J. Phys. Chem. C 2008, 112, 7693–7697
7693
Growth Mechanisms of Quantum Ring Self-Assembly upon Droplet Epitaxy X. L. Li and G. W. Yang* State Key Laboratory of Optoelectronic Materials and Technologies, Institute of Optoelectronic and Functional Composite Materials, School of Physics & Engineering, Zhongshan UniVersity, Guangzhou 510275, People’s Republic of China ReceiVed: February 16, 2008; ReVised Manuscript ReceiVed: March 10, 2008
A thermodynamic approach has been proposed to address the quantum rings (QRs) self-assembly upon droplet epitaxy. It is found that the selective nucleation on the droplet skirt induced by the high surface energy density of droplet leads to the QR formation at the initial deposition stage, and then the QR growth is controlled by the diffusion of the droplet atoms and the trapping of the deposited atoms, which determines the final size and shape of QRs. Taking the GaAs/AlGaAs system as an example, the established theory nicely elucidates the physical mechanisms of the self-assembly of GaAs nanostructures including the single and double QRs and the holed nanostructure upon droplet epitaxy. Theoretical predictions are consistent with experiments, implying that the proposed thermodynamic model could be expected to be a general approach to pursue the nanostructural self-assembly upon droplet epitaxy. Introduction Semiconductor quantum rings (QRs) have become the focus of intensive research owing to their unique applications in mesoscopic physics and fabrication of nanodevices, i.e., QRs not only provide a good system to study the electronic, optical, and magnetic properties in one-dimensional confinement but also are expected to play an important role in functional units in fabricating electronic, optoelectronic, and magnetic storage devices with nanoscale dimensions.1,2 Therefore, to attain various sizes and shapes of QRs, a lot of self-assembly processes have emerged in recent years.3–13 For instance, the droplet epitaxy,3–8 the lithographic techniques,9 the self-organization technology with a cap layer,10,11 and so on.12,13 Among these techniques above, the lithographic technique has a drawback for the fabrication of small rings, and the self-organization method is only suitable for the lattice-mismatched system, and the strain induced by the additional cap layer has effects on the QRs’ electronic, optical, and magnetic properties. Therefore, very recently, the droplet epitaxy has been intensively investigated as an important self-assembly technique of semiconductor QRs in the lattice-matched systems such as GaAs/AlGaAs3–8 and the lattice-mismatched systems such as InGaAs/GaAs,5 due to its suitability for the lattice-matched systems and the latticemismatched systems, the well-defined strain-free rings, and the controlling size for various applications contrast to other methods. Generally, the QR growth is divided into two stages upon droplet epitaxy. The first stage is the formation of liquid droplets, and the second stage is the crystallization of liquid droplets. In detail, in the GaAs/AlGaAs system, the liquid droplets of Ga first form on the AlGaAs substrate by depositing Ga atoms in the absence of As flux at the first stage, and then the Ga droplets crystallize into the GaAs QRs by supply of As flux at the second stage. However, there are some fundamental issues in the QRs self-assembly upon droplet epitaxy. For example, why does the droplet not crystallize into a quantum dot or other nanostruc* To whom correspondence should be addressed. E-mail: stsygw@ mail.sysu.edu.cn.
Figure 1. Gibbs free energy difference (∆G - ∆gVV) as a function of the nucleus volume for three GaAs nucleations. The inset is the schematic diagram of the GaAs nucleation (green region) on the skirt of the Ga droplet (red region).
