1995 mechanism of the step flow to island growth transition during mbe on si(001) 2 × 1


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1995 mechanism of the step flow to island growth transition during mbe on si(001) 2 × 1

  1. 1. i•ii•!i!i•!i••i!i•i!•ii•i•i•i•i•i!i•i•i•1•i!•i•i•i!•i•i•i•i•!i•i•!ii!i•i surface science ELSEVIER Surface Science 331-333 (1995) 473-478 Mechanism of the step flow to island growth transition during MBE on Si(001)-2 X 1 J. van Wingerden *, Y.A. Wiechers, P.M.L.O. Scholte, F. Tuinstra Crystallography Group, Solid State Physics, Department of Applied Physics, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands Received 5 August 1994; accepted for publication 9 December 1994 Abstract During homoepitaxial growth of Si on Si(001) steps act as sinks for the diffusing species on the surface. The ratio between step distance and diffusion length determines, whether growth proceeds by incorporation of the diffusing species into steps, step flow, or by nucleation and growth of islands in between the steps, island growth. For Si(001)-2 X 1 adatom diffusion is extremely anisotropic. However, for many growth conditions incorporating the diffusion perpendicular to the easy diffusion direction is essential in calculating the critical values for the step flow to island growth transition. It will be shown that for usual experimental growth conditions a transition from 1D to 2D diffusion occurs for increasing terrace width, w. As a consequence, the critical flux at marginal step flow scales as w-3 for small terraces and as w-4 for large terraces. Furthermore, it has been investigated, whether a stronger anisotropy for dimer than for adatom diffusion may cause any significant contribution of the dimer diffusion in sustaining step flow. Keywords: Adatoms; Diffusion and migration; Epitaxy; Growth; Models of surface kinetics; Molecular beam epitaxy; Silicon; Single crystal epitaxy; Single crystal surfaces; Surface diffusion 1. Introduction or the growth rate. Thus, critical fluxes, growth temperatures or terrace widths can be defined as During epitaxial deposition on stepped surfaces those values for which marginal step flow occurs, the ratio between the mobility of the diffusing species i.e. the diffusion length of the diffusing species is and the incident flux determines the growth mode. just large enough to prevent island formation. Step flow growth occurs if the mobility of the diffus- On the Si(001)-2 X 1 surface diffusion is ex- ing species is high enough to let the diffusing species tremely anisotropic, with fast diffusion along the stick to the step edge before collisions with other dimer rows and slow diffusion perpendicular to them. adatoms or clusters make them immobile. The island Ab initio estimates [1] of diffusion barriers parallel growth mode can be obtained by decreasing the and perpendicular to the dimer rows yield values of growth temperature or by increasing the terrace width 0.6 and 1.0 eV, respectively. This energy difference of 0.4 eV for the fast and slow diffusion direction will yield a diffusion anisotropy of 10 3 at 670 K and * Corresponding author. Fax: +31 15 783251. at 1000 K the diffusion anisotropy is still 102. Direct 0039-6028/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 0 3 9 - 6 0 2 8 ( 9 5 ) 0 0 3 4 5 - 2
  2. 2. 474 J. van Wingerden et aL / Surface Science 331-333 (1995) 473-478 experimental observations of diffusing adatoms are dimer diffusion to the step flow cannot merely be not available, but an activation energy of 0.67 eV for neglected because of the large difference in mobility diffusion along the dimer rows has been determined along the dimer rows for dimers and adatoms. It will from the temperature-dependent island density [2]. turn out, that in principle a stronger diffusion The diffusion of dimers has been observed directly anisotropy for dimers than for adatoms may compen- with STM [3]. From these observations dimer diffu- sate for the low mobility of the dimers along the sion also turned out to be strongly anisotropic. Be- dimer rows. cause of these large diffusion anisotropies for adatoms and dimers, the diffusion perpendicular to the dimer rows is generally neglected if the dimer 2. Critical terrace widths in case of anisotropic rows are oriented perpendicular to the step edges. diffusion However, we will show that for many growth condi- tions this is not allowed. In this paper two subjects In this model we restrict ourselves to the concerning the transition of step flow to island growth anisotropic diffusion on terraces with the dimer rows will be discussed in view of this diffusion anisotropy. perpendicular to the step edges. This means that the First, the dependence of the critical flux on the fast and the slow diffusion directions are perpendicu- terrace width will be discussed. Second, the possibil- lar and parallel to the step edges, respectively. This ity that dimer diffusion influences the transition from situation is depicted in Fig. 1, as well as the meaning island growth to step flow is discussed. of some variables used in the following derivation. Section 2 will present a derivation of the critical Furthermore, we will assume that for the diffusing terrace widths in case of diffusion anisotropy, where species the sticking coefficient at the step edges is diffusion perpendicular to the dimer rows is not unity. neglected. As shown there, decreasing the terrace To derive a formula for the critical terrace width width will cause a transition from 2D to effectively for marginal step flow we introduce the following 1D diffusion. The scaling of the critical flux with the concepts. The mean free time, zf, of a diffusing terrace width will also change at this transition. The particle is the average time a particle can diffuse on results will be compared with data from literature in the surface before colliding with another particle. Section 3. The mean free area, Af, of that particle is the The dimer diffusion is addressed in Section 4, average surface area which is visited by the particle where it will be emphasized that the contribution of during its mean free time. Te and A f a r e related by dimer rows step edge / slow diffusion // =5 •~ LLLLL~ b]llllllllll',~,,, IIIIII[IkLAIIII,, ,,p,,, , t=0 t='t t=x c (a) (b) (c) Fig. 1. Schematic view of a terrace with dimer rows perpendicular to the step edge. (a) Directly after deposition of an adatom, (b) the average area visited by the adatom during a time ~" after deposition, (c) average area visited during the critical mean free time ~'c. The symbols are explained in the text.
