Nonlocal optical response of a layered high-temperature superconductor slab
We theoretically study the effect of the spatial dispersion on the optical response of a layered hightemperature superconductor slab. The nonlocality of the inherently-anisotropic layered superconductor comes from the wave vector dependence of its average permittivity tensor, and leads to the gener...
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| Date: | 2018 |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2018
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| Cite this: | Nonlocal optical response of a layered high-temperature superconductor slab / S. Cortés-López, F. Pérez-Rodríguez // Физика низких температур. — 2018. — Т. 44, № 12. — С. 1630-1638. — Бібліогр.: 24 назв. — англ. |
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| author | Cortés-López, S. Pérez-Rodríguez, F. |
| author_facet | Cortés-López, S. Pérez-Rodríguez, F. |
| citation_txt | Nonlocal optical response of a layered high-temperature superconductor slab / S. Cortés-López, F. Pérez-Rodríguez // Физика низких температур. — 2018. — Т. 44, № 12. — С. 1630-1638. — Бібліогр.: 24 назв. — англ. |
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| container_title | Физика низких температур |
| description | We theoretically study the effect of the spatial dispersion on the optical response of a layered hightemperature superconductor slab. The nonlocality of the inherently-anisotropic layered superconductor comes
from the wave vector dependence of its average permittivity tensor, and leads to the generation of additional
electromagnetic modes just above the characteristic Josephson plasma frequency, that is in the terahertz range.
We calculate p-polarization optical spectra for a Bi₂Sr₂CaCu₂O₈+δ (Bi2212) superconductor slab, which show
very narrow resonances associated with the quantization of the wave vectors of both long-wavelength electromagnetic modes, having negative dispersion, and short-wavelength additional (nonlocal) modes of positive dispersion.
The dependence of the frequency position and shape of the resonances on the nonlocality parameter, the slab thickness, and the components of the quasiparticle conductivity is analyzed. We have found that the quantized longwavelength modes of negative dispersion, which can only be observed at relatively-large slab thicknesses, give rise
to prominent resonances in the p-polarization reflectivity spectrum. On the other hand, the resonances associated
with quantized additional short-wavelength electromagnetic modes are weak, but they can be clearly observed when
the superconductor slab thickness is smaller than the smallest magnetic-field penetration depth.
Теоретично вивчено вплив просторової дисперсії на оптичний відгук шаруватої високотемпературної надпровідної
пластини. Нелокальність анізотропного шаруватого надпровідника обумовлена залежністю від хвильового вектора
середньої величини тензора діелектричної проникності та
призводить до генерації додаткових електромагнітних мод,
частота яких перевищує характерну джозефсонівську плазмову частоту, що відповідає терагерцовому діапазону. Обчислено р-поляризаційні оптичні спектри в надпровідній
пластині Bi₂Sr₂CaCu₂O₈+δ (Bi2212), які вказують на наявність
дуже вузьких резонансів, пов’язаних з квантуванням хвильових векторів як довгохвильових електромагнітних мод, що
мають від’ємну дисперсію, так і короткохвильових додаткових (нелокальних) мод з позитивною дисперсією. Вивчено
частотну залежність та залежність форми резонансів від параметра нелокальності, товщини пластини, крім того,
проаналізовано поведінку різних складових в провідності
квазічастинок. Встановлено, що квантовані довгохвильові
моди з від’ємною дисперсією, які можуть спостерігатися в
пластинах відносно великої товщини, породжують виражені
резонанси в р-поляризаційному спектрі відбитих хвиль. З
іншого боку, резонанси, пов’язані з додатковими квантовими
електромагнітними модами, слабко виражені, але можуть бути
чітко визначені в разі, коли товщина надпровідної пластини не
перевищує найменшу магнітну довжину проникнення.
Ключові слова: шаруваті надпровідники, купратні надпровідники, метаматеріали, просторова дисперсія, тонкі лінії.
Теоретически изучено влияние пространственной дисперсии на оптический отклик слоистой высокотемпературной
сверхпроводящей пластины. Нелокальность анизотропного
слоистого сверхпроводника обусловлена зависимостью от
волнового вектора средней величины тензора диэлектрической проницаемости и приводит к генерации дополнительных электромагнитных мод, частота которых превышает
характерную джозефсоновскую плазменную частоту, что
соответствует терагерцовому диапазону. Вычислены р-поляризационные оптические спектры в сверхпроводящей пластине Bi₂Sr₂CaCu₂O₈+δ (Bi2212), которые указывают на наличие очень узких резонансов, связанных с квантованием
волновых векторов как длинноволновых электромагнитных
мод, имеющих отрицательную дисперсию, так и коротковолновых дополнительных (нелокальных) мод с положительной
дисперсией. Изучена частотная зависимость и зависимость
формы резонансов от параметра нелокальности, толщины
пластины, кроме того, проанализировано поведение различных составляющих в проводимости квазичастиц. Установлено, что квантованные длинноволновые моды с отрицательной дисперсией, которые могут наблюдаться в пластинах
относительно большой толщины, порождают выраженные
резонансы в р-поляризационном спектре отраженных волн. С
другой стороны, резонансы, связанные с дополнительными
квантованными электромагнитными модами, слабо выражены, но могут быть четко определены в случае, когда толщина
сверхпроводящей пластины не превышает наименьшую магнитную длину проникновения.
|
| first_indexed | 2025-12-01T14:01:14Z |
| format | Article |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 12, pp. 1630–1638
Nonlocal optical response of a layered
high-temperature superconductor slab
S. Cortés-López and F. Pérez-Rodríguez
Instituto de Física, Benemérita Universidad Autónoma de Puebla
Apartado Postal J-48, Puebla, Pue. 72570, México
E-mail: scortesl@ifuap.buap.mx
Received July 4, 2018, published online October 26, 2018
We theoretically study the effect of the spatial dispersion on the optical response of a layered high-
temperature superconductor slab. The nonlocality of the inherently-anisotropic layered superconductor comes
from the wave vector dependence of its average permittivity tensor, and leads to the generation of additional
electromagnetic modes just above the characteristic Josephson plasma frequency, that is in the terahertz range.
