# Electrodynamics of electron doped iron-pnictide superconductors:

Normal state properties

###### Abstract

The electrodynamic properties of Ba(FeCoAs and Ba(FeNiAs single crystals have been investigated by reflectivity measurements in a wide frequency range. In the metallic state, the optical conductivity consists of a broad incoherent background and a narrow Drude-like component which determines the transport properties; only the latter contribution strongly depends on the composition and temperature. This subsystem reveals a behavior in the dc resistivity and scattering rate disclosing a hidden Fermi-liquid behavior in the 122 iron-pnictide family. An extended Drude analysis yields the frequency dependence of the effective mass (with in the static limit) and scattering rate that does not disclose a simple power law. The spectral weight shifts to lower energies upon cooling; a significant fraction is not recovered within the infrared range of frequencies.

###### pacs:

74.25.Gz, 74.25.Jb, 78.20.-e, 75.30.Fv,## I Introduction

The family of the recently discovered iron-pnictides surprises not only because of its extreme electronic properties, but also because of its fine-tuned interplay of electronic, spin and lattice degrees of freedom. The investigation is facilitated by the possibility to vary the systems by chemical substitution in many ways; often changing simultaneously structure and carrier density. However, the important parameters and their influence are not understood by now: in fact, it is not even obvious that the core of the problem has been identified.Xu08 (); Tesanovic09 (); Ishida09 () Such or similar problems are common to unconventional superconductors. In the case of cuprates, it is debated for long how a Mott insulator becomes conducting upon charge-carrier doping, how important the coupling to neighboring planes is, how magnetic and superconducting orders develop, what the influence of fluctuations is on the superconducting ground state and many other issues which remain unresolved. In the case of heavy-fermion materials, the hybridization of localized electrons and conduction electrons causes unusual properties that are subject to intense research even for longer than cuprates.Ott87 () These intermetallic compounds are usually good metals with strong influence of local electron-electron interaction leading to magnetic order at low temperatures; the tuning is not done by doping but by changing the bandwidth compared to the interaction strength. Organic superconductors are even another class of correlated electron matter, which are distinctively different due to their layered structure of molecules with electrons; nevertheless with similar issues relevant for the understanding of superconductivity evolving out of a correlated electronic state.Fukuyama06 () While a common understanding of the overall behavior in those materials has surfaced over the years, the subtleties remain a challenge, and novel surprising facts are discovered regularly.

In the case of iron-pnictides, the parent compounds are semimetals where the conduction electrons order in an antiferromagnetic fashion forming a collinear spin-density-wave (SDW).Cruz08 (); HuangQ08 () A structural phase transition precedes the SDW transition at a slightly higher temperature.Rotter08 () The SDW instability is usually associated with the imperfect interband nesting of itinerant electrons. Doping by electrons or holes suppresses both the structural transition and the magnetic order rapidly and superconductivity is found with above 50 K.Chu09 () However, since the parent iron pnictide compounds are already metallic, the increase in the charge carrier concentration as a result of doping does not seems to be a decisive factor in the development of superconductivity. Physical or chemical pressure causes superconductivity as well, indicating that the influence of the structural arrangement cannot be overestimated. The enormous and inverse isotope effect makes the situation even more puzzling.Liu09 (); Shirage09 (); Yanagisawa09 ()

Band structure calculations agree with angle resolved photoemission spectroscopy (ARPES) indicating that several electronic bands are close to the Fermi level, not all of them well separated in energy and momentum.Zabolotnyy2009 (); Yi2009 (); Terashima09 () This considerably complicates the understanding of these materials. At this point it is not clear how to associate the observed electron correlations with the multiband structure of iron-pnictides.Qazilbash09 () Most experimental methods, in accord with theory,Mazin08 () evidence a fully gapped superconductor with no nodes of the order parameter. Its symmetry might be wave, i.e. reverses sign for electron and hole pockets of the Fermi surface. ARPES resultsEvtushinsky09 (); Terashima09 () yield two gaps – different in energy by a factor of 2 – in certain parts of the Fermi surface which evolve simultaneously below .

Optical investigation turned out to be one of the most efficient and powerful tools to reveal the electronic and dynamic properties of multiband conductors, iron-pnictides in particular.Chen08 (); Chen09 (); Dong08 (); Hu08 (); Akrap09 (); Nakajima09 (); Li08SC (); Hu09c (); Wu09 (); Wu09NP (); BFilm10 (); Bernhard09 (); Dirk09 (); Homes10 () Here we will restrict ourselves to the 122 family for which single crystals of suitable size are readily available. Direct observation of the partial SDW gap was first achieved in SrFeAs and BaFeAs and interpreted by a two-gap structure.Hu08 (); Akrap09 (); Nakajima09 () This, however, seems to be controversial since in EuFeAs Wu09 () and CaFeAs Nakajima09 () and in a more recent study on BaFeAs Pfuner09 () similar features have not been reported.remark1 () Chen et al. Chen09 () performed optical experiment on BaNiAs which undergoes a structural phase transition at 130 K Ronning08 (); Sefat09 () indicates the opening of a gap. However, so far no evidence for a magnetic phase was reported. Interestingly, in contrast to the SDW transition in BaFeAs or EuFeAs, the spectral weight of BaNiAs shifts to much higher frequencies (well above 10 000 cm). The extended Drude analysis of the complex conductivity of EuFeAs yields a linear behavior of the frequency-dependent scattering rate below , indicating an interaction between the charge carriers and spin fluctuations in the spin-density-wave state.Wu09 () From investigating the normal state properties of the superconducting BaKFeAs and by applying a sophisticated model, Yang et al.Yang09 () also concluded that the electron system couples to bosonic excitations. In their analysis it became obvious that it is important to separate the different contributions to the optical conductivity.

In Ref. Wu09NP, we briefly reported on our dc resistivity and optical measurements in a broader context of the observed universal electronic behavior in Ba(FeAs single crystals. Here we will give a full presentation of our optical experiments and thorough discussion of our results in the normal state. A proceeding paper (Ref. PartII, ) will deal with the superconducting state.

