Kwant-discuss February 2019

kwant-discuss@python.org

9 participants
14 discussions

Gate-defined Nanowire Quantum Dot
by Jonathan Fields 15 Sep '22

15 Sep '22

Dear Reader, I am new to KWANT and trying to figure out if it is suitable for calculating transport properties of gate-defined quantum dots made out of nanowires, along the lines of the structure used in https://arxiv.org/abs/cond-mat/0609463. Now, I understand that the mailing list is not a place to ask people to do my work for me, and my question is also not for someone to do this. What I am looking for is some insights into if this is possible (and not all that difficult) with the package. At first sight (and going through the tutorials as well as the APS March Meeting material) this does not seem straightforward; what I struggle with is how to define this type of dot in KWANT. Now, one way would of course be to built the entire 3D geometry of the system, but this will be computationally expensive, and probably also rather tricky to do in terms of defining everything, from the hexagonal wire itself to all of the gates and oxide layers and such. Some abstraction would be preferred, perhaps even reducing the dimensionality and making a 2D system, such as often done in the tutorial. Is this a good idea? My problem is that if one does this, then I run into the problem of how to model the 'half wrap-around' type gates typically used in these devices. In the transport through barrier section of the APS March Meeting material there is a section on how to model realistic potentials, but this would not work all that well for these wrap around type gates. On the other hand, in a way they are perhaps not so different from the QPC in that section. One could model the three gates as something like https://i.imgur.com/WZC983c.png as they do essentially form channels in the wire. Apart from if this sacrifices too much of the nanowire nature in favor of a 2DEG system, I do suppose such a system should in principle be able to be treated as a quantum dot. I haven't been able to confirm this as I am not yet sure how one can apply a source-drain voltage in KWANT; I first thought that this would probably be related to the energy of the modes, but then again I have not been able to produce Coulomb diamonds in this way. The question is getting rather lenghty, and also a bit unclear at this point. Perhaps I should finish up by stating it in a concise form; would you think that it is possible to simulate the transport of such a gate defined nanowire quantum dot device with KWANT, and if so, is the approach I am suggesting above a viable one, or would you go about it very differently? Kind regards Jonathan <http://aka.ms/weboutlook>

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How the spin is positioned in the Smatrix?
by Qingtian Zhang 06 Mar '22

06 Mar '22

Dear all, I am trying square lattice with spin-orbit coupling and Zeeman splititng just like the example of Tutorial 2.3.1. Now I want to get the spin-dependent conductance：Guu,Gud,Gdu,and Gdd, where "u" indicates up spin, d indicates down spin. We can have the scattering matrix through Smatrix=kwant.smatrix(sys, energy), but how the spin is positioned in the scattering matrix? I have checked some simple cases, it seems the Smatrix is organized like this: | r t | S= | t' r' | but how the spin is included? Thanks in advance. Qingtiian Zhang The code is: import kwant from matplotlib import pyplot from numpy import matrix import tinyarray sigma_0 = tinyarray.array([[1, 0], [0, 1]]) sigma_x = tinyarray.array([[0, 1], [1, 0]]) sigma_y = tinyarray.array([[0, -1j], [1j, 0]]) sigma_z = tinyarray.array([[1, 0], [0, -1]]) sys=kwant.Builder() lat=kwant.lattice.square() a=1 t=1.0 alpha=0.5 e_z=0.08 W=2 L=2 sys[(lat(x, y) for x in range(L) for y in range(W))] = \ 4 * t * sigma_0 + e_z * sigma_z # hoppings in x-direction sys[kwant.builder.HoppingKind((1, 0), lat, lat)] = \ -t * sigma_0 - 1j * alpha * sigma_y # hoppings in y-directions sys[kwant.builder.HoppingKind((0, 1), lat, lat)] = \ -t * sigma_0 + 1j * alpha * sigma_x #### Define the left lead. #### lead = kwant.Builder(kwant.TranslationalSymmetry((-a, 0))) lead[(lat(0, j) for j in xrange(W))] = 4 * t * sigma_0 + e_z * sigma_z # hoppings in x-direction lead[kwant.builder.HoppingKind((1, 0), lat, lat)] = \ -t * sigma_0 - 1j * alpha * sigma_y # hoppings in y-directions lead[kwant.builder.HoppingKind((0, 1), lat, lat)] = \ -t * sigma_0 + 1j * alpha * sigma_x #### Attach the leads and return the finalized system. #### sys.attach_lead(lead) sys.attach_lead(lead.reversed()) kwant.plot(sys) sys=sys.finalized() print "Hamiltonian=" print sys.hamiltonian_submatrix() Smatrix=kwant.smatrix(sys, energy=3.) print "The scattering matrix=" print matrix.round((Smatrix.data),2) print "T=",(Smatrix.transmission(1, 0))

