2014 Annual Report Conference on Electrical Insulation and Dielectric Phenomena
Monte Carlo Studies of Hot Electron Transport and
High Field Degradation
Ying Sun
Institute of Materials Science
University of Connecticut, Storrs, Connecticut 06269, USA
[email protected]
Abstract—The problem of hot electron transport and energy loss
at high electric fields in insulators is of considerable interest in
the context of dielectric breakdown and hot carrier induced
degradation. Hot electron transport is discussed in terms of
electron-phonon scattering in polymeric dielectrics. Monte Carlo
(MC) simulation provides the basis for study of hot electron
transport in thin polyethylene (PE) films. Electron trajectories
and spatial evolution of the electron energy distribution are
presented. Possible molecular degradation mechanisms are
discussed.
Keywords— Electons; Monte Carlo; Breakdown; Degradation;
I.
INTRODUCTION
The problem of hot electron transport and energy loss at
high electric field in insulators is of interest in the context of
dielectric breakdown and hot carrier related degradation. The
underlying mechanisms of breakdown and high field
degradation are complex, as many processes are involved, e.g.,
carrier injection, carrier recombination, and the effect of
impurity states in the band gap, impact ionization, etc.
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S. A. Boggs1,2,3and R. Ramprasad1,2,4
Institute of Materials Science, 2Departments of Physics,
Electrical Engineering, 4Materials Science and Engineering
University of Connecticut, Storrs, Connecticut 06269, USA
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eventually, chemical impurity state traps, at which point, the
dominant energy loss is to phonons, i.e., lattice vibration. At
sufficiently high electric field, the electron energy increases
indefinitely until a threshold is reached at which the electron
excites a valance electron across the bandgap, thereby leaving a
hole at valence band. The resulting two electrons near the
conduction band minimum (CBM) are each subject to electric
field acceleration. This process is referred as impact ionization,
and the ensuing carrier multiplication leads to irreversible
damage and breakdown.
Bond cleavage and molecular degradation in polymeric
dielectrics can occur as a result of hot electrons, carrier
recombination, radiation, etc. Recent experiments indicate that
dc field-induced electroluminescence in PE can be interpreted
on the basis of carrier recombination [2]. Ab initio molecular
dynamics simulations demonstrate that excitons localize on a
distorted region of polyethylene weakens significantly nearby
C-H bonds and facilitates C-H bond scission [3]. In the
presence of micro to nano cavities (depending on the field), hot
electrons injected from electrode and accelerated by the electric
field within a cavity can gain sufficient energy to break
chemical bonds and create defects near the cavity wall.
Fig.1 shows an energy diagram which presents carrier
generation, transport, and recombination at high electric field.
In this study, Monte Carlo simulations were used to study hot
electron transport in thin polyethylene (PE) films at high
electric fields based on electron-phonon scattering rate
computed using parameter-free, first principles computational
quantum mechanics. The MC simulation predicts the onset of
avalanche breakdown caused by impact ionization and
provides insight into hot carrier induced degradation.
Fig. 1. Energy diagram of carrier generation, transport, and recombintaion at
high electric field F. Ef is the metal Fermi level, Ec is conducntion band, Ev is
valence band, Ei are the impurity states(dashed lines), and Eg is the bandgap.
The injection of carriers from an electrode depends on the
location of the metal Fermi level in the polymer bandgap and
the proximity of impurity states to the Fermi level. Recent
computations on the interface between PE and Pt indicate that
the Fermi level of the Pt is close to the center of the PE
bandgap, and the barrier to injection of holes or electrons from
the Fermi level into impurity states in the PE (including those
caused by carbonyl at the Pt-PE interface) is in the range of 1
eV, i.e., similar to the measured activation energy for PE [1].
At low fields, the electrons will be trapped by impurity
states caused by chemical (~1 eV) or conformational “abnormalities” (~0.3 eV). With increasing field, the carrier energy
becomes great enough to avoid conformational traps and,
II.