ture?4,7,14 Why does the QR crystallization preferably happen on the skirt of droplets? In despite of much progress in experiments, the growth mechanisms of QRs upon droplet epitaxy have not been understood.3–8 Meanwhile, there have been not any qualitative and quantitative theories to address the QRs self-assembly upon droplet epitaxy. Therefore, for this issue, in this contribution, we establish a thermodynamic theory to pursue the QRs self-assembly upon droplet epitaxy, which brings out the new insights to the understanding of the growth mechanisms of QRs. Our theoretical results reveal that the selective nucleation on the droplet skirt induced by the high surface energy density of droplet leads to the QR formation at the initial deposition stage, and then the QR growth is controlled by the diffusion of the droplet atoms and the trapping of the deposited atoms, which determines the final size and shape of QRs. By use of the GaAs/AlGaAs system as an example, we compare the theoretical predictions with the experimental data and find they are nicely consistent. Thermodynamic Approach A. Thermodynamic Nucleation. On the basis of the experiments3–8 the QR formation is determined by two factors
10.1021/jp801528r CCC: $40.75 2008 American Chemical Society Published on Web 04/15/2008
7694 J. Phys. Chem. C, Vol. 112, No. 20, 2008
Li and Yang
TABLE 1: Values of the Related Interface Energy Densities Used in Our Calculations interface energy densities γGa-Al0.3Ga0.7As ) 7.42 eV/nm γGa-GaAs ) 7.85 eV/nm2 γGaAs-Al0.3Ga0.7As ) 4.0 eV/nm2 2
a
related systems (QD/substrate)
size of QD (radius r and height h)
contact angle θa
ref
Ga/Al0.3Ga0.7As Ga/GaAs GaAs/Al0.3Ga0.7As
r ) 40 nm, h ) 19 nm r ) 75 nm, h ) 45 nm r ) 45 nm, h ) 15 nm
50° 60° 37°
3 23 7
Calculated using the relation: θ ) arccos [(1 - (h/r)2)/(1 + (h/r)2)].
upon droplet epitaxy, i.e., one is the QR nucleation, and another one is the growth of QRs nuclei during the droplet expensing. Therefore, we propose the nucleation thermodynamics and the growth kinetics for the two issues above on the basis of the nucleation thermodynamic theory15,16 in this study. By consideration of the As-Ga phase diagram and Wang et al.7 studies, there are three confirmable nucleation sites as shown in Figure 1. One is on the surface of the Ga droplet (situation A); one is on the skirt of the Ga droplet (situation B); one is on the substrate surface (situation C). Thus, we compare the three nucleations to clarify which nucleation is energetically preferable. On the basis of the nucleation thermodynamics,15,16 the total Gibbs free energy difference of a cluster can be expressed as
∆G(i) ) ∆G(i)1 + ∆G(i)2
(1)
where i ) A, B, C and ∆G(i) is the total Gibbs free energy difference for three situations. ∆G(i)1 and ∆G(i)2 are the interface energy difference and the volume energy difference. Note that ∆G(i)2 can be given by ∆G(i)2 ) ∆gVV and can be considered to be equal for the three situations above. Thus, the comparison of the total Gibbs free energy difference ∆G(i) can be instead made by the comparison of the interface energy difference ∆G(i)1. In the situation A, the nucleation happens on the surface of the Ga droplet, and there is no contact between nucleus and substrate. The interface energy difference can be written as
∆G(A)1 ) (γnl - γlV)S1 + γnV S2
(2)
where γnl, γlv, and γnv are the interface energy densities of the nucleus-liquid, the liquid-vapor, and the nucleus-vapor interfaces and S1 and S2 are the nucleus-liquid interface area and the nuclei exposure area to the vapor. The detailed calculations for situation A have been described by our previous works.17 For situation B, we assume that the nucleus has a spherical cap shape as shown in the inset in Figure 1. The interface energy difference is given by
∆G(B)1 ) (γnvS3 + γnlS4 - γlvS5) + (γns - γsv)S6 + (γns - γls)S7 (3) where γns, γsv, and γls are the nucleus-substrate, the substratevapor, and the liquid-substrate interface energy densities and S3-S7 are shown in the inset in Figure 1. In situation C, the nucleation happens on the substrate surface. The interface energy difference becomes as follows
∆G(C)1 ) (γns - γsv)S8 + γnvS9
(4)
where S8 and S9 are the nucleus-substrate interface area and the nuclei exposure area to the vapor. Then, it is noticed that our model is formulated on the basis of the following assumptions: (i) Both nucleus and droplet have a spherical cap shape; (ii) the size of the droplet is considered to be changeless because the volume of nucleus is too small to influence on the droplet; and (iii) the contact angles between nuclei and substrate or droplet are derived by the minimization of the total energy in our calculations.
