Quantum Transmission Conditions for Diffusive Transport in Graphene with Steep Potentials

Abstract

We present a formal derivation of a drift-diffusion model for stationary electron transport in graphene, in presence of sharp potential profiles, such as barriers and steps. Assuming the electric potential to have steep variations within a strip of vanishing width on a macroscopic scale, such strip is viewed as a quantum interface that couples the classical regions at its left and right sides. In the two classical regions, where the potential is assumed to be smooth, electron and hole transport is described in terms of semiclassical kinetic equations. The diffusive limit of the kinetic model is derived by means of a Hilbert expansion and a boundary layer analysis, and consists of drift-diffusion equations in the classical regions, coupled by quantum diffusive transmission conditions through the interface. The boundary layer analysis leads to the discussion of a four-fold Milne (half-space, half-range) transport problem.

Appendix: Proof of Theorem 4.1

In order to streamline the notation, let us put

$$\begin{aligned} L^i(\varvec{z}) = L^i_s({\varvec{p}}) := F'_{n^i_s}({\varvec{p}}), \end{aligned}$$

(A.1)

and rewrite the Milne problem (4.29) accordingly:

$$\begin{aligned} \left\{ \begin{aligned}&\mu \frac{\partial \theta ^i}{\partial \xi } = L^i {\big \langle {\theta ^i} \big \rangle } - \theta ^i,&(-1)^i&\xi >0, \\&\theta ^i_\mathrm {in}- \mathcal {K}^i\big (\theta ^i_\mathrm {out},\theta ^j_\mathrm {out}\big ) = G^i_\mathrm {in}- \mathcal {K}^i\big (G^i_\mathrm {out},G^j_\mathrm {out}\big ),&\xi =0. \end{aligned} \right. \end{aligned}$$

(A.2)

We recall that in problem (A.2) the y-variable is just a parameter, which shall be omitted throughout the proof.

The proof is inspired by the ideas of Ref. [14] and is divided into three steps for the reader’s convenience.

${ {{Step 1: reduction~to~a }~{\vert {{\varvec{p}}} \vert }{-averaged~problem.}}}$ Let us consider the uncoupled version of (A.2), with assigned inflows $g^i$:

$$\begin{aligned} \left\{ \begin{aligned}&\mu \frac{\partial \theta ^i}{\partial \xi }(\xi ,\varvec{z}) = L^i (\varvec{z}){\big \langle {\theta ^i} \big \rangle } - \theta ^i(\xi ,\varvec{z}),&\quad (-1)^i&\xi >0,\quad \varvec{z}\in \varTheta ,\\&\theta ^i_\mathrm {in}(\varvec{z}) = g^i(\varvec{z}),&\xi =0, \quad \varvec{z}\in \varTheta ^i_\mathrm {in}\end{aligned} \right. \end{aligned}$$

(A.3)

(recall definitions (A.1), (3.23) and (4.19)). We introduce the ${\vert {{\varvec{p}}} \vert }$-average

$$\begin{aligned} {\tilde{\theta }}^i(\xi ,\varphi ,s) := \frac{1}{2\pi \hbar ^2} \int _0^{+\infty } \theta ^i(\xi ,{\vert {{\varvec{p}}} \vert }\cos \varphi ,{\vert {{\varvec{p}}} \vert }\sin \varphi ,s)\,{\vert {{\varvec{p}}} \vert }\,d{\vert {{\varvec{p}}} \vert }, \end{aligned}$$

(A.4)

where the normalisation constant is chosen so that

$$\begin{aligned} {\big \langle {\theta ^i} \big \rangle }(\xi ,s) = \frac{1}{2\pi }\int _0^{2\pi } {\tilde{\theta }}^i(\xi ,\varphi ,s) \,d\varphi \quad \text {and} \quad {\tilde{L}}^i_s = 1, \end{aligned}$$

where ${\left\langle \theta ^i \right\rangle }$ is defined in (3.12). We introduce the analogous of the sets (3.22) for the averaged quantities:

$$\begin{aligned} {\tilde{\varTheta }} := [0,2\pi ) \times \{-1,+1\}, \quad {\tilde{\varTheta }}^i_{\mathrm {in}/\mathrm {out}} := \Big \{ (\varphi ,s) \in \varTheta \mid (-1)^i \cos \varphi \gtrless 0 \Big \} \end{aligned}$$