MONTE CARLO METHOD
A Monte Carlo method was developed to simulate hot
electron transport processes and carrier generation by impact
ionization in thin PE films subjected to high electric field. The
main assumption is that the relevant scattering mechanism for
charge carriers is caused by electron-phonon interactions, i.e.,
electrons gain energy from an external electric field between
successive collisions with phonons. The energy threshold at
which a high energy electron ionizes the lattice which results in
carrier multiplication is the bandgap energy. A summary of the
Monte Carlo method with emphasis on implementation for the
present study is provided in the flowchart of Fig. 2.
The number of electrons injected from the electrode can be
set at the beginning of the program. The initial position of each
electron is selected randomly over the cathode surface of the
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PE film with zero initial energy. During the migration toward
the opposite edge of the film, each electron gains energy from
the electric field and scatters through interactions with phonons
which change its momentum and/or energy.
E ' = E  wql
(5)
The phonon involved in the scattering event, and whether
the scattering event absorbs or emits a phonon is determined
based on probability sampling, where the probability is
P =
g ql
g
, Pql =
gtotal
gtotal
(6)
Fig. 2. Flowchart of 3D Monte Carlo simulation.
Each electron is labeled with (r, k), where r is the position
of the electron and k is its wave vector. The free flight duration
time, Δt(E), is the reciprocal of the electron energy-dependent
electron-phonon scattering rate as shown in Fig.3(a). The
scattering rate is calculated using density functional
perturbation theory (DFPT) based on direct integration of
electronic scattering probabilities over all possible final
states[4].
Dt (E) = 1/ g el - ph (E)
(1)
New r’ and k’ values are calculated according to Newton’s
second law after the free flight Δt. The effective electron mass,
m*, is taken as the free electron mass. The electron energy is
related to its wave vector by the parabolic dispersion relation
E(k)=ħk2/2m*.
k' = k + (eF / m* )Dt
*
(2)
*
r ' = r + (k / m )Dt + (eF / 2m )Dt
2
(3)
At the end of free flight, an electron scatters with a phonon,
which changes its momentum and/or energy. The polar and
azimuthal scattering angles (,) are calculated by isotropic
scattering since isotropic scattering is implicit in the average
electron model [5].
cos q = 2r1 -1, j = 2p r2
(4)
where r1 and r2 are random numbers between 0 and 1. In the
present 3D simulations, the polar angle, , is a solid angle
between initial and final electron wave vectors.
Final energy is determined by energy conservation given
by(5), where the ± sign indicates whether a phonon is absorbed
(+) or emitted (-) during a scattering process. Energy loss ħωqλ
at each scattering event is determined by the phonon dispersion
curve of crystalline PE calculated from DFPT as shown in Fig.
3(b) [6].
Fig. 3. (a) Electron phonon scattering rate of crystalline PE at 300K. The
dashed line and dotted line represent scattering involving phonon absorption
and phonon emission respectively. (b) Phonon dispersion curve and (c)
phonon density of states in crystalline PE calculated from DFPT.
Impaction ionization occurs when the electron energy is
sufficient to elevate an electron from the valence band to the
conduction band (8.8eV in PE), which results in two electrons
near the CB [7], both of which are then subject to acceleration
by the electric field. In the MC simulation, this is modeled by
the generation of a second electron when the energy of an
electron reaches 8.8 eV. When an electron reaches the anode
surface of the film, the MC simulation stops, and the location
and energy after each scattering event are recorded.
III. DISCUSSION OF MC SIMULATIONS
The MC simulation is carried out in three dimensional
spaces and projected onto a plane for display. Fig. 4 presents
electron trajectories of 10 initial electrons in crystalline PE at
various electric fields. No impact ionization occurs at
500MV/m and 1GV/m, as the electron energy distribution
achieves steady state as the energy gain from the external
electric field is balanced by energy loss from collisions with
phonons. The phonon energy ħωqλ lost at each scattering event
is probably caused largely by longitudinal optical phonons
associated with C-H stretching modes shown in Fig.3 (c), since
the LO phonon branch has the greatest phonon energy.
