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Formation and growth of porous silicon
Ndm
Pergaarnon
Mat&k, Vol. 8.No. 4. pp.507-520.1997 EI8cviaScimccLtd Q 1997Ada Mualllugia Inc. RintedintheusA. AlIrights-cd 096.5-9773197 $17.00+ .oo
PII SO9659773(97)00190-6
FORMATION AND GROWTH OF POROUS SILICON R.M. Vadjikar, A.K. Nath and A.N. Chandorkar* Center for Advanced Technology, Indore - 452 013, India *Indian Institute of Technology, Bombay - 400 076, India (AcceptedMay 13,1997) Abstract-The conformalhullfinitediffusionlengthmodel(FDL) simulatesporeformation and growthin siliconby launchingparticlesfrom an isoconcentration profile. Severalfeatures of the aggregationpatternsgenerated by thismodel resemble experimentallyobserved morphology ofporous silicon. In this paper we considersiliconatomdissolutionby a twoparticle aggregation algorithm.Theprobabilitiesofsiteoccupationhavebeen assignedbased on thelocal electricfield. The releaseprobabilityofparticleshasbeen consideredtobe proportionaltotheelectricfield. The incorporationofthesefactorsgeneratesaggregation patterns which are similartothosegenerated by other similar models. Wesuggest thatsuch modificationsgenerate aggregationpatterns that are representativeof porous silicon morphology.0 1997 Acta MetallurgicaInc.
1. INTRODUCTION Nanostructured porous silicon has attracted considerable attention because of the potential for realizing a silicon based luminescence source (1). The demonstrated performance of a new capacitor technology (2) exploits the large surface area associated with porous silicon. The emerging technology of vacuum microelectronics could utilize the nanometer silicon morphology on the silicon emitters (3). Biocompatible implants based on porous silicon hold great promise (4). The nanometer size morphology and photoluminescence of porous silicon has been extensively investigated (5i,6) by Raman spectroscopy. The optical properties of porous silicon have been suggested to be due to silicon complexes (7,8), defect states in nanocrystals (9) and several other explanations (LO,11). The modleling of formation and growth of porous silicon is important (12,13,14) because of the complexity of actual fabrication process and the need for a theoretical framework to understand porous silicon morphology. In spite of considerable attention, the mechanism of formation and pore growth in silicon is not well understood. The finite diffusion length (FDL) model (15,16,17) is a stochastic model, which explains many aspects of experimentally observed features of porous silicon. Some of these features are: the constant pore density in the steady state growth regime, the self-limiting nature of pore growth in the interior regions of pores, and the preferential growth of pores near the tips. 507
508
flid
VADJIKAR,
AK ~TH
AND
AN
hANDOAKAR
In the FDL model, aggregation patterns obtained for small finite diffusion length can be correlated with the experimentally observed cross-sectional transmission electron microscopy observations of nanostructured porous silicon formed on ptype silicon, while the patterns for large finite diffusion length are similar to porous silicon formed on n-type silicon (18). This model is similar to the diffusion limited aggregation @LA) model (19,20,21), which simulates phenomena such as dendritic solidification, electrodeposition, viscous fingering and thin lihn morphology. The result of these growth or aggregation phenomena are fractal clusters, which have power law correlations over a range of length scales. The DLA model is based on simple features such as Brownian motion of particles and aggregate formation and connectivity, and it is able to exhibit long range fractal behavior. Other formation models for porous silicon have been investgated by several researchers. The model suggested by Kang et al. (22) considers pore formation to be due to instability of silicon surfacein theelectrolytetosmallperturbations. Theperturbationcanberepresentedby theFourier component, y = E exp (iox + j3t) where x, o and p are the x-coordinate, the frequency and the amplification factor respectively. It has been shown that some specific values of spatial frequencies correspond to maximum values of the amplification factor. Thus, certain pore separations are predicted for a set of experimental parameters such as electrolyte composition, doping concentration in silicon, and anodic over potential.The distance between pores is predicted to vary as the square root of applied potential. Parlchutik et al. (23) have suggested that pore formation is due to the electric field enhancement at the pore bases resulting in growth of a passive layer. The passive layer then