Numerical Analysis and Innovative Simulation
Techniques for Designing Advanced MRAM
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Chapter 6 Spin-Transfer Torques
in Ultra-Scaled MRAM Cells
As introduced in Section 3.3, the torque generated by the polarization of electrons passing through the ferromagnetic layers in an MTJ stack drives the writing process in modern
MRAM cells. While the standard Slonczewski macro-spin models provide fundamental insight into STT dynamics, they typically assume spatially uniform magnetization or neglect the complex interplay of spin diffusion in
ultra-scaled geometries.
This chapter presents a comprehensive FEM-based modeling and simulation approach for computing spin-transfer torques in ultra-scaled MRAM cells. This approach builds directly upon the computational framework established in
Chapter 4, utilizing the hybrid FE-BEM method for magnetostatics and employing the time integration schemes detailed in Chapter 5 to resolve the stiff dynamics of the coupled system.
The methodology relies on the coupled spin and charge drift-diffusion formalism, which provides a unified framework for computing both the interface-induced Slonczewski torques and the bulk Zhang–Li torques arising from
magnetization textures. The drift-diffusion approach, which accounts only for semiclassical transport properties, is extended to incorporate tunneling across an MTJ via appropriate boundary conditions at the tunnel-barrier
interfaces. Furthermore, to address the complexity of modern SAFs and composite free layers, the solver is extended to include a numerical implementation of IEC.
The presented approach reproduces both the angular and the voltage dependencies expected in MTJs. It also predicts an interdependence between the torque acting in the presence of magnetization gradients and the tunneling
spin-current. The solver is then applied to simulate the magnetization dynamics of ultra-scaled multi-layered structures with composite ferromagnetic layers.
The switching process can be simulated by solving the LLG equation for magnetization dynamics, including a term describing the torque acting on the magnetization. A more complete description of the process, beyond the
Slonczewski approximation, can be obtained by computing the non-equilibrium spin-accumulation \(\mathbf {S}\) across the whole structure. The general expression for the torque term is:
\(\seteqnumber{0}{6.}{0}\)
\begin{equation}
\mathbf {T}_\mathrm {S} = -\frac {D_\mathrm {e}}{\lambda _J^2}\mathbf {m}\times \mathbf {S} - \frac {D_\mathrm {e}}{\lambda _\varphi ^2}\mathbf {m}\times (\mathbf {m}\times \mathbf {S}),
\label {eq:torque-general}
\end{equation}
where \(D_\mathrm {e}\) is the electron diffusion coefficient. The first term describes precession around the exchange field and is characterized by the exchange length \(\lambda _J\). The second term describes the dephasing
process of the spins of the conducting electrons, characterized by the dephasing length \(\lambda _\varphi \).
The spin-accumulation \(\mathbf {S}\) describes the deviation of the polarization of the conducting electrons from the equilibrium configuration created by a charge-current density \(\mathbf {J}_\mathrm {C}\), expressed in
units of the transported magnetic moment (\(\si {\ampere \per \meter }\)). By definition, \(\mathbf {S}\) is non-zero only when an electric current flows through the system. A solution for \(\mathbf {S}\) in all
non-magnetic and ferromagnetic layers of an MRAM cell can be obtained by means of the spin and charge drift-diffusion formalism.
6.1 Spin and Charge Drift-Diffusion Formalism
6.1.1 Derivation of the Transport Equations
Valet and Fert [179] first derived the spin and charge drift-diffusion equations for a spin-valve structure with collinear (parallel or anti-parallel) magnetization in the ferromagnetic layers. Zhang, Levy, and Fert [35]
later reported a formulation of the equations to compute \(\mathbf {S}\) in the presence of arbitrary magnetization orientations.
