Computation of Torques
in Magnetic Tunnel Junctions

Chapter A Analytical Solution

In the presence of uniform current density and magnetization in the ferromagnetic layers, analytical expressions for the spin accumulation entering (6.38) can be derived. The generic form of \(\vb {S}\) in NM layers, with the left boundary located at \(x_\text {L}\) and right boundary at \(x_\text {R}\), is given by

\begin{equation} \label {eq:spin_acc_nm_an} \vb {S} = \vb {A}_\text {L} \exp (-\frac {x-x_L}{\lambda _{\text {sf}}}) + \vb {A}_\text {R} \exp (\frac {x-x_R}{\lambda _{\text {sf}}}) \end{equation}

Here, \(\vb {A}_\text {L}\) and \(\vb {A}_\text {R}\) are vectors of real coefficients to be determined. In a ferromagnetic layer with magnetization pointing along x, the expressions for the three components of S are instead

\begin{gather} S_x = G_{\parallel \text {,L}} \exp (-\frac {x-x_\text {L}}{\lambda _\text {sdl}}) + G_{\parallel \text {,R}} \exp (\frac {x-x_\text {R}}{\lambda _\text {sdl}}) \\ S_y = 2 \, \text {Re} \, \pqty {G_{\perp \text {,L}} \exp (-\frac {x-x_\text {L}}{\lambda _\text {+}})} + 2 \, \text {Re} \, \pqty {G_{\perp \text {,R}} \exp (\frac {x-x_\text {R}}{\lambda _\text {+}})} \\ S_z = 2 \, \text {Im} \, \pqty {G_{\perp \text {,L}} \exp (-\frac {x-x_\text {L}}{\lambda _\text {+}})} + 2 \, \text {Im} \, \pqty {G_{\perp \text {,R}} \exp (\frac {x-x_\text {R}}{\lambda _\text {+}})} \end{gather} Here, \(G_{\parallel \text {,L}}\) and \(G_{\parallel \text {,R}}\) are real coefficients, while \(G_{\perp \text {,L}}\) and \(G_{\perp \text {,R}}\) are complex coefficients, to be determined, \(\lambda _{\text {sdl}}=\lambda _\text {sf}/\sqrt {1-\beta _{\sigma }\beta _{D}}\), and \(\lambda _{+}^{-1}=\sqrt {(1/\lambda _{\text {sf}})^2+(1/\lambda _{\varphi })^2-i(1/\lambda _\text {J})^2}\). The expressions can be generalized to a magnetization pointing in a general direction by multiplication with a rotation matrix. In the absence of a left or right boundary (i.e. in the presence of semi-infinite layers), the corresponding terms can be removed from the equations. The coefficients entering both (A.1) and (A.2) must be obtained by imposing boundary conditions at the interfaces between different materials, for both spin accumulation and spin current. Expressions for the spin current can be derived from \(\vb {S}\) by using (6.38b). The systems of equations presented here were all solved symbolically by employing Mathematica to produce the analytical results reported in the main text.

A.1 Five Layers N1|F1|C|F2|N2

The following equations describe continuity conditions for both the spin accumulation and current in a five layer structure, where N1 and N2 are nonmagnetic contacts, F1 is the reference layer, F2 is the free layer, and C is the middle layer, separating F1 and F2. The magnetization vector points in the x-direction in F1, while the magnetization in F2 lies in the xz-plane, forming an angle \(\theta \) with the one in F1. Spin flipping in the middle layer can be removed by letting \(\lambda _{\text {sf}}^{C} \rightarrow \infty \). The set of 24 equations can be employed to find the 20 unknown coefficients, 16 real and 4 complex.

A.1.1 Interface N1|F1

Continuity equations for the first interface, located at \(x=x_\text {F1}\).

Spin accumulation continuity

\begin{gather} x \qq {:} A_1=G_1 + G_2\exp \left (\frac {x_{\text {F1}}-x_\text {C}}{\lambda _{\text {sdl}}^{\text {F1}}}\right ) \\[14pt] y \qq {:} A_2=2 \, \text {Re} \, \left (G_3\right ) + 2 \, \text {Re} \, \left (G_4 \exp \left (\frac {x_{\text {F1}}-x_\text {C}}{\lambda _{+}^{\text {F1}}}\right )\right ) \\[14pt] z \qq {:} A_3=2 \, \text {Im} \, \left (G_3\right ) + 2 \, \text {Im} \, \left (G_4 \exp \left (\frac {x_{\text {F1}}-x_\text {C}}{\lambda _{+}^{\text {F1}}}\right )\right ) \end{gather}

Spin current continuity

\begin{gather} x \qq {:} -\frac {D_\text {e}^{\text {N1}}}{\lambda _{\text {sf}}^{\text {N1}}}A_1=-\beta _{\sigma }^{\text {F1}} \frac {\mu _B}{e} J_\text {C} + \frac {2 D_{0}^{\text {F1}} \left (1-\beta _{\sigma }^{\text {F1}} \beta _{D}^{\text {F1}}\right )}{\lambda _{\text {sdl}}^{\text {F1}}}\left (G_1-G_2 \exp \left (\frac {x_{\text {F1}}-x_\text {C}}{\lambda _{\text {sdl}}^{\text {F1}}}\right )\right ) \\[14pt] y \qq {:} -\frac {D_\text {e}^{\text {N1}}}{\lambda _{\text {sf}}^{\text {N1}}}A_2=2 D_\text {e}^{\text {F1}} \left ( \, \text {Re} \, \left (\frac {G_3}{\lambda _{+}^{\text {F1}}}\right )- \, \text {Re} \, \left (\frac {G_4}{\lambda _{+}^{\text {F1}}} \exp \left (\frac {x_{\text {F1}}-x_\text {C}}{\lambda _{+}^{\text {F1}}}\right )\right )\right ) \\[14pt] z \qq {:} -\frac {D_\text {e}^{\text {N1}}}{\lambda _{\text {sf}}^{\text {N1}}}A_3=2 D_\text {e}^{\text {F1}} \left ( \, \text {Im} \, \left (\frac {G_3}{\lambda _{+}^{\text {F1}}}\right )- \, \text {Im} \, \left (\frac {G_4}{\lambda _{+}^{\text {F1}}} \exp \left (\frac {x_{\text {F1}}-x_\text {C}}{\lambda _{+}^{\text {F1}}}\right )\right )\right ) \end{gather}

A.1.2 Interface F1|C

Continuity equations for the second interface, located at \(x=x_\text {C}\).

Spin accumulation continuity

\begin{gather} x \qq {:} G_1\exp \left (-\frac {x_\text {C}-x_{\text {F1}}}{\lambda _{\text {sdl}}^{\text {F1}}}\right ) + G_2=A_4+A_5\exp \left (\frac {x_\text {C}-x_0}{\lambda _{\text {sf}}^{C}}\right ) \\[14pt] y \qq {:} 2 \, \text {Re} \, \left (G_3 \exp \left (-\frac {x_\text {C}-x_{\text {F1}}}{\lambda _{+}^{\text {F1}}}\right )\right )+2 \, \text {Re} \, \left (G_4\right )=A_6+A_7\exp \left (\frac {x_\text {C}-x_0}{\lambda _{\text {sf}}^{C}}\right ) \\[14pt] z \qq {:} 2 \, \text {Im} \, \left (G_3 \exp \left (-\frac {x_\text {C}-x_{\text {F1}}}{\lambda _{+}^{\text {F1}}}\right )\right )+2 \, \text {Im} \, \left (G_4\right )=A_8+A_9\exp \left (\frac {x_\text {C}-x_0}{\lambda _{\text {sf}}^{C}}\right ) \end{gather}

Spin current continuity

\begin{gather} \nonumber {x \qq {:} -\beta _{\sigma }^{\text {F1}} \frac {\mu _B}{e} J_\text {C}+\frac {D_\text {e}^{\text {F1}} \left (1-\beta _{\sigma }^{\text {F1}} \beta _{D}^{\text {F1}}\right )}{\lambda _{\text {sdl}}^{\text {F1}}}\left (G_1 \exp \left (-\frac {x_\text {C}-x_{\text {F1}}}{\lambda _{\text {sdl}}^{\text {F1}}}\right )-G_2\right )=} \\ =\frac {D_\text {e}^{C}}{\lambda _{\text {sf}}^{C}} \left (A_4-A_5 \exp \left (\frac {x_\text {C}-x_0}{\lambda _{\text {sf}}^{C}}\right )\right ) \\[14pt] \nonumber {y \qq {:} 2 D_\text {e}^{\text {F1}} \left ( \, \text {Re} \, \left (\frac {G_3}{\lambda _{+}^{\text {F1}}} \exp \left (-\frac {x_\text {C}-x_{\text {F1}}}{\lambda _{+}^{\text {F1}}}\right )\right )- \, \text {Re} \, \left (\frac {G_4}{\lambda _{+}^{\text {F1}}}\right )\right )=} \\ =\frac {D_\text {e}^{C}}{\lambda _{\text {sf}}^{C}} \left (A_6-A_7 \exp \left (\frac {x_\text {C}-x_0}{\lambda _{\text {sf}}^{C}}\right )\right ) \\[14pt] \nonumber {z \qq {:} 2 D_\text {e}^{\text {F1}} \left ( \, \text {Im} \, \left (\frac {G_3}{\lambda _{+}^{\text {F1}}} \exp \left (-\frac {x_\text {C}-x_{\text {F1}}}{\lambda _{+}^{\text {F1}}}\right )\right )- \, \text {Im} \, \left (\frac {G_4}{\lambda _{+}^{\text {F1}}}\right )\right )=} \\ =\frac {D_\text {e}^{C}}{\lambda _{\text {sf}}^{C}} \left (A_8-A_9 \exp \left (\frac {x_\text {C}-x_0}{\lambda _{\text {sf}}^{C}}\right )\right ) \end{gather}

A.1.3 Interface C|F2

Continuity equations for the third interface, located at \(x=x_0\).

