# Terahertz Spectroscopy of Novel Superconductors

Authors: Stefano Lupi
Journal: Advances in Condensed Matter Physics
 (2011)

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## Abstract

Through the coupling of Synchrotron Radiation and Michelson interferometry, one may obtain in the terahertz (THz) range transmittance and reflectivity spectra with a signal-to-noise ratio (S/N) up to 103. In this paper we review the application of this spectroscopic technique to novel superconductors with an increasing degree of complexity: the single-gap boron-doped diamond; the isotropic multiband V3Si, where superconductivity opens two gaps at the Fermi energy; the CaAlSi superconductor, isostructural to MgB2, with a single gap in the hexagonal ab plane and two gaps along the orthogonal c axis.


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## Body

## 1. Introduction

After the discovery of high-Tc cuprates in 1986 that renewed the interest on superconducting materials, many new superconductors with different transport and magnetic properties have been synthesized in recent years. Among them one may cite A-doped (A = K, Rb, Na) C60, NaxCoO2 codoped with H2O; MgB2 and CaAlSi, and Boron-doped diamond and the new Iron pnictides. The magnetic, transport, and electrodynamics properties of a superconductor are intimately related to few parameters: the critical temperature Tc, the value and the symmetry in the k space of the superconducting gap Δ(k), the critical magnetic fields HC1, and HC2, and the penetration depth λ. The main issue about a new superconductor concerns if these properties can be described through the mean-field Bardeen-Cooper-Schrieffer (BCS) theory (which well-captures the properties of conventional superconducting metals), or a different, more exotic, theory should be considered. For instance, strongly covalent bonds, high-concentration of impurities, and high-phonon frequencies make B-doped diamond much different from the conventional metals where the BCS model holds. Moreover, after the discovery of MgB2, multiple-band unconventional-coupling superconductivity has been put forward and the properties of known materials like V3Si, with several electronic bands crossing the Fermi energy (EF), and recently discovered CaAlSi with anisotropic lattice properties, have been investigated in this scenario.The presence of a gap in the electronic excitations near the Fermi level forT<Tc, is one of the main effect induced by superconductivity [1]. If the Cooper pairs are in a spherically symmetric s-state this gap opens along all directions of the Brillouin zone. When more bands cross EF, they can be characterized by different superconducting gaps if a low interband scattering is present. However, if a p or a d type band (or a linear combination) defines the electronic properties of the system, the gap (gaps) opens only along particular k directions. This is the case for high-Tc cuprates and, probably, for iron pnictides.THz spectroscopy is a powerful experimental tool to characterize superconductor materials, as it probes directly, and with the highest spectral resolution, their low-energy bulk electrodynamics. The optical properties of a material can be described through the reflectivityR and/or the transmittance T. In an isotropic metallic superconductor and below Tc, R reaches the value 1 for any frequency ω<2Δ(T) where Δ(T) is the superconducting gap. In the normal state and in the same low-frequency range, R(T) follows a metallic behavior characterized by a plasma frequency ωp and a scattering rate Γ(T). If the metal is in the “dirty” regime defined by Γ(Tc)>2Δ(0), the ratio RS(T<Tc)/RN between the reflectivity in the superconducting state RS and that in the normal state RN exhibits a peak [1] at 2Δ(T) and the same happens for the ratio TS(T<Tc)/TN. Both the value of 2Δ(T) and its temperature dependence can be easily compared with the BCS predictions. For example, in the weak coupling limit of the original BCS model 2Δ(0)=3.53Tc, and for most of superconductors 2Δ(0) ranges in the terahertz region (1 THz = 33 cm−1 = 4 meV = 333 μm).In the experiments, however, one may encounter serious difficulties to measure the small difference betweenRS (TS) andRN (TN), as RN in a good metal may be as high as 0.99 in the range of frequencies ω~2Δ. Therefore, a signal-to-noise ratio on the order of 103 is often needed in the terahertz region to measure the superconducting gap. Nowadays, this strong requirement can be fulfilled with a conventional Michelson interferometric apparatus, when it is coupled to Infrared (IR) and THz Synchrotron Radiation (SR). SR sources with these characteristics are routinely open to users in Europe at the infrared beamlines SISSI [3] and IRIS [4] of the storage rings ELETTRA and BESSY-II, respectively.In this paper we will review the IR and THz properties of three different superconductors with an increasing degree of complexity: theB-doped diamond, where we will show that a phononic BCS theory well-describes its superconducting properties; the V3Si A15-type system, where two main electronic bands contribute to the density of states (DOS) near EF and its electrodynamics properties are associated with two superconducting gaps; finally CaAlSi, a superconductor, isostructural to MgB2, with a single gap in the hexagonal ab plane and two gaps along the orthogonal c axis.


