The optical response of the two-band superconductor MgB$_2$ has been studied in the 0.7-4 THz range on films with very low impurity level. The effect of the high-energy $σ$-gap is observed in the ratio $R_S/R_N$ between the normal and superconducting state reflectance, while in a neutron irradiated film with a slightly higher impurity level mainly the effect of the $π$-gap is evident as reported in previous experiments. At terahertz frequencies, the electrodynamic of MgB$_2$ can be well described by the two-band parallel conductivity model and is dominated by the $π$-bands when the impurity level is only slightly higher than that of an ultra-clean sample.
Thermal conductivity of superconducting MgB2 was studied in both the superconducting and the normal state region. The latter is almost equally determined by the electronic - and the lattice contribution to the total thermal conductivity. In the superconducting state, however, the lattice contribution is larger. The electronic thermal conductivity below Tc was derived from the experimental data considering the Bardeen-Rickayzen-Tewordt theory together with the model of Geilikman. The analysis shows that electron scattering on static imperfections dominates.
The remarkable field dependence of the electronic thermal conductivity observed in MgB_2 can be explained as a consequence of multi-gap superconductivity. A key point is that for moderately clean samples, the mean free path becomes comparable to coherence length of the smaller gap over its entire Fermi surface. In this case, quasiparticle excitations bound in vortex cores can easily be delocalized causing a rapid rise in the thermal conductivity at low magnetic fields. This feature is in marked contrast to that for anisotropic or nodal gaps, where delocalization occurs only on part of the Ferm…
We report on high-order harmonic generation in the two-band superconductor MgB$_2$ driven by intense terahertz electromagnetic pulses. Third- and fifth-order harmonics are resolved in time domain and investigated as a function of temperature and in applied magnetic fields crossing the superconducting phase boundary. The high-order harmonics in the superconducting phase reflects nonequilibrium dynamics of the superconducting order parameter in MgB$_2$, which is probed via nonlinear coupling to the terahertz field. The observed temperature and field dependence of the nonlinear response allows to…
We report the first study of the optical conductivity of MgB2 covering the range of its superconducting energy gap. Terahertz time-domain spectroscopy is utilized to determine the complex, frequency-dependent conductivity of thin films. The imaginary part reveals an inductive reponse due to the emergence of the superconducting condensate. The real part exhibits a strong depletion of oscillator strength near 5 meV resulting from the opening of a superconducting energy gap. The gap ratio of 2Delta/kTc = 1.9 is well below the weak-coupling value, pointing to complex behavior in this novel superco…
The temperature dependence of the real part of the microwave complex conductivity at 17.9 GHz obtained from surface impedance measurements of two c-axis oriented MgB2 thin films reveals a pronounced maximum at a temperature around 0.6 times the critical temperature. Calculations in the frame of a two-band model based on Bardeen-Cooper-Schrieffer (BCS) theory suggest that this maximum corresponds to an anomalous coherence peak resembling the two-gap nature of MgB2. Our model assumes there is no interband impurity scattering and a weak interband pairing interaction, as suggested by bandstructure…
The ab-plane thermal conductivity $κ$ of single-crystalline hexagonal MgB_2 has been measured as a function of magnetic field H with orientations both parallel and perpendicular to the c-axis and at temperatures between 0.5 and 300 K. In the mixed state, $κ(H)$ measured at constant temperatures reveals features that are not typical for common type-II superconductors. The observed behavior may be associated with the field-induced reduction of two superconducting energy gaps, significantly different in magnitude. A nonlinear temperature dependence of the electronic thermal conductivity is observ…
We report data on the thermal conductivity K(T,H) along the basal plane of the hexagonal crystal structure of superconducting Mg{1-y}Al{y}B2 with y=0.02 and 0.07 at temperatures between 0.5 and 50 K and in external magnetic fields between 0 and 70 kOe. The substitution of Al for Mg leads to a substantial reduction of the heat transport via electronic quasiparticles. The analysis of the K(T,H) data implies that the Al impurities provoke an enhancement of the intraband scattering rate, almost equal in magnitude for both the sigma- and the pi band of electronic excitations. This is in contrast wi…
We solve the coupled gap equations for the $σ$- and $π$-bands of MgB$_2$ in the vortex state and calculate the resulting field dependencies of the specific heat coefficient $γ$ and the thermal conductivity $κ$. The crucial parameters of the theory are the interband pairing interaction $λ_{πσ}$ and the ratio $s=ξ_σ/ξ_π$ of the coherence lengths. For reasonably small $λ_{πσ}$ and s, the small gap $Δ_π$ decreases with increasing magnetic field $H$ much faster than the large gap $Δ_σ $. This gives rise to the observed rapid increase of $γ_π$ and $κ_π$ for small fields while $γ_σ$ and $κ_σ$ exhibit…
