The two-fluid model of superconductors

In 1934, Gorter and Casimir provided the two-fluid model for the electric conduction mechanism of superconductors [12,13]. When T >0 K, the two-fluid model can very successfully explain the character of superconductor device under the

condition of superconductivity conduction. According to this theory, one assumes that the total electron density n can be divided into two kinds of conduction electrons, one of which is normal electrons n and another is superconducting electrons _n n . That is _s

n=n_n+ , (2.4.1) n_s and they have different conductivity

σ σ= _n+σ_s , (2.4.2) where σ_n is the conductivity of normal electrons, and σ_s is the conductivity of paired superconductivity electrons. From the Drude model and London equation, we get

( )

If the relaxation time τ → ∞, the conductivity is a complex number, so under the condition of f ≠ , we have 0 In order to satisfy the approximation of the conductivity imaginary part, we assumed the condition is low frequency, so

Therefore according to the two-fluid model the electromagnetic response of a

superconductor can be described in terms of the complex conductivity, σ σ= ₁− jσ₂, where the real part, σ₁, indicating the loss, is contributed by the normal electrons, whereas the imaginary part, σ₂, is due to the superelectrons.

Under the low temperature condition,the imaginary part is expressed as [12,13]

_{2 2}

0

1

L

σ ⁼ωµ λ , (2.4.7)

where the temperature-dependent penetration depth is given by

( )

We shall consider the lossless case, meaning that the real part of the complex conductivity of the superconductor can be neglected and consequently it becomes

2 2 The conditions for a lossless superconductor are well described in Ref. [10,11]. With Eq.

(2.4.4), the relative permittivity as well as its associated index of refraction can be obtained, namely

Chapter 3

Photonic band structure for a

superconductor-dielectric superlattice

3.1 Introduction

It is well known that photonic crystals have photonic band gaps (PBGs) in the photonic dispersion relation. In the PBGs, optical waves with certain frequencies are not allowed to propagate through the crystal [1, 2].The PBGs are analogous to the electronic band gaps in a solid and their physical origin can be ascribed to the Bragg diffraction in a periodic multilayer structure. A simple one-dimensional photonic crystal is, in general, made of alternating layers of material with different permittivities, forming a superlattice with infinite periods. The band structure for a dielectric-dielectric photonic crystal shows that the PBG between the first and second bands widens considerably as the difference in dielectric permittivity is increased [3]. In addition, no low-frequency band gap below the first (lowest) band can be found. In a metallic photonic made of a normal metal and a dielectric, it is however found that a low-frequency (or metallicity) gap may exist.

Contrary to a PBG, this metallicity gap which does not depend on the periodicity, is of the order of the plasma frequency and thus is regarded as a modified effective plasma frequency [35-37].

On the other hand, studies of photonic crystals consisting of a superconducting material and a dielectric have also been reported recently [10-12].The electromagnetic properties of Abrkosov vortex lattice as a photonic crystal were investigated by changing the Ginzburg-Landau parameter and static magnetic field [18]. In addition to a low-frequency band gap below the first band, they also obtained the PBGs for a

superconductor in the presence of vortices. In fact, the issue of a superconducting photonic crystal was first investigated by a group in Singapore [19,20]. They considered a one-dimensional superconductor-dielectric superlattcie. By making use of the transfer matrix method accompanied by the Bloch theorem [19], a low-frequency band gap was seen for both transversal magnetic (TM) and transversal electric (TE) modes. This band gap was found to be about one third of the threshold frequency of a bulk superconducting material. The physical information from this work for TE mode however is quite limited because only the first band is given. As for the other higher bands in addition to the possible PBGs cannot be obtained there. In other words, a full band structure for this one-dimensional superconducting photonic crystal remains unavailable thus far.

A full band structure is a basic and important means for understanding the fundamental physics about electromagnetic wave propagation characteristics in a photonic

crystal. This information is not only of fundamental but also of technical use for a superconducting material. Motivated by this, in this dissertation we shall extend the

work of Ref. [20]. We would like to present the full photonic band structure for TE mode in a superconductor-dielectric photonic crystal. Firstly, we use the Abeles theory for a stratified media to calculate the frequency-dependent transmittance [7]. From the transmittance spectrum, we can clearly learn the locations of all possible pass bands and stop bands. With these in hand, one is able to calculate the band structure from the transcendental equation based on the transfer matrix method together with the Bloch theorem. Then a comparison between the transmittance spectrum and full band structure will be made.