## 2. The Optical Gap of SuperconductingDiamond

The sample here investigated was a 3μm thick film 2.5 × 2.5 mm wide, grown by Chemical Vapor Deposition technique (CVD) and deposited on a pure CVD diamond substrate [5]. This thickness, assuring a zero transmittance in the THz range, provides us to probe the bulk response of the B-doped material.X-ray diffraction patterns collected just after the growth showed that the whole film surface had an (111) orientation, with no appreciable spurious contributions, as already reported for similar samples [6]. The boron concentration was estimated to be nearly 6 × 1021 cm−3, that is over the metal-to-insulator transition limit. The sample magnetic moment M(T), reported in the inset of Figure 1 shows a superconducting transition with an onset at Tc=6 K. Below 6.3 K the zero-resistance regime is already established through a sharp transition, which confirms the good homogeneity of the film (see the dc resistance Rdc in the inset). However, by considering that the magnetization is a bulk quantity like the THz conductivity we assume here Tc=6 K. In order to measure the superconducting gap, the sample was shined by the THz radiation produced when BESSY-II was set in the low-α operation mode [4]. In order to avoid any variation in the sample position and orientation, which may yield frequency-dependent systematic errors in the reflectivity, we cycled the temperature in the 2.6–15 K range without collecting reference spectra. Through this procedure RS(T)/RN(10K)=IS(T)/IN(10K), where IS (IN) is the intensity reflected by the sample in the superconducting (normal) state, and we obtained in the sub-THz range an error δ=IS/IN~0.2%. The ratio RS(T)/RN(10K) is reported in Figure 1. The three curves at T<Tc show a strong frequency dependence in the sub-THz region, with the predicted BCS peak at ω~2Δ(T). As a crosscheck, the data for T>Tc do not show any peak and are equal to 1 within the noise. A first inspection to data at T=2.6 K provides a peak value at ~12 cm−1, which gives 2Δ(2.6K)/kBTc~3. Therefore we fit the data using a BCS approach [7]. The normal state reflectivity was described in terms of a Drude component whose parameters have been determined by the measurement at 10 K. In the superconducting state we used a Mattis-Bardeen modelization of RS with a fixed Tc=6 K and Δ(T) as a free fitting parameter.Figure 1
Reflectivity of aB-doped diamond film in the sub-THz region at different T normalized to its value at 10 K. The lines are fits obtained by assuming a BCS reflectivity below Tc and a Hagen-Rubens model at 10 K. The inset shows on the left scale the magnetic moment of the sample, as cooled either in a 10 Oe field (FC) or in zero field (ZFC), on the right scale its resistance normalized to its value at 12 K. The FC values are multiplied by 10.The main output of the fit is the superconducting gap, which at 4.6, 3.4, and 2.6 K is found to be2Δ=9.5, 10.5, and 11.5 cm−1, respectively. These values furnish an extrapolated value 2Δ(0)=12.5 cm−1, or 2Δ(0)/kBTc=3.0±0.5, in satisfactory agreement with the BCS prediction of 3.53 and ultraviolet photoemission spectroscopy [8] where the authors determined, at nearly 4 K, 2Δ=12.6 cm−1.Afterwards, we measured the absolute reflectivityR (both in the normal and in the superconducting state) up to 20000 cm−1 taking as reference the film itself, coated with a gold or silver layer evaporated in situ.RN at 300 K is represented in the inset of Figure 2(a) and shows a poor metallic behavior consistently with the relatively high dc resistivity value [7]. Through the standard Kramers-Kronig transformations we obtained the optical conductivity. The real part σ1(ω), reported in Figure 2(b), decreases in the THz range for T<Tc, due to the opening of the superconducting gap. At 4.6 K, a residual quasiparticle contribution can still be distinguished at the lowest measured frequencies. At 2.6 K, a nearly zero absorption is attained below ~10 cm−1, a value comparable to 2Δ obtained from the reflectivity ratio. One may notice that the optical conductivity of the superconducting phase σ1S(ω) and that of the normal phase σ1N(ω) in Figure 2(b) coincide for ω~35cm-1~6Δ, indicating a BCS dirty limit behavior [1]. According to the Ferrell-Glover-Tinkham sum rule [1] the area A removed at T<Tc below σ1(ω, T), builds up the collective mode at ω=0. The spectral weight condensed into this peak may be used to extract the penetration depth λ=1/2π(8A)1/2 [1]. We thus estimate λ≃1μm at 2.6 K. Finally, 1/λ2 is plotted in Figure 2(c)  versus  T/Tc, and compared with the BCS prediction for a dirty superconductor [1]. Once again, this model appears to well-describe the behavior here observed in B-doped diamond [7].Infrared behavior of superconducting diamond. (a)R(ω) in the THz range at 10 K. In the inset R(ω) is shown at 300 K up to 20000 cm−1. (b) The real part of the optical conductivity showing a gap opening and a full recovery for ω>6Δ, as expected for a BCS superconductor in the dirty limit. (c) The inverse square of the penetration depth, obtained from missing area in (b) (experimental points), is compared with the BCS dirty superconductor behavior, once normalized to zero temperature (grey line).
(a)(b)(c)