This paper presents a study of the thermal transport properties of MgB$_2$ tapes differing in architecture, stabilization and constituent materials. The temperature and field dependence of thermal conductivity, $κ(T,B)$, was investigated both along the conductor and in the direction perpendicular to the tape. These data provide fundamental input parameters to describe the 3D heat diffusion process in a winding. Thermal transport properties - even in field - are typically deduced using semi-empirical formulas based on the residual resistivity ratio of the stabilizer measured in absence of magne…
The surface resistance (Rs) and the real part (sigma_1) of the microwave complex conductivity of a ~380 nm-thick polycrystalline MgB2 film with the critical temperature (Tc) of 39.3 K were investigated at ~8.5 GHz as a function of temperature. Two coherence peaks were observed in the sigma_1 versus temperature curve at temperatures of ~0.5 Tc and ~0.9 Tc, respectively, providing a direct evidence for the two-gap nature of MgB2. The film appeared to have a pi-band gap energy of 1.8 meV. For the MgB2 film ion-milled down to the thickness of ~320 nm, two coherence peaks were still observed with t…
We present thermal conductivity measurements on very pure and dense bulk samples, as indicated by residual resistivity values as low as 0.5 mW cm and thermal conductivity values higher than 200 W/mK. In the normal state we found that the Wiedemann Franz law, in its generalized form, works well suggesting that phonons do not contribute to the heat transport. The thermal conductivity in the superconducting state has been analysed by using a two-gap model. Thank to the large gap anisotropy we were able to evaluate quantitatively intraband scattering relaxation times of $π$ and $σ$ bands, which de…
We present the results of measurements of the thermal and electrical conductivity in the basal plane of single-crystalline MgB_2 between 2 and 300 K. The analysis of the temperature dependence of the thermal conductivity gives supporting evidence for two different gaps on different sheets of the Fermi surface in the superconducting state. The zero-temperature values of the two gaps are D_1 (0) = 5.3 meV and D_2 (0) = 1.65 meV.
We propose and demonstrate a microscopic way to analyze the frequency-dependent infrared conductivity: extraction of the electron self-energy from the inversion of experimentally measured infrared conductivity through the functional minimization and numerical iterations. The self-energy contains the full information on the coherent and incoherent parts of interacting electrons and, therefore, can describe their charge dynamics even when the quasi-particle concept is not valid. From the extracted self-energy, other physical properties such as the Raman intensity spectrum and the effective inter…
Within a two-band, strong coupling model we analyze SIS-tunneling conductance, Raman scattering and optical conductivity measured on $MgB_2$ samples. We find that features observed in tunneling and Raman spectroscopy at intermediate energies $ω=Δ_σ+Δ_π\simeq 8÷10meV$ can be consistently explained when incoherent scattering between $σ$ and $π$ band is considered. We calculate the optical conductivity and find that strong coupling effects are important for explaining the presently available data, and predict the location and magnitude of interband scattering features for this spectroscopic probe…
The complex conductivity of a MgB2 film has been investigated in the frequency range 4 cm^{-1}< nu < 30 cm^{-1} and for temperatures 2.7 K < T <300 K. The overall temperature dependence of both components of the complex conductivity is reminiscent of BCS-type behavior, although a detailed analysis reveals a number of discrepancies. No characteristic feature of the isotropic BCS gap temperature evolution is observed in the conductivity spectra in the superconducting state. A peak in the temperature dependence of the real part of the conductivity is detected for frequencies below 9 cm^{-1}. The …
The specific heat coefficient gamma_s(H) and the electronic thermal conductivity kappa_{es}(H) are calculated for Abrikosov's vortex lattice by taking into account the effects of supercurrent flow and Andreev scattering. First we solve the gap equation for the entire range of magnetic fields. We take into account vertex corrections due to impurity scattering calculated in the Born approximation. The function gamma_s(H)/gamma_n increases from zero and becomes approximately linear above H/H_{c2} \sim 0.1. The dependence on impurity scattering is substantially reduced by the vertex corrections. T…
The thermal conductivity $κ(H,T)$ of the new superconductor $MgB_2$ was studied as a function of the temperature and a magnetic field. No anomaly in the thermal conductivity $κ(H,T)$ is observed around the superconducting transition in absence or presence of magnetic fields up to 14 Tesla; upon that field the superconductivity of $MgB_2$ persisted. The thermal conductivity in zero-field shows a $T$-linear increase up to 50K. The thermal conductivity is found to increase with increasing field at high fields. We interpret the findings as if there are two subsystems of quasiparticles with differe…