dissolves resulting in pore growth. Erlebacher et al. (24) report on the dynamics of pore formation in silicon by Monte Carlo simulation of an ensemble of electronic holes on a two-dimensional lattice. The motion of holes was biased to the nearest pore tip by the construction of a probability circle. In this paper we present a porous silicon formation model which is based on two particle aggregation for silicon atom dissolution. The basis for this is the mechanism of pore formation and growth in silicon, which suggests that two electrons are involved in interfacial charge transfer leading to silicon atom dissolution (18). The electric field effect has been considered within the isoconcentration profile, and the diffusional field effects are assumed to be important outside this region. The motion of particles within the isoconcentration profile is dependent on the site occupation probability, which is decided by local electric field effects. 2. DESCRIPTION
OF THE MODEL
In this section we discuss the essential features of a simple Monte Carlo technique based porous silicon formation model, which simulates pore growth in silicon by aggregation patterns of particles released from an isoconcentration profile. Initially, the silicon-electrolyte interface is flat and all the surface sites are considered potential sites. An electric field is applied in the region within the isoconcentration profile. We assume that the motion of the particles in this region is governed by local electric field effects. The injection of holes is simulated by release of a single particle from the isoconcentration profile. The release probability from any particular site from the isoconcentration profile is biased by the local electric field. Such biasing of release probability promotes hole injection from abrupt discontinuities in the isoconcentration profile, which have high local electric fields. After the release of a particle from the isoconcentration profile, the
FORMATION ANDGmwr~ OF Pomus SILWN
509
motion of the particle is executed by selection of one site from the set of sites around it. The probabilities of ltbe site selection are biased according to the local electric fields in the direction of each surrounding site. The motion of each particle is terminated when it contacts apotential site. Afta another particle reaches this site it becomes part of the growing aggregate. The nearest surrounding available sites become additional potential sites for subsequent particles. The isoconcentration profile is also modified so that a fmed distance is maintained from the growing aggregate.
3. RESULTS AND DISCUSSIONS The resulls of the simulations performed in planar geometry using 100 X 100 square lattice with periodic boundary conditions in the x-direction are discussed, The computations have been performed for three diffusion lengths of ‘3’, ‘Sand ‘10’. In the initial period of stochastic growth, the region within the isoconcentration profile can be considered to be similar to the concept of ‘migrational envelope’ introduced by Erlebacher ef al. (25). The potential distribution is found within this region by the solution of Laplace’s equation:
V2$ = 0
[ll
The site selection during the process of aggregation is related to the local electric field Ei by the following probability relation:
Pi =Mod(Ei)/CMod(Ei)
i
PI
where Ei = -V+, and 41is the potential obtained from the solution of the Laplace’s equation. The probability that the growth of the aggregate would happen at a particular cluster site ‘k’ is related to the: local electric field at ‘k’. The growth probability ‘G(k)’ can be described by:
[31
where E(k) is the local electric field. The integral is taken over the entire surface ‘s’ of the cluster. The parameter ‘v’ describes different models. The aggregate is maintained at a fixed distance of the chosen finite diffusion length. In the region enclosed by the isoconcentration profile, the released particles move under the influence of an electric field. Outside this region the particles are in the diffusional region and follow random walks. In many respects, our model is similar that reported by Erlebacher ef al. (24); though in their approach the motion of the particles in the high field region is biased so that the trajectories of motion are pointed to the nearest pore tip. This has been achieved by constructing a circle from the probability of motion in each direction.
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Figure l(a). Pattern with two particle aggregation for a finite diffusion length ‘3’.
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Figure l(b). F%ttemwith two particle aggregation for a finite diffusion length ‘5’.
