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Competitive 0 and π states in S/F/S trilayers: Multimode approach

TL;DR: In this article, the authors investigated the behavior of the critical temperature in superconductor/ferromagnet/superconductor (S/F/S) trilayers in the dirty limit as a function of the ferromagnetic layer thickness and the S/F interface transparency.
Abstract: We investigate the behavior of the critical temperature ${T}_{c}$ in superconductor/ferromagnet/superconductor (S/F/S) trilayers in the dirty limit as a function of the ferromagnetic layer thickness ${d}_{f}$ and the S/F interface transparency. We perform ${T}_{c}$ calculations using the general self-consistent multimode approach based on the Usadel equations in Matsubara Green's functions technique, and compare the results with the single-mode approximation, widely used in literature. Both methods produce similar results for sufficiently low interface transparency. For transparent interfaces, we obtain a qualitatively different ${T}_{c}({d}_{f})$ behavior. Using the multimode approach, we observe multiple 0-$\ensuremath{\pi}$ transitions in critical temperature, which cannot be resolved by the single-mode approximation. We also calculate the critical S layer thickness at given ${d}_{f}$ when an S/F/S trilayer still has a nonzero critical temperature. Finally, we establish the limits of applicability of the single-mode approximation.

Summary (5 min read)

Introduction

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  • This results in the dispersion of the impact locations characterizing the ammunition precision error.

II. Maneuver concept

  • The choice of deploying a spoiler during the projectile flight has been particularly motivated by the work of Patel et al. [13] and Simon [14].
  • In their work, Patel et al. demonstrated the effectiveness of micro-spoiler in the capability of modifying the behavior of a projectile for an angle of attack ranging from 0 to 18◦, which is the effective range of the present uncontrolled projectile.
  • A unique deployment instant is chosen for the spoiler.
  • During the projectile flight and until the spoiler deployment, information about the trajectory perturbation would be gathered using an external radar system.
  • Since the spoiler position is time-invariant, the projectile reaches an equilibrium after a short period of time.

III. Surrogate models

  • As estimated from the work of Dietrich [1], the accurate computation of aerodynamic coefficients along the trajectory is incompatible with the optimization framework presented in this study.
  • A short description of the models used in this study is presented here.
  • According to Jones et al. [19], a good initial sampling size counts 10 samples in every dimension which is usually not reachable on high-dimensional problems.
  • From a practical point of view, only the spreading of 3p points will be considered to establish a kriging response surface in the following.
  • The optimization of the points distribution in the design space is achieved considering the Morris and Mitchell criterion [20] which maximizes the minimal distance between the sampling sites.

1. Exploitation and exploration of the Kriging response surface

  • To combine the local and global search of an optimum, Jones et al. [19] proposed the computation of the Expected Improvement (EI) which is the mathematical definition of the probability that an unsampled point performs better than the already best known sampled point of the function.
  • The EI proposes a trade-off between the exploration and the exploitation of the surrogate model.
  • Indeed, values associated to the EI score are high in the regions of the design space where the function values are significant but are also high in the unknown regions of the model where the error associated to the model s(x) is large .

2. Constraints handling from the Kriging model

  • The addition of constraint functions to the initial problem definition leads to the reduction of parameters combinations susceptible to minimize the objective function.
  • They are modelled following the definition of the Probability of Feasibility (PF) introduced by Schonlau [22] and applied successfully to a surrogate-based optimization by Parr et al. [23].
  • In the same way as for the EI computation clarified in the preceding paragraph, the PF takes advantages of the kriging modeling of a function to estimate if constraints are likely to be respected by comparing the estimation of a function to the constraint limit, see eq. (2). P[F(x)] = 1 s(x) √ 2π ∫ ∞ 0 e−[(G(x)−gmin)−ĝ(x)].
  • G(x) is a random variable and s(x) is constraint standard deviation of the Kriging model.

C. Artificial Neural Networks

  • A multi-layer Perceptron is specifically used in this study to model the evolution of the spoiler aerodynamic coefficients (∆CA, ∆CN and ∆Cm) as a function of the aerodynamic conditions (Mach number M∞ and angle of attack α).
  • The training database is composed of more than 1300 CFD evaluations of the spoiler.
  • To avoid an over-fitting of the MLP, resulting in the degradation of the aerodynamic coefficients estimations, the training iterations stop when the coefficient of determination does no improve anymore on a subset of the training database (10% of the evaluations in the present case).
  • The MLP models of each coefficient were implemented using the Python module Scikit-learn [24].
  • It has to be noticed that the best estimations using MLP are obtained when the inputs and outputs are scaled in the same interval of range [−1; 1].

A. Simulations overview

  • A MLP is used to model the variations in the aerodynamic coefficients of the spoiler configurations.
  • The database is composed of RANS-based CFD evaluations.
  • The contribution of the spoiler is isolated by simulating the flow around both controlled and uncontrolled projectile.
  • The uncontrolled projectile has been studied prior to the present work, for instance by Simon et al. [25] and Zeidler et al. [26].
  • The aerodynamic modifications induced by the spoiler deployment are described in the following.

