Electronic properties of graphene

[rizzodex]

I'm taking a graduate course on computational physics which has a project. I'm doing research
on electronic properties of graphene, like quantum hall effect measurements. So, I want to do the project related to my research.
The project shouldn't be too difficult, as I haven't done simulations before. It shouldn't be too easy either, after all it's a graduate level course.

Any idea?

 

[saaskis]

Hi! In case of graphene, I think it is always the best to begin with tight-binding calculations, if you haven't done them yourself yet. After having a program that calculates the TB Hamiltonian for a small graphene lattice of your choice, you could try and include the magnetic field in the hopping elements. With this kind of program, you could study small graphene "quantum dots" and see how the energy levels (and the smoothened density of states) are modified by a magnetic field. But even without a magnetic field, there should be enough programming to do.

And if this is not enough, you could proceed to calculate the conductances of graphene nanoribbons using the Landauer-Büttiker formalism, even in a magnetic field. But as far as I know, getting to QHE-like results computationally is quite difficult.

If this sounds interesting and you need help, just ask :) Good luck with your project!

 

[rizzodex]

Hey
First of all,thank you so much for the reply. It'll be a good idea to take the magnetic field into the hopping Hamiltonian. But, I don't understand why do you think there will be enough programing without the magnetic field. I did it analytically as a undergrad couple of years back, and what I got is something like the following : http://www.iue.tuwien.ac.at/phd/pourfath/node18.html" .

Please correct me if I'm wrong.

 

[saaskis]

Yes, you're correct that the bandstructure of an infinite graphene lattice is easy to determine analytically. In case of a finite lattice, in which the periodicity is broken, you will need a computer to calculate the single-particle energy levels by diagonalizing the Hamiltonian matrix. So I guess that the first step would be to write a subprogram that creates a list of coordinates for a "quantum dot" of your choice. Rectangular dot is easy, but to study e.g. hexagonal dots, you already have to do a little thinking. Given the coordinates, the step next would be to construct the Hamiltonian matrix. And finally, to diagonalize and extract any interesting quantities.

In case of a magnetic field, it would also be interesting to include e.g. next-nearest-neighbor hoppings, since this breaks the particle-hole symmetry and can result in interesting effects in a magnetic field.

http://arxiv.org/find/all/1/all:+AND+dots+AND+graphene+quantum/0/1/0/all/0/1" you can find all the interesting papers about graphene quantum dots in Arxiv.

But these are all just suggestions :)

 

[idursun08]

Hi rizzodex

I am just start to research graphene, like you. I think firstly ı can observe some information about graphene with usıng SIESTA. What do you thınk ?

 

[rizzodex]

Yes, you're correct that the bandstructure of an infinite graphene lattice is easy to determine analytically. In case of a finite lattice, in which the periodicity is broken, you will need a computer to calculate the single-particle energy levels by diagonalizing the Hamiltonian matrix. So I guess that the first step would be to write a subprogram that creates a list of coordinates for a "quantum dot" of your choice. Rectangular dot is easy, but to study e.g. hexagonal dots, you already have to do a little thinking. Given the coordinates, the step next would be to construct the Hamiltonian matrix. And finally, to diagonalize and extract any interesting quantities.

In case of a magnetic field, it would also be interesting to include e.g. next-nearest-neighbor hoppings, since this breaks the particle-hole symmetry and can result in interesting effects in a magnetic field.

http://arxiv.org/find/all/1/all:+AND+dots+AND+graphene+quantum/0/1/0/all/0/1" you can find all the interesting papers about graphene quantum dots in Arxiv.

But these are all just suggestions :)

I'm thinking of doing the band structure calculation by solving the Dirac equation for graphene nanoribbons. I want to show that there are "edge states" for zigzag and no such thing exists for armchair.
This can be done analytically. I'm confused about the scope of computation in this case. Do I solve the Dirac equation numerically? If yes, can the mapped Fourier grid method be useful?
Ref: Phys. Rev. B 54, 17954 (1996)

 

[saaskis]

I'm thinking of doing the band structure calculation by solving the Dirac equation for graphene nanoribbons. I want to show that there are "edge states" for zigzag and no such thing exists for armchair.
This can be done analytically. I'm confused about the scope of computation in this case. Do I solve the Dirac equation numerically? If yes, can the mapped Fourier grid method be useful?
Ref: Phys. Rev. B 54, 17954 (1996)

You should note that Nakada et al. did not solve the Dirac equation, but the tight-binding model. So I'm not sure, which one you want to solve. The TB-model would make a lot more sense for nanoribbons, though. Look, for example, at the bandstructures in Figs. 3 and 4: there is no linear dispersion, except in gapless armchair ribbons.

