theory of lifetime effects in point-contacts: application ... · outline tunneling junction...

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


Credits

Yousef Rohanizadegan, Brock F. Razavi, M. Hajialamdari and M. Reedyk, Brock R. Kremer, MPI & Brock M. Przedborski and K. Samokhin, Brock


Tunneling junction spectroscopy


Tunneling junction spectroscopy

Problems: It is difficult to make good tunneling junctions withsuperconductors which have complicated structure and a shortcoherence length.


Point-contact spectroscopy


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)


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


BTK theory


BTK theory

6: Andreev reflection


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


Experiments

Au-Nb point contact

(a) 10-Ω contact resistance

(b) 3-Ω contact resistance

Note: Experimental curves are broadened BTK curves


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!


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))


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)



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


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


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)



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)



∆ 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


KOs2O6 (Photoemission Spectroscopy)

Shimojima et al. PRL 99, 117003 (2007), using Dynes formula:

2∆

kBTc≥4.56

Tc=9.6 K


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


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



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 (?)



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)!


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 ∆.


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