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Considering a crystal at such a small size (for example, a sphere with diameter about 20 atoms) that the periodic boundary condition assumed in Bloch Theorem is not really a good estimation, I wonder what are other possible ways to compute the band structure efficiently. Is there some quick and neat modification on Bloch Theorem that makes this happen?

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  • $\begingroup$ Not a specialist but I would treat an ensemble of 20 atoms with a molecular approach. $\endgroup$ – Alchimista Dec 6 '19 at 9:03
  • $\begingroup$ Use the zero boundary conditions for the wave function. $\endgroup$ – Alex Trounev Dec 6 '19 at 11:24
  • $\begingroup$ Without periodicity there's no band structure (in the sense of $E(\vec k)$), since quasiwavevector is not conserved. Moreover, due to surface tension, such a nanocrystal will get distorted compared to a large version of this crystal, so even local periodicity will be broken. The band edges will then get fuzzier. I wouldn't hope for Bloch theorem to be too much useful in these conditions. $\endgroup$ – Ruslan Dec 6 '19 at 15:05
  • $\begingroup$ @Ruslan But we can still get a density of state plot out of such system for each orbital? And then there’s still a structure of some kind, ie, energy gap between bands $\endgroup$ – Macrophage Dec 6 '19 at 15:21
  • $\begingroup$ Given that you only have about 20 states per dimension in the allowed band, I'd not call them "bands" — more like "clusters of states". And many of these states will be somewhat localized near the surface due to distortion of geometry. $\endgroup$ – Ruslan Dec 6 '19 at 15:33
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Without periodicity there's no band structure (in the sense of $E(\vec k)$), since quasiwavevector $\vec k$ is not conserved. Moreover, due to surface tension, such a nanocrystal will get distorted compared to a large version of this crystal, so even local periodicity will be broken.

Still, if you want to approximate the nanocrystal by simply a lump of elementary cells of a macrocrystal separated by the same lattice constant, and impose zero Dirichlet boundary conditions somewhere near its surface, you can actually use Bloch's theorem to calculate eigenstates of electron in this system, provided some conditions are fullfilled.

Consider the infinite crystal we would take as the macroscopic material of the same kind as the nanoparticle we are interested in. Bloch eigenfunctions of electron in this crystal are

$$\psi_{\vec k}(\vec r)=u_{\vec k}(\vec r)\exp(i\vec k\vec r),$$

where $u$ is periodic with the periodicity of the crystal's Bravais lattice. These functions are eigenfunctions of the translation operator.

But if the crystal potential has reflection symmetry$^\dagger$, so that e.g. $U(x,y,z)=U(-x,y,z)$, then the degenerate pairs of eigenfunctions corresponding to opposite $k_x$ can be linearly combined in such a way that they combinations will vanish at the reflection plane $x=0$. This would also make the wavefunctions purely real (up to constant phase factor), which then guarantees that they'll regularly attain zeros at some $x=x_0$ as we increase energy eigenvalue (keeping $|k_x|$, $k_y$ and $k_z$ constant). These zeros will define possible values of ($x$ contribution of) energy for which the wavefunction satisfies zero Dirichlet boundary conditions at both planes.

If we now, for e.g. a cubic crystal, define pairs of such planes in each of three symmetry directions and find the corresponding contributions to energy so that our wavefunctions vanish at all these planes, this will yield the eigenstates of our approximation of the (rectangular-shaped) nanocrystal.


$^\dagger$ I'm not sure if this is a necessary condition. It might be that the linear combination talked above can be found for any crystal. It's just easiest to prove for the reflectionally-symmetric potential (because the solutions can be classified by parity).

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  • $\begingroup$ Thank you! This is exactly what I wanted. $\endgroup$ – Macrophage Dec 7 '19 at 18:22
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There is no quick fix. For a cluster of 20 atoms I would use quantum chemistry . Note that you have to cooptimize the geometry as it will deviate considerably from the bulk crystal structure. The result will be bands consisting of 20 states each with considerable irregular energy separation between them, unlike the continuous bands of an infinite crystal.

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  • $\begingroup$ Depends on what you mean by quantum chemistry. If you mean DFT then yeah, it will work (with its usual caveats and pitfalls). If you mean something fancier, then it's unlikely to work at that scale, particularly if you want to re-work the nuclear positions. $\endgroup$ – Emilio Pisanty Dec 6 '19 at 10:05
  • $\begingroup$ Thanks for you answer! I'm think of more atoms, like a sphere of diameter of 20 atoms rather than just a cluster of 20 atoms, and I've read that such nanocrystals could have bulk structure. $\endgroup$ – Macrophage Dec 6 '19 at 13:38
  • $\begingroup$ @EmilioPisanty 20atoms is no problem but the geometry optimisation is quite a challenge. However the OP was thinking more of 8000 atoms. I agree that thus would be challenging in QC. Probably not in DFT. Still there us the challenge of the geometry to which there should be smart solutions by exploiting symmetry and assuming bulk structure beyond, say, three layers. $\endgroup$ – my2cts Dec 6 '19 at 15:41

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