Superconductivity Induced Transparency Coskun Kocabas In this paper I will discuss the effect of the superconducting phase transition on the optical properties of the superconductors. Firstly I will give the basic definition of optical conductivity, spectral weight and Ferrell-Glover-Tinkham sum rule. Conventional superconductors behaves like a normal metal under irradiation of high-frequency radiation, on the other hand high-frequency optical conductivity of high-t c superconductor (cuprates) degreases at temperature below T c. This unusual highfrequency optical activity of high-tc superconductors can be applied to control the optical transmission. I will introduce surface-plasmon (SP) enhanced transmission of visible or near infrared light through superconducting grating. Periodically modulated thick metal surfaces can show extra ordinary transmission at specific wavelength which shows resonance with the surface plasmon on the metal surface. In this paper uses of high-t c superconducting materials, which can be the less absorbing of IR or visible light, will be applied for this purpose.
Introduction Zero resistance superconducting state makes superconductors very interesting material for science and technology. Unfortunately zero resistance superconducting state can not provide extra ordinary optical properties for practical application [1]. In this paper I will introduce a new possible photonic application of superconducting materials which uses the surface plasmon (SP) enhanced optical transmission. Surface plasmons are collective excitation of free electrons on metal surface. Free phonons can couple to the SP and form surface plasmon polariton which can propagate along the metal dielectric interface. Using this structure a new type of optical transmission can be archived [2,3]. This surface plasmon enhanced optical transmission is the new emergent properties of metals. This transmission is 3 orders of magnitude larger than the normal transmission through the metal. There are two main problems for this type of devices, high optical losses and weak tunneling between the interfaces. In this paper I will discuss the solutions for theses problems by using superconducting materials. Superconducting materials can provide less optical absorption and can provide large coupling between the interfaces furthermore the transmission can be controlled by changing the superconducting state to normal state which could provide fast optical switching. However, for high frequency conventional superconductors such as aluminum, lead [1] do not provide any unusual properties when they are in superconducting state. They behave like a normal metal for visible and IR regime. Their optical absorptions do not change with superconducting phase transitions. High-T c superconductors show very unconventional properties and also their optical properties are totally different. Recent results [4-7] show that visible or IR optical properties cuprates could be affected with superconducting transitions. Their optical conductivities decrease when they are cooled below T c. In this paper basic definitions will be given and recent results about optical properties of cuprates will be discussed. Later surface plasmon enhanced optical transmission will be discussed. The advantages and disadvantages of superconducting materials will be presented. If this idea works, it would provide new photonic devices whose optical transmission could be controlled by superconducting phase transitions. In addition to these this system can me used to understand the superconductivity mechanism for high-t c superconductors. The angular dependence of SP enhanced transmission gives the information about the dispersion relation of the surface plasmon which is related with the dielectric function.
Dielectric Function and Optical Conductivity Optical spectroscopy can be used to determine the complex dielectric function of the material which is related with the electronic properties. Electronic excitations and superconducting phase transitions can be studied by measuring the reflectivity of the light from the surface of the superconductor. This kind of methods has been used since the early days of superconductivity [1]. Reflectivity from the surface depends on the dielectric function of the material and can be written as 1/ 2 2ω 1 R p ( ω) = 1 Re cosθ iπσ ( ω) where subscript p denotes the polarization direction and ω is the frequency and θ is the incidence angle. From the frequency dependent reflection spectrum real and imaginary part of dielectric function can be obtained. Optical conductivity is a response function of material which describes the induced current on the material as a function of applied electric filed [8]. Optical conductivity is related with the dielectric function as given below ω σ ( ω) = i [ 1 ε ( ω) ] 4π Optical conductivity is a response function therefore it shows the general properties such as causality which gives the Krammer-Kronig relations. Real part of the optical conductivity σ 1 depicts the absorption of the material which is related with the electronic excitation. Spectral weight in the optical conductivity relates the real part of the optical conductivity σ 1 and number of electrons which involves in the conductivity. We can define spectral weight as ω 2m N ( ω) = σ 1( ω) dω 2 πe 0 When we take the limit ω this quantity will be equal to the total number of the electron per unit volume. If we modify this equation we can get a relation between the total number of electron and plasma frequency ω p. 0 2 πne ω p σ 1( ω) dω = = 2m 8 This equation is known as frequency sum rule or Ferrell-Glover-Tinkham sum rule [9]. For conventional superconductors which can be explained with BCS theory, this sum rule is valid for all temperatures. According to the BCS theory, electrons forms Cooper pairs with binding energy 2, photons with energy less than binding energy 2 can not be absorbed. Therefore below T c optical conductivity is zero for the low frequencies 0< hω< 2. This range is called missing area in the optical conductivity. In order to satisfy the FGT sum rule, condensation produces an additional zero frequency δ function contribution. It is experimentally shown that all missing area can be compensated from zero frequency δ function contribution. Fig.1 shows the transmission ratios of superconducting and normal states of lead film [1]. At low frequency (microwave range) transmission of superconducting state is much larger than the normal state. As frequency increases (>100 kt c /h) this ratio converges to one. This means that a superconducting phase transition does not make any change in optical conductivity and optical absorption at high frequency. Under IR or visible lights conventional superconductors behaves like a normal metal and phase transition can not be used for any switching purpose.