(A.5)

and extend, in the obvious way, to ${\tilde{\theta }}^i$ the notations ${\tilde{\theta }}^i_\mathrm {in}$ and ${\tilde{\theta }}^i_\mathrm {out}$. Taking the ${\vert {{\varvec{p}}} \vert }$-average of (A.3), we obtain that ${\tilde{\theta }}^i_s$ satisfies

$$\begin{aligned} \left\{ \begin{aligned}&\mu \frac{\partial {\tilde{\theta }}^i}{\partial \xi }(\xi ,\varphi ,s) = \frac{1}{2\pi } \int _0^{2\pi } {\tilde{\theta }}^i(\xi ,\varphi ,s)\,d\varphi - {\tilde{\theta }}^i(\xi ,\varphi ,s),&\ (-1)^i&\xi >0,\ (\varphi ,s) \in {\tilde{\varTheta }},\\&{\tilde{\theta }}^i_\mathrm {in}(\varphi ,s) = {\tilde{g}}^i(\varphi ,s),&\xi =0,\ (\varphi ,s) \in {\tilde{\varTheta }}^i_\mathrm {in}, \end{aligned} \right. \end{aligned}$$

(A.6)

where we recall that

$$\begin{aligned} \mu = \frac{cp_x}{{\vert {{\varvec{p}}} \vert }} = c\cos \varphi . \end{aligned}$$

Conversely, it is easy to check that, if ${\tilde{\theta }}^i$ is a solution of (A.6), then

$$\begin{aligned} \theta ^i := L^i{\tilde{\theta }}^i + \left\{ \begin{aligned}&\mathrm {e}^{-\xi /\mu }\left( g^i - L^i {\tilde{g}}^i \right) ,&(-1)^i \mu >0,\\&0,&(-1)^i \mu <0, \end{aligned} \right. \end{aligned}$$

(A.7)

is solution of (A.3). The equations (A.6) are four ($s = \pm 1$, $i = 1,2$) independent Milne problems having the form of a Milne problem for neutron transport, to the extent that the kernel of the collision operator coincides with the functions that are constant with respect to $\varphi$ [2, 15]. About such problem the following facts are known [2, 14, 22]:

• (i) If ${\tilde{g}}^i \in \mathrm{L}^\infty ({\tilde{\varTheta }}^i_\mathrm {in})$, the solution ${\tilde{\theta }}^i$ to problem (A.6) exists and is unique in $\mathrm{L}^\infty \big ( (-1)^i[0,+\infty )\times {\tilde{\varTheta }} \big )$. Moreover, one has the positivity, i.e. ${\tilde{\theta }}^i \ge 0$ if ${\tilde{g}}^i \ge 0$.

• (ii) A constant $n_s^{i,\infty }$ (depending on ${\tilde{g}}^i$) exists such that ${\tilde{\theta }}^i(\xi ,\varphi ,s) \rightarrow n_s^{i,\infty }$, as $\xi \rightarrow (-1)^i\infty$, and the convergence is exponentially fast; in particular

  $$\begin{aligned} \Big \vert \int _0^{2\pi } {\tilde{\theta }}^i(\xi ,\varphi ,s)\, d\varphi - n_s^{i,\infty } \Big \vert \le C\mathrm {e}^{-\alpha {\vert {\xi } \vert }}, \end{aligned}$$

  for some constants $C>0$ and $\alpha >0$. Moreover, $n^{i,\infty } \ge 0$ if ${\tilde{g}}^i \ge 0$.

• (iii) The Albedo operator, that associates the inflow to the outflow, i.e. ${\tilde{\theta }}^i_\mathrm {in}\equiv {\tilde{g}}^i \mapsto {\tilde{\theta }}^i_\mathrm {out}$, is a compact linear operator from $\mathrm{L}^\infty ({\tilde{\varTheta }}^i_\mathrm {in})$ to $\mathrm{L}^\infty ({\tilde{\varTheta }}^i_\mathrm {out})$.

