List of Symbols

factor of proportionality


reciprocal effective permittivity for the oxide/bulk system,


active interface area in the charge pumping


probability per unit time for transition of an electron from level to

,
constants

,
unknown coefficients


quantity in Appendix H

,
constants

BBT
Band-to-Band Tunneling


quantity in Appendix H

,
unknown coefficients


factor of proportionality


normalized ,


depletion-region capacitance per unit area


quasi-static gate capacitance per unit area


gate-channel capacitance


gate-bulk capacitance


inversion-layer capacitance per unit area

CMOS
Complementary MOS technology


oxide capacitance per unit area,


semiconductor capacitance per unit area for the gate, including traps at the gate/oxide interface


semiconductor capacitance per unit area for the bulk, including traps at oxide/bulk interface


interface trap capacitance for electrons (parallel representation)

,
help constants

-
capacitance-voltage


bulk body-factor defined in B.11


width of the localized interface charge


interface trap density ()


average interface trap density across an energy interval


density of acceptor-like interface traps (at the gate/oxide interface)


the maximal value of


density of donor-like interface traps (at the gate/oxide interface)


the maximal value of

DLTS
Deep-Level-Transient-Spectroscopy


density of bulk traps ()


length and width of the single devices in the parallel array model


energy


electric field


elementary charge,


activation energy for an interface trap


energy of conduction band edge


energy of the conduction band edge in an ideal band


electric field strength at the endpoint of the tunneling path


Fermi level


Fermi level in the gate


energy level which corresponds to


quasi-Fermi level for electrons


quasi-Fermi level for electrons at the top level of the gate pulse


quasi-Fermi level for electrons at the bottom level of the gate pulse


quasi-Fermi level for holes


quasi-Fermi level for holes at the top level of the gate pulse


quasi-Fermi level for holes at the bottom level of the gate pulse


silicon band gap


electric field in the gate (the component perpendicular to the interface)


at the gate/oxide interface


band gap of intrinsic Si


direct band gap of Si (the gap at )


intrinsic level in an ideal semiconductor


at the gate contact


midgap energy level for the charge pumping


electron emission level


electron emission level for an abrupt falling edge ()


electron emission level at the transition between the steady-state and the non-steady-state emission mode


maximal field allowed in an insulator (e.g. critical field for breakdown)


mean-square-value of the field in tunneling


for SiO


electric field at the metal surface


at the surface


in the gate


in the oxide


hole emission level


the lower boundary of the energy interval which is active in charge pumping


hole emission level for an abrupt rising edge ()


hole emission level at the transition between the steady-state and the non-steady-state emission mode


in semiconductor at the interface;
electric field strength at the starting point of the tunneling path


at the threshold voltage


surface field which induces


surface field which induces


energy level of traps


the highest energy level completely filled by the hole emission during the gate bottom level


the lowest energy level completely emptied by the electron emission during the gate bottom level


energy level which separates the hole-capture dominant from the electron-emission dominant region during the gate bottom level


the lowest energy level completely emptied by the electron emission during the gate top level


the highest energy level completely filled by the hole emission during the gate top level


energy level which separates the electron-capture dominant from the hole-emission dominant region during the gate top level


energy level corresponding to time


energy of the valence band edge


energy of the valence band edge in an ideal band


electric field component parallel to the interface

,
energy for the motion of particles in direction parallel and perpendicular to the field, respectively


electric field component perpendicular to the interface


effective perpendicular field in the inversion layer


occupancy function; frequency


Fermi integral in the normalized form of order (defined by 2.8)


inverse Fermi integral


steady-state occupancy function

,
steady-state occupancy function at the gate top and bottom level, respectively


oscillation frequency of valence band electrons


occupancy function at the beginning of the interval


initial condition for the occupancy function


occupancy function obtained by the strongly implicit discretization of the trap-dynamics equation


Fourier transformation of

FD
Fermi-Dirac


trap degeneracy


effective (net) electron generation rate


DC component of


effective (net) hole generation rate


DC component of

GIDL
Gate-Induced Drain Leakage


transadmittance


high-frequency transadmittance


quasi-static transconductance


generation (emission) rates of electrons and holes, respectively for the single level traps


interface trap conductance for electrons (parallel representation)

HF
High-Frequency


Planck's constant,


the integral considered in Appendix F


bulk current (DC component or direct current)

,
bulk electron and hole current, respectively (DC components)


bulk current (variable in time)


DC component of


charge-pumping current


for abrupt pulse edges


theoretically maximal


charge-pumping current in virgin device


drain current (DC component or direct current)

,
drain electron and hole current, respectively (DC components)


drain current in virgin device


drain current for device without gate depletion


drain current (variable in time)


source current (DC component or direct current)

,
source electron and hole current, respectively (DC components)


source current (variable in time)

I-V
current-voltage


total current density


imaginar unit


displacement current density

,
electron and hole current density, respectively


correction of the Fermi-level position due to Fermi-Dirac statistics,
expression A.24.


