S D Das Centre of Excellence for Green Energy and Sensor Systems,BESU, India

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1 IOSR Jounal of Electical and Electonics Engineeing (IOSR-JEEE) e-issn: ,p-ISSN: , Volume 9, Issue 1 Ve. IV (Feb. 2014), PP Full wafe 3D modelling of powe distibution duing micowave annealing of doped c-si S D Das Cente of Excellence fo Geen Enegy and Senso Systems,BESU, India Abstact: Application of micowave fo apid annealing puposes have been ecently intoduced fo silicon and apat fom many advantages that it enjoys, it has also povide avenue to apid ecystallisation and eduction of defects. One of the applications of such technology is micowave annealed e-cystallised sola cells. In this wok, modelling of micowave powe spatial distibution in doped cystalline silicon (c-si) is pesented fo advancement of e-cystallised sola cell technology. Helmholtz equation was solved fo detemination of electic field distibution inside the mateial system using 3D exact Finite Diffeence Method. Subsequently, micowave powe distibution was obtained using application of Poynting powe theoem. All esults pesented in this pape coesponds to Industial, Scientific and Medical (ISM) fequency of 2.45 GHz. Keywods: Micowave, c-si, annealing, 3D modeling, efdm I. INTRODUCTION Micowave annealing is useful in semiconducto annealing pocesses because of extemely apid ise in tempeatue to vey high values. A typical apid annealing cycle using micowave adiation last fo few seconds and could aise the tempeatue in the ange of 2000 o C with in faction of a second depending upon fequency and mateial [1]. Othe advantages of micowave annealing in semiconducto pocessing includes selective heating, defect elimination in ion implanted silicon, and dopant activation [2-3]. Micowave annealing is a sustainable technology and it is paticulaly useful fo ecystallised sola cells [4]. Lage bandgap and naow bandgap semiconducto pocessing using micowave annealing has been epoted [5-6], but not widely. Dependence of diffeent paametes, such as dimension and composition of sample, mode of incident adiation etc. have been studied in these woks. A full wafe 3D modelling of system will be pesented in this wok to model the powe distibution duing micowave annealing. This distibution is esponsible fo tempeatue distibution inside a c-si wafe duing micowave annealing which is cucial fo unifom ecystallisation fo ecystallised sola cells. The distibution patten depends on standing wave patten fomation of micowave adiation inside the sample. Fequency of micowave along with dimensions of heating chambe and sample, play cucial pat in this egad. Magnitude of ise in tempeatue and its distibution due to absoption of micowave enegy also depends on local mateial popeties. In case of naow bandgap non-pola semiconducto such as c-si dielectic constant is such mateial popety. Dielectic constant of c-si depends on may diffeent physical mechanisms and which mechanism will dominate is function of fequency. Thus powe distibution is dependent on many diffeent physical paametes and ae sensitive to diffeent paametes by diffeent amount. To model the distibution accuately Helmholtz equation needs to be solved with good degee of accuacy. Many diffeent techniques ae available fo such pupose and one popula method is Finite Diffeence Method (FDM). Most FDMs intoduces tuncation eos. A ecent technique Exact Finite Diffeence Method (efdm) has been intoduced in 2D by Wong et. al. (2011), which do not intoduce tuncation eos [7]. In this wok we investigate dependence of absobed micowave powe distibution in doped (c-si) at ISM fequency of 2.45 GHz at 300K. Fo such pupose 3D efdm technique will be intoduced fist followed by modelling esults. II. THEORY In non-pola semiconductos such as c-si the mechanism of absoption at ISM fequency is pimaily fee-caie absoption which depends upon doping levels of the mateial [8]. Powe absobed is calculated using Poynting powe theoem [9]. Fo non-magnetic mateials micowave powe absobed in a unit volume is given by [10]: Q ω ε o ε c ξ 2 (1) 68 Page