We calculate p-polarization optical spectra for a Bi2Sr2CaCu2O8+δ (Bi2212) superconductor slab, which show
very narrow resonances associated with the quantization of the wave vectors of both long-wavelength electromag-
netic modes, having negative dispersion, and short-wavelength additional (nonlocal) modes of positive dispersion.
The dependence of the frequency position and shape of the resonances on the nonlocality parameter, the slab thick-
ness, and the components of the quasiparticle conductivity is analyzed. We have found that the quantized long-
wavelength modes of negative dispersion, which can only be observed at relatively-large slab thicknesses, give rise
to prominent resonances in the p-polarization reflectivity spectrum. On the other hand, the resonances associated
with quantized additional short-wavelength electromagnetic modes are weak, but they can be clearly observed when
the superconductor slab thickness is smaller than the smallest magnetic-field penetration depth.
Keywords: layered superconductors, cuprate superconductors, metamaterials, spatial dispersion, thin films.
1. Introduction
The electrodynamic properties of layered high-tempera-
ture superconductors is of great interest because of their
applications in the THz frequency range [1–3] and are well
described by using the model of a periodic system with in-
trinsic Josephson junctions in the unit cell [4–9]. As was
demonstrated in several works (see, for example, the re-
views [8,9] and references therein), the gauge-invariant
phase difference of the order parameter in the junctions
obeys sine-Gordon equations, whereas the electric and mag-
netic fields in the laminar superconductor are determined
from the distribution of such a phase difference. Among the
striking phenomena described by sine-Gordon equations, the
stop-light effect and the excitation of Josephson plasma
waves (JPW) have been of particular interest [8].
The JPW can be excited by a p-polarized electromag-
netic wave incident on the high-temperature superconduc-
tor surface, parallel to the ab plane. In the case of small
wave amplitudes (linear regime), the dispersion relation
between the wave vector component ( )s
zk , parallel to the c
axis, and the frequency ω for propagating modes in a lay-
ered superconductor like Bi2212 has two branches at fre-
quencies above the characteristic Josephson plasma fre-
quency, being in the terahertz (THz) range [8,10]:
= /( ).p c ⊥ω λ ε (1)
Here, ⊥λ is the transverse magnetic field penetration depth,
ε is the high-frequency dielectric constant of the insulating
layers alternating with superconducting layers, and c is the
light velocity in vacuum. It turns out that one of the
branches has negative dispersion ( ( )/ < 0s
zk∂ω ∂ ), whereas
the dispersion of the second one is positive ( ( )/ > 0s
zk∂ω ∂ ).
The appearance of the later branch is owing to the effect of
the dynamical breaking of charge neutrality in the layered
superconductor, which is controlled by the capacitive cou-
pling parameter [8],
2
= ,DR
sD
ε
α (2)
where DR is the Debye length for a charge in a superconduc-
tor, s is the thickness of a superconducting layer, and D is the
period of the insulator-superconducting superlattice.
In the long-wavelength regime ( ( )| | 1s
zk D ), the elec-
tromagnetic response of an inherently anisotropic layered
© S. Cortés-López and F. Pérez-Rodríguez, 2018
Nonlocal optical response of a layered high-temperature superconductor slab
high-temperature superconductor can be described with an
average nonlocal permittivity tensor av
↔ε , whose components
depend not only on the frequency ω, but also on the wave
vector ( )s
zk (see Refs. 10, 11). In the limit of charge neutrality,
when the parameter = 0α , the nonlocality of the layered su-
perconductor and, consequently, the second (additional)
branch of the dispersion relation ( ) ( )s
zk ω disappears. In the
later case, the layered superconductor behaves as a hyperbolic
metamaterial with effective negative refraction index. Indeed,
as is shown in Ref. 11, at frequencies ω above the Josephson
plasma frequency pω , the permittivity components, parallel
and perpendicular to the superconducting planes, have differ-
ent sign. Interesting electromagnetic phenomena in layered
high-temperature superconductors have been described within
the local approach. Thus, for example, the dispersion curves
and the excitation of wave-guide [12–14] and surface Joseph-
son plasma [12,14] waves in a superconductor slab, placed
between two identical dielectrics, have been analyzed by
using a local average permittivity tensor. Moreover, the lo-
cal approach has been successfully applied in studying the
resonant optic transmission through different heterostructures,
containing a layered high-temperature superconductor slab, on
which localized modes can be excited [15,16]. The local con-
tinuum limit has also allowed to describe the transmission of
terahertz radiation through periodically modulated slabs of
layered superconductor [17,18].
The Debye length in a superconductor is usually much
smaller than the London penetration depth and, therefore,
the nonlocality parameter α (2) is typically small. Howev-
er, as is shown in Refs. 8, 10, 11, the breaking of the
charge neutrality of the superconducting layers and the
capacitive interlayer coupling can play an important role in
the dispersion properties of the JPWs when the frequency
ω is very close to the Josephson plasma frequency pω (1).
In the present work we shall study the nonlocal elec-
tromagnetic response of a layered high-temperature super-
conductor slab near the Josephson plasma frequency. A
theoretical formalism, based on the use of an average non-
local effective permittivity to calculate the electromagnetic
field inside a superconductor slab, is described in Sec. 2. In
the model, we apply additional boundary conditions, which
allow us to determine the amplitudes of the additional elec-
tromagnetic modes. We calculate and analyze the disper-
sion relations for p-polarized modes and the optical (re-
flectivity) spectra for a Bi2212 superconductor slab in
Secs. 3 and 4. Here, we also study the effect of the non-
locality parameter α upon the resonant structure of the
optical spectra. Our conclusions are written in Sec. 5.