The TB-bandstructure can also be determined analytically, but it's easiest to do it directly numerically. For a zigzag ribbon with N "zigzag lines", there are 2N atoms in the computational unit cell, as explained in Fig. 1, so this is the dimension of the effective Hamiltonian matrix to be diagonalized. You have to construct the tight-binding Hamiltonian matrix for the unit cell and weigh the hopping to the neighbouring unit cells either by e^ik or e^-ik, depending on the direction of hopping (so that the Hamiltonian matrix is hermitian, in the end). For a fixed k, you will get 2N eigenvalues, i.e. bands. Sweeping k over [0,pi] gives you the bandstructure.

If you're not familiar with this kind of TB calculations, you could perhaps take a look at the book by Datta, called "Quantum Transport: Atom to Transistor." Good luck!

 

[rizzodex]

You should note that Nakada et al. did not solve the Dirac equation, but the tight-binding model. So I'm not sure, which one you want to solve. The TB-model would make a lot more sense for nanoribbons, though. Look, for example, at the bandstructures in Figs. 3 and 4: there is no linear dispersion, except in gapless armchair ribbons.

The TB-bandstructure can also be determined analytically, but it's easiest to do it directly numerically. For a zigzag ribbon with N "zigzag lines", there are 2N atoms in the computational unit cell, as explained in Fig. 1, so this is the dimension of the effective Hamiltonian matrix to be diagonalized. You have to construct the tight-binding Hamiltonian matrix for the unit cell and weigh the hopping to the neighbouring unit cells either by e^ik or e^-ik, depending on the direction of hopping (so that the Hamiltonian matrix is hermitian, in the end). For a fixed k, you will get 2N eigenvalues, i.e. bands. Sweeping k over [0,pi] gives you the bandstructure.

If you're not familiar with this kind of TB calculations, you could perhaps take a look at the book by Datta, called "Quantum Transport: Atom to Transistor." Good luck!

Yes,Nakada et al did TB calculation. I gave it as a reference because both TB calclation and Dirac equation solution gives similar results, but in the "high energy" limit.
Yes I'm new to TB calculations. I'll definitely look at the book. Thank you so much.

 

[saaskis]

Yes,Nakada et al did TB calculation. I gave it as a reference because both TB calclation and Dirac equation solution gives similar results, but in the "high energy" limit.
Yes I'm new to TB calculations. I'll definitely look at the book. Thank you so much.

That's interesting, could you perhaps give the reference showing the "similar results"? Or do Nakada et al. state that somewhere in the paper? I don't also quite understand what you mean by the high-energy limit.

But that book by Datta is really nice, I hope it helps!

 

[rizzodex]

That's interesting, could you perhaps give the reference showing the "similar results"? Or do Nakada et al. state that somewhere in the paper? I don't also quite understand what you mean by the high-energy limit.

But that book by Datta is really nice, I hope it helps!

I'm still in the process of "just getting started" :). But I read in some relatively unreliable sources, that based on k.p approximation low energy excitons obey the 2D Dirac-like equation. But I'm not quite sure whether based on this, one can establish similarities between results of TB calculation and Dirac equation.
Ref: Phys. Rev. B, 29 (1984) 1685.

 

[rizzodex]

That's interesting, could you perhaps give the reference showing the "similar results"? Or do Nakada et al. state that somewhere in the paper? I don't also quite understand what you mean by the high-energy limit.

But that book by Datta is really nice, I hope it helps!

Hey, thank you for being so helpful. Now I understand the physics, but have a couple of questions about the computation.
1. If I take N=50, for each k-value within the first Brillouin zone, I have to diagonalize the 2N*2N matrix. I have a 1.6GHz processor with 1Gb RAM and 100Gb available hard disk memory. Do I have enough computation power? If not I can reduce the number N.

2. What is the fastest algorithm for finding ALL the eigenvalues of a Hermitian matrix?

I'm using FORTRAN 90.

 

[saaskis]

1. If I take N=50, for each k-value within the first Brillouin zone, I have to diagonalize the 2N*2N matrix. I have a 1.6GHz processor with 1Gb RAM and 100Gb available hard disk memory. Do I have enough computation power? If not I can reduce the number N.
2. What is the fastest algorithm for finding ALL the eigenvalues of a Hermitian matrix?

