Physical properties of noncentrosymmetric superconductor Ru{}_{7}B{}_{3}

Physical properties of noncentrosymmetric superconductor RuB

Lei Fang, Huan Yang, Xiyu Zhu, Gang Mu, Zhao-Sheng Wang, Lei Shan, Cong Ren and Hai-Hu Wen hhwen@aphy.iphy.ac.cn National Laboratory for Superconductivity, Institute of Physics and National Laboratory for Condensed Matter Physics, Chinese Academy of Sciences, P. O. Box 603, Beijing 100190, P. R. China
July 13, 2019
Abstract

Transition metal boride RuB was found to be a noncentrosymmetric superconductor with equal to 3.3 K. Superconducting and normal state properties of RuB were determined by a self-consistent analysis through resistivity( and ) , specific heat, lower critical field measurement and electronic band structure calculation. It is found that RuB belongs to an s-wave dominated single band superconductor with energy gap 0.5 meV and could be categorized into type II superconductor with weak electron-phonon coupling. Unusual ’kink’ feature is clearly observed in field-broadening resistivity curves, suggesting the possible mixture of spin triplet induced by the lattice without inversion symmetry.

pacs:
74.70.Ad, 74.25.Qt, 74.25.Sv

I Introduction

Unconventional superconductors have been extensively studied during the past decades for the underlying fundamental physics or the potential industrial applications. Some well known examples are heavy fermion superconductorsheavyfermion (), cupratescuprate (), SrRuOSrRuO () and the newly discovered iron arsenideiron (). In recent years superconductors lacking of lattice inversion center have received intensive attentions for the possibility of spin-triplet dominated pairing symmetry. An important example is noncentrosymmetric superconductor CePtSiCePtSi (). Due to the nontrivial antisymmetric spin-orbit coupling(ASOC) effect induced by the heavy Platinum atom and the lacking of inversion symmetry, unconventional superconducting properties are observed, for instance the high upper critical field far beyond the Pauli-Clogston limitpaulilimite (). The subsequent nuclear spin-lattice relaxation rate as well as magnetic penetration depth measurements show line nodes in superconducting gap for CePtSiCePtSinode1 (); CePtSinode2 (). For a material without inversion symmetry the spin degeneracy is lifted by ASOC, under such a condition, orbital angular momentum() and spin angular momentum() are not good quantum numbers any more, thus the strict categorization of even-parity spin singlet and odd-parity spin triplet conformed to Pauli’s exclusion and parity conservation is violated, then spin triplet may be mixed with spin singlet. Up to now, several noncentrosymmetric superconductors have been reported, the development concerning such a scope could be consulted a overview given by Sigrist et. al.noncentrosymmetric (). However strong electronic correlation in some materials complicates the studies on ASOC effect, such as heavy fermion properties of CePtSi. Recently the pairing symmetry of LiPtB has been proved to be with line nodesLiPtBnode (), such a material is not strongly correlated and could be regarded as an appropriate example for ASOC effect. For more comprehension on ASOC, thus, any efforts searching for new noncentrosymmetric superconductors are worthwhile, especially in the absence of correlation effect.

In the present paper we report a new noncentrosymmetric superconductor RuB with equal to 3.3 K. Transport measurements as well as electronic band structure calculation gave a detailed description to the properties. As to our best knowledge, neither measurements nor calculations have been embarked except Matthias mentioned only the as 2.58 K in 1950’sMatthias (). Additionally, for the first time we point out the noncentrosymmetric characterization of RuB.

The paper is organized as follows: Section II describes the sample preparation process and the experimental details; then in section III we provide the structure illustration and the measurement of many transport properties, including AC magnetization, resistivity, magnetoresistance and Hall coefficient; section IV focuses on the specific heat and lower critical field measurements; in the following section V full potential electronic structure calculation is discussed; finally a self-consistent normal state and superconducting parameters are determined in section VI.