Step 2: formulation as a Fredholm problem. Thanks to the explicit formula (A.7), it is easy to extend the above results to the ${\vert {{\varvec{p}}} \vert }$-dependent problem (A.3). Let us define the weighted spaces

$$\begin{aligned} X^i := \mathrm{L}^\infty \Big ( \varTheta , (L^i)^{-1} d{\varvec{p}}\Big ), \ X^i_\mathrm {in}:= \mathrm{L}^\infty \Big (\varTheta ^i_\mathrm {in}, (L^i)^{-1} d{\varvec{p}}\Big ), \ X^i_\mathrm {out}:= \mathrm{L}^\infty \Big (\varTheta ^i_\mathrm {out}, (L^i)^{-1} d{\varvec{p}}\Big ). \end{aligned}$$

Note that $g \in X^i$ implies that $g \in \mathrm{L}^p(\varTheta )$ for all $p \in [0,\infty ]$. Then, from the results of Step 1, we have the following facts about problem (A.3):

• (i) If $g^i \in X^i_\mathrm {in}$, the solution $\theta ^i$ to problem (A.3) exists and is unique in the space $\mathrm{L}^\infty \left( (-1)^i[0,+\infty ), X^i \right)$. Moreover, $\theta ^i \ge 0$ if $g^i \ge 0$.

• (ii) A constant $n_s^{i,\infty }$ (depending on $g^i$) exists such that $\theta ^i(\xi ,\varvec{z}) \rightarrow n_s^{i,\infty } L_s^i({\varvec{p}})$, as $\xi \rightarrow (-1)^i\infty$, and the convergence is exponentially fast; in particular

  $$\begin{aligned} \Big \vert {\big \langle {\theta _s^i} \big \rangle }(\xi ) - n_s^{i,\infty } \Big \vert \le C\mathrm {e}^{-\alpha {\vert {\xi } \vert }}, \end{aligned}$$

  for some constants $C>0$ and $\alpha >0$. Moreover, $n^{i,\infty } \ge 0$ if $g^i \ge 0$.

• (iii) The Albedo operator, associating the inflow $\theta ^i_\mathrm {in}\equiv g^i$ to the outflow $\theta ^i_\mathrm {out}$,

  $$\begin{aligned} \mathcal {A}^i : X^i_\mathrm {in}\rightarrow X^i_\mathrm {out}, \quad \mathcal {A}^i \theta ^i_\mathrm {in}:= \theta ^i_\mathrm {out}, \end{aligned}$$

  is a compact linear operator.

We now come to the coupled Milne problem (A.2). Thanks to the Albedo operator just introduced, we can reformulate (A.2) as a Fredholm problem in $X^1_\mathrm {in}\times X^2_\mathrm {in}$ for the unknown inflow data $(\theta ^1_\mathrm {in},\theta ^2_\mathrm {in})$, namely:

$$\begin{aligned} \begin{pmatrix} \theta ^1_\mathrm {in}\\ \theta ^2_\mathrm {in}\end{pmatrix} -\mathcal {K}\begin{pmatrix} \mathcal {A}^1 \theta ^1_\mathrm {in}\\ \mathcal {A}^2 \theta ^2_\mathrm {in}\end{pmatrix} = \begin{pmatrix} \varGamma ^1\\ \varGamma ^2 \end{pmatrix} \end{aligned}$$

(A.8)

where, recalling definition (4.23),

$$\begin{aligned} \mathcal {K}\begin{pmatrix} \theta ^1_\mathrm {out}\\ \theta ^2_\mathrm {out}\end{pmatrix} = \begin{pmatrix} \mathcal {K}^1\big (\theta ^1_\mathrm {out}, \theta ^2_\mathrm {out}\big ) \\ \mathcal {K}^2\big (\theta ^2_\mathrm {out}, \theta ^1_\mathrm {out}\big ) \end{pmatrix}, \end{aligned}$$

and the components of the non-homogeneous term are

$$\begin{aligned} \varGamma ^i = G^i_\mathrm {in}- \mathcal {K}^i\Big (G^i_\mathrm {out},G^j_\mathrm {out}\Big ). \end{aligned}$$

(A.9)