## 3. The Two-Gap Scenario in V3Si

After the discovery of MgB2 superconductor strong experimental and theoretical efforts have been devoted to multiband superconductivity [9], since several of the unique properties of this system are based on the presence of two bands with two distinct superconducting gaps [10]. In the case of the V3Si system, which belongs to the A15 family contradictory results concerning the number of gaps opening below Tc have been published. Indeed, the study of the electrodynamics response in the microwave region gave evidence of two gaps [11] while muon spin rotation measurements are consistent with a single-gap scenario [12]. THz spectroscopy may help to solve this issue. On this regard we performed both reflectivity and transmittance measurements on high-quality V3Si textured thin films. Details on the film growth by pulsed laser deposition and on their properties are reported elsewhere [13]. We studied two films deposited on LaAlO3 (LAO) (001) 0.5 mm thick substrates, which exhibit preferential (210) orientation along the out-of-plane direction. The first film of thickness d=180 nm (d180 film), characterized through reflectivity data, has good transport properties (resistivity at 300 K close to 200 μ Ω cm, and at 20 K close to 25 μ Ω cm) and Tc=16.1 K. The second film with a thickness of 33 nm (d033 film) used for transmittance measurements, has a slightly lower Tc value (Tc=15.3 K) probably due to the strain induced by the substrate on the film structure. Here, reflectivity and transmittance data in the THz range imposes a strong constraint in the fit parameters and provides a strong evidence of a two-gap scenario for the V3Si system.TheRS(T)/RN ratios (where RN=RN (20 K)) at different T<Tc for the d180 film are reported in Figure 3. These measurements were performed at the SISSI beamline [3] through the use of THz radiation produced by the ELETTRA machine [14], by cycling the temperature in the 6–20 K range, without collecting reference spectra. In this way one avoids any variation in the sample position and orientation, which may yield frequency-dependent systematic errors both in reflectivity R(ω) and transmittance T(ω). We first notice that the RS(T)/RN ratio (Figure 3(a)) increases on decreasing temperature until it reaches a maximum and becomes nearly constant below 20 cm−1. This indicates a superconducting gap ΔA closes to 10 cm−1. Indeed, for ω→0 the reflectance RN of a metallic system tends to 1 in the case of a bulk system, and to a slightly lower value for a thin film. Therefore, as discussed in the introduction, when RS approaches 1 at ω=2Δ, RS/RN exhibits a maximum around 2Δ in the case of a bulk sample, while it remains nearly constant below 2Δ in the case of a thin film. Instead in TS(T)/TN (20 K) data (see Figure 4(a)), a broad maximum develops on decreasing T until, at 6 K, TS(T)/TN exhibits a well-defined peak around 40 cm−1. Therefore, in superconducting films, the combination of transmission and reflection spectroscopy is essential to extract from optical data the superconducting parameters.Figure 3
