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Modification of Stranski–Krastanov growth on the surface of nanowires
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 Nanotechnology 25 435605 (http://iopscience.iop.org/0957-4484/25/43/435605) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 130.207.50.37 This content was downloaded on 11/11/2014 at 21:46
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Nanotechnology Nanotechnology 25 (2014) 435605 (5pp)
doi:10.1088/0957-4484/25/43/435605
Modification of Stranski–Krastanov growth on the surface of nanowires Xinlei Li1 and Guowei Yang2 1
MOE Key Laboratory of Laser Life Science & Institute of Laser Life Science, College of Biophotonics, South China Normal University, Guangzhou 510631, People’s Republic of China 2 State Key Laboratory of Optoelectronic Materials and Technologies, Institute of Optoelectronic and Functional Composite Materials, Nanotechnology Research Center, School of Physics & Engineering, Sun Yat-sen University, Guangzhou 510275, People’s Republic of China E-mail: [email protected] Received 15 July 2014 Accepted for publication 3 September 2014 Published 9 October 2014 Abstract
The heteroepitaxial growth of strained islands on a planar substrate offers an attractive route to the fabrication of quantum dots (QDs). To obtain more functions and superior properties, recent efforts have focused on using nanowires (NWs) as substrates to produce attractive structures that combine QDs with NWs. As the lateral size of an NW is large, it is possible that islands are formed on the side walls of the NW. However, no islands exist, and the lateral surface is rather smooth in thin, core–shell NWs. The existing theoretical models on the growth on planar and patterned substrates are not appropriate for the growth transition on the surface with nanoscale curvature. We thus urgently need to understand the basic physics involved in the strain-induced growth on the surface with nanoscale curvature. Here, we established a theoretical model to study the strain-induced growth on the surface, which showed that the Stranski–Krastanov (SK) mode can change to the Frank–van der Merwe (FM) mode due to the limit of the surface to the island’s lateral growth. Using the model to investigate the heterostructured core/shell nanowires (NWs), we found, in addition to the SK mode on thick NWs and the FM mode on thin NWs, that there is a multiplex mode on medium NWs which includes the initial layer growth, the intermediate islands’ growth and the final layer growth again. The established theoretical model not only explained some puzzling experimental results but also provided useful information to design and control the epitaxial growth on the surface with nanoscale curvature. Keywords: quantum dots, core/shell heterostructure, nanowires, thermodynamics (Some figures may appear in colour only in the online journal) 1. Introduction
NW [8, 9]. Similarly, InAs islands on GaAs NW have also already been observed experimentally [11, 12]. However, no islands exist, and the lateral surface is rather smooth in both thin Si/Ge [8, 9] and GaAs/InAs [11, 12] core–shell NWs due to the effects of the nanoscale surface curvature. These observations strongly suggest that the island’s formation on thick NWs occurs in the SK mode, and the smooth layer growth on thin NWs obeys the Frank–van der Merwe (FM, i.e. layer-by-layer) mode. The island’s formation in the SK mode has been widely studied in theory [15–19]. Furthermore, some models have been established to study the island’s nucleation sites on patterned substrates [20, 21]. However, these theoretical
The heteroepitaxial growth of highly strained islands has been quite attractive as it offers an attractive route to the fabrication of semiconductor quantum dots (QDs). The formation of islands follows a strain-driven Stranski–Krastanov (SK) mechanism because the lattice constant of the epitaxial film is much larger than that of the substrate [1–7]. To obtain more functions and superior properties, recent efforts have focused on using nanowires (NWs) as substrates to produce attractive core/shell structures that combine QDs with NWs [8–14]. For the Si/Ge system, as the lateral size of a Si NW is large, it is possible that Ge islands are formed on the side walls of Si 0957-4484/14/435605+05$33.00
1
© 2014 IOP Publishing Ltd Printed in the UK
Nanotechnology 25 (2014) 435605
X Li and G Yang
models are not appropriate for the SK transition on the surface with nanoscale curvature. Due to the effects of nanoscale curvature, the growth not only keeps the SK mode on thick NWs but also changes to the FM mode on thin NWs. This approach appears intuitively simple, but the underlying physical mechanism is still far from complete. In addition to the transition from the SK mode to the FM mode with the increase in surface curvature, experiments to date have shown some interesting and puzzling behaviors, such as the thicker critical epitaxial layer for the islands’ formation on the NWs’ surface than that on the planar substrate [11, 12] and the reversible growth transition from the islands’ growth to layer-by-layer growth with the increased deposited amount [13]. The experimental results are not adequately explained. Therefore, in this letter, we develop a theoretical model to elucidate the modification of straininduced growth on the surface with nanoscale curvature and answer two critical questions: (1) What is the relationship between the nanoscale surface curvature and the growth mode? (2) How do the critical layer’s thickness and critical islands’ size in their formation on the surface with nanoscale curvature differ from those on the planar substrate?
Figure 1. (a) Energy change per atom to add a Ge layer with the
thickness of θ ML on a Si(001) substrate. The horizontal dotted line, solid line, dash dot line and dash line (i.e. the four horizontal lines from top to bottom) represent the chemical potentials of the critical island for formation, the indefinitely large pyramid island with four {105} facets, the indefinitely large dome island and the indefinitely large island without strain through full relief by misfit dislocations. (b1)–(b6) are the schematic illustrations of the evolution of the layer and the islands during the growth process. During the initial growth process, the layer can grow thicker, and layer-by-layer growth is favorable because μlayer < μisland ((b1)–(b2)). For the late-stage growth, the islands’ formation becomes more favorable when μlayer > μisland . As the islands grow up, μisland decreases, and the layer becomes thin to achieve a thermodynamic equilibrium ((b3)–(b6)).
