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 [1–3]. 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 [6–12] and
spin-valves [13–19], magnetoelectronics [20–22], 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 [1–3,27–30]. 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
[31–49]. 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 [50–53] and critical
temperature T
c
oscillations [54–69]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 [73–78]. 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 [79–81]. 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
[62–64]. 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) [54–69]. 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
/ξ
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
/ξ
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,π
(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,π
(ω
n
) = γ
A
s
γ
b
+ Re B
0,π
f
+ γ
A
s
γ
b
+ B
0,π
f
2
+ γ
γ
b
+ Re B
0,π
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,ω
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
z−1
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,π
(ω
0
), (22)
where we have written W
0,π
(ω
0
) instead of W
0,π
(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,π
= 0, (23)
where the
ˆ
K matrix is defined as
K
0,π
n0
=
W
0,π
(ω
n
) cos
(
0
d
s
/ξ
s
)
−
0
sin
(
0
d
s
/ξ
s
)
ω
n
/πT
cs
+
2
0
, (24a)
K
0,π
nm
=
W
0,π
(ω
n
) +
m
tanh
(
m
d
s
/ξ
s
)
ω
n
/πT
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
![FIG. 6. The comparison between the single-mode approximation and the multimode approach for S/F/S systems. The set of parameters is the same as in Fig. 3. It is clear that for large enough values of γb, both single and multimode approaches perform very close results [(d)–(f)], while there are quantitative and even qualitative differences for small γb [(a), (b)].](/figures/fig-6-the-comparison-between-the-single-mode-approximation-1piwmyg3.webp)
