Download IMPACT IONIZATION AND HIGH FIELD EFFECTS IN

International Journal of High Speed Electronics and Systems
cfWorld Scientic Publishing Company
IMPACT IONIZATION AND HIGH FIELD EFFECTS
IN WIDEBANDGAP SEMICONDUCTORS
M. REIGROTZKI, J. R. MADUREIRA, A. KULIGK, N. FITZER, R. REDMER
Fachbereich Physik, Universitat Rostock
D-18051 Rostock, Germany
and
S. M. GOODNICK, M. DU Ry
Department of Electrical Engineering, Arizona State University
Tempe, Arizona 85287-5706, USA
Received (received date)
Revised (revised date, if applicable)
Accepted (accepted date, if applicable)
Impact ionization plays a crucial role for electron transport in wide-bandgap semiconductors at high electric elds. Therefore, a realistic band structure has to be used in
calculations of the microscopic scattering rate, as well as high eld quantum corrections such as the intercollisional eld eect. Here we consider both, and evaluate the
impact ionization rate for wide-bandgap materials such as ZnS. A pronounced softening of the impact ionization threshold is obtained, as found earlier for materials like Si
and GaAs. This eld dependent impact ionization rate is included within a full-band
ensemble Monte Carlo simulation of high eld transport in ZnS. Although the impact
ionization rate itself is strongly aected, little eect is observed on measurable quantities
such as the impact ionization coecient or the electron distribution function itself.
1. Introduction
High eld transport in semiconductors has long been a concern in relation to the
performance of semiconductor electronic and optoelectronic devices for over three
decades.1 To a high degree of success, nonequilibrium transport has been described
within the context of the semi-classical Boltzmann Transport Equation (BTE), with
instantaneous scattering events described by Fermi's Golden rule, and uncorrelated
scattering processes of carriers with the lattice and one another. Such a framework
is the basis for semiconductor device simulation tools based on moments of the
BTE, or through direct solution of the BTE via particle based techniques such as the
Ensemble Monte Carlo (EMC) method.2 However, the question has always remained
as to the limitations of this approach in terms of the underlying quantum transport
equation, and the role of corrections to the BTE such as collision broadening (CB)
permanent address:
Instituto de Fisica, Universidade Estadual de Campinas, Unicamp, 13083-970
Campinas, S~ao Paulo, Brazil
y permanent address: Siemens GmbH, An der Untergeis 8 D-36251 Bad Hersfeld, Germany
1
2 Impact ionization and high eld eects in semiconductors
and the intercollisional eld eect (ICFE), which become more pronounced at high
elds. In particular, the role of such eects in the performance of submicron Si and
GaAs devices has been uncertain at best. In recent years, interest has developed
in wide-bandgap semiconductors such as ZnS, GaN and SiC for optoelectronic and
high-power, high-frequency electronic applications. In such wide-bandgap materials,
elds in excess of 1 MV/cm are common, which necessitates re-evaluation of the
validity of the BTE for such systems.
A high-eld process of particular interest in such wide-bandgap systems is the
impact ionization rate associated with electron{hole pair excitation due to energetic
hot carriers in the conduction or valence bands. In ZnS and SrS thin lm electroluminescent devices for example, this mechanism is responsible for space charge
formation which may result in eld clamping and suppression of luminescence.3 We
have previously calculated the impact ionization rate for Si and GaAs,4 ZnS,5;6;7
GaN,8 and SrS9 using a full band structure approach, but neglecting the inuence
of the electric eld on the collision term through the ICFE. In these studies, we have
used the local and nonlocal empirical pseudopotential method to calculate the band
structure, which has a pronounced inuence on the numerical results for the impact
ionization rate compared to analytical approximations for the electronic dispersion.
The ICFE should also aect the behavior of the impact ionization rate in
the high-eld regime. For instance, the evaluation of the Barker{Ferry kinetic
equation10 for Si11 has indicated that the threshold energy for impact ionization is
lowered due to the eld because the impacting electron is further accelerated during
the collision. This results in a higher ionization rate near the threshold, whereas
for higher energies of the impacting electron the eld inuence vanishes.
Quade et al.12 applied a density matrix approach to carrier generation in semiconductors. Within the parabolic band approximation, they were able to give an
essentially analytical result for the eld-assisted impact ionization rate which was
evaluated for GaAs and Si. Again, systematic lowering of the threshold energy with
the eld strength has been shown.
