev. B , 155330(2007). D. Segal, Phys. Rev. B , 205415 (2006). Figure Captions
Fig. 1. Schematic diagram of the isulator/quantumdots/vacuum tunnel junction device.Fig. 2. (a) Heat current (b) average occupation num-ber, and (c) differential thermal conductance as a func-tion of ∆ T for various values of E B for a two-QD junc-tion. Γ AR = 0, η A = 0 . E = 200Γ.Fig. 3. (a) Heat current, (b) differential thermal con-ductance and (c) thermal power as a function of ∆ T forvarious values of Γ AR for a three-QD junction.Fig. 4. (a) Heat current, (b) differential thermalconductance and (c) thermal power as functions of ∆ T for various values of E C for a three-QD junction withΓ AR = 0 and η A = 0 . lectrode at T R Electrode at T L Quantum dotsSubstrate
Fig. 1insulatorAB Cvacuum H ea t f l o w
30 -20 -10 0 10 20 300.0000.0040.008 NDTC (c)
Fig2 E B =E F +2 D E/5 E B =E F +4 D E/5 D T C ( k B Q / G ) k B D T/ G (b) N B N A N E B =E F +4 D E/5E B =E F +2 D E/5 (a) Q ( Q )
30 -20 -10 0 10 20 30-3-2-1012 (c) G AR =1 G , G AL =1 G G AR =0.5, G AL =1.5 GG AR =0.1, G AL =1.9 GG AR =0, G AL =2 G Fig3 S ( k B / e ) k B D T/ G (b) D T C ( k B Q / G ) -404 (a) Q ( Q )
30 -20 -10 0 10 20 30-3.2-2.8-2.4-2.0-1.6
Fig4(c) S ( k B / e ) k B D T/ G (b) D T C ( k B Q / G ) E C =E F + D E/5E C =E F +3 D E/5E C =E F +3 D E/5 E C =E F + D E/5E C =E F +3 D E/5 E C =E F + D E/5(a) Q ( Q0