Nodeless vs nodal order parameters in LiFeAs and LiFeP superconductors
High-precision measurements of magnetic penetration depth in clean single crystals of LiFeAs and LiFeP superconductors reveal contrasting low-energy quasiparticle excitations. In LiFeAs the low-temperature shows a flat dependence indicative of a fully gapped state, which is consistent with previous studies. In contrast, LiFeP exhibits a -linear dependence of superfluid density , indicating a nodal superconducting order parameter. A systematic comparison of quasiparticle excitations in the 1111, 122, and 111 families of iron-pnictide superconductors implies that the nodal state is induced when the pnictogen height from the iron plane decreases below a threshold value of Å.
There is growing evidence that the superconducting gap structure is not universal in the iron-based superconductors Stewart11 . In certain materials such as optimally doped (Ba,K)FeAs and Ba(Fe,Co)As, strong evidence for the fully gapped superconducting state has been observed from several low-energy quasiparticle excitation probes including magnetic penetration depth Hashimoto09a ; Luan11 and thermal conductivity measurements Luo09 ; Tanatar10 . In contrast, significant excitations at low temperatures due to nodes in the energy gap have been detected in several Fe-pnictide superconductors. These include LaFePO ( K) Fletcher09 ; Hicks09 ; Yamashita09 , BaFe(As,P) ( K) Hashimoto10a ; Nakai10 ; Yamashita11 , and KFeAs ( K) Fukazawa09 ; Dong10 ; Hashimoto10b . It is quite extraordinary that such distinct pairing states appear in closely related members of the same class of superconductors. To understand the mechanism of superconductivity in iron-based superconductors, it is essential to identify what determines nodal and nodeless states Mazin08 ; Kuroki09 ; Chubukov09 ; Graser09 ; Ikeda10 ; Thomale11 ; Kontani10 .
Theories based on antiferromagnetic spin fluctuations suggest that the pnictogen height above the iron plane (see Fig.1(a)) is an important factor in determining the structure of the superconducting order parameter Kuroki09 ; Graser09 ; Ikeda10 ; Thomale11 . Generally, is much shorter for the P based iron-pnictides in comparison to their As counterparts, so a good test of the theory would be to systematically compare As and P based superconductors. Although this can be achieved in part in the BaFe(As,P) series, the fully As containing end member BaFeAs is a nonsuperconducting antiferromagnet. The same is true for LaFeAsO which is the As analogue of the nodal superconductor LaFePO. Charge doping of the arsenides induces superconductivity, but also introduces disorder which complicates the identification of the pairing state.
The 111 materials, LiFeAs Wang08 ; Tapp08 and LiFeP Deng09 ; Mydeen10 provide a unique route to study this problem as both materials are superconducting ( K and 4.5 K, respectively), nonmagnetic, and importantly very clean, with long electronic mean-free paths. In LiFeAs, antiferromagnetic fluctuations have been observed Jeglic11 ; Taylor11 and fully gapped superconductivity has been demonstrated by several experiments Inosov10 ; Kim11 ; Imai11 ; Tanatar11 ; Li10 ; Borisenko10 , but no information has been reported for the pairing state in LiFeP. Band-structure calculations show that the two materials exhibit similar Fermi surface shapes Singh08 ; Shein10 ; quasi-cylindrical hole sheets near the zone center and two warped electron sheets near the zone corner.
Here we report on precision measurements of the magnetic penetration depth in single crystals, which demonstrate a nodal gap state in LiFeP in sharp contrast to the nodeless state in LiFeAs. Our analysis based on accumulated data in the 1111, 122, and 111 series of superconductors, indicates that the nodal state is induced when is below a threshold value. By comparing calculated electronic band structures of LiFeAs and LiFeP, we discuss the origin of this behavior.
Single crystals of LiFe ( As or P) were grown by a flux method Kasahara11 . The crystal size of LiFeP is up to m, which is smaller than that of LiFeAs. To avoid degradation of the sample due to reaction with air, the crystals were handled in an argon glove box and encapsulated in degassed Apiezon N grease before measurements. Large residual resistivity ratios ( for LiFeAs and for LiFeP) Kasahara11 , observations of de Haas-van Alphen (dHvA) oscillations in magnetic torque Carrington11 , and sharp superconducting transitions (Fig.1(b)) show that the crystals are of very high quality. The temperature dependence of change in the magnetic penetration depth was measured by the tunnel diode oscillator (TDO) technique Fletcher09 ; Hashimoto10b down to . A weak ac field is applied along the axis so that the supercurrent flows in the plane.