Figure 1. Transmission ratios of superconducting and normal states of lead film. At high frequency there is no difference between the normal and superconducting state (Taken from ref.1) Sum Rule and High-T c Superconductors High-T c superconductors are new class of superconductors which can not be explained with BCS theory. For conventional superconductors no change in the high frequency reflectivity was detected, however, in 1998 Basov et al. [4] reported that high frequency (IR or visible range) optical conductivity of high-t c cuprates could be changed when they cooled below T c. They claimed that high-t c superconductors such as Tl 2 Ba 2 CuO 6+x, La 2-x Sr x CuO 4, YBa 2 Cu 3 O 6.6, show an anomalously large energy scale extending up to mid-infrared frequency. They carried out the reflectance measurements with different polarization and they can determine the interlayer optical conductivity. Their observation shows that the missing area in optical conductivity can not be suppressed with the δ function at ω=0. They observed that at least half of the missing area is spread over a large frequency range which could be extended up to IR or visible range. Fig.2 shows the sum rules for conventional and high-t c superconductors. Actually there is no direct evidence for high frequency change in optical conductivity but they explain the discrepancy between the δ function contribution at ω=0 and the missing area could be extended over a large frequency range. Figure 2. Relations between fundamental and partial sum rules (taken from ref.5).
They explain that this unusual behavior can only be observed for materials with incoherent normal state. The cuprate superconductors show anisotropic electrical properties in their normal states. They contain stacks of CuO 2 planes which has high electrical conductivity with in the plane (metallic response) whereas they are like ionic crystal in c axis which is perpendicular to the CuO 2 plane. Therefore in normal state optical conductivity along the c axis is extremely small which does not make any contribution to the sum rule. However in superconducting state supercurrent can flow in any direction and the effective mass decreases under T c and c axis optical conductivity increases dramatically. Ferrell-Glover-Tinkham sum rule is violated [5]. They tested their hypothesis by changing the oxygen doping. The anisotropy in the c axis conductivity can be suppressed by doping and they observed degrease in the discrepancy with increasing the doping. The discrepancy vanishes for YBa 2 Cu 3 O 6.85 which is more isotropic than YBa 2 Cu 3 O 6.53. According to this observation, it can be concluded that cuprates could show different optical properties in superconducting state, they could be less absorbing for IR or visible light. This could make cuprates applicable for photonic devices. However these observations are not direct evidence for high frequency optical activity. In 2002 Molegraaf et al. [6] reported direct observation on Bi 2 Sr 2 CaCu 2 O 8+δ. They used spectroscopic elipsometry and they measured small amount of change in spectral weight in visible range (around 2 ev), furthermore they observed blue shift of the ab-plane plasma frequency when material become superconducting. Fig.3 depicts the measured frequency dependent dielectric function and optical conductivity. The plasma frequency shows clear temperature dependence and derivative kinks around at T c. Their observations are very important because they directly observed the plasma frequency shift and transfer of spectral weight from visible to low frequency interband optical spectrum. Figure 3. Frequency dependent dielectric function and optical conductivity and temperature dependence of ab plane plasma frequency (taken from ref.6)
Surface Plasmons and Extra Ordinary Transmission of Light Surface Plasmons are getting great interest because of its optical application and possibility for nanophotonic devices. With the current technology dimensions for the photonic devices are in millimeter or sometimes centimeter order. High density photonic circuits are not very feasible with such a big device dimension. Fabrication of submicron photonic devices could be possible with using surface plasmons. Recent developments in this area produce a new branch of photonic which is called plasmonics [3]. Surface plasmons (SP) are electromagnetic excitation which can propagate at the metal-dielectric interfaces. SP are coupled collective excitation of photon and free electrons on the surface. Electromagnetic filed of light wave can be coupled to collective excitation of free electrons on the metal and form propagating surface waves and they can decoupled to produce free space light. These surface waves can be guided and controlled with tailoring the metal on dielectric substrate. SP are localized to the surface which allows concentrating light in sub-micron dimension. Fig. 1a shows the electric and magnetic field distribution of SP. SP have transverse magnetic field and can only be excited with transverse magnetic light. Fig 1b shows the evanescent field distribution in perpendicular direction. Decay length into air δ d is of around 100nm and decay length into metal δ m is related with the skin depth and is about 10nm. Figure 4 Electric and magnetic field distribution of surface plasmon. (taken from ref. 3) Fig. 4c shows the dispersion curve for the free space photon and SP. It is clearly seen that there is a mismatched between the momentum and frequency of photon and SP. SP has greater momentum than the free space photon. This mismatched does not allow excitation of SP on flat metal surface with direct irradiation. To overcome this mismatched metal surface should be periodically modulated with a period Λ, and addition 2π/ Λ momentum can satisfy the phase matching condition. SP dispersion relation can be written as, ε dε m k SP = k 0 ε d + ε m where, ε d and ε m are the frequency dependent permittivity of dielectric material and metal, respectively. In order to support SP at the interface permittivity should have different sign, this is valid for metal dielectric (glass) interface.