Ii Experiment

Polycrystalline sample RuB was synthesized by conventional solid state sintering process. Stoichiometric Ruthenium powder(purity 99.9%) and Boron powder(purity 99.95%) were mixed together and grounded thoroughly, then the mixture was pressed into pellet with a pressure 10 MPa. The pellet was wrapped with tantalum foil and sealed into an evacuated silicon tube. 95% high purity Ar mixed by 5% H was selected as protected atmosphere. The tube was warmed up to 1000 C in a muffle furnace with a rate 100 C/h and sintered for 100 hours for composition homogeneity, then the furnace was cooled down to room temperature. The ingot was ceramic-like and with a silver gray color. It was observed that the superconducting quality was stable and not sensitive to humidity or oxygen. X-ray diffraction (XRD) pattern measurement was performed at room temperature employing an M18AHF x-ray diffractometer (MAC Science). Cu was used as the radiation target. Crystallographic orientation and index were determined by Powder-XpowderX (), an software for processing x-ray diffraction data. The AC susceptibility were measured based on an Oxford cryogenic system (Maglab-Exa-12). The resistivity and specific heat were measured on the Quantum Design instrument PPMS with temperature down to 1.8 K and the PPMS based dilution refrigerator (DR) down to 50 mK. The temperatures of both systems have been well calibrated showing consistency with an error below 2% in the temperature range from 1.8 K to 10 K. The temperature dependence of magnetization was measured on the Quantum Design instrument MPMS with temperatures down to 2 K. The lower critical field measurement was based on a two dimensional electron gas(2DEG) micro Hall probe with an active area of 1010 m. All M(H) curves were taken in zero field cooled mode with initial temperature up to 6 K. A low field sweep rate of 60 Oe/min was selected to measure isothermal magnetization curves. Self-consistent band calculations were carried out using the linear muffintin orbital(LMTO) method on full potential plane wave representation(FP-LMTO-PLW)FP-LMTO-PLW ().

Figure 1: (a) a-b projection of RuB structure, the skeleton is consisted of metal tetrahedra and metal octahedra and interstitial Boron anions. From such a projection, the lattice with hexagonal rotation symmetry is obvious. (b) a-c projection of the lattice, two ’chains’ are built up by ruthenium ions at different coordinations.

Iii Structure illustration and transport properties

The crystal structure of RuB was determined by AronssonAronsson () in the late 1950’s. It was found that the lattice is hexagonal with a space group . There are 20 atoms in one unit cell with effective coordinate Ru(6c), Ru(6c), Ru(2b) and B(6c), respectively. Thus two formula units exist in one unit cell. In that paper the author gave a relative comprehensive description to the lattice structure of RuB, however, thanks to the complex structure of transition metal boride, a more detailed illustration should be added as a supplement for better comprehension on it’s properties. As to space group , a distinct characterization of the crystal lattice is without inversion symmetry, for example, the center of Boron atoms’ sub-lattice dislocates the counterpart of Ruthenium atoms along c-axis, thus the inversion symmetry is broken along such direction. For more clear understanding on the crystal lattice, we illustrate the structure along two types projection in Fig. 1. Fig. 1(a) shows the ab projection of RuB structure. It is found that the skeleton consists of metal tetrahedra and metal octahedra and interstitial Boron anions. From such a projection, the lattice with hexagonal rotation symmetry is obvious. Fig. 1(b) is the ac projection illustration. A very interesting phenomenon is that two ’chains’ are built up by ruthenium ions at different coordinations. The metal octahedra at each corner of the lattice is built up by Ru(6c), those octahedras share face along c-axis and thus a zig-zag chain is formed as shown in Fig. 1(b). The rest Ru(6c) and Ru(2b) form two tetrahedras in one unit cell at different (x,y) positions, along c direction two tetrahedras (at the same (x,y) positions) share face and then forming a hexahedra, each hexahedra is connected by Ru(2b) and thus another type of row is formed. The special structure configuration might play an important role in transport properties, also the environment of Ruthenium ions (including the interatomic distance and the number of the nearest boron ions) is very important. Detailed parameters are included in Table 1 as shown below.

atom cite x z No. ()
Ru 6c 0.4563 0.318 4 2.15, 2.15, 2.66, 2.66
Ru 6c 0.1219 0 4 2.15, 2.16, 2.16, 2.86
Ru 2b 1/3 0.818 3 2.20, 2.20, 2.20
B 6c 0.187 0.582 / /
11footnotetext: No. represents the number of nearest Boron for different cites of Ruthenium.22footnotetext: stands for the interatomic distance between Boron ions and Ruthenium ions.
Table 1: Structure parameters of RuB.

