In this research, a Drift-Diffusion model is carried out to calculate includes impact ionization mechanism and can calculate dark current and photocurrent of avalanche photo diode. Poisson equation, electron and hole density continuity equations and electron and hole current equations have been solved simultaneously using Gummel method. Consideration of impact ionization enables the model to completely simulate the carriers flow in high electrical field. The simulation has been done using MATLAB and the results are compared with other reliable results obtained by researchers. Our results show despite of hydrodynamics and Monte Carlo methods which are very complicated we can get the current characteristics of photo detector easily with acceptable accuracy. In addition we can use this method to calculate currents of device in high fields.

1. Fatemeh Roostaie et al Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 7( Version 3), July 2014, pp.102-106
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A Drift-Diffusion Model to Simulate Current for Avalanche Photo Detector 1Fatemeh Roostaie, 2Maryam Raki and 3Hadi Arabshahi 1Department of Electronic Engineering, University of Chamran, Ahvaz, Iran 2Department of Electronic Engineering, Islamic Azad University, Boshehr, Iran. 3Department of Physics, Payame Noor University, Tehran, Iran Abstract In this research, a Drift-Diffusion model is carried out to calculate includes impact ionization mechanism and can calculate dark current and photocurrent of avalanche photo diode. Poisson equation, electron and hole density continuity equations and electron and hole current equations have been solved simultaneously using Gummel method. Consideration of impact ionization enables the model to completely simulate the carriers flow in high electrical field. The simulation has been done using MATLAB and the results are compared with other reliable results obtained by researchers. Our results show despite of hydrodynamics and Monte Carlo methods which are very complicated we can get the current characteristics of photo detector easily with acceptable accuracy. In addition we can use this method to calculate currents of device in high fields. Keywords: Avalanche photo detector (APD), Impact ionization, Drift-Diffusion model (DDM).
I. Introduction
Avalanche photodiodes APDs are crucial components for long wavelength optical communication systems (OCS). Generally, APDs are the first elements in optical receiver that convert optical data to electrical signal [1-3]. Because of impact ionization mechanism in the APD, APDs have an internal gain which causes higher detection sensitivity also there is no need of external circuits of amplifiers due to current gain. For this reason, APD has become an efficient device with broad applications in long distance fiber OCS [4].
To simulate avalanche photodiodes, one need to an efficient method which can describe different mechanisms such as light absorption and impact ionization mechanism are very important occurring in a multiplication region repetitively. Therefore, one should be use a precisely model which covers the mechanism. The drift diffusion model is one of the simplest models that describe the classical transport of charge carriers in a semiconductor [5-7]. It can present simple description for behavior of carriers in through out of the device. The drift-diffusion method is capable of providing an approximate solution of the Boltzmann transport equation (BTE) and provides a description of non-equilibrium carrier transport. Several circuit models of APD have been developed [6-8]. Chen and Liu gave out a p-i-n APD circuit model including the effect of minority carrier diffusion, neglecting the impact of carrier transit time [8]. From two decade ago, different reports presented in the field of manufacturing a optic device. For example GaAs homo and InP/InGaAs/InP hetero structure have been mentioned. The epitaxy technical and changing effect of physical parameters on the output characteristic device are acceptable sections this reports [9]. Dark current characteristic and gain of photodetector are the important parameters having been considered in various reports [10]. Dark current mostly is affected by epitaxy methods while gain has severe dependence to the field of multiplication region and it is a function of reverse bias. Various reports have been presented in the field of modeling and simulation of behavior of photodetector, such as Soroosh et al. [10] calculated dark current by neural networkmodel in a microscopic view. In this model, photodetector has been considered as a system and a pattern for input-output space is established with training the patterns. Also a device has been surveyed microscopy and the effect of parameters such as length, material and the value of impurity of device regions on the dark current and photocurrent have been calculated. Although presenting the picture of dark current, the model does not have good accuracy due to the restriction of the training data. In another report, according to the minority carriers rate a circuit model has been presented calculating current of photodetector in high electric field [11]. Although does not having an good accuracy, the model is very important for its simplicity and integrated ability with other receiving circuits [12]. In the model, the rate has not been presented for dark current, and just the general equation has been presented for describing of the behavior of device. Although the initial model has been extended for SAM structure and extension state, one of the weaknesses of the model is the lack of analysis of dark current. Soroosh and coworkers [10] have presented a Monte Carlo model to simulate
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APD’s photocurrent and its coefficients. This model
having been presented assuming the two valleys for
electrons and the two sub bands for holes with a good
accuracy. The time-taking of simulation and its
complexity are the main weaknesses in this model.
Requiring to advanced hardware, understanding of
physical mechanism of the device and its time-taking
of numerical solution cause the model do not have
suitable performance. High electric field mechanism
and the interference of some mechanism are the most
issue for simulation and modeling. Although
differences efforts have been performed achieving to
suitable accuracy in simulation of high electric field
with simplicity model is the attractive problem. In
this paper a carefully efficient model is presented
which does not have much complexity and can
calculate dark current, photo current and gain of APD
very well. To achieve this purpose is used the DDM.
In presented model, the absorption light is
determined base on photon absorption rate. Also II
mechanism is considered base on an exponential
equation because the model has a suitable accuracy in
high electric field. The Presented model includes a
five nonlinear differential equation with partial
derivation including Poisson equation, electron and
holes density continuity equations and electron and
hole current equations, and a Gammel method is used
for solving equation. In another section, first, the PIN
APD structure and its performance are shortly
introduced. In the following, the DDM and its
numerical solution are presented complicity. In the
final section, the results of the presented model are
compared with others to verify the accuracy of our
model.
II. PIN APD Poto Detectors
A schematic of a PIN APD photodetector and its
absolute value of the electric field are shown in figure
1. The p-side or n-side illuminated light is absorbed
in region and generates several electrons and hole.
Because of high electric field in i region, the
generated carriers are drifted toward the lateral
contacts, and in the during the path cause other
junctions to broke and several electrons and hole are
generated again. This mechanism is called
regeneration which is continuously repeated. Finally,
the number of carriers is increased and the generated
current is called photocurrent. The ratio the increased
photocurrent caused by the initial photocurrent is
called gain. Of course, another current is called dark
current that exists in the lack of light.
Fig 1. Schematic of the PIN APD structure with
absolute value of field.
III. Drift Diffusion Model
Solving the Boltzman transport equations
(BTEs) is the key to simulate the semiconductor
device properties. One way to solve BTEs is DDM.
In DDM, electron and holes continuity equations are
solved simultaneity. This model is better than Monte
Carlo and hydrodynamics in simplicity and the speed
of convergence. Convergence in DDM is obtained by
choosing the proper initial and the boundary
conditions.
This model is included five equations generally.
Poisson, continuity of density carriers equations and
carriers current equations defined with equations 1 to
5 respectively,
.(V)   (1)
(2)
n n
n 1
J U
t q

