Sriram Shastry

UCSC, Santa Cruz, CA

Thermoelectric and Hall transport in Correlated Matter: A new approach

Work supported by

DMR 0408247

UC Irvine, April 18, 2007

Work supported by DOE, BES DE-FG02-

06ER46319

I outline a new approach to the difficult problem of computing transport constants in correlated matter, using a combination of analytical and numerical methods. Our technique exploits the simplifications gained by going to frequencies larger than some characteristic scale, where the Kubo formulas simplify somewhat.

A careful computation of the Kubo type formulas reveal important corrections to known formulas in literature for the thermpower, and thermal conductivity, including new sum rules. Finally these are put together into a numerical computation for small clusters where these ideas are bench marked and several new predictions are made that

should be of interest to materials community.

Mike Peterson

Jan Haerter

References:

1) Strong Correlations Produce the Curie-Weiss Phase of NaxCoO2 J. O. Haerter, M. R. Peterson, and B. S. Shastry Phys. Rev. Lett. 97, 226402 (2006)2) Finite-temperature properties of the triangular lattice t-J modeland applications to NaxCoO2, J. O. Haerter, M. R. Peterson, and B. S. Shastry, Phys. Rev. B 74, 245118 (2006)

3) Sum rule for thermal conductivity and dynamical thermal transport coefficients in condensed matter B. S. Shastry, Phys. Rev. B 73, 085117 (2006)

The Boltzmann theory approach to transport:

A very false approach for correlated matter, unfortunately very

Strongly influential and pervasive.

Need for alternate view point.

Vulcan death grip –Derived from a Star Trek classic episode where a non- existant "Vulcan death grip" was used to fool Romulans that Spock had killed Kirk.

wikipedia

First serious effort to understand Hall constant in correlated matter:

S S, Boris Shraiman and Rajeev Singh, Phys Rev Letts ( 1993)

Introduced object

0* lim lim ( )/H xy

BR B

•Easier to calculate than transport Hall constant

•Captures Mott Hubbard physics to large extent

Motivation: Drude theory has 2( ) (0) /(1 )xy xy i

( ) (0) /(1 )xx xx i

Hence relaxation time cancels out in the Hall resistivity

2( )( ) xy

xxxy

¿xx = q2e~

P´2

x t(~) cy~r+~;¾c~r;¾

¿xy =q2

e~

P~k;¾

d2"~kdk®dk¯

cy~k;¾

c~k;¾

¾®; (! c) =i

~Nsv! c

"

h¿®; i ¡ ~X

n;m

pn ¡ pm

²n ¡ ²m + ~! chnjJ ®jmihmjJ ¯ jni

#

•Very useful formula since

•Captures Lower Hubbard Band physics. This is achieved by using the Gutzwiller projected fermi operators in defining J’s

•Exact in the limit of simple dynamics ( e.g few frequencies involved), as in the Boltzmann eqn approach.

•Can compute in various ways for all temperatures ( exact diagonalization, high T expansion etc…..)

•We have successfully removed the dissipational aspect of Hall constant from this object, and retained the correlations aspect.

•Very good description of t-J model, not too useful for Hubbard model.

•This asymptotic formula usually requires to be larger than J

R¤H = ¡ iN s v

B ~h[J x ;J y ]i

h¿x x i 2

Comparison with Hidei Takagi and Bertram Batlogg data for LSCO showing change of sign of Hall constant at delta=.33 for squar e lattice

We suggest that transport Hall = high frequency Hall constant!!

•Origin of T linear behaviour in triangular lattice has to do with frustration. Loop representation of Hall constant gives a unique contribution for triangular lattice with sign of hopping playing a non trivial role.

2 4 6 8 10

5

10

15

20

25

T

RH

As a function of T, Hall constant is LINEAR for triangular lattice!!

B O( t)3

O( t)4

Triangular lattice

square lattice

Hall constant as a function of T for x=.68 ( CW metal ). T linear over large range 2000 to 4360 ( predicted by theory of triangular lattice transport KS)

T Linear resistivity

STRONG CORRELATIONS & Narrow Bands

Im R_H

Re R_H

Thermoelectric phenomena

£xx = ¡ limkx ! 0

ddkx

[J Qx (kx);K (¡ kx)]

Here we commute the Heat current with the energy density to get the thermal operator

Comment: New sum rule.

Not known before in literature.

Re· (! ) =¼~T

±(! ) ¹DQ +Re· reg(! ) with

Re· reg(! ) =¼

TL

µ1¡ e¡ ¯ !

!

¶ X

²n 6=²m

pnjhnjJ Qx jmij2±(²m ¡ ²n ¡ ~! ); (1)

¹DQ =1L

2

4h£xxi ¡ ~X

²n 6=²m

pn ¡ pm

²m ¡ ²njhnjJ Q

x jmij2

3

5: (2)

Thesumrulefor thereal part of thethermal conductivity (anevenfunctionof ! ) follows Z 1

0Re· (! )d! =

¼2~TL

h£xxi: (1)

· (! c) =i

T~! cDQ +

1TL

Z 1

0dtei! ct

Z ¯

0d¿hJ Q

x (¡ t¡ i¿)J Qx (0)i:

Where

DQ =1L

"

h£xxi ¡ ~X

n;m

pn ¡ pm

²m ¡ ²njhnjJ Q

x jmij2#

:

In normal dissipative systems, the correction to Kubo’s formula is zero, but it is a useful way of rewriting zero, it helps us to find the frequency integral of second term, hitherto unknown!!