Figure 2. Schematic illustration of the kinetic diffusion of Ga (red points) and As (blue points) atoms. rGa is the radius of the Ga droplet (red sphere), and the dashed circle denotes radius of the diffusion region of Ga atoms.
We use the proposed model to address the GaAs QR selfassembly in the typical GaAs/Al0.3Ga0.7 As system to check the validity. The GaAs-vapor and AlAs-vapor interface energy densities are 13.75 eV/nm218 and 17.92eV/nm2,19 respectively. The Al0.3Ga0.7As-vapor interface energy density is thus considered to be 15 eV/nm2 by the relationship of (0.3γAlAs-vapor + 0.7γGaAs-vapor). Note that we assume that the Ga droplet-vapor interface energy density is equal to that of Si (001) (11.8 eV/nm2).20,21 Because of a lack of the Ga droplet-Al0.3Ga0.7As, Ga droplet-GaAs, and GaAsAl0.3Ga0.7 As interface energy densities in experimental and theoretical reports, we can get the approximate interface energy densities according to the Young’s equation.22 Table 1 lists these detailed values of parameters in our calculations. Figure 1 shows a comparison of the Gibbs free energy difference of three nucleations. Clearly, the situation B has the lowest Gibbs free energy difference among three situations. This result definitely indicates that the nucleation on the skirt of the Ga droplet is energetically preferable. Further, the physical origin of the nucleation is that the high surface energy density of droplet causes the nucleation around the droplet, which can depress the total surface energy of droplet. In detail, the nucleation on the skirt of the Ga droplet can depress the total surface energy by the reduction of the high surface energy of droplet. In other words, the change of the surface energy of droplet induced by the nucleation on the skirt of the Ga droplet is larger than that of interface energy. It is noticed that the nucleation on the skirt of the Ga droplet is different from that of the catalyzed growth of nanowires via the vapor-liquid-solid (VLS) mechanism. In the case of the
Growth Mechanisms of QR Self-Assembly
J. Phys. Chem. C, Vol. 112, No. 20, 2008 7695
Figure 3. Schematic illustration of the shape evolution under different DR sizes, lC. The red and green regions represent the Ga droplet and the GaAs nanostructure, respectively. The brown arrows point to the DR boundary.
nanowires growth (taking the growth of Si nanowires as an example), the catalyst such as Au is first deposited on substrates and then forms the Au-Si liquid droplets with Si atoms after Si depositing. Thus, the Au-Si liquid droplets provide a pathway for Si to leave the gas phase by diffusing through or around the droplets and condense on the growing Si nanowire. Therefore, the Au atoms in the liquid droplets can not be exhausted by reacting with Si atoms. However, in the QR case,
Ga atoms in droplets are involved in the reaction with As atoms and the droplets can be exhausted by the reaction. B. Kinetic Growth. Although the thermodynamic nucleation of QRs creates a possibility for the QR self-assembly upon droplet epitaxy, the growth kinetics determines the final assembled shape of nanostructures. For example, if Ga atoms diffuse away from the droplet and crystallize with As atoms on the outside of droplet, the initial shape of the single ring would
7696 J. Phys. Chem. C, Vol. 112, No. 20, 2008
Li and Yang To satisfy the conditions that all the migratory Ga atoms from the droplet in the DR can crystallize with the trapped As atoms and the concentration of Ga atoms in the DR boundary is zero, the amount of the diffusing Ga atoms to the DR boundary should be the equal of the amount of the trapped As adatoms by DR, i.e., NGa ) NAs. From eqs 5-6, the size of DR,lC, is as follows
(
(rGa + lC)ln
(rGa + lC) rGa
)
)
h0D0(Ga)C0υ0 √2πmkTexp(∆E ⁄ kT) a0υ1 P Figure 4. The calculated values of lC/rGa as functions of the BEP of As flux and temperature. Experimental data (shape of nanostructures, temperature T, and BEP of As flux P) are from refs 4-14 (b quantum dots), refs 3 and 4 (2 single rings), refs 3-8 (9 double rings), and ref 8 (f holed nanostructures). In our calculations, rGa ) 40 nm for 200 °C,3 rGa ) 50 nm for 300 and 400 °C,8 υ0 ) υ1, a0 ) h0, D0(Ga) ) 1.4 × 10-4 cm2 s-1,26 and ∆E ) -0.6 eV, which is found by fitting to the experimental results.
