spin_potential_shape.rst 23 KB
 Christoph Groth committed Sep 05, 2013 1 2 3 4 5 More interesting systems: spin, potential, shape ------------------------------------------------ Each of the following three examples highlights different ways to go beyond the very simple examples of the previous section.  Kwant authors committed Jan 10, 2011 6 7 8 9 10 11  .. _tutorial_spinorbit: Matrix structure of on-site and hopping elements ................................................  Joseph Weston committed Jul 17, 2019 12 13 14 15 16 17 .. seealso:: You can execute the code examples live in your browser by activating thebelab: .. thebe-button:: Activate Thebelab  18 19 .. seealso:: The complete source code of this example can be found in  Joseph Weston committed May 07, 2019 20 21 22 23  :jupyter-download:script:spin_orbit .. jupyter-kernel:: :id: spin_orbit  24   Christoph Groth committed Sep 05, 2013 25 26 We begin by extending the simple 2DEG-Hamiltonian by a Rashba spin-orbit coupling and a Zeeman splitting due to an external magnetic field:  Kwant authors committed Jan 10, 2011 27 28 29  .. math::  Christoph Groth committed Aug 29, 2013 30  H = \frac{-\hbar^2}{2 m} (\partial_x^2+\partial_y^2) -  Kwant authors committed Jan 10, 2011 31 32 33 34 35 36 37 38  i \alpha (\partial_x \sigma_y - \partial_y \sigma_x) + E_\text{Z} \sigma_z + V(y) Here :math:\sigma_{x,y,z} denote the Pauli matrices. It turns out that this well studied Rashba-Hamiltonian has some peculiar properties in (ballistic) nanowires: It was first predicted theoretically in Phys. Rev. Lett. 90, 256601 (2003)  Anton Akhmerov committed Dec 22, 2020 39 _ that such a system should  Kwant authors committed Jan 10, 2011 40 41 42 exhibit non-monotonic conductance steps due to a spin-orbit gap. Only very recently, this non-monotonic behavior has been supposedly observed in experiment: Nature Physics 6, 336 (2010)  Anton Akhmerov committed Dec 22, 2020 43 _. Here  Kwant authors committed Jan 10, 2011 44 45 46 47 we will show that a very simple extension of our previous examples will exactly show this behavior (Note though that no care was taken to choose realistic parameters).  Christoph Groth committed Nov 21, 2012 48 49 The tight-binding model corresponding to the Rashba-Hamiltonian naturally exhibits a 2x2-matrix structure of onsite energies and hoppings. In order to  Christoph Groth committed Jul 31, 2013 50 use matrices in our program, we import the Tinyarray package. (NumPy  Anton Akhmerov committed Dec 22, 2020 51 _ would work as well, but Tinyarray is much faster  Christoph Groth committed Nov 21, 2012 52 for small arrays.)  Kwant authors committed Jan 10, 2011 53   Joseph Weston committed May 07, 2019 54 55 56 57 58 59 60 61 62 63 .. jupyter-execute:: :hide-code: # Tutorial 2.3.1. Matrix structure of on-site and hopping elements # ================================================================ # # Physics background # ------------------ # Gaps in quantum wires with spin-orbit coupling and Zeeman splititng, # as theoretically predicted in  Anton Akhmerov committed Dec 22, 2020 64  # https://doi.org/10.1103/PhysRevLett.90.256601  Joseph Weston committed May 07, 2019 65  # and (supposedly) experimentally oberved in  Anton Akhmerov committed Dec 22, 2020 66  # https://doi.org/10.1038/nphys1626  Joseph Weston committed May 07, 2019 67 68 69 70 71 72 73 74 75 76  # # Kwant features highlighted # -------------------------- # - Numpy matrices as values in Builder import kwant # For plotting from matplotlib import pyplot  Joseph Weston committed May 23, 2019 77 78 79 .. jupyter-execute:: boilerplate.py :hide-code:  Joseph Weston committed May 07, 2019 80 81 82 83 .. jupyter-execute:: # For matrix support import tinyarray  Kwant authors committed Jan 10, 2011 84   Anton Akhmerov committed Sep 08, 2013 85 For convenience, we define the Pauli-matrices first (with :math:\sigma_0 the  Kwant authors committed Jan 10, 2011 86 87 unit matrix):  Joseph Weston committed May 07, 2019 88 89 90 91 92 93 94 .. jupyter-execute:: # define Pauli-matrices for convenience 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]])  Kwant authors committed Jan 10, 2011 95   Joseph Weston committed May 