Effect of a cap layer on morphological stability of a strained epitaxial film

Transcription

1 JOURNAL OF APPLIED PHYSICS 97, Eect o a cap layer on morphological tability o a trained epitaxial ilm Hai Liu and Rui Huang a Department o Aeropace Engineering and Engineering Mechanic, The Univerity o Texa, Autin, Texa Received 1 February 2005; accepted 11 April 2005; publihed online 7 June 2005 A trained epitaxial ilm oten undergoe urace roughening during growth and ubequent procee. One poible mean to reduce roughening o a to produce an epitaxial ilm with a lat urace i to depoit an oxide cap layer on the ilm to uppre the kinetic proce o roughening. Thi paper analyze the eect o a cap layer on the tability o an epitaxial ilm and the kinectic o roughening, auming the interace diuion between the ilm and the cap layer a the dominant mechanim o ma tranport. A variational principle i ormulated, which lead to a nonlinear evolution equation coupled with a boundary-value problem o elaticity. A linear perturbation analyi i then perormed, rom which the critical wavelength and the atet growing mode o roughening are obtained. It i ound that both the thickne and the reidual tre o the cap layer play important role in controlling the morphological tability and the roughening kinetic American Intitute o Phyic. DOI: / I. INTRODUCTION It i well known that epitaxially depoited ilm can undergo a tranition rom layer-by-layer growth to orm threedimenional iland. It ha been undertood that thi tranition i due to the preence o elatic tre induced by lattice mimatch between the ilm and the ubtrate. 1 3 While thi phenomenon ha ound important application a a proce to yntheize el-aembled quantum dot or nanoelectronic and optoelectronic device, 4 the rough ilm urace due to the tranition i undeired in other application uch a band-gap engineering or microelectronic device. 5,6 To improve the ilm quality, one procedure ha been recently propoed to depoit a cap layer on the ilm at a relatively low temperature to uppre the tranition proce. 7 The procedure keep the epitaxial ilm at relatively low temperature, allowing limited relaxation by either urace roughening or dilocation ormation. Once the cap layer ha been depoited, the ilm i contrained and thu tabilized during ubequent procee at higher temperature. Experimental evidence o the cap layer eect ha been oberved or a Si cap layer on SiGe/SiGeC ilm 7 and a ZrO 2 cap on a SiGeC ilm. 8 While dilocation ormation may till be a concern or ilm degradation, it may be controlled by everal technique uch a train compenation by carbon incorporation in SiGe alloy. 9,10 Thi paper tudie the eect o a cap layer on the tability and kinetic o urace roughening, auming no dilocation ormation. The morphological intability o a treed olid wa irt tudied by Aaro and Tiller 11 and later independently by Srolovitz 12 and Grineld. 13 Following imilar idea, the morphological intability o epitaxial ilm ha been tudied by many author e.g., Re It wa ound that a trained planar ilm i untable and the intability i manieted by ma tranport mainly via urace diuion. A a Author to whom correpondence hould be addreed; Tel: ; FAX: ; electronic mail: ruihuang@mail.utexa.edu urace chemical potential ha been deined 14,15 and widely ued in numerical imulation o nonlinear evolution o urace proile a well a growth o el-aembled quantum dot. 16,17 Alternatively, a variational principle baed on nonequilibrium thermodynamic provide an equivalent approach, but with a more generic orm that can be extended to more complex ytem. 18,19 The preence o a cap layer on top o a trained epitaxial ilm ha two direct eect on the morphological tability. Firt, it uppree the ma tranport on the otherwie ree urace o the ilm. Intead, interace diuion may take place, but typically at a ubtantially lower rate. Second, the mechanical tine o the cap layer tend to tabilize the ilm. Furthermore, the cap layer i eectively tiened when ubjected to a tenile reidual tre, but otened with a compreive reidual tre. In act, a compreive reidual tre in the cap layer by itel may caue urace intability, a wrinkling o the oxide cale on an aluminum-containing alloy at high temperature. 