Physical Quantities

Symbol	Unit	Description


$a$	$1$	Kinetic exponent in the RD model
$Aab$	$- 1 s$	Electronic matrix element in the Franck-Condon approximation
$Ap$	$V$	Prefactor of the permanent component in the NBTI fit formula
$Ar$	$V$	Prefactor of the recoverable component in the NBTI fit formula
$Ayz$	$m2$	Interface area in the yz-plane
$B$	$1$	Parameter in the empirical relation of the universal recovery
$cn$	$3 -1 m s$	Electron capture coefficient
$c p$	$m3s-1$	Hole capture coefficient
$cNMP p$	$m3s-1$	Hole capture coefficient in the NMP model
$cNMP p,0$	$m3s-1$	Temperature independent prefactor of $cNMP p$
$C ox$	$CV- 1m-2$	Areal gate capacitance
$Dc$	$m2s-1$	Diffusion coefficient in the RDD model
$Dc1D$	$J-1$	Density of states of a one-dimensional, confined electron gas
$Dc3D$	$m-2J-2$	Density of states of a three-dimensional, confined electron gas
$Dit$	$-2 -1 m J$	Interface trap density of states
$Dn$	$-3 -1 m J$	Electron density of states in the conduction band
$D n,1D$	$m-1J-1$	Density of states of a one-dimensional electron gas
$Dn,1D+2D$	$m-3J-2$	Product of $Dn,1D$ and $Dn,2D$
$Dn,2D$	$m-2J-1$	Density of states of a two-dimensional electron gas
$Dox$	$m-2J-1$	Oxide trap density of states
$Dp$	$m-3J-1$	Hole density of state in the valence band
$Dp,c3D$	$m-2J-2$	Density of states of a three-dimensional, confined hole gas
$Dp,1D$	$m-1J-1$	Density of states of a one-dimensional hole gas
$Dp,1D+2D$	$-3 -2 m J$	Product of $Dp,1D$ and $Dp,2D$
$Dp,2D$	$-2 -1 m J$	Density of states of a two-dimensional hole gas
$DX$	$2 -1 m s$	Diffusion coefficient of the species $X$ in the RD model
$e n$	$m3s-1$	Electron emission coefficient
$ep$	$m3s-1$	Hole emission coefficient
$eNpMP$	$m3s-1$	Hole emission coefficient in the NMP model
$eNpM,0P$	$m3s-1$	Temperature independent prefactor of $eNnMP$
$E$	$J$	Electron energy in the band edge energy diagram
$Eact$	$J$	Activation energy
$Eb$	$J$	Electron energy in the conduction/valance band
$E c$	$J$	Conduction band edge energy
$EDcFT$	$J$	Conduction band edge energy in DFT calculations
$Ec,0$	$J$	Conduction band edge energy in the flat band case
$Ed$	$J$	Demarcation energy
$Eds$	$J$	Dissociation barrier in the reaction-limited model
$Edsm$	$J$	Mean value of the dissociation barriers in the reaction-limited model
$Ef$	$J$	Fermi energy
$Ei$	$J$	Energy of the electron system in the state $i$
$E n,i$	$J$	Energy of a channel electron in the quasi-bound state $i$
$Ep,i$	$J$	Energy of a channel hole in the quasi-bound state $i$
$Eq1∕q2$	$J$	Switching trap level for a transition from charge state $q1$ to $q2$
$Et$	$J$	Trap level
$Et,0$	$J$	Trap level in the flat band case
$Et,1$	$J$	Trap level of the oxygen vacancy in the TSM
$E t,2$	$J$	Trap level of the $E′$ center in the TSM
$E ′t$	$J$	Trap level in the NMP model
$Etot,ij$	$J$	Energy of the combined system of electrons and nuclei in the state $ij$
$Ev$	$J$	Valence band edge energy
$DFT Ev$	$J$	Valence band edge energy in DFT calculations
$Ev,0$	$J$	Valence band edge energy in the flat band case
$Ex$	$J$	$x$ -component of electron energy $E$
$Exc$	$J$	Exchange-correlation energy
$faLbSF$	$1$	Lineshape function
