Mater. Res. Soc. Symp. Proc. Vol Materials Research Society. Pattern Evolution of Self-Assembled Quantum Dots Under Biaxial Stresses

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1 Mater. Re. Soc. Symp. Proc. Vol. 9 6 Material Reearch Society 9-T7-8 Pattern volution o Sel-Aembled Quantum Dot Under Biaxial Stree Yaoyu Pang, and Rui Huang Department o Aeropace ngineering and ngineering Mechanic, Univerity o Texa, Autin, TX, 787 ABSTRACT A treed epitaxial ilm can undergo urace intability. The tre ield and the interace interaction have proound eect on the dynamic o urace evolution that lead to elaembled quantum dot. In thi paper, by uing a nonlinear evolution equation, we invetigate pattern evolution o el-aembled quantum dot under general biaxial tree. It i ound that the hape o quantum dot and their patial ordering are trongly inluenced by the relative magnitude o the biaxial tree. Linear perturbation analyi and nonlinear numerical imulation are conducted to elucidate the eect o tre aniotropy on the proce o elaembly that elect dierent pattern. INTRODUCTION Sel-aembled nanotructure in epitaxial ytem are o great interet or both theoretical undertanding and practical application. An epitaxial thin ilm i inherently treed due to lattice mimatch between the ilm and the ubtrate. The competition between urace energy and train energy drive urace intability o an initially lat ilm [-3]. In addition, the interace between the ilm and the ubtrate play an important role in the later tage o urace evolution [4]. Recently, we developed a nonlinear evolution equation [5] taking into account the eect o the econd-order tre ield and a nonlinear wetting potential. Numerical imulation howed that the nonlinear tre ield alone induce blow-up intability, leading to crack-like grooving in D and circular pit-like morphology in 3D. With the wetting potential, the blow-up i uppreed, leading to an array o dicrete iland on top o a thin wetting layer. Under an equi-biaxial tre, the ytem i iotropic (material aniotropy wa ignored), and the model predicted el-aembly o circular iland with no patial ordering (a hown in Figure ). x x t = t = 5 t = t = t = Figure. volution o urace morphology rom a numerical imulation under an equi-biaxial tre ( c = ), tarting rom a random initial perturbation at t =. A bright pot repreent a cret o the urace. The time i normalized by a time cale τ, and the length by a length cale L, both deined in text.

2 It ha been hown that material aniotropy (e.g., crytal elaticity, urace energy, and urace mobility) play a igniicant role in the procee o el-organization and hape tranition o epitaxial quantum dot [6,7]. On the other hand, the eect o aniotropy in the general biaxial tre tate ha received le attention. Recently, Berger et al. [8] and Paret [9] howed that, during a melting-crytallization proce, a biaxially treed emi-ininite olid can develop into a rich variety o pattern, epecially when the applied tre i tenile in one direction and compreive in the orthogonal direction. In thi paper, we invetigate pattern evolution o el-aembled quantum dot under general biaxial tree. A linear analyi and nonlinear numerical imulation are conducted to elucidate the eect o tre aniotropy on the hape o quantum dot and their patial organization. VOLUTION QUATION In a previou tudy [5], a nonlinear evolution equation wa derived or urace evolution o a treed epitaxial ilm, which take into account the contribution rom the econd-order tre ield and a nonlinear wetting potential. A compact orm o the equation can be written a h = Ω M t U h h + U () + U () h + U W. () where h = h( x, x, t) i the evolving thickne proile o the ilm, h = h the urace (n) gradient, h = h the urace curvature to the irt order, U the nth-order elatic train energy denity at the urace, the urace energy denity, U W the wetting potential, Ω the atomic volume, and M the atomic mobility at the urace. A repeated Greek ubcript implie ummation over and or the in-plane coordinate x and x. At the reerence tate, the ilm urace i lat (i.e., h = h ), and the tre i uniorm. Under a general biaxial tre ( σ = σ, σ = σ, and σ ), the train energy denity i = U ( σ + σ ν σ σ ) =, () where i Young modulu o the ilm and ν Poion ratio. The material o the ilm i aumed to be iotropic. In our previou tudy [5], the irt and econd-order train energy denitie were obtained via an aymptotic approach or equi-biaxial tree (i.e., σ = σ ). Under general biaxial tree, the reult are () () () u u U = σ + σ, (3) U () + ν = σ σ h h + σ u () + σ () u (), (4)

