a numerical method for the analysis of nonlinear carrier diffusion in cylindrical semiconductor...

phys. stat. sol. (b) 244, No. 9, 3231–3243 (2007) / DOI 10.1002/pssb.200642524

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Original

Paper

A numerical method for the analysis

of nonlinear carrier diffusion

in cylindrical semiconductor optoelectronic devices

M. S. Shishodia, A. Sharma*, and G. B. Reddy

Physics Department, Indian Institute of Technology Delhi, New Delhi 110 016, India

Received 18 October 2006, revised 11 December 2006, accepted 14 February 2007

Published online 4 April 2007

PACS 73.21.Fg, 73.63.Hs, 85.60.Bt

A numerical method to simulate radial distribution of carrier concentration in cylindrical semiconductor

optoelectronic devices is presented. Method is based on the collocation principle and employs sinusoidal

functions as the basis. The two approaches, evolutionary as well as iterative are presented for solving the

governing differential equations. Coordinate transformation is shown to be extremely advantageous for

enhancing the computational efficiency. To illustrate the versatility of the method, several examples

where the geometry demands different sets of boundary conditions are included. The application of this

technique for analyzing carrier concentration profiles in cylindrical optoelectronic devices, for the first

time, has demonstrated its multi-utility in addition to the established ability in solving electromagnetic

wave equation.

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

Ultra Large Scale Integration (ULSI) of optoelectronic components has resulted in fabrication of very

compact devices. The compactness of the device size has resulted in low threshold current, high modula-

tion speed and lateral mode control. But in such low dimensional devices, the carrier diffusion and sur-

face recombination (SR) may become limiting factors from performance point of view. The influence of

these processes on the performance of devices like Vertical Cavity Surface Emitting Lasers (VCSELs) is

much more severe due to their large surface to volume ratio. For example, it has been reported that in a

typical 5 µm InGaAs–GaAs based VCSEL, more than 50% of injected carriers are lost from the active

zone (AZ) via diffusion process only. Further reduction of the device size to 2 µm results in about 80%

loss [1].

In addition to the lateral diffusion and surface recombination, the injected carriers may be lost through

the bulk recombination processes, viz., Auger recombination (AR) and bi-molecular recombination

(BR). The lateral transport of charge carriers, via diffusion and their loss through the above mentioned

mechanisms in the plane of semiconductor quantum wells, play a crucial role in both the static as well as

dynamic operations of semiconductor lasers [2]. Diffusion characteristics have been shown to affect the

dynamic behavior, modulation response, mode dynamics and selection, beam quality, threshold current,

etc. [2–5]. In order to overcome these problems, techniques such as chemical sidewall passivation, im-

purity induced disordering and semiconductor regrowth are used [5–7]. Determination of the diffusion

controlled carrier profile is of interest because it determines the transverse dependence of the local gain,

and, also it has an influence on the transverse refractive index-profile, which in turn determines the wave-

* Corresponding author: e-mail: [email protected]

3232 M. S. Shishodia et al.: A numerical method for the analysis of nonlinear carrier diffusion

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-b.com

guiding properties of these devices. Considerable attention has been given in the literature for developing

methods for the analysis of the waveguiding properties of devices with cylindrical geometry (e.g., VCSEL),

without giving much attention to their electrical properties. The majority of the reported work treats lateral

transport of carriers analytically, where one has to neglect the effect of either bi-molecular or Auger recom-

bination or both [8–10]. Even in the absence of these nonlinear terms, closed form expressions may be

quite complicated in case of arbitrary current injection profile, thereby losing the advantage of analytical

analysis [9]. Neglect of nonlinear terms may also result in inaccurate estimations due to the complex inter-

play of optical, electrical and thermal effects. Thus, it is important to treat the carrier transport process in

its entirety in a comprehensive simulation tool. Inclusion of the terms representing either BR or AR or

both processes into the transport equation results in a nonlinear differential equation, containing quad-

ratic and cubic nonlinearities. Solving such equations requires the use of numerical techniques.

In this paper, we present a method based on the collocation principle to solve such equations. This

method has successfully been employed for modeling optical wave propagation through linear [11, 12]

and nonlinear media [13]. Recently, this method has been employed to simulate inhomogeneous, time

dependent and nonlinear interdiffusion/intermixing in one-dimensional semiconductor heterostructures

[14], and it has been shown to be numerically more efficient compared to the conventional methods, e.g.,

the finite-difference (FD) methods.