  3. 3. J. van Wingerden et al. / Surface Science 331-333 (1995) 473-478 475 the incoming flux, F, since at most one particle In the case that the diffusion length in the direc- should be deposited on the mean free area during the tion perpendicular to the dimer rows is smaller than mean free time: the jump distance, i.e. F = 1/(Ao-f). (1) A± ( " r ) < v ~ a , (4) If during the mean free time of a particle its the diffusion will be effectively one-dimensional and average diffusion length perpendicular to the step the mean free area is given by edge is equal to the step distance marginal step flow occurs, because each particle can find the step edge A t = AIIV~-a. (5) and stick to it just before colliding with another If A±(~')> V~-a, the diffusion perpendicular to particle. During a certain time ~" the average diffusion the dimer rows should be accounted for. For this situation the mean free area is defined as an ellipse distance along the dimer rows will be with axes All and A l : ( a 2 ) 1/2 All(z)= -~-kllT , (2) Af = ~-AIIA ± . (6) where kll = v e x p ( - E i i / k T ) is the hopping rate in This definition has the advantage that in the limit this direction, with an attempt frequency v and an of vanishing anisotropy (All = A±) the mean free energy barrier Eli. Throughout this paper a is the area reduces to the definition, which is generally lattice parameter of the cubic unit cell of the fcc used in the case of isotropic diffusion. lattice. Therefore, the jump distances along and per- Now, the critical mean free time, ~'c, for marginal pendicular to the dimer rows are ½v~-a and v~-a, step flow follows from the condition that the terrace respectively. In an analogous manner we define the width, w, is equal to the diffusion length along the diffusion length perpendicular to the dimer rows as dimer row during this time: A± ('r) = (2aZk ± ,/.)1/2. (3) All(re) w. = (7) I .-¢1o 8 ............... >.:.,..,,.,!... .......................... ,.................................................................................................................................. 8 i- .............................................. ............... "..I... ....... "'.~.: ................................................. ~.¢. ............................................................................ ~ .................................................... •-- 104 ................................• ~ ;" ............................ i...............~.,.........................................................~ .................................................... % EL 102 lo o i'"< ..................................... 10 -2 10 .4 10 100 w [nml Fig. 2. Critical flux for marginal step flow as a function of the terrace width w for different growth temperatures, attempt rates, and activation energies for diffusion along the dimer rows. (The activation energy for diffusion perpendicular to the dimer rows E 1 = 1.0 eV.) The arrows indicate the transition from effectively 1D to 2D diffusion.