Fig. 2 shows the XRD pattern of the sample RuB, which can be indexed in a hexagonal symmetry with a = b = 7.4629 and c = 4.7141 . The indexed indices slightly deviate from the reported parameters a = b = 7.467 and c = 4.713 . It is clearly found that most diffraction peaks are indexed except one minor peak possibly from un-reacted boron. From the quality of XRD data, it is estimated that the purity of RuB we synthesized is about 95%. In the following specific heat measurement, 90% superconducting component also prove the sample’s purity.

Figure 2: (Color online) X-ray diffraction of RuB, most diffraction peaks are indexed except one minor peak possibly from un-reacted boron, the purity is estimated about 95%. The asterisk marks the peak from impurity phase.

Fig. 3 presents the AC susceptibility measurement under different magnetic fields. It is found that a sharp diamagnetic transition happens at 3.3 K and the diamagnetic value approaches a constant at 3K, 0.3 K transition width indicates the good superconducting quality. Applying magnetic fields, the transition curve moves parallel to low temperatures. For estimating the upper critical field, we take 95% diamagnetic value as criterion and will discuss the result in the following paragraph.

Fig. 4 shows the resistivity data from 2 K to room temperatures, the curve shows a good metallic behavior with a zero temperature resistivity() 9 cm. Such a high conductivity is the common feature of transition metal boride. It is found that the residual resistivity ratio(RRR) is 28, the relative high RRR in polycrystalline sample indicates that impurity scattering is trivial for conductivity. The inset of Fig. 4 shows the enlargement of superconducting transition, the resistivity drops sharply to zero at 3.3 K. Thus the resistivity data and AC susceptibility give a self-consistent T=3.3 K for RuB, which is slightly higher than the preliminary mentioned 2.58 K by MatthiasMatthias (). Considering the high purity of the sample, a quantitative analysis of normal state resistivity is deserved. We try Wilson’s model for transition metalsWilsonmodel ()

(1)

where is a cutoff similar to the Debye temperature and is a constant. Using parameters =500 K and =1350, a good fitting is obtained as shown in Fig. 4.

Figure 3: (Color online) AC susceptibility of RuB, a sharp superconducting transition happens at 3.3 K under zero field, a narrow transition width less than 0.3 K indicates the good superconducting quality.
Figure 4: (Color online) Resistivity of RuB under zero field is measured from 2 K to room temperatures, a good fitting is given by Wilson s-d scattering model. The inset shows the enlargement of superconducting transition, the transition width is less than 0.3 K.