The eld-dependence of the collision integral was also studied by means of
the Green function technique, solving the Kadano{Baym equations13 in various
approximations. Avoiding the conventional gradient expansion or delta-function
approximation for the spectral density, an integral equation was derived for the
EDF taking into account the ICFE.14;15 The Levinson16 or Barker{Ferry transport
equation10 was evaluated within a saddle-point approximation for GaAs at high
eld strengths, taking into account electron{phonon interactions.17 Alternatively, a
gauge-invariant formulation of the Airy representation of the Kadano{Baym theory
was developed.18 The Mori projection operator technique was applied to study nonlinear transport in semiconductors including the ICFE and collision broadening.19
The method of the nonequilibrium statistical operator as developed by Zubarev20
was applied to study both steady-state and transient properties in hot-electron
transport.21;22
In the present work, we derive quantum kinetic equations for the EDF in
Impact ionization and high eld eects in semiconductors 3
semiconductors using the Zubarev approach, and take into account the full elddependence of the collision integral. We then focus on impact ionization processes
and re-derive the general, eld-dependent impact ionization rate given by Quade
12
et al.
within the parabolic band approximation, and the Keldysh formula23 valid
for energies near the threshold. As previously found, a softening of the threshold
is obtained due to the ICFE, which is much more pronounced in the wide-bandgap
materials due to the much higher onset elds required for impact ionization to occur. These eld-dependent ionization rates for ZnS are incorporated into an EMC
simulation for high eld transport to evaluate the eect on observable quantities
such as the electron distribution function, and the impact ionization coecient. In
both cases, the role of the ICFE is found to be minimal even though the eect on
the impact ionization rate itself is substantial.
2. Impact Ionization Rate
2.1. Quantum kinetic equation
Quantum kinetic equations for a semiconductor in a homogeneous electric eld
E~0 (t) can be derived within response theory taking into account the full elddependence of the collision term.24 The distribution function for electrons in the
band 1 with momentum k1 is determined from
@ f (t) + eE~ (t) @ f (t) = J ( k ; t):
(1)
e 1 1
0
@t 1 k1
@ k~1 1 k1
The collision term Je (1 k1 ; t) contains electron{phonon, electron{impurity and
electron{electron scattering via a respective Hamilton operator. We restrict ourselves to electron{electron collisions since we are especially interested in the impact
ionization rate. Dening fi ; ki g i, we get within second-order perturbation
theory a Markovian form for the electron{electron collision term:25
Z t
X
0 "(t ?t) cos[
0 (1; 2; 3; 4)(t; t0)]
Jeel (1; t) = ? 42 jMtot (1; 2; 3; 4)j2 "lim
!0 ?1dt e
h 2;3;4
ff1 (t)f2 (t)[1 ? f3 (t)][1 ? f4(t)] ? f3 (t)f4 (t)[1 ? f1 (t)][1 ? f2 (t)]g :
(2)
0
Mtot is the total matrix element including direct, exchange, and umklapp processes.
The collision term is given by an integral over all former times t0 which is usually
interpreted as memory eect. In the limit of constant electric elds E~ 0 , canonical
momenta transform the distribution functions to a gauge-invariant form:
0 (1; 2; 3; 4)(t; t0) = h1 ("3 + "4 ? "1 ? "2 )(t ? t0 )
!
~0 ~k3 ~k4 ~k1 ~k2
e
E
? 2h m + m ? m ? m (t ? t0 )2 + 31 !F3 (t ? t0 )3 :
3
4
1
2
(3)
4 Impact ionization and high eld eects in semiconductors
The kinetic energies "i are given by the band structure, and mi is the eective mass
of an electron in the band i at wave vector ki . The quantity !F is dened by
2 2
1
1
1
1
e
E
0
3
(4)
!F = 2h m + m ? m ? m :
3
4
1
2
In the process of impact ionization, a conduction band electron impact ionizes a
valence band electron, i. e. 1 + 2 ! 3 + 4, see Fig. 1. The band indices and energies
1; 3; 4 run over the conduction bands, while 2 belongs to the valence bands. Supposing that the semiconductor is not highly excited, the conduction bands are almost
empty so that the Pauli blocking factors are unity, i. e. (1 ? fi ) 1. Furthermore,
the (second) in-scattering term in Eq. (2) can be neglected compared with the (rst)
out-scattering term in the balance for the population of states with momentum k1 .
The collision integral is then simply given by a eld-dependent impact ionization
rate rii (1; E0 ) via Jeii (1; E0 ; t) = ?rii (1; E0 ; t)f1 (k1 ? eE0 t) with
X
rii (1; E0 ; t) = ? 42 jMtot(1; 2; 3; 4)j2
h 2;3;4
"lim
!
