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

  • Branislav K. NikolićEmail author
  • Kapildeb Dolui
  • Marko D. Petrović
  • Petr Plecháč
  • Troels Markussen
  • Kurt Stokbro
Living reference work entry


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 5d 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.


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

The spin-transfer torque (STT) (Ralph and Stiles 2008; Locatelli et al. 2014; Slonczewski 1996; Berger 1996) is a phenomenon in which a spin current of sufficiently large density (∼107 A∕ cm2) 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 Mfixed, and it is directed into the second thin FM layer with magnetization along the unit vector Mfree where transverse (to Mfree) 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).
Fig. 1

Spin-transfer torque basics: (a) in a ferromagnet/nonmagnetic-spacer/ferromagnet setup, with noncollinear magnetizations Mfixed of the fixed FM layer and Mfree of the free FM layer, the transverse spin component of the conduction electrons (red) polarized in the direction of Mfixed is absorbed as they pass through the free layer, thereby generating a torque on Mfree; (b) device applications of STT in (a) are based on torque-induced magnetization dynamics that is converted into resistance variations via the magnetoresistive effects; (c) torques on Mfree where field-like component of current-driven STT, TFL, is orthogonal to the plane spanned by Mfixed and Mfree and competes with the effective-field torque Tfield (present also in equilibrium), while damping-like component of current-driven STT, TDL, is parallel or antiparallel (depending on the current direction) to Gilbert damping torque Tdamping (present also in equilibrium). The illustration in (c) assumes particular case where Mfixed and the effective magnetic field are aligned. (Adapted from Locatelli et al. 2014)

Fig. 2

Schematic view of (a) FM/NM/FM trilayer for calculations of STT in spin valves, (b) FM/HM bilayer for calculations of SOT in the presence of the spin Hall current along the z-axis generated by the HM layer, and (c) FM/monolayer-TMD for calculations of SOT in the absence of any spin Hall current. The semi-infinite FM layers in (a) are chosen as Co(0001), and the spacer in between consists of l (l = 4 in the illustration) monolayers of Cu(111). The trilayer in (a) is assumed to be infinite in the transverse directions to the current flow, so that the depicted supercell is periodically repeated within the yz-plane. The bilayer in (b) consists of Co(0001) and Pt(111), and in (c) it consists of Co(0001) and monolayer MoS2. The bilayers in (b) and (c) are assumed to be infinite in the xy-plane but of finite thickness along the z-axis. Small bias voltage Vb is applied to inject electrons along the positive x-axis, so that particle current is perpendicular to interfaces in (a) and parallel to the interface in (b) and (c)

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 Mfixed 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 5d 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 MX2 (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 MoS2 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 Mfree.

The current-driven (CD) STT and SOT vectors are analyzed by decomposing them into two contributions, TCD = TDL + TFL, 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 Mfree. In the absence of current, displacing Mfree out of its equilibrium position leads to the effective-field torque Tfield which drives Mfree into precession around the effective magnetic field, while Gilbert damping Tdamping acts to bring it back to its equilibrium position. Under nonequilibrium conditions, brought by injecting steady-state or pulse current (Baumgartner et al. 2017), TDL acts opposite to Tdamping for “fixed-to-free” current direction, and it enhances Tdamping for “free-to-fixed” current directions in Fig. 1a. Thus, the former (latter) acts as antidamping (overdamping) torque trying to bring Mfree antiparallel (parallel) to Mfixed (note that at cryogenic temperatures, one finds apparently only antidamping action of TDL for both current directions (Zholud et al. 2017)). The TFL component induces magnetization precession and modifies the energy landscape seen by Mfree. Although |TFL| is minuscule in metallic spin valves (Wang et al. 2008), it can reach 30–40% of |TDL| in MTJs (Sankey et al. 2008; Kubota et al. 2008), and it can become several times larger than |TDL| in FM/HM bilayers (Kim et al. 2013; Yoon et al. 2017). Thus, TFL component of SOT can play a crucial role (Baumgartner et al. 2017; Yoon et al. 2017) in triggering the reversal process of Mfree and in enhancing the switching efficiency. In concerted action with TDL 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 |TDL|∕|TFL| (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 ∼105 A/cm2, 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 Mfree, 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).

However, not all features of STT experiments on MTJs (Wang et al. 2011) can be captured by such NEGF+TBH approach. Furthermore, due to spin-orbit proximity effect, driven by hybridization of wave functions from FM and HM layers (Dolui and Nikolić 2017) or FM and metallic surfaces of three-dimensional (3D) TIs (Marmolejo-Tejada et al. 2017), simplistic Hamiltonians like the Rashba ferromagnetic model (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) or the gapped Dirac model (Ndiaye et al. 2017) are highly inadequate to describe realistic bilayers employed in SOT experiments. This is emphasized by Fig. 3 which shows how spectral function and spin texture on the surface of semi-infinite Co layer can change dramatically as we change the adjacent layer. For example, nonzero in-plane spin texture on the surface of semi-infinite Co layer in contact with vacuum is found in Fig. 3e, despite Co magnetization being perpendicular to that surface. This is the consequence of the Rashba SOC enabled by inversion symmetry breaking (Chantis et al. 2007) where an electrostatic potential gradient can be created by the charge distribution at the metal/vacuum interface to confine wave functions into a Rashba spin-split quasi two-dimensional (2D) electron gas (Bahramy et al. 2012). The surface of semi-infinite Co within the layer embedded into Co/Cu(9 ML)/Co junction illustrated in Fig. 2a does not have any in-plane spin texture in Fig. 3f since the structure is inversion symmetric, but its spectral function in Fig. 3b is quite different from the one on the surface of an isolated semi-infinite Co layer in Fig. 3a. Bringing semi-infinite Co layer in contact with 5 MLs of Pt or 1 ML of MoS2 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.
Fig. 3