2. Theoretical model We consider a two-dimensional layer with the thickness of θ monolayers (ML, number of atomic layers) on a curved surface. The energy required per atom to add the layer from the islands at chemical potential μisland is: Δμ = μlayer − μisland
is the strain mismatch of the layer. The first term represents the effect of the surface curvature, the second term is caused by the surface energy change with the layer thickness [22] and the last term represents the contribution of the elastic strain energy. Next, we will firstly test the applicability of the model for the epitaxial growth on a planar substrate (surface curvature k = 0 ) before investigating the growth on the nanosized curve’s surface. Figure 1 shows the modeling results for the Ge layer on a planar Si (001) substrate, μlayer vs θ . We can find that μlayer increases with the increase in the layer thickness due to the change of the thickness-dependent surface energy of layer. The equilibrium thickness of the Ge layer can be known directly by paying attention to the cross between the layer energy change μlayer and the horizontal line of μisland . The formation and growth of the islands are driven by the elastic relaxation. In other words, the elastic strain energy density of the islands is less than that of the layer, i.e.:
(1)
where μlayer represents the energy change of the epitaxial layer from n atoms to (n + 1) atoms. To minimize the total energy and achieve the stable condition, the system needs to balance the energies of the layer and islands. So, the condition for stability is: μlayer = μisland
(2)
If μlayer < μisland , the layer will grow thicker, while if μlayer > μisland , the layer will not grow, and the islands’ growth will be favorable. In order to determine the values of μlayer and μisland , we choose the chemical potential of the bulk epitaxial material with no strain as reference value μ = 0 . In this case, the energy change of an epitaxial layer from n atoms to (n + 1) atoms is determined by the surface energy and elastic strain energy of the layer, i.e. [22]: μlayer = Ωγ (h) k + Ω
∂γ ( h ) + Ωєlayer ∂h
єisland = єlayer − єrelax
(4)
where єrelax is the elastic relaxation energy of the islands per the unit volume, which can be calculated by [17]:
(3)
where Ω is the atomic volume, γ (h ) is the surface energy density of the layer with the thickness of h (h = θh 0 , h 0 is the thickness of the monolayer) [23], k is the surface curvature and єlayer is the elastic strain energy density. The strain energy density can be calculated by єlayer = 2G [(1 + υ) (1 − υ) ] ε 2 , where G and υ are the shear modulus and Poisson ratio and ε
єrelax = J tan φ
Y (1 + υ) 2 ε 1−υ
(5)
where φ is the tilt angle of the island side facets, Y is the Young’s modulus and J is the numerical shape factor. On the other hand, the islands’ formation results in the increase of the surface area. Therefore, the total energy of a coherent island 2
Nanotechnology 25 (2014) 435605
X Li and G Yang
can be expressed as Eisland = ⎡⎣A1 γf − A2 γ (h ) ⎤⎦ V 2 3 + єisland V . So, μisland can be calculated by: μisland = (2 3) Ω ⎡⎣A1 γf − A2 γ (h) ⎤⎦ V −1 3 + Ωєisland
(6)
where γf represents the surface energy density of the island facets, A1 and A2 are the shape factors of the island and V represents the volume of the island. We can find that μisland decreases and tends to be a critical minimum value as the volume increases due to the decrease of surface/volume ratio −1 3 (V ). The minimum value of μisland corresponds to the case of an indefinitely large island (V → ∞ or V −1 3 → 0), where the contribution of the surface energy can be neglected, as shown by the horizontal solid line in figure 1. Figure 1 explains the origin of the SK growth in the strained-induced heteroepitaxy on the planar substrate. During the initial growth process, μlayer is always less than μisland , which means that the layer can grow thicker, and layer-bylayer growth is favorable. For late-stage growth, μlayer becomes increasingly larger as the thickness of the layer increases. When μlayer > μisland , the islands’ formation becomes more favorable than the layer growth. But only islands that are a size larger than a critical value can form on ′ , the layer. The critical size can be estimated by μisland = μisland ′ where the value of μisland is set as the maximum of μlayer , as shown by the horizontal dot line in figure 1. The critical lateral size for the Ge islands on Si (001) is calculated to be ′ = about 12 nm (V ≈ 60 nm3), according to μisland = μisland 33 meV/atom. The modeling results agree well with the minimum Ge island size, as observed by several different experimental groups [1–4]. From figure 1, the four ML layers are expected to form the first island, which has also been observed in the experiments [4–6]. As the islands grow up, μisland decreases, and the cross point of the μlayer curve line and the μisland line moves to the small thickness of the layer, which suggests that the layer not only fails to capture newly deposited atoms that instead contribute to growth but also releases atoms into the islands and thins to achieve a thermodynamic equilibrium during the islands’ growth process (figures 1(b3)–(b6)). The theoretical results are in good agreement with the experimental observations [3, 7] in which Ge atoms move from the wetting layer into the islands during annealing, and the thickness of the layer decreases with the rising substrate temperature. The strained-induced growth on the surface with the nanoscale curvature is more complex than that on a planar substrate. Because the surface is curved at the nanoscale, the layer-by-layer growth can effectively release the mismatch strain stored in the first few layers [21, 24, 25]. Before the strain is completely released, the strain at the tangential direction of NW has a relation with the layer thickness such that: εt = ε0 − kh (1 − ε0 ) ,
Figure 2. The values of μlayer and μ′island as the function of the Ge layer thickness on the surface of Si NWs with different diameters of (a) 40 nm, (b) 80 nm and (c) 120 nm. The insets in (a), (b) and (c) show the schematic three growth modes, the FM mode, the LIL mode and the SK mode, respectively.
Therefore, the average strain is: ε=
ε0 + εt 2
(8)
The decrease of the surface strain of the layer not only directly affects єlayer but also reduces єisland [25]. In addition to the effect of the curved surface on the strain release, the size of surface can also limit the lateral growth of the islands; in other words, there is a maximal lateral size of the islands. ′ corresponds Therefore, the lowest chemical potential μisland ′ to that of the largest islands. By comparing μisland and μlayer , we can judge which growth mode is more favorable. In par′ , the layer-by-layer growth is favorticular, if μlayer < μisland ′ , the islands are able to form on able, while if μlayer > μisland the NW’s surface.
