Theory of quantum dot formation in Stranski-Krastanov systems

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Applied Surface Science 123/124 (1998) 646 652

Theory of quantum dot formation in Stranski-Krastanov systems H.T. Dobbs ", D.D. Vvedensky a,*, A. Zangwill b ~' The Blackett Laboratory and lnterdisciplinap3, Research Centre fi;r Semiconductor Materials, hnperial College, London SW7 2BZ UK b School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA

Abstract

The formation of coherent three-dimensional (3D) islands during Stranski-Krastanov growth of semiconductors is modelled with rate equations. Effects of strain are included in the growth rate of two-dimensional (2D) islands (atop a wetting layer) and in the transformation of 2D into 3D islands. Comparisons with measured 3D island densities for InP grown on GaP-stabilised GaAs(001) by metalorganic vapor-phase epitaxy are used to parametrize the model and to identity the rate-determining steps for 3D island formation. © 1998 Elsevier Science B.V. PACS: 68.55.

a.; 68.35.Bs

Kevwords: Strain; Semiconductor heteroepitaxy; Self-assembled quantum dots; Rate equations; Vapor-phase epitaxy

1. Introduction

The discovery that the three-dimensional (3D) islands formed during Stranski-Krastanov (SK) heteroepitaxial growth of lattice-mismatched semiconductors are initially dislocation-free or 'coherent' [1] has generated intensive interest because their optical properties suggest potential applications as quantum dots [2a,2b]. This morphology has been demonstrated in a variety of materials systems [3,4], with the islands having characteristic sizes small enough to exhibit quantum confinement effects. Before SK growth can become a robust method of producing quantum dots, it is necessary that a selfcontained and flexible kinetic model of 3D island nucleation be available to facilitate control of the growth process. There are two main issues to ad-

:': Corresponding author. Tel.: +44-171-5947573; fax: +44171-5899463; e-mail: [email protected].

dress. The first is the kinetic pathways by which the 3D islands nucleate and grow. The formation of these islands is thought to begin with the nucleation of two-dimensional (2D) islands on top of the wetting layer that forms initially during SK growth. Calculations based solely on energetic considerations have shown that a 2D island will transform into a pyramidal 3D island whenever it exceeds a certain Limiting size, which in the case of InAs/GaAs(001) corresponds to the observed 3D island size [5]. However, this correspondence must be brought to question, since an island will grow rapidly after the 2 D - 3 D transition because the elastic strain which inhibits the growth of 2D islands [6] is relieved by the 3D geometry. Additionally, the Limiting island size is very large (5000 atoms [7]), and such 2D islands would not be expected to form under typical growth conditions. The second issue is the distribution of island sizes. For optical applications, e.g. surface emitting

0169 4 3 3 2 / 9 8 / $ 1 9 . 0 0 © 1998 Elsevier Science B.V. All rights reserved. PII S 0 1 6 9 - 4 3 3 2 ( 9 7 ) 0 0 4 6 0 - 1

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H.T. Dobbs et al. / A p p l i e d SurJace Science 1 2 3 / 1 2 4 (1998) 646-652

lasers, it is desirable to have as narrow a distribution of sizes as possible. Priester and Lannoo [5] have argued that the narrow 3D island size distribution is due to a narrow e q u i l i b r i u m distribution of 2D islands sizes• But this interpretation must be pursued with caution, since the form of the 2D distribution function is determined by k i n e t i c s [8] rather than simply thermodynamics. More recently [7], a fitted form for the 2D island distribution was used to calculate the number of islands exceeding the limiting size, which were then assumed to spontaneously become three-dimensional. Although this reproduces the sudden onset of 3D nucleation seen in many SK systems [9,10], this treatment is qualitative in that the 2D distribution is chosen to yield the correct 3D density at a particular temperature and growth rate, with no rationale provided to extend the calculation to other growth conditions [ 11 a, 11 b]. In this paper, we present a model that reproduces the observed time dependence of the 3D island density and describes how this density varies with the growth conditions. To demonstrate the utility of this model, we compare our results to extensive measurements of the density of 3D islands formed after the deposition of a fixed amount of InP on a GaP-stabilised GaAs(001) surface by metalorganic vapor phase epitaxy (MOVPE) [12] over a range of temperatures and growth rates. We do not discuss the distribution of 3D island sizes here explicitly, though we can infer some qualitative features of this distribution from the results our calculations.

