Xu Kotliar

8/13/2019 Xu Kotliar

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PHYSICAL REVIEW B 84, 035114 (2011)

High-frequency thermoelectric response in correlated electronic systems

Wenhu Xu, 1 Cedric Weber, 2 and Gabriel Kotliar 11 Department of Physics and Astronomy, Rutgers University, 136 Frelinghuysen Rd., Piscataway, NJ 08854, USA

2Cavendish Laboratories, Cambridge University, JJ Thomson Avenue, Cambridge, United Kingdom(Received 11 March 2011; revised manuscript received 12 May 2011; published 21 July 2011)

We derive a general formalism for evaluating the high-frequency limit of the thermoelectric power of stronglycorrelated materials, which can be straightforwardly implemented in available rst principles LDA + DMFTprograms. We explore this formalism using model Hamiltonians and we investigate the validity of approximatingthe static thermoelectric power S 0 , by its high-temperature limit, S . We point out that the behaviors of S andS 0 are qualitatively different for a correlated Fermi liquid near the Mott transition, when the temperature is in thecoherent regime. When the temperature is well above the coherent regime, e.g., when the transport is dominatedby incoherent excitations, S provides a good estimation of S 0.

DOI: 10.1103/PhysRevB.84.035114 PACS number(s): 71 .10 . w, 71 .15 . m, 72 .15 .Jf

I. INTRODUCTION

Thermoelectric energy harvesting, i.e. the transformationof waste heat into usable electricity, is of great current interest.The main obstacle is the low efciency of materials forconvertingheat to electricity. 1,2 Over thepast decade, there hasbeen a renewed interest on thermoelectric materials, mainlydriven by experimental results. 3

Computing the thermoelectric power (TEP) in correlatedsystems is a highly non-trivial task and several approximationschemes have been used to this intent. The well-known Mott-Heikes formula 4,5 gives an estimate of the high temperaturelimit of TEP 6 in the strongly correlated regime. A generalizedBoltzmann approach including vertex corrections has beendeveloped in Ref. 7 and applied to several materials. Thermo-electric transport at intermediate temperature was carefullyinvestigated in the context of single-band and degenerateHubbard Hamiltonians, by dynamical mean eld theory(DMFT). 810 Kelvin formula was also revisited for variouscorrelated models in Ref. 11 very recently.

The high frequency (AC) limit provides another interestinginsights to gain further understanding of the thermoelectrictransport in correlated materials, and is the main interest of this work. The thermopower in the high frequency limit of adegenerate Hubbard model near half-lling was consideredin Ref. 9, where the authors generalize the thermoelectricresponseto nite frequencies in thehightemperaturelimit. Thesame limit was studied recently by Shastry and collaborators,who have developed a formalism for evaluating the AClimit of thermoelectric response using high temperature seriesexpansion and exact diagonalization. The methodology wasapplied to a single band t-J model on a triangular lattice. 12,13

The authors pointed out that the AC limit of TEP ( S ) is simpleenough that it can be obtained by theoretical calculationswith signicantly less effort, while still provides nontrivialinformations of the thermoelectric properties, and give anestimation of the trend of S 0 .

In this work, we investigate the high frequency limit of TEP, S , by deriving an exact formalism in the context of ageneral multi-band model with local interactions. We showthat S is determined by the bare band structure and thesingle-particle spectral functions. The relation between the

conventional TEP, i.e., obtained at zero frequency ( S 0) andthe AC limit S is discussed from general arguments on the

single particle properties of correlated systems at low andhigh temperatures. The analytical derivation of S is comparedwith the frequency dependent thermopower of the one bandHubbard model, solved by dynamical mean eld theory(DMFT) on the square and triangular lattices. The formalismderived in this work can be conveniently implemented intorst-principles calculations of realistic materials, such as inthe LDA + DMFT framework. 14,15

This paper is organized as follows. In Sec. II A, generalformalismof dynamical thermoelectric transport coefcients issummarized to dene the notation. In Sec. II B, exact formulasto evaluate S are derived for a general tight-binding modelwith local interactions. In Sec. III , we apply the formalism toone-band Hubbard model on square and triangular lattice. Thelow and high temperature limit behaviors of S are discussedand compared to those of S 0 . Numerical results are presentedin Sec. IV. Sec. V summarizes the paper.