0
Z t
?1
dt0 e"(t ?t) cos[
0 (1; 2; 3; 4)(t; t0)]:
0
(5)
The wave-vector dependent impact ionization rate has to be evaluated considering
the full eld dependence and a realistic band structure in the cosine term as well as
the full momentum dependence of the matrix element Mtot including an appropriate
screening function for the Coulomb interaction.
conduction band
1
3
4
2
valence bands
Fig. 1. Schematic impact ionization process for electrons.
2.2. Electron initiated impact ionization at zero elds
The numerical evaluation of Eq. (5) is rather complex. We rst review results for
the impact ionization rate within simple approximations. Neglecting the inuence
of the electric eld in the collision term and considering the Markov limit t ! 1,
the integral over the time t0 gives the energy conserving delta function and we have
X
(6)
rii (1; 0) = ? 42 jMtot(1; 2; 3; 4)j2 ("3 + "4 ? "1 ? "2 ):
h 2;3;4
Impact ionization and high eld eects in semiconductors 5
The integration over the momenta can easily be performed supposing a constant
matrix element and spherical parabolic bands with eective masses for the valence
(m2 = mv ) and conduction bands (m1 = m3 = m4 = mc ). Dening the eective
mass ratio = mc =mv and the parameter = (1+2)=(1+ ), the threshold energy
Eth = Eg is related to the fundamental band gap Eg , and the famous Keldysh
formula23 for impact ionization is derived:
rii(K) ("1 ; 0) = P0 ["1 ? Eth ]2 :
(7)
The prefactor P0 is often used as t parameter for the energy-dependent impact
ionization rate in simulations of high eld transport in semiconductors.26 "1 =
h 2 k12 =(2m1) is the kinetic energy of the impacting electron.
However, previous calculations for Si and GaAs,4;27 ZnS,5;6;7 GaN,8;28;29;30 SrS,8;9
InN,31 and SiC32 have shown that the full band structure has to be considered when
calculating the impact ionization rate via Eq. (6). Pronounced contributions arise
from higher conduction bands, especially in wide-bandgap materials like ZnS, GaN,
or SrS. The empirical pseudopotential method (EPM) is the standard tool to determine the band structure of a semiconductor material. Four to six conduction
bands (and four valence bands) are usually considered for the complete numerical
evaluation of the zero-eld impact ionization rate (6). The inuence of nonlocal
pseudopotentials has been studied for ZnS7 and ab initio band structures have been
used in calculations of the impact ionization rate of SrS.9 However, the EPM represents in most cases a reasonable compromise between desired accuracy and available
computer capacity.
The integrals in Eq. (6) extend over the entire Brillouin zone and are evaluated
using an ecient numerical procedure developed by Sano and Yoshii.27 Making extensive use of symmetry relations imposed by the crystal structure, the integrations
can be restricted to the irreducible wedge of the Brillouin zone where a large number of points can be taken into account for the numerical evaluation. For instance,
using a uniform grid in wave vector space with 152 points in the irreducible wedge
corresponds to 4481 points across the whole Brillouin zone;5;6;7;8;9 228 points in
the irreducible wedge were also considered.28;29;30;31;32 A further increase of the
number of grid points does not aect the calculated rate signicantly, except in the
threshold region.
The interaction between the conduction and valence electrons is described by
a wave-vector dependent dielectric function derived by Levine and Louie.33 The
frequency dependence of the dielectric function becomes more important as the
energy of charge carriers increases. At high energies, we observe that primarily
umklapp processes (which we take into account up to the sixth order) contribute
to the calculated rate and, therefore, due to the large momentum transfer, the
inuence of the carrier energy on the screening function is less important. Jung
34
et al.
have employed a wave-vector and frequency-dependent dielectric function
within the random phase approximation for the calculation of the impact ionization
rate in GaAs using a Monte Carlo integration technique. Their results agree well
6 Impact ionization and high eld eects in semiconductors
with our ndings which were obtained using a static dielectric function.4
The wave-vector dependent impact ionization rate rii (1; 0) is usually averaged
over the entire Brillouin zone to obtain an energy-dependent rate R(E ) via
R(E ) =
X
(E ? "1 )rii (1; 0)
1
X
(E ? "1 );
(8)
1
which is shown in Fig. 2 for the wide-bandgap materials ZnS, GaN, and SrS. The
general behavior is almost the same. The threshold energy relative to the conduction
band minimum is given approximately by the gap energy. The numerical results
according to Eq. (8) are well t by a power law relation of the form
R~ (E ) = C [E ? Eth ]a ;
(9)
which can easily be implemented in full-band Monte Carlo simulations of electron
transport in semiconductors. The prefactor C , the threshold energy Eth , and the
power a are given in Table 1 for a variety of semiconductor materials. Obviously,
the inuence of the band structure manifests itself in values a > 2 compared with
the original Keldysh formula (7) derived for spherical parabolic bands.