be destroyed and changed to other shapes. Therefore, the diffusion of Ga atoms and the trapping of As atoms play key roles in the crystallization and the final morphological feature of QRs.15 To analyze the diffusion of Ga atoms and the trapping of As atoms, we assume that there is a diffusion region (DR) around the droplet where all the migratory Ga atoms from the droplet can crystallize with the trapped As atoms, i.e., the concentration of Ga atoms on the DR boundary is zero, as shown in Figure 2. The DR area is controlled by the diffusivity of Ga atoms away from the droplet and the trapping ability to As atoms. Therefore, the kinetic diffusion is defined to be the random walk of Ga atoms from the high concentration (the Ga droplet) to the low concentration (the boundary of DR) in the DR. According to the Fick’s law, the amount of the diffusing Ga atoms to the boundary per unit time is written as
NGa ) -2πh0DGa
0 - C0 rGa + l ln rGa
( )
(5)
where C0 refers to the concentration of Ga atoms on the droplet boundary that can be assumed to be constant during the diffusion process, h0 is the thickness of a monolayer, and DGa represents the diffusion coefficient of Ga atoms. We noted that the diffusion coefficients of Ga atoms on the substrate surface and on the nuclei surface are considered to be the same, because there is no strain in the nuclei in a match system.24 The component of the substrate surface and the nuclei surface are the same.25 Thus, DGa ) D0(Ga)exp (-EGa/kT), where D0(Ga) is the prefactor, EGa is the diffusion barrier energy, and k is the Boltzmann constant. The amount NAs of the trapped As adatoms by DR per unit time can be calculated as
NAs ) 2π(rGa + l)a0R
υ1 exp[(Ea - Ed) ⁄ kT ] υ0
(6)
where a0 is the space between surface sites, υ0 and υ1 are the thermal vibration frequencies for the upward and lateral directions, Ea and Ed are the adsorption energy and an energy barrier for the hopping between surface sites, and R is the amount of impacting As atoms on the substrate per unit time, which can be written as R ) P/(2πmkT)1/2, in which P is the beam equivalent pressure (BEP) of As flux and m is the mass of an As atom.
