Orbitronics: Orbital currents in solids

Atomic orbitals and their linear combinations present elementary building blocks for electronic Bloch states in solids. Given a specific Bloch vector k, the index n of a Bloch state $\psi_{n\mathbf{k}}$ carries information about two key aspects: valence orbitals of the constituent atoms and intrinsic spin of the electron. Consequently, these two aspects of Bloch states promote degrees of freedom which can be manipulated and transported for novel types of solid-state device applications. In fact, studies of the spin degree of freedom inherent to Bloch states have evolved into a prominent area of spintronics, which addresses generation, detection, and manipulation of the spin information [1–4]. Similarly, one may think of the orbital degree of freedom as a variable which can be controlled in solids, referring to the respective branch of condensed matter physics as orbitronics [5–7]. So far, however, orbital transport has been investigated to a much lesser degree than the spin transport. It is because the orbital degree is often regarded as "frozen" in solids: since the crystal field enforces specific symmetry of the Bloch states, this breaks the continuous rotation symmetry and suppresses the formation of the orbital moment [8]. Thus, for elemental 3d ferromagnets (FMs), the orbital moment is much smaller than the spin moment as it requires spin-orbit coupling (SOC), which is much weaker than the crystal field potential [9]. However, exceptions exist, and it is mainly the orbital moment which contributes to magnetism in some materials [10–13].

In contrast, recent studies have shown that orbital quenching does not preclude prominent nonequilibrium dynamics and transport of the orbital degree of freedom [14, 15], which is a property of the excited state rather than of the ground state. While the crystal field potential has the strongest influence on the properties of electronic states in solids, it also mediates a hybridization between different atomic orbitals, which is a crucial element in orbital dynamics [14]. This implies that the dynamics and transport of orbital information can be driven by external stimuli, such as electric field, regardless of the orbital quenching governing the ground state. Recent studies have indeed demonstrated not only that pure orbital transport, which does not involve other (e.g., spin) currents, takes place in many materials [5, 7, 14, 16–23], but also that orbital currents mediate other transport phenomena such as spin Hall effect (SHE) [14, 24, 25] and valley Hall effect (VHE) [21, 26]. Moreover, recent theories suggest that an injection of the orbital current into a FM can excite magnetization dynamics [15, 27, 28], which is one of the most important functions required for spintronic devices. This possibility has been explored in recent experiments, which demonstrated an alternative way of using the orbital current instead of the spin current in spintronics [29–34].

In this perspective, we introduce the notion of the orbital current in solids and key orbital transport phenomena, which enable an electrical generation and detection of orbital currents. We also explain and develop the idea of utilizing orbital currents and orbital injection as means to manipulate local moments in magnetic materials. Given that the experimental realization and observation of orbital effects have only recently started, we put a strong emphasis on reviewing experimental methods for the detection of the orbital currents. Finally, we discuss challenges and future directions in orbitronics which utilize orbital currents as an information carrier in solid-state devices and resulting effects.

Theoretical description of the orbital current is similar to that of the spin current. However, a clear difference is that an orbital current cannot be defined in the vacuum unlike the spin current which is often schematically depicted as a flow of spin-polarized electrons. The notion of an orbital current is based on the orbital character of the Bloch states in a solid, which comprises many atoms. Thus, it is an emergent concept which does not exist for constituent elements per se [35]. This means that the orbital current needs to be described on the basis of Bloch states with various atomic orbital characters rather than the electronic states in the vacuum. In the wave packet description of p-orbital states, for example, an arbitrary quantum state can be written as

$$\begin{align} \vert{\Psi(t)}\rangle&=\int\textrm{d}\mathbf{k}\left[C_{p_x\mathbf{k}}(t)\vert{\psi_{p_x\mathbf{k}}}\rangle\right.\nonumber\\ &\left.+C_{p_y\mathbf{k}}(t)\vert{\psi_{p_y\mathbf{k}}}\rangle+C_{p_z\mathbf{k}}(t)\vert{\psi_{p_z\mathbf{k}}} \rangle\right], \end{align} \tag{ 1 }$$

where $\vert{\psi_{p_\alpha \mathbf{k}}}\rangle$ is the Bloch state derived from the $p_\alpha\ (\alpha=x,y,z)$ orbital of the atomic state. Here, orbital information is encoded in coefficients $C_{p_\alpha\mathbf{k}}(t)$, which can be controlled by external perturbations. For instance, a wave packet with $\mathrm{arg}(C_{p_y\mathbf{k}}/C_{p_x\mathbf{k}})=\pi/2$ and $C_{p_z\mathbf{k}}=0$ carries a finite orbital angular momentum (OAM) along the z-direction. Note that the coefficients behave as an "internal" degree of freedom of the wave packet, analogously to the case of spin. Thus, we can define the OAM operator in a matrix representation, and the orbital current operator can be written as a second-rank tensor

