A Drift-Diffusion Model to Simulate Current for AvalanchePhoto Detector

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Fatemeh Roostaie et al Int. Journal of Engineering Research and Applications www.ijera.comISSN : 2248-9622, Vol. 4, Issue 7( Version 3), July 2014, pp.102-106

A Drift-Diffusion Model to Simulate Current for AvalanchePhoto 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

AbstractIn 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 acceptableaccuracy. 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. IntroductionAvalanche photodiodes APDs are crucial

components for long wavelength optical

communication systems (OCS). Generally, APDs arethe 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, APDhas become an efficient device with broad

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 havingbeen considered in various reports [10]. Dark

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 asSoroosh et al. [10] calculated dark current by neural

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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 impactionization 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 canpresent 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 circuitmodel 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 heterostructure have been mentioned. The epitaxy technical

networkmodel in a microscopic view. In this model,photodetector has been considered as a

pattern for input-output space is established

training the patterns. Also a device has been surveyedmicroscopy 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

dark current, the model does not have good accuracy

due to the restriction of the training data. In anotherreport, according to the minority carriers rate a circuit

model has been presented calculating current of

photodetector in high electric field

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 thegeneral 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

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

APDs 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 ofphysical 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 tosuitable accuracy in simulation of high electric field

with simplicity model is the attractive problem. In

this paper a carefully efficient model is presentedwhich does not have much complexity and can Fig 1. Schematic of the PIN APD structure with

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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 afive 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 shortlyintroduced. In the following, the DDM and its

numerical solution are presented complicity. In thefinal section, the results of the presented model are

compared with others to verify the accuracy of our

model.

II. PIN APD Poto DetectorsA 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 aregenerated 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.

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)

n n

n 1J U

t q

p pp 1

J Ut q

n n nJ qn E qD n

p p pJ qn E qD 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, U

Up are the sum of the generation (G) and

recombination (R) mechanisms and generally they

are defined as U=G-R, Dn , Dp

coefficients forelectron and hole respectively,

p are the electrical mobilitis, Jn andJpdensity for electron and hole, respectively. Here,

Shockly-Read-Hall and Auge recombination are

considered as recombination rates. Also photon

absorption considered as generation rates.

Fatemeh Roostaie et al Int. Journal of Engineering Research and Applications www.ijera.com

Fatemeh Roostaie et al Int. Journal of Engineering Research and Applications www.ijera.comISSN : 2248-9622, Vol. 4, Issue 7( Version 3), July 2014, pp.102-106

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)pn

ii n p

JJG

q q

Assuming that each photon absorbed generates a pair

of electron and hole, generation rate causedbyabsorption is defined with as,

(7)( )xoptG (x, ) T ( ) ( ) e

T is the light transmission coefficient in the

semiconductor, is the semiconductor absorption

coefficient, is the flow of radial photon, x is the

distance from the semiconductor surface and is the

radial wavelength.

Shockly-Read-Hall recombination,(8)

iSRH

p t n t

np nU

(n n ) (p p )

2

Auger recombination,

(9)2 2 2 2

Aug n i p iU c pn nn c np pn

Where, in above equations, p and nare carriers lifetime, 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,

BTD 2

kx L

q N

And

Bt

k TV V

q

KB is the Boltzmann constant, T is the

temperature of environment and N is the majority

carriersdensity. To solve equations 1 to 5, they are

rewritten so that independent variable v, nand p are

obtained. n and p 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 )

Vin n e

(10-b)p

t

( V)

Vip n e

Assumption equilibrium and eliminating the time

pn

t t

( V)(V )

V Vi A

i

d V qn N N(e e

ndx

2

2

nn

dJqU

dx

pp

dJqU

dx

n

t

(V ) V

n n i tdV d

J q n ( e V (dx dx

p p

t

( V) (

V Vp p i t

dV dJ 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

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 p in nodes is calculated using the potential of nodes

utilizes equations 12 and 13. Then,

placed in equation 11 to correct the

potential. This process is repeated until the maximum

error of nodes to be less than 10

equations 11 to 15 and for calculating J

used Shafter and Gummel method. In this method,the place discretization for Jnand Jp is away sized the

place half-cell with other discritizations. To

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 simulationIn order to investigate the accuracy of our

simulation results and compare them, the parametersof 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, theelectric 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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Assumption equilibrium and eliminating the time

derivative are rewritten equations 1 to 5 as below,

of the i region is bigger than another region.

Fatemeh Roostaie et al Int. Journal of Engineering Research and Applications www.ijera.comISSN : 2248-9622, Vol. 4, Issue 7( Version 3), July 2014, pp.102-106

Table 1. List of the material parameters used for simulations [5-7].

parametervalueparametervalue

Nd

10n

850

Na

1018p

400ni

2*10 n s 5*10

-

Wp

100 p s 3*10-

Wi

200Cn

10-

Wn

100Cp

10-30

Eg

1.423s 12.9

V(ev)0.0850x(cm)

1.3*10-8

(a)

Fig 4. Dark current versus reverse bias for different

multiplication region.

With the increasing a constant voltag

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(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 consideredin 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 otherexperimental jobs [10-12]. Our results are in fair agreement

for voltages near the breakdown.

With the increasing a constant voltag

current is decreased because the electric field in the i

region decreases. Also by increasing reverse bias

close to breakdown voltage. The dark exponential augmentation because in high electricfield 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 = 633 nm. The

justification of photo current graph is

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 increasingmultiplication region at a constant bias voltage, the

electric field decreases and gain decreases, too. So, to

achieve more gain with increasing wavelength thereverse bias has better been increased.

Fatemeh Roostaie et al Int. Journal of Engineering Research and Applications www.ijera.comISSN : 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 andphoto 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 inDDM, 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 equationsusing upper and lower triangular method. After that,

gain and current-voltage characteristic of PIN APD

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voltages. Impact ionization which are happen in highelectric field which has been studied in this research.

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