Handbook of Materials Modeling pp 1-35 | Cite as

# First-Principles Quantum Transport Modeling of Spin-Transfer and Spin-Orbit Torques in Magnetic Multilayers

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## Abstract

A unified approach for computing (*i*) spin-transfer torque in magnetic trilayers like spin valve and magnetic tunnel junction, where injected charge current flows perpendicularly to interfaces, and (*ii*) spin-orbit torque in magnetic bilayers of the type ferromagnet/spin-orbit-coupled material, where injected charge current flows parallel to the interface, was reviewed. The experimentally explored and technologically relevant spin-orbit-coupled materials include 5*d* heavy metals, topological insulators, Weyl semimetals, and transition metal dichalcogenides. This approach requires to construct the torque operator for a given Hamiltonian of the device and the steady-state nonequilibrium density matrix, where the latter is expressed in terms of the nonequilibrium Green’s functions and split into three contributions. Tracing these contributions with the torque operator automatically yields field-like and damping-like components of spin-transfer torque or spin-orbit torque vector, which is particularly advantageous for spin-orbit torque where the direction of these components depends on the unknown-in-advance orientation of the current-driven nonequilibrium spin density in the presence of spin-orbit coupling. Illustrative examples are provided by computing spin-transfer torque in a one-dimensional toy model of a magnetic tunnel junction and realistic Co/Cu/Co spin valve, both of which are described by first-principles Hamiltonians obtained from noncollinear density functional theory calculations, as well as by computing spin-orbit torque in a ferromagnetic layer described by a tight-binding Hamiltonian which includes spin-orbit proximity effect within ferromagnetic monolayers assumed to be generated by the adjacent monolayer transition metal dichalcogenide. In addition, it is shown here how spin-orbit proximity effect, quantified by computing (via first-principles retarded Green’s function) spectral functions and spin textures on monolayers of realistic ferromagnetic material like Co in contact with heavy metal or monolayer transition metal dichalcogenide, can be tailored to enhance the magnitude of spin-orbit torque. Errors made in the calculation of spin-transfer torque are quantified when using Hamiltonian from collinear density functional theory, with rigidly rotated magnetic moments to create noncollinear magnetization configurations, instead of proper (but computationally more expensive) self-consistent Hamiltonian obtained from noncollinear density functional theory.

## Keywords

Magnetic Tunnel Junctions (MTJs) Spin Texture Spin Valve Spin-transfer Torque (STT) Weyl Semimetals (WSM)## 1 What Is Spin Torque and Why Is It Useful?

^{7}A∕ cm

^{2}) injected into a ferromagnetic metal (FM) (For easy navigation, provided is a list of abbreviations used throughout the chapter: 1D, one-dimensional; 2D, two-dimensional; 3D, three-dimensional; BZ, Brillouin zone; CD, current-driven; DFT, density functional theory; DL, damping-like; FL, field-like; FM, ferromagnetic metal; GF, Green’s function; HM, heavy metal; I, insulator; KS, Kohn-Sham; LCAO, linear combination of atomic orbitals; LLG, Landau-Lifshitz-Gilbert; ML, monolayer; MLWF, maximally localized Wannier function; MRAM, magnetic random access memory; MTJ, magnetic tunnel junction; ncDFT, noncollinear DFT; NEGF, nonequilibrium Green’s function; NM, normal metal; PBE, Perdew-Burke-Ernzerhof; scf, self-consistent field; SHE, spin Hall effect; SOC, spin-orbit coupling; SOT, spin-orbit torque; STT, spin-transfer torque; TBH, tight-binding Hamiltonian; TI, topological insulator; TMD, transition metal dichalcogenide; WSM, Weyl semimetal; XC, exchange-correlation.) either switches its magnetization from one static configuration to another or generates a dynamical situation with steady-state precessing magnetization (Locatelli et al. 2014). The origin of STT is the absorption of the itinerant flow of spin angular momentum component normal to the magnetization direction. Figure 1a illustrates a setup with two noncollinear magnetizations which generates STT. This setup can be realized as FM/NM/FM (NM, normal metal) spin valve, exemplified by Co/Cu/Co trilayer in Fig. 2a and employed in early experiments (Tsoi et al. 1998; Myers et al. 1999; Katine et al. 2000), or FM/I/FM (I, insulator) magnetic tunnel junctions (MTJs), exemplified by Fe/MgO/Fe trilayer and employed in later experiments (Sankey et al. 2008; Kubota et al. 2008; Wang et al. 2011) and recent applications (Locatelli et al. 2014; Kent and Worledge 2015). In such magnetic multilayers, injected unpolarized charge current passes through the first thin FM layer to become spin-polarized in the direction of its fixed magnetization along the unit vector

**M**

_{fixed}, and it is directed into the second thin FM layer with magnetization along the unit vector

**M**

_{free}where transverse (to

**M**

_{free}) component of flowing spins is absorbed. The STT-induced magnetization dynamics is converted into resistance variations via the magnetoresistive effect, as illustrated in Fig. 1b, which is much larger in MTJ than in spin valves. The rich nonequilibrium physics arising in the interplay of spin currents carried by fast conduction electrons, described quantum mechanically, and slow collective magnetization dynamics, described by the Landau-Lifshitz-Gilbert (LLG) equation which models magnetization as a classical vector subject to thermal fluctuations (Berkov and Miltat 2008; Evans et al. 2014; Petrović et al. 2018), is also of great fundamental interest. Note that at cryogenic temperatures, where thermal fluctuations are suppressed, quantum-mechanical effects in STT-driven magnetization dynamics can also be observed (Zholud et al. 2017; Mahfouzi and Kioussis 2017; Mahfouzi et al. 2017).

Another setup exhibiting current-induced magnetization dynamics is illustrated in Fig. 2b, c. It utilizes a single FM layer, so that the role of polarizing FM layer with **M**_{fixed} in Figs. 1a and 2a is taken over by strong spin-orbit coupling (SOC) introduced by heavy metals (HMs) (Miron et al. 2011; Liu et al. 2012) (such as 5*d* metals Pt, W, and Ta) as in Fig. 2b, topological insulators (TIs) (Mellnik et al. 2014; Fan et al. 2014; Han et al. 2017; Wang et al. 2017), Weyl semimetals (WSMs) (MacNeill et al. 2017a,b), and atomically thin transition metal dichalcogenides (TMDs) (Sklenar et al. 2016; Shao et al. 2016; Guimarães et al. 2018; Lv et al. 2018). The TMDs are compounds of the type MX_{2} (M = Mo, W, Nb; X = S, Se, Te) where one layer of M atoms is sandwiched between two layers of X atoms, as illustrated by monolayer MoS_{2} in Fig. 2c. The SOC is capable of converting charge into spin currents (Vignale 2010; Sinova et al. 2015; Soumyanarayanan et al. 2016), so that their absorption by the FM layer in Fig. 2b, c leads to the so-called spin-orbit torque (SOT) (Manchon et al. 2018) on its free magnetization **M**_{free}.

