---
abstract: 'We investigate the order parameter of noncentrosymmetric superconductors Li$_2$Pd$_3$B and Li$_2$Pt$_3$B via the behavior of the penetration depth $\lambda(T)$. The low-temperature penetration depth shows BCS-like behavior in Li$_2$Pd$_3$B, while in Li$_2$Pt$_3$B it follows a linear temperature dependence. We propose that broken inversion symmetry and the accompanying antisymmetric spin-orbit coupling, which admix spin-singlet and spin-triplet pairing, are responsible for this behavior. The triplet contribution is weak in Li$_2$Pd$_3$B, leading to a wholly open but anisotropic gap. The significantly larger spin-orbit coupling in Li$_2$Pt$_3$B allows the spin-triplet component to be larger in Li$_2$Pt$_3$B, producing line nodes in the energy gap as evidenced by the linear temperature dependence of $\lambda(T)$. The experimental data are in quantitative agreement with theory.'
author:
- 'H. Q. Yuan'
- 'D. F. Agterberg'
- 'N. Hayashi'
- 'P. Badica'
- 'D. Vandervelde'
- 'K. Togano'
- 'M. Sigrist'
- 'M. B. Salamon'
title: 'S-wave/spin-triplet order in superconductors without inversion symmetry: Li$_2$Pd$_3$B and Li$_2$Pt$_3$B'
---

The crystal structure of most superconducting materials investigated to date includes a center of inversion. The Pauli principle and parity conservation then dictate that superconducting pairing states with even parity are necessarily spin-singlet, while those with odd parity must be spin-triplet [@MUMM; @2003]. In materials that lack inversion symmetry, the tie between spatial symmetry and the Cooper-pair spin may be broken [@Gorkov; @2001; @Frigeri; @2004; @Frigeri; @2005; @Edelstein; @1989; @Levitov; @1985; @Samokhin; @2004]. The absence of inversion symmetry along with parity-violating antisymmetric spin-orbit coupling (ASOC) allows admixture of spin-singlet and spin-triplet components.  Unconventional behavior, including zeroes in the superconducting gap function, is then possible, even if the pair wavefunction exhibits the full spatial symmetry of the crystal.

In this Letter we report the dramatically different electrodynamic behavior of two newly discovered noncentrosymmetric superconductors Li$_{2}$Pd$_{3}$B and Li$%
_{2}$Pt$_{3}$B [@Togano; @2004; @Badica; @2005]. The penetration depth $%
\lambda (T)$ in the former material has the expected exponential temperature dependence of a fully-gapped superconductor while the latter exhibits a linear temperature dependence over the range $0.05\leq T/T_{c}\leq 0.3.$ Inasmuch as the main difference between these two compounds is the larger spin-orbit coupling strength for Pt ((Z$_{Pt}$/Z$_{Pd}$)$^{2}$ = 2.9), we argue that the unconventional behavior is evidence for admixed singlet and triplet order as a consequence of ASOC. Indeed, we show quantitative agreement between the experimental data of $\lambda(T)$ and the theoretical calculations for mixed singlet and triplet states based on ASOC.

Parity-broken superconductivity (SC) was previously discussed in the context of surface superconductors [@Edelstein; @1989] and for dirty bulk materials [@Levitov; @1985]. Recently, the discovery of SC in the magnetic compounds CePt$_{3}$Si [@Bauer; @2004], UIr [@Akazawa; @2004] and CeRhSi$_3$ [@Kimura; @2005](under pressure) has attracted extensive interest in studying SC without inversion symmetry. Unfortunately, in these correlated electron systems the nature of superconductivity is complicated by its coexistence with magnetism, therefore severely restricting the study of parity-broken SC.

Li$_{2}$Pd$_{3}$B and Li$_{2}$Pt$_{3}$B crystallize in a perovskite-like cubic structure composed of distorted octahedral units of BPd$_{6}$ and BPt$%
_{6}$ [@Eibenstein; @1997]. Unlike CePt$_3$Si, CeRhSi$_3$ and UIr, these materials show no evidence of magnetic order or strong correlated-electron effects [@Togano; @2004; @Badica; @2005; @Nishiyama; @2005; @Takeya; @2005] that could lead to unconventional superconducting behavior. Further, the increased spin-orbit coupling in Pt leads to much larger band-splitting in Li$_{2}$Pt$_{3}$B than in Li$_{2}$Pd$_{3}$B [@Lee; @2005], allowing us to study the dependence of superconductivity on the ASOC strength. Therefore, we argue that Li$_{2}$Pd$_{3}$B and Li$_{2}$Pt$_{3}$B provide a *model system* in which to study SC without inversion symmetry.