10
16
10
14
10
12
10
10
−1
electron ionization rate [s ]
Eth=3.7
Eth=3.8
Eth=4.0
10
8
10
6
ZnS (Ref. 5)
GaN (Ref. 8)
SrS (Ref. 9)
GaN (Ref. 28)
4
6
8
electron energy [eV]
10
12
Fig. 2. Electron impact ionization rate for GaN, ZnS, and SrS.
2.3. Hole initiated impact ionization at zero elds
A second possible electron{electron scattering process contributing to impact
ionization is the relaxation of a valence band electron, passing enough energy to a
second valence band electron to be ionized across the gap into the conduction band
and generate an additional free carrier. This scattering process can be understood
as the scattering of two holes in which a hole residing in the conduction band is
ionized across the gap into the valence band, leading to an additional positive charge
in the valence band and, correspondingly, to a negative free charge in the conduction
Impact ionization and high eld eects in semiconductors 7
Table 1. Parameters for the interpolation formula (8) for the electron and hole initiated (see below)
impact ionization rate for various semiconductor materials.35 36
;
Eth
electrons:
Si
GaAs
GaN
ZnS
SrS
holes:
GaN
ZnS
[eV]
[1010eV?a s?1 ]
C
a
0.8
1.8
3.6
3.8
4.0
36.22
93.659
0.00949
5.935
59.723
3.683
4.743
7.434
5.073
3.182
3.4
3.8
0.35
0.71
5.33
6.23
band, see Fig. 3. This scattering process is interpreted as the hole initiated impact
ionization, and due to the above explanations, the corresponding ionization rate
can be determined using the very same numerical scheme as in the calculation of
the ionization rate of electrons described above. The sole dierence is the inversion
of the dispersion relation.
conduction band
4
2
3
1
valence band
Fig. 3. Schematic impact ionization process for holes.
The respective impact ionization rates are calculated for holes in all four valence
bands. Holes in the upper valence band can not initiate ionization processes. In
GaN and ZnS the main contribution arises from the lowest and second lowest band.
In SrS only holes from the lowest valence band are able to initiate ionization events.
The corresponding energy-dependent impact ionization rate is shown in Fig. 4
for the wide-bandgap materials ZnS, GaN, and SrS.35 Previous theoretical results
of Oguzman et al.29 for the threshold region of GaN using another band structure
(B) are included for comparison. The shaded area covers their non-averaged, kdependent rates according to their band structure (B) via "(k). For comparison,
the inset shows our non-averaged, k-dependent rates displayed in the same manner
using the EPM band structures A and B (full and open circles). As can be seen, the
calculated rate is relatively insensitive to modest changes in the band structure itself.
The rates of Oguzman et al. are up to two orders of magnitude higher than ours.
8 Impact ionization and high eld eects in semiconductors
−1
hole ionization rate [s ]
This dierence is probably due to dierent integration schemes or screening models,
and using dierent numbers of k points in the irreducible wedge. The corresponding
parameters of the interpolation formula according to Eq. (9) are given in Table 1.36
10
20
10
18
10
16
10
14
10
12
10
10
14
10
12
10
10
10
10
10
8
10
6
8
3.5
4.5
5.5
6.5
ZnS
GaN (A)
GaN (B)
SrS
GaN (Ref. 29)
2
4
6
8
10 12 14 16
hole energy [eV]
18
20
22
Fig. 4. Hole impact ionization rate for GaN, ZnS, and SrS.35
Compared with the electron initiated impact ionization rates displayed in Fig. 2,
two special features have to be noticed for the hole rates. First, the threshold energy
for hole ionization is in general slightly lower than for electron ionization.9 This
result is due to the at shape of the top of the upper valence bands, which allows
holes to initiate ionization events as soon as they have reached a kinetic energy
equal to the gap energy. In the case of SrS, the width of the upper valence band
is too narrow to allow holes to gain enough kinetic energy to initiate ionization
events due to energy conservation. For this reason, only holes in the lowest valence
band contribute to the ionization rate and, therefore, SrS does not show the usual
threshold behavior near the gap energy. Instead, the ionization rate sets in only
above 14.7 eV and immediately jumps to very high values.
