We are now ready to treat the problem of a grain boundary. This situation is schematized using the Hamiltonian model given in Equation (6). Accordingly, the current density operator does not admit a global definition, while the following piecewise definition:

is required. The form of Equation (9) fully explains the ultimate reasons of the failure of usual boundary conditions in describing the scattering problem defined by the Hamiltonian in Equation (6). Hereafter, we provide a careful explanation of this point. First of all, we explicitly observe that under translational invariance along the y-axis, the y-component of the current density, namely

${\mathcal{J}}_{\mathrm{y}}={\mathsf{\Psi}}^{+}{\widehat{\mathcal{J}}}_{\mathrm{y}}\mathsf{\Psi}$, does not depend on the coordinate y, and thus the quantity

${\partial}_{y}{\mathcal{J}}_{\mathrm{y}}=0$ does not play any role in the continuity equation. Therefore, current density conservation entirely depends on the conservation of the x-component of the current density. The current density conservation problem is related to the invariance of the current density operator under appropriate transformations. This point can be easily understood by observing that, within the transfer matrix formalism, boundary conditions for the spinorial wave function at the

$x=0$ interface can be written as

$\mathsf{\Psi}\left({0}^{+}\right)=\mathcal{M}\mathsf{\Psi}\left({0}^{-}\right)$, with

$\mathcal{M}$ being a

$2\times 2$ matching matrix. Here, we have introduced the notation

${0}^{+}$ and

${0}^{-}$, meaning a spatial position close to

$x=0$ and belonging to the right or left side of the junction, respectively. Within this formalism, the current density on the right (left) side of the interface, namely

${\mathcal{J}}_{\mathrm{R}}$ (

${\mathcal{J}}_{\mathrm{L}}$), takes the form of

${\mathcal{J}}_{\mathrm{R}}={\mathsf{\Psi}}^{+}\left({0}^{+}\right){\widehat{\mathcal{J}}}_{x}\left({0}^{+}\right)\mathsf{\Psi}\left({0}^{+}\right)$ (

${\mathcal{J}}_{\mathrm{L}}={\mathsf{\Psi}}^{+}\left({0}^{-}\right){\widehat{\mathcal{J}}}_{x}\left({0}^{-}\right)\mathsf{\Psi}\left({0}^{-}\right)$), while current density conservation requires the condition

${\mathcal{J}}_{\mathrm{R}}={\mathcal{J}}_{\mathrm{L}}$. Observing that

${\mathcal{J}}_{\mathrm{R}}={\mathsf{\Psi}}^{+}\left({0}^{-}\right){\mathcal{M}}^{+}{\widehat{\mathcal{J}}}_{x}\left({0}^{+}\right)\mathcal{M}\mathsf{\Psi}\left({0}^{-}\right)$ and using the continuity of the current density at the interface, we get the important relation

${\mathcal{M}}^{+}{\widehat{\mathcal{J}}}_{x}\left({0}^{+}\right)\mathcal{M}={\widehat{\mathcal{J}}}_{x}\left({0}^{-}\right)$. Here, we explicitly notice that the current density operator in the absence of lattice mismatching at the interface (i.e.,

$\theta =0$) is globally defined so that

${\widehat{\mathcal{J}}}_{x}\left({0}^{+}\right)={\widehat{\mathcal{J}}}_{x}\left({0}^{-}\right)$, and the usual boundary conditions are recovered. This is not the case for the problem under study where

${\widehat{\mathcal{J}}}_{x}\left({0}^{+}\right)\ne {\widehat{\mathcal{J}}}_{x}\left({0}^{-}\right)$, as evident by direct inspection of Equation (9). According to the above arguments, proper boundary conditions for the scattering problem at the grain boundary interface are assigned once an opportune matching matrix

$\mathcal{M}$ has been identified. Based on the current density conservation law, a matching matrix

$\mathcal{M}$ has to respect the following matrix equation:

for arbitrary choices of the rotation angle,

$\theta $. The matching matrix

$\mathcal{M}$ is completely determined by the properties of the grain boundary local potential

$U\left(x\right)$ to be added to the Dirac Hamiltonian (6). Such potential is a

$2\times 2$ Hermitian operator, which is different from zero only inside the grain boundary region (the GB region in

Figure 1b), while its specific form depends on the realization of the interface between grains at the atomic level. The latter information is clearly not available within the framework of a long wavelength (continuous) model, and thus,

$U\left(x\right)$ has to be meant as a phenomenological potential that is able to reproduce the transport properties of the interface. A useful simplification that will be used in the following discussion parameterizes the interface potential as

$U\left(x\right)=\mathcal{B}\delta \left(x\right)$ with

$\delta \left(x\right)$ being the Dirac delta function and

$\mathcal{B}$ as a

$2\times 2$ Hermitian operator. Once the structure of

$U\left(x\right)=\mathcal{B}\delta \left(x\right)$ is known, the matching matrix

$\mathcal{M}$ can be determined as described in

Section 2.5.