Highly anisotropic energy gap in superconducting Ba(FeCo)As from optical conductivity measurements
We have measured the complex dynamical conductivity, , of superconducting Ba(FeCo)As ( K) at terahertz frequencies and temperatures 2 - 30 K. In the frequency dependence of below , we observe clear signatures of the superconducting energy gap opening. The temperature dependence of demonstrates a pronounced coherence peak at frequencies below 15 cm (1.8 meV). The temperature dependence of the penetration depth, calculated from , shows power-law behavior at the lowest temperatures. Analysis of the conductivity data with a two-gap model, gives the smaller isotropic -wave gap of meV, while the larger gap is highly anisotropic with possible nodes and its rms amplitude is meV. Overall, our results are consistent with a two-band superconductor with an gap symmetry.
pacs:74.25.Gz, 74.25.nd, 74.70.Xa
The issue of the symmetry of the superconducting order parameter in the iron-pnictide superconductors remains unsettled. Shortly after the discovery of superconductivity in these materials kamihara () models, which predict the order parameter to change its sign on different sheets of the Fermi surface (FS), have been put forward. mazin1 (); kuroki1 () In the simplest case of the -wave symmetry, the superconducting gap can be parameterized as . Because the FS of the pnictides consists of two distinct set of sheets, centered at the (hole pocket) and M (electron pocket) points of the Brillouin zone, there are no nodes in this gap.
Many experiments, including e.g. ARPES, are indeed consistent with this picture. christianson (); evtushinsky (); terashima (); williams (); hardy () There is, however, a large body of experimental works, which cannot be easily explained assuming the nodeless gap scenario. martin (); muschler (); salem (); goko () Possibly, the gap function of the iron pnictides is not universal – whether the gap is nodeless or not might depend on the compound and on the doping level. In order to account for this non-universal behavior, models with relatively large intraband Coulomb repulsion within the FS pockets have been proposed, resulting in gap symmetries varying between extended -wave , wave, and wave, all allowing for nodes in the order parameter. chubukov (); moreo (); maier (); kuroki2 ()
Currently, the so-called 122 iron-pnictide family (doped FeAs, = Ba, Sr, Ca) seems to be best suited for experimental investigations due to the availability of relatively large single crystals as well as thin epitaxial films of high quality. sefat (); hiramatsu (); katase (); iida (); lee () In this article we focus on Ba(FeCo)As. Thermal conductivity measurements, performed at temperatures down to 50 mK, indicate a nodeless in-plane gap in this compound. tanatar (); reid () The gap is reported to remain nodeless at all doping levels – from under- to overdoped, but the gap anisotropy increases with doping. This is consistent with APRES results, which systematically show a full gap in all pnictide compounds´. evtushinsky (); terashima (); kondo (); qian ()
Optical spectra of superconductors contain information about the size and the symmetry of the superconducting gaps. Since the last two years, a large number of reports on optical properties of pnictides has been published (see Ref. hu2, for a review). The low-frequency (terahertz-far-infrared) optical properties of the electron-doped 122 pnictides have recently been investigated by a few groups for doping levels near optimal. heumen (); kim (); gorshunov2 (); wu (); perucchi (); nakajima (); lobo () There is, however, neither consensus in these reports about the gap sizes, nor about the number of gaps (although the majority of reports agrees on a two-gap picture). Obviously, the analysis of the optical spectra is challenged by the fact, that the gaps are likely to be different on different sheets of the FS.
Our method allows us to perform not only frequency-dependent measurements, but also measurements of the optical parameters as a function of temperature at fixed frequencies. This gives advantages over standard optical studies, since frequency and temperature dependences of optical parameters can be analyzed simultaneously.
Here, we report on the observation of clear signatures of the superconducting-gap opening in the terahertz optical conductivity. Through the temperature-dependent measurements and theoretical analysis, we show the gap is highly anisotropic with possible nodes on at least one sheet of the FS.
Films of Ba(FeCo)As have been grown by pulsed laser deposition on (001)-orientated (LaSr)(AlTa)O (LSAT) substrates, transparent for terahertz radiation. The Ba(FeCo)As target was ablated with 248 nm KrF radiation under UHV conditions. iida () The phase purity was confirmed by x-ray diffraction in Bragg-Brentano geometry. The axes of the films was normal to the film surface. The thickness of the films, used for the measurements, was measured by ellipsometry. Standard four-probe method has been used to measure the dc resistivity. We have investigated two Ba(FeCo)As films with thicknesses of 20 and 100 nm. The results of our investigations for both films are qualitatively the same. However, the optical transmissivity of the thinner film is obviously higher, leading to a much better signal-to-noise ratio. That is why in this paper we present results for the thinner film only. The resistive onset of the superconducting transition in the film appeared at 25 K (Fig. 1, inset). The substrate was a plane-parallel plate, approximately mm in size with thickness of 1.025 mm.