Spectral function A(E;k, x ∈ML of Co), defined in Eq. (6), plotted along high-symmetry k-path Z-Γ-Y at (a) the surface of semi-infinite Co(0001) in contact with vacuum, (b) ML of semi-infinite Co(0001) in contact with 9 MLs of Cu(111) within Co/Cu(9 ML)/Co spin valve, (c) ML of semi-infinite Co(0001) in contact with 5 MLs of Pt(111), and (d) ML of semi-infinite Co(0001) in contact with 1 ML of MoS2. Panels (e)–(h) plot constant energy contours of A(E − EF = 0;k, x ∈ML of Co) and the corresponding spin textures where the out-of-plane Sx component of spin is indicated in color (red for positive and blue for negative). The magnetization of Co is along the x-axis which is perpendicular to the ML of Co over which the spectral functions and spin textures are scanned. The units for ky and kz are 2πa and 2πb where a and b are the lattice constants along the y- and the z-axis, respectively. The horizontal dashed black line in panels (a)–(d) denotes the position of the Fermi energy EF. Panels (a) and (e) are adapted from Marmolejo-Tejada et al. (2017), and panels (c) and (g) are adapted from Dolui and Nikolić (2017)

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 EF. 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 + V0.

The noncollinear DFT (ncDFT) (Capelle et al. 2001; Eich and Gross 2013; Eich et al. 2013; Bulik et al. 2013) coupled to nonequilibrium density matrix (Stefanucci and van Leeuwen 2013) offers an algorithm to compute spin torque in arbitrary device geometry at vanishing or finite bias voltage. The single-particle spin-dependent Kohn-Sham (KS) Hamiltonian in ncDFT takes the form
$$\displaystyle \begin{aligned} \hat{H}_{\mathrm{KS}} = -\frac{\hbar^2\nabla^2}{2m} + V_{\mathrm{H}}(\mathbf{r}) + V_{\mathrm{XC}}(\mathbf{r}) + V_{\mathrm{ext}}(\mathbf{r}) - \boldsymbol{\sigma} \cdot {\mathbf{B}}_{\mathrm{XC}}(\mathbf{r}), \end{aligned} $$
where VH(r), Vext(r), and VXC(r) = EXC[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 EXC[n(r), m(r)] depends on m(r) pointing in arbitrary direction. The XC magnetic field is then given by BXC(r) = δEXC[n(r), m(r)]∕δm(r).
Once the Hamiltonian of the device is selected, it has to be passed into the formalism of nonequilibrium quantum statistical mechanics. Its central concept is the density matrix ρ 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
$$\displaystyle \begin{aligned} O=\mathrm{Tr}\, [\boldsymbol{\rho} \mathbf{O}], \end{aligned} $$
such as the charge density, charge current, spin current, and spin density of interest to spin torque modeling. These require to insert their operators (in some matrix representation) as 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)
$$\displaystyle \begin{aligned} \mathbf{T} = \frac{d\mathbf{S}}{dt} = \frac{1}{2i} [\boldsymbol{\sigma},{\mathbf{H}}_{\mathrm{KS}}] = \boldsymbol{\sigma} \times {\mathbf{B}}_{\mathrm{XC}}. \end{aligned} $$

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 BXC. 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 Mfree of the free FM layer. Examples of how to evaluate such trace, while using OT in Eq. (2) in different matrix representations, are given as Eqs. (26) and (27) in Sect. 3.

In equilibrium, ρeq is fixed by the Boltzmann-Gibbs prescription, such as ρeq =∑nf(E)| Ψn〉〈 Ψn| 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 En and | Ψn〉 are eigenenergies and eigenstates of the Hamiltonian, respectively. Out of equilibrium, the construction of ρneq 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 ρneq 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 Vb 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).

