Dzyaloshinskii-Moriya interaction from unquenched orbital angular momentum

Before proceeding further with a detailed description of our approach, it is necessary to elucidate the motivation behind our treatment of OAM contribution to the DMI. Our starting point is the non-relativistic non-magnetic Hamiltonian of an orbitally complex system, represented by a ${t_{2g}}$ orbital model on a square lattice with broken inversion symmetry. Starting with this model, our next step is to break the time-reversal symmetry by allowing unquenched local orbital moments, which serve as the primary objects whose canting energetics we are to study. The conventional way to achieve this would be to allow for spin exchange interaction which, when combined with an effect of SOC, results in orbital magnetism. However, here, we choose a different path and explicitly include the effect of the orbital exchange field into consideration, which couples directly to the OAM, see Eq. (3) below, and gives rise to unquenched orbital magnetism without resorting to spin magnetism and SOC. In turn, one can perceive the effect of the orbital exchange field as a combined effect of the two latter phenomena, projected onto the orbital subspace only, which allows for a transparent way to perturb the directions of orbital moments by tracking the energetics of the orbital sub-system.

When thinking of a physical situation in which the effect of the orbital exchange field can be most easily singled out, we can imagine a weakly coupled bilayer type of system, where a nonmagnetic layer develops orbital magnetism via the SOC and the nonlocal exchange coupling with the second layer that is strongly magnetic, e.g. 3$d$ overlayer separated by a spacer from a 5$d$ substrate  [56, 57, 58]. In the latter case, considering that we can freely impose magnetic order in the 3$d$ metal, this will be translated into corresponding orbital order via the effect of SOC and effective orbital field, the energetics of which we can study separately assuming that it would be possible to neglect the spin contributions arising in the 5$d$ layer itself due to smallness of corresponding induced spin moments.

Secondly, we can consider a situation in which the shell of orbitals which drive spin magnetism is separated in energy from a nominally spin-degenerate but orbitally active shell of states residing around the Fermi level, such as the case e.g. for materials incorporating $4f$-ions [59]. Here, again, the SOC-imposed contribution to (e.g. $p$-$d$-$f$) hybridization between the two shells would results in the presence of an effective orbital field. In the latter two cases, for clear identification of the orbital DMI, it seems to be particularly promising to consider antiferromagnetic bilayers and $f$-based compounds, where the net effect of spin splitting on the orbital states can be vanishingly small, and they can be considered practically spin degenerate [59].

Finally, we can turn to a situation of a non-magnetic thin film exposed to a spatially varying external magnetic field which drives a direct interaction with orbital moments in the substrate of the Zeeman type $-$ i.e. orbital Zeeman effect $-$ as given by Eq. (3). The latter case has been considered recently in a situation of orbital pumping, where magnetic field dynamics drives a prominent orbital response [60, 61].

In the following, we provide the details of the ${t_{2g}}$ orbital model and the derivations of the expression of orbital DMI.

We employ a tight-binding model description of a simple two-dimensional square lattice with ${t_{2g}}$ orbitals (${d_{xy},d_{yz},d_{zx}}$) on each site. We operate in terms of Bloch waves $e^{i\mathbf{k}\cdot\mathbf{r}}\left|\varphi_{n\mathbf{k}}\right\rangle=\sum_{\mathbf{R}}e^{i\mathbf{k}\cdot\mathbf{R}}\left|\phi_{n\mathbf{R}}\right\rangle$ as basis, where $\phi_{n\mathbf{R}}$ is the localized state at the Bravais lattice $\mathbf{R}$ with the orbital character $n$, $n=d_{xy},d_{yz},d_{zx}$. The spinless tight-binding Hamiltonian in $\mathbf{k}$-space is written as

where $H_{\text{kin}}(\mathbf{k})$ is the kinetic part of the Hamiltonian arising from hoppings and on-site energies, and $H_{\mathrm{exc}}^{\mathrm{L}}$ describes orbital exchange interaction. $H_{\text{kin}}(\mathbf{k})$ is independent of the spin and its nonzero matrix elements are