(
)
3. Results and discussion
′ Figure 2 shows the values of μlayer and μisland as the function of the Ge layer thickness on the surface of the Si NWs with different diameters. The rapid increase of μlayer is caused by the high decrease rate of the surface energy for a few atomic layers; μlayer subsequently decreases with the increase in the layer thickness because the self-relaxation of the epitaxial layer becomes more effective than the surface energy of the layer. The final, nearly unchanged μlayer is due to the complete strain release and low decrease rate of the surface energy. From figure 2(a), we find that μlayer is always smaller ′ than μisland when the NW’s diameter is 40 nm, which means that it is unfavorable to form islands on such a thin NW; in other words, the FM growth mode is favorable. When ′ dNW = 120 nm , shown in figure 2(c), μisland becomes less than μlayer when the layer exceeds a critical thickness, which
(7)
and the longitudinal strain remains a constant ε0 [24, 25]. 3
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suggests that the islands can form on the layer when the layer reaches a certain thickness. The growth in this case is the typical SK growth mode. However, when dNW = 80 nm , the ′ have two intersection points, as lines of μlayer and μisland shown in figure 2(b). The growth process of the case can be described as follows: In the initial growth stage, the layer-by′ ; when layer growth is more favorable because μlayer < μisland ′ the layer exceeds a critical thickness, μisland becomes less than μlayer , and the islands can form on the layer. A further increase in the layer thickness can cause μlayer to become less than ′ again, and the final growth is layer-by-layer growth, as μisland shown by the schematic growth illustration inserted in figure 2(b). The multiplex growth mode is the transition from ‘layer growth to the islands’ growth to layer growth,’ so we name it the ‘LIL’ mode for short. The modified growth on the surface of the NWs strongly depends on the size of the NWs. The existing theoretical models consider that the nanosized curved surface of the NW could help release the mismatched strain, and the competition between the relaxation of the islands and the self-relaxation of the epitaxial layer determines the final growth behavior [25– 27]. However, these models do not explain the origin of the transition from the islands’ growth to the layer growth (figure 2(b)). Based on the presented model in this paper, we find that only the self-relaxation of the epitaxial layer is the apparent cause of the modified growth, but the real cause is the size limit of NW to the islands’ lateral growth. The major role of the strain release caused by the curved NW surface was to increase the critical size of the islands for their formation. The critical lateral size of the Ge islands on the planar Si substrate is about 12 nm, as described in figure 1. If we do not consider the strain release caused by the curved NW surface, the islands are not able to form on the NW’s surface when the radius of the NWs is smaller than the critical size of the islands. Due to the effects of the strain release of the layer, the critical size of the islands on the NW’s surface becomes larger than that on the planar substrate. The result can be obtained from figure 2. From figure 2(a), the island with a lateral size of 20 nm on the surface of the NW with a diameter of 40 nm always has a higher chemical potential than μlayer , which means the critical size of the islands’ formation should be larger than 20 nm. However, the island cannot grow up to the critical size due to the size limit of the NW. Therefore, layer-by-layer growth is more favorable in this case. When the diameter of NW is large enough, the critical size of the islands’ formation becomes smaller than the NW size due to two reasons: the first is the increase of NW size, and the second is the reduction of the critical size of the islands caused by the decrease of the strain release of the layer with the increase in the NW size. In this situation, the formation of the islands is the highly efficient approach of the strain release, and the SK growth mode is favorable, as shown in figure 2(c). For the multiplex LIL growth mode, as described in figure 2(b), there are two intersection points between the ′ . In the initial growth stage, lines of μlayer and μisland ′ μlayer < μisland because of the high decrease rate of the surface energy of the layer with the increase in its thickness. When
Figure 3. A two-dimensional phase diagram on the (dNW − θ ) plane characterizes the interrelated effects of the NW diameter and the layer thickness on the surface morphology and the growth mode on ′ the NW surface. μlayer < μisland represents a smooth layer on the ′ NW’s surface, and μlayer > μisland suggests the islands’ formation on the NW’s surface.
′ becomes less than the layer exceeds a critical thickness, μisland μlayer , which means the critical size of the islands is less than the NW size. However, as the layer thickness further increases, the mismatched strain of the islands decreases due to the self-relaxation of the layer, which causes the critical size of the islands to become larger than the NW size again. Therefore, the final growth becomes a layer-by-layer mode on the medium NW surface. We can deduce the phase diagram of the surface morphology of the strained-induced growth on the NW surface as a function of NW diameter and layer thickness by calculating ′ . The line of μlayer = μisland ′ μlayer = μisland divides the growth ′ morphology into two types: μlayer < μisland represents a ′ smooth layer on the NW surface, and μlayer > μisland suggests the islands’ formation on the NW surface. For a NW with a certain diameter, the growth mode on its surface can be distinguished into three types, according to the change of the ′ with an increase in the layer relation of μlayer and μisland ′ for a thickness. The first is that μlayer is always less than μisland thin NW, which corresponds to the FM growth mode; the ′ second is that μisland becomes less than μlayer when the layer thickness exceeds a certain value for a thick NW, which means the SK growth mode; the last one is that μlayer is firstly ′ ; then, it becomes larger than μisland ′ . Finally, it less than μisland ′ again for the NW with a medium becomes less than μisland diameter, which corresponds to the transition from the layer growth to the islands’ growth to the layer growth again, i.e. the LIL growth mode. From figure 3, we can find that there is a critical NW diameter of about 70 nm. When the diameter of Si NW is less ′ , and than the critical radius, μlayer is always less than μisland the layer-by-layer growth is favorable. The calculated critical diameter agrees well with the experiments [8, 14] in which Ge 4
Nanotechnology 25 (2014) 435605
X Li and G Yang
islands can only grow on the surface of Si NW with a diameter of 100 nm [8], and the smooth core–shell structure can form when the diameter of the Si NW is less than 50 nm [8, 14]. We also note that the critical thickness of the layer for the formation of the islands on NWs increases as the NW diameter decreases. The change trend of the critical thickness has been observed experimentally in InAs islands on GaAs NWs [11, 12].
