3.1.Aharonov-Bohm-cheat.ipynb

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   "source": [
    "# Aharonov-Bohm effect"
   ]
  },
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   "metadata": {},
   "source": [
    "<img src='images/aharonov-bohm.svg' width=40%/>\n",
    "<small style=\"float:right;\">Image [CC-BY-SA-3.0](https://creativecommons.org/licenses/by-sa/3.0/deed.en), original by [Kismalac](https://commons.wikimedia.org/wiki/File:AharonovBohmEffect.svg)</small>"
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   "source": [
    "In this tutorial, we are going to discuss one of the cornerstone of quantum nanoelectronics,\n",
    "the Aharonov-Bohm effect. We are going to calculate the conductance of a ring through which one applies a magnetic field.\n",
    "One interesting aspect of the Aharonov-Bohm effect is that the magnetic field can actually *vanish* in the sample itself. All that is required is that the potential vector does not. This is a very nice proof that in quantum mechanics the electron motion indeed couples to the potential vector, not the magnetic field. This can be seen from the following Gauge transformation: let us consider the Schrodinger equation $H\\Psi = E\\Psi$ with following Hamiltonian,\n",
    "\n",
    "$$H = \\frac{1}{2m} [ P - eA(R)]^2  + V(R)$$\n",
    "\n",
    "where $B(r) = \\nabla \\times A(r) = 0$ inside the sample itself (but not in the hole of the ring). Then the function\n",
    "\n",
    "$$F(r) = \\int_{r_-}^r A(r).dr $$ is well defined independently of the path from $r_-$ to $r$. Let us define,\n",
    "\n",
    "$$\\tilde \\Psi(r) = e^{iF(r)} \\Psi(r)$$. \n",
    "\n",
    "We find that $\\tilde \\Psi$ obeys the free equation with $H = P^2/2m + V(R)$ to which we add the *boundary condition*,\n",
    "\n",
    "$$\\tilde \\Psi(r_+) = e^{iF(r_+)} \\tilde \\Psi(r_-)$$\n",
    "\n",
    "with $$F(r_+) = \\frac{B S}{\\hbar/e}$$\n",
    "\n"
   ]
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  {
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   "source": [
    "from math import pi\n",
    "\n",
    "%run matplotlib_setup.ipy\n",
    "from matplotlib import pyplot\n",
    "import numpy as np\n",
    "import kwant"
   ]
  },
  {
   "cell_type": "code",
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   "source": [
    "lat=kwant.lattice.square()\n",
    "L,W=30,16\n",
    "def myshape(R): return ( \n",
    "        (R[0]**2 + R[1]**2) > (L-W/2)**2 and                 \n",
    "        (R[0]**2 + R[1]**2) < (L+W/2)**2)\n",
    "\n",
    "#L,W=30,16\n",
    "#def myshape_ellipse(R): return ( \n",
    "#        (R[0]**2/2. + R[1]**2) > (L-W/2)**2 and                 \n",
    "#        (R[0]**2/2. + R[1]**2) < (L+W/2)**2)\n",
    "\n",
    "\n",
    "H=kwant.Builder()\n",
    "\n",
    "H[lat.shape(myshape,(L,0) )]=4\n",
    "\n",
    "#H[lat.shape(myshape_ellipse,(int(L*1.14),0)) ]=4\n",
    "\n",
    "\n",
    "H[lat.neighbors()]=1\n",
    "\n",
    "def Aharonov_Bohm(site1,site2,phi): return np.exp(-2j*pi*phi)\n",
    "    \n",
    "for hop in H.hoppings():\n",
    "    if hop[0].tag[0]==1 and hop[0].tag[1]>0 and hop[1].tag[0]==0: \n",
    "        H[hop]=Aharonov_Bohm\n",
    "\n",
    "sym=kwant.TranslationalSymmetry(lat.vec((1,0)))\n",
    "def lead_shape(R): return abs(R[1]) < W/2 and abs(R[0]) <3\n",
    "Hlead =kwant.Builder(sym)\n",
