Thermoelectric properties of a chain of coupled quantum dots embedded in a nanowire 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: September5, 2012) The thermoelectric properties of a chain of coupled quantum dots (CCQD) embedded in a nanowire are theoretically investigated in the Coulomb blockade regime. An extended Hubbard 2 model is employed to simulate the CCQD nanowire system. The charge and heat currents are 1 calculated in the framework of Keldysh Green’s function technique. We obtained a closed-form 0 Landauer expression for the transmission coefficient of the CCQD system. The electrical conduc- 2 tance (Ge), Seebeck coefficient (S), thermal conductance, and figure of merit (ZT) are numerically p calculated andanalyzed inthelinearresponseregime. Whenthermalconductanceisdominatedby e phonon carriers, the optimization of ZT is determined by the power factor (pF =S2Ge). We find S that off-resonant tunneling processes, asymmetrical interdot electron Coulomb interactions, weak interdot hopping strengths and asymmetrical tunneling rates between QDs and electrodes are not 4 favored in theoptimization of ZT as a result they suppress thepower factor of junction system. ] l l a I. INTRODUCTION nitenumberofcoupledquantumdots(CQDs)embedded h inananowireandconnectedtometallicelectrodes,which - allow carrier injection into the CQDs. Thermoelectric s Recently, considerable studies have been devoted to e properties of a single QD and double QDs embedded in seeking efficient thermoelectric materials with the figure m amatrixconnectedtoelectrodeswerepreviouslystudied ofmerit(ZT)largerthan3becausethere existpotential . by several efforts.10−13 However, in realistic QD junc- t applications of solid state thermal devices such as cool- a ersandpowergenerators.1−7Theoptimizationof(ZT = tions for thermoelectric application, one needs to con- m sider a large number of serially coupled QDs, otherwise S2G T/κ) depends on the thermoelectric response func- - tionse; electrical conductance (G ), Seebeck coefficient it is not easy to keep the large temperature difference d e cross junction, which is crucial in the implementation of n (S), and thermal conductance (κ). T is the equilibrium high efficiency thermoelectric devices.2 o temperature. These thermoelectrical response functions c are related to one another. Mechanisms leading to the Here we consider nanoscale semiconductor QDs, in [ enhancementofpowerfactor(pF =S2Ge)wouldalsoen- which the energy level separations are much larger than 1 hance the thermalconductance. Consequently,itis diffi- their on-site Coulombinteractions and thermal energies. v cult to find ZT above 1 in conventional bulk materials.1 Thus, only one energy level for each quantum dot needs 6 Nanotechnology development provides a possible means to be considered. An N-level Anderson model is em- 0 to achieve highly efficient thermoelectric materials. Re- ployed to simulate the system as shown Fig. 1. Because 5 cently, ithas beendemonstratedthatZT’s ofnanostruc- wetakeintoaccountathickinsulatormatrixbetweentwo 0 . ture composites can reachimpressivevalues (largerthan metallic leads (for instance SiO2 or SiN), direct tunnel- 9 one).8 In particular, quantum dot superlattice (QDSL) ing currents between two electrodes are prohibited. In 0 nanowires exhibit an interesting thermoelectric property general, germanium or silicon QDs can be well growth 2 inthatthepowerfactorandthermalconductancebecome in SiO (SiN) matrix. The system illustrated in Fig. 1 2 : unrelated.8 Based on this property, one can increase the 1 may find promising applications as thermoelectric de- v power factor and decrease the thermal conductance si- vices. When N is infinite, it can be regarded as model i X multaneously to optimize ZT. A ZT value close to 2 in systemtoclarifyfundamentalphysicsofone-dimensional r PbSe/PbTe QDSL system was reported.6 strongly-correlated systems, which is one of the most a The reduction of thermal conductance of QDSL was challengingproblemsincondensatematterphysics.14For attributed to the increase of phononscattering rates,re- finite number of QDs, the transport properties of such sultingfromphononscatteringfromthenanowiresurface QD molecular is complicate due to nonneglecting elec- states and QD interface states.1,2 Nevertheless, the rela- tron Coulomb interactions. As a consequence, it is very tionship between the thermoelectric response functions tedioustodotheoptimizationofZT.Ouranalyticalform and QD quantum confinement effect including intradot of charge and heat formula provides an efficiency means andinterdotCoulombinteractions,electroninterdothop- to deal such difficulties. The derived transport formula ping, and QD size fluctuation remains unclear even still valid in the nonequilibrium regime. In this study, thoughthermoelectricpropertiesofQDSLhavebeenthe- we find that the optimization of ZT value favors the fol- oretically studied by solving the Boltzmann equation.9 lowing conditions:(1) QDs with low energy level fluctu- In ref.