Physical Quantities

Symbol		Unit		Description
$\beta_n$		1		Electron kurtosis
$\beta_\mathrm{Bulk}$		1		Electron kurtosis in the bulk
$\ensuremath {\mathrm{q}}\ensuremath {\chi_\mathrm{S}}$		eV		Electron affinity in the semiconductor
$D_n$		m$^2$s$^{-1}$		Electron diffusion coefficient
$D_p$		m$^2$s$^{-1}$		Hole diffusion coefficient
$E$		Vm$^{-1}$		Electric field
$\ensuremath{E_\mathrm{diel}}$		Vm$^{-1}$		Electric field in the dielectric
${\mathcal{E}}$		eV		Energy
$\ensuremath{{\mathcal{E}}_\mathrm{f}}$		eV		FERMI energy
$\ensuremath {{\mathcal{E}}_\mathrm{c}}$		eV		Conduction band edge energy
$\ensuremath {{\mathcal{E}}_\mathrm{v}}$		eV		Valence band edge energy
${\mathcal{E}}_{\mathrm{c},0}$		eV		Conduction band edge energy in the flat-band case
${\mathcal{E}}_{\mathrm{v},0}$		eV		Valence band edge energy in the flat-band case
$\ensuremath{{\mathcal{E}}_\mathrm{g}}$		eV		Band gap energy
$\ensuremath{{\mathcal{E}}_\mathrm{i}}$		eV		Intrinsic energy
$\ensuremath {{\mathcal{E}}_\mathrm{image}}$		eV		Image force correction energy
${\mathcal{E}}_x$		eV		Energy component in the tunneling direction
${\mathcal{E}}_\rho$		eV		Energy component perpendicular to the tunneling direction
${\mathcal{E}}_i$		eV		Energy eigenvalue
$\ensuremath {{\mathcal{E}}_\mathrm{im}}$		eV		Imaginary part of the energy eigenvalue
$\ensuremath {{\mathcal{E}}_\mathrm{re}}$		eV		Real part of the energy eigenvalue
$\ensuremath{{\mathcal{E}}_\mathrm{T}}$		eV		Trap energy level below the dielectric conduction band
$\ensuremath{{\mathcal{E}}^{\prime}}$		eV		Trap energy
$\phi$		V		Electrostatic potential
$\ensuremath {\phi_\mathrm{surf}}$		V		Surface potential
$\ensuremath {\Phi_\mathrm{f}}$		V		FERMI potential
$\ensuremath {\mathrm{q}}\ensuremath{\Phi_\mathrm{B}}$		eV		Barrier height
$\ensuremath {\mathrm{q}}\ensuremath{\Phi_\mathrm{W}}$		eV		Work function
$\ensuremath {\mathrm{q}}\ensuremath {\Phi_\mathrm{S}}$		eV		Work function of the semiconductor
$\ensuremath {\mathrm{q}}\ensuremath {\Phi_\mathrm{M}}$		eV		Work function of the metal
$\ensuremath {\mathrm{q}}\ensuremath {\Phi_\mathrm{MS}}$		eV		Work function difference between metal and semiconductor
$\ensuremath {\mathrm{q}}\Phi$		eV		Upper edge of a triangular energy barrier
$\ensuremath {\mathrm{q}}\Phi_0$		eV		Lower edge of a triangular energy barrier
$\ensuremath {\mathrm{q}}\ensuremath{\Phi_\mathrm{e}}$		eV		Electron energy barrier
$\ensuremath {\mathrm{q}}\ensuremath{\Phi_\mathrm{h}}$		eV		Hole energy barrier
$\ensuremath{f_\mathrm{P}}$		1		Distribution of phonons in energy
$\ensuremath{f_\mathrm{T}}$		1		Trap occupancy
$g$		m$^{-3}$eV$^{-1}$		Density of states
$\ensuremath{\hbar\omega}$		eV		Phonon energy
$\mathbf{J}$		Am$^{-2}$		Current density