(a) R(T)/RN(20K) ratio for the d180 film at selected temperatures in the THz region. (b) R(6K)/RN(20K) ratio compared with the two-band (2-b) and single-band (1-b) best fit curves. (c) Real part of the optical conductivity in the superconducting state σ1S (in units 104 Ω−1 cm−1) of V3Si from the 2-b model.Figure 4
(a) TS(T)/TN(20K) spectra of the d33 film at selected temperatures in the THz region. (b) TS (6K)/TN(20K) spectrum compared with the two-band (2-b) and single-band (1-b) best fit curves. (c) Absorption coefficient in the superconducting state αS (in units 104 cm−1) of V3Si from 2-b model.This peak is consistent with the presence of a superconducting gap ΔB at about 20 cm−1. We point out that the high-flux synchrotron source allows us a high accuracy in the detection of small effects in the RS (T)/RN spectra and overcomes the problem of the very low intensity transmitted by the film + substrate system in the case of transmission measurements on a conducting film. We also note that TS(T)/TN is affected by the superconducting transition much more than RS(T)/RN as the transmitted intensity is mainly controlled by the strong absorptive processes in the film, which significantly changes below Tc.In order to perform a detailed analysis of the transmittance and reflectivity spectra, the complex refraction index of V3Si has been described in the normal state through the sum of two Drude terms (related to the main conduction bands crossing EF), with parameters ωpA (3.7 ± 0.1 eV), ΓA(T) (800 ± 40 cm−1), and ωpB (1.00 ± 0.05 eV), ΓB(T) (45 ± 4 cm−1).  The superconducting spectra have been described through a Mattis-Bardeen refraction index depending on the gap Δ(T) [1]. Finally the transmittance and reflectance spectra of the film + substrate system in both states can then be evacuated by means of an exact procedure [15] which requires, besides thickness, the values of  for both film and substrate. For LAO, values in the THz region were obtained from previous measurements of transmittance and reflectance [14].By using the previous procedure a single gap description cannot take into account either the shape or theT dependence of RS(T)/RN ratio (blue line in Figure 3(b)). On the other hand RS(T)/RN and TS(T)/TN ratios agree well with a two-gap scenario (black lines in Figures 3(b) and 4(b)), that is also consistent with two main electronic bands crossing the Fermi energy [16].From the reflectivity ratio atT = 0 and from the two-gap fit, one obtains ΔA(0)=10±1 cm−1, and ΔB(0)=19±2 cm−1, while from the transmittance measurements, which are less sensitive to the low energy gap (see [16]), ΔA(0) ranges from 11 to 16 cm−1 and ΔB=21.0±0.5 cm−1. The comparison of 2ΔA(B) with the corresponding scattering rate ΓA=800 cm−1 (ΓB=45 cm−1) shows that V3Si is in a dirty limit regime. Moreover, the gap values correspond to BCS ratios of 2ΔB(0)/kBTc=3.8±0.3 and 2ΔA(0)/kBTc=1.8±0.2, respectively. While the former ratio agrees with a standard BCS weak coupling, the latter, which is instead in agreement with tunneling data [17], indicates an anomalous superconducting weak coupling that are related to the specific electronic density of states in which this small gap opens [16].