23, 2019 96 97 98 99 100 101 102 103 104 and we also define some other parameters useful for constructing our system: .. jupyter-execute:: t = 1.0 alpha = 0.5 e_z = 0.08 W, L = 10, 30  Kwant authors committed Jan 10, 2011 105 106 107 108 Previously, we used numbers as the values of our matrix elements. However, ~kwant.builder.Builder also accepts matrices as values, and we can simply write:  Joseph Weston committed May 07, 2019 109 110 111 .. jupyter-execute:: :hide-code:  Joseph Weston committed Sep 03, 2019 112  lat = kwant.lattice.square(norbs=2)  Joseph Weston committed May 07, 2019 113 114 115 116 117 118 119 120 121 122 123 124 125  syst = kwant.Builder() .. jupyter-execute:: #### Define the scattering region. #### syst[(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 syst[kwant.builder.HoppingKind((1, 0), lat, lat)] = \ -t * sigma_0 + 1j * alpha * sigma_y / 2 # hoppings in y-directions syst[kwant.builder.HoppingKind((0, 1), lat, lat)] = \ -t * sigma_0 - 1j * alpha * sigma_x / 2  Kwant authors committed Jan 10, 2011 126   Joseph Weston committed Sep 03, 2019 127 128 129 Note that we specify norbs=2 when creating the lattice, as each site has 2 degrees of freedom associated with it, giving us 2x2 matrices as onsite/hopping terms.  Kwant authors committed Jan 10, 2011 130 131 132 Note that the Zeeman energy adds to the onsite term, whereas the Rashba spin-orbit term adds to the hoppings (due to the derivative operator). Furthermore, the hoppings in x and y-direction have a different matrix  Anton Akhmerov committed Apr 28, 2013 133 134 135 structure. We now cannot use lat.neighbors() to add all the hoppings at once, since we now have to distinguish x and y-direction. Because of that, we have to explicitly specify the hoppings in the form expected by  Anton Akhmerov committed Mar 14, 2013 136 ~kwant.builder.HoppingKind:  Kwant authors committed Jan 10, 2011 137 138 139 140 141  - A tuple with relative lattice indices. For example, (1, 0) means hopping from (i, j) to (i+1, j), whereas (1, 1) would mean hopping to (i+1, j+1). - The target lattice (where to hop to)  Pierre Carmier committed Feb 15, 2012 142 - The source lattice (where the hopping originates)  Kwant authors committed Jan 10, 2011 143 144 145 146 147 148 149 150 151 152 153  Since we are only dealing with a single lattice here, source and target lattice are identical, but still must be specified (for an example with hopping between different (sub)lattices, see :ref:tutorial-graphene). Again, it is enough to specify one direction of the hopping (i.e. when specifying (1, 0) it is not necessary to specify (-1, 0)), ~kwant.builder.Builder assures hermiticity. The leads also allow for a matrix structure,  Joseph Weston committed May 07, 2019 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177  .. jupyter-execute:: :hide-code: #### Define the left lead. #### lead = kwant.Builder(kwant.TranslationalSymmetry((-1, 0))) .. jupyter-execute:: lead[(lat(0, j) for j in range(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 / 2 # hoppings in y-directions lead[kwant.builder.HoppingKind((0, 1), lat, lat)] = \ -t * sigma_0 - 1j * alpha * sigma_x / 2 .. jupyter-execute:: :hide-code: #### Attach the leads and finalize the system. #### syst.attach_lead(lead) syst.attach_lead(lead.reversed()) syst = syst.finalized()  Kwant authors committed Jan 10, 2011 178 179 180 181  The remainder of the code is unchanged, and as a result we should obtain the following, clearly non-monotonic conductance steps:  Joseph Weston committed May 07, 2019 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 .. jupyter-execute:: :hide-code: # Compute conductance energies=[0.01 * i - 0.3 for i in range(100)] data = [] for energy in energies: smatrix = kwant.smatrix(syst, energy) data.append(smatrix.transmission(1, 0)) pyplot.figure() pyplot.plot(energies, data) pyplot.xlabel("energy [t]") pyplot.ylabel("conductance [e^2/h]") pyplot.show()  Kwant authors committed Jan 10, 2011 197 198 199  .. specialnote:: Technical details  Christoph Groth committed Jul 31, 2013 200 201  - The Tinyarray package, one of the dependencies of Kwant, implements efficient small arrays. It is used internally in Kwant