The general boundary condition for the Dirac equation is a local linear restriction on the components of the spinor wave function at the boundary, $$\psi=M\psi,\;\;M=\begin{pmatrix} n_z&n_x-in_y\\ n_x+in_y&-n_z \end{pmatrix}$$ with ${\mathbf n}=(n_x,n_y,n_z)$ a unit vector. (Check that $M^2=\mathbb{1}$.)

If you are interested in the generalization of the Dirichlet boundary condition, then you will want to ensure that zero current flows through the boundary. This further restricts $M$ to vectors ${\mathbf n}$ that satisfy $({\mathbf n},{\mathbf n}_B)=0$, where ${\mathbf n}_B$ is a unit vector in the $x$-$y$ plane perpendicular to the boundary.

All of this was worked out in the context of graphene some years ago, see arXiv:0710.2723. In that context the different boundary conditions have names: $n_x=0=n_y$ is called the "zigzag" boundary condition and $n_z=0$ is called the "mass" boundary condition.

The general boundary condition for the Dirac equation is a local linear restriction on the components of the spinor wave function at the boundary, $$\psi=M\psi,\;\;M=\begin{pmatrix} n_z&n_x-in_y\\ n_x+in_y&-n_z \end{pmatrix}$$ with ${\mathbf n}=(n_x,n_y,n_z)$ a unit vector. (Check that $M^2=\mathbb{1}$.)

If you are interested in the generalization of the Dirichlet boundary condition, then you will want to ensure that zero current flows through the boundary. This further restricts $M$ to vectors ${\mathbf n}$ that satisfy $({\mathbf n},{\mathbf n}_B)=0$, where ${\mathbf n}_B$ is a unit vector in the $x$-$y$ plane perpendicular to the boundary.

All of this was worked out in the context of graphene, see arXiv:0710.2723. In that context the different boundary conditions have names: $n_x=0=n_y$ is called the "zigzag" boundary condition and $n_z=0$ is called the "mass" boundary condition. There is also something called the "armchair" boundary condition, which you need if you put the Dirac equation on a lattice.

The general boundary condition for the Dirac equation is a local linear restriction on the components of the spinor wave function at the boundary, $$\psi=M\psi,\;\;M=\begin{pmatrix} n_z&n_x-in_y\\ n_x+in_y&-n_z \end{pmatrix}$$ with ${\mathbf n}=(n_x,n_y,n_z)$ a unit vector. (Check that $M^2=\mathbb{1}$.)

If you are interested in the generalization of the Dirichlet boundary condition, then you will want to ensure that zero current flows through the boundary. This further restricts $M$ to vectors ${\mathbf n}$ that satisfy $({\mathbf n},{\mathbf n}_B)=0$, where ${\mathbf n}_B$ is a unit vector in the $x$-$y$ plane perpendicular to the boundary.

All of this was worked out in the context of graphene some years ago, see arXiv:0710.2723

The general boundary condition for the Dirac equation is a local linear restriction on the components of the spinor wave function at the boundary, $$\psi=M\psi,\;\;M=\begin{pmatrix} n_z&n_x-in_y\\ n_x+in_y&-n_z \end{pmatrix}$$ with ${\mathbf n}=(n_x,n_y,n_z)$ a unit vector. (Check that $M^2=\mathbb{1}$.)

If you are interested in the generalization of the Dirichlet boundary condition, then you will want to ensure that zero current flows through the boundary. This further restricts $M$ to vectors ${\mathbf n}$ that satisfy $({\mathbf n},{\mathbf n}_B)=0$, where ${\mathbf n}_B$ is a unit vector in the $x$-$y$ plane perpendicular to the boundary.

All of this was worked out in the context of graphene some years ago, see arXiv:0710.2723. In that context the different boundary conditions have names: $n_x=0=n_y$ is called the "zigzag" boundary condition and $n_z=0$ is called the "mass" boundary condition.