2. Formulation of the problem
2.1. Geometry of the system
The system considered here is a high- cT superconductor
slab of thickness “d ”, specifically Bi2212, whose structure
is inherently layered and periodic. Its superconducting
planes are assumed to be parallel to the x y− plane and the
system is embedded in vacuum (see Fig. 1). Also assuming
that a monochromatic electromagnetic plane wave with p-
polarization is incident on the superconductor-slab surface
at = 0z , the magnetic field in the upper medium ( 0z ≤ )
can be written as
( ) = , 0,u
i r z+ ≤H H H (3)
where the index “u” indicates the upper medium (vacuum),
“i” the incident beam and “r” the reflected one. The ex-
pressions for iH and rH are, respectively,
= (0, ,0)e ,ik x ik z i tx zi iH + − ωH (4)
= (0, ,0)e , 0.ik x ik z i tx zr rH z− − ω ≤H (5)
In these expressions, = sinxk k θ and = coszk k θ are
the components of the incident wave vector ik , where
= /k cω , ω is the frequency, and θ is the incidence angle.
The magnetic field of the transmitted electromagnetic
wave into the lower medium (vacuum) is given by
( )( ) = (0, ,0)e , .ik x ik z d i tt x ztH z d+ − − ω ≥H (6)
2.2. Electromagnetic field in the superconductor slab
Now, to study the propagation of electromagnetic
waves through a high-temperature layered superconductor,
occupying the space 0 z d≤ ≤ , one can exploit the fact that
the superconductor behaves as a uniaxial crystal in the
long-wavelength limit [10,11]. As is shown in such works,
the constitutive equation, relating the displacement vector
D and the electric field E,
av= ,↔εD E (7)
is determined by a nonlocal average permittivity tensor
av
↔ε . The principal values of av
↔ε are functions of the
wave number ( )s
zk and frequency ω as
Fig, 1. Scheme of a high-temperature layered superconducting
slab. ik and rk are the wave vectors of the incident and reflect-
ed light, respectively.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 12 1631
S. Cortés-López and F. Pérez-Rodríguez
2 2
2
4
( ) = ( ) = 1 ,p x
x y
i γ ω πσ ε ω ε ω ε − +
ωω
(8)
( )2 ( )
( )
2
1 2 (1 cos( ))
( , ) = 1 ,
s
p zs
z z
k D
k
ω + α −
ε ω ε − ω
(9)
2
= ,
1 4 z
p
i
ω
ω
ω
− πσ ε
ω
(10)
where pω is the Josephson plasma frequency defined in
Eq. (1), ||= /⊥γ λ λ is the anisotropy parameter given by the
ratio between the transverse ( ⊥λ ) and parallel ( ||λ ) magnet-
ic-field penetration depths, and D is the period of the array
of insulating and superconducting layers. One should men-
tion that expressions (8) and (9) were derived by assuming
that the thickness of the superconducting layers s is much
smaller than the lattice period D (s D ).
In the region of the superconductor slab, we will look
for the solution for Maxwell equations as a p-polarized
plane wave with a magnetic field given by
( )
= (0, ,0)e .
sik z ik x i tz xyH + − ωH (11)
After substituting Eq. (7) and Eq. (11) into Faraday law
( = / )c t∇× ∂ ∂H D and Ampere–Maxwell law for an aniso-
tropic medium ( 1= ( / )ic
↔−ω ε ∇×E H), we can derive the
relations between the nonzero components of the electric
and magnetic fields inside the superconductor:
( ) = ( ) ,s
z y x xck H Eωε ω
( )= ( , ) ,s
x y z z zck H k E−ωε ω (12)
( ) = ( / ) .s
x z z x yk E k E c H− − ω
The dispersion relation for the electromagnetic waves
inside the inherently-anisotropic layered superconductor
can be straightforwardly obtained from the homogeneous
system of algebraic equations (12). We get
2( ) 2 2
( ) 2
( )
= .
( ) ( , )
s
xz
s
x z z
kk
k c
ω
+
ε ω ε ω
(13)
In order to obtain an explicit expression for the disper-
sion relation, we substitute Eq. (9) along with the next
long-wavelength approximation ( ( )| | 1s
zk D ),
( ) ( )2
( ) 2
2 2
2(1 cos( )) 4 (( )/2)sin( ) = ,
s s
s z z
z
k D k Dk
D D
−
≈ (14)
into Eq. (13). Afterwards, the dispersion relation for a p-
polarized electromagnetic wave acquires the form
4 ( ) 2 ( )sin ( /2) sin ( /2) = 0,s s
z za k D b k D c+ + (15)
where
2 2
2
16
= ,p D
a
c
− αω ε
2
2 2 2 2 4
2 4= 4 ( ) 4 ,p x pb D D
c c
ε ω
ω −ω + ε εαω
2 2
2 4 2 2 4
2 4= ( ) .x x x pc k D D
c c
ω ω
ε − ε ε ω −ω
By solving the biquadratic algebraic equation (15), we
can explicitly express the wave number ( )s
zk as a function
of the frequency ω:
2
( ) 2 4= arcsin .
2
s
z
b b ack
D a
− ± − ±
(16)
Thus, four electromagnetic modes can propagate in the
superconducting slab. The wave number of each mode will
be denoted as follows:
( ) , = 1, 2, 3, 4,j
zk j (17)
where (1) (3)=z zk k− and (2) (4)=z zk k− with the restrictions
(1) (2)Im > 0 and Im > 0.z zk k
The latter implies that the first and second ( = 1, 2j ) elec-
tromagnetic modes decay along the positive direction of
the z axis, whereas the third and fourth modes ( = 3, 4j )
decay in the opposite direction.
The total magnetic field inside the superconducting slab
can be expressed as a linear superposition of the four elec-
tromagnetic modes:
( ) ( )= (0, ( ),0)e ,ik x i ts s xyH z − ωH (18)
where
4 ( )( )
=1
( ) = e .
jik zs zy j
j
H z A∑ (19)
Here jA ( = 1, 2, 3, 4j ) are the amplitudes of the plane
waves.