The format of this work is as follows: Section II describes the theoretical approaches to be used in the calculation. The calculated transmittance spectrum and band structure will be given in Section III. Discussion on the PGBs will also be made in Section III. A summary will be addressed in Section 4.

3.2 Theory

A one-dimensional nonmagnetic superconductor-dielectric photonic crystal will be modeled as a periodic superconductor-dielectric multilayer structure with a large number of periods, N >>1. Such an N-period superlattice is shown in Fig. 3.1, where

2 3

a=a + is the spatial periodicity, wherea a₂ is the thickness of the superconducting layer and a denotes the thickness of the dielectric layer. We consider that a TE wave is ₃ incident at an angle θ₁ from the top medium which is taken to be free space with a refractive index, n₁= . The index of refraction of the lossless dielectric is given by 1

3 r3

n = ε , whereε_r3 is its relative permittivity. For the superconductor, the index of refraction can be described on the basis of the conventional two-fluid model [20].

According to the two-fluid model the relative permittivity as well as its associated index of refraction can be obtained, namely According to the Abeles theory, the reflection and transmission coefficients can be determined and are given by¹¹

for the last medium. Both media here are taken to be free space. The reflectance (reflectivity) R, transmittance (transmissivity) T and r , t are related by

R=  , r2 ² 1

T p t

= p^A  . (3.5)

Next, we are going to briefly describe the method used in Ref. [20] for a direct calculation of the band structure in a periodic superconductor-dielectric medium. Based on the basic assumption of translational symmetry and aided by the Floquet (or Bloch) theorem together with the use of transfer matrix method, one can obtain a transcendental equation determining the band structure, namely [8,20]

where K is the Bloch wave number,

k_{x r3} sin² ₁

Equation (3.17) can be numerically solved for ω as a function of K, yielding the so-called photonic band structure or dispersion relation. In Ref. [20], only the first band is given and thus it is not sufficient to explore the whole optical properties in a photonic crystal. In the next section, we shall give other possible higher bands. The higher bands then enable us to study the PGBs.

Before presenting the numerical results we mention that the above theoretical formulations are based on the flat interface model. This is legitimate and widely used to theoretically study the fundamental optical properties in a photonic crystal [3, 19, 20]. In the actual material, some interface issues such as interface roughness, lattice imperfection, and surface discontinuity may arise due to the process of a film growth. To

study surface effect on a photonic crystal, other method such as the plane-wave expansion may be employed and some works are available [38]. A study of interface effect on the photonic crystal is not our interest here.

3.3 Numerical Results and Discussion

3.3.1 Superconductor-dielectric superlattice

Let us now present the numerical results according to the aforementioned equations. Two dimensionless quantities such as Ω ω= a 2πc and Λ=a 2πλ_L will be used as usual in the analysis of photonic bands. We also define the dielectric thickness ratio as r=a a₃ . Figure 3.2 displays the calculated transmittance spectrum (right) and the band structure (left) for the conditions of θ₁=45⁰, ε_r3 =15, Λ=0.05, r=1 3, and N =500. It is seen that the calculated transmission spectrum is in fairly good agreement with that of the calculated band structure. For the sake of convenience, the first six cutoff frequencies (at which T = 0) are denoted by Ω_i, 1, 2, 3 ~ 6i= , as shown in Fig. 3.2. The first band gap, denoted by ∆₁, is equal to Ω₁ =0.017. The first band gap is referred to as the low-frequency (LF) gap [20], which is not seen in the dielectric-dielectric superlattice. This gap size is nearly equal to one third of the cutoff frequency Ω_c for a bulk superconductor which is in value of 0.05 here. Thus, its origin can be regarded as a combined effect of the spatial periodicity and of the addition of dielectric material [20]. The dimensionless bulk cutoff frequency Ω_c =0.05 is equal to a real frequency of ω_c =c λ_L ~ 10¹⁵ s^-1, which is of the same order of plasma frequency for most alkali metals. The dispersion relation for a bulk superconductor is thus recognized as an analogy to the plasma dispersion in metals.⁸ On the other hand, ∆₁ also appears in a metallic photonic crystal but its size is near the plasma cutoff frequency, meaning that it does not depend on the periodicity [18].Thus in the metallic superlattice

∆1 is not a real PBG, whereas it is a true PBG in the superconductor superlattice because

∆1 is indeed related to the periodicity.

