From b67dc8fa6f76ae52588bf97a1ef6e238a6799c4e Mon Sep 17 00:00:00 2001
From: Joseph Weston <joseph@weston.cloud>
Date: Thu, 18 Apr 2019 12:16:35 +0200
Subject: [PATCH] convert quantum well example to jupyter-sphinx
---
doc/source/tutorial/spin_potential_shape.rst | 99 +++++++++++++++++---
1 file changed, 86 insertions(+), 13 deletions(-)
diff --git a/doc/source/tutorial/spin_potential_shape.rst b/doc/source/tutorial/spin_potential_shape.rst
index 3512d3e3..e245bf1e 100644
--- a/doc/source/tutorial/spin_potential_shape.rst
+++ b/doc/source/tutorial/spin_potential_shape.rst
@@ -218,7 +218,29 @@ Spatially dependent values through functions
.. seealso::
The complete source code of this example can be found in
- :download:`quantum_well.py </code/download/quantum_well.py>`
+ :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
Up to now, all examples had position-independent matrix-elements
(and thus translational invariance along the wire, which
@@ -237,22 +259,57 @@ 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:
-.. literalinclude:: /code/include/quantum_well.py
- :start-after: #HIDDEN_BEGIN_ehso
- :end-before: #HIDDEN_END_ehso
+.. jupyter-execute::
+ :hide-code:
+
+ a = 1
+ t = 1.0
+ W, L, L_well = 10, 30, 10
+
+.. jupyter-execute::
+
+ # Start with an empty tight-binding system and a single square lattice.
+ # `a` is the lattice constant (by default set to 1 for simplicity).
+ lat = kwant.lattice.square(a)
+
+ syst = kwant.Builder()
+
+ #### Define the scattering region. ####
+ # Potential profile
+ def potential(site, pot):
+ (x, y) = site.pos
+ if (L - L_well) / 2 < x < (L + L_well) / 2:
+ return pot
+ else:
+ return 0
This function takes two arguments: the first of type `~kwant.builder.Site`,
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
-variables of the surrounding function: `L` and `L_well` are taken from the
-namespace of `make_system`.
+variables `L` and `L_well` that are defined globally.
Kwant now allows us to pass a function as a value to
`~kwant.builder.Builder`:
-.. literalinclude:: /code/include/quantum_well.py
- :start-after: #HIDDEN_BEGIN_coid
- :end-before: #HIDDEN_END_coid
+.. jupyter-execute::
+
+ def onsite(site, pot):
+ return 4 * t + potential(site, pot)
+
+ 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()
For each lattice point, the corresponding site is then passed as the
first argument to the function `onsite`. The values of any additional
@@ -269,9 +326,21 @@ of the lead -- this should be kept in mind.
Finally, we compute the transmission probability:
-.. literalinclude:: /code/include/quantum_well.py
- :start-after: #HIDDEN_BEGIN_sqvr
- :end-before: #HIDDEN_END_sqvr
+.. 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.smatrix`` allows us to specify a dictionary, `params`, that contains
the additional arguments required by the Hamiltonian matrix elements.
@@ -280,7 +349,11 @@ 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:
-.. image:: /code/figure/quantum_well_result.*
+.. jupyter-execute::
+ :hide-code:
+
+ plot_conductance(syst, energy=0.2,
+ welldepths=[0.01 * i for i in range(100)])
Starting from no potential (well depth = 0), we observe the typical
oscillatory transmission behavior through resonances in the quantum well.
--
GitLab