[PPT] - Role of multiple subband renormalization on the thermoelectric PowerPoint Presentation

SLIDE 1

Role of multiple subband renormalization on the thermoelectric effect in correlated

xide superlattices

Andreas Rüegg,

Sebastian Pilgram and Manfred Sigrist Theoretische Physik, ETH Zürich, Switzerland

SLIDE 2

Hvar, 2008

complex oxide heterostructure

2

annular dark field (STEM), Ohtomo et al., Nature (2002)

[SrTiO3]5/[LaTiO3]1

band insulator Mott insulator

SLIDE 3

transition metal oxides - interfaces?

bulk transition-metal oxides:
rich and complex phase diagrams
intrinsic functionalities:

ferroelectricity, superconductivity, CMR, ...

spin, charge and orbital degrees of freedom
strongly correlated electronic systems
artificial nanoscale structures:
novel physics
stabilization of new phases
atomic, electronic and orbital reconstruction
clean doping

3

Tokura, Nagaosa, Science (2000) Dagotto, Science (2005, 2007) Okamoto, Millis, Nature (2004) Chakhalian et al., Science (2007)

cf. semi-

conductors manganites cuprates

SLIDE 4

electronic reconstruction

general: electronic phase at the interface

differs from bulk phase.

important mechanism: transfer of

electronic charge across the interface.

“simple” example: metallic interface

between insulators.

4

SrTiO3 LaTiO3 SrTiO3 LaAlO3 SrTiO3 LaVO3

Ohtomo et al. (2002)

SrVO3 LaVO3

Ohtomo et al. (2004), Thiel et al. (2006), Reyren et al. (2007) Hotta et al. (2007) Sheets et al. (2007)

band/Mott insulators Okamoto and Millis, Nature (2004)

electron energy loss spectra, Ohtomo et al., Nature (2002)

3d-electron charge distribution

metallic

SLIDE 5

MI BI n3d ≈ 1 n3d ≈ 0 BI n3d ≈ 0

single-particle picture

5

momentum space:

quantum confinement (superlattice structure) leads to

multiple subbands in a quasi-two dimensional system.

familiar from (and successfully applied in) semiconductor physics.

MI BI BI

ν 1 2 3

real space:

mutual doping (transfer of electronic charge from Mott to band insulator) leads to metallic interface. atomic limit: appropriate at high temperatures interface states?

how do local correlations modify this picture?

but

kx ky !! !" !# $ # " ! !! !" !# $ # " !

SLIDE 6

quasi-particle dispersion

(i) multi-subband aspect:

electron-like and hole-like contributions, compensation possible.

e.g. thermopower, Hall constant, ...

(ii) correlation effects:

(a) group velocity renormalization (b) renormalization of particle-hole asymmetry (c) indirectly through the multi-subband aspect

semiclassical perspective of transport

6

Ekν = Eν(εk)

Ekν = 0

in-plane momentum non-interacting dispersion

E
∇T

MI BI

N M

MI MI BI (relevant for Drude weight) (relevant for thermopower) αν =

εk

∂2Ekν ∂ε2

k

/

∂Ekν ∂εk

FS

need hybridization of itinerant and (almost) localized degrees

f freedom: interface!

subband index

Zν = ∂Ekν ∂εk

FS

Fermi surface:

SLIDE 7

utline
model:
extended single-orbital Hubbard model
effective low-energy theory (slave-boson mean-field theory)
generic “electronic structure”
multiple subbands
quasiparticle renormalization (group velocity and particle-hole asymmetry)
parallel transport
thermoelectric effects

Seebeck coefficient (thermopower)

free carrier response

Drude weight, optical conductivity

conclusions

7 transport in multilayered nanostructures

J. Freericks

different story: perpendicular transport studied by Okamoto and Freericks, Zlatic, Shvaika...

SLIDE 8

Hvar, 2008

modeling

8

SLIDE 9

geometry

9

BI BI x,y z N M

electronic sites: cubic lattice (Ti sites) counterions (Sr2+ vs. La3+) Mott insulator (LaTiO3; Ti-3d1) band insulator (SrTiO3; Ti-3d0) 3 dimensional model:

BI BI MI

two situations:

quantum well quantum well

MI BI BI

ν 1 2 3 band insulator (SrTiO3; Ti-3d0)

(i) (ii)

N N N M M N

superlattice structure

BI BI BI BI

SLIDE 10

single-orbital model

10

BI BI x,y z N M

Extended single-orbital Hubbard model:

nearest neighbor hopping

n-site

repulsion studied in the current context by:

Okamoto and Millis
Lee and MacDonald
Kancharla and Dagotto
Freericks et al.

