text: ---
  abstract: 'For a noncentrosymmetric superconductor such as Y$_2$C$_3$, we consider a parity-mixing model composed of spin-singlet $s$-wave and spin-triplet $f$-wave pairing components. The $d$-vector in $f$-wave state is chosen to be parallel to the Dresselhaus asymmetric spin-orbit coupling vector. It is found that, the quasiparticle excitation spectrum exhibits distinct nodal structure as a consequence of parity-mixing. Our calculation predict anomalous noninteger power laws for low-temperature nuclear spin-lattice relaxation rate $T_{1}^{-1}$. We demonstrate particularly that such a model can qualitatively account for the existing experimental results of the temperature dependence of $T_{1}^{-1}$ in Y$_2$C$_3$.'
  address: 'School of Physical Sciences, University of Chinese Academy of Sciences - BeiJing 100049 China'
  author:
  - Chongju Chen
  - Biao Jin
  title: 'Nuclear spin-lattice relaxation rate in noncentrosymmetric superconductor Y$_2$C$_3$'
  ---
  
  Noncentrosymmetric superconductor Y$_2$C$_3$ ,Pairing symmetry,Nuclear spin-lattice relaxation rate 74.20.Rp,74.20.-z,74.25.Bt
  
  Introduction {#intro}
  ============
  
  The physics of unconventional superconductivity in materials without inversion symmetry has become a subject of growing interest [@Bauer2012; @Yip2014] since the noncentrosymmetric (NCS) heavy Fermion superconductor CePt$_3$Si was found in 2004 [@Bauer2004]. In these materials the superconducting phase develops in a low-symmetry environment with a missing inversion center. This broken symmetry generates an antisymmetric spin-orbit (SO) coupling and prevent the usual even/odd classification of Cooper pairs according to orbital parity, allowing a mixed-parity superconducting state [@Samokhin2004; @Anderson1984]. This mixture of the pairing channels with different [parities]{} may result in unusual temperature and field dependence of experimentally observed superconducting properties [@Bauer2012; @Yip2014].
  
  For CePt$_3$Si, in particular, where the Rashba-type [@Rashba1960] SO coupling vector $\bm{\gamma_{k}}\propto(
  \hat{k}_{y},-\hat{k}_{x}, 0)$ is generated, various low-energy thermodynamical and transport properties have been extensively investigated from both the experimental and theoretical sides. The NMR relaxation rate [@Yogi2004] $T_{1}^{-1}$, thermal conductivity [@Izawa2005], and London penetration depth [@Bonalde2005] indicate power law behavior at lowest temperatures, suggesting the presence of nodal lines in the quasiparticle excitation spectrum. Besides, the upper critical magnetic field $H_{c2}$ is surprisingly large [@Bauer2004; @Clogston1962], and no change in the Knight shift across the transition temperature $T_{c}$ [@Yogi2006] has been observed. These characteristics are attributed to a spin triplet superconducting order parameter. Theoretically, Frigeri *et al.* have proposed an ($s$+$p$)-wave model [@Frigeri2004] where the $d$-vector of $p$-wave state is chosen to be parallel to the SO coupling vector ($\bm{d_{k}}\propto\bm{g_{k}}$). The gap function of this ($s$+$p$)-wave model has the natural form for a system without inversion symmetry, and exhibits line nodes when the $p$-wave pair potential is larger than that of $s$-wave one. It should be noted that a nonzero $s$-wave pair potential is necessary to get expected line nodes. Hayashi *et al.* [@Hayashi2006] have demonstrated that the presence of line nodes in this ($s+p$)-wave model may account for the experimentally observed low-temperature features of the nuclear spin-lattice relaxation rate $T_{1}^{-1}$ in CePt$_3$Si on a qualitative level.
  
