a new method for solving non-linear carrier diffusion equation in axial direction of broad-area...

INTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS

Int. J. Numer. Model. 2010; 23:492–502Published online 13 January 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/jnm.749

A new method for solving non-linear carrier diffusion equationin axial direction of broad-area lasers

Micha" Szymanski�,y

Institute of Electron Technology, Al. Lotnikow 32/46, 02-668 Warsaw, Poland

SUMMARY

The solution of non-linear carrier diffusion equation associated with the axial direction of broad-area laserhas been achieved by a hybrid asymptotic-numerical method that combines WKB approximation and thevariation of constants method. The non-linearity has been taken into account by using an iterative scheme:carrier lifetime has been linearized and calculated as a function of carrier density known from the previousiteration. Non-uniform photon density has been taken into account. As the new method is based onanalytical solutions, it is very fast and not vulnerable to large gradients of carrier concentration usuallyoccurring in the vicinity of laser facets. Therefore, it can be useful in complicated self-consistent thermalmodels of broad-area lasers. The obtained results are compared with two simplified cases: solution of lineardiffusion equation with constant coefficients and calculations neglecting the diffusion. Copyright r 2010John Wiley & Sons, Ltd.

Received 10 July 2009; Revised 29 October 2009; Accepted 30 October 2009

KEY WORDS: carrier diffusion; diffusion equation; non-linear; optical devices; semiconductor

1. INTRODUCTION

The presence of carriers in the active layer of a broad-area laser plays a crucial role in the deviceoperation. First, it enables the light propagation by ensuring the material transparency and,second, it gives rise to radiative recombination, which is the basic process for lasing. However,carriers often manifest their presence in other ways as well. For example, they lower therefractive index of the semiconductor. In addition, their concentration exhibits localfluctuations. These phenomena result in rather poor beam quality due to the loss of a lateralmode confinement and filamentation [1]. Carriers are also involved in heat-generating processeslike non-radiative recombination, Auger recombination or surface recombination. As aconsequence, the thermal runaway of broad-area lasers is often a big problem. Sometimes itleads to deterioration of the main laser parameters, in other cases it may result in irreversible

*Correspondence to: Micha" Szymanski, Institute of Electron Technology, Al. Lotnikow 32/46, 02-668 Warsaw, Poland.yE-mail: [email protected]

Copyright r 2010 John Wiley & Sons, Ltd.

destruction of the device via catastrophic optical damage (COD) of the mirrors [2]. Toinvestigate all these effects, models allowing for calculation of carrier concentration in the activelayer are absolutely necessary.

A broad-area laser is a p-i-n diode operating under forward bias. In the plane of junction theelectric field is negligible and the movement of the carriers is governed by diffusion. Bimolecularrecombination and Auger process engage two and three carriers, respectively. Such quantitieslike pumping or photon density are spatially inhomogeneous. Far from the pumped region thecarrier concentration falls down to zero level. At the mirrors surface recombination occurs.Taking all these facts into account, one concludes that carrier concentration in the active layercan be described by a non-linear diffusion equation with variable coefficients and mixedboundary conditions. Solving such an equation is really difficult, but the problem can often besimplified. For example if problems of beam quality (divergence or filamentation) are discussed,considering the lateral direction only is a good enough approach.

In this paper I present a new, hybrid asymptotic-numerical method of calculation of carrierconcentration in the active layer of a broad-area laser, which is dedicated to thermal models.I consider the axial direction only. The origin of this simplification is briefly described below.

It has been shown that calculating reliable values of the device temperature requires thesolution of a three-dimensional heat conduction equation [3, 4]. The main heat sources areplaced in the active layer and are carrier-dependent functions. As surface recombination at laserfacets is believed to be a very efficient heating mechanism responsible for COD, calculating thecarrier concentration in axial direction is definitely crucial for thermal analysis. The complexityof the model causes that most of the authors decide to use one-dimensional (axial) diffusionequation [3, 5, 6]. In [4] the lateral diffusion of carriers has additionally been taken into account.In that paper small temperature kinks in the vicinity of the edges of the active layer have beenfound. However, the solution of the two-dimensional diffusion equation has been obtained thereunder the assumption of the averaged carrier lifetime and averaged photon density throughoutthe resonator. The resulting diffusion equation was therefore linear. The one-dimensionaldiffusion equation I considered in this work is dedicated to thermal model described in [3]. It isnon-linear and, due to variable coefficients, can take into account spatially dependent photondensity along the resonator axis.

