6.2: Graphene π-Bands

Graphene is a planar, honeycomb network of carbon atoms adopting the 2-d space group \(p6mm\). The hexagonal unit cell contains two carbon atoms and the lattice points (unit cell corners) are the centers of every hexagon. Also, carbon atoms “1” are bonded solely to three carbon atoms “2” and vice versa.

Figure 6.11: Real space structure of graphene with carbon atoms in one hexagonal unit cell and their nearest neighbors labelled. The lattice points occur at the centers of every hexagon.

Figure 6.12: Reciprocal lattice for hexagonal symmetry. The first Brillouin zone is the shaded hexagonal region around reciprocal lattice point Γ. M and K are important wavevectors for plotting the π-band structure.

The first Brillouin zone for the hexagonal system contains all allowed wavevectors that assign the translational symmetry characteristics of the electronic wavefunctions. Using these allowed wavevectors, a tight-binding (L.C.A.O.-type) electronic structure calculation follows these steps:

(1) FORM Bloch functions for each AO in the basis set of one unit cell:

Each carbon atom utilizes one \(2p_{z}\) AO to form the π-electronic bands. Designate \(\varphi_{1}(\boldsymbol r)\) and \(\varphi_{2}(\boldsymbol r)\) for these AOs. Then, the two general Bloch functions are

\(\psi_{1\boldsymbol k}(\boldsymbol r) = \frac{1}{\sqrt{N_{1}N_{2}}}\sum_{m}^{}{\sum_{n}^{}{e^{i\boldsymbol k \cdot \boldsymbol T_{mn}}\varphi_{1}\left( \boldsymbol r - \boldsymbol T_{mn} \right)}}\);

\(\psi_{2\boldsymbol k}(\boldsymbol r) = \frac{1}{\sqrt{N_{1}N_{2}}}\sum_{m}^{}{\sum_{n}^{}{e^{i\boldsymbol k \cdot \boldsymbol T_{mn}}\varphi_{2}\left( \boldsymbol r - \boldsymbol T_{mn} \right)}}\).

Now, choose the periodic boundary conditions to be \(N_{1} = N_{2} = N\), and replace the vector notation in the phase factors by coordinates. In real space, \(\boldsymbol T_{mn} = m\boldsymbol a_{1} + n\boldsymbol a_{2}\); in reciprocal space, \(\boldsymbol k = k_{1}\boldsymbol a_{1}^* + k_{2}\boldsymbol a_{2}^*\). Then \(\boldsymbol k \cdot \boldsymbol T_{mn} = 2\pi\left( mk_{1} + nk_{2} \right)\). Lastly, substitute \(\varphi_{1}(m,n)\) and \(\varphi_{2}(m,n)\), respectively, for \(\varphi_{1}\left( \boldsymbol r - \boldsymbol T_{mn} \right)\) and \(\varphi_{2}\left( \boldsymbol r - \boldsymbol T_{mn} \right)\). Then

\(\psi_{1\boldsymbol k}(\boldsymbol r) = \frac{1}{N}\sum_{m}^{}{\sum_{n}^{}{e^{2\pi imk_{1}}e^{2\pi ink_{2}}\varphi_{1}(m,n)}}\);

\(\psi_{2\boldsymbol k}(\boldsymbol r) = \frac{1}{N}\sum_{m}^{}{\sum_{n}^{}{e^{2\pi imk_{1}}e^{2\pi ink_{2}}\varphi_{2}(m,n)}}\).

(2) CONSTRUCT the Hamiltonian matrix \(H(\boldsymbol k)\) for a wavevector \(\boldsymbol k\) using these Bloch functions:

For simplicity, use the Hückel approximation, which considers just nearest neighbor orbital overlap. With two basis functions per unit cell, \(H(\boldsymbol k)\) is a \(2\times 2\) matrix. By taking the connectivities of carbon atoms into account, the (on-site) coulomb integrals are

\(\alpha_{p} = \left\langle \varphi_{1}(m,n) \middle| H \middle| \varphi_{1}(m,n) \right\rangle = \left\langle \varphi_{2}(m,n) \middle| H \middle| \varphi_{2}(m,n) \right\rangle \sim\) energy of C \(2p_{z}\) AO,

and the (inter-site) resonance integrals between nearest neighbor atoms are

\(\beta_{zz} = \left\langle \varphi_{1}(m,n) \middle| H \middle| \varphi_{2}(m,n) \right\rangle = \left\langle \varphi_{1}(m,n) \middle| H \middle| \varphi_{2}(m + 1,n) \right\rangle = \left\langle \varphi_{1}(m,n) \middle| H \middle| \varphi_{2}(m,n - 1) \right\rangle\)

\(\beta_{zz} = \left\langle \varphi_{2}(m,n) \middle| H \middle| \varphi_{1}(m,n) \right\rangle = \left\langle \varphi_{2}(m,n) \middle| H \middle| \varphi_{1}(m - 1,n) \right\rangle = \left\langle \varphi_{2}(m,n) \middle| H \middle| \varphi_{1}(m,n + 1) \right\rangle\)

All other inter-site integrals, i.e., integrals for atom pairs that exceed nearest neighbor contacts, are 0. The numerical values of \(\alpha_{p}\) and \(\beta_{zz}\) are negative-valued. Then, the Hamiltonian matrix elements are:

\(\left\langle \psi_{1\boldsymbol k} \middle| H \middle| \psi_{1\boldsymbol k} \right\rangle = \frac{1}{N^{2}}\sum_{m,m'}^{}{\sum_{n,n'}^{}{e^{2\pi i(m - m')k_{1}}e^{2\pi i(n - n')k_{2}}\left\langle \varphi_{1}\left( m',n' \right) \middle| H \middle| \varphi_{1}(m,n) \right\rangle}}\)

\(\left\langle \psi_{1\boldsymbol k} \middle| H \middle| \psi_{1\boldsymbol k} \right\rangle = \frac{1}{N}\sum_{m}^{}{\sum_{n}^{}\left\langle \varphi_{1}(m,n) \middle| H \middle| \varphi_{1}(m,n) \right\rangle} = \alpha_{p}.\)

\(\left\langle \psi_{2\boldsymbol k} \middle| H \middle| \psi_{2\boldsymbol k} \right\rangle = \frac{1}{N}\sum_{m}^{}{\sum_{n}^{}\left\langle \varphi_{2}(m,n) \middle| H \middle| \varphi_{2}(m,n) \right\rangle} = \alpha_{p}.\)

\(\left\langle \psi_{1\boldsymbol k} \middle| H \middle| \psi_{2\boldsymbol k} \right\rangle = \frac{1}{N^{2}}\sum_{m,m'}^{}{\sum_{n,n'}^{}{e^{2\pi i(m - m')k_{1}}e^{2\pi i(n - n')k_{2}}\left\langle \varphi_{1}\left( m'n' \right) \middle| H \middle| \varphi_{2}(mn) \right\rangle}}\)

\(\left\langle \psi_{1\boldsymbol k} \middle| H \middle| \psi_{1\boldsymbol k} \right\rangle = \frac{1}{N}\sum_{m}^{}{\sum_{n}^{}\left\langle \varphi_{1}(m,n) \middle| H \middle| \varphi_{2}(m,n) \right\rangle + e^{2\pi ik_{1}}\left\langle \varphi_{1}(m,n) \middle| H \middle| \varphi_{2}(m + 1,n) \right\rangle + e^{- 2\pi ik_{2}}\left\langle \varphi_{1}(m,n) \middle| H \middle| \varphi_{2}(m,n - 1) \right\rangle}\)

\(\left\langle \psi_{1\boldsymbol k} \middle| H \middle| \psi_{2\boldsymbol k} \right\rangle = \beta_{zz} + \beta_{zz}e^{2\pi ik_{1}} + \beta_{zz}e^{- 2\pi ik_{2}}.\)

\(\left\langle \psi_{2\boldsymbol k} \middle| H \middle| \psi_{1\boldsymbol k} \right\rangle = \beta_{zz} + \beta_{zz}e^{- 2\pi ik_{1}} + \beta_{zz}e^{2\pi ik_{2}} = \left\langle \psi_{1k} \middle| H \middle| \psi_{2k} \right\rangle^{}\).

The Hamiltonian matrix is:

\(H(\boldsymbol k) = \begin{pmatrix} \alpha_{p} & \beta_{zz}\left( 1 + e^{2\pi ik_{1}} + e^{- 2\pi ik_{2}} \right) \\ \beta_{zz}\left( 1 + e^{- 2\pi ik_{1}} + e^{2\pi ik_{2}} \right) & \alpha_{p} \\ \end{pmatrix}\).

(3) DIAGONALIZE \(H(\boldsymbol k)\) to obtain Energies \(E_{n}(\boldsymbol k)\) and Wavefunctions \(\Psi_{n\boldsymbol k}(\boldsymbol r)\):

The two eigenfunctions are linear combinations of the two Bloch functions:

\(\Psi_{1\boldsymbol k}(\boldsymbol r) = c_{11}(\boldsymbol k)\psi_{1\boldsymbol k}(\boldsymbol r) + c_{12}(\boldsymbol k)\psi_{2\boldsymbol k}(\boldsymbol r)\), and

\(\Psi_{2\boldsymbol k}(\boldsymbol r) = c_{21}(\boldsymbol k)\psi_{1\boldsymbol k}(\boldsymbol r) + c_{22}(\boldsymbol k)\psi_{2\boldsymbol k}(\boldsymbol r)\).

The resulting Schrödinger equation becomes

\(H(\boldsymbol k)\Psi_{n\boldsymbol k}(\boldsymbol r) = \begin{pmatrix} \alpha_{p} & \beta_{zz} + \beta_{zz}e^{2\pi ik_{1}} + \beta_{zz}e^{- 2\pi ik_{2}} \\ \beta_{zz} + \beta_{zz}e^{- 2\pi ik_{1}} + \beta_{zz}e^{2\pi ik_{2}} & \alpha_{p} \\ \end{pmatrix}\begin{pmatrix} c_{n1}(\boldsymbol k) \\ c_{n2}(\boldsymbol k) \\ \end{pmatrix}\)

\(H(\boldsymbol k)\Psi_{n\boldsymbol k}(\boldsymbol r) = E_{n}(\boldsymbol k)\begin{pmatrix} c_{n1}(\boldsymbol k) \\ c_{n2}(\boldsymbol k) \\ \end{pmatrix} = E_{n}(\boldsymbol k)\Psi_{n\boldsymbol k}(\boldsymbol r)\),

which is solved by setting the secular determinant to 0:

\(\det\begin{pmatrix} \alpha_{p} - E_{n}(\boldsymbol k) & \beta_{zz}\left( 1 + e^{2\pi ik_{1}} + e^{- 2\pi ik_{2}} \right) \\ \beta_{zz}\left( 1 + e^{- 2\pi ik_{1}} + e^{2\pi ik_{2}} \right) & \alpha_{p} - E_{n}(\boldsymbol k) \\ \end{pmatrix} = 0\).

