Odd Triplet Pairing in clean Superconductor/Ferromagnet heterostructures
Klaus Halterman,1, ∗ Paul H. Barsic,2, † and Oriol T. Valls2, ‡
Physics and Computational Sciences, Research and Engineering Sciences Department,
Naval Air Warfare Center, China Lake, California 93555
School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455
(Dated: October 27, 2018)
We study triplet pairing correlations in clean Ferromagnet (F)/Superconductor (S) nanojunctions,
via fully self consistent solution of the Bogoliubov-de Gennes equations. We consider FSF trilayers,
with S being an s-wave superconductor, and an arbitrary angle α between the magnetizations of
the two F layers. We find that contrary to some previous expectations, triplet correlations, odd in
time, are induced in both the S and F layers in the clean limit. We investigate their behavior as a
function of time, position, and α. The triplet amplitudes are largest at times on the order of the
inverse “Debye” frequency, and at that time scale they are long ranged in both S and F. The zero
temperature condensation energy is found to be lowest when the magnetizations are antiparallel.
PACS numbers: 74.45.+c, 74.25.Bt, 74.78.Fk
The proximity effects in superconductor/ferromagnet
(SF) heterostructures lead to the coexistence of ferromag-
netic and superconducting ordering and to novel trans-
port phenomena[1, 2]. Interesting effects that arise from
the interplay between these orderings have potential tech-
nological applications in fields such as spintronics[3]. For
example, the relative orientation of the magnetizations in
the F layers in FSF trilayers can have a strong influence
on the conductivity[4, 5, 6, 7, 8], making them good spin
valve candidates. Such trilayers were first proposed[9] for
insulating F layers and later for metallic[10, 11] ones.
This interplay also results in fundamental new physics.
An outstanding example is the existence of “odd” triplet
superconductivity. This is an s-wave pairing triplet state
that is even in momentum, and therefore not destroyed
by nonmagnetic impurities, but with the triplet corre-
lations being odd in frequency, so that the equal time
triplet amplitudes vanish as required by the Pauli prin-
ciple. This exotic pairing state with total spin one was
proposed long ago [12] as a possible state in superfluid
3He. Although this type of pairing does not occur there,
it is possible in certain FSF systems[1, 2, 13, 14] with or-
dinary singlet pairing in S. This arrangement can induce,
via proximity effects, triplet correlations with m = 0 and
m = ±1 projections of the total spin. If the magnetiza-
tion orientations in both F layers are unidirectional and
along the quantization axis, symmetry arguments show
that only the m = 0 projection along that axis can exist.
Odd triplet pairing in F/S structures has been studied
in the dirty limit through linearized Usadel-type quasi-
classical equations [2, 13, 14, 15]. In this case, it was
found that m = 0 triplet pairs always exist. They are
suppressed in F over short length scales, just as the sin-
glet pairs. The m = ±1 components, for which the ex-
change field is not pair-breaking, can be long ranged, and
were found to exist for nonhomogeneous magnetization.
For FSF trilayers[2, 16, 17], the quasiclassical methods
predict that the structure contains a superposition of all
FIG. 1: Schematic of FSF junction. The left ferromagnetic
layer F1 has a magnetization oriented at an angle −α/2 in
the x− z plane, while the other ferromagnet, F2, has a mag-
netization orientation at an angle α/2 in the x− z plane.
three spin triplet projections except when the magneti-
zations of the F layers are collinear, in which case the
m = ±1 components along the magnetization axis van-
ish. It is noted in Ref. [1] that the existence of such effects
in the clean limit has not been established and may be
doubted. This we remedy in the present work, where
we establish that, contrary to the doubts voiced there,
induced, long-ranged, odd triplet pairing does occur in
clean FSF structures.
Experimental results that may argue for the existence
of long range triplet pairing of superconductors through
a ferromagnet have been obtained in superlattices[18]
with ferromagnetic spacers, and in two superconduc-
tors coupling through a single ferromagnet[19, 20].
Measurements[19] on a SQUID, in which a phase change
of π in the order parameter is found after inversion, in-
dicate an odd-parity state. Very recently, a Josephson
current through a strong ferromagnet was observed, in-
dicating the existence of a spin triplet state[20] induced
http://arxiv.org/abs/0704.1820v1
by NbTiN, an s-wave superconductor.