In Section 2, we discuss the basic carrier transport equations. We present the method of their numeri-

cal solution in Section 3. Section 4 is devoted to the boundary conditions and the basis functions used for

different cases of practical importance. In Section 5, a coordinate transformation technique is presented

which makes the computations considerably faster. Examples and numerical results, included in Sec-

tion 6, show the applicability and numerical efficiency of the method for various device configurations.

2 Carrier transport equation

The charge carriers entering into the active zone (AZ) are expected to either recombine within AZ or

diffuse out radially. The below threshold distribution of carrier concentration N(r) inside the AZ of the

devices (viz, VCSELs) with cylindrical symmetry is governed by the following rate equation [15]

2 3( ) 1 ( ) ( ) ( )

( ) ( )N r N r N r J r

D r BN r CN rt r r r qdτ

∂ ∂ ∂Ê ˆ= - - - +Ë ¯∂ ∂ ∂, (1)

where the RHS terms describe the radial diffusion, bulk recombination, BR, AR processes and the rate of

injected carriers, respectively. Further, D represents the diffusivity; τ, the bulk lifetime; q, the elementary

charge; d, the effective width of AZ; J(r), the injection current density profile; B, the bimolecular recom-

bination coefficient; and C, the Auger recombination coefficient.

Equation (1) can be written in the following normalized form

2

2 3

2

( ) ( ) 1 ( )( ) ( ) ( ) ( )

N r N r N rUN r VN r WN r K r

X r r r

∂ ∂ ∂= + - - - +

∂ ∂ ∂, (2)

with

, 1/ , / , / , ( ) ( )/X Dt U D V B D W C D K r J r qdDτ= = = = = .

In the steady state Eq. (2) becomes

2

2 3

2

( ) 1 ( )( ) ( ) ( ) ( )

N r N rUN r VN r WN r K r

r r r

∂ ∂+ = + + -

∂ ∂ . (3)

Thus, the equation governing the radial carrier profile is a second order differential equation in the

radial coordinate and it contains second and third order nonlinearities. The solution of this equation by

conventional methods requires linearization of the nonlinear equation and thus obtaining the solution

phys. stat. sol. (b) 244, No. 9 (2007) 3233

www.pss-b.com © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Original

Paper

iteratively. Here, we report a technique that can be used to solve steady state as well as transient transport

equation Eq. (2).

3 Numerical method

To determine the carrier concentration profile N(r), we proceed by writing N(r) as a linear combination

over a set of known orthogonal basis functions φm(r):

0

( ) ( )

M

m m

m

N r p rφ=

=Â , (4)

where pm are the coefficients of series expansion. These unknown coefficients may be determined by

requiring that the governing equation is exactly satisfied at all the sampling points which in the present

case are referred to as collocation points, rj, j = 0, 1, 2, …, M. The collocation points are such that,

φM+1(rj) = 0. Now by writing the carrier transport equation, Eq. (2) at each of these collocation points, we

obtain a set of M + 1 differential equations

2

2 3

2

d 1( ) ( ) ( ) ( ) , = 0,1, 2, , .

dj j

j j j j

r r r r

N N NUN r VN r WN r K r j M

X r r r…

==

∂ ∂Ê ˆ= + - - - +Á ˜Ë ¯∂ ∂

(5)

The set of equations in Eq. (5) can be written in the form of following matrix equation,

( )21

d

dU V W

X= - + + +

NS I N N N Q , (6)

where

[ ]0 1 2col ( ) ( ) ( ) ( ) ,

MN r N r N r N r…=N (7)

0 1

2 2 2

1 2 2 2

1 1 1col ,

Mr r r r r r

N N N N N N

r r r r r r r r r…

= = =

È ˘∂ ∂ ∂ ∂ ∂ ∂Ê ˆ Ê ˆ= + + +Í ˙Á ˜ Á ˜Ë ¯ Ë ¯∂ ∂ ∂ ∂ ∂ ∂Î ˚S (8)

[ ]0 1 2col ( ) ( ) ( ) ( )

MK r K r K r K r…=Q , (9)

I is an identity matrix and ⟨N⟩ represent a diagonal matrix such that the diagonal elements are the corre-

sponding elements of vector N, i.e., ( )i ij i ijijN N rδ δ= ∫N .