2 Full wafe 3D modelling of powe distibution duing micowave annealing of doped c-si z y X Wafe Base Fig.1. Schematic of c-si wafe placed on a micowave tanspaent plate inside a micowave oven. whee, ε o is pemittivity of fee space, ε c is elative imaginay pat of complex dielectic function and ξ is incident micowave electic field at fequency ω. Electic field distibution of incident micowave adiation inside a mateial is govened by Maxwell's equations. The electic field distibution in equation (1) is thus obtained by solving Helmholtz equation [10]: 2 ξ+θ2 ξ = 0 (2) whee, Θ 2 =ω 2 ε o μ o ε is the popagation constant and ε is the elative complex dielectic function. In this study we conside a squae c-si wafe placed on a tanspaent plate inside a micowave oven (Fig. 1). The oven walls ae made of metal and hence we conside electic field to be zeo at the bounday. Thus fo ξ= f (x, y, z) bounday condition can be given by : wee, L coesponds to maximum values in the diection of x, y and z. ξ x, y, z =0, L = 0 (3) III. EXACT FINITE DIFFERENCE METHOD In this section we descibe the numeical method fo solution of (2). efdm scheme enables solution to be found numeically without tuncation eo. The exact FDM scheme is descibed by Wong et. al and is based on Lambe et. al. method of using weight to modify the cental diffeence opeato. Wong et. al. desciption is fo one-dimensional (1D) and two-dimensional (2D) cases. In this wok the mathematics is extended to 3D and adopt bounday condition descibed by (3) instead of adiation bounday condition. The discete equation coesponding to (2) using Finite Diffeence Method is: ξ(x+m, y, z) 2ξ(x, y, z)+ξ(x m, y, z) ξ(x, y+n, z ) 2 ξ(x, y, z)+ξ(x, y n, z ) + m 2 n 2 ξ(x, y, z+o) 2 ξ(x, y, z)+ξ(x, y, z o) + +Θ 2 ξ=0 o 2 wee, standad cental diffeence fomula have been used and intevals in x-, y- & z-diections have m, n and o spatial step sizes, espectively. Hee, we conside the geneal case of m n o. Using the weight facto W in (4) gives: ξ(x+m, y, z)+ξ(x m, y, z ) ξ(x, y+n, z)+ξ(x, y n, z) ξ(x, y, z+o)+ξ(x, y, z o) + + m 2 n 2 o 2 2W [ 1 m 2+ 1 n o 2] ξ+θ 2 ξ=0 Let Θ(x, y, z)=(θ 1, θ 2, θ 3 ) and solution be of the fom ξ=ξ o e i (θ 1x+θ 2 y+θ 3 z), then (5) gives: 2cos(θ 1 m) W m 2 + 2cos(θ 2 n) W n 2 Thus, fom (6) the weight facto is obtained as (4) (5) + 2cos(θ 3 o) W o 2 +Θ 2 =0 (6) W = 2 [n2 o 2 cos(θ 1 m)+m 2 o 2 cos(θ 2 n)+m 2 n 2 cos(θ 3 o)]+(mno Θ) 2 n 2 o 2 +m 2 o 2 +m 2 n 2 (7) 69 Page

3 Full wafe 3D modelling of powe distibution duing micowave annealing of doped c-si Using (5) and (7), we can obtain unique solution fo electic field ξ= f (x, y, z). To establish uniqueness of the solution Uniqueness theoem with poof has been povided below. Uniqueness Theoem: By using efdm scheme the Helmholtz equation in 3D 2 ξ (x, y,z)+θ 2 ξ( x, y, z) t with bounday conditions ξ x, y, z =0, L = 0 has a unique solution. = 0 (8) Poof: Let thee be two solutions w and v. Also, u = w -v such that it satisfies the equation: which can be witten as: u(x+m, y, z)+u(x m, y, z) u(x, y+n, z )+u(x, y n, z) + m 2 n 2 +u(x, y,z +o)+u(x, y, z o) o 2ω[ 1 2 m n ] o u+θ 2 u=0 2 + y 2 + z 2 4ω[ 1 m 2+ 1 n o 2] +Θ 2 =0 foi=1, 2,...,m ; j=1,2... n; k =1,2....o Multiplying (10) by v ijk and summing up ove i, j and k, we get ( 2 + y 2 + z 2 ) v ijk [ ( 4ω ) +Θ2] m n o v ijk =0 (9) (10) (11) Using discete integation by pats, along with conditions : the (11) becomes: 2 v ijk = fo i=1, 2,..., m; m,n, o y 2 v ijk = i, j,k=1 z 2 v ijk = i, j, k=1 j=1,2... n ; k=1,2... o v ijk 1 n,o m x u mjk v mjk (12) j, k =1 m, o y y v ijk 1 n i,k=1 m,n y u ink v ink (13) z z v ijk 1 o z u ijo v ijo (14) i, j=1 v ijk = fo i=1,2...,m ; j=1,2...,n ; k =1,2...,o =0 fo i=0, m ; j=0,n ; k=0, o + m,n, o y y + i, j, k=1 i, j, k=1 z z (15) [ ( 4ω m n 2+ 1o 2) +Θ2] =0 (16) Let v ijk be of the fom v ijk = a 1 mi + a 2 nj + a 3 ok + a 4, then fist tem in (16) becomes: = =a 1 and this is tue fo y and z tems as well in (16). Thus, (a 1 mi+a 2 nj+a 3 ok+a 4 ) n, o =a 1 (u m jk u 0 jk )=0 (17) j, k =1 Futhe, since w and q ae non zeo tems, then fo any positive intege l, [ 4 ω( ) +Θ2] m n o =0 (18) 70 Page