In this paper, we study the induced odd triplet
superconductivity in FSF trilayers in the clean limit
through a fully self-consistent solution of the microscopic
Bogoliubov-de Gennes (BdG) equations. We consider ar-
bitrary relative orientation of the magnetic moments in
the two F layers. We find that there are indeed induced
odd triplet correlations which can include both m = 0
and m = ±1 projections. We directly study their time
dependence and we find that they are largest for times of
order of the inverse cutoff “Debye” frequency. The corre-
lations are, at these time scales, long ranged in both the
S and F regions. We also find that the condensation en-
ergy depends on the relative orientation of the F layers,
being a minimum when they are antiparallel.
To find the triplet correlations arising from the non-
trivial spin structure in our FSF system, we use the BdG
equations with the BCS Hamiltonian, Heff :
Heff =
ψδ(r) +
(iσy)δβ∆(r)ψ
(r) + h.c.]−
(r)(h · σ)δβ ψβ(r)
where ∆(r) is the pair potential, to be determined self-
consistently, ψ
, ψδ are the creation and annihilation op-
erators with spin δ, EF is the Fermi energy, and σ are
the Pauli matrices. We describe the magnetism of the F
layers by an effective exchange field h(r) that vanishes in
the S layer. We will consider the geometry depicted in
Fig. 1, with the y axis normal to the layers and h(r) in
the x − z plane (which is infinite in extent) forming an
angle ±α/2 with the z axis in each F layer.
Next, we expand the field operators in terms of a Bo-
goliubov transformation which we write as:
ψδ(r) =
unδ(r)γn + ηδvnδ(r)γ
, (1)
where ηδ ≡ 1(−1) for spin down (up), unδ and vnδ are the
quasiparticle and quasihole amplitudes. This transforma-
tion diagonalizes Heff : [Heff , γn] = −ǫnγn, [Heff , γ
n. By taking the commutator [ψδ(r),Heff ], and with
h(r) in the x − z plane as explained above, we have the
following:
[ψ↑(r),Heff ] = (He − hz)ψ↑(r)− hxψ↓(r) + ∆(r)ψ
[ψ↓(r),Heff ] = (He + hz)ψ↓(r)− hxψ↑(r)−∆(r)ψ
Inserting (1) into (2) and introducing a set ρ of Pauli-like
matrices in particle-hole space, yields the spin-dependent
BdG equations:
H01̂− hzσz
∆(y)ρx − hx1̂
Φn = ǫnΦn,
where Φn ≡ (un↑(y), un↓(y), vn↑(y), vn↓(y))
T and H0 ≡
−∂2y/(2m) + ε⊥ − EF . Here ε⊥ is the transverse kinetic
energy and a factor of eik⊥·r has been suppressed. In
deriving Eq. (3) care has been taken to consistently use
the phase conventions in Eq. (1). To find the quasiparti-
cle amplitudes along a different quantization axis in the
x−z plane, one performs a spin rotation: Φn → Û(α
′)Φn,
where Û(α′) = cos(α′/2)1̂⊗ 1̂− i sin(α′/2)ρz ⊗ σy.
When the magnetizations of the F layers are collinear,
one can take hx = 0. For the general case shown in
Fig. 1 one has in the F1 layer, hx = h0 sin(−α/2) and
hz = h0 cos(−α/2), where h0 is the magnitude of h,
while in F2, hx = h0 sin(α/2), and hz = h0 cos(α/2).
With an appropriate choice of basis, Eqs. (3) are cast
into a matrix eigenvalue system that is solved itera-
tively with the self consistency condition, ∆(y) = g(y)f3
(f3 =
[〈ψ↑(r)ψ↓(r)〉 − 〈ψ↓(r)ψ↑(r)〉]). In the F layers
we have g(y) = 0, while in S, g(y) = g, g being the usual
BCS singlet coupling constant there. Through Eqs. (1),
the self-consistency condition becomes a sum over states
restricted by the factor g to within ωD from the Fermi
surface. Iteration is performed until self-consistency is
reached. The numerical process is the same that was used
in previous work[24, 25], with now the hx term requiring
larger four-component matrices to be diagonalized.