Let us now show that $\mathcal {K}: X^1_\mathrm {out}\times X^2_\mathrm {out}\rightarrow X^1_\mathrm {in}\times X^2_\mathrm {in}$ is a linear, continuous operator. In fact, for $\varvec{z}\in \varTheta ^i_\mathrm {in}$ we can write

$$\begin{aligned} \frac{\theta ^i_\mathrm {in}(\varvec{z})}{L^i(\varvec{z})} = \frac{\mathcal {K}^i(\theta ^i_{\mathrm {out}},\theta ^j_{\mathrm {out}})(\varvec{z})}{L^i(\varvec{z})} = R^i(\varvec{z}) \frac{\theta ^i_\mathrm {out}({\sim }\varvec{z})}{L^i(\varvec{z})} + T^j(\varvec{z}') \frac{ss'\theta ^j_\mathrm {out}(\varvec{z}')}{L^i(\varvec{z})}, \end{aligned}$$

where $\varvec{z}' \in \varTheta ^j_\mathrm {out}$ is constrained to $\varvec{z}$ by the conservation laws (3.26). Recalling definitions (4.3) and (A.1), and using (4.24) and the identity

$$\begin{aligned} \frac{\mathrm {e}^h}{(\mathrm {e}^h+1)^2} = \frac{\mathrm {e}^{-h}}{(\mathrm {e}^{-h}+1)^2}, \end{aligned}$$

it is not difficult to show that the following relation holds

$$\begin{aligned} \phi _1\big (A(n^i_s)\big )\, L^i_s({\varvec{p}}) = \phi _1\big (A(n^j_{s'})\big ) \, L_{s'}^j({\varvec{p}}'), \end{aligned}$$

(A.10)

for all $\varvec{z}= ({\varvec{p}},s)$ and $\varvec{z}' = ({\varvec{p}}',s')$ related as above. Then, the previous equality can be rewritten as

$$\begin{aligned} \frac{\theta ^i_\mathrm {in}(\varvec{z})}{L^i(\varvec{z})} = R^i(\varvec{z}) \frac{\theta ^i_\mathrm {out}({\sim }\varvec{z})}{L^i(\varvec{z})} + T^j(\varvec{z}') \frac{ss' c^i_s \theta ^j_\mathrm {out}(\varvec{z}')}{c^j_{s'} L^j(\varvec{z}')}, \end{aligned}$$

where

$$\begin{aligned} c^i(\varvec{z}) = c^i_s := \phi _1\big (A(n^i_s)\big ) \end{aligned}$$

(A.11)

are positive constants that only depend on s (and not on ${\varvec{p}}$). Using Jensen inequality we can write

$$\begin{aligned} \Big \vert \frac{\theta ^i_\mathrm {in}(\varvec{z})}{L^i(\varvec{z})} \Big \vert ^2 \le R^i(\varvec{z}) \Big \vert \frac{\theta ^i_\mathrm {out}({\sim }\varvec{z})}{L^i(\varvec{z})} \Big \vert ^2 + T^j(\varvec{z}') \Big \vert \frac{c^i_s}{c^j_{s'}} \Big \vert ^2 \Big \vert \frac{\theta ^j_\mathrm {out}(\varvec{z}')}{ L^j(\varvec{z}')} \Big \vert ^2, \end{aligned}$$

(we adopt the redundant notation ${\vert {\cdot } \vert }^2$ for the square, just to improve readability), which shows the continuity of $\mathcal {K}$, since the scattering coefficients are bounded by 1.

Hence, the composition of $\mathcal {K}$, which is continuous, with the $\mathcal {A}^i$’s, which are compact, is a compact operator on $X^1_\mathrm {in}\times X^2_\mathrm {in}$ and, therefore, (A.8) is a Fredholm equation with compact operator. The proof of Theorem 4.1 is thus reduced to a Fredholm alternative, which will be discussed in the next two steps.

Step 3: the homogeneous problem. We now consider the homogeneous version of the Milne problem (A.2), corresponding to $G=0$:

$$\begin{aligned} \left\{ \begin{aligned}&\mu \frac{\partial \theta ^i}{\partial \xi } = L^i {\big \langle {\theta ^i} \big \rangle } - \theta ^i ,&(-1)^i \xi >0,\\&\theta ^i_\mathrm {in}- \mathcal {K}^i\big (\theta ^i_\mathrm {out},\theta ^j_\mathrm {out}\big ) = 0,&\xi =0. \end{aligned} \right. \end{aligned}$$

(A.12)

