Investigations on liquid phase electroepitaxial growth kinetics of GaAs

Materials Science & Engineering B 112 (2004) 91–95 www.elsevier.com/locate/mseb

Investigations on liquid phase electroepitaxial growth kinetics of GaAs D. Mouleeswaran, R. Dhanasekaran* Crystal Growth Centre, Anna University, Chennai 600025, India Received 21 April 2004; accepted 23 June 2004

Abstract This paper presents a model based on solving a two-dimensional diffusion equation incorporating the electromigration effect by numerical simulation method corresponding to liquid phase electroepitaxial (LPEE) growth of GaAs, whose growth is limited by diffusion and electro migration of solute species. Using the numerical simulation method, the concentration profiles of As in Ga rich solution during the electroepitaxial growth of GaAs have been constructed in front of the growing crystal interface. Using the concentration gradient at the interface, the growth rate and thickness of the epitaxial layer of GaAs have been determined for different experimental growth conditions. The proposed model is based on the assumption that there is no convection in the solution. The results are discussed in detail. # 2004 Elsevier B.V. All rights reserved. Keywords: Numerical simulation; Diffusion; Electromigration; Liquid phase epitaxy; Semiconducting III–V compounds

1. Introduction Liquid phase electroepitaxial (LPEE) growth is a very promising and novel technique for producing device quality epitaxial layer of III–V compound semiconductor. Compound semiconducting epitaxial layers are vital to the electronics industry. This technique provides good control over epitaxial layer thickness, compositional homogeneity, low dislocation density, controlled doping, better surface morphology and defect structure, which are difficult in conventional liquid phase epitaxy [1,2]. LPEE provides encouragable good electronic properties and structural properties of bulk GaAs crystal [3]. The advantages of this technique have kindled interest among researchers on the growth of high quality epitaxial layers. The availability of such device quality epitaxial layers will open new horizons in the fabrication technology of optoelectronic and high power devices. Due to the several advantages of LPEE, the number of modeling studies have been carried our in recent years [4–16]. * Corresponding author. Tel.: +91-44-22352774; fax: +91-44-22352870. E-mail address: [email protected], [email protected] (R. Dhanasekaran). 0921-5107/$ – see front matter # 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.mseb.2004.06.012

LPEE growth is initiated and sustained by passing an electric current through substrate–solution interface. The growth rate is observed in the absence of any intentional furnace cooling. The applied electric current is the sole driving force in this technique. The precise control over the growth rate is achieved by controlling the applied current to the substrate–solution interface, whereas in conventional LPE, controlling the growth rate is still a problem. The liquid phase electroepitaxial growth mainly depends on two factors namely Peltier cooling at the substrate–solution interface and electromigration of solute species towards the growth interface. Peltier effect is a thermoelectric effect occurring when an electric current passes through a junction of two materials having different thermoelectric coefficients, heat is absorbed or evolved depending on the direction of applied electric current. In the case of Peltier cooling, heat is absorbed at the interface and supersaturation in the vicinity of the substrate occurs, resulting in epitaxial growth. In the case of Peltier heating, heat is evolved at the interface and dissolution of the substrate occurs. In III-V solutions and other metallic solutions, electromigration takes place due to electrostatic field forces and electron-momentum exchange.

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This paper presents a numerical simulation of twodimensional mass transport equation during LPEE growth of GaAs, based on diffusion and electromigration of solute species. Using the appropriate boundary conditions, Central difference method has been employed to construct the concentration profiles of solute species in the solution with successive increments of time. The concentration gradient at the interface can be used to calculate the growth velocity and thickness of the epitaxial layer.

2. Growth model Liquid phase electro epitaxy is a process involving the complex interactions of thermo mechanical and electromagnetic fields, heat and mass transport, surface kinetics, and various thermoelectric effects such as Peltier cooling/Peltier heating, Joulean effect, Thomson effect and Soret effect. The aim of this paper is to determine the contributions of Peltier effect and electromigration to the GaAs epitaxial growth. The proposed model considers steady state heat conduction with Peltier effect, steady state electric current flow and incorporation of electromigration to the transient mass transport in the solution.

