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Theory of Lifetime Effects in Point-Contacts:Application to Cd2Re2O7
Bozidar Mitrovi c
Department of PhysicsBrock University
St. Catharines, Ontario, Canada
McMaster, May 24, 2013
B. Mitrovi c Theory of Lifetime Effects in Point-Contacts
Outline
Tunneling junction spectroscopy and point-contact spectroscopyof superconductors
Blonder-Tinkham-Klapwijk (BTK) theory of point-contacts Previous attempts to include the quasiparticle lifetime effects in
the BTK theory BTK theory with self-energy effects Application to point-contact spectroscopy of Cd2Re2O7
B. Mitrovi c Theory of Lifetime Effects in Point-Contacts
Credits
Yousef Rohanizadegan, Brock F. Razavi, M. Hajialamdari and M. Reedyk, Brock R. Kremer, MPI & Brock M. Przedborski and K. Samokhin, Brock
B. Mitrovi c Theory of Lifetime Effects in Point-Contacts
Tunneling junction spectroscopy
B. Mitrovi c Theory of Lifetime Effects in Point-Contacts
Tunneling junction spectroscopy
Problems: It is difficult to make good tunneling junctions withsuperconductors which have complicated structure and a shortcoherence length.
B. Mitrovi c Theory of Lifetime Effects in Point-Contacts
Point-contact spectroscopy
B. Mitrovi c Theory of Lifetime Effects in Point-Contacts
BTK theory
The BTK theory is based on:
1. Bogoliubov equations
− ~2
2m
d2
dx2− µ+ V (x) ∆
~2
2m
d2
dx2+ µ− V (x) ∆
(
u(x, t)v(x, t)
)
= i~∂
∂t
(
u(x, t)v(x, t)
)
∆=0 in N, ∆ 6=0 in S
2. Demers-Griffin model for the N-S interface: V (x) = Hδ(x)
B. Mitrovi c Theory of Lifetime Effects in Point-Contacts
BTK theory
Stationary plane wave solutions(
u(x, t)v(x, t)
)
=
(
u0
v0
)
e~kx−Et/~
E =
√
(~2k2
2m− µ)2 +∆2
u20 =
1
2
[
1 +
√E2 −∆2
E
]
= 1− v20
Density of states N(E) = Re[
(u20 − v20)
−1]
= ReE√
E2 −∆2
B. Mitrovi c Theory of Lifetime Effects in Point-Contacts
BTK theory
B. Mitrovi c Theory of Lifetime Effects in Point-Contacts
BTK theory
6: Andreev reflection
B. Mitrovi c Theory of Lifetime Effects in Point-Contacts
BTK theory
Z =H
~vF
metallic contact: Z=0
tunneling regime: Z ≥5
GNS =dINS
dV= 2N(0)evFA
∫ +∞
−∞
dEdf(E − eV )
dV[1 +A(E)−R(E)]
Fit parameters: ∆ and Z
B. Mitrovi c Theory of Lifetime Effects in Point-Contacts
Experiments
Au-Nb point contact
(a) 10-Ω contact resistance
(b) 3-Ω contact resistance
Note: Experimental curves are broadened BTK curves
B. Mitrovi c Theory of Lifetime Effects in Point-Contacts
Dynes formula and phenomenological extention of theBTK theory
Dynes formula (PRL 41, 1509 (1978)):
ND(E) = ReE − iΓ
√
(E − iΓ)2 −∆2
Eliashberg theory:
N(E) = ReE
√
E2 −∆2(E), ∆(E) = ∆1(E) + ∆2(E)
Mitrovic & Rosema (J. Phys.: Condens. Matter 20, 015215 (2008)):
quasiparticle lifetime Γ = − Im∆(E = ∆0)
When Γ,∆2 ≪ ∆ ND(E) and N(E) give nearly identical results(except near E=0). Nevertheless ND(E) is wrong!
B. Mitrovi c Theory of Lifetime Effects in Point-Contacts
Dynes formula and phenomenological extention of theBTK theory
Phenomenological extension of the BTK theory to include finitequasiparticle lifetime:
(
u(x, t)v(x, t)
)
=
(
u0
v0
)
e~kx−(E−iΓ)t/~
The resulting theory is identical to the BTK theory but with the densityof states given by the Dynes formula.