Acoustic phonons tend to cause little energy change during
scattering and mainly contribute to momentum change which
also impedes electron energy gain, i.e., phonon collisions that
result in an electron traveling “against” the field, being
decelerated by it result in the electron giving up energy to the
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field. At 1.5 and 2 GV/m, electrons gain sufficient energy to
cause impacting ionization, as can be seen from Fig.4 (c) and
(d) that the electrons are multiplied by impact ionization as
they travel through the films. The maximum “generation” of
electrons after impact ionizations is 4 and 9 in Fig.4 (c) and (d),
respectively. The intrinsic breakdown field determined from
this model is in the range of 1.5 GV/m, as impact ionization
occurs at this field. The measured breakdown field for
technical PE is much lower, ~1.6×108V/m [8].The difference is
probably the result of impurity states in the band gap created by
chemical impurities such as carbonyl which are inevitable
during polymerization and sample fabrication.
Fig. 4. Trajectories of ten initial electrons projected onto the zx plane at
electric fields of (a) 500 MV/m, (b)1 GV/m, (c)1.5GV/m, and (d)2 GV/m.
Field-induced electroluminescence experiments [2] indicate
that the energy released during carrier recombination can cause
bond breakage and molecular degradation in polymeric
dielectrics. No net recombination takes place if the carrier
density equals the thermal equilibrium value. Electron-hole
pairs can be created from electrons injected from one side of
the electrode and holes from the other side or from bipolar
injection at a field enhancement subjected to a sufficiently
large AC or time-varying unipolar voltage [9]. Also when
impact ionization occurs, an electron generates a second
conduction electron by excitation across the bandgap, leaving a
hole at the valence band, thereby creating an electron-hole pair.
The rate of carrier recombination is usually proportional to
excess carrier density. A more detailed model would relate the
carrier recombination rate to the rate of bond cleavage.
MC model for technical PE has also been developed.
Technical PE has a semi-crystalline structure with ~50%
crystallinity and a density in the range of 0.95 g/cm3, which
implies ~10% free volume in the amorphous regions. The
presence of micro and nano-cavities is inevitable in polymeric
materials. These may be formed during manufacture by the
evaporation of volatile decomposition products from various
chemicals reactions such as those used for crosslinking and
those associated with antioxidants. They may also be formed
from impurities and additives which decompose, migrate and
outgas from crosslinking inhomogeneities in network
polymers, and from atmospheric gases which have not diffused
out [10]. Morphology variation in technical PE results in
fluctuations in the phonon density of states which affects the
electron-phonon scattering rate. Since phonons are “creatures”
of the polymer backbone, similar electron-phonon scattering
rate is assumed for both amorphous and crystalline regions.
Probably the vibration modes shown in Fig. 3(c) will spread in
amorphous region of PE. The MC model for technical PE
contains 10% volume fraction of nano-cavities. The simulation
assumes no scattering within the cavities, while the electronphonon scattering rate of crystalline PE in assumed for both
crystalline and amorphous regions. Fig. 5 shows electron
trajectories through the film with no cavities and 10% volume
fraction with various cavity radii at 500 MV/m.
Fig. 5. Projections of ten, 3D electron trajectories in the PE film with 10%
volume fraction nanocavities at a field of 500 MV/m. Nanocavities are
indicated by gray circles: (a) no cavities, (b) 1 nm radius, (c) 2.5 nm radius,
and (d) 5 nm radius.
Fig. 6. Electron energy as a function of distance at a field of 500 MV/m. (a)
PE without cavities, (b) PE with 10% volume fraction of 5 nm radius
nanocavities.