[9] authors considered infinite long nanowire and ations, (2) QD energy levels lie above the Fermi level ignored electron Coulomb interactions. In the present of electrodes, (3) Γ < t U , where t , U , and Γ c 0 c 0 ≪ study, we investigate the thermoelectric properties of fi- are electron interdot hopping strength, on-site electron 2 Coulomb interaction, and tunneling rate, respectively, and Planck’s constant, respectively. Obviously, tun- and (4) Γ = Γ with Γ + Γ kept constant, where neling currents are determined by the on-site retarded L R L R Γ (Γ ) is the left (right) tunneling rate. Green’s function (Gr (ǫ)) and lesser Green’s func- L R 1(N),σ tion (G< (ǫ)). It is not trivial to solve N coupled 1(N),σ QDs with electron Coulomb interactions (U and U ) II. FORMALISM ℓ ℓ,j and electron interdot hopping (t = t ). Conventional ℓ,j c Hartree-Fork mean field theory can not resolve the de- The Hamiltonian of N coupled QDs connected to tailed quantum pathes. Our previous approach beyond metallic electrodes can be described by the combina- the mean-field approximationcan obtain all Green func- tion of extended Hubbard model and Anderson model tions in the closed form solutions for N=2 in the limit H =H0+HQD: of tc/U 1.12,13 Based on the same approach, we have ≪ demonstrated that the charge and heat currents of CC- QDs can be expressed as13 H = ǫ a† a + ǫ b† b (1) 0 k k,σ k,σ k k,σ k,σ X X k,σ k,σ + XVk,1d†1,σak,σ+XVk,Nd†N,σbk,σ+c.c J = 2heZ dǫT(ǫ)[fL(ǫ)−fR(ǫ)], (5) k,σ k,σ 2 Q = dǫ (ǫ)(ǫ E e∆V)[f (ǫ) f (ǫ)],(6) wherethefirsttwotermsdescribethefreeelectrongasof hZ T − F − L − R left and right metallic electrodes. a† (b† ) creates an k,σ k,σ electronofmomentumkandspinσ withenergyǫk inthe whereT(ǫ)≡(T1,N(ǫ)+TN,1(ǫ))/2isthetransmissionco- left (right) metallic electrode. Vk,ℓ (ℓ = 1,N) describes efficient. Tℓ,j(ǫ)denotesthetransmissionfunction,which thecouplingbetweenthemetallicelectrodesandthefirst canbeexpressedintermsoftheon-siteretardedGreen’s (N-th) QD. d†ℓ,σ (dℓ,σ) creates (destroys) an electron in functions, even though Tℓ,j(ǫ) should be calculated by the ℓ-th dot. the on-site retarded and lesser Green’s functions. The transmissionfunction inthe weakinterdothoppinglimit (t U) has the following form, ij HQD = Eℓnℓ,σ+ Uℓnℓ,σnℓ,σ¯ (2) ≪ Xℓ,σ Xℓ 2×4N−1 Γ (ǫ)Γm (ǫ) + 21ℓ,Xj,σ,σ′Uℓ,jnℓ,σnj,σ′+ℓX,j,σtℓ,jd†ℓ,σdj,σ, T1,N(ǫ)=−2 mX=1 Γ1(1ǫ)+Γ1,m1N,N(ǫ)ImGr1,m(ǫ), (7) whereImmeanstakingtheimaginarypartofthefunction where E is spin-independent QD energy level, and ℓ n = d† d . Notations U and U describe the in- that follows, and ℓ,σ ℓ,σ ℓ,σ ℓ ℓ,j tradot and interdot Coulomb interactions, respectively. Gr (ǫ)=p /(µ Π Σm ), (8) tℓ,j describes the electron interdot hopping. Note that 1,m 1,m 1− 1,m− 1,N the nearest neighbor interdot hopping and interdot where µ = ǫ E + iΓ /2, Π denotes the sum of Coulomb interaction are taken into account in Eq. (2). 1 − 1 1 1,m Coulomb energies arising from other electron present Using the Keldysh-Green’s function technique,15 the in the first QD and its neighborhood, and Γm (ǫ) = chargecurrentleavingtheleftandrightelectrodesinthe 1,N 2ImΣm (ǫ),whereΣm denotestheselfenergyresult- steady state can be expressed by − 1,N 1,N ing from electron hopping from QD 1 to QD N through e channel m. For electron with spin σ tunneling from the JL = −h Z dǫΓL[2fL(ǫ)ImGr1,σ(ǫ)−iG<1,σ(ǫ)] (3) left electrode into level 1 of N serially coupled QD and exit to the right electrode, we have 2 4N−1 quantum × and pathes(orchannels),sincelevel1canbeeitheremptyor singly occupied (with spin σ) and all other levels can e − J = − dǫΓ [2f (ǫ)ImGr (ǫ) iG< (ǫ)], (4) be empty, singly occupied (with spin up or down), and R h Z R R N,σ − N,σ doublyoccupied. FortheN =2case,theexplicitexpres- sions of probability weights and self energies have been where Γ = Γ and Γ = Γ denote, respectively, L 1 R N worked out in our previous paper.13 Here, we generalize the tunneling rates of the left electrode to the first QD the expressions to a CCQD system with arbitrary num- and the right electrode to the Nth QD, which are as- ber of QDs. We found that for the N-QD system, the sumed to be energy and bias independent for simplicity. probability factors p are determined by the following fL(R)(ǫ) = 1/[e(ǫ−µL(R))/kBTL(R) +1] denotes the Fermi relation 1,m distribution function for the left (right) electrode. The chemicalpotentialdifferenceisgivenbyµL µR =e∆V. N 2×4N−1 − TL(R) denotes the equilibrium temperature of the left (a¯1+¯b1σ¯) (aℓ+bℓσ+bℓσ¯+cℓ)= p1,m, (9) (right) electrode. e and h denote the electron charge Y X ℓ=2 m=1 3 wherea¯ =1 N ,¯b =N ,a =1 N N +c , Gr (ǫ) is obtained from Gr (ǫ) by reversing the roles ℓ − ℓ,σ¯ ℓ,σ¯ ℓ,σ¯ ℓ − ℓ,σ¯− ℓ,σ ℓ N,m 1,m b = N c , b = N c . a , b , b , of QDs 1 and N. Namely, ℓ,σ¯ ℓ,σ¯ ℓ ℓ,σ ℓ,σ ℓ ℓ ℓ,σ¯ ℓ,σ − − and c describe the probability factor for the ℓ-th QD ℓ with no electron, one electron of spin σ¯, one electron Gr (ǫ)=p /(µ Π Σm ), (15) N,m N,m N − N,m− N,1 of spin σ, and two electrons, respectively. N , N , ℓ,σ ℓ,σ¯ and c = n n denote the thermally averaged one- ℓ ℓ,σ¯ ℓ,σ¯ particle ochcupationinumbers for spin σ and σ¯ and two- t2 Σm = N,N−1 particle correlation functions. We note that the sum of N,1 µ Π t2N−1,N−2 Etioqn..