Spin accumulation continuity

\begin{gather} \nonumber {x \qq {:} A_4 \exp \left (-\frac {x_0-x_\text {C}}{\lambda _{\text {sf}}^{C}}\right )+A_5=\left (G_5+G_6\exp \left (\frac {x_0-x_{\text {F2}}}{\lambda _{\text {sdl}}^{\text {F2}}}\right )\right ) \cos \theta +} \\ -\left (2 \, \text {Im} \, \left (G_7\right ) + 2 \, \text {Im} \, \left (G_8 \exp \left (\frac {x_0-x_{\text {F2}}}{\lambda _{+}^{\text {F2}}}\right )\right )\right ) \sin \theta \\[14pt] y \qq {:} A_6\exp \left (-\frac {x_0-x_\text {C}}{\lambda _{\text {sf}}^{C}}\right )+A_7 = 2 \, \text {Re} \, \left (G_7\right ) + 2 \, \text {Re} \, \left (G_8 \exp \left (\frac {x_0-x_{\text {F2}}}{\lambda _{+}^{\text {F2}}}\right )\right ) \\[14pt] \nonumber {z \qq {:} A_8\exp \left (-\frac {x_0-x_\text {C}}{\lambda _{\text {sf}}^{C}}\right )+A_9=\left (2 \, \text {Im} \, \left (G_7\right ) + 2 \, \text {Im} \, \left (G_8 \exp \left (\frac {x_0-x_{\text {F2}}}{\lambda _{+}^{\text {F2}}}\right )\right )\right )\cos \theta +} \\ +\left (G_5+G_6\exp \left (\frac {x_0-x_{\text {F2}}}{\lambda _{\text {sdl}}^{\text {F2}}}\right )\right ) \sin \theta \end{gather}

Spin current continuity

\begin{gather} \nonumber {x \qq {:} \frac {D_\text {e}^{C}}{\lambda _{\text {sf}}^{C}} \left (A_4 \exp \left (-\frac {x_0-x_\text {C}}{\lambda _{\text {sf}}^{C}}\right )-A_5\right )=} \\ \nonumber {=\left (-\beta _{\sigma }^{\text {F2}} \frac {\mu _B}{e} J_\text {C} + \frac {D_\text {e}^{\text {F2}} \left (1-\beta _{\sigma }^{\text {F2}} \beta _{D}^{\text {F2}}\right )}{\lambda _{\text {sdl}}^{\text {F2}}}\left (G_5-G_6 \exp \left (\frac {x_0-x_{\text {F2}}}{\lambda _{\text {sdl}}^{\text {F2}}}\right )\right )\right ) \cos \theta +} \\ -2 D_\text {e}^{\text {F2}} \left ( \, \text {Im} \, \left (\frac {G_7}{\lambda _{+}^{\text {F2}}}\right )- \, \text {Im} \, \left (\frac {G_8}{\lambda _{+}^{\text {F2}}} \exp \left (\frac {x_0-x_{\text {F2}}}{\lambda _{+}^{\text {F2}}}\right )\right )\right ) \sin \theta \\[14pt] y \qq {:} \nonumber {\frac {D_\text {e}^{C}}{\lambda _{\text {sf}}^{C}} \left (A_6 \exp \left (-\frac {x_0-x_\text {C}}{\lambda _{\text {sf}}^{C}}\right )-A_7\right )=} \\ = 2 D_\text {e}^{\text {F2}} \left ( \, \text {Re} \, \left (\frac {G_7}{\lambda _{+}^{\text {F2}}}\right )- \, \text {Re} \, \left (\frac {G_8}{\lambda _{+}^{\text {F2}}} \exp \left (\frac {x_0-x_{\text {F2}}}{\lambda _{+}^{\text {F2}}}\right )\right )\right ) \\[14pt] \nonumber {z \qq {:} \frac {D_\text {e}^{C}}{\lambda _{\text {sf}}^{C}} \left (A_8 \exp \left (-\frac {x_0-x_\text {C}}{\lambda _{\text {sf}}^{C}}\right )-A_9\right )}= \\ \nonumber {=\left (-\beta _{\sigma }^{\text {F2}} \frac {\mu _B}{e} J_\text {C}+\frac {D_\text {e}^{\text {F2}} \left (1-\beta _{\sigma }^{\text {F2}} \beta _{D}^{\text {F2}}\right )}{\lambda _{\text {sdl}}^{\text {F2}}}\left (G_5-G_6 \exp \left (\frac {x_0-x_{\text {F2}}}{\lambda _{\text {sdl}}^{\text {F2}}}\right )\right ) \right ) \sin \theta +} \\ +2 D_\text {e}^{\text {F2}}\left ( \, \text {Im} \, \left (\frac {G_7}{\lambda _{+}^{\text {F2}}}\right )- \, \text {Im} \, \left (\frac {G_8}{\lambda _{+}^{\text {F2}}} \exp \left (\frac {x_0-x_{\text {F2}}}{\lambda _{+}^{\text {F2}}}\right )\right )\right ) \cos \theta \end{gather}

A.1.4 Interface F2|N2

Continuity equations for the third interface, located at \(x=x_\text {F2}\).

Spin accumulation continuity

\begin{gather} x \qq {:} \nonumber {\left (G_5\exp \left (-\frac {x_{\text {F2}}-x_0}{\lambda _{\text {sdl}}^{\text {F2}}}\right )+G_6\right ) \cos \theta } + \\ -\left (2 \, \text {Im} \, \left (G_7 \exp \left (-\frac {x_{\text {F2}}-x_0}{\lambda _{+}^{\text {F2}}}\right )\right )+2 \, \text {Im} \, \left (G_8\right )\right ) \sin \theta =A_{10} \\[14pt] y \qq {:} 2 \, \text {Re} \, \left (G_7 \exp \left (-\frac {x_{\text {F2}}-x_0}{\lambda _{+}^{\text {F2}}}\right )\right )+2 \, \text {Re} \, \left (G_8\right )=A_{11} \\[14pt] z \qq {:} \nonumber {\left (2 \, \text {Im} \, \left (G_7 \exp \left (-\frac {x_{\text {F2}}-x_0}{\lambda _{+}^{\text {F2}}}\right )\right )+2 \, \text {Im} \, \left (G_8\right )\right )\cos \theta } + \\ + \left (G_5\exp \left (-\frac {x_{\text {F2}}-x_0}{\lambda _{\text {sdl}}^{\text {F2}}}\right )+G_6\right )\sin \theta =A_{12} \end{gather}