Second, the hole initiated impact ionization rates vanish in a certain energy
range completely, i. e. between about 5 eV and 12 eV for ZnS and 7 eV and 19.5 eV
for GaN. For higher energies, they set in again immediately as discussed already for
SrS. This special behavior is due to the occurrence of an energy gap between the
upper three valence bands and the lowest one and, therefore, no states are available
for energetic holes.
Thus, the hole initiated impact ionization rate shows a very dierent behavior
compared with the respective electron rate except for the region near the threshold
energy, if hole ionization occurs there at all. Since the hole scattering rates are as
low as the electron rates near the threshold energy, and vanish completely in the
medium energy range, we conclude that hole initiated impact ionization processes
can be neglected compared with the electron processes for both ZnS and SrS.
Impact ionization and high eld eects in semiconductors 9
2.4. Field-dependent impact ionization rate
The evaluation of the eld-dependent impact ionization rate (5) is important for
high eld strenghts when the electrons are further accelerated during the collision.
This is usually denoted as intra-collisional eld eect (ICFE).
Quade et al.12 were able to evaluate the integrations over the momenta in Eq.
(5) essentially analytically in the Markov limit t ! 1 considering the full, statically
screened Coulomb matrix element, but performing the eective mass approximation.
We obtain with the denition of the Airy function
Z 1
?
Ai x
d cos a 3 x
=
(3a)1=3
(3a)1=3
0
the general result:
rii ("1 ; E0 ) = PQ
(Q)
Z
1
dE S ("1; "1 ? E ) 1ii Ai Eth ? "ii1 + E ;
EF
EF
2 1=3
0)
:
EFii = h!Fii ; !Fii = (1 +8m)(eE
h
c
0
(10)
(11)
!Fii is the electro-optical frequency. We follow the denition of the other quantities
in Eq. (11) given in Quade et al.12
Performing the constant matrix element approximation but taking into account
the full eld-dependence of the collision integral (5), a modied impact ionization
rate can be derived:
2
Z 1
1 Ai Eth ? "1 + E :
(12)
r(K) (" ; E ) = P
dE E
ii
1
0
K
0
Eth
EFii
EFii
The zero-eld limit yields the original Keldysh formula for impact ionization (7).
Comparing Eqs. (7), (9), and (12), we have proposed recently24 a new t formula
for impact ionization that considers the inuence of an applied electric eld and the
full band structure according to
a
Z 1
E
1
E
th ? "1 + E
(F)
r (" ; E ) = C
dE
:
(13)
Ai
ii
1
0
0
Eth
EFii
EFii
The parameters C , Eth , and a are already given in Table 1 for the zero-eld electron
initiated impact ionization rates.
We show as an example the eld dependent impact ionization rate for the widebandgap material ZnS in Fig. 5; the behavior of GaN and SrS is very similar.
The ICFE inuences only the direct threshold region and leads to a systematic
lowering of the threshold energy with increasing eld strength. The sensitivity of
this lowering is related to the eective masses which are introduced as material
parameters in the quantity EFii in Eq. (11) for the parabolic band result. The
analytical form of Eq. (11) is also adapted for the t formula (13) that reects the
full band structure via the parameter a and, again, the quantity EFii . About 0.5
10 Impact ionization and high eld eects in semiconductors
13
10
11
-1
rate [s ]
10
9
10
2.0 MV/cm
1.5 MV/cm
1.0 MV/cm
0.5 MV/cm
0.1 MV/cm
Zero Field
7
10
5
10
3.0
3.5
4.0
4.5
5.0
energy [eV]
5.5
6.0
Fig. 5. Field-dependent electron impact ionization rate for ZnS.24
eV (1 eV) above the threshold energy, the impact ionization rate becomes already
independent of the eld for the narrow (wide) band gap materials.
3. Ensemble Monte Carlo Simulation and Impact Ionization Coecient
In order to understand the role that the ICFE plays in high eld transport,
we include the eld dependent rates of the previous section into a full band EMC
simulation for ZnS described in detail elsewhere.7 Basically, an EMC simulation
is a particle based simulation technique where the particle trajectories under the
inuence of external elds and random scattering events are tracked.2 Instantaneous random scattering events in the crystal are generated stochastically using
the random number generator.2;37 As such, the EMC provides a direct solution
to the Boltzmann equation for the one-particle distribution functions for electrons
and holes and macroscopic averages derived from them. The full band dispersion
relation E (k) for ZnS is taken into account using the EPM band structure. In the
present work we only simulate electron transport and neglect the role of any holes
generated by impact ionization processes.