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Figure l(c). Pattern with two particle aggregation for a finite diffusion length ‘10’.
FORMATKIN ANDGROWTH OFPOROUS SUCQN
................
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Figure 2!(a). Pattern with one particle aggregation for a finite diffusion length ‘3’.
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Figure 2(b). Pattern with one particle aggregation for a finite diffusion length ‘5’.
FORMATION ANDGIWWTH OFPOROUS SILICON
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Figure 2(c). Pattern with one particle aggregation for a finite diffusion length ‘10’.
516
RM ~~MKAR, AK NWII AND AN CHANDORI(AR
The results of our work are shown in the aggregationpatterns of Figures l(a), l(b) and l(c). These were generated for three tinite diffusion lengths of ‘3’, ‘5’ and ‘10’. The growth process begins by considering the first row of 100X 100lattice as the set of possible sites. The particles released from the isoconcentration profile move towards the possible sites by the process of site selection under the influence of electric field. When two particles reach a possible site, that site becomes part of the growing cluster. All the available sites around this site become additional possible sites. Figures 2(a), 2(b) and 2(c) are the aggregationpatterns that are generated for finite diffusion lengths of ‘3’, ‘5’ and ‘lo’, in the case of random motion and one particle aggregation algorithm. Figure 3 shows the double log plot of numberof aggregatingparticlesversus distance for two particle aggregation. The central regions correspond to the constant porosity region of the aggregation patterns. This is similar to the plots in Figure 5 for the case of one particle aggregation without considerations of electric field. The slope is approximatelyequal to 1,which suggests that these patterns are non-fractal. The comparison of Figures 4 and 6 shows that the trend is similar in density variation. The constant density values in two particle aggregation show interesting differences from the single particle aggregationprocess. The difference in the constant density is largest for the smaller finite diffusion length of ‘3’. The two particle aggregationalgorithm appears to generate morphologies with more porosity. At the larger finite diffusion length of ‘lo’, the difference in constant density is the least.
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L,, 0 fdl-10 fcV-5 fdl-3 Figure 4. Plot of density versus diskce for two”particleaggregation.
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Figure 6. Plot of density versus distance for one particle aggregation. For a qualitative description of porous silicon growth under the influence of applied electric field, an expression for the hole diffusion length is discussed (26). We consider a constant electric field applied to a n-type sample, which has been doped uniformly. The field is applied normal to the flat faces of the silicon wafer. Optical excitation generates electron-hole pairs at the back surface of the wafer. Theelectric field vector is oriented such that the minority carriers move under the influence of the diffusion and drift field in the direction of the bulk of silicon wafer. The electrons are swept into the back surface contact. Based on the hole flux and continuity equation, we can write:
where Sp is the excess hole concentration, D, is the hole diffusion constant,E is the applied electric field, t.+ is the mobility of the holes, G is the electron hole pair generation rate and rp is the relaxation time of holes. If we assume a steady state optical flux on a strongly absorbing surface, the field induced by the excess carriers will be small in comparison with the applied field. Considering the electric field to be oriented normal to the surface, equation [4] can be written as:
The solution of this equation is given by:
FORMATION AND GRAPH OFPOROUS SIUCON
519
WI When the applied electric field is zero, the solution is given by: 6p(x) = Gp(O)exp[-x / (Dr,$)1’2].
[71
Then a parameter called ‘hole diffusion length’ can be defined: L, = (D~J,)“~
.
PI
The excess hole concentration declines due to recombination with the majority carriers at a characteristic length Lp. This can be considered as the basis for selection of a constant isoconcentration length in the FDL simulations. This is the characteristic length in which the excess holes diffuse and recombine with the majority carrier electrons. It is possible to show that the distribution of excess holes, under the assumption of large electric field, is given by: Sp(x) = 6p(O)exp(-x / j.$,tPE).