In the frequency range 4 - 47 cm (120 - 1400 GHz, 0.5 - 5.8 meV) the measurements have been performed with a spectrometer, which uses backward-wave oscillators (BWOs) as sources of coherent and frequency-tunable radiation. kozlov () A Mach-Zehnder interferometer arrangement of the spectrometer allows to measure both the intensity and the phase shift of the wave transmitted through the Ba(FeCo)As film and the substrate gorshunov (). Using the Fresnel optical formulas for the complex transmission coefficient of the two-layer system, both components of the complex conductivity () of the film have been calculated. The optical parameters of the substrate have been found by measuring a bare substrate without film. This experimental method has been previously applied to a large number of different superconductors. dressel () In addition to the “standard” frequency sweeps at fixed temperatures, we performed temperature sweeps at fixed frequencies in the same way as it has been done e.g. in Ref. pronin, . This allows for a more thorough monitoring of the temperature dependence of and .
Figure 1 shows as-measured transmission and phase-shift spectra as a function of frequency, . The measurements are done with a number of different BWOs covering the range from 4 to 47 cm continuously. Since the major term of the phase shift is proportional to the frequency of the probing radiation, the phase-shift spectra are divided by frequency to eliminate the constant frequency slope. The pronounced fringes in both and are due to the multiple interference inside the substrate, which acts as a Fabry-Perot interferometer. As the complete Fresnel formulas for a two-layer system are used, these fringes are automatically taken into account in the calculations of the complex conductivity.
Figure 2 shows spectra of the real (upper frame) and imaginary (bottom frame) part of the complex conductivity of Ba(FeCo)As at selected temperatures above and below .
The normal-state complex conductivity demonstrates typical Drude behavior, . From simultaneous fit of and at 30 K (dashed line in Fig. 2), we find = 6000 cm and the scattering rate cm ( s), the former is in agreement with our dc measurements, and the later coincides with the scattering rate, obtained via from the complex conductivity spectra ( obtained in this way has, however, a larger error bar due to the large scattering of data points in ). From , we estimate the plasma frequency of the Drude component in the normal state, cm (1.05 eV).
When entering the superconducting state the energy gap opens, as reflected in the lowering of below (upper frame of Fig. 2). For an isotropic -wave superconductor at a sharp frequency onset of is expected at . The absence of such a feature in our frequency window implies that the energy gap (or one of the gaps for a multi-band superconductor) must be larger than cm (2.9 meV). This result is consistent with the majority of optical measurements in Ba(FeCo)As – the feature is usually seen at 50 cm or somewhat higher frequency. heumen (); kim (); nakajima (); lobo ()
The appearance of a -function at in below leads (via the Kramers-Kronig relation) to an -divergence in , clearly visible in the bottom frame of Fig. 2 for K. The strength of this divergence (the pre-factor) is a direct measure of the weight of the superconducting condensate. From the data, one can directly extract the penetration depth via
At 2 K, we find m, which gives for the plasma frequency of the superconducting condensate, , the value of 3600 cm (450 meV), corresponding to the spectral weight of cm. These values are within the margins set by previous reports on terahertz measurements of Ba(FeCo)As films. gorshunov2 (); nakamura ()
For low enough frequencies and temperatures, where normal carriers are not affecting , an analysis of the temperature-dependent penetration depth is meaningful. We have extracted as a function of temperature for a number of frequencies (Fig. 3). For , we observe a clear power law behavior. As can be seen from this figure, the power law has an exponent slightly higher than 2, which is in agreements with microwave measurements. gordon (); prozorov ()
Iv Analysis and Discussion
Let us now turn to the analysis of the conductivity data. As it can be seen from the upper panel of Fig. 2, at frequencies below 15 cm, the temperature dependence of below is non-monotonic. Upon entering the superconducting state, first rises, and then drops. In order to monitor this temperature dependence in more detail, we performed temperature sweeps at fixed frequencies (Fig. 4). At low frequencies, a prominent coherence peak appears in . Multi-gap effects and gap symmetry are known to have crucial influence on the coherence peak appearance. dolgov (); schachinger ()
The analysis of the data is performed in the spirit of Ref. schachinger, . We use an -wave gap schurrer () as a model for the extended -wave, which is one part of the model. We can do this because optics cannot resolve the phase difference, thus for the optics the model just consist of one isotropic -wave gap (, hole pocket) and a second extended -wave gap, which, in our case, is modeled using an -wave symmetric gap when referred to the M-point. This model is also described in greater detail in Ref. carbotte, .
The gap on the electron pocket is parameterized as (here is the -wave component, is the amplitude of the -wave component, and is the polar angle on the FS). An anisotropy parameter has been introduced in such a way that , , and ( is the FS average). In the clean limit, this gap has nodes, if .