Spin current continuity

\begin{gather} \nonumber {x \qq {:} \left (-\beta _{\sigma }^{\text {F2}} \frac {\mu _B}{e} J_\text {C} + \frac {D_\text {e}^{\text {F2}} \left (1-\beta _{\sigma }^{\text {F2}} \beta _{D}^{\text {F2}}\right )}{\lambda _{\text {sdl}}^{\text {F2}}}\left (G_5 \exp \left (-\frac {x_{\text {F2}}-x_0}{\lambda _{\text {sdl}}^{\text {F2}}}\right )-G_6\right ) \right ) \cos \theta +} \\-2 D_\text {e}^{\text {F2}} \left ( \, \text {Im} \, \left (\frac {G_7}{\lambda _{+}^{\text {F2}}} \exp \left (-\frac {x_{\text {F2}}-x_0}{\lambda _{+}^{\text {F2}}}\right )\right )- \, \text {Im} \, \left (\frac {G_8}{\lambda _{+}^{\text {F2}}}\right )\right ) \sin \theta =\frac {D_\text {e}^{\text {N2}}}{\lambda _{\text {sf}}^{\text {N2}}}A_{10} \\[14pt] y \qq {:} 2 D_\text {e}^{\text {F2}} \left ( \, \text {Re} \, \left (\frac {G_7}{\lambda _{+}^{\text {F2}}} \exp \left (-\frac {x_{\text {F2}}-x_0}{\lambda _{+}^{\text {F2}}}\right )\right )- \, \text {Re} \, \left (\frac {G_8}{\lambda _{+}^{\text {F2}}}\right )\right )=\frac {D_\text {e}^{\text {N2}}}{\lambda _{\text {sf}}^{\text {N2}}}A_{11} \\[14pt] \nonumber {z \qq {:} \left (-\beta _{\sigma }^{\text {F2}} \frac {\mu _B}{e} J_\text {C} +\frac {D_\text {e}^{\text {F2}} \left (1-\beta _{\sigma }^{\text {F2}} \beta _{D}^{\text {F2}}\right )}{\lambda _{\text {sdl}}^{\text {F2}}}\left (G_5 \exp \left (-\frac {x_{\text {F2}}-x_0}{\lambda _{\text {sdl}}^{\text {F2}}}\right )-G_6\right ) \right )\sin \theta +} \\+2 D_\text {e}^{\text {F2}} \left ( \, \text {Im} \, \left (\frac {G_7}{\lambda _{+}^{\text {F2}}} \exp \left (-\frac {x_{\text {F2}}-x_0}{\lambda _{+}^{\text {F2}}}\right )\right )- \, \text {Im} \, \left (\frac {G_8}{\lambda _{+}^{\text {F2}}}\right )\right ) \cos \theta =\frac {D_\text {e}^{\text {N2}}}{\lambda _{\text {sf}}^{\text {N2}}}A_{12} \end{gather}

A.2 Tunneling Spin Current

In the presence of a tunneling spin current described by (8.2b), the continuity equations for the spin current at the interfaces with the middle layer need to be modified. The new expressions are reported below.

A.2.1 Interface F1|C

Modified spin current continuity

\begin{gather} \nonumber {x \qq {:} -\beta _{\sigma }^{\text {F1}} \frac {\mu _B}{e} J_\text {C}+\frac {D_\text {e}^{\text {F1}} \left (1-\beta _{\sigma }^{\text {F1}} \beta _{D}^{\text {F1}}\right )}{\lambda _{\text {sdl}}^{\text {F1}}}\left (G_1 \exp \left (-\frac {x_\text {C}-x_{\text {F1}}}{\lambda _{\text {sdl}}^{\text {F1}}}\right )-G_2\right )=} \\ =-\frac {a_\text {RL} \, P_\text {RL}+a_\text {FL} \, P_\text {FL} \, \cos \theta }{ 1+P_\text {RL} \, P_\text {FL} \, \cos \theta } \, \frac {\mu _\text {B}}{e} \, J_\text {C} + \frac {D_\text {e}^{C}}{\lambda _{\text {sf}}^{C}} \left (A_4-A_5 \exp \left (\frac {x_\text {C}-x_0}{\lambda _{\text {sf}}^{C}}\right )\right ) \\[14pt] \nonumber {y \qq {:} 2 D_\text {e}^{\text {F1}} \left ( \, \text {Re} \, \left (\frac {G_3}{\lambda _{+}^{\text {F1}}} \exp \left (-\frac {x_\text {C}-x_{\text {F1}}}{\lambda _{+}^{\text {F1}}}\right )\right )- \, \text {Re} \, \left (\frac {G_4}{\lambda _{+}^{\text {F1}}}\right )\right )=} \\ =-\frac {1/2 \, \left ( P_\text {RL} \, P_\text {RL}^\eta -P_\text {FL} \, P_\text {FL}^\eta \right ) \, \sin \theta }{ 1+P_\text {RL} \, P_\text {FL} \, \cos \theta } \, \frac {\mu _\text {B}}{e} \, J_\text {C} + \frac {D_\text {e}^{C}}{\lambda _{\text {sf}}^{C}} \left (A_6-A_7 \exp \left (\frac {x_\text {C}-x_0}{\lambda _{\text {sf}}^{C}}\right )\right ) \\[14pt] \nonumber {z \qq {:} 2 D_\text {e}^{\text {F1}} \left ( \, \text {Im} \, \left (\frac {G_3}{\lambda _{+}^{\text {F1}}} \exp \left (-\frac {x_\text {C}-x_{\text {F1}}}{\lambda _{+}^{\text {F1}}}\right )\right )- \, \text {Im} \, \left (\frac {G_4}{\lambda _{+}^{\text {F1}}}\right )\right )=} \\ =-\frac {a_\text {FL} \, P_\text {FL} \, \sin \theta }{ 1+P_\text {RL} \, P_\text {FL} \, \cos \theta } \, \frac {\mu _\text {B}}{e} \, J_\text {C} + \frac {D_\text {e}^{C}}{\lambda _{\text {sf}}^{C}} \left (A_8-A_9 \exp \left (\frac {x_\text {C}-x_0}{\lambda _{\text {sf}}^{C}}\right )\right ) \end{gather}

A.2.2 Interface C|F2

Modified spin current continuity

\begin{gather} \nonumber {x \qq {:} -\frac {a_\text {RL} \, P_\text {RL}+a_\text {FL} \, P_\text {FL} \, \cos \theta }{ 1+P_\text {RL} \, P_\text {FL} \, \cos \theta } \, \frac {\mu _\text {B}}{e} \, J_\text {C} + \frac {D_\text {e}^{C}}{\lambda _{\text {sf}}^{C}} \left (A_4 \exp \left (-\frac {x_0-x_\text {C}}{\lambda _{\text {sf}}^{C}}\right )-A_5\right )=} \\ \nonumber {=\left (-\beta _{\sigma }^{\text {F2}} \frac {\mu _B}{e} J_\text {C} + \frac {D_\text {e}^{\text {F2}} \left (1-\beta _{\sigma }^{\text {F2}} \beta _{D}^{\text {F2}}\right )}{\lambda _{\text {sdl}}^{\text {F2}}}\left (G_5-G_6 \exp \left (\frac {x_0-x_{\text {F2}}}{\lambda _{\text {sdl}}^{\text {F2}}}\right )\right )\right ) \cos \theta +} \\ -2 D_\text {e}^{\text {F2}} \left ( \, \text {Im} \, \left (\frac {G_7}{\lambda _{+}^{\text {F2}}}\right )- \, \text {Im} \, \left (\frac {G_8}{\lambda _{+}^{\text {F2}}} \exp \left (\frac {x_0-x_{\text {F2}}}{\lambda _{+}^{\text {F2}}}\right )\right )\right ) \sin \theta \\[14pt] y \qq {:} \nonumber {-\frac {1/2 \, \left ( P_\text {RL} \, P_\text {RL}^\eta -P_\text {FL} \, P_\text {FL}^\eta \right ) \, \sin \theta }{ 1+P_\text {RL} \, P_\text {FL} \, \cos \theta } \, \frac {\mu _\text {B}}{e} \, J_\text {C} + \frac {D_\text {e}^{C}}{\lambda _{\text {sf}}^{C}} \left (A_6 \exp \left (-\frac {x_0-x_\text {C}}{\lambda _{\text {sf}}^{C}}\right )-A_7\right )=} \\ = 2 D_\text {e}^{\text {F2}} \left ( \, \text {Re} \, \left (\frac {G_7}{\lambda _{+}^{\text {F2}}}\right )- \, \text {Re} \, \left (\frac {G_8}{\lambda _{+}^{\text {F2}}} \exp \left (\frac {x_0-x_{\text {F2}}}{\lambda _{+}^{\text {F2}}}\right )\right )\right ) \displaybreak \\[14pt] \nonumber {z \qq {:} -\frac {a_\text {FL} \, P_\text {FL} \, \sin \theta }{ 1+P_\text {RL} \, P_\text {FL} \, \cos \theta } \, \frac {\mu _\text {B}}{e} \, J_\text {C} + \frac {D_\text {e}^{C}}{\lambda _{\text {sf}}^{C}} \left (A_8 \exp \left (-\frac {x_0-x_\text {C}}{\lambda _{\text {sf}}^{C}}\right )-A_9\right )}= \\ \nonumber {=\left (-\beta _{\sigma }^{\text {F2}} \frac {\mu _B}{e} J_\text {C}+\frac {D_\text {e}^{\text {F2}} \left (1-\beta _{\sigma }^{\text {F2}} \beta _{D}^{\text {F2}}\right )}{\lambda _{\text {sdl}}^{\text {F2}}}\left (G_5-G_6 \exp \left (\frac {x_0-x_{\text {F2}}}{\lambda _{\text {sdl}}^{\text {F2}}}\right )\right ) \right ) \sin \theta +} \\ +2 D_\text {e}^{\text {F2}}\left ( \, \text {Im} \, \left (\frac {G_7}{\lambda _{+}^{\text {F2}}}\right )- \, \text {Im} \, \left (\frac {G_8}{\lambda _{+}^{\text {F2}}} \exp \left (\frac {x_0-x_{\text {F2}}}{\lambda _{+}^{\text {F2}}}\right )\right )\right ) \cos \theta \end{gather}