We now define the following time dependent triplet am-
plitude functions in terms of the field operators,
f̃0(r, t) =
[〈ψ↑(r, t)ψ↓(r, 0)〉+ 〈ψ↓(r, t)ψ↑(r, 0)〉] , (4a)
f̃1(r, t) =
[〈ψ↑(r, t)ψ↑(r, 0)〉 − 〈ψ↓(r, t)ψ↓(r, 0)〉] , (4b)
which, as required by the Pauli principle for these s-wave
amplitudes, vanish at t = 0, as we shall verify. Making
use of Eq. (1) and the commutators, one can derive and
formally integrate the Heisenberg equation of the motion
for the operators and obtain:
f̃0(y, t) =
[un↑(y)vn↓(y)− un↓(y)vn↑(y)]ζn(t), (5a)
f̃1(y, t) =−
[un↑(y)vn↑(y) + un↓(y)vn↓(y)]ζn(t),
FIG. 2: (Color online) The real part, f0, of the triplet ampli-
tude f̃0, for a FSF trilayer at 7 different times. We normalize
f0 by the singlet bulk pair amplitude, ∆0/g. The coordinate
y is scaled by the Fermi wavevector, Y ≡ kF y, and time by
the Debye frequency, τ ≡ ωDt. At τ = 0, f0 ≡ 0 as required
by the Pauli principle. The interface is marked by the verti-
cal dashed line, with an F region to the left and the S to the
right. Half of the S region and part of the left F layer are
shown. The inset shows the maximum value of f0 versus τ .
where ζn(t) ≡ cos(ǫnt)− i sin(ǫnt) tanh(ǫn/2T ).
The amplitudes in Eqs. (5) contain all information on
the space and time dependence of induced triplet correla-
tions throughout the FSF structure. The summations in
Eqs. (5) are over the entire self-consistent spectrum, en-
suring that f0 and f1 vanish identically at t = 0 and thus
obey the exclusion principle. Using a non self consistent
∆(y) leads to violations of this condition, particularly
near the interface where proximity effects are most pro-
nounced. Geometrically, the indirect coupling between
magnets is stronger with fairly thin S layers and rela-
tively thick F layers. We thus have chosen dS = (3/2)ξ0
and dF1 = dF2 = ξ0, with the BCS correlation length
ξ0 = 100k
. We consider the low T limit and take
ωD = 0.04EF . The magnetic exchange is parametrized
via I ≡ h0/EF . Results shown are for I = 0.5 (unless
otherwise noted) and the magnetization orientation an-
gle, α, is swept over the range 0 ≤ α ≤ π. No triplet
amplitudes arise in the absence of magnetism (I = 0).
For the time scales considered here, the imaginary
parts of f̃0(y, t) and f̃1(y, t) at t 6= 0 are considerably
smaller than their real parts, and thus we focus on the
latter, which we denote by f0(y, t) and f1(y, t). In Fig. 2,
the spatial dependence of f0 is shown for parallel mag-
netization directions (α = 0) at several times τ ≡ ωDt.
The spatial range shown includes part of the F1 layer
(to the left of the dashed line) and half of the S layer
(to the right). At finite τ , the maximum occurs in the
ferromagnet close to the interface, after which f0 under-
goes damped oscillations with the usual spatial length
FIG. 3: (Color online) Spatial and angular dependence of f1,
at τ = 4 ≈ τc and several α. Normalizations and ranges are
as in Fig. 2. Inset: maxima of f0 and f1 in F1 versus α.
scale ξf ≈ (kF↑ − kF↓)
−1 ≈ k−1F /I. The height of the
main peak first increases with time, but drops off after
a characteristic time, τc ≈ 4, as seen in the inset, which
depicts the maximum value of f0 as a function of τ . As
τ increases beyond τc, the modulating f0 in F develops
more complicated atomic scale interference patterns and
becomes considerably longer ranged. In S, we see imme-
diately that f0 is also larger near the interface. Since
the triplet amplitudes vanish at τ = 0, short time scales
exhibit correspondingly short triplet penetration. The
figure shows, however, that the value of f0 in S is sub-
stantial for τ & τc, extending over length scales on the
order of ξ0 without appreciable decay. In contrast, the
usual singlet correlations were found to monotonically
drop off from their τ = 0 value over τ scales of order
unity.