3 where the irt and econd-order urace diplacement are obtained in the orm o Fourier tranorm: () uˆ = ik C σ hˆ, (5) () () u = C ˆ ϕ + C ˆ φ. (6) ˆ 3 In the above, ϕ = σ h, φ = σ h h, and C ij i a 3-by-3 matrix accounting or the tine o the ubtrate a given in the previou tudy [5]. Baed on a tranition-layer model [], the wetting potential take the orm b U W = π ( b + h, (7) ) where = i the change in the urace energy denity over a tranition layer o thickne b. A two-dimenional pectral method wa developed to olve the evolution equation [5]. Figure how the reult rom a numerical imulation, where the ilm i ubjected to an equibiaxial tre at the reerence tate (i.e., σ = σ = σ ). The length i caled by L =, and σ 3 4 S the time by τ =, where 8 S = i the plane-train modulu o the ubtrate. The 6Ω Mσ ν S parameter ued in the imulation are: / =., / =., h =. L, b =. L, and ν = ν =.5. A dicued beore, in thi cae, the ilm break up into circular iland on a thin wetting layer, with no patial ordering. SYMMTRY BRAKING DU TO STRSS ANISOTROPY For a general biaxial tre, let c = σ σ a the actor o tre aniotropy. A linear analyi o the evolution equation () lead to hˆ t = ( k, k ) hˆ, (8) where h ˆ( k, k, ) i the Fourier tranorm o the thickne proile and t ( k k ) bl, = k. (9) k ( ) ( ) ( ) 4 k + k c k k ck ν k + 3 ν + πh

4 a b c Figure. Contour plot o the growth rate ( k, k ) under dierent tre tate. From let to right: (a) c = (equi-biaxial), (b) c = (uniaxial), and (c) c = - (pure hear). Here the length cale L and the time cale τ (with σ in place o σ ) are ued or normalization, and k = k + k. Thereore, in the linear regime, each Fourier component o the urace proile grow (or decay) exponentially, with the growth rate,, a a unction o the wave vector ( k, k ) in the Fourier pace. Figure plot the growth rate a contour in the plane o ( k, k ). When c =, the ytem i iotropic, and the contour are concentric circle. The growth rate i poitive in an annular region (bounded by the black edge), and the atet growing mode correpond to a circle (dahdotted blue). The rotational ymmetry lead to a chaotic pattern in the early tage o evolution and circular dot in the later tage, a hown in Figure. The ymmetry i broken when c. A hown or c = (uniaxial tre), the atet growing mode correpond to two point (the white pot) located on the axi o k (i.e., k = ). Thi ugget that the initial evolution would develop parallel triped pattern perpendicular to the direction o the uniaxial tre. In general, when > c >, k = or the atet growing mode; when c >, k =. Thereore, the direction o the triped pattern depend on the relative magnitude o the two tre component. Figure 3 how urace evolution rom a nonlinear numerical imulation or c =. A predicted by the linear analyi, a parallel tripe pattern emerge at the initial tage. Ater a long time, however, the triped pattern coaren and eventually break up into elongated iland. The apect ratio o the iland i ound to be dependent upon the parameter c. x x t = t = t = 3 t = t = Figure 3. volution o urace morphology rom a numerical imulation under a uniaxial tre ( c = ).The time i normalized by a time cale τ, and the length by a length cale L, both deined in text.

