Electronic structure of graphene nanoribbons: Tight-binding versus density functional theory methods

The project explores recently discovered graphene nanoribbons (GNRs) by computing their electronic structure as equilibrium property using simple tight-binding method (as implemented in KWANT, PythTB or your own Matlab script) and more advanced density functional theory codes (as implemented in Quantum ESPRESSO or GPAW packages).

Subband structure of armchair GNR

Using the nearest-neighbor tight-binding Hamiltonian with single $p_{z}$ orbital per carbon atom, compute the subband structure of three armchair GNRs whose width is $N_{a}=4,5,30$. The expected result is shown in Lecture 7. Pay attention to select the proper interval of $k_{x}$ values as the first 1D Brillouin zone.

Subband structure of zigzag GNR

Using the nearest-neighbor ($t_{1}=2.7$ eV) tight-binding Hamiltonian with single $p_{z}$ orbital per carbon atom, compute the subband structure of three zigzag GNRs whose width is $N_{z}=4,5,30$. The expected result is shown in Lecture 7. Plot the amplitude squared $|\chi (y)|^{2}$ across $N_{z}=30$ ZGNR as the transverse part of a selected eigenfunction (i.e., conducting channel) whose eigenenergy is close to the Dirac point $E=10^{-3}t_{1}$. This plot should show that probability to find electron peaks around the nanoribbons edges.

Subband structure of AGNR and ZGNR using third-nearest neighbor hoppings

Repeat subband structure calculations for $N_{a}=5$ AGNR and $N_{z}=5$ ZGNR using the tight-binding Hamiltonian which includes up to third-nearest neighbour hoppings whose values are: $t_{1}=2.7$ eV, $t_{2}=0.20$ eV, and $t_{3}=0.18$ eV. Comment on the difference between this result and a) and b).

Subband structure of GNRs with spin-orbit coupling as topological insulator below 0.5 K

At low temperatures, the energy band gap $\approx 0.5K$ due to intrinsic spin-orbit coupling of graphene, as well as the chiral spin-filtered edge states whose subbands pass through the gap, should become visible in experiments. This systems, termed topological insulator where time-reversal invariance ensures the crossing of the energy levels at special points in the Brillouin zone so that their energy spectrum cannot be adiabatically deformed into topologically trivial insulator without such states, can be studied using the following tight-binding model:

${\hat {H}}_{\mathrm {TI} }=-t_{1}\sum _{\langle ij\rangle }{\hat {c}}_{i}^{\dagger }{\hat {c}}_{j}+{\frac {2i}{\sqrt {3}}}t_{\mathrm {SO} }\sum _{\langle \langle ij\rangle \rangle }{\hat {c}}_{i}^{\dagger }{\hat {\mathbf {\sigma } }}\cdot (\mathbf {d} _{kj}\times \mathbf {d} _{ik}){\hat {c}}_{j}$ (1)

where ${\hat {c}}_{i}^{\dagger }=({\hat {c}}_{i\uparrow }^{\dagger },{\hat {c}}_{i\downarrow }^{\dagger })$ are electron creation operators on the honeycomb lattice of GNR and ${\hat {\mathbf {\sigma } }}$ is the vector of the Pauli matrices. The second term in Eq. (1) introduces the intrinsic SO coupling compatible with the symmetries of the honeycomb lattice. The SO coupling acts as spin-dependent next-nearest neighbor hopping where $i$ and $j$ are two next-nearest neighbor sites, $k$ is the only common nearest neighbor of $i$ and $j$, and $\mathbf {d} _{ik}$ is a vector pointing from $k$ to $i$. Compute the band structure of $N_{z}=30$ ZGNR with SO coupling described by Hamiltonian (1) assuming $t_{1}=2.7$ eV and $t_{\mathrm {SO} }=0.03t_{1}$. The value for $t_{\mathrm {SO} }$ is selected to be much larger than the realistic one in order to see clearly opening of the band gap $\Delta _{\rm {SO}}=6{\sqrt {3}}t_{\rm {SO}}$ in your figure. Your result should look the same as Fig. 1 in Phys. Rev. Lett. 95, 226801 (2005).

Subband structure of GNRs using DFT

Using DFT code, Quantum Espresso or GPAW introduced in the Computer Lab, compute subband structure for $N_{a}=5$ AGNR and $N_{z}=5$ ZGNR and comment on difference or similarities with your results in 1., 2., and 3.

References

• Main reference: A. Cresti, N. Nemec, B. Biel, G. Niebler, F. Triozon, G. Cuniberti, and S. Roche, Charge transport in disordered graphene-based low-dimensional materials, Nano Research 1, 361 (2008). [PDF].

• Reference for GNR as 2D topological insulator: C. K. Kane and E. J. Mele, Quantum spin Hall effect in graphene, Phys. Rev. Lett. 95, 226801 (2005).