Boltzmann's constant,


wave vector

, ,
components of the wave vector


normal component of the wave vector,


number of subintervals in the time steps of terminal voltages, which are used to discretize the trap-dynamics equations

LDD
Lightly-Doped-Drain


extrinsic Debye length


effective channel length


charge-pumping effective channel length


gate length


index for time steps


dipole length


tunneling length


effective electron mass in crystal

MB
Maxwell-Boltzmann


number of trap levels for discretization in the energy space


interband matrix element for phonon scattering in indirect tunneling


free electron mass in vacuum,

MOS
Metal(gate)/Oxide/Semiconductor structure

MOSFET
Metal(gate)/Oxide/Semiconductor Field-Effect Transistor


reduced mass in tunneling


interband matrix element for the transition from the valence to the conduction band, between states of the same energy


number of dopant atoms in small volume ;
number of generated pairs by internal field emission in unit time


constant approximation of close to the gate/oxide interface


mathematical expectation of


impurity concentration in the gate area


electron concentration


concentration of ionized acceptors


bulk doping


effective density of states in the conduction band


concentration of ionized donors


interface trap concentration per unit area ()


concentration of acceptor-like


in uniform field, given by the Kane expression [236]


fixed oxide charge concentration per unit area ()

,
concentration of positive and negative , respectively


total concentration of charged sites at the interface


intrinsic concentration of silicon


intrinsic concentration of pure silicon


effective intrinsic concentration of doped silicon

NMOS
MOS technology with -channel MOSFETs


neutral electron concentration


at the gate contact


electron concentration at the surface


surface electron concentration which determines the boundary of the total electron capture


trap concentration at the energy level


trap density at grain boundaries in polysilicon


equivalent volume trap density in polysilicon


total trapped charge


concentration of occupied traps at the energy level


effective density of states in the valence band


tunneling rate in a linearly variable field

,
heavily doped -type and -type silicon, respectively


hole concentration

PIP
Pulsed-Interface-Probing

PMOS
MOS technology with -channel MOSFETs


neutral concentration of holes


hole concentration at the surface


surface hole concentration which determines the boundary of the total hole capture


probability of the penetration of an incident electron from the valence to the conduction band


dipole charge


charge density in the gate per unit area


charge density at the gate/oxide interface per unit area


fixed charge component of





inversion-layer charge density per unit area


total charge at the oxide/bulk interface;
fixed charge at the oxide/bulk interface


peak density of the gaussian distributed fixed charge


charge density in the bulk per unit area


total charge density in the oxide per unit area


equivalent charge density in the oxide reduced to the oxide/bulk interface

QS
Quasi-Static

Q-V
charge-voltage


residual error in the Poisson equation at time


recombination (capture) rates for electrons and holes with respect to single level traps


effective (net) electron recombination rate,


effective (net) hole recombination rate,

RTS
Random-Telegraph-Signal

RTA
Rapid Thermal Annealing


help quantity in Appendix F

SIMOX
Separation by IMplanted OXygen

SRH
Shockley-Read-Hall

SRP
Small-Rectangular-Pulse


temperature; latice temperature; transmission probability per incident particle


time; help variable in Appendix F

TDDB
Time-Dependent Dielectric Breakdown


emission time


fall time of the gate pulse (Figure 3.2)


first fall time of the gate pulse in the three-level techniques (Figure 3.3)


second fall time of the gate pulse (Figure 3.3)


duration of the top level of the gate pulse (Figure 3.2)


duration of the bottom level of the gate pulse (Figure 3.2)


duration of the mid level of the gate pulse (Figure 3.3)

,
emission time of electrons and holes, respectively


an arbitrary time interval


period of oscillations of valence band electrons


initial time for solving the trap equation


gate-oxide thickness


rise time of the gate pulse (Figure 3.2)


transmission probability from the valence to the conduction band for one particle


potential energy


potential energy operator


bulk-source bias


drain-source bias


gate-bulk bias


gate-source bias


corrected gate-source bias defined by 2.31


effective gate-source bias,

,
top and bottom level of the gate pulse, respectively (Figure 3.2)


mid level of the gate pulse in the three-level techniques (Figure 3.3)


voltage drop in the oxide


reverse junction bias


thermal voltage,


threshold voltage


charge-pumping threshold voltage


gate-source bias at the transition between the steady-state and the non-steady-state emission of electrons


gate-source bias at the onset of electron capture


potential difference between quasi-Fermi level for minorities in the channel and Fermi level in the source


equivalent tunneling barrier (for )


