Univiy of Toldo REU Pogam Summ 2002 Th Effc of Shadowing in 2-D Polycyallin Gowh Jff Du Advio: D. Jacqu Ama Dpamn of Phyic, Univiy of Toldo, Toldo, Ohio Abac Th ffc of hadowing in 2-D hin film gowh w udid uing a pu dpoiion modl. Boh coin and unifom angula diibuion w udid. Alo udid w h ffc of an infac diffuion bai. W find n = /3 and b =, wh n i h mound coaning xponn and b i h gowh xponn, in h ca of a coin diibuion wih diffuion bai, and n = b = in all oh ca. W find h fac lngh coaning xponn d = 0.8 fo unifom angula diibuion. In h ca of coin diibuion wihou diffuion bai d = 0.8. Moov, w find a pow law diibuion of id lngh of mound in boh coin and unifom angula diibuion whn a diffuion bai i abn. Ploing a cald gaph of (N)(S av )/N T v. /S av wh N i h numb of id, S av i h avag id lngh, N T i h oal numb of id, and i id lngh. Fo a unifom diibuion, h xponn = 0.7. Fo a coin diibuion, = 0.44. Fo a unifom diibuion boh wih and wihou a diffuion bai a wll a a coin diibuion wihou bai w find h ufac oughning xponn a =.0. In h ca of coin diibuion wih diffuion bai, anomalou caling of id lngh wa found yilding a = 0.88. Mo complx modl in h dimnion, pcially h coin-bai ca, could povid a vy phyical modl of xpimnal pu dpoiion hin film gowh.
I. Inoducion Thin film gowh ha bcom a vy popula fild of ach in h la fw ya, pimaily du o h numou applicaion hof. Thin film gowh fnc a numb of chniqu in which film a gown by adding on aom a a im of a cain lmn o a uba, ofn caing uful im uch a miconduco. Many modl of hin film gowh hav bn udid, and all ag ha, a im pa, h film will dvlop mound ha coan wih im. Fig how xpimnal ul of mound fomaion in Cu. Th mound a ofn an unwand fau, howv. If gowing a miconduco, o caing a hin film wih infac bwn diffn lay of diffn ubanc, a unifom ufac i did. Thi nd fo a mooh ufac ha givn i o h udy of h mchanic of mound gowh in hin film. Mound coaning ha bn udid quaniaivly uing Kinic Mon Calo imulaion. Fom h imulaion, i wa found ha h coaning of a ym could b udid by mauing h m widh W 2 (L,) = <(h- <h>) 2 > () wh L i ym iz, i im, and h i high. Widh ypically vai accoding o im a W(L,) ~ b (2) wh b i h ufac gowh xponn. A wll a mauing b, h imulaion ofn alo mau n, h mound coaning xponn, and d, h id lngh coaning xponn. Almo all pviou Kinic Mon Calo imulaion hav hown h xponn o b bwn /4 and /3.
Howv, n and b hav bn maud xpimnally, uing a pu dpoiion gowh poc, o b 0.7 o 0.8. Th ul w conidd o b du o an ffc calld hadowing. Shadowing occu whn aom a bough o h hin film a lag angl, uling in lag mound ha domina h ym. Fig 2 how an xampl of hadowing. W hav cad a imulaion ha incopoa hadowing o y and ca h xpimnal ul. Fig. Expimnal gowh of Cu film gowh howing mound fomaion. Fig 2. Th lag mound domina h ym, hadowing h mall mound.
II. Mhod Wha balliic and andom dpoiion modl andomly lc a column o i o augmn, ou pogam imula pu dpoiion by gnaing a andom angl hough which h aom avl. Ou pogam u a qua laic o cod h pah of h aom. In od o duc complxiy, h pogam aum h aom occupi h cn of a qua if i ouch any pa of ha qua. Sinc ou modl i a ingl p modl, w aum ha an aom in fligh occupi wo qua: h on i i in and h on dicly blow i. Thi wa ncay o allow h minima of h ym o gow. To duc fini iz ffc, ou imulaion u piodic bounday condiion. In od o all ca, ou modl conain boh unifom angula diibuion, wh all angl a qually pobabl, and coin angula diibuion, wh lag angl a l likly. Moov, ou pogam alo incopoa an infac diffuion bai, which can b und on o off. Fig 3 dmona hi diffuion bai. Figu 3a how dpoiion in h abnc of hi bai whil Figu 3b how dpoiion wih h bai und on. P L = D R /(D R + D L ) D L D R Fig 3a. Wih infac diffuion bai und off, h dpoid paicl go igh o lf wih pobabiliy dmind by i dianc fom ih minima. Fig 3b. Wih h bai und on, h dpoid paicl can only mov down h id o h na minima.