(9Σ)mequiaslsgiovneen,bwyhiachcoinndtiincuaetedsfpraroctbiaobnilityconserva- N−1− N−1,m− µN−2−ΠN−2,m···−µ1−t22Π,11,m, 1,N (16) and p are determined by t2 N,m Σm = 1,2 (10) 1,N µ2−Π2,m− µ3−Π3,m··t·22−,3µNt2N−−Π1N,N,m, (a¯N+¯bNσ¯)N−1(aℓ+bℓσ+bℓ,σ¯+cℓ)=2×4N−1pN,m. (17) Y X ℓ=1 m=1 where Π denotes the sum of Coulomb energies due to ℓm interaction of an electron entering the ℓ-th QD with the For QDs not in direct contact with leads (labeled by otherelectronspresentinthe CCQDinconfigurationm. ℓ=2,N 1), we have − Note that in the uncorrelated limit (U = 0 and U = ℓ ij 0th),etqhueaanbtuomveterxapnrsepsosirotntshrroeduughceatolintehaerecxhaacitnswoliuthtiionnthtoe N = dǫ2×4N−1 Γmℓ,1(ǫ)fL+Γmℓ,N(ǫ)fRImGr (ǫ), nearest-neighbor tight-binding model.16 Therefore, our ℓ −Z π mX=1 Γmℓ,1(ǫ)+Γmℓ,N(ǫ) ℓ,m results are still able to illustrate the transport behavior (18) in the other extreme, U/t 1, since the terms we and ij ≪ ignoredarerelatedto correlationofelectronsindifferent levels,anditisvalidwhentheCoulombU termsbecome c = dǫ 2×4N−1 Γmℓ,1(ǫ)fL+Γmℓ,N(ǫ)fRImGr (ǫ). small. However, this study is restricted in the condition ℓ −Z π Γm (ǫ)+Γm (ǫ) ℓ,m of t /U 1 throughout article. m=4XN−1+1 ℓ,1 ℓ,N i,j ≪ (19) The average occupation numbers N = N and ℓ,σ ℓ,σ¯ Γm (ǫ) = 2ImΣm and Γm (ǫ) = 2ImΣm are the cℓ are determined by solving the on-site lesser Green’s ℓ,1 − ℓ,1 ℓ,N − ℓ,N functions12,13 iG< (ǫ), which take the following form effectivetunnelingratesforelectronsfromtheℓ-thQDto 1,σ the left and right electrodes, respectively. The retarded iG< (ǫ)=2 Γ1(ǫ)fL+Γm1,N(ǫ)fRImGr (ǫ). (11) Green’s function Grℓ,m(ǫ) is given by 1,σ Γ (ǫ)+Γm (ǫ) 1,m Xm 1 1,N p Gr (ǫ)= ℓ,m , (20) Thus, we have ℓ,m µℓ−Πℓ,m−Σmℓ,1−Σmℓ,N N = dǫ Γ1(ǫ)fL+Γm1,N(ǫ)fRImGr (ǫ), wheretheprobabilityfactors,pℓ,maredeterminedbyEq. 1 −Z π Γ (ǫ)+Γm (ǫ) 1,m (9)withtheindies1andℓinterchanged. Theselfenergies Xm 1 1,N Σm and Σm are given by (12) ℓ,1 ℓ,N and t2 c1 =−Z dπǫm2=×4X4NN−−11+1Γ1(Γǫ1)(fǫL)++ΓΓm1m1,,NN((ǫǫ))fRImGr1,m(ǫ), Σmℓ,1 = µℓ−1−Πℓ−1,m− µℓℓ,−ℓ−2−1ΠNt−2ℓ2−,m1,ℓ·−··−2 µ1−t22Π,11,m (21) (13) where 2×4N−1 denotesasumoverconfigurationsob- and m=4N−1+1 tainedPby the product t2 Σm = ℓ,ℓ+1 (22) ℓ,N t2 N1σ¯ N (aℓ+bℓ,σ+bℓ,σ¯+cℓ)= 2×4N−1 p1,m. (14) µℓ+1−Πℓ+1,m− µℓ+2−Πℓ+2ℓ,m+1·,·ℓ·−+2µNt2N−−Π1N,N,m. ℓY=2 m=4XN−1+1 The explicit expressions of p and Π for all levels ℓ,m ℓ,m To calculate the tunneling current from of Eq. (3), we and all configurations of the N=3 case are given in the alsoneedthe retardedGreen’sfunctionGr (ǫ),whichis Appendix. As an example, for a five-dot CCQD with 1,σ given by Gr (ǫ)= Gr (ǫ). configurationdescribedbyp =a b a b c ,wehave 1,σ m 1,m 1,m 1 2σ 3 4,σ¯ 5 NN,σ = NN,σ¯ =PNN and cN have the same forms as theaboveequationswiththeindices1andN exchanged. Gr (ǫ)= 1,m 4 (1−N1,σ¯)b2σa3b4,σ¯c5 κph =κph,0Fs withFs =0.1canexplainwellthephonon t2 thermal conductance of silicon nanowire with surface µ U 1,2 1− 1,2− µ2−U1,2−µ3−U3,4−µ4−U4−t222U,43,5t−23,µ45−Ut524−,52U4,5. ssdctiamatteetnesrsicinoagnlcluewsliastthescdsautbrtyfearctinehgeimffiaprcsuttro-iprtrieiFnsscoiparlreisssuersmfaefcrteohmodde.fp1eh7cotTsnohonef (23) quantum dots.1 It is possible to reduce phonon thermal In the linear response regime, Eqs. (5) and (6) can be conductancebyoneorderofmagnitudewhentheQDsize rewritten as is much smaller than the phonon mean free paths.21,22 Therefore, we adopt F =0.01 as a fixed parameter and s ∆V ∆T J = L11 T +L12 T2 (24) saiszseu.meFs is independentofthe number ofQDsandQD ∆V ∆T Q = + , (25) L21 T L22 T2 III. RESULTS AND DISCUSSION where there are two sources of driving force to yield the charge and heat currents. ∆T = T T is the tem- L R perature difference across the junction−. Thermoelectric A. Three-QD junction response functions , , , and are evaluated 11 12 21 22 L L L L by In this section, we study the transport proper- ties of N=3 case, which were experimentally23,24 and theoretically25,26 investigated in the nonlinear response 2e2T ∂f(ǫ) = dǫ (ǫ)( ) , (26) regime to reveal the coherent and spin-dependent be- L11 h Z T ∂E T F havior of carrier transport. In Ref. [27], the Kondo transport of triple QDs was investigated by using the 2eT2 ∂f(ǫ) slave-bosonmethodto removethe double occupationfor L12 = h Z dǫT(ǫ)( ∂T )EF, (27) each QD. This study is restricted in the Coulomb block- ade regime. So far, few literatures have considered the spin-dependent thermoelectric properties of three cou- pled QDs. In Fig. 2 we plot the electrical conduc- 2eT ∂f(ǫ) L21 = h Z dǫT(ǫ)(ǫ−EF)( ∂E )T, (28) tance (Ge), Seebeck coefficient (S), and electron ther- F mal conductance (κ ) as a function of gate voltage for e and various temperatures. We adopt the following physical parameters U = U = 60Γ , U = 20Γ , t = 1Γ , ℓ 0 0 ℓ,j 0 ℓ,j 0 2T2 ∂f(ǫ) Γ = Γ = Γ = Γ , E = E = E 20Γ , and L22 = h Z dǫT(ǫ)(ǫ−EF)( ∂T )EF. (29) E1 = E3 +30Γ e0V . 