Polycrystalline samples of Li$_2$Pd$_3$B and Li$_2$Pt$_3$B were prepared by arc melting [@Togano; @2004; @Badica; @2005]. Powder x-ray diffraction and metallography identify them as being single phase. The sharp superconducting transitions with a width less than 10% of $T_c$ observed in either bulk magnetization $M(T)$ (see, e.g., the inset of Fig.1), penetration depth $%
\lambda(T)$ or electrical resistivity $\rho(T)$ (not shown) indicate good sample homogeneity. The mean free path [@note], estimated from the rf resistivity $%
\rho$ at $T_c$, coherence length $\xi$ and specific heat coefficient $\gamma$ [@Takeya; @2005] ($\rho=20\mu\Omega$cm, $\xi=9.5$nm, $\gamma=9$mJ/molK$^2$ for Li$_2$Pd$_3$B and $\rho=28 \mu\Omega$cm, $\xi=14.5$nm, $\gamma=7$mJ/molK$%
^2$ for Li$_2$Pt$_3$B), is 24 nm for Li$_2$Pd$_3$B and 42 nm for Li$_2$Pt$_3$B, a few times larger than the corresponding coherence length, indicating clean samples. Precise measurements of penetration depth $%
\Delta\lambda(T)$ were performed utilizing a tunnel-diode based, self-inductive technique at 21 MHz down to 90 mK in a dilution refrigerator. The change in penetration depth $\Delta\lambda(T)$ is proportional to the resonant frequency shift $\Delta f(T)$, i.e., $\Delta \lambda(T)=G\Delta f(T) $, where the factor $G$ is determined by sample and coil geometries [@Chia; @2003]. Due to the uneven sample surface, the uncertainty of the $%
G $-factor can be up to 15%. In this paper, $\Delta\lambda(T)$ is extrapolated to zero at $T=0$, i.e., $\Delta\lambda(T)=\lambda(T)-\lambda_0$. The values of zero temperature penetration depth $\lambda_0$ ($%
\lambda_0=190$ nm for Li$_2$Pd$_3$B and $\lambda_0=364$ nm for Li$_2$Pt$_3$B) are taken from Ref. [@Badica; @2005], determined from the magnetic critical field measurements. The difference of $\lambda_0$ in the two compounds might result from their distinct Fermi surfaces due in part to the spin-orbit coupling [@Lee; @2005]. The magnetization $M(T, H)$ was measured using a commercial SQUID magnetometer (MPMS, Quantum Design).

![(Color online). Temperature dependence of the penetration depth $%
\triangle\protect\lambda(T)$ for Li$_2$Pd$_3$B (\#2) and Li$_2$Pt$_3$B (\#3), showing distinct low-temperature behavior [@Yuan; @2005]. The inset shows the magnetization $M(T)$ for Li$_2$Pt$_3$B (\#3) measured in zero-field-cooling (ZFC) and field-cooling (FC) in a magnetic field of 5 Oe. The values of $T_c$ ($T_c=6.7$ K for Li$_2$Pd$_3$B (\#2) and $T_c=2.43$ K for Li$_2$Pt$_3$B (\#3)) were determined from the mid-points of magnetization drop at $T_c$.](Fig1.eps){width="0.76\columnwidth"}

