Dear Adel,
Thank you for the tutorial script.
Best,
Kuangyia
On Wed, Oct 24, 2018 at 11:33 PM Abbout Adel <abbout.a...@gmail.com> wrote:
> Dear Lee,
>
> All what you need is to express the equation of your path: I mean ky as a
> function of kx on this path.
> Please, find below an example about Graphene band structure. I compare it
> to the analytical result found in literature.
>
> Just be careful, when the lattice vectors are not orthogonal. You need to
> do some transformation to get the vectors in the reciprocal lattice.
>
> I wrote this a while ago, so please check it if there are no mistakes in
> it. It works with kwant 1.3
>
> I hope this helps
> Adel
> ###############################
>
> import numpy as np
> from numpy import pi,sqrt,arccos
> import kwant
> from kwant.wraparound import wraparound,plot_2d_bands
> from matplotlib import pyplot
> import scipy
>
> def make_graphene():
> """Create a builder for a periodic infinite sheet of graphene."""
> lat = kwant.lattice.honeycomb()
> sym = kwant.TranslationalSymmetry(lat.vec((1, 0)), lat.vec((0, 1)))
> sys = kwant.Builder(sym)
> sys[lat.shape(lambda p: True, (0, 0))] = 0
> sys[lat.neighbors()] = 1
> return sys
>
> sys=make_graphene()
> syst = wraparound(sys).finalized()
> kwant.plot(sys)
> plot_2d_bands(syst)
>
>
> # calculate the reciprocal symmetry vectors: these
> # are the columns of 'A'
> sym=sys.symmetry
> B = np.array(sym.periods).T
> A = B.dot(np.linalg.inv(B.T.dot(B)))
> # transforms a momentum in the basis (kx, ky) to the basis of
> # reciprocal symmetry vectors
> def momentum_to_lattice(k):
> k, residuals = scipy.linalg.lstsq(A, k)[:2]
> if np.any(abs(residuals) > 1e-7):
> raise RuntimeError("Requested momentum doesn't correspond"
> " to any lattice momentum.")
> return k
>
> def H_k(kx, ky):
> k = (kx, ky)
> k_prime = momentum_to_lattice(k)
> return syst.hamiltonian_submatrix(args=k_prime)
>
> # This function is the exact result taken from literature
> def function(kx,ky):
> return sqrt(3+2*np.cos(kx)+4*np.cos(kx/2.) *np.cos(sqrt(3)/2*ky))
>
> # Here I am comparing the numerical result with the analytical result.
> with kx varying from -pi to pi
> k_val=np.linspace(-pi,pi,61)
> ky=2*pi/sqrt(3)
> resultat1=[abs(H_k(kx,ky)[0]) for kx in k_val]
> resultat2=[abs(function(kx,ky)) for kx in k_val]
>
> pyplot.plot(k_val,resultat1,'r-')
> pyplot.plot(k_val,resultat2,'go')
> pyplot.show()
>
>
> #
> #
> # $
> # E=\pm t \sqrt{3+2 \cos(\sqrt{3}k_x a)+a \cos(\frac{\sqrt{3}}{2}k_x a)
> \cos(\frac{3}{2} k_y a)}
> # $
> #
> # with $a=1/\sqrt{3}$
> #
> # The point $K$ is: $K(\frac{4\pi}{3\sqrt{3}a},0) $
> # The point $M$ is: $M(\frac{\pi}{\sqrt{3} a},\frac{\pi}{3 a})$
>
> # In[ ]:
>
>
>
> def eigenvalues(kx,ky):
> return np.sort(np.linalg.eigvalsh(H_k(kx, ky)))
>
> nb=30
> d1=4*pi/3 # this is the distance when you go from Gamma to K
> d2=2*pi/3 # this is the distance when you go from K to M
>
>
> Kx1= np.linspace(0,4*pi/3,nb)
> Result1=np.array([eigenvalues(kx,0) for kx in Kx1])
>
> Kx2=[d1 + sqrt((kx - (4*pi/3))**2 + 3*(kx - 4*pi/3)**2) for kx in
> np.linspace(pi,4*pi/3,nb)] #distance Gamma K is added here
> Result2=np.array([eigenvalues(kx,-sqrt(3)*(kx - 4.* pi/3)) for kx in
> np.linspace(pi,4*pi/3,nb)])
>
> Kx3=[d1 + d2 + sqrt((x - pi)**2 + (1/sqrt(3)* x - pi/sqrt(3))**2) for x in
> np.linspace(0,pi,nb)]##3distance Gamma K M is addded here
> Result3=np.array([eigenvalues(kx, 1./sqrt(3)* kx) for kx in
> np.linspace(0,pi,nb)])
>
>
> pyplot.scatter(Kx1,Result1[:,0],color='r')
> pyplot.scatter(Kx2,Result2[:,0],color='g')
> pyplot.scatter(Kx3,Result3[:,0],color='c')
>
> pyplot.scatter(Kx1,Result1[:,1],color='r')
> pyplot.scatter(Kx2,Result2[:,1],color='g')
> pyplot.scatter(Kx3,Result3[:,1],color='c')
>
> pyplot.show()
>
>
>
> On Wed, Oct 24, 2018 at 7:53 PM kuangyia lee <kuangyia....@gmail.com>
> wrote:
>
>> Dear kwant developers and users,
>>
>> I want to calculate and plot the band structure of bulk 2D material along
>> some high-symmetry lines in the Brillouin zone (e.g. Gamma-->K-->M--Gamma),
>> i.e. the 1D band structure.
>>
>> I found that someone already posted a relevant issue and some useful
>> solutions were also given. For example, see
>> https://kwant-discuss.kwant-project.narkive.com/2fT27IJh/kwant-for-bulk-system.
>>
>>
>> However, these solutions are only for calculating and plotting the full
>> 2D band structure in the Brillouin zone.
>>
>> Can anyone give some useful solutions to my concern? Your help is much
>> appreciated.
>>
>> Thank you and best regards,
>> Kuangyia Lee
>>
>>
>>
>
> --
> Abbout Adel
>