1All ener2gy scalFes−are in0 units 3 F 0 g − of Γ , a characteristic energy. A gate voltage is applied 0 Here (ǫ) and f(ǫ)= 1/[e(ǫ−EF)/kBT +1] are evaluated to tune the energy level of E such that the E level can T 3 3 under the equilibrium condition. be varied from being empty to singly occupied (see the If the system is in an opencircuit, the electrochemical insetofFig.2). Thesystemwithspintriplestateasillus- potentialwillbeestablishedinresponsetoatemperature trated in the inset of Fig. 2(a) is in the insulating state gradient; this electrochemical potential is known as the whenE isfarabovetheFermienergy. Theconductance 3 Seebeckvoltage(Seebeckeffect). TheSeebeckcoefficient G reaches a maximum at E = E , whose magnitude e 3 F (amount of voltage generated per unit temperature gra- decreasesasthetemperatureincreases. Meanwhile,G a e dient) is defined as S = ∆V/∆T = −L12/(TL11). To afunctionofVg hasaLorentzshape,whoseFWHMisal- judge whether the system is able to generate or extract most independent of temperature as long as k T/Γ 1. B heat efficiently, we need to consider the figure of merit,1 This is referred to as the nonthermal broadening eff≥ect, which is given by which was also observed in the case of serially coupled quantumdots (SCQD).12,28 The behaviorofSeebeck co- S2G T (ZT) e 0 efficient (S) is illustrated in Fig. 2(b), and it is found ZT = . (30) κe+κph ≡ 1+κph/κe that S vanishes when Ge reaches the maximum, a result attributed to the electron-hole symmetry. Here, holes Here G = /T is the electrical conductance and e 11 are defined as missing electrons in electrodes below E . κ = (( /TL2) S2) is the electron thermal con- F e 22 11 The negative sign of S indicates that electrons are ma- L − L ductance. (ZT) represents the ZT value in the absence 0 jority carries, which diffuse to the right electrode from of phonon thermal conductance, κ . For simplicity, we ph the left electrode through energy levels above E . On F assume κ = κ F .11−13 κ = π2kB2T is the uni- the other hand, holes become majority carriers when S ph ph,0 s ph,0 3h versal phonon thermal conductance arising from acous- turns positive. Thus, the measurement of S can reveal tic phonon confinement in a nanowire,17−19 which was the properties of resonantchannels when comparedwith confirmedinthephononwaveguide.20 Theexpressionof that of G . We see that S vanishes again at k T =1Γ e B 0 5 and eV = 50Γ , where E +U is lined up with E . resonance: E +U =E +U +U =E +U (described g 0 3 J F 1 I 2 I J 3 3 The behavior of electron thermal conductance κ is sim- by p ), which is the Pauli spin blockade process for e 1,10 ilar to that of G and also shows the nonthermal broad- the case of three-QDjunction. The maximum G in this e e ening effect. Like the two-QD junction, the nonthermal spin singlet state is small than the maximum of G in e broadening effect of the three-QD junction can be used spin triple state of Fig. 2. From the results of Figs. (2)- to function as a low temperature filter. (4), we find that the maximum Seebeck coefficients are smaller than one. This is not preferred for the purpose In Fig. 3, we show thermoelectric behaviors of a sys- ofenhancingZT.Sucharesultalsoimpliesthatthefluc- tem with E = E 20Γ , E = E + 10Γ eV , 1 F 0 2 F 0 g − − tuation of QD energy levels in CCQD will suppress ZT. and E = E , as illustrated in the inset of Fig. 3(a). 3 F OtherphysicalparametersarethesameasthoseofFig.2. In Fig. 5, we plot the occupation numbers Nℓ, elec- Now the gate voltage is used to tune the level E2 such trical conductance Ge, Seebeck coefficient S, and elec- thatthe E2 levelvariesfrombeingemptytosinglyoccu- tron thermal conductance κe as functions of gate volt- pied. Meanwhile, the E3 level will be depleted when E2 age at kBT = 1Γ0. We consider identical energy levels is singly occupied. Although the behavior of Ge shown with Eℓ = E0 = EF +30Γ0−eVg and tℓ,j = 6Γ0. All in Fig. 3(a) is very similar to that of Fig. 2(a), we note QD levels are tuned by the gate voltage from far above the FWHM of Ge in Fig. 3(a) is nearly twice as large. EF to far below EF. Other physical parameters are the Furthermore, the nonthermal broadening effect for κe same as those of Fig. 2. Because tℓ,j = tc > kBT and disappears. This indicates that the “effective broaden- tℓ,j = tc > Γ, these thermal response functions (Ge, S, ing” of energy level of dot 2, which is not directly cou- and κe) display structures yielded by the electron hop- pled to electrodes, is different from that of dots 1 and 3. pingeffect. Thethree peakslabeledbyVg1, Vg2,andVg3 The nonthermal broadening effect of Ge is an essential correspondtothreeresonantchannelsatE0 √2tc, E0, − characteristic of resonant junction system. Once kBT is and E0+√2 tc, which are poles of the Green’s function largerthanthetunnelingratesΓ1 =Γ3 =Γ,whichisthe Gr1,m(ǫ) for channel m = 1, in which all three QDs are broadening of energy levels of dots 1 and 3, the broad- empty. The strengths of these peaks are determined by ening of Ge depends mainly on the lifetime of the reso- their probability weights, a¯1a2a3. Another three peaks nance,andbecomesinsensitivetothetemperaturefactor labeled by V , V , and V result from the resonances g4 g5 g6 1/cosh2((ǫ−EF)/(2kBT)). Inaddition,wefindmoreos- corresponding to channel m=28 with p1,m =N1σ¯b2σc3, cillatorypeaksofS inFig.3(b)ascomparedinFig.2(b). in which dots 1 and 3 are doubly occupied and dot 2 For example, when eVg =10Γ0 we have E2 =EF, which occupied with one electron with spin σ. (see Appendix) matches with E3 and E1+UI. This resonance has very A remarkable result of Fig. 5 is the larger enhancement small probability weight and is non-observable in GE, of the maximum Seebeck coefficient. This enhancement whereas it canbe measuredby S. The highly oscillatory of the maximum S is due to the degeneracy of QD lev- behavior of S with respect to Vg indicates that carri- els, instead from larger tc. Note that S is