1. Meshing strategy

  • The 155 mm caliber (cal.) projectile is composed of a 5.6 cal-long body constituted of a 3 cal-long truncated nose.
  • The computational domain external boundary extends at a distance of 50 caliber around the projectile.
  • The spoiler setup on the projectile body is ensured via the chimera method which consists in merging two grids into a unique one as shown in fig.
  • A cell-size ratio close to 1 at the interfaces between the background and the patch grids has been imposed.

2. elsA code

  • The CFD solver used in this study to compute the aerodynamic coefficients of the different projectile configurations is the ONERA’s solver elsA based on a cell-centered finite volume approach for solving the Navier-Stokes equations on structured multiblock grids [27].
  • The computations constituting the initial aerodynamic coefficients database were realized using the Spalart-Allmaras (SA) one-equation turbulence model.
  • Consistently the same turbulence model is used in the computations enhancing this database.
  • It is known that SA model performs poorly in the presence of massively detached flows.
  • A comparison with other turbulence models (k −ω and EARSM not presented here) shows maximum discrepancies of the order of 10% for drag coefficient evaluations which is considered acceptable in the present case, since the authors are more interested in relative performance.

B. Flow around the controlled projectile

  • The flight conditions encountered by the uncontrolled projectile are ranging from supersonic Mach number (at the muzzle exit) to high subsonic, the angle of attack remaining below 6◦.
  • The deployment of the spoiler can occur at any point of the trajectory thus the uncontrolled and controlled projectile aerodynamic models are constructed in the same range of flight parameters.
  • A generic spoiler is located in the middle of the projectile boat-tail to highlight the flow modifications induced by its presence.
  • The detached areas are showed for both projectile configurations at M = 0.9 and α = 0.
  • Moreover, due to the curvature of the streamlines an additional shock appears in the wake which can be designated as recompression shock, for these transonic freestream conditions, which is not present in the case of the uncontrolled projectile.

V. Spoiler optimization

  • The optimization process developed within this study is illustrated by the flowchart presented in fig.
  • Within the 88% of the uncontrolled projectile course, information is gathered on the external perturbations occurring during the flight.
  • A sequential enrichment of the kriging response surfaces based on the selection of optimal design using a genetic algorithm is implemented to converge toward the optimum set of parameters achieving the required course correction, paragraph V.D.
  • The trajectories optimization loop, referred to as inner loop in the following, defines which configuration of the controlled projectile is considered as optimum.
  • This enrichment is the second adaptive sampling.

A. Aerodynamic coefficients estimations

  • The advantage of using a MLP model against a kriging model here has been discussed by the authors in [28].
  • Additionally, the distribution of the CFD evaluations over the parameters space does not follow an optimal design of experiment which eventually leads to an over-fitting of the kriging model.
  • The validation of the MLP response surface training is illustrated in fig.

B. Trajectories initial sampling

  • An initial database of controlled projectiles is spread over the p = 4 dimensional design domain (3 geometrical parameters Xs/D, Hs/D, θs and the roll position φs) using the optimized LHS methodology.
  • The number of samples is directly determined from the rule of thumb stated earlier as Ninit = 34 = 81 configurations.
  • The trajectories are computed using the 6-DOF flight mechanics code Balco [29] and presented in fig.
  • A wide variety of corrections (around the uncontrolled projectile mean impact point at [1;1]) is proposed in this purely exploratory phase of the process.
  • The lateral deviations and range modifications produced by the different spoilers are used to initialize the kriging response surfaces modeling the objective and constraint functions that are 15 described in the next paragraph.

C. Definition of the optimization problem

  • Considering the distribution of the uncontrolled projectile impact points introduced in section II, the objective function is defined as the minimization of the distance between the controlled projectile impact point and the closest point belonging to the 3σ ellipse, eq. (4).
  • It is then assumed that, if the spoiler geometry fulfills this specific case, it will allow producing every course correction, as illustrated in fig.
  • An increase of the projectile range is imposed through eq. (5), illustrated in fig.
  • This constraint translates inot eq. (7) in which the distance between the controlled projectile impact point and the uncontrolled projectile mean impact point is compared to the distance between the uncontrolled projectile mean impact point and the closest point belonging to the 3σ ellipse.

D. Bi-objective optimization

  • An adaptive sampling is developed inside the inner loop in fig.
  • 6, based on the kriging estimations of the objective and constraint functions.
  • A bi-objective optimization is achieved with the aim of maximizing EI and PF through an evolutionary process simulated with the NSGA-II algorithm developed by Deb et al. [30].
  • 10, can be selected to enrich the database.
  • The maximization of EI and PF separately leads to the selection of better candidates than in the case of a single-objective optimization.