In time-dependent situations, the nonequilibrium density matrix is given by (Petrović et al. 2018; Stefanucci and van Leeuwen 2013)
$$\displaystyle \begin{aligned} \boldsymbol{\rho}_{\mathrm{neq}}(t) = {\mathbf{G}}^<(t,t)/i. \end{aligned} $$
In stationary problems 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
$$\displaystyle \begin{aligned} \mathbf{G}(E) = \left[ E- \mathbf{H} - \boldsymbol{\Sigma}_L(E,V_b) - \boldsymbol{\Sigma}_R(E,V_b) \right]^{-1}, \end{aligned} $$
assuming representation in the basis of orthogonal orbitals. In the case of nonorthogonal basis set |ϕn〉, one should make a replacement EED where D is the overlap matrix composed of elements 〈ϕn|ϕm〉. The self-energies (Velev and Butler 2004; Rungger and Sanvito 2008) ΣL,R(E, Vb) 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, μL − μR = eVL − eVR = eVb. The usual assumption about the leads is that the applied bias voltage Vb induces a rigid shift in their electronic structure(Brandbyge et al. 2002), so that ΣL,R(E, Vb) = ΣL,R(E − eVL,R).
In equilibrium or near equilibrium (i.e., in the linear-response transport regime at small eVb ≪ EF), one needs G0(E) obtained from Eq. (5) by setting VL = VR = 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 G0(E)
$$\displaystyle \begin{aligned} A(E;{\mathbf{k}}_\parallel,x)=-\frac{1}{\pi}\mathrm{Im}\,[G_0(E;{{\mathbf{k}}_\parallel};x,x)], \end{aligned} $$
where the diagonal matrix elements G0(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 G0(E)
$$\displaystyle \begin{aligned} \boldsymbol{\rho}_{\mathrm{eq}} = -\frac{1}{\pi} \int\limits_{-\infty}^{+\infty} dE \, \mathrm{Im}\, {\mathbf{G}}_0(E) f(E), \end{aligned} $$
where Im O = (O −O)∕2i.
The nonequilibrium density matrix is determined by the lesser GF
$$\displaystyle \begin{aligned} \boldsymbol{\rho}_{\mathrm{neq}} = \frac{1}{2\pi i} \int\limits_{-\infty}^{+\infty} dE\, {\mathbf{G}}^<(E). \end{aligned} $$
In general, if a quantity has nonzero expectation values in equilibrium, that one must be subtracted from the final result since it is 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
$$\displaystyle \begin{aligned} \boldsymbol{\rho}_{\mathrm{CD}} = \boldsymbol{\rho}_{\mathrm{neq}} - \boldsymbol{\rho}_{\mathrm{eq}}. \end{aligned} $$
Although the NEGF formalism can include many-body interactions, such as electron-magnon scattering (Mahfouzi and Nikolić 2014) that can affect STT (Zholud et al. 2017; Levy and Fert 2006; Manchon et al. 2010) and SOT (Yasuda et al. 2017; Okuma and Nomura 2017), here the focus is on the usually considered and conceptually simpler elastic transport regime where the lesser GF of a two-terminal junction
$$\displaystyle \begin{aligned} {\mathbf{G}}^<(E)=\mathbf{G}(E) \left[i f_L(E) \boldsymbol{\Gamma}_L(E) + i f_R(E) \boldsymbol{\Gamma}_R(E) \right] {\mathbf{G}}^\dagger(E), \end{aligned} $$
is expressed solely in terms of the retarded GF, the level broadening matrices \(\boldsymbol {\Gamma }_{L,R}(E)=i[\boldsymbol {\Sigma }_{L,R}(E) - \boldsymbol {\Sigma }_{L,R}^\dagger (E)]\) determining the escape rates of electrons into the semi-infinite leads and shifted Fermi functions fL,R(E) = f(E − eVL,R).
For purely computational purposes, the integration in Eq. (8) is typically separated [non-uniquely (Xie et al. 2016)] into the apparent “equilibrium” and current-driven “nonequilibrium” terms (Brandbyge et al. 2002; Sanvito 2011)
$$\displaystyle \begin{aligned} \boldsymbol{\rho}_{\mathrm{neq}} &= -\frac{1}{\pi} \int\limits_{-\infty}^{+\infty} dE \, \mathrm{Im}\, \mathbf{G}(E) f(E-eV_R)\\ &\quad + \frac{1}{2 \pi} \int\limits_{-\infty}^{+\infty}dE \, \mathbf{G}(E) \cdot \boldsymbol{\Gamma}_L(E-eV_L) \cdot {\mathbf{G}}^\dagger(E) \left[ f_L(E) - f_R(E) \right]. \end{aligned} $$
The first “equilibrium” term contains integrand which is analytic in the upper complex plane and can be computed via contour integration (Brandbyge et al. 2002; Areshkin and Nikolić 2010; Ozaki 2007; Karrasch et al. 2010), while the integrand in the second “current-driven” term is a nonanalytic function in the entire complex energy plane so that its integration has to be performed directly along the real axis (Sanvito 2011) between the limits set by the window of nonzero values of fL(E) − fR(E). Although the second term in Eq. (11) contains information about the bias voltage [through the difference fL(E) − fR(E)] and about the lead assumed to be injecting electrons into the device (through ΓL), it cannot (Xie et al. 2016; Mahfouzi and Nikolić 2013) be used as the proper ρCD 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 − eVR)] on the arbitrarily chosen reference potential VR 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,
$$\displaystyle \begin{aligned} \boldsymbol{\rho}_{\mathrm{CD}} \stackrel{?}{=} \frac{eV_b}{2 \pi} {\mathbf{G}}_0(E_F) \cdot \boldsymbol{\Gamma}_L(E_F) \cdot {\mathbf{G}}_0^\dagger(E_F), \end{aligned} $$
has often been used in STT literature (Haney et al. 2007; Heiliger et al. 2008) as the putative but improper (due to being gauge-noninvariant, which is marked by “?” on the top of the equality sign) expression for ρCD. Its usage leads to ambiguous (i.e., dependent on arbitrarily chosen VR) nonequilibrium expectation values.
The 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 Un across the active region of the device interpolating between VL and VR. For example, the second term in Eq. (13) traced with an operator gives equilibrium expectation value governed by the states at EF 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 EF together with sampling of k points for the junctions in Fig. 2.
Figure 4 shows an example of a left-right 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 ρCD in Eq. (12) yields linear-response TFL ∝ Vb 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 TFL 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 TFL in Fig. 4b is contaminated by a portion of equilibrium contribution added to it when using improper ρCD in Eq. (12). On the other hand, since TDL has a zero expectation value in equilibrium, both the proper and improper expressions for ρCD give the same result in Fig. 4.
Fig. 4