[7] Medeiros-Ribeiro G, Kamins T I, Ohlberg D A A and Williams R S 1998 Annealing of Ge nanocrystals on Si(001) at 550 °C: metastability of huts and the stability of pyramids and domes Phys. Rev. B 58 3533–6 [8] Pan L, Lew K K, Redwing J M and Dickey E C 2005 StranskiKrastanow growth of germanium on silicon nanowires Nano Lett. 5 1081–5 [9] Wang H, Upmanyu M and Ciobanu C V 2008 Morphology of epitaxial core–shell nanowires Nano Lett. 8 4305–11 [10] Dong Y, Yu G, McAlpine M C, Lu W and Lieber C M 2008 Si/a-Si core/shell nanowires as nonvolatile crossbar switches Nano Lett. 8 386–91 [11] Yan X, Zhang X, Ren X, Huang H, Guo J, Guo X, Liu M, Wang Q, Cai S and Huang Y 2011 Growth of InAs quantum dots on GaAs nanowires by metal organic chemical vapor deposition Nano Lett. 11 3941–5 [12] Yan X, Zhang X, Ren X, Lv X, Li J, Wang Q, Cai S and Huang Y 2012 Formation mechanism and optical properties of InAs quantum dots on the surface of GaAs nanowires Nano Lett. 12 1851–6 [13] Rieger T, Luysberg M, Schäpers T, Grützmacher D and Lepsa M I 2012 Molecular beam epitaxy growth of GaAs/ InAs core–shell nanowires and fabrication of InAs nanotubes Nano Lett. 12 5559–64 [14] Lauhon L J, Gudiksen M S, Wang C L and Lieber C M 2002 Epitaxial core–shell and core-multishell nanowire heterostructures Nature 420 57–61 [15] Tersoff J 1991 Stress-induced layer-by-layer growth of Ge on Si(100) Phys. Rev. B 43 9377–80 [16] Mui D S L, Leonard D, Coldren L A and Petroff P M 1995 Surface migration induced self-aligned InAs islands grown by molecular beam epitaxy Appl. Phys. Lett. 66 1620–2 [17] Shchukin V A, Bimberg D, Munt T P and Jesson D E 2004 Elastic interaction and self-relaxation energies of coherently strained conical islands Phys. Rev. B 70 085416 [18] Lu G-H and Liu F 2005 Towards quantitative understanding of formation and stability of Ge hut islands on Si(001) Phys. Rev. Lett. 94 176103 [19] Li X L, Wang C X and Yang G W 2014 Thermodynamic theory of growth of nanostructures Prog. Mater. Sci. 64 121–99 [20] Hu H, Gao H J and Liu F 2008 Theory of directed nucleation of strained islands on patterned substrates Phys. Rev. Lett. 101 216102 [21] Yang B, Liu F and Lagally M G 2004 Local strain-mediated chemical potential control of quantum dot self-organization in heteroepitaxy Phys. Rev. Lett. 92 025502 [22] Li X L 2013 Selective formation mechanisms of quantum dots on patterned substrates Phys. Chem. Chem. Phys. 15 5238–42 [23] Li X L and Yang G W 2008 Theoretical determination of contact angle in quantum dot self-assembly Appl. Phys. Lett. 92 171902 [24] Li X L, Ouyang G and Yang G W 2007 Thermodynamic theory of nucleation and shape transition of strained quantum dots Phys. Rev. B 75 245428 [25] Li X L and Yang G W 2009 Strain self-releasing mechanism in heteroepitaxy on nanowires J. Phys. Chem. C 113 12402–6 [26] Cao Y Y, Ouyang G, Wang C X and Yang G W 2013 Physical mechanism of surface roughening of the radial Ge-core/Sishell nanowire heterostructure and thermodynamic prediction of surface stability of the InAs-core/GaAs-shell nanowire structure Nano Lett. 13 436–43 [27] Schmidt V, McIntyre P C and Goesele U 2008 Morphological instability of misfit-strained core–shell nanowires Phys. Rev. B 77 235302
4. Conclusions In conclusion, we have established a theoretical model to elucidate the SK transition in the strain-induced growth on the surface with the nanoscale curvature. The theory shows that the SK growth mode can be changed to the FM mode when the islands cannot grow up to the critical size for their formation due to the limit of the nanoscale curved surface. In addition to the SK mode on thick NWs and the FM mode on thin NWs, the presented model predicts that there is a multiplex mode on the NWs with a medium diameter in which the critical size of the islands ranges from smaller to larger than the NWs’ size with the increase in the epitaxial layer thickness. The multiplex mode corresponds to the initial layer growth, the intermediate islands’ growth and the final layer growth again. The multiplex mode provides a new method to fabricate multiplex nanostructures. The established theoretical model in this letter not only explains some puzzling experimental results, such as the transition of the growth mode, but also provides some useful guidelines for future experimental design.
Acknowledgments This work is supported by the National Natural Science Foundation of China (11104084 and 91233203).