Fig. 5 shows the field-broadening resistivity measurement on polycrystalline RuB down to 100 mK. When magnetic field is applied, the onset parts of transition is rounded very similar to the feature of critical fluctuation in superconductors. While adding fields up to 0.6 Tesla, the rounded part of transition evolves into a ’kink’ structure as shown clearly in Fig. 5. It seems that the kinks break the superconducting transition curves into two parts, the lower part moves quickly to low temperatures showing a strong dependence of magnetic fields, while the upper parts respond reversely. It is observed that the zero resistivity point approach zero temperature when sample is bearing 1.1 T magnetic field, while superconductivity is depressed completely at about 5 T. It is thus very interesting to note that the determined upper critical field strongly depends on the selected resistive criterion. As shown in Fig. 6, the criterion of 99% and zero resistivity will lead to two distinct H(0) with a ratio of about 5. A conventional interpretation, as discussed in MgIrBMgIrB (), for field broadening transition is filamentary-like superconductivity along grain boundaries, the stronger scattering reduces the mean free path and consequently influences on the coherence length. However, we argue that unlike the case of MgIrB the clear kink feature in RuB complicates the determination for H, also the kink doesn’t come from sample inhomogeneity or two phases because of the sharp transition in low fields. As to our knowledge, the unconventional ‘kink’ or ‘step’ features in transition curves have been extensively discussed in single crystals MgB. Several reasons have been attributed to this effect, including superconducting fluctuationscfluctuation (), two superconducting gapstwo-gap (), surface barriersSB () and vortex lattice meltingvortexmelting1 (); vortexmelting2 (),etc. However taking account of the very low and H, superconducting fluctuation is believed to be weak. For surface barriers and vortex lattice melting, the polycrystalline quality seems to exclude their possibilities. As to the two-gap scenario, the following specific heat and lower critical field measurements oppose such a point of view. Thus we assumed the possibility that spin triplet induced kink feature in the framework of inversion symmetry is broken. It is known that applying fields broke time reversal symmetry and is detrimental to spin singlet, whereas triplet pairing remains unaffected. So for a superconductor with pairing symmetry mixed by singlet and triplet, the kink feature in field broadening resistivity curves could be plausible on certain extent. In Fig. 6 we plot the phase diagram of RuB, the criteria are taken as below, for resistivity 99% and zero resistance, for magnetization 95% diamagnetic signal and the half position of specific heat anomaly for thermodynamic measurement. Thus the derived d/d equal to -0.43 T/K or -0.277 T/K for criteria 99% and zero resistance, respectively. It is found that except 99% the other three criteria determined data points overlap almost together. Further consideration is that bulk properties provided by specific heat measurement, thus in the following discussion we use 1.1 T as H(0) and -0.277 T/K as d/d near T. We also try to use Ginzburg-Landau formula to fit the data points determined by 99%,

(2)

where t is the normalized temperature T/T, it is found that the fitting curve strongly deviates the 99% points as shown in Fig. 6, indicating the invalidity of Landau twice order phase transition theory in the present material.

Figure 5: (Color online)Field broadening superconducting transition curves of RuB from 0.1 K to 4 K, a clear kink feature appears as the field exceeds 0.6 T. The drop of resistivity is totally suppressed by 5 T magnetic field. Distinct magnetroresistance is observed.
Figure 6: (Color online) Phase diagram of RuB, a non-trivial flux flow area for criteria 99% and zero resistance is formed by kink shape of transition curves. 95% diamagnetic signal from AC susceptibility and one half points of specific heat jumps are also selected as criteria to estimate the intrinsic value of (0). It is shown that except for the criterion of 99% the other points from different criteria overlap together, thus the experimental data 1.1 T is determined as (0). It is found that the estimation of from WHH formula fits experimental data very well, whereas Ginzburg-Landau formula fails to describe the upper bound of .

Another distinct characterization of Fig. 5 is the field induced magnetoresistance. Typically magnetoresistance is used to investigate the electronic scattering process and provide useful information on fermi surface (FS) topology. So detailed studies are needed. In Fig. 7 we present the temperature dependence of resistivity under magnetic fields from 0 to 9 T. It is found that the magnetoresistance(MR) is about 16% [()/] at 5 K, which is one order of magnitude larger than the ratio of recently discovered iron arsenide LaFeAsOFMRLaFeAsO (). The latter was regraded as superconductor with multiple bands. A simple verification for the possibility of multigap effect is the scaling based on Kohler’s rule. The Kohler’s rule is written as =F(H/), where F is a function depending on the nature of the metal itself. For single band metal with symmetric Fermi surface topology Kohler’s law should be conserved. It is shown as Fig. 7(b) that Kohler’s rule is only slightly violated. Unlike typical multi-band superconductor MgBMgB2 () and LaFeAsOF, the breakdown of Kohler’s law is trivial in RuB, indicating a dominated single band behavior. The further specific heat and lower critical field measurements further provide the same conclusion. However we believe that the slightly violation of Kohler’s rule could be induced by noncentrosymmetric structure of RuB, due to ASOC, the degenerate spin-up and spin-down bands are spilt, so Kohler’s rule is slightly broken.

Figure 7: (Color online) (a) Temperature dependence of resistivity of RuB under magnetic fields from 0 to 9 T, (b) the derived magnetroresistance / is 16% at 5 K, it is shown that the scaling of Kohler’s law is slightly violated. The violation is attributed to band splitting by ASOC instead of multiband effect.