The current-driven (CD) STT and SOT vectors are analyzed by decomposing them into two contributions, **T**_{CD} = **T**_{DL} + **T**_{FL}, commonly termed (Ralph and Stiles 2008; Manchon et al. 2018) damping-like (DL) and field-like (FL) torque based on how they enter into the LLG equation describing the classical dynamics of magnetization. As illustrated in Fig. 1c, these two torque components provide two different handles to manipulate the dynamics of **M**_{free}. In the absence of current, displacing **M**_{free} out of its equilibrium position leads to the effective-field torque **T**_{field} which drives **M**_{free} into precession around the effective magnetic field, while Gilbert damping **T**_{damping} acts to bring it back to its equilibrium position. Under nonequilibrium conditions, brought by injecting steady-state or pulse current (Baumgartner et al. 2017), **T**_{DL} acts opposite to **T**_{damping} for “fixed-to-free” current direction, and it enhances **T**_{damping} for “free-to-fixed” current directions in Fig. 1a. Thus, the former (latter) acts as antidamping (overdamping) torque trying to bring **M**_{free} antiparallel (parallel) to **M**_{fixed} (note that at cryogenic temperatures, one finds apparently only antidamping action of **T**_{DL} for both current directions (Zholud et al. 2017)). The **T**_{FL} component induces magnetization precession and modifies the energy landscape seen by **M**_{free}. Although |**T**_{FL}| is minuscule in metallic spin valves (Wang et al. 2008), it can reach 30–40% of |**T**_{DL}| in MTJs (Sankey et al. 2008; Kubota et al. 2008), and it can become several times larger than |**T**_{DL}| in FM/HM bilayers (Kim et al. 2013; Yoon et al. 2017). Thus, **T**_{FL} component of SOT can play a crucial role (Baumgartner et al. 2017; Yoon et al. 2017) in triggering the reversal process of **M**_{free} and in enhancing the switching efficiency. In concerted action with **T**_{DL} and possible other effects brought by interfacial SOC, such as the Dzyaloshinskii-Moriya interaction (Perez et al. 2014), this can also lead to complex inhomogeneous magnetization switching patterns observed in SOT-operated devices (Baumgartner et al. 2017; Yoon et al. 2017; Perez et al. 2014).

By adjusting the ratio |**T**_{DL}|∕|**T**_{FL}| (Timopheev et al. 2015) via tailoring of material properties and device shape, as well as by tuning the amplitude and duration of the injected pulse current (Baumgartner et al. 2017), both STT- and SOT-operated devices can implement variety of functionalities, such as nonvolatile magnetic random access memories (MRAMs) of almost unlimited endurance, microwave oscillators, microwave detectors, spin-wave emitters, memristors, and artificial neural networks (Locatelli et al. 2014; Kent and Worledge 2015; Borders et al. 2017). The key goal in all such applications is to actively manipulate magnetization dynamics, without the need for external magnetic fields that are incompatible with downscaling of the device size, while using the smallest possible current [e.g., writing currents \({\lesssim } 20\) μA would enable multigigabit MRAM (Kent and Worledge 2015)] and energy consumption. For example, recent experiments (Wang et al. 2017) have demonstrated current-driven magnetization switching at room temperature in FM/TI bilayers using current density ∼10^{5} A/cm^{2}, which is two orders of magnitude smaller than for STT-induced magnetization switching in MTJs or one to two orders of magnitude smaller than for SOT-induced magnetization switching in FM/HM bilayers. The SOT-MRAM is expected to be less affected by damping, which offers flexibility for choosing the FM layer, while it eliminates insulating barrier in the writing process and its possible dielectric breakdown in STT-MRAM based on MTJs (Kent and Worledge 2015). Also, symmetric switching profile of SOT-MRAM evades the asymmetric switching issues in STT-MRAM otherwise requiring additional device/circuit engineering. On the other hand, SOT-MRAM has a disadvantage of being a three-terminal device.

## 2 How to Model Spin Torque Using Nonequilibrium Density Matrix Combined with Density Functional Theory Calculations

The absorption of the component of flowing spin angular momentum that is transverse to **M**_{free}, as illustrated in Fig. 1a, occurs (Stiles and Zangwill 2002; Wang et al. 2008) within a few ferromagnetic monolayers (MLs) near NM/FM or I/FM interface. Since the thickness of this interfacial region is typically shorter (Wang et al. 2008) than any charge or spin dephasing length that would make electronic transport semiclassical, STT *requires* quantum transport modeling (Brataas et al. 2006). The essence of STT can be understood using simple one-dimensional (1D) models solved by matching spin-dependent wave function across the junction, akin to elementary quantum mechanics problems of transmission and reflection through a barrier, as provided in Ralph and Stiles (2008), Manchon et al. (2008) and Xiao et al. (2008). However, to describe details of experiments, such as bias voltage dependence of STT in MTJs (Kubota et al. 2008; Sankey et al. 2008) or complex angular dependence of SOT in FM/HM bilayers (Garello et al. 2013), more involved calculations are needed employing tight-binding or first-principles Hamiltonian as an input. For example, simplistic tight-binding Hamiltonians (TBHs) with single orbital per site have been coupled (Theodonis et al. 2006) to nonequilibrium Green’s function (NEGF) formalism (Stefanucci and van Leeuwen 2013) to compute SOT in FM/HM bilayers (Kalitsov et al. 2017) or bias voltage dependence of DL and FL components of STT in MTJs which can describe some features of the experiments by adjusting the tight-binding parameters (Kubota et al. 2008).

_{2}transforms its spectral function from Fig. 3a to the ones in Fig. 3c, d, respectively, while inducing the corresponding spin textures in Fig. 3g, h due to spin-orbit proximity effect signified by the “leakage” of SOC from HM or TMD layer into the FM layer.

Note that the spectral function and spin texture at the Co/Pt interface are quite different from those of the ferromagnetic Rashba Hamiltonian in 2D often employed (Manchon and Zhang 2008; Haney et al. 2013; Lee et al. 2015; Li et al. 2015; Pesin and MacDonald 2012a; Ado et al. 2017; Kalitsov et al. 2017) in the calculations of SOT as the putative simplistic description of the FM/HM interface. When charge current flows within FM monolayer hosting spin textures – such as the ones displayed in Fig. 3e, g, h – more forward-going electron states will be occupied and less the backward-going ones which due to spin-momentum locking leads to nonequilibrium spin density (Edelstein 1990; Aronov and Lyanda-Geller 1989) as one of the principal mechanisms behind SOT (Manchon et al. 2018). The direction of the nonequilibrium spin density is easily identified in the case of simple spin textures, such as the one in Fig. 3e or those associated with simplistic models (Pesin and MacDonald 2012b) like the Rashba Hamiltonian or the Dirac Hamiltonian discussed in Sect. 4. Conversely, for complex spin textures within heterostructures of realistic materials, as exemplified by those in Fig. 3g, h, one needs first principles coupled with electronic transport calculations (Chang et al. 2015; Johansson et al. 2018).

Thus, capturing properties of realistic junctions illustrated by Fig. 3 requires first-principles Hamiltonian as offered by the density functional theory (DFT). In the linear-response regime, appropriate for spin valves or SOT-operated bilayers in Fig. 2, one can also employ first-principles-derived TBH as offered by transforming the DFT Hamiltonian to a basis of orthogonal maximally localized Wannier functions (MLWFs) in a selected energy window around the Fermi energy *E*_{F}. This procedure retains faithfully the overlap matrix elements and their phases, orbital character of the bands, and the accuracy of the original DFT calculations (Marzari et al. 2012). Although Wannier TBH has been used to describe infinite-FM-on-infinite-HM bilayers (Freimuth et al. 2014; Mahfouzi and Kioussis (2018)), its accuracy can be compromised by complicated band entanglement in hybridized metallic systems (Marzari et al. 2012). It is also cumbersome to construct Wannier TBH for junctions in other geometries, like the spin valve in Fig. 2a, or when FM/HM bilayer is attached to leads made of different NM material. In such cases, one needs to perform multiple calculations (Shelley et al. 2011; Thygesen and Jacobsen 2005) (such as on periodic leads, supercell composed of the central region of interest attached to buffer layers of the lead material on both sides, etc.) where one can encounter different MLWFs for two similar but nonidentical systems (Thygesen and Jacobsen 2005), nonorthogonal MLWFs belonging to two different regions (Thygesen and Jacobsen 2005), and Fermi energies of distinct calculations that have to be aligned (Shelley et al. 2011). Also, to compute the current or STT in MTJs at finite bias voltage, one needs to recalculate Hamiltonian in order to take into account self-consistent charge redistribution and the corresponding electrostatic potential in the presence of current flow. Otherwise, without computing them across the device, the current-voltage characteristics violates (Christen and Büttiker 1996; Hernández and Lewenkopf 2013) gauge invariance, i.e., invariance with respect to the global shift of electric potential by a constant, *V* → *V* + *V*_{0}.

