Thermoelectric performance of strongly-correlated quantum impurity models
aa r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Thermoelectric performance of strongly-correlated quantum impurity models
Edward Taylor and Dvira Segal
Chemical Physics Theory Group, Department of Chemistry,University of Toronto, 80 Saint George St. Toronto, Ontario, Canada M5S 3H6 (Dated: July 12, 2018)We derive asymptotically exact expressions for the thermopower and figure of merit of a quantumimpurity connecting two noninteracting leads in the linear response regime where the chemicalpotential and temperature differences between the leads are small. Based on sum rules for thesingle-particle impurity spectral function, these expressions become exact at high temperaturesas well as in the very strongly correlated regime, where the impurity Coulomb repulsion is muchlarger than the temperature. Although modest interactions impede thermoelectric performance, avery large Coulomb scale restores the optimal transport properties of noninteracting electrons, albeitrenormalized to account for the absence of double occupancy in the impurity. As with noninteractingelectrons, the electronic contribution to the figure of merit is limited only by the spectral broadeningthat arises from the coupling between the impurity and the leads.
PACS numbers: 73.50.Lw, 73.23.Hk, 85.65.+h
I. INTRODUCTION
Nearly twenty years ago, Mahan and Sofo elucidatedthe properties that an ideal thermoelectric material—one that can efficiently transform a temperature differ-ential into a voltage—should have . They found thatoptimal thermoelectric conversion efficiency in the lin-ear response regime, as determined by a large figure ofmerit, is realized in systems with a transmission function T ( E ) ∝ δ ( E − ǫ ) exhibiting a Dirac-delta function de-pendence on energy. In such a system, the electron con-tribution κ el to the thermal conductance at zero chargecurrent vanishes and the figure of merit ZT = GS Tκ el + κ ph (1)is only limited by the smallness of κ ph . Here G is theelectrical conductance, S is the Seebeck coefficient (ther-mopower), and κ el and κ ph are the electrical and phononcontributions to the thermal conductance.Although realistic systems do not exhibit such trans-mission, considerable experimental and theoretical workhas concentrated on exploring thermoelectric propertiesof nearly single-level “impurity” systems such as quan-tum dots or molecular junctions . Such systemscan in principle exhibit a transmission that is stronglypeaked about this level and hence, promise large figuresof merit. At the same time, these are small systemsand their coupling to the leads has a significant effecton the transmission function T ( E ) , yielding an extrinsicbroadening Γ. Nonetheless, the phonon contribution tothe thermal conductance is expected to be small in thesesystems as compared to that in bulk ones, and there isstill great interest in exploring their thermoelectric prop-erties. (Recent studies have stressed the importance ofthe phonon contribution κ ph in molecules, however .)Beyond numerical studies of the thermoelectric perfor-mance of impurity models based on numerical renormal-ization group and the nonequilibrium Green’s func- tion approach , simple analytic results for thermoelec-tric coefficients have been found in the “atomic limit”, inwhich the broadening Γ → . In this limit, taking the Coulomb repulsion U to be zero, one recovers the idealized, un-broadenedsingle-level limit of Mahan and Sofo, and correspond-ingly, an infinite electronic contribution to the figure ofmerit (i.e., after setting κ ph = U when two electrons are present in the im-purity. In the atomic limit Γ → U ≫ T , Murphy, Mukerjee, and Moore showedthat the second level becomes unoccupied and hence, ir-relevant for transport, meaning that the system againeffectively reduces to a single-level one with a diverg-ing electronic figure of merit. For intermediate couplingstrength, U ∼ T , both levels are active in transport andthe figure of merit is not large in general.Analogous studies of thermoelectricity in the atomiclimit of bulk systems have been undertaken in a pairof well-known papers by Beni and Chaikin and Beni (see also Ref. 24). As emphasized by Beni , however,it is nontrivial to perturb away from the atomic limit oftransport since transport formally vanishes when Γ = . Even when interactionsare modest, it is challenging to develop a reliable pertur-bation expansion of transport quantities in powers of thecoupling matrix elements between the impurity and leads(effectively, Γ).In this paper, we use sum rules for the impurityelectron spectral function to derive expressions for thethermoelectric coefficients that are asymptotically exactin two regimes for which the broadening is small, butnonzero: First, at high temperatures, greater than U, Γand the bandwidth D that characterizes the leads. Sec-ond, in the very strongly-correlated, “narrow-level” limitwhere U ≫ T ≫ Γ and U > D . Our results in this lat-ter regime reproduce the divergent figure of merit in theatomic limit when U → ∞ . At the same time, this limitis highly singular and we findlim Γ → ZT ( U ≫ T ) ∼ ( ǫ − µ ) Γ D , (2)where ǫ − µ is the difference between the