two-dimensional numerical calculation of solute diffusion in microchannel epitaxy of inp

*Corresponding author. Present address: 1st SH Microcom-puter Development Department Semiconductor & IntegratedCircuits Division, Hitachi Ltd., 5-20-1 Josuihon-cho, Kodaira-shi, Tokyo 187-8588, Japan.

E-mail address: [email protected] (Z. Yan)

Journal of Crystal Growth 209 (2000) 1}7

Two-dimensional numerical calculation of solute di!usion inmicrochannel epitaxy of InP

Zheng Yan*, Shigeya Naritsuka, Tatau Nishinaga

Department of Electronic Engineering, The Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113, Japan

Received 29 July 1999; accepted 1 October 1999Communicated by K.W. Benz

Abstract

To understand the vertical and the lateral growth mechanisms of microchannel epitaxy (MCE) in detail, two-dimensional numerical calculation is carried out employing the di!erent boundary conditions on the top and the sidesurfaces of the MCE island. By solving a two-dimensional di!usion equation, a concentration pro"le of the solute isdetermined as a function of growth time. By calculating the lateral growth rate, the width-to-thickness ratio of the MCEisland (=/¹ ratio) has been derived from the computation. The calculated=/¹ ratio shows a good agreement in bothcooling rate and growth temperature dependence with those obtained from the experiment when the growth temperatureis above 5003C. ( 2000 Elsevier Science B.V. All rights reserved.

PACS: 78.66.Fd; 81.15.Lm

Keywords: Microchannel epitaxy; Solute di!usion; Near-surface di!usion;=/¹ ratio

1. Introduction

Increased research e!orts have been devoted tothe reduction of dislocation in heteroepitaxy. In theprevious research, we have proposed microchannelepitaxy (MCE) which consists of growth througha microchannel cut in SiO

2"lm on substrate and

epitaxial lateral overgrowth (ELO) [1]. This tech-nique was found to be one of the most promisingtechniques to reduce the dislocation density inGaAs grown on Si [2].

In the MCE, as the propagation of dislocationsin the substrate is stopped by SiO

2mask, the

laterally grown area becomes dislocation free. Toget a wide dislocation free area, thin and wide MCElayer should be grown. By choosing the orientationof microchannels in o!-direction to a certain low-index orientation, one can make the side surface ofgrown layer through the microchannel atomicallyrough. Hence, by using the substrate with a low-index plane, it is possible to get faster growth in thelateral direction compared to the vertical direction[1]. The MCE has also been applied to grow InPon Si substrates to obtain the layer with low dislo-cation density [3]. Recently, we also studied thegrowth mechanism of the MCE intensively basedon the experiments, and reported the results of thestep sources [4], the interface supersaturation

0022-0248/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.PII: S 0 0 2 2 - 0 2 4 8 ( 9 9 ) 0 0 5 3 9 - 4

Fig. 1. A schematic illustration of the meshes employed fornumerical calculation. In the "gure, d indicates the size of themeshes.

dependence of the growth [5] and the step velocity[6] in the growth of the InP MCE.

To improve the understandings of the growthmechanism, we have carried out a numerical calcu-lation of the solute di!usion in the MCE of InP byLPE. So far, calculations of the concentrations ingrowth solution employing one-dimensional [7]and two-dimensional [8] di!usion equations havebeen reported. In the current study, we have em-ployed the two-dimensional model and applied itto the MCE.

In this paper, we will describe the details for thealgorithm as well as the results of the calculation.The calculation has been divided into two parts,one is for the MCE that occurs only in the verticaldirection inside the microchannel; the other is forthe MCE that occurs in both the vertical and thelateral directions. We will "rst describe the resultsand the analysis of the concentration pro"les in thegrowth solution. Then, by calculating the growthrate in the lateral direction, we have also deter-mined the width-to-thickness ratio of the MCEisland, which is de"ned as=/¹ ratio. The growthtemperature and the cooling rate dependence of thecalculated=/¹ ratio will be described. Finally, wewill also discuss the comparison of the experi-mental and calculated results.

2. Algorithm for numerical calculation of theconcentration in the MCE growth

A two-dimensional solution has been dividedinto i]j segments of equal width d as shown inFig. 1. The algorithm of the method is to start withan initial concentration C(x, y, t"0)"C

0for all

segments, and to calculate successive concentrationpro"les after each time increments q. C

0denotes

the equilibrium concentration at the growth start-ing temperature. One can write C(x, y, t)"C(i, j, n),where x"di, y"dj and t"qn (i and j are thesegment numbers and n is the number of time cyclesthat have occurred).