$$\begin{eqnarray} &&j^{L_\beta}_\alpha=\frac{1}{2}(v_\alpha L_\beta + L_\beta v_\alpha), \end{eqnarray} \tag{ 2 }$$

where $v_\alpha$ is the α-component of the velocity operator and $L_\beta$ is the β-component of the OAM operator. Note that L refers to the intra-atomic contribution to the OAM, which does not contain the inter-atomic contribution. Except for a particular case [26], a general definition of the orbital current that incorporates the intra- and inter-atomic contributions on an equal footing is not known yet. Theories for transport of OAM which bridge the two pictures (localized and delocalized) need to be developed, which might require a further effort from the side of the Berry phase and general response theories [36–39].

In analogy to the definition of the SHE [3, 40, 41], the orbital Hall effect (OHE) refers to a generation of a transverse orbital current by an external electric field [5, 14, 25]. The OHE serves as a way to electrically generate an orbital current. The reciprocal effect —the inverse OHE— can be used to detect orbital currents electrically.

Mechanisms

Like many other Hall effects, the mechanisms behind the OHE can be divided into intrinsic and extrinsic types. The intrinsic mechanism, which originates from the ground-state electronic states, has been proposed for semiconductors [5], metals [14, 16, 24–25], and 2D materials [17–21, 26, 42]. However, the picture of an extrinsic mechanism, which relies on disorder scattering, has not been developed to date, although it is expected to be present, e.g., in the form of side jump and skew-scattering mechanisms. Only in very particular situations, a consequence of the vertex corrections has been considered. Bernevig et al. showed that vertex corrections from impurity scattering vanish for a low-energy model for hole-doped Si [5]. Vertex corrections in transition metals are in general finite, but Tanaka et al. showed that they are expected to be small [24]. Here we review only the intrinsic mechanism in detail.

The intrinsic contribution depends on whether the spatial inversion symmetry is present in the system or not. Time-reversal and spatial inversion symmetries imply $\langle \mathbf{L} \rangle_\mathbf{k} = -\langle \mathbf{L} \rangle_{-\mathbf{k}}$ and $\langle \mathbf{L} \rangle_\mathbf{k} = \langle \mathbf{L} \rangle_{-\mathbf{k}}$, respectively. In nonmagnetic materials, a combination of time-reversal symmetry and spatial inversion symmetry dictates that OAM is absent for all states at each k in the ground state, which is a "strong" condition of orbital quenching. Thus, in centrosymmetric crystals, OAM has to be induced a priori by an external electric field. Go et al. have demonstrated that an external electric field can drive a hybridization between states with different orbital character, e.g., radial or tangential orbitals, which can induce a finite OAM [14]. As shown in fig. 1(a), an important feature of the orbital hybridization mechanism is that an OAM is induced along the direction of $\mathbf{k}\times \mathbf{E}$ in the lowest order in k, where E is an external electric field. This means that electrons with opposite sign of induced OAM propagate in opposite directions, which is effectively the OHE.

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On the other hand, in noncentrosymetric crystals, the OAM may already be present in equilibrium at each k-point although the total OAM vanishes in the presence of time-reversal symmetry ("weak" condition of orbital quenching). At the same time, it is important to note that the Berry curvature of Bloch states, which gives rise to the anomalous velocity, satisfies the same symmetry constraints as OAM, i.e., the direction of the Berry curvature field is correlated with the direction of the OAM. Thus, the Hall current driven by the Berry curvature is naturally orbitally polarized. This mechanism was pointed out by Bhowal and Satpathy [17, 18]. In transition metal dichalcogenide (TMD) monolayers, orbital polarization is mostly pronounced at $\mathrm{K}$ and $\mathrm{K}'$ valleys, where the sign of the OAM is opposite (fig. 1(b)). In addition, the Berry curvature is also valley dependent, as it is well known from the studies of the valley Hall effect (VHE) [43–46]. Therefore, TMD monolayers can exhibit the OHE, where the mechanism is distinct from that active for centrosymmetric materials. However, we emphasize that both mechanisms can co-exist in noncentrosymmetric materials in general [22].

