Transport Calculations with wannier90

By setting \(\verb#transport#=\verb#TRUE#\), 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 \(\verb#tran_read_ht#=\verb#FALSE#\) the transport properties are calculated using the Hamiltonian in the Wannier function basis of the system found by wannier90. Setting \(\verb#tran_read_ht#=\verb#TRUE#\) allows the user to provide an external Hamiltonian matrix file seedname_htB.dat, from which the properties are found. See Section 2.9 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 \(\verb#tran_read_ht#=\verb#TRUE#\) 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 \(\verb#tran_read_ht#=\verb#FALSE#\), then the transport calculation is performed automatically using the Wannier functions as a basis and the 2c2 geometry described in Section 7.3.

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 \(\verb#tran_read_ht#=\verb#TRUE#\) case are instead built from the Wannier function basis directly. As such, strict rules apply to the system geometry, which is shown in Figure 7.1. 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.

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 reopresentation 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 8.32{reference-type="ref" reference="sec:files_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.

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

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