The angular dependence of the damping-like (i.e., parallel), |TDL|, and field-like (i.e., perpendicular), |TFL|, components of the STT vector in Fig. 1c in an asymmetric (Oh et al. 2009) FM/I/FM MTJ computed in the linear-response regime at zero temperature using (a) the proper gauge-invariant expression Eq. (13) for ρCD and (b) the improper gauge-noninvariant expression Eq. (12) for ρCD. The FM/I/FM trilayer in Fig. 1a is modeled by TBH (Theodonis et al. 2006) defined on an infinite cubic lattice with a single orbital per site and lattice spacing a. Its insulating barrier has thickness 5 MLs with on-site potential εn = 6.0 eV; the left FM layer is semi-infinite, and the right FM layer is 20 MLs thick. Both FM layers have the same exchange field J = 0.5 eV. The symbol \(\Box =a^2\) denotes unit interfacial area. (Adapted from Mahfouzi and Nikolić 2013)

Note that in the left-right symmetric MTJs, TFL ∝ Vb 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 ρCD in Eq. (12), or the proper one in Eq. (13) but with possible software bug, would give nonzero TFL ≠ 0 in symmetric junctions at small applied Vb 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 TFL ≡ 0
$$\displaystyle \begin{aligned} \boldsymbol{\rho}_{\mathrm{CD}} = \frac{eV_b}{4 \pi} {\mathbf{G}}_0(E_F) \cdot [\boldsymbol{\Gamma}_L(E_F) - \boldsymbol{\Gamma}_R(E_F)]\cdot {\mathbf{G}}_0^\dagger(E_F). \end{aligned} $$

This expressions is obtained by assuming (Mahfouzi and Nikolić 2013) the particular gauge VL = −Vb∕2 = −VR. Such special gauges and the corresponding Fermi surface expressions for ρCD ∝ Vb 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).

In the calculations in Fig. 4, \((T^x_{\mathrm {CD}},T^y_{\mathrm {CD}},T^z_{\mathrm {CD}}) = \mathrm {Tr} \, [\boldsymbol {\rho }_{\mathrm {CD}} \mathbf {T}]\) was first computed using the torque operator 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 Mfree ×Mfixed, and DL (or parallel) torque component, \(T_{\mathrm {DL}}=\sqrt {(T^x_{\mathrm {CD}})^2 +(T^z_{\mathrm {CD}})^2}\) in the direction Mfree × (Mfree ×Mfixed). 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.
Fig. 5

(a) Schematic view of a 1D toy model of MTJ consisting of the left and the right semi-infinite chains of carbon atoms separated by a vacuum gap. (b) Comparison of DL component of STT in this MTJ computed via the spin current divergence algorithm (Theodonis et al. 2006; Wang et al. 2008) in Eq. (21) and by using decomposition of the nonequilibrium density matrix into \( \boldsymbol { \rho }_{\mathrm {neq}}^{\mu \nu }\) contributions in Eqs. (15), (16), (18), (19), and (20). The DL component of STT, computed in both algorithms in the linear-response regime, acts on the right carbon chain whose magnetic moments comprise the free magnetization Mfree rotated by an angle θ with respect to the fixed magnetization Mfixed of the left carbon chain. Since MTJ is left-right symmetric, the FL component of the STT vector is zero in the linear-response regime (Wang et al. 2011; Theodonis et al. 2006; Xiao et al. 2008; Oh et al. 2009; Heiliger and Stiles 2008)

