Notation

Symbol	Meaning

$ℝ$	set of real numbers
$∥⋅∥$	norm of the argument
$\|⋅\|$	modulus of the argument
$O(⋅)$	Landau symbol
$Ω$	unit sphere
$θ$	colatitude (polar angle)
$φ$	longitude (azimuth)
$dΩ$	unit sphere surface element, $sinθdθdφ$
$eθ$ , $eφ$	unit vectors w.r.t. $θ$ and $φ$
$∇$	Nabla operator
$Δ$	Laplace operator
$A,B$	matrices (capital letters, bold)
$v,w$	vectors (lowercase, bold)
$Yl,m$	real valued spherical harmonic of major index $l$ and minor index $m$
$Xl,m$	spherical harmonics expansion coefficient to $l,m Y$ of the quantity $X$
$δi,j$	Kronecker delta
$δ(⋅)$	delta distribution
$T$	triangulation of the simulation domain
$B$	Voronoi box grid of $T$
$Bi$	a box from the adjoint box grid $B$
$˜B$	dual box grid of $B$
$˜Bi,j$	a box from the box grid $B$ overlapping $Bi$ and $Bj$
$Ai,j$	interface area between boxes $Bi$ and $Bj$
$Vi,j$	volume fraction of the boxes $Bi$ and $Bj$ associated with $Ai,j$
$ni,j$	surface unit normal vector of box $Bi$ pointing into $Bj$
$N$	number of degrees of freedom in $(x, H )$ -space
$Nx$	number of degrees of freedom in $x$ -space
$NH$	number of discrete total energies $Hi$
$L$	spherical harmonics expansion order

Symbol	Meaning

$ℏ$	Planck constant divided by $2π$
$kB$	Boltzmann constant
$ϵ$	permittivity (not to be confused with kinetic energy $ε$ )
$ρ$	space charge
$\|q\|$	elementary charge
$q$	charge of a carrier (negative for electrons, positive for holes)
$m∗$	effective mass
$ε$	kinetic energy (not to be confused with permittivity $ϵ$ )
$H$	total energy
$t$	time
$x$	spatial coordinate within the device
$k$	wave vector
$p$	momentum vector
$v$	(group) velocity
$F$	force
$E$	electric field
$ψ$	quasi-electrostatic potential
$Q{⋅}$	scattering operator
$Qin{⋅}$	in-scattering operator
$Qout{⋅}$	out-scattering operator
$QVR {⋅}$	velocity-randomizing scattering operator
$η$	index of the scattering process
$s(x, k,k′)$	scattering coefficient
$σ(x,k,k′)$	symmetric scattering coefficient
$n(x,t)$	particle density (electron density)
$Z(ε,θ,φ)$	generalized density of states
$⟨⋅⟩$	moment of the BTE with respect to the argument
$f(x,k,t)$	distribution function
$g(x, k,t)$	generalized energy distribution function
$j l,m$	generalized current density term
$Γ l,m$	spherical harmonics coupling term
$VT$	thermal voltage

BTE	Boltzmann transport equation
CPU	central processing unit
GPU	graphics processing unit
FET	field effect transistor
MEDS	maximum entropy dissipation scheme
MOSFET	metal-oxide-semiconductor field effect transistor
SHE	spherical harmonics expansion
TCAD	technology computer aided design