PHYSICAL REVIEW B 80, 012505 共2009兲
Magnetic screening properties of superconductor-ferromagnet bilayers
Manuel Houzet1 and Julia S. Meyer2,3
1INAC,
SPSMS, CEA, F-38054 Grenoble, France
of Physics, The Ohio State University, Columbus, Ohio 43210, USA
3Université Joseph Fourier, F-38041 Grenoble, France
共Received 12 March 2009; revised manuscript received 18 June 2009; published 14 July 2009兲
2Department
We study theoretically the magnetic screening properties of thin, diffusive superconductor/ferromagnet
bilayers subject to a perpendicular magnetic field. We find that the effective penetration depth characterizing
the magnetic response oscillates with the thickness of the ferromagnetic layer on the scale of the ferromagnetic
coherence length.
DOI: 10.1103/PhysRevB.80.012505
PACS number共s兲: 74.78.Fk, 74.25.Ha, 74.25.Nf, 74.45.⫹c
While superconductor-normal metal 共SN兲 structures have
been intensively studied for decades, superconductorferromagnet 共SF兲 structures have only become accessible recently because of the much-reduced length scales in ferromagnets. Due to their incompatible spin properties, the
proximity effect between a singlet superconductor and a ferromagnet leads to a variety of unusual phenomena.1,2
Through the exchange field acting on the electron spins in
the ferromagnet, Cooper pairs acquire a finite-momentum ␦k
which leads to an oscillatory behavior of the anomalous
Green’s function.3 Observable consequences are, e.g., a nonmonotonic dependence of the transition temperature4–6 and
the density of states at the Fermi level7,8 on the thickness of
the F layer in SF bilayers, and the possibility of ␲-Josephson
junctions at certain thicknesses of the F layer in
superconductor-ferromagnet-superconductor
共SFS兲
trilayers.9–11
While most experiments on hybrid systems use resistive
measurements, screening of an external magnetic field offers
an alternative tool to study the proximity effect. These measurements probe deeply into the superconducting state because they provide both the magnitude and temperature dependence of the effective superfluid density. Various
configurations for the magnetic response can be considered.
The magnetization of SN hybrids with a magnetic field applied parallel to their interface has been addressed theoretically in Ref. 12, with still debated experimental results in the
case of SN cylinders.13,14 Alternatively, the screening properties of thin films can be probed by measuring the mutual
inductance of two coils positioned on opposite sides of the
sample.15,16 The mutual inductance can be related to the
complex conductivity of the film which in turn can be related
to the screening-length ␭ or the superfluid-density ␳S. To be
precise, in SF bilayers, one measures the superfluid-density
␳S ⬀ ␭−2 integrated over the width of the bilayer or an effective screening-length
−2
␭eff
= d−1
S
冕
dF
dx␭−2共x兲,
共1兲
−dS
where dS and dF are the thicknesses of the superconducting
and ferromagnetic layer, respectively, and x is the coordinate
normal to the interface. First experimental results on Nb/Ni
bilayers have been reported using this setup in Ref. 17,
1098-0121/2009/80共1兲/012505共4兲
where a nonmonotonic dependence of the effective
screening-length on the thickness of the Ni layer has been
observed.
In this Brief Report, we study the screening-length ␭eff of
a SF bilayer subject to a weak perpendicular magnetic field.
The main assumptions are that (i) the exchange-field h in the
ferromagnet is much larger than the superconducting orderparameter ⌬, (ii) the system is in the dirty limit and, thus, the
Usadel equation18 can be used, (iii) the screening-length ␭eff
is much larger than the thickness d = dS + dF of the bilayer,
and (iv) the width dS of the superconducting 共S兲 layer is
smaller than the superconducting coherence-length ␰S
= 冑DS / 共2␲Tc0兲, where DS is the diffusion constant and Tc0 is
the transition temperature of the bare S layer. Our main result
is that the screening length displays an oscillatory behavior
with the thickness of the ferromagnet.