In addition to ∆₁, along with the first band from Ω₁=0.017 to Ω₂ =0.165, other higher bands as well as PBGs are also displayed in Fig. 2. The second PBG is denoted by ∆₂ equal to Ω Ω₃− ₂ =0.368 0.165− =0.203 . That is almost twelve times larger than ∆₁. The second band is located from Ω₃=0.368 to Ω₄ =0.455. The third PBG, ∆₃, is Ω Ω₅− ₄ =0.712 0.455− =0.257 in magnitude and is greater than ∆₂ appreciably. The third band is then above Ω₅ and under Ω₆. From the results in Fig. 2 we can deduce that the photonic band structure for a one-dimensional superconducting photonic crystal is quite reminscent of the electronic band structure. Moreover, it has multiple PGBs, instead of having just one lowest band gap as reported in Ref. [20].

Figure 3.3 shows the first five cutoff frequencies and PBGs as a function of penetration depth at the conditions of θ =45⁰, ε_r3 =15, r=1 3, and N=500. The first one, Ω₁, being equal to ∆₁, increases with increasing Λ. The dependence of ∆₂ on Λ is similar to that of ∆₃. Both also increase as Λ increases. The variations in ∆₂ and ∆₃, however, are not as salinet as ∆₁, especially at small values of Λ. Figure 3.4 shows the calculated frequencies and PBGs as a function of angle of incidence at

3 15

εr = , Λ=0.05, r=1 3, and N =500. It is seen that gap ∆₁ essentially does not change with the variation in the angle of incidence, indicating an omnidirectional feature.

In addition, ∆₂ changes slightly as a function of angle of incidence. The change in the third gap size, ∆₃ is appreciable for θ₁ smaller than 20⁰ and becomes nearly linear between 20⁰ and 60⁰. It then approaches a saturation value of about 0.32. In Fig. 3.5, we have plotted the cutoff frequencies versus dielectic constant of dielectric layer for

450

θ = , Λ=0.05, r=1 3, and N =500. All the cutoff frequencies, in general, decrease with increasing dielectric constant. The corresponding first three gap sizes are

depicted in Fig. 3.6, where ∆₁ decreases slowly with increasing dielectric constant. A peak vaule in ∆₂ is attained for ε_r3 = , and then decreases as the dielectric constant 5 increases. As for ∆₃, it also attains a maximum when ε_r3 =10, and ∆₃ is equal to ∆₁ for ε_r3 = . 3

In the above numerical results, the calculated frequency for a superconductor photonic crystal (SPC) is normalized in 1 λ₀ , the sole material parameter of a superconductor involved in the formulation. This indicates that the results are valid for all the possible superconductors described by the two-fluid model [20]. Most high-T_c cuprates have a value of λ₀ ≈200 300− nm, corresponding to infrared region. As for the conventional superconductor such as a typical A15 compound superconductor with Tc above 10 K, λ₀ ≈60 90− nm, it then can work in the yellow to violet region. The feasibility of a SPC has been well discussed by Feng et al [39].

3.3.2 Extraordinary optical properties in near-zero-permittivity operation range

Let us investigate the reflectance in the vicinity of superconductor threshold wavelength at various angle of incidence. To calculate the reflection response, the layer 1 is taken to be the typical high-T_c superconductor, YBa₂Cu₃O₇ (YBCO) with T_c = 92 K and λ₀ =140 nm [40], and the layer 2 is MgO with ε_2r =10 . The operating temperature is T = 77 K in our simulation. The penetration depth λ_L and the permittivity ε_1r of YBCO can be calculated according to Eqs. (3.16) and (3.17). With these material parameters, the threshold wavelength of YBCO is calculated to be

λth=1245 nm. In addition, the superconductor-dielectric superlattice is immersed in free space, i.e, n =₀ n =1. _f

We first consider the conditions of that the thicknesses of YBCO and MgO layers

are set to be d₁=130 nm and d₂ =80 nm, respectively, and the number of periods is wavelength λ_th is contained. Such a PBG is referred to as a near-zero-ngap because within this gap the refractive index of superconductor is much less one and very close to zero. This gap is strongly dependent on the angle of incidence, and increases largely as the angle increases. This additional PBG however cannot be seen for the TE wave. This gap is due to the existence of radial component of the electric field, Eρ. This Eρ interacts with the superelectrons in the superconductor and thus a superpolariton gap is created.