for storing small  Christoph Groth committed Nov 21, 2012 202  vectors and matrices. For performance, it is preferable to define small  Christoph Groth committed Jul 31, 2013 203  arrays that are going to be used with Kwant using Tinyarray. However,  Christoph Groth committed Nov 21, 2012 204 205 206 207 208 209 210 211  NumPy would work as well:: import numpy sigma_0 = numpy.array([[1, 0], [0, 1]]) sigma_x = numpy.array([[0, 1], [1, 0]]) sigma_y = numpy.array([[0, -1j], [1j, 0]]) sigma_z = numpy.array([[1, 0], [0, -1]])  Christoph Groth committed Jul 31, 2013 212  Tinyarray arrays behave for most purposes like NumPy arrays except that  Christoph Groth committed Nov 21, 2012 213  they are immutable: they cannot be changed once created. This is important  Christoph Groth committed Jul 31, 2013 214  for Kwant: it allows them to be used directly as dictionary keys.  Christoph Groth committed Nov 21, 2012 215   Kwant authors committed Jan 10, 2011 216  - It should be emphasized that the relative hopping used for  Anton Akhmerov committed Mar 14, 2013 217  ~kwant.builder.HoppingKind is given in terms of  Kwant authors committed Jan 10, 2011 218 219  lattice indices, i.e. relative to the Bravais lattice vectors. For a square lattice, the Bravais lattice vectors are simply  Christoph Groth committed Nov 21, 2012 220  (a,0) and (0,a), and hence the mapping from  221  lattice indices (i,j) to real space and back is trivial.  Kwant authors committed Jan 10, 2011 222  This becomes more involved in more complicated lattices, where  223  the real-space directions corresponding to, for example, (1,0)  Kwant authors committed Jan 10, 2011 224 225 226 227 228 229 230  and (0,1) need not be orthogonal any more (see :ref:tutorial-graphene). Spatially dependent values through functions ............................................  231 232 .. seealso:: The complete source code of this example can be found in  Joseph Weston committed May 07, 2019 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255  :jupyter-download:script:quantum_well .. jupyter-kernel:: :id: quantum_well .. jupyter-execute:: :hide-code: # Tutorial 2.3.2. Spatially dependent values through functions # ============================================================ # # Physics background # ------------------ # transmission through a quantum well # # Kwant features highlighted # -------------------------- # - Functions as values in Builder import kwant # For plotting from matplotlib import pyplot  256   Joseph Weston committed May 23, 2019 257 258 259 .. jupyter-execute:: boilerplate.py :hide-code:  Kwant authors committed Jan 10, 2011 260 261 262 263 264 265 266 267 268 Up to now, all examples had position-independent matrix-elements (and thus translational invariance along the wire, which was the origin of the conductance steps). Now, we consider the case of a position-dependent potential: .. math:: H = \frac{\hbar^2}{2 m} (\partial_x^2+\partial_y^2) + V(x, y)  Michael Wimmer committed Sep 08, 2013 269 270 271 The position-dependent potential enters in the onsite energies. One possibility would be to again set the onsite matrix elements of each lattice point individually (as in :ref:tutorial_quantum_wire). However,  Kwant authors committed Jan 10, 2011 272 273 274 275 276 changing the potential then implies the need to build up the system again. Instead, we use a python *function* to define the onsite energies. We define the potential profile of a quantum well as:  Joseph Weston committed May 07, 2019 277 278 279 280 281 282 283 284 285 286 .. jupyter-execute:: W, L, L_well = 10, 30, 10 def potential(site, pot): (x, y) = site.pos if (L - L_well) / 2 < x < (L + L_well) / 2: return pot else: return 0  Kwant authors committed Jan 10, 2011 287   Joseph Weston committed Nov 13, 2019 288 This function takes two arguments: the first of type ~kwant.system.Site,  Christoph Groth committed Feb 21, 2013 289 290 from which you can get the real-space coordinates using site.pos, and the value of the potential as the second. Note that in potential we can access  Joseph Weston committed May 07, 2019 291 variables L and L_well that are defined globally.  