II. DYNAMICAL THERMOELECTRIC TRANSPORTFUNCTIONS AND HIGH-FREQUENCY LIMIT

OF THERMOPOWER

A. General formalism

Electrical current can be induced by gradient of electricalpotential and temperature. The phenomenological equationsfor static (DC limit) external elds are 16

J x1 = Lxx11

1T x + L xx12 x

1T

, (1)

J x2 = Lxx21

1T x + L xx22 x

1T

. (2)

We only consider the longitudinal case. J x1 and J x2 are x

component of particle and heat current, respectively. x and x 1T are generalized forces driving J

x1 and J

x2 . = eV ,

in which is chemical potential and V is the electric potential.L xxij are transport coefcients. We follow the denition inRef. 16, which explicitly respects the Onsager relation,

035114-11098-0121/2011/84(3)/035114(9) 2011 American Physical Society
http://dx.doi.org/10.1103/PhysRevB.84.035114http://dx.doi.org/10.1103/PhysRevB.84.035114


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WENHU XU, C EDRIC WEBER, AND GABRIEL KOTLIAR PHYSICAL REVIEW B 84, 035114 (2011)

L xxij = Lxxji . Transport properties can be dened in terms of

L xxij . For example, the electric conductivity , thermoelectricpower S , and the thermal conductivity are

=e2

T L xx11 , (3)

S = 1eT

Lxx12

L xx11, (4)

=1

T 2L xx22

(L xx12 )2

L xx11. (5)

In following context, we use kB = e = h = 1. The practicalvalue of S is recovered by multiplying the factor kB /e =86 .3 V/K , which we use as the unit for thermopower.

In conventional thermoelectric problems, L xxij is theoreti-cally dened and experimentally measured at the DC limit.The extension to dynamical (frequency-dependent) case isabsent in standard textbooks but has been studied in detail in

Ref. 12. Here we give the outlines of the formalism. Borrowedfrom Luttingers derivation, 17 an auxiliary gravitationaleld coupled to energy density is dened. An equivalencebetween the ctitious gravitational eld and the temperaturegradient is proved. Then the transport coefcients L xxij canbe written in terms of correlation functions between particlecurrent and (or) energy current. In Ref. 12, this formalismis generalized to temporally and spatially periodic externalelds, thus the transport coefcients become momentum- andfrequency-dependent functions, L xxij (q , ).

Some interestingremarks can be made on L xxij (q , ).On theone hand, in the DC limit( 0), there are two different waysof taking the thermodynamic limit ( q 0)12,17 because thetwo limits, 0 and q 0 do not commute. If we denev = |q | as the phase velocity of the external perturbationeld, the so-called fast limit is dened as taking q 0before 0, thus leading to v . In the fast limit,the transport thermopower, or, the conventional DC limit of thermopower is obtained. Theslow limit is dened as 0is taken before q 0, thus v 0. Therefore, the perturbationis adiabatic and the charge and energy can redistribute to reachan equilibrium state. The slow limit then gives the Kelvinformula of thermopower discussed in Ref. 11.

Onthe other hand, inthe AC limit ( ), the two limits, and q 0, commute, because the phase velocity vwill be innity in either scenario. This can also be shownfrom the general formalism of L xx

ij (q , ) for nite q and in

Ref. 12.The dynamical transport coefcients with q 0 are

given by,

L xxij () = T

0dte i (+ i 0

+ )t

0d J xj ( t i )J

xi . (6)

For a given Hamiltonian H , the current operators aredened by following the conservation laws, 16

J xi =O xi

t

= i H,O xi . (7)

O xi is the x-component of particle and heat polarizationoperator. Specically,

O x1 =i

R xi n i , (8)

O x2 =i

R xi (h i n i ) , (9)

where n i and h i are local particle and energy density operators.The explicit forms of ni and h i are determined by theHamiltonian of specic models. In next subsection, we willwrite O i and give J i for a general multiband model.