E. Course correction results

  • The adaptive sampling described in the previous section is applied to the projectile course correction problem.
  • The impact points are colored by the spoiler frontal surface associated to the trajectory modification.
  • This means that in the 4-dimensional design space, the configurations producing the required 2-D course correction are located on an hypersurface depending on Hs/D × θs .
  • The choice of any spoiler configuration belonging to this particular hypersurface determines the magnitude of the force induced by the spoiler.
  • The longitudinal position of the spoiler Xs/D influences the magnitude of the pitching moment created by the spoiler which is directly linked to the lateral deviation produced by a spinning projectile while the roll position φs determines toward which direction the force induced by the spoiler is applied.

VI. Flight Mechanics study of the optimum configuration

  • The optimum configuration determined during the process enables the course correction of the projectile.
  • Moreover, the choice of a specific spoiler instant of deployment (12% of the trajectory before the impact in the studied case) in combination with a specific frontal surface value leads to the identification of designs able to produce the required 2-D course correction.
  • In the present course correction problem, this value is lower than the maximum allowed by the parameters range, as indicated by the parameters combination in table 14a.
  • The following iterations of the outer loop are used to increase the accuracy of the MLP database until a convergence state is reached.
  • 13 where the evolution of the optimum projectile coefficients are presented.

VII. Conclusions

  • A surrogate-based algorithm was developed with the aim of optimizing a spoiler geometry deployed during the flight of a spin-stabilized projectile.
  • The selection of an optimum configuration is achieved through several adaptive sampling of the different response surfaces.
  • Kriging response surfaces are enriched in the inner loop of the optimization problem to model the objective and constraint functions while the MLP database is enhanced with additional CFD evaluations of the optimum spoiler aerodynamic coefficients.
  • The controlled projectile is able to reach an increased range and a reduced lateral deviation compared to the uncontrolled projectile trajectory.
  • In the following of this study, multi-fidelity surrogate models could be used to integrate unsteady computations of the optimum configuration in the databases.

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PHYSICAL REVIEW B 100, 104502 (2019)
Competitive 0 and π states in S/F/S trilayers: Multimode approach
T. Karabassov,
1
V. S. Stolyarov,
2,3
A. A. Golubov,
2,4
V. M. Silkin,
5,6,7
V. M. Bayazitov,
8
B. G. Lvov,
1
and A. S. Vasenko
1,9,*
1
National Research University Higher School of Economics, 101000 Moscow, Russia
2
Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Russia
3
Dukhov Research Institute of Automatics (VNIIA), 127055 Moscow, Russia
4
Faculty of Science and Technology and MESA+ Institute for Nanotechnology, University of Twente, 7500 AE Enschede, The Netherlands
5
Donostia International Physics Center (DIPC), Paseo Manuel de Lardizabal 4, San Sebastián/Donostia, 20018 Basque Country, Spain
6
Departamento de Física de Materiales, Facultad de Ciencias Químicas, UPV/EHU, 20080 San Sebastián, Basque Country, Spain
7
IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain
8
N.S. Kurnakov Institute of General and Inorganic Chemistry, Russian Academy of Sciences, 117901 Moscow, Russia
9
I.E. Tamm Department of Theoretical Physics, P.N. Lebedev Physical Institute, Russian Academy of Sciences, 119991 Moscow, Russia
(Received 25 May 2019; revised manuscript received 25 July 2019; published 3 September 2019)
We investigate the behavior of the critical temperature T
c
in superconductor/ferromagnet/superconductor
(S/F/S) trilayers in the dirty limit as a function of the ferromagnetic layer thickness d
f
and the S/F interface
transparency. We perform T
c
calculations using the general self-consistent multimode approach based on
the Usadel equations in Matsubara Green’s functions technique, and compare the results with the single-
mode approximation, widely used in literature. Both methods produce similar results for sufficiently low
interface transparency. For transparent interfaces, we obtain a qualitatively different T
c
(d
f
) behavior. Using the
multimode approach, we observe multiple 0-π transitions in critical temperature, which cannot be resolved
by the single-mode approximation. We also calculate the critical S layer thickness at given d
f
when an S/F/S
trilayer still has a nonzero critical temperature. Finally, we establish the limits of applicability of the single-mode
approximation.
DOI: 10.1103/PhysRevB.100.104502
I. INTRODUCTION
Nowadays, the rates of development in such areas as
spintronics, superconducting logic, and memory circuits in-
crease significantly. In particular, much attention is attracted
to superconductor/ferromagnet (S/F) structures [13]. It is
known that S/F structures are important for rapid single flux
quantum circuits [4], applications for superconducting spin-
tronics [5], and, in particular, memory elements [612] and
spin-valves [1319], magnetoelectronics [2022], qubits [23],
artificial neural networks [24], microrefrigerators [25,26], etc.
Rich physics of S/F systems is based on the proximity
effect in S/F bilayers [13,2730]. It turns out that when a
superconductor and a ferromagnet form a hybrid structure,
superconducting correlations leak into a ferromagnetic metal
over the distance ξ
h
=
D
f
/h, where D
f
is the diffusion co-
efficient and h is the exchange field in the ferromagnetic ma-
terial [1]. As a consequence, it leads to a damped oscillatory
behavior of superconducting correlations in the ferromagnetic
layer, with characteristic lengths of decay and oscillations
given by ξ
h
.
If a ferromagnetic layer serves as a weak link in a
Josephson-type superconductor/ferromagnet/superconductor
(S/F/S) structure, there is a possibility of a π phase state
realization. For small F layer thickness d
f
ξ
h
, the pair wave
function in the F layer is almost constant and the signs of
the superconducting pair potentials in the S layers remain the
*
avasenko@hse.ru
same. In this case, the phase difference between the S layers
is zero (0 phase state). Increasing the F layer thickness up to
d
f
ξ
h
, the pair wave function may cross zero in the center
of the F layer with the π phase shift and different signs of
the superconducting pair potential in the S layers (π phase
state). Further increasing of d
f
may cause subsequent 0-π
transitions due to damped oscillatory behavior of the pair
potential in the F layer. The existence of the π states leads
to a number of striking phenomena. For example, the criti-
cal current in S/F/S Josephson junctions exhibits a damped
oscillatory behavior with increasing the F layer thickness
[3149]. The π state is then characterized by a negative sign
of the critical current. Similarly, 0 to π transitions can also be
observed as density of states oscillations [5053] and critical
temperature T
c
oscillations [5469]inS/F/S trilayers with
increasing d
f
.Zerotoπ transitions were also obtained in
F
/S/F/S structures with a magnetization misalignment in F
and F
layers [70,71]. We also mention that spin-orbit cou-
pling effects can dramatically change spectroscopic signatures
of Josephson S/F/S junctions. For example, it was shown
that in the presence of intrinsic spin-orbit coupling a giant
proximity effect due to spin-triplet Cooper pairs can develop
in diffusive S/F/S junctions in the π phase state [72]. S/F/S
Josephson π junctions have been proposed as elements of
novel superconducting nanoelectronics in many applications
like the aforementioned memory elements and qubits.
The critical temperature also has a nontrivial behavior in
S/F bilayers [7378]. In this case, the transition to the π phase
is impossible, but the commensurability effect between the pe-
riod of the superconducting correlations oscillation (ξ
h
) and
2469-9950/2019/100(10)/104502(9) 104502-1 ©2019 American Physical Society