In the main plot of Fig. 3 we examine the spatial de-
pendence of the real part of the m = ±1 triplet ampli-
tude, f1. Normalizations and spatial ranges are as in
Fig. 2 but now the time is fixed at τ = 4 ≈ τc, and
five equally spaced magnetization orientations are con-
sidered. At α = 0, f1 vanishes identically at all τ , as
expected. For nonzero α, correlations in all triplet chan-
nels are present. As was found for f0, the plot clearly
shows that f1 is largest near the interface, in the F re-
gion. Our geometry and conventions imply (see Fig. 1)
that the magnetization has opposite x-components in the
F1 and F2 regions. The f1 triplet pair amplitude profile
is thus antisymmetric about the origin, in contrast to the
symmetric f0, implying the existence of one node in the
superconductor. Nevertheless, the penetration of the f1
correlations in S can be long ranged. We find that f1 and
f0 oscillate in phase and with the same wavelength, re-
gardless of α. The inset illustrates the maximum attained
values of f0 and f1 in F1 as α varies. It shows that for
FIG. 4: (Color online) The T = 0 condensation energy, ∆E0,
normalized by N(0)∆20 (N(0) is the usual density of states),
vs. the angle α for two values of I . When the two magne-
tizations are antiparallel (α = π) ∆E0 is lowest. The inset
shows the ordinary (singlet) pair potential averaged over the
S region, normalized to the bulk ∆0.
a broad range of α, α . 3π/4, the maximum of f0 varies
relatively little, after which it drops off rapidly to zero at
α = π. This is to be expected as the anti-parallel orienta-
tion corresponds to the case in which the magnetization
is in the x direction, which is perpendicular to the axis
of quantization (see Fig. 1). The rise in the maximum
of f1 is monotonic, cresting at α = π, consistent with
the main plot. At this angle the triplet correlations ex-
tend considerably into the superconductor. At α = π/2
the maxima coincide since the two triplet components
are then identical throughout the whole space because
the magnetization vectors have equal projections on the
x and z axes. At α = π both magnetizations are normal
to the axis of quantization z (see Fig. 1). By making use
of the rotation matrix Û (see below Eq. 3) one can verify
that the m = ±1 components with respect to the axis x
along the magnetizations are zero.
We next consider the condensation energy, ∆E0, cal-
culated by subtracting the zero temperature supercon-
ducting and normal state free energies. The calculation
uses the self consistent spectra and ∆(y), and methods
explained elsewhere [25, 26]. In the main plot of Fig. 4,
we show ∆E0 (normalized at twice its bulk S value) at
two different values of I. The condensation energy results
clearly demonstrate that the antiparallel state (α = π) is
in general the lowest energy ground state. These results
are consistent with previous studies[8] of FSF structures
with parallel and antiparallel magnetizations. The inset
contains the magnitude of the spatially averaged pair po-
tential, normalized by ∆0, at the same values of I. The
inset correlates with the main plot, as it shows that the
singlet superconducting correlations in S increase with
α and are larger at I = 1 than at I = 0.5. The half-
metallic case of I = 1 illustrates that by having a single
spin band populated at the Fermi surface, Andreev reflec-
tion is suppressed, in effect keeping the superconductivity
more contained within S.
Thus, we have shown that in clean FSF trilayers in-
duced odd triplet correlations, with m = 0 and m = ±1
projections of the total spin, exist. We have used a mi-
croscopic self-consistent method to study the time and
angular dependence of these triplet correlations. The
correlations in all 3 triplet channels were found, at times
τ ≡ ωDt & τc, where τc ≈ 4, to be long ranged in both
the F and S regions. Finally, study of the condensation
energy revealed that the ground state energy is always
lowest for antiparallel magnetizations.
This project was supported in part by a grant of HPC
resources from the ARSC at the University of Alaska
Fairbanks (part of the DoD HPCM program) and by the
University of Minnesota Graduate School.
∗ Electronic address: klaus.halterman@navy.mil
† Electronic address: barsic@physics.umn.edu
‡ Electronic address: otvalls@umn.edu; Also at Minnesota
Supercomputer Institute, University of Minnesota, Min-
neapolis, Minnesota 55455
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  We study triplet pairing correlations in clean Ferromagnet (F)/Superconductor
(S) nanojunctions, via fully self consistent solution of the Bogoliubov-de
Gennes equations. We consider FSF trilayers, with S being an s-wave
superconductor, and an arbitrary angle $\alpha$ between the magnetizations of
the two F layers. We find that contrary to some previous expectations, triplet
correlations, odd in time, are induced in both the S and F layers in the clean
limit. We investigate their behavior as a function of time, position, and
$\alpha$. The triplet amplitudes are largest at times on the order of the
inverse ``Debye'' frequency, and at that time scale they are long ranged in
both S and F. The zero temperature condensation energy is found to be lowest
when the magnetizations are antiparallel.