At DC limit, the imaginary part of L xxij ( = 0) is zero, thusS 0 is determined by the real parts. For convenience, dene

L 0ij Re Lxxij (0), (10)

then we have

S 0 Re S ( = 0) = 1T

L 012L 011

. (11)

At AC limit, L xxij () is dominated by the imaginary part,with a O (1/ ) leading order,

Im L xxij () =T

Lij + O 12

. (12)

Using Lehnmans representation, it has been shown that Lij dened above is, up to a factor of i , the expectation values of commutators between current and polarizationoperators, 9,12,13

i.e.,

Lij = i J xj ,O

xj . (13)

Consequently, TEP at AC limit is

S Re S ( ) = 1T

L

12L11

. (14)

Lij can be related to Re L ij (). Applying Kramers-Kronigrelation and keeping the leading order in 1 / , we have

Lij =1

T

d Re L xxij (). (15)

Thus Lij is also connected to the sum rules of dynamicalquantities. For example, L11 is proportional to the sum rule of conductivity. 18,19

L11 =2

0d Re (). (16)

Other sum rules are also derived in Ref. 12 and 13.

B. General formula of L i j Now we explicitly evaluate the commutator in Eq. (13)

for a general tight-binding Hamiltonian with local interaction,which will determine the AC limit of TEP in this system. Westart with the following Hamiltonian

H = ij,

t ij ci cj +

i

ci ci

+i

U ci c

i ci ci . (17)

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HIGH-FREQUENCY THERMOELECTRIC RESPONSE IN . . . PHYSICAL REVIEW B 84, 035114 (2011)

i , j are site indices. , , and denote local orbitals. t ij is the hopping integral, and U is the matrix element forCoulomb interaction between local orbitals. is energy levelof local orbitals. The particle polarization operator is

O x1 =i

R xi

ci ci , (18)

and the heat polarization operator is

O x2 =i

R xi 12

j,

t ij ci cj + t

ji c

j ci

+

U ci c

i ci ci +

( )ci ci .

(19)

The current operators turn out to be

J x1 = i H,Ox1 = i

ij,

R xj Rxi t

ij c

i cj , (20)

and

J x2 = i H,Ox2

=ijl,

i

2t il t

lj R

xj R

xi c

i cj

i

2ij,

t ij Rxj R

xi ( + 2 )c

i cj

i

2ij,

t ij Rxj R

xi

(U U )ci c

j cj cj

+

(U U )ci c

i ci cj . (21)

In the literature, 20 J x2 is also written in a more compactform using the equation of motion in Heisenberg picture,

J x2 = 12

ij,

R xj Rxi t

ij (c

i cj c

i cj ),

in which the dot means the time derivative,

ci = i [H,ci ].

To compute L11 and L12 , we need to further evaluatethe commutators between current operators and polarizationoperators. For L11 , this is simple and straightforward,

L11 =ij,

t ij Rxj R

xi

2ci cj . (22)

However, L12 leads to a complicated formula,

L12 = ijl,

12

t il t lj R

xj R

xi

2ci cj

+12

ij,

t ij Rxj R

xi

2( + 2 ) c

i cj

+12

ij,

t ij Rxj R

xi

2

(U U ) ci c

j cj cj

+

(U U ) ci ci ci cj . (23)

But this formula can be signicantly simplied if we look atthe equation of motion for the following Greenss function,

G ji ( ) = T cj ( )ci . (24)

T is the time-ordering operator in imaginary time. Its equationof motion reads,

G ji ( )

=

j

t jj G j i ( ) ( ) G

ji ( )

(U U )

T cj ( )cj ( )cj ( )c

i . (25)

Taking the 0 limit leads to

(U U ) ci c

j cj cj

= lim 0

G ji ( )

+

j

t jj ci cj

( ) ci cj . (26)

Substituting the last term in Eq. ( 23) by the right hand side of Eq. ( 26), we get

L12 = 12

ijl,

t il t lj R

xj R

xi

2

R xl Rxi

2 R xj R

xl

2ci cj

ij,

t ij Rxj R

xi

2lim

0

G ji ( ). (27)

Using the fact that

ci cj = lim 0G ji ( ), (28)

and performing Fourier transformation in both real space andimaginary time, we get

L11 =1

n

e i n 0

k,2 kk2x

G k (i n ), (29)

and,

L12 =1

n

e i n 0

k, kkx

kkx

+ i n 2 k

k2xG k (i n ). (30)

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k is Fourier transformation of hopping amplitudes,

k =

R

e ikR t (R ), (31)

where we have utilized the translational invariance,

t ij = t (R j R i ). (32)

It is straightforward to convert the Matsubara summationto the integration in real frequencies.