Hall coefficient (R) measurement was done by sweeping temperature at magnetic field 9 T and reversing field (-9 T). For avoiding the possible temperature hysteresis, increasing temperature mode with a moderate rate 1 K/min was adopted for both positive and negative fields. The Hall coefficient is shown in Fig. 8, it is found that the charge carrier of RuB is dominated by hole-like carriers with R 36 from 2 K to 200 K. For verifying the R, we also use the values derived from sweeping field at three temperature points 2 K, 100 K and 200 K. The low temperature R is consistent with the value from sweeping temperature, while error bars exist in high temperatures. The charge carrier density calculated by n=1/(Re) is about 1, which is two order of magnitude larger than low carrier density superconductors, for example, cupratesnecuprate () and hole doped iron arsenide (LaSr)FeAsOneLaSr (). We also notice the nonlinear temperature dependence of R, however, the relative change of R from 2 K to 200 K is small as shown in Fig. 8. Moreover, it is known that the hall effect is very sensitive to the temperature dependent scattering rate, local fermi velocity and complex FS topologyHallsensitive (), thus considering the polycrystalline quality detailed analysis on temperature dependent Hall coefficient would not be discussed here.

Figure 8: (Color online)Hall coefficient measurement of RuB from sweeping temperature under reversing fields, the charge carrier is dominated by hole from low temperature to 200 K, the density of charge carrier is about . The squares represent the data measured by sweeping fields at fixed temperatures.

Iv Specific heat and lower critical field measurement

iv.1 Specific heat

Fig. 9 shows the raw data of specific heat under different magnetic fields from zero to 3 T. With increasing fields the specific heat anomaly near T move quickly to low temperatures leaving a background consistent with that above T at zero field. Thus the normal state specific heat could be extracted easily with the relation , where is the normal state specific heat coefficient and corresponds to phonon contribution. It is found that =0.3732 and = 89.95 , however a residual value 9.8 indicates the existence of about 10% non-superconducting fraction. The non-superconducting fraction could partly come from unreacted boron as inferred from analysis from X-ray diffraction. Thus the normal state Sommerfeld constant could be determined from the relation =- as 80.15 . Using the relation =, where = 6.02 the Avogadro constant, Z=20 the number of atoms in one unit cell, we get the Debye temperature = 470.18 K. In the previous section, we obtained a similar value =500 K from the resistivity curve fitting, the consistent values prove the reliability of two different measurements. It is noticed that comparing with the values of MgIrBMgIrBwen () and LiPtBLiPtBSH () in our measurement is relatively high, thus a prudent checking is necessary. For a type-II superconductor, could be estimated as the following relationgamma-n ():

(3)

where =4479.21, using 0.277 T/K, = 9 , and taking for the weak coupling case, we have = 94 which is very close to the upper bound of the experimental value 89.95 . Further we could estimate the via McMillan equationMcMillan ():

(4)

where is the Coulomb pseudopotential taking 0.11, =470.175 K and T=3.3 K, we obtain =0.48. The value indicates that RuB belongs to a weak coupling superconductors.

Figure 9: (Color online) Raw data of specific heat plotted as vs. . All filled symbols represent the data taken with the DR based on the PPMS at various magnetic fields. The open squares show the data taken with the PPMS at different fields. The thick solid line represents the normal state specific heat which contains both the phonon and the electronic contributions.

For noncentrosymmetric superconductors, novel pairing symmetry could be achieved due to the mixing of spin singlet and spin triplet. Specific heat is a useful tool to investigate the material’s low energy excitation. So, subtracting the contribution of phonon, we present temperature dependence of under magnetic fields up to 3 T in Fig. 10(a). For the convenience of theoretical analysis, we further subtract of zero field data as shown in Fig. 10(b). Thus the weak coupling BCS formula could be used:

(5)

where . In obtaining the theoretical fit we take the implicit relation derived from the weak coupling BCS theory for superconductors with different pairing symmetries: for s-wave, for d-wave, and for p-wave, respectively. The theoretical curve of s wave fits the experimental data very well leading to an isotropic gap value = 0.5 and = 3.22 . The ratio = 1.80 obtained here is quite close to the prediction for the weak coupling limit ( = 1.76). In addition, the specific heat anomaly at T is 1.31 being very close to the theoretical value 1.43 predicted for the case of weak coupling.

