.. _sphx_glr_api_gallery_transport:

Transport
---------

Quantum transport through a finite device attached to semi-infinite
leads: Landauer's formula, in which the conductance is a transmission
probability, and the quantized conductance steps of a quantum point
contact. Leads attached automatically from their geometry, narrow
contacts on a wide cavity, and ragged interfaces (a graphene disc
between zigzag leads) completed with the lead's own sites.

Multi-terminal devices: Landauer-Buttiker conductance matrices and the
quantized resistances of a Hall bar with its edge channels; bond
currents; shot noise and the Fano factor; and the recursive Green's
function for long wires. The scattering matrix from the exact modes of
the leads, the scattering states and the local density of states they
carry. Thermoelectric transport: the Mott formula and the
Wiedemann-Franz law, and their breakdown at a resonance.


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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="A lead is a semi-infinite repetition of a unit cell. To attach it, one needs the device sites it touches and the hoppings \tau between them and the lead&#x27;s first cell. For a straight strip, these are the edge column and the identity times t; for anything else, writing them by hand is error-prone.">

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  .. image:: /api/gallery/transport/images/thumb/sphx_glr_plot_automatic_lead_attachment_thumb.png
    :alt:

  :doc:`/api/gallery/transport/plot_automatic_lead_attachment`

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      <div class="sphx-glr-thumbnail-title">Automatic Lead Attachment: Narrow Contacts on a Wide Cavity</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="attach_lead couples a lead to the outermost of its cells that the device reaches, and that cell must hold every lead orbital with a hopping into the lead. A device whose edge is ragged relative to the lead, such as a disc under a strip, does not end with such a cell.">

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  .. image:: /api/gallery/transport/images/thumb/sphx_glr_plot_completing_lead_interfaces_thumb.png
    :alt:

  :doc:`/api/gallery/transport/plot_completing_lead_interfaces`

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      <div class="sphx-glr-thumbnail-title">Completing Lead Interfaces: a Graphene Disc between Zigzag Leads</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Buttiker (1986) wrote the currents of a phase-coherent conductor with many leads as I_p = \frac{e^2}{h}\sum_q G_{pq}V_q, from the transmissions between leads, and treated voltage probes as leads that draw no net current. In 1988 he applied it to the quantum Hall effect: in a strong field the current flows in \nu chiral edge channels, one per filled Landau level, which carry electrons from each contact to the next one downstream without backscattering. The Landauer-Buttiker equations then give, for any sample shape and any probes,">

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  .. image:: /api/gallery/transport/images/thumb/sphx_glr_plot_hall_bar_edge_channels_thumb.png
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  :doc:`/api/gallery/transport/plot_hall_bar_edge_channels`

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      <div class="sphx-glr-thumbnail-title">Buttiker's Edge Channels: the Quantized Resistances of a Hall Bar</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="R. Landauer (1957), then M. Buttiker (1986), recast electrical conductance as a scattering problem: a phase-coherent conductor between two reservoirs has, at zero temperature and bias,">

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  .. image:: /api/gallery/transport/images/thumb/sphx_glr_plot_landauer_conductance_thumb.png
    :alt:

  :doc:`/api/gallery/transport/plot_landauer_conductance`

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      <div class="sphx-glr-thumbnail-title">The Landauer Formula: Quantized Conductance of a Point Contact</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The Landauer transmission of a device needs its Green&#x27;s function between the two leads. Inverting the whole device matrix costs N^3 for N sites, which rules out the long samples in which localization shows. In 1981 Thouless and Kirkpatrick, Lee and Fisher, and MacKinnon and Kramer cut the sample into slices coupled only to their neighbours and added them one at a time,">

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  .. image:: /api/gallery/transport/images/thumb/sphx_glr_plot_recursive_green_function_thumb.png
    :alt:

  :doc:`/api/gallery/transport/plot_recursive_green_function`

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      <div class="sphx-glr-thumbnail-title">The Recursive Green's Function: Localization Length of Long Disordered Wires</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The Landauer conductance is a sum of transmission probabilities, but the object behind it is the scattering matrix S: the amplitude S_{ab} of every outgoing mode a for every incoming mode b of the leads. Fisher and Lee (1981) related it to the Green&#x27;s function, and Ando (1991) showed how to compute it on a lattice by mode matching. The modes of a lead, \psi_n = \lambda^n\phi in its cell n, solve the generalized eigenproblem of the transfer matrix">

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  .. image:: /api/gallery/transport/images/thumb/sphx_glr_plot_scattering_matrix_thumb.png
    :alt:

  :doc:`/api/gallery/transport/plot_scattering_matrix`

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      <div class="sphx-glr-thumbnail-title">Mode Matching: the Scattering Matrix, Lead Modes and Scattering States</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="A current made of discrete charges fluctuates: for independent electrons crossing a tunnel barrier, Schottky&#x27;s Poisson noise S = 2eI. In a phase-coherent conductor the Pauli principle correlates the electrons, and Lesovik (1989) and Buttiker (1990) found the zero-temperature noise from the transmission eigenvalues T_n of the scattering channels,">

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  .. image:: /api/gallery/transport/images/thumb/sphx_glr_plot_shot_noise_fano_factor_thumb.png
    :alt:

  :doc:`/api/gallery/transport/plot_shot_noise_fano_factor`

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      <div class="sphx-glr-thumbnail-title">Shot Noise: the Fano Factor of Ballistic, Tunnel and Diffusive Conductors</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Sivan and Imry (1986) extended the Landauer formula to heat: a temperature difference across a phase-coherent conductor drives a current, and a bias carries heat. In linear response, every coefficient is a moment of the transmission over the Fermi window,">

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  .. image:: /api/gallery/transport/images/thumb/sphx_glr_plot_thermoelectric_transport_thumb.png
    :alt:

  :doc:`/api/gallery/transport/plot_thermoelectric_transport`

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      <div class="sphx-glr-thumbnail-title">Thermoelectric Transport: the Mott Formula and the Wiedemann-Franz Law</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /api/gallery/transport/plot_automatic_lead_attachment
   /api/gallery/transport/plot_completing_lead_interfaces
   /api/gallery/transport/plot_hall_bar_edge_channels
   /api/gallery/transport/plot_landauer_conductance
   /api/gallery/transport/plot_recursive_green_function
   /api/gallery/transport/plot_scattering_matrix
   /api/gallery/transport/plot_shot_noise_fano_factor
   /api/gallery/transport/plot_thermoelectric_transport