We need to know how the electronic properties of the system are related to the the structural changes we observed. One way of studying the local bonding is to calculate the bond order between two atoms in the unit cell, so as to explain the difference in bond length for the flat and buckled addimers of the line. This is derived from the density matrix. The density m atrix also lets us calculate the charge density of a particular atom (the charge on an atom) and find the overlap of any state with the atoms of the unit cell.
The density matrix is found by evaluating the integral [241]
(5.5)
where |^n) is the normalised eigenstate and |x) is a position eigenstate. We can write l^n) as
(5.6)
ja
where cja (n) is the expansion coefficient of the nth state in the atomic orbital a on atom
PpPja —
^25353
(^)
l^^ja) I (^"^)n ja p0
The density m atrix consists of two terms, a diagonal term {p = j ) th at corresponds to the charge density on an atom j (the occupation) and an off-diagonal term {p ^ j ) that corresponds to the charge density between a pair of atoms j , p (the bond order). This derivation of the density m atrix is for orthogonal states, however as our tight-binding formalism is based on DFT calculations we use non-orthogonal states (atomic orbitals) along with a second reciprocal basis set which uses a mixed tensors basis set [242].
To construct the full wavefunction |^ „ ) we expand in terms of the linearly independent basis vectors \(f>ja) to get
1^„) = ^ C ja (n )|< ^ ja ). (5.8)
We define an overlap m atrix th at has elements
Sjap^ — {<Pja\<Ppp) — (5-9)
We can also construct a dual set from the original basis set vectors
= (5.10)
P/3
which are orthogonal to the original basis set vectors
where is a dummy index. The use of the contravariant index notation is extremely
(4>°‘\H\4,fe) = (5.12) r7
For the case of our DFTB scheme the density matrix is calculated as
= ' £ ‘^ “ [n )c " -'{n )S r^ p » . (5.13)
n T 7 n
If we do not perform the sum over states in equation (5.13) we obtain the crystal orbital overlap population (or COOP) [243, 244]. We can then evaluate the bond order and charge density terms by summing the COOP (5.13) for all occupied states. To evaluate the bond order terms we need to rotate the off-diagonal elements from their original
coordinate frame where the contributions to the a and tt bonds are mixed to a coordinate
frame where these contributions are not mixed and where the cr bonds are aligned along
the addimer direction (along the x direction which is equivalent to the x7 direction). This
is accomplished by the unitary transformation where t/ is a unitary matrix
transformation through the three orthogonal Euler axes [245].
We started by investigating the bond order of the top addimer of the line, summing over all occupied eigenvalues. Why is this addimer shorter when the addimer becomes flat? We find th at the flat addimer of the (7x2) unit cell (the smallest unit cell with a flat addimer) has a shorter bond length than the buckled addimer of the (3x2) or (5x2)
unit cells because there is a stronger a bond and a much stronger tt bond. There are
much stronger ss (0.218 for the (7x2) vs -0.062 for the (3x2)), spx (-0.302 vs 0.004) and
PxPx (-0.521 vs 0.005) overlaps for the (7x2) unit cell. Representative COOP are shown
in Figures 5.6-5.7.
We have also compared the bond order between the top adatom of the line and one of the adatoms of the second adlayer for the buckled addimer case (for the flat addimer case this corresponds to one of the level adatoms). This analysis shows th at as the addimer becomes flat this bond becomes much stronger, with larger a and t t components. There
are much stronger ss (0.200 for the (7x2) unit cell vs 0.076 for the (3x2) unit cell) and
PzPz (0.450 vs 0.253) overlaps for the (7x2) unit cell (see Figures 5.S-5.9). Evaluating the diagonal terms of the density m atrix we find th at associated with this stronger bond
-0.6 Eigenvalue
Figure 5.6: Plot of the COOP for all occupied states as a function of eigenvalue n for the overlap of the s orbitals on the top pair of adatoms of the line. Black indicates the (3x2) unit cell and red the (7x2) unit cell.
Eigenvalue (n) (m H anrees)
Figure 5.7: Plot of the COOP for all occupied states as a function of eigenvalue n for the overlap of the orbitals on the top pair of adatoms of the line. See Figure 5.6 for layout.
between top ad-layer and second ad-layer is the loss of an electron from the top addimer. This electron is transferred to the surface silicon atoms directly below the ad-dimer as shown in Figure 5.4.
Using the density matrix we can also evaluate the overlap of various states on atoms of the unit cell. We calculated the overlap of the HOMO state on all atoms of the surface of the unit cell. We found that for the (3x2) unit cell the HOMO state is quite diffuse, with the largest component on the second-level addimers, and large components on the top adatom of the top adlayer and the silicon atoms on the surface between the second layer addimers.
When we look at the overlap of the HOMO state on the ground-state (nx2) series of reconstructions (see Figure 5.3), we find that the highest overlap is on the top adatom of the top addimer of the line structure (ui). There is a large overlap of the HOMO state with the other adatoms (02,03) of the line and the two silicon atoms of the surface that are in the middle of the line. The HOMO state for this structure is localised on the line part of the (nx2) reconstruction.
- 0.8 - 0 .6 - 0.4 -02
Eigenvalue (n) (Hartrees)
Figure 5.8: Plot of the COOP for all occupied states as a function of eigenvalue n for the overlap of the s orbitals between the top adatom of the line and a second-level adatom. See Figure 5.6 for layout.
-1 - 0.0 - 0.6 - 0.4 -02
Eigenvalue (n) (Kartrees)
Figure 5.9: Plot of the COOP for all occupied states as a function of eigenvalue n for the
overlap of the orbitals between the top adatom of the line and a second-level adatom.
See Figure 5.6 for layout.
For the metastable structures we find that the HOMO state is strongly localised on two of the adatoms, the surface addimer that is flat and weakly bound (see the encircled addimer 05 in Figure 5.3) and the second-level addimer that is nearest this surface addimer (02). The overlap of the HOMO state on other surface and adlayer Si atoms is negligible. The HOMO state resides upon the flat surface addimer. It appears that this surface addimer is responsible for the unit cell acting as a narrow bandgap semiconductor. This (2x1) part of the surface (from now on called the silicon string) may also act as an atomic scale wire.