Figure 10: (Color online) (a) Temperature dependence of for magnetic field up to 3 T, (b) Temperature dependence of - at zero field. The blue solid, green dashed and orange dotted lines are theoretical curves calculated based on BCS model with a gap of s wave, d wave and p wave, respectively.

Condensation energy() is an important parameter for superconductor, thus we calculate with the following process, firstly the entropy difference between normal state and superconducting state could be obtained by , and then E is calculated through . The resulted temperature dependence of E is shown in Fig. 11, the inset is the entropy difference between normal state and superconducting state. The is about 192 mJ/mol at 0 K. Alternatively, could be calculated by the following equation:

(6)

For a BCS s-wave superconductor, 1, taking =80.15 mJ/molK and =0.5 meV, we obtain a value 205 mJ/mol, which is close to experimental value 192 mJ/mol. In addition, the consistence reversely verifies the validity of and determined through our experiment. From condensation energy, the thermodynamic critical field H(0) could be calculated via the relation , yielding H(0)= 612 Oe. For a comparison, H(0) of another noncentrosymmetric superconductor MgIrB is about 300 OeMgIrB ().

Figure 11: (Color online) Superconducting condensation energy of RuB calculated by specific heat, the inset shows entropy difference between the normal and superconducting state.

iv.2 Lower critical field measurement

Lower critical field(H) is an important parameter for a superconductor. According to the Ginzburg-Landau theory, H reflects the superfluid density since H is related to London penetration depth and thus a relation is established that H1/. Moreover, the temperature dependence of H, especially the low temperature features, is always used to investigate the superconducting pairing symmetry and multi-gap effect, for instance the node feature for pairing symmetry and gaps’ value for the latter. In this section we used a two dimensional electron gas(2DEG) micro Hall probe to measure magnetization loops of RuB. For weakening the complex effects of the character of field penetration, such as Bean-Livingston surface barriers and geometrical barriers, we used a low field sweep rate of 60 Oe/min to measure isothermal magnetization curves.

Figure 12: (Color online) The raw data of M(H) curves at different temperature, The red dashed line give the Meissner linear approach. It is found that the value of M is very approaching to that of magnetic field, indicating almost 100% diamagnetization at lower temperatures.

Fig. 12 is the initial isothermal M(H) curves over the temperature range from 1.22 K to 3.2 K. It is found that the low-field parts of those M(H) curves overlap almost on one line (red dash line is guided to the eyes in Fig. 12) with a constant slope, which is attributed to Meissner effect and called as Meissner line. Thus H could be determined as the departure between M(H) curve and Meissner line with the same criterion for all curves. The resulted temperature dependence of H(T)(normalized to H(0)) is shown in Fig. 13. The inset of Fig. 13 shows the criterion for determination of the value of H at 1.4 K, a value of 10 Oe was regarded as error bar due to noise induced uncertainty. According to BCS theory for clean superconductors, the normalized H(T)/H(0) is expressed as followsTinkham ():

(7)

Where is the BCS superconducting energy gap, =1/[exp(-E/T)+1] is the Fermi distribution function, and D(E)=E/[E-(T)] is the quasiparticle density of states. We use above equation to fit the experimental data, H(0) and are fitting parameters. It is found that single gap s-wave pairing could give an appropriate fit with fitting values H(0)=110 Oe and =0.5 meV, the latter is consistent with that of specific heat measurement. Thus the good consistence indicates the reliability of results determined by H measurement, although the lower temperature (less than 1.2 K) experimental data is absent. A possible argument is that the nominal H obtained from experimental data fitting may not reflect the true value due to the inevitable surface barrier and geometrical barrier induced by the polycrystalline quality. Thus in the following paragraph we will do self-consistent checking from superconducting parameters determined by other measurements.

Figure 13: (Color online) The extracted as a function of temperature, isotropic s wave with gap value 0.5 meV could give a good fitting to the experimental data. The inset shows the criterion for determination at temperature 1.4 K.