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Spin polarized transport calculation
by L.M J 06 Mar '22

06 Mar '22

Dear Kwant users and developers I have two questions related to the spin calculation using kwant. 1. I'm wondering if we can do spin-polarized transport in kwant? For example, can we can get the separate Conductance Vs Energy diagram for spin up and spin down terms. I'm thinking one way of doing this is to set up the Rashba Hamiltonian as a whole. And then manually extract the spin up and spin down hamiltonian and do the transport calculation separately. But this will eliminate some extra terms that I don't know whether this is correct anymore. Any suggestions? 2. How can we set up the spin polarization for a particular site? For example, in some of the topological insulator, can we set up spin polarization on the edge as up and the other side of the edge as down, and then do the transport? Or this question is completely wrong? Thanks for advance. Sincerely, Liming

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How to caculate hall resistance
by ZHANG Bing 12 Sep '21

12 Sep '21

Dear Sir, I am a PhD student of Hong Kong University of Science and Technology. I want to use KWANT to caculate Hall resistance of a Hall bar structure.We can get the conductance between 6 electrodes, but how to get hall resistance? Can you give me some help? Thank you very much. Best Regards, Zhang Bing

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Heat and energy currents using tkwant
by RECK Phillipp 08 Mar '19

08 Mar '19

Dear Thomas, dear kwant community, the purpose of this mail is to discuss the technical details to calculate time-dependent heat and energies currents using tkwant (extension of kwant which allows to create a time-dependent Hamiltonian and compute the time-dependent observables like densities and currents) and if/how our additional modules to do so can be integrated into tkwant. As far as I know, tkwant is not yet officially released but publicly available. Upon request of Christoph Groth, we will share our discussion with the kwant community and invite anyone to participate. As far as I know, the kwant mailing list was not (only) meant to be a Q&A platform for the community to ask questions to the developers but also for the whole community to discuss topics. Moreover, with the recent retiring of the developer mailing list, it was recommended to use the general mailing list also for development discussions. Still, this mail is mainly addressed to Thomas, which is why I apologize in advance for code details which are not explained and thus not understandable for everyone. That being said, let's come to the physics. I. Heat and energy currents in time-dependent systems ( https://github.com/PhiReck/heatCurrent ) We want to calculate time-dependent heat currents into a lead. The heat current is defined as I_h = I_E - μ I_N, with I_E being the energy current, μ is the chemical potential in the given lead and I_N is the particle current. Since μ is assumed to be known and I_N(t) can be calculated directly by (t)kwant with the help of the kwant.operator.Current class, the only missing part is the energy current. In a static setup, the energy current is obtained easily by an additional factor of the energy in the integrand of the Landauer-Büttiker formula (e.g. for two leads: I_E =e/h \int dE E T(E) [f_L(E)-f_R(E)] ). However for time-dependent systems, the energy is not conserved anymore which makes things more complicated. I do not want to explain how tkwant works here, since this would take too long. The basic idea is that the lesser Green's function can be expressed in terms of the time-dependent scattering states. The integration over the energy in a Landauer-Büttiker-type formula is done by an instance of the kwant.manybody.Solve class. The provided operator, e.g. kwant.operator.Current or kwant.operator.Density, tells this solver which observable is to be calculated. So far, the kwant.operators available are: - Density: 'bra_i M_i ket_i' - Current: 'bra_i H_ij ket_j - c.c. ' - Source: 'bra_i M_i H_ii ket_i - c.c. where "i" and "j" are sites, H is the Hamiltonian, M denotes an arbitrary onsite Operator (which is a matrix with dimensions given by the number of orbitals) and bra and ket are wave functions. For instance, for spin densities and spin currents, M would be the appropriate Pauli matrix (at least if spin is the only additional degree of freedom). For the purpose of time-dependent heat and energy currents we need additionally: - Energy Current: 'bra_i H_ij H_jk ket_k - c.c.' - Explicit time dependence of hopping: 'bra_i M_i dH_ij/dt ket_j + c.c.' These are of a similar shape as the operators above but different enough such that the old operators cannot be used. We already implemented them ( https://github.com/PhiReck/heatCurrent ) and tested them successfully in comparison to the analytical Keldysh-Green's function approach in the resonant level model. As far as we see, the calculation of the time-dependent wave function takes distinctly more time than the computation of the energy current (factor of >10) such that the calculation time is not really affected by the energy current. (Note that we did not only implement the operators above, but