Symbol		Unit		Description
$\ensuremath{{\mathbf{J_n}}}$		Am$^{-2}$		Electron current density
$\ensuremath{{\mathbf{J_p}}}$		Am$^{-2}$		Hole current density
$k$		m$^{-1}$		Wave number
$\mathbf{k}$		m$^{-1}$		Wave number vector
$k_x$		m$^{-1}$		Wave number component in the tunneling direction
$k_\rho$		m$^{-1}$		Wave number component perpendicular to the tunneling direction
$\ensuremath {k_\mathrm{f}}$		m$^{-1}$		Radius of the FERMI sphere
$\kappa$		AsV$^{-1}$m$^{-1}$		Dielectric permittivity
$\ensuremath{\kappa_\mathrm{diel}}$		AsV$^{-1}$m$^{-1}$		Dielectric permittivity of the dielectric layer
$\ensuremath{\kappa_\mathrm{si}}$		AsV$^{-1}$m$^{-1}$		Dielectric permittivity in silicon
$\ensuremath{\kappa_\mathrm{sio2}}$		AsV$^{-1}$m$^{-1}$		Dielectric permittivity in silicon dioxide
$\ensuremath{\kappa_{\mathrm{high-}\kappa}}$		AsV$^{-1}$m$^{-1}$		Dielectric permittivity in a high-$\kappa$ dielectric
$\mu_n$		m$^2$V$^{-1}$s$^{-1}$		Electron mobility
$\mu_p$		m$^2$V$^{-1}$s$^{-1}$		Hole mobility
$\mu_s$		m$^2$V$^{-1}$s$^{-1}$		Energy flux mobility
$m$		kg		Mass
$\ensuremath{m_\mathrm{diel}}$		kg		Carrier mass in the dielectric
$\ensuremath{m_\mathrm{eff}}$		kg		Carrier effective mass in the semiconductor
$n$		m$^{-3}$		Electron concentration
$\ensuremath {n_\mathrm{i}}$		m$^{-3}$		Intrinsic concentration
$N$		eV		Supply function
$\ensuremath {N_\mathrm{D}}$		m$^{-3}$		Concentration of donors
$\ensuremath {N_\mathrm{A}}$		m$^{-3}$		Concentration of acceptors
$\ensuremath {N_\mathrm{poly}}$		m$^{-3}$		Concentration of dopants in the polysilicon
$\ensuremath {N_\mathrm{c}}$		m$^{-3}$		Effective density of states of the conduction band
$\ensuremath{N_\mathrm{T}}$		m$^{-3}$		Trap concentration
$p$		m$^{-3}$		Hole concentration
$P$		1		Number of phonons
$\Psi$		m$^{-1/2}$		Wave function
$\ensuremath{Q_\mathrm{T}}$		As		Trap charge state
$\mathbf{r}$		m		Space vector
$\rho$		1		Probability density
$R$		s$^{-1}$m$^{-3}$		Net recombination rate
$RC$		1		Reflection coefficient
$\ensuremath{R_\mathrm{tun}}$		m$^{-3}$s$^{-1}$		Additional recombination term due to the tunneling current
$S_n$		J m$^{-2}$ s$^{-1}$		Electron energy flux density
$S$		1		HUANG-RHYS factor
$t$		s		Time
$\tau_{\mathcal{E}}$		s		Energy relaxation time
$\tau_m$		s		Momentum relaxation time
$\tau_s$		s		Energy flux relaxation time
$\tau_\beta$		s		Kurtosis relaxation time

Symbol		Unit		Description
$\ensuremath{\tau_{\mathrm{q}}}$		s		Life time of a quasi-bound state
$\ensuremath{\tau_\mathrm{c}}$		s		Capture time
$\ensuremath{\tau_\mathrm{e}}$		s		Emission time
$\ensuremath{\tau_\mathrm{ca}}$		s		Capture time to the anode
$\ensuremath{\tau_\mathrm{cc}}$		s		Capture time to the cathode
$\ensuremath{\tau_\mathrm{ea}}$		s		Emission time to the anode
$\ensuremath{\tau_\mathrm{ec}}$		s		Emission time to the cathode
$\ensuremath{t_\mathrm{diel}}$		m		Thickness of a dielectric
$\ensuremath {t_\mathrm{sio_2}}$		m		Thickness of a SiO$_2$ dielectric
$\ensuremath {t_{\mathrm{high-}\kappa}}$		m		Thickness of a high-$\kappa$ dielectric
$T$		K		Temperature
$T_\mathrm{L}$		K		Lattice temperature
$T_{n}$		K		Electron temperature
$TC$		1		Transmission coefficient
$v$		ms$^{-1}$		Velocity
$\mathbf{v}$		ms$^{-1}$		Velocity vector
$v_x$		ms$^{-1}$		Velocity component in the tunneling direction
$v_\rho$		ms$^{-1}$		Velocity component perpendicular to the tunneling direction
$\ensuremath{V_\mathrm{poly}}$		V		Voltage drop in the polysilicon
$\ensuremath{V_\mathrm{GS}}$		V		Gate-source voltage
$\ensuremath{V_\mathrm{DS}}$		V		Drain-source voltage
$\ensuremath{V_\mathrm{CG}}$		V		Control gate voltage
$\ensuremath{V_\mathrm{FG}}$		V		Floating gate voltage
$\ensuremath{V_\mathrm{diel}}$		V		Voltage drop in the dielectric
$\ensuremath{V_\mathrm{e}}$		Jm		Overlap integral
$W$		eV		Potential energy
$\ensuremath{W_\mathrm{c}}$		m$^2$s$^{-1}$		Capture rate
$\ensuremath{W_\mathrm{e}}$		m$^2$s$^{-1}$		Emission rate
$\ensuremath{x_\mathrm{T}}$		m		Trap cube side length

A. Gehring: Simulation of Tunneling in Semiconductor Devices