sensitive to ers with high energies can diffuse to the right electrode the fluctuationofQDenergylevels,butnottot .13 This c throughmoreresonantchannels,whicharefaraboveEF. will be further demonstrated in the N = 5 case. Unlike This explains why the tail of κe peak (near eVg = 20Γ0 two-QDjunctionwithidenticalQDs,thespectraofthese and eVg =40Γ0) increases with increasing temperature. thermalresponse functions for the three-QD junction do In Fig. 4, we plot the average occupation number N , notpossesssymmetricbehaviorasaresultoftheinterdot ℓ G and S as functions of the gate voltage V (which is Coulomb interactions. The central dot (dot 2) feels the e g usedtotuneE )forvarioustemperaturesforathree-QD Coulombinteractionsfromboth dots1and2,whiledots 2 junction systemin the spin-blockadeconfiguration,as il- 1 and 3 can only feel the interdot Coulomb interaction lustratedintheinsetofFig.4(b). Here,E =E 10Γ , from the central dot. Such an effect can be observed in 1 F 0 E2 =EF+10Γ0 eVg,E3 =EF 60Γ0,andUℓ,j =−10Γ0. thebehaviorsofoccupationnumbersasfunctionsofeVg. − − OtherphysicalparametersarethesameasthoseofFig.3. The electrical conductance G , Seebeck coefficient S, e This configuration was considered in Ref. [26] within and figure of merit ZT are plotted as functions of de- the framework of Master equation technique for study- tuning energy ∆ = E E with and without interdot ℓ F − ing spin-blockade behavior of three coupled QD in the Coulomb interaction at k T =10Γ in Fig. 6, where we B 0 nonlinear response regime. A spin-blockade can occur, have adopted t =6Γ and Γ =Γ =Γ . We note G ℓ,j 0 l R 0 e becausethesecondelectronappearsindot3mustsatisfy issuppressedbytheinterdotCoulombinteractions. Such thePauliexclusionprinciple. AsseeninFig.4(a),theE an effect becomes weak, when ∆ increases. S is almost 2 levelistunedfrombeingemptytosinglyoccupied,while independent of U and it shows a linear dependence of ℓ,j the E level remains singly occupied even in presence of ∆, roughly described by k ∆/(eT). The behavior of 3 B − the interdot Coulomb interactions. We observe a small ZT as a function of ∆ can be described by the function bump ofG neareV =20Γ . Although both E andE ZT=α∆2/(T3cosh2(∆/2k T)), where α is independent e g 0 1 2 B become resonant at eV = 20Γ , electrons in this reso- of T and ∆. When ∆ k T, κ becomes negligible. g 0 B e ≫ nancestatecannottunneltotherightelectrodethrough Therefore,thethermalconductanceisdominatedbyκ , th E ,due tothe presenceofU arisingindot3. The max- which is assumed to be a linear function of T. ZT is de- 3 J imum G occuring at eV = 30Γ is resulting from the termined by the power factor of S2G . The maximum e g 0 e 6 ZT occurs near ∆max = 3kBT and it is slightly reduced E (3 √5)/2 t , E (3+√5)/2 t ; 0 c 0 c in the presence of U . The effect of U becomes negli- −q − −q ℓ,j ℓ,j giblewhentheQDenergylevelsarefaraboveE . Inthe F followingdiscussion,wewillshowthatelectronCoulomb N =5:ǫ=E +√3t , E +t , E , 0 c 0 c 0 interactions are important when E is above E . F ℓ Fig. 7 shows G , S, and ZT as functions of gate volt- e age at kBT =10Γ0 and Eℓ =E0 =EF +50Γ0−eVg for E0−tc, E0−√3tc. various interdot Coulomb interactions. Other physical Theseparationsbetweentheseresonantchannelsbecome parameters are the same as those of Fig. 3. Compar- small with increasing QD numbers. In addition, these ing with the low-temperature results given in Fig. 5, resonant channels have different broadening. For exam- we see that the six peaks of G at k T = 1Γ now be- e B 0 ple, three resonant channels of the N = 3 case occur at cometwobroadpeaks. Wealsonotethatthepresenceof ǫ=E +√2t , E , and E √2t , and their broadening interdotCoulombinteractions breaksthe structure sym- 0 c 0 0− c widthsareΓ/4,Γ/2,andΓ/4,respectively. Asexpected, metry(takingeV =80Γ asthemidpint,whereE sits g 0 F themaximumZTdecreaseswithincreasingN. Basedon between E and E +U with equal separation). In par- ℓ ℓ ℓ the results of Fig. 8, the reduction of maximum ZT is ticular,ZT (forQDenergylevelsaboveE )isonly max,> F attributedto the reductionof G . IncreasingN, the res- slightly suppressedwith increasinginterdot Coulombin- e onant channels increases whereas probability weights of teractions. On the other hand, ZT (for QD energy max,< these resonant channels decrease. This explains why G levels below E ) is significantly suppressed. The differ- e F is reduced with increasing N. Note that when Coulomb ence in interdot Coulomb interactions between central interactions are turned off, G would become insensitive dot and two outer dots leads to an artificial “QD en- e to N for the case of small t and large ∆ = E E . ergy level fluctuation” which suppresses ZT. Our result c ℓ − F The larger t , the more important the QD number ef- indicates that the optimization of ZT prefers having the c fect becomes. In the high temperature regime, we find QD levels above E , although many studies of individ- F thattheSeebeckcoefficientisnotsensitivetoN [seeFig. ual QDs and two coupled QDs indicated that electron 8(b)], while k reduces significantly as N increases [see Coulomb interactions can enhance ZT when QD levels e Fig. 8(c)]. Thus,k dominatesthethermalconductance are below E .29−32 Authors in reference[29-32] adopted ph F inthelargeN