also utility functions that makes it as easy for the user to calculate particle currents in tkwant as the heat current into/from a lead.) II. A general operator module ( https://github.com/PhiReck/generalOperator ) Moreover, since the operators above all look rather similar, we generalized the operator module and discussed it already on the development mailing list. With this general operator, one is able to calculate any term of the following arbitrary form: - generalOperator: 'bra_a O1_ab O2_bc ... O25_yz ket_z +/-/0 c.c. ' Here, 'a' to 'z' denote again sites where 'z' is used as an arbitrary level of deepness (i.e., 'z' could be equal to 'b', if only one hopping is desired, or 'd' if three hoppings are wished). Moreover, On_xy are user-defined operators (if x==y, it is also an onsite operator). '+/-/0' is meant to imply that the complex conjugate can either be added subtracted or not taken into account at all, depending on what the user wants to do. In practice, we would first initialize each operator as O1 = generalOperator.Operator(syst, O1-FUNCTION, where=where_O1) O2 = generalOperator.Operator(syst, O2-FUNCTION, where=where_O2) ... with On-FUNCTION being the function that defines the operator, e.g. syst.hamiltonian or any other function that takes 1-2 sites and returns a matrix with dimensions given by the additional degrees of freedom (spin, orbitals, ...) at the sites at hand. Moreover, each operator has to know where it is to be considered/computed, specified normally as a list of hoppings (where_On). The product is achieved by prodOp = generalOperator.Op_Product(O1, O2, ..., complex_conj, const_fac) 'complex+conj' takes care of the +/-/0 option above and 'const_fac' is a constant factor which is multiplied to the result (in case of the current, it would be the imaginary unit 1j). To tell tkwant to calculate the matrix product with the wave functions as shown above, it is used the same way as for the kwant.operators: result = prodOp(bra,ket) which returns a list of results for all possible site connections. Speaking of the site connections (or as I call it: the path finding), note that now, the different where_On have to be matched: since we want to have something like "O1_ab O2_bc", we have to make sure that only hoppings in where_O1 are connected to hoppings in where_O2 which have the same site 'b' at the according position. This path finding is done automatically in the initialization of the class generalOperator.Op_Product. The path finding is the only new conceptual thing, but the user does not have to worry about it which is why I don't want to go into details here now. (There is a documentation in the git repository if someone is interested.) How to create a new operator from the generalOperator at the example of the Energy Current : This is a copy of the source code, where I omitted the check_hermiticity option and the sum option for better readability. class offEnergyCurrent: def __init__(self, syst, where1, where2): self.hamil1 = Operator(syst, syst.hamiltonian, where1) self.hamil2 = Operator(syst, syst.hamiltonian, where2) self.offEn = Op_Product(self.hamil1, self.hamil2, complex_conj=-1, const_fac=1j) def __call__(self, bra, ket=None, args=(), *, params=None): return self.offEn(bra, ket, args, params=params) Long story short: With the help of the generalOperator, it is very easy to create new operators (just define the individual operators and take the product via generalOperator.Op_Product). Before, for each operator inheriting from the kwant.operator._Localoperator, an individual "_operate" method was needed, although they are very similar in all 5 cases from above. At the moment, I am not sure if anyone ever needs to create new operators. So far, for kwant, it seems that the 3 kwant.operators have been enough (and would have been enough for our purpose of thermoelectric properties). With t-kwant however, I have the feeling that certain operators will be more complicated as compared to their time-independent counterparts as we have seen in the case of the energy current. Therefore, this general operator might be of use in the future. However, the larger universality of the general Operator comes at a price: Calculations are slower, which is mostly due to the fact that recursive functions are called for each single site combination. For the three examples of the kwant.operators, we see that the __call__ of the same operators initialized with the general operator takes roughly 3-5 times longer. This means that the general operator is not (yet?) suited to replace the 'specialized' ones, but at least it will help potential users to easily define and use new operators: they won't have to write and test an operate method in the 'old' kwant.operator._LocalOperator way. So this is the current state of progress. We hope to merge both the 'specialized' heat/energy current as well as the general operator with tkwant soon. @ Thomas: Do you see any problems or do you have any concerns about merging it with tkwant? Maybe I should have added that all the features of the kwant operators should be able to use, e.g. bind or functions for where. For instance, all the kwant examples from the tutorial also work fine with the general operator. Best regards, Phillipp Reck