Let $(\theta ^1,\theta ^2)$ be a solution of such a problem in the space $X^1\times X^2$. It is convenient to introduce the functions $(\psi ^1,\psi ^2)$ as follows:

$$\begin{aligned} \theta ^i(\xi ,\varvec{z}) = \sqrt{L^i(\varvec{z})} \, \psi ^i(\xi ,\varvec{z}), \quad i = 1,2, \end{aligned}$$

so that $\psi ^i/\sqrt{L^i}$ is bounded and $\psi ^i$ satisfies the equation

$$\begin{aligned} \mu \frac{\partial \psi ^i}{\partial \xi } = \sqrt{L^i} \, {\left\langle \sqrt{L^i} \, \psi ^i \right\rangle } - \psi ^i. \end{aligned}$$

(A.13)

It is immediate to verify that

$$\begin{aligned} \int _{\mathbb {R}^2} \left( \sqrt{L_s^i}\, \Big \langle \sqrt{L_s^i}\, \psi _s^i\Big \rangle - \psi _s^i\right) \psi ^i \, d{\varvec{p}}\le 0, \end{aligned}$$

(A.14)

and that the equality holds if and only if $\psi _s^i$ is in the kernel of the collision operator, i.e. $\psi _s^i = \sqrt{L_s^i} \, \gamma _s$, for some $\gamma _s$ constant with respect to ${\varvec{p}}$. By multiplying by $\psi ^i$ both sides of (A.13), integrating over ${\varvec{p}}\in \mathbb {R}^2$ we obtain

$$\begin{aligned} \frac{1}{2} \frac{\partial }{\partial \xi } \int _{\mathbb {R}^2} {\vert {\psi _s^i(\xi ,{\varvec{p}})} \vert }^2 \mu ({\varvec{p}})\, d{\varvec{p}}= \int _{\mathbb {R}^2} \left( \sqrt{L_s^i}\, \Big \langle \sqrt{L_s^i}\, \psi _s^i \Big \rangle - \psi _s^i\right) \psi _s^i \, d{\varvec{p}}\le 0. \end{aligned}$$

(A.15)

From (ii) of Step 2 we know that $\psi _s^i(\xi ,{\varvec{p}})$, as $\xi \rightarrow (-1)^i\infty$, tends to a function of the form $\sqrt{L_s^i({\varvec{p}})}\, \gamma _s^i(\xi )$ and, therefore, by integrating the previous inequality over $\xi \in (-1)^i[0,+\infty )$ and recalling that $\mu$ is an odd function of ${\varvec{p}}$, we obtain

$$\begin{aligned} \int _{\mathbb {R}^2} {\vert {\psi _s^1(0,{\varvec{p}})} \vert }^2 \mu ({\varvec{p}}) \, d{\varvec{p}}\le 0 \le \int _{\mathbb {R}^2} {\vert {\psi _s^2(0,{\varvec{p}})} \vert }^2 \mu ({\varvec{p}}) \, d{\varvec{p}}. \end{aligned}$$

(A.16)

On the other hand, $(\theta ^1,\theta ^2)$ satisfy the homogeneous KTC

$$\begin{aligned} \theta ^i(0,\varvec{z}) = R^i(\varvec{z}) \theta ^i(0,{\sim }\varvec{z}) + T^j(\varvec{z}') ss' \theta ^j(0,\varvec{z}') \end{aligned}$$

and, correspondingly, $(\psi ^1,\psi ^2)$ satisfy

$$\begin{aligned} \sqrt{L^i(\varvec{z})}\, \psi ^i(0,\varvec{z}) = R^i(\varvec{z}) \sqrt{L^i(\varvec{z})}\, \psi ^i(0,{\sim }\varvec{z}) + T^j(\varvec{z}') ss' \sqrt{L^j(\varvec{z}')}\, \psi ^j(0,\varvec{z}'), \end{aligned}$$

where $\varvec{z}\in \varTheta ^i_\mathrm {in}$ and $\varvec{z}' \in \varTheta ^j_\mathrm {out}$ are constrained by the conservation of energy (3.26). By using (A.10) and (A.11), we obtain

$$\begin{aligned} \sqrt{c^i(\varvec{z})}\, \psi ^i(0,\varvec{z}) = R^i(\varvec{z}) \sqrt{c^i(\varvec{z})}\, \psi ^i(0,{\sim }\varvec{z}) + T^j(\varvec{z}') ss' \sqrt{c^j(\varvec{z}')}\, \psi ^j(0,\varvec{z}'), \end{aligned}$$