  

(3)
p p
p 1
J U
t q

   

Jn  qnnE qDnn (4)
(5) Jp  qnpE  qDpp
Where in this equation, v is the electric potential,
 is the sum of electric charges, n is the free electron
density, p is the hole density, E is the electric field, 
is the dielectric constant in semiconductors, Un and
Up are the sum of the generation (G) and
recombination (R) mechanisms and generally they
are defined as U=G-R, Dn , Dp are diffusion
coefficients for
electron and hole respectively, μn and
μp are the electrical mobilitis, Jn andJp are the currents
density for electron and hole, respectively. Here,
Shockly-Read-Hall and Auge recombination are
considered as recombination rates. Also photon
absorption considered as generation rates.

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Consideration the recombination causes the model to
be able to simulate the current of device in high
electric field. This issue is one of the advantages of
presented model in compared with other drift
diffusion models. This factor expressed as,
(6)
n p
ii n p
J J
G
q q
   
Assuming that each photon absorbed generates a pair
of electron and hole, generation rate caused
byabsorption is defined with as,
( )x (7)
Gopt (x, ) T ( ) ( ) e          
T is the light transmission coefficient in the
semiconductor, α is the semiconductor absorption
coefficient, φ is the flow of radial photon, x is the
distance from the semiconductor surface and λ is the
radial wavelength.
Shockly-Read-Hall recombination,
(8)
i
SRH
p t n t
np n
U
(n n ) (p p )


    
2
Auger recombination,
(9) 2 2 2 2
UAug cn pn nni cp np pni        
   
Where, in above equations, τp and τn are carriers life
time, ni is the intrinsic carrier density, nt and pt are
deficiency density and cn and cp are the auger
constants.
For discretization equations 1 to 5 are used finite
difference method. To achieve convergence must
amount of place cell and steps changing in the
applied voltage according to the following equation,
BT
D 2
k
x L
q N

  
And
B
t
k T
V V
q
  
KB is the Boltzmann constant, T is the
temperature of environment and N is the majority
carrier’s density. To solve equations 1 to 5, they are
rewritten so that independent variable v, φn and φp are
obtained. φn and φp are quasi-Fermi levels of
electrons and hole respectively. To do this, electrons
and holes density are related the quasi- Fermi levels.
(10-a)
n
t
(V )
V
n ni e
  
 
   
(10-b)
p
t
( V)
V
p ni e
   
 
   
Assumption equilibrium and eliminating the time
derivative are rewritten equations 1 to 5 as below,
n p (11)
t t
(V ) ( V)
i V V A D
i
d V qn N N
(e e )
dx n
  

