List of Symbols

Level Set Framework

$F$	Flux on surface $\mathcal{S}$
$F^{src}$	Flux on source plane $\mathcal{P}$
$F^{tot}$	Total flux through plane $\mathcal{P}$
$G$	Re-emission probability function
$H$	Hamiltonian
$\mathcal{L}$	Layer of grid points, $\mathcal{L}\subseteq\mathcal{G}$
$M$	Number of different material regions
$\mathcal{M}$	Material region number
$\hat{n}$	Normal unit vector on $\mathcal{S}$
$\vec{n}_{\mathcal{P}}$	Normal unit vector on $\mathcal{P}$ pointing towards $\mathcal{S}$
$p$	Pressure
$\vec{p}$	Grid point index
$\mathcal{P}$	Source plane
$q$	Particle species
$\hat{r}$	Unit vector in the direction of vector motion
$R$	Surface rate
$\mathcal{S}$	Description of the LS surface
$t$	Process time
$V\left(\vec{x}\right)$	Surface velocity
$\vec{x}$	Point in space, $\vec{x}=\left(x_1,...,x_D\right)\in\mathbb{R}$
$Y^{tot}$	Total sputter rate
$\Gamma$	Flux distribution on $\mathcal{S}$
$\eta$	Surface reaction order
$\kappa$	Mean curvature
$\nu$	Exponent in power cosine distribution
$\rho$	Bulk density
$\Theta$	Surface coverage

Oxidation of Silicon

$B$	Parabolic rate constant
$B\slash A$	Linear rate constant
$C^*$	Equilibrium concentration of oxidants in the ambient
$C_O$	Concentration of oxidants at the oxide surface
$C_S$	Concentration of oxidant at the Si-SiO$_2$ interface
$D$	Diffusivity
$h$	Gas-phase transport coefficient
$H$	Inverse Henry's law constant
$H_{bb}$	Maximum height of the nitride after oxidation
$F_1$	Oxidant flux entering the gas ambient from the bulk
$F_2$	Oxidant flux diffusing through the oxide
$F_3$	Oxidant flux reacting on the silicon surface
$k_B$	Boltsmann constant
$k_s$	Reaction rate at the Si-SiO$_2$ interface
$L_{bb}$	Maximum length of the nitride after oxidation
$N$	Number of oxidant molecules per unit volume of oxide grown
$N_{Si}$	Molecular density of silicon
$N_{SiO_2}$	Molecular density of silicon dioxide
$p$	Partial pressure
$t$	Time
$T$	Temperature
$x_i$	Initial oxide growth
$x_o$	Oxide thickness
$\tau$	Time parameter accounting for $x_i$

AFM Nanolithography

$2a'$	Barrier width between two interstitial sites
$C$	Normalization constant
$D$	Distance of point charge from the silicon surface
$E_0$	Fitting parameter $E_0=45$V/nm
$\vec{E}\left(\vec{p}\right)$	Electric field strength at $\vec{p}$
$h$	Humidity
$h\left(t,V_b\right)$	Nanodot height
$M$	Lorentz distribution $r_{max}$ dependent constant
$N$	Total number of charged dots for a rough needle tip
$q$	Electronic charge of an electron
$Q$	Effective point charge
$Q_{tot}$	Total effective charge after adding all point charges
$r$	Desired distribution along the radius
$R$	Fitting parameter (1.5$\times$10$^3$nm/s
$t$	Pulse time
$V\left(\vec{p}\right)$	Applied potential at point $\vec{p}$
$V_b$	Bias voltage
$w\left(t,V_b\right)$	Nanodot width, full width at half maximum
$x_r$	Value distributed according to the desired distribution
$\Phi\left(r\right)$	CPD function
$\sigma\left(x, y, z \right)$	Surface charge density on the wafer surface
$\sigma_{max}$	maximum surface charge density
$\theta$	Wire orientation with respect to the (010) direction
$\xi$	Evenly distributed value

Spray Pyrolysis

$a_r(t)$	Radial droplet acceleration
$a_v(t)$	Vertical droplet acceleration
$c_{ev}$	Vertical linearization constant for the electric field
$d_r(t)$	Radial droplet displacement
$d_v(t)$	Vertical droplet displacement
$D_{v,f}$	Average diffusion coefficient
$E$	Generated electric field strength
$F_e$	Force due to the applied electric field
$F_g$	Force due to gravity
$F_S$	Stokes force
$F_t$	Thermophoretic force
$q_d$	Droplet charge
$q_{max}$	Maximum droplet charge
$H$	Distance between wafer and atomizing nozzle
$K$	Surface evaporation rate
$l$	Effect of the thermophoretic force on droplet's acceleration
$K_V$	H$\slash$R ratio dependent value
$M_W$	Molecular weight of evaporating liquid
$N_A$	Avogadro's number
$r_d$	Droplet radius
$r_n$	Nozzle outlet radius
$R$	Outer nozzle radius
$\textrm{Re}$	Reynolds number
$S_C$	Schmidt's number
$t_{life}$	Lifetime of a droplet traveling through a heated ambient
$t_{th}$	Thickness of the thermal zone
$T_a$	Air temperature
$T_d$	Droplet temperature
$\nabla T_a$	Air temperature gradient
$v_a$	Air velocity
$v_r(t)$	Radial droplet velocity
$v_v(t)$	Vertical droplet velocity
$W_{crit}$	Critical Weber number
$\gamma$	Liquid-gas surface tension
$\eta_a$	Viscosity of air
$\kappa_a$	Conductivity of air
$\kappa_d$	Conductivity of droplet
$\rho_a$	Density of air
$\rho_d$	Droplet density
$\rho_e$	Fluid charge density
$\Phi_0$	Applied electrical potential
$\Phi^*$	Normalized potential $\left(\Phi\slash\Phi_0\right)$

BiCS Memory Hole Etching

$DR_p\left(\vec{x}\right)$	Deposition rate during SiO$_2$ etching
$ER_p\left(\vec{x}\right)$	Polymer etch rate during SiO$_2$ etching
$F_{Br}\left(\vec{x}\right)$	Bromine flux
$F_{O}\left(\vec{x}\right)$	Oxygen flux
$F_e\left(\vec{x}\right)$	Total etchant flux
$F_i\left(\vec{x}\right)$	Total ion flux
$F_p\left(\vec{x}\right)$	Total polymer flux
$k$	Bromine chemical etch rate constant
$V_{SiO_2}\left(\vec{x}\right)$	SiO$_2$ etch rate
$\beta$	Oxygen chemical etch rate constant
$\Theta_{Br}\left(\vec{x}\right)$	Bromine coverage during silicon etching
$\Theta_{O}\left(\vec{x}\right)$	Oxygen coverage during silicon etching
$\Theta\left(\vec{x}\right)_e$	Etchant surface coverage
$\Theta\left(\vec{x}\right)_e\slash p$	Surface coverage of active sites on polymer
$\Theta\left(\vec{x}\right)_p$	Polymer surface coverage