COMPUTER SIMULATION OF HIGH-FIELD ELECTRON
EMISSION FROM CRYSTALLINE AND NANOSTRUCTURED
SILICON DIOXIDE
V. Kortov, S. Zvonarev
Ural State Technical University, Ekaterinburg, Russia
Abstract. A physical model of the electron emission from charged dielectric films
and near-surface layers of monocrystalline dielectrics is discussed.
The model assumes that free electrons are produced by thermal detrapping via
multiphonon processes and tunneling. The free electrons moving to the surface in the
field of the builit-in charge interact with optical and acoustic phonons. When the
electric field strength is high an important role is played by inelastic electron
scattering such as impact ionization and cascade processes. The model has been
modified considering specific features of nanoscale materials. It takes into account
processes involved in scattering of electrons on nanocrystal boundaries and the
increase in the energy depth of surface trapping centers of charge carriers.
The physical model has been implemented in an algorithm, and a Delphi program has
been developed for computing, electron emission. Emission from nanostructured and
single-crystal silicon dioxides is simulated. It is shown that nanostructured SiO2 is
characterized by a lower emissivity, a wider emitted-electron energy spectrum, and a
higher dielectric strength, as compared to single crystals.
INTRODUCTION
Electron emission from crystalline and amorphous thin SiO2 films and near-surface
layers exposed to an electron beam is the subject of extensive studies. An electron
beam charges the dielectric surface and generates a high-strength electric field in the
near-surface layer. The motion of detrapped electrons to the surface in the strong
electric field leads to the hot-electron emission and, under certain conditions, to the
electrical breakdown [1]. Theoretical and experimental investigations show that the
breakdown threshold field for thin SiO2 films is 10 to 16 MV/cm [2-3].
Nanostructured SiO2 is currently considered as a material with wide potential
applications in micro- and optoelectronics. Nanostructured SiO2 films are also
promising as host materials for electroluminophores with a high quantum yield. To
cause electroluminescence of such films by impact ionization, an external electric
field of 6 to 7 MV/cm strong must be applied [4].
A physical model describing basic processes underlying the motion of an electron to
the surface was modified in our previous studies dealing with simulation of the
transport and the emission of electrons in irradiated crystalline dielectrics.
Specifically, the rates of the electron scattering by optical and acoustic phonons were
refined and processes of impact ionization and cascading were taken into account [57]. The present study deals with the substantiation of the electron transport model
considering processes that take place in charged crystalline and nanostructured
dielectrics. The use of the model for computer simulation of the electron emission in
high-strength electric fields by the example of crystalline and nanostructured silica is
discussed too.
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PHYSICAL MODEL
The model has been developed for calculating the electron emission from charged
near-surface layers in dielectrics exposed to electrons having the energy of (1-10)
keV. The surface of dielectrics is charged positively under electron bombardment,
while at some depth, which depends on the electron energy, trapping of electrons
results in formation of a negative charge. A "plus"-"minus" structure and a strong
electric field as high as a few megavolts [8] appear in the near-surface layer. Trapped
electrons can be detrapped by the tunneling effect, thermo- and photoionization in the
electric field. Their starting energy depends on the field intensity. Electrons drift to
the surface and are scattered by acoustic or optical phonons depending on their
energy; they also participate in processes of impact ionization and cascading. Some
electrons can reach the surface, surmount the energy threshold, and be released to
vacuum. The thermally stimulated electron emission current and the electron energy
distribution are measured in experiments and the measured values can be compared
with the corresponding calculation data.
Formation of free electrons
Electrons are detrapped via mechanisms of multiphonon ionization in an electric field
[9]. In this case, an electron acquires an initial energy determined from the formula
E o  F 2 e 2  2 / 2m  ,
(1)
where F is the electric field intensity, e is the electron charge, m  is the effective
mass of an electron,  is the tunneling time of the nuclear subsystem, which is
calculated from the formula [10]:
1
1 1 