In the case of SOT, TDL ∝Mfree ×f and TDL ∝Mfree × (Mfree ×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 ρCD into contributions whose trace with the torque operator in Eq. (3) directly yields TDL and TFL. Such decomposition was achieved in Mahfouzi et al. (2016), using adiabatic expansion of Eq. (4) in the powers of dMfreedt 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
$$\displaystyle \begin{aligned} \boldsymbol{\rho}^{\mathrm{oo}}_{\mathrm{neq}} & = \frac{1}{8\pi} \int\limits_{-\infty}^{+\infty} dE \, \left[ f_L(E) - f_R(E) \right] \left(\mathbf{G} \boldsymbol{ \Gamma}_L {\mathbf{G}}^\dagger - {\mathbf{G}}^\dagger \boldsymbol{\Gamma}_L \mathbf{G}\right.\\ &\left.\quad- \mathbf{G} \boldsymbol{\Gamma}_R {\mathbf{G}}^\dagger + {\mathbf{G}}^\dagger \boldsymbol{\Gamma}_R \mathbf{G} \right), {} \end{aligned} $$
$$\displaystyle \begin{aligned} \boldsymbol{\rho}^{\mathrm{oe}}_{\mathrm{neq}} & = \frac{1}{8\pi} \int\limits_{-\infty}^{+\infty} dE \, \left[ f_L(E) \,{-}\, f_R(E) \right] \left( \mathbf{G} \boldsymbol{ \Gamma}_L {\mathbf{G}}^\dagger \,{+}\, {\mathbf{G}}^\dagger \boldsymbol{\Gamma}_L \mathbf{G} \,{-}\, \mathbf{G} \boldsymbol{\Gamma}_R {\mathbf{G}}^\dagger \,{-}\, {\mathbf{G}}^\dagger \boldsymbol{\Gamma}_R \mathbf{G} \right), {} \end{aligned} $$
$$\displaystyle \begin{aligned} \boldsymbol{\rho}^{\mathrm{eo}}_{\mathrm{neq}} & = \frac{1}{8\pi} \int\limits_{-\infty}^{+\infty} dE \, \left[ f_L(E) + f_R(E) \right] \left(\mathbf{G} \boldsymbol{ \Gamma}_L {\mathbf{G}}^\dagger - {\mathbf{G}}^\dagger \boldsymbol{\Gamma}_L \mathbf{G}\right. \end{aligned} $$
$$\displaystyle \begin{aligned} &\quad \left.+ \mathbf{G} \boldsymbol{\Gamma}_R {\mathbf{G}}^\dagger - {\mathbf{G}}^\dagger \boldsymbol{\Gamma}_R \mathbf{G} \right) \equiv 0, {} \end{aligned} $$
$$\displaystyle \begin{aligned} \boldsymbol{\rho}^{\mathrm{ee}}_{\mathrm{neq}} & = \frac{1}{8\pi} \int\limits_{-\infty}^{+\infty} dE \, \left[ f_L(E) + f_R(E) \right] \left(\mathbf{G} \boldsymbol{ \Gamma}_L {\mathbf{G}}^\dagger + {\mathbf{G}}^\dagger \boldsymbol{\Gamma}_L \mathbf{G} + \mathbf{G} \boldsymbol{\Gamma}_R {\mathbf{G}}^\dagger + {\mathbf{G}}^\dagger \boldsymbol{\Gamma}_R \mathbf{G} \right).{} \end{aligned} $$
The four terms are labeled by being odd (o) or even (e) under inverting bias polarity (first index) or time (second index) (Mahfouzi et al. 2016). The terms \(\boldsymbol { \rho }^{\mathrm {oo}}_{\mathrm {neq}}\) and \(\boldsymbol {\rho }^{\mathrm {oe}}_{\mathrm {neq}}\) depend on fL(E) − fR(E) and, therefore, are nonzero only in nonequilibrium generated by the bias voltage Vb ≠ 0 which drives the steady-state current. Using an identity from the NEGF formalism (Stefanucci and van Leeuwen 2013), G( ΓL + ΓR)G = i(G −G), reveals that \(\boldsymbol {\rho }^{\mathrm {eo}}_{\mathrm {neq}} \equiv 0\) and
$$\displaystyle \begin{aligned} \boldsymbol{\rho}^{\mathrm{ee}}_{\mathrm{neq}} = -\frac{1}{2\pi} \int\limits_{-\infty}^{+\infty} dE\, [f_L(E) + f_R(E)] \mathrm{Im}\, \mathbf{G}. \end{aligned} $$
Thus, \(\boldsymbol {\rho }^{\mathrm {ee}}_{\mathrm {neq}}\) term is nonzero even in equilibrium where it becomes identical to the equilibrium density matrix in Eq. (7), \(V_b = 0 \Rightarrow \boldsymbol {\rho }^{\mathrm {ee}}_{\mathrm {neq}} \equiv \boldsymbol {\rho }_{\mathrm {eq}}\). Since \(\boldsymbol { \rho }^{\mathrm {oo}}_{\mathrm {neq}}\) is odd under time reversal, its trace with the torque operator in Eq. (3) yields DL component of STT (which depends on three magnetization vectors and it is, therefore, also odd) and FL component of SOT (which depends on one magnetization vector and it is, therefore, also odd). Similarly trace of \(\boldsymbol {\rho }^{\mathrm {oe}}_{\mathrm {neq}}\) with the torque operator in Eq. (3) yields FL component of STT and DL component of SOT (Mahfouzi et al. 2016).