The characteristic feature of the concentration profiles considered here is that they aresmooth in the entire resonator, whereas very large gradients are usually found in the vicinity offacets [3, 5]. Such cases are inconvenient for numerical treatment: non-uniform mesh must beintroduced, calculations become time consuming, etc. The method I developed here is based onan analytical approach, so it is very fast and not vulnerable to the problems mentioned above. Itconsists of two major steps. Firstly, I find the analytical solution of a linearized equation bycombining WKB approximation and the variation of constants method [7]. The non-linearsolution is then found iteratively [8]. The carrier lifetime is linearized (calculated as a function ofcarrier density known from the previous iteration), so it becomes a function of space instead ofcarrier concentration.

The investigated equation has also been solved numerically by a commercial software usingfinite element analysis. Comparison of both methods—their precision and computation time—allow to judge the reasonableness of developing complicated analytical calculations for thisparticular problem.

In addition simplified approaches to the problem of finding the carrier distribution areconsidered. A linear diffusion equation with constant coefficients, used in many works

A NEW METHOD FOR SOLVING NON-LINEAR CARRIER DIFFUSION EQUATION 493

Copyright r 2010 John Wiley & Sons, Ltd. Int. J. Numer. Model. 2010; 23:492–502

DOI: 10.1002/jnm

investigating the problem of facet heating [4–6], is solved. The crudest estimation of carrierconcentration based on the algebraic rate equation, which neglects the diffusion process, is alsopresented. All differences found between methods and approaches are discussed in detail.

2. THE MODEL

2.1. The non-linear diffusion equation

Stationary, one-dimensional diffusion equation for the carrier concentration can be writtenas [3]:

Dd2N

dz2�

c

neffGgðNÞSðzÞ �

N

t1

I

eV¼ 0 ð1Þ

Assuming linear gain gðNÞ¼aðN �NtrÞ, non-linear carrier lifetime tðNÞ¼ðAnr1BN1CAN2Þ�1

and noting the spatial dependence of the photon density distribution S one finds thatEquation (1) is a non-linear (containing quadratic and cubic concentration terms) second-orderdifferential equation with variable coefficients. As the surface recombination is present at thelaser facets, the following boundary conditions must be applied:

DdNð0Þdz

¼ vsurNð0Þ; DdNðLÞdz

¼ �vsurNðLÞ ð2Þ

A way of solving the Equation (1) is to linearize it and make use of an iteration scheme as inRef. [8]. The method will be described below in detail.

Equation (1) can be re-written in the dimensionless form:

e2d2n

dx2�QðxÞn1f ðxÞ ¼ 0 ð3Þ

where

QðxÞ ¼1

tðxÞ1cneff

GSðxÞa

A0; f ðxÞ ¼

IeV

1 cneff

GSðxÞantrn0A0

ð4Þ

and boundary conditions:

dnð0Þdx¼

Lvsur

Dnð0Þ;

dnð1Þdx¼ �

Lvsur

Dnð1Þ ð5Þ

Note that the concentration-dependent function tðNÞ has been replaced by a space-dependentfunction tðxÞ. Thus Equation (3) is a linear equation. It is solved in two steps. First I considerthe homogeneous equation:

e2d2n

dx2¼ QðxÞn ð6Þ

As e� 1, WKB approximation can be used to give [7]:

nðxÞ � C1Q� 1

4ðxÞ exp1

e

Z x ffiffiffiffiffiffiffiffiffiffiffiffiQðx1Þ

pdx1

� �1C2Q

� 14ðxÞ exp

1

e


pdx1

� �ð7Þ

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Then I use the variation of constants method (C1 ! C1ðxÞ,C2 ! C2ðxÞ) to get the generalsolution of Equation (3) in the form:

nðxÞ ¼AQ�14ðxÞ exp

1

e


pdx1

� �1BQ�

14ðxÞ exp �

1

e

Z x ffiffiffiffiffiffiffiffiffiffiffiffiffiQðx1Þ

pdx1

� �

11

2eQ�

14ðxÞ exp

1

e

Z x ffiffiffiffiffiffiffiffiffiffiffiffiffiQðx1Þ

pdx1

� � Z x

Q�14ðx2Þf ðx2Þ exp �

1

e

Z x2 ffiffiffiffiffiffiffiffiffiffiffiffiQðx3Þ

pdx3

� �dx2

�1

2eQ�

14ðxÞ exp �

1

e


pdx1

� � Z x

Q�14ðx2Þf ðx2Þ exp

1

e

Z x2 ffiffiffiffiffiffiffiffiffiffiffiffiQðx3Þ

pdx3

� �dx2

ð8Þ

Constants A and B in (8) are determined by boundary conditions. The solution of the non-linear problem is found with the usage of the linear solution and the iterative proceduredescribed below.

I assume a constant profile of carrier concentration n to get the initial value of the carrierlifetime tðnÞ. Then, using Equation (8), I calculate spatially dependent profile nðxÞ. Thelinearized carrier lifetime tðnðxÞÞ ! tðxÞ is introduced and the use of Equation (8) again leads toa new profile nðxÞ. These calculations are repeated until the convergence is achieved.

Several methods have been developed to assess the accuracy of the obtained solution.Calculating the integral of the left side of non-linear equation:

Om ¼Z 1

0

e2d2nm

dx2�

Anr1cneff

GSðxÞa

A0nm � f ðxÞ �

Bn0

A0n2m �

CAn20

A0n3m

� �dx ð9Þ

allows to check the convergence of the successive iterations. Fulfilling of the boundaryconditions can be verified by parameter p expressed in percents:

p ¼dnð0Þdx �

LvsurD

nð0Þdnð0Þdx

�� 100 ð10Þ

Moreover, the obtained solution of dimensionless version of Equation (1) is compared withconcentration profiles calculated by a commercial software using finite element analysis andaccording to simplified approaches described in the next section.

2.2. Simplified approaches

In broad-area lasers the carrier concentration is determined mainly by driving current andrecombination processes. Therefore, the crudest estimation of that quantity can be obtained byneglecting the diffusion term in Equation (1). Thus an algebraic equation is obtained. In thedimensionless form it can be written as

�CAn

20

A0n3 �

Bn0

A0n2 �

Anr1cneff

GSðxÞa

A0n1f ðxÞ ¼ 0 ð11Þ

It yields a spatially dependent concentration profile due to variable photon density SðxÞ.A much more accurate way of calculation of carrier concentration is to consider a linear

diffusion equation with constant coefficients

e2d2n

dx2�Qcn1fc ¼ 0 ð12Þ



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where averaged carrier lifetime tav and averaged photon density Sav have been assumed,so that

Qc ¼1tav

1 cneff

GSava

A0; fc ¼

IeV

1 cneff

GSavantrn0A0

ð13Þ

The general solution of Equation (12) is

nðxÞ ¼ Ac exp

ffiffiffiffiffiffiQc

pe

x

" #1Bc exp �

ffiffiffiffiffiffiQc

pe

x

" #1

fc

Qcð14Þ

and constants Ac and Bc are determined by the boundary conditions (5).

3. THE RESULTS

In this work I consider a broad-area l ¼ 980 nm continuous-wave AlGaAs/InGaAs/GaAs laserdescribed in [9]. The device is symmetrical, so Figures 1, 2 and 5 present the range 0oxo0:5.Values of the parameters used in the calculations are found in [9] or estimated on the basis ofdata published for similar structures.