Further, we can write the expansion in Eq. (4) at the collocation points as

0

( ) ( ) , 0,1, 2, , .

M

j m m j

m

N r p r j Mφ …

=

= =Â (10)

Equation (10) can be written in the matrix form as

=N FP , (11)

where

0 1

col [ ]M

p p p…=P , (12)

and

0 0 1 0 0

0 1 1 1 1

0 1

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

M

M

M M M M

r r r

r r r

r r r

φ φ φ

φ φ φ

φ φ φ

�

�

� � � �

�

Ê ˆÁ ˜Á ˜=Á ˜Á ˜Ë ¯

F . (13)



Similarly, by differentiating Eq. (4) with respect to r and writing it at the collocation points, Eq. (8)

becomes

1=S GP , (14)

where

0 0 0

1 1 1

0 0 1 1

0 0 1 1

0 0 1 1

1 1 1( ) ( ) ( ) ( ) ( ) ( )

1 1 1( ) ( ) ( ) ( ) ( ) ( )

1 1 1( ) ( ) ( ) ( ) ( ) ( )

M M M

M M

r r r

M M

r r r

M M

r r r

r r r r r r

r r r

r r r r r r

r r r

r r r r r r

r r r

φ φ φ φ φ φ

φ φ φ φ φ φ

φ φ φ φ φ φ

�

�

� � � �

�

Ê ˆ+ + +¢¢ ¢ ¢¢ ¢ ¢¢ ¢Á ˜Á ˜Á ˜+ + +¢¢ ¢ ¢¢ ¢ ¢¢ ¢Á ˜=Á ˜Á ˜Á ˜Á ˜+ + +¢¢ ¢ ¢¢ ¢ ¢¢ ¢Á ˜Ë ¯

G , (15)

where prime denotes differentiation with respect to r. Substituting from Eq. (11), 1-=P F N into

Eq. (14), we get from Eq. (6),

1 2d

( )d

U V WX

⟨ ⟩ ⟨ ⟩-

= - - - +N

GF I N N N Q . (16)

Equation (16), referred to as collocation equation, is an ordinary matrix differential equation. No ap-

proximation is made in arriving at Eq. (16) and it is exactly equivalent to Eq. (2) for M Æ•. This equa-

tion can be solved using any general procedure to solve a system of differential equations such as the

Runge–Kutta method. This would give the time evolution of the carrier density profile.

Generally, one is interested in the steady state and hence, the solutions of Eq. (3) are sought. One

approach to obtain the steady state solutions is to use the time-dependent equation (Eq. (1)) and use the

numerical method discussed above to reach a state beyond which no change occurs in the carrier density

profile. This can be termed as the evolutionary approach. Another approach generally used is an iterative

approach based on Eq. (3). This approach requires linearized iterations. Using the procedure similar to

the one followed above, Eq. (3) can be converted into the following matrix equation

1 2( ) ,U V W⟨ ⟩ ⟨ ⟩-

- - - = -GF I N N N Q (17)

or

,= -SN Q (18)

with

21,U V W

-

= - - -S GF I N N

which is, in general, a nonlinear equation since it contains products of N. To solve such equations an

iterative linearization approach is used, in which, starting from an assumed distribution N(0), the carrier

density profile N is improved iteratively by obtaining N (1), N (2), N (3), …, etc. At each iteration step, the

carrier density profile of the previous step is used to define the matrix S. The iterations are continued till

a suitable convergence in N is achieved.

The iterative approach though generally faster compared to the evolutionary approach, sometimes lead

to divergence, while we have not observed such behavior in the latter approach. The evolutionary ap-

proach is particularly convenient with the collocation method as the nonlinear partial differential equa-

tion is converted to matrix ordinary differential equation which can be easily solved. It does not require

any linearization and hence can be used even with strong nonlinearities. A quantitative comparison of the

computational efficiency and time taken in the two approaches is included in Section 6.

phys. stat. sol. (b) 244, No. 9 (2007) 3235


Original

Paper

J =J0

r=r1r=0 r=r2

J=J0

r=0 r=r2

J=J0 exp [-(r-r1)/r0]

r=r1r=0 r=r2

J=J0

Fig. 1 Schematic of the AZ of a cylindrically symmetric VCSEL. Symmetry axis is at r = 0. (a) Uniform current

density J0 is injected in the region 0 ≤ r ≤ r

1. Radius of the AZ is r

2. (b) Uniform current density J

0 is injected

throughout AZ. Radius of the AZ is r2. (c) Nonuniform current density is injected in the region r

1 ≤ r ≤ r

2 and uni-

form injection in the region r ≤ r1 Radius of the AZ is r

2.