2. Rate equations for self-organized quantum dots Our model [13] is based on the general rate equation theory of epitaxial growth [ 14a, 14b, 14c, 14d]. The primary dynamical variables

(~

are the densities of (Group III) adatoms n I and of 2D and 3D islands, denoted by n and ~, respectively. In this approach all explicit spatial information is omitted, including the spatial distribution of islands and thus and interactions between. But with a suitable description of adatom capture, rate equations can produce quantitative agreement with Monte Carlo simulations for adatom and (2D) island densities [15,16]. We consider five processes that cause nl, n and fi to change (Fig. 1). (i) The deposition of adatoms onto the surface at the flux F. The 'wetting layers' are presumed to have already formed. (ii) The diffusion of adatoms across the surface, with diffusion constant D. Adatoms are taken to be the only mobile species. (iii) The formation of 2D islands. The 2D islands which are composed of not more than i" atoms are assumed to be thermodynamically unstable and to quickly break up into adatoms, while islands with more than i* atoms are able to grow by capturing migrating adatoms. (iv) Adatom attachment and detachment at the edges of 2D and 3D islands. The capture of adatoms by 2D and 3D islands is given in terms of average 'capture numbers' [14a,14b,14c,14d,15,16], which are denoted by cr and 6-, respectively. (v) The transformation of 2D islands into 3D islands at a rate y. Given these processes, the rate equations for n~, n and ~ are

h,=F D[(i*+

1)o-i

ni

+o-n+6-fi]n,-n/z,

h = D ¢ i n i, n I -- y n ,

=,/..

(1)

The initial conditions for the adatom and island densities are n I = n = ~ = 0, with the initial time corresponding to the completion of the wetting lay-

(4)

Fig. I. Schematic illustration of the steps in the rate equations for SK growth: (1) deposition, (2) detachment from edges of 2D islands onto the top of the island, (3) reflection of adatoms atop 2D islands by step-edge barriers, (4) atom mobility on a 3D island to seek sites that maximize strain relief, and (5) attachment barrier for 3D islands (which is also operative for 2D islands).

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H.T. Dobbs et al. /Applied Surface Science 123 / 124 (1998) 646-652

ers. To close this set of equations, we use the Walton relation [17] for the density ni~ of critical nuclei: nl. -~ ( n l ) ' e CE'*,

(2)

where /3 = 1 / k B T , k B is Boltzmann's constant, T is the substrate temperature, and El* is the binding energy of a critical island. Given the kinetic steps we have outlined, the rate equations in (1) are generic. The dependence of island formation kinetics on any particular material system, the degree of strain, and the growth conditions is contained in kinetic coefficients: the flux F, the diffusion constant D, the capture numbers o-, 6- and o-i*, the adatom escape rate from 2D islands, r -~, and the 2 D - 3 D transition rate, 3'. The adatom diffusion constant is D = [2/(/3h)]e -/~E~, where E s is the barrier to hop between surface sites and the spacing between these sites is our unit of length. As adatoms migrate, they collide to form stable islands, which then grow by capturing other migrating adatoms. In our contracted rate theory, we focus on an 'average' 2D island composed of s atoms with radius r and average 3D islands with radius T. We assume circular s-atom 2D islands with radius r = ( s / r r ) ~/2 and piano-convex 3D islands with a (height/diameter) aspect ratio of 0.2 [9]. Although, for the system at hand [4] the 3D islands grow as truncated pyramids, the details of island shape do not significantly alter our results. The equations for the average sizes of 2D and 3D islands can be derived [13] from the rate equations in Eq. (1). Attachment to 2D islands occurs at the rate Do-n t , which defines the capture numbers for these islands. Capture numbers for critical islands, o-i*, and for 3D islands, 6-, are defined similarly. A constant energy barrier E~ is presumed to augment the diffusion barrier E s for adatoms attaching to all islands. This is important in MOVPE where step decoration by reaction products is expected inhibit attachment [ 18]. A size- and shape-dependent enhancement of the attachment barrier can also be included to mimic the effects of strain near 2D and 3D islands [13], but we have not done so here. Since small 2D islands are expected to relieve strain more effectively than larger 2D islands, we include a strain-dependent detachment rate w from

2D islands which are significantly larger than the critical island size: w = 2rrrDe ~ej(,,),

(3)

where the detachment barrier Ed(S) is taken to be of the form [19] In r Ed(s) = Ed(ZC) + Eo -

(4)

r

We set Ed(ZC)= 0, since this barrier is expected to be small for large islands. Detachment rates are expected to increase with increasing strain, so we expect E 0 be a decreasing function of strain. We omit detachment from 3D islands, since atoms are assumed to migrate to other sites on the same island to maximize strain relief. We can now calculate the capture numbers for 2D and 3D islands and the escape rates of adatoms from islands. Capture numbers for 2D islands that take into account both attachment barriers (E~) and detachment are given by [15,16] 2rrrXt( r/,~ )

o- = ~ X , ( ~/~ ) + ~Ko( r / ~ ) '

(5)

where oe ~ = e Cu0 - 1 , K 0 and K~ are zeroth-order and first-order modified Bessel functions, respectively, and ~: is determined self-consistently from ~-2 = (i* + 1)o-i.n,~ + o-n + 6"ft.