Based on the experimentally found existence of two gaps in MgB2 (one gap associated to the boron sigma-states and the other to the boron pi-states), the different contributions to the transport properties, electrical conductivity and Hall coefficient, were studied using the full potential-linearized augmented plane wave method and the generalized gradient approximation. MgB2 doping was analyzed in the rigid band approximation. This permitted the study of the partial substitution of magnesium for aluminium (Mg1-xAlxB2) as well as other substitutions such as AB2 (A=Be, Zr, Nb and Ta). The sigma-…
In this paper we present the resistivity, the Seebeck effect, and the thermal conductivity measurements on a MgB2 sintered sample. Such transport properties highlight the role of the junctions between the grains to a different extent. In particular, the temperature dependence of resistivity may be explained with the assumption that grain boundaries have a resistance, independent of temperature, in series with the resistance of grains. Also the behaviour of the Seebeck effect, as long as the scattering at the grain boundaries is elastic, is not affected at all by granularity. Thus, the thermopo…
We present resistivity and thermal conductivity measurements on bulk samples, prepared either by a standard method or by a one-step technique. The latter samples, due to their high density and purity, show residual resistivity values as low as 0.5 mW cm and thermal conductivity values as high as 215 W/mK, higher than the single crystal ones. Thermal and electrical data of all the samples are analysed in the framework of the Bloch-Gruneisen equation giving reliable parameter values. In particular the temperature resitivity coefficient, obtained both from resistivity and thermal conductivity, in…
The possibility of multi-band conductivity and multi-gap superconductivity is explored in oriented V3Si thin films by means of reflectance and transmittance measurements at terahertz frequencies. The temperature dependence of the transmittance spectra in the normal state gives evidence of two bands contributing to the film conductivity. This outcome is consistent with electronic structure calculations performed within density functional theory. On this basis, we performed a detailed data analysis and found that all optical data can be consistently accounted for within a two-band framework, wit…
We investigated the complex conductivity spectrum of a Co-doped BaFe$_2$As$_2$ epitaxial thin film in the THz region. In the normal state, the complex conductivity shows a Drude-type frequency dependence, while in the superconducting state, the frequency dependence of the complex conductivity changes to that of a typical superconducting materials. We estimated the magnetic penetration depth at absolute zero to be 710 nm and the superconducting gap energy to be 2.8 meV, which is considered to be the superconducting gap opened at the electron-type Fermi surface near the M point. We succeeded in …
We comment on the model proposed by Orenstein and Dodge in arXiv:1506.06758v1, which describes time-domain terahertz measurements of transiently generated, high-electron-mobility (or superconducting) phases of solids. The authors' main conclusion is that time-domain terahertz spectroscopy does not measure a response function that is mathematically identical to the transient optical conductivity. We show that although this is correct, the difference between the measured response function and the microscopic optical conductivity is small for realistic experimental parameters. We also show that f…
The morphological, transport and terahertz optical properties of DC magnetron sputtered granular molybdenum thin-films with nano-grains embedded in an amorphous matrix have been studied in the normal and superconducting states. The superconducting transition temperatures of these films are much higher than that of bulk molybdenum. The optical properties of these thin-films have been studied using terahertz time-domain spectroscopy. Their properties have been compared with the existing materials used for the development of radiation detectors. The resistivity of the films lies in >100 micro-Ohm…
According to the crystal structure of MgB$_{2}$ and band structure calculations quasi-two-dimensional (2D) boron planes are responsible for the superconductivity. We report on critical fields and resistance measurements of 30 nm thick MgB$_{2}$ films grown on MgO single crystalline substrate. A linear temperature dependence of the parallel and perpendicular upper critical fields indicate a 3D-like penetration of magnetic field into the sample. Resistivity measurements, in contrast, yield a temperature dependence of fluctuation conductivity above T$_{c}$ which agrees with the Aslamazov-Larkin t…
The electrical conductivity anisotropy of the sigma-bands is calculated for the (Mg,Sc)B2 system using a virtual crystal model. Our results reveal that anisotropy drops with relatively little scandium content (< 30%); this behaviour coincides with the lowering of Tc and the reduction of the Kohn anomaly. This anisotropy loss is also found in the Al and C doped systems. In this work it is argued that the anisotropy, or 2D character, of the sigma-bands is an important parameter for the understanding of the high Tc found in MgB2.