V Electronic structure calculation

In this section we present density of states(DOS) and band dispersion results based on full potential linear-muffin-tin-orbital program LMTART by SavrasovFP-LMTO-PLW (); Savrasov (). Full potential approximation, Plane wave expansion(PLW), is selected and believed to give the adequate accuracy. Fig. 14 is the DOS calculation of RuB. The total DOS curve have numerous van Hove singularities, the feature is very similar to that of MgIrBMgIrB-DOS (), in that paper the author attributed the characterization as large numbers of atom in the unit cell and various interatomic distances. In RuB only 20 atoms exist in one unit cell, thus the numerous van Hove singularities could stem from the various interatomic distances complicated by the lacking of a inversion center. Another feature is that electronic structure is dominated by Ruthenium 4d states, boron 2p state contributes weakly. It is reasonable to understand from structure aspect that the lattice of RuB is mainly consists of metal tetrahedra and metal octahedra or ’chains’ along c-direction, thus charge carries naturally favor those special channels in a crystal lattice. The total DOS at chemical potential for RuB is 20.988 state/eV per formula unit. For checking the calculated DOS at chemical potential, we could simply estimate the DOS from in the framework of free electron gas.

(8)

where is selected as 80.15 , k is Boltzmann constant=1.380658 J/K, N(0) represents the density of state. The obtained N(0) is about 17 state/cell per formula, which is close the calculated value 20.988. Fig. 15 is the band dispersion curves near Fermi energy. A distinct feature is that the all bands are doubly accompanied, the feature is due to asymmetric spin-orbit coupling effect, thus the degeneracy of spin-up and spin-down is lifted. It is noticed that at some k-points with high symmetry splitting instead of degeneracy also exists, which could come from problems such as un-adequate optimized parameters at initialization during computation. Nevertheless, it is believed that such stigma can not affect the main results of the present paper.

Figure 14: (Color online) Density of states of RuB calculated by full potential PLW-LMTO, the Dos at chemical potential is about 20.988 state/(eV cell spin). It is shown that electronic structure is dominated by Ruthenium 4d states, boron 2p state contributes weakly.
Figure 15: (Color online) Band dispersion curves of noncentrosymmetric materials RuB, the bands are doubly bound due to the ASOC induced spin-up and spin-down splitting.

Vi Self-consistence among superconducting parameters

In the discussion of magnetoresistance, the Kohler’s rule was found to be only slightly broken, indicating a symmetric Fermi surface topology. Thus we could deduce the Fermi-wave number() from charge carrier density(n)kF-from-ne () assuming a single spherical Fermi surface, =(3=6.6527 nm, where n=0.995 cm. The effective mass is estimated as = 17m, where m is bare electron mass and molar volume =136.89 cm/mol. Then the Fermi velocity = / is about 0.4710 m/s. The mean-free-path is evaluated as ==31.36 nm. The superconducting penetration depth is 214 nm. So the coherence length could be estimated using the BCS expression as 19.5 nmTinkham (). Thus the above superconducting parameters could give a stringent checking on experimental data, such as and H and . In the upper critical fields measurement, H(0) is determined as 1.1 T, so using =, where is flux quanta, coherence length is 17.3 nm, such a value is very close to the deduced 19.5 nm. For checking on H, the following formula is usedMgIrB ():

(9)

yielding = 90.2 Oe, where and are deduced values from charge carrier density. In our lower critical field measurement we obtain H=110 Oe, which is larger than the estimated value 90.222 Oe. The thermodynamic critical field H could be obtained from the following formulaMgIrB ():

(10)

using H=11000 Oe, H=90.2 Oe, =214 nm and =17.3 nm, H is given as 628 Oe, which is very close to the value of 612 Oe determined by specific heat. For further checking on the experimental value of H, firstly the experimental values (110 0e) and (17.306 nm) are used into Equation 9, the solved =189.3 nm, then using that value H could be estimated by Equation 10, yielding H=711 Oe, which is about 100 Oe larger than that obtained from specific heat measurement. Therefore it is safe to conclude that the intrinsic value of H is about 90 Oe. So in Table II we list the superconducting and normal state properties of noncentrosymmetric material RuB.

