Mailman 3 June 2016 - Kwant-discuss

QHE code
by T.C. Wu June 15, 2016

June 15, 2016

Hi Kwanters, I recently come across the file in http://topocondmat.org/w3_pump_QHE/Laughlinargument.html. When I try to run the code and plot pho_xx (which is E_x in the code) against B, I can't get a similar plot with the experimental data. Maybe it's a silly question, but do anyone have any clue? Thanks. Best, Johnny

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Leads with greater than NN hopping
by Matthew Daniels June 14, 2016

June 14, 2016

Hello all, I’m interested in simulating a tight-binding system with matrix elements beyond nearest neighbor hopping (which I’ll call extended hopping, for short). I initially realized that the InfiniteSystem finalizer did not allow this construction when I tried to plot the band structure of an infinite lead. It is not clear to me why this restriction exists; I can see that the code in physics/dispersion.py only accounts for nearest neighbor hopping anyway (which could easily be changed), but since the error I encountered occurs in builder.py, I suspect that there’s a deeper reason for disallowing extended hopping terms. Forging ahead, it occurred to me that I could simply enlarge the unit cell so that all of my hopping terms only extended by a single unit cell away. Unfortunately, this also has the effect of band folding (quite a lot in my case), so the dispersion plot generated by Kwant is somewhat useless for me. But that’s of little consequence, because I’m more interested in transport. Now, since the error I encountered happens during finalization, I imagine I cannot even attach a lead with extended hopping to my finite sample. I *could* enlarge the unit cell (i.e. enlarge the translational symmetry vector) like I did for the band structure calculation, and I suspect that Kwant would happily (i.e. without runtime errors) carry out the calculation in this case. However, I worry that my artificial restriction of the leads’ translational symmetry could have some unintended consequences in the transport calculation. I’d appreciate any comments on the codebase or from experience that might suggest whether this is a safe manipulation to make or not. I’m also somewhat curious why this restriction exists, if anyone has insight to that. Thanks! Cheers, Matthew Daniels

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Kwant kkghosh
by kamal ghosh June 14, 2016

June 14, 2016

Dear Joseph Weston, Very many thanks for your instruction to run my code successfully. But, it did not work, instead it sent message "run time error" after showing the system plot. Yes, I simply changed to kwant.plotter.bands(sys.leads[0],show=False). It is a question to me how the "sys" in my code was finite? You also agree that the system "sys" is attached with leads and thereby it turns into infinite system. Is it not possible to draw band diagrams only in the Quantum Well which is only the central part of the entire system ( composed of leads + that central part ). To be honest again, how then I can find by kwanting the band diagrams of only the qwantum well itself?? All the best Dear Joe. Happy Kwanting.

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Re: [Kwant] kkghosh
by Matthew Daniels June 13, 2016

June 13, 2016

Hi Kamal, It seemed to open fine for me. By the way, you also asked about changing the size of the system along the confinement direction. I think that to do this, you’ll need to create multiple finalized system objects by calling your make_system() function multiple times with different widths. For instance, for width in range (10,50,10): sys=make_system(L=width) # finalize system and calculate bands here Cheers, Matthew > On Jun 13, 2016, at 11:16 AM, kk ghosh <kkghosh44(a)gmail.com> wrote: > > Dear members, > I suppose the program which I sent through rediffmail.com <http://rediffmail.com/> may not be opened. due to some technical error from my side.I therefore resend it via gmail.com <http://gmail.com/>. > Attached mail. > Thanks. > K.K.Ghosh > <copy QW_1.py>

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kkghosh
by kk ghosh June 13, 2016

June 13, 2016

Dear members, I suppose the program which I sent through rediffmail.com may not be opened. due to some technical error from my side.I therefore resend it via gmail.com. Attached mail. Thanks. K.K.Ghosh

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kkghosh
by kamal ghosh June 13, 2016