A.3 Ballistic Spin Current

When employing (8.11), the continuity equations need to updated to take the additional spin current terms, depending on the momentum relaxation path \(\lambda \), into account. The expressions for the spin accumulation remain the same, with the following change of parameters:

\begin{gather} \lambda _{+}^{-1} = \sqrt {\frac {k_\varphi }{\lambda _\text {d}^2} - \frac {k_\text {J}}{\lambda _\text {J}^2} - i \pqty {\frac {k_\varphi }{\lambda _\text {J}^2} + \frac {k_\text {J}}{\lambda _\text {d}^2}} } \\ k_\text {J} = \pqty {\frac {\lambda }{\lambda _\text {J}}}^2, \qquad k_\varphi = 1 + \pqty {\frac {\lambda }{\lambda _\varphi }}^2, \qquad \lambda _\text {d}^{-1} = \sqrt {\frac {1}{\lambda _{\text {sf}}^2} + \frac {1}{\lambda _{\varphi }^2}} \end{gather}

The updated continuity equations for the spin current, which take the tunneling contributions into account, are reported below.

A.3.1 Interface N1|F1

Ballistic spin current continuity

\begin{gather} x \qq {:} -\frac {D_\text {e}^{\text {N1}}}{\lambda _{\text {sf}}^{\text {N1}}}A_1=-\beta _{\sigma }^{\text {F1}} \frac {\mu _B}{e} J_\text {C} + \frac {2 D_{0}^{\text {F1}} \left (1-\beta _{\sigma }^{\text {F1}} \beta _{D}^{\text {F1}}\right )}{\lambda _{\text {sdl}}^{\text {F1}}}\left (G_1-G_2 \exp \left (\frac {x_{\text {F1}}-x_\text {C}}{\lambda _{\text {sdl}}^{\text {F1}}}\right )\right ) \\[14pt] y \qq {:} \nonumber {-\frac {D_\text {e}^{\text {N1}}}{\lambda _{\text {sf}}^{\text {N1}}}A_2 = 2 D_\text {e}^{\text {F1}} \left ( \, \text {Re} \, \left (\frac {G_3}{\lambda _{+}^{\text {F1}}}\right )- \, \text {Re} \, \left (\frac {G_4}{\lambda _{+}^{\text {F1}}} \exp \left (\frac {x_{\text {F1}}-x_\text {C}}{\lambda _{+}^{\text {F1}}}\right )\right )\right )\frac {k_\varphi ^\text {F1}}{(k_\varphi ^\text {F1})^2+(k_\text {J}^\text {F1})^2}} + \\ - 2 D_\text {e}^{\text {F1}} \left ( \, \text {Im} \, \left (\frac {G_3}{\lambda _{+}^{\text {F1}}}\right )- \, \text {Im} \, \left (\frac {G_4}{\lambda _{+}^{\text {F1}}} \exp \left (\frac {x_{\text {F1}}-x_\text {C}}{\lambda _{+}^{\text {F1}}}\right )\right )\right ) \frac {k_\text {J}^\text {F1}}{(k_\varphi ^\text {F1})^2+(k_\text {J}^\text {F1})^2} \\[14pt] z \qq {:} \nonumber {-\frac {D_\text {e}^{\text {N1}}}{\lambda _{\text {sf}}^{\text {N1}}}A_3=2 D_\text {e}^{\text {F1}} \left ( \, \text {Re} \, \left (\frac {G_3}{\lambda _{+}^{\text {F1}}}\right )- \, \text {Re} \, \left (\frac {G_4}{\lambda _{+}^{\text {F1}}} \exp \left (\frac {x_{\text {F1}}-x_\text {C}}{\lambda _{+}^{\text {F1}}}\right )\right )\right )\frac {k_\text {J}^\text {F1}}{(k_\varphi ^\text {F1})^2+(k_\text {J}^\text {F1})^2}} + \\ + 2 D_\text {e}^{\text {F1}} \left ( \, \text {Im} \, \left (\frac {G_3}{\lambda _{+}^{\text {F1}}}\right )- \, \text {Im} \, \left (\frac {G_4}{\lambda _{+}^{\text {F1}}} \exp \left (\frac {x_{\text {F1}}-x_\text {C}}{\lambda _{+}^{\text {F1}}}\right )\right )\right ) \frac {k_\varphi ^\text {F1}}{(k_\varphi ^\text {F1})^2+(k_\text {J}^\text {F1})^2} \end{gather}

A.3.2 Interface F1|C

Ballistic spin current continuity

\begin{gather} \nonumber {x \qq {:} -\beta _{\sigma }^{\text {F1}} \frac {\mu _B}{e} J_\text {C}+\frac {D_\text {e}^{\text {F1}} \left (1-\beta _{\sigma }^{\text {F1}} \beta _{D}^{\text {F1}}\right )}{\lambda _{\text {sdl}}^{\text {F1}}}\left (G_1 \exp \left (-\frac {x_\text {C}-x_{\text {F1}}}{\lambda _{\text {sdl}}^{\text {F1}}}\right )-G_2\right )=} \\ =-\frac {a_\text {RL} \, P_\text {RL}+a_\text {FL} \, P_\text {FL} \, \cos \theta }{ 1+P_\text {RL} \, P_\text {FL} \, \cos \theta } \, \frac {\mu _\text {B}}{e} \, J_\text {C} + \frac {D_\text {e}^{C}}{\lambda _{\text {sf}}^{C}} \left (A_4-A_5 \exp \left (\frac {x_\text {C}-x_0}{\lambda _{\text {sf}}^{C}}\right )\right ) \\[14pt] \nonumber {y \qq {:} 2 D_\text {e}^{\text {F1}} \left ( \, \text {Re} \, \left (\frac {G_3}{\lambda _{+}^{\text {F1}}} \exp \left (-\frac {x_\text {C}-x_{\text {F1}}}{\lambda _{+}^{\text {F1}}}\right )\right )- \, \text {Re} \, \left (\frac {G_4}{\lambda _{+}^{\text {F1}}}\right )\right ) \frac {k_\varphi ^\text {F1}}{(k_\varphi ^\text {F1})^2+(k_\text {J}^\text {F1})^2} - } \\ \nonumber {+ 2 D_\text {e}^{\text {F1}} \left ( \, \text {Im} \, \left (\frac {G_3}{\lambda _{+}^{\text {F1}}} \exp \left (-\frac {x_\text {C}-x_{\text {F1}}}{\lambda _{+}^{\text {F1}}}\right )\right )- \, \text {Im} \, \left (\frac {G_4}{\lambda _{+}^{\text {F1}}}\right )\right ) \frac {k_\text {J}^\text {F1}}{(k_\text {J}^\text {F1})^2+(k_\text {J}^\text {F1})^2} = } \\ =-\frac {1/2 \, \left ( P_\text {RL} \, P_\text {RL}^\eta -P_\text {FL} \, P_\text {FL}^\eta \right ) \, \sin \theta }{ 1+P_\text {RL} \, P_\text {FL} \, \cos \theta } \, \frac {\mu _\text {B}}{e} \, J_\text {C} + \frac {D_\text {e}^{C}}{\lambda _{\text {sf}}^{C}} \left (A_6-A_7 \exp \left (\frac {x_\text {C}-x_0}{\lambda _{\text {sf}}^{C}}\right )\right ) \\[14pt] \nonumber {z \qq {:} 2 D_\text {e}^{\text {F1}} \left ( \, \text {Re} \, \left (\frac {G_3}{\lambda _{+}^{\text {F1}}} \exp \left (-\frac {x_\text {C}-x_{\text {F1}}}{\lambda _{+}^{\text {F1}}}\right )\right )- \, \text {Re} \, \left (\frac {G_4}{\lambda _{+}^{\text {F1}}}\right )\right ) \frac {k_\text {J}^\text {F1}}{(k_\varphi ^\text {F1})^2+(k_\text {J}^\text {F1})^2} + } \\ \nonumber {+ 2 D_\text {e}^{\text {F1}} \left ( \, \text {Im} \, \left (\frac {G_3}{\lambda _{+}^{\text {F1}}} \exp \left (-\frac {x_\text {C}-x_{\text {F1}}}{\lambda _{+}^{\text {F1}}}\right )\right )- \, \text {Im} \, \left (\frac {G_4}{\lambda _{+}^{\text {F1}}}\right )\right ) \frac {k_\varphi ^\text {F1}}{(k_\text {J}^\text {F1})^2+(k_\text {J}^\text {F1})^2} = } \\ =-\frac {a_\text {FL} \, P_\text {FL} \, \sin \theta }{ 1+P_\text {RL} \, P_\text {FL} \, \cos \theta } \, \frac {\mu _\text {B}}{e} \, J_\text {C} + \frac {D_\text {e}^{C}}{\lambda _{\text {sf}}^{C}} \left (A_8-A_9 \exp \left (\frac {x_\text {C}-x_0}{\lambda _{\text {sf}}^{C}}\right )\right ) \end{gather}

