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update example in documentation
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Libor Vojacek committed Oct 10, 2024
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47 changes: 44 additions & 3 deletions docs/build/html/_sources/example.rst
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Original file line number Diff line number Diff line change
Expand Up @@ -16,6 +16,11 @@ There is also a time-complexity study in ``./examples/time_complexity.ipynb``.
chain = LinearChain(N=100, eps_0=0.0, t=1.0, a=1.0, H_impurity=None, plot_dispersion=True)
.. image::
./_images/clean_wire.png
:width: 370px
:align: center

.. image::
./_images/linear_chain_dispersion_analytic_vs_diagonalized.png
:width: 550px
Expand All @@ -34,6 +39,12 @@ There is also a time-complexity study in ``./examples/time_complexity.ipynb``.
chain.add_H_impurity(H_impurity, plot_dispersion=False)
chain.plot_onsite_energy()
.. image::
./_images/single_middle_shifted_more.png
:width: 450px
:align: center


.. image::
./_images/impurity_single_middle.png
:width: 450px
Expand All @@ -53,6 +64,7 @@ This can be demonstrated by calculating the occupation function profile.
Without any relaxation mechanism, strong resonances arise, making the potential drop unclear.

.. code-block:: python
# no relaxation
D0_phase = 0.00 * t**2
D0_phase_momentum = 0.00 * t**2
Expand All @@ -61,7 +73,7 @@ Without any relaxation mechanism, strong resonances arise, making the potential
.. image::
./_images/single_imp_middle_no_relaxation.png
:width: 350px
:width: 380px
:align: center

3.2. Phase relaxation
Expand All @@ -70,6 +82,7 @@ Without any relaxation mechanism, strong resonances arise, making the potential
Phase relaxation attenuates the Fabry-Pérot resonances.

.. code-block:: python
# only phase
D0_phase = 0.09 * t**2
D0_phase_momentum = 0.00 * t**2
Expand All @@ -78,7 +91,7 @@ Phase relaxation attenuates the Fabry-Pérot resonances.
.. image::
./_images/single_imp_middle_phase_relaxation.png
:width: 350px
:width: 380px
:align: center


Expand All @@ -88,6 +101,7 @@ Phase relaxation attenuates the Fabry-Pérot resonances.
With an additional momentum relaxation, the potential drop is partially distributed over the whole channel length.

.. code-block:: python
# phase and momentum
D0_phase = 0.09 * t**2
D0_phase_momentum = 0.03 * t**2
Expand All @@ -96,7 +110,7 @@ With an additional momentum relaxation, the potential drop is partially distribu
.. image::
./_images/single_imp_middle_phase_and_momentum_relaxation.png
:width: 350px
:width: 380px
:align: center


Expand Down Expand Up @@ -141,6 +155,33 @@ Let us now compare the transmission for several cases.
4. With many distributed impurities of equivalent total strength :math:`\Sigma_i U_i = -2.0 t`, the transmission starts to resemble the clean limit :math:`T(E) = 1.0 = \mathrm{const.}`