From Eq. (19) and Faraday law, the x and z compo-
nents of the electric field can be written in the form
4 ( )( ) ( )
=1
( ) = e ,
jik zs j zx j z
x j
cE z A k
ωε ∑ (20)
4 ( )( )
( )
=1
( ) = e .
( , )
jj ik zs x zz j
j z z
Ack
E z
k
−
ω ε ω
∑ (21)
1632 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 12
Nonlocal optical response of a layered high-temperature superconductor slab
2.3. Boundary conditions
In order to calculate the amplitudes jA in Eq. (19), as
well as the amplitudes of the reflected ( rH ) and transmitted
( tH ) electromagnetic waves, the well-known Maxwell
boundary conditions should be applied, namely, the continu-
ity of the tangential components of the electric and magnetic
fields at the surfaces. These conditions for the vacuum-
superconductor interfaces at = 0z and =z d are given by
( ) ( ) ( ) ( )(0) = (0), ( ) = ( ),u s s t
x x x xE E E d E d
( ) ( ) ( ) ( )(0) = (0), ( ) = ( ).u s s t
y y y yH H H d H d (22)
However, the number of unknown amplitudes is six ( jA ,
= 1, 2, 3, 4j ; rH and tH ) and the Maxwell boundary
equations (22) are only four. Therefore, it is necessary to
derive two additional boundary conditions (ABC) to calcu-
late all the amplitudes. As in Ref. 10, we will derive the
ABCs by taking into account the fact that the surface Jo-
sephson junctions have only one neighboring junction. In
other words, there are no superconducting planes outside
the slab. It means that the average of the polarization com-
ponent, parallel to the growth direction of the layered su-
perconductor, over the width of imaginary Josephson junc-
tions just outside the slab should be equal to the
polarization of the external medium.
For the anisotropic layered superconductor having a
nonlocal dielectric response, the polarization vector can be
written as
( ) ( ) ( )= ( ( ),0, ( ))e .ik x i ts s s xx zP z P z − ωP (23)
Here
( ) ( )
,( ) = ( ),s s
x e x xP z E zχ (24)
where , = ( 1)/4e x xχ ε − π, ( ) ( )s
xE z has the form (20) and
4 ( )( ) ( ) ( ) ( )
,
=1
( ) = ( ) ( )e ,
jik zs j s j zz e z z z z
j
P z k E kχ∑ (25)
with ( ) ( )
, ( ) = ( ( ) 1)/4j j
e z z z zk kχ ε − π and
( ) ( )
( )( ) = .
( , )
js j x
z z j
z z
Ack
E k
k
−
ω ε ω
(26)
Since the external medium is vacuum, the polarization
z -component, averaged over the width ( D≈ ) of imaginary
Josephson junctions outside the sample, should vanish.
Hence, the ABCs at = 0z and =z d can be written as
0
( )1 ( ) = 0,s
z
D
P z dz
D
−
∫ (27)
( )
( )1 ( ) = 0.
d D
s
z
d
P z dz
D
+
∫ (28)
Let us expand ( ) ( )s
zP z into the Taylor series. We get
0 0 ( )
( ) ( )
=0
1 1( ) (0)
s
s s z
z z
D D z
PP z dz P z dz
D D z
− −
∂ ≈ + =
∂
∫ ∫
( )
( )
=0
1= (0) ,
2
s
s z
z
z
PP D
z
∂
−
∂
(29)
and
( )
( ) ( )
=
1 1( ) ( )
2
d D s
s s z
z z
d z d
PP z dz P d D
D z
+ ∂
≈ +
∂∫ . (30)
In this way, the additional boundary conditions are
( )
( ) 1(0) (0) = 0,
2
s
s z
z
PP D
z
∂
−
∂
(31)
( )
( ) 1( ) ( ) = 0.
2
s
s z
z
PP d D d
z
∂
+
∂
(32)
Applying these ABCs together with the Maxwell
boundary conditions (22), the reflectivity ( 2= | / |r iR H H )
and transmissivity ( 2= | / |t iT H H ) spectra for the layered
superconductor slab in the far infrared can be calculated.
3. Results
In this section, the theoretical formalism above present-
ed is applied to study the propagation of p-polarization
waves with = 75θ in a Bi2212 superconducting slab. The
effects on the electromagnetic response of the supercon-
ducting slab due to the variation of the nonlocality pa-
rameter, the energy dissipation parameters, and slab
thickness are all analyzed here. First, we have studied the
effect of the nonlocality parameter on the dispersion rela-
tion for the electromagnetic propagating modes and far-
infrared reflectivity spectrum. Three cases are described
and discussed below.
3.1. Quasi-local case
The dispersion relation for the propagating modes
when the nonlocality parameter α is almost zero is shown
in Fig. 2(a), and the corresponding reflectivity is presented
in Fig. 2(b). The superconductor parameters used in the
calculations are [11]: 12= 10pω rad/s, = 500γ , = 12.0ε .
Other parameters are 7= 10−α , =d δ, where δ is the
smallest of the penetration depths for the anisotropic su-
perconductor ( = = / ( ) = 173.20pcδ λ γω ε
nm), and
= 15.35D Å. In order to compare these results with our
previous calculations for the local case ( = 0α ) and without
dissipation, published in Ref. 13, we have considered not
just a very small value of α but also very small energy
losses determined by the parallel and perpendicular con-
ductivities of the normal state quasiparticles. Specifical-
ly, the conductivities are, respectively, 5= 3.6 10x p
−σ ⋅ ω
and 7= 1.8 10z p
−σ ⋅ ω .