A.3.3 Interface C|F2

Ballistic spin current continuity

\begin{gather} \nonumber {x \qq {:} -\frac {a_\text {RL} \, P_\text {RL}+a_\text {FL} \, P_\text {FL} \, \cos \theta }{ 1+P_\text {RL} \, P_\text {FL} \, \cos \theta } \, \frac {\mu _\text {B}}{e} \, J_\text {C} + \frac {D_\text {e}^{C}}{\lambda _{\text {sf}}^{C}} \left (A_4 \exp \left (-\frac {x_0-x_\text {C}}{\lambda _{\text {sf}}^{C}}\right )-A_5\right )=} \\ \nonumber {=\left (-\beta _{\sigma }^{\text {F2}} \frac {\mu _B}{e} J_\text {C} + \frac {D_\text {e}^{\text {F2}} \left (1-\beta _{\sigma }^{\text {F2}} \beta _{D}^{\text {F2}}\right )}{\lambda _{\text {sdl}}^{\text {F2}}}\left (G_5-G_6 \exp \left (\frac {x_0-x_{\text {F2}}}{\lambda _{\text {sdl}}^{\text {F2}}}\right )\right )\right ) \cos \theta +} \\ \nonumber {-2 D_\text {e}^{\text {F2}} \left (\left ( \, \text {Re} \, \left (\frac {G_7}{\lambda _{+}^{\text {F2}}}\right )- \, \text {Re} \, \left (\frac {G_8}{\lambda _{+}^{\text {F2}}} \exp \left (\frac {x_0-x_{\text {F2}}}{\lambda _{+}^{\text {F2}}}\right )\right )\right )\frac {k_\text {J}^\text {F2}}{(k_\text {J}^\text {F2})^2+(k_\text {J}^\text {F2})^2} + \right .} \\ \left . \left ( \, \text {Im} \, \left (\frac {G_7}{\lambda _{+}^{\text {F2}}}\right )- \, \text {Im} \, \left (\frac {G_8}{\lambda _{+}^{\text {F2}}} \exp \left (\frac {x_0-x_{\text {F2}}}{\lambda _{+}^{\text {F2}}}\right )\right )\right ) \frac {k_\text {J}^\text {F2}}{(k_\varphi ^\text {F2})^2+(k_\text {J}^\text {F2})^2} \right )\sin \theta \displaybreak \\[14pt] \nonumber {y \qq {:} -\frac {1/2 \, \left ( P_\text {RL} \, P_\text {RL}^\eta -P_\text {FL} \, P_\text {FL}^\eta \right ) \, \sin \theta }{ 1+P_\text {RL} \, P_\text {FL} \, \cos \theta } \, \frac {\mu _\text {B}}{e} \, J_\text {C} + \frac {D_\text {e}^{C}}{\lambda _{\text {sf}}^{C}} \left (A_6 \exp \left (-\frac {x_0-x_\text {C}}{\lambda _{\text {sf}}^{C}}\right )-A_7\right )=} \\ \nonumber {= 2 D_\text {e}^{\text {F2}} \left ( \, \text {Re} \, \left (\frac {G_7}{\lambda _{+}^{\text {F2}}}\right )- \, \text {Re} \, \left (\frac {G_8}{\lambda _{+}^{\text {F2}}} \exp \left (\frac {x_0-x_{\text {F2}}}{\lambda _{+}^{\text {F2}}}\right )\right )\right )\frac {k_\varphi ^\text {F2}}{(k_\varphi ^\text {F2})^2+(k_\text {J}^\text {F2})^2} + } \\ - 2 D_\text {e}^{\text {F2}}\left ( \, \text {Im} \, \left (\frac {G_7}{\lambda _{+}^{\text {F2}}}\right )- \, \text {Im} \, \left (\frac {G_8}{\lambda _{+}^{\text {F2}}} \exp \left (\frac {x_0-x_{\text {F2}}}{\lambda _{+}^{\text {F2}}}\right )\right )\right )\frac {k_\text {J}^\text {F2}}{(k_\varphi ^\text {F2})^2+(k_\text {J}^\text {F2})^2} \\[14pt] \nonumber {z \qq {:} -\frac {a_\text {FL} \, P_\text {FL} \, \sin \theta }{ 1+P_\text {RL} \, P_\text {FL} \, \cos \theta } \, \frac {\mu _\text {B}}{e} \, J_\text {C} + \frac {D_\text {e}^{C}}{\lambda _{\text {sf}}^{C}} \left (A_8 \exp \left (-\frac {x_0-x_\text {C}}{\lambda _{\text {sf}}^{C}}\right )-A_9\right )}= \\ \nonumber {=\left (-\beta _{\sigma }^{\text {F2}} \frac {\mu _B}{e} J_\text {C}+\frac {D_\text {e}^{\text {F2}} \left (1-\beta _{\sigma }^{\text {F2}} \beta _{D}^{\text {F2}}\right )}{\lambda _{\text {sdl}}^{\text {F2}}}\left (G_5-G_6 \exp \left (\frac {x_0-x_{\text {F2}}}{\lambda _{\text {sdl}}^{\text {F2}}}\right )\right ) \right ) \sin \theta +} \\ \nonumber {+2 D_\text {e}^{\text {F2}}\left (\left ( \, \text {Re} \, \left (\frac {G_7}{\lambda _{+}^{\text {F2}}}\right )- \, \text {Re} \, \left (\frac {G_8}{\lambda _{+}^{\text {F2}}} \exp \left (\frac {x_0-x_{\text {F2}}}{\lambda _{+}^{\text {F2}}}\right )\right )\right )\frac {k_\text {J}^\text {F2}}{(k_\varphi ^\text {F2})^2+(k_\text {J}^\text {F2})^2} + \right .} \\ \left . + \left ( \, \text {Im} \, \left (\frac {G_7}{\lambda _{+}^{\text {F2}}}\right )- \, \text {Im} \, \left (\frac {G_8}{\lambda _{+}^{\text {F2}}} \exp \left (\frac {x_0-x_{\text {F2}}}{\lambda _{+}^{\text {F2}}}\right )\right )\right )\frac {k_\varphi ^\text {F2}}{(k_\varphi ^\text {F2})^2+(k_\text {J}^\text {F2})^2}\right )\cos \theta \end{gather}