Odd Triplet Pairing in clean Superconductor/Ferromagnet heterostructures
Klaus Halterman,1, ∗ Paul H. Barsic,2, † and Oriol T. Valls2, ‡
Physics and Computational Sciences, Research and Engineering Sciences Department,
Naval Air Warfare Center, China Lake, California 93555
School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455
(Dated: October 27, 2018)
We study triplet pairing correlations in clean Ferromagnet (F)/Superconductor (S) nanojunctions,
via fully self consistent solution of the Bogoliubov-de Gennes equations. We consider FSF trilayers,
with S being an s-wave superconductor, and an arbitrary angle α between the magnetizations of
the two F layers. We find that contrary to some previous expectations, triplet correlations, odd in
time, are induced in both the S and F layers in the clean limit. We investigate their behavior as a
function of time, position, and α. The triplet amplitudes are largest at times on the order of the
inverse “Debye” frequency, and at that time scale they are long ranged in both S and F. The zero
temperature condensation energy is found to be lowest when the magnetizations are antiparallel.
PACS numbers: 74.45.+c, 74.25.Bt, 74.78.Fk
The proximity effects in superconductor/ferromagnet
(SF) heterostructures lead to the coexistence of ferromag-
netic and superconducting ordering and to novel trans-
port phenomena[1, 2]. Interesting effects that arise from
the interplay between these orderings have potential tech-
nological applications in fields such as spintronics[3]. For
example, the relative orientation of the magnetizations in
the F layers in FSF trilayers can have a strong influence
on the conductivity[4, 5, 6, 7, 8], making them good spin
valve candidates. Such trilayers were first proposed[9] for
insulating F layers and later for metallic[10, 11] ones.
This interplay also results in fundamental new physics.
An outstanding example is the existence of “odd” triplet
superconductivity. This is an s-wave pairing triplet state
that is even in momentum, and therefore not destroyed
by nonmagnetic impurities, but with the triplet corre-
lations being odd in frequency, so that the equal time
triplet amplitudes vanish as required by the Pauli prin-
ciple. This exotic pairing state with total spin one was
proposed long ago [12] as a possible state in superfluid
3He. Although this type of pairing does not occur there,
it is possible in certain FSF systems[1, 2, 13, 14] with or-
dinary singlet pairing in S. This arrangement can induce,
via proximity effects, triplet correlations with m = 0 and
m = ±1 projections of the total spin. If the magnetiza-
tion orientations in both F layers are unidirectional and
along the quantization axis, symmetry arguments show
that only the m = 0 projection along that axis can exist.
Odd triplet pairing in F/S structures has been studied
in the dirty limit through linearized Usadel-type quasi-
classical equations [2, 13, 14, 15]. In this case, it was
found that m = 0 triplet pairs always exist. They are
suppressed in F over short length scales, just as the sin-
glet pairs. The m = ±1 components, for which the ex-
change field is not pair-breaking, can be long ranged, and
were found to exist for nonhomogeneous magnetization.
For FSF trilayers[2, 16, 17], the quasiclassical methods
predict that the structure contains a superposition of all
FIG. 1: Schematic of FSF junction. The left ferromagnetic
layer F1 has a magnetization oriented at an angle −α/2 in
the x− z plane, while the other ferromagnet, F2, has a mag-
netization orientation at an angle α/2 in the x− z plane.
three spin triplet projections except when the magneti-
zations of the F layers are collinear, in which case the
m = ±1 components along the magnetization axis van-
ish. It is noted in Ref. [1] that the existence of such effects
in the clean limit has not been established and may be
doubted. This we remedy in the present work, where
we establish that, contrary to the doubts voiced there,
induced, long-ranged, odd triplet pairing does occur in
clean FSF structures.
Experimental results that may argue for the existence
of long range triplet pairing of superconductors through
a ferromagnet have been obtained in superlattices[18]
with ferromagnetic spacers, and in two superconduc-
tors coupling through a single ferromagnet[19, 20].