(long range) electron- electron interaction Wij = EC |ri − rj| electron-ion interaction

Vi = −

j

EC

ri − rion

i

W ion

ij

= EC

rion

i

− rion

j

(long range) ion-

ion interaction parameters at T = 0:

hopping
on-site
long-range
geometry:

MI: BI:

EC = e2 a Ur = U − EC ≥ 0 t N M ˆ H = −t

ijσ
ˆ

c†

iσˆ

cjσ + h.c.

+ U
i

ˆ ni↑ˆ ni↓ +

i

Viˆ ni + 1 2

i=j

ˆ niWijˆ nj + 1 2

i=j

W ion

ij

BI MI BI

SLIDE 11

i) layer-resolved hopping renormalization: ii) free energy density: iii) saddle-point equations: maximize with respect to minimize with respect to

KR-slave-bosons and heterostructures

11

Kotliar, Ruckenstein, PRL (1986) Brinkmann, Rice, PRB (1970) KR-slave-boson mean-field theory provides a way to find the local (low-energy) self-energy Gutzwiller, PRL (1963) effective 1D Schrödinger equation: AR, Pilgram, Sigrist, PRB (2007) zl = zl(nl, dl) =

(1 − nl + d2

l )(nl − 2d2 l ) + dl

nl − 2d2

l

nl(1 − nl/2)
z2

l εk + λl

ψkν(l) − t
γ=±1

zlzl+γψkν(l + γ) = Ekνψkν(l).

hopping renormalization subband index in-plane momentum non-interacting in- plane dispersion Lagrange multiplier layer index electronic density (amplitude of) double occupancy

λ n, d f(n, d, λ) = − 2 βN||

kν

log

1 + e−βEkν(n,d,λ)

+ Ur

l

dl

2 + 1

2

ll′

nlWll′nl′ −

l

(λl − Vl)nl λ n, d f(n, d, λ) = − 2 βN||

kν

log

1 + e−βEkν(n,d,λ)

+ Ur

l

dl

2 + 1

2

ll′

nlWll′nl′ −

l

(λl − Vl)nl

SLIDE 12

typical solution

12

20 40 60 80 0.5 1 nl N = M = 10, Ur = 22t, Ec = 0.8t 20 40 60 80 5 10 λl 20 40 60 80 0.1 0.2 l dl

BI BI BI charge density single-particle potential double occupancy (amplitude) MI MI MI MI N M

(T = 0)

EC/t

Ur/t

λTF ∝
t

EC a

SLIDE 13

!!" !# " # !" " "$% "$& "$' "$( ! l n, ncoh

) )

*+,- *+,- .,-/ .,-/

ncoh n

(coherent) charge density

13

Mott regime ncoh n ω ω A(ω) A(ω)

(two-site) DMFT data courteously taken from Okamoto and Millis (2004)

metallic interfaces AR, Pilgram, Sigrist, PRB (2007) Okamoto and Millis, PRB (2004) layer-resolved spectral density: Ur = 23t U = 16t

SLIDE 14

Hvar, 2008

“electronic structure”

14

SLIDE 15

subband dispersion

15

quantum well

MI BI BI ν 1 2 3

z2

l εk + λl

ψkν(l) − t
γ=±1

zlzl+γψkν(l + γ) = Ekνψkν(l)

N = 8 Ur = 24t EC = 0.8t AR, Pilgram, Sigrist, PRB (2007)

*Ueda, Rice, PRL (1985)

partially filled subbands

!! !" # " ! !$ !! !" # " ! $ % εk/t Ekν/t

ν = 1, 2

SLIDE 16

envelope wave functions

z2

l εk + λl

ψkν(l) − t
γ=±1

zlzl+γψkν(l + γ) = Ekνψkν(l)

