Thermal rectification properties of multiple-quantum-dot junctions David M.-T. Kuo1 and Yia-chung Chang2 1Department of Electrical Engineering and Department of Physics, National Central University, Chungli, 320 Taiwan and 2Research Center for Applied Sciences, Academic Sinica, Taipei, 115 Taiwan (Dated: January 8, 2010) Itisillustratedthatsemiconductorquantumdots(QDs)embeddedintoaninsulatingmatrixcon- nectedwith metallicelectrodes andsomevacuumspacecanlead tosignificant thermalrectification 0 effect. AmultilevelAndersonmodelisusedtoinvestigatethethermalrectificationpropertiesofthe 1 multiple-QDjunction. Thechargeandheatcurrentsinthetunnelingprocessarecalculated viathe 0 Keldysh Green’s function technique. We show that pronounced thermal rectification and negative 2 differential thermal conductance (NDTC) behaviors can be observed for the multiple-QD junction with asymmetrical tunnelingrates and strong interdot Coulomb interactions. n a J Records of thermal rectification date back to 1935 ] 8 wtiohnesncSatnardrisdpilsacyovaerethdertmhaatl cdoiopdpeerboexhiadvei/ocr.o1ppReercejnutnlcy-, Q= −h2XZ dǫγℓ(ǫ)ImGrℓ,σ(ǫ)(ǫ−EF−e∆V)fLR(ǫ), (2) ll thermal rectification effects have been predicted to oc- ℓ ha curinonedimensionalphononjunctionsystems.2−5Such where γℓ(ǫ) = ΓΓℓℓ,L,L((ǫǫ))+ΓΓℓ,ℓR,R(ǫ(ǫ)) is the transmission a thermal rectification effect is crucial for heat storage. factor. f (ǫ) = f (ǫ) − f (ǫ) and f (ǫ) = - LR L R L(R) s Scheibner and coworkers have experimentally observed 1/(exp(ǫ−µL(R))/(kBTL(R)) + 1) is the Fermi distribution e m the asymmetrical thermal power of the two-dimensional function for the left (right) electrode. The chemical po- electrongasinQDunderhighmagneticfields.6Sofarthe tential difference between these two electrodes is related t. rectificationmechanismofasingleQDis stillambiguous to the bias difference µ − µ = e∆V created by the a L R m owingtothe unclearrelationbetweenthe thermalpower temperature gradient. TL(TR) denotes the temperature and the thermal rectification effect. This inspires us to maintainedattheleft(right)lead. E =(µ +µ )/2de- F L R d- investigate whether the QD junction system can act as notesthe averageFermienergyofthe electrodes. Γℓ,L(ǫ) n a thermal rectifier. A useful thermal diode to store so- and Γℓ,R(ǫ) [Γℓ,β = 2π k|Vℓ,β,k|2δ(ǫ−ǫk)] denote the o lar heating energy not only requires a high rectification tunneling rates from thPe QDs to the left and right elec- c efficiency but also high heat flow. The later requires a trodes, respectively. e and h denote the electron charge [ high QD density in the QD junction thermal diode. The and Plank’s constant, respectively. For simplicity, these 1 main goal of this study is to illustrate that the multiple tunneling rates are assumed to be energy- and bias- v QDsembeddedintoaninsulatorconnectedwithmetallic independent. Eqs. (1) and (2) have been employed to 4 electrodesandwithavacuumlayerinsertcangiveriseto study the thermal properties of single-level QD in the 0 significant thermal rectification and negative differential Kondo regime.9 Here, our analysis is devoted to the 2 thermalconductance(NDTC)effectsinthenonlinearre- multiple-QD system in the Coulomb blockade regime. 1 . sponse regime. We also clarify the relation between the The expression of the retarded Green function for dot 1 thermal power and the rectification effect. ℓ of a multi-QD system, Gr (ǫ) can be found in Ref. [7] 0 The proposed isulator/quantum dots/vacuum (IQV) To study the direction-ℓd,σependent heat current, we 0 double barrier tunnel junction system (as illustrated in let T = T + ∆T/2 and T = T − ∆T/2, where 1 L 0 R 0 : Fig. 1)canbe adequatelydescribedby a multi-levelAn- T0 =(TL+TR)/2 is the equilibrium temperature of two v dersonmodel.7 Here,thevacuumlayerservesasablock- sideelectrodesand∆T =T −T isthetemperaturedif- i L R X inglayerforphononcontributionstothermalconduction, ference. Becausetheelectrochemicalpotentialdifference, r while allowing electrons to tunnel through. We assume e∆V yieldedbythethermalgradientcouldbesignificant, a thattheenergylevelseparationbetweenthegroundstate it is important to keep track the shift of the energy level andthe firstexcitedstatewithineachQDismuchlarger of each dot according to ǫ = E +η ∆V/2, where η ℓ ℓ ℓ ℓ thankBT,whereT isthetemperatureofconcern. There- is the ratio of the distance between dot ℓ and the mid fore,thereareonlyoneenergylevelforeachQD.Wehave plane of the QD junction to the junction width. Here ignoredtheinterdothoppingtermsduetothehighpoten- we set η = η = 0. A functional thermal rectifier re- B C tialbarrierseparatingQDs. The key effects includedare quires a good thermal conductance for ∆T > 0, but a the intradot and interdot Coulomb interactions and the poor thermal conductance for ∆T < 0. Based on Eqs. couplingbetweentheQDswiththemetallicleads. Using (1) and (2), the asymmetrical behavior of heat current the Keldysh-Green’sfunctiontechnique,8 the chargeand with respect to ∆T requires not only highly asymmetric heat currents through the junction can be expressed as coupling strengthes between the QDs and the electrodes but also strong electron Coulomb interactions between −2e J = dǫγ (ǫ)ImGr (ǫ)f (ǫ), (1) dots. To investigate the thermal rectification behavior, e h XZ ℓ ℓ,σ LR wehavenumericallysolvedEqs. (1)and(2)formultiple- ℓ 2 QD junctions involving two QDs and three QDs for var- η =(Q(∆T =30Γ)−|Q(∆T =−30Γ)|)/Q(∆T =30Γ). Q ious systemparameters. We firstdetermine ∆V by solv- We obtain η = 0.86 for E = E + 2∆E/5 and Q B F ing Eq. (1) with J = 0 (the open circuit condition) for 0.88 for E = E + 4∆E/5. Fig. 2(c) shows DTC e B F a given ∆T, T and an initial guess of the average one- in units of Q k /Γ. It is found that the rectification 0 0 B particle and two-particle occupancy numbers, N and c behavior is not very sensitive to the variation of E . ℓ ℓ B for each QD. Those numbers are then updated accord- DTC isroughlylinearlyproportionalto ∆T inthe range ing to Eqs. (5) and (6) in Ref. [7] until self-consistency −20Γ< k ∆T < 20Γ. In addition, we also find a small B is established. For the open circuit, the electrochemical negative differential thermal conductance (NDTC) for potentialwillbeformedduetochargetransfergenerated E = E +4∆E/5. Similar behavior was reported in B F by the temperature gradient. This electrochemical po- the phonon junction system.10 tential is known as the Seebeck voltage (Seebeck effect). Fig. 3 shows the heat current, differential thermal Once ∆V is solved, we then use Eq. (2) to compute the conductance and thermal power (S = e∆V/k ∆T) as B heat current. functions of temperature difference ∆T for a three-QD Fig. 2 shows the heat currents, occupation numbers, case for various values of Γ , while keeping Γ = AR B(C),R anddifferentialthermalconductance(DTC) forthe two- Γ = Γ. Here, we adopt η = |Γ −Γ |/(2Γ) B(C),L A AL AR QDcase,inwhichtheenergylevelsofdotAanddotBare insteadoffixingη at0.3toreflectthecorrelationofdot A E =E −∆E/5 and E =E +α ∆E, where α is A F B F B B position with the asymmetric tunneling rates. We as- tunedbetween0and1. Theheatcurrentsareexporessed sume that the three QDs are roughlyalignedwith