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130 lines
7.2 KiB
ReStructuredText
.. _Transport:
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Transport calculations
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============================
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Formalism
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---------
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The conductivity and the Seebeck coefficient in direction :math:`\alpha\beta` are defined as [#transp]_:
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.. math::
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\sigma_{\alpha\beta} = \beta e^{2} A_{0,\alpha\beta} \ \ \ \text{and} \ \ \ S_{\alpha\beta} = -\frac{k_B}{|e|}\frac{A_{1,\alpha\beta}}{A_{0,\alpha\beta}},
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in which the kinetic coefficients :math:`A_{n,\alpha\beta}` are given by
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.. math::
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A_{n,\alpha\beta} = N_{sp} \pi \hbar \int{d\omega \left(\beta\omega\right)^n f\left(\omega\right)f\left(-\omega\right)\Gamma_{\alpha\beta}\left(\omega,\omega\right)}.
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Here :math:`N_{sp}` is the spin factor and :math:`f(\omega)` is the Fermi function. The transport distribution :math:`\Gamma_{\alpha\beta}\left(\omega_1,\omega_2\right)` is defined as
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.. math::
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\Gamma_{\alpha\beta}\left(\omega_1,\omega_2\right) = \frac{1}{V} \sum_k Tr\left(v_{k,\alpha}A_{k}(\omega_1)v_{k,\beta}A_{k}\left(\omega_2\right)\right),
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where :math:`V` is the unit cell volume. In multi-band systems the velocities :math:`v_{k}` and the spectral function :math:`A(k,\omega)` are matrices in the band indices :math:`i` and :math:`j`.
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The frequency depended optical conductivity is given by
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.. math::
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\sigma(\Omega) = N_{sp} \pi e^2 \hbar \int{d\omega \Gamma_{\alpha\beta}(\omega+\Omega/2,\omega-\Omega/2)\frac{f(\omega-\Omega/2)-f(\omega+\Omega/2)}{\Omega}}.
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Prerequisites
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-------------
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First perform a standard :ref:`DFT+DMFT calculation <full_charge_selfcons>` for your desired material and obtain the
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real-frequency self energy by doing an analytic continuation.
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.. warning::
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This package does NOT provide an explicit method to do an **analytic continuation** of
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self energies and Green functions from Matsubara frequencies to the real frequency axis!
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There are methods included e.g. in the :program:`ALPS` package, which can be used for these purposes.
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Keep in mind that all these methods have to be used very carefully. Especially for optics calculations
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it is crucial to perform the analytic continuation in such a way that the obtained real frequency self energy
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is accurate around the Fermi energy as low energy features strongly influence the final results!
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Besides the self energy the Wien2k files read by the transport converter (:meth:`convert_transport_input <dft.converters.wien2k_converter.Wien2kConverter.convert_transport_input>`) are:
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* :file:`.struct`: The lattice constants specified in the struct file are used to calculate the unit cell volume.
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* :file:`.outputs`: In this file the k-point symmetries are given.
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* :file:`.oubwin`: Contains the indices of the bands within the projected subspace (written by :program:`dmftproj`) for each k-point.
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* :file:`.pmat`: This file is the output of the Wien2k optics package and contains the velocity (momentum) matrix elements between all bands in the desired energy
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window for each k-point. How to use the optics package is described below.
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* :file:`.h5`: The hdf5 archive has to be present and should contain the dft_input subgroup. Otherwise :meth:`convert_dft_input <dft.converters.wien2k_converter.Wien2kConverter.convert_dft_input>` needs to be called before :meth:`convert_transport_input <dft.converters.wien2k_converter.Wien2kConverter.convert_transport_input>`.
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Wien2k optics package
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---------------------
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The basics steps to calculate the matrix elements of the momentum operator with the Wien2k optics package are:
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1) Perform a standard Wien2k calculation for your material.
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2) Run `x kgen` to generate a dense k-mesh.