L11 =

d

k,

2 kk2x

f ()Ak (), (33)

and

L12 =

d

k,

kkx

kkx

+ 2 k

k2x

f ()Ak (). (34)

f () = 1/ (1 + exp( )) is the Fermi function. Ak () =

1 G

k () is the spectral function.Eqs. ( 29), (30), (33), and ( 34) are main results in this

work. They arederived from a general formalism of dynamicalthermoelectric transport outlined in Sec. II A and a multibandHamiltonian, Eq. ( 17). The equation of motion is exact and noapproximation is assumed in the derivation. These equationsindicate that L11 and L

12 , and thus S are determined by the

non-interacting band structure and the single-particle spectralfunction.

III. S0 AND S IN A ONE-BAND HUBBARD MODEL

In this section, we discuss S 0 and S of one-band Hubbardmodel in thescenario of dynamical mean eld theory (DMFT),using the formalism we presented in previous sections.

The Hamiltonian of one-band Hubbard model is

H = ij,

t ij ci cj + U

i

n i n i . (35)

In DMFT, it is mapped to a single-impurity Anderson model 21

supplemented by the self-consistent condition, which reads,

1i n + (i n ) (i n )

=k

G k(i n ). (36)

On the left hand side is the local Greens function on theimpurity. (i n ) is the hybridization function of the impurity

model. On the right hand side,G

k(i

n ) is the Greens functionof lattice electrons,

G k(i n ) =1

i n + k (i n ),

with k the non-interacting dispersion relation of the latticemodel, and (i n ) the self energy for both local and latticeGreens function in the self-consistent condition. In DMFT,both coherent and incoherent excitations in a correlated metalare treated on the same footing. 22

In DMFT, the evaluation of transport coefcients, e.g.,Eq. ( 6), can be signicantly simplied. Because the k-dependence falls solely on the non-interacting dispersion k ,thevertexcorrections vanishes. 23 Consequently, Re L xxij ()can

be written in terms of single-particle spectral function in realfrequency.

Re L xxij () = T k,

kkx

2

d +

2

i + j 2

f ( ) f ( + )

Ak( )Ak( + ).

(37)

Notice that here the dependence of ReL ij () on the single-particle spectral function is generally approximate for a nite-dimensional system, which is achieved due to the vanishing of vertex corrections exact only in innite dimensions. But thedependence of Lij on single-particle spectral function is exact,as pointed out at the end of Sec. II A.

Another question is on the sum rule of the approximateRe L ij (), i.e., if we substitute Eq. (37) into the denition of Lij , Eq. (15), whether or not it will give the same form of Lij as we have derived in last section. The answer to thisquestion is yes and we have a brief proof for this one-bandcase in the Appendix but the extension to multiband case isstraightforward. This means that ignoring vertex correctionwill modify the distribution of weight in ReL ij (), but willnot change the integrated weight.

The DC limit of Re L ij (), L 0ij can be obtained by takingthe limit 0, which gives,

L 0ij = T k,

kkx

2

d i + j 2

f ()

Ak()2 .

(38)

Therefore in the framework of DMFT, S 0 is computed fromEq. (38). The AC limit, S can be computed from Eqs. ( 29) and

(30), or Eqs. ( 33) and ( 34). In principle, Matsubara frequencyand integrationover real frequency give identicalresults.But inpractice, especially in numerical computations on correlatedsystems, correlation functions in Matsubara frequencies aremore easily accessible. For example, among various impuritysolvers in DMFT, quantum Monte Carlo method (QMC), i.e.,Hirsch-Fye method 24 and recently developed continuous timeQMC 25,26 are implemented in imaginary time. To get corre-lation functions in real frequencies, numerical realization of analytical continuation has to be employed, such as maximumentropy method, which is an involved procedure and usuallyrequires special care. In this case, a formula in Matsubarafrequencies will signicantly simplify the calculation.

Due to the bad convergence of the series, Eqs. (29) and (30)are not appropriate for direct implementation into numericalcomputations. Following standard recipe (separating and ana-lytically evaluating the badly convergent part), we transformthem into a form more friendly to numerics. For the one-bandHubbard model,

L11 =k,

2 k k2x

1

n

Re G k(i n ) 12

, (39)

and

L12 =k,

kkx

2 1

n

Re G k(i n ) [1 + 2n Im G k(i n )].

