.. _sphx_glr_api_gallery_hall_effects:

Hall effects
------------

Hall responses at any Fermi level, from the Berry curvature of the bands:
the anomalous Hall conductivity of the Haldane model (its quantized
plateau in the gap, and its non-quantized value in the bands), the same
plateau surviving disorder in a real-space Kubo-Bastin (kernel polynomial)
calculation on a torus, and the intrinsic spin Hall conductivity of the
Kane-Mele model with its departure from quantization under Rashba coupling.
The same curvature, weighted by entropy or by energy, drives the anomalous
Nernst effect (with the Mott relation) and the thermal Hall effect (with
the Wiedemann-Franz law and its quantized Chern plateau).


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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="The Berry-phase theory of the anomalous Hall effect is a statement about Bloch bands, but a real sample has impurities, and no Brillouin zone. The Kubo-Bastin formula needs neither: it is a trace over the whole sample,">

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  .. image:: /api/gallery/hall_effects/images/thumb/sphx_glr_plot_anomalous_hall_disorder_thumb.png
    :alt:

  :doc:`/api/gallery/hall_effects/plot_anomalous_hall_disorder`

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      <div class="sphx-glr-thumbnail-title">The Anomalous Hall Effect in Real Space: a Disordered Chern Insulator</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="In 1954 Karplus and Luttinger found that the electrons of a ferromagnet acquire an anomalous velocity transverse to an applied electric field, a Hall current with no magnetic field. Half a century later it was recognized as a Berry-phase effect (Jungwirth, Niu and MacDonald 2002; Haldane 2004): a band&#x27;s velocity picks up -\dot{\mathbf{k}}\times \boldsymbol\Omega_n(\mathbf{k}), and the intrinsic Hall conductivity is the Berry curvature summed over the occupied states,">

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  .. image:: /api/gallery/hall_effects/images/thumb/sphx_glr_plot_anomalous_hall_effect_thumb.png
    :alt:

  :doc:`/api/gallery/hall_effects/plot_anomalous_hall_effect`

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      <div class="sphx-glr-thumbnail-title">The Anomalous Hall Effect: from the Anomalous Velocity to the Berry Phase</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="A temperature gradient drives a transverse charge current in a ferromagnet, with no magnetic field: the anomalous Nernst effect, j_x = \alpha_{xy}(-\partial_yT). Xiao, Yao, Fang and Niu (2006) showed that its intrinsic part is the Berry curvature again, weighted this time by the entropy of each state,">

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  .. image:: /api/gallery/hall_effects/images/thumb/sphx_glr_plot_anomalous_nernst_effect_thumb.png
    :alt:

  :doc:`/api/gallery/hall_effects/plot_anomalous_nernst_effect`

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      <div class="sphx-glr-thumbnail-title">The Berry-Phase Anomalous Nernst Effect</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="An electric field along y can drive a spin current along x with no charge current at all: up and down spins are deflected to opposite sides. Murakami, Nagaosa and Zhang (2003) and Sinova et al. (2004) showed that spin-orbit coupling alone produces it, as a property of the band structure -- the spin analogue of the intrinsic anomalous Hall effect. With the spin current j^s_x = \{s_z, v_x\}/2, the Kubo formula gives the spin Hall conductivity (~tbkit.kspace.KSpace.spin_hall_conductivity), in units of e/2\pi.">

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  .. image:: /api/gallery/hall_effects/images/thumb/sphx_glr_plot_intrinsic_spin_hall_effect_thumb.png
    :alt:

  :doc:`/api/gallery/hall_effects/plot_intrinsic_spin_hall_effect`

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      <div class="sphx-glr-thumbnail-title">The Intrinsic Spin Hall Effect: Kane-Mele Plateau and Rashba Coupling</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="A temperature gradient also drives a transverse heat current, j^Q_x = \kappa_{xy}(-\partial_yT). Computing it from the Kubo formula is subtle: part of the heat current circulates as a bound magnetization current and does not flow. Qin, Niu and Shi (2011) subtracted it and found the intrinsic electronic thermal Hall conductivity as one more weighted integral of the Berry curvature,">

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  .. image:: /api/gallery/hall_effects/images/thumb/sphx_glr_plot_thermal_hall_effect_thumb.png
    :alt:

  :doc:`/api/gallery/hall_effects/plot_thermal_hall_effect`

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      <div class="sphx-glr-thumbnail-title">The Intrinsic Thermal Hall Effect and the Wiedemann-Franz Law</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /api/gallery/hall_effects/plot_anomalous_hall_disorder
   /api/gallery/hall_effects/plot_anomalous_hall_effect
   /api/gallery/hall_effects/plot_anomalous_nernst_effect
   /api/gallery/hall_effects/plot_intrinsic_spin_hall_effect
   /api/gallery/hall_effects/plot_thermal_hall_effect