## Ii Experimental Details

Single crystals of Ba(FeCoAs and Ba(FeNiAs, i.e. close to optimal doping, were grown using FeAs as self-flux dopants Shuai (); Chu09 (); Li09 () and characterized by x-ray, EDX-microanalysis, transport and susceptibility measurements. The platelets with a typical size of have naturally flat surfaces. The temperature-dependent dc resistivity was obtained by standard four-probe technique, using silver paint as contacts. Samples were characterized by measuring the magnetic susceptibility down to 4 K by a Quantum Design SQUID system. Applied magnetic field of 5 Oe was perpendicular to the plane.Wu09NP ()

The temperature dependent optical reflectivity (in plane) was measured in a wide frequency range from 20 to 37 000 cm using a coherent-source spectrometer in the THz range ( cm),Gorshunov07 (); Dressel08 () two infrared Fourier transform spectrometers (Bruker IFS 66v/s and IFS 113v) ( cm) and a Woollam variable-angle spectroscopic ellipsometer extending up to the ultraviolet ( cm, restricted to room temperature). Part of the experiments were performed by employing an infrared microscope Bruker Hyperion. Using different CryoVac and homemade He cryostats we could continuously cool down to approximately 10 K. As reference we used either freshly evaporated aluminum or gold mirrors; alternatively gold was deposited onto the sample in situ and the measurements repeated at each temperature. While the accuracy in the absolute value might not be better than 1%, the uncertainty in the relative change with temperature is significantly better. The low-frequency extrapolation was done according to the dc conductivity measured on the same specimens. In the superconducting state we have smoothly extrapolated without introducing artificial kinks or steps. The complex optical conductivity was calculated from the reflectivity spectra using Kramers-Kronig analysis.DresselGruner02 ()

## Iii Electrical resistivity

The dc resistivity is measured to characterize our pnictides single crystals and to determine the energy scales which enter the problem. On one hand, resistivity is a bulk probe and therefore quite sensitive to extrinsic disorder or inclusion phases. On the other hand, it is a property that results from integration over the whole Fermi surface, and it is therefore sensitive to most of changes in the intrinsic behavior of the electronic system. Several samples of each compound are measured, showing temperature dependences identical to those presented in Fig. 1. The high data reproducibility indicates that samples are single phased and together with the observation of a sharp superconducting transition of K suggest that they are uniformly doped and of good quality.Barisic08 ()

The absolute value of the room-temperature dc resistivity is and for Ba(FeCoAs and Ba(FeNiAs, respectively, indicating a fairly good metal.remark4 () Indeed, by lowering the temperature from 300 K down to the superconducting transition at and 20 K, the resistivity decreases monotonously. Following previous attempts, we could fit equally the temperature dependence of the resistivity by the power-law

(1) |

with for Ba(FeCoAs and for the Ni compound, or by the sum of a linear term and a quadratic term

(2) |

with , for Ba(FeCoAs and , and Ba(FeNiAs.

A quadratic temperature dependence is expected for electron-electron scattering that is supposed to be dominant for correlated electron systems at low temperatures and subject to Landau’s theory of a Fermi liquid.Abrikosov63 (); Pines66 () The observation of a non-Fermi liquid exponent () or linear temperature dependence of the resistivity is often taken as the hallmark of a quantum critical regime.Sachdev01 (); Lohneysen07 () Similar behaviors were observed and recently discussed in heavy-fermions systems, organic superconductors or in high-temperature superconductors.Doiron09 () Quantum critical phenomena were initially investigated using a scaling analysis, based on an extension of the Landau theory of phase transitions, Landau () which – in the original version – describes the classical critical behavior near a second-order phase transition at . Theories for the continuous phase transitions give the so-called scaling laws.Fisher67 (); Kadanoff67 () The corresponding low-temperature dependences of the resistivity, for the spin-fluctuation models, result – depending on the dimensionality of the system – in the power-law coefficient which between 1 and 1.5 in the case of antiferromagnetic fluctuations.Hertz76 (); Millis93 (); Moriya95 ()

Thus it is not surprising that the proximity of the magnetic phase and the observation of the non-Fermi Liquid behavior initiated arguments that the superconducting ground state is a consequence of an underlying quantum criticality.Chu09 (); Gooch09 (); Jiang09 ()

Based on our analysis of the optical spectra of a large number of iron-pnictides,Wu09NP () we suggest an alternative approach. Instead of analyzing the data using Matthiessen’s rule, where different scattering contributions are treated additively, we introduce subsystems which add up their contributions to the conductivity. As will be discussed below in Sec. V.3, our findings suggest a two-component metal with a temperature-independent background () and an essentially Fermi-liquid channel () that determines the electronic properties of the pnictides.

## Iv Reflectivity and optical conductivity

### iv.1 Overall behavior

The electrodynamic properties of both compounds under investigation are very similar, which can serve as an indication that the behavior is intrinsic for the normal state of all members of the 122-family. The optical reflectivity exhibits a metallic behavior at all temperatures . It drops almost linearly with frequency up to more than 1000 cm where it changes slope around and continues to decrease again linearly up to the visible. In the logarithmic plots of Figs. 2(b) and (f), this becomes a shoulder-like feature which has been previously reported in the 122 family.Wu09 () It is attributed to an interband transition.Singh08 (); remark2 () Significant changes with temperature are seen in the far-infrared, where rises by decreasing temperature as depicted in Figs. 2(a) and (e). Correspondingly, the mid-infrared reflectivity exhibits the opposite temperature behavior: rises with the temperature; the isobetic point in is at cm and 700 cm, for Ba(FeCoAs and Ba(FeNiAs, respectively.

In Figs. 2(d) and (h) the corresponding optical conductivity of Ba(FeCoAs and Ba(FeNiAs is plotted for different temperatures in the metallic state. As drops, a Drude-like contribution evolves in accord with the rising dc conductivity. Its width decreases on reducing temperature; a few degrees above , at and 35 K, the scattering rate is around cm (for a more detailed analysis, see below). In the range of cm the conductivity decreases upon cooling, implying that the spectral weight is significantly reduced in this frequency range as the temperature is decreased. As discussed in the subsequent Section, part of it shifts to lower frequencies; however a sizeable fraction is not recovered within the spectral range considered here. For both compounds the conductivity around 1000 cm reaches a more-or-less temperature independent local minimum with a value of .

As mentioned above, the shoulder in the reflectivity plot corresponds to an interband transition that peaks around 4400 cm and a second near-infrared feature around 15 800 cm. The mid-infrared maximum is more pronounced for Ba(FeNiAs compared to the cobalt analogue.

