Mass-renormalization and superconductivity in n-doped SrTiO3

D van der Marel, J.L.M. van Mechelen et al, in prep (2011)

J.L.M. van Mechelen, DvdM, C. Grimaldi, A.B. Kuzmenko, N.P. Armitage, N. Reyren, H. Hagemann, I.I. Mazin, PRL

100, 226403 (2008)

J. T. Devreese, S. N. Klimin, J. L. M. van Mechelen, and DvdM, PRB 81 (2010) 125119

W. Meevasana, X. J. Zhou, B. Moritz, C.-C. Chen, R. H. He, S.-I. Fujimori, D. H. Lu, S-K Mo, R. G. Moore, F.

Baumberger, T. P. Devereaux, DvdM, N. Nagaosa, J. Zaanen and Z.-X. Shen, NJP 12, 023004 (2010)

Characteristics of the charge transport in n-type STO

Are the charge carriers Fermions?

What is the relevant Fermi temperature ?

Wat is the energy scale and strength of fermion-fermion interactions ?

What causes that Tc 0 at low doping ?

Contents

DC resistivity STO/LAO

SrTiO3

-20

0

20

40

60

80

100

120

140

[0,0,][0,0,0]

Ener

gy (

meV

)

Momentum

SrTi1-xNbxO3

EF

I.I. Mazin et al, unpublished

X=0.02

ma = mb ~ mc / 20

X=0.001

DC transport in n-type SrTi1-xNbxO3

0 50 100 150 200 250 3000

10

20

30

40

50

60

DC

resi

stiv

ity (

cm

)

Temperature (K)

x = 0.001 0.002 0.009 0.020

J.L.M. van Mechelen. Ph D thesis, Univ. de Genève (2010)

Hall effect

Optical conductivity

HWHM

0 2 4 6 8 100

10

20

30

40

50

60SrTi1-xNbxO3

T = 7 K x

0.001 0.002 0.009 0.020

Energy (meV)

(kS/

cm)

J.L.M. van Mechelen et al , PRL 100, 226403 (2008)

Free carrier scattering rate

0 20 40 60 80 1000

2

4

6

8

10

/

(meV

)

T (K)

x 0.020 0.009 0.002 0.001

SrTi1-xNbxO3

Mean free path

0 20 40 60 80 1000

5

10

15

20

25

30

35

40

45

50

k Fl

T (K)

0.020 0.009 0.002 0.001

SrTi1-xNbxO3

**

**

2/

2FF

F

FFF lkvl

vk

Free carrier scattering rate

0.0 0.5 1.0 1.5 2.00

1

2

3

4

5

6

7

/

(meV

)

(T / 50 K)2

x 0.020 0.009 0.002 0.001

22

0

TAkB

SrTi1-xNbxO3

0.000 0.005 0.010 0.015 0.0200.00

0.05

0.10

0.15

0.20

A

(meV

-1)

x0.000 0.005 0.010 0.015 0.020

0.0

0.5

1.0

1.5

2.0

/ 0

(m

eV)

x

22

0

TAkB

impnl0

1

SrTi1-xNbxO3

Charge transport in n-type STO:

Mobile charge carriers

3 intersecting bands

Each band: ma = mb ~ mc / 20

T2 type inelastic scattering

Are the charge carriers Fermions?

What kind of ?

Mass renormalization obtained from ARPES:2.2

1*

*

mm

vv

LDA

F

F

W. Meevasana et al, New Journal of Physics 12 (2010) 023004

LDAARPES

Elecron doped Sr1-xLaxTiO3

0

f.c.r.Coherent

f.c.r. Incoherent

Electrons coupled to phonons

*

2

2

2

01

2 :s.w. charges free ofpart Coherent

2 : weightspectral charge free Total

2 :sumrule-f

mnπe

mnπe

mnπed

f

b

f

e

Interband

*

2*

0 28')'(

mn

d fpcoh

Coherent free carrier spectral weight

0 2 4 6 8 100

10

20

30

40

50

60SrTi1-xNbxO3

T = 7 K x

0.001 0.002 0.009 0.020

Energy (meV)

(kS/

cm)

0 0.005 0.010 0.015 0.0200

0.2

0.4

0.6

Free carriers per unit cell

2 p2 (e

V2 )