3. Governing equations A horizontal growth cell is described using the Cartesian coordinates (X, Y), the Y-axis being perpendicular to the interface and is oriented towards the solution. We have followed a model presented by Kimura et al. [17], who have developed a two-dimensional kinetic model based on diffusion of As atoms for the growth of GaAs epitaxial layer in a conventional LPE system using the numerical simulation method. In the present work, the effect due to electromigration has been incorporated to the diffusion equation and the concentration profiles of solute species in the solution have been simulated. 3.1. Steady state heat conduction equation During the LPEE growth process, the overall furnace temperature is maintained constant. The temperature distribution for a given substrate is governed by the steady state heat conduction equation  kX

@2 T @X 2



 2  @ T þ kY ¼0 @Y 2

(1)

where kX and kY are the thermal conductivities in the X and Y directions. In LPEE growth, the Joulean heat is neglected since the growth time is long compared to the onset of the equilibrium. The Joulean heat plays a dominant role when the growth time is low compared to the onset of the

equilibrium. The boundary conditions applied to the steady state heat conduction are (i) @T/@X = 0 at distance along the substrate (ii) T = T0 at outside walls The condition at the substrate–solution interface is     @T @T (2) k Ys  k Yl ¼ QP @Y @Y where kYs and kYl are the thermal conductivities of the substrate and solution perpendicular to the plane of interface. QP is the Peltier cooling at the substrate–solution interface. The electric current passes directly through the substrate– solution interface and electric current density remains uniform throughout the solution. 3.2. Mass transport equation In LPEE, the growth proceeds by Peltier cooling and electromigration of solute species towards the substrate due to the applied electric current to the solution. The Peltier cooling at the interface induces the concentration gradient in the solution resulting in supersaturation at the interface. In addition, the electromigration of solute species towards the substrate induces the supersaturation at the interface. The two dimensional transient mass transport equation for the electroepitaxial system can be written as  2    @ Ci @2 Ci @Ci @Ci @Ci @Ci @Ci Di þv  mi E þ þ 2 þu ¼ 2 @X @Y @X @Y @X @Y @t (3) where Di is the diffusion coefficient, mi the mobility, Ci the concentration of the solute species (i = As), u and v are the velocity components in X and Y direction in the solution. X is the direction parallel to the plane of interface towards the solution and Y is the direction perpendicular to the plane of interface towards the solution. The sign of the electromigration term is determined by the direction of the current flow. It is well known from the LPEE experiments that the epitaxial film growth rate is low compared to the mean drift velocity of the solute particles in the liquid phase [6] u and n  mE

(4)

So neglecting the velocity terms in Eq. (3), the equation becomes  2    @ Ci @2 Ci @Ci @Ci @Ci Di þ þ 2  mi E ¼ @X 2 @Y @X @Y @t

(5)

First term on the left hand side of Eq. (5) contributes to the diffusion, the second term contributes to the electromigration. The mass transport equation is subjected to the

D. Mouleeswaran, R. Dhanasekaran / Materials Science & Engineering B 112 (2004) 91–95

following boundary conditions: eq

at t ¼ 0; C i ¼ C0 for all X and Y eq at t > 0; C i ¼ C0 for X ¼ 30e

(6.a) (6.b)

Y ¼ 30e ðabsence of convectionÞ or at

t > 0 and X ¼ 0; eq Ci

Y ¼ 0;

tration profile in the solution, one can write Ci (X, Y, t) = Ci (j, k, n), where X = ej and Y = ek, t = tn, t is the successive time increment, j and k are the segment numbers in the X and Y directions and n is the number of time cycles that have occurred. The solution in front of the crystal interface has been divided into 30  30 segments of equal width e. The initial concentration of As in the Ga rich solution has been calculated from [11].

(6.c)

¼ C1 ðgrowth follows the liquidus lineÞ

C0 is taken as the equilibrium concentration of As in the GaAs solution at a given growth temperature and e the width of the segment. Using the phase diagram, C1 is determined for the interface temperature. The governing transient mass transport equation with appropriate boundary conditions is solved numerically by central difference scheme. To construct the solute concen-

93

C ¼ 4:82  104 exp

  8:42  ð1:32  104Þ T

(7)

4. Growth rate Growth rate (R) depends on the number of solute atoms flowing towards the growth interface and is given by, in the

Fig. 1. Concentration profile surfaces of As in the Ga rich solution at different Peltier cooling conditions: (a) 1 8C; (b) 2 8C; (c) 3 8C; (d) 4 8C.