4 Full wafe 3D modelling of powe distibution duing micowave annealing of doped c-si By mathematical induction fo l =1,2,3,... l v ijk =0 (19) 1 (l+1) 2 x v l+1 ijk =0 (20) 1 (l+1)(l+2) 2 x v l+2 ijk =0 (21) l v ijk =0 (22) Thus, it can be concluded that = 0, completing the poof of uniqueness theoem. IV. RESULTS AND DISCUSSIONS The spatial distibution of absobed micowave powe inside a metal cavity (micowave oven) was obtained by solving the 3D discete equation, (5), inside the metal cavity. Full system modeling was caied out by using spatial filteing on 3D matix, which defines the system. The wok was caied out on a 32-bit compute. Fo implementation the system matix was conveted to vecto and an indexing technique defined by: l=i ( j 1)+i+( IJ )(k 1), l=1,2,3... IJK (23) was used. Hee, i, j and k ae indices fo x-, y- and z-diections espectively and I, J & K ae coesponding maximum values. This appoach eliminates nested loop equiement fo 3D systems and educes complexity of the algoithm. Assumption was made that micowave souce delives powe which has a spectum defined by: P ( f )=A exp( f f o 2 σ ) 2 (24) Micowave souce spectum vay fom instument to instument and hence theoetical distibution was consideed fo quick modeling. Howeve, autho consides measued spectum [9] to be the best souce fo accuate modeling. The theoetical spectum defined by (24) is shown in Fig.2, whee, f = 2.45 GHz, σ 2 =8.86x10 10 and A = W. The distibution coesponds to 800W of total powe. The initial spatial distibution of available powe fo a given fequency was assumed unifom inside the cavity and the incident 2 flux was obtained using I =0.5cε o ξ o. Fig.2 Theoetical powe distibution of micowave souce. Micowave powe distibution inside an ai field cavity with dimensions cm 3 fo diffeent fequencies ae shown in Fig. 3. Results include hoizontal (Fig. 3 1(a) 4(a)) and vetical coss sections (Fig. 3 1(b) 4(b)) of the excited cavity. These esults ae fo 60 iteations with 15 points pe wavelength accuacy. The spatial distibution is not same fo all fequencies suggesting not all fequencies ae well suppoted by the cavity. Fig.4 povides peak powe (P p ) spectum of the excited cavity, showing fo some fequencies powe is moe concentated in spatial distibution than othes. 71 Page

5 Full wafe 3D modelling of powe distibution duing micowave annealing of doped c-si Fo micowave annealing puposes in a multi-mode micowave (such as house hold micowave oven), modeling only sample which is to be heated with appopiate bounday conditions would not poduce accuate esults of heating distibution inside a sample. Standing wave pattens inside a cavity is guided by the dimensions and popeties of the cavity and sample placed inside it [9]. Hence, in this wok full system modeling was caied out. A system of dielectic slab placed at the bottom cente of the cavity was consideed (Fig. 1). Dimensions of the oven cavity was kept as ealie, while dimensions of a slab was cm 3. Results of modeling ae shown in Fig. 4, fo two cases whee complex elative dielectic constant (ε c, en ) of cavity envionment is less than complex elative dielectic constant of slab (ε c, slab. Clealy, fo heating pupose ε c, en <ε c, slab ) and ε c, en geate than ε c, slab must be maintained, othe wise most micowave enegy will be concentated outside the slab. Unde this condition sample get heated without heating the ambient. This condition has significant impact on the choice of envionment with in the cavity fo heating. Choice of envionment thus have to be both non-oxidising to the sample and satisfy the above condition fo efficient >ε c, slab silicon pocessing. On the othe hand fo ε c, en most of micowave enegy will be outside the sample. This condition will heat the sample but with less efficiency. 1 (a) 1 (b) 2 (a) 2 (b) 3 (a) 3 (b) 4 (a) 4 (b) Fig. 3 Micowave powe distibution in (a) hoizontal and (b) vetical coss section of an ai filled cavity fo (1) f = 2.4 GHz, (2) f = 2.45 GHz, (3) f = 2.47 GHz and (4) f = 2.5 GHz. 72 Page