(40)

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A. Low temperature limit

At low temperatures (low-T), the derivative of Fermifunction, ( f ()/ ) in the integrand of Eq. ( 38) becomesDirac- function-like, thus only the low energy part of thespectral weight near the Fermi surface contributes to theintegral. The low energy part of the self energy of a Fermiliquid () can be approximated by a Taylor expansion interms of and T .

Re () 1 1Z

,

(41)Im ()

0Z 2

(2 + 2T 2) +1

Z 3(a 13 + a 2T 2).

Previous studies 8,27 showed that at low-T limit, L 011 Z2/T

and L 012 ZT , thus S 0 = L012 / (T L

011 ) T /Z .

Since we are interested in the relation between S 0 ad S ,it would be convenient to write L12 and L

11 in terms of the conventional transport function, ( k /k x )2 . This can beachieved by performing integration by part on the summation

over k in Eqs. (33) and (34), then we haveLij = L

ij,I + L

ij,II ,

with

Lij,I =k,

kkx

2

d f () i + j 2

1

Im [G k()Z ()] , (42)

Lij,II =k,

kkx

2

df () 1 Im G ( , ) (

i + j 2(1 Z ())) ,

(43)

where we have dened

Z () =1

1 ()/. (44)

We introduced the function Z (), which is dependenton the derivative of self energy with respect to energy .The integrand in Lij,I [Eq. (42)] also has the derivative of Fermi function. Also notice that at low-T, Z ( = 0) = Z ,which is the renormalization factor of correlated Fermiliquid. Then Lij,I resembles L

012 except for the power of

Im G k(). Low temperature expansion show that L11 ,I Z ,and L12,I T

2 . Therefore, if L 11 ,I I and L

12,I I were absent,S = (T L12,I )/L

11 ,I T /Z , which is similar to the low-Tbehavior of S 0 .

However, L11 ,I I and L

12,I I do not vanish ingeneral at low-Tlimit. First, at low-T limit, the integral over in Eq. ( 43)

df () is replaced by

0

d.

Then both the real and imaginary part of G k() and Z ()below the Fermi surface have to contribute to the leadingorder of Lij,II , unless () is independent, or at least weaklydependent on , leading Z () 1, and then the integrand in

Lij,II would vanish. But this in general cannot be true. Forexample, in a correlated Fermi liquid phase near the Motttransition of Hubbard model, () contains the informationof coherent quasiparticles at the Fermi surface as well as thatof incoherent excitations in high-energy Hubbard bands, thus

() will depend on inverydifferent ways at these separatedenergy scales. At low energy scale, Z ( 0) Z , and Z issignicantly less than 1 near Mott transition. Therefore, atlow-T limit, Lij,II will exhibit a nite value at low-T limit. Sothe total value of L 12 will be dominated by L

12,I I instead of the T 2 contribution from L12,I . The niteness of L

11 can bealso justied by the general sum rule Eq. ( 33), which indicatesthat L11 is proportional to the kinetic energy. Consequently,S will diverge 1 /T -like at low-T limit for a correlated Fermiliquid.

There are somecircumstancesin which Z () = 1and L ij,II vanishes. One example is that in a static mean eld theory,such as Hartree-Fock approximation, () is independent on , thus in static mean eld theory, it is possible that S canshow a similar behavior to that of S 0 at low temperature.

B. High temperature limit

In the literature, the high temperature limit of thermo-power, 4 or known as Mott-Heikes formula, has been widelyused as a benchmark for thermoelectric capability 6 for corre-lated materials. Here we discuss the high temperature limit of S implied from the formulas we have derived.

The high temperature limit relevant for correlated systemswas approached by rst taking the limit U , whichexcludes the double-occupancy in hole-doped systems or thevacancy in electron-doped systems, then taking the high tem-perature limit T 0. This leads to two major simplications.First, by denition in thermodynamics,

T

= sN E,V

.

Here s is the entropy and N is number of electrons. scan be calculated by counting all possible occupation statessatisfying the U limit. It turns out that T is a constantdetermined by the electron density. Thus is proportional toT at high temperature. The second simplication is that at hightemperature, we can approximate the single particle spectralfunction by a rigid band picture, namely,

Ak() = Ak( ). (45)

Ak() is a function of but independent of temperature andchemical potential. Applying these simplication to Eqs. (33)and ( 34), and keeping the leading order in T , we have

L11 =1

1 + e d k, 2

k

k2x Ak(),

L 12 =

1 + e d k, 2

k

k2x Ak().