Fig. 6 shows the electron energy as a function of distance at
a field of 500 MV/m. Without cavities, the electron energy
distribution achieves steady state, as the energy gain from the
external electric field is balanced by energy loss from collisions
with phonons as shown in Fig. 6 (a). The energy of electrons
passing through cavities increases linearly without scattering
with phonons as shown in Fig.6 (b). Note in Fig.6 (b) that the
energetic electrons sometimes move against the field during the
process of losing energy after passing through a cavity. The
black reference line shows electrons often have energy greater
than 3 to 4 eV, comparable to C-H and C-C bond energies in
organic molecules, which is likely to cause bond cleavage [11].
In the linear hydrocarbon n-C36H74, strong evidence was found
that the injection of electrons with kinetic energy larger than 4
eV leads to chemical changes of alkane films, while no
changes were observed over long period for electrons with
kinetic energy smaller than 3.5 eV. Electrons with energy
above 4 eV generate deep electron traps in the bandgap of the
organic films, the carriers within which can be exited into
conduction band optically with sub-bandgap light (ħω<Eg/2)
[12].
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IV. MOLECULAR DEGRADATION MECHANISMS
Various processes of high field molecular degradation can
lead to breakdown. Pyrolysis and radiolysis are two
mechanisms that could contribute to molecular degradation in
polymeric materials [13]. Pyrolysis is related to thermal
degradation and is not highly relevant to high field degradation.
The main contribution to field-induced degradation appears to
come from radiolysis, which is caused mainly by electron-hole
recombination and by hot electrons injected into a polymer
from an electrode or traps. The change in molecular weight and
accumulation of degradation products (e.g. C=O, C-O-C, CH3,
C=C, and OH groups) in a polymeric dielectrics exposed to the
electric field provide evidence for radiolysis [13].
A range of chemical reactions can cause polymer bond
cleavage. Initial step of molecular degradation involves energy
absorption of polymer molecular. For example, interaction of a
photon generated by carrier recombination with a polymer can
inject one or more hot electrons. At ambient temperature, ionelectron recombination occurs quickly and results in highly
excited states (P*) and cations restructure. The excited states
dissipate some of their excess energy by bond scission to give
free radicals. The scission of C-H bonds is favored over C-C
backbone scission [14].
Energy absorption
Polymer(P) 
 e- , P + Electron ejection
e + nP  nP + (n+1)e
-
+
e - + P +  P*
P*  P   P 

P  P H
*

-
(7)
electric field. The existence of 5nm radius cavities results in
large portion of electrons above the threshold energy at which
they could break chemical bonds (3-4 eV), which leads to high
field degradation in the vicinity of cavity walls.
High field degradation of polymeric dielectrics is preceded
by energy absorption of polymer molecules, which may come
from the energy released in electron-hole recombination or
excess kinetic energy of hot electrons. Bond scission occurs
and the free radicals formed lead to other chemical products,
which introduce deep traps in the bandgap.
Future work should include “extrinsic” factors other than
nanocavities, such as the effect of density fluctuations in
amorphous polymers. The bond cleavage rate should be related
to carrier recombination rate and the ratio of hot electrons
above threshold energy. The deep traps induced by chemical
products during aging will also be included. All these effects
should be included statistically to investigate high field
degradation and engineering breakdown of insulating films.
REFERENCES
[1]
[2]
Secondary electron ejection (8)
Excited state formation
C-C scission
(9)
(10)
C-H scission
(11)
Recent ab initio molecular dynamics simulations
demonstrate that the electron and hole states of the exciton
localize on a distorted region of polyethylene which weakens
significantly nearby C-H bonds and facilitates C-H bond
scission. The resulting bond cleavage occurs very rapidly upon
exciton injection [3]. The free radicals produced in reactions
(10) and (11) lead to many chemical products and are
accelerated by oxygen. For example, hydrogen atoms may
abstract from the polymer chain to give molecular hydrogen
and alkyl radicals, and the combination of alkyl radicals or
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eV (vinyl group and double bond) [15].
V. CONCLUSION
From the Monte Caro model, the breakdown behavior of
crystalline PE can be predicted. The inevitable nanocavities
existed in technical polymers increase the number of high
energy electrons. As a result radiation damage induced by hot
electrons may happen in technical polymers at a much lower
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