Effect of a dc magnetic field on the anomalous dispersion of localized Josephson plasma modes in layered superconductors
We study theoretically the propagation of Josephson plasma waves (JPWs) localized on a slab of layered superconductor in the presence of an external dc magnetic field. The slab is sandwiched between two dielectric half-spaces and the wave modes propagate across the layers. We derive analytic express...
Saved in:
| Published in: | Физика низких температур |
|---|---|
| Date: | 2018 |
| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
| Published: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2018
|
| Subjects: | |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/176166 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Effect of a dc magnetic field on the anomalous dispersion of localized Josephson plasma modes in layered superconductors / T. Rokhmanova, S.S. Apostolov, N. Kvitka, V.A. Yampol’skii// Физика низких температур. — 2018. — Т. 44, № 6. — С. 711-720. — Бібліогр.: 29 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860219411364839424 |
|---|---|
| author | Rokhmanova, T. Apostolov, S.S. Kvitka, N. Yampol’skii, V.A. |
| author_facet | Rokhmanova, T. Apostolov, S.S. Kvitka, N. Yampol’skii, V.A. |
| citation_txt | Effect of a dc magnetic field on the anomalous dispersion of localized Josephson plasma modes in layered superconductors / T. Rokhmanova, S.S. Apostolov, N. Kvitka, V.A. Yampol’skii// Физика низких температур. — 2018. — Т. 44, № 6. — С. 711-720. — Бібліогр.: 29 назв. — англ. |
| collection | DSpace DC |
| container_title | Физика низких температур |
| description | We study theoretically the propagation of Josephson plasma waves (JPWs) localized on a slab of layered superconductor in the presence of an external dc magnetic field. The slab is sandwiched between two dielectric half-spaces and the wave modes propagate across the layers. We derive analytic expressions for the dispersion relations of the localized JPWs and present the numerical simulation for the effect of the external dc magnetic field on the dispersion. The anomalous dispersion of localized JPWs is predicted for a wide range of frequencies, wave vectors, and dc fields. Also, we discuss the possibility of the internal reflection of the localized modes in the inhomogeneous dc magnetic field. This phenomenon can find application in the terahertz electronics for the control of the localized mode propagation.
|
| first_indexed | 2025-12-07T18:17:26Z |
| format | Article |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6, pp. 711–720
Effect of a dc magnetic field on the anomalous dispersion
of localized Josephson plasma modes
in layered superconductors
T. Rokhmanova1,2, S.S. Apostolov1,2, N. Kvitka2, and V.A. Yampol’skii1,2
1A.Ya. Usikov Institute for Radiophysics and Electronics NASU, Kharkiv 61085, Ukraine
2V.N. Karazin Kharkiv National University, Kharkiv 61077, Ukraine
E-mail: Rokhmanova@ieee.org
Received December 27, 2017, published online April 25, 2018
We study theoretically the propagation of Josephson plasma waves (JPWs) localized on a slab of layered su-
perconductor in the presence of an external dc magnetic field. The slab is sandwiched between two dielectric
half-spaces and the wave modes propagate across the layers. We derive analytic expressions for the dispersion
relations of the localized JPWs and present the numerical simulation for the effect of the external dc magnetic
field on the dispersion. The anomalous dispersion of localized JPWs is predicted for a wide range of frequencies,
wave vectors, and dc fields. Also, we discuss the possibility of the internal reflection of the localized modes in
the inhomogeneous dc magnetic field. This phenomenon can find application in the terahertz electronics for the
control of the localized mode propagation.
PACS: 74.72.–h Cuprate superconductors;
73.20.Mf Collective excitations (including excitons, polarons, plasmons and other charge-density
excitations);
52.35.Mw Nonlinear phenomena: waves, wave propagation, and other interactions.
Keywords: layered superconductors, localized Josephson plasma waves, dc magnetic field, anomalous dispersion.
1. Introduction
Layered superconductors are periodic materials, where
thin superconducting layers are separated by thicker insulat-
ing ones and are electrodynamically related to each other by
means of the intrinsic Josephson effect (see Fig. 1). The ex-
perimental confirmation of such an electrodynamic model
for layered superconductors can be found in Refs. 1, 2.
High-temperature superconductors based on Bi, La, or Y
with CuO2 superconducting layers belong to this family of
materials. The essential property of layered superconductors
is the considerable anisotropy of their current-carrying capa-
bility. The current along the layers is of the same nature as
in the bulk superconductors and is much stronger than the
current across the layers caused by the Josephson effect.
Thus, the so-called Josephson plasma is formed in layered
superconductors. This strongly anisotropic Josephson solid-
state plasma supports the propagation of the specific excita-
tions in layered superconductors, the Josephson plasma
waves (JPWs). These waves belong to the terahertz frequen-
cy range, which makes layered superconductors interesting
for terahertz electronics (see, e.g., Ref. 3). In turn, the te-
rahertz technologies are promising for various applications,
including medical diagnostics and security control (see, e.g.,
Ref. 4). Furthermore, the study of the interaction of strong
terahertz pulses with layered superconductors (see experi-
Fig. 1. Sketch of the setup. Here D is the thickness of the sample,
H0 is the external dc magnetic field, k is the wave vector.
© T. Rokhmanova, S.S. Apostolov, N. Kvitka, and V.A. Yampol’skii, 2018
T. Rokhmanova, S.S. Apostolov, N. Kvitka, and V.A. Yampol’skii
mental works Refs. 5,6,7) may reveal new possibilities for
high-temperature superconductive state control.
In the Josephson plasma, various interesting electro-
dynamic phenomena can be observed, both common and
uncommon to the other types of plasmas. As was theoreti-
cally demonstrated in Refs. 8, 9, the surface JPWs can pro-
pagate along the interface between the layered superconduc-
tor and external dielectric, similarly to surface plasmon–
polaritons in usual plasmas. The excitation of these waves
leads to various resonant phenomena [9,11,12] similar to the
Wood anomalies well known in optics (see Refs. 13–15).
However, contrary to usual plasmas, the surface JPWs can
propagate with frequencies not only below the plasma fre-
quency but also above it [9]. As was shown in Ref. 16, the
phenomena similar to the Anderson localization and the
formation of a transparency window for THz waves can be
observed in layered superconductors with randomly–
fluctuating value of the maximum Josephson current.
As was described in Ref. 17, when the layers are perpen-
dicular to the slab boundaries and to the direction of JPWs
propagation, the anomalous dispersion of the localized waves
can be observed in layered superconductors in a certain range
of frequencies and wave numbers. It is caused, in particular,
by the different signs of the longitudinal and transversal
components of the effective permittivity tensor in layered
superconductors. A system containing material with anoma-
lous dispersion can have negative refractive index.
Refs. 9, 10 present evidence of the negative refractive index
for the surface JPWs in layered superconductors above the
plasma frequency. Although the negative index materials
were mentioned earlier in the literature (see Ref. 18), they
started to attract great attention after 2000, when a theoreti-
cal prediction of a perfect lens creation using such materials
was presented in Ref. 19. Since then a great amount of
works have been carried out resulting rapid development of
this field. Some reviews of the recent advances can be found
in Refs. 20, 21.