Therefore, at high temperature limit,

S = L12

T L11=

T

. (46)

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FIG. 1. (Color online) Frequency-dependent transport coefcients and thermoelectric power of a hole-doped one-band Hubbard modelon square lattice. U = 1.75D and n = 0.85. (a) ReL 11 () and Im L 11 () at T = 0.125D . (b) ReL 12 () and Im L 12 () at T = 0.125 D .(c) The evolution of ReL 12 () with temperature. (d) ReS () at T = 0.0625 D and T = 0.0875 D . The inset blows up the regionnear = 0.

This is the same result to the high temperature limit of S 0 inRef. 4. Thus the leading order of S is identical to the leadingorder of S 0 at high temperature.

IV. NUMERICAL RESULTS

In this section, we compute the dynamical thermoelectricpower S () by dynamical mean eld theory (DMFT). Weuse exact diagonalization (ED) as the impurity solver. Theadvantage of the ED solver is that the Greens functionscan be computed simultaneously in real and Matsubarafrequencies. Thus we have two approaches to compute theAC limit S . The rst one is to substitute the Greensfunction in Matsubara frequencies into Eqs. ( 29) and (30).The second method starts from compsuting ReL 11 () andRe L 12 () from spectral functions Ak() using Eq. (37) fora wide range of , Kramers-Kronig relation implementedto compute Im L 11 () and Im L 12 (), and nally with thevalue of L11 and L

12 obtained by tting Eq. ( 12) at the limit. The second method is more laborious buthere we use it as a check for our formulas in Matsubarafrequencies.

We study one-bandHubbard model onsquare andtriangularlattices and consider only the hopping between nearestneighboring sites.

A. Square lattice

In this section, we compute the thermoelectric transportcoefcients and TEP for a hole-doped Hubbard model onsquare lattice. We use the bandwidth D as the unit forfrequency , temperature T and interaction strength U . Forsquare lattice, D = 8| t | , t is the hopping constant.

In Fig. 1, we show the frequency-dependent quantitiesfor U = 1.75D and n = 0.85. Fig. 1(a) and 1(b) show thethermoelectric transport coefcients L 11 () and L 12 () bytheir real (red line) and imaginary part (black line). The realparts are computed from Eq. ( 37). The imaginary parts arecomputed from Kramers-Kronig relation. Three contributionsare recognizable in ReL 11 : (i) the low frequency peak dueto transition within the resonance peak of quasiparticles, (ii)the transition between quasiparticles and the lower Hubbardband, which accounts for the hump at 0.5D , (iii) theweight around U , which is due to the incoherent exci-tations between Hubbard bands. Same features also exist inRe L 12 (), but the feature near 0 i.e., transition betweenquasiparticles, and the transition between quasiparticles andlower Hubbard band, are much less obvious. This is becausethe DC limit L 012 is dominated by the particle-hole asymmetryof the band velocity k /k x and the spectral function A k(),due to the i + j 2 = term in the integrand of Eq. (38) forL 012 . Thus at small , ReL 12 () is signicantly impaired,compared to Re L 11 (). Therefore the transition by incoherentexcitations around U takes a major part in the total

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FIG. 2. (Color online) (a) Temperature dependence of S 0 . (b) Temperature dependence of S obtained from real frequencies (lled circles)and from Matsubara frequencies (open circles). Square lattice.

weight in Re L 12 (), and the sum rule of Re L 12 (), i.e., L12 ,is also dominated by the incoherent excitations. Im L 11 ()and Im L 12 () are odd functions of and vanish at = 0.

It is evident that thereal parts approachto zero much fasterthanthe imaginary parts at AC limit ( ). Figure. 1(c) showsthe evolution of ReL 12 () as temperatures. The dominanceof the incoherent excitations is robust as the variation of temperature. Figure 1(d) shows the real part of thermoelectricpower, Re S ()for T = 0.0625 D and T = 0.0875 D . The insetblows up the region near = 0, indicating that S 0 displays +or signs at different temperatures.