    "Hlead[lat.shape(lead_shape,(0,0) )]=4\n",
    "Hlead[lat.neighbors()]=1\n",
    "H.attach_lead(Hlead)\n",
    "H.attach_lead(Hlead.reversed())\n",
    "kwant.plot(H);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   "source": [
    "\n",
    "Hf=H.finalized()\n",
    "data = []\n",
    "phis = np.linspace(0,1.,50)\n",
    "for phi in phis:\n",
    "    smatrix = kwant.smatrix(Hf, 3.3,args=[phi])\n",
    "    data.append(smatrix.transmission(1, 0))\n",
    "pyplot.plot(phis, data,'o');\n",
    "pyplot.xlabel('$\\phi = BS/(h/e)$')\n",
    "pyplot.ylabel('g in unit of $(2e^2/h)$');\n",
    "pyplot.title('Aharonov-Effect')"
   ]
  },
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   "source": [
    "We see that the Aharonov-Bohm effect contains several harmonics\n",
    "$$ g = g_0 + g_1 cos(\\phi) + g_2 cos(2\\phi) + ...$$\n",
    "* How can we get just one harmonics (as in most experiments)? \n",
    "* Try L = 100 and W= 12, what do you see?\n",
    "* The results should not depend on the position of the gauge transform, can you check that?\n",
    "* Can you make a different sample with an assymetric shape, say a rectangular shape?"
   ]
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   "source": [
    "# Real magnetic field\n",
    "\n",
    "In real life it is difficult to put the field only in the hole so experimentalists tend to\n",
    "apply a uniform field everywhere. Let us modify the script to just do that. You need now to pick up a\n",
    "phase $\\phi$ on each small square. This can be done with the following piece of code - try to modify your code to incorporate this new pieces."
   ]
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   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "from math import pi\n",
    "from matplotlib import pyplot\n",
    "import kwant\n",
    "\n",
    "lat=kwant.lattice.square()\n",
    "L,W=100,12\n",
    "\n",
    "def myshape(R): return ( \n",
    "        (R[0]**2 + R[1]**2) > (L-W/2)**2 and                 \n",
    "        (R[0]**2 + R[1]**2) < (L+W/2)**2)\n",
    "H=kwant.Builder()\n",
    "H[lat.shape(myshape,(L,0) )]=4\n",
    "\n",
    "def Field(site1,site2,phi):\n",
    "    x1,y1=site1.pos\n",
    "    x2,y2=site2.pos\n",
    "    return -np.exp(-0.5j * phi * (x1 - x2) * (y1 + y2))\n",
    "\n",
    "H[lat.neighbors()] = Field\n",
    "\n",
    "sym=kwant.TranslationalSymmetry(lat.vec((1,0)))\n",
    "def lead_shape(R): return abs(R[1]) < W/2 and abs(R[0]) <3\n",
    "Hlead =kwant.Builder(sym)\n",
    "Hlead[lat.shape(lead_shape,(0,0) )]=4\n",
    "Hlead[lat.neighbors()]=Field\n",
    "H.attach_lead(Hlead)\n",
    "H.attach_lead(Hlead.reversed())\n",
    "kwant.plot(H);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "Hf=H.finalized()\n",
    "data = []\n",
    "phis = np.linspace(0.,0.0005,50)\n",
    "for phi in phis:\n",
    "    smatrix = kwant.smatrix(Hf, 3.3,args=[phi])\n",
    "    data.append(smatrix.transmission(1, 0))\n",
    "pyplot.plot(phis, data);\n",
    "pyplot.xlabel('$\\phi = Ba^2/(h/e)$')\n",
    "pyplot.ylabel('g in unit of $(2e^2/h)$');\n",
    "pyplot.title('Aharonov-Effect')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Do you understand why the x - scale is so much smaller?\n",
    "* What should happen at higher field?"
   ]
  }
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