III. Rul A ym of 3,000 igh wa gown o 0,000 lay in ach of h fou ca. Daa wa akn fo id lngh, widh, and high vy 0 lay and id-idcolaion, id lngh diibuion, and h G2 colaion funcion vy 2000 lay. IIIA. No Bai Fig 4 how h daa akn fom h unifom diibuion ca wihou diffuion bai. Fig4a how ha af only a fw lay a gown, h film ach ciical inabiliy. Fi fo ufac widh and fac iz af 000 lay how boh n = and b =, ha i W ~ and x~. Fuhmo, hi gaph how ha h avag lngh of on fac id i gowing wih im accoding o h xponn d = 0.84. A pow law diibuion of id lngh wa fac dicovd a hown in Fig4b. Th ul cal accoding o of (N)(S av )/N T v. /S av wh N i h numb of id, S av i h avag id lngh, N T i h oal numb of id, and i id lngh, a can b n in Fig4c. I yild an xponn = 0.7. Fig4d i h gaph of h G 2 [] colaion funcion. A xpcd, w find ha h ufac oughn wih im and ha h ufac oughning xponn a =. Fig5 how h daa fo h coin diibuion wihou a bai. Fig5a how ha, whil h coin diibuion ak long o bcom ciically unabl, i ill do o. Af 000 lay a gown, a fi how n and b again qual o on. Alo h avag lngh of a fac id i gowing wih im again accoding o d = 0.78. Again, hi lad o a pow law in id lngh diibuion, a in how in Fig5b and Fig5c. In h coin ca w find = 0.44. Thi valu i low han h unifom ca, which i o
b xpcd du o h lighly low d valu fo hi angula diibuion. Fig5d how ha a again qual. Fig6 how picu akn of boh coin and unifom diibuion ym wihou bai. Boh ym conain 2000 i, bu h picu a akn a diffn im. Fig6a how a coin diibuion akn a = 600 lay. Th picu how ha h film i bcoming vy ough, wih a fw vy wll dfind mound. Alo vy appan in hi picu i ha h avag id lngh i gowing, lnding fuh vidnc o a pow law diibuion of id lngh. Fig6b i a picu of a unifom diibuion akn a = 600 lay. Th ym i bcoming vy ough, wih on lag mound hadowing val mall on. Th gowh in avag id lngh i appan. Alo appan i h fac ha h avag id lngh i lighly low in h unifom ca han in h coin diibuion. Fig6a. Coin diibuion, no infac diffuion bai, L = 2000, = 600 lay Fig6b. Unifom diibuion, no infac diffuion bai, L = 2000, = 600 lay
IIIB. Bai W now com o h ca of an infac diffuion bai. Fi w will look a h unifom angula diibuion ca. Fig7a how ha h film bcom ju a unabl a h no bai ca fo unifom diibuion. Fi fo n and b af 000 lay how hm o b. W find ha in h ca of an infac diffuion bai, h avag lngh of a fac id gow mo lowly, in hi ca d = 0.24. Du o h wak id lngh gowh, a pow law diibuion wa no lookd fo. Fig7b how ha a ill qual. Finally, Fig8 how h ul fom h coin-bai imulaion. Thi ca i h mo lik xpimnal hin film gowh. I ha h mo ining ul. Fig 8a how ha, unlik all of h pviou ca, n = /3. Moov, in aly im b = /3. Howv, af 3000 lay, a fi fo b val ha i wa achd. W d ym of val diffn iz o inu ha hi wa no a fini iz ffc and in fac a al popy of h film. Whn w xamin h ufac of h film, w find gion of lag lop and gion of low lop. Thi man ha h mound gowh angl i no long fixd, and hfo n no long mu qual b. Thi alo xplain why h avag widh i gowing bu h mound coan i no. Again w find ha, in h pnc of an infac diffuion bai, h avag lngh of on id do no gow; in hi ca, d = 0. Howv, w do find anomalou caling in hi ca, a hown in Fig8b. Th gaph of G 2 [] do cal accoding o G 2 []/ v., wh i qual o im, in lay, a hown in Fig8c. An ining ul fom hi gaph i ha a = 0.8, no. Fig 4d, a gaph of G 2 [] v., how G 2 [] ha a pow law going G 2 [] ~ D. Thi fuh pov h anomalou caling.
