revealing the fflo phase by the in- plane critical field
TRANSCRIPT
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A. Buzdin University of Bordeaux I and Institut Universitaire de France
Revealing the FFLO phase by the in-plane critical field anisotropy in layered
superconductors
in collaboration with M. Croitoru
ECRYS-2014, August 11-23, 2014 Cargèse , France
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1. Singlet superconductivity destruction by the magnetic field: - The main mechanisms - Origin of FFLO state. 2. Experimental evidences of FFLO state. 3. Quasi-2D superconductors: in-plane anisotropy of the critical field due to FFLO modulation. 4. Quasi-1D superconductors
Outline
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• Orbital effect (Lorentz force)
B
-p FL
p
FL
• Paramagnetic effect (singlet pair)
Sz=+1/2 Sz=-1/2
μBH~Δ~Tc ( ) cTsSI ≈⋅
Electromagnetic mechanism
(breakdown of Cooper pairs by magnetic field
induced by magnetic moment)
Exchange interaction
1. Singlet superconductivity destruction by the magnetic field.
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Pc
orbc
HH
2
22≡α 1~ <<∆
Fεα
Superconductivity is destroyed by magnetic field
Orbital effect (Vortices)
Zeeman effect of spin (Pauli paramagnetism)
12
χN H 2 =12
N(0)∆2
χN =12
(gµ B )2 N (0) B
Pc g
Hµ
∆=
22
20
2 2πξΦ
=orbcH
Maki parameter
Usually the influence of Pauli paramagnetic effect is negligibly small
B
-p FL
p
FL
μBH~Δ~Tc
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Superconducting order parameter behavior under paramagnetic effect
Standard Ginzburg-Landau functional:
...2222 +Ψ∇+Ψ∇−Ψ= ηγaF
The minimum energy corresponds to Ψ=const
The coefficients of GL functional are functions of the Zeeman field h= μBH !
Modified Ginzburg-Landau functional ! :
422
241
Ψ+Ψ∇+Ψ=b
maF
The non-uniform state Ψ~exp(iqr) will correspond to minimum energy and higher transition temperature
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q
F
q0
242 )( qqqaF Ψ+−= ηγ
Ψ~exp(iqr) - Fulde-Ferrell-Larkin-Ovchinnikov state (1964). Only in pure superconductors and in the rather narrow region.
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FFLO inventors
Fulde and Ferrell
Larkin and Ovchinnikov
P. Fulde, R. A. Ferrell, Phys.Rev. 135, A550 (1964) A. I. Larkin, Yu. N. Ovchinnikov, JETP 47, 1138 (1964)
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E
k kF +δkF
kF -δkF
The total momentum of the Cooper pair is -(kF -δkF)+ (kF -δkF)=2 δkF
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ky kx
E
Conventional pairing
FFLO pairing
( k ,-k )
( k ,-k+q )
pairing between Zeeman split parts of the Fermi surface Cooper pairs have a single non-vanishing center of mass momentum
k
-k
-k k
ky kx
E
q
q~gµBH/vF
-k+q q
-k+q
k
k
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Pairing of electrons with opposite spins and momenta unfavourable :
But :
• the upper critical field is increased
• Sensivity to the disorder and to the orbital effect:
(clean limit)
At T = 0, Zeeman energy compensation is exact in 1d, partial in 2d and 3d.
if
0.2 0.4 0.6 0.8 1 cTT /0.56
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
cTT /
∆/HBµ1d SC 2d SC 3d SC
0.56
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•Anisotropies of the Fermi surface and the gap function can stabilize the FFLO state. Stability of FFLO state crucially depends on the dimensionality of the system.
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Requirements for the formation of the FFLO sate
• Strongly type-II superconductors with very large Ginzburg-Landau parameter
• Large Maki parameter, such that the upper critical field can easily approach the Pauli paramagnetic limit:
pairing • Very clean, , since the FFLO state is sensitive to the presence of impurities.
1D SC 2D SC 3D SC
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FFLO phase in the case of paramagnetic and orbital effect (3D BCS limit) – upper critical field
Lowest m=0 Landau level solution, Gruenberg and Gunter, 1966
Pc
orbc
HH
2
22≡α FFLO exists for Maki parameter α>1.8.
For Maki parameter α>9 the highest Landau level solutions are realized – Buzdin and Brison, 1996.