impurity energyand Fermi levels. This result emphasizes the difficulty ofstudying transport in this limit and also the crucial roleplayed by the broadening in the thermoelectric perfor-mance of impurity models with strong interactions.We start in Sec. II by introducing the linear-responseformulae for thermoelectric transport coefficients interms of the single-particle impurity spectral function.In Sec. III, we introduce two sum rules for this spectralfunction—one that integrates all spectral weight and an-other which removes the irrelevant weight in the upperHubbard peak—and show how these can be used to de-rive results for transport that are asymptotically exactin the regimes elucidated above. We then calculate ex-plicit forms for these sum rules for the Anderson impu-rity model in Sec. IV, and use these results to discussthermoelectric performance in Secs. V and VI. Finally,in Sec. VII, we summarize our main results and concludewith a discussion of the implications of our results forquantum dot and molecular junction systems. II. TRANSPORT COEFFICIENTS
Our starting point is the (generalized) Landauer-likeexpressions for the charge J = e π ∫ ∞−∞ dE T (
E, T L , T R )[ f L ( E ) − f R ( E )] (3)and heat J Q = π ∫ ∞−∞ dE T ( E, T L , T R )( E − µ L )[ f L ( E ) − f R ( E )] (4)currents through an impurity connecting two leads. f ν ( E ) ≡ { exp [ β ν ( E − µ ν )] + } − is the Fermi functionat the left ( ν = L ) and right ( ν = R ) leads, with temper-ature T ν ≡ β − ν and chemical potential µ ν . Unless speci-fied otherwise, throughout this paper we set ̵ h = k B = A ( E ) of the impurity electrons : T ( E, T L , T R ) = ( πγ / ) Γ ( E ) A ( E − µ ) , (5) where Γ ≡ Γ L + Γ R (6)is the sum of the broadenings [assumed in (5) to be purelyreal] at the left and right leads, γ ≡ L Γ R / Γ (7)is an asymmetry parameter that deviates from unitywhen Γ L ≠ Γ R , and A ( E ) ≡ − π ∑ σ Im G ret σ ( E ) (8)is the spectral function for both spin species σ of impu-rity electrons with retarded Green’s function G ret ( E ) . Ittrivially follows from the analysis in Ref. 27 that (4) isalso exact with the identification in (5) .The precise form of the spectral function A ( E ) is spec-ified by the details of the coupling between the impurityand the leads. To be specific, supposeˆ V = ∑ k νσ [ V k ν ˆ c † k νσ ˆ d σ + h . c . ] (9)couples the single-level impurity electrons characterizedby ˆ d σ to the non-interacting ν = L, R leads, andˆ H l = ∑ k νσ ( ǫ k ν − µ ν ) ˆ n k νσ (10)describes noninteracting electrons with momentum dis-tribution ˆ n k νσ ≡ ˆ c † k νσ ˆ c k νσ in these leads. For this model, A ( E ) is given by A ( E ) = ∑ σ ∑ a,b ( P a + P b )⟨ b ∣ ˆ d † σ ∣ a ⟩⟨ a ∣ ˆ d σ ∣ b ⟩ δ ( E − E b + E a ) , (11)and the broadening isΓ ν ( E ) = π ∑ k δ ( E − ǫ k ν )∣ V k ν ∣ . (12)The states ∣ a ⟩ , ∣ b ⟩ in the spectral representation (11) arethe exact eigenstates of the many-body grand-canonicalhamiltonian, ˆ H ∣ a ⟩ = E a ∣ a ⟩ . Within linear response µ L ≃ µ R and T L ≃ T R , P a ≡ exp ( − βE a )/ Z , with Z the grandcanonical partition function. ˆ H includes (9) and (10)as well as the as-yet-unspecified impurity hamiltonian.We will restrict ourselves in what follows to the linearresponse regime.Equations (3), (4), and (5) lead to the following ex-pressions for the transport coefficients in the linearresponse regime for an energy-independent broadeningΓ ν ( E ) = Γ ν (momentarily restoring ̵ h and k B ): G = e γ Γ16 ̵ hk B T M , (13) κ el = γ Γ16 ̵ hk B T ( M − M M ) , (14) S = eT M M , (15)and ZT = M M ( M − M / M ) , (16)defined in terms of the integral expression M n ≡ ∫ ∞−∞ dEE n sech ( βE / ) A ( E ) . (17)In arriving at these expressions, we have shifted E → E + µ in the linear-response limit µ L = µ R = µ of the expressions(3) and (4) for the currents, meaning that the spectralfunction A ( E ) is the one defined in (11). An energy-independent broadening is appropriate for leads that ex-hibit good metallic behaviour with a broad bandwidth,generally much larger than Γ. Had we not made this as-sumption, the broadening Γ ( E ) would have entered theintegral expression (17). (As we explain below, this situ-ation can also be dealt with using the methods developedin this paper, although the resulting calculations wouldbe more complicated.) The above expressions will formthe basis of our analysis in the remainder of this paper.Before closing this section, we briefly comment on themajor result of Mahan and Sofo who, as noted earlier,showed that the electronic contribution to the figure ofmerit would diverge in a system described by a single, un-broadened energy level. In the language of this section,this corresponds to having a Dirac-delta function spectralfunction A ( E ) ∝ δ ( E − ǫ ) . (18)Using this in the above, the thermopower is (restoring k B ) S = ( k B / e )( ǫ / T ) . Crucially, using (18) in (17), onefinds M = M M , (19)and the electronic contribution to the thermal conduc-tance vanishes identically at all temperatures , with theresult that the figure of merit is divergent at all temper-atures.Moving away from the limit (18) of having a singleun-broadened energy level, a small thermal conductancecan more generally be understood as expressing a smallvariance