- Chapter 3: Stochastic Model of the Resistive Switching

A _{e}dimensionless coefficient for electrons

A _{i}dimensionless coefficient for ions

a localization radius

E _{a}annihilation energy of the mth vacancy V

_{o}E _{f}threshold energy for the mth vacancy V

_{o}ΔE difference between the energies of an electron positioned at sites n and m

ΔE′ difference in energy of an ion after and before hopping

d x-dimension of system

ℏ reduced Planck constant

I current generated by hopping

k _{B}Boltzmann constant

q _{e}electron charge

R _{nm}hopping distance for electrons

T local temperature

t the time spent for moving a single particle (electron or ion)

U applied voltage

Δx difference between the x coordinate of the sites (vacancies) m and n

α coefficients of the boundary conditions on the cathode

β coefficients of the boundary conditions on the anode

Γ _{nm}hopping rates from site n to site m for electrons

Γ _{0m}^{iC}hopping rates from an electrode site 0 to an oxygen vacancy m

Γ _{m0}^{oC}hopping rates from an oxygen vacancy m to an electrode site 0

Γ _{(2N)m}^{iA}hopping rates from an electrode site 2N to an oxygen vacancy m

Γ _{m(2N)}^{oA}hopping rates from an oxygen vacancy m to an electrode site 2N

Γ _{n}′hopping rates for ions

- Chapter 4: Magnetoresistive Memory

E _{b}energy barrier that separates the two magnetization state

e electron charge

A _{cs}cross-section area of the free layer

H _{K}effective anisotropy field

H _{K}^{in-plane}effective anisotropy field for in-plane MTJ

H _{K}^{perp}effective anisotropy field for perpendicular MTJ

ℏ reduced Planck constant

I _{c}switching (critical) current

I _{c}^{in-plane}critical currents for in-plane MTJ

I _{c}^{perp}critical currents for perpendicular MTJ

k _{B}Boltzmann constant

M _{S}saturation magnetization of the free layer

R _{AP },R_{P }resistances in HRS and LRS, respectively

T temperature

V volume of the free layer

α Gilbert damping parameter

Δ thermal stability factor

Δ _{in-plane}thermal stability factor for in-plane MTJ

Δ _{perp}thermal stability factor for perpendicular MTJ

η polarizing factor

μ _{0}magnetic constant

ρ _{AP },ρ_{P }resistivities in HRS and LRS, respectively

σ _{AP },σ_{P }conductivities in HRS and LRS, respectively

- Chapter 5: Macro- and Micromagnetic Approach

A material exchange constant

a(j) time-dependent current-proportional function for the in-plane torque

b(j) time-dependent current-proportional function for the perpendicular torque

d thickness of the free layer

E _{ani}magnetocrystalline anisotropy energy

E _{current}energy associated with Ampere field

E _{demag}demagnetization energy

E _{exch}exchange energy

E _{ext}energy associated with external magnetic field

E _{ms}energy associated with magnetostatic coupling between the reference layer and the free layer

E _{th}energy thermal field

e electron charge

g g-factor

g(Θ) Slonczewski’s expression for spin-torque

H _{ani}magnetocrystalline anisotropy field

H _{current}Ampere field

H _{demag}demagnetization field

H _{eff}effective magnetic field

H _{exch}exchange field

H _{ext}external magnetic field

H _{ms}magnetostatic coupling between the reference layer and the free layer

H _{th}thermal field

ℏ reduced Planck constant

J current density vector

j current density

K _{1},K_{2}material anisotropy coefficient

M magnetization of the free layer

M _{S}saturation magnetization of the free layer

M _{Sp}saturation magnetization of the reference layer

m magnetization unit vector of the free layer

m _{e}electron mass

m _{x},m_{y},m_{z}projection of m on x, y, and z axes, respectively

p magnetization unit vector of the reference layer

T temperature

V volume of the free layer

α Gilbert damping parameter

β coefficient of the perpendicular torque

γ gyromagnetic ratio

δ Dirac delta function

ε energy density

η polarizing factor

Θ angle between direction of magnetization of the free and fixed layer

λ phenomenological damping parameter

μ _{0}magnetic constant

μ _{B}Bohr magneton

ρ volume magnetic charge

σ surface magnetic charge

τ spin transfer torque term

Φ _{d}scalar potential of the stray field

- Chapter 6: Numerical Implementation of Micromagnetic Approach

A material exchange constant

E _{ani}(i,j,k)magnetocrystalline anisotropy energy of the cell (i,j,k)

E _{cell}(i,j,k)full energy of the cell (i,j,k)

E _{current}(i,j,k)energy of the cell (i,j,k) associated with Ampere field

E _{demag}(i,j,k)demagnetization energy of the cell (i,j,k)

E _{exch}(i,j,k)exchange energy of the cell (i,j,k)

E _{ext}(i,j,k)energy of the cell (i,j,k) associated with external magnetic field

E _{ms}(i,j,k)energy associated with magnetostatic coupling between the reference layer and the cell (i,j,k)

E _{th}(i,j,k)energy of the cell (i,j,k) associated with thermal field

E _{tot}total energy of the free layer

H _{ani}(i,j,k)magnetocrystalline anisotropy field of the cell (i,j,k)

H _{current}(i,j,k)Ampere field of the cell (i,j,k)

H _{demag}(i,j,k)demagnetization field of the cell (i,j,k)

H _{eff}(i,j,k)effective magnetic field of the cell (i,j,k)

H _{exch}(i,j,k)exchange field of the cell (i,j,k)

H _{ext}(i,j,k)external magnetic field influences on the cell (i,j,k)

H _{ms}(i,j,k)magnetostatic coupling between the reference layer and the cell (i,j,k)

H _{th}(i,j,k)thermal field influences on the cell (i,j,k)

J(i,j,k) current density vector, current flows through the cell (i,j,k)

K _{1},K_{2}material anisotropy coefficient

k _{B}Boltzmann constant

M magnetization of the free layer

M _{S}saturation magnetization of the free layer

M _{Sp}saturation magnetization of the reference layer

m(i,j,k) magnetization unit vector of the cell (i,j,k)

m _{x}, m_{y}, m_{z}projection of m on x, y, and z axes, respectively

N _{x}, N_{y}, N_{x}grid dimension in x, y, z direction, respectively

p magnetization unit vector of the reference layer

T temperature

Δt time step

ΔV volume of the cell

Δx, Δy, Δz x, y, z dimensions of the cell

μ _{0}magnetic constant

σ(i,j,k) Gaussian random uncorrelated function with standard deviation equal 1