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 12 1633
S. Cortés-López and F. Pérez-Rodríguez
According to Eq. (15), for a non-zero α there are al-
ways two additional electromagnetic modes. At the specif-
ic value of the nonlocality parameter 7= 10−α , the branch
with negative dispersion ( (1) ( )zk ω ) practically coincides
with that of the local case ( = 0α ). On the other hand, the
additional electromagnetic modes ( (2) ( )zk ω ) turn out to be
evanescent because the imaginary part (2)
zkℑ is much larg-
er than the real part (2)
zkℜ (see Fig. 2(a)). The black dots in
the figure stand for the frequency positions where the
Fabry–Perot condition is satisfied ( (1)| |= ,zk d nℜ π
= 1, 2n ), and the corresponding resonances appear in
the far infrared spectrum for 7= 10−α (Fig. 2(b)). This
spectrum has resonances at the same frequencies as in re-
flectivity and transmissivity spectra for the local case with
= 75θ (see Fig. 3 in Ref. 13).
3.2. Weak nonlocality
In Fig. 3, the calculations were carried out with
= 0.0015α , whereas the other parameters are the same as
in Fig. 2. As is seen, the dispersion relation of the addi-
tional electromagnetic modes (the branch for (2) ( )zk ω ) now
possesses a pass band of positive dispersion just above the
Josephson plasma frequency. For this reason, the reflectivi-
ty spectrum (Fig. 3(b)) exhibits Fabry–Perot resonances
associated not only with the modes of negative dispersion
(1)( ( )zk ω branch), but also with the additional modes. The
number of the later resonances is rather large because the
wave number (2) ( )zk ω , being almost real in the pass band,
rapidly increases with frequency ω until it reaches the bor-
der of the first Brillouin zone ( (2) / = 1zk Dℜ π or, equiva-
lently, (2) / = 112.83zk dℜ π ). Notice that the additional
Fabry–Perot resonances are weaker and narrower than the
resonances associated with the electromagnetic modes with
negative dispersion.
3.3. Strong nonlocality
The dispersion relation ( ) ( )s
zk ω for the case when the
nonlocality parameter has a realistic value ( = 0.05α
[10,11,19]) for a Bi2212 superconductor is shown in
Fig. 4(a), and the respective p-polarization reflectivity for
a layered high-temperature slab with thickness =d δ is
presented in Fig. 4(b). In the numerical calculations of the
curves we used very small conductivities 5= 3.6 10x p
−σ ⋅ ω
and 7= 1.8 10z p
−σ ⋅ ω , producing a rather small energy
dissipation in the layered high-temperature superconduc-
tor. Although the value = 0.05α could be considered
Fig. 2. (Color online) (a) Dispersion relation ( ) ( )s
zk ω for p -
polarized modes in a Bi2212 superconductor at = 75θ in the
quasi-local case ( 7= 10−α ). (b) Reflectivity spectrum for a
Bi2212 superconductor slab of thickness =d δ with the parame-
ters 5= 3.6 10x p
−σ ⋅ ω and 7= 1.8 10z p
−σ ⋅ ω .
Fig. 3. (Color online) (a) Dispersion relation ( ) ( )s
zk ω for p -
polarized modes in a Bi2212 superconductor at = 75θ in the
case of weak nonlocality ( = 0.0015α ). (b) Reflectivity spectrum
for a Bi2212 superconductor slab of thickness =d δ with the
parameters 5= 3.6 10x p
−σ ⋅ ω and 7= 1.8 10z p
−σ ⋅ ω .
1634 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 12
Nonlocal optical response of a layered high-temperature superconductor slab
small, the nonlocality in this case is well developed and
sufficiently strong. Indeed, with = 0.05α , the resonances
associated with the quantization of the wave vectors, corre-
sponding to the branch of positive dispersion (additional
modes), appear in a wide frequency interval and are clearly
separated from each other (see Fig. 4).
In the panels of Fig. 5 we show p-polarization reflec-
tivity spectra for a superconductor slab as that of Fig. 4,
but with a typical large value of the in-plane component of
the quasiparticle conductivity 4= 3.6 10x pσ ⋅ ω [11,20–22]
and two different small values of the perpendicular com-
ponent: 7= 1.8 10z p
−σ ⋅ ω (panel (a)) and 4= 1.8 10z p
−σ ⋅ ω
(panel (b)). Interestingly, the large value of the conductivi-
ty component xσ increases the light absorption in the su-
perconductor and, consequently, the reflectivity has broad
minima around the first resonances ( = 1, 2, 3n ) of the
quantized modes with negative dispersion. As it is seen,
the resonances associated with the quantized additional
modes are practically unaffected by xσ . The later reso-
nances are smoothed out with increasing the perpendicular
component of the conductivity, namely zσ (compare pa-
nels (a) and (b)).
In panel (a) of Fig. 6 we show the p-polarization reflec-
tivity spectra for Bi2212 superconductor slabs of thick-
nesses =d δ, = 4d δ and = 8d δ, which were calculated by
using realistic values for both in-plane ( 4= 3.6 10x pσ ⋅ ω
[11,20–22]) and transverse ( 3= 1.8 10z p
−σ ⋅ ω [11,22,23])
conductivities. The dispersion relation for the electromag-
netic modes in the layered superconductor is plotted in the
subfigure 6(b). There, the black dots indicate the frequency
and wave vector on the dispersion relation curve, where
the Fabry–Perot condition for a superconductor slab of
thickness = 8d δ is satisfied. The dots positions are in good
agreement with the observed resonances in the correspond-
ing reflectivity spectrum shown in the panel (a). Notice
that only the resonances of electromagnetic modes with
negative dispersion are well-resolved. For this reason, the
dips of the resonances in the reflectivity spectra are shifted
towards higher frequencies when the thickness slab is in-
creased (compare the curves in Fig. 6(a)).
Figure 7(a) exhibits the p-polarization reflectivity spec-
tra for superconductor slabs of thicknesses smaller than the
skin depth δ : = 0.10d δ, = 0.15d δ and = 0.25d δ . The
panel (b) of Fig. 7 shows the dispersion relation curve with
the positions of the frequencies where the Fabry–Perot
condition is satisfied in a slab of thickness = 0.10d δ. Be-
cause of the small slab thickness, only the wave vectors of
Fig. 4. (Color online) (a) Dispersion relation ( ) ( )s
zk ω for p -
polarized modes in a Bi2212 superconductor at = 75θ in the
case of strong nonlocality ( = 0.05α ). (b) Reflectivity spectrum
for a Bi2212 superconductor slab of thickness =d δ with the
parameters 5= 3.6 10x p
−σ ⋅ ω and 7= 1.8 10z p
−σ ⋅ ω .