*V*

_{H}(

**r**),

*V*

_{ext}(

**r**), and

*V*

_{XC}(

**r**) =

*E*

_{XC}[

*n*(

**r**),

**m**(

**r**)]∕

*δn*(

**r**) are the Hartree, external, and exchange-correlation (XC) potentials, respectively, and \(\boldsymbol {\sigma }=(\hat {\sigma }_x,\hat {\sigma }_y,\hat {\sigma }_z)\) is the vector of the Pauli matrices. The extension of DFT to the case of spin-polarized systems is formally derived in terms of total electron density

*n*(

**r**) and vector magnetization density

**m**(

**r**). In the collinear DFT,

**m**(

**r**) points in the same direction at all points in space, which is insufficient to study magnetic systems where the direction of the local magnetization is not constrained to a particular axis or systems with SOC. In ncDFT (Capelle et al. 2001), XC functional

*E*

_{XC}[

*n*(

**r**),

**m**(

**r**)] depends on

**m**(

**r**) pointing in arbitrary direction. The XC magnetic field is then given by

**B**

_{XC}(

**r**) =

*δE*

_{XC}[

*n*(

**r**),

**m**(

**r**)]∕

*δ*

**m**(

**r**).

*describing quantum many-particle system at finite temperature in equilibrium, or in the presence of external static or time-dependent fields which drive the system out of equilibrium. The knowledge of*

**ρ***makes it possible to compute the expectation value of any observable*

**ρ****O**into Eq. (2), where a notation in which bold letters denote matrix representation of an operator in a chosen basis is used. For the KS Hamiltonian in ncDFT in Eq. (1), the torque operator is given by the time derivative of the electronic spin operator (Haney et al. 2007; Carva and Turek 2009)

Its trace with * ρ* yields the spin torque vector while concurrently offering a microscopic picture (Haney et al. 2007) for the origin of torque – misalignment of the nonequilibrium spin density of current carrying quasiparticles with respect to the spins of electrons comprising the magnetic condensate responsible for nonzero

**B**

_{XC}. This causes local torque on individual atoms, which is summed by performing trace in Eq. (2) to find the net effect on the total magnetization

**M**

_{free}of the free FM layer. Examples of how to evaluate such trace, while using

**O**↦

**T**in Eq. (2) in different matrix representations, are given as Eqs. (26) and (27) in Sect. 3.

In equilibrium, **ρ**_{eq} is fixed by the Boltzmann-Gibbs prescription, such as **ρ**_{eq} =∑_{n}*f*(*E*)| Ψ_{n}〉〈 Ψ_{n}| in grand canonical ensemble describing electrons with the Fermi distribution function *f*(*E*) due to contact with a macroscopic reservoir at chemical potential *μ* and temperature *T*, where *E*_{n} and | Ψ_{n}〉 are eigenenergies and eigenstates of the Hamiltonian, respectively. Out of equilibrium, the construction of **ρ**_{neq} is complicated by the variety of possible driving fields and open nature of a driven quantum system. For example, the Kubo linear-response theory has been used to obtain **ρ**_{neq} for small applied electric field in infinite-FM-on-infinite-HM bilayer geometry (Freimuth et al. 2014; Mahfouzi and Kioussis (2018)). However, for arbitrary junction geometry and magnitude of the applied bias voltage *V*_{b} or injected pulse current, the most advantageous is to employ the NEGF formalism (Stefanucci and van Leeuwen 2013). This requires to evaluate its two fundamental objects – the retarded GF, \(G^{\sigma \sigma '}_{\mathbf {nn}'}(t,t')=-i \Theta (t-t') \langle \{\hat {c}_{\mathbf {n}\sigma }(t) , \hat {c}^\dagger _{\mathbf {n}'\sigma '}(t')\}\rangle \), and the lesser GF, \(G^{<,\sigma \sigma '}_{\mathbf {nn}'}(t,t')=i \langle \hat {c}^\dagger _{\mathbf {n}'\sigma '}(t') \hat {c}_{\mathbf {n} \sigma }(t)\rangle \) – describing the density of available quantum states and how electrons occupy those states, respectively. The operator \(\hat {c}_{\mathbf {n}\sigma }^{\dagger }\) (\(\hat {c}_{\mathbf {n}\sigma }\)) creates (annihilates) electron with spin *σ* at site **n** (another index would be required to label more than one orbital present at the site), and 〈…〉 denotes the nonequilibrium statistical average (Stefanucci and van Leeuwen 2013).

**G**and

**G**

^{<}depend only on the time difference

*t*−

*t*

^{′}and can, therefore, be Fourier transformed to depend on energy

*E*instead of

*t*−

*t*

^{′}. The retarded GF in stationary situations is then given by

*ϕ*

_{n}〉, one should make a replacement

*E*↦

*E*

**D**where

**D**is the overlap matrix composed of elements 〈

*ϕ*

_{n}|

*ϕ*

_{m}〉. The self-energies (Velev and Butler 2004; Rungger and Sanvito 2008)

**Σ**

_{L,R}(

*E*,

*V*

_{b}) describe the semi-infinite leads which guarantee continuous energy spectrum of devices in Fig. 2 required to reach the steady-state transport regime. The leads terminate at infinity into the left (L) and right (R) macroscopic reservoirs with different electrochemical potentials,

*μ*

_{L}−

*μ*

_{R}=

*eV*

_{L}−

*eV*

_{R}=

*eV*

_{b}. The usual assumption about the leads is that the applied bias voltage

*V*

_{b}induces a rigid shift in their electronic structure(Brandbyge et al. 2002), so that

**Σ**

_{L,R}(

*E*,

*V*

_{b}) =

**Σ**

_{L,R}(

*E*−

*eV*

_{L,R}).

*eV*

_{b}≪

*E*

_{F}), one needs

**G**

_{0}(

*E*) obtained from Eq. (5) by setting

*V*

_{L}=

*V*

_{R}= 0. The spectral functions shown in Fig. 3a–d can be computed at an arbitrary plane at position

*x*within the junction in Fig. 2a using

**G**

_{0}(

*E*)

*G*

_{0}(

*E*;

**k**

_{∥};

*x*,

*x*) are obtained by transforming the retarded GF from a local orbital to a real-space representation. The spin textures in Fig. 3e–h within the constant energy contours are computed from the spin-resolved spectral function. The equilibrium density matrix can also be expressed in terms of

**G**

_{0}(

*E*)

**O**= (

**O**−

**O**

^{†})∕2

*i*.

*unobservable*in transport experiments. This is exemplified by spin current density in time-reversal invariant systems (Nikolić et al. 2006); spin density, diamagnetic circulating currents, and circulating heat currents in the presence of external magnetic field or spontaneous magnetization breaking time-reversal invariance; and FL component of STT (Theodonis et al. 2006). Thus, the current-driven part of the nonequilibrium density matrix is defined as

*f*

_{L,R}(

*E*) =

*f*(

*E*−

*eV*

_{L,R}).