References [1] Steinfort A J et al 1996 Strain in nanoscale germanium hut clusters on Si(001) studied by x-ray diffraction Phys. Rev. Lett. 77 2009–12 [2] Jesson D E, Kästner M and Voigtländer B 2000 Direct observation of subcritical fluctuations during the formation of strained semiconductor islands Phys. Rev. Lett. 84 330–3 [3] Liu C P, Gibson J M, Cahill D G, Kamins T I, Basile D P and Williams R S 2000 Strain evolution in coherent Ge/Si islands Phys. Rev. Lett. 84 1958–61 [4] Vailionis A, Cho B, Glass G, Desjardins P, Cahill D G and Greene J E 2000 Pathway for the strain-driven twodimensional to three-dimensional transition during growth of Ge on Si(001) Phys. Rev. Lett. 85 3672–5 [5] Eaglesham D J and Cerullo M 1990 Dislocation-free StranskiKrastanow growth of Ge on Si(100) Phys. Rev. Lett. 64 1943–6 [6] Costantini G, Rastelli A, Manzano C, Acosta-Diaz P, Katsaros G, Songmuang R, Schmidt O G, Känel H V and Kern K 2005 Pyramids and domes in the InAs/GaAs(001) and Ge/Si(001) systems J. Cryst. Growth 278 38–45
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Modification of Stranski–Krastanov growth on the surface of nanowires
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 Nanotechnology 25 435605 (http://iopscience.iop.org/0957-4484/25/43/435605) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 130.207.50.37 This content was downloaded on 11/11/2014 at 21:46
Please note that terms and conditions apply.
Nanotechnology Nanotechnology 25 (2014) 435605 (5pp)
doi:10.1088/0957-4484/25/43/435605
Modification of Stranski–Krastanov growth on the surface of nanowires Xinlei Li1 and Guowei Yang2 1
MOE Key Laboratory of Laser Life Science & Institute of Laser Life Science, College of Biophotonics, South China Normal University, Guangzhou 510631, People’s Republic of China 2 State Key Laboratory of Optoelectronic Materials and Technologies, Institute of Optoelectronic and Functional Composite Materials, Nanotechnology Research Center, School of Physics & Engineering, Sun Yat-sen University, Guangzhou 510275, People’s Republic of China E-mail: [email protected] Received 15 July 2014 Accepted for publication 3 September 2014 Published 9 October 2014 Abstract
The heteroepitaxial growth of strained islands on a planar substrate offers an attractive route to the fabrication of quantum dots (QDs). To obtain more functions and superior properties, recent efforts have focused on using nanowires (NWs) as substrates to produce attractive structures that combine QDs with NWs. As the lateral size of an NW is large, it is possible that islands are formed on the side walls of the NW. However, no islands exist, and the lateral surface is rather smooth in thin, core–shell NWs. The existing theoretical models on the growth on planar and patterned substrates are not appropriate for the growth transition on the surface with nanoscale curvature. We thus urgently need to understand the basic physics involved in the strain-induced growth on the surface with nanoscale curvature. Here, we established a theoretical model to study the strain-induced growth on the surface, which showed that the Stranski–Krastanov (SK) mode can change to the Frank–van der Merwe (FM) mode due to the limit of the surface to the island’s lateral growth. Using the model to investigate the heterostructured core/shell nanowires (NWs), we found, in addition to the SK mode on thick NWs and the FM mode on thin NWs, that there is a multiplex mode on medium NWs which includes the initial layer growth, the intermediate islands’ growth and the final layer growth again. The established theoretical model not only explained some puzzling experimental results but also provided useful information to design and control the epitaxial growth on the surface with nanoscale curvature. Keywords: quantum dots, core/shell heterostructure, nanowires, thermodynamics (Some figures may appear in colour only in the online journal) 1. Introduction
NW [8, 9]. Similarly, InAs islands on GaAs NW have also already been observed experimentally [11, 12]. However, no islands exist, and the lateral surface is rather smooth in both thin Si/Ge [8, 9] and GaAs/InAs [11, 12] core–shell NWs due to the effects of the nanoscale surface curvature. These observations strongly suggest that the island’s formation on thick NWs occurs in the SK mode, and the smooth layer growth on thin NWs obeys the Frank–van der Merwe (FM, i.e. layer-by-layer) mode. The island’s formation in the SK mode has been widely studied in theory [15–19]. Furthermore, some models have been established to study the island’s nucleation sites on patterned substrates [20, 21]. However, these theoretical
The heteroepitaxial growth of highly strained islands has been quite attractive as it offers an attractive route to the fabrication of semiconductor quantum dots (QDs). The formation of islands follows a strain-driven Stranski–Krastanov (SK) mechanism because the lattice constant of the epitaxial film is much larger than that of the substrate [1–7]. To obtain more functions and superior properties, recent efforts have focused on using nanowires (NWs) as substrates to produce attractive core/shell structures that combine QDs with NWs [8–14]. For the Si/Ge system, as the lateral size of a Si NW is large, it is possible that Ge islands are formed on the side walls of Si 0957-4484/14/435605+05$33.00
1
© 2014 IOP Publishing Ltd Printed in the UK
Nanotechnology 25 (2014) 435605
X Li and G Yang
models are not appropriate for the SK transition on the surface with nanoscale curvature. Due to the effects of nanoscale curvature, the growth not only keeps the SK mode on thick NWs but also changes to the FM mode on thin NWs. This approach appears intuitively simple, but the underlying physical mechanism is still far from complete. In addition to the transition from the SK mode to the FM mode with the increase in surface curvature, experiments to date have shown some interesting and puzzling behaviors, such as the thicker critical epitaxial layer for the islands’ formation on the NWs’ surface than that on the planar substrate [11, 12] and the reversible growth transition from the islands’ growth to layer-by-layer growth with the increased deposited amount [13]. The experimental results are not adequately explained. Therefore, in this letter, we develop a theoretical model to elucidate the modification of straininduced growth on the surface with nanoscale curvature and answer two critical questions: (1) What is the relationship between the nanoscale surface curvature and the growth mode? (2) How do the critical layer’s thickness and critical islands’ size in their formation on the surface with nanoscale curvature differ from those on the planar substrate?
