Dirac nodal lines and induced spin Hall effect in metallic rutile oxides

TL;DR

This paper identifies Dirac nodal lines in metallic rutile oxides and links them to a large spin Hall effect, suggesting these materials' potential for spintronics applications.

Contribution

It reports the discovery of Dirac nodal lines in rutile oxides and elucidates their role in inducing a significant spin Hall effect, advancing understanding of topological materials in spintronics.

Findings

Dirac nodal lines found in IrO₂, OsO₂, RuO₂

Large spin Hall conductivity linked to nodal lines

Two types of DNLs with different SOC behaviors

Abstract

We have found Dirac nodal lines (DNLs) in the band structures of metallic rutile oxides IrO$_2$, OsO$_2$, and RuO$_2$ and revealed a large spin Hall conductivity contributed by these nodal lines, which explains a strong spin Hall effect (SHE) of IrO$_2$ discovered recently. Two types of DNLs exist. The first type forms DNL networks that extend in the whole Brillouin zone and appears only in the absence of spin-orbit coupling (SOC), which induces surface states on the boundary. Because of SOC-induced band anti-crossing, a large intrinsic SHE can be realized in these compounds. The second type appears at the Brillouin zone edges and is stable against SOC because of the protection of nonsymmorphic symmetry. Besides reporting new DNL materials, our work reveals the general relationship between DNLs and the SHE, indicating a way to apply Dirac nodal materials for spintronics.

Tables

Table 1. Table 1: SHC for IrO 2 , OsO 2 , and RuO 2 . The Fermi energy is set to the charge neutral point. The SHC is in units of ( ℏ / e ) ​ ( Ω ⋅ c ​ m ) − 1 Planck-constant-over-2-pi 𝑒 superscript ⋅ Ω 𝑐 𝑚 1 (\hbar/e)(\Omega\cdot cm)^{-1} .

	IrO2	OsO2	RuO2
${\sigma }_{xy}^z$	8	9	83
${\sigma }_{zx}^y$	–253	–311	–238
${\sigma }_{yz}^x$	–161	–541	–284

Equations

$${\sigma }_{ij}^k=e\hslash {\int }_{{}_{BZ}}\frac{d\vec{k}}{(2\pi {)}^3}\sum _n{f}_{n\vec{k}}{\Omega }_{n,ij}^k(\vec{k}),$$

$${\Omega }_{n,ij}^k(\vec{k})=-2Im\sum _{n^{\prime }\ne n}\frac{<n\vec{k}|J_i^k|n^{\prime }\vec{k}><n^{\prime }\vec{k}|v_j|n\vec{k}>}{(E_{n\vec{k}}-E_{n^{\prime }\vec{k}}{)}^2}$$

$$E,m_z,P,n_x\equiv \{m_x|\vec{\tau }\},n_{4z}\equiv \{c_{4z}|\vec{\tau }\},$$

$$H(\vec{k})|{\varphi }_{\pm }(\vec{k})>$$

$$m_z|{\varphi }_{\pm }(\vec{k})>$$

$$m_z(TP|{\varphi }_{\alpha }(\vec{k}))>=TP(m_z|{\varphi }_{\alpha }(\vec{k})>)$$

$$=TP(i\alpha |{\varphi }_{\alpha }(\vec{k})>)=-i\alpha (TP|{\varphi }_{\alpha }(\vec{k})>)$$

$$(x,y,z)$$

$$\stackrel{m_z}{\to }(-x+\frac{1}{2},y+\frac{1}{2},-z-\frac{1}{2})$$

$$(x,y,z)$$

$$\stackrel{n_x}{\to }(-x+\frac{1}{2},y+\frac{1}{2},-z+\frac{1}{2})$$

$$m_z{n}_x$$

$$m_z{n}_x=e^{i{k}_z}(-1)n_x{m}_z=-n_x{m}_z$$

$$(x,y,z)$$

$$\stackrel{P{n}_x}{\to }(x-1,y,z)$$

$$<{\varphi }_{\alpha }(\vec{k})|TP{n}_x|{\varphi }_{\alpha }(\vec{k})>$$

$$=<(TP{n}_x{)}^2{\varphi }_{\alpha }(\vec{k})|TP{n}_x|{\varphi }_{\alpha }(\vec{k})>$$