The possibility of the anomalous dispersion manipula-
tion is promising for various applications. dc magnetic
field is one of the tools that can flexibly change the elec-
tromagnetic properties of layered superconductors. In
Refs. 22, 23, the effect of the weak external dc magnetic
field on the Terahertz waves transmission, reflection, and
polarization transformation in layered superconductors was
studied theoretically. It turned out that even relatively
weak magnetic field can significantly change the condi-
tions for the waves propagation. Therefore, the external dc
magnetic field turns up to be an interesting tool to control
the localized JPWs in layered superconductors.
In the present work, we study theoretically how the
relatively weak external dc magnetic field affects the dis-
persion properties of the localized JPWs. The paper is
organized as follows. In the second section of the paper,
the studied model is presented. There are presented the
geometry of the problem and the main equations for
romagnetic fields. The third section is devoted to the der-
ivation of the dispersion relations for the localized modes
in the WKB approximation and in the exact form in terms
of the special Legendre functions. The fourth section con-
tains analysis of the obtained relations, where we consid-
er the effect of the external dc magnetic field on the dis-
persion curves. In the fifth section, we discuss the
possibility of the internal reflection of the localized
modes in the inhomogeneous dc magnetic field. This
phenomenon can find application in the terahertz elec-
tronics for the control of the localized mode propagation.
The obtained results are summarized in the conclusions.
2. Model
We study the linear localized JPWs propagating in a slab
of layered superconductor sandwiched between two dielec-
tric half-spaces (see Fig. 1). The layers are perpendicular to
the boundaries of the slab. The coordinate system is chosen
in such a way that the z-axis is directed across the supercon-
ducting layers, i.e., along the crystallographic c-axis, and
parallel to the boundaries of the slab. The x- and y-axes are
directed along the superconducting layers, i.e., along the ab-
plane. The x-axis is perpendicular to the slab boundaries,
while the y-axis is parallel to them. The slab of the thickness
D is located at | |< /2x D , where the upper and lower die-
lectric half-spaces with the permittivity dε occupy the re-
gions > /2x D and < /2x D− , respectively. Thus, the plane
= 0x is in the middle of the slab and divides the system into
two symmetrical parts. The external dc magnetic field H0 is
directed along the y-axis and is uniformly distributed outside
the slab of layered superconductor.
We consider the localized JPWs of the following po-
larization:
{ }( , , , ) = 0, ( ),0 exp( ),y zx y z t H x ik z i t− ωH
{ }( , , , ) = ( ),0, ( ) exp( ),x z zx y z t E x E x ik z i t− ωE (1)
where ω is the frequency of the localized mode that prop-
agates along the z-axis, i.e., = 0yk .
2.1. Electromagnetic field in the dielectric
The JPWs are localized near the slab and evanesce far
from the slab in the dielectric half-spaces. Thus, from
Maxwell equation, we can obtain expressions for the com-
ponents yH and zE of the electromagnetic wave,
( ) = exp[ ( /2)],y dH x H k x D± ±
( ) = exp[ ( /2)],d
z d
d
ick
E x H k x D± ±
ε ω
(2)
where superscripts “ + ” and “ − ” mean the upper ( > /2)x D
and lower ( < /2)x D− half-spaces, respectively, H ± is the
amplitude of the magnetic field. The decrement kd,
712 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6
Effect of a dc magnetic field on the anomalous dispersion of localized Josephson plasma modes in layered superconductors
2 2 2 2= / > 0,d z dk k c− ε ω (3)
determines how quickly the localized mode evanesces
from the slab.
2.2. Main equations for the layered superconductor
We assume that the period d of the layered structure of
the superconductor is much smaller than the wavelength
across the layers, 1.zk d Therefore, we can present the
electrodynamic equations for layered superconductors in
the continual limit. We use wave equation for the vector
potential A, which is common for macroscopic electrody-
namics (some more explanations one can find in Ref. 24),
2
2 2
4graddiv = ,s
cc t
ε ∂ π
− ∆ − +
∂
AA A J (4)
where sε is dielectric constant of the insulating layers in
the superconductor. The components xJ and yJ of the
current density J along the layers are described within the
London model,
2 2= , = ,
4 4
x x y y
ab ab
c cJ A J A− −
πλ πλ
(5)
where abλ is the London penetration depth in the c-axis
direction, while the current density zJ across the layers is
described by the Josephson relation,
= sin .z cJ J ϕ (6)
Here cJ is the maximum Josephson current density and ϕ
is the gauge invariant interlayer phase difference [25] be-
tween neighboring layers.
The vector potential A is related to the electric E and
magnetic H fields by the standard equations,
1= rot , = ,
c t
∂
−
∂
AH A E (7)
and the scalar potential is supposed to be equal to zero.
The z-component of the electric field produces the break-
down of electro-neutrality of superconducting layers, which
causes additional, so-called capacitive, interlayer coupling.
According to Ref. 26, the capacitive coupling is substantial
only for longitudinal JPWs with frequencies close to the Jo-
sephson plasma frequency Jω . In this paper, we can neglect
the capacitive coupling due to the smallness of the capacitive
coupling parameter, 2= / 1,D sR sdα ε where DR is Debye
length for a charge in the superconductor, s is the thickness
of the superconducting layers ( 1 nm).s Then the following
relation between the gauge invariant interlayer phase differ-
ence ϕ and the z-component of vector potential is valid:
0=
2zA
d
Φ
− ϕ
π
, (8)
where 0 = /c eΦ π is the magnetic flux quantum, e is the
elementary charge, and c is the speed of light.
It should be noted that from the wave Eq. (4), using
Eqs. (5) and (6), one can obtain well-known coupled sine-
Gordon equation which is widely used in electrodynamic
description of layered superconductors (see, e.g. Ref. 25).
For sufficiently small frequencies, ,Jω γω this equation
takes the following form,
2 2 2
2 2
2 2 2 2
11 sin = 0,ab c
Jz t x
∂ ∂ ϕ ∂ ϕ
− λ + ϕ − λ
∂ ω ∂ ∂
(9)
where = /( )c J scλ ω ε is the London penetration depth
along the layers, = 8 /J c sedJω π ε is the Josephson
plasma frequency, = /c abγ λ λ is the anisotropy parameter.
2.3. dc magnetic field in the layered superconductor
Here we describe how the external dc magnetic field pen-
etrates inside the slab of the layered superconductor. In this
paper, we study the case of relatively small magnetic fields,
0 0 0< = / ,cH dΦ π λ when the Josephson vortices do not
penetrate into the superconductor. For estimations, the value
of 0 for Bi2Sr2CaCu2O8+δ (with 7= 1.5 10 cmd −⋅ and
3= 4 10 cm)c
−λ ⋅ is about 100 Oe. In addition, we suppose
that the superconducting slab is sufficiently thick,
exp( / ) 1.cD λ (10)
In this case, the dc magnetic field penetrates into the lay-
ered superconductor over small distances in the form of the
tails of two fictitious vortices, each near the corresponding
interfaces.