dot A in units of Q = Γ2/(2h) through out this article. The 0 in the middle. The energy levels of dots A, B and C are intradotandinterdotCoulombinteractionsusedareU = ℓ chosen to be E = E − ∆E/5, E = E + 2∆E/5 A F B F 30k T and U = 15k T . The tunneling rates are B 0 AB B 0 and E = E + 3∆E/5. U = U = 15k T , C F AC BA B 0 Γ = 0, Γ = 2Γ, and Γ = Γ = Γ. k T is AR AL BR BL B 0 U = 8k T , U = 30k T , and all other parameters BC B 0 C B 0 chosen to be 25Γ throughout this article. Here, Γ = are kept the same as in the two-dot case. The thermal (Γ +Γ )/2 is the average tunneling rate in energy AL AR rectification effect is most pronounced when Γ = 0. AR units, whose typical values of interest are between 0.1 as seen in Fig. 4(a). (Note that the heat current is and 0.5 meV. The dashed curves are obtained by using not very sensitive to U ). In this case, we obtain a BC a simplified expression of Eq. (2) in which we set the small heat current Q = 0.068Q at ∆T = −30Γ, but a 0 average two particle occupation in dots A and B to zero large heat current Q = 0.33Q at ∆T = 30Γ and the 0 (resulting from the large intradot Coulomb interactions) rectificationefficiency η is 0.79. However,the heatcur- Q and taking the limit that Γ ≪ k T so the Lorentzian B 0 rent for Γ = 0 is small. For Γ = 0.1Γ, we obtain AR AR function of resonantchannels can be replaced by a delta Q = 1.69Q at ∆T = −30Γ,Q = 5.69Q at ∆T = 30Γ, 0 0 function. We have and η = 0.69. We see that the heat current is sup- Q pressed for ∆T < 0 with decreasing Γ . This im- Q/γ = π(1−N )[(1−2N )(E −E )f (E )(3) AR B B A B F LR B plies that it is important to blockade the heat current + 2N (E +U −E )f (E +U )], A B AB F LR B AB through dot A to observe the rectification effect. Very clear NDTC is observedin Fig. 3(b) for the Γ =0.1Γ HereN istheaverageoccupancyindotA(B).There- AR A(B) case,whileDTCissymmetricwithrespectto∆T forthe fore, it is expected that the curve corresponding to Γ =Γ case. E = E + 4∆E/5 obtained with this delta function AR AL B F approximation is in good agreement with the full solu- From the experimental point of view, it is easier to tion, since E is far away from the Fermi energy level. measurethethermalpowerthanthedirection-dependent B For cases when E is close to E , the approximation is heat current. The thermal power as a function of ∆T is B F notasgood,butitstillgivesqualitativelycorrectbehav- showninFig. 3(c). Allcurvesexceptthedash-dottedline ior. Thus,itisconvenienttousethissimpleexpressionto (whichisforthesymmetricaltunnelingcase)showhighly illustrate the thermal rectification behavior. The asym- asymmetrical behavior with respect to ∆T, yet it is not metrical behavior of N with respect to ∆T is mainly easyatalltojudgetheefficiencyoftherectificationeffect A resulted from the condition Γ = 0 and Γ = 2Γ. from S for small |∆T| (k |∆T|/Γ<10). Thus, it is not AR AL B The heat current is contributed from the resonant chan- sufficienttodeterminewhetherasingleQDcanactasan nel with ǫ = E , because the resonant channel with efficientthermalrectifierbasedonresultsobtainedinthe B ǫ=E +U is too high in energy compared with E . linear response regime of ∆T/T ≪1.6 According to the B AB F 0 The sign of Q is determined by f (E ), which indi- thermalpowervalues,theelectrochemicalpotentiale∆V LR B rectly depends on Coulomb interactions, tunneling rate can be very large. Consequently, the shift of QD energy ratioandQDenergylevels. Therectificationbehaviorof levelscausedby∆V isquiteimportant. Toillustratethe Qisdominatedbythefactor1−2N ,whichexplainswhy importance of this