*f*

_{L}(

*E*) −

*f*

_{R}(

*E*). Although the second term in Eq. (11) contains information about the bias voltage [through the difference

*f*

_{L}(

*E*) −

*f*

_{R}(

*E*)] and about the lead assumed to be injecting electrons into the device (through

**Γ**

_{L}), it cannot (Xie et al. 2016; Mahfouzi and Nikolić 2013) be used as the proper

**ρ**_{CD}defined in Eq. (9). This is due to the fact that the second term in Eq. (9), expressed in terms of the retarded GF via Eq. (7), does not cancel the gauge-noninvariant first term in Eq. (11) which depends explicitly [through

*f*(

*E*−

*eV*

_{R})] on the arbitrarily chosen reference potential

*V*

_{R}and implicitly on the voltages applied to both reservoirs [through

**G**(

*E*)]. Nevertheless, the second term in Eq. (11), written in the linear-response and zero-temperature limit,

*improper*(due to being gauge-noninvariant, which is marked by “?” on the top of the equality sign) expression for

**ρ**_{CD}. Its usage leads to ambiguous (i.e., dependent on arbitrarily chosen

*V*

_{R}) nonequilibrium expectation values.

*proper*gauge-invariant expression was derived in Mahfouzi and Nikolić (2013) which is given here at zero temperature so that it can be contrasted with Eq. (12). The second and third term in Eq. (13), whose purpose is to subtract any nonzero expectation value that exists in thermodynamic equilibrium, make it quite different from Eq. (12) while requiring to include also electrostatic potential profile

*U*

_{n}across the active region of the device interpolating between

*V*

_{L}and

*V*

_{R}. For example, the second term in Eq. (13) traced with an operator gives equilibrium expectation value governed by the states at

*E*

_{F}which must be removed. The third term in Eq. (13) ensures the gauge invariance of the nonequilibrium expectation values while making the whole expression non-Fermi-surface property. The third term also renders the usage of Eq. (13) computationally demanding due to the requirement to perform integration from the bottom of the band up to

*E*

_{F}together with sampling of

**k**

_{∥}points for the junctions in Fig. 2.

*asymmetric*MTJ, FM/I/FM

^{′}, whose FM and FM

^{′}layers are assumed to be made of the same material but have different thicknesses. This setup allows us to demonstrate how application of improper

**ρ**_{CD}in Eq. (12) yields linear-response

**T**

_{FL}∝

*V*

_{b}in Fig. 4b that is

*incorrectly*an order of magnitude smaller than the correct result in Fig. 4a. This is due to the fact that

**T**

_{FL}in MTJs possess both the nonequilibrium CD contribution due to spin reorientation at interfaces, where net spin created at one interface is reflected at the second interface where it briefly precesses in the exchange field of the free FM layer, and equilibrium contribution due to interlayer exchange coupling (Theodonis et al. 2006; Yang et al. 2010). The ambiguity in Fig. 4 arises when this equilibrium contribution is improperly subtracted, so that current-driven

**T**

_{FL}in Fig. 4b is contaminated by a portion of equilibrium contribution added to it when using improper

**ρ**_{CD}in Eq. (12). On the other hand, since

**T**

_{DL}has a zero expectation value in equilibrium, both the proper and improper expressions for

**ρ**_{CD}give the same result in Fig. 4.

*symmetric*MTJs,

**T**

_{FL}∝

*V*

_{b}vanishes. Since this is a rather general result which holds for both MTJs and spin valves in the linear-response regime (Theodonis et al. 2006; Xiao et al. 2008; Heiliger and Stiles 2008) and it has been confirmed in numerous experiments (Wang et al. 2011; Oh et al. 2009), one can use it as a validation test of the computational scheme. For example, the usage of improper

**ρ**_{CD}in Eq. (12), or the proper one in Eq. (13) but with possible software bug, would give nonzero

**T**

_{FL}≠ 0 in symmetric junctions at small applied

*V*

_{b}which contradicts experiments (Wang et al. 2011; Oh et al. 2009). In the particular case of symmetric junction, one can actually employ a simpler expression (Mahfouzi and Nikolić 2013; Stamenova et al. 2017) than Eq. (13) which guarantees

**T**

_{FL}≡ 0

This expressions is obtained by assuming (Mahfouzi and Nikolić 2013) the particular gauge *V*_{L} = −*V*_{b}∕2 = −*V*_{R}. Such special gauges and the corresponding Fermi surface expressions for **ρ**_{CD} ∝ *V*_{b} in the linear-response regime do exist also for asymmetric junctions, but one does not know them in advance except for the special case of symmetric junctions (Mahfouzi and Nikolić 2013).

**T**akin to Eq. (3) but determined by the TBH of the free FM layer. These three numbers are then used to obtain FL (or perpendicular) torque component, \(T_{\mathrm {FL}}=T^y_{\mathrm {CD}}\) along the direction

**M**

_{free}×

**M**

_{fixed}, and DL (or parallel) torque component, \(T_{\mathrm {DL}}=\sqrt {(T^x_{\mathrm {CD}})^2 +(T^z_{\mathrm {CD}})^2}\) in the direction

**M**

_{free}× (

**M**

_{free}×

**M**

_{fixed}). In MTJs angular dependence of STT components stems only from the cross product, so that \(\propto \sin \theta \) dependence (Theodonis et al. 2006; Xiao et al. 2008) for both FL and DL components is obtained in Figs. 4 and 5.

**T**

_{DL}∝

**M**

_{free}×

**f**and

**T**

_{DL}∝

**M**

_{free}× (

**M**

_{free}×

**f**), where the direction specified by the unit vector

**f**is determined dynamically once the current flows in the presence of SOC. Therefore,

**f**is not known in advance (aside from simplistic models like the Rashba ferromagnetic one where

**f**is along the

*y*-axis for charge current flowing along the

*x*-axis, as illustrated in Fig. 9). Thus, it would be advantageous to decompose

**ρ**_{CD}into contributions whose trace with the torque operator in Eq. (3) directly yields

**T**

_{DL}and

**T**

_{FL}. Such decomposition was achieved in Mahfouzi et al. (2016), using adiabatic expansion of Eq. (4) in the powers of

*d*

**M**

_{free}∕

*dt*and symmetry arguments, where \(\boldsymbol {\rho }_{\mathrm {neq}} = \boldsymbol {\rho }_{\mathrm {neq}}^{\mathrm {oo}}+\boldsymbol { \rho }_{\mathrm {neq}}^{\mathrm {oe}}+\boldsymbol {\rho }_{\mathrm {neq}}^{\mathrm {eo}}+\boldsymbol {\rho }_{\mathrm {neq}}^{\mathrm {ee}}\) is the sum of the following terms

*f*

_{L}(

*E*) −

*f*

_{R}(

*E*) and, therefore, are nonzero only in nonequilibrium generated by the bias voltage

*V*

_{b}≠ 0 which drives the steady-state current. Using an identity from the NEGF formalism (Stefanucci and van Leeuwen 2013),

**G**(

**Γ**

_{L}+ Γ

_{R})

**G**

^{†}=

*i*(

**G**−

**G**

^{†}), reveals that \(\boldsymbol {\rho }^{\mathrm {eo}}_{\mathrm {neq}} \equiv 0\) and

In the linear-response regime, pertinent to calculations of STT in spin valves and SOT in FM/spin-orbit-coupled-material bilayers, *f*_{L}(*E*) − *f*_{R}(*E*) → (−*∂f*∕*∂E*)*eV*_{b}. This confines integration in \(\boldsymbol {\rho }^{\mathrm {oo}}_{\mathrm {neq}}\) and \(\boldsymbol {\rho }^{\mathrm {oe}}_{\mathrm {neq}}\) expressions to a shell of few *k*_{B}*T* around the Fermi energy, or at zero temperature these are just matrix products evaluated at the Fermi energy, akin to Eqs. (12), (13), and (14). Nevertheless, to compute \(\boldsymbol {\rho }_{\mathrm {CD}}=\boldsymbol { \rho }_{\mathrm {neq}}^{\mathrm {oo}}+\boldsymbol {\rho }_{\mathrm {neq}}^{\mathrm {oe}}+\boldsymbol {\rho }_{\mathrm {neq}}^{\mathrm {eo}}+\boldsymbol { \rho }_{\mathrm {neq}}^{\mathrm {ee}} - \boldsymbol {\rho }_{\mathrm {eq}}\), one still needs to perform the integration over the Fermi sea in order to obtain \(\boldsymbol { \rho }_{\mathrm {neq}}^{\mathrm {ee}} - \boldsymbol {\rho }_{\mathrm {eq}}\), akin to Eq. (13), which can be equivalently computed as \([\boldsymbol { \rho }_{\mathrm {neq}}^{\mathrm {ee}}(V_b) - \boldsymbol {\rho }_{\mathrm {neq}}^{\mathrm {ee}}(-V_b)]/2\) using some small *V*_{b}.