We report an infrared spectroscopy study of a 200 nm thick FeSe$_{0.5}$Te$_{0.5}$ film grown on LaAlO$_3$ with T$_c$=13.7 K. We analyze the 20 K normal state absolute reflectance R$_N$ measured over a broad infrared range and the reflectance ratio R$_S$/R$_N$, R$_S$ being the superconducting state reflectance, measured at 6 K in the terahertz range down to 12 cm$^{-1}$. We show that the normal state model conductivity is given by two Drude components, one of which much broader and intense than the other. In the superconducting state, we find that a gap $Δ$=37$\pm$3 cm$^{-1}$ opens up in the na…
The flux flow resistivity associated with purely viscous motion of vortices in high-quality MgB_2 was measured by microwave surface impedance. Flux flow resistivity exhibits unusual field dependence with strong enhancement at low field, which is markedly different to conventional s-wave superconductors. A crossover field which separates two distinct flux flow regimes having different flux flow resistivity slopes was clearly observed in H//ab-plane. The unusual H-dependence indicates that two very differently sized superconducting gaps in MgB_2 manifest in the vortex dynamics and almost equally…
We show that strong electric fields of ~ 30 kV cm^(-1) at terahertz frequencies can significantly weaken the superconducting characteristics of cuprate superconductors. High-power terahertz time-domain spectroscopy (THz-TDS) was used to investigate the in-plane conductivity of YBa2Cu3O7-delta (YBCO) with highly intense single-cycle terahertz pulses. Even though the terahertz photon energy (~ 1.5 meV) was significantly smaller than the energy gap in YBCO (~ 20-30 meV), the optical conductivity was highly sensitive to the field strength of the applied terahertz transients. Possibly, this is due …
We present thermal conductivity and specific heat measurements on MgB2 and Mg-AlB2 samples. Thermal properties have been analysed by using a two-gap model in order to estimate the gap amplitudes, D(0)p and D(0)s and the intra-band scattering rates, Gss and Gpp. As a function of Al doping and disorder D(0)s rapidly decreases, while D(0)p is rather constant. Gss and Gpp are increased by the disorder, being Gpp more affected than Gss.
Temperature dependent optical conductivities and DC resistivity of c-axis oriented superconducting (Tc = 39.6 K) MgB2 films (~ 450 nm) have been measured. The normal state ab-plane optical conductivities can be described by the Drude model with a temperature independent Drude plasma frequency of ω_{p,D}=13,600 +/- 100 cm-1 or 1.68 +/- 0.01 eV. The normal state resistivity is fitted by the Bloch-Gruneisen formula with an electron-phonon coupling constant λ_{tr} = 0.13 +/- 0.02. The optical conductivity spectra below T_c of these films suggest that MgB2 is a multi-gap superconductor.