(A.17)

and then, using Jensen inequality,

$$\begin{aligned} c^i(\varvec{z}) {\vert {\psi ^i(0,\varvec{z})} \vert }^2 \le R^i(\varvec{z}) c^i(\varvec{z}) {\vert {\psi ^i(0,{\sim }\varvec{z})} \vert }^2 + T^j(\varvec{z}') c^j(\varvec{z}') {\vert {\psi ^j(0,\varvec{z}')} \vert }^2 \end{aligned}$$

or, equivalently,

$$\begin{aligned}&c^i(\varvec{z}) \left( {\vert {\psi ^i(0,\varvec{z})} \vert }^2- {\vert {\psi ^i(0,{\sim }\varvec{z})} \vert }^2\right) \\&\quad \le - T^i(\varvec{z}) c^i(\varvec{z}) {\vert {\psi ^i(0,{\sim }\varvec{z})} \vert }^2 + T^j(\varvec{z}') c^j(\varvec{z}'){\vert {\psi ^j(0,\varvec{z}')} \vert }^2. \end{aligned}$$

We now multiply both sides by $\mu (\varvec{z})$, $\varvec{z}\in \varTheta ^i_\mathrm {in}$, which is negative (or zero) for $i = 1$ and positive (or zero) for $i = 2$, so that

$$\begin{aligned} c^1(\varvec{z}) \left( {\vert {\psi ^1(0,\varvec{z})} \vert }^2- {\vert {\psi ^1(0,{\sim }\varvec{z})} \vert }^2\right) \mu (\varvec{z})\ge & {} - T^1(\varvec{z}) c^1(\varvec{z}) {\vert {\psi ^1(0,{\sim }\varvec{z})} \vert }^2\mu (\varvec{z})\\&+ T^2(\varvec{z}') c^2(\varvec{z}'){\vert {\psi ^2(0,\varvec{z}')} \vert }^2\mu (\varvec{z}),\\ c^2(\varvec{z}) \left( {\vert {\psi ^2(0,\varvec{z})} \vert }^2- {\vert {\psi ^2(0,{\sim }\varvec{z})} \vert }^2\right) \mu (\varvec{z})\le & {} - T^2(\varvec{z}) c^2(\varvec{z}) {\vert {\psi ^2(0,{\sim }\varvec{z})} \vert }^2\mu (\varvec{z})\\&+ T^1(\varvec{z}') c^1(\varvec{z}'){\vert {\psi ^1(0,\varvec{z}')} \vert }^2\mu (\varvec{z}). \end{aligned}$$

If we now integrate the first inequality over $\varvec{z}\in \varTheta ^1_\mathrm {in}$, and the second one over $\varvec{z}\in \varTheta ^2_\mathrm {in}$, by following the same passages as in the proof of Proposition 3.1 we arrive at

$$\begin{aligned} \int _\varTheta c^1(\varvec{z}) {\vert {\psi ^1(0,\varvec{z})} \vert }^2 \mu (\varvec{z}) d\varvec{z}\ge & {} \int _{\varTheta ^1_\mathrm {out}} T^1(\varvec{z}) c^1(\varvec{z}) {\vert {\psi ^1(0,\varvec{z})} \vert }^2 \mu (\varvec{z})\,d\varvec{z}\\&+ \int _{\varTheta ^2_\mathrm {out}} T^2(\varvec{z}') c^2(\varvec{z}') {\vert {\psi ^2(0,\varvec{z}')} \vert }^2 \mu (\varvec{z}')\,d\varvec{z}'\\ \int _\varTheta c^2(\varvec{z}) {\vert {\psi ^2(0,\varvec{z})} \vert }^2 \mu (\varvec{z}) d\varvec{z}\le & {} \int _{\varTheta ^2_\mathrm {out}} T^2(\varvec{z}') c^2(\varvec{z}') {\vert {\psi ^2(0,\varvec{z}')} \vert }^2 \mu (\varvec{z}')\,d\varvec{z}'\\&+ \int _{\varTheta ^1_\mathrm {out}} T^1(\varvec{z}) c^1(\varvec{z}) {\vert {\theta ^1(0,\varvec{z})} \vert }^2 \mu (\varvec{z})\,d\varvec{z}\end{aligned}$$