T. KARABASSOV et al. PHYSICAL REVIEW B 100, 104502 (2019)
FIG. 1. Geometry of the considered system. A single symmet-
ric S/F/S trilayer is considered in our model. It can also be an
elementary unit of the infinite periodic S/F multilayer system. The
transparency parameter γ
b
is proportional to the resistance across the
S/F interface.
the F layer thickness leads to a nonmonotonic T
c
(d
f
) depen-
dence. For the transparent S/F interface, T
c
decays monoton-
ically, vanishing at finite d
f
. With decreasing interface trans-
parency, the critical temperature demonstrates a reentrant be-
havior: It vanishes in a certain interval of d
f
and is finite oth-
erwise. At sufficiently low interface transparency, T
c
decays
nonmonotonically to a finite value exhibiting a minimum at a
particular d
f
[73]. Nonmonotonic T
c
(d
f
) dependencies were
also observed in F/F
/S and F/S/F
spin valves with a mag-
netization misalignment in F and F
layers [7981]. Depairing
currents in S/F proximity structures were studied in Ref. [82].
As already mentioned, in contrast to bilayers, the S/F/S
trilayers may exhibit more complex behavior, with the compe-
tition of 0 and π phase states. The purpose of this paper is to
provide a quantitative model of the critical temperature T
c
(d
f
)
behavior in a symmetric S/F/S trilayer structure as a function
of the ferromagnetic layer thickness d
f
and the S/F interface
transparency. Such a structure may be also considered as a
single unit of the infinite periodic S/F multilayer system as
shown in Fig. 1. The total S/F multilayer can then be in the
0orinπ state, depending on the state of a single S/F/S
unit. It should be noted that the S/F multilayer system may
host states, corresponding to different, more complex con-
figurations of the distribution of superconducting correlations
[6264]. We did not consider these more exotic states in our
paper, which could be a subject of future work.
Previously the T
c
(d
f
) behavior of the S/F/S trilayers
was studied only in the so-called single-mode approximation
(SMA) [5469]. In this paper, we calculate the T
c
(d
f
) de-
pendence, using the multimode approach (MMA), considered
to be an exact method for solving this problem. We also
compare the results of the multimode approach with the SMA,
setting the limits for the latter approximate method (see the
Appendix). In our paper, we do not consider nonequilib-
rium effects [83], and use the Matsubara Green’s functions
technique, which has been developed to describe many-body
systems in equilibrium at finite temperature [84].
The paper is organized as follows. In Sec. II,weformulate
the theoretical model and basic equations. In Secs. III and IV,
SMAs and MMAs are formulated, correspondingly. The re-
sults are presented and discussed in Sec. V and concluded in
Sec. VI.
II. MODEL
We consider the S/F/S trilayer depicted in Fig. 2 con-
sisting of a ferromagnetic layer of thickness d
f
and two
superconducting layers of thickness d
s
along the x direction.
The structure is symmetric and its center is placed at x = 0.
To calculate the critical temperature T
c
(d
f
) of this struc-
ture, we assume the diffusive limit and use the framework
of the linearized Usadel equations for the S and F layers
in Matsubara representation [84,85]. Near T
c
, the normal
Green’s function is G = sgn ω
n
, and the Usadel equation for
the anomalous Green’s function F takes the following form.
In the S layers (d
f
/2 < |x| < d
s
+ d
f
/2) it reads
ξ
2
s
πT
cs
d
2
F
s
dx
2
−|ω
n
|F
s
+ = 0. (1)
In the F layer (d
f
/2 < x < d
f
/2), the Usadel equation can
be written as
ξ
2
f
πT
cs
d
2
F
f
dx
2
(|ω
n
|+ih sgn ω
n
)F
f
= 0. (2)
Finally, the self-consistency equation reads [84]
ln
T
cs
T
= π T
ω
n
|ω
n
|
F
s
. (3)
In Eqs. (1)–(3), ξ
s
=
D
s
/2πT
cs
, ξ
f
=
D
f
/2πT
cs
, D
s
is the
diffusion coefficient in the S layers, ω
n
= 2π T (n +
1
2
), where
n = 0, ±1, ±2,...are the Matsubara frequencies, h is the ex-
change field in the ferromagnet, T
cs
is the critical temperature,
and is the pairing potential in the S layers, and F
s( f )
denotes
FIG. 2. Schematic behavior of the real part of the pair wave function. For a thin enough ferromagnetic layer, the system is in the 0 phase
state (solid red line), while for larger d
f
the system can be in the π state (dashed black line). Only one of these states is realized, depending on
the F layer thickness.
104502-2