$$=-<{\varphi }_{\alpha }(\vec{k})|TP{n}_x|{\varphi }_{\alpha }(\vec{k})>=0$$

Full text

I Introduction

Topological semimetals are an emerging topological phase that has attracted great attention of the condensed-matter community in recent years. Their conduction and valence bands cross each other through robust nodal points or nodal lines in the momentum space Wan et al. (2011); Volovik (2003); Burkov et al. (2011). Distinct from normal semimetals or metals, resultant Fermi surfaces can be characterized by nontrivial topological numbers, giving rise to exotic quantum phenomena such as Fermi arcs on the surface Wan et al. (2011), chiral anomaly in the bulk Nielsen and Ninomiya (1983); Son and Spivak (2013), anomalous Hall effect (AHE) Yang et al. (2011); Burkov (2014), and spin Hall effect (SHE) Sun et al. (2016).

Dirac and Weyl semimetals are typical three-dimensional topological nodal point semimetals, where nodal points and surface Fermi arcs have been recently discovered in real materials Yan and Felser (2016), for example, Na3Bi Wang et al. (2012); Liu et al. (2014); Xu et al. (2015a), TaAs Weng et al. (2015a); Huang et al. (2015); Lv et al. (2015); Xu et al. (2015b); Yang et al. (2015), and MoTe2 Soluyanov et al. (2015); Sun et al. (2015); Deng et al. (2016); Jiang et al. (2017); Huang et al. (2016); Tamai et al. (2016). Beyond Dirac and Weyl points, new-type symmetry-protected nodal points with three-, six- and eight-band crossings in nonsymmorphic space groups Wieder et al. (2016); Bradlyn et al. (2016) and nodal points with triple degeneracy in symmorphic space groups Zhu et al. (2016); Winkler et al. (2016); Weng et al. (2016a, b) have been proposed with potential material candidates, such as MoP Zhu et al. (2016); Lv et al. (2016). Furthermore, nodal lines (Burkov et al., 2011; Chiu and Schnyder, 2014; Zeng et al., 2015; Fang et al., 2015; Chiu et al., 2016, and references therein) can exist in two classes of systems, according to the absence and presence of spin-orbit coupling (SOC) Fang et al. (2015, 2016). The first type without SOC has been reported in many systems (e.g., LaN)  Zeng et al. (2015); Weng et al. (2015b); Xie et al. (2015); Kim et al. (2015); Yu et al. (2015); Chen et al. (2015); Chan et al. (2016); Li et al. (2016); Hirayama et al. (2016). The inclusion of SOC will either gap or split the nodal lines Weng et al. (2015a); Zeng et al. (2015). The second type requires the protection of additional symmetries such as nonsymmorphic symmetries Fang et al. (2015); Young and Kane (2015); Schoop et al. (2016); Bzdušek et al. (2016) and mirror reflection Chiu and Schnyder (2014); Weng et al. (2015c); Ali et al. (2014); Bian et al. (2016). Nodal lines can cross the whole Brillouin zone (BZ) in a line shape, form closed rings inside the BZ, or form a chain containing connected rings Bzdušek et al. (2016). The topological nature of a nodal line can be characterized by a quantized Berry phase $\pi$ along a Wilson loop enclosing the nodal line Burkov et al. (2011); Yu et al. (2011); Soluyanov and Vanderbilt (2011); Fang et al. (2012); Chan et al. (2016). On the surface, nodal line materials were predicted to host drumhead-like surface states Weng et al. (2015c). Theoretical search for exotic nodal phases and corresponding materials launches a race for discovering novel topological states in experiments.

Since topological nodal semimetals or metals commonly exhibit nontrivial Berry phases and strong SOC, they are expected to reveal a strong SHE as an intrinsic effect from the band structure Sun et al. (2016). The intrinsic SHE, in which the charge current generates the transverse spin current, is intimately related with the Berry phase and SOC Sinova et al. (2015). Four of us recently found that the first type of Dirac nodal lines (DNLs) (without SOC) can induce a strong intrinsic SHE when turning on SOC, for example, in the TaAs-family Weyl semimetals Sun et al. (2016). Furthermore, it provokes us to search for topological nodal systems among known SHE (or AHE) materials. Therefore, our attention has been drawn to SHE material IrO2 discovered recently Fujiwara et al. (2013), where its thin films act as efficient spin detectors Fujiwara et al. (2013); Qiu et al. (2015) via the inverse SHE that converts the spin current to the electric voltage. However, the microscopic understanding of the SHE is still missing for this metallic oxide. Furthermore, the strong SOC of the Ir-$5d$ orbitals and nonsymmorphic symmetries of its rutile crystal structure imply that IrO2 may host topological nodes in the band structure. Additionally, this oxide has been used for electrodes in various applications such as catalysts in water splitting (Tilley et al., 2010, and references therein) and ferroelectric memories for a long time Scott (2000).