[91
This shows that the characteristic diffusion length, in the presence of applied field, is larger and is equal to pr,r,E. When the applied field is not very large the diffusion length lies in between these two extremes. Further improvements in the FDL model are needed for incorporation of these fundamental issues to explain porous silicon growth
4. CONCLUSIONS The incorporation of two particle aggregation during growth of aggregation patterns leads to different values of constant density. The difference in density is largest for the smaller finite length. These modifications in the finite diffusion length algorithm are expected to provide a better understanding of dynamic processes such aspore growth during porous silicon formation. The two particle aggregation algorithm considers capturing of two holes by a silicon atom, which are required for its dissolution, and, hence, the morphologies generated by this algorithm would be more representative of porous silicon formation and growth.
ACKNOWLEDGMENTS The authors thank Dr. D.D. Bhawalkar, Center for Advanced Technology, Indore, India, for support.
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REFERENCES 1.
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.
Collins, R.T., Fauchet, PM., and Tischler, M.A., Physics Today, Jan. 1997.24. Lehmann, V., Honlein, W., Reisinger, H., Spitzer, A., Wendt, H., and Wiler, J., Solid State Technology. No.v 1995,99. Boswell. E.C., Huang, M., Smith, G.D.W., and Wilshaw, P.R., Journal of Vacuum Science und Technology, 1996, B 14(3), 1895. Knott, M., New Scientist, 1997,29,36. Deb, S.K., Indian Journal of Pure and Applied Physics, 1994,32,638. Sood, A.K., Jayaram, K., and Muthu, D.V.S., Journal ofApplied Physics, 1993,73, 1%3. Narsimhan, K.L., Banerjee, S., Srivastava, A.K., and Sardesai, A., Applied Physics Letters, 1993, 62,331. Banerjee, S., Narsimhm, K.L., Ayyub, P., Srivastava, A.K., and Sardesai, A., Solid State Communications, 1992,84,691. Sinha, S., Banerjee, S. and Arora, B.M., Physical Review, 1994, B(49), 5706. Lehman, V., and Gosele, U., Applied Physics Letters, 1991,58,856. Brand& B.S., Breitschwerdt, A., Fuchs, H.D., Hopner, A., Rosenbauer, M., Stutzmanu, M. and Weber, J., Applied Physics, 1992, A54,567. Weng, Y.M., Qiu, J.Y., Zhou, Y.H., Zong, X.F., Journal ofVacuum Science and Technology, 1996, B 14(4), 2505. John, G.C. and Singh, V.A., Physical Review, 1995, B52,1125. John, G.C., and Sir@, V.A., Physical Review, 1994, B50.5329. Smith, R.L., Chuang, S.F. and Collins, S.D., Journal of Electtvnic Materials, 1988,17,533. Vadjikar, R.M., Chandorkar, A.N., Sharma, D. and Venkataehalam, S., Nunostructured Materials, 19955,273. Smith, R.L. and Collins, S.D., Physical Review, 1989, A39,5409. Smith, R.L. and Collii S.D., Journul of Applied Physics, 1992,71, Rl. Witten, T.A. and Sander, L.M., Physical Review Letters, 1981,47,1400. Meakin, P., Physical Review, 1983, A 27,604. Witten, T.A. and Sander, L.M., Physical Review, 1983, B 27,5686. Kang, Y., and Jome, J., Journal of Electrochemical Society, 1993,140,2258. Parkhutik, V., and Shershulsky, V., Journal of Physics D: Applied Physics, 1992,25,1258. Erlebacher, J., Sieradzki, K., and Searson, PC., Journal of Applied Physics, 1994.76.182. Erlebacher, J., Searson, P.C., and Sieradzki, K., Physical Review Letters, 1993,71,3311. Wolfe, C.M., Holonyak, N., aud Stillman, G.E., Phy sical Properties of Semiconductors, PrenticeHall, Inc., New Jersey, 1989, p. 253.