Each vortex tail can be described by the well-known so-
lution [27] of the sine-Gordon equation (9),
0( ) = 4arctan[exp( )],±ϕ ξ ξ ± ξ (11)
where subscripts “ + ” and “ − ” mean the upper ( = /2)x D
and lower ( = /2)x D− interfaces, respectively, near which
the vortex tails exist, and = / cxξ λ is normalized coordi-
nate. The constant 0ξ corresponds to the center of the fic-
titious vortex and is defined by the normalized magnitude
0h of the external dc magnetic field and the normalized
half-thickness δ of the slab,
1
0 0 0 0 0= arccosh( ), = /2 , = / .ch D h H−ξ δ + δ λ (12)
2.4. Electromagnetic field in the layered superconductor
In order to describe the wave propagation in the slab of
layered superconductor, we present ϕ as a sum of static
solutions ( )±ϕ ξ , caused by the dc magnetic field,
Eq. (11), and a small additive ( , , )lm z tϕ ξ induced by the
localized mode,
( , , ) = ( , , ) ( ) ( ).lmz t z t + −ϕ ξ ϕ ξ + ϕ ξ + ϕ ξ (13)
We seek ( , , )lm z tϕ ξ in the form of the wave propagating
along the z-axis,
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6 713
T. Rokhmanova, S.S. Apostolov, N. Kvitka, and V.A. Yampol’skii
( , , ) = ( )exp[ )].lm zz t a ik z i tϕ ξ ξ − ω (14)
Excluding the x- and y-components of the vector poten-
tial from the wave equation (4) and linearizing sin ,ϕ ≈ ϕ
we derive the equation for the amplitude ( )a ξ ,
2
( ) [ ( ) 1] ( ) = 0
s
a u aξ′′− + ξ − ξ
κ
, (15)
where the prime denotes derivative with respect to ξ,
2 1
2 2
0 0
2 2( ) = (1 )
( ) ( )cosh cosh
u −
ξ − Ω +
ξ − ξ ξ + ξ
, (16)
and = / JΩ ω ω is the normalized frequency. Parameter sκ
represents the normalized x-projection of the wave vector
in the absence of the dc magnetic field,
2
2 2
2 2= ( 1) 1 ,
1 /
z
s
κ
κ Ω − +
− Ω γ
(17)
and =z z abkκ λ is the normalized z-projection of the wave
vector.
Using Eq. (7), we can express the components s
yH and
s
zE of the electromagnetic field in the slab via the function
( )a ξ ,
0 2 2 2
( )( ) =
1 /(1 / )
s
y
z
aH ξ′ξ
+ κ − Ω γ
,
0( ) = ( )s
z
iE aΩ
ξ − ξ
ε
. (18)
In the next section we present the analytic solution of
Eq. (15) and derive the dispersion relations for the locali-
zed modes.
3. Dispersion relations
In order to derive the dispersion relation for the local-
ized JPWs in the slab of layered superconductor, we match
the tangential components of the electric and magnetic
fields at the interfaces of the slab,
= =
s
z z
s
y y
E E
H H
ξ ±δ
±
±
ξ ±δ
= . (19)
Rewriting this equation in terms of the amplitude ( )a ξ ,
see Eqs. (2) and (18), we achieve the following relations:
1 2 2
2
( = ) =
( = ) ( 1)
s
d
a
a
−ε Ω κξ ±δ′ ±
ξ ±δ Ω − κ
(20)
where = /s dε ε ε and dκ represents the normalized spa-
tial decrement for the dielectric half-spaces, see Eq. (3),
2 2 2 1 2= > 0.d z
−κ γ κ − ε Ω (21)
It should be noted that the symmetry of the studied sys-
tem implies the symmetry of the localized modes, symmet-
ric and antisymmetric with respect to the magnetic field.
Therefore, we can use relation (20) only for upper inter-
face, = ,ξ +δ but impose additional conditions in the mid-
dle of the slab,
(0) = 0 or (0) = 0a a′ (22)
for symmetric or antisymmetric mode, respectively.
Differential equation (15) with condition (20) at =ξ +δ
and one of conditions (22) define the spectrum of the local-
ized modes. In the following subsections 3.1 and 3.2 we
present the asymptotic and exact solutions of Eq. (15), re-
spectively.
3.1. Dispersion relations within the WKB approximation
In this subsection, we solve Eq. (15) asymptotically.
We restrict our study to the relatively low frequency range,
< Jω γω . On the one hand, in this frequency range all the
features of the anomalous dispersion affected by dc mag-
netic field can be observed. On the other hand, the high
frequency range is hardly attained in the experiment be-
cause of destroying the superconducting state. It should be
emphasized that Eq. (15) resembles the one-dimensional
Schrödinger equation with 1 standing instead of the total
energy and with ( )u ξ instead of the potential energy.
Therefore, in the case
1,sκ (23)
we can solve this equation by means of the WKB (quasi-
classical) approximation. In turn, inequality Eq. (23) is
satisfied under following conditions:
2 2 21 / , | 1 | 1.zκ − Ω γ Ω − (24)
3.1.1. Frequencies higher than Jω
We start our analysis from the case of relatively high
frequencies, 1 < <Ω γ , when the parameter sκ is positive
(see Eq. (17)) and the potential energy ( )u ξ is negative
(see Eq. (16)). In this case, the classical turning points are
absent and the WKB solution of (15) can be presented as
W
sym( ) = sin[ ( )],
( )
KB
s
a
a b
b
ξ κ ξ
ξ′
W
asym( ) = cos[ ( )],
( )
KB
s
a
a b
b
ξ κ ξ
ξ′
(25)
for the symmetric and antisymmetric with respect to mag-
netic field modes. Here W
asym
KBa and W
sym
KBa are the integra-
tion constants,
0
( ) = 1 ( )b du
ξ
ξ ξ′− ξ′∫ . (26)
714 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6
Effect of a dc magnetic field on the anomalous dispersion of localized Josephson plasma modes in layered superconductors
In order to derive the dispersion relations, we substitute
solutions (25) into Eq. (20). As a result, we get the disper-
sion relations,
cot[ ( )] = ,sbκ δ β tan[ ( )] = ,sbκ δ −β (27)
for the symmetric and antisymmetric localized modes.
Here
1/21 2 2
0
2 2
2
= 1
( 1) 1
s
d
h
−− ε Ω κ
β +
Ω − κ Ω −
. (28)
If the z-projection of the wave vector is sufficiently large,
1/2 / ,z
−κ ε Ω γ then the parameter β is small,
1( ) 1.−β εγ In this case, we can simplify the dispersion
relation to the form ( ) = ( 2)/2,sb nκ δ π − where integer
= 3, 4,n numerates the dispersion curves from bottom to
top (see section 4 and Fig. 4). The odd numbers = 3, 5,n
correspond to the symmetric modes whereas even
= 4, 6,n describe the antisymmetric ones. Note that we
start the numeration from = 3n because the dispersion
curves with numbers = 1n and = 2n are located in the
lower frequency range, < 1Ω . The last implicit dispersion
relation can be rewritten in the explicit form for ( )zκ Ω ,
2 2
2
2 2 2
[ ( 2)/2]( ) = 1 1 .