parameters RuB
(mJ/mol K) 80.15
(state/eV cell spin) 20.988
(m eV) 0.5
(Oe) 11000
(Oe) 90
(nm) 17.3
(nm) 215
12.4
(0) (Oe) 612628
/kT 1.80
/T 1.31
(mJ/mol K) 0.3735
(K) 470.175
(mJ/mol) 192205
17m
(cm) 110
(nm) 31.36
( cm)(4 K) 9
Table 2: Superconducting and normal state properties of RuB

In the final part of self-consistent checking on physical parameters of RuB, a brief discussion on the possible exotic properties due to ASOC is necessary. Generally, two criteria for novel superconductivity in the framework of noncentrosymmetry have been established. The first one is Pauli-Clogston limiting fieldpaulilimite (), which could be expressed as , for RuB with 3.3 K, yielding T. In our experiment the upper bound of H(99%) is found to be about 5 T, the experimental value is obvious below Pauli limit, indicating that H is still determined by orbital depairing fields. Nevertheless, the observed kink feature is still a puzzle under the common thinking, such as two bands or superconducting fluctuation scenarios, et. al. Another criterion is the presence of line or point node in the superconducting gap. The good fitting of specific heat data with isotropic s-wave has provided a strong evidence that spin singlet dominated the condensate. The lower critical field measurement also give the same conclusion even though the lower temperature data (less than 1.2 K) is unfortunately absent. Thus a safe conclusion could be given that noncentrosymmetric superconductor RuB is dominated by s-wave pairing symmetry, minor spin triplet could admix among the majority of spin singlet.

Figure 16: (Color online) Normal state magnetization measurement of RuB from 5 K to 150 K. (a) FC(field cooling) mode, a rounded hump happens at about 40 K, (b) ZFC(zero field cooling) mode, the rounded hump on FC curve changes into peak and moves to higher temperature about 55 K.

In the end of experiment and discussion, we show magnetization of RuB from 5 K to 150 K bearing the magnetic field modes of FC(field cooling) and ZFC(zero field cooling) in Fig. 16 . It is found that there is a hump at 40 K on the FC curves, while for ZFC mode the hump changes into a peak and moves to higher temperature at about 55 K. For now it is still hard for us to comprehend the normal state magnetization of RuB, especially more subtle features adding on ZFC curves as shown in Fig. 16(b). For comparison we measured room temperature magnetization of Ruthenium element(not shown here), it is found that Ruthenium element is paramagnetic at 300 K, lowering temperature an antiferromagnetic transition happens at 150 K on the paramagnetic background. The density states calculation have shown that ruthenium contributes most DOS on FS. Thus from such a point view, the normal state magnetization of RuB could be similar to that of ruthenium element. Furthermore for the ruthenium element the outer shell electron configuration is , thus in RuB the ruthenium cations with higher spin angular momentum could be anticipated. However, we could not exclude the possibility of impurity induced magnetism due to the 10% residual for specific heat measurement. Nevertheless, the studies on magnetism of RuB is worthwhile, for example the possible antiferromagnetic fluctuation induced superconductivity has been a hot issue in MgCNicava (), for RuSrGdCuOTallon () the interplay between ferromagnetism and superconductor is also very attractive.

Vii Summary

Transition metal boride RuB was found to be a noncentrosymmetric superconductor with equal to 3.3 K. Superconducting and normal state properties of RuB were determined by a self-consistent analysis through the results of resistivity( and ) , specific heat, lower critical field measurement and electronic structure calculation. It is found that RuB belongs to an s-wave dominated single band superconductor with energy gap 0.5 meV and could be categorized into type II superconductor with weak electron-phonon coupling. Unusual ’kink’ features are clearly observed in field-broadening resistivity curves, possibly indicating the admixture of spin singlet and spin triplet due to the absence of lattice inversion symmetry.

Viii Acknowledgments

This work is supported by the National Science Foundation of China, the Ministry of Science and Technology of China (973 project: 2006CB601000 and 2006CB921802), and the Knowledge Innovation Project of the Chinese Academy of Sciences (ITSNEM). The author thanks to T. Xiang for helpful discussion and Dr. L. Tang for technical support on electronic structure calculations. Appreciation also give to C. Dong for the help of structure analysis.

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