4 Boundary conditions and the basis functions

An important aspect of the collocation method is the choice of the basis functions, φm(r), and that de-

pends on the boundary conditions and the symmetry (e.g., planar, cylindrical, or spherical) of the device

under consideration. For the present case of cylindrically symmetric devices, we introduce the following

basis functions

2( ) cos( ) for 0,1, 2, 3, , ,m mr r m Mφ ν …= = (19)

which are orthogonal to each other. For the present analysis we choose the cases as shown in Fig. 1.

Several cases may arise depending upon the injection profile and the processing of the device. If the

injection radius is very small compared to the AZ radius, then domain may be referred to as unbounded,

and bounded otherwise. For the present, we consider the following three kinds of injection profiles

(shown schematically in Fig. 1):

Case 1: 0 1

1 2

( )0 ,

J r rJ r

r r r

£Ï= Ì

£ £Ó (Fig. 1a) (20a)

Case 2: 0 2

( ) for 0J r J r r= £ £ , (Fig. 1b) (20b)

and

Case 3: 0 1

0 1 0 1 2

( )exp [ ( ) / ] .

J r rJ r

J r r r r r r

£Ï= Ì

- - £ £Ó (Fig. 1c) (20c)

Accurate treatments of these different situations demand the use of appropriate boundary con-

ditions. For both systems, bounded or unbounded, the equation must satisfy, the axial symmetry condi-

(a) (b)

(c)



tion [15]:

0

0

r

N

=

∂=

∂r (21)

and the second boundary condition will depend upon the type of device under consideration.

For the unbound domain the following Dirichlet boundary is suitable

( ) 0 as .N r rÆ Æ• (22)

For the bounded domain with no sidewall recombination, the following Neumann boundary is suitable

2

0 .

r r

N

=

∂=

∂r (23)

For the bounded domain with significant sidewall recombination the following mixed boundary is

suitable

2 2

,

r r r r

N sN

D= =

∂= -

∂r (24)

where s denotes the surface recombination velocity. The constants νm’s of the basis functions in Eq. (19)

are obtained from the appropriate boundary conditions. Thus, we obtain νm as 2

2((2 1)π/2 ),m r+

2

2( π/ )m r

and 2

2( / )

mrχ , respectively for the boundary conditions given by Eqs. (22)–(24). Here, χm denotes the m-

th root of the transcendental equation, χ tan (χ) = sr2/2D. The collocation points rj for each of these cases

are then, respectively, given by:

1/ 2

2

2 1

2 3

jr

M

+Ê ˆË ¯+

,

1/ 2

2

2 1

2 2

jr

M

+Ê ˆË ¯+

and

1/ 2

2

1

(2 1) π/2, for 0,1, 2, , .

M

jr j M

χ…

+

+Ê ˆ =Á ˜Ë ¯ (25)

5 Coordinate transformation

The computational efficiency of the above mentioned approaches can be improved by using a suitable

coordinate transformation. From Eq. (25), it is clear that the sampling points are not equally spaced.

Since their spacing varies as the square root of the radial position, the density of points close to the axis

is low compared to that close to the edge. This results in the large number of points being outside the

injection region and, thus an accurate modeling of carrier density profile would require a larger number

of sampling points. It would be computationally advantageous if the sampling points are redistributed in

such a way that the density of points in the injection region is increased. The redistribution is possible

with the help of a suitable coordinate transformation,

r = h(σ) , (26)

where σ is the new radial coordinate. Equation (2), can now be written as

2

2 3

2 2 3

1 1,

N N h NUN VN WN K

X h hh hσ σ

∂ ∂ ∂¢¢Ê ˆ= + - - - - +Ë ¯∂ ∂ ∂¢ ¢ ¢ (27)

where

1/ , / , / , ( )/ ,U D V B D W C D K J r qdDτ= = = =

h¢ and h¢¢represents the first and second derivative of h. Here, we employ the transformation used earlier

in [16]:

( ) sin ( )h hσ σ= (28)

phys. stat. sol. (b) 244, No. 9 (2007) 3237


Original

Paper

0 1 2 3 4 5 6

1

2

w ith o u t t r a n s fo rm a t io n ( M = 1 5 0 )

w i th tr a n s fo rm a t io n ( M = 6 0 )A

rbitr

ary

Axi

s

R a d ia l P o s i t io n ( µ m )

Fig. 2 Distribution of collocation points with and without coordinate transformation.

The effect of coordinate transformation can be understood from Fig. 2, which shows the distribution

of collocation points with and without transformation. In untransformed case, out of one hundred fifty

one points, only seventeen lie within 2 µm (injection region), while in the transformed case twenty points

lie in the injection region out of a total sixty-one points only. This result in the significant reduction in

the sizes of the matrices involved, thereby improving the computational efficiency of the method. The

boundary conditions (Eq. (21) to (24)) take the following form in the transformed coordinate σ re-

spectively,

0

0 ,N

σσ

=

∂=

∂ (29)

( ) 0 and 0 as ,N

N σ σ

σ

∂Æ Æ Æ•

∂ (30)

L

0 ,N

σ σσ

=

∂=

∂ (31)

L L

( ) ,N sN

hD

σ σ σ σ

σ

σ= =

∂= - ¢

∂ (32)

where σL denotes the outer edge in terms of transformed variable and corresponds to r2 in the untrans-

formed coordinates. Thus the form of all the boundary conditions remains the same except a slight change

in the mixed boundary case. Equation (27) can be solved using any of the previously described approach.

6 Results and discussion

We simulate the carrier profiles in different kinds of device geometries (see, Fig. 1), using evolutionary

and iterative approaches discussed in Section 3. The following four cases are analyzed in the present

work.

Case 1: The device with a strong confinement of injection current (e.g., oxide confined or ion implanted

VCSELs), with current injection as depicted in Fig. 1a and the sidewall boundary condition de-

fined by Eq. (22) (or, Eq. (30)).



Case 2(a): The device with weak confinement of injection current (e.g., etched post VCSELs with large

carrier concentration at the sidewall of the AZ), with current injection as depicted in Fig. 1b

and the sidewall boundary condition defined by Eq. (23) (or, Eq. (31)).

Case 2(b): The device with weak confinement of injection current (e.g., etched post VCSELs with large

carrier concentration at the sidewall of the AZ and significant surface recombination), with

current injection as depicted in Fig. 1b and the sidewall boundary condition defined by

Eq. (24) (or, Eq. (32)).

Case 3: The device with the injection current defined by Eq. (20c) and depicted in Fig. 1c and the side-

wall boundary condition defined by Eq. (22) (or, Eq. (30)).

The diffusion equation admits closed form solutions for Cases 1, 2(a), and 2(b) in the absence of BR

and AR processes. These solutions are given in the Appendix for these three cases. No simple closed

form solutions exist for the case 3.

In order to validate the present method, first, we neglected the contributions from BR and AR pro-

cesses and computed the carrier profiles for above mentioned cases (cases 1, 2(a) and 2(b)) using this

method. The profiles so obtained are then compared with those obtained from closed form expressions

(see Appendix). Next, in order to understand the effect of AR and BR processes on the carrier profile, we

computed the profiles including their contributions for different cases using evolutionary as well as itera-

tive approaches. The parameter values used in the calculations has been taken from the literature and are

typical for semiconductor lasers. For example, D = 5 × 108 µm2/s, and τ = 2.63 × 10–9 s have been used

by Zhang et al. to analyze AlGaAs–GaAs based 850 nm VCSEL [17]. Further, D = 1.0 × 108 µm2/s and

1.0 × 109 µm2/s have been used in [1], where authors have addressed the issue of lateral carrier confine-

ment in VCSELs. The value of bi-molecular recombination coefficient, B = 100 µm3/s is typical of the

material systems used for semiconductor laser fabrication and has widely been used by the researchers

[17–19]. The Auger recombination coefficient, C = 5 × 10–6 µm6/s and 3.5 × 10–6 µm6/s have also widely

been used; we have used the former to show the validity of the method for stronger nonlinearities [17–

19].