$faαbβ$	$1$	Franck-Condon factor
$feq,i,n$	$1$	Equilibrium occupancy for the trap $n$
$f i$	$1$	Occupancy of a trap in state $i$
$fi,n$	$1$	Occupancy for the trap $n$ in state $i$
$fit$	$1$	Interface trap occupancy
$fn$	$1$	Electron occupancy
$f ox$	$1$	Oxide trap occupancy
$fp$	$1$	Hole occupancy
$ft$	$1$	Trap occupancy
$eq ft$	$1$	Equilibrium trap occupancy
$ft,2$	$1$	Electron occupancy of the $′ E$ center in the TSM
$f FD$	$1$	Fermi-Dirac distribution
$Fox$	$Vm -1$	Oxide field
$Fc$	$Vm -1$	Reference field for the MPFAT mechanism
$Fox,r$	$Vm -1$	Oxide field during relaxation
$Fox,s$	$Vm -1$	Oxide field during stress
$F$	$1$	Trap occupancy for a set of states
$F i$	$N$	Force on atom $i$
$gH$	$m-3J-1$	Hydrogen trap density of states in the RDD model
$gν$	$1$	Degeneracy of the valley $ν$
$H$	$-3 m$	Hydrogen concentration in the RDD model
$H$	$-1 Am$	Magnetic field strength
$H$	$J$	General hamilton operator in the derivation of Fermi’s golden rule
$Hc$	$m-3$	Free hydrogen concentration in the RDD model
$Hch$	$J$	Hamilton operator of the channel in the derivation of Fermi’s golden rule
$^Htot$	$J$	Hamilton operator of an atomic system
$ID$	$A$	Drain current
$ID0$	$A$	Initial drain current
$k$	$m-1$	Electron wavevector
$k$	$1$	Coupling constant between $τcap$ and $τem$ in Yang’s model
$kf$	$- 1 s$	Forward rate of a reaction/transition
$kf,0$	$- 1 s$	Attempt frequency in the reaction-limited model
$k r$	$s- 1$	Reverse rate of a reaction/transition
$kx∕y∕z$	$m-1$	$x∕y∕z$ -component of the electron wavevector
$Lx$	$m$	Length of the substrate in the $x$ -direction
$mn$	$m$	Effective mass of the electrons
$mn,ν$	$m$	Effective mass of the electrons for the valley index $ν$
$m p$	$m$	Effective mass of the holes
$mp,ν$	$m$	Effective mass of the holes for the valley index $ν$
$mt$	$m$	Tunneling mass of the charge carriers
$M$	$kg$	Oscillator mass in the harmonic approximation
$Ma αbβ$	$J$	Matrix element for a vibronic transition from state $aα$ to state $bβ$
$M if$	$J$	Matrix element for a transition from state $i$ to state $f$
$n$	$m-3$	Electron density
$np$	$1$	Power law exponent of permanent component in the NBTI fit formula
$nr$	$1$	Power law exponent of recoverable component in the NBTI fit formula
$nRD$	$1$	Power law exponent in the RD model
$N a$	$m-3$	Acceptor concentration
$Nc$	$m-3$	Effective conduction band weight
$Nc,H$	$m-3$	Effective density of conduction states in the RDD model
$Nd$	$m-3$	Donor concentration
$Nit$	$m-2$	Concentration of interface states per area
$Nit,0$	$-2 m$	Initial concentration of interface states per area
$Nox$	$-2 m$	Number of oxide charges per area
$N t$	$m-3$	Trap density
$Nv$	$m-3$	Effective valence band weight
$Nx$	$1$	Number of states for a free electrons/holes gas in the $x$ -direction
$Nyz$	$1$	Number of states for a free electrons/holes gas in the $yz$ -plane
$Nxyz$	$1$	Number of states for a free electrons/holes gas in the $xyz$ -volume
$p$	$-3 m$	Hole density
$pij$	$- 1 s$	Transition probability from state $i$ to state $j$
$q$	$m$	Configuration coordinate
$qi$	$m$	Configuration coordinate at the energy minimum $i$
$Qit$	$Cm -2$	Areal interface charges
$Qox$	$-2 Cm$	Areal oxide charges