A.3.4 Interface F2|N2

Ballistic spin current continuity

\begin{gather} \nonumber {x \qq {:} \left (-\beta _{\sigma }^{\text {F2}} \frac {\mu _B}{e} J_\text {C} + \frac {D_\text {e}^{\text {F2}} \left (1-\beta _{\sigma }^{\text {F2}} \beta _{D}^{\text {F2}}\right )}{\lambda _{\text {sdl}}^{\text {F2}}}\left (G_5 \exp \left (-\frac {x_{\text {F2}}-x_0}{\lambda _{\text {sdl}}^{\text {F2}}}\right )-G_6\right ) \right ) \cos \theta +} \\ \nonumber {-2 D_\text {e}^{\text {F2}} \left ( \left ( \, \text {Re} \, \left (\frac {G_7}{\lambda _{+}^{\text {F2}}} \exp \left (-\frac {x_{\text {F2}}-x_0}{\lambda _{+}^{\text {F2}}}\right )\right )- \, \text {Re} \, \left (\frac {G_8}{\lambda _{+}^{\text {F2}}}\right )\right )\frac {k_\text {J}^\text {F2}}{(k_\varphi ^\text {F2})^2+(k_\text {J}^\text {F2})^2} + \right .} \\ \left . + \left ( \, \text {Im} \, \left (\frac {G_7}{\lambda _{+}^{\text {F2}}} \exp \left (-\frac {x_{\text {F2}}-x_0}{\lambda _{+}^{\text {F2}}}\right )\right )- \, \text {Im} \, \left (\frac {G_8}{\lambda _{+}^{\text {F2}}}\right )\right )\frac {k_\varphi ^\text {F2}}{(k_\varphi ^\text {F2})^2+(k_\text {J}^\text {F2})^2} \right )\sin \theta =\frac {D_\text {e}^{\text {N2}}}{\lambda _{\text {sf}}^{\text {N2}}}A_{10} \\[14pt] y \qq {:} \nonumber {2 D_\text {e}^{\text {F2}} \left ( \, \text {Re} \, \left (\frac {G_7}{\lambda _{+}^{\text {F2}}} \exp \left (-\frac {x_{\text {F2}}-x_0}{\lambda _{+}^{\text {F2}}}\right )\right )- \, \text {Re} \, \left (\frac {G_8}{\lambda _{+}^{\text {F2}}}\right )\right )\frac {k_\varphi ^\text {F2}}{(k_\varphi ^\text {F2})^2+(k_\text {J}^\text {F2})^2} + } \\ - 2 D_\text {e}^{\text {F2}} \left ( \, \text {Im} \, \left (\frac {G_7}{\lambda _{+}^{\text {F2}}} \exp \left (-\frac {x_{\text {F2}}-x_0}{\lambda _{+}^{\text {F2}}}\right )\right )- \, \text {Im} \, \left (\frac {G_8}{\lambda _{+}^{\text {F2}}}\right )\right )\frac {k_\text {J}^\text {F2}}{(k_\varphi ^\text {F2})^2+(k_\text {J}^\text {F2})^2} =\frac {D_\text {e}^{\text {N2}}}{\lambda _{\text {sf}}^{\text {N2}}}A_{11} \displaybreak \\[14pt] \nonumber {z \qq {:} \left (-\beta _{\sigma }^{\text {F2}} \frac {\mu _B}{e} J_\text {C} +\frac {D_\text {e}^{\text {F2}} \left (1-\beta _{\sigma }^{\text {F2}} \beta _{D}^{\text {F2}}\right )}{\lambda _{\text {sdl}}^{\text {F2}}}\left (G_5 \exp \left (-\frac {x_{\text {F2}}-x_0}{\lambda _{\text {sdl}}^{\text {F2}}}\right )-G_6\right ) \right )\sin \theta +} \\ \nonumber {+2 D_\text {e}^{\text {F2}} \left (\left ( \, \text {Re} \, \left (\frac {G_7}{\lambda _{+}^{\text {F2}}} \exp \left (-\frac {x_{\text {F2}}-x_0}{\lambda _{+}^{\text {F2}}}\right )\right )- \, \text {Re} \, \left (\frac {G_8}{\lambda _{+}^{\text {F2}}}\right )\right )\frac {k_\text {J}^\text {F2}}{(k_\varphi ^\text {F2})^2+(k_\text {J}^\text {F2})^2} \right .} \\ \left . + \left ( \, \text {Im} \, \left (\frac {G_7}{\lambda _{+}^{\text {F2}}} \exp \left (-\frac {x_{\text {F2}}-x_0}{\lambda _{+}^{\text {F2}}}\right )\right )- \, \text {Im} \, \left (\frac {G_8}{\lambda _{+}^{\text {F2}}}\right )\right )\frac {k_\varphi ^\text {F2}}{(k_\varphi ^\text {F2})^2+(k_\text {J}^\text {F2})^2} \right )\cos \theta =\frac {D_\text {e}^{\text {N2}}}{\lambda _{\text {sf}}^{\text {N2}}}A_{12} \end{gather}

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List of Publications

Journal Articles

• • S. Fiorentini, J. Ender, S. Selberherr, R. L. de Orio, W. Goes, and V. Sverdlov, “Comprehensive evaluation of torques in ultra-scaled MRAM devices,” Solid-State Electron., vol. 199, p. 108491, 2023, doi: 10.1016/j.sse.2022.108491.

• • S. Fiorentini, M. Bendra, J. Ender, R. L. de Orio, W. Goes, S. Selberherr, and V. Sverdlov, “Spin and charge drift-diffusion in ultra-scaled MRAM cells,” Sci. Rep., vol. 12, no. 1, p. 20958, Dec. 2022, doi: 10.1038/s41598-022-25586-4.

• • S. Fiorentini, J. Ender, S. Selberherr, W. Goes, and V. Sverdlov, “Spin transfer torque evaluation based on coupled spin and charge transport: A finite element method approach,” J. Syst. Cyb. Inf., vol. 20, no. 4, pp. 40–44, 2022, doi: 10.54808/JSCI.20.04.40.

• • M. Bendra, S. Fiorentini, W. Goes, S. Selberherr, and V. Sverdlov, “Interface Effects in Ultra-Scaled MRAM Cells,” Solid-State Electron., vol. 194, p. 108373, 2022, doi: 10.1016/j.sse.2022.108373.

• • J. Ender, S. Fiorentini, R. L. de Orio, T. Hadámek, M. Bendra, W. Goes, S. Selberherr, and V. Sverdlov, “Advances in modeling emerging magnetoresistive random access memories: From finite element methods to machine learning approaches,” Proc. SPIE Int. Soc. Opt. Eng., vol. 12157, pp. 1215708-1–1215708-14, 2022, doi: 10.1117/12.2624595.

• • J. Ender, R. L. de Orio, S. Fiorentini, S. Selberherr, W. Goes, and V. Sverdlov, “Reinforcement learning to reduce failures in SOT-MRAM switching,” Microelectron. Reliab., vol. 135, pp. 1–5, 2022, doi: 10.1016/j.microrel.2022.114570.

• • T. Hadámek, S. Fiorentini, M. Bendra, J. Ender, R. L. de Orio, W. Goes, S. Selberherr, and V. Sverdlov, “Temperature increase in STT-MRAM at writing: A Fully three-dimensional finite element approach,” Solid-State Electron., vol. 193, pp. 108269-1–108269-7, 2022, doi: 10.1016/j.sse.2022.108269.

• • N. Jørstad, S. Fiorentini, W. J. Loch, W. Goes, S. Selberherr, and V. Sverdlov, “Finite element modeling of spin-orbit torques,” Solid-State Electron., vol. 194, pp. 108323-1–108323-4, 2022, doi: 10.1016/j.sse.2022.108323.

• • W. J. Loch, S. Fiorentini, N. Jørstad, W. Goes, S. Selberherr, and V. Sverdlov, “Double reference layer STT-MRAM structures with improved performance,” Solid-State Electron., vol. 194, pp. 108335-1–108335-4, 2022, doi:
  10.1016/j.sse.2022.108335.

• • J. Ender, S. Fiorentini, R. L. de Orio, W. Goes, V. Sverdlov, and S. Selberherr, “Emerging CMOS compatible magnetic memories and logic,” IEEE J. Electron Devices Soc., vol. 9, pp. 456–463, 2021, doi: 10.1109/JEDS.2021.3066679.

• • J. Ender, R. L. de Orio, S. Fiorentini, S. Selberherr, W. Goes, and V. Sverdlov, “Improving failure rates in pulsed SOT-MRAM switching by reinforcement learning,” Microelectron. Reliab., vol. 126, pp. 114231-1–114231-5, 2021, doi: 10.1016/j.microrel.2021.114231.

• • J. Ender, R. L. de Orio, S. Fiorentini, S. Selberherr, W. Goes, and V. Sverdlov, “Reinforcement learning approach for deterministic SOT-MRAM switching,” Proc. SPIE Int. Soc. Opt. Eng., vol. 11805, pp. 1180519-1–1180519-8, 2021, doi: 10.1117/12.2593937.

• • S. Fiorentini, J. Ender, S. Selberherr, R. L. de Orio, W. Goes, and V. Sverdlov, “Coupled spin and charge drift-diffusion approach applied to magnetic tunnel junctions,” Solid-State Electron., vol. 186, pp. 108103, 2021, doi:
  10.1016/j.sse.2021.108103.

• • R. L. de Orio, J. Ender, S. Fiorentini, W. Goes, S. Selberherr, and V. Sverdlov, “Numerical analysis of deterministic switching of a perpendicularly magnetized spin-orbit torque memory cell,” IEEE J. Electron Devices Soc., vol. 9, pp. 61–67, 2021, doi: 10.1109/JEDS.2020.3039544.

• • R. L. de Orio, J. Ender, S. Fiorentini, W. Goes, S. Selberherr, and V. Sverdlov, “Optimization of a spin-orbit torque switching scheme based on micromagnetic simulations and reinforcement learning,” Micromachines, vol. 12, pp. 443, 2021, doi: 10.3390/mi12040443.

• • R. L. de Orio, J. Ender, S. Fiorentini, W. Goes, S. Selberherr, and V. Sverdlov, “Two-pulse switching scheme and reinforcement learning for energy efficient SOT-MRAM simulations,” Solid-State Electron., vol. 185, pp. 108075, 2021, doi: 10.1016/j.sse.2021.108075.