20,21 To develop a quantitative undertanding o thee eect, we employ the variational approach to analyze the urace intability and the roughening kinetic o a trained epitaxial ilm covered by an elatic cap layer. The ret o the paper i organized a ollow. Section II ormulate the variational principle, which lead to a nonlinear evolution equation coupled with a boundary-value problem o elaticity. In Sec. III a linear perturbation analyi i perormed. The eect o the cap layer on the critical wavelength o perturbation and the atet growing mode are dicued in Sec. IV. Section V conclude with a ummary o the reult. II. FORMULATION Figure 1 illutrate the model tructure o the preent tudy, coniting o a trained epitaxial ilm andwiched between a thick ubtrate and a thin cap layer. The ilm and the ubtrate are ingle crytal and orm a coherent interace /2005/9711/113537/8/$ , American Intitute o Phyic

2 H. Liu and R. Huang J. Appl. Phy. 97, E p h U p = p 241 v 2 w,xx + w,yy 2 21 v p w,xx w,yy p w 2,xy ds + E ph p p w 2 21 v,x + w 2,y ds, 2 p where h p i the thickne o the cap layer, E p i Young modulu, v p i Poion ratio, a comma in the ubcript denote partial dierentiation with repect to the ubequent variable, and S i an arbitrary plane parallel to the lat interace at the reerence tate. A part o the thin-plate approximation, we have ignored the in-plane diplacement o the cap layer. At the reerence tate, the train i uniorm in the ilm and zero in the ubtrate. Upon roughening, the train become nonuniorm in both the ilm and the ubtrate. The change o the repective train energy are U =0 h 1 2 ij ij dz E 2 h 1 v ds, 3 FIG. 1. Schematic o the model tructure: a the reerence tate and b the tate ater roughening. U = 1 20 ij ij dz ds, 4 The cap layer, on the other hand, i typically an amorphou oxide. At the reerence tate Fig. 1a both the ilm and the cap layer are lat. The ilm i ubjected to an equibiaxial in-plane train due to the lattice mimatch with the ubtrate, and the cap layer in general i ubjected to a biaxial reidual train p ; both train can be either tenile or compreive, depending on the material and the depoition procee. The train energy tored in uch a ytem may be relaxed by variou mechanim. 22 Thi paper conider urace roughening by interace diuion between the ilm and the cap layer. A Carteian coordinate ytem ha been et up in Fig. 1 with the x-y plane coinciding with the ilm ubtrate interace and the z axi a the upward normal o the interace. A. Energetic Let hx,y repreent the proile o the ilm cap interace meaured rom the ilm ubtrate interace. At the reerence tate, hx,y=h, which i a contant. A the interace roughen, the atom o the epitaxial ilm diue along the interace, and the cap layer deorm concomitantly. The roughening induce a change to the total ree energy G o the trilayer ytem, coniting o the urace/interace energy and the elatic train energy in the ilm U, the ubtrate U, and the cap layer U p, namely, G = U + U + U p +. Conider the elatic energy irt. Aume an iotropic, elatic cap layer, modeled a a thin plate undergoing a vertical diplacement, wx,y, relative to the reerence tate. The train energy in the cap layer conit o two part, aociated with bending and in-plane deormation, 23 repectively, 1 where E and v are the Young modulu and Poion ratio o the ilm, ij and ij are the tre and train tenor, and the upercript and denote the ilm and the ubtrate, repectively. A repeated Latin ubcript i or j implie ummation over the three coordinate x, y, and z. Both the ilm and the ubtrate are aumed to be iotropic in the preent tudy. The nonuniorm tre and train ield mut be determined by olving a boundary-value problem a decribed in a latter ection. Following the thin plate model or the cap layer, the upper and lower ace o the cap layer are aumed to remain parallel. For mooth urace with mall lope everywhere, the change o the urace energy i approximately = h,x 2 + h 2,y ds, 5 where 1 i the interace energy denity o the ilm cap interace and 2 i the urace energy denity o the cap layer. The urace and interace energie are aumed to be iotropic. B. Variational principle The change o the total ree energy in the model ytem can be aociated with two procee. One i the ma tranport, i.e., the atomic diuion at the ilm cap interace or the preent tudy. The divergence o the atomic relocation at the interace reult in the change o the interace proile, which lead to, by ma conervation, h = I,, where i the atomic volume and I i the atomic relocation vector, with denoting the in-plane coordinate x or y.a repeated Greek ubcript implie ummation over x and y. The other proce i the mechanical diplacement in the ilm 6

3 H. Liu and R. Huang J. Appl. Phy. 97, u i, the ubtrate u i, and the cap layer w. Auming that the interace remain bonded, the compatibility require that u i S = u i at the ilm ubtrate interace z=0, and w = h + u z at the ilm cap interace z=h. Taking the variation o Eq. 2 to 5, we obtain that 7 8 = S 2 hhds, 9 U p D p =S 4 w N p 2 wwds, U =S ij u i n j ds 1 +S 2 V 1 +S 1 2 ij ij hds, ij,j u i dv U S =S S ij u S i n j ds ij,j 2 V S u S i dv, 12 where 2 = 2 /x /y 2, = 1 + 2, D p =E p h 3 p /121 v 2 p, N p =E p h p p /1 v p, V dv= V h 0 dz ds, V dv = V 0 dz ds, S 1 and S 2 are the ilm cap interace and the ilm ubtrate interace, repectively, and n j i the normal vector o the correponding interace. Applying the compatibility relation in 7 and 8 lead to the variation o the total ree energy G =D p 4 h + N p 2 h ij ij z=hhds D p + 4 h N p 2 h + 3j n j z=h u z ds + j n j z=h u a ds 3j + z=0 3j z=0 u j ds V ij,j u i dv ij,j u V i dv 13 In deriving Eq. 13 we have approximately taken wh h under the aumption o mall deormation. O the two procee, the ma tranport i uually much lower than the mechanical diplacement. Conequently, in the time cale o ma tranport, it i uicient to aume that the ytem maintain mechanical equilibrium. Under thi condition, the variational principle dictate that the variation o the ree energy vanihe or arbitrary variation in mechanical diplacement, which lead to ij,j ij,j 3j j 3j =0 V =0 V n j = D p 4 h + N p 2 h n j =0 = 3j z = h z =0 z = h 14 Equation 14 decribe a boundary-value problem or the ilm ubtrate tructure ubjected to a urace traction due to the cap layer. Together with the contitutive relation or the ubtrate and the ilm, the boundary-value problem can be olved to determine the tre and train ield. On the other hand, the ytem i thermodynamically in-equilibrium, a the variation o the ree energy with repect to ma tranport drive interace diuion. The thermodynamic driving orce P i deined a G = P I ds. 15 By comparing Eq. 13 and 15 and applying the mechanical equilibrium condition in Eq. 14 and the ma conervation relation in Eq. 6, we obtain P = D p 4 h + N p 2 h x i i z=h. 16 When the cap layer i abent i.e., D p =N p =0, Eq. 16 i reduced to the amiliar driving orce or urace diuion, namely, the gradient o the chemical potential at a olid urace. 15 The preence o a cap layer thereore modiie the chemical potential at the interace. A imilar driving orce wa deined or interace diuion between a trained oxide cale and an aluminum alloy ubtrate, 20 in which the urace energy and the train energy in the ubtrate were ignored. C. Kinetic The kinetic o interace diuion i oten complex and diicult to characterize experimentally. For implicity, we aume a linear kinetic law o that the atomic lux rate i proportional to the thermodynamic driving orce, namely, J = MP, 17 where M i a contant characterizing the atomic mobility at the ilm cap interace. It i noted that the atomic mobility at an interace trongly depend on the cap layer and i typically maller than that at a ree urace. The divergence o the atomic lux change the interace proile, and the ma conervation require that h t = J,. 18 Subtitution o Eq. 16 into Eq. 17 and then into Eq. 18 lead to h t = M2 2 D p 4 h + N p 2 h ü ü z=h. 19 Equation 19 decribe the evolution o the interace proile, which couple with the boundary-value problem decribed