In Fig. 2(a) and 2(b) we show S 0 and S at varioustemperatures. Ontheone side, in Fig. 2(a) , S 0 presents multiplechanges of sign with temperature increased. The sign changeat lower temperature( T 0.1D ) demonstrates the crossoverfrom the low-temperature hole-like coherent quasiparticles toincoherent excitations at intermediate temperature. AroundT = 0.2D , S 0 reaches its maximum positive value, where thecoherent quasiparticles have almost diminished. The secondsign change around T = 0.6D indicates a subtle competitionbetween the spectral weight of lower and higher Hubbardband. As temperature increases, the asymmetry betweenthe two Hubbard bands near Fermi surface becomes lesssignicant because more spectral weight from the higherHubbard band takes part into the transport and the sign of S 0is determined by the difference between the weight of lowerand higher Hubbard. This crossover is thus considered to beresponsible for the second sign change 9 and also has beenobserved experimentally. 28 Therefore, above T = 0.6D , thetransport is completely dominated by incoherent excitationsfrom both Hubbard bands. On the other side, in Fig. 2(b) , thesituation for S is quite different. S does not change signand keeps negative in the shown temperature range. Towardslow temperature, S blows up, consistent with our argumentbased on a Fermi liquid self energy in Sec. IIIA . Towardshigh temperature, i.e., when the temperature is well above thecoherence regime, S 0 and S have the same sign and similarmagnitude. We notice that S 0 in Fig. 2(a) does not convergeto the value predicted by the Mott-Heikes formula in thecorrelated regime( S MH 1.04 kB /e , from Eq. (11 ) in Ref. 4).This is because in our case, with U = 1.5D , the requirementfor |t | T U can not be satised for a wide range of temperature. Thus at high temperature, e.g., when T > 0.6D ,

the states with double occupancy can not be excluded and theyare responsible for the second sign change in S 0 as discussedabove.

In Fig. 2(b) , we show S

obtained by the two methods men-tioned at the beginning of Sec. IV. The solid circles representsS by tting Im L 11 () and Im L 12 () in real frequency at limit. The open circles represent S computed usingEqs. (29) and ( 30). The values of S at open and closed circlesare very close, indicating the consistency between the real andMatsubara frequency approach to calculate S .

The dependence on electron density of S 0 and S is morenon-trivial, which is difcult to tell from analytical formulas.Figure 3 shows S 0 and S at various densities for U = 1.75D .S 0 changes sign from positive at half lling to negative aselectron density decreases, while S remains negative. Thebehavior of S 0 here is also due to the breakdown of coherenceas the evolution of spectral weight. In a doped Mott insulator,the quasiparticle peak gradually diminishes as the systemis doped away from half-lling. 29 Thus near half-lling,the transport is dominated by the coherent excitations nearFermi surface. But when the doping is heavy enough to killquasiparticles, transport is carried by incoherent excitations inthe Hubbard bands. Therefore S 0 turns to a same sign with S ,since S is dominated by the Hubbard bands (see Fig. 1(c) and

FIG. 3. (Color online) Doping dependence of S 0 and S for U =1.75D . S was obtained from real frequencies (lled circles) andMatsubara frequencies (open circles). The temperature here is T =0.125 D . Square lattice.

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FIG. 4. (Color online) (a) and (b) Density dependence of S 0and S for U = 1.25D and U = 0.5D . S was obtained from theMatsubara frequencies. Triangular lattice.

discussion there). In Fig. 3 we also put the results of S by realand Matsubara frequency approach.

B. Triangular lattice

Recent intereston thermoelectric performance of correlatedsystems was attributed to the discovery of TEPenhancement inhighly electron doped cobaltates. 30 The Co atoms in the CoO 2layers form a triangular lattice. The physics behind the largeTEP in Na x Co O 2 is highly non-trivial. For example, the Napotential is crucial to induce the correlation in Na 0.7Co O 231 ,and the spin and orbital degrees of freedom are argued to bea key factor for the enhancement. 6,32 These complexities arebeyond a single band Hubbard on a triangular lattice. Herewe only focus on some qualitative features of S 0 and S in

a electron-doped single band Hubbard model on triangularlattice.In triangular lattice, U = 12 | t | and we use a positive t. S

in this section is solely computed by using Eqs. ( 29) and ( 30).Figure 4(a) and 4(b) shows thedensitydependence of S 0 and