COMPETITIVE 0 AND π STATES IN S/F/S PHYSICAL REVIEW B 100, 104502 (2019)
the anomalous Green’s function in the S(F) region (we assume
¯h = k
B
= 1). We note that ξ
h
= ξ
f
2πT
cs
/h.
Equations (1)–(3) should be complemented by the
Kupriyanov-Lukichev boundary conditions at the S/F bound-
aries (x d
f
/2) [86]:
ξ
s
dF
s
(±d
f
/2)
dx
= γξ
f
dF
f
(±d
f
/2)
dx
, (4a)
ξ
f
γ
b
dF
f
(±d
f
/2)
dx
F
s
(±d
f
/2) F
f
(±d
f
/2). (4b)
In Eqs. (4), the dimensionless parameter γ = ξ
s
σ
n
f
σ
s
determines the strength of suppression of superconductivity
in the S layers near the S/F interfaces compared to the
bulk (inverse proximity effect). No suppression occurs for
γ = 0, while strong suppression takes place for γ 1. Here
σ
s(n)
is the normal-state conductivity of the S(F) layer. The
dimensionless parameter
γ
b
= R
b
σ
n
f
(5)
describes the effect of the interface barrier [86,87]. Here R
b
is
the resistance of the S/F boundary (we suppose the symmetric
structure with same resistance R
b
for x d
f
/2).
According to the definition, Eq. (5), γ
b
= 0 for a fully
transparent interface. It follows from Eq. (4b) that the anoma-
lous Green’s functions (pair wave functions) F
s( f )
are con-
tinuous at the interface in this case. In the regime of low-
barrier transparency (tunnel junction), γ
b
1 and F
s( f )
have
discontinuities at the interface (see Fig. 2, where finite dis-
continuity is shown). Lambert et al. have shown that the
condition Eq. (4b) is exact in two limits of high and low barrier
transparency, γ
b
1 and γ
b
1, correspondingly. They have
also found corrections at the intermediate values of γ
b
1
which, however, do not exceed 10% [88].
At the borders of the S layers with a vacuum, we naturally
have
dF
s
(±d
s
± d
f
/2)
dx
= 0. (6)
The solution of the Usadel equation in the F layer depends
on the phase state of the structure. In the 0 phase state, the
anomalous Green’s function is symmetric relative to x = 0
(see Fig. 2, left panel) [73],
F
0
f
= C(ω
n
) cosh(k
f
x), (7)
while in the π phase state the anomalous Green’s function is
antisymmetric (see Fig. 2, right panel),
F
π
f
= C
(ω
n
)sinh(k
f
x), (8)
where
k
f
=
1
ξ
f
|ω
n
|+ih sgn ω
n
πT
cs
. (9)
In Eqs. (7) and (8), the C(ω
n
) and C
(ω
n
) are proportionality
coefficients to be found from the boundary conditions.
To solve the boundary value problem Eqs. (1)–(6), we use
the method proposed in Ref. [73]. At the right S/F boundary
(x = d
f
/2) from Eqs. (4) we obtain
ξ
s
dF
s
(d
f
/2)
dx
=
γ
γ
b
+ B
f
(ω
n
)
F
s
(d
f
/2), (10)
where B
f
(ω
n
) can acquire one of two different values, depend-
ing on phase state. In the 0 phase state [73],
B
0
f
= [k
f
ξ
f
tanh(k
f
d
f
/2)]
1
, (11)
while in π phase state from Eq. (8) we obtain
B
π
f
= [k
f
ξ
f
coth(k
f
d
f
/2)]
1
. (12)
A similar boundary condition can be written at x =−d
f
/2.
The boundary condition Eq. (10) is complex. To rewrite it
in a real form, we use the following relation:
F
±
= F (ω
n
) ± F (ω
n
). (13)
According to the Usadel Eqs. (1)–(3), there is a symmetry
relation F (ω
n
) = F
(ω
n
), which implies that F
+
is a real
while F
is a purely imaginary function.
Thus we can consider only positive Matsubara frequencies
and express the self-consistency Eq. (3) via the symmetric
function F
+
s
:
ln
T
cs
T
= π T
ω
n
>0
2
ω
n
F
+
s
. (14)
The problem of determining T
c
can be then formulated in a
closed form with respect to F
+
s
. Using the boundary condition
Eq. (10) we arrive at the effective boundary conditions for F
+
s
at the boundaries of the right S layer,
ξ
s
dF
+
s
(d
f
/2)
dx
= W
0
(ω
n
)F
+
s
(d
f
/2), (15a)
dF
+
s
(d
s
+ d
f
/2)
dx
= 0, (15b)
where we used the notations
W
0
(ω
n
) = γ
A
s
γ
b
+ Re B
0
f
+ γ
A
s
γ
b
+ B
0
f
2
+ γ
γ
b
+ Re B
0
f
,
A
s
= k
s
ξ
s
tanh(k
s
d
s
), k
s
=
1
ξ
s
ω
n
πT
cs
. (16)
Similar boundary conditions can be written at the boundaries
of the left S layer.
The self-consistency Eq. (14) and boundary conditions
Eqs. (15), together with the Usadel equation for F
+
s
,
ξ
2
s
πT
cs
d
2
F
+
s
dx
2
ω
n
F
+
s
+ 2 = 0, (17)
will be used for finding the critical temperature of the S/F/S
structure both in 0 and π phase states. In general, this problem
should be solved numerically.
III. SINGLE-MODE APPROXIMATION
In this section, we present the SMA method. The solution
of the problem Eqs. (14)–(17) can be searched in the form of
the following anzatz:
F
+
s
(x
n
) = f (ω
n
) cos
x d
s
d
f
/2
ξ
s
, (18a)
(x) = δ cos
x d
s
d
f
/2
ξ
s
, (18b)
104502-3

