.. _sphx_glr_api_gallery_moire:

Supercells and Moiré Structures
-------------------------------

Supercells of periodic models: band unfolding and the spectral function
of disordered supercells, and the commensurate moiré cells of twisted
bilayers, down to the flat bands of magic-angle twisted bilayer graphene.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="Disorder, defects and superstructures need a supercell, and a supercell folds the bands into a smaller Brillouin zone: N times more bands, most of them spectators. W. Ku, T. Berlijn and C.-C. Lee (2010), and V. Popescu and A. Zunger, unfolded them back: each supercell state |\mathbf{K}J\rangle gets the weight W_J(\mathbf{k}) = \sum_o|\langle\mathbf{k}o|\mathbf{K}J\rangle|^2 at every primitive \mathbf{k} folding onto \mathbf{K}, and the spectral function A(\mathbf{k},\omega) = \sum_J W_J(\mathbf{k}) \delta(\omega-E_J) -- what angle-resolved photoemission measures -- is drawn in the primitive zone again.">

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  .. image:: /api/gallery/moire/images/thumb/sphx_glr_plot_band_unfolding_thumb.png
    :alt:

  :doc:`/api/gallery/moire/plot_band_unfolding`

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      <div class="sphx-glr-thumbnail-title">Band Unfolding: the Effective Band Structure of a Disordered Supercell</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Twist two graphene sheets by a small angle \theta and a moiré pattern appears, with a period a/(2\sin\frac\theta2) of a hundred lattice constants. R. Bistritzer and A. H. MacDonald (2011) showed that the interlayer tunnelling w slows the Dirac electrons down,">

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  .. image:: /api/gallery/moire/images/thumb/sphx_glr_plot_magic_angle_twisted_bilayer_thumb.png
    :alt:

  :doc:`/api/gallery/moire/plot_magic_angle_twisted_bilayer`

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      <div class="sphx-glr-thumbnail-title">Magic-Angle Twisted Bilayer Graphene: Flat Moiré Bands</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /api/gallery/moire/plot_band_unfolding
   /api/gallery/moire/plot_magic_angle_twisted_bilayer