  The cubic Pu$_{2}$C$_{3}$-type sesquicarbide compound Y$_2$C$_3$ is a NCS superconductor known for its relatively high superconducting transition temperature [@Amano2004] ($T_{c}\sim$18K ). Different from the CePt$_3$Si case, the Dresselhaus [@Dresselhaus1955] SO coupling vector $\bm{\gamma_{k}}\propto ( \hat{k}_{x} (\hat{k}_{y}^{2}-\hat{k}_{z}^{2}), \hat{k}_{y} (\hat{k}_{z}^{2}
  -\hat{k}_{x}^{2}), \hat{k}_{z} (\hat{k}_{x}^{2}
  -\hat{k}_{y}^{2}))$ is relevant to Y$_2$C$_3$.
  
  Even many years after its discovery, the nature and symmetry of the superconducting gap function in Y$_2$C$_3$ appears to be full of contradiction. While the specific heat measurement [@Akutagawa2007] and tunneling experiment [@Ekino2013] are interpreted as a fully gapped isotropic $s$-wave state, the nuclear spin-lattice relaxation rate [@Harada2007] $T_{1}^{-1}$ and muon spin rotation [@Kuroiwa2008] ($\mu$SR) measurements on Y$_2$C$_3$ are qualitatively fitted with a nodeless two-gap model similar to MgB$_{2}$. On the other hand, Chen *et al.* [@Chen2011] have measured the magnetic penetration depth as a function of temperature and found a weak linear dependence at very low temperatures. They also reanalysed the NMR data reported in Ref. [@Harada2007] and claimed that, where $T_{1}^{-1}\sim T^{3}$ at $T<3$K, as a matter of fact. Such behavior seems to support the existence of line nodes rather than a fully opened gap in the superconducting state of Y$_2$C$_3$. In addition, the upper critical magnetic field $H_{c2}$ is found to be compatible with the paramagnetic limiting field [@Clogston1962; @Chen2011], and [the]{} Knight shift in NMR [@Harada2007] is decreased to approximately 2/3 of its normal-state value. These features are again incompatible with the single gap or two-gap $s$-wave pictures. It is expected that line nodes (or point nodes of second-order) would be generated due to parity-mixing, similar to the case of CePt$_3$Si mentioned above. In order to shed light on these controversy, further experimental and theoretical studies on the superconducting properties of Y$_2$C$_3$ are required.
  
  In this work, we theoretically investigate the nuclear spin-lattice relaxation rate [@Harada2007] $T_{1}^{-1}$ on the basis of ($s$+$f$)-wave model, where the $d$-vector in $f$-wave state is chosen to be parallel to the Dresselhaus-type asymmetric SO coupling vector. We analyse various possible nodal structures which can be generated by the effect of parity-mixing. In particular, the temperature dependence of the nuclear spin-lattice relaxation rate $T_{1}^{-1}$ is calculated and compared with the experimental result obtained in Ref. [@Harada2007] for Y$_2$C$_3$.
  
  Model Hamiltonian
  =================
  
  Our starting point is the following mean-field $(s+f)$-wave pairing Hamiltonian
  
  $$\label{H}
  H=H_{0}+H_{int}.$$
  
  The Hamiltonian $H_{0}$ describes the noninteracting conduction electrons in a NCS crystal,
  
  $$H_{0}=\sum_{\bm{k}}\sum_{\alpha,\beta}(\epsilon_{\bm{k}}\sigma_{0}+ \gamma_0\bm{\gamma}_{\bm{k}}\cdot\bm{\sigma})_{\alpha\beta}c^\dagger_{\bm{k}\alpha}c_{\bm{k}\beta},\label{H0}$$
  