Relation to other Hall effects

The OHE can play a crucial role in other Hall effects. One of the most important examples of this is the fact that the SHE results from the OHE. We emphasize that the mechanism of the OHE does not require SOC in general [5, 14, 25]. This means that the OHE can be finite in materials with negligible SHE [16]. When the SOC is taken into account, the OAM and spin become entangled and the OHE is accompanied by the SHE [14, 24]. The relative sign between the OHE and SHE generally depends on the correlation $\langle \mathbf{L}\cdot\mathbf{S}\rangle$, where S is the operator of spin, as shown in fig. 2(a) [14]. This explains the overall trend in sign and magnitude of the SHE in 4d and 5d transition metals [24–25]. Since the SHE originates from the OHE, i.e., the SHE vanishes in the absence of the OHE, the strength of the SHE is fundamentally limited by the strength of the OHE. Thus, finding large-SHE materials, which is one of important challenges in spintronics, is closely tied with finding materials which exhibit a large OHE. Meanwhile, Kontani et al. demonstrated that the OHE plays a critical role for the anomalous Hall effect as it originates from the SHE when occupations of majority and minority electrons differ due to the exchange splitting [3, 47].

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In 2D materials with honeycomb lattice, the VHE arises due to sublattice symmetry breaking, which is achieved, e.g., in TMD monolayers or graphene grown on hexagonal BN. As discussed in fig. 1(b), the valley-dependent Berry curvature is entangled with the OAM, and thus the VHE is tied to the OHE. However, the valley information is not a physically measurable quantity and its definition is ambiguous, depending on an arbitrary cutoff of the k-space integral. For this reason, Bhowal and Vignale proposed that the VHE is better described by the OHE [26]. However, for disentangling the OHE and the VHE, Cysne et al. proposed to use TMD bilayers where the inversion symmetry is present, which eliminates the valley-dependent Berry curvature and eventually the VHE while keeping the OHE finite [21].

Materials

So far, classes of materials for which the OHE has been investigated are very limited. Bernevig et al. theoretically found that OHE is sizeable in hole-doped Si despite negligible SOC, which originates in the p orbital character of the valence bands [5]. For transition metals, Jo et al. studied 3d elements and found that the orbital Hall conductivities (OHCs) are gigantic although the spin Hall conductivities (SHCs) are much smaller due to small SOC [16] (fig. 3(a)). Tanaka and Kontani et al. investigated 4d and 5d transition metals and found that the OHCs are also huge (fig. 3(b)), which leads to sizeable SHCs [24, 25].

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The OHE in 2D materials such as TMDs has been also investigated by Canonico et al. [19, 20] independently of refs. [17, 18]. In particular, it was found that the OHC is finite even within the energy gap of TMDs (fig. 4(a)) [20]. This is particularly intriguing since the Z₂ topological invariant was found to be trivial in the latter case, as reflected in a zero SHC within the energy gap. Soon after, Cysne et al. showed that the orbital Hall insulating phase is characterized by a nonzero orbital Chern number [21] by generalizing the method developed for computing the spin Chern number [48]. They also calculated electronic states in the TMD nanoribbon geometry with a zigzag boundary. The result seems to indicate that a nonzero orbital Chern number in the bulk corresponds to existence of the edge states which are orbital polarized (fig. 4(b)) [21]. However, it is still an open question whether an orbital Hall insulator is just another manifestation of an already-known topological phase or whether it goes beyond the current paradigm of topological insulators, and its physical meaning and consequences for orbital transport need to be investigated further in depth. We also note that Tokatly investigated the OHE in hole-doped graphene and found that it is related to the Berry phase of the degeneracy point at $\mathbf{k}=0$ [42].

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Experimental detection

While the OHE is a mechanism prominent in the bulk of materials, the orbital Rashba effect (ORE), found at surfaces and interfaces, can also result in current-induced OAM accumulation [6, 56–58]. The ORE denotes a Rashba-type coupling between L and k, which leads to orbital-dependent energy splitting and chiral OAM textures in k-space [6, 59, 60]. The ORE emerges from the formation of electric dipole moment $\sim \mathbf{L}\times\mathbf{k}$, which couples to a potential gradient present at a surface or at an interface (fig. 5(a)) [59]. In the presence of a chiral OAM texture, a shift of the occupation function by an external electric field results in a finite OAM, which is called the orbital Edelstein effect (OEE) (fig. 5(b)) [6, 56–58].

