# Re: [Kwant] Symmetry and leads

```Hi, I would like to seek clarification regarding a few things about your
proposed solution:```
```
> My idea is to define a different unit cell for the graphene lattice
> which allows lattice vectors to be defined in the x and y real space
> directions. So, instead of using the standard graphene unit cell (which
> I shall denote with a symbol | which is supposed to indicate the bond
> joining the two carbon atoms) I used a unit cell that looks like \_/
> which joins four carbon atoms in a short chain. Repeating this unit cell
> will generate a graphene flake with armchair edges along the horizontal
> direction, and zigzag edges along the vertical direction.

If I have understood correctly you have essentially rotated the lattice
by a right angle. If this is the case, and if you have attached
leads on the left and right as you say then you are no longer simulating
the same system as before, surely? Before, the horizontal edges were
zigzag and now they are armchair.

As I understand it the transport properties of graphene nanoribbons with
armchair or zigzag edges are fundamentally different, so will
this not affect any quantities calculated?

I would  be interested to see a concrete example using the technique
that Anton proposed.

Joseph

On 10/25/14 08:15, David Abergel wrote:
> Dear Anton,
>
>
> I actually found a slightly different solution, which I shall document
> here in case it helps anyone else in the future.
>
> My idea is to define a different unit cell for the graphene lattice
> which allows lattice vectors to be defined in the x and y real space
> directions. So, instead of using the standard graphene unit cell (which
> I shall denote with a symbol | which is supposed to indicate the bond
> joining the two carbon atoms) I used a unit cell that looks like \_/
> which joins four carbon atoms in a short chain. Repeating this unit cell
> will generate a graphene flake with armchair edges along the horizontal
> direction, and zigzag edges along the vertical direction.
>
> If a is the length of the lattice vector of the original (two-atom) unit
> cell, then the lattice vectors of the new unit cell are:
> (sqrt(3)*a,0), and (0,a).
>
> The positions of the four atoms within the unit cell are:
> (0,a/2),
> (a/(2*sqrt(3), 0),
> (sqrt(3)*a/2, 0), and
> (2*a/sqrt(3),a/2).
>
> This can be easily defined using the kwant.lattice.general() method as
> shown in the examples in the documentation. When leads are attached to
> the left and right ends (with translational symmetry (Âħsqrt(3)*a,0)) of
> a rectangular scattering region formed in this way, they do so in the
> vertical direction, as desired.
>
> Hope that helps.
>
> David
>
>
>
> On 23/10/14 19:47, Anton Akhmerov wrote:
>> Dear David,
>>
>> Currently a Kwant symmetry stores the extra directions that describe
>> how it handle various lattices. If you don't do anything, the choice
>> of the extra directions happens automatically, and a Bravais lattice
>> vector is chosen. However you can manually override this by using the
>> add_site_family method of the TranslationalSymmetry (see
>>
>> for details). I know several people have used this to solve exactly
>> the problem that you currently encounter.
>>
>> Best,
>> Anton
>>
>> On Thu, Oct 23, 2014 at 4:40 PM, David Abergel
>> <david.aber...@nordita.org> wrote:
>>> Dear all,
>>>
>>> I am having a problem defining a system which matches what I want.
>>>
>>> I start by defining a rectangular graphene lattice of size 0<=x<L in
>>> the x
>>> direction and 0<=x<W in the y direction. (By "rectangular" I mean in the
>>> real space coordinates, not the crystallographic coordinates.)
>>>
>>> I want to attach a lead to the left-had end of this rectangle, going to
>>> minus infinity. Therefore, I define a lead with translational symmetry
>>> (-1,0) and the appropriate hopping. I attach the lead and plot the
>>> system.
>>>
>>> When I plot the system, I find that the lead has been attached along the
>>> (0,1) crystallographic direction (so, that is along the (1/2, sqrt(3)/3)
>>> real space vector). A triangle of extra sites have been added for x<0
>>> (real
>>> space) so that the total shape of the scattering region is now not
>>> rectangular.
>>>
>>> happens on
>>> the right of the sample so that my scattering region is now a
>>> parallelogram.
>>>
>>> As I understand it, this should not affect the physics in any way,
>>> since the
>>> lead is semi-infinite. But, if I want to draw pictures, plot
>>> functions over
>>> the scattering region, and gain physical understanding, it is a bit of a
>>> pain. So, my question is whether there is any way to make the lead
>>> attach
>>> along the (0,1) real space direction (which is the same as the (-1,2)
>>> crystallographic direction) and yet maintain the (-1,0) translational
>>> symmetry?
>>>
>>> If you require a sample program which reproduces this behavior then I
>>> can
>>> easily provide that, but I thought I should not extend an already
>>> long post
>>> unnecessarily.
>>>
>>>
>>> David
>>>
>
> --
> David Abergel
> Nordita
> david.aber...@nordita.org
> @David_Abergel

```

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