Figure2(a) depicts the low-temperature variation of the in-plane penetration depth . The data for LiFeAs is completely flat within the experimental error of nm below . This demonstrates negligible quasiparticle excitations at low temperatures, indicating a fully gapped state. This result is fully consistent with previous results in LiFeAs Kim11 ; Imai11 . In sharp contrast to this, the data for LiFeP exhibits much steeper temperature dependence of at low temperatures. When we use a power law fit to this data below , we obtain a small value of . In iron-based superconductors, a power law dependence with can be expected even in the dirty full gap case when the sign changing state is considered Vorontsov09 , and indeed a tendency of the exponent decrease from to with increased impurity scattering has been observed experimentally Hashimoto09a ; Kim10 . However, the small power cannot be explained by such a dirty nodeless state, and it is rather a strong indication that the superconducting gap has line nodes. Indeed, our data can also be fitted to , which is applicable to the nodal case with small impurity scattering Hirschfeld93 . The obtained low value of K indicates a clean nodal behavior and is consistent with the other measures of sample quality described above.
We also analyze the normalized superfluid density (Fig.2(b)). To do this we need the value of , which we cannot directly determine from the TDO measurements. The small angle neutron scattering measurements of LiFeAs reveal nm Inosov10 . To estimate the value for LiFeP, we consider the difference of the effective mass in these two superconductors whose carrier number (Fermi surface volume) is quite similar. The effective masses determined by the dHvA oscillations Carrington11 as well as the electronic specific heat coefficients have a factor of difference ( and mJ/Kmol for LiFeP Deng09 and LiFeAs Lee10 , respectively), from which we estimate nm. The extracted temperature dependence of shows contrasting behaviors for As and P cases at low temperatures again: flat dependence for As and steeper dependence for P. The expanded view at low temperatures (Fig.2(c)) demonstrates a wide temperature range of -linear dependence, which clearly indicates the energy-linear density of state of quasiparticles and hence the existence of line nodes in the energy gap.
The strength of the electron-electron correlations can be measured by the mass enhancement which is closely related to the value. The larger for LiFeAs than for LiFeP suggests weaker correlations in the P case, which is reinforced by the smaller value of the Fermi-liquid coefficient in the dependence of resistivity Kasahara11 ; Heyer10 and smaller quasiparticle mass enhancements measured by quantum oscillations Carrington11 . Strong correlations usually promote sign change in the superconducting order parameter Hashimoto10b , which leads to the gap nodes in single-band superconductors. In the present multiband case with separated Fermi surface sheets, however, the seemingly opposite trend that LiFeP has nodes but is weakly correlated suggests that other factors are also important for node formation.
To obtain further insights, we gather the available data for the low-energy quasiparticle excitations in several iron-pnictide superconductors including 1111 Fletcher09 ; Hicks09 ; Hashimoto09b ; Malone09 , 122 Hashimoto10a ; Hashimoto09a ; Hashimoto10b ; Hashimoto11 ; Luan11 , and 111-based materials Kim11 ; Imai11 . The amount of thermally excited quasiparticles is directly related to the change in the penetration depth. Thus we quantify as a measure of excitations so that we avoid ambiguity resulting from uncertainties in . Among the available data in the literature, we select only the data which shows either or in the power-law approximation, because the power-law dependence with cannot distinguish the dirty nodeless and nodal states as discussed previously. A plot of as a function of pnictogen height in Fig.3 suggests that there is a threshold value of Å, below which all the superconductors exhibit significant quasiparticle excitations (with ) characteristic of a nodal state. Above the threshold, most of the materials are nodeless with the exception of the highly hole-doped compound, KFeAs. This particular material is unusual in that it lacks electron sheets and thus there is no interband nesting. In addition, the quasiparticle effective mass is strongly enhanced Hashimoto10b ; Sato09 , and is very low ( K) so superconductivity may have a different origin to that in the other materials. Therefore, our analysis strongly suggests that the pnictogen height is an important parameter that determines the gap structure in the iron-pnictide superconductors having significant interband scattering. One may also ask about the Fe--Fe bond angle, but the nodal LiFeP has a closer angle (108.6) to the perfect tetrahedron value of 109.47 than the nodeless LiFeAs (102.8), from which we do not find any simple correlation between the bond angle and gap structure in iron-pnictide superconductors.