Pergaarnon
Mat&k, Vol. 8.No. 4. pp.507-520.1997 EI8cviaScimccLtd Q 1997Ada Mualllugia Inc. RintedintheusA. AlIrights-cd 096.5-9773197 $17.00+ .oo
PII SO9659773(97)00190-6
FORMATION AND GROWTH OF POROUS SILICON R.M. Vadjikar, A.K. Nath and A.N. Chandorkar* Center for Advanced Technology, Indore - 452 013, India *Indian Institute of Technology, Bombay - 400 076, India (AcceptedMay 13,1997) Abstract-The conformalhullfinitediffusionlengthmodel(FDL) simulatesporeformation and growthin siliconby launchingparticlesfrom an isoconcentration profile. Severalfeatures of the aggregationpatternsgenerated by thismodel resemble experimentallyobserved morphology ofporous silicon. In this paper we considersiliconatomdissolutionby a twoparticle aggregation algorithm.Theprobabilitiesofsiteoccupationhavebeen assignedbased on thelocal electricfield. The releaseprobabilityofparticleshasbeen consideredtobe proportionaltotheelectricfield. The incorporationofthesefactorsgeneratesaggregation patterns which are similartothosegenerated by other similar models. Wesuggest thatsuch modificationsgenerate aggregationpatterns that are representativeof porous silicon morphology.0 1997 Acta MetallurgicaInc.
1. INTRODUCTION Nanostructured porous silicon has attracted considerable attention because of the potential for realizing a silicon based luminescence source (1). The demonstrated performance of a new capacitor technology (2) exploits the large surface area associated with porous silicon. The emerging technology of vacuum microelectronics could utilize the nanometer silicon morphology on the silicon emitters (3). Biocompatible implants based on porous silicon hold great promise (4). The nanometer size morphology and photoluminescence of porous silicon has been extensively investigated (5i,6) by Raman spectroscopy. The optical properties of porous silicon have been suggested to be due to silicon complexes (7,8), defect states in nanocrystals (9) and several other explanations (LO,11). The modleling of formation and growth of porous silicon is important (12,13,14) because of the complexity of actual fabrication process and the need for a theoretical framework to understand porous silicon morphology. In spite of considerable attention, the mechanism of formation and pore growth in silicon is not well understood. The finite diffusion length (FDL) model (15,16,17) is a stochastic model, which explains many aspects of experimentally observed features of porous silicon. Some of these features are: the constant pore density in the steady state growth regime, the self-limiting nature of pore growth in the interior regions of pores, and the preferential growth of pores near the tips. 507
508
flid
VADJIKAR,
AK ~TH
AND
AN
hANDOAKAR
In the FDL model, aggregation patterns obtained for small finite diffusion length can be correlated with the experimentally observed cross-sectional transmission electron microscopy observations of nanostructured porous silicon formed on ptype silicon, while the patterns for large finite diffusion length are similar to porous silicon formed on n-type silicon (18). This model is similar to the diffusion limited aggregation @LA) model (19,20,21), which simulates phenomena such as dendritic solidification, electrodeposition, viscous fingering and thin lihn morphology. The result of these growth or aggregation phenomena are fractal clusters, which have power law correlations over a range of length scales. The DLA model is based on simple features such as Brownian motion of particles and aggregate formation and connectivity, and it is able to exhibit long range fractal behavior. Other formation models for porous silicon have been investgated by several researchers. The model suggested by Kang et al. (22) considers pore formation to be due to instability of silicon surfacein theelectrolytetosmallperturbations. Theperturbationcanberepresentedby theFourier component, y = E exp (iox + j3t) where x, o and p are the x-coordinate, the frequency and the amplification factor respectively. It has been shown that some specific values of spatial frequencies correspond to maximum values of the amplification factor. Thus, certain pore separations are predicted for a set of experimental parameters such as electrolyte composition, doping concentration in silicon, and anodic over potential.The distance between pores is predicted to vary as the square root of applied potential. Parlchutik et al. (23) have suggested that pore formation is due to the electric field enhancement at the pore bases resulting in growth of a passive layer. The passive layer then dissolves resulting in pore growth. Erlebacher et al. (24) report on the dynamics of pore formation in silicon by Monte Carlo simulation of an ensemble of electronic holes on a two-dimensional lattice. The motion of holes was biased to the nearest pore tip by the construction of a probability circle. In this paper we present a porous silicon formation model which is based on two particle aggregation for silicon atom dissolution. The basis for this is the mechanism of pore formation and growth in silicon, which suggests that two electrons are involved in interfacial charge transfer leading to silicon atom dissolution (18). The electric field effect has been considered within the isoconcentration profile, and the diffusional field effects are assumed to be important outside this region. The motion of particles within the isoconcentration profile is dependent on the site occupation probability, which is decided by local electric field effects. 2. DESCRIPTION