Kwant authors committed Jan 10, 2011 292   Christoph Groth committed Jul 31, 2013 293 Kwant now allows us to pass a function as a value to  Kwant authors committed Jan 10, 2011 294 295 ~kwant.builder.Builder:  Joseph Weston committed May 07, 2019 296 297 .. jupyter-execute::  Joseph Weston committed May 23, 2019 298 299 300  a = 1 t = 1.0  Joseph Weston committed May 07, 2019 301 302 303  def onsite(site, pot): return 4 * t + potential(site, pot)  Joseph Weston committed Sep 03, 2019 304  lat = kwant.lattice.square(a, norbs=1)  Joseph Weston committed May 23, 2019 305 306  syst = kwant.Builder()  Joseph Weston committed May 07, 2019 307 308 309 310 311 312 313 314 315 316 317 318 319 320  syst[(lat(x, y) for x in range(L) for y in range(W))] = onsite syst[lat.neighbors()] = -t .. jupyter-execute:: :hide-code: #### Define and attach the leads. #### lead = kwant.Builder(kwant.TranslationalSymmetry((-a, 0))) lead[(lat(0, j) for j in range(W))] = 4 * t lead[lat.neighbors()] = -t syst.attach_lead(lead) syst.attach_lead(lead.reversed()) syst = syst.finalized()  Kwant authors committed Jan 10, 2011 321   Joseph Weston committed Feb 18, 2013 322 For each lattice point, the corresponding site is then passed as the  323 first argument to the function onsite. The values of any additional  Joseph Weston committed Feb 18, 2013 324 parameters, which can be used to alter the Hamiltonian matrix elements  Anton Akhmerov committed Aug 24, 2013 325 at a later stage, are specified later during the call to smatrix.  326 Note that we had to define onsite, as it is  Joseph Weston committed May 05, 2016 327 not possible to mix values and functions as in syst[...] = 4 * t +  Kwant authors committed Jan 10, 2011 328 329 330 331 332 333 334 335 336 potential. For the leads, we just use constant values as before. If we passed a function also for the leads (which is perfectly allowed), this function would need to be compatible with the translational symmetry of the lead -- this should be kept in mind. Finally, we compute the transmission probability:  Joseph Weston committed May 07, 2019 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 .. jupyter-execute:: def plot_conductance(syst, energy, welldepths): # Compute conductance data = [] for welldepth in welldepths: smatrix = kwant.smatrix(syst, energy, params=dict(pot=-welldepth)) data.append(smatrix.transmission(1, 0)) pyplot.figure() pyplot.plot(welldepths, data) pyplot.xlabel("well depth [t]") pyplot.ylabel("conductance [e^2/h]") pyplot.show()  Kwant authors committed Jan 10, 2011 352   Joseph Weston committed Feb 15, 2019 353 354 355 356 357 358 kwant.smatrix allows us to specify a dictionary, params, that contains the additional arguments required by the Hamiltonian matrix elements. In this example we are able to solve the system for different depths of the potential well by passing the potential value (remember above we defined our onsite function that takes a parameter named pot). We obtain the result:  Kwant authors committed Jan 10, 2011 359   Joseph Weston committed May 07, 2019 360 361 362 363 364 .. jupyter-execute:: :hide-code: plot_conductance(syst, energy=0.2, welldepths=[0.01 * i for i in range(100)])  Kwant authors committed Jan 10, 2011 365 366 367 368 369 370 371 372 373 374 375 376 377  Starting from no potential (well depth = 0), we observe the typical oscillatory transmission behavior through resonances in the quantum well. .. warning:: If functions are used to set values inside a lead, then they must satisfy the same symmetry as the lead does. There is (currently) no check and wrong results will be the consequence of a misbehaving function. .. specialnote:: Technical details - Functions can also be used for hoppings. In this case, they take  Joseph Weston committed Nov 13, 2019 378  two ~kwant.system.Site's as arguments and then an arbitrary number  Joseph Weston committed Feb 18, 2013 379  of additional arguments.  Kwant authors committed Jan 10, 2011 380   Joseph Weston committed Nov 13, 2019 381  - Apart from the real-space position pos, ~kwant.system.Site has also an  382  attribute tag containing the lattice indices of the site.  Kwant authors committed Jan 10, 2011 383 384 385 386 387 388  .. _tutorial-abring: Nontrivial shapes .................  