6 Full wafe 3D modelling of powe distibution duing micowave annealing of doped c-si 1 (a) 1 (b) 2 (a) 2 (b) Fig. 4 Absobed micowave powe distibution in (a) hoizontal and (b) vetical coss section of an ai filled cavity with a dielectic slab at bottom cente at f = 2.4 GHz fo (1) ε c, en <ε c, slab and (2) ε c, en >ε c, slab. Fig. 5 Peak powe (P p ) spectum of an cavity of dimension cm 3 filled with ai. Fom Fig. 5, two fequencies coesponding to highest peak powe (f = 2.45GHz) and lowest peak powe ( f = 2.4 GHz) was chosen and it was found that absobed powe distibution patten does not vay significantly with fequency with in the sample (Fig.6). This is due to vey small diffeence in peak powe values, which is of the ode of 0.1 times the peak powe values. Howeve, the distibution patten vaies significantly with position of the slab, with dimensions of cavity kept constant. Fom Fig. 6 and Fig. 7, it is evident pattens inside the sample ae significantly influenced by the standing wave pattens due to eflection fom cavity walls. Futhe, dimensions of the sample plays cucial pat in distibution patten as well. Compaing Fig. 6 and Fig. 8, this fact is diectly obsevable. Sample dimensions fo Fig. 6 esult is cm 3 while that of Fig. 8 is cm 3. Fig.9 shows vaiation of powe distibution with input total powe. The distibution is not significantly influenced by the input powe. Howeve, amplitude of the peak powe is dependent on the total input powe and thus total input powe will be significant fo contolling tempeatue inside the sample. The complex dielectic function ε fo c-si is dependent on fee-caie absoption and is given by Dude's model [8, 11]: ε=ε +i ε c =ε coe + 4π i [ n e e 2 τ e ω m e (1 i ω τ e ) + n h e 2 τ h )] (25) m h (1 i ω τ h Fo n-type c-si with, suffix 'e' stands fo majoity caie electon while suffix 'h' stands fo minoity caie hole. In (25), n is fee-caie concentation, m is effective mass and τ is elaxation time constant. Fig. 10 shows dependence of imaginay pat of dielectic function ( ε c ) on doping density (N D ) at 300K. The doping species fo this case is phosphoous. At 300K, powe absoption distibution fom full system modeling of ai filled cavity with a c-si wafe ( cm 3 and N D = 1x10 20 cm 3 ) on boosilicate glass slab ( Page

7 Full wafe 3D modelling of powe distibution duing micowave annealing of doped c-si cm 3 ) at bottom cente is shown in Fig. 11. The simulation was caied out with 15x15x1000 points pe wavelength. Fig. 12 shows esults fo simila conditions as Fig. 11 but with wafe oientation changed fom hoizontal flat position to vetical upight position. With vetical up ight position the powe distibution inside the wafe is significantly non unifom as compaed to hoizontal flat position. 1 (a) 1 (b) 2 (a) 2 (b) Fig. 6 Absobed micowave powe distibution in (a) hoizontal and (b) vetical coss section of an ai filled cavity with a dielectic slab at bottom cente at (1) f = 2.45 GHz and (2) f = 2.4 GHz. 1 (a) 1 (b) 2 (a) 2 (b) Fig. 7 Absobed micowave powe distibution in (a) hoizontal and (b) vetical coss section of an ai filled cavity with a dielectic slab at (1) bottom font and (2) bottom left. f = 2.45 GHz. (a) (b) (c) Fig. 8 Absobed micowave powe distibution in (a) hoizontal, (b) vetical coss sections of an ai filled cavity with a dielectic slab ( cm 3 ) at bottom cente. (c) zoomed in vetical coss section of slab. f = 2.45 GHz. 74 Page

