5238 Phys. Chem. Chem. Phys., 2013, 15, 5238--5242 This journal is c the Owner Societies 2013

Cite this: Phys. Chem.Chem.Phys.,2013,15, 5238

Selective formation mechanisms of quantum dots onpatterned substrates†

Xinlei Li*

We have presented a theoretical model to elucidate the formation mechanisms of quantum dots (QDs)

on patterned substrates, which introduced the perspective that the preferred formation site of QDs is

determined by the nonuniformity of wetting layer thickness caused by surface potential. Two different

preferred formation sites, low surface curvature or high surface curvature, can be switched through

controlling growth temperature or selecting appropriate patterns. The model explains some interesting

and puzzling experimental observations, which implies that the established approach could be

applicable to the physical understanding of the QDs on patterned substrates.

1. Introduction

Self-assembled quantum dots (QDs) have attracted extensiveinterest because of their original properties and potentialapplications.1–3 In general, QDs usually have a random spatialdistribution due to the spontaneous nature of self-assembledgrowth. To improve the device applications, the fabrication ofQDs with highly ordered arrangement is intensively desired.4,5

Growth on patterned substrates is one of the most widely usedtypical approach for controlling the spatial distribution of self-assembled QDs.6–16 For example, strained QDs with well spatialorder have been grown on a hole-patterned substrate.6–9 Inparticular, different formation sites, i.e. on the inside of the holeversus on the terrace between holes, can be switched by controllinggrowth temperature6,7 or selecting appropriate patterns.8,9 Experi-mental results are not adequately explained by existing theories.

Generally, the surface with negative curvature, such asv-groove structure, is usually the preferred nucleation site uponthe epitaxy growth from the point of view of nucleation thermo-dynamics.17 However, for the Stranski-Krastanov growth, i.e.firstly layer-by-layer growth of wetting layer and then three-dimensional (3D) QD growth, the formation site of QDs isstrongly influenced by the thickness of the wetting layer. In otherwords, QDs formed on the wetting layer only when the thicknessof the wetting layer exceeds a critical value. Therefore, the initialwetting layer growth may be an important factor for the formation

site of QDs. Additionally, Cao and Yang established a thermo-dynamic theory to address the thermal stability of QDs self-assembly by introducing thermal fluctuations.3,18,19 The thermalfluctuations are determined by the growth temperature, whichalso influences the formation of QDs. In our model, we do notconsider the influences of thermal fluctuations but the thicknessof the wetting layer.

In this contribution, we developed a theoretical modelto elucidate QD formation on a curved patterned substratesurface, as shown in Fig. 1. Firstly, we studied the morphologiesat the initial stages of two-dimensional (2D) epitaxial wettinglayer growth and found that the thickness of the wetting layer isstrongly influenced by the surface curvature. Subsequently wecompared the QD formation energies on different surfacecurvatures and revealed that the nonuniformity of the wettinglayer thickness determines the formation sites of QDs.

2. Theoretical model

During the initial stage of the epitaxial deposition, because thesubstrate acts as a seed crystal, the epitaxial layer may lock intoone or more crystallographic orientations with respect to thesubstrate crystal. Therefore, the ability of epitaxial growth isgreatly affected by the substrate, mainly including sufferingfrom a mismatch strain induced by the substrate, and changeof chemical potential caused by the surface curvature of thesubstrate. Therefore, the chemical potential along the surfaceof an uneven substrate usually is written as16,20

m = m0 + Ogk + OEstr (1)

where m0 is the thermodynamic driving force of the crystal-lization process which can be expressed as the difference

MOE Key Laboratory of Laser Life Science & Institute of Laser Life Science, College

of Biophotonics, South China Normal University, Guangzhou 510631, China.

E-mail: xlli@scnu.edu.cn

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c3cp43890b

Received 2nd November 2012,Accepted 13th February 2013

DOI: 10.1039/c3cp43890b

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between the chemical potential of the supersaturated ambientphase and that of the infinitely large crystal (also called as‘‘supersaturation’’), O is the atomic volume, g is the surfaceenergy density, k is the local surface curvature, and Estr is themismatch strain energy density stored in epitaxial material. Thestrain energy density can be calculated by Estr = 2G[(1 + u)/(1� u)]e0

2,where G and u are shear modulus and Poisson ratio, e0 is the strainmismatch between epitaxial material and substrate.