From optical data Ab initio (LDA)

Coherent free carrier spectral weight

J.L.M. van Mechelen et al , PRL 100, 226403 (2008)

)(4 :LDA 2

2

,,

222

x

k,j,σ

σjkk,j,σp,x k

εTn

Vπeω

0.000 0.005 0.010 0.015 0.020 0.025 0.0300.0

0.5

1.0

From specific heat Ab initio (LDA)

DO

S(E F)

(eV

-1)

carriers per unit cell

Specific Heat

*22

3mDπk

TC

FB

0.0 0.5 1.0 1.5 2.0 2.50

1

2

3

4

m*

/ mLD

A

Carriers per unit cell (%)

Elecron doped SrTiO3

optics

Specific heatARPES

The charge carriers are Fermions

Mass renormalization: m*/mLDA~2.5

0.000 0.005 0.010 0.015 0.020 0.0250

20

40

60

80

100

F (LDA)

F*=Fm/m*

F (

meV

)

x

*03xx-1 :0.02)(x ONbSrTi

F

ε*F < ω0 : Anti-adiabatic limit

Expansion parameter: ε*F / ω0

(i) Solve e-ph interaction for 1 electron in an empty lattice (polaron: Landau, Feynman)

(ii) Liquid of composite fermions

(iii) Residual interactions

2

2CE eP

Polaron

Self-trapping by e-phonon coupling

(path-integral)

Polaron effective massα = e-ph coupling parameter

0 20 40 60 80 1000

1

2

(kS/

cm)

SrTiO3

021.02115

1

1,

1,

L

T

46.05821

2

2,

2,

L

T

58.19968

3

3,

3,

L

T

J. T. Devreese, et al., Physical Review B 81 (2010) 125119

αj’s from the optical phonons of SrTiO3

2/1,0

2/1

3

4 1112 jl

bj

me

J. T. Devreese, et al., Physical Review B 81 (2010) 125119

Ab initio many-polaron theory of σ(ω)

Anti-adiabatic limit (ε*F < ω0 )

|ε*k -ε*F| < ω0 : No energy transfer to ω0 phonons

Instead: Virtual phonon exchange

1/τ ~ λFF2 T2/ε*F

m**/m*=1+ λFF

Tc~ n1/2 exp(-1/λFF)

0.000 0.005 0.010 0.015 0.020 0.0250

10

20

30

40

(m

eV)

x

The energy scale of the fermion-fermion interactions

*2

~ ; 2F

BFF

FF

Tk

0.000 0.005 0.010 0.015 0.020 0.0250.0

0.1

0.2

0.3

0.4

0.5 F

F

x

The strength of the fermion-fermion interactions

2 2

TkBFF

FF

Superconductivity

N. Reyren et al.,Science 317, 1196

(2007)

0 0.005 0.010 0.015 0.020 0.0250.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

T c (

K)

Free carriers per unit cell

C.S. Koonce, M.L. Cohen, J.F. Schooley, W.R. Hosler and E.R. Pfeiffer, Phys. Rev. 163, 380 (1967).

3D SC

SuperconductivityY. Lee et al, PRL 106, 136809 (2011)

What causes Tc0 for x0 ?

• Weak non-retarded interaction between polarons• Weak coupling BCS theory applies

Superconductivity

0)(:0

0 :

)(/ :

2tanh

2

0

0

22

22

N

V

NV

TkV

k

kpk

pairkpk

l B

kk

kk

kkpp

130.1

,

2tanh

21

/10

*

0*

*0

0

*

pair

F

eTk

Tk

dTk

FcB

FcB

B

pair

F

DvdM, Physica C 165 (1990) 35.

0.000 0.005 0.010 0.015 0.020 0.0250.0

0.1

0.2

0.3

0.4

0.5 from experimental 1/

from experimental Tc

FF

x

0 00n *

/10

*

cBF

FcB

Tk

eTk pair

What causes Tc0 for x0 ?

Anti-adiabatic electron-phonon coupling in Sr1-xNbxTiO3

T2- resistivity observed

A polaron-liquid forms in n-doped STO

Mechanism for superconductivity:

Fermion-fermion interaction λFF < 0.4

Quasi-instantaneous pairing-interaction: λpair ~ 0.1

Conclusions