In a magnetic multilayer with the current perpendicular to the plane of the layer, along the \(x\)-direction, the linear response of the current to the electric field can be written in spinor form as:
\(\seteqnumber{0}{6.}{1}\)
\begin{equation}
\hat {j} = \hat {\sigma }\hat {E} + \hat {D}\frac {\partial \hat {n}}{\partial x}, \label {eq:spinor-current}
\end{equation}
where \(\hat {E}\) is the electric field, which can be expressed in terms of the electric potential as \(\hat {E} = -\partial \hat {V}/\partial x\), and \(\hat {j}\), \(\hat {\sigma }\), \(\hat {D}\), and \(\hat {n}\)
are the spinor matrices representing the current density, the conductivity, the diffusion coefficient, and the accumulation at a given position, respectively. For a metal, the diffusion constant and the conductivity are connected via
the Einstein relation \(\hat {D} = \hat {\sigma }/e^2\hat {N}\), with \(\hat {N}\) the spin-dependent density of states at the Fermi level.
These matrices can be expressed in terms of the vector of
Pauli matrices \(\bm {\sigma } = (\sigma _1, \sigma _2, \sigma _3)\):
\(\seteqnumber{0}{6.}{2}\)
\begin{align}
\hat {n} &= nI + \mathbf {S} \cdot \bm {\sigma }, \label {eq:spinor-n}\\ \hat {\sigma } &= \sigma I + \bm {\sigma }_\sigma \cdot \bm {\sigma }, \label {eq:spinor-sigma}\\ \hat {D}
&= D_\mathrm {e} I + \mathbf {D} \cdot \bm {\sigma }, \label {eq:spinor-D}
\end{align}
where \(n\) is the charge accumulation, \(\sigma \) is related to the conductivity, and \(D_\mathrm {e}\) is related to the electron diffusion coefficient. The vectors \(\bm {\sigma }_\sigma \) and \(\mathbf {D}\) correspond
to the different conductivities and diffusion constants for the majority and minority electrons, denoted \(\sigma _+\), \(\sigma _-\), \(D_+\), and \(D_-\), respectively. This difference is characterized by the conductivity
polarization parameter \(\beta _\sigma = (\sigma _+ - \sigma _-)/(\sigma _+ + \sigma _-)\) and the diffusion polarization parameter \(\beta _D = (D_+ - D_-)/(D_+ + D_-)\).
6.1.2 Three-Dimensional Formulation
Generalizing the one-dimensional formulation to three dimensions yields the following expressions for the charge-current density vector \(\mathbf {J}_\mathrm {C}\) and the spin-current tensor \(\tilde {\mathbf {J}}_\mathrm
{S}\):
\(\seteqnumber{0}{6.}{5}\)
\begin{align}
\mathbf {J}_\mathrm {C} &= \sigma \mathbf {E} + \frac {e}{\mu _\mathrm {B}}\beta _D D_\mathrm {e}(\nabla \mathbf {S})^\mathrm {T}\mathbf {m}, \label {eq:charge-current-3d}\\ \tilde
{\mathbf {J}}_\mathrm {S} &= -\frac {\mu _\mathrm {B}}{e}\beta _\sigma (\mathbf {m} \otimes \sigma \mathbf {E}) - D_\mathrm {e}\nabla \mathbf {S}, \label {eq:spin-current-3d-simplified}
\end{align}
where \(\otimes \) denotes the outer product, \(\mathbf {m}\) is the unit magnetization vector, and the components of \(\tilde {\mathbf {J}}_\mathrm {S}\) indicate the flow of the \(i\)-th component of spin polarization
in the \(j\)-th direction. The term \((\nabla \mathbf {S})^\mathrm {T}\mathbf {m}\) is a vector with components \(\sum _i m_i \partial S_i/\partial x_j\). The factor \(\mu _\mathrm {B}/e\) converts from the
units of electric charge to the units of the spin polarization current density.