The Monte Carlo simulation includes polar optical phonon scattering, scattering
due to acoustic phonons, optical deformation potential scattering, ionized impurity
scattering, and band-to-band impact ionization using the modied rates of the previous section. As discussed elsewhere,7 we employ deformation potential scattering
only above the intervalley threshold, where the density of states is used to renormalize the scattering rate to account for full band eects in the scattering rate. Two
deformation potential constants are used to model the scattering rate due to optical
phonons: D(1; 2) = (1 109; 9 108) eV/cm.36 Simulations are performed for an
electron concentration of 1016 cm?3 in the conduction band and a temperature of
300 K.
Impact ionization and high eld eects in semiconductors 11
Fig. 6 compares the electron distribution function versus energy calculated from
the EMC simulation for three dierent eld strengths above 1 MV/cm. As can be
seen, there appears to be no signicant eect due to the inclusion of the ICFE at
any eld strength.
electron distribution function
3
10
without ICFE
with ICFE
2.0
2
10
1.5
1
10
1.0
0
10 0
1
2
3
4
5
electron energy [eV]
6
7
8
Fig. 6. Electron distribution function for three dierent electric eld strengths (in MV/cm) with
(solid lines) and without (broken lines) the inclusion of the ICFE.
A measurable quantity directly associated with the impact ionization rate is
the impact ionization coecient which represents the number of electron{hole pairs
created by an energetic particle per unit path length in the crystal. The impact
ionization coecient is directly related to carrier multiplication in a reverse biased
junction, and hence can be obtained from current multiplication versus bias data. In
the EMC simulation, this quantity is calculated by tabulating the number of impact
ionization events per carrier traversing a 0.5 m thick layer of ZnS. Fig. 7 shows the
calculated result with and without the inclusion of the ICFE. The result without
the ICFE corresponds to the results reported earlier by us for this quantity.35 As
can be seen from this gure, there is only little dierence in the impact ionization
coecient despite the rather dramatic eect on the impact ionization rate near the
threshold energy shown in Fig. 5.
4. Discussion and Conclusions
The main results from the present study are the calculated eect of the ICFE on
the impact ionization rate in ZnS and other wide-bandgap materials, and its subsequent eect on the electron distribution function and impact ionization coecient.
While the eect on the bare impact ionization rate is substantial, the resulting eect
on the distribution function itself and the associated impact ionization coecient
is negligible.
The reason for the lack of any substantial inuence on transport may be understood from Fig. 5. When the eld is relatively low (below 0.1 MV/cm) the ICFE
is negligible. At high elds close to the threshold for impact ionization to occur,
12 Impact ionization and high eld eects in semiconductors
4
impact ionization coefficient [10 /cm]
10
8
without ICFE
with ICFE
6
4
2
1
0.50
0.52
0.54
0.56
0.58
0.60
0.62
0.64
inverse electric field [cm/MV]
Fig. 7. Calculated impact ionization coecient versus inverse electric eld strength for EMC
simulations with (solid line) and without (broken line) the inclusion of the ICFE.
the threshold energy is broadened, but only for the region of energy over which the
scattering rate is relatively low, below 1010 s?1 . The strongest contributions to the
total scattering must come from the higher energy tail of the distribution where
the impact ionization rate is comparable to the electron{phonon scattering rate,
for any eect to be observed in the distribution function itself. However, the eect
of the ICFE in this high energy region is small. Likewise, in terms of the impact
ionization coecient itself, the signicant contributions to band-to-band impact
ionization come from electrons well above threshold where the scattering rate is
large. Conversely, in this region, there is little eect again due to the ICFE, hence
the inuence on the impact ionization coecient is negligible.
The role of collision broadening on the impact ionization rate has been studied
recently using nonequilibrium Green functions for the derivation of an appropriate
kinetic equation.38 It has been shown that collisions between the particles lead
to a broadened one-particle spectral function which is of non-Lorentzian shape in
contrast to former assumptions.11 A further lowering of the threshold energy for
impact ionization is obtained, and the rate itself is increased in the threshold region.
Collision broadening is already eective for eld strengths below 0.5 MV/cm, while
the ICFE determines the behavior of the impact ionization rate for higher elds.