### iv.2 Spectral weight analysis

In order to get a more quantitative picture of the spectral weight redistribution with temperature, in Fig. 3 we plot the integrated conductivity DresselGruner02 ()

(3) |

as a function of cut-off frequency , where is the density of carriers participating in optical transitions below the frequency . Up to cm no obvious step or saturation evidences a plasma frequency; this is in accord with the gradual decrease in the reflectivity to higher frequencies [Fig. 2(b)]. With temperature dropping from to 35 K, the spectral weight around 2500 cm is reduced by approximately 5% for both compounds, an effect which is significant and beyond the uncertainty of our experiments. To elucidate this point, we have normalized the spectral weight to its room temperature behavior: SW()/SW() and plotted the ratio in Figs. 3 (c) and (d) for an intermediate ( K) and low temperature (30 K, 35 K). The overall behavior is very similar for both compounds. As is reduced, there is an accumulation of spectral weight at low frequencies that is partially compensated in the range of cm, leading to SW()/SW(. Albeit the ratio goes through a minimum around 2500 cm and increases for higher frequencies, the relative spectral weight does not reach unity within the infrared range. As the temperature is lowered, the spectral-weight loss gets more pronounced. The surprising fact that the spectral weight is not recovered up to 5000 cm is a strong argument that correlation effects are of importance, as suggested previously.Qazilbash09 (); Chen09c () At this point, we cannot make well-founded statements on the recovery of the spectral weight in the energy range of the interband transitions, since our ellipsometric measurement were confined to ambient temperature. The necessary experiments are on our agenda.

To determine the plasma frequency, we have chosen cm as a suitable cut-off frequency. This leaves out the high frequency interband transition which starts to dominate the spectral weight above 2500 cm and considers mainly the itinerant electrons. According to Eg.3, we obtain cm for Ba(FeCoAs and 15 000 cm for the Ni analogue. The values are comparable to findings of other groups in compounds of the 122 family.Hu08 (); Li08SC (); Hu09c (); Pfuner09 (); Wu09 (); Nakajima09 () From the right hand side of the Eq. (3) we can estimate the carrier density for Ba(FeCoAs and Ba(FeNiAs to and , respectively, assuming . The values are summarized in Tab. 1.

### iv.3 Generalized Drude analysis

The extended Drude analysis has matured as a common tool to get insight into the low-energy excitations and the relevant scattering mechanisms of correlated electron systems.DresselGruner02 () The frequency dependent scattering rate and frequency dependent effective mass is calculated from the complex conductivity spectra which is given by:

(4) |

Here is the real part of a complex frequency dependent scattering rate . The imaginary part is related to the frequency dependent mass enhanced compared to the bandmass . From the complex conductivity one can obtain expressions for and in terms of and in following forms:

(5a) | |||||

(5b) |

In Figs. 4 and 5 the scattering rate 2 and normalized mass are plotted as a function of frequency for various temperatures in the metallic state.

For all measured temperatures, we find (below 800 cm) a steady increase in the effective mass, indicating deviations from a simple Drude response. The mass enhancement becomes stronger as the temperature is lowered which is usually attributed to correlation effects. For it reaches for both materials. It should be noted that the magnitude depends on the reference values of in Eq. (5). As argued above, we have chosen plasma frequencies and 15 000 cm.remark6 () Below a certain frequency, which is approximately 750 cm in the case of Ba(FeCoAs and 900 cm for Ba(FeNiAs, we find that the effective mass starts to exhibit a temperature dependence.

(m cm) | (cm) | (cm) | (s) | (nm) | ||||

Ba(FeCoAs | 0.415 | 12 000 | 1 | 5 | 53 | |||

Ba(FeNiAs | 0.385 | 15 000 | 1 | 4 | 37 |

For Ba(FeCoAs the scattering rate reveals a linear frequency dependence at most temperatures [Fig. 4(b)], where the slope increases on decreasing . A similar behavior was reported for the non-superconducting parent compound EuFeAs and for hole-doped BaFeAs.Wu09 (); Yang09 () In general this behavior is assumed to be a signature of bosonic excitations, like spin fluctuation. However, for Ba(FeNiAs the behavior is distinctly different: we can find a linear dependence of the scattering rate on frequency for the limited range between 200 and 600 cm, with the slope decreasing with in the same way as it does for Ba(FeCoAs. This, however, does not hold over an extended range. Also a dependence does not describe the entire frequency dependence satisfactorily, as demonstrated in Fig. 5(c). While for elevated temperatures the agreement is rather good, the deviations increase as drops. Interestingly, a fit by a power law yields an exponent over quite some range similar to what we obtained from the fit of by Eq. (1). This, however, might be a coincidence and requires additional investigations before entering further discussion.

The reduction of when lowering the temperature is somewhat expected for a metal. More surprising is the fact that the scattering rate increases above 400 cm as decreases. A similar behavior is observed in other strongly correlated electron systems, like high- superconductors, UPdAl, but also chromium.Dressel02 (); Basov02 () It was pointed out by Basov et al. Basov02 () in the context of high-temperature superconductors that reproduces the main features of the density of states including the gap or pseudogap followed by a sharp peak. Thus, a shift of weight in scattering rate from low to high frequencies might be interpreted as a signature of a gap or pseudogap leading to a reduction in the electronic density of states. Interestingly, a very similar behavior is found in the parent compound EuFeAs,Wu09 () where it was, again, attributed to the development of the SDW gap at low temperatures. However, from our susceptibility measurements we do not see any indications of a magnetically ordered state. Also the dc resistivity does not give any hint of a SDW transition in our electron doped Ba(Fe)As. Similarly, we do not find indications of gap-like features in the reflectivity data. Furthermore, the energy scale in frequency and temperature seems to be much to high for a SDW transition. All those arguments indicate that a change in the density of states is not the origin of the unusual increase in scattering rate above 400 cm. Importantly, it cannot be ruled out, that the shape of is just a consequence of a more complicated conduction mechanism. Multiple bands may induce a complex form of intra- and inter-band (cross) scattering, which may even vary with temperature. As a consequence, the single-component analysis – as performed here – leads to features which are of no specific meaning. Nevertheless, the agreement in between the parent compounds and the doped iron pnictides for measured frequencies seems unlikely to be a coincidence, since not only the overall behavior but also the energy scales are identical. This, again, can be an indication that, the doping has only small influence on the scattering mechanism.