In this work the effect of Carbon and Aluminium doping on the multiband MgB2 superconductor is analyzed. Using the rigid band and virtual crystal approximations (RBA and VCA), it was found that the main effect of doping on the band structure is band filling and a relative band-shift. If this band-shift is eliminated with an appropriate change of scale, then the RBA provides a good description of the band structure as function of doping. With this procedure both the inplane electrical conductivity of the C- and Al-doped MgB2 and the superconducting critical temperature follow the same curve. Th…
The microwave properties of polycrystalline MgB2 thin films prepared by the so-called in-situ method are investigated. The characterization of the films at microwave frequencies was obtained by a coplanar resonator technique. The analysis of the experimental data results in the determination of penetration depth, surface impedance and complex conductivity. The aim of this work is to set the experimental results in a consistent framework, involving the two-band model in the presence of impurity scattering. The energy gaps are calculated and the contribution of intra- and inter-band scattering i…
We consider the far-infrared and the microwave conductivities of a two-band superconductor with non-magnetic impurities. The strong coupling expressions for the frequency and temperature dependent conductivity of a two-band superconductor are developed assuming isotropic bands and interactions. Our numerical results obtained using realistic interaction parameters for MgB$_{2}$ are compared with experiments on this compound. We find that the available experimental results for the far-infrared conductivity of MgB$_{2}$ are consistent with multi-band superconductivity in the presence of a suffici…
We show that a two-band model with ${\bf k}$-dependent superconducting gaps well describes the transmission and optical conductivity measured for MgB$_2$ thin films. It is also shown that the two-band anisotropic model consistently describes the specific-heat jump and thermodynamic critical magnetic field $H_c$. A single-gap anisotropic model is shown to be insufficient to understand consistently optical and thermodynamic behaviors. In our model, the pairing symmetry in each band has an anisotropic characteristic which is determined almost uniquely; the superconducting gap in the $σ$-band has …
We show how intrinsic material properties modify light transmission through subwavelength hole arrays on thin metallic films in the THz regime. We compare the temperature-dependent transmittance of Au films and MgB$_{2}$ films. The experimental data is consistent with analytical calculations, and is attributed to the temperature change of the conductivity of both films. The transmission versus conductivity is interpreted within the open resonator model when taking the skin depth into consideration. We also show that the efficiency of this temperature control depends on the ratio of the transmi…
Optical conductivity of a moderately disordered superconducting NbN film was investigated by terahertz time-domain spectroscopy in external magnetic field applied along the film plane. The film thickness of about 5 nm was comparable with the coherence length, so vortices should not form. This was confirmed by the fact that no marked difference between the spectra with terahertz electric field set perpendicular and parallel to the external magnetic field was observed. Simultaneous use of Maxwell-Garnett effective medium theory and the model of optical conductivity by Herman and Hlubina proved t…
We report the growth of high-quality epitaxial MgB2 thin films on (1102) Al2O3 and (001) SrTiO3 substrates by using a pulsed laser deposition technique. The thin films grown on Al2O3 substrates show a Tc of 39 K with a sharp transition width of ~ 0.7 K. The critical current density in zero field is ~ 8 x 10^6 A/cm^2 at 5 K, suggesting that this compound has great potential for industrial applications. Also, Al2O3 is a very promising substrate in terms of superconducting device applications because of its high thermal conductivity and small dielectric constant. For the films deposited on Al2O3,…
The optical conductivity of MgB$_2$ has been determined on a dense polycrystalline sample in the spectral range 6 meV - 4.6 eV using a combination of ellipsometric and normal incidence reflectivity measurements. $σ_{1}(ω)$ features a narrow Drude peak with anomalously small plasma frequency (1.4 eV) and a very broad "dome" structure, which comprises the bulk of the low-energy spectral weight. This fact can be reconciled with the results of band structure calculations by assuming that charge carriers from the 2D $σ$-bands and the 3D $π$-bands have principally different impurity scattering rates…