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Transmission matrix and momenta
by Luca Lepori 19 Feb '19

19 Feb '19

Dear all, I am trying to obtain the coefficients of the transmission matrix (fine, so far.....) and to related them to the modes of the leads. In particular, I would like to know the momenta of these modes (supposing periodic conditions along the directions orthogonal to the leads). Is it possible ? Thank you very much for the help. L.

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Berry phase of a system periodic in 1 direction
by Srilok Srinivasan 18 Feb '19

18 Feb '19

Dear Kwant users and developers, Is there a way to calculate the integral(< u_nk | d_k(u_nk)>dk) across the 1D BZ of a system periodic in one direction. Here u_nk is the periodic part of the Bloch wave function corresponding to band n. I am trying to compute this for a simple system of graphene nanoribbon. I am trying compute the gauge independent value of the above integral using a "1D Wilsoon loop" where the integral is approximately equal to the product of <u_nk|u_n(k+1) > for all k across the BZ. I defined the unit cell of the nanoribbon as a lead and tried to get the u_nk as the propagating modes using leads.modes() method (I have pasted the code below) but the resolution in k is too large - only 4 momenta across the BZ. How can we get u_n(k) with a good resolution of k, say for 100 points in the BZ ? Is this something which can be easily done in Kwant ? Code: def make_1D_zigzag(N=7,t=1): syst = kwant.Builder(kwant.TranslationalSymmetry(Zigzag.prim_vecs[0])) syst[Zigzag.shape((lambda pos: pos[1] >0 and pos[1] < get_width(N)),(0,0))] = 2*t syst[Zigzag.neighbors()] = -t return syst lead = make_1D_zigzag(N=N,t=2); lead = lead.finalized() prop_modes=lead.modes()[0] u_nk = prop_modes.wave_functions Thanks a lot for your help! Srilok Srinivasan Graduate Student Mechanical Engineering Iowa State University, Ames, IA

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Updating Kwant
by Marc Vila 15 Feb '19