In my calculations I assume that the laser is driven by current I ¼ 1A and the output powerfrom the front mirror is Pout ¼ 0:3W [9]. The photon density in the vicinity of the mirror can beassessed according to [6]

S0 ¼ð11Rf ÞGPout

wdað1� Rf Þh cl

cneff

ð15Þ

Both mirrors are uncoated, so Rf ¼ 0:3, S0 ¼ 2 � 1015 cm�3 and Sðx ¼ 0:5Þ � 0:85S0. Theorigin of the last assumption will be explained in Section 3.1.

On the basis of References [11, 12] and [12, 13] the following temperature-dependentfunctions are used:

BðTÞ ¼4:93 � 10�8

T� 4:54 � 10�11; CAðTÞ ¼ 2:64 � 10�26 � exp

�0:168eVkBT=e

� �ð16Þ

which yields BðT ¼ 300KÞ ¼ 1:2 � 10�10 cm3=s and CAðT ¼ 300KÞ ¼ 4 � 10�29 cm6=s.

3.1 Parabolic approximation for QðxÞ and f ðxÞ

Calculation of the non-linear carrier density can be significantly simplified if QðxÞ and f ðxÞ areinterpolated by second-order polynomials. This approximation will be explained briefly below.

Neglecting the non-linear term describing the spontaneous emission and assuming constantgain along the resonator axis, the photon rate equations can be written as

dSf

dx¼ GSf ;

dSb

dx¼ �GSb ð17Þ

At the facets Sf ðx ¼ 0Þ ¼ RfSbðx ¼ 0Þ and Sbðx ¼ 1Þ ¼ RbSf ðx ¼ 1Þ. Note that the gain isrelated to mirror reflectivities through the equality G ¼ ln½1=ðRfRbÞ�. Analytical solutions ofEquation (17) yield the expression for the total photon density:

SðxÞ ¼ Sf ðxÞ1SbðxÞ ¼S0Rf

Rf11½expðGxÞ1RbexpðGð2� xÞÞ� ð18Þ

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which is presented graphically in Figure 1 for three cases of the symmetrical resonator. It can beseen that all profiles are parabolic-like with the minimum localized in the centre of theresonator. In the case of uncoated facets the difference between the photon density extrema is15%. This difference increases with decreasing facet reflectivity. Note that profiles of photondensity presented here are very similar to those presented in Figure 8 in [3], obtained accordingto self-consistent model taking into account axially non-uniform gain and non-linear term dueto spontaneous emission.

Looking at (4) it becomes clear that f(x) can be interpolated by a second-order polynomial: itdepends on S(x), while the rest of parameters is constant. Similar interpolation referring to QðxÞis not so obvious as the function depends additionally on tðxÞ(or—more precisely—on tðnðxÞÞ).SðxÞ determines parabolic-like shape of QðxÞ in the entire part of resonator, where nðxÞ is almostconstant. In the vicinity of facets large gradients of n cause rapid variations of carrier lifetime

Table I. Nomenclature and parameter values for InGaAs laser.

Symbol Description

Anr ¼ 1 � 108 s�1 Non-radiative recombination coefficientA0 ¼ 1010 s�1 Scaling constantBðTÞ Bi-molecular recombination coefficientCAðTÞ Auger recombination coefficientda¼8 nm Active layer thicknessD ¼ 11 cm2=s Diffusion coefficientg Linear gainG Constant gainI Driving currentL ¼ 700mm [9] Resonator lengthN Carrier concentrationNtr ¼ 1:5 � 1018 cm�3 [10] Transparency carrier concentrationn ¼ N=n0 Dimensionless carrier concentrationntr ¼ 1:5 [10] Dimensionless transparency carrier concentrationn0 ¼ 1018 cm�3 Scaling constantneff ¼ 3:5 Effective refractive indexPout Output powerRf ;Rb Power reflectivity of the front and back mirrorS Total photon densitySf ;Sb Photon density of the forward and backward travelling waveSav Averaged photon densityT TemperatureV Volume of the active layervsur ¼ 5 � 105 cm=s Surface recombination velocityw ¼ 100 mm [9] Contact widthz Spatial coordinate (0pzpL)a ¼ 15 � 10�16 cm2 [10] Differential gainG ¼ 0:03 Confinement factor

e ¼ 1L

ffiffiffiffiDA0

qConstant

m Iteration numberx Dimensionless spatial coordinate (0pxp1)t Carrier lifetimetav Averaged carrier lifetimec; e; h; kB Physical constants: light velocity, elementary charge,

Planck and Boltzmann constants, respectively.