Due to the normalization-condition ĝ2 = 1 of the quasiclassical Usadel Green’s-function ĝ, it can be parametrized by an
angle ␪ such that the normal Green’s-function G = cos ␪
whereas the anomalous Green’s-function F = sin ␪. The system is then described by four coupled equations in terms of
the angles ␪S on the S side of the SF interface, ␪0 on the
ferromagnetic 共F兲 side of the SF interface, and ␪F at the
ferromagnet-vacuum interface.
The Usadel equation of the F layer, −DFⵜ2␪ + 2ih sin ␪
= 0, can be integrated to yield
2冑iy =
冕
␪0
␪F
d␪
1
冑cos ␪F − cos ␪ ,
共2兲
where y = dF / ␰F and ␰F = 冑DF / h is the ferromagnetic coherence length, with the diffusion-constant DF of the F layer.
The boundary condition imposing current conservation at the
SF interface19 can be expressed as
sin共␪S − ␪0兲 = 2冑i␤冑cos ␪F − cos ␪0 ,
共3兲
where ␤ = Rb␴F / ␰F. Here Rb is the interface resistance per
square, and ␴F is the conductivity of the F layer. In the limit
dS Ⰶ ␰S, the Usadel equation of the S layer, −DSⵜ2␪
+ 2␻ sin ␪ = 2⌬ cos ␪, where ␻ is a fermionic Matsubara frequency, can be simplified by an expansion in small spatial
variations of the angle ␪ across the S layer combined with
the boundary condition 共3兲. One obtains
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©2009 The American Physical Society
PHYSICAL REVIEW B 80, 012505 共2009兲
BRIEF REPORTS
␻ sin ␪S + 2冑i␣冑cos ␪F − cos ␪0 = ⌬ cos ␪S ,
共4兲
where ␣ = DS␴F / 共2␴SdS␰F兲 and ␴S is the conductivity of the
S layer. Finally, the self-consistency equation for the orderparameter ⌬ reads
⌬ = ␲T␭BCSR
冋兺 册
␻
sin ␪S ,
共5兲
where ␭BCS is the Bardeen-Cooper-Schrieffer coupling constant.
In diffusive superconductors, the screening length ␭ describes the local 共London兲 current response20 to a vector potential, j = −1 / 共␮0␭2兲A, where ␭−2 = 共2␲T␮0␴S / ប兲兺␻F2 is
proportional to the superfluid density. Here ␮0 is the vacuum
permeability. In SF bilayers, the effective screening length is
related to the angles ␪ through the equation
1
2
␭eff
=
2 ␲ T ␮ 0␴ S
R
ប
冋兺 冉
␻
sin2 ␪S + ␥
冕
y
0
dx sin2 ␪共x兲
冊册
␻ sin ␪S + 2˜␣ sin
冋
␪S共0兲
⌬
+
= R ln tan
2
⌬0
冕
␪S共0兲
0
册
d␪SF⬘共␪S兲sin ␪S ,
−
= ⌬ cos ␪S . 共8兲
16
R关˜␣e−2ỹ兴关3 ln共1 + 冑2兲 + 4 − 5冑2兴.
3
共9兲
This solution describes gapless superconductivity with a finite density-of-states ␯共0兲 at the Fermi level in the superconductor that oscillates with the thickness of the ferromagnet:
␯共0兲 = −␯0R关␦␪S兴, where ␯0 is the density of states in normal
state. The equation for the screening length takes the form
␭20
= 1 − 共a1 − a2共cos 2y + sin 2y兲e−2y兲
,
+
冉
␣
⌬0
冊
2 冑2
␣
␥ 1 + a3ye−2y cos 2y − a4
,
3␲
⌬0
共10兲
where ␭−2
0 = ␲␮0␴S⌬0 / ប is the inverse screening length of the
bare S layer at zero temperature, and the coefficients ai are
positive.22 Both the contributions to the effective screening
length from the S layer and from the F layer 共⬀␥兲 oscillate
on the length scale of the ferromagnetic coherence length.
If, on the other hand dF Ⰶ ␰F, the variation of the angle ␪
is small across the F layer and, thus, ␪S − ␪F Ⰶ 1. Using Eq.
共2兲, one obtains ␪S = ␪F cosh ỹ. Inserting this relation into the
Usadel equation of the S layer results in
冉
冊
4
␻ + 2i␣y + ␣y 3 cos ␪S sin ␪S = ⌬ cos ␪S .