help potential in Appendix E


drain potential


flat-band potential


charge-pumping flat-band potential


gate-bulk bias at the transition between the steady-state and the non-steady-state emission of holes


gate-bulk bias at the onset of hole capture


gate potential


volume in the impulse space

,
impulse-space intervals for and , respectively


help potential perturbation; small volume in silicon


drift velocity

,
average thermal velocity of electrons and holes, respectively


channel width

WKBJ
Wentzel-Kramers-Brillouin-Jeffreys (sometimes called WKB or WBK)


interband matrix element for the potential energy operator


coordinate parallel to the oxide/bulk interface in MOSFETs


boundary of the total carrier capture area in the charge pumping

,
the edges of the charge-pumping active interface area at the drain and source side, respectively

,
and in virgin device


-coordinate of the metallurgical junction (gate/drain overlap region)


coordinate of the middle of the interface charge-sheet


relative coordinate with respect to

,
classical turning points in WKBJ approach (for )


origin of the local coordinate system in solving the line-charge problem


coordinate perpendicular to the oxide/bulk interface in MOSFETs


width of the depletion region in the bulk and the gate


coordinate of the oxide/bulk interface


complex variable

1D
one-dimensional

2D, 3D
two-dimensional and three-dimensional


reduced Planck's constant (Dirac's constant),

,
eigenfunctions in the conduction and valence band, respectively in the presence of the electric field


coefficient of the linear field variation


Gamma function


body-effect like factor; Euler constant


Laplace operator; voltage drop in the gate


voltage drop in the gate calculated by Fermi-Dirac statistics


voltage drop in the gate at the onset of gate inversion


voltage drop in the gate calculated by Maxwell-Boltzmann statistics


voltage drop in the gate at the flat-band conditions in the bulk


voltage drop in the gate calculated quantum-mechanically


voltage drop in the gate at the threshold in the bulk


width of the transition region of the occupancy function


change in due to hot-carrier degradation


differential charge-pumping current,


change in due to hot-carrier degradation


charge in the interface trap concentration due to hot-carrier degradation


gate-pulse amplitude; shift of the gate bias in Appendix F


extreme value of in Appendix F


back-shift in the charge-pumping characteristics after compensation of the trapped holes by a short electron injection


small increment of the gate-source bias


shift in due to hot-carrier degradation


shift in the charge-pumping threshold voltage


time step between and ,


electron net generation rate due to interface traps at


hole net generation rate due to interface traps at


surface potential variation related to in Appendix F


spatial shift of caused by fixed oxide charge


Dirac's delta ``function''


change in the energy position of an interface trap


rigid downward shift of the conduction band due to heavy-doping effects


at the gate-contact


total electrical band-gap narrowing,


energy difference between adjacent levels in the trap discretization


rigid upward shift of the valence band due to heavy-doping effects


at the gate-contact


difference of between two devices


increase in due to an enhanced recombination


surface band-bending


permittivity of insulator


permittivity of metal


absolute permittivity,


permittivity of the gate oxide


permittivity of the polysilicon gate


permittivity of Si


relative field in Appendix E; kinetic energy of electron in band

,
energy of electron in the conduction and valence band, respectively


energy associated with the motion perpendicular to the field in tunneling


for which tunneling is possible


the ratio ; the parameter in the definitions of and


Nabla operator


the ratio

,
electron and hole mobility, respectively


electron mobility as minorities


surface mobility solely due to scattering on the surface charge


surface low-field mobility


surface low-field mobility including the scattering on surface charge


width of the intersection of the Brillouin zone in the field direction


circular frequency; probability per unit time of the transition of one electron from the valence to the conduction band


volume of unit cell in semiconductor


potential

,
wave functions in the conduction and valence band, respectively


potential perturbation


potential of the gate contact with respect to Fermi level in the source


extreme potential perturbation


surface potential


surface potential at flat-band


surface potential at threshold


surface potential corresponding to


surface potential in the channel with respect to the potential at the source channel-end


potential difference between the conduction band edge and Fermi level


potential difference between Fermi level and the valence band edge


space charge density


density of space charge in the oxide


standard deviation in the gaussian distributed or ;
specific conductance


standard deviation in the number of dopant atoms in


standard deviation in the concentration

,
capture cross-section for electrons and holes, respectively


surface charge density due to charge trapped at the level


time constant


electron and hole capture time constant, respectively


dielectric relaxation time


electron and hole emission time constant, respectively


Fermi barrier in the bulk,


Fermi barrier corresponding to


Fermi barrier in the gate


relative surface potential in the MOSFET channel

Nomenclature for currents and voltages:

voltage between the points and (reference), variable in time

voltage between and , DC component

AC component of


terminal current variable in time (by definition the current is positive when flowing into device)

terminal current, DC component of or direct current

AC component of