In the linear-response regime, pertinent to calculations of STT in spin valves and SOT in FM/spin-orbit-coupled-material bilayers, fL(E) − fR(E) → (−∂f∂E)eVb. This confines integration in \(\boldsymbol {\rho }^{\mathrm {oo}}_{\mathrm {neq}}\) and \(\boldsymbol {\rho }^{\mathrm {oe}}_{\mathrm {neq}}\) expressions to a shell of few kBT 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 Vb.

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 fp(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 Emin (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 Emin 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.

The algorithm based on Eqs. (15), (16), (18), (19), and (20) is compared to often employed spin current divergence algorithm (Theodonis et al. 2006; Wang et al. 2008; Manchon et al. 2008) using a toy model of 1D MTJ, prior to its application to STT calculations in Sect. 3 and SOT calculations in Sect. 4. The model, illustrated in Fig. 5a where the left and right semi-infinite chains of carbon atoms are separated by a vacuum gap, is described by the collinear DFT Hamiltonian implemented in ATK package (Atomistix Toolkit 2017) using single-zeta polarized (Junquera et al. 2001) orbitals on each atom, Ceperley-Alder (Ceperley and Alder 1980) parametrization of the local spin density approximation for the XC functional (Ceperley and Alder 1980), and norm-conserving pseudopotentials accounting for electron-core interactions. In the absence of spin-flip processes by impurities and magnons or SOC, the STT vector at site 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
$$\displaystyle \begin{aligned} T^\alpha_{\mathrm{CD}} = \sum_{n=1}^{N_{\mathrm{free}}} (I_{n-1,n}^{S_\alpha} - I_{n,n+1}^{S_\alpha}) = I_{0,1}^{S_\alpha} - I_{N_{\mathrm{free}},N_{\mathrm{free}}+1}^{S_\alpha}. \end{aligned} $$
Here \(I_{0,1}^{S_\alpha }\) is the local spin current, carrying spins pointing in the direction 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 Nfree 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, Nfree → 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 ρCD
$$\displaystyle \begin{aligned} I_{n,n+1}^{S_\alpha} =\frac{i}{2} \mathrm{Tr}\, \left[ \boldsymbol{\sigma}_\alpha \left({\mathbf{H}}_{n,n+1} \boldsymbol{\rho}_{\mathrm{CD}}^{n+1,n} - \boldsymbol{ \rho}_{\mathrm{CD}}^{n,n+1} {\mathbf{H}}_{n+1,n} \right) \right]. \end{aligned} $$
Here Hn,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 2Norbital × 2Norbital (2 is for spin and Norbital 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 Mfree and Mfixed. 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 ρCD 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

First-principles quantum transport modeling of STT in spin valves (Haney et al. 2007; Wang et al. 2008) and MTJs (Stamenova et al. 2017; Heiliger and Stiles 2008; Jia et al. 2011; Ellis et al. 2017) is typically conducted using an assumption that greatly simplifies computation – noncollinear spins in such systems are described in a 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 Mfixed and Mfree (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 Bxc 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
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} {\mathbf{B}}_{\mathrm{XC}}^x &\displaystyle =&\displaystyle 2 \cdot \mathrm{Re} \, {\mathbf{H}}^{\uparrow \downarrow} , \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathbf{B}}_{\mathrm{XC}}^y &\displaystyle =&\displaystyle -2 \cdot \mathrm{Im} \, {\mathbf{H}}^{\uparrow \downarrow}, \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathbf{B}}_{\mathrm{XC}}^z &\displaystyle =&\displaystyle {\mathbf{H}}^{\uparrow\uparrow} - {\mathbf{H}}^{\downarrow\downarrow}. \end{array} \end{aligned} $$
Since LCAO basis sets |ϕn〉 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
$$\displaystyle \begin{aligned} {\mathbf{T}}_{\mathrm{CD}}=\mathrm{Tr}\, [\boldsymbol{\rho}_{\mathrm{CD}} \boldsymbol{\sigma} \times {\mathbf{D}}^{-1}{\mathbf{B}}_{\mathrm{XC}}], \end{aligned} $$
requires to use the identity operator \(\mathbf {1} = \sum _{ij}|\phi _i\rangle D_{ij}^{-1} \langle \phi _j|\) which inserts 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 meq(r) = 〈r|ρeqσ|r〉 in ncDFT is replaced by mCD(r) = 〈r|ρCDσ|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 meq ×BXC(r) can be nonzero locally which gives rise to equilibrium torque on the free FM layer that has to be removed by subtracting ρeq to obtain ρCD in Eq. (9) and plug it into Eq. (27).
Fig. 6

(a), (b) Transmission function, defined in Eq. (28), for Co/Cu(9 ML)/Co spin valve illustrated in Fig. 2a at two selected transverse wave vectors k, as well as (c) summed, as in Eq. (29), over all k vectors within the 2D BZ. (d)–(f) The corresponding layer-resolved Cartesian components of the STT vector at an angle θ = 90 between the magnetizations Mfixed of the left Co layer acting as a polarizer and Mfree of the right Co layer receiving the torque. The mesh of k equally spaced points employed in (c) is 51 × 51, and in (f) it is 201 × 201. The symbol \(\Box =a^2\) denotes unit interfacial area, where a = 2.52 Å is the lattice constant of a common 2D hexagonal unit cell of Co/Cu bilayer (with a lattice mismatch of about 0.65%)