Figure 3 shows the carrier density distribution for Case 1, determined using (i) the present method and

(ii) the closed form expressions (Eq. (A1)) by neglecting the effect of nonlinear processes. Also shown

0 2 4 6 8 10 12 14 16 18 200

1x1018

2x1018

3x1018

4x1018

5x1018

6x1018

7x1018

Present MethodAnalytical

Nonlinear

(a)

(b)

Linear

(a)

(b)

Car

rier-

Den

sity

(cm

-3)

Radial Position (µm)

Fig. 3 Computed carrier density profiles in Case-1 with and without AR and BR effects (linear &

nonlinear conditions respectively) for (a) D = 1 × 109 µm2/s, τ = 1.5 × 10–9 s; (b) D = 5 × 108 µm2/s,

τ = 2.63 × 10–9 s. For linear case, results obtained using present method and those from closed form ex-

pressions are compared. Other parameters used for the computation are: B = 100 µm3/s, C = 5 × 10–6 µm6/s,

r1 = 5 µm, r

2 = 20 µm, J = 3 × 10–5 A/µm2, d = 0.08 µm. Results are found to be identical for both, the

evolutionary as well as iterative approach.

phys. stat. sol. (b) 244, No. 9 (2007) 3239


Original

Paper

0 1 2 3 4 5 60

1x1018

2x1018

3x1018

4x1018 Linear

Nonlinear(a)

(a)

(b)

(b) Present MethodAnalytical

Car

rier-

Den

sity

(cm

-3)


Fig. 4 Computed carrier density profiles in Case-2(a) with and without AR and BR effects (linear and




r1 = 2 µm, r

2 = 6 µm, J = 3 × 10–5 A/µm2, d = 0.08 µm. Results are found to be identical for both, the evo-

lutionary as well as iterative approach.

are the respective carrier density distribution in the presence of nonlinear processes like BR and AR.

The calculations were performed for two typical sets of diffusion coefficient, D (5 × 108 µm2/s,

1 × 109 µm2/s) and lifetime, τ (2.63 × 10–9 s, 1.5 × 10–9 s). The figure shows that, in the linear case, the

carrier density profiles obtained using the analytical expressions and those obtained using the present method

0 1 2 3 4 5 60

1x1018

2x1018

3x1018

4x1018

5x1018

6x1018

7x1018

8x1018

Present MethodAnalytical

Nonlinear

Linear

(a)

(b)

(a)

(b)

Car

rier-

Den

sity

(cm

-3)


Fig. 5 Computed carrier density profiles in Case-2(b) with and without AR and BR effects (linear and




s = 1 × 1010 µm/s, r1 = 2 µm, r

2 = 6 µm, J = 3 × 10–5 A/µm2, d = 0.08 µm. Results are found to be identical

for both, the evolutionary as well as iterative approach.



0 20 40 60 800.0

8.0x1017

1.6x1018

2.4x1018

3.2x1018

(b)

(a)C

arrie

r-D

ensi

ty(c

m-3)

Radial Position(µm)

Fig. 6 Computed carrier density distribution for a graded current density profile described in Case-3 for

(a) D = 1 × 109 µm2/s, τ = 1.5 × 10–9 s; (b) D = 5 × 108 µm2/s, τ = 2.63 × 10–9 s. Other parameters used for

the computation are: B = 100 µm3/s, C = 5 × 10–6 µm6/s, r1 = 5 µm, r

2 = 100 µm, r0 = 0.5 µm, d = 0.08 µm,

J = 3 × 10–5 A/µm2.

completely overlap. Next, we use the present method to solve carrier transport equation in the presence

of nonlinear processes such as BR and AR using both the evolutionary as well as iterative procedures

implemented using coordinate transformation approach described in Section 5. Same results are achieved

without transformation also (not shown here), but that would require a larger number of collocation

points, thus resulting in a larger matrix size and hence making the calculations computationally expen-

sive. All the computational and material parameters for nonlinear calculations are taken to be the same as

those for linear calculations. It can be seen that the inclusion of nonlinear processes results in the ex-

pected reduction of carrier density due to the carrier loss through these processes. Similar calculations for

