The Hydrogen Chain: Practical Aspects of the Band Structure Unfolding

In the following we discuss some practical details on the basis of a few examples shown in Fig.11.5.

Example a: Figure 11.5a shows a perfectly periodic 10-atom hydrogen chain. In this example the band structure can be unfolded exactly.

In thePCBZthe spectral weight, Eq.11.12, is either zero or one.

Example b: Figure 11.5b shows a distorted geometry with broken trans-lational symmetry. The overall shape of the band structure is very similar to that of example a, but close to theΓpoint the effect of the distortion can be seen. In this example, the difference between an atom in thePClat positionρl and an atom at positionρl⁰ in thel⁰thPCis not a lattice vector of thePC. The spectral weight reflects the strength of the symmetry breaking by reaching a value between zero and one.

However, for very strong distorted geometry, where the displacement is at the order of a lattice translation the approximation to the overlap matrix Eq.11.14would break down. In this case, all overlap matrix elements[_S(K)]_µ;µ⁰_l⁰ would need to be calculated explicitly.

Example c: In this example the hydrogen chain is perfectly periodic except

Figure 11.5.: The unfolded band structures of four different 10-atom hydrogen chains are plotted along the PCBZpath M to Γ to M. Dashed lines indicate theSCBZ. The corresponding atomic structure is shown at the bottom. a.: A perfectly periodic hydrogen chain, b.: a hydrogen chain with geometric distortions, c.: a defect in thePCl=6 in an otherwise perfectly periodic chain and d.: two chains with different periodicity.

for one defect in the PC l = 6. Part of the original hydrogen band can still be observed, but the symmetry breaking is clearly seen in Fig.11.5c by small gap openings. To evaluate Eq.11.12, the overlap matrix of atomµ⁰inPCl⁰and atomµinPCl =6 is required, but the overlap is not calculated withinFHI-aims, because in the PC l = 6 contains no basis functions. This can be circumvent by introducing a ghost atom that consists of the same basis functions but without any mass or charge.

Example d: The system contains two hydrogen chains, a 10-atomic and a 2-atomic one separated by 2.5 Å. It is not clear howN _{should be} chosen. We are interested in the effect of the additional chain on the 10-atomic hydrogen chain, therefore

Figure11.5d shows the unfolded band structure. The band originating from the 10-atomic hydrogen experiences a small gap opening at the Fermi level. The second 5-atomic chain is too far away for any chemical bonding between the chains, which is reflected by the almost undisturbed band, but introduces a periodic potential. Additional states appear in particular some very weak states at the Fermi level.

What cannot conclusively be determined from the plot is which states originate from which chain. In this system we face a similar problem as in example c. We need to calculate the overlap of the two atoms in PCl andl⁰, as depicted in Fig.11.5d. ThePC l⁰contains just one atom, as a consequence the overlap between the basis function inl andl⁰ is not fully available. In this particular system, ghost atoms can again be used to unfold the combined system. However, in general the full (N_basis×N_CN_basis)overlap matrix defined in Eq.11.13is required to unfold an interface system formed by two subsystems with different periodicities.

Figure 11.6.: The unfolded band structure of the two chain system introduced in Example 11.3d. The band structure is projected on the 10-atomic chain marked by a grey box.

In summary, the unfolding procedure intro-duced in this section facilitates the unfold-ing of the band structure of an arbitrary system by imposing a translational sym-metry. However, because of the overlap matrix [S(K)]_µ⁰_;µl Eq. 11.13 a first approx-imation was introduced, see Example11.3b.

For small displacements of atoms in thePC the changes of the respective overlap matrix element are smalss, such that the approx-imation holds. As shown in Example11.3c there is a workaround by introducing mass-and chargeless basis functions at the posi-tion of “missing atoms”. For interface sys-tems this workaround is only applicable in a few special cases.

For an interface system formed by two sub-systems with different periodicities, we are generally interested in the band structure projected on the subsystem of interest. In aLCAObasis a straight forward projection is given by a Mulliken partitioning [222].

In Eq.11.12the sum over basis functionsµ should than run over all basis functionsµs located in the subsystem and not over all basis functions. In essence the projection splits the eigenvectorC^p in two parts:

1. Eigenvector entries belonging to the subsytemC^p

µ⁰s

2. Eigenvector entries belonging to the restC^p

µ⁰r

If the sum over basis functions in Eq.11.16is split into these two contribu-tions the eigenvector is no longer normalised, because the contribution from the off-diagonal blocksC_µ^∗_s^p[S]_µs;µrC_µ^p_r andC^∗p

µr⁰ [S]_µ⁰

r;µ⁰_sC^p

µ⁰s is not included in the sum. As a result, the eigenvector norm in Eq.11.16 has to be introduced

explicitly by a rescaling factorN_sub^p such that 1

N_sub^p

N_basis^proj

∑

µs,µ⁰_s

C_µ^∗_s^p(K) [S(K)]_µs;µ⁰

sC^p

µ⁰s(K)≡1 (11.19) the weight normalisation (Eq.11.15) is ensured. In analog to Eq.11.19a normalisation factor for the rest N_rest^p can be calculated. With

1−(N_sub^p +N_rest^p ) = J^p (11.20) we can then determine how strongly the subsystem and the rest are coupled for a state p. For a combined system in which the two subsystems are infinitely separated, the overlap matrix becomes a block matrix and the coupling coefficientJ^p =0. When interpretingJ^pone has to keep in mind that J^p strongly depends on the chosen basis set used for the projection and there is no well defined upper limit of J^pin cases of strong coupling.

However, it can be used to visualise general trends within one system.

In Figure11.6the band structure of the system introduced in Example11.3d is shown. The two subsystems are well separated, the overlap matrix is close to a block matrix and the coupling coefficients are very small. For the band structure plotted in Fig.11.6the maximal coupling coefficient is J^p=0.038 for the statep=5 meV and thek-point at which the band touches the Fermi level. At this very point the projected band structure shows a very small gap opening at the Fermi level in an otherwise undisturbed band.

12 The Electronic Structure of Epitaxial