In Fig. 1 the penetration depth change $\Delta \lambda (T)$ is shown for Li$%
_{2}$Pd$_{3}$B and Li$_{2}$Pt$_{3}$B, respectively. The nearly $T$-independence of $\lambda (T)$ at low temperatures for the Pd-compound is characteristic of fully gapped behavior, consistent with NMR experiments [@Nishiyama; @2005] and specific heat measurements [@Takeya; @2005]. However, the penetration depth $\lambda (T)$ of Li$_{2}$Pt$_{3}$B follows a linear temperature dependence [@Yuan; @2005]. Such a $T$-linear behavior of $\lambda(T)$ can be theoretically interpreted by (a) phase fluctuations among Josephson-coupled grains [@Ebner; @1983] and (b) line nodes in the superconducting energy gap. The former one can be ruled out in this context. The importance of phase fluctuations depends inversely on grain size, which is large ($>
$100 $\mu $m) in both the Pt- and Pd-samples [@Togano; @2004].If phase fluctuations dominate, the Pd sample, with comparable normal-state resistivity and grain size, should also show a strong linear temperature dependence. Further, the transition temperature $T_c$ is strongly dependent on the normal state resistivity in the phase-fluctuation regime. We find that the $T_c$ varies by less than 10% among samples that have normal-state resistivities that differ by a factor of three or more. Finally we have reanalyzed the specific heat of Li$_{2}$Pt$%
_{3}$B reported in Ref.[@Takeya; @2005] and find that $C_{el}/T\sim T$  is a much better representation of those data at low temperature than is an exponential dependence, further supporting the existence of line nodes.

Before describing our model, we explore possible non s-wave states that might exhibit line nodes in Li$_2$Pt$_3$B. Weak-coupling theory of SC, justified by the low $%
T_{c}$, permits only the following three:\
(i) $\Delta_{+}(\mathbf{k})\simeq\Delta_{-}(\mathbf{k})=(k_x^2-k_y^2)(k_y^2-k_z^2)(k_z^2-k_x^2)$,\
(ii)$\Delta_{+}(\mathbf{k},z)\simeq\Delta_{-}(\mathbf{k},z)=e^{iqz}k_z[k_y(k_y^2-k_z^2)+ik_x(k_z^2-k_x^2)]$,\
(iii) $\Delta_{+}(\mathbf{k},z)\simeq\Delta_{-}(\mathbf{k},z)=e^{iqz}k_z(k_x+ik_y)$.\
In the latter two cases, broken parity and time reversal symmetries combine to destabilize the spatially uniform state, giving rise to the spatial dependence in the gap functions. The former two states are unlikely in any theory that is based on local interactions (like the single band Hubbard model). Since the above three states are not s-wave pairing states, and (as argued below) Li$_{2}$Pd$_{3}$B appears to be s-wave, a phase transition in the pairing state of Li$_{2}$(Pd$_{1-x}$Pt$_{x}$)$_{3}$B with varying $x$ would have to occur for any of these states to exist in Li$_{2}$Pt$_{3}$B. Furthermore, these states should be extremely sensitive to impurities and $%
T_{c}$ should be strongly suppressed when $x$ is varied away from 1. These are in contrast with the experimentally observed smooth evolution of $T_{c}$ with $x$ [@Badica; @2005]. Given these arguments against unconventional superconductivity, we attribute the dramatic difference between these two compounds to the variation of ASOC.

When parity symmetry is violated, the ASOC that breaks the spin degeneracy of each band takes the form $\alpha\mathbf{g(k)\cdot S(k)}/\hbar$, where $%
\alpha$ denotes the spin-orbit coupling strength, $\mathbf{S(k)}$ is the spin of an electron with momentum $\hbar \mathbf{k}$, and $\mathbf{g(k)}$ is a dimensionless vector ($\mathbf{g(-k)=-g(k)}$ to preserve time reversal symmetry). This ASOC leads to an energy splitting of the originally degenerate spin states and results in spin-eigenstates that are polarized parallel or anti-parallel to $\mathbf{g(k)}$. The ASOC plays a crucial role in the determination of the superconducting state. The key point is that if a spin-triplet contribution to the superconducting gap function is to emerge, its characteristic d-vector $\mathbf{d(k)}$ must be parallel to $%
\mathbf{g(k)}$ (provided that the ASOC is sufficiently large) [@Frigeri; @2004; @Frigeri; @2005]. This leads to two gap functions $\Delta_{\pm}(\mathbf{k}%
)=\psi\pm t \mid\mathbf{g(k)}\mid$, where each gap is defined on one of the two bands formed by the degeneracy lifting of the ASOC; $\psi$ and $t$ are the singlet and triplet order parameters respectively. For a range of values of $\nu=\psi/t$, $\Delta_{-}(\mathbf{k})$ can change sign and nodes may exist in the superconducting gap.