The eigenvalues (π-band energies) are:

\(E_{1}(\boldsymbol k) = E_{1}\left( k_{1},k_{2} \right) = \alpha_{p} + \beta_{zz}\sqrt{3 + 2\cos 2\pi k_{1} + 2\cos 2\pi k_{2} + 2\cos 2\pi\left( k_1 + k_2 \right)}\)

\(E_{2}(\boldsymbol k) = E_{2}\left( k_{1},k_{2} \right) = \alpha_{p} - \beta_{zz}\sqrt{3 + 2\cos 2\pi k_{1} + 2\cos 2\pi k_{2} + 2\cos 2\pi\left( k_1 + k_{2} \right)}\).

Since \(\beta_{zz} < 0\), \(E_{1}(\boldsymbol k) \leq \alpha_{p}\) and \(E_{2}(\boldsymbol k) \geq \alpha_{p}\). The coefficients of the Bloch functions for each eigenfunction are \(c_{11}(\boldsymbol k) = c_{12}(\boldsymbol k) = \frac{1}{\sqrt{2}}\) and \(c_{21}(\boldsymbol k) = {- c}_{22}(\boldsymbol k) = \frac{1}{\sqrt{2}}\):

\(\Psi_{1\boldsymbol k}(\boldsymbol r) = \frac{1}{\sqrt{2}}\psi_{1\boldsymbol k}(\boldsymbol r) + \frac{1}{\sqrt{2}}\psi_{2\boldsymbol k}(\boldsymbol r) = \frac{1}{N}\sum_{m}^{}{\sum_{n}^{}{e^{2\pi imk_{1}}e^{2\pi ink_{2}}\frac{1}{\sqrt{2}}\left( \varphi_{1}(m,n) + \varphi_{2}(m,n) \right)}}\)

\(\Psi_{1\boldsymbol k}(\boldsymbol r) = \frac{1}{N}\sum_{m}^{}{\sum_{n}^{}{e^{2\pi imk_{1}}e^{2\pi ink_{2}}\pi_{1}(m,n)}} \equiv \Pi_{\boldsymbol k}(\boldsymbol r)\)

\(\Psi_{2\boldsymbol k}(\boldsymbol r) = \frac{1}{\sqrt{2}}\psi_{1\boldsymbol k}(\boldsymbol r) - \frac{1}{\sqrt{2}}\psi_{2\boldsymbol k}(\boldsymbol r) = \frac{1}{N}\sum_{m}^{}{\sum_{n}^{}{e^{2\pi imk_{1}}e^{2\pi ink_{2}}\frac{1}{\sqrt{2}}\left( \varphi_{1}(m,n) - \varphi_{2}(m,n) \right)}}\)

\(\Psi_{1\boldsymbol k}(\boldsymbol r) = \frac{1}{N}\sum_{m}^{}{\sum_{n}^{}{e^{2\pi imk_{1}}e^{2\pi ink_{2}}\pi_{2}(m,n)}} \equiv \Pi_{\boldsymbol k}^{}(\boldsymbol r)\)

Both eigenfunctions of the Hamiltonian \(H(\boldsymbol k)\) are also Bloch functions, but they are formed by two molecular orbitals in a single unit cell: \(\pi_{1}(\boldsymbol r)\) is the in-phase, \(\pi\)-bonding combination of \(p_{z}\) AOs; and \(\pi_{2}(\boldsymbol r)\) is the out-of-phase, \(\pi\)-antibonding combination of \(p_{z}\) AOs:¹

Figure 6.13: \(\pi_{1}(0,0) = \frac{1}{\sqrt{2}}\left( \varphi_{1}(0,0) + \varphi_{2}(0,0) \right)\)π1(0,0)=21(φ1(0,0)+φ2(0,0))

Figure 6.14: \(\pi_{2}(0,0) = \frac{1}{\sqrt{2}}\left( \varphi_{1}(0,0) - \varphi_{2}(0,0) \right)\)π2(0,0)=21(φ1(0,0)−φ2(0,0))

As a result, we “rename” \(\Psi_{1\boldsymbol k}(\boldsymbol r)\) and \(\Psi_{2\boldsymbol k}(\boldsymbol r)\) by \(\Pi_{\boldsymbol k}(\boldsymbol r)\) and \(\Pi_{\boldsymbol k}^{}(\boldsymbol r)\), respectively, to emphasize that they are derived from the \(\pi\)-bonding and \(\pi\)-antibonding combinations of \(p_{z}\) AOs.