.. code-block:: python
# RELAXATION
D0_phase = 0.12 * t**2
D0_phase_momentum = 0.03 * t**2
N_sc = 90
# IMPURITY HAMILTONIANS
H_imp_clean = np.zeros((N, N))
H_imp_single = np.zeros((N, N))
H_imp_single[N // 2, N // 2] = -2 * t
...
H_impurities = [H_imp_clean, H_imp_single, H_imp_double, H_imp_multiple]
# CALCULATE AND PLOT
fig, axes = plt.subplots(5,len(H_impurities))
for i, H_imp in enumerate(H_impurities):
chain = LinearChain(N, eps_0, t, a, H_impurity=H_imp, plot_dispersion=False)
chain.plot_onsite_energy(axes[0, i])
chain.plot_occupation(
D0_phase=0, D0_phase_momentum=0, E_to_plot=E_to_plot, N_sc=N_sc, ax=axes[1, i]
)
...
.. image::
./_images/example_fuNEGF.png
:width: 950px
Expand Down
54 changes: 51 additions & 3 deletions docs/build/html/example.html
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Original file line number Diff line number Diff line change
Expand Up @@ -332,6 +332,7 @@ <h2>1. Linear-chain tight-binding Hamiltonian<a class="headerlink" href="#linear
<span class="n">chain</span> <span class="o">=</span> <span class="n">LinearChain</span><span class="p">(</span><span class="n">N</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">eps_0</span><span class="o">=</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">a</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">H_impurity</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">plot_dispersion</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
</pre></div>
</div>
<a class="reference internal image-reference" href="_images/clean_wire.png"><img alt="_images/clean_wire.png" class="align-center" src="_images/clean_wire.png" style="width: 370px;" /></a>
<a class="reference internal image-reference" href="_images/linear_chain_dispersion_analytic_vs_diagonalized.png"><img alt="_images/linear_chain_dispersion_analytic_vs_diagonalized.png" class="align-center" src="_images/linear_chain_dispersion_analytic_vs_diagonalized.png" style="width: 550px;" /></a>
</div>
<div class="section" id="impurity-potential">
Expand All @@ -345,6 +346,7 @@ <h2>2. Impurity potential<a class="headerlink" href="#impurity-potential" title=
<span class="n">chain</span><span class="o">.</span><span class="n">plot_onsite_energy</span><span class="p">()</span>
</pre></div>
</div>
<a class="reference internal image-reference" href="_images/single_middle_shifted_more.png"><img alt="_images/single_middle_shifted_more.png" class="align-center" src="_images/single_middle_shifted_more.png" style="width: 450px;" /></a>
<a class="reference internal image-reference" href="_images/impurity_single_middle.png"><img alt="_images/impurity_single_middle.png" class="align-center" src="_images/impurity_single_middle.png" style="width: 450px;" /></a>
</div>
<div class="section" id="chemical-potential-with-impurities">
Expand All @@ -354,17 +356,38 @@ <h2>3. Chemical potential with impurities<a class="headerlink" href="#chemical-p
<div class="section" id="no-relaxation">
<h3>3.1. No relaxation<a class="headerlink" href="#no-relaxation" title="Permalink to this heading">#</a></h3>
<p>Without any relaxation mechanism, strong resonances arise, making the potential drop unclear.</p>
<a class="reference internal image-reference" href="_images/single_imp_middle_no_relaxation.png"><img alt="_images/single_imp_middle_no_relaxation.png" class="align-center" src="_images/single_imp_middle_no_relaxation.png" style="width: 350px;" /></a>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># no relaxation</span>
<span class="n">D0_phase</span> <span class="o">=</span> <span class="mf">0.00</span> <span class="o">*</span> <span class="n">t</span><span class="o">**</span><span class="mi">2</span>
<span class="n">D0_phase_momentum</span> <span class="o">=</span> <span class="mf">0.00</span> <span class="o">*</span> <span class="n">t</span><span class="o">**</span><span class="mi">2</span>

<span class="n">plot_onsite_and_occupation</span><span class="p">(</span><span class="n">E_to_plot</span><span class="p">,</span> <span class="n">D0_phase</span><span class="p">,</span> <span class="n">D0_phase_momentum</span><span class="p">,</span> <span class="n">N_sc</span><span class="p">)</span>
</pre></div>
</div>
<a class="reference internal image-reference" href="_images/single_imp_middle_no_relaxation.png"><img alt="_images/single_imp_middle_no_relaxation.png" class="align-center" src="_images/single_imp_middle_no_relaxation.png" style="width: 380px;" /></a>
</div>
<div class="section" id="phase-relaxation">
<h3>3.2. Phase relaxation<a class="headerlink" href="#phase-relaxation" title="Permalink to this heading">#</a></h3>
<p>Phase relaxation attenuates the Fabry-Pérot resonances.</p>
<a class="reference internal image-reference" href="_images/single_imp_middle_phase_relaxation.png"><img alt="_images/single_imp_middle_phase_relaxation.png" class="align-center" src="_images/single_imp_middle_phase_relaxation.png" style="width: 350px;" /></a>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># only phase</span>
<span class="n">D0_phase</span> <span class="o">=</span> <span class="mf">0.09</span> <span class="o">*</span> <span class="n">t</span><span class="o">**</span><span class="mi">2</span>
<span class="n">D0_phase_momentum</span> <span class="o">=</span> <span class="mf">0.00</span> <span class="o">*</span> <span class="n">t</span><span class="o">**</span><span class="mi">2</span>

