Nodeless two-gap superconductivity in stoichiometric iron pnictide LiFeAs
The variations of in- and inter- plane London penetration depths, , were measured using a tunnel diode resonator in single crystals of the intrinsic pnictide superconductor LiFeAs. This compound appears to be in the clean limit with a residual resistivity of 4 () to 8 () cm and of 65 to 35, respectively. The superfluid density, , is well described by the self-consistent two-gap model. Together with the previous data, our results support the universal evolution of the superconducting gap from nodeless to nodal upon departure from optimal doping. We also conclude that pairbreaking scattering plays an important role in the deviation of the low-temperature behavior of from exponential in Fe-based compounds.
Studies of the superconducting gap structure play an important role in the determination of the mechanism responsible for superconducting pairing. In iron-based superconductors Kamihara2008 (), the situation regarding the gap structure remains controversial. In the ”1111” RFeAs(O,F) compounds (R = rare earth) at optimal doping tunneling Chen2008 () and angle-resolved photoemission spectroscopy (ARPES) Kondo2008 () experiments have found a full superconducting gap. An exponential low-temperature variation of the penetration depth, , was reported in compounds with magnetic rare earths Sm1111 Malone2009 () and Pr1111 Hashimoto2009a (), but the non-magnetic La1111 has shown a close to power-law behavior, incompatible with a clean, full gap Martin2009 (). A similar power-law behavior of was found in optimally electron -doped Ba(FeT)As (”BaT122”, T = transition metal) Gordon2009a (); Williams2009 (); Martin2010a (); Kim2010b () and hole-doped BaKFeAs (”BaK122”) Hashimoto2009 (); Martin2009a (). It was suggested that such a non-exponential behavior comes from a full gap with pair-breaking scattering Dolgov2009 (); Bang2009 (); Mishra2009a (); Vorontsov2009 (); Kogan2009 (); Hashimoto2009 (); Gordon2010 (); Kim2010b (). On the other hand, in clean (as suggested by observations of quantum oscillations Carrington2009 (); Shishido2010 (); Terashima2010 ()) isoelectronic P-doped BaFe(AsP) (”BaP122”) Hashimoto2010a (), and low- stoichiometric LaFePO Fletcher2009 () and KFeAs (”K122”) Dong2010 (); Hashimoto2010 (), the gap appears to be nodal, which was suggested to be an intrinsic behavior of clean Fe-pnictides. However, nodal behavior was observed also in dirty systems. Doping-dependent gap anisotropy Luo2009 (); Tanatar2010 () and nodes Martin2010a (); Reid2010 () were reported for 122 crystals. It was suggested that a full gap at optimal doping evolves into a 3D nodal structure when doped toward the edges of the superconducting“dome”. Hereinafter, when we refer to a “full gap”, we cannot exclude the possibility of some angular variation (e.g., see Refs. [Zeng2010, ; Chubukov2010, ; Vorontsov2010, ]), and only mean that such variation is smaller than the gap magnitude.
Since doping inevitably introduces scattering Kemper2009 (), which is pairbreaking in iron pnictides Dolgov2009 (); Bang2009 (); Vorontsov2009 (); Kogan2009 (); Gordon2010 (); Kim2010b (), measurements of stoichiometric intrinsic superconductors become of utmost importance. LiFeAs with high K Wang2008 (); Tapp2008 (); Chu2009 (); Song2010 () is among very few such compounds. It is one of the cleanest systems with a high residual resistivity ratio () of about 50 Song2010 (), much higher than BaP122 (5 to 8 for different doping) Kasahara2010 (), BaCo122 (3 to 4) Ni2008 () and BaK122 (7 to 10) Luo2008 (), pure Ba122 (7 to 10 under pressure) Colombier2009 (), though still below pure K122 (over 1000) Terashima2010 (). Since of LiFeAs decreases with pressure Chu2009 (); Gooch2009 (), which is observed only in optimally and overdoped compounds Colombier2010 (), we can assign its ”equivalent” doping level as slightly overdoped, as opposed to underdoped NaFeAs, whose goes through a maximum with pressure Zhang2009 () and heavily overdoped K122. This doping assignment is consistent with the temperature-dependent resistivity, discussed later. With the much reduced effect of pairbreaking scattering, comparison of these stoichiometric compounds can bring an insight into the intrinsic evolution of the superconducting gap.