(6)

Capture numbers for critical 2D islands and 3D islands can be calculated from these formulae with appropriate values of the radii. The escape rate of adatoms from 2D islands is [16] r'=

wo-

e/3e",

(7)

27rr where w is given by Eq. (3). Note that the escape rate in Eq. (7) is different from the detachment rate in Eq. (3) because detaching atoms may be recaptured immediately by their parent islands [16]. As a 2D island grows, it becomes increasingly likely that a second layer will be initiated on top of it. To estimate the rate for this process, we suppose that the mechanism for this nucleation is atoms detaching from perimeter sites onto the top of the island. If the island radius small compared to the diffusion length, the density of adatoms p on top of the island is uniform and the rate at which these

H.T. Dobbs et al./ Applied Surface Science 123/124 (1998) 646-652

adatoms fall off the island edge is 2 r r r D p e -~E~, where E~ is an additional (step-edge) barrier for atoms to cross the step edge. In the steady state, p = ea[E~ E,<,~], so the nucleation rate for the second layer is

4.0

/J - . 3.0

We take y as the rate at which 2D islands transform into 3D islands. This 2 D - 3 D transition mechanism is different from that based on a limiting 2D island size [5,7], although the rate y is still strongly size dependent due to the barrier Ed(s).

/

E

--

/

q~

55 / /

(8)

y = 7"i'r2Dp i*+l e IzE~* .

649

2.0 ¢....

~ t,c~j 1.o / 0.0

3.0

4.0

5.0

6.0

Coverage (ML)

3. Comparison with measured 3D island densities Our model has six parameters: E s, E i . , E 0, E~, E~ and i *. To make meaningful comparisons with particular materials systems requires an extensive set of measurements for the 3D island density as a function of coverage, temperature and growth rate. The data used here were taken from the growth of on InP on GaP-stabilized GaAs(001) using M O V P E [20]. W e have determined the values that give the best fit to the experimental data for the dependence of the 3D density on growth rate and temperature. Both these data and the results of the rate equations are shown in Fig. 2. Typical errors for the measured 3D density are 4-5.0 × l0 s cm 2 so the comparison

8.0

6.0

/" 4.0

?

[- j J"

t

//

-J .... jJ~

"5 e~D

~ 853K l i 883K /', 913K

2.0

o.o [E 0.0

2.0

4.0

61o

Growth Rate (ML/s) Fig. 2. Measured (symbols) and calculated (solid line) 3D island density after 3.5 ML deposition. The optimized model parameters arei* =6, Es - l . 0 4 e V , E i. =0.87eV, Eo=3.28eV,E e=0.17 eVand Ea=0.10eV.

Fig. 3. Measured (symbols) and calculated (solid line) 3D island density vs coverage at a growth temperature of 853 K and a growth rate of 0.5 ML s i. between the experimental data and the model is acceptable. For low growth rates the 3D density is an increasing function of the growth rate and a decreasing function of temperature. In this limit the density of 3D islands appears to be controlled by the density of 2D islands that nucleate on the wetting layer. This increases as the growth rate increases or the temperature decreases. For large enough growth rates (at a fixed temperature), the 3D density decreases because although the number of 2D islands continues to increase with increasing growth rate, the average size of these islands eventually falls below the value at which the 2 D - 3 D conversion rate I/ to 3D is appreciable. Such a decrease at large growth rates has in fact already been observed [21]. In Fig. 3 we compare the observed time development of the 3D island density with the results of the model using the same parameters as those used to generate Fig. 2. Although the parameters were fitted from data for the density at an InP coverage of 3.5 ML, the model gives a good description of the variation of the 3D density with coverage. This provides an additional level of confidence that our model captures many of the essential competing kinetic effects of 3D island formation process. The shape of this curve is a result of the time variation of the 2D island size. Just after the completion of the wetting layer, only small 2D islands have nucleated, so the conversion rate y is small. By approximately

H.T. Dobbs et al. /Applied Surface Science 1 2 3 / 1 2 4 (1998) 646-652

650

2.3 ML the islands reach a size at which conversion may begin, and the 3D density starts to increase, with an initial exponential, rather than power law [9], time dependence. As soon as 3D islands nucleate, they act as traps not only for the adatoms that are deposited but also for atoms that detach from 2D islands. Thus, the 2D island size suddenly decreases, and y becomes small again so that 3D nucleation abruptly stops.