<span class="n">plot_onsite_and_occupation</span><span class="p">(</span><span class="n">E_to_plot</span><span class="p">,</span> <span class="n">D0_phase</span><span class="p">,</span> <span class="n">D0_phase_momentum</span><span class="p">,</span> <span class="n">N_sc</span><span class="p">)</span>
</pre></div>
</div>
<a class="reference internal image-reference" href="_images/single_imp_middle_phase_relaxation.png"><img alt="_images/single_imp_middle_phase_relaxation.png" class="align-center" src="_images/single_imp_middle_phase_relaxation.png" style="width: 380px;" /></a>
</div>
<div class="section" id="phase-and-momentum-relaxation">
<h3>3.3. Phase and momentum relaxation<a class="headerlink" href="#phase-and-momentum-relaxation" title="Permalink to this heading">#</a></h3>
<p>With an additional momentum relaxation, the potential drop is partially distributed over the whole channel length.</p>
<a class="reference internal image-reference" href="_images/single_imp_middle_phase_and_momentum_relaxation.png"><img alt="_images/single_imp_middle_phase_and_momentum_relaxation.png" class="align-center" src="_images/single_imp_middle_phase_and_momentum_relaxation.png" style="width: 350px;" /></a>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># phase and momentum</span>
<span class="n">D0_phase</span> <span class="o">=</span> <span class="mf">0.09</span> <span class="o">*</span> <span class="n">t</span><span class="o">**</span><span class="mi">2</span>
<span class="n">D0_phase_momentum</span> <span class="o">=</span> <span class="mf">0.03</span> <span class="o">*</span> <span class="n">t</span><span class="o">**</span><span class="mi">2</span>

<span class="n">plot_onsite_and_occupation</span><span class="p">(</span><span class="n">E_to_plot</span><span class="p">,</span> <span class="n">D0_phase</span><span class="p">,</span> <span class="n">D0_phase_momentum</span><span class="p">,</span> <span class="n">N_sc</span><span class="p">)</span>
</pre></div>
</div>
<a class="reference internal image-reference" href="_images/single_imp_middle_phase_and_momentum_relaxation.png"><img alt="_images/single_imp_middle_phase_and_momentum_relaxation.png" class="align-center" src="_images/single_imp_middle_phase_and_momentum_relaxation.png" style="width: 380px;" /></a>
</div>
</div>
<div class="section" id="transmission-function">
Expand Down Expand Up @@ -396,6 +419,31 @@ <h3>4.2. Different impurity distributions<a class="headerlink" href="#different-
</ol>
</dd>
</dl>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># RELAXATION</span>
<span class="n">D0_phase</span> <span class="o">=</span> <span class="mf">0.12</span> <span class="o">*</span> <span class="n">t</span><span class="o">**</span><span class="mi">2</span>
<span class="n">D0_phase_momentum</span> <span class="o">=</span> <span class="mf">0.03</span> <span class="o">*</span> <span class="n">t</span><span class="o">**</span><span class="mi">2</span>
<span class="n">N_sc</span> <span class="o">=</span> <span class="mi">90</span>

<span class="c1"># IMPURITY HAMILTONIANS</span>
<span class="n">H_imp_clean</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">N</span><span class="p">,</span> <span class="n">N</span><span class="p">))</span>
<span class="n">H_imp_single</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">N</span><span class="p">,</span> <span class="n">N</span><span class="p">))</span>
<span class="n">H_imp_single</span><span class="p">[</span><span class="n">N</span> <span class="o">//</span> <span class="mi">2</span><span class="p">,</span> <span class="n">N</span> <span class="o">//</span> <span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="mi">2</span> <span class="o">*</span> <span class="n">t</span>
<span class="o">...</span>

<span class="n">H_impurities</span> <span class="o">=</span> <span class="p">[</span><span class="n">H_imp_clean</span><span class="p">,</span> <span class="n">H_imp_single</span><span class="p">,</span> <span class="n">H_imp_double</span><span class="p">,</span> <span class="n">H_imp_multiple</span><span class="p">]</span>

<span class="c1"># CALCULATE AND PLOT</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="nb">len</span><span class="p">(</span><span class="n">H_impurities</span><span class="p">))</span>

<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">H_imp</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">H_impurities</span><span class="p">):</span>
<span class="n">chain</span> <span class="o">=</span> <span class="n">LinearChain</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">eps_0</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">H_impurity</span><span class="o">=</span><span class="n">H_imp</span><span class="p">,</span> <span class="n">plot_dispersion</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="n">chain</span><span class="o">.</span><span class="n">plot_onsite_energy</span><span class="p">(</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">i</span><span class="p">])</span>
<span class="n">chain</span><span class="o">.</span><span class="n">plot_occupation</span><span class="p">(</span>
<span class="n">D0_phase</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">D0_phase_momentum</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">E_to_plot</span><span class="o">=</span><span class="n">E_to_plot</span><span class="p">,</span> <span class="n">N_sc</span><span class="o">=</span><span class="n">N_sc</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span>
<span class="p">)</span>
<span class="o">...</span>
</pre></div>
</div>
<a class="reference internal image-reference" href="_images/example_fuNEGF.png"><img alt="_images/example_fuNEGF.png" class="align-center" src="_images/example_fuNEGF.png" style="width: 950px;" /></a>
</div>
</div>
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