effect, we plot in Fig. 4 the heat cur- A theenergylevelofdot-AshouldbechosenbelowE and rent for various values of E for the case with Γ =0, F C AR the presence of interdot Coulomb interactions is crucial. U =10k T andη =0.3. Otherparametersarekept BC B 0 A The negative sign of Q in the regime of ∆T < 0 indi- the same as those for Fig. 3. The solid (dashed) curves cates that the heat current is from the right electrode to are obtained by including (excluding) the energy shift theleftelectrode. Wedefinetherectificationefficiencyas η ∆V/2. ItisseenthattheshiftofQDenergylevelsdue A 3 to ∆V canleadtosignificantchangeinthe heatcurrent. current density versus ∆T is given by Figs. 3 and 4 It is found that NDTC is accompanied with low heat with the units Q replaced by N Q , which is approxi- 0 2d 0 current for the case of E =E +∆E/5 [see Fig. 4(b)]. mately 965W/m2 if we assume Γ = 0.5meV. Similarly, C F Even though the heat currents exhibits rectification ef- theunitsforDTCbecomesN k Q /Γ,whichisapprox- 2d B 0 fect for E = E +∆E/5 and E = E +3∆E/5, the imately 34W/0Km2. Since the phonon contribution can C F C F thermalpowershaveverydifferentbehaviors. FromFigs. be blocked by the vacuum layer in our design, this de- 3(c)and4(c),weseethattheheatcurrentisahighlynon- vice should have practical applications near 1400K with linear function of electrochemical potential ∆V. Conse- (k T ≈12.5meV). If we choose a higher tunneling rate B 0 quently, the rectification effect is not straightforwardly Γ>1meV and Coulombenergy >300meV (possible for related to the thermal power in this system. QDswithdiameterlessthan1nm),thenitispossibleto Comparing the heat currents of the three-dot case chieve room-temperature operation. (shown in Figs. 3 and 4) to the two-dot case (shown in Insummary,wehavereportedadesignofmultiple-QD Fig. 2), we find that the rectification efficiency is about junction which can have significant thermal rectification the samefor bothcases,while the magnitude ofthe heat effect. The thermal rectification behavior is sensitive to current can be significantly enhanced in the three-dot the coupling between the QDs and the electrodes, the case. For practical applications, we need to estimate the electron Coulomb interactions and the energy level dif- magnitude of the heat current density and DTC of the ferences between the dots. IQV junction device in order to see if the effect is signif- Acknowledgments icant. We envision a thermal rectification device made ThisworkwassupportedbyAcademiaSinica,Taiwan. of an array of multiple QDs (e.g. three-QD cells) with a Email-address: [email protected]; yi- 2D density N = 1011cm−2. For this device, the heat [email protected] 2d 1 C. Starr, J. 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Electrode at T R vacuum C B Quantum dots A insulator w o l Electrode at T f L t a e H Substrate Fig. 1 0.3 (a) E =E +2D E/5 ) 0 0.2 B F Q E =E +4D E/5 ( 0.1 B F Q 0.0 0.5 (b) 0.4 N A N 0.3 0.05 N B 0.00 ) GGGG/ (c) E =E +2D E/5 0 0.008 Q B F E =E +4D E/5 B k 0.004 B F ( NDTC C 0.000 T D -30 -20 -10 0 10 20 30 kD DDD TG /GGG Fig2 B (a) 4 ) 0 Q 0 ( Q -4 G =1G , G =1G AR AL ) GGGG/ (b) 0 0.2 Q B k 0.1 G =0.5, G =1.5G ( AR AL C 0.0 T D 2 1 ) (c) e 0 / B -1 G =0, G =2G k G =0.1, G =1.9G ( -2 AR AL AR AL S -3 -30 -20 -10 0 10 20 30 kD DDD TG /GGG Fig3 B 1.2 (a) ) 0 E =E +D E/5 Q 0.8 C F ( 0.4 E =E +3D E/5 Q C F 0.0 ) 0.08 GGGG/ 0 (b) Q E =E +D E/5 B 0.04 C F k E =E +3D E/5 ( NDTC C F C 0.00 T D -1.6 -2.0 ) (c) e E =E +D E/5 / -2.4 B C F k E =E +3D E/5 -2.8 ( C F S -3.2 -30 -20 -10 0 10 20 30 kD DDD TG /GGG Fig4 B