4. Roles of model parameters Our rate theory is designed to describe 3D island evolution in just enough detail to reliably predict average quantities. Five 'atomistic' energy parameters have been defined, but it should be clear that these are effective quantities in a severely contracted model that makes no attempt to describe the extremely complex MOVPE process in microscopic detail. All of these parameters are expected to depend on strain and the choice of materials. The quantities E s [22] and E 0 [23] can be expressed in terms of the

strain and material-dependent quantities or calculated explicitly, but for E~ a specific atomistic mechanism for inter-layer hopping must be assumed (e.g. atomic exchange at the step edge). Moreover, the fit in Fig. 2 is sensitive only to the sum of the parameters E s and E~. (with small variations in the other parameters), a situation that is familiar from the conventional 2D nucleation problem for submonolayer hom o e p i t a x y a n a l y z e d with rate equations [14a,14b,14c,14d]. Acceptable fits can also be achieved for all values of i* > 2 (but with corresponding variations in the other parameters). This reflects the rather weak dependence of our model on this parameter, which arises perhaps from the known effect of an attachment barrier on the i *-dependence of island densities for the conventional problem [18]. Thus, of the six free parameters in our model, accurate determination of only four is necessary, at least over the range of experimental data shown in Figs. l and 2. In Fig. 4 we show the effect of changing each of the parameters on the density of 3D islands at the highest temperature in Fig. 2 (853 K). In Fig. 4a, E s has been reduced by 5% but E~ has been increased

10.0

10.0

8.0

8.0

6.0

6.0 4.0

HE 4.0 "~O2. 0

(b)

/

Es+i*E i. fixed

O ,i/-- - ~ 0.0 >, 0.0

I

I

I

2.0

4.0

6.0

~10.0

2.0 0.0

0.0

I

I

J

2.0

4.0

6.0

10.0

ID

a a.0 O

(c)

(d)/"

8.0

03 6.0

6.0

4.0

4.0

/

= i+o-

2.0 0.0

,

0.0

I

2.0

r

t

4.0

~

I

6.0

2.0 0.0

0.0

AEa/E a = -5% I

2.0

,

I

4.0

,

k

6.0

Growth Rate (ML/s) Fig. 4. The effect of variations of model parameters on the 3D island density for T = 853 K. The solid line is the same as that in Fig. 2 and the broken line is the corresponding density calculated with the indicated parameter changed, but the others held fixed. (a) E s decreased by 5 ~ but E,~ increased accordingly to maintain their sum fixed. (b) E 0 decreased by 5%. (c) E~ decreased by 5%. (d) E~ decreased by 5%.

H.T. Dobbs et al./Applied Surfdce Science 123/124 (1998) 646-652

accordingly to maintain the sum E s + i * E i constant. The effect on the 3D island density is negligible at the lowest growth rates but increases appreciably with increasing growth rate. Reducing E s decreases the density of 2D islands but increasing E i enhances the stability of critical islands, which has a compensating effect of the density, as suggested by (2). Decreasing E 0, makes detachment from 2D island edges easier, which promotes the 2 D - 3 D transition (Fig. 4b). This effect is also most evident for the greater growth rates. The effect of decreasing the step-edge barrier on 2D islands E e is quite dramatic and extends over all growth rates, as shown in Fig. 4c. The decrease in island density results from the corresponding reduction in adatom density on top of 2D islands because of the diminished barrier to hopping down which, according to (8) decreases the 2 D - 3 D transformation rate. Finally, decreasing the attachment barrier E, to 2D and 3D islands results in an appreciable increase of 3D island density over most of the range of growth rates (Fig. 4d). With a reduced attachment barrier 2D islands are able to grow more quickly to sizes where the 2 D - 3 D transformation rate is appreciable. One of the main conclusions we can draw from Fig. 4 is that although varying the parameters changes the 3D density, the decrease in this quantity at sufficiently high growth rates is a generic feature of the model.

5. Summary and conclusions There is evidence in s o m e SK systems that the total material contained in 3D islands exceeds that deposited after the wetting layer forms [9,24,25]. This implies that atoms that detach from the wetting layer are readily accommodated by the 3D islands. We have performed additional calculations using a scheme that includes a dynamic wetting layer of this sort [13] and found no significant change in the calculated density of 3D islands except possibly at the highest temperature and lowest growth rate. The major difference is for the predicted volume of 3D islands, for which we do not have reliable data in the present case. It is also conceivable that such detachment processes depend on the growth technique used, with the presence of reaction products at step edges during MOVPE being an inhibiting factor.

651

Finally, we comment on the implications of our results for the conspicuous narrowness observed in experimental distributions of 3D island sizes. This effect arises quite naturally in a model of irreversible aggregation to 3D islands with barriers to attachment. Such a barrier reduces the net flux of adatoms to the island (compared to the situation when the barrier is absent) and thus acts similarly to detachment processes acting alone. Monte Carlo simulations of homoepitaxy have demonstrated that increasing rates of detachment do in fact lead to a progressive narrowing of the island size distributions [26].

Acknowledgements Work performed at Georgia Tech was supported by the US Department of Energy under Grant No. DE-FG05-88ER45369.

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