Figure 1. (a) Energy change per atom to add a Ge layer with the
thickness of θ ML on a Si(001) substrate. The horizontal dotted line, solid line, dash dot line and dash line (i.e. the four horizontal lines from top to bottom) represent the chemical potentials of the critical island for formation, the indefinitely large pyramid island with four {105} facets, the indefinitely large dome island and the indefinitely large island without strain through full relief by misfit dislocations. (b1)–(b6) are the schematic illustrations of the evolution of the layer and the islands during the growth process. During the initial growth process, the layer can grow thicker, and layer-by-layer growth is favorable because μlayer < μisland ((b1)–(b2)). For the late-stage growth, the islands’ formation becomes more favorable when μlayer > μisland . As the islands grow up, μisland decreases, and the layer becomes thin to achieve a thermodynamic equilibrium ((b3)–(b6)).
2. Theoretical model We consider a two-dimensional layer with the thickness of θ monolayers (ML, number of atomic layers) on a curved surface. The energy required per atom to add the layer from the islands at chemical potential μisland is: Δμ = μlayer − μisland
is the strain mismatch of the layer. The first term represents the effect of the surface curvature, the second term is caused by the surface energy change with the layer thickness [22] and the last term represents the contribution of the elastic strain energy. Next, we will firstly test the applicability of the model for the epitaxial growth on a planar substrate (surface curvature k = 0 ) before investigating the growth on the nanosized curve’s surface. Figure 1 shows the modeling results for the Ge layer on a planar Si (001) substrate, μlayer vs θ . We can find that μlayer increases with the increase in the layer thickness due to the change of the thickness-dependent surface energy of layer. The equilibrium thickness of the Ge layer can be known directly by paying attention to the cross between the layer energy change μlayer and the horizontal line of μisland . The formation and growth of the islands are driven by the elastic relaxation. In other words, the elastic strain energy density of the islands is less than that of the layer, i.e.:
(1)
where μlayer represents the energy change of the epitaxial layer from n atoms to (n + 1) atoms. To minimize the total energy and achieve the stable condition, the system needs to balance the energies of the layer and islands. So, the condition for stability is: μlayer = μisland
(2)
If μlayer < μisland , the layer will grow thicker, while if μlayer > μisland , the layer will not grow, and the islands’ growth will be favorable. In order to determine the values of μlayer and μisland , we choose the chemical potential of the bulk epitaxial material with no strain as reference value μ = 0 . In this case, the energy change of an epitaxial layer from n atoms to (n + 1) atoms is determined by the surface energy and elastic strain energy of the layer, i.e. [22]: μlayer = Ωγ (h) k + Ω
∂γ ( h ) + Ωєlayer ∂h
єisland = єlayer − єrelax
(4)
where єrelax is the elastic relaxation energy of the islands per the unit volume, which can be calculated by [17]:
(3)
where Ω is the atomic volume, γ (h ) is the surface energy density of the layer with the thickness of h (h = θh 0 , h 0 is the thickness of the monolayer) [23], k is the surface curvature and єlayer is the elastic strain energy density. The strain energy density can be calculated by єlayer = 2G [(1 + υ) (1 − υ) ] ε 2 , where G and υ are the shear modulus and Poisson ratio and ε
єrelax = J tan φ
Y (1 + υ) 2 ε 1−υ
(5)
where φ is the tilt angle of the island side facets, Y is the Young’s modulus and J is the numerical shape factor. On the other hand, the islands’ formation results in the increase of the surface area. Therefore, the total energy of a coherent island 2
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can be expressed as Eisland = ⎡⎣A1 γf − A2 γ (h ) ⎤⎦ V 2 3 + єisland V . So, μisland can be calculated by: μisland = (2 3) Ω ⎡⎣A1 γf − A2 γ (h) ⎤⎦ V −1 3 + Ωєisland
(6)
where γf represents the surface energy density of the island facets, A1 and A2 are the shape factors of the island and V represents the volume of the island. We can find that μisland decreases and tends to be a critical minimum value as the volume increases due to the decrease of surface/volume ratio −1 3 (V ). The minimum value of μisland corresponds to the case of an indefinitely large island (V → ∞ or V −1 3 → 0), where the contribution of the surface energy can be neglected, as shown by the horizontal solid line in figure 1. Figure 1 explains the origin of the SK growth in the strained-induced heteroepitaxy on the planar substrate. During the initial growth process, μlayer is always less than μisland , which means that the layer can grow thicker, and layer-bylayer growth is favorable. For late-stage growth, μlayer becomes increasingly larger as the thickness of the layer increases. When μlayer > μisland , the islands’ formation becomes more favorable than the layer growth. But only islands that are a size larger than a critical value can form on ′ , the layer. The critical size can be estimated by μisland = μisland ′ where the value of μisland is set as the maximum of μlayer , as shown by the horizontal dot line in figure 1. The critical lateral size for the Ge islands on Si (001) is calculated to be ′ = about 12 nm (V ≈ 60 nm3), according to μisland = μisland 33 meV/atom. The modeling results agree well with the minimum Ge island size, as observed by several different experimental groups [1–4]. From figure 1, the four ML layers are expected to form the first island, which has also been observed in the experiments [4–6]. As the islands grow up, μisland decreases, and the cross point of the μlayer curve line and the μisland line moves to the small thickness of the layer, which suggests that the layer not only fails to capture newly deposited atoms that instead contribute to growth but also releases atoms into the islands and thins to achieve a thermodynamic equilibrium during the islands’ growth process (figures 1(b3)–(b6)). The theoretical results are in good agreement with the experimental observations [3, 7] in which Ge atoms move from the wetting layer into the islands during annealing, and the thickness of the layer decreases with the rising substrate temperature. The strained-induced growth on the surface with the nanoscale curvature is more complex than that on a planar substrate. Because the surface is curved at the nanoscale, the layer-by-layer growth can effectively release the mismatch strain stored in the first few layers [21, 24, 25]. Before the strain is completely released, the strain at the tangential direction of NW has a relation with the layer thickness such that: εt = ε0 − kh (1 − ε0 ) ,
Figure 2. The values of μlayer and μ′island as the function of the Ge layer thickness on the surface of Si NWs with different diameters of (a) 40 nm, (b) 80 nm and (c) 120 nm. The insets in (a), (b) and (c) show the schematic three growth modes, the FM mode, the LIL mode and the SK mode, respectively.