$raαbβ$	$- 1 s$	Vibronic transition rate from state $aα$ to state $bβ$
$rcap,e$	$- 1 s$	Electron capture rate
$r cap,h$	$s- 1$	Hole capture rate
$re$	$s- 1$	Electron tunneling rate
$req,ij,n$	$s- 1$	Transition rate $rij$ of a trap $n$ during equilibrium conditions
$re,q1∕q2$	$s- 1$	Electron tunneling rate from the charge state $q1$ to $q2$ in the LSM
$rem,e$	$s- 1$	Electron emission rate
$rem,h$	$s- 1$	Hole emission rate
$rij$	$s- 1$	Rate for the transition $Ti→j$
$rh$	$s- 1$	Hole tunneling rate
$rh,q1∕q2$	$- 1 s$	Hole tunneling rate from the charge state $q1$ to $q2$ in the LSM
$Ri$	$1$	Ratio of the vibration frequencies in an NMP transition
$ri$	$m$	Spatial coordinates of electron $i$
$R$	$s- 1$	Transition rate matrix
$Ri$	$m$	Spatial coordinates of atom $i$
$s$	$-1 -1 K V m$	Scaling factor for stress/relaxation curves
$s$	$K-1V -1m$	Scaling factor $s$ during relaxation
$\text{[math]}$	$\text{[math]}$	Scaling factor $\text{[math]}$ during stress
$\text{[math]}$	$\text{[math]}$	Prefactor of the scaling factor $\text{[math]}$
$\text{[math]}$	$\text{[math]}$	Action in the WKB method
$\text{[math]}$	$\text{[math]}$	Huang-Rhys factor for an NMP transition $\text{[math]}$
$\text{[math]}$	$\text{[math]}$	Time
$\text{[math]}$	$\text{[math]}$	Thickness of the insulator
$\text{[math]}$	$\text{[math]}$	Relaxation time
$\text{[math]}$	$\text{[math]}$	Stress time
$\text{[math]}$	$\text{[math]}$	First measurement point during stress
$\text{[math]}$	$\text{[math]}$	Temperature
$\text{[math]}$	$\text{[math]}$	Transmission coefficient for tunneling through the conduction band
$\text{[math]}$	$\text{[math]}$	Transmission coefficient for tunneling through the valence band
$\text{[math]}$	$\text{[math]}$	Temperature during equilibrium conditions
$\text{[math]}$	$\text{[math]}$	Transition from state $\text{[math]}$ to state $\text{[math]}$
$\text{[math]}$	$\text{[math]}$	Transitions between the states $\text{[math]}$ to state $\text{[math]}$
$\text{[math]}$	$\text{[math]}$	Chain of the transitions $\text{[math]}$ and $\text{[math]}$
$\text{[math]}$	$\text{[math]}$	Temperature during relaxation
$\text{[math]}$	$\text{[math]}$	Temperature during stress
$\text{[math]}$	$\text{[math]}$	Electron kinetic energy operator
$\text{[math]}$	$\text{[math]}$	Nucleus kinetic energy operator
$\text{[math]}$	$\text{[math]}$	Adiabatic potential of an atomic system
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$ for charge state $\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$ in the minimum configuration of state $\text{[math]}$
$\text{[math]}$	$\text{[math]}$	Effective electron potential
$\text{[math]}$	$\text{[math]}$	Thermal velocity of the electrons
$\text{[math]}$	$\text{[math]}$	Thermal velocity of the holes
$\text{[math]}$	$\text{[math]}$	Velocity of atom $\text{[math]}$
$\text{[math]}$	$\text{[math]}$	Defect potential
$\text{[math]}$	$\text{[math]}$	Channel potential in the derivation of Fermi’s golden rule
$\text{[math]}$	$\text{[math]}$	Gate bias during equilibrium conditions
$\text{[math]}$	$\text{[math]}$	Interatomic empirical potential
$\text{[math]}$	$\text{[math]}$	Gate bias