In this article, we have theoretically investigated the topology of the band structure of metallic rutile oxide IrO2 and similar oxides RuO2 and OsO2. We observe two types of DNLs in their band structures. The first type extends the whole BZ and forms a square-like DNL network in the absence of SOC, resulting in surface states. Joint points of the network are six- and eight-band-crossing points at the center and boundary of the BZ, respectively. These DNLs become gapped and lead to a strong SHE when SOC exists. The second type is stable against SOC and appears at the edges of the tetragonal BZ, which is protected by nonsymmorphic symmetries.

II Methods

To investigate the band structure and the intrinsic SHE, we have performed $ab~{}initio$ calculations based on the density-functional theory (DFT) with the localized atomic orbital basis and the full potential as implemented in the full-potential local-orbital (FPLO) code Koepernik and Eschrig (1999). The exchange-correlation functionals were considered at the generalized gradient approximation (GGA) level Perdew et al. (1996). We adopted the experimentally measured lattice structures for $R$O2 ($R$ = Ir, Os, and Ru) compounds. By projecting the Bloch states into a highly symmetric atomic orbital like Wannier functions (O-$p$ and $R$-$d$ orbitals), we constructed tight-binding Hamiltonians and computed the intrinsic spin Hall conductivity (SHC) by the linear-response Kubo formula approach in the clean limit Sinova et al. (2015); Xiao et al. (2010),

[TABLE]

where $f_{n\vec{k}}$ is the Fermi–Dirac distribution for the $n$-th band. The spin current operator is J_{i}^{k}=\frac{1}{2}\left\{\begin{array}[]{cc}{v_{i}},&{s_{k}}\end{array}\right\}, with the spin operator ${s}$, the velocity operator ${v_{i}}=\frac{1}{\hbar}\frac{\partial{H}}{\partial k_{i}}$, and $i,j,k=x,y,z$. $|n\vec{k}>$ is the eigenvector for the Hamiltonian ${H}$ at the eigenvalue $E_{n\vec{k}}$. $\Omega_{n,ij}^{k}(\vec{k})$ is referred as the spin Berry curvature as analogy to the ordinary Berry curvature. A $500\times 500\times 500$ $k$-grid in the BZ was used for the integral of the SHC. The SHC $\sigma_{ij}^{k}$ refers to the spin current ($j_{i}^{s,k}$), which flows along the $i$-th direction with the spin polarization along $k$, generated by an electric field ($E_{j}$) along the $j$-th direction, i.e., $j_{i}^{s,k}=\sigma_{ij}^{k}E_{j}$.

III Results and Discussions

III.1 Nonsymmorphic symmetry

Three compounds $R$O2 ($R$ = Ir, Os, and Ru) share the rutile-type lattice structure with space group $P4_{2}/mnm$ (No. 136), as shown in Fig. 1. A primitive unit cell contains two $R$ atoms that sit at the corner and center of the body-centered tetragonal lattice, respectively. One $R$ atom is surrounded by six O atoms that form a distorted octahedron. For space group No. 136, we have the following generator operations,

[TABLE]

where $m_{x,z}$ are mirror reflections, $c_{4z}$ is the four-fold rotation, $P$ is the inversion symmetry, $n_{x}$ and $n_{4z}$ represent nonsymmorphic symmetries, and $\vec{\tau}=(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ is the translation of one-half of a body diagonal. Additionally; the time-reversal symmetry $T$ also appears for $R$O2 systems.