4 H. Liu and R. Huang J. Appl. Phy. 97, by Eq. 14. The coupled problem can be olved a ollow. At a given intance, the interace proile hx,y,t i known. Solve the boundary-value problem to determine the tre and train at the ilm cap interace. Then, ubtitute the tre and train into Eq. 19 and integrate over time to update the interace proile. Repeat the procedure to evolve the interace over time. In general, a numerical method i required to olve the boundary-value problem and to integrate the evolution equation. In the ollowing we purue analytical olution by a linear perturbation analyi to illutrate the eect o the cap layer. III. LINEAR PERTURBATION ANALYSIS An arbitrary interace proile hx,y can be repreented by the ummation o many Fourier component o dierent wavelength along dierent direction. For linear perturbation analyi, we conider a ingle component, i.e., a inuoidal perturbation with a contant wavelength. Since the model tructure i iotropic in the x-y plane, any direction o the inuoidal wave i equivalent, and we chooe the direction to coincide with the x coordinate without loing any generality. Thu, we write hx,t = h + Atin kx, 20 where A i the perturbation amplitude and k i the wave number. The perturbation induce the change o the tre and train ield in the ilm and the ubtrate, which can be determined by two tep conidering the eect o ma relocation and the interaction with the cap layer eparately. Firt, auming no cap layer, the ma relocation at the urace o the ilm change the morphology. The aociated change in the tre ield can be obtained by olving an equivalent problem with a ditributed hear traction acting on the urace o a lat ilm, a decribed in Re. 2. The correponding hear traction i proportional to the lope o the urace, namely, zx z = h = E 1 v ka co kx. 21 Next, the cap layer upon delection exert a normal traction on the urace o the ilm, i.e., zz z = h = D p 4 h + N p 2 h. Subtituting Eq. 20 into Eq. 22, we obtain zz z = h = D p k 4 N p k 2 A in kx Equation 21 and 23 repreent the linear approximation o the boundary condition at the ilm urace z=h in Eq. 14 or mall perturbation. The olution to the boundary-value problem i given in the Appendix. In particular, under the hear and normal traction in Eq. 21 and 23, the in-plane diplacement at the ilm urace i u x z = h = 1+v E 1 2 D p k 3 E 1 v + N p ka cokx, 24 where 1 and 2 are given in Eq. A18 and A19. For a mall perturbation rom the reerence tate, Eq. 19 i reduced to h t = M2 2 D p 4 h + N p 2 h + E 1 v u z. 25 xz=h Subtitution o Eq. 20 and 24 into Eq. 25 lead to da dt = A, 26 where = M 2 1+v k2 1 v 2 1E 2 k v 1 v 2 N p k v 1 v 2D p k Thereore, the amplitude o the perturbation a a unction o time i At=A 0 expt, where A 0 i the initial amplitude. The perturbation either grow or decay, depending on the ign o. The irt term in the bracket o Eq. 27 i poitive or both tenile and compreive ilm train, which drive roughening to relax the train energy. The econd term repreent the penalty due to the increae o urace energy and, in addition, the tretching o the cap layer. The reidual train in the cap layer can be either tabilizing N p 0 or detabilizing N p 0, depending on it ign. The third term urther penalize the roughening due to the lexural tine o the cap layer. The competition among the three term lead to two length cale. A comparion between the irt two term deine a length E l 1 = 2 1+v, 28 0 where 0 i the biaxial ilm tre at the reerence tate, i.e., 0 =E /1 v. Thi length cale ha been ued previouly to characterize the competition between the urace energy and the train energy. Similarly, a comparion between the irt and the third term lead to another length l 2 = E 1/3 D p 1+v 0 2, 29 which characterize the eect o the bending tine o the cap layer. Rewrite Eq. 27 with the length l 1 and l 2 a