• • S. Fiorentini, J. Ender, M. Mohamedou, S. Selberherr, R. Orio, W. Goes, and V. Sverdlov, “Comprehensive modeling of coupled spin-charge transport and magnetization dynamics in STT-MRAM cells,” Proc. SPIE Int. Soc. Opt. Eng., vol. 11470, pp. 114701B-1–114701B-7, 2020, doi: 10.1117/12.2567480.

• • S. Fiorentini, R. Orio, S. Selberherr, J. Ender, W. Goes, and V. Sverdlov, “Analysis of switching under fixed voltage and fixed current in perpendicular STT-MRAM,” IEEE J. Electron Devices Soc., vol. 8, pp. 1249–1256, 2020, doi: 10.1109/JEDS.2020.3023577.

• • R. L. de Orio, A. Makarov, W. Goes, J. Ender, S. Fiorentini, and V. Sverdlov, “Two-pulse magnetic field-free switching scheme for perpendicular SOT-MRAM with a symmetric square free layer,” Physica B, vol. 578, pp. 411743, 2020, doi: 10.1016/j.physb.2019.411743.

• • R. L. de Orio, A. Makarov, S. Selberherr, W. Goes, J. Ender, S. Fiorentini, and V. Sverdlov, “Robust magnetic field-free switching of a perpendicularly magnetized free layer for SOT-MRAM,” Solid-State Electron., vol. 168, pp. 107730-1–107730-7, 2020, doi: 10.1016/j.sse.2019.107730.

Book Contributions

• • T. Hadámek, M. Bendra, S. Fiorentini, J. Ender, R. L. de Orio, W. Goes, S. Selberherr, and V. Sverdlov, “Temperature increase in MRAM at writing: A finite element approach,” in Proc. EUROSOI-ULIS Conf., pp. 1–4, 2021, doi: 10.1109/EuroSOI-ULIS53016.2021.9560669.

• • S. Fiorentini, J. Ender, S. Selberherr, R. L. de Orio, W. Goes, and V. Sverdlov, “Comprehensive modeling of coupled spin and charge transport through magnetic tunnel junctions,” in Proc. EUROSOI-ULIS Conf., pp. 1–4, 2020, doi: 10.1109/EUROSOI-ULIS49407.2020.9365497.

• • S. Fiorentini, R. L. de Orio, S. Selberherr, J. Ender, W. Goes, and V. Sverdlov, “Influence of current redistribution in switching models for perpendicular STT-MRAM,” in Adv. CMOS-Comp. Semicond. Devices 19, vol. 97, no. 5, pp. 159–164, 2020, doi: 10.1149/09705.0159ecst.

• • R. L. de Orio, J. Ender, S. Fiorentini, W. Goes, S. Selberherr, and V. Sverdlov, “Reduced current spin-orbit torque switching of a perpendicularly magnetized free layer,” in Proc. EUROSOI-ULIS Conf., pp. 1–4, 2020, doi: 10.1109/EUROSOI-ULIS49407.2020.9365497.

• • R. L. de Orio, A. Makarov, S. Selberherr, W. Goes, J. Ender, S. Fiorentini, and V. Sverdlov, “Efficient magnetic field-free switching of a symmetric perpendicular magnetic free layer for advanced SOT-MRAM,” in Proc. EUROSOI-ULIS Conf., pp. 1–4, 2019, doi: 10.1109/EUROSOI-ULIS45800.2019.9041920.

Conference Contributions (with Proceedings-Entry)

• • S. Fiorentini, W. J. Loch, M. Bendra, N. Jørstad, J. Ender, R. L. de Orio, T. Hadámek, W. Goes, V. Sverdlov, and S. Selberherr, “Design analysis of ultra-scaled MRAM cells,” in Proc. ICSICT Conf., 2022.

• • V. Sverdlov, M. Bendra, S. Fiorentini, J. Ender, R. L. de Orio, T. Hadámek, W.J. Loch, N. Jørstad, W. Goes, and S. Selberherr, “Modeling advanced spintronic based magnetoresistive memory,” in Book of Abstracts IRPhE Conf., 2022.

• • S. Fiorentini, M. Bendra, J. Ender, R. L. de Orio, W. Goes,
  S. Selberherr, and V. Sverdlov, “Spin torques in ultra-scaled MRAM devices,” in
  Proc. ESSDERC Conf., 2022, pp. 348–351, doi:
  10.1109/ESSDERC55479.2022.9947196.

• • S. Fiorentini, J. Ender, R. L. de Orio, S. Selberherr, W. Goes, and V. Sverdlov, “Comprehensive evaluation of torques in ultra scaled MRAM devices,” in Book of Abstracts SISPAD Conf., 2022, pp. 11–12.

• • S. Fiorentini, J. Ender, R. L. de Orio, S. Selberherr, W. Goes, and V. Sverdlov, “Spin transfer torque evaluation based on coupled spin and charge transport: A finite element method approach,” in Proc. WMSCI Conf., 2022, pp. 12–15, doi: 10.54808/WMSCI2022.02.12.

• • V. Sverdlov, W. J. Loch, M. Bendra, S. Fiorentini, J. Ender, R. L. de Orio, T. Hadámek, N. Jørstad, W. Goes, and S. Selberherr, “Modeling Approach to Ultra-Scaled MRAM Cells,” in Book of Abstracts ASETMEET Conf., 2022, pp. 7–8.

• • T. Hadámek, S. Fiorentini, M. Bendra, R. L. de Orio, W. J. Loch, N. Jorstad, S. Selberherr, W. Goes, and V. Sverdlov, “Temperature modeling in STT-MRAM: A fully three-dimensional finite element approach,” in Book of Abstracts NANO Conf., 2022.

• • N. Jørstad, S. Fiorentini, S. Selberherr, W. Goes, and V. Sverdlov, “Modeling interfacial and bulk spin-orbit torques,” in Book of Abstracts NANO Conf., 2022.

• • R. L. de Orio, J. Ender, W. Goes, S. Fiorentini, S. Selberherr, and V. Sverdlov, “About the switching energy of a magnetic tunnel junction determined by spin-orbit torque and voltage-controlled magnetic anisotropy,” in Proc. LAEDC Conf., 2022, pp. 1–4, doi: 10.1109/LAEDC54796.2022.9908222.

• • V. Sverdlov, M. Bendra, S. Fiorentini, J. Ender, R. Orio, T. Hadámek, W. J. Loch, N. Jørstad, W. Goes, and S. Selberherr, “Emerging devices for digital spintronics,” in Proc. GECNN Conf., 2022, pp. 32–33.

• • M. Bendra, S. Fiorentini, J. Ender, R. Orio, T. Hadámek, W. J. Loch, N. Jørstad, S. Selberherr, W. Goes, and V. Sverdlov, “Spin transfer torques in ultra-scaled MRAM cells,” in Proc. MIPRO Conf., 2022, pp. 129–132.

• • M. Bendra, S. Fiorentini, J. Ender, R. Orio, T. Hadámek, W. J. Loch, N. Jørstad, W. Goes, and S. Selberherr, “Interface effects in ultra-scaled MRAM cells,” in Book of Abstracts EUROSOI-ULIS Conf., 2022.

• • V. Sverdlov, M. Bendra, S. Fiorentini, J. Ender, R. Orio, T. Hadámek, W. J. Loch, N. Jørstad, and S. Selberherr, “Modeling advanced magnetoresistive Memory: A journey from finite element methods to machine learning approaches,” in Proc. Global Webinar on Nanosc. and Nanotec., 2022.

• • N. Jørstad, S. Fiorentini, W. Goes, V. Sverdlov, “Efficient finite element method approach to model spin orbit torque MRAM,” in Book of Abstracts MOS-AK Workshop, 2021, pp. 1.

• • S. Fiorentini, M. Bendra, J. Ender, R. L. de Orio, S. Selberherr, W. Goes, and V. Sverdlov, “Design support for ultra-scaled MRAM cells,” in Proc. IEDM Conf. (Special MRAM Poster Session), 2021.

• • S. Fiorentini, R. L. de Orio, S. Selberherr, J. Ender, W. Goes, and V. Sverdlov, “Spin and Charge Drift-Diffusion Approach to Torque Computation in Spintronic Devices,” in Book of Abstracts WINDS Conf., 2021.

• • J. Ender, S. Fiorentini, R. L. de Orio, T. Hadámek, M. Bendra, W. Goes,
  S. Selberherr, and V. Sverdlov, “Advanced modeling of emerging MRAM: From finite element methods to machine learning approaches,” in Proc. ICMNE Conf., 2021.