3
共11兲
We find ␦␪S = −2i␣y / ⌬0 and ␦⌬ = − ␲3 ␣y 3. Note that because
␪S共0兲 = ␲2 + i␾, where ␾ real, the density of states possesses a
gap in this regime. Using Eq. 共11兲 to convert the integral
over ␻ to an integral over ␪S, the screening length is given by
␭20
2
␭eff共0兲
共7兲
where ⌬0 is the zero-temperature gap of the bare S layer, a
−2
.
result which can then be used to compute ␭eff
In the following, we provide analytical results for the effective screening length in two limits, namely 共i兲 for a system without barrier ␤ = 0 and (ii) for a system with a strong
barrier ␤ Ⰷ 1. In both cases, solutions are presented for small
parameters ␣. For convenience, we introduce the notation x̃
= 共1 + i兲x for x = ␣ , ␤ , y.
In the absence of a barrier ␤ = 0, the boundary condition
共3兲 imposes that the angles on both sides of the SF interface,
␪0 and ␪S, are the same.
If dF Ⰷ ␰F, the angle ␪F is small, and Eq. 共2兲 yields ␪F
␪
= 8 tan 4S exp关−ỹ兴. Thus, we can simplify Eq. 共4兲 to yield
冣
␦⌬ = − 2␣关冑2 − ln共1 + 冑2兲兴
共6兲
ln
冢
Treating ␣ Ⰶ ⌬0 perturbatively, one finds ␪S共0兲 = ␲2 + ␦␪S,
where ␦␪S = −冑2共˜␣ / ⌬0兲关1 − 16e−2ỹ共3 − 2冑2兲兴, and
2
␭eff
共0兲
where ␥ = ␴F␰F / 共␴SdS兲. Using the Usadel equation of the F
layer, the integral over x can be traded for an integral over ␪,
namely dx = 21冑i 共cos ␪F − cos ␪兲−1/2d␪, ranging from ␪F to ␪0.
In general the set of Eqs. 共2兲–共5兲 can be solved numerically only, but in some limiting cases an analytical solution is
possible. At T = 0, a simplification occurs because the sums
over ␻ can be replaced by integrals, and subsequently the
integration over ␻ can be traded for an integration over ␪S
using the Usadel equation.21 It is then sufficient to solve the
Usadel equation at ␻ = 0 for ␪S共0兲. In particular, using Eqs.
共2兲 and 共3兲, the Usadel equation of the S layer 共4兲 can be
brought into the form 关␻ + F共␪S兲兴sin ␪S = ⌬ cos ␪S, yielding
d␻ = −共⌬ / sin2 ␪S + F⬘共␪S兲兲d␪S. Using this trick, the zerotemperature gap ⌬ is given as
␪S
2
␪S
4
1 − 8e−2ỹ
␪S
sin2
2
tan2
=1−
冉
冊
16 ␣ 3
␲
1+
y + ␥y.
3
3␲2 ⌬0
共12兲
−2
as long as dF
Equation 共12兲 predicts an increase in ␭eff
2
⬍ ␰F / ␰S before it starts to decrease. The regime dF ⬃ ␰F connecting the results Eqs. 共10兲 and 共12兲 is treated numerically
共see below兲.
In the opposite limit of a strong barrier, ␤ Ⰷ 1, both ␪0 and
␪F are small, if the F layer is not too thin, y Ⰷ ␤−1. Equation
共2兲 then yields ␪0 = ␪F cosh ỹ, and, using the boundary condition, the Usadel equation of the S layer can be rewritten as
冉
␻+
冊
␣
␣
cos ␪S sin ␪S = ⌬ cos ␪S . 共13兲
−
␤ 冑2i␤2 tanh ỹ
Equation 共13兲 yields ␦␪S = −␣ / 共␤⌬0兲, and thus no gap in the
density-of-states, ␯共0兲 = ␯0␣ / 共␤⌬0兲, while
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PHYSICAL REVIEW B 80, 012505 共2009兲
BRIEF REPORTS
␦⌬ = −
␣ ␲␣
+
R关共1 − i兲coth ỹ兴.