limit. Consequently,thebehaviorofZTis the mean field approximation, which truncates the high essentiallydeterminedbythepowerfactor(pF =S2G ), order Green’s functions arising from electron Coulomb e andthe trendofZTwithrespecttoincreasingN issimi- interactions. lartothatofG . Wealsonotethatthedependenceofall e thermoelectric functions on N saturates once N reaches 5 for the weak hopping strength considered, t = 3Γ . B. Effect of number of QDs in CCQD c 0 Thus, it is sufficient to model the thermoelectric behav- iors of a CCQD with large N by using N = 5. How- Since in realistic QD junctions for thermoelectric ever, for larger t , the saturation behavior would occur application, one needs to consider a large number of c at larger N. serially coupled QDs in order to accommodate rea- In order to find the optimization condition of ZT, we sonable temperature gradient across the junction, it is plot in Fig. 9 the occupation number of each QD, elec- important to know the effect of number of QDs (N) on trical conductance, Seebeck coefficient, and ZT as func- the thermoelectric properties of a CCQD system for a tionsofgate voltageforvarioustemperaturesfor a5-dot given t . To clarify the effect of N on ZT, we plot G , c e CCQD. Other physical parameters are the same as the S, κ , and ZT as functions of temperature for various e N = 5 case shown in Fig. 8. From Fig. 9(a) we see a values of N by using the following physical parameters: small difference in the occupation number between ex- E = E = E + 30Γ , U = U = 30Γ , t = 3Γ . ℓ 0 F 0 ℓ 0 0 c 0 terior dots (N ,N ) and interior dots (N ,N ,N ) even The interdotCoulombinteractionshavebeen turnedoff. 1 5 2 3 4 thoughthe interdotCoulombinteractionsareturnedoff. Increasing the number of QDs would change the density This phenomena also occurs in the N = 3 and N = 4 of states (DOS) of the CCQD system from atomic limit cases (not shown here). The occupation number fluc- to the band limit. For N varying from 2 to 5, in the tuation was also reported in Ref. [25] for the N = 3 absence of electron Coulomb interactions, we have the case by solving the Master equation. For eV = 25Γ , following energy spectra (under the condition t /Γ 1) g 0 c ≫ electrons prefer to occupy the outer QDs (N1 and N5). For eV = 75Γ , electrons prefer to accumulate in the g 0 interior QDs (N ,N ,N ). The maximum electrical N =2:ǫ=E +t , E t ; 2 3 4 0 c 0 c − conductance is suppressed with increasing temperature, while the peak width becomes wider. Such a behavior N =3:ǫ=E +√2t ,E , E √2t ; is no different from a single QD with multiple energy 0 c 0 0 c − levels.11 However, we note that G does not have e,max a Lorentzian shape even though each resonant channel N =4:ǫ=E + (3+√5)/2t , E + (3 √5)/2t , has a Lorentzianshape. This is mainly attributed to the 0 q c 0 q − c 7 formation of a ”miniband” with the probability weight timum value at t 2Γ and then decreases for higher C 0 ≈ (1 N )(1 N )(1 N )(1 N )(1 N ). Note t . The reduction of ZT at higher t arises from the 1,σ¯ 2σ¯ 3,σ¯ 4,σ¯ 5,σ¯ c c − − − − − that once U = 0, there are only two kind of proba- faster reductionofS2 in comparisonwiththe increaseof ℓ,j bility weights (1 N ) and N for each QD.33This G . To reveal how the asymmetrical coupling between ℓ,σ¯ ℓ,σ¯ e − characteristiccanalsobe demonstratedfromthe expres- the QDs and the electrodes influences ZT, we also plot sions for N=3 given in the appendix. When the applied ZT versus V for the detuning energy ∆ = 30Γ for dif- g 0 gate voltage continue to increase, the CCQD turns into ferent tunneling rates in Fig. 10(d). (See curves marked a Mott-insulator at half-filling N = N = N = 0.5, by filled triangles and diamonds) The maximum ZT is ℓ ℓ,σ ℓ,σ¯ resulting from Coulombband gap.33 (The electricalcon- suppressed when the ratio of Γ /Γ is far away from 1, L R ductance almost vanishes for each QD with one elec- while keeping the same average value, Γ +Γ =2Γ . L R 0 tron). The upper ”miniband” with probability weight N N N N N arises for electrons hopping be- 1,σ¯ 2σ¯ 3,σ¯ 4,σ¯ 5,σ¯ tween energy levels at E0+U0. The Seebeck coefficient IV. SUMMARY AND CONCLUSIONS showninFig.9(c)goesthroughzeroatV ,V ,andV , g1 g2 g3 respectively. At these applied gate voltages, the CCQD In summary, we have theoretically investigated the hasanelectron-holesymmetry. NotethattheSeebeckco- thermoelectric effects of CCQD embedded in a nanowire efficient vanishes at the half-filling case. We see that the and connected to metallic electrodes. The length of the electrical conductance and Seebeck coefficient are very CCQD nanowire is finite, and shorter than the electron sensitive to QD energy levels. Due to the electron-hole mean free path. Thus, the electron-phonon interaction symmetry, the G and ZT curves are symmetric (while e effect can be ignored in this study. We have derived an S curve is antisymmetric) with respect to the mid point expression for the transmission coefficient (ǫ) in terms at eVg = 45Γ0. Thus, ZT curve has two maxima, one of the on-site retarded Green’s functionsTand effective for QD energylevels aboveE and the other for QD en- F tunneling rates involving electron hopping and Coulomb ergy levels below E . The maximum ZT at k T = 5Γ F B 0 interactions,whichkeepsthesameformastheLandauer