  where $c^\dagger_{\bm{k}\alpha} (c_{\bm{k}\alpha})$ creates (annihilates) an electron with wave vector $\bm{k}$ and spin $\alpha$, $\bm{\sigma}=(\sigma_{x}, \sigma_{y}, \sigma_{z})$ denotes the vector of Pauli matrices, $\sigma_{0}$ is the $2\times2$ unit matrix, $\epsilon_{\bm{k}}$ is the parabolic bare band dispersion measured relative to the chemical potential restricted to $\mid\epsilon_{\bm{k}}\mid<\omega_{c}$, with $\omega_{c}$ being the usual cutoff energy. Furthermore, $\bm{\gamma_{k}}=(\hat{k}_x(\hat{k}_z^2-\hat{k}_y^2), \hat{k}_y(\hat{k}_z^2-\hat{k}_x^2), \hat{k}_z(\hat{k}_x^2-\hat{k}_y^2))$, with $\hat{k}_{x}=\sin\theta_{\bm{k}}\cos\phi_{\bm{k}}$, $\hat{k}_{y}=\sin\theta_{\bm{k}}\sin\phi_{\bm{k}}$, and $\hat{k}_{z}=\cos\theta_{\bm{k}}$, is the asymmetric ($\bm{\gamma}_{\bm{k}}=-\bm{\gamma}_{\bm{-k}}$) Dresselhaus SO coupling vector considered to be relevant for Y$_2$C$_3$ and La$_2$C$_3$. The strength of SO coupling is denoted by $\gamma_{0}$.
  
  The second term in Eq. (\[H\]) represents the pairing interaction:
  
  $$\begin{aligned}
  H_{int}=\frac{1}{2}\sum_{\bm{k}}\sum_{\alpha,\beta}[\Delta_{\bm{k},\alpha\beta}c^\dagger_{\bm{k}\alpha}c^\dagger_{\bm{-k}\beta}
  +\Delta^\dagger_{\bm{k},\alpha\beta}c_{\bm{-k}\alpha}c_{\bm{k}\beta}\nonumber+\Delta_{\bm{k},\alpha\beta}F^\dagger_{\bm{k},\beta\alpha}], \label{Hint}\end{aligned}$$
  
  with the anomalous averages $F_{\bm{k},\alpha\beta}=\langle c_{\bm{k}\alpha}c_{\bm{-k}\beta}\rangle$, and the gap function defined by [@Mineev1999]
  
  $$\Delta_{\bm{k},\alpha\beta}=-\sum_{\bm{k^{'}}}\sum_{\lambda,\mu}V_{\beta\alpha,\lambda\mu}(\bm{k},\bm{k^{'}})F_{\bm{k^{'}},\lambda\mu},\label{geq}$$
  
  where $V_{\alpha\beta,\lambda\mu}(\bm{k},\bm{k^{'}})$ is the pairing potential. In this work, we will adopt $V_{\alpha\beta,\lambda\mu}(\bm{k},\bm{k^{'}})$ [as]{} the phenomenological one [@Frigeri2006]:
  
  V\_[,]{}(,) = -(i\_[y]{})\_(i\_[y]{})\^\_ -(i\_[y]{})\_(i\_[y]{})\^\_\
  +\[ (i\_[y]{})\_(i\_[y]{})\^\_ +(i\_[y]{})\_(i\_[y]{})\^\_\],
  
  where the first two terms represent the interaction in the $s$-wave pairing channel and in the spin-triplet $f$-wave pairing channel, respectively, and the last term describes the scattering between the two channels. In the following, we will chose the interaction parameters $V_{s}$, $V_{f}$, and $V_{m}$ to be positive, and take for simplicity $V_{m}=\sqrt{V_{s}V_{f}}$ which yields $\Delta_s(T)/\Delta_f(T)=$const. [@Frigeri2006].
  