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(see Figure 2). To simplify the calculations I decide to neglect this region and use theapproximated profile of Q. The validity of such an apparently drastic approximation will bejustified a posteriori by error analysis.

3.2. Accuracy of the non-linear solution

The non-linear solution of diffusion equation obtained according to the method described inSection 2.1 may be inaccurate because of several reasons. The main are:

� usage of WKB approximation;� problems with convergence of successive iterations;� parabolic interpolation of QðxÞ introduced in Section 3.1.

Figure 1. Total photon density along the laser axis calculated for Rf ¼ Rb ¼ 0:3(solid line), 0.12 (dashedline) and 0.05 (dotted line).

Figure 2. Profiles of Q: real (dashed line) and approximated by second-order polynomial (solid line).

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To investigate this problem I plot error integral vs iteration number (please cf. (9)) for carrierdensities calculated for different input data. In addition I check the accuracy of fulfilling theboundary condition according to parameter (10). All my calculations start from the guessed,constant concentration profile n ¼ 4:6. Up to T ¼ 215K good convergence is found. For thetemperatures above that value, however, strong oscillations appeared and the boundaryconditions were fulfilled with much poorer accuracy. To overcome those difficulties. I assumethe low initial temperature and increase it in each iteration step by a small value. Example plotsof O illustrating poor (circles) and good (squares) convergence are shown in Figure 3. Note thatthe higher value of Om correlates with higher value of p, i.e. larger deviation from the boundarycondition.

The investigations described above allow me to conclude that convergence of the methoddepends mainly on the temperature via the non-linear, exponentially growing Auger term C(T).Slightly decreasing bi-molecular term B(T) is of less importance in this aspect.

Note the relatively small value of p even in the cases of poor convergence (Figure 3). This factgives evidence in favour of the validity of the parabolic approximation for QðxÞ.

3.3. Calculation of axial carrier concentration according to different methods

A comparison of two different methods of solution to the non-linear diffusion equation isillustrated in Figure 4. The curves were obtained according to: (1) the hybrid asymptotic-numerical method described in Section 2.1 and (2) with the help of a commercial software basedon finite element analysis. The inset in the figure shows the relative difference between bothconcentration profiles.

It is well seen that these two methods yield convergent results except the region close to thefacet. This finding is not a surprising one. It reveals the expected numerical problems occurringwhen generally smooth functions locally exhibit rapid changes. In the considered case the carrierconcentration is smooth in the entire resonator and drastically decreased at the mirror due tosurface recombination.

Figure 3. The illustration of good and poor convergence.



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To investigate the problem of accuracy of each method, calculations of nðxÞ presented inFigure 4 have been repeated for several other reliable values of vsur. In each case the parameterp was checked. It has been found that p� 1:6% for the hybrid method, while 6opo9% for thepurely numerical method. It is noteworthy that this result has been obtained despite applyingthe non-uniform mesh in the commercial software.

Another important issue while comparing both methods is the computation time. Thecommercial software was providing the result in about 40 s, while the hybrid method was about20 times faster. Such difference may seem not to be very significant. However, one should keepin mind that thermal models require self-consistent calculations, i.e. constitutive equations aresolved in loops many times (see for example [3]). On the other hand simultaneous solutions ofsets of differential equation with the usage of finite element analysis is a time-consuming task aswas reported in [4]. Therefore, finding fast algorithms based on analytical solutions can be animprovement of existing thermal models.

The profiles nðxÞ calculated according to the physical models of different complexity (seeSection 2), have been plotted in Figure 5. All three approaches will be discussed below.

The dotted line shows the simplest solution based on Equation (11). The clearly visible slopeis a consequence of bi-molecular recombination enhanced by the photons, density of whichgrows towards laser facet. Note that the profile is not smoothed by diffusion, because thisapproach does not take into consideration that mechanism. Also the Equation (11) does notdescribe the surface recombination, so no rapid decrease of carrier concentration in the vicinityof a facet can be observed.