(54) In this problem, \(E_{n}(k)\) are analytical expressions of wavevector. Generally, determining the eigenvalues and eigenvectors involves diagonalizing \(H(k)\) for selected wavevectors in the irreducible wedge of the first Brillouin zone. After this is accomplished, the next step is to

(4) PLOT the Band Structure and Electronic Density of States (DOS) Curves:

Figure 6.15

Wavevectors for the hexagonal system are \(\boldsymbol k = k_{1}\boldsymbol a_{1}^* + k_{2}\boldsymbol a_{2}^* \equiv \left( k_{1},k_{2} \right)\), and the irreducible wedge is the triangular region identified by a reciprocal lattice point \(\Gamma = (0,0)\), a corner \(\text K = (⅓,⅓)\), and an edge center \(\text M = (½,0)\). The electronic band structure shows the eigenvalues as a function of wavevector along the borders of the irreducible wedge: \(\text M\)–\(\Gamma\)–\(\text K\)–\(\text M\). The density of states (DOS) curve is determined by evaluating the eigenvalues for a dense set of wavevectors uniformly distributed throughout the irreducible wedge. The dashed line indicates the Fermi level \(E_{F}\), which is the highest energy of the occupied states in the DOS. Since each C \(2p_{z}\) AO is assigned one electron, then this two-band system contains two electrons and \(E_{F} = \alpha_{p}\). According to the band structure, the Fermi wavevector is \(\text K\). As will be revealed from analysis of the eigenfunctions, states below \(E_{F}\) are \(\pi\)-bonding and states above \(E_{F}\) are \(\pi\)-antibonding.

Figure 6.16

The band structure also reveals that the two \(\pi\)-bands cross at the wavevector \(\text K\). From the rotational symmetry of \(E_{n}(\boldsymbol k)\) in the vicinity of \(\text K\), the energy bands form two cones with their tips meeting at \(E_{n}(\text K)\). This feature of the band structure is called Dirac cones, which lead to unusual electronic transport properties of graphene and other topological insulators.² Dirac cones occur for certain 2-d materials and exhibit linear dispersion of electronic energy with wavevector near the contact points in reciprocal space. For these cases, electronic conduction is explained by massless charge carriers, which are revealed by applying relativistic effects to the Hamiltonian operator.

Figure 6.17

The accompanying density of states curve shows symmetrical distributions of \(\pi\)-electronic states around the Fermi level \(\alpha_{p}\). Because the density of states at \(\alpha_{p}\) goes to zero, graphene is called a zero-gap semiconductor. A small analogue to graphene is the hexagonal molecule benzene C₆H₆. The result of a Hückel-type calculation of the π MOs of benzene yields a pattern of 6 MOs with energies: \(\alpha_{p} + 2\beta_{zz}\) (totally bonding; 1\(\times\)); \(\alpha_{p} + \beta_{zz}\) (net bonding; 2\(\times\)); \(\alpha_{p} - \beta_{zz}\) (net antibonding; 2\(\times\)); and \(\alpha_{p} - 2\beta_{zz}\) (totally antibonding; 1\(\times\)). The density of states curve for the \(\pi\)-bands of graphene shows strong similarities to the MO energy pattern of benzene. For example, the doubly degenerate MOs of benzene line up exactly with the two largest peaks in the density of states curve of graphene. Six π-electrons fill all three bonding π MOs and leave all three antibonding π MOs empty.

The exercise to calculate the \(\pi\)-band structure of graphene illustrates the significance of Bloch functions as useful starting sets of wavefunctions to calculate the electronic structure of a crystalline solid. Since Bloch functions are eigenfunctions of translational symmetry operations, the logical question arises, how does rotational symmetry influence the outcome? The answer is embedded in the diagonalization of the Hamiltonian \(H(\boldsymbol k)\), a procedure which yields the complete wavefunctions and must be eigenfunctions of all symmetry operations of the group of the wavevector \(\mathcal G(\boldsymbol k)\). These characteristics of the wavefunctions produce the degeneracy in the \(\pi\)-band structure at wavevector \(\text K\). In the following, we incorporate rotational symmetry of graphene to identify the IRs for the \(\pi\)-bands in the band structure.

Graphene \(\pi\)-Bands - Symmetry Analysis

The atomic structure of graphene adopts the 2-d space group \(p6mm\), which is the product of the following sets:

\[\mathcal R = \begin{Bmatrix} \left( 6 \middle| 0 \right),\left( 6^{2} \middle| 0 \right),\left( 6^{3} \middle| 0 \right),\left( 6^{4} \middle| 0 \right),\left( 6^{5} \middle| 0 \right),\left( 1 \middle| 0 \right), \\ \left( m_{10} \middle| 0 \right),\left( m_{01} \middle| 0 \right),\left( m_{11} \middle| 0 \right),\left( m_{21} \middle| 0 \right),\left( m_{12} \middle| 0 \right),\left( m_{\overline{1}1} \middle| 0 \right) \\ \end{Bmatrix} = \text{point group} \nonumber \]

and

\[\mathcal L = \{\left( 1 \middle| \boldsymbol T_{mn} \right):\boldsymbol T_{mn} = m\boldsymbol a_{1} + n\boldsymbol a_{2};m,n = \text{integer};a_{1} = a_{2},\alpha_{3} = 120{^\circ}\} = \text{Bravais lattice group} \nonumber \]

Each eigenfunction of \(H(\boldsymbol k)\) transforms according to one of the IRs of \(\mathcal G(\boldsymbol k)\), which is a subgroup of \(p6mm\). Because \(p6mm\) is symmorphic, these IRs are products of the IRs of \(\mathcal L\) and \(\mathcal G_{0}(\boldsymbol k)\):

\[D^{(\boldsymbol k\mu)}\left( R_{i} \middle| \boldsymbol T_{mn} \right) = e^{- \boldsymbol k \cdot \boldsymbol T_{mn}}D_{0}^{(\boldsymbol k\mu)}\left( R_{i} \right). \nonumber \]

We will demonstrate this outcome for selected wavevectors in the band structure diagram. To accomplish this, we use the form of Bloch’s theorem that relates the wavefunction for any unit cell positioned at \(\boldsymbol T_{mn}\) with respect to the Bloch function at the origin cell \(\boldsymbol T_{00}\):