!!" !# " # !" !$% !&% "% &% $% '% !$% !&% "% &% $% l [λl + (z2

l - 1)εk]/t

% %

N = 10 Ur = 14t εk/t = - 4 εk/t = 4

effective potential square of envelope wave function

single quantum well double well

SLIDE 17

subband dispersion

17

quantum well

MI BI BI ν 1 2 3

!! !" # " ! !$ !! !" # " ! $ % εk/t Ekν/t

ν = 1, 2

“band insulator” interface “Mott insulator” hybridization of almost localized (correlated) and itinerant degrees

f freedom

reminiscent of heavy- fermion systems* high thermopower?

z2

l εk + λl

ψkν(l) − t
γ=±1

zlzl+γψkν(l + γ) = Ekνψkν(l)

N = 8 Ur = 24t EC = 0.8t AR, Pilgram, Sigrist, PRB (2007)

*Ueda, Rice, PRL (1985)

partially filled subbands

SLIDE 18

! " # $ % !! !" & &'# ! ν Zν ! " # $ % !! !" !( & ( ) ν αν N = 8 Ur = 24t

renormalization

18

quantum well

MI BI BI ν 1 2 3

“band insulator” interface “Mott insulator” subbands

z2

l εk + λl

ψkν(l) − t
γ=±1

zlzl+γψkν(l + γ) = Ekνψkν(l).

Zν = ∂Eνk ∂εk

FS

suppressed in “Mott insulator” quasi-particle weight: αν =

εk

∂2Eνk ∂ε2

k

/

∂Eνk ∂εk

FS

enhanced at interface!

particle-hole asymmetry:

SLIDE 19

Hvar, 2008

thermoelectric effects

19

SLIDE 20

Seebeck coefficient

20

E = Q∇T Q = −π2 3 k2

BT

e ∂ ∂E log Φ(E)

E=0

thermopower of a metal: transport distribution function: Φ(E) = e2 4π2¯ h

ν

τν(E)¯ vν(E)Sν(E)

Q =

ν Qνσν

σ multiple subbands: electron-like ( ) and hole- like ( ) contributions lead to partial cancellation Qν > 0 Qν < 0

total electrical conductivity subband conductivity

transport life-time non-interacting band structure (filling) el.-el. interaction particle-hole asymmetry scattering

subband contribution:

kx ky !! !" !# $ # " ! !! !" !# $ # " !

thermoelectric effects: Peltier, Seebeck, Thomson Qν = − π2 3 kB e kBT Zνε

αν + ε

τ

ν

τν + N

v

Nv

FS

Qν = − π2 3 kB e kBT Zνε

αν + ε

τ

ν

τν + N

v

Nv

FS

Fermi-surface area

SLIDE 21

thermopower - total

21

τν(E) = τ τν(E)¯ vν(E) = Λ

constant relaxation time (CRT)
constant mean free path (CMP)

the sharper the interface the bigger the (absolute) thermopower

short screening length,
strong on-site interaction,
large

EC t Ur t N

from microscopic model

s-wave scattering + T

matrix

approximation

[AR,Pilgram, Sigrist, PRB (2008)]

(no effect of ) αν (full effect of ) αν

!"# $ $"# !$"# !$ !!"# ! !"#

EC/t Q [kB/e]

% %

&'( &)* N = 15 M = 5 Ur = 22t kBT = 0.01t

smooth interface sharp interface screening length: λTF ∝

t

EC a

SLIDE 22

!"# $ $"# !$% !$! !& !' !( !% ! % EC/t Qn [kB/e]

) )

*+ ,+ +- N = 15 M = 5 Ur = 22t kBT = 0.01t

thermopower - contributions

22

interface “band insulator” “Mott insulator” sharp interface smooth interface

enormous contribution from interface!

constant relaxation time (CRT) τMI, τBI, τIF

how to get close to “=”?

Q =

ν

Qνσν σ |Q| ≤ |Qνmax|

but

, ZT = 1

SLIDE 23

conclusions

parallel transport using semiclassical approach and SBMF
generic electronic structure (normal state)
1. several partially filled subbands
2. correlation effects characterized by

reduction of Fermi velocity enhancement of particle-hole asymmetry

Drude weight
key quantity: Fermi velocity renormalization
thermopower
key quantity: particle-hole asymmetry
large contribution from interface!
where to go?
design of efficient thermoelectric material. more realistic scattering,...
broken-symmetry states: magnetism, superconductivity...
...

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