15 Feb '19

Dear Kwant developers, I want to implement the latest development of Kwant since I want to try to calculate Kubo conductivities with the KPM method (as it appears in the development version of the documentation). I tried to use Conda to update the packages (conda install -c kwant kwant) but I still get the message "AttributeError: module 'kwant.kpm' has no attribute 'conductivity'". Maybe this is not the way to go? Thanks a lot in advance! Bests, Marc ------------------------ Marc Vila Tusell La Caixa - Severo Ochoa PhD in the Theoretical and Computational Nanoscience Group Catalan Institute of Nanoscience and Nanotechnology (ICN2) Barcelona Institute of Science and Technology (BIST) Additional information: https://scholar.google.es/citations?user=h2V4iNIAAAAJ&hl=es http://icn2.cat/en/theoretical-and-computational-nanoscience-group https://www.researchgate.net/profile/Marc_Vila_Tusell https://www.becarioslacaixa.net/marc-vila-tusell-BI00042?nav=true https://orcid.org/0000-0001-9118-421X

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dephasing effect and self-energy
by 张金龙 14 Feb '19

14 Feb '19

Dear Colleagues, I want to investigate the dephasing effect in the coherent junction, so I want to couple a phenomenological term $i\frac{\Gamma}{2}$ to the Hamiltonian $H(\phi)$. I want plot the figure that describe the conductance via the phase $\phi$ at different dephasing strength $\Gamma$. I try it, but the total Hamiltonian becomes non-hermitian. The program got broken. How to realize this by Kwant? Can you give more ways or some examples. -- Jinlong Zhang jlzhang(a)163.com

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Attaching identical leads
by Sergio Castillo Robles 14 Feb '19

14 Feb '19

Hello everyone, i would appreciate any suggestion you could give me to solve this I'm trying to attach a couple of leads to my structure, I have the code for both, but I want to know how to make that both leads begin with the same atom, In the code below you can see that the left lead starts in a gray atom, while the lead in the right side starts in the couple of atoms blue and yellow, if I increase the size of the structure it only creates another primitive cell and the case is the same Is there a way to make that the right lead starts in a gray atom as the left lead, like only creating half primitive cell? Thanks in advance Here is the code import kwant from matplotlib import pyplot %matplotlib tk PrimVec = [(3.176, 0, 0), (1.588, 2.7504, 0), (0, 0, 17.49)] base = [(0, 0, 0), (1.588, 0.9168, 0.8745), (0, 0, 1.749)] lat = kwant.lattice.general(prim_vecs = PrimVec, basis = base) d, e, f= lat.sublattices def make_cuboid(t=1.0, a=15, b=10, c=5): def cuboid_shape(pos): x, y, z = pos return 0 <= x < a and 0 <= y < b and 0 <= z < c syst = kwant.Builder() syst[lat.shape(cuboid_shape, (0, 0, 0))] = 4 * t hoppings = (((1, 0, 0), d, e), ((0, 0, 0), d, e),((0, 1, 0), d, e), ((0, 0, 0), f, e), ((1, 0, 0), f, e), ((0, 1, 0), f, e)) syst[[kwant.builder.HoppingKind(*hopping) for hopping in hoppings]] = -6 lead = kwant.Builder(kwant.TranslationalSymmetry((-3.176, 0, 0))) def lead_shape(pos): return 0 <= pos[1] < b and 0 <= pos[2] < c lead[lat.shape(lead_shape, (0, 0, 0))] = 4 * t lead[[kwant.builder.HoppingKind(*hopping) for hopping in hoppings]] = -6 syst.attach_lead(lead) syst.attach_lead(lead.reversed()) return syst def main(): syst = make_cuboid(a=30, b=14, c=4) def family_colors(site): if site.family == d: return 'yellow' elif site.family == e: return 'gray' else: return 'blue' kwant.plot(syst, site_size=0.25, site_lw=0.025, hop_lw=0.05, site_color=family_colors) syst = syst.finalized() if __name__ == '__main__': main()

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Kwant-discuss February 2019