To evade singularities on the real axis caused by the poles of the retarded GF in the matrix integral of the type \(\int\limits _{-\infty }^{+\infty } dE \, \mathbf {G} f_p(E)\) appearing in Eqs. (7), (13), and (20), such integration can be performed along the contour in the upper half of the complex plane where the retarded GF is analytic. The widely used contour (Brandbyge et al. 2002) consists of a semicircle, a semi-infinite line segment, and a finite number of poles of the Fermi function *f*_{p}(*E*). This contour should be positioned sufficiently far away from the real axis, so that **G** is smooth over both of these two segments, while also requiring to select the minimum energy *E*_{min} (as the starting point of semicircular path) below the bottom of the band edge which is not known in advance in DFT calculations. That is, in self-consistent calculations, incorrectly selected minimum energy causes the charge to erroneously disappear from the system with convergence trivially achieved but to physically incorrect solution. By choosing different types of contours (Areshkin and Nikolić 2010; Ozaki 2007; Karrasch et al. 2010) [such as the “Ozaki contour” (Ozaki 2007; Karrasch et al. 2010) employed in the calculations in Fig. 9] where residue theorem leads to just a sum over a finite set of complex energies, proper positioning of *E*_{min} and convergence in the number of Fermi poles, as well as selection of sufficient number of contour points along the semicircle and contour points on the line segment, are completely bypassed.

*n*within the right chain can be computed from the divergence (in discrete form) of spin current (Theodonis et al. 2006), \(T_n^\alpha = -\nabla I^{S_\alpha } = I_{n-1,n}^{S_\alpha } - I_{n,n+1}^{S_\alpha }\). Its sum over the whole free FM layer gives the total STT as

*α*∈{

*x*,

*y*,

*z*}, from the last site inside the barrier (which is the last site of the left carbon chain in Fig. 5a) toward the first site of the free FM layer (which is the first site of the right carbon chain in Fig. 5a). Similarly, \(I_{N_{\mathrm {free}},N_{\mathrm {free}}+1}\) is the local spin current from the last site inside the free FM layer and the first site of the right lead. Thus, Eq. (21) expresses STT on the free FM layer composed of

*N*

_{free}sites as the difference (Wang et al. 2008) between spin currents entering through its left and exiting through its right interface. In the case of semi-infinite free FM layer,

*N*

_{free}→

*∞*and \(I_{N_{\mathrm {free}},N_{\mathrm {free}}+1} \rightarrow 0\). The nonequilibrium local spin current can be computed in different ways (Wang et al. 2008), one of which utilizes NEGF expression for

**ρ**_{CD}

**H**

_{n,n+1}and \(\boldsymbol {\rho }_{\mathrm {CD}}^{n,n+1}\) are the submatrices of the Hamiltonian and the current-driven part of the nonequilibrium density matrix, respectively, of the size 2

*N*

_{orbital}× 2

*N*

_{orbital}(2 is for spin and

*N*

_{orbital}is for the number of orbitals per each atom) which connect sites

*n*and

*n*+ 1.

Combining Eqs. (14), (21), and (22) yields \(T^x_{\mathrm {CD}}\) and \(T^z_{\mathrm {CD}}\) from which we obtain \(T_{\mathrm {DL}}=\sqrt {(T^x_{\mathrm {CD}})^2 +(T^z_{\mathrm {CD}})^2}\) in the linear-response regime plotted in Fig. 5b as a function of angle *θ* between **M**_{free} and **M**_{fixed}. Alternatively, evaluating the trace of the product of \(\boldsymbol { \rho }_{\mathrm {neq}}^{\mathrm {oo}}\) and the torque operator in Eq. (3) yields a vector with two nonzero components, which turn out to be identical to \(T^x_{\mathrm {CD}}\) and \(T^z_{\mathrm {CD}}\) computed from the spin current divergence algorithm, as demonstrated in Fig. 5b. The trace of \(\boldsymbol { \rho }_{\mathrm {neq}}^{\mathrm {oe}}\) with the torque operator gives a vector with zero *x*- and *z*-components and nonzero *y*-component which, however, is canceled by adding the trace of \(\boldsymbol {\rho }_{\mathrm {neq}}^{\mathrm {ee}} - \boldsymbol {\rho }_{\mathrm {eq}}\) with the torque operator to finally produce zero FL component of the STT vector. This is expected because MTJ in Fig. 5a is left-right symmetric.

We emphasize that the algorithm based on the trace of the torque operator with the current-driven part of the nonequilibrium density matrix **ρ**_{CD} is a *more general approach* than the spin current divergence algorithm since it is valid even in the presence of spin-flip processes by impurities and magnons or SOC (Haney et al. 2007). In particular, it can be employed to compute SOT (Freimuth et al. 2014; Mahfouzi and Kioussis 2018) in FM/spin-orbit-coupled-material bilayers where spin torque *cannot* (Haney and Stiles 2010) be expressed any more as in Eq. (21).

## 3 Example: Spin-Transfer Torque in FM/NM/FM Trilayer Spin-Valves

*rigid*approximation where one starts from the collinear DFT Hamiltonian and then rotates magnetic moments of either fixed or free FM layer in the spin space in order to generate the relative angle between

**M**

_{fixed}and

**M**

_{free}(as it was also done in the calculations of STT in 1D toy model of MTJ in Fig. 5). On the other hand, obtaining true ground state of such system requires noncollinear XC functionals (Capelle et al. 2001; Eich and Gross 2013; Eich et al. 2013; Bulik et al. 2013) and the corresponding self-consistent XC magnetic field

**B**

_{xc}introduced in Eq. (1). For a given self-consistently converged ncDFT Hamiltonian represented in the linear combination of atomic orbitals (LCAO) basis, we can extract the matrix representation of \({\mathbf {B}}_{\mathrm {xc}}^\alpha \) in the same basis using

*ϕ*

_{n}〉 are typically nonorthogonal (as is the case of the basis sets (Junquera et al. 2001; Ozaki 2003; Schlipf and Gygi 2015) implemented in ATK (Atomistix Toolkit 2017) and OpenMX (Openmx) packages employed in the calculations of Figs. 5, 6, 7, and 8), the trace leading to the spin torque vector

**D**

^{−1}matrix into Eq. (26) where all matrices inside the trace are representations in the LCAO basis. In the real-space basis spanned by the eigenstates |

**r**〉 of the position operator, the same trace in Eq. (26) becomes This is a nonequilibrium generalization of the equilibrium torque expression found in ncDFT (Capelle et al. 2001) where