T. KARABASSOV et al. PHYSICAL REVIEW B 100, 104502 (2019)
where δ and do not depend on ω
n
. The above solution
automatically satisfies boundary condition Eq. (15b)atx =
d
s
+ d
f
/2. Substituting expression Eqs. (18) into the Eq. (17)
we obtain
f (ω
n
) =
2δ
ω
n
+
2
πT
cs
. (19)
To determine the critical temperature T
c
, we have to sub-
stitute Eqs. (18)–(19) into the self-consistency Eq. (14)at
T = T
c
. Then it is possible to rewrite the self-consistency
Eq. (14) in the following form:
ln
T
cs
T
c
= ψ
1
2
+
2
2
T
cs
T
c
ψ
1
2
, (20)
where ψ is the digamma function:
ψ (z)
d
dz
ln (z),(z) =
0
η
z1
e
η
dη. (21)
Boundary condition Eq. (15a)atx = d
f
/2 yields the fol-
lowing equation for :
tan
d
s
ξ
s
= W
0
(ω
0
), (22)
where we have written W
0
(ω
0
) instead of W
0
(ω
n
)on
the right-hand side, because the left-hand side must be ω
n
-
independent and only zero Matsubara frequency ω
0
should be
taken into account.
The critical temperature T
c
is determined by Eqs. (20) and
(22) for both 0 and π phase states. These equations extend
the model of Ref. [73], taking into account the possibility of
π phase state realization in the considered structure. Although
SMA is popular, it is often used without pointing out the limits
of its applicability. We derive these limits in the Appendix.
IV. MULTIMODE APPROACH
The SMA implies that one takes the (only) real root of
Eq. (20). An exact multimode method for solving the problem
Eqs. (14)–(17) is obtained if we also take imaginary roots into
account (there is infinite number of these, but numerically we
take some finite number). The MMA was applied for the first
time considering the problem of T
c
in an S/N bilayer [89].
We do not present here the derivation of the MMA.
We refer the reader to Ref. [73] [Sec. III, Eqs. (19)–(26)
therein] and use similar notations. The solution of the problem
Eqs. (15)–(17) within the MMA reduces then to the equation
det
ˆ
K
0
= 0, (23)
where the
ˆ
K matrix is defined as
K
0
n0
=
W
0
(ω
n
) cos
(
0
d
s
s
)
0
sin
(
0
d
s
s
)
ω
n
T
cs
+
2
0
, (24a)
K
0
nm
=
W
0
(ω
n
) +
m
tanh
(
m
d
s
s
)
ω
n
T
cs
2
m
, (24b)
where n = 0, 1, ..., N is the index of a Matsubara frequency
and m = 1, 2, ..., M is the index of an imaginary root
m
[
0
is the (only) real root]. We take M = N. The roots
n
are
determined by the following equation, obtained from Eq. (14)
FIG. 3. T
c
(d
f
) dependencies for the S/F/S structure in the π
phase state, calculated by the multimode approach. T
c
is normalized
by T
cs
, which is the critical temperature of superconductor in the
absence of ferromagnetic layer. We also normalize d
f
by the ξ
f
. Each
curve corresponds to particular value of transparency parameter γ
b
.
Other parameters are mentioned in the text.
at T = T
c
:
ln
T
cs
T
c
= ψ
1
2
+
2
n
2
T
cs
T
c
ψ
1
2
. (25)
The MMA is considered to be much more accurate com-
pared to the SMA, and it was shown in previous studies
that in some cases, SMA and MMA perform significantly
different qualitative behavior for 0 phase state junctions in S/F
bilayers [73]. In the following, using the MMA, we provide
calculations of the critical temperature for various parameters
of the S/F/S structure both in 0 and π phase states.
V. R E S U LT S
In this section, we present the results obtained by numeri-
cal calculations for 0 and π phase states using both the SMA
and MMA. We provide complete theory for T
c
(d
f
) behavior
description in the general case, where systems can be in 0 or π
phase states, depending on the F layer thickness d
f
. Moreover,
comparison between the SMA and MMA is also presented.
The accuracy of calculations was checked by choosing suffi-
ciently large matrix
ˆ
K dimensions in MMA. Here and below,
we have used in our calculations the same parameters as in
Ref. [73], i.e., γ = 0.15, h = 6.8π T
cs
, d
s
= 1.24ξ
s
.
A. T
c
in S/F/S structures in π phase state
In Fig. 3, the critical temperature T
c
(d
f
) dependencies
on ferromagnetic layer thickness d
f
in the π phase state
calculated by the MMA are shown. This situation corresponds
to an S/F/S structure enclosed in a ring, where the π phase
shift can be fixed by applying the magnetic flux quantum for
any d
f
. Different curves correspond to various values of γ
b
,
104502-4