Here, $E_{\|}$ and $E_{\perp}$ are on-site energies for the in-plane ($d_{xy}$) and out-of-plane ($d_{yz}$ and $d_{zx}$) orbitals, respectively, and $t_{\pi}$ and $t_{\delta}$ are the nearest neighbor hopping amplitudes between $t_{2g}$ orbitals via the $\pi$ and $\delta$ bondings, respectively. The inversion symmetry breaking at the surface is equivalent to the potential gradient along the surface-normal direction ($z$), which promotes hybridization between $d_{xy}$ and $d_{yz}$, as well as $d_{xy}$ and $d_{zx}$ states, with hopping integral $\gamma$ characterizing the strength of the symmetry breaking. The orbital exchange interaction is written as

where $\mathbf{L}$ is the atomic OAM operator, $\hat{\mathbf{M}}^{\mathrm{L}}$ denotes the direction of the orbital exchange field, and $J_{L}$ denotes the coupling strength of the orbital exchange interaction. For $t_{2g}$ orbitals, $\bm{L}=\left(L_{x},L_{y},L_{z}\right)$ becomes

in the matrix representation using the basis states $\varphi_{d_{xy}\mathbf{k}},\varphi_{d_{yz}\mathbf{k}}$, and $\varphi_{d_{zx}\mathbf{k}}$.

A spinfull model Hamiltonian is constructed by adding SOC $H_{\rm{soc}}$, and spin exchange interaction $H_{\mathrm{exc}}^{\mathrm{S}}$. The SOC is given by

where $\mathbf{S}$ is the spin angular momentum operator and $\lambda_{\text{soc}}$ is the SOC strength. The spin exchange interaction is written as

where $\hat{\mathbf{M}}^{\mathbf{S}}$ denotes the direction of the spin exchange field, and $J_{S}$ denotes the spin exchange interaction strength. The spin angular momentum operator is represented by the vector of the Pauli matrices $\bm{\sigma}=(\sigma_{x},\ \sigma_{y},\ \sigma_{z})$ within each orbital

where $\sigma,\sigma^{\prime}$ are spin indices (up or down).

The values of the model parameters used in the calculation are $E_{\|}=1.6,E_{\perp}=1.0,t_{\pi}=0.5,t_{\sigma}=0.1,\gamma=0.02,\lambda_{\text{soc }}=$ $0.04,J_{L}=1.0,J_{S}=1.0$, all in unit of eV. The lattice constant is set $a=1$ Å. All parameters are set as above unless specified otherwise.

We define the orbital DMI as the energy contribution to the free energy functional by the spiralization of the orbital exchange field $\hat{\mathbf{M}}^{\mathrm{L}}$,

where $\hat{\mathbf{r}}_{i}$ is the unit vector in the $i$ direction, and $\partial_{r_{j}}$ is the partial derivative in the $j$ direction ($i,j=x,y,z$). We derive a microscopic expression for the orbital DMI coefficient $D_{ij}^{\mathrm{L}}$ by means of the Berry phase theory, which has been applied in the past to calculate the spin DMI [55, 62, 63]. In this method, $D_{ij}^{\mathrm{L}}$ can be quantified by expanding the free energy in terms of small spatial gradients of orbital magnetization direction within quantum mechanical perturbation theory.

Given the orbital exchange interaction in the spinless Hamiltonian in Eq. (3), we define the torque operator on the orbital exchange field due to the OAM by

Following the detailed derivation in Ref. [62], the first-order perturbation by the spiralization of $\hat{\mathbf{M}}^{\mathrm{L}}$ starting from a collinear configuration of $\hat{\mathbf{M}}^{\mathrm{L}}$ gives rise the the following expression in terms of electronic states

where $f\left(\varepsilon_{n\mathbf{k}}\right)=1/[1+e^{\beta(\varepsilon_{n\mathbf{k}}-\mu)}]$ is the Fermi-Dirac distribution function for the unperturbed band energy $\varepsilon_{\mathrm{k}n}$, $\mu$ is the electrochemical potential, and $\beta=1/k_{\mathrm{B}}T$ for the Boltzmann constant $\mathbf{k}_{\mathrm{B}}$ and the temperature $T$. The quantities $A_{ij}^{n}(\mathbf{k})$ and $B_{ij}^{n}(\mathbf{k})$ are given by the correlation between the orbital torque operator and the velocity operator,

and

where $u_{n\mathbf{k}}$ denotes the lattice-periodic part of the unperturbed Bloch wave function of band $n$ at $\mathbf{k}$ and $v_{j}=\partial_{k_{j}}H_{\mathrm{kin}}/\hbar$ is the $j$-th component of the velocity operator.