8 Full wafe 3D modelling of powe distibution duing micowave annealing of doped c-si (a) (b) (c) Fig. 9 Absobed micowave powe distibution in an ai filled cavity with a dielectic slab ( cm 3 ) at bottom cente. Total input powe (a) 800 W, (b) 1000 W and (c) 10,000 W. All the figues show zoomed esults. f = 2.45 GHz. Fig. 10 Dependence of imaginay pat of dielectic function ( ε c ) on dono density (N D ) fo c-si at 300K. (a) (b) Fig. 11 Absobed micowave powe distibution in an ai filled cavity with a c-si wafe on boosilicate glass slab at bottom cente. (a) Hoizontal and (b) vetical (zoomed) coss sections. f = 2.45 GHz. (a) (b) Fig. 12 Absobed micowave powe distibution in an ai filled cavity with a c-si wafe on boosilicate glass slab at bottom cente with wafe in vetical position. (a) Hoizontal (zoomed) and (b) vetical coss sections. f = 2.45 GHz.. 75 Page

9 Full wafe 3D modelling of powe distibution duing micowave annealing of doped c-si V. CONCLUSION In conclusion, 3D modeling esults fo micowave powe distibution in phosphoous doped c-si wafe has been pesented fo ISM fequency of 2.45 GHz at 300K. A new 3D modeling technique, efdm, has been intoduced fo such pupose which has no tuncation eo. Results fom modeling show inefficient powe distibution with in the sample fo ectangula cavity (house hold micowave oven). This shows cavity shape and dimension needs to be edesigned to make the annealing pocess efficient and unifom fo e-cystallized sola cell wok. Acknowledgments This wok was caied out as Reseach Associate at CEGESS, BESU unde the DST Poject entitled 'Sola Photovoltaic Hub at BESU''. Autho would like to thank institute, univesity and funding agency fo thei full suppot and encouagement. REFERENCES [1] M. V. Rao, Ulta-fast Micowave Heating fo Lage Bandgap Semiconducto Pocessing, Advances in Induction and Micowave Heating of Mineals & Oganic Mateials ISBN: , 2011, 459. [2] R. N. P. Vemui, M. J. Gade, N. D. Theodoe, and T. L. Alfod, Dopant Activation in Asenic -Implanted Si by Suscepto- Assisted Low-Tempetue Micowave Anneal, IEEE Electon Device Lettes 32, 2011, [3] S. C. Fong, C. Y. Wang, T. H. Chang, and T. S. Chin, Cystallization of amophous Si film by micowave annealing with SiC susceptos, Applied Physics Lettes 94, 2009, [4] S. D. Das, Pospects of Micowave Heating in Silicon Sola Cell Fabication A Review, IOSR-JEEE, 6(3), 2013, [5] J. N. Lee, Y. W. Choi, B. J. Lee and B. T. Ahn, Micowave-induced low-tempeatue cystallization of amophous silicon thin films, Jounal of Applied Physics, 82(6), 1997, [6] Md. R. Hossan, D-Y Byun, P. Dutta, Analysis of micowave heating fo cylindical shaped objects, Intenational Jounal of Heat and Mass Tansfe, 53, 2010, [7] Y. S. Wong and G. Li, Exact Finite Diffeence Schemes Fo Solving Helmholtz Equation At Any Wavenumbe, Intenational J. of Numeical Analysis and Modelling, Seies B, 2 (1), 2011, [8] D. C. Dibben, Numeical and Expeimental Modelling of Micowave applicatos (Univesity of Cambidge, 1995). [9] A. K. Shukla, A. Mondal and A. Upadhyay, Numeical Modelling of Micowave Heating, Science of Sinteing, 42, 2010, 99. [10] T Basak, Role of lateal and adial iadiations on efficient micowave pocessing of food cylindes, Chemical Engineeing Science, 62, 2007, [11] T. Sameshima, H. Hayasaka and T. Haba, Analysis of Micowave Absoption Caused by Fee Caies in Silicon, Japanese Jounal of Applied Physics, 48, 2009, Page