  4. 4. 476 J. van Wingerden et al. / Surface Science 331-333 (1995) 473-478 From this equation, together with Eq. (2), it fol- edges. For the dimer row orientation parallel to the lows that the critical mean free time step edges they included the diffusion perpendicular to the dimer rows, resulting in an expression similar 7"c= 2W2/(a2kll). (8) to Eq. (11). The diffusion perpendicular to the dimer For 1D diffusion the critical mean free area fol- rows was neglected in their derivation of the critical lows from Eqs. (5) and (7): flux in case of an orthogonal orientation of dimer rows and step edges. Therefore, they found an ex- a c = wv/-2a. (9) pression similar to Eq. (10), which is only correct if Using Eqs. (1), (8), and (9), the critical flux for the terrace width is smaller than the transition value this extremely anisotropic case, where diffusion is from Eq. (12). effectively 1D, is akll 3. Comparison with experimental results F ~ - 2v~-w3. (10) The previous section discussed the critical terrace From Eqs. (1), (3), (8), and the critical mean free widths for anisotropic diffusion on terraces with the surface area A c = (Tr/4)wh±, it follows that the dimer rows perpendicular to the step edges. For the critical flux in case of two-dimensional anisotropic Si(001)-2 × 1 surface this applies to the so-called diffusion type-B terraces of a double domain surface as well a2 / as to the terraces of a single domain surface if the (11) step edges are close to the <ll0>-direction. For F~ = 1 / ( a c r c ) = ~ V k i rough step edges with large terrace width fluctua- tions the local terrace width should be considered. Thus, in case of 2D (an)isotropic diffusion (Eq. However, calculations with these local terrace widths (11)), the critical flux scales with the terrace width as only make sense if the wavelength of the step edge w -4 and in the case of 1D diffusion (Eq. (10)) as W -3" fluctuations is larger than the diffusion length per- pendicular to the dimer rows, h± (To). In our treat- From the condition (4) and Eqs. (3) and (8) can ment the non-unity sticking coefficient at one of the be derived that the transition from 2D to effectively step edges is neglected, as this will only influence 1D diffusion takes place for terrace widths the proportionality constant of the variation of the 1 ~/_~a/ ktl critical time, re, with the squared terrace width, w 2, (12) in Eq. (8). w='2 V k± From the STM measurements published by Mo et The critical flux as a function of the terrace width al. [5] conditions for the transition from step flow to is plotted in Fig. 2 for several relevant parameters. island growth can be determined. They showed (Figs. The critical flux has been multiplied by a2/2 to 3a and 3b from Ref. [5]) that increasing the growth obtain the flux in units of ML/s. From this figure it temperature from 563 to 593 K at a growth rate of can be seen that changing the attempt frequency ~, 1/400 M L / s caused a complete disappearance of does not change the transition from 1D to 2D diffu- islands on the terrace with the dimer rows perpendic- sion although the critical flux changes strongly. ular to the step edge. Thus, growth at 1/400 M L / s However, increasing the temperature or decreasing and 593 K causes marginal step flow on a terrace, the anisotropy of the activation energies for diffusion which is estimated to be about 150 nm wide. (E± - E H) will reduce the terrace width at which the Here, we will use these conditions to estimate the transition occurs. diffusion barrier along the dimer rows by solving Eq. Similar formulas for the critical fluxes have also (11) for Ell. Using v = 1013 and E± = 1.0 eV, the been derived by Lu and Metiu [4], who determined diffusion barrier along the dimer rows should be the critical fluxes for both dimer rows oriented paral- Ell = 0.73 eV. Neglecting the diffusion perpendicular lel and dimer rows oriented perpendicular to the step to the dimer rows and using Eq. (10) for 1D diffu-
  5. 5. J. van Wingerden et al. / Surface Science 331-333 (1995) 473-478 477 sion, an activation energy Ell of 0.89 eV is found. free time on the mean free area to form a second This value is certainly incorrect, because then the dimer which can collide with the first one. adatom motion would be frozen out at room temper- To evaluate Eq. (13) a few assumptions are made. ature, contradicting the fact that as yet no adatoms First, the attempt frequencies for the hopping of have been observed by STM. Thus, including the adatoms and dimers are assumed to be identical. diffusion perpendicular to the dimer rows is essential Second, for the energy barriers for adatom diffusion, in explaining the observed growth phenomena. This Ell and E l , we use 0.7 and 1.0 eV, respectively. is confirmed by the diffusion length perpendicular to Third, for dimer diffusion the values Eib = 1.0 and the dimer rows (Eq. (3)), which is 21 nm. E l = 1.2 eV are used. These values for the dimer Concerning the transition in scaling of the critical motion were found experimentally by Dijkkamp et flux with the terrace width it is easy to show that it al. [3]. The value for the barrier perpendicular to the can be observed in practical situations. If, e.g., acti- dimer rows is only a minimum value deduced from vation energies of 0.73 and 1.0 eV are assumed for the fact that at 340 K the diffusion perpendicular to the fast and slow diffusion, respectively, the transi- the dimer rows was at least a factor 103 lower than tion from 1D to 2D diffusion occurs at a terrace along the rows. This yields an energy barrier differ- width of 10 nm for a growth temperature of 480 K ence of at least 0.2 eV. (see Eq. (12)). Calculation of the ratio of the critical fluxes of adatoms and dimers with