π-bonding: \(\Pi_{\boldsymbol k}\left(\boldsymbol r +\boldsymbol T_{mn} \right) = e^{i\boldsymbol k \cdot \boldsymbol T_{mn}}\Pi_{\boldsymbol k}(r) = e^{2\pi imk_{1}}e^{2\pi ink_{2}}\Pi_{\boldsymbol k}(\boldsymbol r)\),

π-antibonding: \(\Pi_{\boldsymbol k}^{}\left(\boldsymbol r + \boldsymbol T_{mn} \right) = e^{i\boldsymbol k \cdot \boldsymbol T_{mn}}\Pi_{\boldsymbol k}^{}(r) = e^{2\pi imk_{1}}e^{2\pi ink_{2}}\Pi_{\boldsymbol k}^{}(\boldsymbol r)\).

For each wavevector, we must identify \(\mathcal G_{0}(\boldsymbol k)\) to determine the appropriate IR for each wavefunction. The nodal characteristics of each crystal orbital will rationalize its energy relative to \(\alpha_{p}\). This analysis starts by recognizing how the rotational symmetry operations of \(6mm\) are oriented in the real space structure and the corresponding reciprocal lattice:

@ >p(- 2) * >p(- 2) * @

() REAL Space & RECIPROCAL Space
() () REAL Space & RECIPROCAL Space
()

Figure 6.18 &

Figure 6.19
()

(56) \(\Gamma\)-point \((0,0)\)

All operations of \(\mathcal G_{0}\) keep \((0,0)\) invariant, so \(\mathcal G_{0}(\Gamma)\) is \(6mm\). The character table indicates 6 classes divided into the proper rotations and two sets of reflections. For order 12, there are four 1-d and two 2-d IRs. The phase factors for Bloch-type wavefunctions are unity along both \(a_{1}\) and \(a_{2}\):

\[e^{2\pi i(0m + 0n)} = (1)^{m}(1)^{n} \nonumber \]

@ >p(- 14) * >p(- 14) * >p(- 14) * >p(- 14) * >p(- 14) * >p(- 14) * >p(- 14) * >p(- 14) * @

() 6mm & 1 &

6
6⁵

&

3
3²

&

2

& & & Basis Functions³
() () 6mm & 1 &

6
6⁵

&

3
3²

&

2

& & & Basis Functions
() a₁ & 1 & 1 & 1 & 1 & 1 & 1 & cos 6\(\theta\)
a₂ & 1 & 1 & 1 & 1 & –1 & –1 & sin 12\(\theta\)
b₁ & 1 & –1 & 1 & –1 & 1 & –1 & sin 3\(\theta\)
b₂ & 1 & –1 & 1 & –1 & –1 & 1 & cos 3\(\theta\)
e₁ & 2 & 1 & –1 & –2 & 0 & 0 & cos \(\theta\), sin \(\theta\)
e₂ & 2 & -1 & –1 & 2 & 0 & 0 & cos 2\(\theta\), sin 2\(\theta\)
()

@ >p(- 2) * >p(- 2) * @

() \(\Pi_{\Gamma} (\boldsymbol r+\boldsymbol T_{mn})=(1)^{m}(1)^{n} \Pi_{\Gamma} (\boldsymbol r)\) & \(\Pi_{\Gamma}^{*} (\boldsymbol r+\boldsymbol T_{mn})=(1)^{m}(1)^{n} \Pi_{\Gamma}^{*} (\boldsymbol r)\)
() () \(\Pi_{\Gamma} (\boldsymbol r+\boldsymbol T_{mn})=(1)^{m}(1)^{n} \Pi_{\Gamma} (\boldsymbol r)\) & \(\Pi_{\Gamma}^{*} (\boldsymbol r+\boldsymbol T_{mn})=(1)^{m}(1)^{n} \Pi_{\Gamma}^{*} (\boldsymbol r)\)
()

Figure 6.20 &

Figure 6.21
\(E_{1} (\Gamma)= \alpha_{p} +3\beta_{zz}\) & \(E_{2} (\Gamma)= \alpha_{p} -3\beta_{zz}\)
IR = \(a_{1}\): Totally bonding crystal orbital; overall lowest energy of the π-bands. & IR = \(b_{1}\): Totally antibonding crystal orbital; overall highest energy of the π-bands.
()

The star of \(\Gamma\) contains one wavevector, so these crystal orbitals are each nondegenerate.

(57) \(\Sigma\)-point \(\left( k_{1},0 \right)\)

Only \(m_{01}\) keeps \(\left( k_{1},0 \right)\) invariant, so \(\mathcal G_{0}(\Sigma)\) is \(m = \left\{ 1,m_{01} \right\}\). There are two 1-d IRs, which are either symmetric or antisymmetric with respect to \(m_{01}\). The phase factors for Bloch-type wavefunctions vary along \(\boldsymbol a_{1}\) and are unity along \(\boldsymbol a_{2}\). At \((¼,0)\), wavefunctions alternate between real and imaginary values along \(\boldsymbol a_{1}\):

\[e^{2\pi i(¼m + 0n)} = \left( e^{\frac{i\pi}{2}} \right)^{m}(1)^{n} = (i)^{m}(1)^{n} \nonumber \]

() m	1	m01	Basis Functions
() () m	1	m01	Basis Functions
() a’	1	1	cos \(\theta\)
a’ ’	1	-1	sin \(\theta\)
()