(7)
where ∆E ) -EGa - Ea+Ed. The DR size represents the balance of the diffusivity of Ga atoms and the trapping ability to As atoms and intensely affects the shape of the final nanostructure. On the basis of the discussions above, one can see that the diffused Ga atoms are confined and react with As atoms to produce GaAs in a given region. In detail, the size of the given region is smaller than that of DR at the initial stage of supply of As flux and approached to the size of DR in the subsequently steady state. According to the foregoing thermodynamic analysis, the nucleation of the resultant GaAs on the skirt of the Ga droplet is energetically preferable and creates an initial single ring. From the point of view of kinetics, under the conditions that the reaction of Ga and As atoms and the nucleation of GaAs happen in the DR, the GaAs located outside of droplet can nucleate when the inner nuclei form on the edge of droplet. Thus, the size of DR intensely affects the shape of the final nanostructure. Figure 3 shows the schematic evolution of nanostructures under the different sizes of DR. When the DR size is almost as large as that of the Ga droplet (lC is very small), Ga atoms hardly diffuse away from the droplet, and the crystallization of Ga and As atoms always occurs on the edge of droplet. Thus, this result would cause the nuclei to grow inside and finally create the formation of quantum dots with the same or a little bit bigger size than that of original droplet as shown in Figure 3a. When lC or lC/rGa becomes rather large, i.e., the DR size is slightly bigger than that of the Ga droplet, the diffusing Ga atoms away from the droplet crystallize with the trapped As atoms in the vicinity of droplets. Therefore, the nuclei on the droplet skirts grow mainly to outside at the preliminary stage, whereas these rings grow to inside with decreasing the droplet size. Finally, this growth mode may lead to the appearance of a single QR (Figure 3b). When lC is larger than the radius of the initial Ga droplet, Ga atoms can diffuse away from the formerly formed nuclei and nucleate second with the trapped As atoms; meanwhile the formerly formed nuclei may incorporate and grow into a ring. Once the second nuclei grow up densely enough, most of As atoms are trapped by the second nuclei by reacting with the diffusing Ga atoms from the droplet and the crystallization mainly occurs on the boundary of DR. Thus, the outer ring can form when the second nuclei start to incorporate. Meanwhile, the inner ring stops growing due to the absence of As atoms. Therefore, this growth mode would lead to the formation of the concentric quantum double rings (Figure 3c). When lC is much larger than the radius of the initial Ga droplet, Ga atoms can diffuse and nucleate with As atoms far away from the droplet. In this case, the holed nanostructures7 would form (Figure 3d). Figure 4 shows the values of lC/rGa as functions of the temperature T and the BEP of As fluxP. Clearly, we can see that the high temperature and the low pressure lead to the lC/ rGa increasing, and contrarily the low temperature and the high
Growth Mechanisms of QR Self-Assembly pressure result in the lC/rGadecreasing. Importantly, the physical mechanisms are that the high (low) temperature induces a strong (weak) diffusion of Ga atoms, and the high (low) pressure of As flux causes the strong (weak) trapping ability to As atoms. Thus, we divide the growth condition (BEP of As flux P and temperature T) into the four zones according to the values of lC/rGa. For the formation of quantum dots, the values of lC/rGa should be less than 0.2. When the values of lC/rGa are in the range of 0.2-1.0, the formation of single ring is kinetically preferable. The zone of 1.0-6.0 of lC/rGa belongs to the forming region of double rings. Additionally, the holed nanostructures could form in the region that the values of lC/rGa are more than 6.0. More importantly, these theoretical results are well consistent with experiments,3,4,8,14 in which all kinds of the shapes of nanostructures locate in the corresponding zones as shown in Figure 4. For example, a supply of As flux with intensity of 2 × 10-4 Torr BEP leads to the quantum dots formation,4 the single rings form under 1 × 10-5 Torr and 8 × 10-6 Torr BEP, and the concentric double rings form under 2 × 10-6 Torr BEP at 200 °C. Additionally, Lee et al.8 reported an interesting phenomenon that double rings form under the 6.4 × 10-6 Torr BEP, the single ringlike structures appear when the BEP decreases to 9 × 10-7 Torr at 400 °C. In fact, the single ringlike structures are the holed nanostructures in our case. Accordingly, our studies provide a phase diagram for the nanostructures selfassembly upon droplet epitaxy. Conclusion In summary, we have established the nucleation thermodynamics and growth kinetics to elucidate the QR self-assembly upon droplet epitaxy. The theoretical treatments indicated that the selective nucleation on the skirt of droplet leads to the ring structure formation, and then the diffusion of droplet atoms and the trapping of deposited atoms determine the evolution of the initial nuclei in the growing process. Acknowledgment. NSFC (50525206 and U0734004) and the Ministry of Education (106126) supported this work.