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The ORE emerges even in the absence of SOC, but the ORE complemented by SOC results in the "spin" Rashba effect [59, 61], with a hierarchical relation similar to that of OHE and SHE. Thus, the ORE provides a coherent physical description of the electronic states with finite OAM and spin polarization. The ORE has been investigated in a wide range of systems, such as surfaces of metals [62], oxides [61], topological insulators [63], and oxide interfaces [58]. Experimentally, the ORE has been verified by circular dichroism in angle-resolved photoelectron spectroscopy [61–66] and momentum microscopy [67].

However, studies of transport effects in the presence of ORE are limited and further research in this direction is required. For the OEE, Salemi et al. investigated bulk collinear antiferromagnets Mn₂Au and CuMnAs [57], and Johansson et al. studied SrTiO₃ interfaces [58]. While these works consider only the atomic contribution, Niu et al. investigated the OEE in mixed topological semimetals using Berry phase theory which consistently incorporates intra-atomic and inter-atomic contributions, observing a drastic variation in OEE for different angles of the magnetization in response to changes in the topology of the bands [68]. For thin films, there can be an orbital current or diffusion of the OAM along the direction perpendicular to the surface/interface, which has been investigated for a surface-oxidized Cu [69].

Since the orbital current carries angular momentum as the spin current does, an injection of an orbital current into a FM can induce a torque on the magnetization. This type of torque induced by the orbital injection is named orbital torque (OT) [27]. Phenomenologically, the OT is similar to the spin-transfer torque (ST) generated by spin injection [70, 71], but there are clear differences in microscopic mechanisms. Because the OAM cannot interact with the local magnetic moment directly, the OT requires SOC in the FM. Thus, within the mechanism of the OT driven by the orbital current injected into the FM, e.g., via the OHE in the NM, spin-orbit entangled states exert a torque on the magnetization (fig. 6) [27]. In contrast, within the mechanism of the ST, the spin current alone is absorbed by the FM without the need for SOC.

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An important feature of the OT is that it is expected to depend sensitively on the chosen FM as the OT depends on the SOC in the FM [27, 28], which is not expected for the ST [70–71]. In a magnetic heterostructure consisting of a NM layer and a FM layer, the NM generates both OHE and SHE, which result in the OT and ST, respectively. As a result, the sum of the OT and ST effectively acts on the magnetization. The relative sign of the OT and ST depends not only on the spin-orbit correlation in the NM, which determines the relative sign between the OHE and SHE (fig. 2(a)), but also on that in the FM. The latter determines the "orbital-to-spin" conversion [27, 28]. For example, the spin-orbit correlation is expected to be positive for typical 3d FMs (Fe, Co, and Ni) but negative for Gd. Utilizing the OT instead of, or together with, the ST the may help to overcome material limitations as the OHE has a much higher efficiency than the SHE, and even small SOC of the FM may result in a sizeable OT. Thus, it is crucial to explore novel mechanisms of harnessing orbital currents, which have not been exploited so far, for the magnetization control in spintronics.

Experimental evidence

Remarkably, soon after the theoretical prediction, evidence of the OT has been presented for a few systems. One of them is a surface-oxidized Cu film. In 2016, An et al. discovered that by naturally oxidizing the Cu layer in Cu/permalloy (Py) structures, the torque efficiency is strongly enhanced [72]. This result was initially found to be quite surprising, as a material like Cu with negligible SOC is not expected to produce significant spin or its currents. After a few years a very prominent ORE was found theoretically in a surface-oxidized Cu film (fig. 7(a)), which was shown to generate not only a significant orbital accumulation but also a large orbital current flowing perpendicular to the interface [69]. Experimentally, the orbital current mechanism was confirmed by Ding et al. in CuO_x /Pt/TmIG, where the orbital current generated from CuO_x is efficiently converted into the spin current in an ultrathin $(\sim 1.5\ \mathrm{nm})$ Pt layer (fig. 7(b)) [29]. Meanwhile, Kim et al. independently discovered that AlO_x /Cu/FM structures show a very large torque efficiency after a partial oxidization by AlO_x  [30]. An interesting feature found in this experiment is that the orbital transparency depends significantly on the crystallinity of the Cu/FM interface, which was theoretically speculated to be very important for the OT mechanism [27]. We note that Tazaki et al. also found evidence of OT in CuO_x /FM structures, where the torque efficiency exhibits different sign depending on the FM choice [34].