## V Discussion

### v.1 One-component analysis

From our transport and optical measurement of Ba(FeCoAs and Ba(FeNiAs single crystals in highly-conducting -plane, we have deduced several characteristic parameters listed in Table 1. From the spectral weight analysis the carrier density was estimated to and for Ba(FeCoAs and Ba(FeNiAs, respectively, assuming a bandmass . Taking into account that the volume of the unit cell ,Rotter08 () one find approximately 0.32 and 0.5 itinerant electrons per unit cell, respectively; although there are two Fe ions per unit with a nominal valence of Fe meaning six electrons per Fe atom. A similar value of the charge carrier density was obtained for the parent compound EuFeAs,Wu09 () which – together with the observation that chemical or physical pressure can also cause superconductivity Torikachvili08 (); Jiang09 (); Ren09 (); remark7 () – leads to conclusion that crystal structure rather than carrier doping needs to be fine-tuned to provide the superconductivity. The fact that only a fraction of the Fe electrons contribute to conductivity suggests a distinction of localized and itinerant charge carriers. The effect of local magnetism of the Fe atoms was pointed by Yildirim and others.Yildirim08 (); Moon09 () In this regard, a comparison with the completely substituted compounds BaNiAs might be of interest where the carrier density is much higher.Chen09 () No magnetic order was detected and no superconductivity, but a structural distortion which causes a slight reduction in conduction carriers.

The extended Drude analysis shown in Figs.4(b) and 5(b) gives scattering rates which can be compared with rates obtained from the dc resistivity. In the limit the value of the is and for Ba(FeCoAs and Ba(FeNiAs, respectively. The classical expression for the conductivity yields and , correspondingly. This is in reasonable agreement with the values obtained from our extended Drude analysis. The calculated mean free path , assuming a Fermi velocity m/s of a typical metal Kittel () is of nm.

Up to now our approach was very conservative using only a classical concept (extended Drude analysis) which does not make assumptions that imply a microscopic interpretation. In the subsequent steps, we consider the electronic response due to two more-or-less independent electronic subsystems with some very distinct properties. Such an analysis is in part motivated by the multiband nature of iron-pnictides and by observation of two distinct superconducting gaps attributed to two distinct parts of Fermi-surface. Furthermore, such decomposition not only nicely fits the data but it is also the simplest one. Importantly, it leads to an unexpectedly simple description of the temperature dependent part of optical conductivity in terms of non or weakly interacting Fermi liquid.

### v.2 Decomposition of optical conductivity

From Fig. 6 it becomes obvious that the spectra of various 122 materials are very similar at elevated temperatures, in the normal state, no matter which ground state they eventually enter when cooled down. While the parent compounds BaFeAs, SrFeAs, CaFeAs and EuFeAs undergo a SDW transition below 200 K,Wu09 (); Hu08 (); Nakajima09 () the electron doped materials Ba(FeCoAs and Ba(FeNiAs as well as the electron doped BaKFeAs become superconducting at , 20 and 37 K, respectively.Wu09NP (); Li08SC () The completely substituted BaNiAs is again non-superconducting.Chen09 () There, the conductivity is significantly larger because of the increased carrier concentration; Ni donates twice as many electrons per ions compared to Cu.

Following the widely used procedure, we fit the complex optical conductivity by a sum of Drude and Lorentz terms, keeping the number of contributions as small as possible:DresselGruner02 ()

(6) | |||||

Here is the number of contributing carriers, and are their electronic charge and mass. They define the plasma frequency: . is the center frequency of the Lorentz oscillator, is the relaxation rate describing the scattering or damping of the excitations.remark3 ()

For all members of the 122 family there seems to be a constant background in the conductivity that does not change very much with temperature and composition. On the top of it there are additional contributions, which are specific for the different compounds and exhibit distinct temperature dependences. This is demonstrated in the Fig. 7 for two compounds, Ba(FeCoAs and Ba(FeNiAs which have been extensively studied here. The temperature independent part of the conductivity is indicated by the gray shaded area, and it consist at low frequencies from a single relatively broad Drude-like term (medium hatched orange area) and two oscillators that mimic sufficiently well the high frequency contributions (magenta and blue hatched area). Significantly, the remaining temperature dependent part of conductivity Fig. 7(d) in the normal state can be described with a single Drude term . The overall quality of the fit is shown in Fig. 7(c) for the highest measured temperature and temperature just above the superconducting transition. It was shown that identical decomposition also holds for the normal state of EuFeAs.Wu09 ()

However, already at elevated temperatures upon passing through the SDW transition in the case of BaFeAs, SrFeAs, EuFeAs, or BaNiAs, for instance, a gap in the excitation spectrum opens since parts of the Fermi surface are removed. Consequently, spectral weight in shifts from low-frequencies to above approximately 1000 cm. Nevertheless, a small Drude-like peak (accounting for the metallic dc properties) remains and it becomes very narrow as the temperature is reduced.Wu09 (); Hu08 (); Nakajima09 (); Chen09 () On the other hand, the metallic state of the superconductors close to optimal doping, such as Ba(FeCoAs, Ba(FeNiAs or BaKFeAs does not exhibit any trace of the SDW pseudogap. Upon cooling down to the superconducting transition, narrows and grows; spectral weight shifts to lower frequencies and the overall spectral weight [Eq. 3] is slightly reduced, as discussed in Sec. IV.2.

The suggested decomposition seems to be universal for the normal state of the whole 122 family directly, which follows from Fig. 6. The conductivity of nine different compounds (four parent, three electron doped, one hole doped and one in which the Fe is entirely substituted Ni) is decomposed in the above mentioned four terms, fitting satisfactorily the data. Furthermore, it is important to note that one of the spectra belongs to the 90 nm Ba(FeCoAs thin film. Thus, we can rule out the flux contribution in the optical spectra. Consequently the existence of two electronic subsystems is most likely an intrinsic property of 122 systems. In addition, it is worth of mentioning that a recent study on the Co doped series of Ba(FeCo)As, where is varied between 0 and 0.08, obey the suggested decomposition in the normal state regardless of doping.Uchida ()

Notably, there seems to be no considerable transfer of spectral weight between both subsystems upon cooling within the error bars ( 2 for each contribution) of our experiment and fits. We are inclined to ascribe the two low frequency Drude-like components to contributions from different bands crossing the Fermi surface. This immediately implies that they have a distinct influence on the electronic properties of these compounds. The coherent narrow component contains only a quarter of the observed optic spectral weight compared to the incoherent background but it turns out that it is of superior importance to their low energy behavior.