in the single-particle energy. Defining ⟪ ⋯ ⟫ F ≡ ∫ dE ( ⋯ ) A ( E ) F ( E ) ∫ dEA ( E ) F ( E ) , (20)we see that M − M M = M [⟪ E ⟫ F − ⟪ E ⟫ F ] , (21)with F = sech ( βE / ) . Hence, the ratio κ el GT = e T (⟪ E ⟫ F − ⟪ E ⟫ F ) F = sech ( βE / ) , (22) which defines the Lorenz number in bulk systems, pro-vides a direct measure of the variance in the single-particle energies clustered within T of the Fermi level(because of the form of F ). It is thus clear why a systemwith a single un-broadened energy level would be opti-mal, having zero variance in its energy levels; see alsoEq. (18) in Mahan and Sofo . This result also presagesour main conclusion: Even though an extremely large U can eliminate the energy-level variance due to interac-tions, unless the broadening Γ due to coupling with theleads can be made smaller than the temperature (notcurrently the situation in experiments ), the energyvariance will be greater that T , and one should onlyexpect modest thermoelectric performance. III. SUM RULE EXPRESSIONS FORTRANSPORT COEFFICIENTS
The evaluation of the spectral function A ( E ) that en-ters the expressions for the thermoelectric coefficients isa challenging many-body problem. In this paper, we pro-pose an alternative, evaluating instead M Tn ≡ ∫ ∞−∞ dEE n A ( E ) (23)and ˜ M n ( Ω c ) ≡ ∫ Ω c −∞ dEE n A ( E ) . (24)The first of these is a well-known sum rule , and canstraightforwardly be evaluated in terms of commutatorsinvolving electron creation and annihilation operatorsand the hamiltonian (see e.g., Refs. 30,31). Integratingthe product of E n and (11) leads to (see Appendix A) M Tn = ∑ σ ⟨{ ˆ d † σ , [ ˆ d σ , ˆ H ] n }⟩ . (25)Here [ ˆ d σ , ˆ H ] n is a nested commutator with [ ˆ d σ , ˆ H ] = ˆ d σ the zeroth-order commutator, [ ˆ d σ , ˆ H ] = [ ˆ d σ , ˆ H ] thefirst-order commutator, [ ˆ d σ , ˆ H ] = [[ ˆ d σ , ˆ H ] , ˆ H ] , and soon.The second expression, (24), involving an energy cutoffΩ c , is evaluated by projecting out the states with energyabove Ω c . Taking Γ ≪ Ω c ≪ U , this amounts to de-riving an effective operator d σ,ǫ for the lower Hubbardpeak. The resulting sum rule for the spectral weight con-tained in the lower Hubbard peak of a single-level impu-rity model is ˜ M n = ∑ σ ⟨{ d † σ,ǫ , [ d σ,ǫ , ˆ H ] n }⟩ . (26)The precise choice of operator d σ,ǫ depends on the modelunder consideration; its derivation for the Anderson im-purity model coupled to two leads will be given in thenext section. We could equally derive a sum rule for thespectral weight contained in the upper Hubbard peak, A ! E " A ! E " (a)(b) FIG. 1: Schematic plot of the spectral function A ( E ) at hightemperatures T ≫ U, Γ (a) and for strong correlations Γ ≲ T ≪ U (b) relative to the lead Fermi level (denoted hereby E = ( βE / ) that enters the moments M n ;the dotted lines show the weight functions—unity and thestep function Θ ( Ω c − E ) —involved with the sum rules. as would be relevant for the situation when the chemicalpotential is close to this peak. In what follows, we willassume that the Fermi level is close to the lower Hubbardpeak, however.The sum rule (23) provides a rigorous upper bound on M n when n is even: M n ≤ M Tn n = , , ... ∀ T, (27)since A ( E ) ≥ ∀ E and sech ( x ) ≤ ∀ x . This immedi-ately allows us to write down upper bounds on the chargeconductance G ≤ e Γ8 ̵ hT M T (28)and, making use of the fact that M / M ≥ , thermalconductance κ el at zero charge current: κ el ≤ Γ2 ̵ hT M T . (29)Related bounds for the thermal conductance due tobosonic excitations (e.g., phonons) were derived inRef. 33.Because they involve a ratio of moments, (27) does notobviously translate to bounds for the Seebeck coefficient S and figure of merit ZT . The usefulness of the sum rulesis that they provide asymptotically exact approximationsto the transport coefficients at high-temperatures, in-cluding the asymptotic limit T ≫ U, Γ , D , and also inthe strongly-correlated regime Γ ≲ T ≪ U . In Fig. 1, we show two schematic plots of the impurity spectral func-tion along with the weight function sech ( βE / ) thatenters the exact transport moments M n . In Fig. 1(a),the temperature is much larger than the Coulomb repul-sion U as well as Γ; the majority of the spectral weightarises in a Lorentzian centred around the Fermi level ofwidth ∼ Γ. A small peak—the weight of which vanisheswith decreasing U / T —arises at U corresponding to theenergy of a doubly occupied impurity. In this regime, thesech ( βE / ) function is essentially unity wherever thereis nonzero spectral weight and one can replace M n by thehigh-temperature asymptotes M Tn , with the result thatthe thermoelectric transport coefficients calculated usingthese sum rules become asymptotically exact:lim T ≫ U, Γ ,D { G, κ el , S, ZT } [ M n ] = { G, κ el , S, ZT } [ M Tn ] . (30)We formally require that T ≪ D be much larger thanthe lead bandwidth since the spectral function scales as A ( E ) ∝ E − for large E , meaning that M T and M T cannot well-approximate M and M unless D ≲ T .Fig. 1(b) shows the spectral function in the strongly-correlated regime Γ ≪ T ≪ U . Here two Hubbard peaksdevelop, the lower one centred at some single-particleenergy scale, ǫ , and the upper one centred at ǫ + U .When the Fermi