Note : The system with elliptic Fermi surface can be tranformed by scaling transformation to ihe isotropic one. Sure the direction of the magnetic field will be changed.
FFLO and orbital effect
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FFLO phase in 2D superconductors in the tilted magnetic field - upper critical field
Highest Landau level solutions are realized – Bulaevskii, 1974; Buzdin and Brison, 1996; Houzet and Buzdin, 2000.
θ B
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3. Experimental evidences of FFLO state.
•Unusual form of Hc2(T) dependence •Change of the form of the NMR spectrum •Anomalies in altrasound absorbtion •Unusual behaviour of magnetization •Change of anisotropy ….
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Layered organic superconductor
Layered structure suppression of orbital effects in H parallel to the planes.
BEDT-TTF layer
Cu[N(CN)2]Br layer
∼ 15 Å
C S H or D
C S H or D
BEDT-TTF (donor molecule)
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Possible FFLO state in Heavy Fermion superconductors
This 2nd order phase transition is characterized by a structural transition of the flux line lattice (FLL).
Ultrasound and NMR results are consistent with the FFLO state which predicts a segmentation of the flux line lattice
Y. Matsuda and H. Shimahara, J. Phys. Sos. Japan 76, 051005 (2007)
Pauli paramagnetically limited superconducting state
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Field induced superconductivity (FISC) in an organic compound
S. Uji et al., Nature 410 908 (2001)
L. Balicas et al., PRL 87 067002 (2001)
FISC
Metal
Insulator AF
λ-(BETS)2FeCl4
c-axis (in-plane) resistivity
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Other example:
Temperature [K]
Crit
ical
fiel
d H
c2 [T
esla
]
Jaccarino-Peter effect
Eu-Sn Molybdenum chalcogenide
H. Meul et al, 1984
Zeeman energy Exchange energy between conduction electrons in the BETS layers and magnetic ions Fe3+ (S=5/2)
For some reason J > 0 : the paramagnetic effect is suppressed at
S
S (Eu0.75Sn0.25Mo6S7.2Se0.8)
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Anomalous in-plane anisotropy of the onset of SC in (TMTSF)2ClO4
S.Yonezawa, S.Kusaba, Y.Maeno, P.Auban-Senzier, C.Pasquier, K.Bechgaard, and D. Jerome, Phys. Rev. Lett. 100, 117002 (2008)
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Model system –quasi 2 metal
We assume that the corrugation of the Fermi surface is small
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Eilenberger equation for layered SC
Here:
Taking into account that the system is near the second-order phase transition, the linearized Eilenberger equation describing layered superconducting systems acquires the form
The order parameter is defined self-consistently as (s-wave)
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Choice of the solution The solution of the Eilenberger equation can be chosen without loss of generality as a Bloch function
At the same time, the order parameter can be expanded as
This expansion takes into account the possibility for the formation of the pairing state with finite center-of-mass momentum:
• The direction of the FFLO modulation vector is fixed by the symmetry of the Fermi surface.
• |q| is taken that maximizes Hc2 calculated in the Pauli paramagnetic limit.
M. Croitoru, M. Houzet, and A. I. Buzdin, Phys. Rev. Lett. 108, 207005 (2012).
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Solution
Here:
Making use of the self-consistency relation we finally obtain
General case:
Resonance:
Out of resonance:
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Orbital correction (out of resonance) Normalized orbital correction of the superconducting onset temperature as a function of in-plane magnetic field H for several angles between magnetic field and the FFLO modulation vector q.
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Absolute value of the wave vector q of the FFLO phase (dashed lines) and of the wave vectors Q (solid lines) versus the reduced temperature calculated for several values of Fermi velocity .
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Orbital correction (resonance)
Resonance condition:
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Resonance condition • Vector potential of the parallel magnetic field results in a modulation of the interlayer coupling;
Resonance condition:
Conducting layer
• The period of this modulation may interfere with the in-plane FFLO modulation leading to the anomalies in the critical field behavior;
Conducting layer
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Intrinsic vortex pinning in LOFF phase for parallel field orientation
Δn= Δ0cos(qr+αn)exp(iφn(r))
t – transfer integral
Josephson coupling between layers is modulated
Fn,n+1=[-I0cos(αn_- αn+1)+I2cos(qr) cos(αn_+ αn+1)] cos(φn- φn+1)
φn- φn+1=2πxHs/Φ0 + πn s-interlayer distance, x-coordinate along q
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Intrinsic Josephson vortex pinning in the FFLO phase for parallel field orientation
S. Uji et al., Phys.Rev.Lett. 97, 157001 (2006) L. Bulaevskii et al., Phys.Rev.Lett. 90, 067003 (2006)
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Normalized superconducting transition temperature, as a function of the angle between the directions of the applied magnetic field and the vector q for several values of .