### v.3 Hidden Fermi liquid

No question, the optical conductivity can be decomposed in several ways. The analysis presented here, however, is the simplest one;remark5 () in addition a closer inspection of the temperature dependence of the narrow Drude component, as we will discussed now, makes obvious that some fundamental issues are uncovered, evidencing a deep physical consistency of our decomposition.

In order to reveal the temperature dependence of the dc resistivity of the subsystem described by , we follow the same logic as in the case of the optical conductivity. Consequently we decompose the conductivity at into the sum of and , we restrain ourselves from assuming any temperature dependence for the later one:

(7) |

For each compound, we simply deduced from the Drude-Lorentz fits displayed in Figs. 6 and 7. This procedure gives a surprising but very simple result, namely that

(8) |

all the way up to room temperature, as shown in Fig. 8(a). Importantly, no kink induced by a gap or pseudogap is observed in contrast to the parent compound, also suggesting that the SDW correlations are weak or absent in the samples close to the optimal doping.

We strongly feel that the observation of a simple power law in dc resistivity is not a pure coincidence. In fact it suggests strongly that the decomposition of the optical conductivity presented in Sec. V.2 is correct. An independent confirmationremark8 () is obtained from the scatting rate of Ba(FeAs: fitting the optical conductivity by the Drude model

(9) |

yields , as presented in Fig. 8(b). Note, both approaches do not just get the same temperature dependence, also the slope for Ba(FeCoAs is always larger compared to Ba(FeNiAs. From our fit we obtain for Ba(FeCoAs and for Ba(FeNiAs.

The dependence of scattering rate and the dc resistivity is generally associated with a Fermi liquid. Examples of similar temperature dependence of dc resistivity up to elevated temperatures can be found in: Ruthenate superconductors,Maeno97 () organic layered superconductors,Bulaevskii88 () transition metal dichalcogenides,Julien91 () High temperature superconductors.Nakamae03 (); Taillefer09 (); Manako92 (); Alloul89 () In Fermi liquids the weak Umklapp electron-electron scattering processes are dominant in the scattering rate which corresponds to

(10) |

where is a dimensionless constant. Using the same numbers for the Fermi energy as in Ref. Rullier09, , we found that . This is consistent with the weak coupling limit behind the above expression, which assumes that the effective Umklapp is small with respect to the Fermi energy. It is worth of noting that a possibly relevant complication in estimating occurs for strongly interacting electron systems. The term “normal Fermi system” should then refer only to quasi particles which obey the Fermi-Dirac statistics. This produces an ambiguity in the definition of the appropriate .

Notably, the proximity of the SDW and the SC phase, above which the Fermi liquid state is found, evokes the phase diagram of the organic superconductors. Those materials are understood in terms of Umklapp process, normal scattering and nesting.Emery () If the Umklapp process is weak with respect to , Fermi liquid behavior follows from Eq. (10) at low temperatures. Nevertheless, as well known, though the normal electron-electron scattering makes no contribution to the resistance, when attractive it can produce superconductivity as in the case of organic conductors. On the other side, if Umklapp is enhanced due to nesting, the system scales towards the strong coupling regime in which SDW is a possible ground state. Such reasoning opens an interesting avenue for understanding of iron-pnictides.

Our finding of a hidden behavior is in qualitative agreement with the conclusion drawn from comprehensive Hall-effect studies on the series Ba(FeCo)As by Rullier-Albenque et al. Rullier09 () It was pointed out there for the first time that the iron-pnictides are archetypical Fermi liquids. However, it should be mentioned that our way to a hidden differs from that followed in the analysis of the Hall coefficient where a change in electron concentration of approximately was assumed with temperature. In contrast, our decomposition of optical conductivity reveals that the spectral weight of remains unchanged to the precision of the experiment, which is less than a few percent. Furthermore it seems that the coefficient A4 estimatedRullier09 () from Hall-effect seems somewhat large to be consistent with the weak coupling limit.

It is also worth mentioning that doping does not change the slope of the resistivity curves, but only the intercept .Chu09 (); Rullier09 () Thus it can be asserted that the Fermi-liquid behavior extends through the complete doping series, i.e. the whole paramagnetic phase of these compounds. Furthermore other iron-pnictides behave in a very similar way, as demonstrated on Ba(FeNiAs in Fig. 8(b), confirming our general view. Notably, a behavior of the resistivity is also found in 1111 compoundssefatprb (); Suzuki09 () while a recent optical conductivity study revealed the Fermi-liquid behavior in FeTeSe compoundHomes10 (). However, while the values extracted in 1111 compound are comparable with moderate heavy-fermion compounds, such as CePd and UIn,KadowakiWoods86 (); Jacko09 () we obtain values one order of magnitude smaller in the 122 family. This places the 122 iron-pnictides right between the cuprates and conventional metals, as far as the correlation strength is concerned. Our finding is in very good accord with conclusions drawn from comparison of the spectral weight with band structure calculations.Qazilbash09 (); Mazin09 ()

Since 122 compounds exhibit a Fermi liquid behavior it is of interest to check whether the empirical Kadowaki-Woods relation is obeyed. This relation links the resistivity and the specific heat giving universally a constant ratio cm(mole K/mJ). is the Sommerfeld coefficient of the linear term in the temperature dependence of the specific heat. Typically in iron-pnictides it varies between 7 and 35 mJ/moleK.Welp09 (); Rotter08 (); Ni08 () As shown in Fig. 9 the agreement between the universal relation and the data is remarkable which only reinforces further our conclusions.

## Vi Conclusions

We have measured the electrodynamic properties of Ba(FeCoAs and Ba(FeNiAs single crystals in a wide frequency range as a function of temperature and presented a detailed analysis of their behavior in the metallic state above and 20 K, respectively. In a one-component picture, the extended Drude analysis yields the frequency dependence of the effective mass and scattering rate. is enhanced at low temperatures and frequencies by approximately a factor of 5. The low-frequency scattering rate decreases on cooling but turns into the opposite behavior above 400 cm. The spectral weight shifts to lower energies with decreasing temperature; a small but significant fraction is not recovered within the infrared range of frequencies.

Comparing the optical spectra a numerous 122 iron-pnictides, it becomes obvious that the frequency-dependent conductivity can be universally decomposed into a broad temperature-independent background and a narrow Drude-like component which solely determines the transport properties. The conducting subsystem reveals a behavior in the dc resistivity up to elevated temperatures, indicating a hidden Fermi liquid behavior of the these iron-pnictide superconductors. We conclude that the superconducting state evolves out of a Fermi liquid state where correlations are important to a moderate extend; we find no indications of a quantum critical behavior.