For convenience, one can introduce the parameter , which gives the percentage of the -wave component in the -wave admixture. The temperature dependence of the superconducting gaps has been modeled by the standard mean-field BCS temperature dependence with K (the mid-point of the transition). The complex conductivity is a sum of two weighted components, corresponding to the two FS pockets: , and .
In our analysis we concentrated on the 10.6 cm data set following the procedure outlined by Schachinger and Carbotte. schachinger () The size of the two gaps, the -wave contribution () to the anisotropic gap, and the weights and were adjusted in order to reproduce the temperature dependence of the normalized conductivity as good as possible. We obtained the best fit with meV = 24 cm, meV = 64 cm, and , corresponding to and indicating the existence of nodes. The weights of the FS pockets were found to be for the hole and for the electron pocket, the normal-state elastic scattering rate being equal to 22 meV (180 cm). wu1 ()
The model calculations for the other three frequencies (14.6, 43, and 46 cm) are found to follow the experimental data without any further adjustment of the primary parameters. The theory predicts, assuming an instantaneous transition, that the 43 and 46 cm data sets should also show a coherence peak at K. This temperature, however, is already well within the superconducting transition of the film and thus, such peaks cannot be observed experimentally. What is left as a very convincing result is the correct reproduction of the coherence peak for 10.6 and 14.6 cm and a very good agreement with the data measured at 43 and 46 cm for K.
We calculated, moreover, using the model discussed previously the low temperature variation of the penetration depth in the zero frequency limit. modre () The result is presented in Fig. 3 as the dashed line. At , the agreement between the model and the experimental data is perfect. lambda ()
Another explanation for the low-temperature dependence of the penetration depth was given by Vorontsov et al. vorontsov () These authors studied the superconducting symmetry state and concluded that the strong interband interaction (scattering) is strongly pair-breaking and that the superconductor is driven into a gapless state. For this state the authors prove that the penetration depth will be at low temperatures. A similar result was reported by Gordon et al. gordon2 () who studied a highly anisotropic single -wave superconductor with a high concentration of pair-breaking impurities which will also drive the superconductor into a gapless state. They argued that because of the assumed high anisotropy this model must not necessarily be in contradiction to the thermal conductivity results of Tanatar et al. tanatar ()
Authors of recent optical measurements of Ba(FeCo)As argue for interpretation of the conductivity data in terms of an state with strong pair-breaking scattering. lobo (); aguilar () Nevertheless, it is necessary to point out that Nicol and Carbotte nicol () studied the optical properties of a classical -wave superconductor with magnetic impurities. They come to the conclusion that in the gapless state (strong pair-braking) a coherence peak becomes strongly suppressed which is in clear contradiction to our results presented in Fig. 4. Likely, as it has been proposed in Ref. lobo, , both effects, the large gap anisotropy and the pair-braking scattering have to be taken into account in order to describe the optical conductivity in the pnictides. No detailed model for such a calculation has been developed yet. The analysis performed in Ref. lobo, , however, suggests that only a minor part of in the superconducting state below the gap can be due to the pair-breaking effects, while the major contribution is due to the nodes or at least strong gap anisotropy.
Our observation of a superconducting gap with nodes is, at a first glance, at odds with the results of Tanatar et al. tanatar () However, the -wave model used by us to describe the anisotropic superconducting gap around the -point of the Brillouin zone provides a mechanism called “lifting of the nodes”, first discussed by Mishra et al. mishra () and later on studied explicitly in Ref. carbotte, . The model allows a lifting of the nodes in the superconducting gap given a high enough -wave contribution to the -wave admixture and additional elastic scattering. This mechanism is also temperature dependent and it is more likely to lift existing nodes the lower the temperature. Thus, it is possible within this model that at very low temperatures, like the ones used by Tanatar et al., tanatar () a small spectral gap can develop in the real part of the optical conductivity. Increasing the temperature will smear out this small gap and will show nodal behavior at higher temperatures. Most recent thermal conductivity measurements give evidence of a small gap (almost nodes) in regions of the FS which contribute significantly to in-plane conduction. reid ()
We found the superconducting energy gap in Ba(FeCo)As to be highly anisotropic. This anisotropy is responsible for the broad and pronounced coherence peak observed in which is in clear contradiction to a gapless superconducting state. A two-gap model – isotropic -wave gap for the hole pocket and an -wave gap for the electron pocket of the FS (Ref. schachinger, ) – has been applied to mimic the gap symmetry. The model provides a very good description of the curves measured at different frequencies. In the framework of this model, we have found the value of 3 meV for the isotropic gap and an rms amplitude of 8 meV for the second gap which is also found to be nodal. It was possible using this model to reproduce consistently not only the energy and temperature dependence of in the terahertz region but also the low temperature variation of the penetration depth.
We would like to thank Jules P. Carbotte and Ilya Eremin for useful discussions and Mykola Vinnichenko for the film-thickness measurements. Part of this work was supported by EuroMagNET II (EU contract No. 228043).
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