• • J. Ender, R. L. de Orio, S. Fiorentini, S. Selberherr, W. Goes, and V. Sverdlov, “Improving failure rates in pulsed SOT-MRAM Switching by
  reinforcement learning,” in Proc. ESREF Conf., 2021, pp. 1–4,
  doi: 10.1016/j.microrel.2021.114231.

• • M. Bendra, J. Ender, S. Fiorentini, T. Hadámek, R. L. de Orio, W. Goes,
  S. Selberherr, and V. Sverdlov, “Finite element method approach to
  MRAM modeling,” in Proc. MIPRO Conf., 2021, pp. 70–73,
  doi: 10.23919/MIPRO52101.2021.9597194.

• • J. Ender, R. L. de Orio, S. Fiorentini, S. Selberherr, W. Goes, and V. Sverdlov, “Reinforcement learning approach for sub-critical current SOT-MRAM switching materials,” in Proc. SISPAD Conf., 2021, pp. 150–154, 10.1109/SISPAD54002.2021.9592561.

• • S. Fiorentini, J. Ender, R. L. de Orio, S. Selberherr, W. Goes, and V. Sverdlov, “Spin and charge drift-diffusion approach to torque computation in magnetic tunnel junctions,” in Proc. SISPAD Conf., 2021, pp. 155–158, 10.1109/SISPAD54002.2021.9592561.

• • J. Ender, R. L. de Orio, S. Fiorentini, S. Selberherr, W. Goes, and V. Sverdlov, “Reinforcement learning to reduce failures in SOT-MRAM switching,” in Proc. IPFA Conf., 2021, doi: 10.1109/IPFA53173.2021.9617362

• • R. L. de Orio, J. Ender, S. Fiorentini, W. Goes, S. Selberherr, and V. Sverdlov, “Deterministic spin-orbit switching scheme for an array of perpendicular MRAM cells suitable for large scale integration,” in Proc. TMAG Conf., 2021.

• • T. Hadámek, M. Bendra, S. Fiorentini, J. Ender, R. L. de Orio, W. Goes, S. Selberherr, and V. Sverdlov, “Temperature increase in MRAM at writing: A finite element approach,” in Book of Abstracts EUROSOI-ULIS Conf., 2021, pp. 133–134.

• • J. Ender, S. Fiorentini, V. Sverdlov, W. Goes, R. L. de Orio, and S. Selberherr, “Reinforcement learning approach for deterministic SOT-MRAM switching,” in Proc. SPIE Conf., 2021, pp. 11805-53.

• • J. Ender, S. Fiorentini, S. Selberherr, W. Goes, and V. Sverdlov, “Advanced modeling of emerging nonvolatile magnetoresistive devices,” in Book of Abstracts IWCN Conf., 2021, pp. 45–46.

• • S. Fiorentini, J. Ender, R. L. de Orio, S. Selberherr, W. Goes, and V. Sverdlov, “Spin drift-diffusion approach for the computation of torques in multi-layered structures,” in Book of Abstracts IWCN Conf., 2021, pp. 51–52.

• • J. Ender, M. Mohamedou, S. Fiorentini, R. L. de Orio, S. Selberherr,
  W. Goes, and V. Sverdlov, “Efficient demagnetizing field calculation for disconnected complex geometries in STT-MRAM cells,” in Proc. SISPAD Conf., 2020, pp. 213–216, doi: 10.23919/SISPAD49475.2020.9241662.

• • S. Fiorentini, J. Ender, M. Mohamedou, R. L. de Orio, S. Selberherr,
  W. Goes, and V. Sverdlov, “Computation of torques in magnetic tunnel junctions through spin and charge transport modeling,” in Proc. SISPAD Conf., 2020, pp. 209–212, doi: 10.23919/SISPAD49475.2020.9241657.

• • R. L. de Orio, A. Makarov, W. Goes, J. Ender, S. Fiorentini, S. Selberherr, and V. Sverdlov, “Switching of a perpendicularly magnetized free-layer by spin-orbit-torques with reduced currents,” in Proc. WMSCI Conf., 2020, pp. 58–61.

• • R. L. de Orio, J. Ender, S. Fiorentini, W. Goes, S. Selberherr, and V. Sverdlov, “Reduced current spin-orbit torque switching of a perpendicularly magnetized free layer,” in Book of Abstracts EUROSOI-ULIS Conf., 2020, pp. 123–124.

• • S. Fiorentini, J. Ender, M. Mohamedou, V. Sverdlov, W. Goes, R. L. de Orio, and S. Selberherr, “Comprehensive modeling of coupled spin-charge transport and magnetization dynamics in STT-MRAM cells,” in Proc. SPIE Conf., 2020, pp. 11470–44.

• • S. Fiorentini, R. L. de Orio, S. Selberherr, J. Ender, W. Goes, and V. Sverdlov, “Influence of current redistribution in switching models for perpendicular STT-MRAM,” in Book of Abstracts ECS Conf., 2020, doi: 10.1149/MA2020-01241389mtgabs

• • S. Fiorentini, J. Ender, S. Selberherr, R. L. de Orio, W. Goes, and V. Sverdlov, “Comprehensive modeling of coupled spin and charge transport through magnetic tunnel junctions,” in Book of Abstracts EUROSOI-ULIS Conf., 2020, pp. 112–113.

• • S. Fiorentini, R. L. de Orio, S. Selberherr, J. Ender, W. Goes, and V. Sverdlov, “Perpendicular STT-MRAM switching at fixed voltage and at fixed current,” in Proc. EDTM Conf., 2020, pp. 341–344, doi: 10.1109/EDTM47692.2020.9117985.

• • V. Sverdlov, S. Fiorentini, J. Ender, W. Goes, R. L. de Orio, and S. Selberherr, “Emerging CMOS compatible magnetic memories and logic,” in Proc. LAEDC Conf., 2020, doi: 10.1109/LAEDC49063.2020.9073332.

• • R. L. de Orio, A. Makarov, J. Ender, S. Fiorentini, W. Goes, S. Selberherr, V. Sverdlov, “A dynamical approach to fast and reliable external field free perpendicular magnetization reversal by spin-orbit torques,” in Proc. IEDM Conf. (Special MRAM Poster Session), 2019.

• • S. Fiorentini, R. Orio, S. Selberherr, J. Ender, W. Goes, V. Sverdlov, “Comprehensive modeling of switching in perpendicular STT-MRAM,” in Proc. WINDS Conf., 2019, pp. 107–108.

• • R. Orio, S. Selberherr, J. Ender, S. Fiorentini, W. Goes, and V. Sverdlov, “Robustness of the two-pulse switching scheme for SOT-MRAM,” in Book of Abstracts WINDS Conf., 2019, pp. 54–55.

• • R. L. de Orio, A. Makarov, S. Selberherr, W. Goes, J. Ender, S. Fiorentini, and V. Sverdlov, “Switching speedup of the magnetic free layer of advanced SOT-MRAM,” in Proc. ESSDERC Conf., 2019, pp. 146–149, doi: 10.1109/ESSDERC.2019.8901780.

• • S. Fiorentini, R. L. de Orio, W. Goes, J. Ender, and V. Sverdlov, “Comprehensive comparison of switching models for perpendicular spin-transfer torque MRAM cells,” in Proc. SISPAD Conf., 2019, pp. 57–60, doi:
  10.1109/SISPAD.2019.8870359.

• • R. L. de Orio, A. Makarov, S. Selberherr, W. Goes, J. Ender, S. Fiorentini, and V. Sverdlov, “Robust magnetic field free switching scheme for perpendicular free layer in advanced spin orbit torque magnetoresistive random access memory,” in Book of Abstracts IWCN Conf., 2019, pp. 69–71.

• • R. L. de Orio, A. Makarov, W. Goes, J. Ender, S. Fiorentini, and V. Sverdlov, “Two-pulse magnetic field free switching scheme for advanced perpendicular SOT-MRAM,” in Book of Abstracts HMM Conf., 2019, p. 34.

• • R. L. de Orio, A. Makarov, S. Selberherr, W. Goes, J. Ender, S. Fiorentini, and V. Sverdlov, “Efficient magnetic field free switching of symmetric perpendicular magnetic free layer for advanced SOT-MRAM,” in Book of Abstracts EUROSOI-ULIS Conf., 2019, pp. 152–153.

Conference Contributions (no Proceedings-Entry)

• • J. Ender, R. Orio, S. Fiorentini, W. Goes, and V. Sverdlov, “Large-scale finite element micromagnetics simulations using open source software,” Poster at EMRS Conf., 2019.

• • S. Fiorentini, R. L. de Orio, W. Goes, J. Ender, and V. Sverdlov, “Comprehensive comparison of switching models for perpendicular spin-transfer torque MRAM cells,” Poster at EMRS Conf., 2019.