At zero temperature Eq. (10) becomes

By utilizing $\bm{T}^{\mathrm{L}}=\hat{\mathbf{M}}^{\mathrm{L}}\times\partial_{\hat{\mathbf{M}}^{\mathrm{L}}}H_{\mathrm{tot}}$, Eq. (13) becomes

Note the similarity of this expression to the mixed Berry curvature

Up to date, the properties of orbital mixed Berry curvature in solid state systems have been not explored. According to Eq. (II.3), the orbital spiralization, in analogy to spin spiralization and modern theory of orbital magnetization [64, 65, 66], comprises two terms: one, “itinerant” in the language of orbital magnetization, proportional to the orbital mixed Berry curvature and associated with the $B_{ij}^{n}$-part in Eq. (13), and another one, which should be treated as the “self-rotation” contribution, driven by the $A_{ij}^{k}$-part in Eq. (13). While the second part of the orbital DMI is arising from the local breaking of inversion symmetry, the itinerant mixed Berry curvature part can be attributed to the breaking of inversion symmetry at the boundary of a finite sample taken to the thermodynamic limit.

In considering the spin, as mentioned above, the Hamiltonian for spinfull systems necessitates the inclusion of both spin-orbit coupling and spin exchange interaction. In this case, the spin torque operator can be represented as $\bm{\mathcal{T}}^{\mathrm{S}}=(J_{S}/\hbar)\hat{\mathbf{M}}^{\mathrm{S}}\times\mathbf{S}$. From this, we can obtain the spin spiralization $D_{ij}^{\mathrm{S}}$ through the above equations by replacing the orbital exchange field $\hat{\mathbf{M}}^{\mathrm{L}}$ by the spin exchange field $\hat{\mathbf{M}}^{\mathrm{S}}$.

First, let us unravel the emergence of an intrinsic orbital texture of $t_{2g}$ orbitals without considering the effect of SOC and orbital/spin exchange interaction. In Figs. 1(a)-(b) we present the band structure of the model along the high symmetry directions and near $\Gamma$, respectively. We can see clearly in Fig. 1(b) the effect of orbital hybridization mediated by inversion symmetry breaking, inducing avoided crossings of the bands with different orbital characters. In Figs. 1(c)-(e), we show the distribution of the expectation value of OAM in $\mathbf{k}$-space near the $\Gamma$ point for the bands $E_{1,\mathbf{k}}$, $E_{2,\mathbf{k}}$, and $E_{3,\mathbf{k}}$, which are labeled in the increasing order of energy. We observe that the OAM exhibits a chiral texture around the $\Gamma$ point, specifically, of clockwise, counter-clockwise, and zero character around the $\Gamma$ point, respectively. The chiral behavior of the $t_{2g}$ orbital textures is driven by the orbital Rashba effect of the form $\hat{z}\cdot(\mathbf{k}\times\mathbf{L})$, which is consistent with previous studies [28]. Investigating the influence of SOC reveals that the spin Rashba effect plays a crucial role in driving the spin texture’s chirality appearing on top of the orbital distribution. As a result, the spin textures are closely tied to the local orientation of the OAM in the Brillouin zone, manifesting as either parallel or antiparallel alignment with the underlying OAM (See supplementary section S1 [67]).

Next, we consider the effect of the orbital Zeeman effect $H_{\mathrm{exc}}^{\mathrm{L}}$ due to the orbital exchange field $\hat{\mathbf{M}}^{\mathrm{L}}$ (keeping the value of $J_{L}$ at 1.0 eV). In the two-dimensional square lattice with an out-of-plane inversion symmetry breaking, the orbital DMI spiralization is described by an antisymmetric tensor due to the four-fold rotational symmetry around the axis normal to the film, which amounts to $D_{xx}^{\mathrm{L}}=D_{yy}^{\mathrm{L}}=0$ and $D_{yx}^{\mathrm{L}}=-D_{xy}^{\mathrm{L}}$. We compute the magnitude of $D_{yx}^{\mathrm{L}}$ and present its magnitude in Fig. 2(a) as a function of band filling as given by the position of the Fermi level $E_{F}$, without SOC and with the orbital exchange field pointing out of the plane. The computed oscillatory behavior is strongly reminiscent of the Fermi energy dependence of the spin DMI typical for thin films of transition metals [63]. Remarkably, we observe that the magnitude of orbital DMI spiralization $D_{yx}^{\mathrm{L}}$ within our model reaches a large value of $16.8$ meV/Å  for $E_{F}=2.7$ eV, which is comparable to the theoretical magnitude of spin DMI in typical transition-metal systems, such as e.g. Co/Pt, Co/Ir and Co/Au thin films [62, 63]. This finding serves as compelling evidence that significant orbital DMI can manifest in a system lacking SOC but with inversion symmetry breaking when this system possesses a strong orbital exchange interaction.