High chemical reactivity leading to quick degradation in air makes LiFeAs challenging to study. A single, fully isotropic gap Inosov2010 (), as well as two-gap pairing Borisenko2010 () were reported from ARPES. Two-gap superconductivity is also supported by magnetization Sasmal2010 (); Song2010a (), specific heat Wei2010 () and nuclear magnetic resonance (NMR) Li2010 ().
In this Letter, we report on the in-plane London penetration depth, , in single crystals of LiFeAs measured using a tunnel diode resonator (TDR) Degrift1975 (). The superfluid density can be well fitted with the self-consistent clean two-gap model Kogan2009a (). Our results imply that the ground state of pnictide superconductors in the clean limit at optimal doping is given by symmetry with two distinct gaps, and .
Single crystals of LiFeAs were grown in a sealed tungsten crucible using Bridgeman method Song2010 (); Song2010a () and were transported in sealed ampoules. Immediately after opening, mm pieces of the same crystal (all surfaces cleaved in Apiezon N grease) were used for TDR, transport and magnetization measurements. Small resistance contacts () were tin-soldered Tanatar2010a () and resistivity was measured using a four probe technique in Quantum Design PPMS. The transition temperature, , was determined at the maximum of the derivative , Table 1. The London penetration depth was measured with the TDR technique (for review, see Prozorov2006 ()). The sample was inserted into a 2 mm inner diameter copper coil that produced an rf excitation field (at MHz) with amplitude mOe, much smaller than . Measurements of the in-plane penetration depth, , were done with -axis, while with we measured that contains a linear combination of and Martin2010a (). The shift of the resonant frequency, , where is the differential magnetic susceptibility, is a constant, is the demagnetization factor, is the sample volume and is the coil volume. The constant was determined from the full frequency change by physically pulling the sample out of the coil. With the characteristic sample size, , , from which can be obtained Prozorov2000 (); Prozorov2006 ().
The main panel in Fig. 1 shows the temperature-dependent resistivity, (left axis), and skin depth, (right axis). up to room temperature is shown in the top inset. The residual resistivity ratio, and it reaches the value of 65 when extrapolated to using a 2 order polynomial. This behavior is consistent with the -dependent resistivity of BaCo122 in the overdoped regime Doiron-Leyraud2009 (). The calculated skin depth, , compares well with the TDR data for , where , Hardy1993 () when we use =250 cm, the lowest directly measured value among our crystals. A very good quantitative match of two independent measurements gives us a confidence in both resistivity data and the TDR calibration.
To check for degradation effects, a sample was intentionally exposed to air for an hour and the measurements were repeated, as shown in the lower inset in Fig. 1. After the exposure, the sample surface lost its shiny metallic gloss, but the transition temperature and width remained the same. At the same time, the total frequency shift through the transition (proportional to the sample volume) decreased, suggesting that the surface had degraded without introducing any change in the bulk. This experiment clearly supports bulk uniform superconductivity of our samples.
in three LiFeAs crystals is shown up to in Fig. 2 by solid dots. was analyzed using (1) power-law, (with and being free parameters) and (2) exponential BCS form, (with and as free parameters). The best fit results for sample #1 are shown with solid (power-law) and dashed (exponential) lines. The fit residuals are shown in the inset. The exponential fit quality is as good as the power-law, although is smaller than the value of 1.76 expected for a conventional single fully-gapped s-wave pairing and nm is somewhat larger than the experimental nm Pratt2009 (); Inosov2010 (). This is naturally explained by two-gap superconductivity in LiFeAs. The superconducting and best fit parameters (obtained from fitting up to ) for all samples are summarized in Table 1. was defined at 90% of the rf susceptibility variation over the transition: the mean was defined at the maximum of and . from the single-gap exponential BCS behavior. The power-law coefficient was obtained with the exponent as a free parameter, while was obtained with a fixed (average of 3 samples). Crosses in Fig. 2 show for sample #2. A clear saturation of at low temperatures suggests exponential behavior of .