(where in the last integral of both inequalities, $\varvec{z}$ and $\varvec{z}'$ are constrained by $E(\varvec{z}) = E(\varvec{z}') + {\delta V}$), which immediately leads to

$$\begin{aligned} \int _\varTheta c^1(\varvec{z}) {\vert {\psi ^1(0,\varvec{z})} \vert }^2 \mu (\varvec{z}) d\varvec{z}\ge \int _\varTheta c^2(\varvec{z}) {\vert {\psi ^2(0,\varvec{z})} \vert }^2 \mu (\varvec{z}) d\varvec{z}. \end{aligned}$$

(A.18)

If we now come back to (A.16), multiply the right and the left sides by the positive constants $c^1(\varvec{z}) = c^1_s$ and $c^2(\varvec{z}) = c^2_s$, respectively, and sum up with respect to s, we obtain

$$\begin{aligned} \int _\varTheta c^1(\varvec{z}) {\vert {\psi ^1(0,\varvec{z})} \vert }^2 \mu (\varvec{z}) d\varvec{z}\le 0 \le \int _\varTheta c^2(\varvec{z}) {\vert {\psi ^2(0,\varvec{z})} \vert }^2 \mu (\varvec{z}) d\varvec{z}. \end{aligned}$$

(A.19)

By comparing (A.18) with (A.19) we see that, necessarily,

$$\begin{aligned} \int _\varTheta c^1(\varvec{z}) {\vert {\psi ^i(0,\varvec{z})} \vert }^2 \mu (\varvec{z}) d\varvec{z}= 0, \qquad i = 1,2. \end{aligned}$$

Multiplying (A.15) by $c^i(\varvec{z}) = c^i_s$, summing up with respect to s and integrating with respect to $\xi$ yields, therefore,

$$\begin{aligned} \int _0^{(-1)^i\infty } \int _\varTheta c^i \left( \sqrt{L^i}\, \Big \langle \sqrt{L^i}\, \psi ^i \Big \rangle - \psi ^i\right) \psi ^i \, d\varvec{z}\,d\xi = 0. \end{aligned}$$

Since $c^i$ are positive constants that only depend on s, and the integrals with respect to ${\varvec{p}}$ are definite in sign (see (A.14)), this implies that

$$\begin{aligned} \int _{\mathbb {R}^2} \left( \sqrt{L_s^i}\, \Big \langle \sqrt{L_s^i}\, \psi _s^i \Big \rangle - \psi _s^i\right) \psi _s^i \,d{\varvec{p}}= 0 \end{aligned}$$

for all $\xi \in (-1)^i[0,+\infty )$. This equality can only hold when $\psi ^i(\xi ,\cdot )$ is in the kernel of the collision operator, which implies that $\theta ^i(\xi ,{\varvec{p}})$ is necessarily of the form

$$\begin{aligned} \theta ^i(\xi ,\varvec{z}) = \theta _s^i(\xi ,{\varvec{p}}) = L_s^i({\varvec{p}}) \gamma _s^i(\xi ). \end{aligned}$$

Finally, the substitution of this expression in the first of equations (A.12) immediately yields that $\gamma _s^i$ is constant, so that

$$\begin{aligned} \theta _s^i(\xi ,{\varvec{p}}) = L_s^i({\varvec{p}}) \gamma _s^i. \end{aligned}$$

(A.20)

Substituting (A.20) in the second of equations (A.12) and using (A.10) leads to the following necessary and sufficient condition for (A.20) to be solution of the homogeneous Milne problem (A.12):

$$\begin{aligned} c_{s'}^j\gamma _s^i = ss'c_s^i \gamma _{s'}^j \end{aligned}$$

(A.21)

where $c^i_s$ is defined by (A.11), and i, j and $s,s'$ are related as usual. Recalling that the conservation of energy is satisfied by three couples $(s,s')$ if ${\delta V}\not = 0$ and just by two couples in the case if ${\delta V}= 0$ (see Remark 3.1), we notice that (A.21) is a rank-3 condition if ${\delta V}\not = 0$ and a rank-2 condition if ${\delta V}= 0$.