###### Acknowledgements.

We thank D.N. Basov, N. Drichko, B. Gorshunov and N.L. Wang for helpful discussions. The contributions of J. Braun, A. Faridian, P. Kallina, X. Lin and E.S. Zhukova to the experiments are appreciated. D.W and N.B. acknowledge their fellowship by the Alexander von Humboldt Foundation. The work at Zhejiang University was funded by NSF of China.## References

- (1) C. Xu and S. Sachdev, Nature Phys. 4, 898 (2008).
- (2) Tesanovic, Z. Are iron pnictides new cuprates? Physics 2, 60 (2009).
- (3) K. Ishida, Y. Nakai, and H. Hosono, J. Phys. Soc. Jpn. 78, 062001 (2009).
- (4) H.R. Ott, Prog. Low Temp. Phys. 11, 215 (1987); N. Grewe and F. Steglich, in: Handbook on the Physics and Chemistry of Rare Earths, Vol. 14, ed. by K.A. Gscheidner Jr. and L. Eyring (Elsevier, Amsterdam, 1991), p. 343; P. Fulde, P. Thalmeier, and G. Zwicknagl, in: Solid State Physics, Vol. 60, ed. by H. Ehrenreich and F. Spaepen (Elsevier, Amsterdam, 2006) p. 1.
- (5) H. Fukuyama, J. Phys. Soc. Jpn. 75, 051001 (2006); A. G. Lebed, The Physics of Organic Superconductors and Conductors (Springer-Verlag, Berlin, 2008); M. Dressel and N. Drichko, Chem. Rev. 104, 5689 (2004).
- (6) Clarina de la Cruz, Q. Huang, J. W. Lynn, Jiying Li, W. Ratcliff, J. L. Zarestky, H. A. Mook, G. F. Chen, J. L. Luo, N. L. Wang and Pengcheng Dai, Nature, 453, 647 (2008)
- (7) Q. Huang, Y. Qiu, Wei Bao, M. A. Green, J. W. Lynn, Y. C. Gasparovic, T. Wu, G. Wu, and X. H. Chen, Phys. Rev. Lett. 101, 257003(2008)
- (8) M. Rotter, M. Tegel, D. Johrendt, I. Schellenberg, W. Hermes, and R. Pottgen Phys. Rev. B 78,020503(R) (2008).
- (9) J.H. Chu, J. G. Analytis, C. Kucharczyk, I. R. Fisher, Phys. Rev. B 79, 014506 (2009).
- (10) R. H. Liu, T. Wu, G. Wu, H. Chen, X. F. Wang, Y. L. Xie, J. J. Ying, Y. J. Yan, Q. J. Li, B. C. Shi, W. S. Chu, Z. Y. Wu and X. H. Chen, Nature 459, 64 (2009); D. G Hinks, Nature Physics 5, 386 (2009).
- (11) P. M. Shirage, K. Kihou, K. Miyazawa, C. H. Lee, H. Kito, H. Eisaki, T. Yanagisawa, Y. Tanaka, and A. Iyo, Phys. Rev. Lett. 103, 257003 (2009).
- (12) T. Yanagisawa, K. Odagiri, I. Hase, K. Yamaji, P. M. Shirage, Y. Tanaka, A. Iyo, and H. Eisaki, J. Phys. Soc. Jpn. 78, 094718 (2009).
- (13) V. B. Zabolotnyy, D. S. Inosov, D. V. Evtushinsky, A. Koitzsch, A. A. Kordyuk, G. L. Sun, J. T. Park, D. Haug, V. Hinkov, A. V. Boris, C. T. Lin, M. Knupfer, A. N. Yaresko, B. Büchner, A. Varykhalov, R. Follath, and S. V. Borisenko Nature 457, 569 (2009).
- (14) M. Yi, D. H. Lu, J. G. Analytis, J.-H. Chu, S.-K. Mo, R.-H. He, R. G. Moore, X. J. Zhou, G. F. Chen, J. L. Luo, N. L. Wang, Z. Hussain, D. J. Singh, I. R. Fisher, and Z.-X. Shen Phys. Rev. B, 80, 024515 (2009).
- (15) K. Terashima, Y. Sekiba, J. H. Bowen, K. Nakayama, T. Sato, P. Richard, Y. -M. Xu, L. J. Li, G. H. Cao, Z. -A. Xu, H. Ding, and T. Takahashi, PNAS 106, 7330 (2009).
- (16) M.M. Qazilbash, J.J. Hamlin, R.E. Baumbach, L. Zhang, D.J. Singh, M.B. Maple, D.N. Basov, Nature Physics 5, 629 (2009).
- (17) I.I. Mazin, D.J. Singh, M.D. Johannes, M.H. Du, Phys. Rev. Lett. 101, 057003 (2008).
- (18) D. V. Evtushinsky, D. S. Inosov, V. B. Zabolotnyy, A. Koitzsch, M. Knupfer, B. Büchner, M. S. Viazovska, G. L. Sun, V. Hinkov, A. V. Boris, C. T. Lin, B. Keimer, A. Varykhalov, A. A. Kordyuk, and S. V. Borisenko, Phys. Rev. B, 79, 054517 (2009).
- (19) G.F. Chen, Z. Liu, W.Z. Hu, J. Dong, X.D. Zhang, P. Zheng, N.L. Wang, and J.L. Luo, Chin. Phys. Lett. 25, 3403 (2008).
- (20) Z. G. Chen, G. Xu, W. Z. Hu, X. D. Zhang, P. Zheng, G. F. Chen, J. L. Luo, Z. Fang, and N. L. Wang, Phys. Rev. B 80, 094506 (2009).
- (21) J. Dong, H.J. Zhang, G. Xu, Z. Li, G. Li, W.Z. Hu, D. Wu, G.F. Chen, X. Dai, J.L. Luo, Z. Fang, N.L. Wang, Europhys. Lett. 83, 27006 (2008).
- (22) W.Z. Hu, J. Dong, G. Li, Z. Li, P. Zheng, G.F. Chen, J.L. Luo, and N.L. Wang, Phys. Rev. Lett. 101, 257005 (2008).
- (23) A. Akrap, J. J. Tu, L. J. Li, G. H. Cao, Z. A. Xu, and C. C. Homes, Phys. Rev. B 80, 180502(R) (2009)
- (24) M. Nakajima, S. Ishida, K. Kihou, Y. Tomioka, T. Ito, C.-H. Lee, H. Kito, A. Iyo, H. Eisaki, K. M. Kojima, and S. Uchida, Physica C (2010); doi:10.1016/j.physc.2009.11.071; M. Nakajima, S. Ishida, K. Kihou, Y. Tomioka, T. Ito, Y. Yoshida, C. H. Lee, H. Kito, A. Iyo, H. Eisaki, K. M. Kojima, and S. Uchida, arXiv:1003.5038 (2010)