The dashed line illustrates the solution (14). To obtain it I calculated Sav ¼R 10 SðxÞ dx and

tav ¼ ð1=5ÞP

i tðNiÞ using carrier density obtained according to (11) in 5 points of the resonator.In this approach surface recombination is taken into account due to boundary conditions (5).Hence, in the nearest vicinity of facet a rapid decrease of carrier concentration is present.Beyond this region the calculated profile is flat, which is the consequence of assumptionS ¼ Sav ¼ const.

Figure 4. Solution of the non-linear diffusion equation with variable coefficients according to thehybrid method (solid line) and finite element analysis (dotted line). The inset shows the relative difference

between both curves.

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The solid line shows the solution of the non-linear diffusion equation described in Section2.1. In this approach both surface recombination and non-homogeneous photon densitydistribution throughout the resonator are taken into account. Therefore, the curve demonstratesrapid decrease of the carrier concentration near the facet as well as the slope in the inner part ofthe laser. Note that the slope is smaller comparing to solution (11), which can be explained bythe diffusion process.

Figure 5 shows that a crude estimation of the carrier concentration in the active layer can bedone simply by solving the algebraic Equation (11). However, for thermal modelling, wherephenomena in the vicinity of facets are crucial due to possible COD processes, a diffusionequation must be solved.

In many works (see for example [4–6]) the averaged values of photon densities and carrierlifetimes are assumed and the linear diffusion equation with constant coefficients (12) is solved.Such an approach seems to be a good approximation for a typical broad-area laser, which isalmost axially homogeneous device in the sense that the depression of the photon density(Figure 1) is not very deep or temperature differences along the resonator are not so big todramatically change the non-linear recombination terms B and C. Note that solution ofEquation (11) can be used for assessing tav.

The non-linear diffusion equation with variable coefficients is useful in all cases where theabove-mentioned axial homogeneity is perturbed. In particular, the approach is suitable forbroad-area lasers with modified regions close to facets. These modifications are meant to achievemirror temperature reduction through placing current blocking layers [14], producing non-injected facets (so called NIFs) [15] or generating a larger band gaps [16]. However, analysingsuch devices by the method described in this work may not be straightforward. The reason liesin the behaviour of carrier concentration profiles, which exhibit rapid variations on the distanceslarger than in standard devices. Thus, my analytical approximations for QðxÞ and f ðxÞ asdescribed in Section 3.1 become questionable. A solution to this problem seems to be in makingthe method ‘more numerical’ by introducing to Equation (8) appropriate algorithms forintegration of QðxÞ and f ðxÞ.

Figure 5. Axial carrier concentration calculated according to algebraic equation (dotted line), lineardiffusion equation with constant coefficients (dashed line) and non-linear diffusion equation with variable

coefficients (solid line).



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4. SUMMARY AND PERSPECTIVES

A new method of solution of non-linear diffusion equation for the carrier concentration in theaxial direction of broad-area laser was presented. It is based on analytical solution of the lineardiffusion equation with variable coefficients and numerical iterative scheme for taking intoaccount the non-linearity. The same problem has been solved by a commercial software basedon finite element analysis. The comparison showed that the new method is fast, accurate and notvulnerable to very large carrier concentration gradients occurring in the vicinity of facets. Inaddition, the simplified models of carrier diffusion have been investigated. It has been shownthat broad-area lasers of typical geometry can be analysed by a combination of modelsneglecting the diffusion and linear diffusion equation with constant coefficients. The non-lineardiffusion equation with variable coefficients is necessary for analysis of axially modified devicesdesigned to achieve mirror temperature reduction for preventing COD processes. Such non-standard broad-area lasers will be the subject of my further investigations.

ACKNOWLEDGEMENTS

The author would like to thank Professor Magdalena Zaluska-Kotur for discussions about diffusionprocesses and for advices concerning mathematical methods.

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a new method for solving non-linear carrier diffusion equation in axial direction of broad-area...

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