Step 4: the inhomogeneous problem. Let us finally return to the complete, inhomogeneous problem (A.2). Let $(\theta ^1,\theta ^2)$ be a bounded solution of (A.2). The integration in ${\varvec{p}}$ of the first equation in (A.2) yields

$$\begin{aligned} \frac{d}{d\xi } {\left\langle \mu \theta _s^i \right\rangle } = 0, \end{aligned}$$

and the integration in ${\varvec{p}}$ after multiplication by $\mu$ yields

$$\begin{aligned} \frac{d}{d\xi } {\left\langle \mu ^2 \theta _s^i \right\rangle } = - {\left\langle \mu \theta _s^i \right\rangle }. \end{aligned}$$

Hence, ${\left\langle \mu \theta _s^i \right\rangle }$ is a constant, and this constant must be zero, otherwise ${\vert {{\left\langle \mu ^2 \theta _s^i \right\rangle }} \vert }$ would grow linearly with $\xi$, in contradiction with the boundedness assumption. So we have

$$\begin{aligned} \int _{\mathbb {R}^2} \theta ^i_s(\xi ,{\varvec{p}})\,\mu ({\varvec{p}})\,d{\varvec{p}}= 0, \end{aligned}$$

(A.22)

for all $\xi \in (-1)^i[0,+\infty )$, $i =1,2$ and $s = \pm 1$. If we now rewrite the boundary conditions as

$$\begin{aligned} \theta ^i_\mathrm {in}- G^i_\mathrm {in}- \mathcal {K}^i\Big (\theta ^i_\mathrm {out}- G^i_\mathrm {out},\theta ^j_\mathrm {out}- G^j_\mathrm {out}\Big ) = 0, \end{aligned}$$

then, from Proposition 3.1 (that applies also to the linear KTC), we have that the conservation of charge flux holds:

$$\begin{aligned} \int _\varTheta s(\theta ^1(0,\varvec{z})- G^1(\varvec{z})) \mu (z)\,d\varvec{z}= \int _\varTheta s(\theta ^2(0,\varvec{z})- G^2(\varvec{z})) \mu (\varvec{z})\,d\varvec{z}. \end{aligned}$$

But then, since (A.22) implies that the charge flux associated to $\theta ^i$ vanishes, we obtain that the flux conservation for $G^i$ must hold:

$$\begin{aligned} \int _\varTheta s\,G^1(\varvec{z}) \mu (\varvec{z})\,d\varvec{z}= \int _\varTheta s\, G^2(\varvec{z}) \mu (\varvec{z})\,d\varvec{z}. \end{aligned}$$

(A.23)

Equation (A.23) is therefore a necessary condition for the existence of a bounded solution to the Milne problem (A.2) or, equivalently, to the Fredholm problem (A.8) when $\varGamma ^i$ has the form (A.9). Using (4.14), it is immediate to verify that the $G^i$’s, given by (4.18), satisfy this condition if and only if (4.30) holds.

Now, from Step 3 we know that the kernel of the Fredholm operator at the left-hand side of (A.8) is spanned by the functions of the form $L^i_s({\varvec{p}})\gamma ^i_s$, with $\gamma ^i_s$ satisfying (A.21), and is therefore a subspace of $X^1_\mathrm {in},\times X^2_\mathrm {in}$ of dimension d, where $d = 1$ if ${\delta V}\not = 0$, and $d = 2$ if ${\delta V}= 0$. Hence, the range of the Fredholm operator is a closed subspace of codimension d. But (A.23) also defines a subspace of codimension d, and then it describes the condition of existence of the solution to the Fredholm equation when $\varGamma ^i$ is of the form (A.9). We conclude that (4.30) is a necessary and sufficient condition for the existence of a solution to the Fredholm equation (A.8) (and, therefore, to the Milne problem (A.2)), up to a solution of the associated homogeneous problem. This proves the first part of Theorem 4.1.

The second part of the theorem, that is the existence of the asymptotic densities $n_s^{i,\infty }$ and the exponential estimate (4.32), follows from (ii) of Step 2. In fact, once the coupled Milne problem is solved, the inflow of each component $\theta ^i_s$ is determined (up to the addition of a term of the form (A.20)–(A.21)), and point (ii) of Step 2 applies.⁷