OF THE MODEL
In this section we discuss the essential features of a simple Monte Carlo technique based porous silicon formation model, which simulates pore growth in silicon by aggregation patterns of particles released from an isoconcentration profile. Initially, the silicon-electrolyte interface is flat and all the surface sites are considered potential sites. An electric field is applied in the region within the isoconcentration profile. We assume that the motion of the particles in this region is governed by local electric field effects. The injection of holes is simulated by release of a single particle from the isoconcentration profile. The release probability from any particular site from the isoconcentration profile is biased by the local electric field. Such biasing of release probability promotes hole injection from abrupt discontinuities in the isoconcentration profile, which have high local electric fields. After the release of a particle from the isoconcentration profile, the
FORMATION ANDGmwr~ OF Pomus SILWN
509
motion of the particle is executed by selection of one site from the set of sites around it. The probabilities of ltbe site selection are biased according to the local electric fields in the direction of each surrounding site. The motion of each particle is terminated when it contacts apotential site. Afta another particle reaches this site it becomes part of the growing aggregate. The nearest surrounding available sites become additional potential sites for subsequent particles. The isoconcentration profile is also modified so that a fmed distance is maintained from the growing aggregate.
3. RESULTS AND DISCUSSIONS The resulls of the simulations performed in planar geometry using 100 X 100 square lattice with periodic boundary conditions in the x-direction are discussed, The computations have been performed for three diffusion lengths of ‘3’, ‘Sand ‘10’. In the initial period of stochastic growth, the region within the isoconcentration profile can be considered to be similar to the concept of ‘migrational envelope’ introduced by Erlebacher ef al. (25). The potential distribution is found within this region by the solution of Laplace’s equation:
V2$ = 0
[ll
The site selection during the process of aggregation is related to the local electric field Ei by the following probability relation:
Pi =Mod(Ei)/CMod(Ei)
i
PI
where Ei = -V+, and 41is the potential obtained from the solution of the Laplace’s equation. The probability that the growth of the aggregate would happen at a particular cluster site ‘k’ is related to the: local electric field at ‘k’. The growth probability ‘G(k)’ can be described by:
[31
where E(k) is the local electric field. The integral is taken over the entire surface ‘s’ of the cluster. The parameter ‘v’ describes different models. The aggregate is maintained at a fixed distance of the chosen finite diffusion length. In the region enclosed by the isoconcentration profile, the released particles move under the influence of an electric field. Outside this region the particles are in the diffusional region and follow random walks. In many respects, our model is similar that reported by Erlebacher ef al. (24); though in their approach the motion of the particles in the high field region is biased so that the trajectories of motion are pointed to the nearest pore tip. This has been achieved by constructing a circle from the probability of motion in each direction.
510
RM VADJIKAR,
AK NATHAND AN
&tANDORlWl
* : : . : : : : : : : : : : : :
l.
.A Lee
.* se”” l
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. .. . .
A
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. *-
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.
lt
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.
.
: . T.:.
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.
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. .
. . ..
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. . 60
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Figure l(a). Pattern with two particle aggregation for a finite diffusion length ‘3’.