It is known that electronic bands are doubly degenerate (considering spin and SOC) at every $k$-point of the BZ owing to the coexistence of $P$ and $T$. Furthermore, a generic nonsymmorphic symmetry leads to new band crossings and thus higher degeneracies at the BZ boundary. Therefore, the coexistence of $P$, $T$, and nonsymmorphic symmetries guarantees four-fold or even larger degeneracies at some $k$-points of the BZ boundary. As discussed in the following, IrO2 exhibits a four-fold degeneracy at the BZ edge lines, $X(Y)-M$ and $M-A$ (also see Fig. 1d). Consequently, one requires $4n$ electrons for filling these bands to obtain a band insulator. However, a primitive unit cell contains two IrO2 formula units and in total 42 valence electrons, failing to satisfy the precondition of a band insulator in this nonsymmorphic space group Parameswaran et al. (2013); Bradlyn et al. (2016). Therefore, IrO2 is constrained by the lattice symmetry to be a band metal in the weak interaction case. Recently, it has been reported that IrO2 cannot become a Mott insulator because of a large bandwidth Kahk et al. (2014); Kim et al. (2016), while several seemingly similar iridates (e.g., Sr2IrO4) are known as $5d$ Mott insulators with the $J_{eff}=1/2$ state Kim et al. (2008, 2009).

III.2 Dirac Nodal lines without SOC

We first investigate the band structures without including SOC. As shown in Fig. 1c, six-band and eight-band crossing points (including spin) appear at the $\Gamma-Z$ and $M-A$ axes, respectively, which are noted as a hexatruple point (HP) and an octuple point (OP), respectively. There is a DNL connecting neighboring HP and OP in the BZ, forming a network in the $k$-space, as indicated by **Figs. 1b ** and 3a. Two layers of networks are present above and below the $k_{z}=0$ plane respectively, and can be transformed to each other by $P$ or $T$. DNLs exist inside the (110) and ($\bar{1}$10) mirror planes and originate from crossing of $d_{x^{2}-y^{2}}$ and $d_{xz,yz}$ bands. Here, $d_{x^{2}-y^{2}}$ and $d_{xz,yz}$ bands are all doubly degenerate owing to $P$ and $T$ and exhibit opposite eigenvalues $-1$ and $+1$, respectively, for each mirror reflection. The mirror symmetry protects the four-band-crossing in the absence of SOC.

The topology of a DNL is characterized by the nontrivial Berry phase (or winding number) along a closed path that includes the DNL. We choose a loop, along which the system is fully gapped as indicated in Fig. 1b Yu et al. (2011); Soluyanov and Vanderbilt (2011). The Berry phase for all “occupied” bands is found to be a quantized value, $\pi$. This nonzero Berry phase further leads to surface states. When projecting to the (001) surface, the nodal band structure exhibits a local energy gap between $\bar{M}$ and $\bar{X}$ in the surface BZ. Two layers of DNL networks overlap in the (001) surface projection. Thus, one can observe two sets of surface states (spinless) connecting OPs between two adjacent $\bar{M}$ points inside the gap in Fig. 2a.

When SOC is included and the $SU(2)$ symmetry is broken, these DNLs including HPs and OPs are gapped (see Fig. 1d), since there is no additional symmetry protection. On the surface, original spinless surface states split into two Rashba-like spin channels (see Fig. 2b). Because of the lack of robust symmetry protection (e.g., $P$ is commonly breaking on the surface), these surface states may appear or disappear according to the surface boundary condition.

III.3 Dirac nodal lines with SOC

The $k_{x}=\pi$ and $k_{y}=\pi$ planes are actually Dirac nodal planes in the absence of SOC. The presence of SOC gaps these planes and only leaves DNLs along some high-symmetry lines, $X$-$M$ and $M$-$A$. By taking the DNL along $X$-$M$ as an example, we can understand the four-fold degeneracy by considering time reversal symmetry $T$, point group symmetries, $m_{z}$, $P$, and nonsymmorphic symmetry $n_{x}$. Since $[m_{z},H]=0$ in the $k_{z}=0$ plane, we can choose the eigenstates of Hamiltonian $H$ with definite mirror parity for $\vec{k}$ along $X$-$M$,