Fig. 5. (Color online) Effect of energy dissipation on the p -
polarization reflectivity spectrum for a Bi2212 superconductor
slab of thickness =d δ with nonlocality parameter = 0.05α and
at = 75θ as in Fig. 4. Curve in panel (a) was calculated with
4= 3.6 10x pσ ⋅ ω and 7= 1.8 10z p
−σ ⋅ ω . In panel (b),
4= 3.6 10x pσ ⋅ ω and 4= 1.8 10z p
−σ ⋅ ω .
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 12 1635
S. Cortés-López and F. Pérez-Rodríguez
the additional electromagnetic modes are quantized, lead-
ing to the appearance of discernible resonances in the p-
polarization reflectivity spectrum (see panel (a)). As is also
seen, the dips of the resonances in the reflectivity spectrum
are shifted towards higher frequencies when the slab thick-
ness is decreased. There is another remarkable feature of
the reflectivity spectra for thin slabs: the position of the
reflectivity resonances turn out to be slightly shifted to
lower frequencies with respect to the frequencies where the
Fabry–Perot condition is satisfied. In fact, in Fig. 7(a) the
small lines next to the numbers, labeling the resonances,
indicate the frequencies where the Fabry–Perot condition is
really satisfied. This shift of the resonances is attributed to
the type of the additional boundary conditions, (31) and
(32), used in our calculations since the optical spectra of
nonlocal media depend on the ABCs.
4. Discussion of the results
Because of the nonlocality of the optical response of a
layered high-temperature superconductor slab, i.e., as a
result of the wave vector dependence of its average permit-
tivity tensor ( )
av ( )s
zk↔
ε (Eqs. (7)–(9)), for a given frequen-
cy four p-polarized electromagnetic modes can propagate
through the sample. For this reason, to calculate their am-
plitudes it was necessary to apply the Maxwell boundary
conditions (22) together with the ABCs derived in subsec-
tion 2.3, namely Eqs. (31) and (32). Using the classifica-
tion of ABCs, which is employed for other nonlocal media
such as excitonic ones, the above derived ABCs corre-
spond to the generalized ABCs [24]:
( ) ( )
,0 ,0(0) (0)/ = 0,s s
z z z zP P zα +β ∂ ∂
( ) ( )
, ,( ) ( )/ = 0,s s
z d z z d zP d P d zα +β ∂ ∂ (33)
with ,0 ,= = 1z z dα α and ,0 ,= = /2z z d Dβ −β − . In our case,
the applied here ABCs came from the absence of Joseph-
son junctions just outside the superconductor sample. The
choice of the ABCs can qualitatively change the optical
properties of nonlocal systems, e.g., the reflectivity and
transmissivity, since these nonlocal spectra are sensitive to
the microstructure of the sample surfaces. Therefore, to
properly describe the nonlocal response of a superconduc-
tor, the parameters ,0zα , ,z dα , ,0zβ , and ,z dβ in the gener-
alized ABCs (33) can be fitted to experimental optical
spectra.
Fig. 6. (Color online) (a) p -polarization reflectivity spectra for
Bi2212 superconductor slabs of thickness = 1d , 4, 8δ at = 75θ .
The quasiparticle conductivities used are: 4= 3.6 10x pσ ⋅ ω and
3= 1.8 10z p
−σ ⋅ ω . (b) Dispersion relation ( ) ( )s
zk ω for p -pola-
rized modes in a Bi2212 superconductor.
Fig. 7. (Color online) (a) p -polarization reflectivity spectra
for Bi2212 superconductor slabs of thickness = 0.1d , 0.15,
0.25δ at = 75θ . The quasiparticle conductivities used are:
4= 3.6 10x pσ ⋅ ω and 3= 1.8 10z p
−σ ⋅ ω . (b) Dispersion relation
( ) ( )s
zk ω for p -polarized modes in a Bi2212 superconductor.
1636 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 12
Nonlocal optical response of a layered high-temperature superconductor slab
Our results presented in previous section demonstrate
that, even for realistic large in-plane ( xσ ) and transverse
( )zσ components of quasiparticle conductivity, the p-
polarization reflectivity spectra for Bi2212 superconductor
slabs with thickness d larger than the skin depth
= = /( )pcδ λ γω ε
( >d δ) have well-resolved resonances
associated with the quantization of the wave vector for
electromagnetic modes with negative dispersion. In con-
trast, when the thickness d is smaller than δ ( <d δ) the
optical spectra exhibit separated resonances originated by
Fabry–Perot resonances of the additional (short-wave length)
electromagnetic modes inside the superconductor slab.
In Ref. 13, where a local average permittivity tensor
was considered, it was demonstrated that the quantized
electromagnetic modes in a layered superconductor slab
are quasi-longitudinal because of the large anisotropy of its
dielectric response, i.e., a strong contrast between the per-
mittivity components (| | | |x zε ε ). Indeed, as follows
from Eq. (13), we obtain
( )| |
1,
s
xz
x z
k
k
−ε
≈
ε
(34)
and from the Maxwell equation = 0∇⋅D , we get
( )
( )| | = | | .
s
x x z
z x x xs
xz z
k kE E E E
kk
− ε
≈
ε
(35)
Hence, the z component of the electric field is much larger
than its x component. This conclusion is also valid for the
additional short-wavelength electromagnetic modes gener-
ated in the nonlocal case because they have even larger
wave numbers ( )s
zk ( ( )| |s
z xk k ) and the inequality (35) is
fulfilled.