Fig. 7

The DL component of STT vector in symmetric Co/Cu(9 ML)/Co spin valve in Fig. 2a driven by current at small bias voltage Vb as a function of angle θ between the magnetizations Mfixed of the left Co layer and Mfree of the right Co layer. Its value at θ = 90 is the sum of layer-resolved STT shown in Fig. 6f. The non-scf curve is computed from Eq. (26) using the rigid approximation (Haney et al. 2007; Wang et al. 2008; Heiliger and Stiles 2008; Jia et al. 2011), where the spin valve is described by collinear DFT Hamiltonian and magnetic moments of the fixed Co layer are rotated by an angle θ with respect to magnetic moments in the free Co layer. Conversely, in the computation of scf curve, the DFT Hamiltonian from the rigid approximation serves as the first iteration leading toward converged self-consistent ncDFT Hamiltonian of the spin valve

Fig. 8

(a) The arrangement of Bi and Se atoms within a supercell of Bi2Se3 thin film (infinite in the x- and the y-directions) consisting of five quintuple layers (one such layer contains three Se layers strongly bonded to two Bi layers in between) and total thickness ≈5 nm along the z-axis. (b) The vector field of current-driven nonequilibrium spin density SCD(r) within selected planes shown in (a), generated by injection of unpolarized charge current along the x-axis. The planes 1 and 3 correspond to the top and bottom metallic surfaces of Bi2Se3 thin film, while plane 2 resides in the bulk at a distance d ≈ 0.164 nm away from plane 1. (c) The vector fields in (b) projected onto each of the selected planes in (a). The real-space grid of points in panels (b) and (c) has spacing ≃ 0.4 Å. (Adapted from Chang et al. 2015)

Fig. 9

(a) Schematic view of an ultrathin FM layer comprised of few MLs, modeled as infinite square tight-binding lattices (with lattice constant a) in the xy-plane. The bottom layer 0 is assumed to be in direct contact with a spin-orbit-coupled material like TMD. (b)–(g) The layer-resolved FL and DL components of SOT vector per unit area \(\Box =a^2\) for the device in (a) where FM film is 4 MLs thick in all panels except (d) where the thickness is 10 MLs. The bottom layer 0 is assumed to host the Rashba SOC of strength γSO = 0.2 eV, induced by the spin-orbit proximity effect from monolayer-TMD (which is assumed to be insulating and, therefore, not explicitly considered), as illustrated in Fig. 3h for Co/monolayer-MoS2. This is twice as strong as the exchange field in the FM layer, γSOJ = 2 in Eq. (31). In panels (e) and (f) the Rashba SOC was also added in layers l > 0 with strength 50% smaller than in layer l − 1 (i.e., γSO = 0.2, 0.1, 0.05, and 0.025 eV in layers 0–3, respectively). Panel (g) plots FL and DL components of SOT summed over all layers as a function of the strength of a homogeneous on-site potential Un = U0 in Eq. (33) added into the central region of the thickness of 5 sites along the x-axis. The green square and dot in panel (g) mark SOT components at U0 = 0.5 eV whose layer-resolved values are shown in panels (c) and (f), respectively. The temperature is set at T = 300 K, Fermi energy is EF = 1 eV, and a grid of \(N_{k_y}=2000\) equally spaced ky-points was used to sample periodically repeated lattice in the y-direction. The red and blue arrows in the insets of panels (b), (c), (e), and (f) denote Mfree ×ey and Mfree × (Mfree ×ey) unit vectors along the direction of FL and DL components of SOT, respectively. The quad-graphs in the insets of panels (b), (c), (e), (f), and (g) signify the presence or absence of the Rashba SOC within layers 0–3 using the shade of the corresponding line

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 Mfree, 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.

The propagating vs. evanescent states are identified by finite (Fig. 6b) vs. vanishing (Fig. 6a), respectively, k-resolved transmission function obtained from the Landauer formula in terms of NEGFs (Stefanucci and van Leeuwen 2013)
$$\displaystyle \begin{aligned} T(E,{\mathbf{k}}_\parallel) = \mathrm{Tr}\, [\boldsymbol{\Gamma}_R(E,{\mathbf{k}}_\parallel) {\mathbf{G}}_0(E,{\mathbf{k}}_\parallel) \boldsymbol{\Gamma}_L(E,{\mathbf{k}}_\parallel) {\mathbf{G}}_0^\dagger(E,{\mathbf{k}}_\parallel)], \end{aligned} $$
where the transverse wave vector k is conserved in the absence of disorder. The total transmission function per unit interfacial area is then evaluated using ( ΩBZ is the area of sampled 2D BZ)
$$\displaystyle \begin{aligned} T(E) = \frac{1}{\Omega_{\mathrm{BZ}}} \int\limits_{\mathrm{2D \ BZ}} T(E,{\mathbf{k}}_\parallel), \end{aligned} $$
as shown in Fig. 6c. The transmission function in Fig. 6a at 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 Mfixed and Mfree 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, ρCD 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 AlOx) 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.

