Shape control in nanocrystal growth

Secondly, each crystal has a surface Gibbs free energy that can be minimized by adopting the shape that is energetically most favorable.

Surface energies of crystal planes are related to their Miller indices, which is why these can help predict the equilibrium shape of a certain nanocrystal.

[3] High concentration, low temperatures and short aging times favor the kinetic regime, whereas low concentration, high temperatures and long aging times favor the thermodynamic regime.

It equals half the energy per unit area needed for cutting a larger piece of solid in two parts along the surface under examination.

The type of plane is most easily described using the orientation of the surface with respect to a given unit cell that is characteristic of the material.

The orientation of a plane with respect to the unit cell is most conveniently expressed in terms of Miller indices.

[2][7] Qualitatively, this follows from the fact that for higher Miller indices, on average more surface atoms are at positions at a corner instead of a terrace, as can be seen in the figure.

However, the comparison is in fact somewhat more complex, as the surface energy as function of the Miller indices also depends on the structure of the crystal lattice (e.g., bcc or fcc) and bonds between non-next nearest neighbours play a role as well.

[1][8] Experimental research on noble metals (copper, gold and silver), shows that for these materials, the surface energy is well-approximated by taking only the nearest neighbours into account.

This is also visible in the following table, which lists computer simulated surface energies of some planes in copper (Cu), silver (Ag) and gold (Au).

is the number of broken bonds between nearest neighbours created when making the surface, being 3 for the (111) plane.

[7] The table below shows examples of computer simulated surface energies of (hk0) planes in a NiO crystal (with

In this case, the unit cell has a multi-atom basis, as there are two types of atoms that make up the crystal (nickel and oxygen).

From this table, it is clearly visible that the trend between surface energy and Miller indices is not as straightforward in this case as for the noble metals discussed above.

Planes with low surface energies are relatively stable and thus tend to be predominantly present in the thermodynamic equilibrium shape of a crystal.

However, a crystal's thermodynamic equilibrium shape typically does not only consist of planes with the lowest possible surface energy.

Research on this topic is mainly centered around nanocrystals, as their synthesis is not as straightforward as that of bulk materials and thus requires a deeper understanding of types of crystal growth.

Due to the high surface-volume ratio and the resulting instability, nanocrystals most easily show the difference between the thermodynamic and kinetic regime.

In the kinetic regime, the addition of monomers happens so rapidly that the crystal continues growing at the corners.

In this case, the formed product is not at a global minimum of the free energy, but is in a metastable anisotropic state.

This corresponds to the shape with a global minimum in Gibbs free energy, which can be obtained via the Wulff construction.

Raising the temperature has a similar effect because the extra thermal energy increases the mobility of the atoms on the surface, making rearrangements easier.

Due to these, the system is driven by lowering the volume Gibbs free energy, which decreases rapidly upon monomer consumption.

Minimization of the surface Gibbs free energy is of less relevance to the system and the shape evolution is controlled by reaction rates instead.

Thus the product obtained in this regime is a metastable state, with a local minimum in Gibbs free energy.

Kinetic control is obtained when there is not enough time for atoms on the surface to diffuse to an energetically more favorable state.

[4] Conditions that favor kinetic control are low temperatures (to ensure thermal energy is smaller than activation energy of the thermodynamic reaction) and high monomer concentration (in order to obtain high growth rates).

The band gap as well as the density of states of nanoparticles depend significantly on their shape and size.

Studying different shapes of nanoparticles can improve the understanding of quantum confinement effects.

By elongating an axis in certain spherical nanoparticles (quantum dots), degeneracies in the energy levels can be resolved.