5. Conclusions
As a result of the spatially-dispersive (nonlocal) optical
response of layered high-temperature superconductors,
additional electromagnetic modes are generated in the p-
polarization geometry. The calculated p-polarization THz
spectra for a Bi2212 superconductor slab show very nar-
row resonances associated with the quantization of the
wave vectors of long-wavelength electromagnetic modes,
having negative dispersion, and short-wavelength addi-
tional modes of positive dispersion, in the frequency inter-
val just above the characteristic Josephson plasma frequen-
cy of the superconductor. The frequency positions of the
resonances are determined by the nonlocality parameter,
the slab thickness and the angle of incidence. In the case
when the thickness slab is larger than the skin depth
( > )d δ , the discernible resonances in reflectivity spectra
are due to the excitation of electromagnetic modes with
anomalous dispersion and, therefore, they undergo a shift
towards higher frequencies as the slab thickness is in-
creased. At slab thicknesses smaller than the skin depth
( <d δ), the dips of the resonances are mainly associated
with quantized additional electromagnetic modes. Because
of their positive dispersion, such resonances are shifted
towards higher frequencies as the slab thickness is de-
creased. We have found that the quantized electromagnetic
modes are quasi-longitudinal because of the strong anisot-
ropy in the nonlocal optical response of the high-
temperature superconductor.
Acknowledgments
This work was partially supported by CONACYT
(grant CB-2012-01-183673) and VIEP-BUAP (grant
100160855-VIEP2018).
________
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___________________________
Нелокальний оптичний відгук шаруватої
високотемпературної надпровідної пластини
S. Cortés-López, F. Pérez-Rodríguez
Теоретично вивчено вплив просторової дисперсії на оп-
тичний відгук шаруватої високотемпературної надпровідної
пластини. Нелокальність анізотропного шаруватого над-
провідника обумовлена залежністю від хвильового вектора
середньої величини тензора діелектричної проникності та
призводить до генерації додаткових електромагнітних мод,
частота яких перевищує характерну джозефсонівську плаз-
мову частоту, що відповідає терагерцовому діапазону. Об-
числено р-поляризаційні оптичні спектри в надпровідній
пластині Bi2Sr2CaCu2O8+δ (Bi2212), які вказують на наявність
дуже вузьких резонансів, пов’язаних з квантуванням хвильо-
вих векторів як довгохвильових електромагнітних мод, що
мають від’ємну дисперсію, так і короткохвильових додатко-
вих (нелокальних) мод з позитивною дисперсією. Вивчено
частотну залежність та залежність форми резонансів від па-
раметра нелокальності, товщини пластини, крім того,
проаналізовано поведінку різних складових в провідності
квазічастинок. Встановлено, що квантовані довгохвильові
моди з від’ємною дисперсією, які можуть спостерігатися в
пластинах відносно великої товщини, породжують виражені
резонанси в р-поляризаційному спектрі відбитих хвиль. З
іншого боку, резонанси, пов’язані з додатковими квантовими
електромагнітними модами, слабко виражені, але можуть бути
чітко визначені в разі, коли товщина надпровідної пластини не
перевищує найменшу магнітну довжину проникнення.
Ключові слова: шаруваті надпровідники, купратні надпро-
відники, метаматеріали, просторова дисперсія, тонкі лінії.
Нелокальный оптический отклик слоистой
высокотемпературной сверхпроводящей
пластины
S. Cortés-López, F. Pérez-Rodríguez
Теоретически изучено влияние пространственной диспер-
сии на оптический отклик слоистой высокотемпературной
сверхпроводящей пластины. Нелокальность анизотропного
слоистого сверхпроводника обусловлена зависимостью от
волнового вектора средней величины тензора диэлектриче-
ской проницаемости и приводит к генерации дополнитель-
ных электромагнитных мод, частота которых превышает
характерную джозефсоновскую плазменную частоту, что
соответствует терагерцовому диапазону. Вычислены р-поля-
ризационные оптические спектры в сверхпроводящей пла-
стине Bi2Sr2CaCu2O8+δ (Bi2212), которые указывают на на-
личие очень узких резонансов, связанных с квантованием
волновых векторов как длинноволновых электромагнитных
мод, имеющих отрицательную дисперсию, так и коротковол-
новых дополнительных (нелокальных) мод с положительной
дисперсией. Изучена частотная зависимость и зависимость
формы резонансов от параметра нелокальности, толщины
пластины, кроме того, проанализировано поведение различ-
ных составляющих в проводимости квазичастиц. Установле-
но, что квантованные длинноволновые моды с отрицатель-
ной дисперсией, которые могут наблюдаться в пластинах
относительно большой толщины, порождают выраженные
резонансы в р-поляризационном спектре отраженных волн. С
другой стороны, резонансы, связанные с дополнительными
квантованными электромагнитными модами, слабо выраже-
ны, но могут быть четко определены в случае, когда толщина
сверхпроводящей пластины не превышает наименьшую маг-
нитную длину проникновения.
Ключевые слова: слоистые сверхпроводники, купратные
сверхпроводники, метаматериалы, пространственная диспер-
сия, тонкие линии.