( 1) ( )
z
n
b
Ω π − κ Ω − −
γ Ω − δ
(29)
3.1.2. Frequencies lower than Jω
Now we proceed to the low frequency range, < 1Ω . In
this case, the electromagnetic field in the layered super-
conductor evanesces across the slab, and the wave in the
slab can be represented as two weakly coupled surface
modes localized near interfaces = /2x D and = /2x D− .
Therefore, the spectrum of such modes nearly coincides
with the spectrum of the surface modes localized on the
interface between the half-infinite layered superconductor
and the half-infinite dielectric. The spectrum of these sur-
face waves was studied in Ref. 27.
In the low frequency range, the parameter sκ is nega-
tive, see Eq. (17), and the potential energy ( )u ξ is posi-
tive, see Eq. (16). This means that, under conditions
2
2 20
022
0
4 exp( 2 )
< 1 < 2
1 1( )
h
h
h
− δ
− Ω
+ −
, (30)
there exist classical turning points, = tpξ ±ξ , defined by
the equation ( = ) = 1tpu ξ ±ξ , or, according to Eq. (16),
2
0 2
2( ) =cosh
1
tpξ − ξ
− Ω
. (31)
Here we keep only one summand 2
0( )cosh tp
− ξ − ξ in
Eq. (16) because the other summand, 2
0( )cosh tp
− ξ + ξ , is
exponentially small. It should be noted that, under assump-
tions (10) and (24), the left-hand inequality in Eq. (30) is
satisfied for arbitrary 0h .
The WKB solution of Eq. (15) with classical turning
point tpξ can be presented in the following form:
W
W
cos[| | ( ) /4], < < ,
( )
( ) =
/2 exp[ | ( ) |], 0 < < ,
( )
KB
s tp
KB
s tp
a b
b
a
a b
b
κ ξ − π ξ ξ δ
ξ′ξ
− κ ξ ξ ξ ξ′
(32)
where WKBa is the integration constant and
( ) = .( ) 1
tp
b du
ξ
ξ
ξ ξ′ξ −′∫ (33)
The first line in Eq. (32) corresponds to the classically al-
lowed region, < <tpξ ξ δ , while the second line describes
the classically forbidden zone, 0 < < tpξ ξ . Formula (32)
presents the field near upper interface =ξ +δ only. This
solution is valid when exp( 2 | (0) |) 1,sb− κ and we can
neglect the weak coupling with the field near lower inter-
face, =ξ −δ , writing “ exp ” instead of “ cosh ” or “ sinh ”
in the classically forbidden zone.
Applying solution (32) in Eq. (20), we derive the dis-
persion relation
tan[| | ( ) /4] =s bκ δ − π −β (34)
for weakly coupled modes localized near interfaces =ξ ±δ .
Here β is defined by Eq. (28). If the z-projection of the wave
vector is sufficiently large, 1/2 / ,z
−κ ε Ω γ the parameter
β is small, 1.β In this case, we can simplify the disper-
sion relation to | | ( ) = ( 1/4),s b mκ δ π + where = 0, 1, 2,m ,
and rewrite it in the explicit form for ( ).zκ Ω
2 2 2
2
2 2 2
( 1/ 4)( ) = 1 1
(1 ) ( )
z
m
b
Ω π +
κ Ω − −
γ − Ω δ
. (35)
The number m in this equation is used in Sec. 4 to numer-
ate the corresponding pairs of the dispersion curves with
numbers = 2 1n m + and = 2 2n m + (see the lower inset in
Fig. 5). In particular, the dispersion curve numbered by
= 0m is actually the pair of curves with numbers = 1n
and = 2n for antisymmetric and symmetric localized
modes, respectively, that are close to each other.
Derived dispersion relations (27) and (34) are valid in
relatively wide range of the frequencies and wave vectors,
and in this range they reveal the anomalous dispersion of
the localized modes (see Sec. 4 for details). Though the
WKB approximation predicts the non-monotonicity of the
dispersion curves, this feature is located near light line,
1/2= zΩ ε γκ , beyond the formal applicability of this ap-
proximation. Moreover, the WKB approximation does not
capture the behavior of the dispersion curves in the fre-
quency range close to Jω , | 1 | 1.Ω − In the following
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6 715
T. Rokhmanova, S.S. Apostolov, N. Kvitka, and V.A. Yampol’skii
subsection, we present the exact solution that is devoid of
these drawbacks.
3.2. Exact solution
One can obtain exact solution of Eq. (15) in terms of
the associated Legendre functions, see Appendix A. The
symmetric and antisymmetric with respect to magnetic
field solutions can be presented in the following forms:
0 0
sym
0 0
( ) ( )
( ) =
( ) ( )
p q
p q
f f
a a
f f
ξ − ξ ξ − ξ
ξ −
ξ ξ
,
0 0
asym
0 0
( ) ( )
( ) =
( ) ( )
p q
p q
f f
a a
f f
ξ − ξ ξ − ξ
ξ −
ξ ξ′ ′
, (36)
respectively. Here syma and asyma are integration con-
stants, and
( ) = [tanh( )], ( ) = [tanh( )],p qf P f Qµ µ
ν νξ ξ ξ ξ (37)
where [ ]Pµ
ν τ and [ ]Qµ
ν τ are associated Legendre func-
tions of the first and second kind, respectively, with
2 1 22 1 = , = .8( 1) 1 ss i−ν + µ κΩ − κ + (38)
The dispersion relations are defined by Eq. (20) taken at
=ξ +δ with ( )a ξ substituted from Eqs. (36).
Now we show the transition of the exact solutions to the
case of absence of the external dc magnetic field. When the
dc field is close to zero, 0 0h → , we can simplify Eqs.
(36) in the following way. Firstly, we expand the argument
of the associated Legendre functions,
2
0 0
1tanh ( ) 1 exp[2( )]
2
hξ − ξ ≈ − ξ − δ . (39)
Then, the associated Legendre functions with argument τ
close to 1 can be presented as linear combinations of func-
tions /2(1 )±µ− τ , see Eq. (A14) in Apendix A. Therefore,
taking into account that = siµ κ , we get
0 1( ) exp[ ( )],p sf c iξ − ξ ≈ − κ ξ − δ
0 2 3( ) exp[ ( )] exp[ ( )]q s sf c i c iξ − ξ ≈ − κ ξ − δ + κ ξ − δ
where 1c , 2c and 3c are constants. Hence, the symmetric
and antisymmetric solutions (36) obviously take the form
sym( ) = sin ,sa aξ κ ξ asym( ) = cos ,sa aξ κ ξ (40)
respectively, leading to the corresponding dispersion
relations,
1 2
2cot = ,
( 1)
s
s
d
−ε Ω κ
κ δ
Ω − κ
1 2
2tan = ,
( 1)
s
s
d
−ε Ω κ
κ δ −
Ω − κ
(41)
which are valid in the absence of external dc magnetic
field (see Ref. 17 for details).