@ >p(- 2) * >p(- 2) * @

() \(\Pi_{\Sigma} (\boldsymbol r+\boldsymbol T_{mn})=(i)^{m}(1)^{n} \Pi_{\Sigma} (\boldsymbol r)\) & \(\Pi_{\Sigma}^{*} (\boldsymbol r+\boldsymbol T_{mn})=(i)^{m}(1)^{n} \Pi_{\Sigma}^{*} (\boldsymbol r)\)
() () \(\Pi_{\Sigma} (\boldsymbol r+\boldsymbol T_{mn})=(i)^{m}(1)^{n} \Pi_{\Sigma} (\boldsymbol r)\) & \(\Pi_{\Sigma}^{*} (\boldsymbol r+\boldsymbol T_{mn})=(i)^{m}(1)^{n} \Pi_{\Sigma}^{*} (\boldsymbol r)\)
()

Figure 6.22 &

Figure 6.23
\(E_{1} (\Sigma)= \alpha_{p}+ \sqrt {5} \beta_{zz}\) & \(E_{2} (\Sigma)= \alpha_{p}- \sqrt {5} \beta_{zz}\)
IR = \(a'\): Net bonding crystal orbital. Dashed orbitals indicate imaginary functions. & IR = \(a'\): Net antibonding crystal orbital. Dashed orbitals indicate imaginary functions.
()

The star of \(\Sigma\) contains six wavevectors, so these crystal orbitals are each 6-fold degenerate.

(58) \(M\)-point \((½,0)\)

This boundary point has one equivalent point at \(( - ½,0)\). The operations taking \((½,0)\) into either \((½,0)\) or \(( - ½,0)\) are \(m_{01},2,\) and \(m_{21}\). Therefore, \(\mathcal G_{0}(\text M)\) is \(2mm = \left\{ 1,2,m_{01},m_{21} \right\}\). The phase factors for Bloch-type wavefunctions alternate signs along \(\boldsymbol a_{1}\) and are unity along \(\boldsymbol a_{2}\):

\[e^{2\pi i(½m + 0n)} = \left( e^{i\pi} \right)^{m}(1)^{n} = ( - 1)^{m}(1)^{n} \nonumber \]

() 2mm	1	2	m01	m21	Basis Functions
() () 2mm	1	2	m01	m21	Basis Functions
() a1	1	1	1	1	cos 2\(\theta\)
a1	1	1	-1	-1	sin 2\(\theta\)
b1	1	-1	1	-1	cos \(\theta\)
b2	1	-1	-1	1	sin \(\theta\)
()

@ >p(- 2) * >p(- 2) * @

() \(\Pi_{\text M} (\boldsymbol r+\boldsymbol T_{mn})=(-1)^{m}(1)^{n} \Pi_{\text M} (\boldsymbol r)\) & \(\Pi_{\text M}^{*} (\boldsymbol r+\boldsymbol T_{mn})=(-1)^{m}(1)^{n} \Pi_{\text M}^{*} (\boldsymbol r)\)
() () \(\Pi_{\text M} (\boldsymbol r+\boldsymbol T_{mn})=(-1)^{m}(1)^{n} \Pi_{\text M} (\boldsymbol r)\) & \(\Pi_{\text M}^{*} (\boldsymbol r+\boldsymbol T_{mn})=(-1)^{m}(1)^{n} \Pi_{\text M}^{*} (\boldsymbol r)\)
()

Figure 6.24 &

Figure 6.25
\(E_{1} (\text M)= \alpha_{p}+ \beta_{zz}\) & \(E_{2} (\text M)= \alpha_{p}- \beta_{zz}\)
IR = \(b_{1}\): Net bonding crystal orbital; bonding along \(\boldsymbol a_{2}\), antibonding along \(\boldsymbol a_{1}\). & IR = \(a_{1}\): Net antibonding crystal orbital; antibonding along \(\boldsymbol a_{2}\), bonding along \(\boldsymbol a_{1}\).
()

The star of \(\text M\) contains three wavevectors, so these crystal orbitals are each 3-fold degenerate.

(59) \(\mathrm{\Delta}\)-point \(\left( k_{1},k_{1} \right)\)

Only \(m_{\overline{1}1}\) keeps \(\left( k_{1},k_{1} \right)\) invariant, so \(\mathcal G_{0}(\mathrm{\Delta})\) is \(m = \left\{ 1,m_{\overline{1}1} \right\}\). There are two 1-d IRs, which are either symmetric or antisymmetric with respect to \(m_{\overline{1}1}\). The phase factors for Bloch-type wavefunctions vary along \(\boldsymbol a_{1}\) and \(\boldsymbol a_{2}\). At \((¼,¼)\), wave-functions alternate between real and imaginary values along \(\boldsymbol a_{1}\) and \(\boldsymbol a_{2}\):

\[e^{2\pi i\left( \frac{m}{4} + \frac{n}{4} \right)} = \left( e^{\frac{i\pi}{2}} \right)^{m}\left( e^{\frac{i\pi}{2}} \right)^{n} = (i)^{m}(i)^{n} \nonumber \]

() \(m\)	1	\(m_{\overline{1}1}\)	Basis Functions
() () \(m\)	1	\(m_{\overline{1}1}\)	Basis Functions
() a’	1	1	cos (\(\theta - \frac{\pi}{6}\))
a’ ’	1	-1	sin (\(\theta - \frac{\pi}{6}\))
()