In 1998 Ebbesen et al. [2] reported unusual optical transmission through subwavelength hole array patterned on metal film. Fig 2 shows the transmission spectrum of optically thick silver film. Two dimensional periodic arrays of holes with sub-wavelength diameter were fabricated on metal film. Figure 5. Zero-order transmission spectrum of an Ag array and SEM image (taken from ref.2). Fig.5 shows the 2-D hole array with 150nm diameter and 600 nm periodicity on 200 nm thick silver which is optically thick. Electromagnetic theory predict that the transmission through sub-wavelength aperture scale as (r/λ) 4. On the other hand transmission through periodic holes on metal film shows sharp peaks at wavelengths which are in resonance with surface plasmons on the metal surface. The transmission is 3 orders of magnitude larger than the classical predictions. This extra ordinary transmission is related with the excitation of the SP on the metal-air interface. The free space photons are coupled to the SP via the 2-D grating and then SP on one interface tunnel to the other interface through the holes. After tunneling, the SP out-coupled to free space photon and extra ordinary transmission occurs. The total transmission is only about 5% because of the losses through the metal film and week tunneling between the interfaces. The transmission spectrum can be changed by changing the periodicity of the array and the angle of the irradiation. Fig. 6 shows three hole arrays with different periodicity and their transmission spectrum. This image shows the flexibility of this structure, it can be easily tuned to desired wavelength. Figure 6 Transmission spectrum of three different hole array with different periodicity. (taken from ref.3 )
After discovery, plasmon enhanced optical transmission has become very interesting research area and this phenomena has been applied to many different areas such as quantum optics [10], nano-lithography [11]. Furthermore, simple version of this transmission can be observed on 1D metal grating with certain polarization. The only difference between the hole array and grating is coupling condition. There are two main problem for this kind of structure one is the high optical loses through the metal film, second is the low coupling between the two interface. These two problems can be solved by introducing superconducting materials such as cuprates. As we discussed cuprates could be less absorbing in the superconducting state and the optical transmission can be control by changing from superconducting state to normal state. Conclusion As a conclusion, I have introduced the possibility of superconducting materials for photonic device application. IR and visible optical properties of high-t c superconductors have been discussed. Resent results on violation of Ferrell-Glover-Tinkham sum rule makes the high-t c superconductors promising and unconventional materials for photonic devices. Their optical properties could be adjusted by changing the superconducting state to normal state. Surface plasmon enhanced extra ordinary optical transmission can be improved with superconducting materials. Superconducting materials could be less absorbing for IR or visible light. Furthermore this system can provide some information about the mechanism of the high-t c superconductors. References [1] R. E. Glover and M. Tinkham, Phys. Rev. 104, 844 (1956); ibid. 108, 243 (1957). [2] Ebbesen, T. W., Lezec, H. J., Ghaemi, H. F., Thio, T. & Wolff, P. A. Nature 391, 667 669 (1998). [3] Ebbesen, T. W, Nature 424, 824, 2003, [4] D. N. Basov et al., Science 283, 49 (1999). [5] M. V. Klein and G. Blumberg,Science 283, 42, (1999) [6] H. J. A. Molegraaf et al. Science, 295, 2239 (2002) [7] J. E. Hirsch Science, 295, 226 (2002) [8] Fausto Patricio Mena, Optics and magnetism, 1999 University Library Groningen [9] R. A. Ferrell and R. E. Glover, ibid. 109, 1398 (1958); M. Tinkham and R. A. Ferrell, Phys. Rev. Lett. 2, 331 (1959). [10] E. Altewischer, M. P. van Exter & J. P. Woerdman, Nature 418, 304, 2002, [11] X. Luo and T. Ishihara Opt. Express 12, 3055-3065 (2004),