level (denoted by the zero in theseplots) is within ∼ T of e.g. the lower Hubbard peak,the sech ( βE / ) function excludes the upper Hubbardpeak. Because there is negligible spectral weight outsidethe two peaks when U ≫ Γ , in this regime M n can bereplaced by ˜ M n ( Ω c ) , where Γ ≪ Ω c ≪ U and the trans-port coefficients again admit asymptotically exact sumrule values:lim Γ ≪ T ≪ U { G, κ el , S, ZT } [ M n ] = { G, κ el , S, ZT } [ ˜ M n ( Γ ≪ Ω c ≪ U )] . (31)Even though (31) is asymptotically exact in the limit Γ ≪ T , it is evident that these results will remain qualitativelyvalid for Γ ≲ T .As noted earlier, we could also consider the thermo-electric performance when the Fermi level is close to theupper Hubbard peak by considering the sum rule forthis peak. Apart from effects that break particle-holesymmetry (enhanced by molecular vibrational modes ),the thermoelectric performance will be the same in bothcases.In Figure 2, we plot the regimes in the parameter spacespanned by U / T, Γ / T where our results hold. Also shownis the regime studied by Murphy, Mukerjee, and Moore ,as well as that studied using NRG by Costi and Zlati´c ,although NRG can study the entire region shown in thisfigure. Even though it might seem that we are dealingwith a highly restricted region of parameter space—andone that is moreover far from most current experimentsinvolving molecular junctions for which Γ ∼ O ( T ) —we re¨emphasize that it is only when Γ ≪ T that a sub-stantial figure of merit can be realized. We also note that
MolecularjunctionsThis workRef. ! " Ref. ! " ZT large ZT large ! T U T FIG. 2: Plot of the regimes where our results hold ( U ≫ T ≳ Γand T ≫ U, Γ , D ), in comparison to the regimes studied inthe sequential tunnelling approximation of Ref. 21 (all U ,Γ / T →
0) and the numerical renormalization group calcula-tions of Ref. 3 ( U / Γ =
8, all Γ). Note that the latter calcu-lation could have been extended to any region shown above.Also indicated are the regimes (Γ , U ≪ T and Γ ≪ T, U ≫ T )where the electronic contribution to the figure of merit be-comes large as well as the parameter regime of typical molec-ular junction experiments. Although the value of U / T is un-known, Γ is typically ≳ T . this regime is well away from the Kondo regime T ≲ T K ,since T K ≪ Γ In the remainder of this paper, we investigate the im-plications of (30) and (31) for an archetypal model ofstrongly correlated “impurity” electrons, the Andersonimpurity model, generalized to include the coupling be-tween an impurity and two leads. These sum-rule resultshave been derived assuming a constant broadening func-tion, as is almost always taken to be the case. One canalso consider energy-dependent broadenings: for simplepower-law forms such as Γ ( E ) = Γ E l with l and integer,the resulting transport coefficients simply involve higher-moment sum rules ( n → n + l ). For more complicated de-pendencies, one must resort to the operator-product ex-pansion technique of Wilson, Kadanoff, and Polyakov. IV. SUM RULES FOR THE ANDERSONIMPURITY MODEL
The single-level Anderson impurity model (AIM) foran impurity coupled to two leads isˆ H = ˆ H d + ˆ H l + ˆ V . (32)Here ˆ H d = ( ǫ − µ ) ∑ σ ˆ d † σ ˆ d σ + U ˆ n ↑ ˆ n ↓ (33)is the impurity hamiltonian and H l and ˆ V are given by(10) and (9), the former with µ ν = µ , as appropriate inthe linear response regime. A. Sum rules for the entire spectrum
The results of a straightforward calculation using (25)and (32) are M T = , (34) M T = ( ǫ − µ + U n d / ) , (35)and M T = ( ǫ − µ ) + ( ǫ − µ ) U n d + U n d + ∑ k ν ∣ V k ν ∣ . (36)Here n d ≡ ∑ σ n σ is the total occupancy of the impurity.The factor of two in (34) (and elsewhere in the above)counts the number of single-particle states: two for thesingle-level AIM with two spins. Taking the V k ν to beindependent of momentum, ∑ k ν ∣ V k ν ∣ = ∑ ν ∣ V ν ∣ N ∫ D − D dE = DN ∑ ν ∣ V ν ∣ . (37)Here we have taken the lead DOS N σ ( E ) = N per spin tobe constant (and independent of spin) with D the half-bandwidth. Making the same assumptions to evaluatethe broadening (32), it becomesΓ ν = πN ∣ V ν ∣ , (38)and thus ∑ k ν ∣ V k ν ∣ = D Γ / π. (39)As noted in (21), the combination M T − ( M T ) / M T = M T [⟪ E ⟫ F = − ⟪ E ⟫ F = ] provides a measure of the vari-ance in the single-particle energies, in this case over theentirety of the spectral weight since there is no thermalweighting function sech ( βE / ) . In order to have a smallelectronic contribution to the thermal conductance andhence, a substantial figure of merit, the energy variancein the system needs to be small. Equations (34)-(36) give M T − ( M T ) M T = U n d ( − n d ) + ∑ k ν ∣ V k ν ∣ . (40)The first term on the right-hand side gives the variancein the interaction energy per particle (proportional to U times the compressibility), while the second term de-scribes the variance in the kinetic energy per particle.One can understood this result in terms of Fig. 1: withtwo peaks separated by U , the energy variance is clearly ∼ U . The variance in a single band (either one) however,is ∼ ∑ k ν ∣ V k ν ∣ ∼ Γ D .At temperatures greater than both Γ and U , the vari-ance is of course large, and (36) indicates a correspond-ingly large κ el . On the other hand, for lower tempera-tures, T ≪ U , assuming the Fermi level is close to thelower Hubbard peak, the sech ( βE ) factor in M n is sen-sitive only to the spectral weight in that peak. Corre-spondingly, κ el is only sensitive to the variance ∼ Γ D inthis peak. This notion can be made precise by projectingout the spectral weight contained in the upper Hubbardpeak from the sum rules, as in (24). We turn to this now. B. Projected sum rules for the lower Hubbard peak