@ >p(- 2) * >p(- 2) * @

() \(\Pi_{\Delta} (\boldsymbol r+\boldsymbol T_{mn})=(i)^{m}(i)^{n} \Pi_{\Delta} (\boldsymbol r)\) & \(\Pi_{\Delta}^{*} (\boldsymbol r+\boldsymbol T_{mn})=(i)^{m}(i)^{n} \Pi_{\Delta}^{*} (\boldsymbol r)\)
() () \(\Pi_{\Delta} (\boldsymbol r+\boldsymbol T_{mn})=(i)^{m}(i)^{n} \Pi_{\Delta} (\boldsymbol r)\) & \(\Pi_{\Delta}^{*} (\boldsymbol r+\boldsymbol T_{mn})=(i)^{m}(i)^{n} \Pi_{\Delta}^{*} (\boldsymbol r)\)
()

Figure 6.26 &

Figure 6.27
\(E_{1} (\Delta)= \alpha_{p}+ \beta_{zz}\) & \(E_{2} (\Delta)= \alpha_{p}- \beta_{zz}\)
IR = \(a'\): Net bonding crystal orbital. Dashed orbitals indicate imaginary functions. & IR = \(a'\): Net antibonding crystal orbital. Dashed orbitals indicate imaginary functions.
()

The star of \(\mathrm{\Delta}\) contains six wavevectors, so these crystal orbitals are each 6-fold degenerate.

(60) \(K\)-point \((⅓,⅓)\)

This boundary point has two equivalent points at \(( - ⅔,⅓)\) and \((⅓, - ⅔)\). The operations of \(\mathcal G_{0}\) that take \((⅓,⅓)\) into itself or one of its equivalent points are members of the point group \(\mathcal G_{0}(\text K) = 3m = \left\{ 1,3,3^{2},m_{\overline{1}1},m_{21},m_{12} \right\}\), which has three classes. There are two 1-d and one 2-d IRs. The phase factors for Bloch-type wavefunctions vary along both \(\boldsymbol a_{1}\) and \(\boldsymbol a_{2}\):

\[e^{2\pi i\left( \frac{m}{3} + \frac{n}{3} \right)} = \left( e^{\frac{2\pi i}{3}} \right)^{m}\left( e^{\frac{2\pi i}{3}} \right)^{n} = \left( e^{\frac{2\pi i}{3}} \right)^{m + n} \nonumber \]

() 3m	1			Basis Functions
() () 3m	1			Basis Functions
() a1	1	1	1	sin 3\(\theta\)
a2	1	1	-1	cos 3\(\theta\)
e	2	-1	0	cos \(\theta\), sin \(\theta\)
()

@ >p(- 2) * >p(- 2) * @

() \(\Pi_{\text K} (\boldsymbol r+\boldsymbol T_{mn})=(e^{2\pi i / 3})^{m+n} \Pi_{\text K} (\boldsymbol r)\) & \(\Pi_{\text K}^{*} (\boldsymbol r+\boldsymbol T_{mn})=(e^{2\pi i / 3})^{m+n} \Pi_{\text K}^{*} (\boldsymbol r)\)
() () \(\Pi_{\text K} (\boldsymbol r+\boldsymbol T_{mn})=(e^{2\pi i / 3})^{m+n} \Pi_{\text K} (\boldsymbol r)\) & \(\Pi_{\text K}^{*} (\boldsymbol r+\boldsymbol T_{mn})=(e^{2\pi i / 3})^{m+n} \Pi_{\text K}^{*} (\boldsymbol r)\)
()

Figure 6.28 &

Figure 6.29
\(E_{1} (\text K)= \alpha_{p}\) & \(E_{2} (\text K)= \alpha_{p}\)
IR = \(e\): Nonbonding crystal orbital (real part only). & IR = \(e\): Nonbonding crystal orbital (real part only).
()

From this visualization of the real parts of these Bloch functions, it may not be apparent that they are nonbonding. Because any linear combination of degenerate wavefunctions creates new degenerate wavefunctions, construct the new pair \(\Pi_{\text K}(\boldsymbol r) \pm \Pi_{\text K}^*(\boldsymbol r)\), which yields functions showing zero nearest neighbor orbital interactions:

@ >p(- 2) * >p(- 2) * @

() \(\Pi_{\text K}(\boldsymbol r)+\Pi_{\text K}^{*}(\boldsymbol r)\) & \(\Pi_{\text K}(\boldsymbol r)-\Pi_{\text K}^{*}(\boldsymbol r)\)
() () \(\Pi_{\text K}(\boldsymbol r)+\Pi_{\text K}^{*}(\boldsymbol r)\) & \(\Pi_{\text K}(\boldsymbol r)-\Pi_{\text K}^{*}(\boldsymbol r)\)
()

Figure 6.30 &

Figure 6.31
()

The star of \(\text K\) contains two wavevectors, so these crystal orbitals are 4-fold degenerate.