An example of first-principles quantum transport modeling of the Edelstein effect is shown in Fig. 8 for the case of the metallic surface of Bi2Se3 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
$$\displaystyle \begin{aligned} {\mathbf{S}}_{\mathrm{CD}} = \frac{\hbar}{2} \mathrm{Tr}\, [\boldsymbol{\rho}_{\mathrm{CD}} \boldsymbol{\sigma}]. \end{aligned} $$
In the case of simplistic Hamiltonians – such as the Rashba one for 2D electron gas (Winkler 2003), \(\hat {H}_{\mathrm {Rashba}}=(\hat {p}_x^2+\hat {p}_y^2)/2m^* + \frac {\alpha _{\mathrm {SO}}}{\hbar }(\hat {\sigma }_x \hat {p_y} - \hat {\sigma }_y \hat {p_x})\), or the Dirac one for the metallic surface of 3D TI, \(\hat {H}_{\mathrm {Dirac}}= v_F (\hat {\sigma }_x \hat {p_y} - \hat {\sigma }_y \hat {p_x})\) – the direction and the magnitude of SCD 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 SCD in the direction transverse to current flow. The same effect is substantially enhanced (Pesin and MacDonald 2012b), by a factor ħvFαSO ≫ 1 where vF is the Fermi velocity in TI and αSO 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 SCD ∝ Vb 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 Bi2Se3 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 SCD(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).

In order to be able to controllably switch SOC on and off in different MLs or to add other types of single-particle interactions, a setup is described in Fig. 9a using TBH defined on an infinite cubic lattice of spacing a with a single orbital per site located at position n = (nxa, nza)
$$\displaystyle \begin{aligned} \hat{H} = \sum_{\mathbf{n}} \varepsilon_{\mathbf{n}} \hat{c}^\dagger_{\mathbf{n}} \hat{c}_{\mathbf{n}} +\sum_{\langle \mathbf{n} \mathbf{n}'\rangle} \hat{c}^\dagger_{\mathbf{n}} {\mathbf{t}}_{\mathbf{n}\mathbf{n}'} \hat{c}_{\mathbf{n'}} - J \sum_{\mathbf{n}} \hat{c}^\dagger_{\mathbf{n}} {\mathbf{M}}_{\mathrm{free}} \cdot{\boldsymbol \sigma} \hat{c}_{\mathbf{n}}. \end{aligned} $$
Here \(\hat {c}_{\mathbf {n}}^\dagger =(\hat {c}^\dagger _{\mathbf {n}\uparrow } \ \ \hat {c}^\dagger _{\mathbf {n}\downarrow })\) is the row vector of operators which create electron at site 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
$$\displaystyle \begin{aligned} {\mathbf{t}}_{\mathbf{n}\mathbf{n}'} = \left\{ \begin{array}{ll} -t {\mathbf{I}}_{2}, & \mathrm{for}\ \ \mathbf{n} = \mathbf{n}' + a{\boldsymbol e}_{z},\\ -t {\mathbf{I}}_{2} - i\gamma_{\mathrm{SO}}\sigma_y, & \mathrm{for}\ \ \mathbf{n} = \mathbf{n}' + a{\boldsymbol e}_{x},\\ \end{array} \right. \end{aligned} $$
where γSO = αSO∕2a measures the strength of the Rashba SOC on the lattice (Nikolić et al. 2006), I2 is the unit 2 × 2 matrix, and eα are the unit vectors along the axes of the Cartesian coordinate system. The on-site energy
$$\displaystyle \begin{aligned} \varepsilon_{\mathbf{n}} = \left[U_{\mathbf{n}} - 2t\cos \left(k_y a\right)\right] {\mathbf{I}}_{2} - 2\gamma_{\mathrm{SO}} \sigma_x \sin (k_y a), \end{aligned} $$
includes the on-site potential energy Un (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 ky-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 γSO chosen in all three regions. Thus, γSO is homogeneous within a given ML, and always present γSO = 0.2 eV on layer 0, but it can take different values in other MLs. The Fermi energy is set at EF = 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 Mfree ×ey 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 γSO ≠ 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 γSO on MLs above layer 0 generates additional nonequilibrium spin density SCD ∝ey 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 Un = U0 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 SCD in the direction Mfree ×ey and the corresponding DL component of SOT ∝Mfree × (Mfree ×ey) 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 Un = U0 (except when U0 ≃ 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 (γSO ≠ 0 only on layer 0) and dashed (γSO ≠ 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.



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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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Branislav K. Nikolić
    • 1
    Email author
  • Kapildeb Dolui
    • 1
  • Marko D. Petrović
    • 2
  • Petr Plecháč
    • 2
  • Troels Markussen
    • 3
  • Kurt Stokbro
    • 3
  1. 1.Department of Physics and AstronomyUniversity of DelawareNewarkUSA
  2. 2.Department of Mathematical SciencesUniversity of DelawareNewarkUSA
  3. 3.Synopsys QuantumWiseCopenhagenDenmark

Section editors and affiliations

  • Stefano Sanvito
    • 1
  1. 1.Department of PhysicsTrinity CollegeDublinIreland

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