Download Formation of Magnetic Impurities and Pair

May 13-14 (2011) GriffinFest, Toronto, Canada
Formation of Magnetic Impurities, π-Junctions,
and a Spontaneous Current State
in a Superfluid Fermi Gas
Yoji Ohashi1,3, Takashi Kashimura1 , and Shunji Tsuchiya2,3
1:Department of Physics, Keio University, Japan
2:Department of Physics, Tokyo University of Science, Japan
3:CREST (JST), Japan
introduction
idea to introduce magnetic impurities to a
superfluid Fermi gas
physical properties around magnetic impurity
local density of states
localized excited (bound) states
superfluid/ferromagnet/superfluid-junction,
and spontaneous current state
1991.4~1994.3 (PhD course : Tokyo Institute of Technology)
“Antiferromagnetic spin fluctuations in high-Tc cuprates”
1991.4~1994.3 (PhD course : Tokyo Institute of Technology)
“Antiferromagnetic spin fluctuations in high-Tc cuprates”
1991~1994 (PhD course : Tokyo Institute of Technology)
1994.4~1995.3 (PD: Osaka University)
“Boundary effects on d-wave superconductors”
1991~1994 (PhD course : Tokyo Institute of Technology)
1994.4~1995.3 (PD: Osaka University)
“Boundary effects in d-wave superconductors”
1991~1994 (PhD course : Tokyo Institute of Technology)
1994.4~1995.3 (PD: Osaka University)
1995.4~2006.3 (Tsukuba University)
“Carlson-Goldman mode and Josephson plasma
in high-Tc cuprates”
1991~1994 (PhD course: Tokyo Institute of Technology)
1994.4~1995.3 (PD: Osaka University)
1995.4~2006.3 (Tsukuba University)
“Carlson-Goldman mode and Josephson plasma
in high-Tc cuprates”
1991~1994 (PhD course : Tokyo Institute of Technology)
1994.4~1995.3 (PD: Osaka University)
1995.4~2006.3 (Tsukuba University)
2001.8~2002.6 (visiting researcher, Toronto University)
“BCS-BEC crossover in a gas of Fermi atoms with a Feshbach
resonance”
2004.9~2004.11 (visiting researcher,
Toronto University)
2006.4~ (Keio University)
2011.5.13 and 14 (Toronto!)
May 13-14 (2011) GriffinFest, Toronto, Canada
Formation of Magnetic Impurities, π-Junction,
and Spontaneous Current State
in a Superfluid Fermi Gas
Yoji Ohashi1,3, Takashi Kashimura1 , and Shunji Tsuchiya2,3
1:Department of Physics, Keio University, Japan
2:Department of Physics, Tokyo University of Science, Japan
3:CREST (JST), Japan
Introduction
Idea to introduce magnetic impurities to a
superfluid Fermi gas
Properties around pseudo-magnetic impurity
density of states
localized excited (bound) states
superfluid/ferromagnet/superfluid-junction,
π-junction, and spontaneous current state
40K and 66Li Fermi gases
Superfluid 40
“quantum simulator” to study various physical properties of superconductivity
BCS-BEC crossover by tunable interaction
40K
strong-coupling effects beyond mean-field theory
pseudogap (preformed pairs)
Fermi SF
Bose SF
optical lattice
C. A. Regal, et al.
PRL 92 (2004) 040403.
| 9 / 2, −7 / 2 >
| 9/ 2, −9/ 2 >
band effect
low-dimensional effect
Hubbard model and strong correlation
population imbalance
superconductivity under magnetic field
FFLO
“Magnetic” effect
superfluid
Fermi gas
metallic
superconductivity
La3-xGdxIn
40K
Tc
Tc0
C. A. Regal, et al.
PRL 92 (2004) 040403.
| 9 / 2, −7 / 2 >
| 9/ 2, −9/ 2 >
“pseudospin”
Crow et al., Phys.
Lett. 21, 378 (1966).
magnetic impurity concentration
?
“magnetic”
impurity
“spin”
Magnetic effects on superconductivity
pairing mechanism
n
spi tions
ua
t
c
flu
depairing effect
pa
r
e
p
Coo
ir
magnetic
impurity
High-Tc cuprates:
antiferromagnetic spin fluctuations
Superfluid liquid 3He:
ferromagnetic spin fluctuations
pair-breaking effect by magnetic impurities
Satori, Shiba (1992)
Shiba(1968)
Abrikosov, Gor’kov (1961)
bound state
below the gap
suppression of SC state
by magnetic scattering
competition between
SC and Kondo effect
gapless SC
Nonmagnetic impurities
do not affect s-wave SC.