these values shows that the importance of the dimer diffusion relative to the 4. Dimer diffusion at marginal step flow adatom diffusion increases with increasing tempera- ture, but is still less than 2% at 1000 K. At first sight, one could argue why any influence of dimer A dimer is formed if two adatoms on the surface diffusion could be expected at all, because the diffu- collide. In analogy to the critical adatom flux (Eq. sion barrier along the dimer rows is much higher for (11)) a critical flux for the diffusion of these dimers dimers than for adatoms. However, the relative im- can be determined. This is the maximum flux of portance of the dimer diffusion depends strongly on incoming adatoms for which the diffusion of dimers the size of the critical mean free area of the dimers is large enough to reach the step edge before collid- compared to that of the adatoms. A strong anisotropy ing with an incoming adatom. This critical dimer of the dimer diffusion can limit the diffusion range flux is found by using the hopping rates kll and k± perpendicular to the dimer rows, thus reducing the for dimer diffusion in the expression for the critical mean free area and consequently the chance of a flux (Eq. (11)). collision with another dimer or adatom. This reduced Now, the influence of the dimer diffusion in mean free area yields an increase of the mean free sustaining step flow is determined from the ratio of time, which can compensate for the low mobility of the critical fluxes for adatom and dimer diffusion: the dimer. Therefore, the influence of the dimer motion on the step flow cannot be ruled out just by fc,adatom = ~( kll,adatom k 3_,dimer 1 )3( (13) comparing the mobilities along the dimer rows. Fc,dimer KII,dimer k 3_,adatom ' Now, it turns out that if we account for the diffusion perpendicular to the dimer rows, the energy where it has been assumed that for adatom as well as barrier difference of 0.2 eV is insufficient to make dimer diffusion Eq. (11) for 2D diffusion can be dimer diffusion of any importance for step flow. applied. Furthermore, the fact that a collision be- Even if the barrier for perpendicular diffusion is tween the dimer and another dimer may be necessary much higher (e.g., 1.6 eV), the critical adatom flux to form an immobile cluster has been neglected. If a will be more than 5 times higher than the critical collision with an adatom does not influence the dimer flux at 1000 K. mobility of the dimer, the critical flux for dimer If the terrace widths are reduced, the dimer diffu- diffusion will be twice as large. This is due to the sion may become 1D, whereas the adatom diffusion fact that 2 adatoms need to be deposited in the mean is still 2D. In that case, Eq. (10) should be used to
  6. 6. 478 J. van Wingerden et al. / Surface Science 331-333 (1995) 473-478 calculate the critical dimer flux. In this case, the ratio This explains why our results for large terrace widths of the critical adatom flux and the critical dimer flux with dimer rows perpendicular to the step edges are depends on the terrace width. Unless the dimer equivalent to those of Lu and Metiu for the dimer diffusion anisotropy is extremely large, terrace widths rows parallel to the step edges. need to be unrealistically small to make the critical Concerning the influence of dimer diffusion on dimer flux comparable to the critical adatom flux. the step flow no final answer can be obtained, be- cause the activation energy difference for dimer dif- fusion parallel and perpendicular to the dimer rows 5. Conclusions is not available. To make the critical fluxes for adatoms and dimers comparable, however, strong It has been shown in Section 3 that the experi- anisotropy will be needed as compensation for the mental observation of marginal step flow by Mo and low dimer mobility along the dimer rows. Lagally [5] can be explained with the model for anisotropic 2D diffusion from Section 2 by using a reasonable value for the activation energy for diffu- Acknowledgements sion along the dimer rows. Taking into account the uncertainty in the attempt frequency, v, the agree- One of the authors (J.v.W.) would like to thank ment between the calculated activation energy and Professor D.D. Vvedensky, H.J.W. Zandvliet and Z. literature results, Refs. [1,2], is quite good. Zhang for their stimulating discussion and useful Comparing our results to those of Lu and Metiu remarks during the material growth workshop of the [4] for the dimer rows perpendicular to the step International School of Materials Science in The edges yields equivalent results for terrace widths Netherlands. smaller than the width at which the transition from 1D to 2D diffusion takes place. For many growth experiments, however, terrace widths are longer and References the diffusion perpendicular to the dimer rows can no longer be neglected. [1] G. Brocks, P.J. Kelly and R. Car, Phys. Rev. Lett. 66 (1991) Now, the results of Lu and Metiu [4] for the dimer 1729. rows oriented parallel to the step edge can easily be [2] Y.W. Mo, J. Kleiner, M.B. Webb and M.G. Lagally, Phys. understood by considering this situation as one with Rev. Lett. 66 (1991) 1998. very long dimer rows oriented perpendicular to the [3] D. Dijkkamp, E.J. van Loenen and H.B. Elswijk, Proc. 3rd NEC Symp. on Fundamental Approach to New Material step edge. This can be justified by the fact that on Phases, Vol. 17 of Springer Series in Materials Science real surfaces with the dimer rows parallel to the step (Springer, Berlin, 1992) p. 85. edge the dimer rows will have a limited length due [4] Y.-T. Lu and H. Metiu, Appl. Phys. Lett. 59 (1991) 3054. to waviness or a slight misorientation of the steps. [5] Y.-W. Mo and M.G. Lagally, Surf. Sci. 248 (1991) 313.