A (100 x 100)-matrix is still a piece of cake for today's computers. If the matrix is of dimension 10 000, the diagonalization will start to take a macroscopic amount of time. But you should be able to find the relevant physics already for much smaller systems. You may test the scaling using e.g. Matlab's timer-function and by running a loop over different sizes of matrices. The number of k-points is not really relevant for non-interacting problems, since the computation time scales only linearly as a function of them. As for the diagonalization algorithms, I have unfortunately no idea. I have only blindly used EIG functions found in linear algebra libraries.

 

[element4]

@ rizzodex

Take a look at this paper http://arxiv.org/abs/cond-mat/9809260v2" . They do with some detail, the things you are interested in. Actually they almost derive the matrices you need to diagonalize (including magnetic field). See equation 2.9 and 2.11.If you have managed to do all this, you can easily make a simple modification and see some very interesting physics. Besides the tight-binding term $$H_0 = \sum_{<i,j>}t_{ij}c^{\dagger}_ic_j$$ (spin included), try to add a spin-orbit term like this $$i\lambda_{SO}\sum_{<<i,h>>}\nu_{ij}c_i^{\dagger}s^zc_j$$ but no magnetic field (it's important to preserve time-reversal symmetry). Here <<i,j>> stand for second-nearest neighbor, and $$\nu_{ij}=\pm 1$$ depending on you are going from right or left on the honeycomb lattice. See more detail in this paper http://arxiv.org/abs/cond-mat/0506581" . You can use any geometry for this, take for example a zig-zag strip (finite in one direction only).

You will see that for some values of $$\lambda_{SO}$$ the system is a usual insulator with a band gap. But for other values a Dirac cone emerges in the gap, the states described by the cone is localized at the edge of the material. So the bulk is gapped, but with gapless edge states. This is a topological phase transition, from a trivial insulator to a topological insulator (also called Quantum Spin Hall effect in 2D).

The edge states will come in pairs of counter-propagating spin up and spin down states at each edge, you can see this by looking at the eigenvectors of the matrix you are diagonalizing (see figure 2 http://arxiv.org/abs/cond-mat/0411737" ). Furthermore these edge states are topologically robust, meaning that they survive any perturbation (that dosn't break time-reversal symmetry). So if you choose any geometry, zigzag, armchair or anything else, they will still be there. And if you put in (non-magnetic) impurities, the edge states won't feel anything! (In comparison, the edge states in the pure tight-binding model is only present for zigzag edges and they will be destroyed by small perturbations!)

This is a very simple manifestation of a new and exotic phase of matter, called time-reversal invariant topological insulator (TRS TI). This is on everybody's lips in the condensed matter physics today, even some people working on particle physics and string theory find this interesting since the low-energy physics of these phases are described by topological field theory related to aspects of high energy physics (Chern-Simons theory, Axion fields and so on). (This phase has been found experimentally in other materials in both 2D and 3D. In 3D, the physics is even more exotic.)

All this exiting physics, comes out of adding a simple term to your Hamiltonian and run your program.

Good luck

 

[rizzodex]

A (100 x 100)-matrix is still a piece of cake for today's computers. If the matrix is of dimension 10 000, the diagonalization will start to take a macroscopic amount of time. But you should be able to find the relevant physics already for much smaller systems. You may test the scaling using e.g. Matlab's timer-function and by running a loop over different sizes of matrices. The number of k-points is not really relevant for non-interacting problems, since the computation time scales only linearly as a function of them. As for the diagonalization algorithms, I have unfortunately no idea. I have only blindly used EIG functions found in linear algebra libraries.

Yeah, that's true. I found all the eigenvalues for one value of k. So, if I have intersecting subbands, when I sweep the value of the k, how do I keep track of the corresponding eigen values of different subbands?

 

[rizzodex]

Oh sorry for posting the stupid question about how to "track" eigen numbers. I kind of been able to reproduce some of Nakada et al. results. Thanks for all you guys, especially saaskis, for your kind help. Though my results for small number of atoms are not in good agreement with Nakada et al. I'll try to figure it out.
Hope this much computation will satisfy the instructor. Otherwise, I've to try including the magnetic field or something else.
Thank you element4, for all your ideas and help. I'm more interested in doing experiments with graphene. Computation was a nice experience though.