␤ 8␤2
The screening length is given as
冉 冊 冉
Λeff 2dF 冊
2 ␣
␣
␲ 2
=1− 1+
+
+
R关共1
2
2
␲ ␤⌬0
8 3␲ ␤ ⌬0
␭eff共0兲
␭20
冋
Λ0 2
共14兲
1.0
册
4iy + 共1 + i兲sinh 2ỹ
␥
R
.
− i兲coth ỹ兴 −
16␤2
sinh2 ỹ
0.8
0.155
0.6
0.145
共15兲
0.4
Again the effective screening length oscillates on the scale of
␰F. However, these oscillations are suppressed due to the
large barrier. For y Ⰷ 1, the oscillatory function in the contribution of the S layer has the same form as the one in Eq. 共10兲
whereas the oscillating part of the contribution from the F
layer is proportional to ye−2y sin共2y兲.
In the case of a very thin F layer, y Ⰶ ␤−1, the variation of
the angle ␪ is small across the F layer 共see above兲. The
boundary condition at the SF interface then simplifies, and
the Usadel equation of the S layer yields
0.2
共␻ + 2i␣y + 4␣␤y 2 cos ␪S兲sin ␪S = ⌬ cos ␪S .
共16兲
We find ␦␪S = −2i␣y / ⌬0 and ␦⌬ = −␲␣␤y 2. As for the case
without a barrier, the density of states in the thin-film regime
is gapped. Using Eq. 共16兲 to convert the integral over ␻ to an
integral over ␪S, the screening length is given by
␭20
2
␭eff
共0兲
冉
=1−␲ 1+
冊
16 ␣␤ 2
y + ␥y.
3␲2 ⌬0
␪0
=
2
冑
1
1 + cot2
冉
␪F 2
␪F
cn ỹ,cos2
2
2
冊
0.140
2.0
3.0
3.5
4.0
0.5
1.0
1.5
2.0
2.5
3.0
ΞF
FIG. 1. 共Color online兲 Oscillation of the inverse screening2
length 1 / ␭eff
as a function of dF at temperature T = 0.1Tc0. Here we
use the parameters ␣ = 1.2, ␤ = 1, and ␥ = 0.6. The inset magnifies the
weak maximum at dF ⬇ 2.5␰F.
length mirror the oscillations of the critical temperature as
well as the slope of the screening length close to Tc.
In the vicinity of the critical-temperature Tc an analytic
solution is possible for all parameter values. For the SF bilayer, the critical temperature is given by the solution of the
equation1
ln
冉冊 冋冉
1
1
1
Tc
=⌿
−R ⌿ +
2
2 2 ␲ T c␶ s
Tc0
冊册
,
共19兲
where the 共complex兲 relaxation-time ␶s reads
Λ0 2
Λeff 2 T
0.20
共18兲
0.10
where cn is the Jacobi elliptic function. Using Eq. 共18兲, the
boundary condition 共3兲 yields ␪S as a function of ␪F. Inserting this solution into the other equations, the set of coupled
Eqs. 共4兲 and 共5兲 can then be solved numerically. The thickness dependence of the low-temperature screening length is
displayed in Fig. 1. The minimum at dF ⬃ ␰F is clearly visible
whereas further oscillations at larger dF are very small.
Furthermore, the numeric solution allows one to describe
the temperature dependence of the screening length. Figure 2
shows the finite temperature curves for different values of
dF / ␰F. The oscillations of the zero-temperature screening
2.5
dF
0.0
0.0
0.15
,
0.150
共17兲
The inverse screening length increases in the very narrow
regime dF ⬍ ␰F3 / 共␤␰2S兲.
Thus, we find oscillations of the screening length both in
the absence of a barrier and in the presence of a strong barrier. The amplitude of oscillations in the latter case is suppressed, however. In both cases, analytic results can be found
for small-thicknesses y ⬍ y ⴱ and large-thicknesses y ⬎ y ⴱ,
where y ⴱ ⬃ min兵1 , ␤−1其 denotes the position of the first strong
minimum. The vicinity of this minimum is not accessible to
analytic solution.