Therefore, the average strain is: ε=
ε0 + εt 2
(8)
The decrease of the surface strain of the layer not only directly affects єlayer but also reduces єisland [25]. In addition to the effect of the curved surface on the strain release, the size of surface can also limit the lateral growth of the islands; in other words, there is a maximal lateral size of the islands. ′ corresponds Therefore, the lowest chemical potential μisland ′ to that of the largest islands. By comparing μisland and μlayer , we can judge which growth mode is more favorable. In par′ , the layer-by-layer growth is favorticular, if μlayer < μisland ′ , the islands are able to form on able, while if μlayer > μisland the NW’s surface.
(
)
3. Results and discussion
′ Figure 2 shows the values of μlayer and μisland as the function of the Ge layer thickness on the surface of the Si NWs with different diameters. The rapid increase of μlayer is caused by the high decrease rate of the surface energy for a few atomic layers; μlayer subsequently decreases with the increase in the layer thickness because the self-relaxation of the epitaxial layer becomes more effective than the surface energy of the layer. The final, nearly unchanged μlayer is due to the complete strain release and low decrease rate of the surface energy. From figure 2(a), we find that μlayer is always smaller ′ than μisland when the NW’s diameter is 40 nm, which means that it is unfavorable to form islands on such a thin NW; in other words, the FM growth mode is favorable. When ′ dNW = 120 nm , shown in figure 2(c), μisland becomes less than μlayer when the layer exceeds a critical thickness, which
(7)
and the longitudinal strain remains a constant ε0 [24, 25]. 3
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suggests that the islands can form on the layer when the layer reaches a certain thickness. The growth in this case is the typical SK growth mode. However, when dNW = 80 nm , the ′ have two intersection points, as lines of μlayer and μisland shown in figure 2(b). The growth process of the case can be described as follows: In the initial growth stage, the layer-by′ ; when layer growth is more favorable because μlayer < μisland ′ the layer exceeds a critical thickness, μisland becomes less than μlayer , and the islands can form on the layer. A further increase in the layer thickness can cause μlayer to become less than ′ again, and the final growth is layer-by-layer growth, as μisland shown by the schematic growth illustration inserted in figure 2(b). The multiplex growth mode is the transition from ‘layer growth to the islands’ growth to layer growth,’ so we name it the ‘LIL’ mode for short. The modified growth on the surface of the NWs strongly depends on the size of the NWs. The existing theoretical models consider that the nanosized curved surface of the NW could help release the mismatched strain, and the competition between the relaxation of the islands and the self-relaxation of the epitaxial layer determines the final growth behavior [25– 27]. However, these models do not explain the origin of the transition from the islands’ growth to the layer growth (figure 2(b)). Based on the presented model in this paper, we find that only the self-relaxation of the epitaxial layer is the apparent cause of the modified growth, but the real cause is the size limit of NW to the islands’ lateral growth. The major role of the strain release caused by the curved NW surface was to increase the critical size of the islands for their formation. The critical lateral size of the Ge islands on the planar Si substrate is about 12 nm, as described in figure 1. If we do not consider the strain release caused by the curved NW surface, the islands are not able to form on the NW’s surface when the radius of the NWs is smaller than the critical size of the islands. Due to the effects of the strain release of the layer, the critical size of the islands on the NW’s surface becomes larger than that on the planar substrate. The result can be obtained from figure 2. From figure 2(a), the island with a lateral size of 20 nm on the surface of the NW with a diameter of 40 nm always has a higher chemical potential than μlayer , which means the critical size of the islands’ formation should be larger than 20 nm. However, the island cannot grow up to the critical size due to the size limit of the NW. Therefore, layer-by-layer growth is more favorable in this case. When the diameter of NW is large enough, the critical size of the islands’ formation becomes smaller than the NW size due to two reasons: the first is the increase of NW size, and the second is the reduction of the critical size of the islands caused by the decrease of the strain release of the layer with the increase in the NW size. In this situation, the formation of the islands is the highly efficient approach of the strain release, and the SK growth mode is favorable, as shown in figure 2(c). For the multiplex LIL growth mode, as described in figure 2(b), there are two intersection points between the ′ . In the initial growth stage, lines of μlayer and μisland ′ μlayer < μisland because of the high decrease rate of the surface energy of the layer with the increase in its thickness. When
Figure 3. A two-dimensional phase diagram on the (dNW − θ ) plane characterizes the interrelated effects of the NW diameter and the layer thickness on the surface morphology and the growth mode on ′ the NW surface. μlayer < μisland represents a smooth layer on the ′ NW’s surface, and μlayer > μisland suggests the islands’ formation on the NW’s surface.
′ becomes less than the layer exceeds a critical thickness, μisland μlayer , which means the critical size of the islands is less than the NW size. However, as the layer thickness further increases, the mismatched strain of the islands decreases due to the self-relaxation of the layer, which causes the critical size of the islands to become larger than the NW size again. Therefore, the final growth becomes a layer-by-layer mode on the medium NW surface. We can deduce the phase diagram of the surface morphology of the strained-induced growth on the NW surface as a function of NW diameter and layer thickness by calculating ′ . The line of μlayer = μisland ′ μlayer = μisland divides the growth ′ morphology into two types: μlayer < μisland represents a ′ smooth layer on the NW surface, and μlayer > μisland suggests the islands’ formation on the NW surface. For a NW with a certain diameter, the growth mode on its surface can be distinguished into three types, according to the change of the ′ with an increase in the layer relation of μlayer and μisland ′ for a thickness. The first is that μlayer is always less than μisland thin NW, which corresponds to the FM growth mode; the ′ second is that μisland becomes less than μlayer when the layer thickness exceeds a certain value for a thick NW, which means the SK growth mode; the last one is that μlayer is firstly ′ ; then, it becomes larger than μisland ′ . Finally, it less than μisland ′ again for the NW with a medium becomes less than μisland diameter, which corresponds to the transition from the layer growth to the islands’ growth to the layer growth again, i.e. the LIL growth mode. From figure 3, we can find that there is a critical NW diameter of about 70 nm. When the diameter of Si NW is less ′ , and than the critical radius, μlayer is always less than μisland the layer-by-layer growth is favorable. The calculated critical diameter agrees well with the experiments [8, 14] in which Ge 4
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islands can only grow on the surface of Si NW with a diameter of 100 nm [8], and the smooth core–shell structure can form when the diameter of the Si NW is less than 50 nm [8, 14]. We also note that the critical thickness of the layer for the formation of the islands on NWs increases as the NW diameter decreases. The change trend of the critical thickness has been observed experimentally in InAs islands on GaAs NWs [11, 12].