(61) \(M\)-point \((½,½)\)

Figure 6.32

The final piece of the band structure plots energies along the \(\text K\)–\(\text M\) direction. Rather that proceed from \((⅓,⅓)\) to \((½,0)\), the calculation continues along the \(\mathrm{\Delta}\)-line beyond \(\text K\) to \((½,½)\), which is equivalent to \(( - ½,½)\) as pointed out to the right by the dashed blue line:

The point \(( - ½,½)\) is a member of the star of \(\text M\), and \(\mathcal G_{0}(\text M) = 2mm = \left\{ 1,2,m_{11},m_{\overline{1}1} \right\}\), which has 4 classes and 4 1-d IRs. The phase factors for Bloch-type wavefunctions alternate signs along both \(\boldsymbol a_{1}\) and \(\boldsymbol a_{2}\):

\[e^{2\pi i\left( \frac{m}{2} + \frac{n}{2} \right)} = ( - 1)^{m}( - 1)^{n} \nonumber \]

@ >p(- 10) * >p(- 10) * >p(- 10) * >p(- 10) * >p(- 10) * >p(- 10) * @

() 2mm & 1 & 2 & \(m_{11}\) &

\(m_{\overline{1}1}\)

& Basis Functions
() () 2mm & 1 & 2 & \(m_{11}\) &

\(m_{\overline{1}1}\)

& Basis Functions
() a₁ & 1 & 1 & 1 & 1 & cos \(2 (\theta - \frac{\pi}{6})\)
a₂ & 1 & 1 & -1 & -1 & sin \(2 (\theta - \frac{\pi}{6})\)
b₁ & 1 & -1 & 1 & -1 & sin \((\theta - \frac{\pi}{6})\)
b₂ & 1 & -1 & -1 & 1 & cos \((\theta - \frac{\pi}{6})\)
()

() \(\Pi_{\text M} (\boldsymbol r+\boldsymbol T_{mn})=(-1)^{m}(-1)^{n} \Pi_{\text M} (\boldsymbol r)\)	\(\Pi_{\text M}^{*} (\boldsymbol r+\boldsymbol T_{mn})=(-1)^{m}(-1)^{n} \Pi_{\text M}^{*} (\boldsymbol r)\)
() () \(\Pi_{\text M} (\boldsymbol r+\boldsymbol T_{mn})=(-1)^{m}(-1)^{n} \Pi_{\text M} (\boldsymbol r)\)	\(\Pi_{\text M}^{*} (\boldsymbol r+\boldsymbol T_{mn})=(-1)^{m}(-1)^{n} \Pi_{\text M}^{*} (\boldsymbol r)\)
() Figure 6.33	Figure 6.34
\(E_{1} (M)= \alpha_{p}- \beta_{zz}\)	\(E_{2} (M)= \alpha_{p}+ \beta_{zz}\)
IR = \(a_{1}\): Net antibonding crystal orbital although it arises from the \(\pi\)-bonding MO in one unit cell.	IR = \(b_{1}\): Net bonding crystal orbital although it arises from the \(\pi\)-antibonding MO in one unit cell.
()

NOTE: The \(\Pi_{\boldsymbol k}(\boldsymbol r)\) and \(\Pi_{\boldsymbol k}^*(\boldsymbol r)\) crystal orbitals remain orthogonal along the entire \(\Gamma\)–\(\text K\)–\(\text M\) direction in reciprocal space. As a result, these two bands may cross each other, which they do at \(\text K\) as determined by the hexagonal symmetry of the graphene structure. Therefore, the unusual transport characteristics of graphene arise from its structural symmetry.

Figure 6.35

Graphene Band Structure and Density of States

(62) Carbon atom valence orbitals are \(2s\), \(2p_{x}\), \(2p_{y}\) and \(2p_{z}\) AOs, the first three of which build \(\sigma\)-bands and provide the strongest interatomic attractions. The graphene electronic band structure calculated by density functional theory indicates that the \(\pi\)-bands cross the \(\sigma\)-bands near the \(\Gamma\) point. In fact, the totally \(\pi\)-bonding crystal orbital at \(\Gamma\) drops below the 2-fold degenerate \(\sigma\)-bonding crystal orbital formed by the \(2p_{x}\) and \(2p_{y}\) AOs. Along both \(\Gamma\)–\(\text M\) and \(\Gamma\)–\(\text K\) directions, the 2-fold degeneracy breaks. Although symmetry allows the \(\sigma\)-bands and \(\pi\)-bands to cross in reciprocal space, the Fermi level occurs at the energy of the \(\pi\)-bands at \(\text K\). Both the band structure and the density of states curve illustrate that the electronic states within 5 eV of the Fermi level arise from the \(\pi\)-bands exclusively. Also, the density of states goes to zero at \(E_{F}\).

Figure 6.36

Graphene Phonon Dispersion

(63) Normal mode analysis using the harmonic approximation yields the phonon dispersion and density of states curves shown here.⁴ There are 6 modes per unit cell because there are 2 C atoms per unit cell and 3 independent modes per atom. At the \(\Gamma\) point, 3 acoustic modes at \(\omega\) = 0 cm^–1 and 3 optical modes at \(\omega\) > 0 cm^–1 occur. Like the electronic band structure, phonon modes perpendicular to the graphene plane (\(z\)-direction) cross the in-plane modes (\(x,y\)-directions) because these functions are orthogonal. Also note that the out-of-plane phonon dispersions, shown in red, mimic the \(\pi\)-band structure.

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1. In the crystal orbital figures for the graphene π-bands, only p-AO lobes on one side of the plane are illustrated.↩
2. I.N. Yakovkin, Crystals, 2016, 6, 143; S. Das Sarma, S. Adam, E.H. Hwang, E. Rossi, Rev. Mod. Phys. 2011, 83, 407-470.↩
3. Theta = angle from lattice vector \(a_{1}\)a1.↩
4. Based on the results of Falkovsky, L.A., Phys. Lett. A 2008, 372, 5189-5192.↩