**m**

_{eq}(

**r**) = 〈

**r**|

**ρ**_{eq}

*|*

**σ****r**〉 in ncDFT is replaced by

**m**

_{CD}(

**r**) = 〈

**r**|

**ρ**_{CD}

*|*

**σ****r**〉. Note that in thermodynamic equilibrium the integral in Eq. (27) must be zero when integration is performed over all space, which is denoted as “zero-torque theorem” (Capelle et al. 2001), but

**m**

_{eq}×

**B**

_{XC}(

**r**) can be nonzero locally which gives rise to equilibrium torque on the free FM layer that has to be removed by subtracting

**ρ**_{eq}to obtain

**ρ**_{CD}in Eq. (9) and plug it into Eq. (27).

We employ ATK package to compute STT in Co/Cu(9 ML)/Co spin valve illustrated in Fig. 2a using ncDFT Hamiltonian combined with Eq. (26). Prior to DFT calculations, the interface builder in the VNL package (Virtual Nanolab 2017) is employed to construct a common unit cell for Co/Cu bilayer. In order to determine the interlayer distance and relaxed atomic coordinates, DFT calculations using VASP (Vienna; Kresse and Hafner 1993; Kresse and Furthmüller 1996) with Perdew-Burke-Ernzerhof (PBE) parametrization (Perdew et al. 1996) of the generalized gradient approximation for the XC functional and projected augmented wave (Blöchl 1994; Kresse and Joubert 1999) description of electron-core interactions are performed. The cutoff energy for the plane wave basis set is chosen as 600 eV, while *k*-points were sampled on a 11 × 11 surface mesh. In ATK calculations, PBE XC functional, norm-conserving pseudopotentials for describing electron-core interactions and SG15 (medium) LCAO basis set (Schlipf and Gygi 2015) are employed. The energy mesh cutoff for the real-space grid is chosen as 100 Hartree.

The layer-resolved Cartesian components of STT vector within the free Co layer are shown in Fig. 6d–f. The contribution from a propagating state oscillates as a function of position without decaying in Fig. 6e with a spatial period \(2\pi /(k^\downarrow _\zeta - k^\uparrow _\eta )\) where *ζ* (*η*) denotes different sheets (Wang et al. 2008) of the Fermi surface for minority (majority) spin. This is due to the fact that noncollinear spin in Fig. 2a entering the right Co layer is not an eigenstate of the spin part of the Hamiltonian determined by **M**_{free}, and it is, therefore, forced into precession. However, since the shapes of the Fermi surface for majority and minority spin in Co are quite different from each other (Wang et al. 2008), the spatial periods of precession can vary rapidly for different **k**_{∥} within the 2D Brillouin zone (BZ). Thus, summation of their contributions leads to cancelation and, therefore, fast decay of STT away from the interface (Stiles and Zangwill 2002; Wang et al. 2008), as demonstrated by plotting such sum to obtain the total STT per ML of Co in Fig. 6f.

**k**

_{∥}-resolved transmission function obtained from the Landauer formula in terms of NEGFs (Stefanucci and van Leeuwen 2013)

**k**

_{∥}is conserved in the absence of disorder. The total transmission function per unit interfacial area is then evaluated using ( Ω

_{BZ}is the area of sampled 2D BZ)

**k**

_{∥}= (0, 0) vanishes at the Fermi energy, signifying evanescent state which

*cannot*carry any current across the junction. Nonetheless, such states can contribute (Ralph and Stiles 2008; Stiles and Zangwill 2002; Wang et al. 2008) to STT vector, as shown in Fig. 6d. Thus, the decay of STT away from Cu/Co interface in Fig. 6f arises both from the cancelation among contributions from propagating states with different

**k**

_{∥}and the decay of contributions from each evanescent state, where the latter are estimated (Wang et al. 2008) to generate ≃10% of the total torque on the ML of free Co layer that is closest to the Cu/Co interface in Fig. 2a.

Since the considered Co/Cu(9 ML)/Co spin valve is left-right symmetric, the FL component of the STT vector is zero. The DL component, as the sum of all layer-resolved torques in Fig. 6f, is plotted as a function of the relative angle *θ* between **M**_{fixed} and **M**_{free} in Fig. 7. The angular dependence of STT in spin valves does not follow \(\propto \sin \theta \) dependence found in the case of MTJs in Figs. 4 and 5.

Although similar analyses have been performed before using collinear DFT Hamiltonian and rigid rotation of magnetic moments in fixed Co layer (Haney et al. 2007; Wang et al. 2008), in Fig. 7 an error made in this approximation was additionally quantified by computing torque using ncDFT Hamiltonian. The rigid approximation is then just the first iteration of the full self-consistent field (scf) calculations leading to the converged ncDFT Hamiltonian. The difference between scf and non-scf calculations in Fig. 7 is rather small due to large number of spacer MLs of Cu, but it could become sizable for small number of spacer MLs enabling coupling between two FM layers.

## 4 Example: Spin-Orbit Torque in FM/Monolayer-TMD Heterostructures

The calculation of SOT driven by injection of unpolarized charge current into bilayers of the type FM/HM shown in Fig. 2b, FM/monolayer-TMD shown in Fig. 2c, FM/TI, or FM/WSM can be performed using the same NEGF+ncDFT framework combining the torque operator **T**, **ρ**_{CD} expressed in terms of NEGFs and ncDFT Hamiltonian that was delineated in Sect. 2 and applied in Sect. 3 to compute the STT vector in spin valves. Such first-principles quantum transport approach can also easily accommodate possible third capping insulating layer (such as MgO or AlO_{x}) employed experimentally to increase (Kim et al. 2013) the perpendicular magnetic anisotropy which tilts the magnetization out of the plane of the interface. However, the results of such calculations are not as easy to interpret as in the case of a transparent picture (Stiles and Zangwill 2002; Wang et al. 2008) in Fig. 6d–f explaining how spin angular momentum gets absorbed close to the interface in junctions which exhibit conventional STT. This is due to the fact that several microscopic mechanisms can contribute to SOT, such as the spin Hall effect (SHE) (Vignale 2010; Sinova et al. 2015) within the HM layer (Freimuth et al. 2014; Mahfouzi and Kioussis 2018) with strong bulk SOC and around FM/HM interface (Wang et al. 2016); current-driven nonequilibrium spin density – the so-called Edelstein effect (Edelstein 1990; Aronov and Lyanda-Geller 1989) – due to strong interfacial SOC; spin currents generated in transmission and reflection from SO-coupled interfaces within 3D transport geometry (Zhang et al. 2015; Kim et al. 2017); and spin-dependent scattering of impurities (Pesin and MacDonald 2012a; Ado et al. 2017) or boundaries (Mahfouzi et al. 2016) in the presence of SOC within FM monolayers. This makes it difficult to understand how to optimize SOT by tailoring materials combination or device geometry to enhance one or more of these mechanisms.