Measurements[19] on a SQUID, in which a phase change
of π in the order parameter is found after inversion, in-
dicate an odd-parity state. Very recently, a Josephson
current through a strong ferromagnet was observed, in-
dicating the existence of a spin triplet state[20] induced
http://arxiv.org/abs/0704.1820v1
by NbTiN, an s-wave superconductor.
In this paper, we study the induced odd triplet
superconductivity in FSF trilayers in the clean limit
through a fully self-consistent solution of the microscopic
Bogoliubov-de Gennes (BdG) equations. We consider ar-
bitrary relative orientation of the magnetic moments in
the two F layers. We find that there are indeed induced
odd triplet correlations which can include both m = 0
and m = ±1 projections. We directly study their time
dependence and we find that they are largest for times of
order of the inverse cutoff “Debye” frequency. The corre-
lations are, at these time scales, long ranged in both the
S and F regions. We also find that the condensation en-
ergy depends on the relative orientation of the F layers,
being a minimum when they are antiparallel.
To find the triplet correlations arising from the non-
trivial spin structure in our FSF system, we use the BdG
equations with the BCS Hamiltonian, Heff :
Heff =
ψδ(r) +
(iσy)δβ∆(r)ψ
(r) + h.c.]−
(r)(h · σ)δβ ψβ(r)
where ∆(r) is the pair potential, to be determined self-
consistently, ψ
, ψδ are the creation and annihilation op-
erators with spin δ, EF is the Fermi energy, and σ are
the Pauli matrices. We describe the magnetism of the F
layers by an effective exchange field h(r) that vanishes in
the S layer. We will consider the geometry depicted in
Fig. 1, with the y axis normal to the layers and h(r) in
the x − z plane (which is infinite in extent) forming an
angle ±α/2 with the z axis in each F layer.
Next, we expand the field operators in terms of a Bo-
goliubov transformation which we write as:
ψδ(r) =
unδ(r)γn + ηδvnδ(r)γ
, (1)
where ηδ ≡ 1(−1) for spin down (up), unδ and vnδ are the
quasiparticle and quasihole amplitudes. This transforma-
tion diagonalizes Heff : [Heff , γn] = −ǫnγn, [Heff , γ
n. By taking the commutator [ψδ(r),Heff ], and with
h(r) in the x − z plane as explained above, we have the
following:
[ψ↑(r),Heff ] = (He − hz)ψ↑(r)− hxψ↓(r) + ∆(r)ψ
[ψ↓(r),Heff ] = (He + hz)ψ↓(r)− hxψ↑(r)−∆(r)ψ
Inserting (1) into (2) and introducing a set ρ of Pauli-like
matrices in particle-hole space, yields the spin-dependent
BdG equations:
H01̂− hzσz
∆(y)ρx − hx1̂
Φn = ǫnΦn,
where Φn ≡ (un↑(y), un↓(y), vn↑(y), vn↓(y))
T and H0 ≡
−∂2y/(2m) + ε⊥ − EF . Here ε⊥ is the transverse kinetic
energy and a factor of eik⊥·r has been suppressed. In
deriving Eq. (3) care has been taken to consistently use
the phase conventions in Eq. (1). To find the quasiparti-
cle amplitudes along a different quantization axis in the
x−z plane, one performs a spin rotation: Φn → Û(α
′)Φn,
where Û(α′) = cos(α′/2)1̂⊗ 1̂− i sin(α′/2)ρz ⊗ σy.
When the magnetizations of the F layers are collinear,
one can take hx = 0. For the general case shown in
Fig. 1 one has in the F1 layer, hx = h0 sin(−α/2) and
hz = h0 cos(−α/2), where h0 is the magnitude of h,
while in F2, hx = h0 sin(α/2), and hz = h0 cos(α/2).
With an appropriate choice of basis, Eqs. (3) are cast
into a matrix eigenvalue system that is solved itera-
tively with the self consistency condition, ∆(y) = g(y)f3
(f3 =
[〈ψ↑(r)ψ↓(r)〉 − 〈ψ↓(r)ψ↑(r)〉]). In the F layers
we have g(y) = 0, while in S, g(y) = g, g being the usual
BCS singlet coupling constant there. Through Eqs. (1),
the self-consistency condition becomes a sum over states
restricted by the factor g to within ωD from the Fermi
surface. Iteration is performed until self-consistency is
reached. The numerical process is the same that was used
in previous work[24, 25], with now the hx term requiring
larger four-component matrices to be diagonalized.