ln
  / 2k BT
(2)
2 1  1  
In the formula (2) the parameter  means the electron-phonon coupling constant,
which depends on the ratio between the optical and the thermal depth of the trap, k B
is the Boltzmann constant, T is the temperature, and  is the local oscillation
frequency of the center of the subsystem.
Scattering of electrons by longitudinal optical phonons
Electrons, which become detrapped by tunneling or ionization, move to the surface of
dielectrics and interact with phonons. The scattering by longitudinal optical phonons
influences the transport of electrons having the energy of fractions of an electron-volt
to a few electron-volts. The rate of scattering of an electron having the energy E by
optical phonons is calculated in terms of the Fröhlich theory [11]:

f LO

e2
40  2
1  1   LO / E
1 1
m  1 1 
  n LO    

    LO  ln
,
2 2  2E     

 1  1   LO / E
(3)
where the signs (+) and () denote the generation and the annihilation of an optical
phonon respectively; o,  and  are the absolute, optical and static permittivity
respectively; ћLO is the energy of optical phonons; E is the electron energy; nLO is
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the Bose distribution of the number of phonons over modes ћLO at the crystal lattice
temperature T, which is calculated from the formula [12]:
1
n LO 
,
(4)
exp(  LO / k BT )  1
Scattering of electrons by acoustic phonons
The scattering of electrons by acoustic phonons begins dominating when the electron
energy is units to tens of electron-volts. The rate of scattering by acoustic phonons
depends on the energy at the edge of the Brillouin zone EBZ [13]:
3(m  ) 3 / 2 C12 k B T
f ac 
 E when E  EBZ / 2 ,
(5)
2C S2  4
3/ 2
8 3  2 N 2   E  
1 1
(6)
f  

  n BZ    when E  EBZ / 2 ,
m M BZ  E BZ  
2 2
where C1 is the deformation potential constant;  is the dielectric density; CS is the
sound velocity; M is the mass of the heaviest atom in a unit cell; nBZ is determined
from the formula (4) at the acoustic phonon frequency BZ; N is the concentration of
lattice atoms determined as:
2
 4 N 2 q 2
(7)
Sq 
 ,
m2
where S q is the coupling constant and q is the phonon wave-vector.
The sound velocity is calculated considering three (two transverse and one
longitudinal) branches of the acoustic phonon spectrum [13]:
CS  3 /2 / CT  1/ CL  ,
(8)
where CT and CL are the transverse and longitudinal velocities of the sound
respectively.

ac
Impact ionization
When moving to the surface, electrons acquire some energy on account of electronphonon interactions and acceleration in the electric field. As a result, their energy can
be as high as the ionization energy E th , which is found from the relationship [14]:
Eth 


2  mVB
/ mCB
 Eg ,


1  mVB
/ mCB
(9)

In the formula (9) the effective mass of the valence band hole mVB
and the conduction


 10m0 and
band electron mCB
depend on the electron mass at rest mo : mVB

mCB
 m0 respectively; E g is the forbidden gap width.
When the required energy is reached, electrons begin participating in the impact
ionization. In this case their scattering rate is [14]:
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
 ( E / Eth  1)
E 
f ii  Cii 
ln(
) when E  E th ,
2
Eth 
1  Dii E / Eth
(10)
where Cii is the impact ionization coupling constant, Dii is the impact screening
parameter, and  is a constant.
Cascading
When involved in the impact ionization, drifting electrons can generate new electrons
by cascading. The probability that a secondary electron is formed depends on the
electron drift length z and the electric field intensity F [15]:
H
 A ( z, F )  exp( f ) z   exp 0 exp( ) z  ,
F 