The expression above considers the surface energy as aconstant during epitaxial growth process. However, thoughthe surface area remains constant, the surface energy densityof epitaxial layer varies with its thickness.21–23 In other words,the surface energy density of the epitaxial layer tends to that ofthe substrate surface when its thickness approaches zero, andtends to that of its own bulk material when its thickness isinfinite. Therefore, the change in the surface energy densityalso drives the growth of epitaxial layer to implement a thickerextension. If we defined the expression for the surface energydensity of epitaxial layer as glayer = g(h), where h is its thickness,the driving force from surface energy change can be expressedas Oqg(h)/qh. Therefore, considering the contribution of thesurface energy change, the chemical potential for the epitaxiallayer growth becomes

m ¼ m0 þ Ogkþ OEstr þ O@gðhÞ@h

(2)

From the thermodynamic views, the deposited atoms diffusefrom the regions with high chemical potential to the ones withlow chemical potential, which implies that the thickness ofepitaxial layer at the region with low chemical potential is largerthan that at the region with high chemical potential. Therefore,we can use the value of chemical potential to analyze thethickness of the epitaxial layer.

To formulate the surface energy density of the epitaxiallayer, we use an exponential term to approximately expressthe variation, i.e. g(h) = gNB + (gNA � gNB )(1 � e�h/h0Z),23 where gNAis the surface energy density of an infinite-thickness epitaxiallayer, gNB is that of the substrate surface, h0 is the thickness of amonolayer, and Z is a dimensionless parameter that dependson the interactions between the layers. The exponential changeof the surface energy density has been described correctly to

Ge–Si23 and InAs–GaAs systems.24 Here we take a Ge epitaxiallayer on patterned Si substrate as an example to investigate theinfluences of layer thickness and surface curvature on thechemical potential. In the calculation, G = 41 GPa, u = 0.26,e0 = 0.0402, Z = 1, and h0 = 0.1415 nm. In order to simplify ourmodel, we assume that the surface energy density for differentsurface orientation have the same value as (001) surface. There-fore, the surface energy densities of Si and Ge are chosen asfollows: gNSi = 8.71 eV nm�2, and gNGe = 6.05 eV nm�2.25 Fig. 2shows the values of chemical potential as a function of layerthickness and surface curvature. We can find that the values ofthe chemical potential increase with increasing layer thicknessand surface curvature, which suggests that the thickness ofepitaxial layer at the region with low surface curvature is largerthan that at the region with high surface curvature.

At the subsequent stage, as the epitaxial layer continues togrow, the lattice mismatch-induced strain stored in the epitaxiallayer cannot be maintained and needs to be released. Thus, inorder to effectively release the strain, three-dimensional (3D)QDs can form on the epitaxial wetting layer when its thicknessexceeds a critical value. The growth mode transition from the 2Dlayer growth to the formation of 3D QDs is thus determined bythe formation energy of QDs, i.e. the total energy difference

Fig. 1 Schematic illustrations of QD formation on patterned substrates. (a) Top view of patterned substrates; (b) sectional view of the surface with a negativecurvature (k o 0, line b in (a)); (c) sectional view of the surface with a positive curvature (k 4 0, line c in (a)).

Fig. 2 The values of chemical potential as a function of epitaxial layer thicknessunder different surface curvatures for Ge layer on Si substrate. The inset is theamplificatory part for the thickness of the epitaxial layer from 3.6 to 4.4.