canreach8 for V near 10Γ and80Γ . When κ domi- g 0 0 ph formula for charge and heat currents. Closed-form ex- nates the thermalconductance, the maximumZT values pressions (which are correct in the limit t /U 1) for are correlated to the maximum power factor. The max- ij ≪ the retarded Green’s functions of a CCQD system with imum ZT occurs at neither good conducting state nor arbitrarynumberofQDsarederived. Suchananalytical insulating state, because the maximum G (good con- e form is quite needed for finding the best ZT values. In ductance)is accompanied by a poor Seebeck coefficient, addition, our formula still works in the nonequilibrium and vice versa. Note that the results shownin Fig. 9 are regime. for the case with no interdot Coulomb interactions, i.e. In the linear response regime, electrical conductance U = 0. Once U are turned on, the spectra of G ℓ,j ℓ,j e G , Seebeck coefficient S, electron thermal conductance becomes somewhat complicated to analyze. Meanwhile, e κ ,andfigureofmeritZTarecalculatedforfinite length it would significantly lower the maximum ZT value for e CCQDswithQDnumberrangingfrom2to5. The ther- the peak with E >E . F 0 moelectric properties of spin-dependent configurations Electroninterdothoppingstrengtht isakeyparame- (spintriple andsingletstates)arestudied. The nonther- c terindeterminingthebandwidthoftheminiband,which malbroadeningbehaviorofG ismaintainedinthethree- e wouldaffectthethermoelectricproperties. Fig.10shows QDcaseinthepresenceofelectronCoulombinteractions, G , S, (ZT) and ZT as functions of t for various de- because such a behavior is the essential feature of reso- e 0 c tuning energies ∆ = E E at k T = 10Γ . Here, nant tunneling junction system.12,28 Note that the non- ℓ F B 0 − we consider the case with on-site Coulomb interaction thermalbroadeningeffectofG cannotbeillustratedby e U = 60Γ and zero interdot Coulomb interaction. Note usingthemean-fieldapproximation.30−32 WhenQDsare 0 that (ZT) corresponds to ZT in the absence of phonon notidentical,thesethermoelectricresponsefunctionsare 0 thermal conductance. When t is smaller than Γ , G weaker,resultingfromsymmetrybreakinginthejunction c 0 e increases quickly with respect to t . This behavior can structure. Unlike the case of double QDs,10 the interdot c beexplainedbythefactofthatthetransmissionfactoris Coulomb interactions lead to considerable suppression proportional to t8 for the 5-dot case when t approaches of the maximum ZT value in the three-QD case when c c zero. Once t is larger than Γ , G becomes almost sat- the degenerate QD energy levels are below E . This is c 0 e F urated. G values at∆=10Γ areveryclose to those at caused by the symmetry breaking via interdot Coulomb e 0 ∆=20Γ . Suchabehaviorisalsoseeninthedottedline interactions in the CCQD junction. 0 ofFig.9(b),inwhichtheQDenergylevelistunedbythe We find that for a givent , increasing the QDnumber c gate voltage. We notice that S is rather insensitive to t N inthe CCQDwouldsuppressthemaximumZTvalue, c intheweakhopinglimit,t /Γ 1(S k ∆/eT). In and the reduction quickly saturates once N reaches 5 in c 0 B ≪ ≈− the absence of phonon thermal conductance, (ZT) di- the weak hopping limit (t /U 1). However, for larger 0 c ≪ vergesintheweakhopinglimit,becausetheLorenznum- t , the saturation behavior would occur at larger N. In c ber L=κ /(G T) approacheszero. From Fig. 10(d), we addition, we illustrate that the QD energy level fluctu- e e seethatZTincreaseswithincreasingt ,reachinganop- ation suppresses the maximum ZT by using a three-QD C 8 junction. We find that the Seebeck coefficient is insen- p =a¯ b b ;Π =U ,Π =U +U ,Π =U +U . 1,6 1 2,σ¯ 3,σ¯ 1,6 I 2,6 2 J 3,6 3 J sitive to t and U (U ) when QD energy levels are far c ℓ ℓ,j above E , and it can be approximated by a simple lin- F p =a¯ b b ;Π =U ,Π =U +U ,Π =2U . ear expression, S = k ∆/eT, where ∆ = E E 1,7 1 2,σ¯ 3,σ 1,7 I 2,7 2 J 3,7 J B ℓ F − − and T is the equilibrium temperature. This character- iHstuibcboafrdSc=hai−nkwBit∆h/neaTrrwowasbaalnsdowriedptohr.t3e4d−3i6nTahneifneafitnuitree p1,8 =a¯1b2,σ¯c3;Π1,8 =UI,Π2,8 =U2+2UJ,Π3,8 =U3+2UJ. of S = ∆V/∆T k ∆/eT can be utilized to realize B a temperature sen≈so−r.37 Therefore, the CCQD system is p1,9 =a¯1b2,σa3;Π1,9 =UI,Π2,9 =UI,Π3,9 =0. morepromisingthanthesingleQDjunctionformeasure- ment of larger temperature difference. p =a¯ b b ;Π =U ,Π =U +U ,Π =U . Acknowledgment 1,10 1 2,σ 3,σ¯ 1,10 I 2,10 I J 3,10 3 This work was supported in part by National Science Council, Taiwan under Contract Nos. NSC 101-2112-M- p =a¯ b b ;Π =U ,Π =U +U ,Π =U . 1,11 1 2,σ 3,σ 1,11 I 2,11 I J 3,11 J 008-014-MY2and NSC 101-2112-M-001-024-MY3. † E-mail address: [email protected] ∗ E-mail address: [email protected] p1,12 =a¯1b2,σc3;Π1,12 =UI,Π2,12 =UI+2UJ,Π3,12 =U3+UJ. p =a¯ c a ;Π =2U ,Π =U +U ,Π =U . 1,13 1 2 3 1,13 I 2,13 2 I 3,13 J Appendix A: Green’s functions of N=3 p =a¯ c b ; In this appendix, we give the detailed expression of 1,14 1 2 3,σ¯ for the N=3 case. For simplicity, we assume the 1,3 T same interdot hopping constant between any two QDs, Π =2U ,Π =U +U +U ,Π =U +U . 