More insight can be gained from a band-projected analysis of orbital DMI at $E_{F}=2.7$ eV, as shown in Fig. 2(b), where the color marks the value of projected orbital DMI onto each band. We observe that significant orbital DMI can be observed predominantly in the vicinity of the avoided band crossing between $\Gamma$ and $\mathrm{M}$ points in the Brillouin zone. As shown in Fig. 2(c), the $\mathbf{k}$-resolved orbital DMI displays a very spiky behavior in this region, which corresponds to the region of orbital hybridization associated with inversion symmetry breaking. At this Fermi energy, another significant contribution to the orbital DMI can be seen between M and X in the region of another band anti-crossing along that path. This underlines the crucial importance of the band-crossing points for achieving large values of the orbital DMI in realistic systems.

Given a close relation between the orbital DMI and orbital mixed Berry curvature, it is insightful to compare the two quantities directly. In Fig. 3 we plot the $\mathbf{k}$-resolved distribution of the orbital spiralization $D_{yx}^{\mathrm{L}}$ and of the mixed Berry curvature $\Omega_{M_{y}^{\mathrm{L}}k_{x}}$, which are summed over the occupied bands below $E_{F}=1.0$ eV and $E_{F}=2.7$ eV, respectively. At $E_{F}=1.0$ eV, both $\Omega_{M_{y}^{\mathrm{L}}k_{x}}$ and $D_{yx}^{\mathrm{L}}$ exhibit complex patterns with significant background contributions arising from broad areas of the Brillouin zone. Moreover, they show relatively small values throughout the entire Brillouin zone, consistent with the low values of the orbital DMI at this energy as visible in Fig. 1(a). In contrast, when $E_{F}=2.7$ eV, both $\Omega_{M_{y}^{\mathrm{L}}k_{x}}$ and $D_{yx}^{\mathrm{L}}$ are sharply peaked in narrow regions of the Brillouin zone, corresponding to quasi-nodal lines arising due to near degenerate bands crossing the Fermi level. Clearly, the nodal lines serve as prominent sources of both orbital mixed Berry curvature and orbital DMI, with the two quantities being directly correlated in sign and magnitude.

Next, we explore the impact of SOC on orbital DMI. Namely, we perform calculations to examine the energy dependence, band dispersion, and $\mathbf{k}$-space distribution of orbital DMI while considering the presence of SOC. Remarkably, we find that the qualitative nature of orbital DMI remains unaffected by SOC, see supplementary sections S2 and S3 [67]. Our findings align with previous studies on other orbital effects, such as the orbital Rashba effect and orbital Hall effect [4, 20], indicating that SOC does not play a decisive role in the emergence of orbital DMI, although it impacts its magnitude via the influence on the electronic structure details, see also discussion below.

We analyze in great detail to reveal the anatomy of the orbital DMI. At this point, by keeping the SOC strength at a constant value of 0.04 eV, we study the behavior of the orbital DMI in response to changing the key parameters of the system $-$ the strength of orbital exchange coupling $J_{L}$, the strength of inversion symmetry breaking $\gamma$, and the angle $\theta$ that the orbital exchange field makes with the $z$-axis $-$ presenting the results in Fig. 4. In Fig. 4(a), the orbital DMI $D_{yx}^{\mathrm{L}}$ is shown as a function of the Fermi energy $E_{F}$, for increasing orbital exchange coupling strength $J_{L}$. It is noteworthy that with increasing $J_{L}$, both the amplitude and position of the peak exhibit distinct deviations. Specifically, the height of the peak exhibits a positive correlation with $J_{L}$, whereas the peak position undergoes an upward shift with increasing $J_{L}$. This observation underscores the critical significance of orbital exchange coupling in determining the magnitude of the orbital DMI. When $J_{L}$ is sufficiently small, the orbital DMI assumes a significantly diminished value, aligning with the general scenario in practical materials where the orbital Zeeman effect is negligibly small and consequently conceals the manifestation of the orbital DMI.