511
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l
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_________________________---________________-------------_____---.------.-------_-.---.______._ 100 I lo 40 40 lo $I lo to L b
Figure l(b). F%ttemwith two particle aggregation for a finite diffusion length ‘5’.
512
RM VA~WUR,AK NmdAND AN &lANCQRKAR
Figure l(c). Pattern with two particle aggregation for a finite diffusion length ‘10’.
FORMATKIN ANDGROWTH OFPOROUS SUCQN
................
I
1._________.____.________________________________________________________________________________
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lo
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Figure 2!(a). Pattern with one particle aggregation for a finite diffusion length ‘3’.
514
RM VADJIKAR,
AK NAM AND AN
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Figure 2(b). Pattern with one particle aggregation for a finite diffusion length ‘5’.
FORMATION ANDGIWWTH OFPOROUS SILICON
515
-70
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Figure 2(c). Pattern with one particle aggregation for a finite diffusion length ‘10’.
516
RM ~~MKAR, AK NWII AND AN CHANDORI(AR
The results of our work are shown in the aggregationpatterns of Figures l(a), l(b) and l(c). These were generated for three tinite diffusion lengths of ‘3’, ‘5’ and ‘10’. The growth process begins by considering the first row of 100X 100lattice as the set of possible sites. The particles released from the isoconcentration profile move towards the possible sites by the process of site selection under the influence of electric field. When two particles reach a possible site, that site becomes part of the growing cluster. All the available sites around this site become additional possible sites. Figures 2(a), 2(b) and 2(c) are the aggregationpatterns that are generated for finite diffusion lengths of ‘3’, ‘5’ and ‘lo’, in the case of random motion and one particle aggregation algorithm. Figure 3 shows the double log plot of numberof aggregatingparticlesversus distance for two particle aggregation. The central regions correspond to the constant porosity region of the aggregation patterns. This is similar to the plots in Figure 5 for the case of one particle aggregation without considerations of electric field. The slope is approximatelyequal to 1,which suggests that these patterns are non-fractal. The comparison of Figures 4 and 6 shows that the trend is similar in density variation. The constant density values in two particle aggregation show interesting differences from the single particle aggregationprocess. The difference in the constant density is largest for the smaller finite diffusion length of ‘3’. The two particle aggregationalgorithm appears to generate morphologies with more porosity. At the larger finite diffusion length of ‘lo’, the difference in constant density is the least.
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Figure 6. Plot of density versus distance for one particle aggregation. For a qualitative description of porous silicon growth under the influence of applied electric field, an expression for the hole diffusion length is discussed (26). We consider a constant electric field applied to a n-type sample, which has been doped uniformly. The field is applied normal to the flat faces of the silicon wafer. Optical excitation generates electron-hole pairs at the back surface of the wafer. Theelectric field vector is oriented such that the minority carriers move under the influence of the diffusion and drift field in the direction of the bulk of silicon wafer. The electrons are swept into the back surface contact. Based on the hole flux and continuity equation, we can write:
where Sp is the excess hole concentration, D, is the hole diffusion constant,E is the applied electric field, t.+ is the mobility of the holes, G is the electron hole pair generation rate and rp is the relaxation time of holes. If we assume a steady state optical flux on a strongly absorbing surface, the field induced by the excess carriers will be small in comparison with the applied field. Considering the electric field to be oriented normal to the surface, equation [4] can be written as:
The solution of this equation is given by:
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Then a parameter called ‘hole diffusion length’ can be defined: L, = (D~J,)“~
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The excess hole concentration declines due to recombination with the majority carriers at a characteristic length Lp. This can be considered as the basis for selection of a constant isoconcentration length in the FDL simulations. This is the characteristic length in which the excess holes diffuse and recombine with the majority carrier electrons. It is possible to show that the distribution of excess holes, under the assumption of large electric field, is given by: Sp(x) = 6p(O)exp(-x / j.$,tPE).