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5240 Phys. Chem. Chem. Phys., 2013, 15, 5238--5242 This journal is c the Owner Societies 2013

between the two growth modes, the formation of a 3D QD on thewetting layer and the further 2D growth, which can be written as24,26

DE = [gsA1V2/3 � g(h)A2V2/3] � oA3e02V � A[g(h0 + V/A) � g(h0)]

(3)

where gs is the surface energy density of the QD’s side facet, g(h)and h represent the surface energy density and thickness of thewetting layer which is covered by the QD, i.e. underneath theQD, V is the volume of QD, o is a elastic constant given by o =Y(1 + u)/(1 � u) in which Y and u are Young’s modulus andPoisson ratio,27 A1, A2 and A3 are the shape factors which aredetermined by the shape of QD, A is the area of occupancy by asingle QD which is named after occupancy area of QD and isequal to the capture area by a single QD (a detailed explanationcan be found in the ESI†). Occupancy area of QD is determinedby the density of QDs (the number of QDs per unit area) and 1/Arepresents the density of QDs. g(h0) and g(h0 + V/A) represent theaverage surface energy densities of wetting layer in occupancyarea of QD with the average thickness of h0 and (h0 + V/A), where

(h0 + V/A) represents the thickness of wetting layer in the case ofthe 2D growth mode where the further deposition has the samevolume as that of QD. Note that, the first two terms in eqn (3)represent the change of energy caused by the formation of QD,and the last term represents the change of energy in the case ofthe 2D layer growth mode.

3. Results and discussion

We can compare the values of DE for different sites to judgewhich one is more favorable for QD formation using eqn (3).When we calculate the value of DE for a given site, g(h)represents the surface energy density of the wetting layercovered by the QD, g(h0) represent the average surface energydensity of the wetting layer in the occupancy area by the QD. Wecan note that the average surface energy density, g(h0), is relatedto the size of occupancy area if the substrate surface ispatterned. More in detail, the chemical potentials at a differentsite on a patterned substrate are different, which leads tothe nonuniformity of wetting layer thickness. Therefore, theaverage surface energy density of the wetting layer is determinedby the size of occupancy area if the location of QD is fixed. Herewe will discuss the formation of QD in two typical cases, whenthe occupancy area is larger than the periodicity of the patternedsubstrate surface, and when the surface energy densities of thewetting layer in occupancy area are uniform.

When the occupancy area is larger than the periodicity ofthe patterned substrate surface, the average surface energydensities of the wetting layer in occupancy area can be con-sidered to have the same value as that with deposited thickness.Furthermore, the all average surface energy densities of thewetting layer for different formation sites are equal. Therefore,the main difference of QD formation energy (eqn (3)) ondifferent sites is determined by the surface energy density ofthe wetting layer covered by the QD (g(h)). In this case, the sitewith high surface energy density, i.e. small thickness and highsurface curvature, is more favorable for QD formation. Fig. 3compares the QD formation energy as a function of volume onthree different substrate surface curvature, �0.01 nm�1, 0, and0.01 nm�1. In our calculation, the thickness of the wetting layeron planar substrate (surface curvature k = 0) is set to 4.0 ML. Inthis case, in order to achieve the equality of chemical potentialat each region, the thickness of the wetting layer on the concavesubstrate surface with k =�0.01 nm�1 is about 4.2 ML, and that onthe convex substrate surface with k = 0.01 nm�1 is about 3.84 ML,as shown the inset in Fig. 2. We can find that the formation energyof QD on a convex substrate surface is lower than that on a planarand concave substrate surface, which suggests it is more favorablefor QD to form on a convex substrate surface.

However, when the occupancy area by QD is smaller thanthe periodicity of the patterned substrate surface, the averagesurface energy densities of the wetting layer in occupancy areacan not be considered to have the same value for different sites.We take an extraordinary instance as an example, where thesubstrate surface curvature in occupancy area is near to constant,i.e. the surface energy densities in occupancy area are uniform.

Fig. 3 QD formation energy (DE in eqn (3)) as a function of QD volume on threedifferent substrate surface curvature, �0.01 nm�1 (blue line), 0 (black line), and0.01 nm�1 (red line). (a) Occupancy area by QD is larger than the periodicity ofthe patterned substrate surface; (b) occupancy area by QD is much smaller thanthe periodicity of the patterned substrate surface. In our calculation, the shape ofQD is pyramid and contact angle a = 11.31. According to the geometricalrelations, shape factors, A1, A2 and A3, can be given by A2 = (6/tana)2/3,A1 = A2/cosa, A3 = 0.9922 tana.27

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In this case, each occupancy area can be considered as aseparate region to calculate the change of energy. Therefore,the average thickness of the wetting layer in occupancy area(h0 in eqn (3)) is equal to the thickness of the wetting layercovered by the QD (h in eqn (3)). Under the same depositedamount as that in Fig. 3(a) (4.0 ML on planar substrate), the QDformation energy on a concave substrate surface is lower thanthat on a planar and convex substrate surface, which is quitecontrary to the case of large occupancy area, as shown inFig. 3(b). The results suggest that the formation of QD on aconcave substrate surface is more favorable than that on planarand convex substrate surface in the case of small occupancy area.