  Owing to the lack of inversion symmetry, the superconducting gap function Eq. (\[geq\]) generally contains an admixture of even-parity spin-singlet and odd-parity spin-triplet pairing states,
  
  $$\Delta_{\bm{k},\alpha\beta}=[\psi_{\bm{k}}i\sigma_{y}+\bm{d_{k}}\cdot\bm{\sigma}i\sigma_{y}]_{\alpha\beta}\label{gap},$$
  
  where $\psi_{\bm{k}}=\psi_{-\bm{k}}$ and $\bm{d_{k}}=-\bm{d_{-k}}$ represent the spin-singlet and spin-triplet components, respectively. The direction of the $\bm{d_{k}}$ (the $d$-vector) is assumed to be parallel to $\bm{\gamma_{k}}$, as for this choice the antisymmetric SO interaction is not destructive for spin-triplet pairing[@Frigeri2004]. Hence, we parametrize the $\bm{d}$-vector as $\bm{d_{k}}=\Delta_{f}\bm{\gamma_{k}}$. For the spin-singlet component we assume $s$-wave pairing $\psi_{\bm{k}}=\Delta_{s}$, and choose the amplitudes $\Delta_{s}$ and $\Delta_{f}$ to be real and positive.
  
  Using the vector operator $\Psi_{\bm{k}}=(c_{\bm{k}\uparrow},c_{ \bm{k}\downarrow},c^\dagger_{\bm{-k}\uparrow},c^\dagger_{\bm{-k}\downarrow})^t$, where $(\cdots)^t$ stands for the transposing operation, we can write the Hamiltonian in a more compact form:
  
  $$\begin{aligned}
      H&=\frac{1}{2}\sum_{\bm{k}}\Psi^\dagger_{\bm{k}}\check{H}_{\bm{k}} \Psi_{\bm{k}}+\sum_{\bm{k}}\epsilon_{\bm{k}}
      +\frac{1}{2}\sum_{\bm{k}} \sum_{\alpha,\beta} \Delta_{\bm{k},\alpha\beta}F^{\dagger}_{\bm{k},\beta\alpha},\label{hamiltonian}\end{aligned}$$
  
  where
  
  $$\begin{aligned}
   \check{H}_{\bm{k}}=\left( \begin{array}{cc} \hat{M}_{\bm{k}} &  \hat{\Delta}_{\bm{k}}
          \\ \hat{\Delta}^\dagger_{\bm{k}} & -\hat{M}^*_{\bm{-k}}
      \end{array}
      \right),\label{HM}\end{aligned}$$
  
  with
  
  $$\begin{aligned}
      \hat{M}_{\bm{k}}=&\epsilon_{\bm{k}}\sigma_0 + \gamma_0\bm{\gamma}_{\bm{k}}\cdot\bm{\sigma}, \nonumber\\
      \hat{\Delta}_{\bm{k}}=&(\Delta_s+\Delta_{f}\bm{\gamma_{k}}\cdot\bm{\sigma})(i\sigma_y).\label{Delta}\end{aligned}$$
  
  Nodal structures
  ================
  
  The Bogoliubov-de Gennes quasiparticle excitation spectrum $E(\bm{k})$ can be obtained readily by diagonalizing the matrix $\check{H}_{\bm{k}}$ above. One can find four solutions, namely, $E^{(e)}_{\pm}(\bm{k})$ and $E^{(h)}_{\pm}(\bm{k})$, with $E^{(h)}_{\pm}(\bm{k})=-E^{(e)}_{\pm}(\bm{k})$. We have
  
  $$\begin{aligned}
      E^{(e)}_{\pm}(\bm{k})&=\sqrt{(\epsilon_{\bm{k}}\pm\gamma_0|\bm{\gamma_k}|)^2+(\Delta_s\pm \Delta_f |\bm{\gamma_k}|)^2}\equiv E^{\bm{k}}_{\pm}, \label{Ek}\end{aligned}$$
  
  corresponding to two sheets of Fermi surfaces with the energy gaps given by $\Delta_{\bm{k}+}=\Delta_s+\Delta_f|\bm{\gamma_k}|$ and $\Delta_{\bm{k}-}=\Delta_s-\Delta_f|\bm{\gamma_k}|$, respectively. Zeros of $E^{\bm{k}}_{\pm}$ determine the nodal structure of the superconducting state in momentum space. Here let us assume a sufficiently large value of the cutoff energy $\omega_{c}$ $(\omega_{c}\gg\gamma_0, \Delta_s, \Delta_f)$. It is apparent that the upper branch $E^{\bm{k}}_{+}$ is positive definite. Therefore, here we focus on the zeros of the lower branch $E^{\bm{k}}_{-}$.
  