Figure 3(a) shows vs. with which is the average exponent for three samples. The dependence of the parameters, and , on the fitting temperature range is summarized in Fig. 3 (b) and (c), respectively. As expected, the exponent is more scattered for the shortest fit interval, otherwise and do not depend much on the fitting range from base temperature to 6 K and give for all samples, with the average value . With fixed at this average value, we determined the prefactor m/K.
The superfluid density, is the quantity to compare with the calculations for different gap structures. Figure 4 shows for crystal #1 calculated with nm Pratt2009 (); Inosov2010 (). A noticeable positive curvature above is similar to other Fe-based superconductors Prozorov2009 () and MgB Fletcher2005 (). This suggests multigap superconductivity in LiFeAs, which we analyze in the framework of the self-consistent model Kogan2009a (). Since LiFeAs is a compensated metal and its bandstructure is fairly two-dimensional Borisenko2010 (), we may introduce the mass ratio on two cylindrical bands, , whence the partial density of states of the first band, . The total superfluid density is with . We also use the Debye temperature of 240 K Wei2010 () to calculate , which allows fixing one of the interaction constants, . This leaves three free fit parameters: the in-band, , and interband, , interaction and the mass ratio, . Figure 4 shows that can be well described in the entire temperature range by this clean-limit weak-coupling BCS model. In the fitting, the two gaps were calculated self-consistently (which is the major difference between this one and the popular, but not self-consistent, - model Bouquet2001 ()) and the self-consistent and are shown in the upper inset in Fig. 4, while the gap ratio is shown in the lower inset indicating strong non-single-gap-BCS behavior of the small gap. The best fit, using Levenberg-Marquardt nonlinear simplex optimization in Matlab, gives and . As expected, one of the gaps is larger and the other is smaller than the single-gap value of 1.76, which is always the case for a self-consistent two-gap solution. The best fit parameters are: , , and . The determined mass ratio gives and . This is consistent with bandstructure calculations that yield and Mazin2010a (), and ARPES experiments that find Borisenko2010 (). The effective coupling strength, , is not far from estimated for 122 Boeri2010 () and for 1111 Boeri2008 () pnictides. The electron band with a smaller gap gives about 1.5 times larger contribution to the total resulting in a crossing of the partial densities at low temperatures. Similar result was obtained from magnetization measurements Sasmal2010 (). We stress that while , and (hence, , and ) and are unique self-consistent solutions describing the data, the coupling matrix is not unique. There are other combinations that could produce similar results, so it seems that has to be calculated from first principles Mazin2010a ().
In conclusion, we find that in the clean limit, optimally-doped Fe-based superconductors are fully gapped, but most measurements are affected by pairbreaking scattering Dolgov2009 (); Bang2009 (); Mishra2009a (); Vorontsov2009 (); Kogan2009 (); Hashimoto2009 (); Gordon2010 (); Kim2010b (). This conclusion is in line with studies of thermal conductivity (which is not so sensitive to scattering) in BaCo122 Tanatar2010 (); Reid2010 (). On the other hand, intrinsic K122 reveals a nodal gap Dong2010 (); Hashimoto2010 (), which is also found in heavily overdoped BaCo122 Reid2010 (). Overall, this establishes a common trend for all Fe-based superconductors to have a superconducting gap that evolves from full to nodal when moving toward the edge of the superconducting dome.
We thank S. Borisenko, S. Bud’ko, P. Canfield, A. Chubukov, D. Evtushinsky, P. Hirschfeld, V. Kogan, Y. Matsuda and I. Mazin for useful discussions. Work at Ames was supported by the U.S. Department of Energy, Office of Basic Energy Science, Division of Materials Sciences and Engineering. Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under Contract No. DE-AC02-07CH11358. Work at SKKU was partially supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (2010-0007487). R.P. acknowledges support from the Alfred P. Sloan Foundation.
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