$\text{[math]}$	$\text{[math]}$	Energy of the system in state $\text{[math]}$ in the TWM and the CDW
$\text{[math]}$	$\text{[math]}$	Initial energy of the system in state $\text{[math]}$ in the TWM
$\text{[math]}$	$\text{[math]}$	Gate bias during relaxation
$\text{[math]}$	$\text{[math]}$	Gate bias during stress
$\text{[math]}$	$\text{[math]}$	Threshold voltage
$\text{[math]}$	$\text{[math]}$	Initial threshold voltage
$\text{[math]}$	$\text{[math]}$	Threshold voltage extracted from OTF measurements
$\text{[math]}$	$\text{[math]}$	Initial threshold voltage extracted from OTF measurements
$\text{[math]}$	$\text{[math]}$	Trap potential in the derivation of Fermi’s golden rule
$\text{[math]}$	$\text{[math]}$	Operator for the electron-nucleus Coulomb interactions
$\text{[math]}$	$\text{[math]}$	Operator for the electron-electron Coulomb interactions
$\text{[math]}$	$\text{[math]}$	Operator for the nucleus-nucleus Coulomb interactions
$\text{[math]}$	$\text{[math]}$	WKB factor for tunneling through the conduction band
$\text{[math]}$	$\text{[math]}$	WKB factor for tunneling through the valence band
$\text{[math]}$	$\text{[math]}$	Direction perpendicular to the substrate-insulator interface
$\text{[math]}$	$\text{[math]}$	Position of the substrate-insulator interface
$\text{[math]}$	$\text{[math]}$	Characteristic tunneling length for electrons
$\text{[math]}$	$\text{[math]}$	Characteristic tunneling length for holes
$\text{[math]}$	$\text{[math]}$	Position of the trap
$\text{[math]}$	$\text{[math]}$	Position of the tunneling hole front
$\text{[math]}$	$\text{[math]}$	Density of the diffusing species $\text{[math]}$ in the RD model
$\text{[math]}$	$\text{[math]}$	Direction parallel to the substrate-insulator interface
$\text{[math]}$	$\text{[math]}$	Direction parallel to the substrate-insulator interface
$\text{[math]}$	$\text{[math]}$	Charge state of the diffusing species $\text{[math]}$ in the RD model
$\text{[math]}$	$\text{[math]}$	Charge state of the diffusing species in the RDD model
$\text{[math]}$	$\text{[math]}$	Characteristic exponent in the reaction-limited model
$\text{[math]}$	$\text{[math]}$	Parameter in the empirical relation of the universal recovery
$\text{[math]}$	$\text{[math]}$	Field dependence factor of the thermal barrier in the TSM
$\text{[math]}$	$\text{[math]}$	Stress parameter in the TWM
$\text{[math]}$	$\text{[math]}$	Stress parameter in the CDW model
$\text{[math]}$	$\text{[math]}$	Time step in molecular dynamics
$\text{[math]}$	$\text{[math]}$	Change in the interface trap density of states
$\text{[math]}$	$\text{[math]}$	Change in the trap occupancy in state $\text{[math]}$
$\text{[math]}$	$\text{[math]}$	Change in the oxide trap occupancy
$\text{[math]}$	$\text{[math]}$	Change in the trap occupancy
$\text{[math]}$	$\text{[math]}$	Energy barrier for charge trapping in the model of Kirton and Uren
$\text{[math]}$	$\text{[math]}$	Energy barrier for hole capture of the oxygen vacancy in the TSM
$\text{[math]}$	$\text{[math]}$	Energy barrier for hole capture of the $\text{[math]}$ center in the TSM
$\text{[math]}$	$\text{[math]}$	Relaxation barrier to the oxygen vacancy in the TSM
