5.4: Electronic Wavefunctions and Energies in 1-d

At each atomic site, we assign a valence s AO, i.e., \(\varphi_{n}(x) \equiv\) \(\varphi_{s}(x)\), which is a spherically symmetric real function. The Bloch function for wavevector \(k\) is:

\[\psi_{sk}(x) = \frac{1}{\sqrt{N}}\sum_{m}^{}{e^{ikma}\ \varphi_{s}(ma)}. \nonumber \]

Now, we examine these Bloch functions and eigenvalues \(E_{n}(k)\) at three wavevectors:

Zone Center \(\Gamma,\ k = 0\):

\(\psi_{s\Gamma}(x) = \frac{1}{\sqrt{N}}\sum_{m}^{}{e^{i0ma}\ \varphi_{s}(ma)} = \frac{1}{\sqrt{N}}\sum_{m}^{}{( + 1)^{m}\ \varphi_{s}(ma)}\)

\(\psi_{s\Gamma}(x) = \frac{1}{\sqrt{N}}\left\lbrack \varphi_{s}(0) + \varphi_{s}(a) + \varphi_{s}(2a) + \varphi_{s}(3a) + \varphi_{s}(4a)\cdots + \varphi_{s}\left( (N - 1)a \right) \right\rbrack:\)

Figure 5.24

\[E_{s}(\Gamma) = \left\langle \psi_{s\Gamma}(x) \middle| H \middle| \psi_{s\Gamma}(x) \right\rangle = \alpha_{s} + 2\beta_{ss} \nonumber \]

The open circles represent \(+ \varphi_{s}(x)\). This wavefunction has the complete translational symmetry of the structure because there is no phase change in the atomic basis function between adjacent unit cells. As a result, there are no nodes between AOs on adjacent atomic sites. Therefore, this crystal orbital is totally bonding so that \(\beta_{ss} < 0\), and it has the lowest energy of the band.

Zone Boundary \(X,\ k = \frac{\pi}{a}\):

\(\psi_{sX}(x) = \frac{1}{\sqrt{N}}\sum_{m}^{}{e^{im\pi}\ \varphi_{s}(ma)} = \frac{1}{\sqrt{N}}\sum_{m}^{}{( - 1)^{m}\ \varphi_{s}(ma)}\)

\(\psi_{(s)\Gamma}(x) = \frac{1}{\sqrt{N}}\left\lbrack \varphi_{s}(0) - \varphi_{s}(a) + \varphi_{s}(2a) - \varphi_{s}(3a) + \varphi_{s}(4a)\cdots + ( - 1)^{N - 1}\varphi_{s}\left( (N - 1)a \right) \right\rbrack:\)

Figure 5.25

\[E_{s}(X) = \left\langle \psi_{sX}(x) \middle| H \middle| \psi_{sX}(x) \right\rangle = \alpha_{s} - 2\beta_{ss} \nonumber \]

The open circles represent \(+ \varphi_{s}(x)\); the filled circles represent \(- \varphi_{s}(x)\). Therefore, there are nodes in the wavefunction between every pair of adjacent AOs, and this crystal orbital is totally antibonding. It has the highest energy of the band.

General Point, e.g., \(k = \frac{\pi}{2a}\) (all wavevectors \(0 < k < \frac{\pi}{a}\) are labelled \(\Delta\)):

\[\psi_{s\mathrm{\Delta}}(x) = \frac{1}{\sqrt{N}}\sum_{m}^{}{e^{\frac{im\pi}{2}}\ \varphi_{s}(ma)} = \frac{1}{\sqrt{N}}\sum_{m}^{}{( + i)^{m}\ \varphi_{s}(ma)} \nonumber \]

\[\psi_{s\Gamma}(x) = \frac{1}{\sqrt{N}}\left\lbrack \varphi_{s}(0) + i\varphi_{s}(a) - \varphi_{s}(2a) - i\varphi_{s}(3a) + \varphi_{s}(4a)\cdots + ( + i)^{N - 1}\varphi_{s}\left( (N - 1)a \right) \right\rbrack: \nonumber \]

This wavefunction is complex with real and imaginary parts:

• Real Part: \(\mathfrak{R}\left( \psi_{s\mathrm{\Delta}}(x) \right) = \frac{1}{\sqrt{N/2}}\left\lbrack \varphi_{s}(0) - \varphi_{s}(2a) + \varphi_{s}(4a) - \varphi_{s}(6a)\cdots \right\rbrack\):
• Imag. Part: \(\mathfrak{I}\left( \psi_{s\mathrm{\Delta}}(x) \right) = \frac{1}{\sqrt{N/2}}\left\lbrack + i\varphi_{s}(a) - i\varphi_{s}(3a) + i\varphi_{s}(5a) - i\varphi_{s}(7a)\cdots \right\rbrack:\)

Figure 5.26

\(\mathfrak{R}:\)R:

Figure 5.27

\(\mathfrak{T}:\)T:

\[E_{s}\left( \frac{\pi}{2a} \right) = \left\langle \psi_{s\Delta}(x) \middle| H \middle| \psi_{s\Delta}(x) \right\rangle = \alpha_{s} \nonumber \]

In this crystal orbital, real functions alternate with imaginary functions along the lattice. According to the illustrations, both the real and imaginary parts of the Bloch function has nodes at alternating lattice (atomic) sites. Therefore, this wavefunction is nonbonding and its eigenvalue is the energy of the \(s\ AO\).