Recent band structure calculations for these compounds [@Lee; @2005] provide information about $\mid\mathbf{g(k)}\mid$. These results indicate that $\alpha$ is a large energy scale relative to the bandwidth and that $%
\mid\mathbf{g(k)}\mid$ is highly anisotropic. To capture these results in a model we take:$\mathbf{g(k)}=a_1\mathbf{k}-a_2[\widehat{\mathbf{x}}k_x(k_y2+k_z2)+%
\widehat{\mathbf{y}}k_y(k_z2+k_x2)+\widehat{\mathbf{z}}k_z(k_x2+k_y2)]$,with $a_2/a_1=3/2$, $\mathbf{k}$, a unit vector, and the spherical average of $\mid\mathbf{g(k)}\mid^2$ equal to unity. This form of $\mid\mathbf{g(k)}%
\mid$ is the simplest that is consistent with cubic symmetry and allows for anisotropy on a model, spherical Fermi surface.

![(Color online). The temperature dependence of (a) the normalized penetration depth $\protect\lambda(T)/\protect\lambda_0$ and (b) the corresponding superfluid density $\protect\rho_s(T)$ for Li$_2$Pd$_3$B, in which $T_c=7$ K, $G=0.42$ nm/Hz for sample \#1 and $T_c=6.7$ K, $G=0.63$ nm/Hz for sample \#2. The symbols, as described in the figure, represent the experimental data and the solid line is a theoretical fit with parameters $%
\protect\delta=0.1$ and $\protect\nu=4$. The insets in the upper panel and the lower panel show a 3-dimensional (3D) polar plot of the gap function $%
\Delta_{-}(\mathbf{k})$, and the temperature dependence of the order parameter components $\protect\psi$ (spin singlet) and $t$ (spin triplet), respectively. ](Fig2.eps){width="0.75\columnwidth"}

![(Color online). The temperature dependence of (a) the normalized penetration depth $\protect\lambda(T)/\protect\lambda_0$ and (b) the corresponding superfluid density $\protect\rho_s(T)$ for Li$_2$Pt$_3$B, in which $T_c=2.43$ K, $G=1$ nm/Hz for sample \#3 and $T_c=2.3$ K, $G=0.41$ nm/Hz for sample \#5. The fitting parameters are $\protect\delta=0.3$ and $%
\protect\nu=0.6$. In Li$_2$Pt$_3$B, the spin-triplet component $t$ is the dominant order parameter (the inset of Fig. 3(b)). In order to clearly show the line nodes, a small constant is added to the gap function $\Delta_{-}(%
\mathbf{k})$ (the inset of Fig. 3(a)). Six circle-like line nodes can be seen along the large lobes as marked by the dark lines. $\Delta_{-}(\mathbf{k%
})$ changes sign from the large lobes (+) to the small lobes (-) in the 3D polar plot.](Fig3.eps){width="0.75\columnwidth"}