In order to gain a deeper understanding of the trend of orbital DMI with respect to the orbital exchange coupling strength $J_{L}$, we conduct band-projected as well as $\mathbf{k}$-resolved calculations of orbital DMI. In Fig. 5 we compare two cases $-$ $J_{L}=0.3$ eV and $J_{L}$ $=0.7$ eV $-$ with the Fermi energy set to the peak position for each case. It is evident from Figs. 5(a) and (b) that as $J_{L}$ increases, the position of the band avoided crossing region moves up in energy in response to band dynamics. This is directly reflected in a shift of the peak in orbital DMI as seen in Fig. 4(a). Furthermore, by comparing Figs. 5(c) and (d), we observe an increase in the absolute value of $D_{yx}^{\mathrm{L}}(k)$ in the $\mathbf{k}$-region associated with orbital hybridization, resulting in an enhanced peak height in Fig. 4(a) with increasing $J_{L}$. The analysis for other values of $J_{L}$ in presented in the supplementary section S4 [67].

In Fig. 4(b) we show the variation of the orbital DMI $D_{yx}^{\mathrm{L}}$ with the inversion symmetry breaking strength as given by parameter $\gamma$. As mentioned above, the strength of inversion symmetry breaking directly influences the “intensity” of orbital hybridization, ultimately resulting in the formation of chiral orbital textures. We observe that the orbital DMI monotonically increases with the gradual increase of $\gamma$, and zero $\gamma$ results in vanishing orbital DMI. This highlights the decisive role of inversion symmetry breaking as a dominant factor for orbital DMI. The presence of ISB, along with the induced orbital hybridization and chiral orbital textures, governs the existence of orbital DMI, consistent with previous studies on the orbital Hall effect [4].

The analysis of band-projected and $\mathbf{k}$-resolved orbital DMI for $\gamma=0.005$ eV and $\gamma=0.015$ eV (results for other values of $\gamma$ can be found in the supplementary section S5 [67]), shown in Figs. 6(a) and 6(b), reveals that the position of the orbital hybridization points, which contribute predominantly to the orbital DMI, remains unaffected as $\gamma$ undergoes variation within the considered range of values. Consequently, there is no discernible shift of the peak position in the orbital DMI curve, as evident from Fig. 4(b). As shown in Figs. 6(c) and 6(d), it is also directly evident that the increase in the degree of inversion symmetry breaking results in an amplification of orbital hybridization at the respective hybridization points, as reflected in the degree of orbital mixing [68]. This leads to an enhancement of the local orbital DMI, and an increase in the peak height of the orbital DMI curve in Fig. 4(b).

At last, we present in Figs. 4(c) and 4(d) the dependence of $D_{yx}^{\mathrm{L}}$ on the angle of the orbital exchange field $\theta$. Specifically, the range of $\theta$ from $0^{\circ}$ to $45^{\circ}$ is depicted in Fig. 4(c), while the range from $45^{\circ}$ to $90^{\circ}$ is shown in Fig. 4(d). We can clearly observe that for $\theta$ less than $45^{\circ},D_{yx}^{\mathrm{L}}$ gradually decreases with increasing $\theta$ at around $E_{F}=2.7\mathrm{eV}$, while in other energy regions the DMI values stay stable. On the other hand, for $\theta$ values above $45^{\circ}$, the situation is reversed, and it is the DMI between 0 and 1 eV which is increasing rapidly with increasing $\theta$ on the background of relative stability at other energies. Such behavior is a manifestation of strong anisotropy of orbital DMI with respect to the direction of the orbital exchange field. To understand this effect better, we plot the band-projected and $\mathbf{k}$-resolved orbital DMI for two orientations of the orbital exchange field: the out-of-plane direction ($\theta=0^{\circ}$), and the in-plane direction ($\theta=90^{\circ}$). As shown in Fig. 7, drastic changes in the band structure are observed as the orbital exchange field direction varies. As a result, the position of the band avoided crossing region that mainly contributes to the orbital DMI experiences alteration. Simultaneously, the degree of orbital hybridization also exhibits pronounced variation, manifested as changes in the magnitude of orbital DMI at the hybridization positions. These factors combined lead to modifications in both the peak position and peak height of the orbital DMI curve as the orbital exchange field direction transitions from the out-of-plane to the in-plane configuration, showcasing a strong anisotropy, as depicted in Figs. 4 (c) and 4(d).

