Growing Ordered and Stable Nanostructures on Polyhedral Nanocrystals
L. X. Lu1, M. S. Bharathi1, M. Upmanyu2, Y. W. Zhang1,a)
1Department of Engineering Mechanics, Institute of High Performance Computing, Singapore 138632, Singapore
2Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02115, USA
PACS number: 61.46.Df, 81.16.-c
S.1 A general energetic model
In this section, we would like to extend the simplified energetic model to a more general framework by considering the different properties for the core and shell. Assuming the surface/interface energy as , and the corresponding surface/interface area as , the total volume as , then the surface/interface energy per unit volume can be written:
(S1)
where, , and . Comparing Eq. (S1)and Eq. (1) in the main text, we see that is not only dependent on the geometry but also on the surface/interface energy.
Similarly, by considering the different elastic properties for the core and shell, we can rewrite Eq. (2) as:
(S2)
Now is not only dependent on geometry, but also on the elastic constants of both core and shell. Combining the above two free energies, we can also obtain the following critical diameter that controls the transition of NCs from one configuration to another:
(S3)
where, and are the differences of coefficients and for the two configurations. Using typical values for Si/Ge system:, , , , and , we have calculated and plotted (Fig. S1) the energy density curves ofthe four different shell configurations grown on the cubic core (Fig. 1) as a function of core size. From Fig. S1, it is seen that the critical core size for the transition fromConf a to Conf b is about 28 nm, the critical core size for the transition fromConf b to Conf c is about 106 nm, andConf d is still energetically unfavorable. It is concluded that by considering the different properties of the core and shell, only are the critical core sizes slightly shifted, and there is no change in our main conclusion.
Fig. S1. Energy density curves of the different configurations as a function of with , , , , and . Dashed blue, red, black and purple lines correspond to Conf a, Conf b, Conf c and Conf d, respectively, and the solid line is the minimum energy density curve.
S.2 Phase field model
In our phase field model, three long-range order parameters: , andare employed to denote the core-shell system, with r being the position vector of a point. The total free energy is expressed as a function of these order parameters, e.g. in the vacuum phase,in the shell phase and in the core phase, with constraint: in the whole simulation region. The coarse-grained Ginzburg-Landau free energy of the multiphase system, including bulk free energy, surface energies, interfacial energy, and elastic energy, is modelled using the following free-energy function [1]:
(S4)
where are constants related to the height of double-well potential, are the gradient energy coefficients, these two parameters together determine the surface/interface energy density as:
(S5)
The term in Eq. S4 is the total elastic energy, which can be expressed by the following expression:
(S6)
where is the 4th-order elastic tensor, is the total strain, which is calculated by using the PFM model[2] based on the Eshelby’s equivalency approach[3], is the total eigenstrain. Since we have assumed that the core and the shell have identical elastic constants, the position-dependent elastic tensor can be expressed as:
(S7)
where is the elastic tensor of shell, which can be reduced into 3 independent constants: ,and, and is the material density function, which is equal to 1 in solid phases and zero in vacuum phase. The expression of is given by the following equation:
(S8)
In the present work, only one shell layer is considered, so the eigenstrain is a function ofonly:
(S9)
where is the Kronecker delta, and is the mismatch strain between the shell and the core.
To simplify the evolution equation, we assume that the phase field describing the evolution of NC is static, i.e., it does not evolve with time. Since , we choose as an independent phase field and its evolution follows the Cahn-Hilliard equation:
(S10)
where, is a parameter that controls the growth rate, is a random distribution function and is the position-dependent mobility, which is given by:
(S11)
where is the maximum mobility at the interface between the vacuum and the solid.
S.3Numerical algorithm
Equation (S10) is solved usingFourier-Spectral method. Equations(S12)-(S14) show the detailed algorithm used in the present work.
(S12)
(S13)
(S14)
where, , and represent the Fourier transformation and inverse Fourier transformation, respectively.
To obtain the elastic energy, we assume that the speed of elastic motion is much faster than the speed of atomic diffusion, so during all diffusion steps, elastic deformation is always in its equilibrium state.In practice, we add an extra iteration into the main iteration of Eq.(S12). In this extra iteration, we solve the elastic displacements using an “equivalent” method described in [2]. The extra iteration executes until all the mesh pointsreach to relative equilibriumpositions according to a givencriterion. Subsequently, we save the elastic information of the current step and return to the main iteration. We then solve the new distribution of and start a new diffusion step. According to the core size, we use different mesh sizes and time increments in order to ensure calculation accuracy. Typically, the space resolution changes from 0.5nm to 10 nm and the reduced time increment changes from 10-6 to 10-3.Based on existing references of the diffusion coefficient of Ge on Si (111) surface [4], we can obtain the relationship between the reduced simulation time and the real time. The real time scale strongly depends on the growth temperature. For example, at 700K, a unit simulation time roughly corresponds to 103-104 s of the real time, and at 800K, it corresponds to about 1-10 s of the real time.
In the phase field simulations, before deposition, we add an initial thin layer with average thickness of two grids and a random perturbation with amplitude of 1 grid to consider the wetting effectand the surface roughness. The deposition rate is set to be 0.8 nm per reduced time and per area of deposition surface, which is slow enough to allow sufficient diffusion of shell material according to the energy landscape.
S.4Additional three-dimensional (3D) phase field results
Fig. S2.Kinetic evolution of shell grown on the surface of cubic NCs with different values of : (a) 14 nm, (b) 60 nm, and (c) 110 nm. Color in (a1)-(a3) represents the radius while in (b1)-(b3) and (c1)-(c3) represents the shell thickness.
Fig. S3.3D phase field simulations of shell grown on large size polyhedral cores.(a) Cubic core with 270 nm; (b) Tetrahedral core with 180 nm.(c) Octahedral core with 300 nm. (d) Dodecahedral core with 400 nm. (e) Icosahedral core with 400 nm. QDs are seen on allover the facets, cornersand ridges. Color represents the shell thickness.
Fig. S4.Overgrowth of shell material deposited on octahedral core with 100 nm. Coarsening is observed, which results in the formation of some interesting morphologies, such as: (a) dish-like,(b) dumbbell-like, and (c) spiral-like morphologies. Color represents the shell thickness.
Fig. S5.Overgrowth of shell material deposited on the different NC cores:(a) for dodecahedral core with 90 nm, and (b) for icosahedral core with 120 nm. Coarsening between shell QDs occurs, which resultsthe formation of thedumbbell-like morphologyas shown in (a) and the paired QDmorphology as shown in (b). Color represents the shell thickness.
Fig. S6.Effect of annealing to a shell configuration grown on a dodecahedral core with240 nm.(a) Morphology before annealing, growth time: 60,(b) Morphology after a long time annealing, annealing time: 250.
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