[7] Medeiros-Ribeiro G, Kamins T I, Ohlberg D A A and Williams R S 1998 Annealing of Ge nanocrystals on Si(001) at 550 °C: metastability of huts and the stability of pyramids and domes Phys. Rev. B 58 3533–6 [8] Pan L, Lew K K, Redwing J M and Dickey E C 2005 StranskiKrastanow growth of germanium on silicon nanowires Nano Lett. 5 1081–5 [9] Wang H, Upmanyu M and Ciobanu C V 2008 Morphology of epitaxial core–shell nanowires Nano Lett. 8 4305–11 [10] Dong Y, Yu G, McAlpine M C, Lu W and Lieber C M 2008 Si/a-Si core/shell nanowires as nonvolatile crossbar switches Nano Lett. 8 386–91 [11] Yan X, Zhang X, Ren X, Huang H, Guo J, Guo X, Liu M, Wang Q, Cai S and Huang Y 2011 Growth of InAs quantum dots on GaAs nanowires by metal organic chemical vapor deposition Nano Lett. 11 3941–5 [12] Yan X, Zhang X, Ren X, Lv X, Li J, Wang Q, Cai S and Huang Y 2012 Formation mechanism and optical properties of InAs quantum dots on the surface of GaAs nanowires Nano Lett. 12 1851–6 [13] Rieger T, Luysberg M, Schäpers T, Grützmacher D and Lepsa M I 2012 Molecular beam epitaxy growth of GaAs/ InAs core–shell nanowires and fabrication of InAs nanotubes Nano Lett. 12 5559–64 [14] Lauhon L J, Gudiksen M S, Wang C L and Lieber C M 2002 Epitaxial core–shell and core-multishell nanowire heterostructures Nature 420 57–61 [15] Tersoff J 1991 Stress-induced layer-by-layer growth of Ge on Si(100) Phys. Rev. B 43 9377–80 [16] Mui D S L, Leonard D, Coldren L A and Petroff P M 1995 Surface migration induced self-aligned InAs islands grown by molecular beam epitaxy Appl. Phys. Lett. 66 1620–2 [17] Shchukin V A, Bimberg D, Munt T P and Jesson D E 2004 Elastic interaction and self-relaxation energies of coherently strained conical islands Phys. Rev. B 70 085416 [18] Lu G-H and Liu F 2005 Towards quantitative understanding of formation and stability of Ge hut islands on Si(001) Phys. Rev. Lett. 94 176103 [19] Li X L, Wang C X and Yang G W 2014 Thermodynamic theory of growth of nanostructures Prog. Mater. Sci. 64 121–99 [20] Hu H, Gao H J and Liu F 2008 Theory of directed nucleation of strained islands on patterned substrates Phys. Rev. Lett. 101 216102 [21] Yang B, Liu F and Lagally M G 2004 Local strain-mediated chemical potential control of quantum dot self-organization in heteroepitaxy Phys. Rev. Lett. 92 025502 [22] Li X L 2013 Selective formation mechanisms of quantum dots on patterned substrates Phys. Chem. Chem. Phys. 15 5238–42 [23] Li X L and Yang G W 2008 Theoretical determination of contact angle in quantum dot self-assembly Appl. Phys. Lett. 92 171902 [24] Li X L, Ouyang G and Yang G W 2007 Thermodynamic theory of nucleation and shape transition of strained quantum dots Phys. Rev. B 75 245428 [25] Li X L and Yang G W 2009 Strain self-releasing mechanism in heteroepitaxy on nanowires J. Phys. Chem. C 113 12402–6 [26] Cao Y Y, Ouyang G, Wang C X and Yang G W 2013 Physical mechanism of surface roughening of the radial Ge-core/Sishell nanowire heterostructure and thermodynamic prediction of surface stability of the InAs-core/GaAs-shell nanowire structure Nano Lett. 13 436–43 [27] Schmidt V, McIntyre P C and Goesele U 2008 Morphological instability of misfit-strained core–shell nanowires Phys. Rev. B 77 235302
4. Conclusions In conclusion, we have established a theoretical model to elucidate the SK transition in the strain-induced growth on the surface with the nanoscale curvature. The theory shows that the SK growth mode can be changed to the FM mode when the islands cannot grow up to the critical size for their formation due to the limit of the nanoscale curved surface. In addition to the SK mode on thick NWs and the FM mode on thin NWs, the presented model predicts that there is a multiplex mode on the NWs with a medium diameter in which the critical size of the islands ranges from smaller to larger than the NWs’ size with the increase in the epitaxial layer thickness. The multiplex mode corresponds to the initial layer growth, the intermediate islands’ growth and the final layer growth again. The multiplex mode provides a new method to fabricate multiplex nanostructures. The established theoretical model in this letter not only explains some puzzling experimental results, such as the transition of the growth mode, but also provides some useful guidelines for future experimental design.
Acknowledgments This work is supported by the National Natural Science Foundation of China (11104084 and 91233203).
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