View source for Electronic structure of graphene and topological insulator nanowires

The project explores recently discovered graphene nanoribbons (GNRs) by computing their electronic structure as  equilibrium property using simple tight-binding method (as implemented in [https://kwant-project.org/ KWANT], [http://www.physics.rutgers.edu/pythtb/ PythTB] or your own Python script).

== Subband structure of armchair GNR ==
Using the nearest-neighbor tight-binding Hamiltonian with single <math> p_z </math> orbital per carbon atom, compute the subband structure of three armchair GNRs whose width is <math> N_a=4,5, 30</math>. The expected result is shown in [[Media:gnr_and_cnt.pdf|Lecture 7]]. Pay attention to select the proper interval of <math> k_x </math> values as the first 1D Brillouin zone.

== Subband structure of zigzag GNR ==
Using the nearest-neighbor (<math> t_1=2.7 </math> eV) tight-binding Hamiltonian with single <math> p_z </math> orbital per carbon atom, compute the subband structure of three zigzag GNRs whose width is <math> N_z=4,5, 30</math>. The expected result is shown in [[Media:gnr_and_cnt.pdf|Lecture 7]]. Plot the amplitude squared <math> |\chi(y)|^2 </math> across <math> N_z=30</math> ZGNR as the transverse part of a selected eigenfunction  (i.e., conducting channel)  whose eigenenergy is close to the Dirac point <math> E=10^{-3} t_1 </math>. 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 <math> N_a=5 </math> AGNR and <math> N_z=5 </math> ZGNR using the tight-binding Hamiltonian which includes up to third-nearest neighbour hoppings whose values are: <math> t_1=2.7 </math> eV,  <math> t_2 = 0.20 </math> eV, and <math> t_3 = 0.18</math> 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 <math> \approx  0.5 K </math> 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:

<math> \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 </math> (1) 

where <math> \hat{c}^\dagger_i = (\hat{c}^\dagger_{i\uparrow}, \hat{c}^\dagger_{i\downarrow}) </math> are electron creation operators on the honeycomb lattice of GNR and <math> \hat{\mathbf{\sigma}} </math> 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 <math> i </math> and <math> j </math> are two next-nearest neighbor sites, <math> k </math>  is the only common nearest neighbor of <math> i </math> and <math> j </math>, and <math> \mathbf{d}_{ik} </math> is a vector pointing from  <math> k </math> to <math> i </math>. Compute the band structure of <math> N_z=30</math> ZGNR with SO coupling described by Hamiltonian (1) assuming <math> t_1 = 2.7 </math> eV and  <math> t_{\mathrm{SO}}=0.03 t_1 </math>. The value for <math> t_{\mathrm{SO}} </math> is selected to be much larger than the realistic one in order to see clearly opening of the band gap  <math> \Delta_{\rm SO}=6\sqrt{3} t_{\rm SO} </math> in your figure. Your result should look the same as Fig. 1 in  [http://link.aps.org/doi/10.1103/PhysRevLett.95.226801 Phys. Rev. Lett. '''95''', 226801 (2005)]. Plot the amplitude squared <math> |\chi(y)|^2 </math> across ZGNR, which is the transverse part of a selected eigenfunction, whose eigenenergy is within the gap. Alternatively, you can plot local density of states at the same energy inside the gap.

== Subband structure of nanowires carved from graphene/WSe2 van der Waals heterostructure ==
Using first-principles tight-binding Hamiltonian for graphene/WSe2 van der Waals heterostructure derived in  [https://journals.aps.org/prb/abstract/10.1103/PhysRevB.93.155104 Phys. Rev. B '''93''', 155104 (2016)], which gives parameters in TABLE I for honeycomb lattice of graphene only with spin-orbit coupling induced by the presence of WSe2, compute band structure of zigzag nanowire made of graphene/WSe2 heterostructure. Comment if this is topologically trivial or nontrivial system.

== 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'', [https://link.springer.com/content/pdf/10.1007%2Fs12274-008-8043-2.pdf  Nano Research '''1''', 361 (2008)].
* '''Reference for GNR as 2D topological insulator:''' C. K. Kane and E. J. Mele, ''Quantum spin Hall effect in graphene'', [http://link.aps.org/doi/10.1103/PhysRevLett.95.226801 Phys. Rev. Lett. '''95''', 226801 (2005)].
* '''Reference for graphene/WSe2 nanowires:''' M. Gmitra, D. Kochan, P. Högl, and J. Fabian, ''Trivial and inverted Dirac bands and the emergence of quantum spin Hall states in graphene on transition-metal dichalcogenides'', [https://doi.org/10.1103/PhysRevB.93.155104 Phys. Rev. B '''93''', 155104 (2016)].