(1
where  0 and H are constants, which depend on the material. In SiO2
 0  6.5  1011 1/cm and H  1.8  10 8 V/cm [15].
The electron drift length was determined as the projection on the normal to the surface
of the mean free path l after each electron-phonon or electron-electron interaction
taking into account the scattering angle. The length passed by an electron between
interactions was calculated by the formula
2E
t ,
m
where  t is the time interval between interactions.
The initial energy of a cascading-generated electron depends on the energy loss by the
primary electron during the impact ionization. The minimum energy loss by an
electron required for generation of another electron during the impact ionization is
determined by the forbidden gap width and equals E min  E g  9 eV for SiO2. If their initial
l 
energy is sufficiently high, secondary electrons can be accelerated in the electric field and generate so-called
tertiary electrons, leading to the avalanche formation of electrons. Cascading of electrons is time-limited and lasts
for about 10-14 s [16].
Energy and angular scattering of electrons
Electron trajectories were calculated by the Monte Carlo method with simulation of at
least 10000 histories of free electrons detrapped by the action of different
mechanisms. The type of the interaction with phonons was determined using the
Monte Carlo method with the random number generation.
Because of the scattering by phonons and the acceleration in the electric field, the
electron energy E changes after each electron-phonon interaction according to the
formula [17]:
E j  E j 1  l j F cos  j   ,
where lj is the free path; j is the scattering angle;  is the phonon frequency. In this
case, the electron energy increases and decreases due to the phonon annihilation and
generation respectively. If the electron does not interact with a photon, its energy is
influenced by the electric field only.
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(1
If the electron is scattered by phonons, not only the electron energy, but also the
electron direction changes after each electron-phonon interaction. The angle 
between the electron directions before and after its interaction with a phonon is
calculated from the formula [17]:
( E  Ek )
(14)
cos  
 (1  A R )  A R ,
1/ 2
2( EEk )
( E  E k  2( EEk )1 / 2 )
,
(15)
A
E  E k  2( EEk )1 / 2
where E' is the electron energy after its interaction with a phonon and R is a random
number from 0 to 1.
The angle  between the electron direction and the normal to the surface is
determined as
(16)
cos i  cos  i cos  i 1  sin i 1 sin  i cos i ,
where i is the interaction number and  is the isotropic azimuthal scattering angle
calculated from the formula   2R .
Trapping of electrons by holes near the surface
It was already noted that the charge distribution in depth of the near-surface layer has
a "+/-" structure. For this reason, an electron passes a region of a positive charge as it
moves to the surface. Holes, which are located near the surface, can capture drifting
electrons. The probability that electrons are captured by holes between electronphonon collisions during their motion in the zone of the positive charge is defined by
the relationship [18]:
Pj (l )  1  exp( n0 li ) ,
(17)
where n0   / e ,  is the volume density of the positive charge, and  is the
trapping cross-section.
The probability that electrons are captured by holes in the entire zone of the positive
charge should be taken into account by the following formula:
P(l )   Pj (l )
(18)
j
Release of electrons to vacuum
An electron, which reaches the surface, can escape from the surface if its energy is
larger than the potential barrier  taking into account the scattering angle :

