phys. stat. sol. (b)
244
, No. 9, 3231–3243 (2007) /
DOI
10.1002/pssb.200642524
© 2007 WILEYVCH Verlag GmbH & Co. KGaA, Weinheim
OriginalPaper
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 multiutility in addition to the established ability in solving electromagnetic wave equation.
© 2007 WILEYVCH 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 modulation speed and lateral mode control. But in such low dimensional devices, the carrier diffusion and surface 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
µ
m 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
µ
m 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 bimolecular 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 indexprofile, which in turn determines the wave
*
Corresponding author: email:
asharma@physics.iitd.ac.in
3232 M. S. Shishodia et al.: A numerical method for the analysis of nonlinear carrier diffusion
© 2007 WILEYVCH Verlag GmbH & Co. KGaA, Weinheim
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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 bimolecular or Auger recombination 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 quadratic 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 onedimensional semiconductor heterostructures [14], and it has been shown to be numerically more efficient compared to the conventional methods, e.g., the finitedifference (FD) methods. In Section 2, we discuss the basic carrier transport equations. We present the method of their numerical 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 Section 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 r t r r r qd
τ
∂ ∂ ∂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 recombination coefficient; and
C
, the Auger recombination coefficient. Equation (1) can be written in the following normalized form
22 32
( ) ( ) 1 ( )( ) ( ) ( ) ( )
N r N r N r UN 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 qdD
τ
= = = = =
. In the steady state Eq. (2) becomes
22 32
( ) 1 ( )( ) ( ) ( ) ( )
N r N r UN 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
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OriginalPaper
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
φ
m
(
r
):
0
( ) ( )
M m mm
N r p r
φ
=
=
Â
, (4) where
p
m
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,
r
j
,
j
= 0, 1, 2,
…
,
M
. The collocation points are such that,
φ
M
+1
(
r
j
) = 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
22 32
d1()()()(), = 0,1,2,,.d
j j
j j j j r r r r
N N N UN 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
dd
U V W X
=  + + +
N S I N N N Q
, (6) where
[ ]
0 1 2
col ( ) ( ) ( ) ( ) ,
M
N r N r N r N r
…
=
N
(7)
0 1
2 2 21 2 2 2
1 1 1col ,
M
r 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 2
col ( ) ( ) ( ) ( )
M
K 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 corresponding elements of vector
N
, i.e.,
( )
i ij i ij ij
N N r
δ 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 M
φ
…
=
= =
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 00 1 1 1 10 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 ˆ Á ˜ Á ˜ =Á ˜ Á ˜ Ë ¯
F
. (13)
3234 M. S. Shishodia et al.: A numerical method for the analysis of nonlinear carrier diffusion
© 2007 WILEYVCH Verlag GmbH & Co. KGaA, Weinheim
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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 01 1 1
0 0 1 10 0 1 10 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 2
d( )d
U V W X
〈 〉 〈 〉

=    +
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 equation 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 timedependent 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 equation 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.
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© 2007 WILEYVCH Verlag GmbH & Co. KGaA, Weinheim
OriginalPaper
J
=
J
0
r
=
r
1
r
=0
r
=
r
2
J=J
0
r
=0
r=r
2
J
=
J
0
exp [(
r

r
1
)/
r
0
]r=
r
1
r
=0
r
=
r
2
J
=
J
0
Fig. 1
Schematic of the AZ of a cylindrically symmetric VCSEL. Symmetry axis is at
r
= 0. (a) Uniform current density
J
0
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
r
2
. (c) Nonuniform current density is injected in the region r
1
≤
r
≤
r
2
and uniform injection in the region
r
≤
r
1
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,
φ
m
(
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 m
r r m M
φ 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 11 2
( )0 ,
J r r J r r r r
£Ï= Ì£ £Ó
(Fig. 1a) (20a) Case 2:
0 2
( ) for 0
r J r r
= £ £
, (Fig. 1b) (20b) and Case 3:
0 10 1 0 1 2
( )exp [ ( )/ ] .
J r r J r 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)