- (25) G. Li, W. Z. Hu, J. Dong, Z. Li, P. Zheng, G. F. Chen, J. L. Luo, and N. L. Wang, Phys. Rev. Lett. 101, 107004 (2008).
- (26) W. Z .Hu, Q. M. Zhang, and N. L. Wang, Physica C 469, 545 (2009).
- (27) D. Wu, N. Barišić, N. Drichko, S. Kaiser, A. Faridian, M. Dressel, S. Jiang, Z. Ren, L.J. Li, G.H. Cao, Z.A. Xu, H.S. Jeevan, and P. Gegenwart, Phys. Rev. B 79, 155103 (2009).
- (28) D. Wu, N. Barišić, P. Kallina, A. Faridian, B. Gorshunov, N. Drichko, L. J. Li, X. Lin, G. H. Cao, Z. A. Xu, N. L. Wang, and M. Dressel, Phys. Rev. B 81, 100512(R) (2010).
- (29) B. Gorshunov, D. Wu, A. A. Voronkov, P. Kallina, K. Iida, S. Haindl, F. Kurth, L. Schultz, B. Holzapfel, and M. Dressel, Phys. Rev. B 81, 060509(R) (2010).
- (30) K. W. Kim, M. R¨ossle, A. Dubroka, V. K. Malik, T. Wolf, and C. Bernhard, arXiv: 0912.0140
- (31) E. van Heumen, Y. Huang, S. de Jong, A.B. Kuzmenko, M.S. Golden, and D. van der Marel, arXiv: 0912.0636
- (32) C. C. Homes, A. Akrap, J. S. Wen, Z. J. Xu, Z. W. Lin, Q. Li, and G. D. Gu, arXiv: 1002.4846
- (33) F. Pfuner, J.G. Analytis, J.-H. Chu, I.R. Fisher, and L. Degiorgi, Eur. Phys. J. B 67, 513 (2009).
- (34) There are indications that some details of the optical properties might actually depend in the particular system under investigation.
- (35) F. Ronning, N. Kurita, E. D. Bauer, B. L. Scott, T. Park, T. Klimczuk, R. Musshovich, J.D. Thompson, J. Phys. Condens. Matter 20, 342208 (2008).
- (36) A. S. Sefat, M. A. McGuire, R. Jin, B. C. Sales, D. Mandrus, F. Ronning, E. D. Bauer, Y. Mozharivskyj, Phys. Rev. B 79, 094508 (2009).
- (37) J. Yang, D. Hüvonen, U. Nagel, T. Rõõm, N. Ni, P.C. Canfield, S.L. Bud’ko, J.P. Carbotte, and T. Timusk, Phys. Rev. Lett., 102, 187003 (2009).
- (38) D. Wu, N. Barišić, B. Gorshunov, M. Dressel, L. J. Li, G. H. Cao, and Z. A. Xu, (in preparation).
- (39) S. Jiang, Y. K. Luo, Z. Ren, Z. W. Zhu, C. Wang, X. F. Xu, Q. Tao, G. H. Cao, Z-A. Xu, New J. Phys. 11, 025007 (2009).
- (40) L.J. Li, Y.K. Luo, Q.B. Wang, H. Chen, Z. Ren, Q. Tao, Y.K. Li, X. Lin, M. He, Z.W. Zhu, G.H. Cao, and Z-A. Xu, New J. Phys. 11, 025008 (2009).
- (41) B. Gorshunov, A. Volkov, I. Spektor, A. Prokhorov, A. Mukhin, M. Dressel, S. Uchida, and A. Loidl, Int. J. of Infrared and Millimeter Waves, 26, 1217 (2005).
- (42) M. Dressel, N. Drichko, B. Gorshunov, and A. Pimenov, IEEE J. Select. Topics Quantum Electronics 14, 399 (2008).
- (43) M. Dressel and G. Grüner, Electrodynamics of Solids (Cambridge University Press, Cambridge, 2002).
- (44) N. Barišić, Y. Li, X. Zhao, Y.-C. Cho, G. Chabot-Couture, G. Yu, and M. Greven, Phys Rev. B 78, 054518 (2008).
- (45) The room temperature resisitivity is very close to cm of the undoped parent compound EuFeAs, measured previously.Wu09 () In that compound, however, the initial slope is smaller.
- (46) A. A. Abrikosov, L. P. Gor’kov, and I. Y. Dsyaloshinskii, Methods of Quantum Field Theory in Statistical Physics (Prentice-Hall, Englewood, 1963).
- (47) D. Pines and P. Nozières, The Theory of Quantum Liquids, Vol. 1 (Addison-Wesley, Reading, 1966).
- (48) S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, 2001).
- (49) H. v. Löhneysen, A. Rosch, M. Vojta, and P. Wölfle, Rev. Mod. Phys. 79, 1015 (2007).
- (50) N. Doiron-Leyraud, P. Auban-Senzier, S. Rene de Cotret, A. Sedeki, C. Bourbonnais, D. Jérome, K. Bechgaard, and L. Taillefer, arXiv:0905.0964.
- (51) L.D. Landau and E.M. Lifshitz, Course of Theoretical Physics, Vol. V (Statistical Physics, Part 1), 3rd edition (Butterworth-Heinemann, Oxford, 1991); Chap. XIV.
- (52) M. E. Fisher, Rep. Prog. Phys. 30, 615 (1967).
- (53) L. Kadanoff, W. Götze, D. Hamblen, R. Hecht, E.A. Lewis, V.V. Palciauskas, M. Rayl, J. Swift, D. Aspnes, and J. Kane, Rev. Mod. Phys. 39, 395 (1967).
- (54) J. Hertz, Phys. Rev. B 14, 1165 (1976).
- (55) A.J. Millis, Phys. Rev. B 48, 7183 (1993).
- (56) T. Moriya and T. Takimoto, J. Phys. Soc. Jpn. 64, 960 (1995).
- (57) M. Gooch, B. Lv, B. Lorenz, A. M. Guloy, and C. W. Chu, Phys. Rev. B 79, 104504 (2009).
- (58) S. Jiang, Y. K. Luo, Z. Ren, Z. W. Zhu, C. Wang, X. F. Xu, Q. Tao, G. H. Cao, Z-A. Xu, New J. Phys. 11, 025007 (2009); S. Jiang, H. Xing, G. Xuan, C. Wang, Z. Ren, C. Feng, J. Dai, Z. Xu, and G. Cao, J. Phys.: Condens. Matter 21, 382203 (2009).
- (59) D. J. Singh, Phys. Rev. B 78, 094511 (2008).
