Scattering Region Convergence

The open boundary condition of NEGF rests on a single assumption: the few layers at both ends of the device are already indistinguishable from the bulk electrode. The electrode self-energy plants at that boundary the condition "from here on, perfect bulk continues infinitely", so if the scattering region (the buffer section from the impurity/molecule to the boundary) is too short, the potential and charge density meet the self-energy without having reached their bulk values at the boundary. That mismatch appears as artificial scattering and contaminates $T(E)$. This document organizes the three standard criteria for judging whether "the device is long enough" and the length-series procedure.

Three criteria

Criterion 1 — Boundary potential matching

The local electronic structure of the device boundary layers must match the bulk electrode. The most practical indicator is the Hartree (electrostatic) potential.

1. In the device calculation, save the *.VH grid with SaveElectrostaticPotential T.
2. Average over the plane perpendicular to the transport axis (z) to build the planar average $\bar V(z)$.
3. Since $\bar V(z)$ oscillates with the atomic period, apply one more moving-average window whose length equals the electrode repeat period to obtain the macroscopic average $V_{\mathrm{mac}}(z)$ (Baldereschi–Baroni–Resta averaging). For our example, the window length is the electrode C4 cell's $c = 5.16$ Å.
4. If $V_{\mathrm{mac}}$ in the electrode sections at both ends of the device is flat and matches the value from a separately computed bulk electrode, the test passes. It means the potential perturbation created by the impurity has decayed sufficiently before reaching the boundary.

As an auxiliary indicator, you can also overlay the PDOS of boundary-layer atoms with the bulk electrode DOS — this catches cases where local states remain even though the potential matches.

Criterion 2 — Device-length convergence of $T(E)$

The final verdict is made with the result quantity itself. Build a series with stepwise-increased scattering region length, overlay the $T(E)$ curves, and adopt the minimum length at which the curve no longer changes.

Made concrete with our example: with the N impurity at the center, extend the buffer C on both sides in electrode-cell units (4 atoms, 5.16 Å) to build the C11N, C15N, C19N, C23N series. Keeping the electrode, k-grid, $\eta$, and energy grid all identical, run 0 V TranSIESTA + TBtrans and overlay the $T(E)$ curves in one figure. Judgment points:

• Does the entire curve (including resonance positions and widths) overlap — do not look at only the single point $T(E_F)$.
• If a resonance position keeps shifting with length, that is a signal the impurity level is still interacting with the boundary.

Check η before judging length convergence

If the numerical imaginary part $\eta$ of the Green function is too large, the broadened resonance tails decay differently with length, producing an artificial length dependence. In that state, $T(E)$ may appear not to converge no matter how much you extend the length. Before starting the length series, first pass the $\eta$ convergence test of the η broadening artifact page.

Criterion 3 — Fix the electrode lattice constant by bulk-only relaxation

Determine the electrode lattice from the bulk equilibrium value obtained by relaxing the electrode alone, and use that value verbatim when assembling the device. Avoid variants such as relaxing the entire device and then back-porting the end-region lattice into the electrode — even if they pass the NEGF consistency conditions (the electrode cell and the device end buffer layers must be atomically identical; the electrode must couple only to its nearest-neighbor cells), you lose the grounds for claiming "this lattice is the true bulk equilibrium".

If the lattice of the electrode calculation and the lattice at the device boundary differ even slightly, the seam itself becomes a scatterer. This artificial scattering drags $T$ below the integer plateaus even in a perfect chain, so it shows up immediately in the ballistic verification (step 5 below). When device relaxation is required, apply constraints that fix the electrode-side buffer sections to preserve lattice matching.

Length-series procedure (example)

1. Bulk electrode relaxation — converge the transport-direction lattice with the electrode unit cell (C4) alone. This value is the reference for all subsequent structures.
2. Assemble the series — build the C11N / C15N / C19N / C23N devices with the fixed lattice. Extend in integer multiples of the electrode cell (to preserve boundary matching).
3. Identical-condition calculations — share the electrode TSHS, and run 0 V + TBtrans with identical $\eta$ (e.g., 0.001 eV), energy grid, and convergence criteria. The only input difference among the compared cases must be the length.
4. Judge — check both the boundary flatness of $V_{\mathrm{mac}}$ (criterion 1) and the $T(E)$ overlap (criterion 2). Confidence is high when both criteria pass at the same length.
5. Ballistic control — run the same length series with the pristine chain without the impurity and confirm that $T(E)$ is length-independent with integer plateaus. If length dependence appears here, the problem is the setup ($\eta$, lattice mismatch, boundary), not impurity physics.
6. Adopt — use the minimum passing length for production. Taking it longer than necessary only inflates the $O(N^3)$ cost.

Summary check

Criterion	Deliverable	Pass condition
Boundary potential matching	$V_{\mathrm{mac}}(z)$ from *.VH	Flat in the boundary section + matches the bulk electrode value
$T(E)$ length convergence	Length-series $T(E)$ overlay	Curves (including resonance positions) overlap
Electrode lattice fixed	Bulk-only relaxed lattice	Electrode cell = device buffer layers, atomically identical

References

• A. Baldereschi, S. Baroni, and R. Resta, Phys. Rev. Lett. 61, 734 (1988) — macroscopic averaging. DOI 10.1103/PhysRevLett.61.734
• M. Brandbyge et al., "Density-functional method for nonequilibrium electron transport", Phys. Rev. B 65, 165401 (2002). DOI 10.1103/PhysRevB.65.165401
• N. Papior et al., "Improvements on non-equilibrium and transport Green function techniques", Comput. Phys. Commun. 212, 8 (2017). DOI 10.1016/j.cpc.2016.09.022