The superconducting transition temperature, T$_{C}$, the residual resistivity $ρ_{0}$ and the slope of resistivity curve at high temperature, d$ρ$/dT, have been measured in a series of MgB$_{2}$ samples that have been chemically substituted to varying degree with Li or Cu at the Mg-site and by Li or Cu at the Mg-site along with C substitution at the B-site. DC resistivity and ac susceptibility measurements were employed to extract the above parameters. T$_{C}$ versus the electron count (estimated from simple chemical valence count arguments) shows a universal behaviour, with T$_{C}$ being cons…
We demonstrate the effect of bandstructure on the superconducting properties of Pb by calculating the strong-coupling features in the optical conductivity, $σ(ω)$, due to the electron-phonon interaction. The importance of momentum dependence in the calculation of the properties of superconductors has previously been raised for MgB$_2$. Pb resembles MgB$_2$ in that it is a two band superconductor in which the bands' contributions to the Fermi surface have very different topologies. We calculate $σ(ω)$ by calculating a memory function which has been recently used to analyze $σ(ω)$ of Bi$_2$Sr$_2…
We present the anisotropic optical conductivity of MgB$_{2}$ between 0.1 and 3.7 eV at room temperature obtained on single crystals of different purity by the spectroscopic ellipsometry and reflectance measurements. The bare (unscreened) plasma frequency $ω_{p}$ is almost isotropic and equal to 6.3 eV, which contrasts some earlier reports of a very small value of $ω_{p}$. The data suggests that the $σ$-bands are characterized by a stronger electron-phonon coupling $λ_ {tr}$ but smaller impurity scattering $γ_{imp}$, compared to the $π$-bands. The optical response along the boron planes is mark…
We performed terahertz magneto-optical spectroscopy of FeSe thin film to elucidate the charge carrier dynamics. The measured diagonal (longitudinal) and off-diagonal (Hall) conductivity spectra are well reproduced by two-carrier Drude model, from which the carrier densities, scattering times and effective masses of electron and hole carriers are determined in a wide range of temperature. The hole density decreases below the structural transition temperature while electron density increases, which is attributed to the band structure modification in the electronic nematic phase. The scattering t…
Time-resolved terahertz time-domain spectroscopy (THz-TDS) is an ideal tool for probing photoinduced nonequilibrium metallic and superconducting states. Here, we focus on the interpretation of the two-dimensional response function $Σ(ω;t)$ that it measures, examining whether it provides an accurate snapshot of the instantaneous optical conductivity, $σ(ω;t)$. For the Drude model with a time-dependent carrier density, we show that $Σ(ω;t)$ is not simply related to $σ(ω;t)$. The difference in the two response functions is most pronounced when the momentum relaxation rate of photocarriers is long…
We measured reflectivity on static stripe-ordered La$_{1.84-y}$Nd$_y$Sr$_{0.16}$CuO$_4$ (LNSCO) $y$\,=\,0.1, 0.2, 0.3, and 0.4 by means of reflective terahertz time-domain spectroscopy (THz-TDS) with electric field polarized along c-axis, in which one can obtain lower frequency information than the conventional Fourier transform type spectrometer. Recently, two-dimensional superconducting (2DSC) property \cite{tajima,schafgans,li}, intra-layer perfect conductivity not accompanied with inter-layer superconducting behavior, was reported on LNSCO ($x$\,=0.15\,,\,$y$\,$\geq$\,0.2). We observed the…
In this Letter we present resonance properties in terahertz metamaterials consisting of a split-ring resonator array made from high temperature superconducting films. By varying the temperature, we observed efficient metamaterial resonance switching and frequency tuning with some features not revealed before. The results were well reproduced by numerical simulations of metamaterial resonance using the experimentally measured complex conductivity of the superconducting film. We developed a theoretical model that explains the tuning features, which takes into account the resistive resonance damp…
We report on time-resolved linear and nonlinear terahertz spectroscopy of the two-band superconductor MgB$_2$ with the superconducting transition temperature $T_c \approx 36$ K. Third-harmonic generation (THG) is observed below $T_c$ by driving the system with intense narrowband THz pulses. For the pump-pulse frequencies $f=$ 0.3, 0.4, and 0.5 THz, temperature-dependent evolution of the THG signals exhibits a resonance maximum at the temperatures with the resonance conditions $2f=2Δ_π(T)$ fulfilled, for the dirty-limit superconducting gap $2Δ_π$. In contrast, for $f=$ 0.6 and 0.7 THz with $2f>…
We investigated a high-quality MgB$_{2}$ thin film with a thickness of $\sim$1000 nm on an Al$_{2}$O$_{3}$ substrate using optical spectroscopy. We measured the reflectance spectra of the film at various temperatures both below, and above, the superconducting transition temperature, $T_c$ $\simeq$ 40 K. An earlier study showed that when the sample surface is exposed to air the optical properties of the surface change immediately, however, the saturated change is negligibly small in the far-infrared region. The optical conductivity spectrum in the normal state shows two (narrow and broad) Drude…