S for two different interaction strength. Here we present thefull range for electron doping. Here S is from the summationover Matsubara frequency. For U = 1.25D [Fig. 4(a) ], S 0is negative near half-lling and changes to positive after asmall amount of doping. As the density approaches to bandinsulator ( n = 2), the merging of S 0 and S is very evident.Forsmaller interaction strength, i.e., U = 0.5D , S 0 and S alsodisplaysimilar trend throughthe range of electrondensity. Thisbehavior is similar to the case on square lattice, Fig. 3. Thediscrepancy between S 0 and S is most evident for U = 1.25Dand near half-lling ( n = 1.0), since around this regime thecoherent quasiparticles take a signicant role in transport. Forelectron density larger than 1 .5, which is the range of interestfor cobaltate, the trend of S shows that it is a reasonableapproximation to S 0 .

V. SUMMARY

Using the formulas derived in Sec. II, we investigate towhat extent the AC limit of TEP, S , can be a reasonableapproximation to the DC limit, S 0 . Analytical and numericalresults on a single-band Hubbard model show that below and

around coherent temperature, i.e., when the spectral weightaround quasiparticle peak dominates in the thermoelectrictransport, the behaviors of S 0 and S are signicantly different.Specically, S 0 displays multiple sign changes around thecoherent temperature, but S does not. But when the tem-perature is well beyond the coherent regime, thus the transportproperties are dominated by the incoherent excitations, S

shows same sign and similar magnitude to S 0 and can givereasonable prediction on the behavior of S .

Our work suggest that a realistic implementation of Eqs. (29) and ( 30) in LDA + DMFT codes can serve as ausefulguide for the search of high performance thermoelectricmaterials among the strongly correlated electron systems,which have a very broad temperature regime characterizedby incoherent transport.

At the time of writing, we are aware of a recent work byM. Uchida et al. ,33 in which the incoherent thermoelectrictransport over a wide temperature range is studied in a typicaldensity-driven Mott transition system La 1 x Sr x VO 3 and thevalidity of Mott-Heikes formula for real strongly correlated

materials is veried.

ACKNOWLEDGMENTS

This work was supported by the NSF under NSF grantDMR-0906943. C.W. was supported by the Swiss Founda-tion for Science (SNF). Useful discussions with K. Haule,V. Oudovenko, and J. Tomczak are gratefully acknowledged.

APPENDIX : SUM RULES FOR Re L12 ( ) ANDRe L11 ( ) IN DMFT

In this appendix, we compute L11 and L

12 in the framework of dynamical mean eld theory and show they also obey thegeneral formulas, Eqs. ( 33) and ( 34).

In terms of retarded current-current correlations,

Re L xxij () = 1

Im

dte i (+ i 0

+ )[ i (t ) [J j (t ),J i ] ] ,

(A1)

which can be computed in Matsubara frequencies by standarddiagrammatic techniques. 16 In the innite dimension limit,a signicant simplication is achieved because all nonlocalirreducible vertex collapse and only the rst bubble diagramsurvives. 19,23 This simplication leads to

Re Lxx

ij () = T k,

kkx

2

d +

2

i + j 2

f ( ) f ( + )

Ak( )Ak( + ).

(A2)

Now we calculate Lij . Using Eq. (15),

L12 =k,

kkx

2

dd + 2

f ( ) f ( + )

Ak( )Ak( + ). (A3)

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Changing variables by

1 = + , 2 = ,

leads to

L12 =k,

kkx

2

d 1d 2f (2)Ak(1)Ak(2)

+ 2k,

kkx

2

d1d 2 21 2 f (2)Ak(1)Ak(2). (A4)

The sum rule d1Ak(1) = 1 simplies the rst term tok,

kkx

2

d 2f (2)Ak(2).In the second term, Kramer-Kronig relation can be used toeliminate the integral over 1 , i.e.,

d1

Ak(1)1 2 = Re G k(2).

Then we use the fact that

2Re G k()Im G k() = Im G 2k()

and

kxG k() = G 2k()

kk x

,

to simplify the second term on the right hand side of Eq. ( A4) to

k,

kkx d22f (2) 1 kx Im G k(2).

Applying integration by part over k, it turns out to be

k,

2 kk2x d 22f (2)Ak(2).

Combined with the rst term, we have

L12 =k, d kk x

2

+ 2 kk2x

f ()Ak().

(A5)

The calculation for L11 is similar and straightforward,which results in

L11 =k, d

2k

k2xf ()Ak(). (A6)

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