We now define the following time dependent triplet am-
plitude functions in terms of the field operators,
f̃0(r, t) =
[〈ψ↑(r, t)ψ↓(r, 0)〉+ 〈ψ↓(r, t)ψ↑(r, 0)〉] , (4a)
f̃1(r, t) =
[〈ψ↑(r, t)ψ↑(r, 0)〉 − 〈ψ↓(r, t)ψ↓(r, 0)〉] , (4b)
which, as required by the Pauli principle for these s-wave
amplitudes, vanish at t = 0, as we shall verify. Making
use of Eq. (1) and the commutators, one can derive and
formally integrate the Heisenberg equation of the motion
for the operators and obtain:
f̃0(y, t) =
[un↑(y)vn↓(y)− un↓(y)vn↑(y)]ζn(t), (5a)
f̃1(y, t) =−
[un↑(y)vn↑(y) + un↓(y)vn↓(y)]ζn(t),
FIG. 2: (Color online) The real part, f0, of the triplet ampli-
tude f̃0, for a FSF trilayer at 7 different times. We normalize
f0 by the singlet bulk pair amplitude, ∆0/g. The coordinate
y is scaled by the Fermi wavevector, Y ≡ kF y, and time by
the Debye frequency, τ ≡ ωDt. At τ = 0, f0 ≡ 0 as required
by the Pauli principle. The interface is marked by the verti-
cal dashed line, with an F region to the left and the S to the
right. Half of the S region and part of the left F layer are
shown. The inset shows the maximum value of f0 versus τ .
where ζn(t) ≡ cos(ǫnt)− i sin(ǫnt) tanh(ǫn/2T ).
The amplitudes in Eqs. (5) contain all information on
the space and time dependence of induced triplet correla-
tions throughout the FSF structure. The summations in
Eqs. (5) are over the entire self-consistent spectrum, en-
suring that f0 and f1 vanish identically at t = 0 and thus
obey the exclusion principle. Using a non self consistent
∆(y) leads to violations of this condition, particularly
near the interface where proximity effects are most pro-
nounced. Geometrically, the indirect coupling between
magnets is stronger with fairly thin S layers and rela-
tively thick F layers. We thus have chosen dS = (3/2)ξ0
and dF1 = dF2 = ξ0, with the BCS correlation length
ξ0 = 100k
. We consider the low T limit and take
ωD = 0.04EF . The magnetic exchange is parametrized
via I ≡ h0/EF . Results shown are for I = 0.5 (unless
otherwise noted) and the magnetization orientation an-
gle, α, is swept over the range 0 ≤ α ≤ π. No triplet
amplitudes arise in the absence of magnetism (I = 0).
For the time scales considered here, the imaginary
parts of f̃0(y, t) and f̃1(y, t) at t 6= 0 are considerably
smaller than their real parts, and thus we focus on the
latter, which we denote by f0(y, t) and f1(y, t). In Fig. 2,
the spatial dependence of f0 is shown for parallel mag-
netization directions (α = 0) at several times τ ≡ ωDt.
The spatial range shown includes part of the F1 layer
(to the left of the dashed line) and half of the S layer
(to the right). At finite τ , the maximum occurs in the
ferromagnet close to the interface, after which f0 under-
goes damped oscillations with the usual spatial length
FIG. 3: (Color online) Spatial and angular dependence of f1,
at τ = 4 ≈ τc and several α. Normalizations and ranges are
as in Fig. 2. Inset: maxima of f0 and f1 in F1 versus α.
scale ξf ≈ (kF↑ − kF↓)
−1 ≈ k−1F /I. The height of the
main peak first increases with time, but drops off after
a characteristic time, τc ≈ 4, as seen in the inset, which
depicts the maximum value of f0 as a function of τ . As
τ increases beyond τc, the modulating f0 in F develops
more complicated atomic scale interference patterns and
becomes considerably longer ranged. In S, we see imme-
diately that f0 is also larger near the interface. Since
the triplet amplitudes vanish at τ = 0, short time scales
exhibit correspondingly short triplet penetration. The
figure shows, however, that the value of f0 in S is sub-
stantial for τ & τc, extending over length scales on the
order of ξ0 without appreciable decay. In contrast, the
usual singlet correlations were found to monotonically
drop off from their τ = 0 value over τ scales of order
unity.