For impact energies of about 1 eV above threshold, the collision broadening is
almost negligible similar to the behavior found for the ICFE, see Fig. 5. Therefore,
collision broadening has also no pronounced inuence on the EDF and the ionization
coecient.
Acknowledgements
We thank K. Brennan (Atlanta), D. Ferry (Tempe), M. Fischetti (Yorktown Heights),
T. Kuhn (Munster), V. Morozov (Moscow), W. Schattke (Kiel), E. Scholl (Berlin)
Impact ionization and high eld eects in semiconductors 13
and P. Vogl (Munchen) for stimulating discussions. This work was supported by the
Deutsche Forschungsgemeinschaft under contract No. RE 882/9-2 and by DARPA
under the Phosphor Technology Center of Excellence, Grant No. MDA 972-93-10030.
1. Quantum Transport in Semiconductors, edited by C. Jacoboni, L. Reggiani, and D. K.
Ferry, Plenum Press, New York, 1992.
2. C. Jacoboni and P. Lugli, The Monte Carlo Method for Semiconductor Device Simulation, Springer-Verlag, Berlin, 1989.
3. W. M. Ang, S. S. Pennathur, L. Pham, J. F. Wager, S. M. Goodnick, and A. A. Douglas, \Evidence for band-to-band impact ionization in evaporated ZnS:Mn alternatingcurrent thin-lm electroluminescent devices", J. Appl. Phys. 77, 2719 (1995).
4. M. Stobbe, R. Redmer, and W. Schattke, "Impact ionization rate in GaAs", Phys. Rev.
B 49, 4494 (1994).
5. M. Reigrotzki, M. Stobbe, R. Redmer, and W. Schattke, \Impact ionization rate in
ZnS", Phys. Rev. B 52, 1456 (1995).
6. M. Reigrotzki, R. Redmer, I. Lee, S. S. Pennathur, M. Dur, J. F. Wager, S. M. Goodnick,
P. Vogl, H. Eckstein, and W. Schattke, \Impact ionization rate and high-eld transport
in ZnS with nonlocal band structure", J. Appl. Phys. 80, 5054 (1996).
7. M. Dur, S. M. Goodnick, S. S. Pennathur, J. F. Wager, M. Reigrotzki, and R. Redmer,
\High-eld transport and electroluminescence in ZnS phosphor layer", J. Appl. Phys.
83, 3176 (1998).
8. M. Reigrotzki, M. Dur, W. Schattke, N. Fitzer, R. Redmer, and S. M. Goodnick, \HighField Transport and Impact Ionization in Wide Bandgap Semiconductors", phys. stat.
sol. (b) 204, 528 (1997).
9. M. Dur, S. M. Goodnick, R. Redmer, M. Reigrotzki, M. Stadele, and P. Vogl, Journal
of the Society for Information Displays (accepted for publication).
10. J. R. Barker and D. K. Ferry, \Self-Scattering Path-Variable Formulation of highField, time-Dependent, Quantum Kinetic Equations for Semiconductors in the FiniteCollision-Duration Regime", Phys. Rev. Lett. 42, 1779 (1979); see also D. K. Ferry,
Semiconductors, Macmillan, New York, 1991, Chapter 15.
11. J. Bude, K. Hess, and G. J. Iafrate, \Impact ionization in semiconductors: Eects of
high electric elds and high scattering rates", Phys. Rev. B 45, 10958 (1992).
12. W. Quade, E. Scholl, F. Rossi, and C. Jacoboni, \Quantum theory of impact ionization
in coherent high-eld semiconductor transport", Phys. Rev. B 50, 7398 (1994).
13. L. P. Kadano and G. Baym, Quantum Statistical Mechanics, Benjamin, New York,
1962.
14. A. P. Jauho and J. W. Wilkins, \Theory of high-electric-eld quantum transport for
electron-resonant impurity systems", Phys. Rev. B 29, 1919 (1984).
15. S. K. Sarker, \Quantum transport theory for high electric elds", Phys. Rev. B 32,
743 (1985).
16. I. B. Levinson, Zh. Eksp. Teor. Fiz. 47, 660 (1969) [Sov. Phys.-JETP 30, 362 (1970)].
17. P. Lipavsky, F. S. Khan, F. Abdosalami, and J. W. Wilkins, \High-eld transport in
semiconductors. I. Absence of the intra-collisional-eld eect", Phys. Rev. B 43, 4885
(1991).
18. R. Bertoncini and A. P. Jauho, \Quantum transport theory for electron-phonon systems
in strong electric elds", Phys. Rev. Lett. 68, 2826 (1992).