(Anderson’s theorem)
impurity
band
bound state energy /Δ
ρimp × S ( S + 1)ui2mp N (0) / Tc0
superconducting DOS
spin
doublet
Tc
Tc0
Kondo
singlet
Kondo
singlet
TK
= 0.3
∆
TK / ∆
π-junction
Insulator
SC
SC
ferromagnet
SC
SC
∆ ( x)
∆( x)
x
0
“0”-junction
x
0
“π”-junction
Nb
ferromagnet
∆ ( x)
SC
Nb
Al/Al2O3/PdNi
T. Kontos et al., PRL 89, 137007 (2002)
π-junction
Insulator
SC
SC
ferromagnet
SC
SC
∆ ( x)
∆( x)
x
0
x
0
“π”-junction
“0”-junction
Nb
∆ ( x)
F
SC
SC
Nb
Al/Al2O3/PdNi
T. Kontos et al., PRL 89, 137007 (2002)
We theoretically discuss an idea to introduce magnetic impurities and
magnetic junction to a superfluid Fermi gas.
Phase sepration in a superfuid Fermi gas with population imbalance
pseudo-spin ↑
pseudo-spin ↓
↑−↓ > 0
↑
↑
N SF N
↑↓
G. Partridge et al., PRL 97, 190407 (2006)
We theoretically discuss an idea to introduce magnetic impurities and
magnetic junction to a superfluid Fermi gas.
Phase sepration in a superfuid Fermi gas with population imbalance
pseudo-spin ↑
π-junction
∆( x)
pseudo-spin ↓
x
↑−↓ > 0
N SF N
↑↓
G. Partridge et al., PRL 97, 190407 (2006)
Super
fluid
N
Ferro
magnet
↑
Super
fluid
↑
↑
↑↓
SF
↑↓
SF
We theoretically discuss an idea to introduce magnetic impurities and
magnetic junction to a superfluid Fermi gas.
Phase sepration in a superfuid Fermi gas with population imbalance
pseudo-spin ↑
pseudo-spin ↓
↑−↓ > 0
↑
polarized magnetic impurities
↑
N SF N
↑↓
↑
↑
↑
↑
G. Partridge et al., PRL 97, 190407 (2006)
small nonmagnetic
impurity potentials
We theoretically discuss an idea to introduce magnetic impurities and
magnetic junction to a superfluid Fermi gas.
Phase sepration in a superfuid Fermi gas with population imbalance
Ferro
magnet
↑
↑
↑
↑
Super
fluid
N
Super
fluid
↑
↑↓
SF
↑↓
SF
To examine these ideas in a simple manner, we consider a Hubbard
model at T=0. We self-consistently determine the order parameter,
particle density, and polarization, around impurity potential and
barrier within the mean-field level. We examine the possibilities of
magnetization of impurities, SFS-, and π-junction.
Formulation
two-component 2D Fermi gas
population imbalance
−t
σ =↑, ↓
N↑ > N↓
−U
H = −t ∑ ⎡⎣ci†σ c jσ + h.c ⎤⎦ − U ∑ ni↑ ni↓ + ∑ Vi niσ − µ↑ N ↑ − µ↓ N ↓
i, j
i
i ,σ
“nonmagnetic” potential
Vi
[t ]
impurity
junction
trap
Formulation
H = −t ∑ ⎡⎣ci†σ c jσ + h.c ⎤⎦ − ∑ ∆ i ⎡⎣ci†↑ ci†↓ + h.c.⎤⎦ − U ∑ niσ ni −σ
i, j
i
i ,σ
+ ∑ Vi niσ − µ↑ N ↑ − µ↓ N ↓ + EG
i ,σ
We solve the BdG equations to self-consistently determine ∆ i , niσ , µσ
for given Nσ .
ni↑ + ni↓ ≅ 0.36
N↑ − N↓
P=
N↑ + N↓
phase diagram of 2D uniform Hubbard model
Formation of pseudo-magnetic impurity
[t ]
Γ2
Vi = V0
(R − R imp ) 2 + Γ 2
V0 / t = 0.25
Γ /t =1
U /t = 6
N ↑ = 300, N ↓ = 300
per 41× 41sites
∆ is not destroyed by impurity
potential (Anderson’s theorem).
N ↑ = 301, N ↓ = 300
per 41× 41sites
∆ is remarkably damaged by impurity
potential.
Formation of pseudo-magnetic impurity
S z = ni↑ − ni↓ Local magnetization
∆N = N ↑ − N ↓ = 0
∆N = 1
∆N = 5
Nonmagnetic potential is magnetized by excess atoms!
Magnetization of nonmagnetic potential
“pseudo”-magnetic impurity
To minimize the condensation energy loss
by the depairing effect, excess atoms are
localized around the region where ∆ is
small from the begining.
magnetic impurity in metal
Strong electron correlation excludes
double occupancy of ↑ and↓spins.
