I'm interested in looking at current flow across a nanoscale junction, specifically a pair of electrodes linked by a molecular bridge. How is this sort of problem typically approached?

I'm vaguely familiar with the idea of using Green's functions for this purpose, but don't know much in detail beyond that. In terms of the device, is it more common to treat the electrodes as a large cluster or employ periodic boundary conditions for the whole system?


Here is a paper reviewing charge transport in molecular junctions: J. Chem. Phys. 148, 030901 (2018), http://dx.doi.org/10.1063/1.5003306

A typical way to treat such a problem is to couple the Non-equilibrium Green's function (NEGF) formalism with density-functional-theory (DFT), short NEGF-DFT. You need to solve $$ (E-H-\Sigma^{R,B}(E))\cdot G^R(E) = I, $$ $$ G^\lessgtr(E) = G^R(E)\cdot\Sigma^{\lessgtr,B}(E)\cdot G^A(E), $$ where $E$ is the electron energy, $H$ the Hamiltonian obtained from DFT, $G$ are Green's functions, $\Sigma$ are self-energies and $I$ is the identity matrix. The superscripts denote the retarded, advances, lesser and greater $G$ and $\Sigma$. The self-energies come from the open boundary conditions (OBC) and couple the device region perturbatively to the leads. These semi-infinite leads are essentially a periodic continuation of the electronic structure of the electrodes. The OBC enable particles to enter and leave the simulation domain. Once the equations are solved for all energies of interest, obeservable quantities such as the current and charge density are computed from the $G^\lessgtr(E)$. Injecting charge into the device will affect the potential and therefore also the Hamiltonian. This creates a dependence of DFT on NEGF and vice-versa, which needs to be resolved self-consistently.

A detailed review of NEGF-DFT can be found here: Proceedings of the IEEE, vol. 101, no. 2, pp. 518-530, Feb. 2013, http://dx.doi.org/10.1109/JPROC.2012.2197810

NEGF-DFT can also treat thermal transport, and coupled electro-thermal transport though the latter comes at considerable computational cost. In the presence of strong electron-electron or electron-phonon coupling NEGF can break down and more sophisticated but also far more costly simulations are needed such as the hierarchical quantum master equation (HQME) framework [C. Schinabeck, R. Härtle, and M. Thoss, Phys. Rev. B 94, 201407(R) (2016)].

EDIT 2: The Hamiltonian, $G$ and $\Sigma$ depend on the electron momentum $k$. If there is periodicity in the transverse directions, these equations have to be solved for multiple $k$-vectors.

EDIT 3: more on OBC

The Hamiltonian has the following form (requires a localized basis and correct ordering of the atoms): $$ H = \begin{pmatrix} % \ddots & & & \\[0.2em] H_{11} & H_{12} & & \\[0.2em] H_{21} & H_{22} & H_{23} & \\[0.2em] & H_{32} & H_{33} & \ddots \\[0.2em] & & \ddots & \ddots \\[0.2em] \end{pmatrix} $$ The $H_{nn}$ are matrices corresponding to slabs sorted along the transport axis. Plug this into the stationary Schrödinger equation $ (IE-H)\Psi = 0$ to find $$(IE-H_{nn})\Psi_n - H_{nn+1}\Psi_{n+1} - H_{nn-1}\Psi_{n-1}=0,$$ where $\Psi_n$ is the wave function in the $n$-th slab. Assuming that the electrode material is periodically continued, we know that $H_{11}$, $H_{12}$, and $H_{21}$ should be repeated, i.e. $H_{00}=H_{11}$ etc. If the potential is homogeneous a plane-wave ansatz can be used for $\Psi$ to compute $\Sigma_{11}$, which contains the "impact" the lead $H_{00}$ has on $H_{11}$. This $\Sigma$ is the self-energy in the NEGF equations. A thorough derivation can be found in [Phys. Rev. B 74, 205323 (2006)] https://doi.org/10.1103/PhysRevB.74.205323.

The content of $H_{nn}$ depends on the electronic structure in the transverse directions. It will be different in the case of a nanowire without periodicity than in a 2D material or in bulk. But the tri-diagonal structure of the Hamiltonain remains the same in all cases. Therefore, the procedure to obtain the boundary self-energy is the same.

Edit: typo

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    $\begingroup$ +1. This is a great answer. I would personally be very appreciative if you could comment a bit more on the idea of open boundary conditions and the semi-infinite leads. How are OBC's included into a calculation? I had seen some usage of semi-infinite leads in the literature, but didn't understand it when thinking in the context of PBCs. Can we actually treat an electrode in such a way where two dimensions are infinitely periodic, but the third only repeats in the direction away from the device region? $\endgroup$ – Tyberius May 7 '20 at 22:08

Here's another approach for strongly correlated systems: Al-Hassanieh et al. Phys. Rev. B 73 195304 2006.

In this case, they use time-dependent DMRG. They treat the wire as a chain of sites occupied by spinless Fermions and the junction as an impurity in the wire. They can't observe a steady state current because their system has hard boundaries (the wires just end). The current therefore 'bounces off' the boundary and goes back the other way and then sloshes back and forth. With long enough leads, the middle of a slosh looks like the steady state result.

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    $\begingroup$ Should that not be "It has no OBC", instead of "It has OBC"? $\endgroup$ – Fabian May 8 '20 at 7:37
  • $\begingroup$ OBC = open boundary conditions, so "it has open boundary conditions," is grammatical. Is there some other meaning of OBC? $\endgroup$ – taciteloquence May 8 '20 at 8:37
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    $\begingroup$ The sloshing back and forth usually happens if there are NO OBC. WITH OBC you can converge to a steady-state solution. $\endgroup$ – Fabian May 8 '20 at 9:05
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    $\begingroup$ I think this might be a difference definition. I used OBC to mean that the wires just end, so there is nowhere for the charge to go (like a wire that just ends). It seems like you mean open as in "charge is not conserved." It's very possible both definitions are right. In any case, I'll edit my answer to make my meaning unambiguous. $\endgroup$ – taciteloquence May 8 '20 at 9:50
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    $\begingroup$ @taciteloquence my understanding would be that sloshing back and worth would imply closed boundary conditions. Generally, I think open boundary conditions is used analogous to open systems, which can exchange heat, particles, etc with their surroundings. $\endgroup$ – Tyberius May 8 '20 at 16:24

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