List of Symbols

Chapter 3: Stochastic Model of the Resistive Switching

Ae	dimensionless coefficient for electrons
Ai	dimensionless coefficient for ions
a	localization radius
Ea	annihilation energy of the mth vacancy V o
Ef	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
kB	Boltzmann constant
qe	electron charge
Rnm	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
Γnm	hopping rates from site n to site m for electrons
Γ0miC	hopping rates from an electrode site 0 to an oxygen vacancy m
Γm0oC	hopping rates from an oxygen vacancy m to an electrode site 0
Γ(2N)miA	hopping rates from an electrode site 2N to an oxygen vacancy m
Γm(2N)oA	hopping rates from an oxygen vacancy m to an electrode site 2N
Γn′	hopping rates for ions

Chapter 4: Magnetoresistive Memory

Eb	energy barrier that separates the two magnetization state
e	electron charge
Acs	cross-section area of the free layer
HK	effective anisotropy field
HKin-plane	effective anisotropy field for in-plane MTJ
HKperp	effective anisotropy field for perpendicular MTJ
ℏ	reduced Planck constant
Ic	switching (critical) current
Icin-plane	critical currents for in-plane MTJ
Icperp	critical currents for perpendicular MTJ
kB	Boltzmann constant
MS	saturation magnetization of the free layer
RAP ,RP	resistances in HRS and LRS, respectively
T	temperature
V	volume of the free layer
α	Gilbert damping parameter
Δ	thermal stability factor
Δin-plane	thermal stability factor for in-plane MTJ
Δperp	thermal stability factor for perpendicular MTJ
η	polarizing factor
μ0	magnetic constant
ρAP ,ρP	resistivities in HRS and LRS, respectively
σAP ,σP	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
Eani	magnetocrystalline anisotropy energy
Ecurrent	energy associated with Ampere field
Edemag	demagnetization energy
Eexch	exchange energy
Eext	energy associated with external magnetic field
Ems	energy associated with magnetostatic coupling between the reference layer and the free layer
Eth	energy thermal field
e	electron charge
g	g-factor
g(Θ)	Slonczewski’s expression for spin-torque
Hani	magnetocrystalline anisotropy field
Hcurrent	Ampere field
Hdemag	demagnetization field
Heff	effective magnetic field
Hexch	exchange field
Hext	external magnetic field
Hms	magnetostatic coupling between the reference layer and the free layer
Hth	thermal field
ℏ	reduced Planck constant
J	current density vector
j	current density
K1,K2	material anisotropy coefficient
M	magnetization of the free layer
MS	saturation magnetization of the free layer
MSp	saturation magnetization of the reference layer
m	magnetization unit vector of the free layer
me	electron mass
mx,my,mz	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
μ0	magnetic constant
μB	Bohr magneton
ρ	volume magnetic charge
σ	surface magnetic charge
τ	spin transfer torque term
Φd	scalar potential of the stray field

Chapter 6: Numerical Implementation of Micromagnetic Approach

A	material exchange constant
Eani(i,j,k)	magnetocrystalline anisotropy energy of the cell (i,j,k)
Ecell(i,j,k)	full energy of the cell (i,j,k)
Ecurrent(i,j,k)	energy of the cell (i,j,k) associated with Ampere field
Edemag(i,j,k)	demagnetization energy of the cell (i,j,k)
Eexch(i,j,k)	exchange energy of the cell (i,j,k)
Eext(i,j,k)	energy of the cell (i,j,k) associated with external magnetic field
Ems(i,j,k)	energy associated with magnetostatic coupling between the reference layer and the cell (i,j,k)
Eth(i,j,k)	energy of the cell (i,j,k) associated with thermal field
Etot	total energy of the free layer
Hani(i,j,k)	magnetocrystalline anisotropy field of the cell (i,j,k)
Hcurrent(i,j,k)	Ampere field of the cell (i,j,k)
Hdemag(i,j,k)	demagnetization field of the cell (i,j,k)
Heff(i,j,k)	effective magnetic field of the cell (i,j,k)
Hexch(i,j,k)	exchange field of the cell (i,j,k)
Hext(i,j,k)	external magnetic field influences on the cell (i,j,k)
Hms(i,j,k)	magnetostatic coupling between the reference layer and the cell (i,j,k)
Hth(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)
K1,K2	material anisotropy coefficient
kB	Boltzmann constant
M	magnetization of the free layer
MS	saturation magnetization of the free layer
MSp	saturation magnetization of the reference layer
m(i,j,k)	magnetization unit vector of the cell (i,j,k)
mx, my, mz	projection of m on x, y, and z axes, respectively
Nx, Ny, Nx	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
μ0	magnetic constant
σ(i,j,k)	Gaussian random uncorrelated function with standard deviation equal 1