Fit parameters: ∆, Z and Γ
(Plecennık et al., PRB 49, 10016 (1994); de Wilde et al., Physica B218, 165 (1996))
B. Mitrovi c Theory of Lifetime Effects in Point-Contacts
BTK theory with self-energy in S
McMillan, Phys. Rev. 175, 559 (1968): Eliashberg version ofBogoliubov Equations
[− ~2
2m
d2
dx2− µ]τ3 +Σ(x,E)
(
u(x,E)v(x,E)
)
= E
(
u(x,E)v(x,E)
)
Σ(x,E) = (1− z(x,E))τ0 + φ(x,E)τ1 , ∆(x,E) =φ(x,E)
z(x,E)
B. Mitrovi c Theory of Lifetime Effects in Point-Contacts
BTK theory with self-energy in S
The resulting theory is identical to the BTK theory but with complexand energy dependent gap ∆(E)
GNS =dINS
dV= 2N(0)evFA
∫ +∞
−∞
dEdf(E − eV )
dV[1 +A(E)−R(E)]
A(E) =|u|2|v|2|γ|2
R(E) =[|u|4 + |v|4 − 2Re(u2v2)]z2(z2 + 1)
|γ|2γ = u2 + (u2 − v2)z2
u =1√2
√
1 +√
E2 −∆2(E)/E
v =1√2
√
1−√
E2 −∆2(E)/E .
(Y. Rohanizadegan, MSc. Thesis, Brock University (2013))B. Mitrovi c Theory of Lifetime Effects in Point-Contacts
BTK theory with self-energy in S
For the energies close to the gap edge ∆ the fit parameters are:
∆, Z and ∆2–the imaginary part of gap at the gap edge
Note: The temperature enters via ∆ and ∆2
B. Mitrovi c Theory of Lifetime Effects in Point-Contacts
Application to Cd2Re2O7
Razavi, Rohanizadegan, Hajialamdari, Reedyk, Mitrovic and Kremer,submitted to PRL (May, 2013)
-1.0 -0.5 0.0 0.5 1.00.75
1.00
1.25
1.50
1.75
2.00
Temperature 0.36(2) K 0.45(5) K 0.571(2) K 0.580(1) K 0.646(2) K 0.744(1) K 0.831(4) K 0.874(4) K 0.945(6) K 0.976(1) K 1.015(2) K 1.207(3) K
Nor
mal
ized
con
duct
ance
Voltage (mV)
B. Mitrovi c Theory of Lifetime Effects in Point-Contacts
Application to Cd2Re2O7
Fits:
T=0.831 K: A–with ∆2, B–with Γ
T=0.360 K: C–with ∆2, D–with Γ
0.96
0.98
1.00
1.02
1.04
1.06
A
Nor
mal
ized
con
duct
ance
B
D
-0.002 -0.001 0.000 0.0010.7
0.8
0.9
1.0
1.1
1.2
C
Nor
mal
ized
Con
duct
ance
Voltage (V)
-0.002 -0.001 0.000 0.001 0.002
Voltage (V)
B. Mitrovi c Theory of Lifetime Effects in Point-Contacts
Application to Cd2Re2O7
∆ and ∆2 ∆ and Γ
0.2 0.4 0.6 0.8 1.00.00
0.05
0.10
0.15
0.20
0.25
0.2 0.4 0.6 0.8 1.0
E
nerg
y G
ap (m
eV)
Temperature (K)
Temperature (K)
2∆
kBTc=5.0(1) Tc=1.02 K
B. Mitrovi c Theory of Lifetime Effects in Point-Contacts
KOs2O6 (Photoemission Spectroscopy)
Shimojima et al. PRL 99, 117003 (2007), using Dynes formula:
2∆
kBTc≥4.56
Tc=9.6 K
B. Mitrovi c Theory of Lifetime Effects in Point-Contacts
Model of a rattler
Mitrovic and Nicol (unpublished):
α2F is a cutoff Lorentzianat ΩR =2.2 meV
λR=3
Tc=6 K kBTc/ωln=0.24
2∆
kBTc=5.73
B. Mitrovi c Theory of Lifetime Effects in Point-Contacts
Comparison with other experiments
NMR:
Vyaselev et al., PRL 89, 017001(2002)
Allen & Rainer, Nature 349, 396(1991)
A large NMR coherence peak ⇒ Cd2Re2O7 is a BCS superconductor
with2∆
kBTc=3.68
B. Mitrovi c Theory of Lifetime Effects in Point-Contacts
Comparison with other experiments
Specific heat:
Hiroi & Hanawa, J. Phys. Chem. Solids 63, 1021 (2002):
γexpγband
=2.63 ⇒ λ=1.63
Razavi et al., submitted to PRL
Note:There is a kink at T =80 % Tc!
∆Ce
γTc=1.15 < the BCS value of 1.43
⇓
anisotropic/multiband supercond. (?)
or
kBTc/ωln > 0.24, i.e. extreme strongcoupling (?)
B. Mitrovi c Theory of Lifetime Effects in Point-Contacts
Comparison with other experiments
Far-IR:
Hajialamdari et al., J. Phys.: Condens. Matter 24, 505701 (2012)
New peaks appear in the superconducting state at T =0.5 K (< 0.8K)!
B. Mitrovi c Theory of Lifetime Effects in Point-Contacts
Possible scenario
There is a structural transition in Cd2Re2O7 below Tc (at 0.8 K)similar to the transition in KOs2O6. The new low frequency phononmodes appear which couple strongly to the electrons leading to alarge low temperature ∆.
B. Mitrovi c Theory of Lifetime Effects in Point-Contacts