Relevant to the case shown in Fig. 1(b), we now cal-culate the sum rule for the lower Hubbard peak (LHP),obtained by projecting out the doubly-occupied statesthat give rise to the upper Hubbard peak (UHP). Ourapproach here follows closely that used in studies of thebulk electron Hubbard model and also the AIMitself by Schrieffer and Wolff . To remove the spec-tral weight associated with the UHP, we need a spectraldecomposition for the impurity electron operator of theform d σ = d σ,ǫ + d σ,ǫ + U . (41)Here d σ,E describes an annihilation process that lowersthe impurity energy by ∼ E modulo Γ, i.e., ⟨ b ∣ d σ,E ∣ a ⟩ = E a − E b = E + O ( Γ ) . Once a suitable expression isfound for d σ,ǫ , one can proceed to calculate the projectedsum rules given by (26).When Γ =
0, the decomposition is particularly simplesince the occupancies ˆ n σ are good quantum numbers,and, defining ¯ σ = − σ , d σ,ǫ = ( − ˆ n ¯ σ ) ˆ d σ (42)and d σ,ǫ + U = ˆ n ¯ σ ˆ d σ . (43)When Γ ≠
0, we seek a renormalized operator ˜ d σ suchthat ˜ n σ is a good quantum number of the bare hamilto-nian ˆ H , and [ ˜ n σ , ˆ H ] =
0. This allows us to undertake asimilar spectral decomposition as (41) and identify therenormalized operator d σ,ǫ for the LHP needed for thesum rules.Since ˜ n σ commutes with ˜ H ≡ ˆ H ( ˆ d σ → ˜ d σ , ˆ c k νσ → ˜ c k νσ ) , where ˆ H ≡ ˆ H d − µ ∑ k ν ˆ n k νσ (44)is the total hamiltonian less all kinetic energies, [ ˜ n σ , ˆ H ] = [ ˜ H , ˆ H ] =
0. Relating the renormalized opera-tors to the original operators by a unitary transformation(not to be confused with the thermopower S , but in keep-ing with widely-used notation) ˆ d σ = e S ˜ d σ e − S andˆ c k νσ = e S ˜ c k νσ e − S , this last condition amounts to findinga transformation operator S that satisfies [ ˆ H, ˜ H ] = ⇒ [ e S ˜ He − S , ˜ H ] = . (45) Here, and in the remaining, we use a tilde to indicate atransformed operator.The fact that ˜ n σ is a good quantum number meansthat—analogous to (41)—we can decompose any oper-ator ˜ O = ˜ O ( ˜ d σ , ˜ c k νσ ) in terms of the eigenstates ˜ n d = ∑ σ ˜ n σ = , ,
2, or equivalently, the corresponding energychanges E = ± ǫ , ± ( ǫ + U ) that they effect, modulo cor-rections O ( Γ ) : ˜ O = ˜ O ± ǫ + ˜ O ± ( ǫ + U ) . (46)˜ O ± E operators are related by hermitian conjugacy:˜ O − E = ˜ O † E . (47)Equation (41) is a special example of this more generalspectral decomposition. Because the eigenvalues ˜ n d of ˜ H are good quantum numbers, so are the energy changes E defined above, with the result that [ ˜ O E , ˜ H ] = E ˜ O E + O ( Γ ) , E = ± ǫ , ± ( ǫ + U ) . (48)Equation (48) allows us to determine S and d σ,ǫ to aspecified order in V /( ǫ + U ) , where V is the characteristicsize of the in-general momentum dependent coupling V k ν .We expand ˆ H = exp ( S ) ˜ H exp ( − S ) as ˆ H = ˜ H + [ S, ˜ H ] + ⋯ and S = S ( ) + S ( ) + ⋯ in powers of V /( ǫ + U ) and usethis in (45). Requiring that S eliminates the ˜ H ± ( ǫ + U ) terms in the spectral decomposition (46) of ˜ H (therebyremoving the associated spectral weight) and using (48),(45) reduces to [ S ( ) , ˜ H d ] = − ˜ T ǫ + U − ˜ T − ǫ − U . (49)Here ˜ T E are the terms in the spectral decomposition ofthe kinetic energy contribution ˜ T ≡ ˜ H − ˜ H to the hamil-tonian; note that these are the only terms in the hamilto-nian that can change the impurity energy by ∼ ± ( ǫ + U ) :˜ H ± ( ǫ + U ) = ˜ T ± ( ǫ + U ) . Explicitly,˜ T ǫ = ∑ k νσ ǫ k ν ˜ n k νσ + ∑ k νσ [ V k ν ˜ c † k νσ ˜ d σ ( − ˜ n ¯ σ ) + H . c . ] (50)and ˜ T ǫ + U = ∑ k νσ V ∗ k ν ˜ n ¯ σ ˜ d † σ ˜ c k νσ . (51)Using (51) and (47) in (49), one finds S ( ) = − ǫ + U [ ˜ T − ǫ − U − ˜ T ǫ + U ] . (52)This is essentially just the Schrieffer–Wolff transforma-tion used by Schrieffer and Wolff to derive the Kondohamiltonian as the effective low-energy description of theAIM . Apart from the presence of the additional single-particle energy scale ǫ in this expression, it is also thesame result as obtained for the bulk Hubbard model, with˜ V replaced by inter-site tunnelling; see e.g., Eq. (7) inRef. 40 and Eq. (10) in Ref. 37.Having determined the form of S ( ) , we can now ob-tain an expression for the renormalized LHP operator ap-pearing in (41). By expanding ˆ d σ = exp ( S ) ˜ d σ exp ( − S ) = ˜ d σ + [ S ( ) , ˜ d σ ] + ⋯ and taking the spectral decompositionof both sides of the resulting expression, one finds per-turbative expressions for the spectral decomposition d σ,ǫ and d σ,ǫ + U of the original operator; c.f. (41). Using (49)and equating terms of the same order in E gives d σ,ǫ = ( − ˜ n ¯ σ ) ˜ d σ + ǫ + U [ ˜ T ǫ + U , ˜ d σ, − ǫ − U ] − ǫ + U [ ˜ T − ǫ − U , ˜ d σ, − ǫ − U ] , (53)for the LHP operator. Explicit evaluation of this expres-sion results in d σ,ǫ = ( − ˜ n ¯ σ ) ˜ d σ − ǫ + U ( ˜ n σ + ˜ n ¯ σ ) ˜ d σ ∑ k ν V ∗ k ν ˜ d † ¯ σ ˜ c k ν ¯ σ − ǫ + U ∑ k ν V ∗ k ν ˜ n ¯ σ ˜ c k νσ + O [ V /( ǫ + U )] . (54)In the limit U → ∞ , (54) reduces to (42).Using (54) to evaluate the first few sum rules (26),making use of the fact that there are no doubly-occupiedstates in the transformed basis, ˜ d † σ ˜ d † ¯ σ =