_{2}Se

_{3}as the prototypical 3D TI (Bansil et al. 2016). Such materials possess a usual bandgap in the bulk, akin to conventional topologically trivial insulators, but they also host metallic surfaces whose low-energy quasiparticles behave as massless Dirac fermions. The spins of such fermions are perfectly locked to their momenta by strong SOC, thereby forming spin textures in the reciprocal space (Bansil et al. 2016). In general, when charge current flows through a surface or interface with SOC, the presence of SOC-generated spin texture in the reciprocal space, such as those shown in Figs. 3e, g, and h, will generate nonequilibrium spin density which can be computed using

**S**

_{CD}are easily determined by back-of-the-envelope calculations (Pesin and MacDonald 2012b). For example, spin texture (i.e., expectation value of the spin operator in the eigenstates of a Hamiltonian) associated with \(\hat {H}_{\mathrm {Rashba}}\) consists of spin vectors locked to momentum vector along the two Fermi circles formed in the reciprocal space at the intersection of the Rashba energy-momentum dispersion (Winkler 2003) and the Fermi energy plane. Thus, current flow will disturb balance of momenta to produce

**S**

_{CD}in the direction transverse to current flow. The same effect is substantially enhanced (Pesin and MacDonald 2012b), by a factor

*ħv*

_{F}∕

*α*

_{SO}≫ 1 where

*v*

_{F}is the Fermi velocity in TI and

*α*

_{SO}is the strength of the Rashba SOC, because spin texture associated with \(\hat {H}_{\mathrm {Dirac}}\) consists of spin vectors locked to momentum vector along a

*single*Fermi circle formed in the reciprocal space at the intersection of the Dirac cone energy-momentum dispersion (Bansil et al. 2016) and the Fermi energy plane. This eliminates the compensating effect of the spins along the second circle in the case of \(\hat {H}_{\mathrm {Rashba}}\). Note that nonzero total

**S**

_{CD}∝

*V*

_{b}generated by the Edelstein effect is allowed only in nonequilibrium since in equilibrium

**S**changes sign under time reversal and, therefore, has to vanish (assuming the absence of external magnetic field or magnetization).

In the case of a thin film of Bi_{2}Se_{3} described by ncDFT Hamiltonian, unpolarized charge current injected along the *x*-axis generates \({\mathbf {S}}_{\mathrm {CD}} = (0,S^y_{\mathrm {CD}},S^z_{\mathrm {CD}})\) on the top surface of the TI, marked as plane 1 in Fig. 8. The in-plane component \(S^y_{\mathrm {CD}}\), expected from back-of-the-envelope calculations sketched in the preceding paragraph, is an order of magnitude larger than the ouf-of-plane component \(S^z_{\mathrm {CD}}\) arising due to hexagonal warping (Bansil et al. 2016) of the Dirac cone on the TI surface. The spin texture on the bottom surface of the TI, marked as plane 3 in Fig. 8, has opposite sign to that shown on the top surface because of opposite direction of spins wounding along single Fermi circle on the bottom surface. In addition, a more complicated spin texture in real space (on a grid of points with ≃0.4 Å spacing), akin to noncollinear intra-atomic magnetism (Nordström and Singh 1996) but driven here by current flow, emerges within ≃2 nm thick layer below the TI surfaces. This is due to the penetration of evanescent wave functions from the metallic surfaces into the bulk of the TI, as shown in Fig. 8 by plotting **S**_{CD}(**r**) within plane 2.

The conventional unpolarized charge current injected into the HM layer in Fig. 2b along the *x*-axis generates transverse spin Hall currents (Vignale 2010; Sinova et al. 2015; Wang et al. 2016) due to strong SOC in such layers. In 3D geometry, spin Hall current along the *y*-axis carries spins polarized along the *z*-axis, while the spin Hall current along the *z*-axis carries spins polarized along the *y*-axis (Wang et al. 2016). Thus, the effect of the spin Hall current flowing along the *z*-axis and entering FM layer resembles STT that would be generated by a fictitious polarizing FM layer with fixed magnetization along the *y*-axis and with charge current injected along the *z*-axis. While this mechanism is considered to play a major role in the generation of the DL component of SOT (Liu et al. 2012), as apparently confirmed by the Kubo-formula+ncDFT modeling (Freimuth et al. 2014; Mahfouzi and Kioussis 2018), the product of signs of the FL and DL torque components is negative in virtually all experiments (Yoon et al. 2017) (except for specific thicknesses of HM = Ta (Kim et al. 2013) layer). Although HM layer in Fig. 2b certainly generates spin Hall current in its bulk, such spin current can be largely suppressed by the spin memory loss (Dolui and Nikolić 2017; Belashchenko et al. 2016) as electron traverses HM/FM interface with strong SOC. Thus, in contrast to positive sign product in widely accepted picture where SHE is most responsible for the DL component of SOT and Edelstein effect is most responsible for the FL component of SOT, negative sign product indicates that a single dominant mechanism could be responsible for *both* DL and FL torque.

In order to explore how such single mechanism could arise in the absence of spin Hall current, one can calculate torque in a number of specially crafted setups, such as the one chosen in Fig. 9a where a single ultrathin FM layer (consisting of 4 or 10 MLs) is considered with the Rashba SOC present either only in the bottom ML (marked as 0 in Fig. 9a) or in all MLs but with decreasing strength to mimic the spin-orbit proximity effect exemplified in Fig. 3g, h. The setup in Fig. 9a is motivated by SOT experiments (Sklenar et al. 2016; Shao et al. 2016; Guimarães et al. 2018; Lv et al. 2018) on FM/monolayer-TMD heterostructures where SHE is absent due to atomically thin spin-orbit-coupled material. In such bilayers, clean and atomically precise interfaces have been achieved, while back-gate voltage (Lv et al. 2018) has been employed to demonstrate control of the ratio between the FL and DL components of SOT. Note that when bulk TMD or its even-layer thin films are centrosymmetric, its monolayer will be noncentrosymmetric crystal which results in lifting of the spin degeneracy and possibly strong SOC effects (Zhu et al. 2011).

*a*with a single orbital per site located at position

**n**= (

*n*

_{x}

*a*,

*n*

_{z}

*a*)

**n**in spin-

*↑*or spin-

*↓*state, and \(\hat {c}_{\mathbf {n}}=(\hat {c}_{\mathbf {n}\uparrow } \ \ \hat {c}_{\mathbf {n}\downarrow })\) is the column vector of the corresponding annihilation operators. The spin-dependent nearest-neighbor hoppings in the

*xz*-plane form a 2 × 2 matrix in the spin space

*γ*

_{SO}=

*α*

_{SO}∕2

*a*measures the strength of the Rashba SOC on the lattice (Nikolić et al. 2006),

**I**

_{2}is the unit 2 × 2 matrix, and

**e**

_{α}are the unit vectors along the axes of the Cartesian coordinate system. The on-site energy

*U*

_{n}(due to impurities, insulating barrier, voltage drop, etc.), as well as kinetic energy effectively generated by the periodic boundary conditions along the

*y*-axis which simulate infinite extension of the FM layer in this direction and require

*k*

_{y}-point sampling in all calculations. The infinite extension along the

*x*-axis is taken into account by splitting the device in Fig. 9a into semi-infinite left lead, central region of arbitrary length along the

*x*-axis, and semi-infinite right lead, all of which are described by the Hamiltonian in Eq. (31) with the same values for

*t*= 1 eV,

*J*= 0.1 eV, and the

*γ*

_{SO}chosen in all three regions. Thus,

*γ*

_{SO}is homogeneous within a given ML, and always present

*γ*

_{SO}= 0.2 eV on layer 0, but it can take different values in other MLs. The Fermi energy is set at

*E*

_{F}= 1.0 eV to take into account possible noncircular Fermi surface (Lee et al. 2015) effects in realistic materials.

The SOT is often studied (Manchon and Zhang 2008; Haney et al. 2013; Lee et al. 2015; Li et al. 2015; Pesin and MacDonald 2012a; Ado et al. 2017; Kalitsov et al. 2017) using the Rashba ferromagnetic model in 2D, which corresponds to just a single layer in Fig. 9a. In that case, only FL torque component is found (Manchon and Zhang 2008; Kalitsov et al. 2017) due to the Edelstein effect and in the absence of spin-dependent disorder (Pesin and MacDonald 2012a; Ado et al. 2017). This is also confirmed in the 3D transport geometry in Fig. 9b, d, e where SOT vector, \(\frac {2J}{\hbar } {\mathbf {S}}_{\mathrm {CD}} \times {\mathbf {M}}_{\mathrm {free}}\), on layer 0 has only FL component pointing in the **M**_{free} ×**e**_{y} direction, as long as the device is infinite, clean, and homogeneous. In addition, nonzero FL component of SOT in Fig. 9b, d was also found on layers above layer 0 despite the fact that only layer 0 hosts *γ*_{SO} ≠ 0. This is due to vertical transport along the *z*-axis in 3D geometry of Fig. 9a, but in the absence of the Rashba SOC on other layers, such effect decays fast as we move toward the top layer, as shown in Fig. 9d for 10 MLs thick FM film. The presence of the Rashba SOC with decreasing *γ*_{SO} on MLs above layer 0 generates additional nonequilibrium spin density **S**_{CD} ∝**e**_{y} on those layers and the corresponding enhancement of the FL component of SOT on those layers in Fig. 9e.