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3) Run `x lapw1`.
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4) For metals change TETRA to 101.0 in :file:`case.in2`.
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5) Run `x lapw2 -fermi`.
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6) Run `x optic`.
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Additionally the input file :file:`case.inop` is required. A detail description on how to setup this file can be found in the Wien2k user guide [#userguide]_ on page 166.
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The optics energy window should be chosen according to the window used for :program:`dmftproj`. Note that the current version of the transport code uses only the smaller
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of those two windows. However, keep in mind that the optics energy window has to be specified in absolute values and NOT relative to the Fermi energy!
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You can read off the Fermi energy from the :file:`case.scf2` file. Please do not set the optional parameter NBvalMAX in :file:`case.inop`.
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Furthermore it is necessary to set line 6 to "ON" and put a "1" in the following line to enable the printing of the matrix elements to :file:`case.pmat`.
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Using the transport code
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------------------------
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First we have to read the Wien2k files and store the relevant information in the hdf5 archive::
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from pytriqs.applications.dft.converters.wien2k_converter import *
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from pytriqs.applications.dft.sumk_dft_tools import *
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Converter = Wien2kConverter(filename='case', repacking=True)
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Converter.convert_transport_input()
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SK = SumkDFTTools(hdf_file='case.h5', use_dft_blocks=True)
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The converter :meth:`convert_transport_input <dft.converters.wien2k_converter.Wien2kConverter.convert_transport_input>`
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reads the required data of the Wien2k output and stores it in the `dft_transp_input` subgroup of your hdf file.
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Additionally we need to read and set the self energy, the chemical potential and the double counting::
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ar = HDFArchive('case.h5', 'a')
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SK.set_Sigma([ar['dmft_output']['Sigma_w']])
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chemical_potential,dc_imp,dc_energ = SK.load(['chemical_potential','dc_imp','dc_energ'])
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SK.set_mu(chemical_potential)
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SK.set_dc(dc_imp,dc_energ)
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del ar
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As next step we can calculate the transport distribution :math:`\Gamma_{\alpha\beta}(\omega)`::
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SK.transport_distribution(directions=['xx'], Om_mesh=[0.0, 0.1], energy_window=[-0.3,0.3],
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with_Sigma=True, broadening=0.0, beta=40)
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Here the transport distribution is calculated in :math:`xx` direction for the frequencies :math:`\Omega=0.0` and :math:`0.1`.
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To use the previously obtained self energy we set with_Sigma to True and the broadening to :math:`0.0`.
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As we also want to calculate the Seebeck coefficient we have to include :math:`\Omega=0.0` in the mesh.
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Note that the current version of the code repines the :math:`\Omega` values to the closest values on the self energy mesh.
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For complete description of the input parameters see the :meth:`transport_distribution reference <dft.sumk_dft_tools.SumkDFTTools.transport_distribution>`.
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The resulting transport distribution is not automatically saved, but this can be easily achieved with::
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SK.save(['Gamma_w','Om_meshr','omega','directions'])
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You can retrieve it from the archive by::
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SK.Gamma_w, SK.Om_meshr, SK.omega, SK.directions = SK.load(['Gamma_w','Om_meshr','omega','directions'])
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Finally the optical conductivity :math:`\sigma(\Omega)` and the Seebeck coefficient :math:`S` can be obtained with::
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SK.conductivity_and_seebeck(beta=40)
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SK.save(['seebeck','optic_cond'])
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It is strongly advised to check convergence in the number of k-points!
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References
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----------
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.. [#transp] `V. S. Oudovenko, G. Palsson, K. Haule, G. Kotliar, S. Y. Savrasov, Phys. Rev. B 73, 035120 (2006) <http://link.aps.org/doi/10.1103/PhysRevB.73.0351>`_
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.. [#userguide] `P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka, J. Luitz, ISBN 3-9501031-1-2 <http://www.wien2k.at/reg_user/textbooks/usersguide.pdf>`_
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