Case 2(a) and Case 2(b) are also presented in Figs. 4 and 5, respectively. In these cases also an excellent

agreement is found between the results obtained from analytical expressions and those obtained using

0 2 4 6 8 10 12 14 16 18 200.0

5.0x1017

1.0x1018

1.5x1018

2.0x1018

2.5x1018

(steady state)800

400300

200

Car

rier-

Den

sity

(cm

-3)


Fig. 7 Evolution of the carrier-concentration profiles determined using evolutionary approach for differ-

ent Runge–Kutta (R–K) propagation steps. All other parameters are same as for Fig. 3 (nonlinear case,

curve a). The number in front of arrow represents the number of evolutionary steps (R–K steps). Iterative

approach cannot give this transient evolution of the carrier density profiles.

phys. stat. sol. (b) 244, No. 9 (2007) 3241


Original

Paper

the present method (Eqs. (A2), (A3)) in the linear case. In the corresponding nonlinear cases also, same

results are obtained from evolutionary and iterative approach. An excellent agreement between the re-

sults obtained using present method and those from closed form analytical expressions in all above dis-

cussed cases validates the suitability of the present method for analysis of such devices.

Next, we consider the situation depicted in Fig. 1c, where current distribution is given by Eq. (20c).

This situation is close to reality and its linear analysis has been carried out in [9]. Even in the linear case,

the closed form expression available is too complicated for analytical investigation. However, both linear

as well as nonlinear cases can be dealt with easily using either evolutionary or iterative variants of the

present method. Results for the same are shown in Fig. 6 for the boundary condition given in Eq. (22).

Finally, Fig. 7 shows the temporal evolution of the carrier profiles for Case 1 before attaining steady

state. These calculations are made using the evolutionary approach. Here, we start the calculations from

an arbitrarily chosen initial carrier profile and determine the profiles after a chosen time interval using

the fourth order Runge–Kutta method. After a certain time, the carrier concentration profile becomes

stationary. This corresponds to the steady state distribution of carriers.

Comparison of the required number of iterations/Runge–Kutta steps and CPU time in both the ap-

proaches (evolutionary and iterative) is shown in Table 1. It can bee seen that the evolutionary approach

takes marginally higher computational time compared to that required in the iterative approach. On the

other hand the evolutionary approach is found to be more stable compared to the iterative procedure, as

the latter may produce divergent solutions in some cases as shown in the Table 1. We have not observed

such behavior in the evolutionary approach. Also, a transient analysis of the carrier density distribution is

possible only by the evolutionary approach.

It may be added that we have used, in our examples, the values of the parameters, which are typical

for semiconductor optoelectronic devices. However, the method can be used equally well to model diffu-

sion process in other type of devices. We have not seen any instability of computation in these examples;

Table 1 Comparison of CPU time and computational resources involved in different approaches.

accuracy without transformation with transformation data

set iterative evolutionary iterative evolutionary

case

no. of

itera-

tions

time

(s)

no. of

R.K.

steps

time

(s)

no. of

itera-

tions

time

(s)

no. of

R.K.

steps

time

(s)

(a) 1 × 10–5

1 × 10–6

21

24

0.281

0.313

2570

4130

0.906

1.390

20

23

0.063

0.078

657

1002

0.125

0.437

1

(b) 1 × 10–5

1 × 10–6

11

13

0.219

0.250

2769

4383

1.000

1.437

10

12

0.047

0.063

799

1104

0.141

0.187

(a) 1 × 10–5

1 × 10–6

12

14

0.250

0.269

16560

31691

5.125

9.562

11

13

0.063

0.078

2791

4546

0.453

0.593

2(a)

(b) 1 × 10–5

1 × 10–6

7

8

0.203

0.218

19742

36394

5.950

10.700

6

7

0.047

0.063

2450

3977

0.343

0.515

(a) 1 × 10–5

1 × 10–6

20

23

1.687

1.734

17670

25803

7.250

9.250

20

23

0.672

0.703

1816

2353

0.843

0.906

(b) 1 × 10–5

1 × 10–6

10

12

1.656

1.671

25373

39729

9.810

13.400

10

12

0.656

0.662

2817

3760

0.953

1.093

2(b)