4. Numerical analysis
In this section we present the obtained analytical results
in the graphic form and describe the effect of the dc mag-
netic field on the dispersion curves.
4.1. The distribution of the dc and ac fields
Figure 2 shows the spatial distributions of the normal-
ized dc magnetic field ( )dch ξ (the red dashed curves) and
the magnetic field ( )yH ξ of the antisymmetric localized
Josephson plasma modes (the solid blue curves) at
= 0.98Ω in the low frequency range (main panel) and
= 1.12Ω in the high frequency range (inset). The exter-
nal dc magnetic field is uniform in the dielectric,
( ) =dch ξ 0 0 0= /h H , and penetrates into the layered
superconductor in the form of the tails of the fictitious
vortices, Eq. (11),
0 0
1 1( ) = .
cosh( ) cosh( )dch ξ +
ξ − ξ ξ + ξ
(42)
The ac magnetic field ( )s
yH ξ is defined by Eq. (18) with
solutions (36) and is plotted in Fig. 2 in arbitrary units.
For the case of high frequencies, > 1Ω , the localized
JPWs oscillate across the slab, while, for the low frequen-
cies, < 1Ω , the electromagnetic fields oscillate near the
interfaces and evanesce deep into the slab. In both cases,
the dc field causes the change of the amplitude and wave-
length of the ac field oscillations near the interfaces.
Fig. 2. (Color online) The spatial distribution of the ac magnetic
field ( )s
yH ξ (blue solid curves), in arb. units, and the normalized
dc magnetic field (red dashed curves), hdc(ξ), for Ω = 0.98 (main
panel) and Ω = 1.12 (inset). Parameters: γ = 5, ε = 4, κ = 10, δ = 5,
h0 = 0.9.
716 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6
Effect of a dc magnetic field on the anomalous dispersion of localized Josephson plasma modes in layered superconductors
4.2. Effect of the dc magnetic field and anomalous
dispersion
Firstly, we study the dispersion curves for relatively
small 1zκ in two frequency ranges, the low frequen-
cies, < 1Ω , and the high frequencies, > 1Ω . For these
cases, we plot the curves according to the exact solution
obtained in Subsec. 3.2 and analyze their variation by
changing the value 0h of the external dc magnetic field.
Secondary, we focus on the frequencies close to the Jo-
sephson plasma frequency Jω , | 1 | 1Ω − , and the wider
range over zκ . Here, for certain value of 0h , we examine
the transition from > 1Ω to < 1Ω in the behavior of the
dispersion curves and compare the exact solution with the
result obtained within the WKB approximation.
In order to simplify the following explanations, we nu-
merate the dispersion curves from the bottom to top by
= 1,2,3,n and study the shift of each curve by variation
of the normalized amplitude 0h of the external dc magnetic
field.
We start the description from the low frequency range,
< 1Ω . Figure 3 shows two lowest dispersion curves with
numbers = 1n (thin solid lines for antisymmetric localized
mode) and = 2n (thick dashed lines for symmetric localized
mode) at 0 = 0h , 0 = 0.6h , 0 = 0.9h , 0 = 0.98h , and
0 = 0.999h . As seen, the dispersion curves shift towards the
lower frequencies and increase their curvature when increas-
ing the external dc magnetic field. The curves with = 1n and
= 2n become close to each other with the increase of 0h .
This occurs because the symmetric and antisymmetric local-
ized modes can be represented as two weakly coupled sur-
face modes localized near interfaces = /2x D and
= /2x D− , and the coupling becomes weaker for the smaller
values of Ω . Moreover, when 0 1h → , both curves can be
described asymptotically by the same dispersion relation,
2 2 1 2
0/3 [1 ( ) ] = 1z d h−κ + Ω + εκ − , and converge to the
point = 0zκ , = 0Ω . Here dκ is defined by Eq. (21).
The dispersion curves presented in Fig. 3 (with the excep-
tion of the curve with = 1n at 0 = 0),h are non-monotonous
and consist of the parts with normal (where / > 0)z∂Ω ∂κ
and anomalous (where / < 0)z∂Ω ∂κ dispersions. There-
fore, the curves have maximums, where the group velocity
vanishes, / = 0z∂Ω ∂κ . These maximums appear near the
light line, 1/2= zΩ ε γκ and shift when changing the ampli-
tude of the external dc magnetic field. The possible applica-
tion of this phenomenon is discussed in Subsec. 5.
Now we examine the high frequency range, > 1Ω . The
corresponding dispersion curves are plotted in Fig. 4 by the
solid ( = 3, 5,n for antisymmetric localized modes) and
dashed ( = 4, 6,n for symmetric localized modes) lines
for 0 = 0h , 0 = 0.6h , 0 = 0.9h , and 0 = 1h . Similarly to
the case of low frequencies, the increase of the external dc
magnetic field shifts the curves towards the lower frequen-
cies (see the arrows in Fig. 4). The curves are non-
monotonous and have the parts with normal and anomalous
dispersions. Comparing the curves for the intermediate
fields 0 = 0.6h and 0 = 0.9h for different n, one can see
that the shift due to the increase of 0h is non-uniform and
Fig. 3. (Color online) The dispersion curves with n = 1 (thin solid
lines, antisymmetric localized mode) and n = 2 (thick dashed lines,
symmetric localized mode) for Ω < 1 and the dc magnetic field
h0: 0.999. The filled area above the light line, Ω = ε1/2γκz, presents
the forbidden zone, where the localized modes do not exists. Other
parameters: γ = 5, δ = 5, ε = 4.
Fig. 4. (Color online) The dispersion curves for Ω > 1 and the dc
magnetic field h0: 0, 0.6, 0.9, and 1 plotted by solid (n = 3, 5,…,
antisymmetric localized modes) and dashed (n = 4, 6,…, symmetric
localized modes) lines. The arrows show the increase of the field h0.
The other parameters and notations are the same as in Fig. 3.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6 717
T. Rokhmanova, S.S. Apostolov, N. Kvitka, and V.A. Yampol’skii
depends on n. When increasing n, the curves with 0 0.6h ≤
come close to each other, while the distance between curves
with 0 0.6h ≥ increases. Figure 4 shows the curves for suf-
ficiently small zκ and do not capture the interesting feature
induced by the external dc magnetic field: at finite 0 > 0h
and sufficiently large zκ , all the dispersion curves with
= 3, 4,n intersect the line = 1Ω and end at = 0.Ω
To study the mentioned feature, we proceed to the fre-
quency range | 1 | 1Ω − close to the Josephson plasma
frequency. The main panel in Fig. 5 shows the behavior of
two curves with = 3n and = 4n at 0 = 0.9h in this fre-
quency range for sufficiently large zκ while the upper inset
in Fig. 5 shows the curves with = 3, 4, 5, 6n in more nar-
row range of zκ . The solid lines represent the exact disper-
sion curves, in accordance with Eqs. (20) and (36), while the
dashed and dotted lines describe the dispersion curves ob-
tained in the WKB approximation. The dashed curves are
plotted using Eqs. (27) and (34), where we leave β in the
right-hand side. Though the WKB approximation formally
is not valid close to the light line, 1/2= zΩ ε γκ where the
maximums appear, one can see a good agreement between
the solid and dashed curves for Ω not very close to 1. The
dotted curves are plotted using Eqs. (29) and (35), where we
neglect β in the right-hand side. The dotted curves are close
to the solid ones for sufficiently great zκ , where the WKB
approximation is applicable, see condition (24).