To find a solution in this regime, we note that Eq. 共2兲
yields a general relation between ␪0 and ␪F, namely,
sin
0.160
0.05
0.00
0.00
dF 0.5 ΞF
dF 0.9 ΞF
dF 1.3 ΞF
dF 3. ΞF
0.05
0.10
0.15
0.20
0.25
0.30
0.35
T
Tc0
−2
FIG. 2. 共Color online兲 Temperature dependence of ␭eff
. The parameters used are the same as in Fig. 1, and four different thicknesses are shown: dF / ␰F = 0.5, 0.9, 1.3, and 3. The corresponding
critical temperatures are Tc共0.5兲 = 0.36Tc0, Tc共0.9兲 = 0.17Tc0 共close
to the minimum兲, Tc共1.3兲 = 0.23Tc0, and Tc共0.5兲 = 0.31Tc0 共close to
the asymptotic value for dF / ␰F Ⰷ 1兲.
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PHYSICAL REVIEW B 80, 012505 共2009兲
BRIEF REPORTS
␶s−1 =
冑2i␣ tanh共冑2iy兲
.
1 + 冑2i␤ tanh共冑2iy兲
Equations 共19兲 and 共20兲 are obtained by linearizing 共2兲–共5兲 in
small ␪ close to the transition. For ␶s−1 small, Tc = Tc0
− ␲4 R关␶s−1兴. For 兩␶s−1兩 = ⌬0 / 2, the transition temperature vanishes according to Eq. 共19兲. Note, however, that for large ␶s−1
the transition typically becomes first order.23
Expansion of Eqs. 共2兲–共5兲 up to cubic order then yields
the temperature dependence of the screening-length ␭eff close
to Tc. Namely,
1
2
␭eff
共T兲
=
␮ 0␴ S 2
⌬ 共T兲R
ប␲Tc
⫻⌿共1兲
冉
冋冉
1+
1
1
+
2 2 ␲ T c␶ s
␥
2ỹ + sinh共2ỹ兲
4冑2i 共cosh ỹ + ˜␤ sinh ỹ兲2
冊册
.
冊
共21兲
−2
displays an
We see that the contribution of the F layer to ␭eff
oscillatory dependence on its thickness. Furthermore, both Tc
and ⌬共T兲 oscillate. In particular,
⌬2共T兲 4␲Tc共1 − R关共2␲Tc␶s兲−1⌿共1兲共z兲兴兲
=
,
Tc − T − R关⌿共2兲共z兲 − f共␶s, ␤,y兲⌿共3兲共z兲兴
共22兲
at T ⬍ Tc, where z = 21 + 共2␲Tc␶s兲−1 and
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2 F.
f共␶s, ␤,y兲 =
共20兲
1
1
共2␲Tc␶s兲−1
48
共1 + ˜␤ tanh ỹ兲3
冉
冊
2ỹ + sinh共2ỹ兲
⫻ 4共1 + 2˜␤ tanh ỹ兲2 −
.
sinh ỹ cosh3 ỹ
共23兲
Note that the simple-relation ␭−2 ⬀ ⌬2 does not hold in the
−2
close to Tc has its
presence of the F layer. The slope of ␭eff
own dependence on the thickness of the F layer and the
relaxation-rate ␶s−1.
In conclusion, we have shown that the screening length of
SF bilayers displays an oscillatory dependence on the thickness of the F layer. Analytic solutions have been found in
various regimes and a general solution has been determined
numerically. The obtained nonmonotonic dependence of the
screening length has been observed experimentally.17 Our
method can be easily extended to more complicated situations such as multilayers where unusual features of the proximity effect have been predicted.1,2
We would like to acknowledge A. Buzdin and T. Lemberger for many discussions. J.S.M. thanks the CEA/INAC/
SPSMS for hospitality.
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K. D. Usadel, Phys. Rev. Lett. 25, 507 共1970兲.
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See, e.g., P. G. De Gennes, Superconductivity Of Metals And
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22 The numerical values of the coefficients are a = 1.77, a = 3.82,
1
2
a3 = 6.44, and a4 = 3.19.
23 S. Tollis, Phys. Rev. B 69, 104532 共2004兲.
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