- (60) This is supported by -axis reflectivity measurements on EuFeAs (Ref. Wu09, ) where the interband transition is also seen.
- (61) Z. G. Chen, R. H. Yuan, T. Dong, and N. L. Wang, arXiv:0910.1318.
- (62) The choice of the plasma frequency already indicates that for the analysis presented here, we consider the total conductivity. The subtraction of a frequency independent background conductivity severely influence the numbers, albeit not the overall behavior. The subtraction of some finite-frequency contributions on the frequency dependence of the scattering rate and mass was also discussed by Yang et al. (Ref. Yang09, ). We also restrained ourselves from subtracting from our data, often used to summarily account for higher frequency excitations. It bring some abiguity into the analysis, but does not significantly change the frequency dependence.
- (63) C. Kittel, Introduction to Solid State Physics, 8th edition (J. Wiley & Sons, 2005).
- (64) M. Dressel, N. V. Kasper, K. Petukhov, B. P. Gorshunov, G. Grüner, M. Huth, and H. Adrian, Phys. Rev. Lett. 88, 186404 (2002); M. Dressel, N. V. Kasper, K. Petukhov, D. N. Peligrad, B. P. Gorshunov, M. Jourdan, M. Huth, and H. Adrian, Phys. Rev. B 66, 035110 (2002).
- (65) D. N. Basov, E. J. Singley, and S. V. Dordevic, Phys. Rev. B 65, 054516 (2002); S. V. Dordevic and D. N. Basov, Ann. Phys. (Leipzig) 15, 545 (2006).
- (66) M.S. Torikachvili, S. L. Bud’ko, N. Ni, and P.C. Canfield, Phys. Rev. Lett. 101, 057006 (2008).
- (67) Z. Ren, Q. Tao, S. Jiang, C. Feng, C. Wang, J. Dai, G. Cao, and Z. Xu, Phys. Rev. Lett. 102, 137002 (2009).
- (68) A similar conclusion was obtained from systematic studies of a doping series on 1111 compounds by Susuki et al. (Ref. Suzuki09, ).
- (69) S. Suzuki, S. Miyasaka, S. Tajima, T. Kida, and M. Hagiwara, J. Phys. Soc. Jpn. (2010); arXiv0910.1711.
- (70) T. Yildirim, Phys. Rev. Lett. 101, 057010 (2008).
- (71) C.-Y. Moon, S. Y. Park, and H. J. Choi, Phys. Rev. B 80, 054522 (2009).
- (72) For our analysis, the reflectivity and conductivity are fitted simultaneously.
- (73) M. Nakajima, S. Ishida, K. Kihou, Y. Tomioka, T. Ito, Y. Yoshida, C. H. Lee, H. Kito, A. Iyo, H. Eisaki, K. M. Kojima, S. Uchida, arXiv:1003.5038
- (74) Other decompositions of the optical conductivity than suggested here might be tempting, too. For example, the next simplest one is to fit the low-frequency conductivity by a Drude and a Lorentzian terms. However, there is not really any physical meaning of the Lorentzian, and furthermore its position, width and amplitude strongly change with temperature which is highly anomalous.
- (75) Although we have utilized the dc resistivity for the extrapolation of the low-frequency reflectivity, the determination of the Drude roll-off in the optical conductiivty is to a large extend independent from this procedure.
- (76) Y. Maeno, K. Yoshida, H. Hashimoto, S. Nishizaki, S. Ikeda, M. Nohara, T. Fujita, A. P. Mackenzie1, N. E. Hussey, J. G. Bednorz and F. Lichtenberg, J. Phys. Soc. Jpn. 66, 1405 (1997).
- (77) L. N. Bulaevskii, Adv. Phys. 37, 443(1988).
- (78) C. Julien, J. Ruvalds, A. Virosztek, and O. Gorochov, Solid State Communications 79, 875 (1991).
- (79) S. Nakamae, K. Behnia, N. Mangkorntong, M. Nohara, H. Takagi, S. J. C. Yates, and N. E. Hussey, Phys. Rev. B 68, 100502(R) (2003).
- (80) L. Taillefer, J. Phys.: Condens. Matter 21, 164212 (2009).
- (81) T. Manako, Y. Kubo, and Y. Shimakawa, Phys. Rev. B 46, 11019 (1992).
- (82) H. Alloul, T. Ohno, and P. Mendels, Phys. Rev. Lett. 63, 1700 (1989)
- (83) F. Rullier-Albenque, D. Colson, A. Forget, and H. Alloul, Phys. Rev. Lett 103, 057001 (2009).
- (84) V. J. Emery, R. Bruinsma, S. Barišić, Phys. Rev. Lett 48, 1039 (1982).
- (85) A. S. Sefat, M. A. McGuire, B. C. Sales, R. Jin, J. Y. Howe, and D. Mandrus, Phys. Rev. B 77, 174503 (2008).
- (86) K. Kadowaki and S. B. Woods, Solid State Commun. 58, 507 (1986).
- (87) A. C. Jacko, J. O. Fjærestad, and B. J. Powell, Nature Physics 5, 422 (2009).
- (88) I.I. Mazin, arXiv:0910.4117.
- (89) U. Welp, G. Mu, R. Xie, A.E. Koshelev, W.K. Kwok, H.Q. Luo, Z.S. Wang, P. Cheng, L. Fang, C. Ren, H.-H. Wen, Physica C 469, 575 (2009).
- (90) N. Ni, S. L. Budko, A. Kreyssig, S. Nandi, G. E. Rustan, A. I. Goldman, S. Gupta, J. D. Corbett, A. Kracher, and P. C. Canfield, Phys. Rev. B 78, 014507 (2008).