5 H. Liu and R. Huang J. Appl. Phy. 97, where = 1 kl kl 1 2 kl 2 3, 1 =1+ 1+v 1 v 2 N p, 2 =1+ 1+v 1 v 2, = 3 E 4 1+v 4 M The parameter 1 can be either poitive or negative, characterizing the eect o the reidual tre in the cap layer. Equation 33 deine a time cale or the evolution proce, which i identical to the time cale or the evolution o a ree urace, 1,2 except that the atomic mobility at the interace or the preent cae i typically much maller. Note that, while Eq. 30 appear to take a polynomial orm in term o the wave number k, the actual dependence o the growth rate on the wave number i more complicated ince the parameter 1 and 2 are, in general, unction o the wave number a given in Eq. A18 and A19. The eect o the elatic tine o the ubtrate i alo included through the deinition o 1 and 2. IV. RESULTS AND DISCUSSIONS Compared to the previou tudie on ilm with no cap layer, Eq. 30 apparently include two additional term that repreent the eect o the cap layer. To make the dicuion more concrete, we conider a peciic ytem with an epitaxial Si 0.5 Ge 0.5 ilm andwiched between a Si100 ubtrate and a SiO 2 cap layer. The Young modulu o Si 0.5 Ge 0.5 and Si are 116 and 130 GPa, repectively. The Poion ratio i taken to be 0.25 or both the ilm and the ubtrate. The mimatch train in the ilm i The cap layer ha a Young modulu o 71 GPa and a Poion ratio o Variou thickne and reidual tree in the cap layer will be conidered. Taking a typical value o 1 J/m 2 or the urace energy denity, the length cale l 1 deined in Eq. 28 i then 9.7 nm. The other length cale l 2 i proportional to the thickne o the cap layer, l 2 =3.89h p or the preent ytem. The time cale deined in Eq. 33, however, i more diicult to etimate due to the uncertainty o the atomic mobility at the interace. Roughly, the time cale trongly depend on the temperature and i igniicantly longer than that or the evolution o a ree urace. Figure 2 plot the normalized growth rate a a unction o the wave number kl 1 with and without a cap layer. A noted in previou tudie, without a cap layer h p =0, the lat ilm i untable; there exit a critical wave number, below which the perturbation grow. The preence o a cap layer with no reidual tre tend to tabilize the ilm, leading to a maller critical wave number longer wavelength and a lower growth rate. Both the critical wave number and the FIG. 2. Normalized growth rate a a unction o the wave number with and without a cap layer. growth rate decreae a the thickne o the cap layer increae. The ytem, however, remain untable at the long wavelength end. The critical wave number alo depend on the ilm thickne and the tine o the ubtrate, a hown in Fig. 3 or the cae with no cap layer. Similar plot were reported previouly. 1,2 Two point are noted here. Firt, or a given tine ratio, the critical wavelength i bounded between two limit. For thick ilm h /l 1 3 the eect o the ubtrate i negligible, and the critical wavelength approache that or a treed olid in the hal plane, which i = 1 v l On the other hand, or very thin ilm h /l 1 0 the ubtrate eect dominate, and the critical wavelength again approache that or a treed hal plane but now with the ubtrate tine, i.e., FIG. 3. The critical wavelength a a unction o the ilm thickne or variou tine ratio between the ubtrate and the ilm with no cap layer. The dahed line i or a Si 0.5 Ge 0.5 ilmonasi100 ubtrate. The open circle are the olution or limiting cae with very thin ilm.

6 H. Liu and R. Huang J. Appl. Phy. 97, FIG. 4. Eect o an elatic cap layer on the critical wavelength. E 0 = l 1, 35 1 ve a denoted by the open circle in Fig. 3 or variou tine ratio. For an arbitrary ilm thickne, the critical wavelength i in between. When the ubtrate and the ilm have the ame tine, the critical wavelength i independent o the ilm thickne. For SiGe ilm on Si ubtrate, the tine ratio i cloe to unity and the critical wavelength weakly depend on the ilm thickne, a hown by the dahed line in Fig. 3 orasi 0.5 Ge 05 ilm. The econd point to note i that a tier ubtrate igniicantly increae the critical wavelength or thin ilm h /l 1 1. At the limit o a rigid ubtrate, the critical wavelength approache ininity or the ilm below a critical thickne. Thee reult agree with previou tudie. 