0, one finds aftersome straightforward but laborious algebra˜ M = − n d − ǫ + U ∑ k νσ [ V ∗ k ν ⟨ ˜ d † σ ˜ c k νσ ⟩ + H . c . ] +O ( Vǫ + U ) , (55)˜ M = ( ǫ − µ ) ˜ M + ∑ k νσ V ∗ k ν ⟨ ˜ d † σ ˜ c k νσ ⟩ + O ( V ǫ + U ) , (56)and˜ M = ( ǫ − µ ) ˜ M + ( − n d ) ∑ k ν ∣ V k ν ∣ − ∑ k νσ V ∗ k ν ( ǫ k ν − µ )⟨ ˜ d † σ ˜ c k νσ ⟩ + ⟨( ∑ k νσ V ∗ k ν ˜ d † σ ˜ c k νσ ) ( ∑ k ′ ν ′ σ ′ V k ′ ν ′ ˜ c † k ′ ν ′ σ ′ ˜ d σ ′ )⟩ + O ( V ǫ + U ) . (57)The analogue of (55) for the bulk Hubbard model hasbeen derived in Refs. 31,37,38. Note that e.g. ∑ k νσ V ∗ k ν ⟨ ˜ d † σ ˜ c k νσ ⟩ = ∑ k νσ V k ν ⟨ ˜ c † k νσ ˜ d σ ⟩ (58)is purely real in equilibrium, as required to have no chargeor heat current. Definingˆ V ≡ ∑ k νσ V k ν ˆ c † k νσ ˆ d σ , (59) such that ˆ V ≡ ˆ V + ˆ V † is the lead-impurity coupling (9),these give the following result for the variance in theLHP:˜ M − ˜ M ˜ M = ( − n d ) ∑ k ν ∣ V k ν ∣ − ∑ k νσ V ∗ k ν ( ǫ k ν − µ )⟨ ˜ d † σ ˜ c k νσ ⟩ + ⟨ ˜ V ⟩ − ( − n d ) ⟨ ˜ V ⟩ . (60)Comparing the full sum rules (34)-(36) with the pro-jected ones (55)-(57) and also the variances, (21) and(60), one sees that the projection has two primary effects:First, it removes spectral weight associated with the UHPand the large Coulomb repulsion U . Second, it renormal-izes the spectral weight from 2 (the total spectral weight)to ∼ − n d , the low-energy spectral weight in the LHP.The variance (60) is essentially the variance of a non-interacting single level , broadened by ∑ k ν ∣ V k ν ∣ and ⟨ ˜ V ⟩ ;the effects of interactions only enter through the spectralweight renormalization 2 → − n d and their effect on thecoherence ⟨ ˜ d † σ ˜ c k νσ ⟩ . To the orders in V /( ǫ + U ) shown,one can replace the transformed operators ˜ d σ , ˜ c k νσ in theabove with the original operators, ˆ d σ , ˆ c k νσ , mindful, how-ever, of the restriction that there be no doubly-occupiedstates.We now use the above results to give insight into thethermoelectric performance of strongly correlated ma-terials in the high-temperature and strongly-correlatedregimes. V. THERMOELECTRIC PERFORMANCE ATHIGH TEMPERATURES T ≫ U Using the results from the previous section, we nowevaluate the exact high-temperature limiting values ofthe thermopower S and figure of merit ZT for the An-derson impurity model. In the high temperature limit, n d ( T ≫ U, Γ ) →
1. Using this in, (34)-(36), the high-temperature values for (15) and (16) become S ( T → ∞ ) = ǫ − µ + U / eT ≡ V g T (61)and, further using (39), ZT ( T → ∞ ) = V g U + D Γ / π . (62)Here, we have introduced the gate voltage V g ≡ ( ǫ − µ + U / )/ e. (63)Taking D = =
0, (61) and (62) reproduce the an-alytic high-temperature limiting values found in Ref. 21[specifically, their Eqs. (19) and (21)] using the sequen-tial tunnelling approximation for Γ ≃
0. From (62),one sees that the diverging figure of merit of Mahan andSofo (valid for all temepratures) is realized in the non-interacting U ≪ ǫ and un-broadened limit D, Γ ≪ ǫ .The high- T limiting form (61) for the Seebeck coefficienthas the same form as the “Mott-Heikes” approximation S ( T → ∞ ) = [ µ ( T = ) − µ ]/ eT —well-known in studies ofbulk narrow-band systems —with the identification µ ( T = ) = ǫ + U /
2, which is indeed the zero-temperaturechemical potential at half-filling, n d = results in atransmission function (single-particle Green’s function)that satisfies the sum rules and hence, (61) and (62).As we have already noted, the sequential tunnelling ap-proximation used in Ref. 21 satisfies these asymptoticresults at high temperatures. Our results also give an-alytic expressions for the high-temperature asymptotesfound numerically in Ref. 3 using numerical renormaliza-tion group . VI. THERMOELECTRIC PERFORMANCE INTHE STRONGLY CORRELATED REGIME U ≫ T . In the strong-correlation, narrow-level regime U ≫ T ≳ Γ, (55)-(57) give S ( U ≫ T ≳ Γ ) = eT [ ǫ + ⟨ ˆ V † ⟩ − n d − µ + O ( V ǫ + U )] (64)for the thermopower and ZT = ( ǫ − µ ) ( − n d ) + ( ǫ − µ )⟨ ˆ V † ⟩( − n d ) + ⟨ ˆ V † ⟩ ( − n d ) ∑ k ν ∣ V k ν ∣ + ( − n d )[⟨ ˆ V † ˆ V ⟩ − ⟨ ˜ V † ξ ⟩] − ⟨ ˆ V † ⟩ + O ( Vǫ + U ) , (65)for the figure of merit, where we have defined ( ξ ≡ ǫ − µ ) ⟨ ˜ V † ξ ⟩ ≡ ∑ k σν V ∗ k νσ ( ǫ k ν − µ )⟨ ˆ d † σ ˆ c k νσ ⟩ . (66)Note that we are using the bare electron operators in theabove and ⟨ ˆ V ⟩ is defined using (59). Although (64) and(65) are asymptotically exact in the limit where thereis a large separation between energy scales, Γ ≪ T ≪ U (also the limit where thermoelectric performance isoptimized), as noted earlier, we expect them to providea reasonable approximation when Γ ≲ T .The thermopower (64) in the strongly-correlatedregime and intermediate temperatures ( T ≳ Γ) againhas the Mott–Heikes form, but with a renormalized low-energy measure ǫ + ⟨ ˆ V † ⟩/( − n d ) of the single-particleenergy that does not include the Coulomb energy scale.In fact, apart from the renormalization factor 2 − n d ac-counting for the absence of doubly-occupied states, this G (cid:144) T S @ k B (cid:144) e D G (cid:144) T ZT FIG. 3: Dependence of the Seebeck coefficient (top) and theelectron contribution to the figure of merit (bottom) on thebroadening Γ for different lead bandwidths: D / T =