1,14 I 2,14 2 I J 3,14 3 J i.e. t = t , and we denote U = U U , U = ℓ,j c 1,2 2,1 I 2,3 ≡ U U , and µ = ǫ E +iΓ /2; ℓ = 1,2,3. Note 3,2 J ℓ ℓ ℓ ≡ − that Γ2 =0, because QD 2 is not directly coupled to the p1,15 =a¯1c2b3,σ; electrodes. Eq. (5) becomes Π =2U ,Π =U +U +U ,Π =2U . 32 Γ (ǫ)Γm (ǫ) 1,15 I 2,15 2 I J 3,15 J (ǫ)= 2 1 1,3 ImGr (ǫ), (A1) T1,3 − Γ (ǫ)+Γm (ǫ) 1,m mX=1 1 1,3 p =a¯ c c ; 1,16 1 2 3 where p Π =2U ,Π =U +U +2U ,Π =U +2U . Gr (ǫ)= 1,m , (A2) 1,16 I 2,16 2 I J 3,16 3 J 1,m µ Π Σm 1− 1,m− 1,3 p =N a a ;Π =U ,Π =U ,Π =0. 1,17 1,σ¯ 2 3 1,17 1 2,17 I 3,17 t2 Σm = c , (A3) 1,3 µ2−Π2,m− µ3−tΠ2c3,m p1,18 =N1,σ¯a2b3,σ¯;Π1,18 =U1,Π2,18 =UI+UJ,Π3,18 =U3. andΓm = 2ImΣm denotestheeffectivetunnelingrate ofthe1e,3lectr−onfrom1,t3hefirstQDenergyleveltotheright p1,19 =N1,σ¯a2b3,σ;Π1,19 =U1,Π2,19 =UI+UJ,Π3,19 =UJ. electrode via channel m. The probability weights (p ) 1,m and Coulomb energies (Πℓ,m) are given by p1,20 =N1,σ¯a2c3; p =a¯ a a ;Π =Π =Π =0. 1,1 1 2 3 1,1 2,1 3,1 Π =U ,Π =U +2U ,Π =U +U . 1,20 1 2,20 I J 3,20 3 J p =a¯ a b ;Π =0,Π =U ,Π =U . 1,2 1 2 3,σ¯ 1,2 2,2 J 3,2 3 p =N b a ; 1,21 1,σ¯ 2,σ¯ 3 p =a¯ a b ;Π =0,Π =Π =U . 1,3 1 2 3,σ 1,3 2,3 3,3 J Π =U +U ,Π =U +U ,Π =U . 1,21 1 I 2,21 2 I 3,21 J p1,4 =a¯1a2c3;Π1,4 =0,Π2,4 =2UJ,Π3,4 =U3+UJ. p1,22 =N1,σ¯b2,σ¯b3,σ¯; p =a¯ b a ;Π =U ,Π =U ,Π =U . Π =U +U ,Π =U +U +U ,Π =U +U . 1,5 1 2,σ¯ 3 1,5 I 2,5 2 3,5 J 1,22 1 I 2,22 2 I J 3,22 3 J 9 p1,23 =N1,σ¯b2,σ¯b3,σ; where Σm2,1 = µ1−tΠ2c1,m and Σm2,3 = µ3−tΠ2c3,m. The proba- bilityfactors,p andCoulombenergiesΠ ;ℓ=1,2,3 2,m ℓ,m are given by Π =U +U ,Π =U +U +U ,Π =2U . 1,23 1 I 2,23 2 I J 3,23 J p =a¯ a a ;Π =Π =Π =0. 2,1 2 1 3 1,1 2,1 3,1 p =N b c ; 1,24 1,σ¯ 2,σ¯ 3 p =a¯ a b ;Π =0,Π =U ,Π =U . 2,2 2 1 3,σ¯ 1,2 2,2 J 3,2 3 Π =U +U ,Π =U +U +2U ,Π =U +2U . 1,24 1 I 2,24 2 I J 3,24 3 J p =a¯ a b ;Π =0,Π =Π =U . 2,3 2 1 3,σ 1,3 2,3 3,3 J p =N b a ; 1,25 1,σ¯ 2,σ 3 p =a¯ a c ;Π =0,Π =2U ,Π =U +U . 2,4 2 1 3 1,4 2,4 J 3,4 3 J Π =U +U ,Π =2U ,Π =0. 1,25 1 I 2,25 I 3,25 p =a¯ b a ;Π =U ,Π =U ,Π =0. 2,5 2 1,σ¯ 3 1,5 1 2,5 I 3,5 p =N b b ; 1,26 1,σ¯ 2,σ 3,σ¯ p =a¯ b b ;Π =U ,Π =U +U ,Π =U . 2,6 2 1,σ¯ 3,σ¯ 1,6 1 2,6 I J 3,6 3 Π =U +U ,Π =2U +U ,Π =U . 1,26 1 I 2,26 I J 3,26 3 p =a¯ b b ;Π =U ,Π =U +U ,Π =U . 2,7 2 1,σ¯ 3,σ 1,7 1 2,7 I J 3,7 J p =N b b ; 1,27 1,σ¯ 2,σ 3,σ p =a¯ b c ;Π =U ,Π =U +2U ,Π =U +U . 2,8 2 1,σ¯ 3 1,8 1 2,8 I J 3,8 3 J Π =U +U ,Π =2U +U ,Π =U . 1,27 1 I 2,27 I J 3,27 J p =a¯ b a ;Π =U ,Π =U ,Π =0. 2,9 2 1,σ 3 1,9 I 2,9 I 3,9 p =N b c ; 1,28 1,σ¯ 2,σ 3 p =a¯ b b ;Π =U ,Π =U +U ,Π =U . 2,10 2 1,σ 3,σ¯ 1,10 I 2,10 I J 3,10 3 Π =U +U ,Π =2U +2U ,Π =U +U . 1,28 1 I 2,28 I J 3,28 3 J p2,11 =a¯2b1,σb3,σ;Π1,11 =UI,Π2,11 =UI+UJ,Π3,11 =UJ. p1,29 =N1,σ¯c2a3; p2,12 =a¯2b1,σc3;Π1,12 =UI,Π2,12 =UI+2UJ,Π3,12 =U3+UJ. Π =U +2U ,Π =U +2U ,Π =U . p =a¯ c a ;Π =U +U ,Π =2U ,Π =0. 1,29 1 I 2,29 2 I 3,29 J 2,13 2 1 3 1,13 1 I 2,13 I 3,13 p1,30 =N1,σ¯c2b3,σ¯; p2,14 =a¯2c1b3,σ¯; Π1,30 =U1+2UI,Π2,30 =U2+2UI+UJ,Π3,30 =U3+UJ. Π1,14 =U1+UI,Π2,14 =2UI +UJ,Π3,14 =U3. p =a¯ c b ; p =N c b ; 2,15 2 1 3,σ 1,31 1,σ¯ 2 3,σ Π =U +U ,Π =2U +U ,Π =U . Π =U +2U ,Π =U +2U +U ,Π =2U . 1,15 1 I 2,15 I J 3,15 J 1,31 1 I 2,31 2 I J 3,31 J p =a¯ c c ; 2,16 2 1 3 p =N c c ; 1,32 1,σ¯ 2 3 Π =U +U ,Π =2U +2U ,Π =U +U . 1,16 1 I 2,16 I J 3,16 3 J Π =U +2U ,Π =U +2U +2U ,Π =U +2U . 1,32 1 I 2,32 2 I J 3,32 3 J For the retarded Green’s unction of QD 2, we have p =N a a ;Π =U ,Π =U ,Π =U . 2,17 2,σ¯ 1 3 1,17 I 2,17 2 3,17 J p Gr (ǫ) = 2,m , (A4) 2,m µ Π Σm Σm p =N a b ;Π =U ,Π =U +U ,Π =U +U . 2− 2,m− 2,1− 2,3 2,18 2,σ¯ 1 3,σ¯ 1,18 I 2,18 2 J 3,18 3 J 10 p =N a b ;Π =U ,Π =U +U ,Π =2U . p =N b b ; 2,19 2,σ¯ 1 3,σ 1,19 I 2,19 2 J 3,19 J 2,27 2,σ¯ 1,σ 3,σ p =N a c ; 2,20 2,σ¯ 1 3 Π =2U ,Π =U +U +U ,Π =2U . 1,27 I 2,27 2 I J 3,27 J Π =U ,Π =U +2U ,Π =U +2U . 1,20 I 2,20 2 J 3,20 3 J p =N b c ; 2,28 2,σ¯ 1,σ 3 p =N b a ; 2,21 2,σ¯ 1,σ¯ 3 Π =2U ,Π =U +U +2U ,Π =U +2U . 1,28 I 2,28 2 I J 3,28 3 J Π =U +U ,Π =U +U ,Π =U . 1,21 1 I 2,21 2 I 3,21 J p =N c a ; 2,29 2,σ¯ 1 3 p =N b b ; 2,22 2,σ¯ 1,σ¯ 3,σ¯ Π =U +2U ,Π =U +2U ,Π =U . 1,29 1 I 2,29 2 I 3,29 J Π =U +U ,Π =U +U +U ,Π =U +U . 1,22 1 I 2,22 2 I J 3,22 3 J p =N c b ; 2,30 2,σ¯ 1 3,σ¯ p =N b b ; 2,23 2,σ¯ 1,σ¯ 3,σ Π =U +2U ,Π =U +2U +U ,Π =U +U . 1,30 1 I 2,30 2 I J 3,30 3 J Π =U +U ,Π =U +U +U ,Π =2U . 1,23 1 I 2,23 2 I J 3,23 J p =N c b ; p =N b c ; 2,31 2,σ¯ 1 3,σ 2,24 2,σ¯ 1,σ¯ 3 Π1,24 =U1+UI,Π2,24 =U2+UI+2UJ,Π3,24 =U3+2UJ. Π1,31 =U1+2UI,Π2,31 =U2+2UI +UJ,Π3,31 =2UJ. p2,25 =N2,σ¯b1,σa3; p2,32 =N2,σ¯c1c3; Π =2U ,Π =U +U ,Π =U . 1,25 I 2,25 2 I 3,25 J Π =U +2U ,Π =U +2U +2U ,Π =U +2U . 1,32 1 I 2,32 2 I J 3,32 3 J p2,26 =N2,σ¯b1,σb3,σ¯; Gr3,m(ǫ) is obtained from Gr1,m(ǫ) by exchanging the in- dices 1 and 3. The average occupation numbers N , ℓ,σ N , and c are determined by solving Eqs. (18) and ℓ,σ¯ ℓ Π =2U ,Π =U +U +U ,Π =U +U . (19). 1,26 I 2,26 2 I J 3,26 3 J 1 A.J.Minnich,M.S.Dresselhaus,Z.F.Ren,andG.Chen, D.Wang,Z.Ren,J.P.Fleurial,andP.Gogna,Adv.Mater. Energy Environ Sci 2 466 (2009). 19, 1043 (2007). 2 M. Zebarjadi, K. Esfarjania, M.S. Dresselhaus, Z.F. Ren 9 Y. M. Lin, and M. S. Dresselhaus, Phys. Rev. B 68, 0 and G. Chen, Energy Environ Sci5 5147 (2012). 75304 (2003). 3 G.Mahan,B.SalesandJ.Sharp,PhysicsToday50(3)42 10 P. Murphy, S. Mukerjee, and J. Moore, Phys. Rev. B 78, (1997). 161406 (2008). 4 R. Venkatasubramanian, E. Siivola, T. 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