1,2 The eect o the cap layer on the critical wavelength i hown in Fig. 4. With no reidual tre, the lexural tine o the cap layer diavor roughening. Conequently, the critical wavelength increae with the thickne o the cap layer. A tenile reidual tre p 0 in the cap layer urther tien the layer againt roughening, leading to a igniicantly longer critical wavelength. The epitaxial ilm i thereore eectively tabilized. On the other hand, a compreive reidual tre p 0 detabilize the ilm becaue roughening relaxe the compreive tre in the cap layer. Thi lead to a horter critical wavelength or a thin cap layer. However, a the thickne o the cap layer increae, the tabilizing eect due to the lexural tine eventually overcome the detabilizing eect due to compreion, and the critical wavelength then increae. Thereore, a minimum thickne i required or a compreively treed cap layer to tabilize the epitaxial ilm. Figure 2 how that the cap layer igniicantly aect the kinetic o urace roughening. At the initial tage o roughening, the atet growing mode dominate. Both the wavelength and the growth rate o the atet growing mode are inluenced by the cap layer. Generally peaking, the wavelength increae and the growth rate decreae with the cap layer, a hown in Fig. 5. In act, the cap layer uppree the kinetic proce o roughening. Recall that interace diuion FIG. 5. a The wavelength and b the growth rate o the atet growing mode a unction o the ilm thickne with and without a cap layer. The dahed line in a i the critical wavelength with no cap layer. i typically much lower than urace diuion, and thereore the eect o the cap layer on the growth rate i even more ubtantial. The reidual tre in the cap layer alo ha a trong eect on the kinetic, a illutrated in Fig. 6. A tenile tre enhance the tabilizing eect o the cap layer, leading to longer wavelength and lower growth rate. A compreive tre, however, detabilize the ytem, leading to horter wavelength and ater growth rate. Thi i not urpriing becaue a compreed cap layer by itel tend to buckle to relax the compreive tre. The competition between the compreive reidual tre and the tine o the cap layer lead to a minimum wavelength and a maximum growth rate at a peciic cap layer thickne. Thereore, care mut be taken to determine the thickne when uing a compreively treed cap layer to tabilize the epitaxial ilm. V. SUMMARY In thi paper, a variational approach i ormulated to analyze the eect o a cap layer on morphological tability and roughening kinetic o a trained epitaxial ilm. Atomic diuion at the ilm cap interace i conidered. The thermodynamic driving orce i deined with the preence o the cap layer. The derived evolution equation couple with a boundary-value problem o elaticity. A linear perturbation

7 H. Liu and R. Huang J. Appl. Phy. 97, xx = C 1 cohkz + C 2 inhkz + C 3 2 inhkz + kz cohkz + C 4 2 cohkz + kz inhkzin kx, A1 zz = C 1 cohkz + C 2 inhkz + C 3 kz cohkz + C 4 kz inhkzin kx, A2 zx = C 1 inhkz + C 2 cohkz + C 3 cohkz + kz inhkz + C 4 inhkz + kz cohkzco kx, A3 u x = 1+v C1 cohkz + C2 inhkz + C 3 kz cohkz +21 v inhkz co kx, E k + C 4 kzinhkz +21 v cohkz u z = 1+v C1 inhkz + C2 cohkz + C 3 2v 1cohkz + kz inhkz in kx. E k + C 4 2v 1inhkz + kz cohkz A4 A5 For the ubtrate o ininite thickne 0z, the olution i reduced to xx = D 1 + D 2 2+kzexpkzin kx, A6 FIG. 6. a The wavelength and b the growth rate o the atet growing mode a unction o the cap layer thickne. analyi i then perormed, baed on which the eect o the cap layer i dicued. The lexural tine, which cale with the cube o it thickne, tend to tabilize the ilm, leading to longer critical wavelength and lower growth rate. A tenile reidual tre in the cap layer urther enhance the tabilizing eect. A compreive reidual tre, however, detabilize the ilm. It i uggeted that the thickne be careully elected when uing a compreively treed cap layer to tabilize the epitaxial ilm. ACKNOWLEDGMENTS The author are grateul or the upport by NSF Grant No. CMS and the Texa Advanced Material Reearch Center. R.H. thank Proeor