[TABLE]

where $i$ is from the spin. First, we know $[TP,m_{z}]=0$. Therefore, we have

[TABLE]

where $\alpha=\pm 1$. Therefore, $TP|\phi_{\alpha}(\vec{k})>$ is also an eigenstate at $\vec{k}~{}(TP\vec{k}=\vec{k})$ with the mirror parity $-i\alpha$, where $-$ sign is from the complex conjugate in $T$. Next, we consider the commutation between $m_{z}$ and the glide mirror symmetry $n_{x}$, both of which act in real space $(x,y,z)$ and the spin space simultaneously. In real space, we have

[TABLE]

where $T_{(0,0,-1)}=e^{ik_{z}}$ is the translation operator when acting on the Bloch wavefunction. In the spin space, $m_{z}=i\sigma_{z}$ and $n_{x}=i\sigma_{x}$ and thus $m_{z}n_{x}=-n_{x}m_{z}$. By combining the real space and the spin space, we obtain

[TABLE]

Therefore, $n_{x}|\phi_{\alpha}(\vec{k})>$ is also an eigenstate at $\vec{k}~{}(n_{x}\vec{k}=\vec{k})$, but with a mirror parity $-i\alpha$. Further, the combination of $TPn_{x}$ leads to one more state $TPn_{x}|\phi_{\alpha}(\vec{k})>$ with a mirror parity $+i\alpha$.

In total, we can have four eigenstates, $|\phi_{\alpha}(\vec{k})>$, $TP|\phi_{\alpha}(\vec{k})>$, $n_{x}|\phi_{\alpha}(\vec{k})>$, and $TPn_{x}|\phi_{\alpha}(\vec{k})>$ for $\vec{k}~{}(\pi,k_{y},0)$ along $X$-$M$. The mirror parities of $m_{z}$ are $+i\alpha$ for $|\phi_{\alpha}(\vec{k})>$ and $TPn_{x}|\phi_{\alpha}(\vec{k})>$, and $-i\alpha$ for $TP|\phi_{\alpha}(\vec{k})>$ and $n_{x}|\phi_{\alpha}(\vec{k})>$. Next, we will prove that they are orthogonal to each other, i.e., two eigenstates with the same mirror parity are orthogonal, $<\phi_{\alpha}(\vec{k})|TPn_{x}|\phi_{\alpha}(\vec{k})>=0$. This requires that $TPn_{x}$ is anti-unitary. In real space, we have

[TABLE]

In the spin space, $P=1$ and $n_{x}=i\sigma_{x}$. Therefore, we have $(Pn_{x})^{2}=-e^{ik_{x}}$. Considering $T^{2}=-1$ for spinful fermions, we obtain $(TPn_{x})^{2}=e^{ik_{x}}=-1$ for $\vec{k}=(\pi,k_{y},0)$. Since $TPn_{x}$ is an anti-unitary operator that satisfies $<\phi|\psi>=<TPn_{x}\psi|TPn_{x}\phi>$, we have

[TABLE]

which means the states $TP|\phi_{\alpha}(\vec{k})>$ and $n_{x}|\phi_{\alpha}(\vec{k})>$ are orthogonal to each other.

Therefore, we prove that there are four degenerate orthogonal eigenstates along the $X$-$M$ line. We point out that the nonsymmorphic symmetry is crucial to protect the four-fold degeneracy, while only $P,T$, and $m_{z}$ cannot stabilize DNLs. Likewise, we can also understand the four-fold degeneracy along the $M$-$A$ line considering $m_{z}$, $P$, $T$, and another nonsymmorphic symmetry $n_{4z}$.

These DNLs protected by nonsymmorphic symmetries are similar to those DNLs observed at the BZ edges in ZrSiS and HfSiS (e.g., Schoop et al. (2016); Chen et al. (2017)). We find that the Berry phase along a loop (indicated in Fig. 1b) including such a DNL is zero. Therefore, we do not expect apparent topological surface states related to these DNLs. It is still interesting to point out that t he density of states scales linearly to the energy for DNLs. Then one can expect different correlation effects in DNL semimetals from a nodal point semimetal and a normal metal Huh et al. (2016); Fang et al. (2015). For IrO2, such DNLs appear at the Fermi energy, while they stay far below the Fermi energy in ZrSiS-type compounds. These DNLs may be responsible for the high conductivity and large magneroresistance Lin et al. (2004); Ryden et al. (1972).