1638 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 12
https://doi.org/10.1088/1367-2630/15/2/023040
https://doi.org/10.1063/1.4894323
https://doi.org/10.1103/PhysRevB.67.024510
https://doi.org/10.1103/PhysRevLett.77.735
https://doi.org/10.1023/A:1022536120728
https://doi.org/10.1103/PhysRevB.68.134504
https://doi.org/10.1103/PhysRevB.68.134504
https://doi.org/10.1103/PhysRevLett.82.5345
1. Introduction
2. Formulation of the problem
2.1. Geometry of the system
2.2. Electromagnetic field in the superconductor slab
2.3. Boundary conditions
3. Results
3.1. Quasi-local case
3.2. Weak nonlocality
3.3. Strong nonlocality
4. Discussion of the results
5. Conclusions
Acknowledgments
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| id | nasplib_isofts_kiev_ua-123456789-176459 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-01T14:01:14Z |
| publishDate | 2018 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Cortés-López, S. Pérez-Rodríguez, F. 2021-02-04T17:47:03Z 2021-02-04T17:47:03Z 2018 Nonlocal optical response of a layered high-temperature superconductor slab / S. Cortés-López, F. Pérez-Rodríguez // Физика низких температур. — 2018. — Т. 44, № 12. — С. 1630-1638. — Бібліогр.: 24 назв. — англ. 0132-6414 https://nasplib.isofts.kiev.ua/handle/123456789/176459 We theoretically study the effect of the spatial dispersion on the optical response of a layered hightemperature superconductor slab. The nonlocality of the inherently-anisotropic layered superconductor comes from the wave vector dependence of its average permittivity tensor, and leads to the generation of additional electromagnetic modes just above the characteristic Josephson plasma frequency, that is in the terahertz range. We calculate p-polarization optical spectra for a Bi₂Sr₂CaCu₂O₈+δ (Bi2212) superconductor slab, which show very narrow resonances associated with the quantization of the wave vectors of both long-wavelength electromagnetic modes, having negative dispersion, and short-wavelength additional (nonlocal) modes of positive dispersion. The dependence of the frequency position and shape of the resonances on the nonlocality parameter, the slab thickness, and the components of the quasiparticle conductivity is analyzed. We have found that the quantized longwavelength modes of negative dispersion, which can only be observed at relatively-large slab thicknesses, give rise to prominent resonances in the p-polarization reflectivity spectrum. On the other hand, the resonances associated with quantized additional short-wavelength electromagnetic modes are weak, but they can be clearly observed when the superconductor slab thickness is smaller than the smallest magnetic-field penetration depth. Теоретично вивчено вплив просторової дисперсії на оптичний відгук шаруватої високотемпературної надпровідної пластини. Нелокальність анізотропного шаруватого надпровідника обумовлена залежністю від хвильового вектора середньої величини тензора діелектричної проникності та призводить до генерації додаткових електромагнітних мод, частота яких перевищує характерну джозефсонівську плазмову частоту, що відповідає терагерцовому діапазону. Обчислено р-поляризаційні оптичні спектри в надпровідній пластині Bi₂Sr₂CaCu₂O₈+δ (Bi2212), які вказують на наявність дуже вузьких резонансів, пов’язаних з квантуванням хвильових векторів як довгохвильових електромагнітних мод, що мають від’ємну дисперсію, так і короткохвильових додаткових (нелокальних) мод з позитивною дисперсією. Вивчено частотну залежність та залежність форми резонансів від параметра нелокальності, товщини пластини, крім того, проаналізовано поведінку різних складових в провідності квазічастинок. Встановлено, що квантовані довгохвильові моди з від’ємною дисперсією, які можуть спостерігатися в пластинах відносно великої товщини, породжують виражені резонанси в р-поляризаційному спектрі відбитих хвиль. З іншого боку, резонанси, пов’язані з додатковими квантовими електромагнітними модами, слабко виражені, але можуть бути чітко визначені в разі, коли товщина надпровідної пластини не перевищує найменшу магнітну довжину проникнення. Ключові слова: шаруваті надпровідники, купратні надпровідники, метаматеріали, просторова дисперсія, тонкі лінії. Теоретически изучено влияние пространственной дисперсии на оптический отклик слоистой высокотемпературной сверхпроводящей пластины. Нелокальность анизотропного слоистого сверхпроводника обусловлена зависимостью от волнового вектора средней величины тензора диэлектрической проницаемости и приводит к генерации дополнительных электромагнитных мод, частота которых превышает характерную джозефсоновскую плазменную частоту, что соответствует терагерцовому диапазону. Вычислены р-поляризационные оптические спектры в сверхпроводящей пластине Bi₂Sr₂CaCu₂O₈+δ (Bi2212), которые указывают на наличие очень узких резонансов, связанных с квантованием волновых векторов как длинноволновых электромагнитных мод, имеющих отрицательную дисперсию, так и коротковолновых дополнительных (нелокальных) мод с положительной дисперсией. Изучена частотная зависимость и зависимость формы резонансов от параметра нелокальности, толщины пластины, кроме того, проанализировано поведение различных составляющих в проводимости квазичастиц. Установлено, что квантованные длинноволновые моды с отрицательной дисперсией, которые могут наблюдаться в пластинах относительно большой толщины, порождают выраженные резонансы в р-поляризационном спектре отраженных волн. С другой стороны, резонансы, связанные с дополнительными квантованными электромагнитными модами, слабо выражены, но могут быть четко определены в случае, когда толщина сверхпроводящей пластины не превышает наименьшую магнитную длину проникновения. This work was partially supported by CONACYT (grant CB-2012-01-183673) and VIEP-BUAP (grant 100160855-VIEP2018). en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Електронні властивості низьковимірних систем Nonlocal optical response of a layered high-temperature superconductor slab Нелокальний оптичний відгук шаруватої високотемпературної надпровідної пластини Нелокальный оптический отклик слоистой высокотемпературной сверхпроводящей пластины Article published earlier |
| spellingShingle | Nonlocal optical response of a layered high-temperature superconductor slab Cortés-López, S. Pérez-Rodríguez, F. Електронні властивості низьковимірних систем |
| title | Nonlocal optical response of a layered high-temperature superconductor slab |
| title_alt | Нелокальний оптичний відгук шаруватої високотемпературної надпровідної пластини Нелокальный оптический отклик слоистой высокотемпературной сверхпроводящей пластины |
| title_full | Nonlocal optical response of a layered high-temperature superconductor slab |
| title_fullStr | Nonlocal optical response of a layered high-temperature superconductor slab |
| title_full_unstemmed | Nonlocal optical response of a layered high-temperature superconductor slab |
| title_short | Nonlocal optical response of a layered high-temperature superconductor slab |
| title_sort | nonlocal optical response of a layered high-temperature superconductor slab |
| topic | Електронні властивості низьковимірних систем |
| topic_facet | Електронні властивості низьковимірних систем |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/176459 |
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