E
,
cos 2 
In this case, the energy of the electron release to vacuum is calculated from the
formula:
E  Ei   ,
where Ei is the electron energy after the interaction.
Specific features of the electron transport in nanostructures
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Processes of the electron transport in nanostructured dielectrics should be simulated
taking into account the change of the phonon spectrum of nanoparticles, quantum
limitations of the electron free path, porosity and other factors typical of the
nanostructured state. However, of greatest importance in nanomaterials are processes
involving the passage of electrons through numerous boundaries between
nanoparticles. In this connection, the physical model used for calculation of basic
processes of the electron transport in crystals was modified considering specific
features of nanostructured dielectrics.
The objects of study were model nanostructures with crystals 5 to 20 nm in size. The
structure of hexagonal crystalline nanoparticles with closely fitting faces was
analyzed. Analogous opal-like thin-film structures were prepared in experiments (19).
With the chosen model nanostructure, it was possible to disregard the porosity and
use, as the first approximation, known parameters of SiO2 crystals necessary for
calculation of electron-phonon interactions. Also, the model could be conveniently
used for estimating the contribution from processes involved in crossing of
nanocrystal boundaries by moving electrons.
The model and the calculation algorithm took into account that a free electron could
be produced during photoionization either in the bulk or at the boundary of a
nanocrystal located in the emission layer. As the electron was moving in this layer, its
energy changed not only upon the interaction with phonons, but also upon crossing
the nanocrystal boundaries. A potential barrier  B at the boundary of a nanoparticle
can be surmounted if the electron energy Ei after the i-th electron-phonon interaction
is larger than  B with account taken of the scattering angle  i (see formula (19)).
Having crossed the boundary, an electron can continue moving, with the scattering by
phonons and penetration through boundaries of other nanocrystals. The motion stops
when the electron becomes thermalized (E = 3/2kT) and is captured in a trap. If the
energy is insufficient for surmounting the barrier at the nanocrystal boundary, the
electron reflects from the boundary, is scattered by phonons, and is thermalized.
RESULTS OF SIMULATION AND DISCUSSION
An algorithm was constructed and a program in the Delphi language for calculating
the electron emission was written on the basis of the physical model described above.
The concentration of detrapped electrons diffusing to the surface is controlled by
thermal ionization and tunneling processes. Figure 1 shows free-electron
concentrations as a function of the electric field strength for single-crystal and
nanostructured SiO2. The contributions of the electron detrapping mechanisms
mentioned above to the function corresponding to nanostructured silicon dioxide are
also shown as an illustration. In relatively weak fields (less than 2 MV/cm), free
electrons are produced mainly by thermal ionization, and their concentration
gradually increases with the electric field. This result is explained by the Poole–
Frenkel theory, which predicts that a strong electric field reduces the energy depth of
a trap. As the field strength exceeds 2.5 MV/cm, tunneling begins to contribute to
electron detrapping. The concentration of tunneling electrons increases exponentially
with the field strength and thermal ionization can be neglected at 5 MV/cm.
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1
Nf , cm
-3
12
10
11
10
10
10
9
10
8
10
7
10
6
10
5
10
4
10
3
10
2
10
1
10
0
10
2
4
3
0
2
4
6
8
10
F, MV/cm
Fig. 1. Concentration of free electrons generated in the near-surface layer vs. the
electric field strength: ( 1 ) single-crystal SiO2 ; ( 2 ) nanostructured SiO2 . Curves for
the nanostructured material correspond to different mechanisms: ( 3 ) tunneling
detrapping; ( 4 ) thermal ionization.
The transport of electrons in bulk crystalline and nanostructured samples of SiO2 was
studied by the Monte Carlo simulation of N = 10000 histories of free electrons, which
had the initial energy of 4 eV after photoionization of traps. The parameters necessary
for calculation of electron-phonon interactions at T = 300 K were assigned universally
adopted values [11, 14]. The potential barrier at the boundaries between
nanocrystallites was taken equal to  B =0.1 eV.
Electron trajectories in crystalline and nanostructured SiO2 with particles 20 nm in
size were calculated. It was assumed that free electrons started at a depth of 50 nm.
Some typical electron trajectories are shown in fig. 2. It is seen that the electron
trajectories in the crystal and the nanostructured sample are considerably different. As
electrons are moving in the crystal, they go through numerous interactions with
phonons before their thermalization and move to a distance of more than 100 nm from
the start point. The electron trajectories in the nanostructured sample are shorter. The
electrons get thermalized after they have crossed 3-5 boundaries between
nanocrystals. Therefore, the distance to their start point is not over 30 nm.