5 BIFURCATION OF GROWTH MOD When c <, a biurcation occur at a critical value. A hown in Figure (c) or c =, the atet growing mode correpond to our point located at an angle ± π / 4 ; the two white pot in (b) plit into our in (c). Deine an angle θ or the wave vector o that k = k coθ and k = k inθ. By etting θ =, we obtain that ( c )( [ ν )( + c) ν ( c) coθ ] inθ coθ =. vidently, under an equi-biaxial tre ( c = ), θ = or all angle. Thu, the growth rate i independent o the angle, a hown by the circular contour in Figure (a). When c, the growth rate depend on the angle and reache extreme value at peciic angle atiying q.. The angle o the atet growing mode can then be determined by examining the econd derivative o the growth rate, which i plotted in Figure 4 a a unction o the tre aniotropy. There exit three cae: () When > c > ( ν ), in θ = or the atet growing mode, which give θ = or equivalently θ = ± π. () When c > or c < ( ν ), the atet growing mode correpond to co θ =, which give θ = ±π /. Cae () i equivalent to Cae () by imply witching the role o σ and σ. (3) When ( ν ) < c < ( ν ), the angle o the atet growing mode i given by ( + c)( ν ) ( c) ν co θ =. () Conequently, a biurcation o the atet growing mode i predicted. When c =, the biaxial tre i equivalent to a pure hear tre, and the angle θ = ±π / 4. Figure 5 how the evolution o urace morphology rom a numerical imulation with c =. A predicted by the linear analyi, the initial growth elect the atet growing mode, in thi cae, with the angle θ = ±π / 4. A diamond type pattern emerge and enue until the ilm break up into iland, a predicted by Berger et al. [8]. Subequent evolution beyond the linear regime how coarening o the iland and eventually ormation o a tilted tripe pattern. The competition o the two tilting direction lead to co-exiting o long and hort (broken) tripe. Figure 4: The angle o the atet growing mode at the initial tage a a unction o the tre aniotropy ( ν =. 5).

6 x x Figure 5. volution o urace morphology rom a numerical imulation with c =. The time i normalized by a time cale τ, and the length by a length cale L, both deined in text. SUMMARY t = t = t = t = t = We have theoretically hown that, under general biaxial tree ( c ), the rotational ymmetry in an otherwie iotropic ytem i broken, leading to parallel triped pattern in the early tage o evolution and elongated iland in the late tage. Furthermore, a biurcation o the growth mode i predicted when ( ν ) < c < ( ν ), in which cae a diamond pattern and tilted tripe are predicted. Thi oer a rich variety o pattern or el-aembled urace tructure that can be achieved by controlling the tre tate. A o today, limited experimental evidence have been reported in exploring tre aniotropy (a well a other approache o ymmetry breaking) or el-aembled nanotructure [-3]. Further tudie are in progre to experimentally implement tre aniotropy (e.g., by uing elatically aniotropic ubtrate urace uch a Si(3), by patterned urace template, or by train engineering). ACKNOWLDGMNTS The author are grateul or the upport by Department o nergy through Grant D-FG- 5R463. RFRNCS. R.J. Aaro and W.A. Tiller, Metall. Tran. 3, (97).. M.A. Grineld, Sov. Phy. Dokl. 3, (986). 3. D.J. Srolovitz, Acta Metall. 37, 6-65 (989). 4. T.V. Savina, P.W. Voorhee, and S.H. Davi, J. Appl. Phy. 96, (4). 5. Y. Pang and R. Huang, ubmitted (5). Preprint available at 6. Y.W. Zhang, Phy. Rev. B 6, (). 7. S.P.A. Gill, Thin Solid Film 43, (3). 8. P. Berger, P. Kohlert, K. Kaner, C. Mibah, Phy. Rev. Lett. 9, 763 (3). 9. J. Paret, Phy. Rev. 7, 5 (5).. B.J. Spencer, Phy. Rev. B 59, -7 (999).. T. Ogino, et al., Surace Science 54, -9 ().. M. Hupalo and M.C. Tringide, Phy. Rev. B 73, 445 (6). 3. Y. Chen, et al., Appl. Phy. Lett. 76, ().