  The amplitude of the Dresselhaus SO coupling $|\bm{\gamma_k}|$ (see Fig. \[fig1\]) becomes zero at 14 points (such as the south and north poles), possesses 24 saddle points at ($\theta_{k}=\arctan 2\sqrt{2}$, $\phi_{k}=\arccos\sqrt{2}/4$), etc. with $|\bm{\gamma_k}|=2\sqrt{2}/9$, and attains its maximum value 0.5 at 12 points ($\theta_{k}=\pi/2$, $\phi_{k}=\pi/4$), etc. on the Fermi surface. Therefore, one encounters different nodal topology depending on the ratio $\kappa\equiv\Delta_{s}/\Delta_{f}$. When $\kappa=0$ ($\kappa=0.5$), $E^{\bm{k}}_{-}$ shows 14 (12) nodal points of first-order (second-order), while exhibits line nodes for $0<\kappa<0.5$ as displayed in Fig. \[fig2\]. For $\kappa>0.5$, however, we always have $\Delta_{\bm{k}-}\neq0$, and thus the quasiparticle excitation spectrum is gapped.
  
  Nuclear magnetic relaxation rate
  ================================
  
  Let us consider the temperature dependence of the nuclear magnetic relaxation rate $T_{1}^{-1}$ defined as $$\begin{aligned}
  \frac{1}{T_1T}\propto\sum_{\bm{q}}\frac{\Im[\chi_{-+}(\bm{q},i\omega_n\to \omega+i0^+)]}{\omega}\Bigg|_{\omega\to 0},\end{aligned}$$ where $\Im$ denotes the imaginary part. The dynamical susceptibility in imaginary time is given by $$\begin{aligned}
      \chi_{-+}(\bm{q},i\omega_n)=\int_{0}^{1/T} d\tau\sum_{\bm{k}\bm{k^\prime}}\left\langle \hat{T}c_{\bm{k^\prime-\bm{q}\downarrow}}^{\dagger}(\tau)c_{\bm{k^\prime}\uparrow}(\tau)c_{\bm{k}+\bm{q}\uparrow}^{\dagger}(0)c_{\bm{k}\downarrow}(0)\right\rangle e^{i\omega_n\tau},\end{aligned}$$ where $\hat{T}$ denotes the time-ordering operator, $\omega_n=(2n+1)\pi T$ is the Matsubara frequency, and $c_{\bm{k}}(\tau)=e^{iH\tau}c_{\bm{k}}e^{-iH\tau}$. We obtain an explicit expression for $T_{1}^{-1}$ as
  
  \_\_[,=]{} (1+),
  
  where $\epsilon_{\pm,\bm{k}}=\epsilon_{\bm{k}}\pm\gamma_0 |\bm{\gamma_k}|$. The temperature dependence of $\Delta_{s}$ and $\Delta_{f}$ are determined by the self-consistent gap equations: $$\begin{aligned}
      \Delta_{s}&=\sum_{\bm{k},\ell}\frac{\tanh( E^{\bm{k}}_{\ell}/2T)}{4E^{\bm{k}}_{\ell}}\Delta_{\ell,\bm{k}}
      (~V_{s}+\ell V_{m}|\bm{\gamma_k}|~), \nonumber\\
      \Delta_{f}&=\sum_{\bm{k},\ell}\frac{\tanh( E^{\bm{k}}_{\ell}/2T)}{4E^{\bm{k}}_{\ell}}
      \Delta_{\ell,\bm{k}}(~V_{m}+\ell V_{f}|\bm{\gamma_k}|~). \label{gap3}\end{aligned}$$
  