19. J. Y. Ryu and S. D. Choi, \Quantum-statistical theory of high-eld transport phenomena", Phys. Rev. B 44, 11 328 (1991).
20. D. N. Zubarev, Nonequilibrium Statistical Thermodynamics, Consultants Bureau, New
14 Impact ionization and high eld eects in semiconductors
York, 1974.
21. D. Y. Xing and C. S. Ting, \Green's-function approach to transient hot-electron transport in semiconductors under a uniform electric eld", Phys. Rev. B 35, 3971 (1987).
22. D. Y. Xing, P. Hu, and C. S. Ting, \Balance equations for steady-state hot-electron
transport in the approach of the nonequilibrium statistical operator", Phys. Rev. B 35,
6379 (1987).
23. L. V. Keldysh, \Kinetic theory of impact ionization in semiconductors", Zh. Exp. Theor.
Phys. 37, 713 (1959) [Sov. Phys.{JETP 37, 509 (1960)].
24. R. Redmer, J. R. Madureira, N. Fitzer, S. M. Goodnick, W. Schattke, and E. Scholl,
\Field eect on the impact ionization rate in semiconductors", J. Appl. Phys. 87, 781
(2000).
25. D. N. Zubarev, V. Morozov, and G. Ropke, Statistical Mechanics of Nonequilibrium
Processes, Vol. 1: Basic Concepts, Kinetic Theory, Akademie-Verlag, Berlin, 1996.
26. M. V. Fischetti and S. E. Laux, \Monte Carlo analysis of electron transport in small
semiconductor devices including band-structure and space-charge eects", Phys. Rev.
B 38, 9721 (1988).
27. N. Sano and A. Yoshii, \Impact-ionization theory consistent with a realistic band structure of silicon", Phys. Rev. B 45, 4171 (1992).
28. J. Kolnik, I. H. Oguzman, K. F. Brennan, R. Wang, and P. P. Ruden, \Calculation of
the wave-vector-dependent interband impact-ionization transition rate in wurtzite and
zinc-blende phases of bulk GaN", J. Appl. Phys. 79, 8838 (1996).
29. I. H. Oguzman, E. Bellotti, K. F. Brennan, J. Kolnik, R. Wang, and P. P. Ruden,
\Theory of hole initiated impact ionization in bulk zincblende and wurtzite GaN", J.
Appl. Phys. 81, 7827 (1997).
30. E. Bellotti, K. F. Brennan, R. Wang, and P. P. Ruden, \Calculation of the electron
initiated impact ionization transition rate in cubic and hexagonal phase ZnS", J. Appl.
Phys. 82, 2961 (1997).
31. E. Bellotti, B. K. Doshi, K. F. Brennan, J. D. Albrecht, and P. P. Ruden, \Ensemble
Monte Carlo study of electron transport in wurtzite InN", J. Appl. Phys. 85, 916
(1999).
32. E. Bellotti, H.-E. Nilsson, K. F. Brennan, P. P. Ruden, and R. Trew, \Monte Carlo
calculation of hole initiated impact ionization in 4H phase SiC", J. Appl. Phys. 87,
3864 (2000).
33. Z. H. Levine and S. G. Louie, \New model dielectric function and exchange-correlation
potential for semiconductors and insulators", Phys. Rev. B 25, 6310 (1982); see also
M. S. Hybertson and S. G. Louie, \First-principles theory of quasiparticles: Calculation
of band gaps in semiconductors and insulators", Phys. Rev. Lett. 55, 1418 (1985);
\Electron correlation in semiconductors and insulators: Band gaps and quasiparticle
energies", Phys. Rev. B 34, 5390 (1986).
34. H. K. Jung, K. Taniguchi, and C. Hamaguchi, \Impact ionization model for full band
Monte Carlo simulation in GaAs", J. Appl. Phys. 79, 2473 (1996).
35. M. Reigrotzki, R. Redmer, N. Fitzer, S. M. Goodnick, M. Dur, and W. Schattke, \Hole
initiated impact ionization in wide band gap semiconductors", J. Appl. Phys. 86, 4458
(1999).
36. M. Reigrotzki, PhD thesis, University of Rostock, 1998.
37. C. Moglestue, Monte Carlo Simulations of Semiconductor Devices, Chapman and Hall,
New York, 1993.
38. J. R. Madureira, D. Semkat, M. Bonitz, and R. Redmer, \Impact ionization rates of
semiconductors in an electric eld: The eect of collisional broadening", in preparation.