## 4. The Anisotropic Gaps of CaAlSi

CaAlSi is a recently discovered superconductor characterized by aTc of 7.7 K and hole transport [18]. It has attracted wide interest for its properties including the hexagonal crystal structure similar to that of MgB2, where however the carriers are electrons. Like MgB2, CaAlSi has two Fermi-surface sheets. However, it is not clear if one should expect one gap or two gaps like in MgB2. CaAlSi has an s-wave anisotropic symmetry [19] probably related to the hexagonal crystal structure. Angle resolved photoemission (ARPES) [20], within the energy resolution, distinguished in CaAlSi a single isotropic gap of about 1.2 meV = 4.2 kBTc. Muon spin relaxation (μR) data [2] indicated a highly anisotropic gap, or possibly two distinct gaps like in MgB2. Finally penetration depth measurements [19] support weakly anisotropic s-wave gap, but not two distinct gaps. THz spectroscopy, which allows one to measure the gap directly and with a resolution in energy higher than in ARPES, may help to solve this issue, provided that one attains a suitable signal-to-noise ratio. Here we measured a CaAlSi single crystal with a size of 2 × 4.5 × 3 mm3 [18]. Its magnetic moment M(T) (see the upper inset in Figure 7) shows the superconducting transition with an onset at 6.7 K.In Figure5 we represent the reflectivity in the hexagonal plane measured at different T in the normal phase from 50 to 30000 cm−1. At variance with B-doped diamond (see Figure 2(a)) Rab(ω) looks like that of a good metal. Below the plasma edge around 10000 cm−1, and at any T>Tc, the reflectivity can be well described by a conventional Drude model with a bare plasma frequency ωp(ab)=20000 cm−1 independent of T and a scattering rate Γab(T)=480cm−1 and 850 cm−1 at 10 and 300 K, respectively. The corresponding ωp(ab) for MgB2 (see the inset of the same figure) is instead nearly 45000 cm−1 [21].Figure 5
Reflectivity spectra of CaAlSi in the hexagonal plane at selected temperatures from the THz to the visible spectral region. In the inset a comparison between the reflectivity at 300 K of MgB2 [2] and CaAlSi is shown.In Figure6 we show instead the c-axis reflectivity at various T in the normal state. At variance with the ab reflectivity the low-energy properties along the c-axis cannot be described in terms of a single Drude term. Indeed we observed interband transitions at 4000, 9000, and 12000 cm−1 superimposed to a Drude component with a bare ωp(c)=9000 cm−1 and a Γc(T) of 180 cm−1 at 10 K, 235 cm−1 at 300 K [22]. By comparing Figures 5 and 6 we can note that CaAlSi presents a quite strong normal-state anisotropic behavior which will be also observed in the superconducting state (see below).Figure 6
Reflectivity spectra of CaAlSi along thec-axis at selected temperatures from 50 to 30000 cm−1.Figure 7
Ratio between the sub-THz reflectivity of the ab planes belowTc and in the normal phase at 10 K. The lines are fits based on a Mattis-Bardeen reflectivity below Tc and on a Drude model at 10 K (see text). The upper inset shows the magnetic moment measured by cooling the sample in a field of 10 Oe, showing the superconducting transition at Tc=6.7 K. The lower inset shows a check on data reproducibility, performed by dividing two subsequent spectra of 256 scans.The ratioRab(T)/Rab(10 K) is reported in Figure 7 for the radiation polarized in the hexagonal plane at different T, both below and above Tc. The curves at T<Tc exhibit the expected peak at 2Δab(T), while for T>Tc (12 K), the above ratio is equal to 1 at any ω within the noise. In order to extract the gap the reflectivity ratios have been fitted by using the Drude parameters ωp(ab) and Γab(T) determined from the normal state reflectivity and using the Mattis-Bardeen model for the superconducting state with a fixed Tc=6.7 K and Δab(T) as free parameter. The curves Rab(T)/Rab(10K) calculated in this way are also reported in Figure 7 (thin lines). The fit is good at the three temperatures below Tc and provides 2Δab=15 and 17.5 cm−1at 4.5 and 3.3 K, respectively. This leads to an extrapolated value 2Δab(0)=19.0±1.5 cm−1 or 2Δab(0)/kBTc=4.1±0.4, a value which is in substantial agreement with ARPES data [18]. The actual 2Δab(0)/kBTc ratio suggests that CaAlSi is a BCS superconductor with moderately strong electron-phonon coupling. On the basis of our