III.4 Dirac nodal lines and spin Hall effect

The first type of DNLs, DNL networks without SOC, indicate the existence of a strong SHE in $R$O2. It is known that band anti-crossing induced by SOC can lead to a large intrinsic SHC. To maximize the SHC, one needs to increase the number of band anti-crossing points, i.e., nodal points in the absence of SOC. Therefore, a DNL, an assemble of continual nodal points in the BZ, can induce a strong SHE. The DNL networks that constitute many DNLs will further enhance the SHE. Because such type of DNLs without SOC are usually protected by the mirror symmetry, it will be insightful to look for SHE materials in space groups that host many mirror or glide mirror planes. This argument seems consistent with the fact that current best SHE materials are usually Pt and W metals(e.g. Tanaka et al. (2008); Guo et al. (2008)) from the high-symmetry space groups.

Next, we compute the intrinsic SHC for $R$O2 compounds following Eq. 1 based on the band structure including SOC. The SHC $\sigma_{ij}^{k}$ is a second-order tensor with 27 elements. The number of independent nonzero elements of the SHC tensor is constrained by the symmetry of the system. We have only three independent nonzero elements, $\sigma_{xy}^{z}=-\sigma_{yx}^{z}$, $\sigma_{zx}^{y}=-\sigma_{zy}^{x}$, and $\sigma_{yz}^{x}=-\sigma_{xz}^{y}$. We list their values in Table I for all the three compounds. The SHC of OsO2 is larger than that of RuO2, and Os-$5d$ bands exhibit much stronger SOC than Ru-$4d$ bands. The SHC is dependent on the Fermi energy of the system. Because Ir has one more electron than Os/Ru, the Fermi surface of IrO2 is higher than that of OsO2 and RuO2 although their band structures look very similar (see Figs. 1d-1f). Thus, IrO2 displays a smaller SHC than OsO2 although Ir has stronger SOC. For all the three compounds, the SHC is very anisotropic, as can be seen from Table I. Here, the largest SHC of $R$O2 is still smaller than that of Pt [$\sigma^{z}_{xy}\sim 2000(\hbar/e)(\Omega\cdot cm)^{-1}$]. However, $R$O2 may exhibit a large spin Hall angle (the ratio of the SHC over the charge conductivity) owing to their high resistivity compared to a pure metal, as already found in IrO2  Fujiwara et al. (2013).

Here, we also demonstrate the direct correspondence between DNLs and the SHC. Since the SHC is obtained by integrating the spin Berry curvature over the BZ, we show the distribution of $\Omega^{x}_{n,yz}(\vec{k})$ in the (110) mirror plane that hosts DNLs. In the band structure along DNLs in Fig. 3, the SOC clearly gaps a DNL that connects a HP and an OP. Correspondingly, one can find two “hot lines” of the spin Berry curvature, which is the anti-crossing region of the DNLs. Thus, it is clear that the first-type DNLs contribute to a large SHC for IrO2-type materials. In contrast, the second-type DNLs at the BZ edges (e.g., that along $M$-$A$ in Fig. 3c) show a slight contribution to the SHC.

IV Conclusions

To summarize, we found two types of DNLs in metallic rutile oxides IrO2, OsO2, and RuO2. First-type DNLs form networks that extend in the whole BZ and appear only in the absence of SOC, which induces surface states at the boundary. The second type of DNLs is stable against SOC because of the protection of nonsymmorphic symmetry. Because of the SOC-induced gap for first-type DNLs, a large intrinsic SHE can be realized in these compounds. This explains the strong SHE observed in IrO2 in the previous experiment. Moreover, our calculation suggests that OsO2 will behave even better than IrO2 in SHE devices, as OsO2 shows a larger intrinsic SHC. Our work implies that first-type DNLs (or nodal points that can be gapped by SOC) and the SHE maybe commonly related to each other, indicating new guiding principles to search for DNLs in SHE materials or enhance the SHE by DNLs in the band structure. For example, it will be insightful to look for SHE materials in high-symmetry space groups with many mirror or glide mirror planes, which can induce the first type of DNLs.

Acknowledgements.

We thank Jakub Zelezny and Lukas Müchler for fruitful discussions. This work was financially supported by the ERC (Advanced Grant No. 291472 “Idea Heusler”). Y.Z. and B.Y. acknowledge German Research Foundation (DFG) SFB 1143.