The two-dimensional di!usion equation is given as

RCRt "DA

R2CRx2

#

R2CRy2 B, (1)

where D denotes the di!usion coe$cient of phos-phorus in the indium solution. Thus, the change inthe concentration in i, jth segment during time q canbe written as

Ci,j,n`1

!Ci,j,n

q"DA

Ci~1,j,n

!2Ci,j,n

#Ci`1,j,n

(*x)2

#

Ci,j~1,n

!2Ci,j,n

#Ci,j`1,n

(*y)2 B(2)

or when *x"*y"d as,

Ci,j,n`1

"Ci,j,n

#

Dqd2

(Ci~1,j,n

#Ci`1,j,n

#Ci,j~1,n

#Ci,j`1,n

!4Ci,j,n

). (3)

Dq/d2 is a factor determining the computationalerror. It is necessary to choose the values of q andd for a given value of D in such a way as to makeDq/d2(1

4[9].

Let C%

and q denote the equilibrium concentra-tion of phosphorus in indium solution as a functionof the growth temperature and the #ux into thecrystal derived from the vertical growth rate of the

2 Z. Yan et al. / Journal of Crystal Growth 209 (2000) 1}7

Fig. 2. A contour pro"le showing the calculated concentrationof the solute in the MCE where growth occurs only inside themicrochannel. The growth temperature, the cooling rate and thegrowth time were 5503C, 0.13C /min and 120 s, respectively.

MCE, respectively. The boundary conditions forthe di!usion equation can be expressed as follows.

At the boundaries,(a) between the solution and a crystal with rough

surface,

Ci,j,n

"C%,

(b) between the solution and a crystal withfaceted surface,

DRCRy"q,

(c) between the solution and the oxide mask,

DRCRy"0,

should be held. From the phase diagram of phos-phorus in indium solution [10] and from the verti-cal growth rate measured experimentally, C

%and

q can be derived respectively. By employing thesevalues in the calculation, the two-dimensional dif-fusion equation can be solved numerically and theconcentration pro"le of phosphorus in indiumsolution can be obtained as a function of growthtime.

In the following, we will describe how to deter-mine the lateral growth rate. The lateral growthoccurs on a rough side surface in the current MCE,and its growth rate can be derived from the concen-tration gradient near the surface. The #ux of thesolute towards the growing interface, q, is given by

q"DAdC

dxBx/0

"lLC

4(4)

or

lL"

D

C4

Ci"#1, j!Ci

", j

d, (5)

where lL

and i"

denote, respectively, the lateralgrowth rate and the i-axis index for the segmentsadjacent to the lateral growing surface. By compar-ing the growth rates in the lateral and verticaldirections, the=/¹ ratio can be determined.

The size of the mesh, the di!usion coe$cient ofphosphorous in the indium solution, and the value

of the successive time increment were chosen, resp-ectively, as d"0.5 or 1 lm, D"5]10~5 cm2 s~1

[11] and q"1]10~5 s in the calculation.

3. Growth within microchannel

In the initial growth stage of the MCE, only thevertical growth in the microchannel occurs. A con-tour diagram of the concentration pro"les in thesolution after 120 s of real time growth is shown inFig. 2. The growth temperature and the coolingrate were chosen to be 5503C and 0.13C/min, re-spectively. From the experimental result with thesame conditions, the vertical growth rate or the #uxq entering the facet surface of the MCE layer wasdetermined and used as a parameter in the calcu-lation. Since the direction of the #ux is parallel tothe gradient of the concentration, the concentrationcontour shows that solute di!uses in the solution tothe growing surface. From the contour pro"le closeto the microchannel, a #ux from the edge to thecenter of the microchannel can be noted, which isillustrated by the arrow in Fig. 2.

Z. Yan et al. / Journal of Crystal Growth 209 (2000) 1}7 3

Fig. 3. (a) A contour pro"le showing the concentration of thesolute in the MCE where the growth occurs in vertical andlateral directions. The growth temperature, the cooling rate andthe growth time were selected as 5003C, 0.13C/min and 500 s,respectively. (b) A schematic illustration of the near-surfacedi!usion at the growth interfaces.

The formation of a facet inside the microchannelmeans that the growth velocity has the same valueeverywhere in the microchannel. When a certainpoint on the facet has an excess supply, instead ofextra growth, the supply has to be di!used away. Inthe selective area growth such as MCE, excesssupply occurs near the edges of the growing area,which is known as the edge e!ect [8]. This excesssupply near the edges will be transported to thecenter where the supply has a lower value. The #owseen in Fig. 2 is of this sort of transportation.