One can see from Fig. 5 that the curves with = 3n and
= 4n intersect the line = 1Ω . Moreover, these curves
come close to each other, when < 1Ω , due to the weak cou-
pling between two interfaces. The same behavior is revealed
by each pair of the dispersion curves, antisymmetric mode
with = 2 1n m + and symmetric mode with = 2 2n m + ,
with the same = 1, 2, 3,m . The curves with = 1, 2, 3, 4m
are plotted in the lower inset of Fig. 5 for < 1Ω . In this fre-
quency range, all the curves end at = 0Ω .
5. Internal reflection of the localized modes in the
inhomogeneous dc magnetic field
In this section, we predict the internal reflection phe-
nomenon that is related to the non-monotonous dispersion
affected by the external dc magnetic field.
The dispersion curves discussed in the previous section
are non-monotonous as functions zκ for fixed value of h0.
Therefore, for fixed value of Ω , there are two values of
zκ on each dispersion curve which correspond to the parts
with the normal and anomalous dispersions. This means
that the dispersion curves presented as functions 0( )z hκ
for fixed value of Ω should be two-valued. This feature is
shown in Fig. 6, where the dispersion curves with numbers
= 3n (dash-dotted line), = 4n (dashed line), and = 5n
(solid line) are plotted as functions 0( )z hκ for fixed va-
lues = 1.07Ω , = 1.25Ω , and = 1.5Ω , respectively.
Fig. 5. (Color online) The dispersion curves with n = 3, 4 (main
panel) and n = 3, 4, 5, 6 (upper inset) obtained for h0 = 0.9. Solid
lines are described by exact solution, Eqs. (20) and (36); dashed
lines are plotted using the WKB formulas (27) and (34); dotted
lines correspond to the simplified expressions Eqs. (29) and (35).
The lower inset shows the pairs (numbered by m = 1, 2, 3, 4) of
the dispersion curves with numbers n = 2m + 1 and n = 2m + 2,
which are close to each other. The other parameters and notations
are the same as in Fig. 3.
Fig. 6. (Color online) Dispersion curves with numbers n = 3
(dash-dotted line), n = 4 (dashed line), and n = 5 (solid line) as
functions κz(h0) for the fixed frequencies Ω = 1.07, Ω = 1.25, and
Ω = 1.5, respectively. The arrow shows the variation of κz along
the mode propagation in the non-homogeneous dc magnetic field,
while the solid and empty circles correspond to the initial (h = 0,
κz = 0.23) and critical (h0 = hmax = 0.31, κz = 0.28) points of this
variation. Other parameters are the same as in Fig. 3.
718 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6
Effect of a dc magnetic field on the anomalous dispersion of localized Josephson plasma modes in layered superconductors
Now we presume that the localized mode propagates
along the slab of the layered superconductor and that the
external dc magnetic field is non-homogeneous and
smoothly increases along the z-axis from 0 = 0h to
0 = 0.5h . Considering, for example, the mode frequency
= 1.5Ω and the initial z-projection of the wave vector
= 0.23zκ , we examine the solid curve in Fig. 6. When
the external dc magnetic field increases along the propa-
gation, zκ traces the curve (along the arrow in Fig. 6) from
the initial point (marked as a solid circle) to the critical point
(marked as an empty circle) with 0 max= = 0.31h h and
= 0.28zκ . After the critical point, when 0h continues to
increase, the wave vector should become imaginary, so the
wave attenuates along the z-axis. Thus, in this point the
localized mode should reflect, i.e., the phenomenon simi-
lar to the total internal reflection occurs. This phenome-
non is of particular interest and can be used to control the
localized mode propagation.
6. Conclusions
In this work, the effect of the external dc magnetic
field on the Josephson plasma modes localized on a slab
of layered superconductor is studied theoretically. The
dispersion relations for the localized modes are obtained
analytically within the WKB approximation and in the
exact form in terms of the Legendre functions. The analy-
sis is performed both for the frequencies ω higher and
lower than the Josephson plasma frequency Jω . It is
shown, that for the wide range of frequencies and wave
numbers, the anomalous dispersion can be observed. The
symmetry of the studied system implies the symmetry of
the localized modes, symmetric and antisymmetric with
respect to the ac magnetic field. For high and low fre-
quency ranges, the localized modes have the different
behavior. For > ,Jω ω the external dc magnetic field
shifts relatively slightly the dispersion curves towards the
lower frequencies. For < ,Jω ω the shift caused by the
dc field is more significant. In this range, the localized
modes evanesce deep into the slab, and waves near two
interfaces are coupled weakly. Therefore, the dispersion
curves for the symmetric and antisymmetric modes nearly
coincide. Additionally, due to the dc magnetic field, all
the dispersion curves starting at > Jω ω descend to
< Jω ω at sufficiently great values of the wave vectors
and end at = 0ω . The variation of the dispersion curves,
when changing the dc magnetic field, reveals the way to
control of the localized modes. In particular, we discuss
the internal reflection of the localized modes propagating
in the external inhomogeneous dc magnetic field. Dis-
cussed phenomenon can be applied in terahertz electron-
ics and photonics for manipulation by the localized Jo-
sephson plasma modes.
We dedicate this paper to the memory of outstanding
physicist-theoretician Alexei Alexeevich Abrikosov on the
occasion of his 90th birthday. Gone, but not forgotten.
We gratefully acknowledge partial support from the
grant of State Fund For Fundamental Research of Ukraine
(Project No. 76/33683)
Appendix A: Associated Legendre functions
Here we describe how the solutions (36) can be derived
in terms of the Legendre functions.
Firstly, we consider Eq. (15) with ( )u ξ taken in the form
2 1
2
2(1 )( ) = .