6.1.3 Spin-Accumulation Equation
The equation of motion for the spin-accumulation in a ferromagnet is determined by the interaction of the local magnetic moments and the spin of the itinerant electrons. This interaction is described by the Hamiltonian term:
\(\seteqnumber{0}{6.}{7}\)
\begin{equation}
H_\mathrm {sd} = -J_\mathrm {sd}\,\mathbf {m}\cdot \bm {\sigma }, \label {eq:sd-hamiltonian}
\end{equation}
where \(J_\mathrm {sd}\) is the coupling strength between local moments and itinerant electrons. From this interaction, the equation of motion for the spin-accumulation is derived as follows:
\(\seteqnumber{0}{6.}{8}\)
\begin{equation}
\frac {d\mathbf {S}}{dt} = -\frac {\mathbf {S}}{\tau _\mathrm {sf}} - \frac {D_\mathrm {e}}{\lambda _J^2}\mathbf {m}\times \mathbf {S}, \label {eq:spin-eom}
\end{equation}
where \(\tau _\mathrm {sf}\) is the spin-flip relaxation time of the conduction electrons. The second term describes the precessional motion of the accumulation due to exchange interaction, which becomes important when the
non-equilibrium spin-accumulation and the local magnetization are not parallel.
The typical time-scales for the spin-accumulation and the magnetization differ significantly: while the former is of the order of picoseconds, the latter is of the order of nanoseconds [35]. For the computation of the spin torque
to be added to the LLG equation, it is thus sufficient to consider a steady-state expression for the spin-accumulation. With \(\partial \mathbf {S}/\partial t = 0\), the equation describing the spin-accumulation becomes:
\(\seteqnumber{0}{6.}{9}\)
\begin{equation}
-\nabla \cdot \tilde {\mathbf {J}}_\mathrm {S} - D_\mathrm {e}\frac {\mathbf {S}}{\lambda _\mathrm {sf}^2} + D_\mathrm {e}\frac {\mathbf {m}\times \mathbf {S}}{\lambda _J^2} = 0, \label
{eq:spin-accumulation-steady}
\end{equation}
where the spin-flip length \(\lambda _\mathrm {sf}\) and the exchange length \(\lambda _J\) are defined as \(\lambda _\mathrm {sf} = \sqrt {D_\mathrm {e}\tau _\mathrm {sf}}\) and \(\lambda _J = \sqrt
{D_\mathrm {e}\hbar /(2J_\mathrm {sd})}\).
6.1.4 Spin Dephasing and Complete Transport Equations
Dephasing arises when, after propagating over a certain distance, different spins have undergone unequal amounts of precession, causing their transverse components to partially cancel. In the context of spin-transfer torque, this
effect originates either from variations in electron velocities across the Fermi surface or from spins that precess at the same rate but reach a given position at different times because of scattering. The combined influence of
precession and dephasing on spin-accumulation has been formulated within the Continuous Random Matrix Theory (CRMT) framework, and the equivalence between CRMT and the spin- and charge-drift-diffusion formalism has
been established.
This equivalence between CRMT and the spin and charge drift-diffusion formalism yields the following relations between the characteristic length scales [180, 181]:
\(\seteqnumber{0}{6.}{10}\)
\begin{align}
\sigma &= \frac {1}{\rho _*(1-\beta ^2)}, \label {eq:conductivity}\\ \lambda _\mathrm {sf}^2 &= \frac {l_\mathrm {sf}^2}{1-\beta ^2}, \label {eq:sf-length}\\ \lambda _J^2 &= \frac
{l_* l_\mathrm {L}}{1-\beta ^2}, \label {eq:exchange-length}\\ \lambda _\varphi ^2 &= \frac {l_\perp }{l_\mathrm {L}}\lambda _J^2, \label {eq:dephasing-length}\\ \lambda ^2 &= \frac
{l_*^2}{1-\beta ^2}, \label {eq:momentum-length}
\end{align}
where \(l_\mathrm {sf}\) is the spin-flip diffusion length, \(l_*\) is the average mean free path of majority and minority electrons defined as \(1/l_* = 1/l_\uparrow + 1/l_\downarrow \), \(l_\mathrm {L}\) is the Larmor
spin precession length, \(l_\perp \) is the transverse spin coherence length, \(\rho _*\) is the spin-averaged resistivity, \(\lambda _\varphi \) is the spin dephasing length, \(\lambda = \sqrt {D_\mathrm {e}\tau }\) is the
momentum relaxation length, \(\tau \) is the time for momentum relaxation, and \(\beta = (l_\uparrow - l_\downarrow )/(l_\uparrow + l_\downarrow )\) is the spin asymmetry parameter with \(l_\uparrow \) and
\(l_\downarrow \) the mean free paths for majority and minority electrons, respectively.