These results can be summarized in two diagrams:

Figure 5.28

• The band structure illustrates the \(s\)-band energy as a function of wavevector \(k\) from the Brillouin zone center \(\Gamma = 0\) to the zone boundary \(X = \frac{\pi}{a}\). It is also called an energy dispersion curve. The \(s\)-band energy increases with increasing wavevector because the crystal orbital at \(\Gamma\) is totally bonding and the crystal orbital at \(X\) is totally antibonding. The middle of the energy band is the energy of the atomic \(s\ \text{AO}\) \(\alpha_{s}\). The energy difference between the top and bottom of the band is called the bandwidth, which is \(4\left| \beta_{ss} \right|\) in this case. The size of the bandwidth depends on the strength of the orbital interaction between adjacent atomic sites.
• The DOS curve shows the relative number of electronic states in a small, fixed energy window plotted as a function of energy. Since each wavevector identifies an eigenfunction and there are \(N\) equally spaced wavevectors in the first Brillouin zone (\(\frac{- \pi}{a} < k < \frac{\pi}{a}\)) arising from the periodic boundary condition, then the density of states in reciprocal space is the total number of states divided by the size (length in 1-d) of the first Brillouin zone, i.e.,

\[\frac{dN(k)}{dk} = \frac{N}{\left( \frac{2\pi}{a} \right)} = \frac{Na}{2\pi}\ . \nonumber \]

The energy density of states plotted in the DOS curve is the number of states per energy unit:

\[\frac{dN(E)}{dE} = \frac{\frac{dN(k)}{dk}}{\left| \frac{dE(k)}{dk} \right|} = \frac{Na}{2\pi} \cdot \frac{1}{2\beta_{ss}a\sin{ka}} = \frac{N}{4\pi\beta_{ss}} \cdot \frac{1}{\sin{ka}}\ . \nonumber \]

The energy DOS curve is the solid-state analogue of a molecular orbital energy diagram. For the 1-d chain, it shows peaks at the bottom and top and a minimum value at the center \(E = \alpha_{s}\). Integrating the entire DOS curve gives the total number of states in the first Brillouin zone \(N\). Since each crystal orbital (“state”) can hold 2 electrons (spin up and spin down), then the \(s\)-band in the first Brillouin zone can hold at most \(2N\) electrons or 2 electrons per unit cell in real space. Therefore, the number of energy bands in the band structure corresponds to the number of AO basis functions in a single unit cell.

If the chain consists of H atoms, each atom has 1 valence electron and the entire chain after applying the periodic boundary condition has \(N\) valence electrons. As a result, only \(\frac{N}{2}\) crystal orbitals of the chain are occupied, which means that the highest occupied crystal orbital has \(k = \frac{\pi}{2a}\) or \(E_{s}(k) = \alpha_{s}\). In this case, the wavevector \(k = \frac{\pi}{2a}\) is called the Fermi wavevector \(k_{F}\) and the corresponding energy \(E_{s}\left( k_{F} \right) \equiv E_{F}\) is called the Fermi energy or Fermi level.

We repeat the determination of Bloch functions for \(p\) AOs on each atomic site, an exercise that demonstrates how band dispersion is related to the nodal characteristics of the AOs. One of the \(p\) AOs is directed along the chain direction and engages in \(\sigma\)-type overlap (\(p_{x}\)), while the other two \(p\) AOs are oriented perpendicular to the chain and engage in \(\pi\)-type overlap (\(p_{y},p_{z}\)). Since these three AOs are mutually orthogonal, the resulting crystal orbitals are also mutually orthogonal, and the eigenfunctions of the Hamiltonian are the Bloch functions:

\[\psi_{p_{j}k}(x) = \frac{1}{\sqrt{N}}\sum_{m}^{}{e^{ikma}\ \varphi_{p_{j}}(ma)};\ \ j = x,y,z. \nonumber \]

The corresponding wavevector-dependent energies (eigenvalues) are

• \(p\sigma\) band: \[E_{p_{x}}(k) = \alpha_{p} + 2\beta_{xx}\cos{ka} \nonumber \]

and

• \(p\pi\) bands: \[E_{p_{y}}(k) = \alpha_{p} + 2\beta_{yy}\cos{ka} = E_{p_{z}}(k) = \alpha_{p} + 2\beta_{zz}\cos{ka}. \nonumber \]

These results are summarized in the following band structure and DOS curve:

Figure 5.29

The band structure contains one \(p\sigma\) band intersecting two degenerate \(p\pi\) bands. This effect in the band structure, called a symmetry-allowed crossing, occurs because the \(p\) AOs have different rotational symmetry characteristics with respect to the chain direction. Furthermore, the nodal characteristics of the \(p\) AOs lead to \(\beta_{xx} > 0\) and \(\beta_{yy} = \beta_{zz} < 0\). Therefore, the \(p\sigma\) Bloch function \(\psi_{p_{x}k}(x)\) at the zone center \(k = 0\) is totally antibonding because a positive lobe on one atomic site overlaps with a negative lobe on an adjacent site. Likewise, \(\psi_{p_{x}k}(x)\) at the zone boundary is totally bonding. Therefore, the \(p\sigma\)-band decreases in energy from \(k = 0\) to \(\frac{\pi}{a}\). On the other hand, the \(p\pi\)-bands increase in energy from the zone center to the zone boundary, just like the \(s\)-band. The band center of all bands is the energy of the \(p\) AOs \(\alpha_{p}\), and the bandwidth of the \(p\sigma\)-band exceeds that of the \(p\pi\)-bands: \(4\left| \beta_{xx} \right| > 4\left| \beta_{zz} \right|\). Thus, the entire DOS curve shows 4 distinct peaks and can accommodate up to 6 electrons per unit cell.

Combining \(s\)- and \(p\)-Bands: Realistic band structures become complicated as the number of basis AOs in the unit cell increases. With this greater complexity, interactions can arise between different Bloch functions. To demonstrate this effect, consider the 1-d chain of atoms using \(s\) and \(p\) AOs as the atomic basis functions.

These 4 AOs are mutually orthogonal on each atomic site, but there is an additional nearest neighbor \(\sigma\)-interaction between \(s\) and \(p_{x}\) AOs (see figure to the right). With a basis set of 4 Bloch functions, the Hamiltonian \(H(k)\) is a \(4 \times 4\) matrix. The diagonal elements are the expressions for the \(s\)-, \(p\sigma\)-, and two \(p\pi\)-bands, and there is one off-diagonal element between the \(s\) and \(p_{x}\) AOs:

\[\left\langle \psi_{sk}(x) \middle| H \middle| \psi_{p_{x}k}(x) \right\rangle = \left\langle \frac{1}{\sqrt{N}}\sum_{m'}^{}{e^{ikm'a}\ \varphi_{s}\left( m'a \right)} \middle| H \middle| \frac{1}{\sqrt{N}}\sum_{m}^{}{e^{ikma}\ \varphi_{p_{x}}(ma)} \right\rangle \nonumber \]

\[\left\langle \psi_{sk}(x) \middle| H \middle| \psi_{p_{\sigma}k}(x) \right\rangle = \frac{1}{N}\sum_{m'}^{}{\sum_{m}^{}{e^{ika\left( m - m' \right)}\ \left\langle \varphi_{s}\left( m'a \right) \middle| H \middle| \varphi_{p_{x}}(ma) \right\rangle}} \nonumber \]

\[\left\langle \psi_{sk}(x) \middle| H \middle| \psi_{p_{\sigma}k}(x) \right\rangle = \frac{1}{N}\left\lbrack Ne^{ika}\ \beta_{sp} - Ne^{- ika}\ \beta_{sp} \right\rbrack = i\left( 2\beta_{sp}\sin{ka} \right). \nonumber \]

The resulting Hamiltonian matrix that must be diagonalized to give eigenvalues and eigenvectors is

\[H(k) = \begin{pmatrix} \alpha_{s} + 2\beta_{ss}\cos{ka} & i\left( 2\beta_{sx}\sin{ka} \right) & 0 & 0 \\ - i\left( 2\beta_{sx}\sin{ka} \right) & \alpha_{p} + 2\beta_{xx}\cos{ka} & 0 & 0 \\ 0 & 0 & \alpha_{p} + 2\beta_{yy}\cos{ka} & 0 \\ 0 & 0 & 0 & \alpha_{p} + 2\beta_{zz}\cos{ka} \\ \end{pmatrix}, \nonumber \]

and the results are illustrated in the following band structure and DOS curve:

Figure 5.30

The band structure shows an avoided crossing between the \(s\)- and \(p\sigma\)-bands nearer to the zone boundary \(X\) because these two crystal orbitals have identical symmetry characteristics when the wavevector \(k\) is between \(0\) and \(\frac{\pi}{a}\). According to the off-diagonal Hamiltonian matrix element \(H_{sp_{x}}(k)\), there is no \(s\)-\(p\sigma\) interaction for \(k = 0\) (zone center \(\Gamma\)) and \(\frac{\pi}{a}\) (zone boundary \(X\)). Examination of the crystal orbitals at these two points of the band structure indicates that the \(s\)- and \(p\sigma\)-bands “cross” because the bonding function \(\psi_{p_{x}X}(x)\) is lower in energy than the antibonding function \(\psi_{sX}(x)\). The two \(p\pi\)-bands are orthogonal to the \(s\)- and \(p\sigma\)-bands.