100 (blacksolid line), D / T =
20 (blue dashed line), and D / T =
10 (reddotted line). Curves are shown for U = T and ǫ − µ = T although the behaviour shown here is not sensitive to theprecise choice of values. is precisely the single-particle energy of a noninteractingelectron in an impurity coupled to leads by (9). Consis-tent with the findings of Ref. 21, by eliminating the pos-sibility of having a doubly occupied impurity, extremelystrong correlations have effectively reduced the transportproblem to that of a single-level. As a result, transportis effectively that of noninteracting electrons, albeit witha reduced Hilbert space ( n d ≠ ⟨ ˆ V ⟩ ≳ ǫ − µ , thefigure of merit (65) is generically of order unity. It is onlywhen the coupling becomes much smaller than the detun-ing, ⟨ ˆ V ⟩ ≪ ǫ − µ , that one obtains the diverging electroniccontribution to the figure of merit found in Ref. 21. Out-side this limit, even though the strong Coulomb repul-sion has reduced the problem to an effectively single-levelone, the variance in the single-particle energy (roughly,the greater of ∼ ∑ k ν ∣ V k ν ∣ and ⟨ ˆ V † ˆ V ⟩ − ⟨ ˆ V † ⟩ ) preventsthe system from attaining optimal thermoelectric perfor-mance.All terms in the denominator of (65) vanish in theΓ → or D → , D . To check this, in Fig. 3 we show (64) and(65) as functions of Γ using Hartree–Fock (HF) theoryand (39) for a range of bandwidths. Details of the HFcalculations are given in Appendix B. Although HF isnot a quantitatively reliable theory when interactions arestrong, we emphasize that we are only applying HF the-ory to evaluate the asymptotically exact (conserving) re-sults (64) and (65), and we expect it to give qualita-tively reliable results. We are not presenting the resultsof a (non-conserving) HF theory of transport, the re-sults of which would be completely unreliable. Whilethe thermopower is generically of order unity (in unitsof k B / e ) for ǫ − µ = T , the figure of merit is singularat small broadenings and is well-described by the scal-ing in (2). For comparison with this plot, in molecularsystems, Γ ∼ ( . → . ) eV or, Γ = ( . → ) T room ,with T room = K . For leads constructed of a goodmetal such as gold, D ∼ ∼ T room .Fig. 3 emphasizes that either Γ or the lead bandwidth D need to be much smaller than temperature in order toreproduce the substantial figure of merit found in Ref. 21for U → ∞ . Reinforcing the fact that the thermoelec-tric transport properties in the very strongly correlatedregime U ≫ T is equivalent to that of a renormalized non-interacting system, we note that Fig. 3 closely resemblesthe U = VII. DISCUSSION
Much of the interest in studying the thermoelectricperformance of quantum dots and molecules originatesin the likely small phonon contribution to the thermalconductance in these systems and also the fact that theycan be well-approximated as single-level systems. For asingle-level noninteracting impurity system, the figure ofmerit diverges in the limit that the extrinsic broadeningarising from the coupling to the leads is much smallerthan the temperature. Turning on a Coulomb repulsion,the appearance of a satellite Hubbard peak leads to anincrease in the variance of the single-particle energy leveland hence, a diminished figure of merit.Somewhat counterintuitively, when the Coulomb re-pulsion is very strong, however, the new satellite Hub-bard level becomes unimportant, as it is too high an en-ergy scale to be relevant for transport. We have usedasymptotically exact sum rule expressions in this limitto show that the thermoelectric transport coefficients as-sume essentially the same form as for a noninteractingsystem, slightly renormalized to account for the absenceof doubly occupied states. The thermopower (64) in par- ticular assumes the well-known Mott–Heikes form, butwith an effective noninteracting chemical potential. Aswith a noninteracting system, the transport propertiesin this strong interaction regime are limited only by thebroadening due to the coupling between the leads andthe impurity.This bodes well for using small impurity-type sys-tems such as quantum dots and molecules to achievesubstantial thermoelectric efficiencies. Even though thebare (un-screened) Coulomb energy scale is very large( U ≫ T ) in these systems, it is precisely its large sizethat means it does not lead to a diminishment of trans-port, as would happen if it was of moderate strength, say U ∼ T . At the same time, the highly singular dependenceof the figure of merit on the broadening and also the leadbandwidth [c.f. (2)] means that it will be essential tohave a small broadening arising from lead-impurity cou-pling or to tailor the bandwidth properties of the leadsin order to realize the potential of impurity systems. Acknowledgments
This work was funded by the Natural Sciences and En-gineering Research Council of Canada Discovery Grantand the Canada Research Chair Program.
