Berry phase theory of Dzyaloshinskii–Moriya interaction and spin–orbit torques

Abstract

Recent experiments on current-induced domain wall motion in chiral magnets suggest important contributions both from spin-orbit torques (SOTs) and from the Dzyaloshinskii-Moriya interaction (DMI). We derive a Berry phase expression for the DMI and show that within this Berry phase theory DMI and SOTs are intimately related, in a way formally analogous to the relation between orbital magnetization (OM) and anomalous Hall effect (AHE). We introduce the concept of the twist torque moment, which probes the internal twist of wave packets in chiral magnets in a similar way like the orbital moment probes the wave packet's internal self rotation. We propose to interpret the Berry phase theory of DMI as a theory of spiralization in analogy to the modern theory of OM. We show that the twist torque moment and the spiralization together give rise to a Berry phase governing the response of the SOT to thermal gradients, in analogy to the intrinsic anomalous Nernst effect. The Berry phase theory of DMI is computationally very efficient because it only needs the electronic structure of the collinear magnetic system as input. As an application of the formalism we compute the DMI in Pt/Co, Pt/Co/O and Pt/Co/Al magnetic trilayers and show that the DMI is highly anisotropic in these systems. Broken inversion symmetry in chiral magnets, such as B20 compounds, (Ga,Mn)As and asymmetric bior trilayers opens new perspectives for current-induced magnetization control via so-called spin-orbit torques (SOTs) [1][2][3][4][5]. Notably, magnetization switching by SOTs in single collinear ferromagnetic layers has been demonstrated experimentally [6,7]. In addition to SOTs also the Dzyaloshinskii-Moriya interaction (DMI) arises from the interplay of broken inversion symmetry and spinorbit interaction (SOI) in magnetic systems [8,9]. Recent experiments and simulations suggest that both SOTs and DMI substantially influence current-induced domain-wall motion in chiral magnets [10][11][12] and that their combination may lead to a very efficient coupling of domain-wall motion to the applied current. Additionally, relations between SOTs and DMI have been proposed theoretically based on model calculations [13].Expanding the micromagnetic free energy density F (r) at position r in terms of gradients ∂n/∂r j of magnetization directionn(r), we obtain in first order of the gradientswhere the Dzyaloshiskii vectors D j (n) will generally depend on magnetization directionn(r). Within ab initio density-functional theory (DFT) methods, DMI is often computed by adding SOI perturbatively to spirals with finite wave vectors q and extracting D j from the q-linear term in the dispersion E(q) [14][15][16]. Alternative methods for the calculation of DMI are based on multiplescattering theory [17,18] or a tight-binding representation of the electronic structure [19].In the present work we develop a Berry phase theory of DMI. Our approach is based on expanding the free energy in terms of small gradients of magnetization direction within quantum mechanical perturbation ...

Path to collapse for an isolated Néel skyrmion

“…For a thin film, d ij = d(û ij ×ẑ) 2,29 , withû ij the unit vector between sites i and j and z the normal to the plane. The third term is the uniaxial anisotropy, with constant k, the fourth term is the dipolar coupling and the last term is the Zeeman energy in field H. The parameters are: µ at = S i = 2.1 µ B /atom [30][31][32][33] , J = 29 meV/bond 30 , d = 1.5 meV/bond [25][26][27] , k = 0.4 meV/atom 31,33 (including the shape anisotropy 34 , the effective anisotropy is 0.276 meV/atom in good agreement with literature 31,33 ). The sample is limited by free boundary conditions and the size has been chosen so that an isolated skyrmion is not affected by the edges (no morphology nor energy changes seen for larger calculation box sizes).…”

Higher-dimensional Wannier Interpolation for the Modern Theory of the Dzyaloshinskii–Moriya Interaction: Application to Co-based Trilayers

“…Following the detailed derivation in Ref. 23, we find that the spiralization at finite temperatures T amounts to…”

“…23,24,27) The antidamping spin-orbit torques are ascribed to the non-trivial geometry of the mixed phase space of k andm in terms of the so-called mixed Berry curvature Ωm k i j of all occupied states: 23)…”

Higher-dimensional Wannier functions of multiparameter Hamiltonians

“…Recently, the mixed Berry curvature in (k,m)-space has been found to be important for spin-orbit torques, for the Dzyaloshinskii-Moriya interaction and for the charge of skyrmions [9,10,12,14]. This mixed Berry curvature is given by…”