We note that the Kubo-Bastin formula (Bastin et al. 1971) adapted for SOT calculations (Freimuth et al. 2014) predicts actually nonzero DL component of SOT vector for the Rashba ferromagnetic model in 2D due to the change of electronic wave functions induced by an applied electric field termed the “Berry curvature mechanism” (Lee et al. 2015; Kurebayashi et al. 2014; Li et al. 2015). Despite being apparently intrinsic, i.e., insensitive to disorder, this mechanism can be completely canceled in specific models when the vertex corrections are taken into account (Ado et al. 2017). It also gives positive sign product (Lee et al. 2015; Li et al. 2015) of DL and FL components of SOT contrary to majority of experiments where such product is found to be negative (Yoon et al. 2017). It is emphasized here that no electric field can exist in the ballistic transport regime through clean devices analyzed in Fig. 9b, d, e, for which the Kubo-Bastin formula also predicts unphysical divergence (Freimuth et al. 2014; Mahfouzi and Kioussis 2018; Lee et al. 2015; Kurebayashi et al. 2014; Li et al. 2015) of the FL component of SOT. Adding finite voltage drop within the central region, which is actually unjustified in the case of infinite, clean, homogeneous device, results in nonzero DL component of SOT also in the NEGF calculations (Kalitsov et al. 2017). However, the same outcome can be obtained simply by introducing constant potential *U*_{n} = *U*_{0} on each site of the central region acting as a barrier which reflects incoming electrons, as demonstrated in Fig. 9c, f, g. In the presence of both SOC and such barrier, spin-dependent scattering (Pesin and MacDonald 2012a; Ado et al. 2017) is generated at the lead/central-region boundary which results in nonzero component of **S**_{CD} in the direction **M**_{free} ×**e**_{y} and the corresponding DL component of SOT ∝**M**_{free} × (**M**_{free} ×**e**_{y}) acting on edge magnetic moments (Mahfouzi et al. 2016). This will, therefore, induce inhomogeneous magnetization switching which starts from the edges and propagates into the bulk of FM layer, as observed in experiments (Baumgartner et al. 2017) and micromagnetic simulations (Baumgartner et al. 2017; Mikuszeit et al. 2015).

Interestingly, Fig. 9g also demonstrates that the signs of the DL and FL component are opposite to each other for almost all values of *U*_{n} = *U*_{0} (except when *U*_{0} ≃ *J*), as observed in the majority of SOT experiments (Yoon et al. 2017). Importantly, the spin-orbit proximity effect within the MLs of FM layer close to FM/spin-orbit-coupled-material interface, as illustrated in Figs. 3g, h and mimicked by introducing the Rashba SOC of decaying strength within all MLs of FM thin film in Fig. 9a, enhances both FL and DL components of SOT. This is demonstrated by comparing solid (*γ*_{SO} ≠ 0 only on layer 0) and dashed (*γ*_{SO} ≠ 0 on all layers 0–3) lines in Fig. 9g. This points out at a knob that can be exploited to enhance SOT by searching for optimal combination of materials capable to generate penetration of SOC over long distances within the FM layer (Marmolejo-Tejada et al. 2017). In fact, in the case of FM/TI and FM/monolayer-TMD heterostructures, proximity SOC coupling within the FM layer is crucial for SOT efficiency (Wang et al. 2017) where it has been considered (Mellnik et al. 2014) that applied current will be shunted through the metallic FM layer and, therefore, not contribute to nonequilibrium spin density generation at the interface where SOC and thereby induced in-plane spin textures are naively assumed to reside.

## 5 Conclusions

This chapter reviews a unified first-principles quantum transport approach, implemented by combining NEGF formalism with ncDFT calculations, to compute both STT in traditional magnetic multilayers with two FM layers (i.e., the polarizing and analyzing FM layers with fixed and free magnetizations, respectively) and SOT in magnetic bilayers where only one of the layers is ferromagnetic. In the latter case, the role of the fixed magnetization of the polarizing FM layer within spin valves or MTJs is taken over by the current-driven nonequilibrium spin density in the presence of strong SOC introduced by the second layer made of HM, 3D TI, WSM, or monolayer-TMD. This approach resolves recent confusion (Freimuth et al. 2014; Kalitsov et al. 2017) in the literature where apparently only the Kubo formula, operating with expressions that integrate over the Fermi sea in order to capture change of wave functions due to the applied electric field and the corresponding interband electronic transitions (Freimuth et al. 2014; Lee et al. 2015; Li et al. 2015), can properly obtain the DL component of SOT. In addition, although the Kubo formula approach can also be integrated with first-principles calculations (Freimuth et al. 2014; Mahfouzi and Kioussis 2018), it can only be applied to a single device geometry (where infinite FM layer covers infinite spin-orbit-coupled-material layer while current flows parallel to their interface) and in the linear-response transport regime. In contrast, NEGF+ncDFT approach reviewed in this chapter can handle arbitrary device geometry, such as spin valves and MTJs exhibiting STT or bilayers of the type FM/spin-orbit-coupled-material which are made inhomogeneous by attachment to NM leads, at vanishing or finite applied bias voltage. In contrast to often employed 2D transport geometry (Manchon and Zhang 2008; Haney et al. 2013; Lee et al. 2015; Li et al. 2015; Pesin and MacDonald 2012a; Ado et al. 2017; Kalitsov et al. 2017; Ndiaye et al. 2017) for SOT theoretical analyses, it is emphasized here the importance of 3D transport geometry (Kim et al. 2017; Ghosh and Manchon 2018) to capture both the effects at the FM/spin-orbit-coupled-material interface and those further into the bulk of the FM layer. Finally, ultrathin FM layers employed in SOT experiments can hybridize strongly with the adjacent spin-orbit-coupled material to acquire its SOC and the corresponding spin textures on all of the FM monolayers. Such “hybridized ferromagnetic metals” can have electronic and spin structure (Fig. 3) which is quite different from an isolated FM layer, thereby requiring usage of both 3D geometry and first-principles Hamiltonians [of either tight-binding (Freimuth et al. 2014; Mahfouzi and Kioussis 2018) or pseudopotential-LCAO-ncDFT (Theurich and Hill 2001) type] to predict the strength of SOT in realistic systems and optimal materials combinations for device applications of the SOT phenomenon.

## Notes

### Acknowledgements

We are grateful to K. D. Belashchenko, K. Xia, and Z. Yuan for illuminating discussions and P.-H. Chang, F. Mahfouzi, and J.-M. Marmolejo-Tejada for the collaboration. B. K. N. and K. D. were supported by DOE Grant No. DE-SC0016380 and NSF Grant No. ECCS 1509094. M. P. and P. P. were supported by ARO MURI Award No. W911NF-14-0247. K. S. and T. M. acknowledge support from the European Commission Seventh Framework Programme Grant Agreement IIIV-MOS, Project No. 61932, and Horizon 2020 research and innovation program under grant agreement SPICE, Project No. 713481. The supercomputing time was provided by XSEDE, which is supported by NSF Grant No. ACI-1548562.

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