a
125
distance (nm)
40
1
20nm
30
2
20
10
3
100
75
50
25
vacuum
vacuum
0
b
150
distance (nm)
50
0
25
surface
50
75
0
-25
0
surface
start
25
50
75
100
125
150
start
depth (nm)
depth (nm)
Fig. 2. Trajectories (1, 2, 3) of electrons in nanostructured (a) and crystalline (b)
silica.
The energy spectrum of electrons emitted from crystalline silicon dioxide was
calculated taking the following parameters: the electron start depth of 30 nm; the
electric field intensity of 0.6 MV/cm; the electron affinity on the surface of the sample
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equal to 0.5 eV; the initial energy of electrons at the start equal to kT; and the
temperature of 300 K (Fig. 3a). In experiments the electron energy was measured by
the retarding potential method. It is seen that the calculated and experimental spectra
of the emitted electrons are in satisfactory agreement, pointing to the validity of the
constructed physical model.
The energy spectrum of the released electrons was also calculated for nanostructured
silicon dioxide with particles 20 nm in size (Fig. 3b). The calculation parameters were
as follows: the electric field intensity of 1.9 MV/cm; the electron affinity of 0.1 eV;
the potential barrier at the nanoparticle boundary taken equal to 0.1 eV; the initial
energy of electrons at the start equal to kT; and the temperature of 300 K. Contrary to
the crystalline sample, the energy distribution in nanostructured SiO2 had a
pronounced "tail" of the high-energy component. This result can be explained by
specific features of the transport and the emission of electrons in these samples, which
are disregarded in the proposed model. The calculated energy spectrum is similar in
its shape to the Maxwell distribution. The calculated values of the mean and the most
probable energy of the released electrons are in good agreement with the experimental
data.
0,20
6
a
calculation
experiment
5
0,15
dn/dN (arb. units)
dn/dN (arb. units)
calculation
experiment
0,10
0,05
b
4
3
2
1
0,00
0,0
0,5
1,0
0
0,0
0,5
1,0
1,5
E(eV)
E(eV)
Fig. 3. Energy distribution of electrons emitted from crystalline (a) and
nanostructured (b) silicon dioxide.
Cascading processes significantly contribute to the electron emission from nearsurface layers of dielectric materials in high electric fields. Figure 4 shows the
concentration ( N c ) of free electrons produced by cascading processes as a function of
the field strength computed for the single-crystal and nanostructured samples. The
electron concentration in the avalanche region produced by cascading processes is
higher for the single crystal as compared to the nanostructured material, even though
the cascading thresholds are similar for both structures. This is primarily due to the
higher free-electron concentration at the same field strength for crystal as compared to
the nanostructure. The results obtained in this study demonstrate that the dielectric
strength of the nanostructured material is higher than that of the crystal. According to
our simulations, the breakdown threshold field for single-crystal SiO2 is
approximately 10 MV/cm, which agrees with the literature data [2,3]. For
nanostructured SiO2, the breakdown threshold field is estimated for the first time and
is found to be approximately 12 MV/cm. Note also that the slope of the electron
concentration in the avalanche region as a function of the field strength is lower in the
latter case as compared to a single crystal.
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5x10
10
4x10
10
3x10
10
2x10
10
1x10
10
1
Nc , cm
-3
2
0
8
10
12
14
16
18
F, MV/cm
Fig. 4. Concentration of electrons produced by cascading processes vs. the electric
field strength: (1) single-crystal SiO2; (2) nanostructured SiO2.
CONCLUSION
A computer simulation study of the electron emission from near-surface layers in
SiO2 was performed taking into account two basic characteristics of nanostructures,
namely, multiple grain boundaries and a high concentration of surface electron traps.
Our computations showed that the emissivity of nanostructured materials is lower
than that of single crystals under comparable conditions. The nanoscale structure
strongly modifies stimulated electron detrapping and the electron transport across
charged layers in the materials under study. The degree of modification can be even
higher if other characteristics of nanostructured materials are taken into account
(changes in dielectric constants, potential energy wells, porosity, etc.).
It is found that nanostructured SiO2 is characterized by a higher dielectric strength as
compared to single crystals. If supported by experimental evidence, this result can be
of interest for microelectronics technologies. It is also important that the higher
dielectric strength of electroluminophores with nanostructured SiO2 films as host
materials can be used to generate higher electric fields, opening up prospects for
development of highly efficient light-emitting devices. It should be emphasized that
the physical model does not involve any restrictions that would make it impossible to
perform analogous computations for different inorganic dielectrics. Therefore the
laws found for SiO2 can be expected to hold for other nanostructured dielectric
materials.
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