Figure 3.8 shows the calculated bandedge frequencies as a function of angle of incidence at the conditions of d₁=130 nm, d₂ =80 nm and the number of periods is N =10. It is seen that there is an additional PBG appears near λ_th in the oblique-incidence case.

Except the additional PBG, the other PBG gradually disappear or appear as the angle increases. When the angle of incidence increases, this additional gap which is at λ_th doesn’t disappear and increases largely.

Next, we shall investigate the PBG which the threshold wavelength of YBCO, λth=1245 nm, is located within. To reach this end, the thicknesses of YBCO and MgO layers are also changed to be d₁=100mm and d₂ =140mm and the number of periods

20

N = is taken in our calculation. In Fig. 3.9, we see that the threshold wavelength of YBCO λ_th is located within the PBG. We can see that there are dips near λ_th within the PBG at oblique incidence for TM wave. The dip in TM wave is shallower compared with the TE wave. The appearance of such dips in reflectance is mainly due to the field component H_ρ of TM wave, which, in fact, does not show up in the PBR in the normal-incidence case. The deep dip in TE wave enables us to design a transmission

narrowband filter or resonator without introducing any physical defect. Moreover, a multi-resonance filter is also possible because of the presence of the multiple dips in the reflection response.

3.4 Summary

By using the Abeles theory for a stratified medium and two-fluid model for a superconductor, we have calculated the TE mode transmittance spectrum for a superconductor-dielectric superlattice. We have also presented the photonic band structure based on the transfer matrix method together with the Bloch theorem. Results show excellent agreement for both methods. From the calculated results, some conclusions can be drawn as follows: For a one dimensional superconducting photonic crystal, the band structure shows a multiple-PBG structure, not just the first band as shown previously in Ref. [11]. The fundamental difference is the existence of the low-frequency band gap which is not shown in all-dielectric photonic crystals. This gap is a true PBG, whereas it is not a PBG for a metallic photonic crystal. Besides the first band gap, we also have investigated the second and third PBGs as a function of penetration depth, angle of incidence, and permittivity of dielectric. The results reveal more basic information for the electromagnetic response of superconductor and it could be of technical use in superconducting electronics. Furthermore, we find that there are an additional PBG or dips appear near λ_th in the oblique-incidence case.

θ

₁

θ

₁

n

₁

n

₁

n

₂

n

₃

n

₂

n

₃

Superconductor

Superconductor Dielectric

Dielectric

a

₂

a

₃

a

Fig. 3.1. A superconductor-dielectric periodic layered structure. A transversal electric mode optical wave is incident obliquely from the top medium at an angle of incidence θ1 on the plane superconductor boundary. The media are characterized by distinct indices of refraction n , ₁ n , and ₂ n , respectively. The period is a and the thicknesses of ₃ superconductor and dielectric layeres are denoted by a , and ₂ a , respectively. ₃

K a / 2 π

Fig. 3.2. The calculated transmittance spectrum (right) and the band structure (left). The horizontal dash lines mark the first six cutoff frequencies denoted by Ω_i, 1, 2, 3 ~ 6i= . Excellent agreement is achieved in both results. The conditions are θ₁=45⁰, ε_r3 =15,

Λ=0.05, r=1 3, and N =500

0.0 0.2 0.4 0.6 0.8 1.0

Ω = ω a / 2 π c

0.0 0.2 0.4 0.6 0.8

Λ = a / 2 π λ

_L

Ω

₅

Ω

₁

= ∆

₁

Ω

₄

Ω

₃

Ω

₂

∆

₂

= Ω

₃

− Ω

₂

∆

₃

= Ω

₅

− Ω

₄

Fig. 3.3. Calculated five cutoff frequencies (solid lines) as well as the first three PBGs (dotted lines) as a function of penetration depth. The conditions are θ₁=45⁰, ε_r3 =15,

1 3

r= , and N =500.

Angle of Incidence, θ

₁

0 20 40 60 80

Ω = ω a / 2 π c

0.0 0.2 0.4 0.6 0.8

Ω

₁

= ∆

₁

Ω

₅

Ω

₄

Ω

₃

Ω

₂

∆

₂

= Ω

₃

- Ω

₂

∆

₃

= Ω

₅

- Ω

₄

Fig. 3.4. Calculated five cutoff frequencies (solid lines) as well as the first three PBGs (dotted lines) as a function of angle of incidence. The conditions are ε_r3 =15, Λ=0.05,

1 3

r= , and N =500.