The contrary formation sites for the two cases are deter-mined by the occupancy area of QD. When the occupancy areais larger than the periodicity of the patterned substrate surface,the changes of energy caused by 2D layer growth mode (the lastterm of eqn (3)) for different formation sites are equal to eachother. Therefore, the main difference of QD formation energieson different sites is the surface energy density of the wettinglayer covered by the QD (g(h) in eqn (3)). Because the thicknessof the epitaxial wetting layer on the surface with high curvatureis smaller than that on the surface with low curvature, thesurface energy density of the wetting layer on the surface withhigh curvature is larger than that on the surface with lowcurvature. So the decrease of surface energy of the wetting layercaused by coverage of QD on the surface with high curvature islarger than that on the surface with low curvature, which resultsin that the QD formation energy on the surface with highcurvature is always lower than that on the surface with lowcurvature. However, when the occupancy area by QD is smallerthan the periodicity of the patterned substrate surface to enablethe surface curvature in occupancy area near to a constant, thesurface energy densities of the wetting layer in occupancy areacan be considered to have a uniform value. In this case, eachoccupancy area can be considered as a separate region tocalculate the change of energy. The change of energy causedby 2D layer growth plays a key role for the total energy, and thewetting layer on the surface with low curvature first achieves thecritical thickness for QD formation due to its larger thicknessthan that on the surface with high curvature. Therefore, thesurface with low curvature is more favorable for QD formation.

We can apply our theoretical results to explain some inter-esting and puzzling experimental observations. For example, inthe system of Ge QDs on hole-patterned Si substrates, Ge QDsprefer to form on the inside of holes at 550 1C, but only form onthe terraces between holes at higher temperature (higher than700 1C).6,7 The high (low) temperature means a large (small)occupancy area by QD.28 Therefore, Ge QDs form on the inside ofholes (surface curvature k o 0) at low temperature, but form onthe terraces surface with high surface curvature (k E 0) at hightemperature. Furthermore, it has also been reported that Ge QDsform at more stable points such as convex structures at highergrowth temperatures on patterned Si(001).10 These experimentalobservations are consistent with our theoretical results.

Our model shows that the preferred formation sites stronglydepend on the comparison between the occupancy area by QD

and the periodicity of the patterned substrate surface. Ingeneral, if occupancy area is larger than the periodicity of thepatterned substrate surface, then QDs form on the surface withhigh surface curvature; if the periodicity of the patternedsubstrate surface is much larger than occupancy area by QD,then QDs form on the surface with low surface curvature. Thesetrends may help to explain the puzzling effect of the distancebetween the pattern holes on changing the QD formationsite.8,9 For the large distance between the pattern holes case,the periodicity of the patterned substrate may be much largerthan the occupancy area by QD, which leads to QD formationon the surface with low surface curvature. However, when thedistance between the pattern holes is reduced, the occupancyarea may become larger than the periodicity of the patternedsubstrate. In this case, QD would prefer to form on the surfacewith high surface curvature. This explains the experimentalsuccess in directing QD formation on the inside of holes in thecase of large distance between the pattern holes, or on theterrace between holes when the distance between the patternholes is reduced.8,9

4. Conclusion

In summary, we have presented a thermodynamic model toelucidate the formation of QDs on patterned substrates. Thetheoretical analyses show that the preferred formation sitesstrongly depend on the comparison between the occupancyarea by QD and the periodicity of the patterned substratesurface. The model explains some interesting and puzzlingexperimental observations, which implies that the establishedthermodynamic approach could be applicable to the physicalunderstanding of the QDs on patterned substrates.

Acknowledgements

This research was supported by National Natural Science Founda-tion of China (Grant No. 11104084) and Natural Science Founda-tion of Guangdong Province (Grant No. S2011040003245).

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