fits, which take into account both the measured data in the normal and in the superconducting state, a single gap well-describes the superconducting transition in the hexagonal planes.Let us now consider the optical response of CaAlSi along thec-axis. Figure 8 shows the ratio Rc(T)/Rc(10 K), as measured with the radiation polarized along the c direction. A peak appears below Tc, but its different shape with respect to those in Figure 7. Such a shape suggests that either only a fraction of the carriers contributes to the optical conductivity of the superconducting phase or there are, at least, two distinct gaps. The former case is observed, for example, in high-Tc cuprates, where the order parameter has nodes in the k space due to its d-wave symmetry, but this should be excluded in CaAlSi where the main conducting electronic bands around EF have a highly symmetric character. The latter case has been observed in MgB2, where a two-gap model well-accounts for the reflectivity data belowTc. In CaAlSi, the two gaps should come out from different regions of the Fermi surface, which are topologically disconnected along the kz direction [23]. A two-gap model well-fit the Rc(T < Tc)/Rc(10 K) data of Figure 8 (solid lines) and the results for the larger gap are 2Δc,1=22 cm−1 at 4.5 K, 26 cm−1 at 3.3 K, and 28 cm−1 extrapolated to 0 K; instead the fit provides for the smaller gap: 2Δc,2=2,6,and8 cm−1 at 4.5, 3.3 and 0 K, respectively. The very low value for the Δc,2 may be related to a specific electronic density of states in which this small gap opens.Figure 8
Ratio between the sub-THz reflectivity along thec-axis below Tc and in the normal phase at 10 K. The lines are fits based on a BCS reflectivity below Tc and on a Drude-Lorentz model at 10 K (see text).Finally, the comparison between the scattering rate at 10 KΓab(T)=480cm-1(Γc(T)=180cm-1) and 2Δab(0)=19.0cm-1(2Δc(0)=28cm-1) shows that, both in the ab plane, and along the c axis, CaAlSi is also in the dirty limit regime.In this experimental review we have shown some applications of frequency-domain THz spectroscopy in superconductivity. THz radiation extracted from the III generation synchrotron machines ELETTRA and BESSY-II coupled with Michelson interferometry, provides the possibility to investigate the bulk properties of novel superconductors in their normal and superconducting state. In particular we have determined the superconducting gap for theB-doped diamond, V3Si (A15 type), and CaAlSi (MgB2 type) superconductors. This technique provides a signal-to-noise ratio of the order of 103, which is needed to appreciate the weak increase (decrease) in the reflectivity (transmittance) across the critical temperature Tc. We have shown that B-doped diamond is an isotropic superconductor characterized by a ratio 2Δ(0)/kBTc=3.0±0.5, in satisfactory agreement with the theoretical BCS prediction of 3.53. In CaAlSi, which has a crystal structure similar to that of MgB2, we have determined the superconducting gap both in the hexagonal ab planes and along the orthogonal c-axis. Here 2Δab(0)<2Δc(0) showing that CaAlSi is a BCS anisotropic s-wave superconductor. We have also addressed the debated problem of multiband, multigap nature of A15 type V3Si superconductor by means of reflectance and transmittance measurements in the THz region. Our experimental results indicate that in the two main electronic bands crossing EF two gaps of different magnitude open below Tc. This result is of current interest, especially after the discovery of the Fe-based superconductors, as it addresses multigap superconductivity in a system where complications coming from magnetism and nodal symmetries do not arise.However, frequency domain spectroscopy does not exhaust all potentialities of THz radiation. Indeed the recent development of strong, femtosecond, THz pulses [24, 25] may open a new route in THz spectroscopy. These pulses should be used in time resolved THz-pump THz-probe investigations to probe the intrinsic dynamics of Cooper pairs and their relaxation towards a steady state. A new fascinating scenario could be also open through the control and the manipulation of the superconducting transition by using the intense electric and magnetic fields associated with the femto-second THz pulses [26].



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