J. Phys. Chem. C, Vol. 112, No. 20, 2008 7697 References and Notes (1) Bayer, M.; Korkusinski, M.; Hawrylak, P.; Gutbrod, T.; Michel, M.; Forchel, A. Phys. ReV. Lett. 2003, 90, 186801. (2) Fo¨ldi, P.; Ka´lma´n, O.; Benedict, M. G.; Peeters, F. M. Phys. ReV. B 2006, 73, 155325. (3) Mano, T.; Kurada, T.; Sanguinetti, S.; Ochiai, T.; Tateno, T.; Kim, J.; Noda, T.; Kawabe, M.; Sakoda, K.; Kido, G.; Koguchi, N. Nano Lett. 2005, 5, 425. (4) Mano, T.; Koguchi, N. J. Cryst. Growth 2005, 278, 108. (5) Gong, Z.; Niu, Z. C.; Huang, S. S.; Fang, Z. D.; Sun, B. Q.; Xia, J. B. Appl. Phys. Lett. 2005, 87, 093116. (6) Huang, S. S.; Niu, Z. C.; Fang, Z. D.; Ni, H. Q.; Gong, Z.; Xia, J. B. Appl. Phys. Lett. 2006, 89, 031921. (7) Wang, Z. M.; Holmes, K.; Skultz, J. L.; Salamo, G. J. Phys. Status Solidi A 2005, 8, R85. (8) Lee, J. H.; Wang, Z. M.; Abuwaar, Z. Y.; Strom, N. W.; Salamo, J. G. Nanotechnology 2006, 17, 3973. (9) Keyser, U. F.; Fu¨hner, C.; Borck, S.; Haug, R. J.; Bichler, M.; Abstreiter, G.; Wegscheider, W. Phys. ReV. Lett. 2003, 90, 196601. (10) Granados, D.; Garcia, J. M. Appl. Phys. Lett. 2003, 82, 2401. (11) Hanke, M.; Mazur, Y. I.; Marega, E.; AbuWaar, Z. Y., Jr.; Salamo, G. J.; Scha¨fer, P.; Schmidbauer, M. Appl. Phys. Lett. 2007, 91, 043103. (12) Yu, L. W.; Chen, K. J.; Song, J.; Xu, J.; Li, W.; Li, X. F.; Wang, M.; Huang, X. F. Phys. ReV. Lett. 2007, 98, 166102. (13) Robinson, J. T.; Evans, P. G.; Liddle, J. A.; Dubon, O. D. Nano Lett. 2007, 7, 2009. (14) Watanabe, K.; Koguchi, N.; Gotoh, Y. Jpn. J. Appl. Phys. 2000, 39, L79. (15) Yang, G. W.; Liu, B. X. Phys. ReV. B 2000, 61, 4500. (16) Wang, C. X.; Yang, G. W. Mater. Sci. Eng. R 2005, 49, 157. (17) Wang, C. X.; Wang, B.; Yang, Y. H.; Yang, G. W. J. Phys. Chem. B 2005, 109, 9966. (18) Guyer, J. E.; Barnett, S. A.; Voorhees, P. W. J. Cryst. Growth 2000, 217, 1. (19) Mirbt, S.; Moll, N.; Kley, A.; Joannopoulos, J. D. Surf. Sci. 1999, 422, L177. (20) Nagata, T.; Ahmet, P.; Sekiguchi, T.; Chikyow, T. J. Cryst. Growth 2005, 285, 328. (21) Zhang, J. M.; Ma, F.; Xu, K. W.; Xin, X. T. Surf. Interface Anal. 2003, 35, 805. (22) Sun, C. Q. Prog. Solid State Chem. 2007, 35, 1. (23) Wang, Z. M.; Liang, B. L.; Sablon, K. A.; Salamo, G. J. Appl. Phys. Lett. 2007, 90, 113120. (24) Zhu, R.; Pan, E.; Chung, P. W. Phys. ReV. B 2007, 75, 205339. (25) It was reported that the first monolayer of Ga is consumed for the formation of GaAs layer and that the rest of Ga form droplets on the GaAs layer as in ref 4. (26) Zinke-Nlmang, M. Scanning Microsc. 1990, 4, 523.
JP801528R