( )cosh
u
−− Ω
ξ
ξ
(A1)
Introducing new variable = tanh( )τ ξ we can rewrite
Eq. (15) as
2
2
2(1 ) ( ) 2 ( ) ( 1) ( ) = 0,
1
a a a
µ
− τ τ − τ τ + ν ν + − τ′′ ′
− τ
(A2)
where
2 1 2 2 2( 1) = 2(1 ) , = .s s
−ν ν + − Ω κ µ −κ (A3)
The solutions of Eq. (A2) are ( )Pµ
ν τ and ( )Qµ
ν τ , the
associated Legendre functions (see, e.g., [29]) of the first
and second kinds, respectively,
1 2( ) = ( ) ( ),a C P C Qµ µ
ν ντ τ + τ (A4)
or, returning to variable ξ ,
1 2( ) = [tanh( )] [tanh( )].a C P C Qµ µ
ν νξ ξ + ξ (A5)
It should be noted that associated Legendre functions
can be represented in the following form:
/21 1( ) = [ , 1;1 ;(1 )/2],
(1 ) 1
P F
µ
µ
ν
+ τ τ −ν ν + − µ − τ Γ − µ − τ
(A6)
2 /2
1 1
( 1) (1 )( ) =
2 ( 3 / 2)
Q
µ
µ
ν ν+ ν+µ+
π Γ ν + µ + − τ
τ ×
Γ ν + τ
2
1 2 3 1, ; ;
2 2 2
F ν + µ + ν + µ + × ν + τ
(A7)
where ( )zΓ is the Euler gamma-function,
1
0
( ) = ez xz x dx
∞
− −Γ ∫ , (A8)
and [ , ; ; ]F a b c z is the hypergeometric function that is a
solution (regular at z = 0) of Eulers hypergeometric diffe-
rential equation,
[ ](1 ) ( 1) = 0,z z F c a b z F abF− + − + + −′′ ′ (A9)
where prime denotes derivative with respect to z . The
hypergeometric function can be also defined by power series,
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6 719
T. Rokhmanova, S.S. Apostolov, N. Kvitka, and V.A. Yampol’skii
=0
( ) ( )
( , ; ; ) = ,
( ) !
n
n n
nn
a b zF a b c z
c n
∞
∑ (A10)
where ( )nq is the rising Pochhammer symbol
1, = 0,
( ) =
( 1) ( 1), = 1,2, .n
n
q
q q q n n
+ + −
(A11)
Now we use general solution (A5) to find the symmet-
ric and antisymmetric solutions of Eq. (15) with ( )u ξ de-
fined by Eq. (16). For this purpose, we consider this equa-
tions only for 0 < <ξ δ with one of conditions (22). Since
we presume relatively thick slabs, exp( / ) 1,cD λ we
can reduce ( )u ξ to
2 1
2
0
2(1 )( ) =
( )cosh
u
−− Ω
ξ
ξ − ξ
, (A12)
because the second term in the square brackets in Eq. (16)
is negligible. Therefore, the general solution of Eq. (15)
can be presented in the same form as Eq. (A4) but with
changing ξ to 0ξ − ξ ,
1 0 2 0( ) = [tanh( )] [tanh( )]a C P C Qµ µ
ν νξ ξ − ξ + ξ − ξ . (A13)
Finally, we apply one of conditions (22) to the last equa-
tion and achieve the solutions presented in Eqs. (36).
It worth noticing that the associated Legendre functions
display the following asymptotic behavior,
/2
/22[ ] (1 ) , 0 < 1 1,
(1 )
P
µ
µ −µ
ν τ ≈ − τ − τ
Γ − µ
/2
/22 cot( )[ ] (1 )
2 (1 )
Q
µ
µ −µ
ν
π πµ
τ ≈ − τ +
Γ − µ
2 /2
/22 csc( ) csc[ ( )] (1 ) .
2 (1 ) ( ) (1 )
−µ
µπ πµ π µ + ν
+ − τ
Γ + µ Γ −µ − ν Γ − µ + ν
(A14)
These asymptotic expressions are used in the main text to
examine the case of weak dc magnetic fields, see Eq. (40).
_______
1. R. Kleiner, F. Steinmeyer, G. Kunkel, and P. Müller, Phys.
Rev. Lett. 68, 2394 (1992).
2. R. Kleiner and P. Müller, Phys. Rev. B 49, 1327 (1994).
3. L. Ozyuzer, A. E. Koshelev, C. Kurter, N. Gopalsami, Q. Li,
M. Tachiki, K. Kadowaki, T. Yamamoto, H. Minami,
H. Yamaguchi, T. Tachiki, K.E. Gray, W.K. Kwok, and
U. Welp, Science 318, 1291 (2007).
4. M. Tonouchi, Nature Photon. 1, 97 (2007).
5. A. Dienst, E. Casandruc, D. Fausti, L. Zhang, M. Eckstein,
M. Hoffmann, V. Khanna, N. Dean, M. Gensch, S. Winnerl,
W. Seidel, S. Pyon, T. Takayama, H. Takagi, and A. Cavalleri,
Nature Mater. 12, 535 (2013).
6. S. Rajasekaran, E. Casandruc, Y. Laplace, D. Nicoletti, G.D.
Gu, S.R. Clark, D. Jaksch, and A. Cavalleri, Nature Phys. 12,
1012 (2016).
7. Y. Laplace, A. Cavalleri, Adv. Phys: X 1, No. 3, 387 (2016).
8. S. Savel’ev, V. Yampol’skii, and F. Nori, Phys. Rev. Lett.
95, 187002 (2005).
9. V.A. Golick, D.V. Kadygrob, V.A. Yampol’skii, A.L.
Rakhmanov, B.A. Ivanov, and F. Nori, Phys. Rev. Lett. 104,
187003 (2010).
10. A.L. Rakhmanov, V.A. Yampol’skii, J.A. Fan, F. Capasso,
and F. Nori, Phys. Rev. B 81, 075101 (2010).
11. V.A. Yampol’skii, A.V. Kats, M.L. Nesterov, A.Yu. Nikitin,
T.M. Slipchenko, S. Savel'ev, and F. Nori, Phys. Rev. B 76,
224504 (2007).
12. D.V. Kadygrob, N.M. Makarov, F. Perez-Rodriguez, T.M.
Slipchenko, and V.A. Yampol'skii, New J. Phys. 15, 023040
(2013).
13. V.M. Agranovich and D.L. Mills, Surface Polaritons, V.M.
Agranovich and D.L. Mills (eds.), NorthHolland, Amsterdam
(1982).
14. H. Raether, Surface Plasmons, Springer-Verlag, New York
(1988).
15. R. Petit, Electromagnetic Theory of Gratings, Springer,
Berlin (1980).
16. V.A. Yampol’skii, S. Savel’ev, O.V. Usatenko, S.S. Mel’nik,
F.V. Kusmartsev, A.A. Krokhin, and F. Nori, Phys. Rev. B
75, 014527 (2007).
17. S.S. Apostolov, V.I. Havrilenko, Z.A. Maizelis, and V.A.
Yampol’skii, Fiz. Nizk. Temp. 43, 360 (2017) [Low Temp.
Phys. 43, 296 (2017)].
18. V.G. Veselago, Usp. Phys. Nauk 92, 517 (1967).
19. J.B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
20. W.J. Padilla, D.N. Basov, and D.R. Smith, Mater. Today 9,
28 (2006).
21. A. Goswami, S. Aravindan, P.V. Rao, and M. Yoshino, Crit.
Rev. Solid State Mater. Sci. 41, 367 (2016).
22. S.S. Apostolov, Z.A. Maizelis, N.M. Makarov, F. Pérez-
Rodríguez, T. N. Rokhmanova, and V.A. Yampol'skii, Phys.
Rev. B 94, 024513 (2016).
23. T.N. Rokhmanova, S.S. Apostolov, Z.A. Maizelis, and V.A.
Yampol’skii, Fiz. Nizk. Temp. 42, 1167 (2016) [Low Temp.
Phys. 42, 916 (2016)].