[91
This shows that the characteristic diffusion length, in the presence of applied field, is larger and is equal to pr,r,E. When the applied field is not very large the diffusion length lies in between these two extremes. Further improvements in the FDL model are needed for incorporation of these fundamental issues to explain porous silicon growth
4. CONCLUSIONS The incorporation of two particle aggregation during growth of aggregation patterns leads to different values of constant density. The difference in density is largest for the smaller finite length. These modifications in the finite diffusion length algorithm are expected to provide a better understanding of dynamic processes such aspore growth during porous silicon formation. The two particle aggregation algorithm considers capturing of two holes by a silicon atom, which are required for its dissolution, and, hence, the morphologies generated by this algorithm would be more representative of porous silicon formation and growth.
ACKNOWLEDGMENTS The authors thank Dr. D.D. Bhawalkar, Center for Advanced Technology, Indore, India, for support.
520
RhdVAWIKAR, AK Nmi
AND
AN CHANDORKAR
REFERENCES 1.
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.
Collins, R.T., Fauchet, PM., and Tischler, M.A., Physics Today, Jan. 1997.24. Lehmann, V., Honlein, W., Reisinger, H., Spitzer, A., Wendt, H., and Wiler, J., Solid State Technology. No.v 1995,99. Boswell. E.C., Huang, M., Smith, G.D.W., and Wilshaw, P.R., Journal of Vacuum Science und Technology, 1996, B 14(3), 1895. Knott, M., New Scientist, 1997,29,36. Deb, S.K., Indian Journal of Pure and Applied Physics, 1994,32,638. Sood, A.K., Jayaram, K., and Muthu, D.V.S., Journal ofApplied Physics, 1993,73, 1%3. Narsimhan, K.L., Banerjee, S., Srivastava, A.K., and Sardesai, A., Applied Physics Letters, 1993, 62,331. Banerjee, S., Narsimhm, K.L., Ayyub, P., Srivastava, A.K., and Sardesai, A., Solid State Communications, 1992,84,691. Sinha, S., Banerjee, S. and Arora, B.M., Physical Review, 1994, B(49), 5706. Lehman, V., and Gosele, U., Applied Physics Letters, 1991,58,856. Brand& B.S., Breitschwerdt, A., Fuchs, H.D., Hopner, A., Rosenbauer, M., Stutzmanu, M. and Weber, J., Applied Physics, 1992, A54,567. Weng, Y.M., Qiu, J.Y., Zhou, Y.H., Zong, X.F., Journal ofVacuum Science and Technology, 1996, B 14(4), 2505. John, G.C. and Singh, V.A., Physical Review, 1995, B52,1125. John, G.C., and Sir@, V.A., Physical Review, 1994, B50.5329. Smith, R.L., Chuang, S.F. and Collins, S.D., Journal of Electtvnic Materials, 1988,17,533. Vadjikar, R.M., Chandorkar, A.N., Sharma, D. and Venkataehalam, S., Nunostructured Materials, 19955,273. Smith, R.L. and Collins, S.D., Physical Review, 1989, A39,5409. Smith, R.L. and Collii S.D., Journul of Applied Physics, 1992,71, Rl. Witten, T.A. and Sander, L.M., Physical Review Letters, 1981,47,1400. Meakin, P., Physical Review, 1983, A 27,604. Witten, T.A. and Sander, L.M., Physical Review, 1983, B 27,5686. Kang, Y., and Jome, J., Journal of Electrochemical Society, 1993,140,2258. Parkhutik, V., and Shershulsky, V., Journal of Physics D: Applied Physics, 1992,25,1258. Erlebacher, J., Sieradzki, K., and Searson, PC., Journal of Applied Physics, 1994.76.182. Erlebacher, J., Searson, P.C., and Sieradzki, K., Physical Review Letters, 1993,71,3311. Wolfe, C.M., Holonyak, N., aud Stillman, G.E., Phy sical Properties of Semiconductors, PrenticeHall, Inc., New Jersey, 1989, p. 253.