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absence of convection [9], R¼

  DTP dCi Di 1=2  mi E CS  Ci ð0; tÞ dT pt L   Ci ð0; tÞ  CS  Ci ð0; tÞ

(8)

The first term on the right hand side of Eq. (8) represents the contribution of the Peltier effect and the second term represents the electromigration of the solute atoms to the growth velocity, both terms depend linearly on the current density through DTP and E, respectively. CS is the concentration in the solid and Ci (0, t) is the concentration of solute species at the interface. The thickness of the epitaxial layer at any given time can be obtained by integrating the growth rate.

5. Results and discussions

Fig. 2. Growth rate for a –1 8C Peltier cooling at various growth times.

Numerical analysis is a very promising tool and thus reduces the number of experiments and consequently the cost of the optimising process. Simulating the concentration profiles of solute species in the solution during the LPEE growth is essential to understand the growth kinetics of this technique. Using the phase diagram, functions have been generated to describe the equilibrium concentration of the solute atoms as a function of temperature. The initial growth temperature is taken as 800 8C. Concentration of solute atoms at successive increments of time t is calculated using the computer simulation technique. To simulate the numerical values, the value of t has been chosen as one second for satisfying the stability conditions and for the accuracy of the numerical analysis. The solution width is 0.6 cm along X-axis and 0.6 cm along Y-axis. The diffusion coefficients and mobility of As in Garich melt are [11] DAs = 6  105 cm2/s and mAs = 2.7  102 cm2/V s, respectively. Fig. 1a–d show the concentration profile surfaces of As in Ga rich solution during the electroepitaxial growth of GaAs at different Peltier cooling conditions. The temperature at substrate–solution interface decreases due to Peltier cooling. With the growth of the epilayer due to Peltier cooling, there is a decrease in the concentration of the solute species near the interface. In addition to Peltier cooling, the epitaxial growth depends on the electromigration of the solute species towards the substrate by the applied electric current. Along the substrate–solution interface, the temperature decreases from the outer wall to center of the growth cell results a nonuniform temperature distribution at the growing interface [5,17,18]. Fig. 2 shows growth rate for a 1 8C Peltier cooling at various growth times along the substrate. The concentration gradients present along the interface due to Peltier effect and electromigration cause non-uniform growth due to the nonuniform temperature distribution along the substrate. The

growth rate at the cell wall is lower than that at the cell center and both decrease with time. The growth rate attains saturation after a certain time, which indicates that the electromigration is dominant. Fig. 3 shows the growth rate along the substrate for different Peltier cooling at the interface. As the Peltier cooling is increased at the interface, the growth rate of the epitaxial layer is enhanced because of the strong concentration gradient at the interface. Fig. 4 shows the thickness of the epitaxial layer as a function of growth time. At the center, the thickness of the epitaxial layer is constant due to electromigration dominant growth. Fig. 5 shows the growth rate for different current densities. A linear variation in the growth rate of the epitaxial

Fig. 3. Growth rate vs. the distance along the substrate at various Peltier coolings.

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6. Conclusion Investigation on the growth kinetics of GaAs during the liquid phase electroepitaxy has been carried out by developing a model based on the two dimensional diffusion and electromigration of the solute species. A numerical simulation technique has been employed to construct the concentration profiles of the solute species in the solution with the influence of Peltier cooling and electromigration. In the absence of convection, mass transfer by electromigration plays an important role for the growth of the epitaxial layer. The non-uniform growth rate over the substrate is obtained due to non-uniform temperature distribution in the solution. The resulting growth velocity is proportional to the current density. The growth rate determined is compared with the experimentally reported values.

Fig. 4. Calculated thickness of the epitaxial layer as a function of growth time.

Fig. 5. Growth rate for various current densities J.

layer is observed due to increase in the applied current density. The present theoretical findings have been compared with experimental results of Bryskiewicz and co-workers [3,19] which show a good agreement.

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