24. S.I. Khankina, V.M. Yakovenko, and V.A. Yampol’skii, Fiz.
Nizk. Temp. 38, 245 (2012) [Low Temp. Phys. 38, 193 (2012)].
25. S. Savel’ev, V.A. Yampol’skii, A.L. Rakhmanov, and
F. Nori, Rep. Prog. Phys. 73, 026501 (2010).
26. Ch. Helm and L.N. Bulaevskii, Phys. Rev. B 66, 094514
(2002).
27. V.A. Yampol’skii, D.R. Gulevich, Sergey Savel’ev, and
Franco Nori, Phys. Rev. B 78, 054502 (2008).
28. S. Savel’ev, A.L. Rakhmanov, V.A. Yampol’skii, and F.
Nori, Nature Phys. 2, 521 (2006).
29. A. Ango, Mathematics for Electro- and Radioengineers,
Science, Moscow (1967).
___________________________
720 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6
https://doi.org/10.1103/PhysRevLett.68.2394
https://doi.org/10.1103/PhysRevLett.68.2394
https://doi.org/10.1103/PhysRevB.49.1327
https://doi.org/10.1126/science.1149802
https://doi.org/10.1038/nphoton.2007.3
https://doi.org/10.1038/nmat3580
https://doi.org/10.1038/nphys3819
https://doi.org/10.1080/23746149.2016.1212671
https://doi.org/10.1103/PhysRevLett.95.187002
https://doi.org/10.1103/PhysRevLett.95.187002
https://doi.org/10.1103/PhysRevLett.104.187003
https://doi.org/10.1103/PhysRevB.81.075101
https://doi.org/10.1103/PhysRevB.76.224504
https://doi.org/10.1088/1367-2630/15/2/023040
https://www.osapublishing.org/ol/viewmedia.cfm
https://doi.org/10.1063/1.4789809
https://doi.org/10.1007/978-3-642-81500-3
https://doi.org/10.1103/PhysRevB.75.014527
https://doi.org/10.1063/1.4977740
https://doi.org/10.1063/1.4977740
https://doi.org/10.3367/UFNr.0092.196707d.0517
https://doi.org/10.1103/PhysRevLett.85.3966
https://doi.org/10.1016/S1369-7021(06)71573-5
https://doi.org/10.1080/10408436.2015.1135413
https://doi.org/10.1080/10408436.2015.1135413
https://doi.org/10.1103/PhysRevB.94.024513
https://doi.org/10.1103/PhysRevB.94.024513
https://doi.org/10.1063/1.4966244
https://doi.org/10.1063/1.4966244
https://doi.org/10.1063/1.3691528
https://doi.org/10.1088/0034-4885/73/2/026501
https://doi.org/10.1103/PhysRevB.66.094514
https://journals.aps.org/prb/issues/78/5
https://www.nature.com/
https://docslide.com.br/
1. Introduction
2. Model
2.1. Electromagnetic field in the dielectric
2.2. Main equations for the layered superconductor
2.3. dc magnetic field in the layered superconductor
2.4. Electromagnetic field in the layered superconductor
3. Dispersion relations
3.1. Dispersion relations within the WKB approximation
3.1.1. Frequencies higher than
3.1.2. Frequencies lower than
3.2. Exact solution
4. Numerical analysis
4.1. The distribution of the dc and ac fields
4.2. Effect of the dc magnetic field and anomalous dispersion
5. Internal reflection of the localized modes in the inhomogeneous dc magnetic field
6. Conclusions
Appendix A: Associated Legendre functions
|
| id | nasplib_isofts_kiev_ua-123456789-176166 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T18:17:26Z |
| publishDate | 2018 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Rokhmanova, T. Apostolov, S.S. Kvitka, N. Yampol’skii, V.A. 2021-02-03T19:42:16Z 2021-02-03T19:42:16Z 2018 Effect of a dc magnetic field on the anomalous dispersion of localized Josephson plasma modes in layered superconductors / T. Rokhmanova, S.S. Apostolov, N. Kvitka, V.A. Yampol’skii// Физика низких температур. — 2018. — Т. 44, № 6. — С. 711-720. — Бібліогр.: 29 назв. — англ. 0132-6414 PACS: 74.72.–h, 73.20.Mf, 52.35.Mw https://nasplib.isofts.kiev.ua/handle/123456789/176166 We study theoretically the propagation of Josephson plasma waves (JPWs) localized on a slab of layered superconductor in the presence of an external dc magnetic field. The slab is sandwiched between two dielectric half-spaces and the wave modes propagate across the layers. We derive analytic expressions for the dispersion relations of the localized JPWs and present the numerical simulation for the effect of the external dc magnetic field on the dispersion. The anomalous dispersion of localized JPWs is predicted for a wide range of frequencies, wave vectors, and dc fields. Also, we discuss the possibility of the internal reflection of the localized modes in the inhomogeneous dc magnetic field. This phenomenon can find application in the terahertz electronics for the control of the localized mode propagation. We gratefully acknowledge partial support from the
 grant of State Fund For Fundamental Research of Ukraine
 (Project No. 76/33683) en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Специальный выпуск К 90-летию со дня рождения A.A. Абрикосова Effect of a dc magnetic field on the anomalous dispersion of localized Josephson plasma modes in layered superconductors Article published earlier |
| spellingShingle | Effect of a dc magnetic field on the anomalous dispersion of localized Josephson plasma modes in layered superconductors Rokhmanova, T. Apostolov, S.S. Kvitka, N. Yampol’skii, V.A. Специальный выпуск К 90-летию со дня рождения A.A. Абрикосова |
| title | Effect of a dc magnetic field on the anomalous dispersion of localized Josephson plasma modes in layered superconductors |
| title_full | Effect of a dc magnetic field on the anomalous dispersion of localized Josephson plasma modes in layered superconductors |
| title_fullStr | Effect of a dc magnetic field on the anomalous dispersion of localized Josephson plasma modes in layered superconductors |
| title_full_unstemmed | Effect of a dc magnetic field on the anomalous dispersion of localized Josephson plasma modes in layered superconductors |
| title_short | Effect of a dc magnetic field on the anomalous dispersion of localized Josephson plasma modes in layered superconductors |
| title_sort | effect of a dc magnetic field on the anomalous dispersion of localized josephson plasma modes in layered superconductors |
| topic | Специальный выпуск К 90-летию со дня рождения A.A. Абрикосова |
| topic_facet | Специальный выпуск К 90-летию со дня рождения A.A. Абрикосова |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/176166 |
| work_keys_str_mv | AT rokhmanovat effectofadcmagneticfieldontheanomalousdispersionoflocalizedjosephsonplasmamodesinlayeredsuperconductors AT apostolovss effectofadcmagneticfieldontheanomalousdispersionoflocalizedjosephsonplasmamodesinlayeredsuperconductors AT kvitkan effectofadcmagneticfieldontheanomalousdispersionoflocalizedjosephsonplasmamodesinlayeredsuperconductors AT yampolskiiva effectofadcmagneticfieldontheanomalousdispersionoflocalizedjosephsonplasmamodesinlayeredsuperconductors |