Final Project: Si band structure using Quantum Espresso


1) Si band structure results


The resulting Si band structure is shown in the following figure:


Figure 1: Si band structure calculated using PWSCF QE

We can see the high-symmetry points in the above figure. The band gap is seen in the energy axis: the difference between the minimum conduction band (near X point) and the maximum valence band (near point). The following figure shows the Si band structure calculated using VASP software. We see there are similarities between them.

Figure 1

Figure 2: VASP software simulated Si band diagram (doi:10.1209/0295-5075/82/48004)

2) Lattice constant dependence on Band Structure

In the following table we can see a sweep simulation that I run manually. It consists in changing the value "celldm(1)=10.2625" to different values and see the effect on the output. This value refers to the lattice constant "a" of the unit cell in Bohr units. It must be changed both in "" and "" (please see the setup page for how to change those files).




In the above table we can see the change in the band structure depending on different lattice constants from 8.3 to 24. The marked one is the Si lattice constant known and used before in the simulations. Also the approximately Ge lattice constant case is indicated following the Si one. We see the change in the band-gap until 11-13 Bohr units of the lattice constant. When we continue to increase the lattice constant we can observe two thinner branches of energies. For a lattice constant of 24 we have almost two discrete states (isolated atoms).

We can see in figure 3 (left) the relation between the previous diagrams and the energy band diagram with lattice constant dependence. This diagram indicates how the band diagram for (C) diamond changes around the a0 lattice constant forming the "band gap", to the more isolated atoms when the lattice constant is larger. Indicated with arrows are C, Si, and Ge atoms and their respective position in the periodic table (right). We can see the band gap of the diamond (very large) and the semiconductors (Si, Ge) shorter band-gaps when the lattice constant increases between atoms. It is worthwhile to say that this is a qualitative comparison showing the connection between the results and the effect of a periodic arrangement on the electron energy levels.



Figure 3: (left) the energy band of (C) diamond type vs. lattice constant. Zeghbroec, Principles of Semiconductor Devices (Right) 14th column of the periodic table. (Wikipedia)

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