Fig9 how picu akn of boh coin and unifom diibuion ym wih bai. Again, boh ym conain 2000 i wih picu akn a h am im a in Fig6. Fig9a how h coin diibuion. Th picu how l hap, dfind mound. Alo vidn in h picu i ha h avag id lngh in gowing lik in h no bai ca. Fig9a illua h ffc of boh low and lag lop gion on h ufac of h film. Fig9b i a picu of a unifom diibuion. I alo how ha h mound a l hap han h no bai ca. On lag mound i alo no a dominan a in h no bai ca. Th avag mound gowh i alo claly no gowing a fa. Howv, h mall, hadowd mound ough, which may xplain why n and b a lag han in h unifom-no bai ca. Fig 9a. Coin diibuion, L = 2000, = 600 lay, infac diffuion bai Fig9b. Unifom diibuion, L = 2000, = 600 lay, infac diffuion bai
IV. Concluion Ou imulaion m o how ha, in h pnc of hadowing du o pu dpoiion, hin film bcom ciically unabl, wih oughning and gowh xponn qual o. Moov, in h ca of coin diibuion wihou an infac diffuion bai, w find many ining, unxpcd ul. Fuh wok on hi ca, paiculaly looking a h id lngh diibuion, h anomalou caling of id lngh, and h aly im gim in which b co ov fom /3 o, could pov uful. Thi wok wa uppod by an NSF gan fo h Univiy of Toldo Summ 2002 REU pogam. Th auho would lik o hank all of ho a UT who aidd hi wok, paiculaly D. Jacqu Ama.
Fig8c. Scald gaph of G 2 [] / ), ( 2 G 0. 0.0 0.00 Coin diibuion Bai L = 3k a = 0.80 0.000 0-5 G2/-cov2k G2/-cov4k G2/-cov6k G2/-cov8k G2/-cov0k 0 00 Fig8d. Gaph of D ), ( 2 G 0 4 000 00 =5 =50 =99 Coin Diibuion Baio L = 3k D =.35 0 000 0 4 Thickn(lay)
Fig8a. Gaph of n, b, and d z i 000 u a F 00 0 0. w x w/x id Coin diibuion Baio n = /3 L = 3k b =.0 d = 0.0 0.0 0 00 000 0 4 Thickn(lay) Fig8b. Gaph of G 2 [] 000 ), ( 2 G 00 0 G2-cov0 G2-cov2k G2-cov4k G2-cov6k G2-cov8k G2-cov0k a = 0.88 Coin diibuion Baio L = 3k 0. 0 00
Fig5b. Gaph of id lngh diibuion d i f o b m u n 00 0 cov2k cov4k cov6k cov8k cov0k Coin diibuion No baio L = 32k = 0.44 0 00 000 id lngh Fig5c. Scald gaph of id lngh diibuion 0.3 Coin diibuion No bai L = 3k N / v 0.2 S a N =.44 cov2k cov4k cov6k 0. cov8k 0.09 cov0k 0.08 0.4 0.5 0.6 0.7 0.8 0.9 /S av
Fig4b. Gaph of id lngh diibuion d i f o b m u n 0 Unifom diibuion No baio L = 32k cov2k cov4k cov6k cov8k, cov0k = 0.65 0 00 000 id lngh Fig4c. Scald gaph of id lngh diibuion N / S a v 0. Unifom diibuion No bai L = 32k = 0.7 N cov2k cov4k cov6k cov8k cov0k 0.0 0. /S av
Fig5a. Gaph of n, b, and d z i S 000 u a F 00 0 w x w/x id Coin diibuion No bai L = 3k b =.0 n =.0 d = 0.78 0. 0 00 000 0 4 Thickn(lay) Fig5d. Gaph of G 2 [] ), ( 2 G 0 4 000 00 0 Coin diibuion No baio L = 3k a =.0 0. G2-cov0 G2-cov2k G2-cov4k G2-cov6k G2-cov8k G2-cov0k 0.0 0 00
Fig7a. Gaph of n, b, and d z i 0 4 000 u a F 00 0 w Unifom diibuion x Bai w/x L = 3k n =.2 id b =. d = 0.23 0. 0 00 000 Thickn(lay) Fig7b. Gaph of G 2 [] ), ( 2 G 0 4 000 00 0 Unifom diibuion Baio L = 32k a =.0 0. G2-cov00 G2-cov2k G2-cov4k G2-cov6k G2-cov8k G2-cov0k 0.0 0 00
Fig4a. Gaph of n, b, and d z i u a F 0 4 000 00 0 w x w/x id Unifom diibuion No Bai n = 0.97 L = 3k b =.0 d = 0.84 0. 0 00 000 Thickn(lay) Fig4d. Gaph of G 2 [] ), ( 2 G 0 4 000 00 0 Unifom diibuion No baio L = 32k a =.0 G2-cov00 G2-cov2k G2-cov4k G2-cov6k G2-cov8k G2-ov0k 0 00