$\text{[math]}$	$\text{[math]}$	Thermal barrier for the hydrogen transition in the TSM
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$ referenced to the substrate valence bandedge
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$ referenced to the substrate valence bandedge
$\text{[math]}$	$\text{[math]}$	Change in the number of interface charges per area
$\text{[math]}$	$\text{[math]}$	Change in the number of oxide charges per area
$\text{[math]}$	$\text{[math]}$	Change in the areal interface charges
$\text{[math]}$	$\text{[math]}$	Change in the areal oxide charges
$\text{[math]}$	$\text{[math]}$	Change in the threshold voltage
$\text{[math]}$	$\text{[math]}$	Change in the threshold voltage during relax
$\text{[math]}$	$\text{[math]}$	Change in the threshold voltage during stress
$\text{[math]}$	$\text{[math]}$	Change in threshold voltage extracted from OTF measurements
$\text{[math]}$	$\text{[math]}$	NMP barrier for the transition $\text{[math]}$
$\text{[math]}$	$\text{[math]}$	Electric permittivity
$\text{[math]}$	$\text{[math]}$	Energy level of the bound state $\text{[math]}$
$\text{[math]}$	$\text{[math]}$	Kohn-Sham eigenvalue
$\text{[math]}$	$\text{[math]}$	Energy barrier for the transition $\text{[math]}$ in the NMP model
$\text{[math]}$	$\text{[math]}$	Thermodynamic trap level for a transition from charge state $\text{[math]}$ to $\text{[math]}$
$\text{[math]}$	$\text{[math]}$	Energy difference between the state $\text{[math]}$ and $\text{[math]}$ in the NMP model
$\text{[math]}$	$\text{[math]}$	Square root of the WKB factor
$\text{[math]}$	$\text{[math]}$	Mean value of the distribution of $\text{[math]}$ in Yang’s model
$\text{[math]}$	$\text{[math]}$	Mobility in the RD model
$\text{[math]}$	$\text{[math]}$	Mobility of the diffusing species $\text{[math]}$ in the RDD model
$\text{[math]}$	$\text{[math]}$	Valley index
$\text{[math]}$	$\text{[math]}$	Attempt frequency
$\text{[math]}$	$\text{[math]}$	Normalized relaxation time
$\text{[math]}$	$\text{[math]}$	Occupation probability for being in state $\text{[math]}$
$\text{[math]}$	$\text{[math]}$	Occupation probability for set of states
$\text{[math]}$	$\text{[math]}$	Equilibrium occupation probability for set of states
$\text{[math]}$	$\text{[math]}$	Charge density
$\text{[math]}$	$\text{[math]}$	Density of traps
$\text{[math]}$	$\text{[math]}$	Trapped hydrogen density in the RDD model
$\text{[math]}$	$\text{[math]}$	Ground state charge density
$\text{[math]}$	$\text{[math]}$	Spread of the distribution of $\text{[math]}$ in Yang’s model
$\text{[math]}$	$\text{[math]}$	Spread of dissociation barrier in the reaction-limited model
$\text{[math]}$	$\text{[math]}$	Cross section for an electon transition in the TSM model
$\text{[math]}$	$\text{[math]}$	Cross section for an electron tunneling in the ETM
$\text{[math]}$	$\text{[math]}$	Temperature-independent prefactor of $\text{[math]}$
$\text{[math]}$	$\text{[math]}$	Cross section for a hole transition in the SRH model
$\text{[math]}$	$\text{[math]}$	Cross section for a hole transition in the eNMP model
$\text{[math]}$	$\text{[math]}$	Cross section for a hole transition in the TSM model