The complete set of three-dimensional spin and charge drift-diffusion equations, including the dephasing term, takes the form [32]:
\(\seteqnumber{0}{6.}{15}\)
\begin{gather}
\mathbf {J}_\mathrm {C} = \sigma \mathbf {E} + \frac {e}{\mu _\mathrm {B}}\beta _D D_\mathrm {e}(\nabla \mathbf {S})^\mathrm {T}\mathbf {m}, \label {eq:full-charge-current}\\[5pt] \tilde
{\mathbf {J}}_\mathrm {S} - \frac {\lambda ^2}{\lambda _J^2}[\mathbf {m}]_\times \tilde {\mathbf {J}}_\mathrm {S} - \frac {\lambda ^2}{\lambda _\varphi ^2}[\mathbf {m}]_{\times \times }\tilde
{\mathbf {J}}_\mathrm {S} = -\frac {\mu _\mathrm {B}}{e}\beta _\sigma \mathbf {m}\otimes (\sigma \mathbf {E}) - D_\mathrm {e}\nabla \mathbf {S}, \label {eq:full-spin-current}\\[5pt] \nabla
\cdot \mathbf {J}_\mathrm {C} = 0, \label {eq:charge-continuity}\\[5pt] -\nabla \cdot \tilde {\mathbf {J}}_\mathrm {S} - D_\mathrm {e}\frac {\mathbf {S}}{\lambda _\mathrm {sf}^2} + D_\mathrm
{e}\frac {\mathbf {m}\times \mathbf {S}}{\lambda _J^2} + D_\mathrm {e}\frac {\mathbf {m}\times (\mathbf {m}\times \mathbf {S})}{\lambda _\varphi ^2} = 0. \label {eq:full-spin-accumulation}
\end{gather}
where \([\mathbf {m}]_\times \) and \([\mathbf {m}]_{\times \times }\) denote the matrices associated with the cross-product and double cross-product with unit magnetization vector \(\mathbf {m}\):
\(\seteqnumber{0}{6.}{19}\)
\begin{equation}
[\mathbf {m}]_\times = \begin{pmatrix} 0 & -m_z & m_y \\ m_z & 0 & -m_x \\ -m_y & m_x & 0 \end {pmatrix}, \quad [\mathbf {m}]_{\times \times } = \begin{pmatrix} m_x^2-1
& m_x m_y & m_x m_z \\ m_x m_y & m_y^2-1 & m_y m_z \\ m_x m_z & m_y m_z & m_z^2-1 \end {pmatrix}. \label {eq:cross-matrices}
\end{equation}
The result \(\nabla \cdot \mathbf {J}_\mathrm {C} = 0\) stems from the absence of electric current sources within the metallic layers, due to the rapid redistribution of any charge imbalance. Equation (6.19) differs from the formulation without dephasing by the inclusion of the spin dephasing term dependent on \(\lambda _\varphi \).
The present work focuses on a finite element implementation that excludes the additional terms on the left-hand side of Equation (6.17), which mix the orthogonal spin-current components depending on the local magnetization orientation. These terms originate from the underlying ballistic nature
of the transverse spin precession or dephasing and rely mainly on the ratio between the momentum relaxation length and the transverse absorption lengths \(\lambda _J\) and \(\lambda _\varphi \). In transition-metal
ferromagnets, these length scales are of the same order of magnitude. This implementation demonstrates that an appropriate treatment of the tunneling layer and tuning of the system parameters reproduce the most crucial torque
properties expected in MTJs, while retaining the ability to obtain all torque contributions across several ferromagnetic layers from a unified formalism.