It is worth pointing out here that, although sur-face di!usion is not expected to occur in solutiongrowth like LPE, a bulk di!usion of the soluteoccurs from the points with higher supersaturationto those with lower supersaturation in the region ofthe solution near the growing surface. We namethis di!usion as `near-surface di!usiona. The di!u-sion coe$cient in bulk solution has a value of theorder of 10~4}10~5 cm2 s~1, which is approxim-ately 103}104 times higher than those of surfacedi!usions reported in vapor growth. Therefore, thenear-surface di!usion is more e$cient in keepingthe homogeneity of the supply at the growing inter-face. As a result, atomically #at surfaces can beobtained in the LPE-MCE.

4. Growth of MCE islands

When the grown layer becomes thicker than theoxide "lm, the lateral growth will commence. Byapplying di!erent boundary conditions on the topand the side surfaces of the MCE island, the calcu-lation for anisotropical growth of the MCE be-comes possible. In this section, "rst, we will discussthe concentration pro"le in the growth solutionwhen lateral growth is taken into consideration.Second, we will describe the result of the calculatedgrowth rate in the lateral direction. Third, a com-parison of the=/¹ ratio with those obtained fromthe experiments will be described and discussed.

4.1. Concentration proxle of phosphorus in indiumsolution

Fig. 3a shows the contour pro"le of concentra-tion distribution in the growth solution after 500 s

of real time growth. The growth temperature andcooling rate were selected as 5003C and 0.13C/min,respectively. The density of the contour lines islarger in the region close to the lateral interfacethan that close to the vertical interface, which indi-cates a faster growth in the lateral direction. More-over, from the concentration pro"le close to theinterfaces, the near-surface di!usion has been foundagain, however, it has an opposite direction to thatdescribed in Section 3, where the growth occursonly in the vertical direction. In the current MCE,the near-surface di!usion has a direction fromthe facet to the rough side surfaces, as depicted inFig. 3b. This should be attributed to the fastergrowth in the lateral than in the vertical directions.


Fig. 4. Position dependence of the growth amount on the sidesurface, where the inset shows how the positions were indexed.The size of the mesh was 0.5 lm. The growth temperature andthe cooling rate were 5003C and 0.13C/min, respectively. Themarks of star and dot stand for the calculations after 500 and1000 s of the growth, respectively.

Fig. 5. An SEM photograph of the cross section of MCE island.Excess amount of growth can be observed on the side surfacenear the top corner.

Because the side surface is atomically rough,interface supersaturation is almost zero. Thismeans that the lateral growth is governed only bydi!usion. Therefore, with the aid of the near-surfacedi!usion that brings solute from the smooth surfaceto the rough surface, the growth in the lateral andthe vertical directions will be, respectively, en-hanced and reduced. It is apparent that the near-surface di!usion contributes to the enhancement ofthe=/¹ ratio.

4.2. Lateral growth rate

By employing Eq. (5), the growth rate in thelateral direction can be determined from the cal-culated concentration pro"le. Fig. 4 shows the re-sult of the position dependence of the growthamount on the side surface, where the inset showshow the positions were indexed in the lateral direc-tion. The growth amount has been found to in-crease with the growth time. It can also be foundfrom Fig. 4 that the growth amount is not uniformon the side surface. The growth has a larger growthamount in the region near the top surface than thatclose to the oxide mask. The higher growth rate

near the top surface is due to the high supply fromtwo kinds of di!usions. One is the near-surfacedi!usion that brings solute from the faceted surfaceto the side surface; the other one is the bulk di!u-sion towards the edge of unmasked area in normalmask epitaxy [8].

This result agrees very well with the experiment.Fig. 5 shows an SEM photograph of the crosssection of a MCE island obtained from the experi-ments. Excess growth near the corner of the topsurface can be seen clearly.

4.3. Calculation of W/T ratio

In this section, the cooling rate dependence of thecalculated=/¹ ratio will be described "rst. Then,we will discuss the growth temperature dependenceof the =/¹ ratio.