(c) 1 × 10–5

1 × 10–6

DNC

DNC

–

–

5151

6818

3.250

3.843

DNC

DNC

–

–

501

618

0.671

0.687

(a) D = 5 × 108 µm2/s, τ = 2.63 × 10–9 s; (b) D = 1 × 109 µm2/s, τ = 1.5 × 10–9 s; (c) D = 1 × 108 µm2/s,

τ = 1.5 × 10–8 s. DNC stands for solution does not converge.



however, one has to carefully choose the computation parameters to avoid numerical instability, which

may occur if the variation in the concentration is very strong either with time or in the space. Further, we

have dealt with only cylindrically symmetric cases which are common in practice; however, the method

can be extended to cases where there is an azimuthal variation of concentration, albeit at a considerable

increase in computational cost.

7 Conclusion

A new method based on the collocation principle using sinusoidal functions is presented for simulating

diffusion and surface recombination of charge carriers in cylindrical devices. Two different approaches,

namely evolutionary and iterative have been developed to solve the carrier transport equation for differ-

ent geometries and injection profiles of practical importance. Coordinate transformation is shown to be

useful in improving the computational efficiency of both the approaches. The method has been imple-

mented to simulate carrier density distribution profiles in the devices with cylindrical symmetry such as

VCSELs. The method is validated by comparing the results obtained using the present method with those

obtained from analytical expressions after neglecting the contributions due to AR and BR processes.

Further, the present method is also implemented using the evolutionary as well as the iterative ap-

proaches to solve nonlinear diffusion equation in the presence of AR and BR processes. Comparison of

the computational efforts involved in both the approaches with/without coordinate transformation is also

made. The evolutionary approach is found to be more stable and versatile as it can be used to determine

steady state as well as transient solutions of the diffusion equation. Also, it does not produce any diverg-

ing solutions, while this is a real possibility in any of the iterative methods.

Acknowledgement One of the authors (Manmohan Singh Shishodia) acknowledges the financial support provided

by the CSIR as a Senior Research Fellow.

Appendix: Closed form analytical solutions

Here, the closed form solutions in three cases are given in terms of the following parameters:

1 1 2 2 0

/ , / , and / .r D r D r D J J qdρ τ ρ τ ρ τ τ= = = =

Case 1: In case of 0 < r <• , linear diffusion equation under consideration admits following closed

form solutions

1 0 1

2 0 1

( )( )

( ) ,

J C I r rN r

C K r r

ρ

ρ

- £Ï= Ì

>Ó (A1)

where, ( ) and ( )m mI Kρ ρ are the m-th order modified Bessel functions of the first and second kind. The

values of C1 and C2 are obtained using the unbound domain boundary conditions (Eqs. (21) and (22)) as:

[ ]1 1 1 0 1 1 1 0 1 1 1 2 1 1 1 1 1/ ( ) ( )/ ( ) ( )/ ( ) and ( )/ ( )C J I I I K K C C I Kρ ρ ρ ρ ρ ρ ρ= + = .

Case 2(a): In this case the carrier profiles satisfy the boundary conditions given by Eqs. (21) and (23),

and it can be written as

1 0 1

2 0 3 0 1 2

( )

( ) ( ) .

J C I r rN

C I C K r r r

ρ

ρ ρ

- <Ï= Ì

+ < <Ó (A2)

phys. stat. sol. (b) 244, No. 9 (2007) 3243


Original

Paper

The values of C1, C2 and C3 are:

3

0 1 1 1 1 2 0 1 0 1 1 2

0 1

1 1 1 2 1 2

,( ) ( ) ( ) ( ) ( ) ( )

( )( ) ( ) ( )

JC

I K K I I KK

I I I

ρ ρ ρ ρ ρ ρρ

ρ ρ ρ

=È ˘

- + +Í ˙Î ˚

2 3 1 2 1 2

( )/ ( )C C K Iρ ρ= ,

1 1 1 2

1 3

1 1 1 2

( ) ( )

( ) ( )

K KC C

I I

ρ ρ

ρ ρ

È ˘= -Í ˙

Î ˚ .

Case 2(b): In this case the carrier profiles satisfy the boundary conditions given by Eqs. (21) and (24),

and it can be written as

1 0

( )N J C I ρ= - (A3)

where

1

1 2 0 2( ) ( )

JsC

DI sIρ ρ=

+

.

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