COMPETITIVE 0 AND π STATES IN S/F/S PHYSICAL REVIEW B 100, 104502 (2019)
(a) (b)
(c) (d)
(e) (f)
FIG. 4. Plots of T
c
(d
f
) dependencies in both 0 and π phase states
calculated by the multimode approach. Solid black lines correspond
to the 0 phase state, while dashed red lines to the π phase state. Each
plot corresponds to a particular value of transparency parameter γ
b
:
(a) γ
b
= 0, (b) γ
b
= 0.02, (c) γ
b
= 0.05, (d) γ
b
= 0.07, (e) γ
b
= 0.1,
(f) γ
b
= 0.5.
which is proportional to resistance across the S/F interface
[see Eq. (5)] and can be determined from the experiment [78].
For fully transparent S/F interfaces, γ
b
= 0, the critical
temperature appears at d
f
ξ
h
(we note that in our case
ξ
h
= 0.54ξ
f
, since h = 6.8πT
cs
), reaches a maximum at a
particular d
f
, and with further increase in d
f
eventually drops
to zero. If we consider γ
b
= 0.001, we may see the reentrant
behavior of the π phase state, as it first also vanishes but then
reappears at larger d
f
with exponentially dumped amplitude,
and finally saturates at small finite value (see also Fig. 5 in
logarithmic scale) (we note that at γ
b
= 0, we do not observe
the reentrant behavior due to vanishingly small amplitude of
the reentrant π phase state). We can explain this behavior
as follows. At γ
b
= 0, all electronic transport through the
structure is governed only by the Andreev reflections. In this
case, the critical temperature T
c
vanishes when the π phase
state becomes energetically unfavorable in a certain interval of
d
f
, and at d
f
ξ
f
the T
c
eventually tends to zero. At larger γ
b
,
the Andreev reflections mix with normal reflections and the
inverse proximity effect becomes less pronounced. Therefore,
the critical temperature T
c
at each d
f
is larger than T
c
at γ
b
=
0. Still, at moderately small γ
b
, we observe similar behavior:
FIG. 5. Illustration of the possibility of multiple 0-π transitions
in case of γ
b
= 0.001. Calculations are made by the multimode
approach.
T
c
(d
f
) reaches a maximum at a particular d
f
and then decays
nonmonotonically and saturates to some value, depending on
γ
b
(we note that the oscillatory behavior for large d
f
can not be
seen due to vanishingly small amplitudes of the oscillations).
For γ
b
= 0.1 ÷ 0.2, one can see the dip on T
c
(d
f
) curve when
the π phase state is energetically unfavorable. For larger γ
b
,
this minimum is not resolved due to large contribution of
normal reflections at S/F interfaces and strong suppression
of the inverse proximity effect in S layers.
B. T
c
in S/F/S structures: 0-π transitions
To provide complete behavior of the critical temperature
in S/F/S trilayers, we calculate T
c
(d
f
) dependencies in both
0 and π phase states by using the MMA and show them on
the same plot, see Fig. 4. Both dependencies are calculated
for the same set of parameters mentioned above. In S/F/S
trilayers, only the state with highest T
c
is realized at certain
d
f
, i.e., when increasing d
f
the dashed red line appears above
the solid black line, the 0-π transition occurs, and the structure
switches to the π phase state. With further increase of d
f
, one
can see subsequent 0-π transitions in the T
c
(d
f
) curve, and
in the limit of long F layer the T
c
(d
f
) saturates at some finite
value, depending on γ
b
.
The critical temperature T
c
(d
f
) dependencies in the π
phase state (shown by dashed red lines in Fig. 4) were already
discussed above. Let us discuss now the critical temperature
behavior in the 0 phase state (shown by solid black lines
in Fig. 4). First of all, one can see the reentrant behavior
in the 0 phase state only in the case of highly transpar-
ent S/F interfaces [Figs. 4(a)4(c)]. At d
f
ξ
h
, the critical
temperature vanishes and then reappear at larger d
f
with
exponentially dumped amplitude. This is similar to the case
of S/F bilayers [73]. As was mentioned above for small γ
b
,
the Andreev reflections at S/F interfaces are dominant, while
for larger γ
b
they mix with normal reflections and the inverse
proximity effect is suppressed. Therefore, with increasing γ
b
,
104502-5