4.3.1. Cooling rate dependence of W/T ratioKeeping the growth temperature at 5503C, we

have calculated the =/¹ ratio as a function ofcooling rate. Fig. 6 shows the result. The calculated=/¹ ratio has been found to increase with thedecrease of the cooling rate. This dependence issimilar to that obtained from the experiment. Be-cause the lateral growth on rough surface is mainlycontrolled by the di!usion process, at high temper-ature like 5503C, the calculation agrees very well


Fig. 6. Cooling rate dependence of the calculated =/¹ ratio.The circles and the squares denote the calculation and theexperiment, respectively.

Fig. 7. Growth temperature dependence of the=/¹ ratio. Thecircles and the squares denote the calculation and the experi-ment, respectively.

Fig. 8. Growth temperature dependence of the growth rate inthe lateral and vertical lateral directions. The circles and thesquares denote the calculated and the experimental growthrate in the lateral direction, respectively. The triangles show theexperimental growth rate in the vertical direction.

with the real growth. In the next section, we will seea di!erent result when growth temperature de-creases.

4.3.2. Growth temperature dependence of the W/Tratio

Keeping the cooling rate at 0.13C/min, wechanged growth temperature and calculated the=/¹ ratio. Fig. 7 shows the growth temperature de-pendence of the =/¹ ratio obtained from the cal-culation and the experiment. The calculated =/¹ratio has been found to increase signi"cantly withthe decrease of the growth temperature.

Comparing with the experimental data, we can seethat the calculation agrees well with the experimentin the high-temperature region; while, the =/¹ratio obtained from experiment becomes smallerthan the calculated results when the growth tem-perature is lower than 5003C. To "nd out the rea-son for the di!erence between the experiment andthe calculation, we have separately plotted the lat-eral and the vertical growth rates in Fig. 8. Inthe "gure, the lateral growth rate obtained from thecalculation can be regarded as independent of thegrowth temperature. However, the experimentalgrowth rates both in the lateral and the verticaldirections decrease with the decrease of temper-ature. In the low growth temperature range, thelateral growth rate obtained from the experimentbecomes much smaller than that from the calcu-lation.

In our model of the calculation, the lateralgrowth is assumed being governed by the bulkdi!usion, which is correct only when the growthtemperature is high. When the growth temperaturebecomes low, the growth kinetic begins to play animportant role, and the lateral growth rate becomessmall. However, in the current calculation, thegrowth kinetics was not taken into consideration;as a result, the calculation disagrees with the ex-periment in the low-temperature range. We canconclude that, above 5003C, the lateral growth is


mainly controlled by bulk di!usion, however, whenthe temperature drops, the contribution of thegrowth kinetics becomes signi"cant.

5. Conclusions

In this paper, we have described the numericalcalculation of the solute concentration in the solu-tion of InP MCE by LPE. First, the modeling toimplement the calculation has been described. Byemploying di!erent boundary conditions on thetop and the side surfaces of the MCE island, a two-dimensional di!usion equation has been solved nu-merically and a concentration pro"le of the growthsolution has been derived.

Second, from the concentration pro"le, the direc-tion of solute #ux has been determined. It wasfound that a strong di!usive #ow is induced fromthe top to the side surfaces of the MCE island tocompensate the di!erence in interface supersatura-tion.

Third, the distribution of the growth rate on theside surface has been calculated. It was found thatthe growth rate has a larger value in the region nearthe corner of the top surface of the MCE island. Wesuggest that this should be attributed to the exist-ence of the near-surface di!usion as well as the edgee!ect of bulk di!usion.

Finally, the cooling rate and the growth temper-ature dependences of the calculated =/¹ ratiohave been described. It was found that the coolingrate dependence of the calculated=/¹ ratio agreeswell with those obtained from the experiments.However, the growth temperature dependence ofthe calculated=/¹ ratio disagrees with the experi-ments when growth temperature is low. This isattributed to the fact that under low temperature,the contribution of the growth kinetics becomes

signi"cant, which has not been taken into accountin the present calculation.

Acknowledgements

This work was supported by JSPS Research forthe Future Program in the Area of Atomic-ScaleSurface and Interface Dynamics under the project`Self-assembling of Nanostructure and its Con-trolsa and Scienti"c Research (B) `Growth of Dis-location Free GaAs on Si by Microchannel Epitaxyand Fabrication of Laser Diodea No. 10555119from the Ministry of Education, Science, Sportsand Culture of Japan. The authors would like tothank Dr. M. Tanaka for his discussion. One of theauthors (Z. Yan) would like to acknowledge thesupport of the Research Fellowship of the JapanSociety for the Promotion of Science for YoungScientists.

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two-dimensional numerical calculation of solute diffusion in microchannel epitaxy of inp

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