June 13, 2016

Dear all members of Kwant community: My objective is to write a Kwant coding for E-k dispersion relation of 2D electron gas. I have followed the relevant code of quantum well. But, I cannot insert the proper placing of the method" bands" in that code so that I can get the profile of: E=(H bar**2/2*m)*k(transverse)**2 + (H bar**2/2*m)*(n*pi/W)**2. Barrier height and well depth are synonymous I understand. Also, potential at the two extreme barriers are set to zero while that in the well is set to negative as I understand. I want to know the effect of varying the dimension of confinement direction on the energy; in addition the number of subbands also. Physics is understood but I cannot incorporate these properly in the code. I would benefited in my work with Kwant application if you please extend your wise knowledge. I send the code which I use. Thanks to all members. K.K.Ghosh kk_ghosh(a)rediffmail.com

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Energy spectrum w/o translational invariance
by LaGasse, Samuel June 10, 2016

June 10, 2016

Hi Everyone, I am searching for a way to calculate the energy spectrum of a system without translational invariance using Kwant. I am familiar with how to calculate E-k relationships for systems with translational invariance (such as a lead) using Kwant's infinite systems. What is unclear to me is how to approach doing this for a system which breaks this invariance, by for example, applying an electric field via a gate or source-drain bias. In the tutorial section I have gone through the tutorial for calculating the magnetic flux vs. energy spectrum for a quantum dot, but it is unclear to me how one would modify this example to investigate the E-k relationship. I have seen a few examples of this type of calculation performed in the literature, but often the numerical details are left to the reader. A few of these are below: (1) http://arxiv.org/abs/cond-mat/0610237 - This paper shows several figures for differently shaped quantum wells formed along the transport direction in a graphene ribbon. (2) http://arxiv.org/abs/1512.02144 - It appears to me that the lower panel of Fig. 1 essentially has an electric field, although I could be wrong. This example was actually done with Kwant. (3) http://arxiv.org/abs/cond-mat/0603594 - Their Fig. 1 shows the spectrum in the quantum hall regime w/o an electric field applied (this I am able to reproduce w/Kwant). In their Fig. 2 an electric field is applied in the x-direction of a graphene ribbon- this in particular I would like to be able to implement- in general, not necessarily for graphene. I have attached a code I am working on which generates a square lattice ribbon with leads on the left/right sides. A potential well is formed in the ribbon in a similar fashion to (1) above. I can extract the Hamiltonian of the channel and calculate its eigenvalues, but it is unclear to me how to get the E-k relationship. I believe I need to add some phase factor across the Hamiltonian which accounts for different k values. I apologize if this is more of a physics question than a Kwant question. Indeed, Kwant trivializes what I imagine is the hard part- generating the lattice and turning that into a tight binding Hamiltonian. Hopefully this topic is of interest to the readers of the mailing list- I believe it would be an interesting addition to the Kwant tutorials in the future! Thanks so much! Sam LaGasse

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Kwant for bulk system
by Weiyuan Tong June 8, 2016

June 8, 2016

Dear all, I want to calculate the conductance and the band structure of a bulk graphene system.(infinite in y direction) I have checked the source code for the calculation of band structure. https://github.com/kwantproject/kwant/blob/7c55b0cb2d0dec163e5483dea8ffdbc8… It seems that we can get the sys.cell_hamiltonian and hopping between the sys.cell_hamiltonian in kwant. In y direction, we can also have sys.cell_hamiltonian and hopping, then it can be extended to infinite in y direction: V^\dagger e^{iky} + H + Ve^{-iky}. I am not familiar with kwant coding. Can anyone help me to write some code lines to do these calculations? Or you can also change the Tutorial 2.2.3 as an example. Thanks in advance! Best wishes Weiyuan Tong

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Query about transmission matrices always vanishing
by 张银寒 June 2, 2016

