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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I.I. Mazin et al, unpublished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
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