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Frequently Asked Questions (15)
Q1. What are the contributions in this paper?

In this paper, the authors compared SMA and MMA in the case of S/F/S trilayers for different values of transparency parameter γb and derived the limits of applicability of SMA for both 0 and π states. 

One of the interesting problems would be to extend the MMA to the nonequilibrium case by using the KeldyshUsadel Green ’ s function approach [ 84 ]. It is also interesting to study more complex phases in S/F multilayers in the MMA, extending the results of Ref. [ 63 ] obtained in the SMA. 

In S/F/S trilayers, only the state with highest Tc is realized at certain d f , i.e., when increasing d f the dashed red line appears above the solid black line, the 0-π transition occurs, and the structure switches to the π phase state. 

With decrease of the S-layer thickness ds in S/F/S trilayers, the critical temperature is suppressed due to the inverse proximity effect, which becomes more profound in the case104502-6of small ds. 

For fully transparent S/F interfaces, γb = 0, the critical temperature appears at d f ∼ ξh (we note that in their case ξh = 0.54ξ f , since h = 6.8πTcs), reaches a maximum at a particular d f , and with further increase in d f eventually drops to zero. 

Other possible extensions will include spin-orbit coupling effects in equilibrium [72] and nonequilibrium cases [90] and considering Tc in S/F/S junctions in the presence of an equilibrium supercurrent [91]. 

In this case, the critical temperature Tc vanishes when the π phase state becomes energetically unfavorable in a certain interval of d f , and at d f ξ f the Tc eventually tends to zero. 

2s πTcs d2Fs dx2− |ωn|Fs + = 0. (1) In the F layer (−d f /2 < x < d f /2), the Usadel equation can be written asξ 2f πTcs d2Ff dx2− (|ωn| + ih sgn ωn)Ff = 0. 

(13) According to the Usadel Eqs. (1)–(3), there is a symmetry relation F (−ωn) = F ∗(ωn), which implies that F+ is a real while F− is a purely imaginary function. 

At the borders of the S layers with a vacuum, the authors naturally havedFs(±ds ± d f /2) dx= 0. (6) The solution of the Usadel equation in the F layer depends on the phase state of the structure. 

The self-consistency Eq. (14) and boundary conditions Eqs. (15), together with the Usadel equation for F+s ,ξ 2s πTcs d2F+s dx2− ωnF+s + 2 = 0, (17) will be used for finding the critical temperature of the S/F/S structure both in 0 and π phase states. 

This situation corresponds to an S/F/S structure enclosed in a ring, where the π phase shift can be fixed by applying the magnetic flux quantum for any d f . 

Thus the authors confirm the importance of using the MMA in a wide range of parameters in the case of S/F/S trilayers, where 0-π phase transitions are possible. 

Using the boundary condition Eq. (10) the authors arrive at the effective boundary conditions for F+s at the boundaries of the right S layer,ξs dF+s (d f /2)dx = W 0,π (ωn)F+s (d f /2), (15a)dF+s (ds + d f /2) dx= 0, (15b) where the authors used the notationsW 0,π (ωn) = γ As( γb + Re B0,πf ) + γ As∣∣γb + B0,πf ∣∣2 + γ (γb + Re B0,πf ) , As = ksξs tanh(ksds), ks = 1ξs√ ωnπTcs . 

To provide complete behavior of the critical temperature in S/F/S trilayers, the authors calculate Tc(d f ) dependencies in both 0 and π phase states by using the MMA and show them on the same plot, see Fig.