June 2, 2016

Dear Sir, I am trying to reproduce the Hall conductance in a Graphene-hexagonal boron-nitride structure by using Kwant, referring to (https://arxiv.org/abs/1401.4401). The five-terminal Hall bar geometry is considered to study the Hall conductance. My problem is that the smatrix.transmission always vanishes in my code. How can I fix the problem? I pasted my KWANT code. Any suggestions will be greatly appreciated. Thanks Yinhan # Physical background #------------------------------------- # Calculate the Hofstadter's Butterfly of G-BN from __future__ import division # so that 1/2==0.5, and not 0 from math import pi, sqrt,cos,sin from cmath import exp import numpy as np import kwant #For plotting from matplotlib import pyplot #Define the graphnee lattice sin_30, cos_30=(1 / 2, sqrt(3) / 2) #Define the scale a_tb=1.0 gamma=5 L=20*gamma W=5*gamma*sqrt(3) ab=4*(pi/sqrt(3))*gamma En=0.05 graphene=kwant.lattice.general([(1, 0),(sin_30, cos_30)],\ [(0, -1/(4*cos_30)),(0, 1/(4*cos_30))]) a, b = graphene.sublattices class SimpleNamespace(object): def __init__(self, **kwargs): self.__dict__.update(kwargs) def make_system(L, W, mu_lead=2): #### Define the scattering region. ########### # slab scattering region def slab_shape(pos): (x, y)=pos return 0 <= x <= L and 0 <= y <= W # Define on-site energy def potentiala(site): (x, y)=site.pos da1=cos((-sqrt(3)*x/2+y/2)*ab)+cos(-y*ab)+\ cos((sqrt(3)*x/2+y/2)*ab) da2=sin((-sqrt(3)*x/2+y/2)*ab)+sin(-y*ab)+\ sin((sqrt(3)*x/2+y/2)*ab) return (2*pi/gamma)*(da1-sqrt(3)*da2)*En def potentialb(site): (x, y)=site.pos da1=cos((-sqrt(3)*x/2+y/2)*ab)+cos(-y*ab)+\ cos((sqrt(3)*x/2+y/2)*ab) da2=sin((-sqrt(3)*x/2+y/2)*ab)+sin(-y*ab)+\ sin((sqrt(3)*x/2+y/2)*ab) return (2*pi/gamma)*(da1+sqrt(3)*da2)*En def onsitea(site,p): return potentiala(site)-p.mu def onsiteb(site,p): return potentialb(site)-p.mu sys = kwant.Builder() sys[a.shape(slab_shape,(1,1))]=onsitea sys[b.shape(slab_shape,(1,1))]=onsiteb # specify the hoppings of the graphene lattice in the # fomat expected by builder.HoppingKind def Ax(x,y): result=-En*2*(pi/gamma)*(cos((-sqrt(3)*x/2+y/2)*ab)+\ cos((sqrt(3)*x/2+y/2)*ab)-2*cos(-y*ab)) return result def Ay(x,y): result=-En*2*sqrt(3)*(pi/gamma)*(cos((-sqrt(3)*x/2+y/2)*ab)\ -cos((sqrt(3)*x/2+y/2)*ab)) return result def hopping1(site1,site2,p): (x1,y1)=site1.pos result=(-p.t+2*Ax(x1,y1)/3)\ *exp(1j * 2 * pi * 0.4 * x1/(gamma * gamma)) return result def hopping2(site1,site2,p): (x1,y1)=site2.pos result=-p.t+Ay(x1-0.25,y1-sqrt(3)/4)/sqrt(3)\ -Ax(x1-0.25,y1-sqrt(3)/4)/3 return result def hopping3(site1,site2,p): (x1,y1)=site2.pos result=-p.t-Ay(x1+0.25,y1-sqrt(3)/4)/sqrt(3)\ -Ax(x1+0.25,y1-sqrt(3)/4)/3 return result kind1=kwant.builder.HoppingKind((0,0),a,b) sys[kind1]=hopping1 kind2=kwant.builder.HoppingKind((0,1),a,b) sys[kind2]=hopping2 kind3=kwant.builder.HoppingKind((-1,1),a,b) sys[kind3]=hopping3 #### Define the leads. #### # left lead0 sym0 = kwant.TranslationalSymmetry(graphene.vec((1, 0))) def lead0_shape(pos): x, y = pos return (W/2-2.5*gamma <= y <= W/2+2.5*gamma) def lead_hopping(site1,site2,p): return -p.t def lead_onsite(site1,p): return mu_lead lead0 = kwant.Builder(sym0) hoppings = (((0, 0), a, b), ((0, 1), a, b), ((-1, 1), a, b)) lead0[graphene.shape(lead0_shape, (0, W/2))] = lead_onsite lead0[[kwant.builder.HoppingKind(*hopping) for hopping in hoppings]] =lead_hopping #sys.attach_lead(lead0) # down lead1 sym1 = kwant.TranslationalSymmetry(graphene.vec((1, -2))) def lead1_shape(pos): x, y= pos return (L/4-2.5*gamma <= x <= L/4 + 2.5*gamma ) lead1 =kwant.Builder(sym1) lead1[graphene.shape(lead1_shape, (L/4-2.5*gamma+1,0))] = lead_onsite lead1[[kwant.builder.HoppingKind(*hopping) for hopping in hoppings]] = lead_hopping #sys.attach_lead(lead1) # right lead2 lead2=lead0.reversed() #sys.attach_lead(lead2) # right-up lead3 sym3= kwant.TranslationalSymmetry(graphene.vec((-1,2))) def lead3_shape(pos): x, y = pos return (0.75*L-2.5*gamma <= x <= 0.75*L +2.5*gamma) lead3=kwant.Builder(sym3) lead3[graphene.shape(lead3_shape,(0.75*L-2.5*gamma+1,0))] = lead_onsite lead3[[kwant.builder.HoppingKind(*hopping) for hopping in hoppings]]= lead_hopping #sys.attach_lead(lead3) # left-up lead4 lead4=lead1.reversed() #sys.attach_lead(lead4) #print("ab",ab) return sys, [lead0,lead1,lead2,lead3,lead4] def calculate_sigmas(G): # reduce by one dimension G -> G[temp, temp] temp = [0, 1, 3, 4] G = G[temp, :] G = G[:, temp] # invert R = G^-1 # find out whether it is a numpy object r = np.linalg.inv(G) # Voltages follow: V = R I[temp] V = r.dot(np.array([1, -1, 0, 0])) # Completely solved ihe six terminal system. # Consider the 2x2 conductance now: Use I = sigma U #E_x = V[1] - V[0] E_x=(V[3]-V[2]) #print("V=",V) #print("V[0]=",V[0],"V[1]=",V[1],"V[2]=",V[2],"V[3]=",V[3]) #E_x=(V[3]-V[2])/L #E_y = V[1] - V[3] E_y=V[3] #E_y=V[3]/W # formula above sigma_xx = E_x / (E_x**2 + E_y**2) sigma_xy = E_y / (E_x**2 + E_y**2) return sigma_xx, sigma_xy def plot_Hallconductance(sys,p): # Compute conductance Bs = np.linspace(0.5, 1, 10) num_leads=len(sys.leads) def G(sys, p): smatrix = kwant.smatrix(sys, args=[p]) print("trans",smatrix.transmission(0,1)) Bs = np.linspace(0.5, 1, 10) G = [[smatrix.transmission(i, j) for i in range(num_leads)] for j in range(num_leads)] print(G) G -= np.diag(np.sum(G, axis=0)) return calculate_sigmas(G) sigmasxx, sigmasxy = np.array([G(sys, p) for p.B in Bs]).T pyplot.figure() pyplot.plot(Bs,sigmasxy) pyplot.xlabel("B [t]") pyplot.ylabel("sigma_{xy} [e^2/h]") pyplot.show() def main(): t0=2/(sqrt(3)*ab) p=SimpleNamespace(t=t0, mu=0.25, B=None) sys, leads = make_system(L, W, mu_lead=2) # To highlight the two sublattices of graphene, we plot one with # a filled, and the other one with an open circle: def family_colors(site): return 0 if site.family == a else 1 # Then, plot the system with leads. #kwant.plot(sys, site_color=family_colors, site_lw=0.1, # lead_site_lw=0, colorbar=False) # Attach the leads to the system. for lead in leads: sys.attach_lead(lead) # Then, plot the system with leads. #kwant.plot(sys, site_color=family_colors, site_lw=0.1, # lead_site_lw=0, colorbar=False) # Finized the system sys=sys.finalized() # Plot conductance # Compute conductance plot_Hallconductance(sys,p) if __name__ == '__main__': main()

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