S. K. Banerjee or helpul dicuion. zz = D 1 + D 2 kzexpkzin kx, A7 zx = D 1 + D 2 1+kzexpkzco kx, A8 u x = 1+v E k D 1 + D 2 2 2v + kzexpkzco kx, A9 u z = 1+v E k D 1 D 2 1 2v kzexpkzin kx. A10 The ix coeicient are determined by the boundary condition at the ilm urace z=h and the continuity condition at the ilm ubtrate interace z=0, i.e., zx z = h = B 1 co kx, zz z = h = B 2 in kx, zx z =0 = zx z =0, zz z =0 = zz z =0, A11 A12 A13 A14 u x z =0 = u x z =0, A15 APPENDIX Conider a lat elatic ilm o thickne h on an ininitely thick elatic ubtrate ubjected to a periodic traction normal and hear on the urace. The plane train problem can be olved by uing the tre and diplacement potential. 24 The tre component and the diplacement in the ilm are u z z =0 = u z z =0, A16 where B 1 and B 2 are the amplitude o the hear and normal traction acting on the urace, repectively. Ater obtaining the coeicient, the diplacement at the ilm urace can be determined. In particular, the hear diplacement at the urace i given by

8 H. Liu and R. Huang J. Appl. Phy. 97, u x x,z = h = 1+v E k 1B B 2 co kx, where A17 1 = 21 v 2 inh2kh + 3 coh2kh + 4 kh coh2kh + 3 inh2kh + 4 kh 2, A18 2 = 2v 1 2 coh2kh + 2v 1 3 inh2kh + 4 kh coh2kh + 3 inh2kh + 4 kh 2, A19 and 1 = p 13 4v + p8v 2 12v +5, u x = 1+v E k 21 vb 1 + 2v 1B 2 co kx, A25 2 = 1+p 2 3 4v +2p1 2v 2, 3 =8p1 v 2, 4 =2p 1p +3 4v, 5 = p v1 2v, A20 with p=e /E. In the above olution we have aumed v =v =v to impliy the reult. The above olution can be reduced in everal limiting cae. Firt, or a rigid ubtrate i.e., p, A18 and A19 are reduced to 3 4vinh2kh +2kh 1 = 1 v 3 4vcoh 2 kh + kh 2 + 2v 1 2, A21 2 = 3 4v2v 1inh2 kh + kh 2 3 4vcoh 2 kh + kh 2 + 2v 1 2, A22 which are identical to the olution or an elatic layer with a ixed boundary at the bottom given in Re. 24. The olution may be urther reduced or incompreible material v =0.5. At the oppoite limit when the ubtrate tine i approaching zero p 0, we have 1 = 1 v inh2kh 2kh inh 2 kh kh 2, A23 2 = 2v 1inh2 kh kh 2 inh 2 kh kh 2, A24 which correpond to the olution or an elatic layer with no ubtrate contraint, i.e., a traction-ree urace at the bottom. For an ininitely thick elatic ilm i.e., kh, the olution i independent o the ubtrate and Eq. A17 reduce to which i the olution or an elatic hal plane. 11 In the other limit when the elatic ilm i very thin i.e., kh 0, the olution i reduced to u x = 1+v E k 21 vb 1 + 2v 1B 2 co kx, A26 which i again the olution or an elatic hal plane, but now with the ubtrate tine. The two olution, thereore, bound the general olution or elatic ilm o arbitrary thickne. In the pecial cae when the ilm and the ubtrate have the ame elatic modulu i.e., p=1, the two bound collape and the olution i independent o the thickne. 1 B. J. Spencer, P. W. Voorhee, and S. H. Davi, Phy. Rev. Lett. 67, L. B. Freund and F. Jontottir, J. Mech. Phy. Solid 41, H. Gao and W. D. Nix, Annu. Rev. Mater. Sci. 29, B. Yang, F. Liu, and M. G. Lagally, Phy. Rev. Lett. 92, M. Yang, J. C. Sturm, and J. Prevot, Phy. Rev. B 56, Z. H. Shi, D. Onongo, and S. K. Banerjee, Appl. Sur. Sci. 224, G. S. Kar, A. Dhar, L. K. Bera, S. K. Ray, S. John, and S. K. Banerjee, J. Mater. Sci.: Mater. Electron. 13, R. Mahapatra, S. Maikap, J.-H. Lee, G. S. Kar, A. Dhar, D.-Y. Kim, D. Bhattacharya, and S. K. Ray, J. Vac. Sci. Technol. A 21, H. J. Oten, Mater. Sci. Eng., B 36, A. C. Mocuta and D. W. Greve, J. Vac. Sci. Technol. A 17, R. J. Aaro and W. A. Tiller, Metall. Tran. 3, D. J. Srolovitz, Acta Metall. 37, M. A. Grineld, J. Nonlinear Sci. 3, C. H. Wu, J. Mech. Phy. Solid 44, L. B. Freund, J. Mech. Phy. Solid 46, Y. W. Zhang and A. F. Bower, J. Mech. Phy. Solid 47, C.-H. Chiu, Appl. Phy. Lett. 75, A. C. F. Cock and S. P. A. Gill, Acta Mater. 44, Z. Suo, Adv. Appl. Mech. 33, Z. Suo, J. Mech. Phy. Solid 43, V. K. Tolpygo and D. R. Clarke, Acta Mater. 46, J. Tero and F. K. LeGoue, Phy. Rev. Lett. 72, S. Timohenko and S. Woinowky-Krieger, Theory o Plate and Shell, 2nd ed. McGraw-Hill, New York, R. Huang, J. Mech. Phy. Solid 53,