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Evidence for OT was also found in NM/FM bilayers, where the orbital current is generated by the OHE in the NM bulk. Lee et al. investigated various Cr-based heterostructures, where the variation in the sign and magnitude of the torque efficiency can be explained by the OT or the competition between the OT and ST [31] (note that the OHE and SHE have an opposite sign in Cr [16]). Similar to ref. [29], it was found that the torque efficiency is significantly modified in Cr/Pt/CoFeB by the orbital-to-spin conversion in a Pt layer, when compared to Cr/CoFeB. Meanwhile, Lee et al. showed that the OT can be comparable to the ST even when the NM is a heavy element such as Ta, whose OHC is an order of magnitude larger than the SHC with opposite sign [32]. It has been shown that the torque efficiency changes sign in Ta/FM structures depending on the choice of the FM, which does not occur for Pt/FM structures since the OHC and SHC have same sign in Pt [32]. We note that the current-induced torque measured in Zr/CoFeB raised a possibility of the OHE from Zr because the SHE in Zr is far too small to explain the measured torque [73]. Meanwhile, a possible role of the ORE for current-induced torque in Pt/Co/SiO₂ was reported, where the inversion asymmetry leads to distortion of orbitals at the interface [74].

An additional class of effects that relies on the transfer of angular momentum and related to OT are magnetoresistance effects such as the spin Hall magnetoresistance (SMR) [75, 76], which results from the interaction of a spin current and the magnetization. Considering that an orbital current also interacts with the magnetization, a light metal or metal oxide where no significant SHE is expected can be used. This has been recently realized in a Py/CuO_x  [33]. Here it was shown that the resistance of a the multilayer depends on the magnetization direction of Py with an angular dependence reminiscent of SMR. Given the negligible spin current generated from CuO_x , this could be ascribed to the orbital current induced by the ORE, which then interacts with the magnetization in Py via SOC. This effect was coined as orbital Rashba-Edelstein magnetoresistance and it is expected to occur in a wide range of systems which can host orbital currents.

As the field of orbital transport is in its infancy, fundamental physical principles and novel transport effects await to be discovered. One of the most important tasks is to establish clear characterization tools in experiments. To date, no transport measurement scheme to detect the orbital currents, which does not involve spin currents, has been developed. While Xiao et al. proposed a magnetoresistance measurement with a FM lead for the detection of the OHE [77], it requires the SOC for the orbital-to-spin conversion, which is hard to disentangle from the signal by the SHE. One of the main problems here is that no "orbital polarizer", which plays a role of a FM for the spin current, is known.

Another important phenomenon that needs to be addressed is the relaxation and dephasing of the orbital current. So far, there is no consensus on how far the orbital current can travel in different systems and setups and which microscopic processes lead to orbital relaxation. A recent theory suggests that the orbital current can propagate over much longer distances than the spin current in FMs, whose dephasing mechanism is completely different from that of the spin current [15]. Interestingly, recent magnetoresistance measurements in CuO_x /Py found that the dephasing length extracted by assuming the spin current model is much longer than the known value in Py [33]. Considering that the orbital current is expected to be efficiently generated at the interface of CuO_x , the result suggests that the orbital current is absorbed more slowly than the spin current in FMs.

Considering the interaction of electrons with other quasi-particles, one of the exciting directions to pursue lies with effects due to interaction of orbital currents with phonons and magnons. We remark that phononic analog of the OHE has been theoretically proposed recently [78], and various mechanisms for inducing phonon OAM are being actively investigated [79–82]. In magnonics, it was suggested that the magnon Hall effect can be orbitally polarized via spin excitations which carry spin chirality, with the magnon current "dragging" the electronic OAM [83]. Also, a concept of magnonic OAM was proposed recently [84]. We remark that there exists an elementary excitation of orbital wave in an orbitally ordered state [85], and such "orbitons" were measured in LaMnO₃ [86]. We believe that studying orbital transport effects in transition metal oxides with orbital ordering may shed new light on the orbital physics and correlated phenomena [87, 88].

To summarize, the orbital current holds promises for new types of transport effects, which may possibly lead to new types of device applications that employ the orbital degree of freedom as an information carrier. The field is evolving rapidly in part due to an intensive interaction with neighboring research areas such as phononics, spintronics and valleytronics, where the orbital current provides new twists and solutions to existing problems with spin and valley currents. We look forward to further explorations in this exciting area of physics in the near future.

This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) − TRR173 − 268565370 (projects A01, B02 and A11), TRR288 − 422213477 (project B06). DJ was supported by the Global PhD Fellowship Program by National Research Foundation of Korea (Grant No. 2018H1A2A1060270). DJ and H-WL acknowledge the financial support from the Samsung Science and Technology Foundation (Grant No. BA-1501-51). MK acknowledges support by the Research Council of Norway through its Centers of Excellence funding scheme, project No. 262633 "QuSpin".