£xx

~T =1dC¹ v2

ef f

So, what does£ looklikeandwhat isitsvalue? Answer ismodel dependent,and inbrief, £ isthespeci c heat timesavelocity

Thermo-power follows similar logic:

< J x >= ¾(! )Ex + °(! )(¡ r T)

then the thermopower is

S(! ) =°(! )¾(! )

:

©xx = ¡ limk! 0

ddkx

[J x(kx);K (¡ kx)]:

This is the thermo electric operator

°(! c) =i

~! cTL

"

< ©xx > ¡ ~X

n;m

pn ¡ pm

²n ¡ ²m +~! chnjJ x jmihnjJ Q

x jmi

#

:

D° =1L

"

< ©xx > ¡ ~X

n;m

pn ¡ pm

²m ¡ ²nhnjJ x jmihnjJ Q

x jmi

#

:

°(! c) =i

~! cTD° +

1TL

Z 1

0dtei ! c t

Z ¯

0d¿hJ x(¡ t ¡ i¿)J Q

x (0)i ;

High frequency limits that are feasible and sensible similar to R*

Hence for any model system, armed with these three operators, we can compute the Lorentz ratio, the thermopower and the thermoelectric figure of merit!

L¤ =h£xxi

T2h¿xxi(1)

Z¤T =h©xxi2

h£xxih¿xxi: (2)

S¤ =h©xxiTh¿xxi

: (3)

So we naturally ask

•what do these operators look like

•how can we compute them

• how good an approximation is this?

In the paper: several models worked out in detail

•Lattice dynamics with non linear disordered lattice

•Hubbard model

•Inhomogenous electron gas

•Disordered electron systems

•Infinite U Hubbard bands

•Lots of detailed formulas: we will see a small sample for Hubbard model and see some tests…

H =X

j

H j

H j =

"~p2

j

2mj+Uj

#

; Uj =12

X

i6=j

Vj ;i ; (1)

Anharmonic Lattice example

J Ex (~k) =

14

X

i ;j

(X i ¡ X j )mi

ei kx X i©px;i ;F x

i ;j

ª

~F i ;j = ¡ ~@@~ui

Vi ;j

£ xx = (~! 20a2

0)

"1m

X

i

pi pi+1 +ks

4

X

i

(ui ¡ 1 ¡ ui+1)2

#

:

h£xxi = L(~2! 30a

20)

Z ¼

0

dk¼

·(12

+1

e ! k ¡ 1)½

! k

! 0cos(k)+

! 0

! ksin2(k)

¾;

Thermo power operator for Hubbard model

©xx = ¡qe

2

X

~; ~0;~r

(´x +´0x)2t(~)t(~0)cy

~r+~+ ~0;¾c~r ;¾¡ qe¹

X

~

´2xt(~)cy

~r+~;¾c~r ;¾+

qeU4

X

~r ;~

t(~)(´x)2(n~r ;¹¾+n~r+~;¹¾)(cy~r+~;¾c~r;¾+ cy

~r ;¾c~r+~;¾): (1)

This object can be expressed completely in Fourier space as

©xx = qe

X

~p

@@px

©vx

p("~p ¡ ¹ )ª

cy~p;¾c~p;¾ (2)

+qeU2L

X

~l;~p;~q;¾;¾0

@2

@l2x

n"~l + "~l+~q

ocy~l+~q;¾

c~l;¾cy~p¡ ~q; ¹¾0

c~p; ¹¾0: (3)

¿xx =q2e

~

X´2

x t(~) cy~r+~;¾c~r;¾ or (1)

=q2e

~

X

~k

d2"~kdk2

xcy~k;¾

c~k;¾ (2)

Free Electron Limit and Comparison with the Boltzmann Theory

It iseasytoevaluatethevariousoperatorsinthelimit of U ! 0, andthisexerciseenables us toget a feel for themeaningof thesevarious somewhatformal objects. Wenotethat

h¿xxi =2q2

eL

X

pn~p

ddpx

£vx~p¤

h£xxi =2L

X

pn~p

ddpx

£vx~p("~p¡ ¹)2

¤

h©xxi =2qeL

X

pn~p

ddpx

£vx~p("~p¡ ¹)

¤: (1)

At low temperatures, we use the Sommer¯eld formula after integrating byparts, and thus obtain the leading low T behaviour:

h¿xx i = 2q2e½0(¹ )h(vx

~p)2i ¹

h£ xx i = T22¼2k2B

3½0(¹ )h(vx

~p)2i ¹ (1)

h©xx i = T22qe¼2k2B

3

·½0

0(¹ )h(vx~p)2i ¹ + ½0(¹ )

dd¹

h(vx~p)2i ¹

¸; (2)

Wemay form the high frequency ratios

S¤ = T¼2k2

B

3qe

dd¹

log£½0(¹ )h(vx

~p)2i ¹¤

L¤ =¼2k2

B

3q2e

: (1)

It isthereforeclear that thehigh frequency result givesthesameLorentznumberaswell asthethermopower that theBoltzmann theory gives in itssimplest form.

The thermal conductivity cannot be found from this approach, but basically the formula is the same as the Drude theory with i/ ->

Some new results for strong correlations and triangular lattice:

Thermopower formula to replace the Heikes-Mott-Zener formula

Results from this formalism:

Prediction for t>0 material

Comparision with data on absolute scale!

Magnetic field dep of S(B) vs data

RH =T ranspor t = R¤H +

R10

=m[RH (! )]=! d! =¼

T-t’-J model, typical frequency dependence is very small . This is very encouraging for the program of x_eff

SSS Remarkably, a small t’ of either sign shifts the zero crossing of Hall constant significantly!

Conclusions:

•New and rather useful starting point for understanding transport phenomena in correlated matter

•Kubo type formulas are non trivial at finite frequencies, and have much structure

•We have made several successful predictions for NCO already

•Can we design new materials using insights gained from this kind of work?

http://physics.ucsc.edu/~sriram/sriram.html

Useful link for this kind of work: