mirror of https://github.com/triqs/dft_tools
129 lines
6.3 KiB
ReStructuredText
129 lines
6.3 KiB
ReStructuredText
.. index:: Transport
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.. _Transport:
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Transport calculations
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======================
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.. index:: Theory
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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\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\right)` is defined as
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.. math::
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\Gamma_{\alpha\beta}\left(\omega\right) = \frac{1}{V} \sum_k Tr\left(v_{k,\alpha}A_{k}(\omega)v_{k,\beta}A_{k}\left(\omega\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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.. index:: Prerequisite
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Prerequisites
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-------------
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First perform a standard DFT+DMFT calculation for your desired material (see :ref:`DFTDMFTmain`) and obtain the real-frequency self energy by doing an
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analytic continuation.
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.. note::
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It is crucial to perform the analytic continuation in such a way that the obtained real-frequency self energy is accurate around the Fermi energy as only its
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low energy structure influences the final results!
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Besides the self energy the Wien2k files read by the transport converter 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 hdf file has to be present and should contain the dft_input subgroup. Otherwise :class:`convert_dft_input` needs to be called before :class:`convert_transport_input`.
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.. index:: Wien2k optics package
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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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Here the energy window can be chosen according to the window used for :program:`dmftprj`. However, keep in mind that energies have to be specified in absolute values! Furthermore it is important
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to set line 6 to ON for printing the matrix elements to the :file:`.pmat` file.
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.. index:: Using the transport code.
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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 hdf-file::
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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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Additionally we need to read and set the self energy, the chemical potential and the double counting::
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ar = HDFArchive('case_Sigma.h5', 'a')
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SK.put_Sigma(Sigma_imp = [ar['dmft_transp_output']['Sigma_w']])
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SK.chemical_potential = ar['dmft_transp_output']['chemical_potential']
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SK.dc_imp = ar['dmft_transp_output']['dc_imp']
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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=[0.00, 0.02], energy_window=[-0.3,0.3],
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with_Sigma=True, broadening=0.0, beta=40)
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The parameters are:
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* `directions`: :math:`\alpha` and :math:`\beta` (e.g. xx, yy, xz, ...)
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* `Om_mesh`: :math:`\Omega`-mesh for the optical conductivity. Note that the code repines this mesh to the closest values on the self energy mesh! The new mesh is stored in `Om_meshr`.
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The Seebeck coefficient is only calculated if :math:`\Omega=0.0` is included.
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* `energy_window`: Limits for the integration over :math:`\omega` (Due to the Fermi functions the integrand is only of considerable size in a small
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window around the Fermi energy). For optical conductivity calculations the window is automatically enlarged by :math:`\Omega` .
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* `with_Sigma`: If this parameter is set to False then Sigma is set to 0 (i.e. the DFT band structure :math:`A(k,\omega)` is taken).
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* `broadening`: The numerical broadening should be set to a finite value for with_Sigma = False.
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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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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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.. index:: References
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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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