Dielectric Constant, ε

_r

5 10 15 20 25 30

Ω = ω a / 2 π c

0.0 0.2 0.4 0.6 0.8 1.0

Ω

₅

Ω

₄

Ω

₃

Ω

₂

Fig. 3.5. Calculated four cutoff frequencies as a function of dielectric constant. The conditions are θ₁=45⁰, Λ=0.05, r=1 3, and N =500.

Col 1 vs Col 2

Dielectric Constant, ε

_r

5 10 15 20 25 30

Ω = ω a / 2 π c

0.00 0.05 0.10 0.15 0.20 0.25 0.30

∆

₁

= Ω

₁

∆

₂

= Ω

₃

− Ω

₂

∆

₃

= Ω

₅

− Ω

₄

Fig. 3.6. The corresponding first three PBGs as a function of dielectric constant calculated from Fig. 3.5 at the same conditions.

Fig. 3.7. Calculated TM wave reflectance spectra of a superconductor-dielectric superlattice at different incident angle (a) θ = , (b) 0⁰ θ =15⁰, (c) θ =30⁰, (d) θ =45⁰ and (e) θ =60⁰, respectively, under the conditions of d₁ =130nm, d₂ =80nm and

10 N = .

Fig. 3.8. Calculated bandedge frequencies as a function of angle of incidence under the conditions of d₁=130 nm, d₂ =80 nm and the number of periods is N =10.

Fig. 3.9. Calculated TM wave reflectance spectra of a superconductor-dielectric superlattice for different incident angle (a) θ =0, (b) θ =0.1, (c) θ =0.2, (d) θ =0.3, (e) θ =0.4, (f) θ =0.5, (g) θ =0.6, (h) θ =0.7, (i) θ =0.8, (j) θ =0.9, (k) θ =1.0 and (l) θ =1.1, respectively, under the conditions of d₁=100nm, d₂ =140nm and N =20.

Chapter 4

Optical properties of a superconducting annular periodic multilayer structure

4.1. Introduction

The study of a Bragg reflector (BR) or one-dimensional photonic crystal (1DPC) has been the interesting subject and has attracted lots of attention in recent years. There have been lots of reports on the calculations of the photonic band structures in 1DPCs thus far [41-43]. It is known that PCs have photonic band gaps (PBGs) at which the electromagnetic waves cannot propagate through the layered structures. Materials with PBGs have been playing an important role in modern photonic science and technology.

In the earlier stage, the PBG structures were mainly fabricated by using the usual dielectrics, semiconductors and metals as well. Recently, the studies of the photonic band structures in a periodic multilayer structure consisting of superconducting and dielectric materials have also been reported [18-21]. Such a superconducting planar Bragg reflector (SPBR) has some basic distinctions compared to an all-dielectric plane Bragg reflector.

For example, there exists a low-frequency PBG due to the combined effects of periodicity and of incorporating superconducting materials [19-21]. This low-frequency PBG is further tunable as a function of the temperature and the applied static magnetic field as well. This tunable feature comes from the temperature- and field-dependent penetration length of a superconductor. Moreover, in the region near the threshold frequency of the bulk superconductor, which plays a similar role as the plasma frequency in metal, some extraordinary optical properties in a SPBR can be seen [41].

A Bragg reflector with an annular geometric structure shown in Fig. 1 has now

been achievable with the advance of modern fabrication techniques. In our analysis we use the two-fluid model for the superconductor [9,10] together with the transfer matrix method for the cylindrical waves developed by Kaliteevski et al. [11]. With the fact that the field solutions of the cylindrical waves are closely related to the azimuthal mode number, denoted by m, for both the TE and TM waves, optical properties at different m-number will be examined. In this paper, we have found that an additional

been achievable with the advance of modern fabrication techniques. In our analysis we use the two-fluid model for the superconductor [9,10] together with the transfer matrix method for the cylindrical waves developed by Kaliteevski et al. [11]. With the fact that the field solutions of the cylindrical waves are closely related to the azimuthal mode number, denoted by m, for both the TE and TM waves, optical properties at different m-number will be examined. In this paper, we have found that an additional

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