  ![Evolution of nodal structure with the parameter $\kappa$ ($0\leq\kappa\leq0.5$). At $\kappa=0$ ($\kappa=0.5$), $E^{\bm{k}}_{-}$ shows 14 (12) nodal points of first-order (second-order), while exhibits line nodes for $0<\kappa<0.5$.[]{data-label="fig2"}](fig2.eps)
  
  ![The exponent of temperature as a function of $\kappa$, calculated numerically according to $n=d\ln(1/T_1T)/d\ln T$.[]{data-label="fignk"}](fig4)
  
  It turns out that for given values of $V_{s}, V_{f}$, and $\omega_{c}$, $1/T_{1}T$ depends on the $\Delta_{s}$ and $\Delta_{f}$ only through the ratio $\kappa$, and is independent of the strength of SO coupling $\gamma_{0}$, similar to the case of Ref. [@Hayashi2006]. It is interesting to see how the low-temperature power law behaviour for the nuclear spin-lattice relaxation rate $1/T_{1}T\propto T^{n}$ is changed with the ratio $\kappa$. Plotted in Fig. \[fignk\] is the exponent of temperature, $n$, as a function of $\kappa$ at $T=0.04T_{c}$ calculated numerically according to $n=d\ln(1/T_1T)/d\ln T$ [@Mazidian2013a]. As we see, the exponent $n$ attains its maximum $n=4$ at $\kappa=0$ (point node of first-order), decreases oscillatorily with increasing $\kappa$, and end with $n\approx2.2$ at $\kappa=0.5$. It is worth noting that, the exponent $n$ is not necessarily to be an integer here, similar to the cases in Ref. [@Mazidian2013a]. For $\kappa>0.5$, however, the gap is open and $1/T_{1}T$ decays exponentially in nature. We present in Fig. \[fig3\] the temperature dependence of $1/T_{1}T$ obtained experimentally [@Harada2007] by Harada *et al.* for Y$_2$C$_3$, together with the calculated results for $\kappa$=0.47, 0.50, and 0.53 for comparison. Shown in the inset of Fig. \[fig3\] is the detailed temperate dependence of $\Delta_{s}$ and $\Delta_{f}$ obtained by solving the gap equations Eq. (\[gap3\]) for $\kappa=0.53$. As can be seen form Fig. \[fig3\], there is a fair agreement between our simple theory and experimental results. However, further experimental measurements at low temperatures $T/T_{c}<0.15$ are needed to obtain a decisive information about the pairing symmetry and to test the prediction of our theory.
  
  ![Comparison of the experimental data in Ref. [@Harada2007] with the calculated temperature dependence of $1/T_{1}T$ for $\kappa=$ 0.47, 0.5, and 0.53. Inset shows the temperature dependence of $\Delta_{s}$ and $\Delta_{f}$ at $\kappa=0.53$.[]{data-label="fig3"}](fig3.eps)
  
  Summary
  =======
  
  In summary, we have calculated the temperature dependence of the nuclear magnetic relaxation rate $1/T_{1}T$ in the Dresselhaus-type noncentrosymmetric superconductor Y$_2$C$_3$. We have considered the ($s+f$)-wave parity-mixing model where the $d$-vector is chosen to be parallel to the Dresselhaus SO coupling vector. It is found that various types of nodal structures can be generated due to the effect of parity-mixing, depending on the value of $\kappa$. We also find that, for $\kappa\sim0.5$, the ($s+f$)-wave model can explain the experimental results fairly well over a wide range of temperatures. However, accurate measurements of $1/T_{1}T$ at lower temperatures would be crucial to the further clarification of pairing symmetry and gap structure in Y$_2$C$_3$.
  
  Acknowledgement
  ===============
  
  This work is partially supported by Science Research fund of GUCAS (No. Y25102BN00).
  
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