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In-plane anisotropy of the onset of superconductivity
Resonance case Non-resonance case
Croitoru, Buzdin, PRB, 2012
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Anomalous in-plane anisotropy of the onset of SC in (TMTSF)2ClO4
S.Yonezawa, S.Kusaba, Y.Maeno, P.Auban-Senzier, C.Pasquier, K.Bechgaard, and D. Jerome, Phys. Rev. Lett. 100, 117002 (2008)
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Normalized correction of the superconducting onset temperature as a function of in-plane magnetic field H for several angles between magnetic field and the FFLO modulation vector q.
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Orbital correction (d-wave)
Orbital correction FFLO modulation vector
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Normalized superconducting transition temperature, as a function of the angle between the directions of the applied magnetic field and the vector q for several values of .
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In-plane anisotropy of the onset of superconductivity (d-wave)
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Phase diagram with FFLO • Normalized correction of the superconducting onset temperature as a function of (i) the reduced temperature, (ii) of the magnetic field direction; • Polar plots of .
0.0 0.2 0.4 0.6 0.8 1.0-2.0
-1.5
-1.0
-0.5
0.0
α = 0°
α = 20°
α = 70°
α = 45°
∆TcP
/TcP
T2 c0
/t2 c0
T/Tc0
mx/my = 1, 10 η = 5.5
0 45 90 135 180-1.6
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
η = 5.5
T/Tc = 0.05
T/Tc = 0.1
T/Tc = 0.2
T/Tc = 0.4
T/Tc = 0.6T/Tc = 0.8
my/mx = 1
∆TcP
/TcP
T2 c0
/t2 c0
angle (°)
0.6
0.8
1.00
20
40
60
80
100
120
140
160180200
220
240
260
280
300
320
340
0.6
0.8
1.0
Croitoru, Houzet, Buzdin, PRL, 2012
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Quasi 1D superconductors
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Croitoru, Buzdin, PRB, 2014
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Conclusions
• There are strong experimental evidences of the existence of the the FFLO state in organic layered superconductors.
• The interplay between FFLO modulation and orbital effect in
layered superconductor in tilted field results in the very special oscillatory-like angular dependence of the critical field.
• The careful studies of the in-plane critical field anisotropy
could provide the information about the direction and the absolute value of the FFLO wave-vector.
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System of coupled equations
Substituting one gets system of coupled equations
where
Assumptions: We keep term up to:
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FFLO phase in 2D superconductors in the tilted magnetic field - upper critical field
Highest Landau level solutions are realized – Bulaevskii, 1974; Buzdin and Brison, 1996; Houzet and Buzdin, 2000.
θ B
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4. Vortices in FFLO state.
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Order parameter distribution for the asymmetric and square lattices with Landau level n=1.
The dark zones correspond to the maximum of the order parameter and the white zones to its minimum.
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Exotic vortex lattice structures in tilted magnetic field
Near the tricritical point, the characteristic length is
Microscopic derivation of the Ginzburg-Landau functional :
Instability toward FFLO state
Instability toward 1st order transition
Validity:
• large scale for spatial variation of ∆ : vicinity of T *
small orbital effect, introduced with
• we neglect diamagnetic screening currents (high-κ limit)
Generalized Ginzburg-Landau functional
Next orders are important :
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• 2nd order phase transition at
→ higher Landau levels
• Near the transition: minimization of the free energy with solutions in the form
gauge
ζ Parametrizes all vortex lattice structures at a given Landau level N
is the unit cell
All of them are decribed in the subset :
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__ 1st order transition __
2nd order transition __ 2nd order transition in the
paramagnetic limit
Temperature -2 -1 0
Mag
netic
fiel
d
0
1
2
3
Tricritical point
n=1
n=0
n=2
Analysis of phase diagram :
• cascade of 2nd and 1st order transitions between S and N phases
• 1st order transitions within the S phase
• exotic vortex lattice structures
At Landau levels n > 0, we find structures with several points of vanishment of the order parameter in the unit cell and with different winding numbers w = ± 1, ± 2 …