Coulomb
repulsion
∆( x)
Vimp
H = − J Sz σ z
(classical spin)
(no exchange term)
x
H = − JS ⋅ σ
(quantum spin)
Bound states by off-diagonal pair-potential well
V ( x)
∆N = 5
pair-potential well ∆(R)
bound states
potential well V(R)
We can expect that this well structure of off-diagonal pair potential ∆(Rxx,Ryy)
induces bound states around the impurity potential, as in the case of ordinary
(diagonal) potential well V(Rxx,Ryy) .
local density of states ρ↑
↑(ω,R)
BCS gap
uniform Fermi gas
N↑ = N↓
N ↑ = 301, N ↓ = 300
Ry
×
impurity
ρ↑(ω,R)
×
Rx
In-gap states appear around
“pseudo-magnetic” impurity.
(γ / t = 0.05)
ω /t
bound state wavefunctions below the energy gap
| Ψ particle ( Rx , Ry ) |2
L=1(p)
L=2 (d)
L=3 (f)
2nd lowest states
3rd lowest states
4th lowest states
L=0 (s)
Ry
lowest bound state
Rx
The pair-potential well ∆(Rxx,Ryy) works as a cylindrical potential well,
so that bound states can be classified by angular momentum L.
Superfluid/ferromagnet/superfluid (SFS) junction
V (R )
Ry
Rx
S z (R ) = ni↑ − ni↓
Population imbalance
N↑ − N↓
P=
= 0.11
N↑ + N↓
Ry
Rx
S
F
S
U /t = 7
N = 99
∆N = N ↑ − N ↓ = 11
π-phase in a superfluid Fermi gas
S z (R )
N↑ − N↓
P=
= 0.11
N↑ + N↓
Ry
S
∆( Rx , Ry )
π-junction
S
F
∆( Rx , Ry )
_
+
0-junction
Ry
Rx
E / t = −875.00
+
+
Ry
Rx
E / t = −874.78
stability of π-junction
−1
[(tLy ) ]
Eπ − junction − E0− junction
π-junction
0-junction
S z / Ly
The π-junction becomes stable when magnetization is large to some extent.
Effects of trap potential 2
2D cigar trap
S z ( Rx , Ry )
N↑ = 32
N↓ = 28
U = 4t
Vx = 0.1t
∆ ( Rx , Ry )
＋
Vy = 0.001t
−
VB = 1.0t
σ =6
spontaneous current state in a ring trap
Because of the single-valueness of ∆, the
π-junction twists the phase θ of the order
parameter by π along the ring.
N↑ > N↓
Spontaneous current
+∆
πi
−∆ = ∆e
J ~ ∇θ > 0
nonmagnetic potential barrier
+
localized excess ↑-spin atoms
||
π-junction
∆( x)
+
_
Rx
“corner-junction”
spontaneous current state in a ring trap
J ~ ∇θ > 0
1D-ring trap
( N ↑ , N ↓ ) = (41, 40)
+∆
−∆ = ∆eπ i
nonmagnetic potential barrier
+
localized excess ↑-spin atoms
||
π-junction
∆( x)
+
_
Rx
θ(x)/π
N↑ > N↓
| ∆( x) | eiθ ( x )
δθ ~ π
spontaneous flow
Rx
Summary
We have discussed an idea to introduce magnetic impurities and magnetic junction to
a superfluid Fermi gas. Using the phase separation of a polarized Fermi superfluid, we
have shown that nonmagnetic potential is magnetized in the sense that some of excess
atoms are localized around the potential.
pseudo-magnetic impurities
polarized classical spin
suppression of ∆ around impurity
in-gap states
“quantum dot”
gapless Fermi superfluid
S
superfluid/ferromagnet/superfluid junction
∆( Rx , Ry )
π-junction
FFLO
_
+
Ry
Rx
F
S
∆ FFLO ( x)
S
π
spontaneous supercurrent
Summary
T. Kashimura, S. Tsuchiya, Y. Ohashi, Phys. Rev. A, 82, 033617 (2010)
Y. Ohashi, Phys. Rev. A (2011), in press, cond-mat/1103.1942.
T. Kashimura, S. Tsuchiya, and Y. Ohashi, in preparation
pseudo-magnetic impurities
polarized classical spin
suppression of ∆ around impurity
in-gap states
“quantum dot”
gapless Fermi superfluid
S
superfluid/ferromagnet/superfluid junction
∆( Rx , Ry )
π-junction
FFLO
_
+
Ry
Rx
F
S
∆ FFLO ( x)
S
π
spontaneous supercurrent