In the main plot of Fig. 3 we examine the spatial de-
pendence of the real part of the m = ±1 triplet ampli-
tude, f1. Normalizations and spatial ranges are as in
Fig. 2 but now the time is fixed at τ = 4 ≈ τc, and
five equally spaced magnetization orientations are con-
sidered. At α = 0, f1 vanishes identically at all τ , as
expected. For nonzero α, correlations in all triplet chan-
nels are present. As was found for f0, the plot clearly
shows that f1 is largest near the interface, in the F re-
gion. Our geometry and conventions imply (see Fig. 1)
that the magnetization has opposite x-components in the
F1 and F2 regions. The f1 triplet pair amplitude profile
is thus antisymmetric about the origin, in contrast to the
symmetric f0, implying the existence of one node in the
superconductor. Nevertheless, the penetration of the f1
correlations in S can be long ranged. We find that f1 and
f0 oscillate in phase and with the same wavelength, re-
gardless of α. The inset illustrates the maximum attained
values of f0 and f1 in F1 as α varies. It shows that for
FIG. 4: (Color online) The T = 0 condensation energy, ∆E0,
normalized by N(0)∆20 (N(0) is the usual density of states),
vs. the angle α for two values of I . When the two magne-
tizations are antiparallel (α = π) ∆E0 is lowest. The inset
shows the ordinary (singlet) pair potential averaged over the
S region, normalized to the bulk ∆0.
a broad range of α, α . 3π/4, the maximum of f0 varies
relatively little, after which it drops off rapidly to zero at
α = π. This is to be expected as the anti-parallel orienta-
tion corresponds to the case in which the magnetization
is in the x direction, which is perpendicular to the axis
of quantization (see Fig. 1). The rise in the maximum
of f1 is monotonic, cresting at α = π, consistent with
the main plot. At this angle the triplet correlations ex-
tend considerably into the superconductor. At α = π/2
the maxima coincide since the two triplet components
are then identical throughout the whole space because
the magnetization vectors have equal projections on the
x and z axes. At α = π both magnetizations are normal
to the axis of quantization z (see Fig. 1). By making use
of the rotation matrix Û (see below Eq. 3) one can verify
that the m = ±1 components with respect to the axis x
along the magnetizations are zero.
We next consider the condensation energy, ∆E0, cal-
culated by subtracting the zero temperature supercon-
ducting and normal state free energies. The calculation
uses the self consistent spectra and ∆(y), and methods
explained elsewhere [25, 26]. In the main plot of Fig. 4,
we show ∆E0 (normalized at twice its bulk S value) at
two different values of I. The condensation energy results
clearly demonstrate that the antiparallel state (α = π) is
in general the lowest energy ground state. These results
are consistent with previous studies[8] of FSF structures
with parallel and antiparallel magnetizations. The inset
contains the magnitude of the spatially averaged pair po-
tential, normalized by ∆0, at the same values of I. The
inset correlates with the main plot, as it shows that the
singlet superconducting correlations in S increase with
α and are larger at I = 1 than at I = 0.5. The half-
metallic case of I = 1 illustrates that by having a single
spin band populated at the Fermi surface, Andreev reflec-
tion is suppressed, in effect keeping the superconductivity
more contained within S.
Thus, we have shown that in clean FSF trilayers in-
duced odd triplet correlations, with m = 0 and m = ±1
projections of the total spin, exist. We have used a mi-
croscopic self-consistent method to study the time and
angular dependence of these triplet correlations. The
correlations in all 3 triplet channels were found, at times
τ ≡ ωDt & τc, where τc ≈ 4, to be long ranged in both
the F and S regions. Finally, study of the condensation
energy revealed that the ground state energy is always
lowest for antiparallel magnetizations.
This project was supported in part by a grant of HPC
resources from the ARSC at the University of Alaska
Fairbanks (part of the DoD HPCM program) and by the
University of Minnesota Graduate School.
∗ Electronic address: klaus.halterman@navy.mil
† Electronic address: barsic@physics.umn.edu
‡ Electronic address: otvalls@umn.edu; Also at Minnesota
Supercomputer Institute, University of Minnesota, Min-
neapolis, Minnesota 55455
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