We compute the penetration depth (the superfluid density) on the basis of the formula described in [@Hayashi; @2005]. These fits provide estimates for $\nu $ (defined at $T\rightarrow T_{c}$) and $\delta $, the ratio of the relative density of states between the spin-orbit-split bands. The resulting fits are shown in Figs.2(a) and 3(a), respectively. Li$_{2}$Pd$_{3}$B is nearly a pure spin-singlet state, with a large value of $\nu \simeq 4$. We note that the preliminary fit of two-band model in Li$_{2}$Pd$_{3}$B with a fraction of 4% from the small energy gap [@Yuan; @2005] treated data only for $T<0.3T_{c}$, while the present analysis covers the whole temperature range. As argued above, Li$_{2}$Pt$_{3}$B clearly evidences line nodes, meaning that $\Delta _{-}(\mathbf{k})$ changes sign for a range of wavevectors. The best fit for Li$_{2}$Pt$_{3}$B has $\nu =0.6$ and $\delta
=0.3$, indicating that the spin-triplet component is dominant. We expect $%
\delta $ to be proportional to the strength $\alpha $ of the ASOC, which in turn varies as the square of the atomic number, as above. The obtained value of $\delta (Pt)/\delta (Pd)=0.3/0.1=3$ is consistent with the expectations. In the insets to Figs. 2(a) and 3(a) we show polar plots of $\Delta _{-}(%
\mathbf{k})$ for the two compounds; for Li$_{2}$Pt$_{3}$B, the existence of line nodes appears in the form of circular bands. For Li$_{2}$Pd$_{3}$B, both $\Delta _{+}(\mathbf{k})$ and $\Delta
_{-}(\mathbf{k})$ are non-zero, but anisotropic, over the entire Fermi surface. It is noted that the gap functions $\Delta
_{+}(\mathbf{k})$ and $\Delta _{-}(\mathbf{k})$ possess cubic symmetry (see the insets to Figs. 2(a) and 3(a)) and the pairing states break only gauge invariance symmetry, i.e., an s-wave orbital symmetry for both Li$_{2}$Pd$_{3}$B and Li$_{2}$Pt$_{3}$B. However, $\Delta _{-}(\mathbf{k})$ exhibits a sign change in Li$_{2}$Pt$_{3}$B indicated by dark circles in Fig. 3(a). Figures 2(b) and 3(b) present the superfluid density $\rho _{s}(T)$ obtained from the penetration depth ($\rho _{s}(T)=\lambda
_{0}^{2}/\lambda ^{2}(T)$), along with calculated curves. The agreement is satisfactory. One notes that the weak tail in the experimental $\rho _{s}(T)$ as $T\rightarrow T_{c}$ is mainly due to the influence of rf skin depth upon approaching $T_{c}$. The insets to Figs. 2(b) and 3(b) show the calculated temperature dependences of the order parameters $\psi $ and $t$. Obviously, the spin-singlet component is dominant in the order parameter of Li$_{2}$Pd$_{3}$B, but it is not the case in Li$_{2}$Pt$_{3}$B. For the latter compound, the spin-triplet component $t$ is sufficiently large to give rise to the existence of line nodes in the superconducting energy gap. The existence of a spin-triplet state may be stabilized by the inter-parity" coupling (termed $e_{m}$ in Ref.[@Frigeri; @2005]) between singlet and triplet channels as allowed by broken inversion symmetry. This interaction can arise from el.-ph. (and el.-el.) coupling and may dominate in Li$_{2}$Pt$_{3}$B because of the large ASOC [@Lee; @2005]. We note that while our model (spherical Fermi surface and isotropic spin-singlet gap) predicts that spin-triplet component is larger than the spin-singlet component, this needs not be the case in reality. In particular, if the spin-singlet gap is anisotropic, then the Fermi surface average of the magnitude of the spin-triplet component required to produce line nodes can be significantly decreased.

In addition to the profound effect on the pairing state in Li$_2$Pt$_3$B, broken parity symmetry has other non-trivial consequences. For example, the cubic symmetry allows for a novel contribution to the Ginzburg Landau (GL) free energy density of the form $\varepsilon\mathbf{B \cdot j_{so}}$, where $%
\mathbf{j_{so}}$ is the supercurrent as defined in the usual GL theory and $%
\varepsilon$ is a constant. As a consequence, the condensate wavefunction will not be spatially uniform along the direction of the applied magnetic field as it usually is. Near the upper critical field, it will develop a finite center of mass momentum that is parallel to the applied field [Kaur 2005]{}. This helical structure of the order parameter is similar to that of a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconductor [@Fulde; @1964; @Larkin; @1965]. However, in contrast to the FFLO phase, a non-zero center of mass momentum exists at all temperatures. In the vortex state, this coupling term causes the magnetization to develop a transverse component that is parallel to the supercurrent. This may be observable through small angle neutron scattering experiments.

In summary, our observations have demonstrated that superconductors lacking inversion symmetry exhibit qualitatively distinct properties from those with an inversion center. The existence of unconventional properties (e.g., line nodes) for an s-wave type superconductor, found in Li$_2$Pt$_3$B, provides an alternative way to study unconventional SC, especially that arising from phonon pairing mechanism. Indeed, the absence of parity symmetry coupled with strong spin-orbit coupling, which results in an admixture of spin-singlet and spin-triplet pairing, requires a complete reconceptualization of Cooper pairs and the nature of the superconducting state.

We thank P. Frigeri, R. Kaur, I. Milat and H. Takeya for useful discussions. This work is supported by the National Science Foundation under awards No. NSF-EIA0121568 and NSF-DMR0318665, the Department of Energy under award No. DEFG02-91ER45439, the Swiss National Science Foundation and Petroleum Research Funds. We also acknowledge supports from ICAM (HQY), the 2003-PFRA program of Japan Society for the Promotion of Science (NH) and the CEEX program of Romanian Ministry of Education and Research (PB).

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