The importance of the pnictogen height on the superconducting order parameter in iron-pnictides has been suggested in theoretical considerations based on the antiferromagnetic spin fluctuation mechanism Kuroki09 ; Graser09 ; Ikeda10 ; Thomale11 . When is low, one of the hole bands with orbital character, which is located near the position in the unfolded Brillouin zone (BZ), tends to sink below the Fermi level. The disappearance of this Fermi surface makes interband electron-hole scattering weaker and hence the importance of the scattering between electron sheets relatively greater, promoting a sign change of the superconducting gap (and hence nodes) on the electron sheets.
To check this, we have performed band-structure calculations based on density functional theory (DFT) including spin-orbit coupling, by using the wien2k package WIEN2k and the experimental lattice constants and internal positions Tapp08 ; Kasahara11 . The obtained Fermi surfaces (Fig.4) are similar to previous calculations Shein10 . Although angle-resolved photoemission results have suggested quite different Fermi surfaces with no interband nesting in LiFeAs Borisenko10 , more recent bulk measurements of dHvA oscillations reveal quasi-nested hole and electron sheets Carrington11 in a good agreement with the calculations. Importantly, the calculations show that the hole sheet, which in the folded BZ is the outermost hole sheet at , is present in both compounds. This indicates that the absence of the hole sheet is not a requisite for the nodal state.
A more detailed comparison between LiFeAs and LiFeP reveals that the size of the outer hole sheet shrinks and its relative weight of the orbital contribution is significantly suppressed for the P case. Moreover, the middle hole sheet in LiFeP has rather mixed and contributions. Such differences in the orbital character in hole sheets may affect relative importance of the interband hole-electron scattering compared with the scattering between electron bands. dHvA measurements in LiFeP Carrington11 imply that the middle hole sheets has less mass enhancement than the electron sheets, suggesting that scattering between electron sheets is stronger than the interband electron-hole scattering. This could lead to the formation of line nodes in the electron sheets. We note that the extended- state with line nodes in the electron sheets has been discussed as the most likely nodal gap structure of BaFe(As,P) Yamashita11 . The strong -linear dependence of superfluid density in LiFeP is consistent with the nodes being on electron bands containing high Fermi velocity parts, which almost coincide with the -dominated regions (yellow parts of the electron sheets in Fig.4(b)) Kasahara11 . To determine the exact node locations in LiFeP, however, other measurements are necessary including angle-resolved probes of low-energy quasiparticle excitations.
Our results that the nodal state is favored for low support the trend that the spin-fluctuation theory predicts, but there remains challenging issues including the fact that the emergence of nodes is not directly caused by the disappearance of the hole sheet. It has also been theoretically suggested that a competition between the orbital fluctuations and spin fluctuations generates nodes in the electron sheets Kontani10 . The difference in the orbital character in hole sheets would also change the orbital fluctuations, which may affect the competition and hence the nodal gap structure. Further quantitative calculations of the pnictogen-height effect based on these theories will help clarify the mechanism of iron-based superconductivity.
In summary, we have measured the penetration depth in clean crystals of LiFeAs and LiFeP. We found a -linear superfluid density for LiFeP indicating a nodal order parameter, in strong contrast to the fully-gapped state found for LiFeAs. A comparison of low-energy excitations across the different iron-pnictide superconductors suggests that the nodal state is induced when the pnictogen height is shortened below a threshold value.
We thank K. Cho, A. V. Chubukov, A. I. Coldea, P. J. Hirschfeld, H. Kontani, K. Kuroki, I. I. Mazin, and R. Prozorov for discussions. This work is supported by KAKENHI from JSPS, Grant-in-Aid for GCOE program “The Next Generation of Physics, Spun from Universality and Emergence” from MEXT, Japan, and EPSRC in the UK.
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