  

2
2
(12) n
n
dJ
qU
dx
 
(13) p
p
dJ
qU
dx

n n (14)
t t
(V ) (V )
V V
n n i t
dV d
J q n ( e V (e ))
dx dx
 
    
p p (15)
t t
( V) ( V)
V V
p p i t
dV d
J q n ( e V (e ))
dx dx
   
    
First, equation 11is solved in equilibrium. With
initial guess for nodes potential is obtained the value
of nodes potential in the during device for one repeat
recursively. The maximum difference between the
previous value and the verbal value of nodes
potential is considered error.
This process is repeated until the maximum error
of nodes for two successive iterations to be less than
10-3. Thus the initial potential of nodes is obtained for
non-equilibrium. In the following, the value of φn and
φp in nodes is calculated using the potential of nodes
utilizes equations 12 and 13. Then, this values are
placed in equation 11 to correct the value of nodes
potential. This process is repeated until the maximum
error of nodes to be less than 10-3. For solving
equations 11 to 15 and for calculating Jn and Jp are
used Shafter and Gummel method. In this method,
the place discretization for Jn and Jp is away sized the
place half-cell with other discritizations. To calculate
electric field at each point must know the difference
of the potential between the adjacent points. Hence
the discretization has been selected.
IV. The results of simulation
In order to investigate the accuracy of our
simulation results and compare them, the parameters
of PIN APD detector are shown in table 1.
Fig. 3 shows the potential distribution and electric
field of the device for reverse voltage biased 2, 6 and
10 Volt. The gradient of potential is constant in i
region and is less in p an n regions. Also the gradient
of potential expresses the electric field. In Fig. 3b, the
electric field is more in i region and is less in p and n
regions. The three regions p, n and i can be
considered the series resistances which the resistance
of the i region is bigger than another region.

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Table 1. List of the material parameters used for simulations [5-7].
value parameter value parameter
Nd
1018 μn
850
Na
1018 μp
400
ni
2*106   n 5*10  s -9
Wp
  100 p 3*10  s -3
Wi
Cn 200
10-40
Wn
Cp 100
10-30
Eg
1.423
s 12.9 
V(ev)
x(cm) 0.0850
1.3*10-8
(a)
(b)
Fig 3. Distribution (a) and electrical field (b) in the electric
potential for voltages of 2, 6 and 10 volt.
To calculate dark current, the equation 7 is not considered
in simulation. In other word, light radiation is eliminated.
Figure 4 is shown the dark current obtained from the model
in comparison with the experimental data for the biased
voltage and the different widths of the multiplication
region. Our results are in good agreement with other
experimental jobs [10-12]. Our results are in fair agreement
for voltages near the breakdown.
Fig 4. Dark current versus reverse bias for different
multiplication region.
With the increasing a constant voltage, the dark
current is decreased because the electric field in the i
region decreases. Also by increasing reverse bias
close to breakdown voltage. The dark current has an
exponential augmentation because in high electric
field electron and hole coefficients are related as
exponential with electric field of multiplication
region.
Figure 5 shows photocurrent versus reverse bias
for three different multiplication regions 100, 200
and 500 nm and Pin = 35 mw and λ = 633 nm. The
justification of photo current graph is similar to dark
current graph.
The gain graph as a function of reverse bias is
shown in Fig 6. As shown, with the increasing of
reverse bias, the intensity of electric field also
increases and then gain increases. By increasing
multiplication region at a constant bias voltage, the
electric field decreases and gain decreases, too. So, to
achieve more gain with increasing wavelength the
reverse bias has better been increased.

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ISSN : 2248-9622, Vol. 4, Issue 7( Version 3), July 2014, pp.102-106
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Fig 5. Photo current versus reverse bias for different
multiplication region.
Fig 6. Gain versus reverse bias.
With survey graphs 3 to 6, we can understand
that presented DDM can calculate dark current and
photo current of PIN APD detector with good
accuracy. The simplicity of the presented model, the
ability of simulation in high electric field and suitable
convergence are characteristics of DDM. Hence, the
development of this model is proposed for other
structures of APD.
V. Conclusions
In this paper, employing Gummel method in
DDM, we could simulate PIN APD photodetector at
high voltage. In the proposed model, potential
distribution and electric field characteristic of the
detector were calculated by discretization Poisson
equation, electron and hole concentrations continuity
equations and electron and hole current equations
using upper and lower triangular method. After that,
gain and current-voltage characteristic of PIN APD
were obtained for different widths of absorption
layer. Comparing our results with other experimental
data, demonstrates the accuracy of our model at low
voltages. Impact ionization which are happen in high
electric field which has been studied in this research.
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