Appendix A: Spectral properties of thesingle-particle Green’s function
In this Appendix, we give an example of the derivationof the general result (25) for n = M T ≡ ∫ ∞−∞ dEEA ( E ) = ∑ σ ∑ a,b ( P a + P b )⟨ a ∣ ˆ d σ ∣ b ⟩⟨ b ∣ ˆ d † σ ∣ a ⟩( E b − E a ) = ∑ σ ∑ a,b P a [⟨ a ∣ ˆ d σ ˆ H ∣ b ⟩⟨ b ∣ ˆ d † σ ∣ a ⟩ − ⟨ a ∣ ˆ H ˆ d σ ∣ b ⟩⟨ b ∣ ˆ d † σ ∣ a ⟩] + ∑ σ ∑ a,b P b [⟨ b ∣ ˆ d † σ ∣ a ⟩⟨ a ∣ ˆ d σ ˆ H ∣ b ⟩ − ⟨ b ∣ ˆ d † σ ∣ a ⟩⟨ a ∣ ˆ H ˆ d σ ∣ b ⟩] = ∑ σ ∑ a P a [⟨ a ∣ ˆ d σ ˆ H ˆ d † σ ∣ a ⟩ − ⟨ a ∣ ˆ H ˆ d σ ˆ d † σ ∣ a ⟩] + ∑ σ ∑ b P b [⟨ b ∣ ˆ d † σ ˆ d σ ˆ H ∣ b ⟩ − ⟨ b ∣ ˆ d † σ ˆ H ˆ d σ ∣ b ⟩] = ∑ σ [⟨ ˆ d σ ˆ H ˆ d † σ ⟩ − ⟨ ˆ H ˆ d σ ˆ d † σ ⟩ + ⟨ ˆ d † σ ˆ d σ ˆ H ⟩ − ⟨ ˆ d † σ ˆ H ˆ d σ ⟩] = ∑ σ ⟨{ ˆ d † σ , [ ˆ d σ , ˆ H ]}⟩ . (A1) Appendix B: Hartree–Fock theory for the AIMmodel
In this Appendix, we describe how to calculate theimpurity occupation n d , lead-impurity coherence ⟨ ˆ V ⟩ = ⟨ ˆ V † ⟩ , and ⟨ ˜ V † ξ ⟩ , defined in (66) using Hartree–Fock (HF)theory. Within this approximation, ⟨ ˆ V † ˆ V ⟩ = ⟨ ˆ V ⟩ andhence, these three quantities are sufficient to evaluatethe figure of merit (65).Momentarily generalizing the AIM hamiltonian (32) toallow for different chemical potentials and couplings foreach spin, µ → µ σ and V k ν → V k νσ , one has n σ ≡ ⟨ ˆ n σ ⟩ = − ( ∂ Ω ∂µ σ ) T (B1)and ⟨ ˆ d σ ˆ c † k νσ ⟩ = − ⟨ ˆ c † k νσ ˆ d σ ⟩ = − ( ∂ Ω ∂V k νσ ) T , (B2)where Ω is the free energy. At the end of our calcula-tion, we set µ ↑ = µ ↓ = µ and likewise with V k νσ . (B1) isa standard thermodynamic identity, while (B2) followsfrom an application of the Hellmann–Feynman formula ( ∂ Ω / ∂λ ) T,N = ⟨ ∂ ˆ H / ∂λ ⟩ with λ = V k νσ to (32).Since it is an effective single-particle theory, the HFfree energy is given by the trace of the logarithm of theHF Green’s function G σ ( iω n ) for spin σ impurity elec-trons: Ω = − β − ∑ nσ e iω n + ln [ − G − σ ( iω n )] , yieldingΩ = − β ∑ nσ e iω n + ln [ − iω n + ǫ − µ σ + U n ¯ σ + ∑ k ν ∣ V k νσ ∣ iω n − ξ k νσ ] . (B3)Here ω n = ( n + )/ β are Fermi Matsubara frequencies,with integer n , and ξ k νσ ≡ ǫ k ν − µ σ . The exp ( iω n + ) factor with positive infinitesimal 0 + ensures conver-gence. Making the same assumptions (constant leadDOS, momentum-independent coupling V k νσ ) that weused to arrive at (38), (B3) reduces toΩ =− β ∑ nσ e iω n + ln [ − iω n + ǫ − µ σ + U n ¯ σ − ( Γ4 π ) ln ( iω n − Diω n + D )] . (B4)We will not make the usual approximation in the AIMliterature of taking D → ∞ , in which case ln [( iω n − D )/( iω n + D )] reduces to π sgn ( ω n ) (see e.g., Sec. 5.2in Ref. 35).Applying (B1) and (B2) to (B3) [for the coherence(B2) we need to take the derivative of the free energywith respect to V k νσ before assuming the coupling to be constant], one arrives at the results n σ = β ∑ n e iω n + iω n − ǫ + µ σ − U n ¯ σ + ( Γ4 π ) ln ( iω n − Diω n + D ) (B5)and ⟨ ˆ d σ ˆ c † k νσ ⟩ = − β ∑ n e iω n + ( iω n − ξ k σ ) × V ∗ k νσ [ iω n − ǫ + µ σ − U n ¯ σ + ( Γ4 π ) ln ( iω n − Diω n + D )] (B6)Using this last result to evaluate ⟨ ˆ V ⟩ and (66) and re-placing the momentum integration with one over energy[assuming again, a constant lead DOS as in (37)] gives ⟨ ˆ V ⟩ = β ∑ n e iω n + ( Γ2 π ) ln ( iω n − Diω n + D ) iω n − ǫ + µ σ − U n ¯ σ + ( Γ4 π ) ln ( iω n − Diω n + D ) (B7)and ⟨ ˜ V † ξ ⟩ = β ∑ n e iω n + ( Γ2 π )[ D + iω n ln ( iω n − Diω n + D )] iω n − ǫ + µ σ − U n ¯ σ + ( Γ4 π ) ln ( iω n − Diω n + D ) , (B8)To obtain the results shown in Fig. 3, we solve (B5)self-consistently for n σ , evaluating the Matsubara fre-quency sum numerically by summing between n = − → . 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