$\text{[math]}$	$\text{[math]}$	Cross section for hole tunneling in the ETM
$\text{[math]}$	$\text{[math]}$	Temperature-independent prefactor of $\text{[math]}$
$\text{[math]}$	$\text{[math]}$	Temperature-independent prefactor of $\text{[math]}$
$\text{[math]}$	$\text{[math]}$	Temperature-independent prefactor of $\text{[math]}$
$\text{[math]}$	$\text{[math]}$	Transition time constant in the reaction-limited and the Yang model
$\text{[math]}$	$\text{[math]}$	Hole capture time constant over the state $\text{[math]}$ in the eNMP model
$\text{[math]}$	$\text{[math]}$	Hole capture time constant over the state $\text{[math]}$ in the eNMP model
$\text{[math]}$	$\text{[math]}$	Capture time constant
$\text{[math]}$	$\text{[math]}$	Electron capture time constant
$\text{[math]}$	$\text{[math]}$	Hole capture time constant
$\text{[math]}$	$\text{[math]}$	Minimal $\text{[math]}$ in the eNMP model
$\text{[math]}$	$\text{[math]}$	Minimal $\text{[math]}$ in the eNMP model
$\text{[math]}$	$\text{[math]}$	Emission time constant
$\text{[math]}$	$\text{[math]}$	Hole emission time constant over the state $\text{[math]}$ in the eNMP model
$\text{[math]}$	$\text{[math]}$	Hole emission time constant over the state $\text{[math]}$ in the eNMP model
$\text{[math]}$	$\text{[math]}$	Electron emission time constant
$\text{[math]}$	$\text{[math]}$	Hole emission time constant
$\text{[math]}$	$\text{[math]}$	Minimal $\text{[math]}$ in the eNMP model
$\text{[math]}$	$\text{[math]}$	Minimal $\text{[math]}$ in the eNMP model
$\text{[math]}$	$\text{[math]}$	Electron capture time constant as defined in the TSM model
$\text{[math]}$	$\text{[math]}$	Hole capture time constant as defined in the ETM
$\text{[math]}$	$\text{[math]}$	Hole capture time constant as defined in the eNMP model
$\text{[math]}$	$\text{[math]}$	Hole capture time constant as defined in the SRH model
$\text{[math]}$	$\text{[math]}$	Hole capture time constant as defined in the TSM model
$\text{[math]}$	$\text{[math]}$	Reduced electron capture time in the ETM
$\text{[math]}$	$\text{[math]}$	Reduced hole capture time in the ETM
$\text{[math]}$	$\text{[math]}$	Wavefunction of the combined system of $\text{[math]}$ electrons and $\text{[math]}$ nuclei
$\text{[math]}$	$\text{[math]}$	Electric potential
$\text{[math]}$	$\text{[math]}$	Surface potential
$\text{[math]}$	$\text{[math]}$	Potential energy of electrons/holes
$\text{[math]}$	$\text{[math]}$	Left/Right border of the tunnel barrier
$\text{[math]}$	$\text{[math]}$	Wavefunction of an $\text{[math]}$ nuclei system in state $\text{[math]}$
$\text{[math]}$	$\text{[math]}$	Single-electron wavefunction in state $\text{[math]}$
$\text{[math]}$	$\text{[math]}$	Kohn-Sham orbitals
$\text{[math]}$	$\text{[math]}$	Electron channel wavefunction of the quasi-bound state $\text{[math]}$
$\text{[math]}$	$\text{[math]}$	Hole channel wavefunction of the quasi-bound state $\text{[math]}$
$\text{[math]}$	$\text{[math]}$	Wavefunction of an $\text{[math]}$ electron system in state $\text{[math]}$
$\text{[math]}$	$\text{[math]}$	Wavefunction of an $\text{[math]}$ electron system in the ground state
$\text{[math]}$	$\text{[math]}$	Oscillator frequency in the harmonic approximation