Finally, we consider the case when in addition to the orbital exchange and SOC the spin exchange interaction as given by $H_{\mathrm{exc}}^{\mathrm{S}}$ is also present in the Hamiltonian. We keep the values of $J_{L}$ and $J_{S}$ at 1 eV, while keeping the out-of-plane directions of the spin and orbital exchange fields, and compute both spin and orbital DMI by following the Berry phase theory outlined above. We focus specifically on the dependence of the spin and orbital DMI on the SOC strength, presenting the results in Fig. 8. First, we consider the regime of small $\lambda_{\rm soc}$, Figs. 8(a-b). We observe that in this regime the orbital DMI exhibits minimal variations, whereas the spin DMI consistently increases with increasing the SOC strength, vanishing identically without spin-orbit interaction. More importantly, we observe that for the energies above +1.5 eV the qualitative behavior of the spin and orbital DMI is quite similar in that both quantities develop large peaks at about 2.2 and 3.3 eV, albeit of opposite sign in the case of spin.

To understand the origin of this correlation, we scrutinize the band-resolved contributions to the spin and orbital DMI at the value of $\lambda_{\rm soc}=0.04$ eV and $E_{F}=2.2$ eV, shown in Figs. 8(e-f). We observe that in the discussed region of energy both falvors of DMI come from the same anticrossings in the electronic structure along $\Gamma$M. These anticrossings are nothing else but the ISB-driven hybridization points between orbitally-different bands $-$ discussed in depth above and shifted in energy by spin exchange splitting $-$ which induce orbital DMI without SOC. We thus come to a fundamental conclusion that the orbital DMI is the effect which is parent to spin DMI. As in the case of a relation between the orbital Hall effect and spin Hall effect [4], the spin DMI is “pulled” by the orbital DMI via SOC. In the case when DMI-driving hybridizations are well-defined in energy and $\mathbf{k}$-space, such as e.g. for energies above $+1.5$ eV, the behavior of spin and orbital DMI can be very similar but not necessarily identical: for example, in the case considered here, the switch in sign correlation between $D_{yx}^{\mathrm{L}}$ and $D_{yx}^{\mathrm{S}}$ at $+2.2$ and $+3.3$ eV can be explained by the opposite sign of the spin-orbit correlation $\langle LS\rangle$ of participating bands [4] at these energies, similarly to the case of Hall effects.

As a result, when at a given energy the DMI contributions are small and spread over several bands with different orbital character filling larger areas of $\mathbf{k}$-space, as it is the case below $+1$ eV, Figs. 8(a-b), the correlation between $D_{yx}^{\mathrm{L}}$ and $D_{yx}^{\mathrm{S}}$ is much less pronounced, or not at all present. This is ultimately the reason why it is difficult to observe a direct relation between $D_{yx}^{\mathrm{L}}$ and $D_{yx}^{\mathrm{S}}$ for the case of larger SOC strength, shown in Figs. 8(c-d). In the latter case, the strongly modified by SOC bands develop larger areas in $\mathbf{k}$-space where the effect of SOC on the DMI is active. This is also consistent with the saturation in the values of the spin DMI with increasing SOC towards the values comparable to the strength of exchange interaction. It is especially worth mentioning that when the strength of orbital exchange interaction $J_{L}$ is similar to the magnitude of spin exchange $J_{S}$, the values of the orbital DMI on average exceed those of the spin DMI by an order of magnitude. This underlines the fact that the orbital angular momentum in solids is much more prone to the effects of chiralization when compared to spin, due to its stronger coupling to the lattice and qualitatively different energetics of orbital dynamics relying on crystal field structure rather than spin-orbit interaction.