389 390 .. seealso:: The complete source code of this example can be found in  Joseph Weston committed May 07, 2019 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419  :jupyter-download:script:ab_ring .. jupyter-kernel:: :id: ab_ring .. jupyter-execute:: :hide-code: # Tutorial 2.3.3. Nontrivial shapes # ================================= # # Physics background # ------------------ # Flux-dependent transmission through a quantum ring # # Kwant features highlighted # -------------------------- # - More complex shapes with lattices # - Allows for discussion of subtleties of attach_lead (not in the # example, but in the tutorial main text) # - Modifcations of hoppings/sites after they have been added from cmath import exp from math import pi import kwant # For plotting from matplotlib import pyplot  420   Joseph Weston committed May 23, 2019 421 422 423 .. jupyter-execute:: boilerplate.py :hide-code:  Kwant authors committed Jan 10, 2011 424 425 426 427 Up to now, we only dealt with simple wire geometries. Now we turn to the case of a more complex geometry, namely transport through a quantum ring that is pierced by a magnetic flux :math:\Phi:  Joseph Weston committed May 23, 2019 428 .. image:: /figure/ab_ring_sketch.*  Kwant authors committed Jan 10, 2011 429 430 431 432 433 434 435 436 437 438 439 440 441  For a flux line, it is possible to choose a gauge such that a charged particle acquires a phase :math:e\Phi/h whenever it crosses the branch cut originating from the flux line (branch cut shown as red dashed line) [#]_. There are more symmetric gauges, but this one is most convenient to implement numerically. Defining such a complex structure adding individual lattice sites is possible, but cumbersome. Fortunately, there is a more convenient solution: First, define a boolean function defining the desired shape, i.e. a function that returns True whenever a point is inside the shape, and False otherwise:  Joseph Weston committed May 07, 2019 442 443 444 445 446 447 448 449 .. jupyter-execute:: r1, r2 = 10, 20 def ring(pos): (x, y) = pos rsq = x ** 2 + y ** 2 return (r1 ** 2 < rsq < r2 ** 2)  Kwant authors committed Jan 10, 2011 450   451 Note that this function takes a real-space position as argument (not a  Joseph Weston committed Nov 13, 2019 452 ~kwant.system.Site).  Kwant authors committed Jan 10, 2011 453 454 455 456 457  We can now simply add all of the lattice points inside this shape at once, using the function ~kwant.lattice.Square.shape provided by the lattice:  Joseph Weston committed May 07, 2019 458 459 .. jupyter-execute::  Joseph Weston committed May 23, 2019 460 461 462  a = 1 t = 1.0  Joseph Weston committed Sep 03, 2019 463  lat = kwant.lattice.square(a, norbs=1)  Joseph Weston committed May 23, 2019 464 465  syst = kwant.Builder()  Joseph Weston committed May 07, 2019 466 467  syst[lat.shape(ring, (0, r1 + 1))] = 4 * t syst[lat.neighbors()] = -t  Kwant authors committed Jan 10, 2011 468   469 Here, lat.shape takes as a second parameter a (real-space) point that is  470 inside the desired shape. The hoppings can still be added using  Anton Akhmerov committed Apr 28, 2013 471 lat.neighbors() as before.  Kwant authors committed Jan 10, 2011 472 473 474 475 476  Up to now, the system contains constant hoppings and onsite energies, and we still need to include the phase shift due to the magnetic flux. This is done by **overwriting** the values of hoppings in x-direction along the branch cut in the lower arm of the ring. For this we select  477 all hoppings in x-direction that are of the form (lat(1, j), lat(0, j))  Kwant authors committed Jan 10, 2011 478 479 with j<0:  Joseph Weston committed May 07, 2019 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 .. jupyter-execute:: # In order to introduce a flux through the ring, we introduce a phase on # the hoppings on the line cut through one of the arms. Since we want to # change the flux without modifying the Builder instance repeatedly, we # define the modified hoppings as a function that takes the flux as its # parameter phi. def hopping_phase(site1, site2, phi): return -t * exp(1j * phi) def crosses_branchcut(hop): ix0, iy0 = hop[0].tag # builder.HoppingKind with the argument (1, 0) below # returns hoppings ordered as ((i+1, j), (i, j)) return iy0 < 0 and ix0 == 1 # ix1 == 0 then implied # Modify only those hopings in x-direction that cross the branch cut def hops_across_cut(syst): for hop in kwant.builder.HoppingKind((1, 0), lat, lat)(syst): if crosses_branchcut(hop): yield hop syst[hops_across_cut] = hopping_phase  Kwant authors committed Jan 10, 2011 504 505 506 507 508 509 510 511  Here, crosses_branchcut is a boolean function that returns True for the desired hoppings. We then use again a generator (this time with an if-conditional) to set the value of all hoppings across the branch cut to fluxphase. The rationale behind using a function instead of a constant value for the hopping is again that we want to vary the flux through the ring, without constantly rebuilding the system -- instead the flux is governed  Joseph Weston committed Feb 18, 2013 512 by the parameter phi.  Kwant authors committed Jan 10, 2011 513   514 For the leads, we can also use the lat.shape-functionality:  Kwant authors committed Jan 10, 2011 515   Joseph Weston committed May 07, 2019 516 517 518 .. jupyter-execute:: #### Define the leads. ####  Joseph Weston committed May 23, 2019 519 520  W = 10  Joseph Weston committed May 07, 2019 521 522 523  sym_lead = kwant.TranslationalSymmetry((-a, 0)) lead = kwant.Builder(sym_lead)  Joseph Weston committed May 23, 2019 524   Joseph Weston committed May 07, 2019 525 526 527 528 529 530  def lead_shape(pos): (x, y) = pos return (-W / 2 < y < W / 2) lead[lat.shape(lead_shape, (0, 0))] = 4 * t lead[lat.neighbors()] = -t  Kwant authors committed Jan 10, 2011 531   Michael Wimmer committed Aug 30, 2013 532 533 534 535 Here, the shape must be compatible with the translational symmetry of the lead sym_lead. In particular, this means that it should extend to infinity along the translational symmetry direction (note how there is no restriction on x in lead_shape) [#]_.  Kwant authors committed Jan 10, 2011 536 537 538  Attaching the leads is done as before:  Joseph Weston committed May 07, 2019 539 540 541 542 543 544 .. jupyter-execute:: :hide-output: #### Attach the leads #### syst.attach_lead(lead) syst.attach_lead(lead.reversed())  Kwant authors committed Jan 10, 2011 545 546 547 548  In fact, attaching leads seems not so simple any more for the current structure with a scattering region very much different from the lead shapes. However, the choice of unit cell together with the  549 translational vector allows to place the lead unambiguously in real space --  Kwant authors committed Jan 10, 2011 550 551 the unit cell is repeated infinitely many times in the direction and opposite to the direction of the translational vector.  Christoph Groth committed Jul 31, 2013 552 Kwant examines the lead starting from infinity and traces it  Kwant authors committed Jan 10, 2011 553 554 555 556 back (going opposite to the direction of the translational vector) until it intersects the scattering region. At this intersection, the lead is attached:  Joseph Weston committed May 23, 2019 557 .. image:: /figure/ab_ring_sketch2.*  Kwant authors committed Jan 10, 2011 558 559 560  After the lead has been attached, the system should look like this:  Joseph Weston committed May 07, 2019 561 562 563 564 .. jupyter-execute:: :hide-code: kwant.plot(syst);  Kwant authors committed Jan 10, 2011 565 566 567 568  The computation of the conductance goes in the same fashion as before. Finally you should get the following result:  Joseph Weston committed May 07, 2019 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590  .. jupyter-execute:: :hide-code: def plot_conductance(syst, energy, fluxes): # compute conductance normalized_fluxes = [flux / (2 * pi) for flux in fluxes] data = [] for flux in fluxes: smatrix = kwant.smatrix(syst, energy, params=dict(phi=flux)) data.append(smatrix.transmission(1, 0)) pyplot.figure() pyplot.plot(normalized_fluxes, data) pyplot.xlabel("flux [flux quantum]") pyplot.ylabel("conductance [e^2/h]") pyplot.show() # We should see a conductance that is periodic with the flux quantum plot_conductance(syst.finalized(), energy=0.15, fluxes=[0.01 * i * 3 * 2 * pi for i in range(100)])  Kwant authors committed Jan 10, 2011 591 592 593 594 595 596  where one can observe the conductance oscillations with the period of one flux quantum. .. specialnote:: Technical details  Christoph Groth committed Jul 31, 2013 597  - Leads have to have proper periodicity. Furthermore, the Kwant  Kwant authors committed Jan 10, 2011 598 599 600 601 602 603 604 605  format requires the hopping from the leads to the scattering region to be identical to the hoppings between unit cells in the lead. ~kwant.builder.Builder.attach_lead takes care of all these details for you! In fact, it even adds points to the scattering region, if proper attaching requires this. This becomes more apparent if we attach the leads a bit further away from the central axis o the ring, as was done in this example:  Joseph Weston committed May 07, 2019 606 607 608 609 610 611 612 613 614  .. jupyter-kernel:: :id: ab_ring_note1 .. jupyter-execute:: :hide-code: import kwant from matplotlib import pyplot  Joseph Weston committed May 23, 2019 615 616 617 618 619 620  .. jupyter-execute:: boilerplate.py :hide-code: .. jupyter-execute:: :hide-code:  Joseph Weston committed May 07, 2019 621 622 623 624 625  a = 1 t = 1.0 W = 10 r1, r2 = 10, 20  Joseph Weston committed Sep 03, 2019 626  lat = kwant.lattice.square(norbs=1)  Joseph Weston committed May 07, 2019 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646  syst = kwant.Builder() def ring(pos): (x, y) = pos rsq = x**2 + y**2 return ( r1**2 < rsq < r2**2) syst[lat.shape(ring, (0, 11))] = 4 * t syst[lat.neighbors()] = -t sym_lead0 = kwant.TranslationalSymmetry((-a, 0)) lead0 = kwant.Builder(sym_lead0) def lead_shape(pos): (x, y) = pos return (-1 < x < 1) and ( 0.5 * W < y < 1.5 * W ) lead0[lat.shape(lead_shape, (0, W))] = 4 * t lead0[lat.neighbors()] = -t lead1 = lead0.reversed() syst.attach_lead(lead0) syst.attach_lead(lead1) kwant.plot(syst);  Kwant authors committed Jan 10, 2011 647 648 649 650 651 652 653 654 655  - Per default, ~kwant.builder.Builder.attach_lead attaches the lead to the "outside" of the structure, by tracing the lead backwards, coming from infinity. One can also attach the lead to the inside of the structure, by providing an alternative starting point from where the lead is traced back::  Joseph Weston committed May 05, 2016 656  syst.attach_lead(lead1, lat(0, 0))  Kwant authors committed Jan 10, 2011 657 658 659 660  starts the trace-back in the middle of the ring, resulting in the lead being attached to the inner circle:  Joseph Weston committed May 07, 2019 661 662 663 664 665 666 667 668 669  .. jupyter-kernel:: :id: ab_ring_note2 .. jupyter-execute:: :hide-code: import kwant from matplotlib import pyplot  Joseph Weston committed May 23, 2019 670 671 672 673 674 675  .. jupyter-execute:: boilerplate.py :hide-code: .. jupyter-execute:: :hide-code:  Joseph Weston committed May 07, 2019 676 677 678 679 680  a = 1 t = 1.0 W = 10 r1, r2 = 10, 20  Joseph Weston committed Sep 03, 2019 681  lat = kwant.lattice.square(a, norbs=1)  Joseph Weston committed May 07, 2019 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700  syst = kwant.Builder() def ring(pos): (x, y) = pos rsq = x**2 + y**2 return ( r1**2 < rsq < r2**2) syst[lat.shape(ring, (0, 11))] = 4 * t syst[lat.neighbors()] = -t sym_lead0 = kwant.TranslationalSymmetry((-a, 0)) lead0 = kwant.Builder(sym_lead0) def lead_shape(pos): (x, y) = pos return (-1 < x < 1) and ( -W/2 < y < W/2 ) lead0[lat.shape(lead_shape, (0, 0))] = 4 * t lead0[lat.neighbors()] = -t lead1 = lead0.reversed() syst.attach_lead(lead0) syst.attach_lead(lead1, lat(0, 0)) kwant.plot(syst);  Kwant authors committed Jan 10, 2011 701 702 703 704 705 706 707 708 709  Note that here the lead is treated as if it would pass over the other arm of the ring, without intersecting it. .. rubric:: Footnotes .. [#] The corresponding vector potential is :math:A_x(x,y)=\Phi \delta(x) \Theta(-y) which yields the correct magnetic field :math:B(x,y)=\Phi \delta(x)\delta(y).  Michael Wimmer committed Aug 30, 2013 710 711 .. [#] Despite the "infinite" shape, the unit cell will still be finite; the ~kwant.lattice.TranslationalSymmetry takes care of that.