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Transport Calculations with wannier90

By setting transport = TRUE, wannier90 will calculate the quantum conductance and density of states of a one-dimensional system. The results will be written to files seedname_qc.dat and seedname_dos.dat, respectively.

The system for which transport properties are calculated is determined by the keyword transport_mode.

transport_mode = bulk

Quantum conductance and density of states are calculated for a perfectly periodic one-dimensional conductor. If tran_read_ht = FALSE the transport properties are calculated using the Hamiltonian in the Wannier function basis of the system found by wannier90. Setting tran_read_ht = TRUE allows the user to provide an external Hamiltonian matrix file seedname_htB.dat, from which the properties are found. See Section Post-Processing for more details of the keywords required for such calculations.

transport_mode = lcr

Quantum conductance and density of states are calculated for a system where semi-infinite, left and right leads are connected through a central conductor region. This is known as the lcr system. Details of the method is described in Ref. 1.

In wannier90 two options exist for performing such calculations:

  • If tran_read_ht = TRUE the external Hamiltonian files seedname_htL.dat, seedname_htLC.dat, seedname_htC.dat, seedname_htCR.dat, seedname_htR.dat are read and used to compute the transport properties.

  • If tran_read_ht = FALSE, then the transport calculation is performed automatically using the Wannier functions as a basis and the 2c2 geometry described in Section Automated lcr Transport Calculations: The 2c2 Geometry.

Automated lcr Transport Calculations: The 2c2 Geometry

Calculations using the 2c2 geometry provide a method to calculate the transport properties of an lcr system from a single wannier90 calculation. The Hamiltonian matrices which the five external files provide in the tran_read_ht = TRUE case are instead built from the Wannier function basis directly. As such, strict rules apply to the system geometry, which is shown in Figure below. These rules are as follows:

  • Left and right leads must be identical and periodic.

  • Supercell must contain two principal layers (PLs) of lead on the left, a central conductor region and two principal layers of lead on the right.

  • The conductor region must contain enough lead such that the disorder does not affect the principal layers of lead either side.

  • A single k-point (Gamma) must be used.

lcr_2c2
Schematic illustration of the supercell required for 2c2 lcr calculations, showing where each of the Hamiltonian matrices are derived from. Four principal layers (PLs) are required plus the conductor region.

In order to build the Hamiltonians, Wannier functions are first sorted according to position and then type if a number of Wannier functions exist with a similar centre (eg. d-orbital type Wannier functions centred on a Cu atom). Next, consistent parities of Wannier function are enforced. To distingiush between different types of Wannier function and assertain relative parities, a signature of each Wannier function is computed. The signature is formed of 20 integrals which have different spatial dependence. They are given by:

\[ I=\frac{1}{V}\int_V g(\mathbf{r})w(\mathbf{r})d\mathbf{r} \label{eq:sig_ints} \]

where \(V\) is the volume of the cell, \(w(\mathbf{r})\) is the Wannier function and \(g(\mathbf{r})\) are the set of functions:

\[ \begin{aligned} g(\mathbf{r})=&\left\lbrace1,\sin\left(\frac{2\pi (x-x_c)}{L_x}\right), \sin\left(\frac{2\pi (y-y_c)}{L_y}\right), \sin\left(\frac{2\pi (z-z_c)}{L_z}\right), \sin\left(\frac{2\pi (x-x_c)}{L_x}\right) \sin\left(\frac{2\pi (y-y_c)}{L_y}\right),\right.\nonumber \\ &\left.\sin\left(\frac{2\pi (x-x_c)}{L_x}\right) \sin\left(\frac{2\pi (z-z_c)}{L_z}\right), ... \right\rbrace \label{eq:g(r)} \end{aligned} \]

upto third order in powers of sines. Here, the supercell has dimension \((L_x,L_y,L_z)\) and the Wannier function has centre \(\mathbf{r}_c=(x_c,y_c,z_c)\). Each of these integrals may be written as linear combinations of the following sums:

\[ S_n(\mathbf{G})=\displaystyle{e^{i\mathbf{G.r}_{c}}\sum_{m}U_{mn}\tilde{u}_{m\Gamma}^{*}(\mathbf{G})} \]

where \(n\) and \(m\) are the Wannier function and band indexes, \(\mathbf{G}\) is a G-vector, \(U_{mn}\) is the unitary matrix that transforms from the Bloch representation of the system to the maximally-localised Wannier function basis and \(\tilde{u}_{m\Gamma}^{*}(\mathbf{G})\) are the conjugates of the Fourier transforms of the periodic parts of the Bloch states at the \(\Gamma\!\) -point. The complete set of \(\tilde{u}_{m\mathbf{k}}(\mathbf{G})\) are often outputted by plane-wave DFT codes. However, to calculate the 20 signature integrals, only 32 specific \(\tilde{u}_{m\mathbf{k}}(\mathbf{G})\) are required. These are found in an additional file (seedname.unkg) that should be provided by the interface between the DFT code and wannier90 . A detailed description of this file may be found in Section seedname.unkg.

Additionally, the following keywords are also required in the input file:

  • tran_num_ll : The number of Wannier functions in a principal layer.

  • tran_num_cell_ll : The number of unit cells in one principal layer of lead

A further parameter related to these calculations is tran_group_threshold.

Tutorial of how 2c2 calculations are preformed can be found in the wannier90 Tutorial.

  1. Marco Buongiorno Nardelli. Electronic transport in extended systems: application to carbon nanotubes. Phys. Rev. B, 60:7828, 1999. ↩