# Homogenization of 2D materials in the Thomas-Fermi-von Weizsäcker theory

Saad Benjelloun<EMAIL_ADDRESS>Makhbar Mathematical Sciences
Research Institute, Casablanca, Morocco  Salma Lahbabi<EMAIL_ADDRESS>College of Computing, Université Mohamed 6 Polytechnique, Benguerir, Morocco
EMAMI, LRI, ENSEM, Hassan II University of Casablanca, Morocco  Abdelqoddous
Moussa<EMAIL_ADDRESS>College of Computing, Université Mohamed 6
Polytechnique, Benguerir, Morocco

###### Abstract

We study the homogenization of the Thomas-Fermi-von Weizsäcker (TFW) model for
2D materials introduced in [3]. It consists in considering 2D-periodic nuclear
densities with periods going to zero. We study the behavior of the
corresponding ground state electronic densities and ground state energies. The
main result is that these three dimensional problems converge to a limit model
that is one dimensional, similar to the one proposed in [9]. We also
illustrate this convergence with numerical simulations and estimate the
converging rate for the ground state electronic densities and the ground state
energies.

Keywords— Thomas-Fermi-von Weizsäcker model, Homogenization, 2D materials,
Periodic models, Crystals

###### Contents

  1. 1 Introduction and main results
    1. 1.1 Thomas-Fermi-von Weizsäcker model for finite systems
    2. 1.2 Thomas-Fermi-von Weizsäcker model for 2D crystals
    3. 1.3 Homogenization of 2D materials
  2. 2 Hartree interaction
  3. 3 Homogenization procedure: proof of Theorem 1.5
  4. 4 Numerical illustration

##  1 Introduction and main results

In recent years, 2D materials became an active research field [8] due to their
promising physical, electrical, chemical, and optical properties that their 3D
materials do not have [16, 7, 10]. Electronic structure simulations are highly
useful in the discovery of these properties and their tuning for the potential
applications [17, 11, 15]. Thus the need of mathematical models and simulation
algorithms tailored for 2D materials [5, 14, 18].

Density Functional Theory is one of the most widely used simulation tool in
electronic structure calculations. It consists in describing the electrons by
their density $\rho$ and the energy of the system by a functional of $\rho$. A
famous model in this class is the (orbital free) Thomas-Fermi-Von Weizsäcker
(TFW) model [12, 19]. From a mathematical point of view, an important result
in the study of crystals is the thermodynamic limit problem. It consists in
proving that when a finite cluster converges to some periodic perfect crystal,
the corresponding ground state electronic density and ground state energy per
unit volume converge to the periodic equivalent. This program has been carried
out for the Thomas-Fermi-von Weizsäcker model for three dimensional (3D)
crystals in [6] and for one and two dimensional (1D and 2D) crystals in [3].

In this paper, we investigate the homogenization of the TFW model for 2D
crystals. Our goal is to find a homogeneous material equivalent to a 2D
crystal in the limit when the lattice parameter goes to zero. This corresponds
to putting more and more (normalized) nuclei in the unit cell, or equivalently
looking at the crystal from further and further away. It turns out that the
homogenized material can be described by a 1D model, in the same spirit as in
[9], which allows to reduce computational time and resources required to
simulate a 2D material, at least in a zero order approximation. Our proof can
be generalized if we substitute the $\int\rho^{5/3}$ term in the kinetic
energy by $\int\rho^{p}$ for some $\dfrac{3}{2}<p$. Note that the strict
convexity of the energy functional gives the uniqueness of the ground state,
which plays an important role in the proof.

Up to our knowledge, the closest work to ours is [4], where the authors use
the 2D TFW model [3] to derive macroscopic features of a crystal from the
microscopic structure in the presence of an external electric field. The
crystal is modeled in the band ${\mathbb{R}}^{2}\times[-1,1]$ and micro-macro
limit is taken when the ration between the atomic spacing and the size of the
crystal goes to zero.

The article is organized as follows. We start by recalling the Thomas-Fermi-
von Weizsäcker model for finite systems in Section 1.1, and for 2D crystals in
Section 1.2. In Section 1.3, we define the homogenization process, along with
the limit problem, and state our main result, whose proof is detailed in
Section 3. Intermediate results about the 2D Coulomb interaction are presented
in Section 2. Finally, numerical illustrations are gathered in Section 4.

### Acknowledgments

The research leading to these results has received funding from OCP grant AS70
“Towards phosphorene based materials and devices”. Prof. S. Lahbabi thanks the
CEREMADE for hosting her during the final writing of this article.

###  1.1 Thomas-Fermi-von Weizsäcker model for finite systems

We present in this section the TFW model for finite systems. Let $m\in
L^{1}({\mathbb{R}}^{3})$ be a finite nuclear charge density. The state of the
electrons is described by a non negative electronic density $\rho\in
L^{1}({\mathbb{R}}^{3})$. The energy functional is given by

${\mathcal{E}}^{m}(\rho)=\int_{\mathbb{R}^{3}}\left|\nabla\sqrt{\rho}\right|^{2}+\int_{\mathbb{R}^{3}}\rho^{5/3}+\frac{1}{2}D(\rho-m,\rho-m),$
(1)

where the first two terms represent the kinetic energy of the electrons and
$D(f,g)$ is the Coulomb interaction between charge densities $f$ and $g$ in
the Coulomb space
${\mathcal{C}}=\left\\{f\in{\mathcal{S}}^{\prime}({\mathbb{R}}^{3}),\;\frac{\widehat{f}}{\left|\cdot\right|}\in
L^{2}({\mathbb{R}}^{3})\right\\}$. It is defined by

$D(f,g)=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\dfrac{f(x)g(y)}{\left|x-y\right|}\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y=4\pi\int_{{\mathbb{R}}^{3}}\frac{\overline{\widehat{f}(k)}\widehat{g}(k)}{\left|k\right|^{2}}\mathop{}\\!\mathrm{d}k,$

where $\hat{f}(k)$ denotes the $k-$th Fourier coefficient for $f$. The ground
state is given by the following minimization problem

$I^{m}=\inf\left\\{{\mathcal{E}}^{m}(\rho),\;\rho\geqslant 0,\;\sqrt{\rho}\in
H^{1}(\mathbb{R}^{3}),\int_{\mathbb{R}^{3}}\rho=\int_{\mathbb{R}^{3}}m\right\\}.$
(2)

It is well known that problem (2) has a unique minimizer $\rho$ (see for
instance [2]). $u=\sqrt{\rho}$ is the unique solution of the corresponding
Euler-Lagrange equation

$\left\\{\begin{array}[]{rll}-\Delta u+\frac{5}{3}u^{7/3}+u\Phi&=&\lambda
u,\\\ -\Delta\Phi&=&4\pi(u^{2}-m)\end{array}\right.,$ (3)

where $\displaystyle\lambda\in{\mathbb{R}},\mbox{ and
}\Phi=(u^{2}-m)*\frac{1}{\left|\cdot\right|}$ is the mean-field potential.

###  1.2 Thomas-Fermi-von Weizsäcker model for 2D crystals

We present in this section the TFW model for 2D crystals introduced in [3]. 2D
crystals are characterized by a nuclear density $m$ that has the periodicity
of a 2D lattice ${\mathcal{R}}=a_{1}{\mathbb{Z}}+a_{2}{\mathbb{Z}}$, where
$(a_{1},a_{2})$ are two linearly independent vectors in ${\mathbb{R}}^{2}$
(see Figure 1), namely

$m(x_{1}+k_{1},x_{2}+k_{2},x_{3})=m(x_{1},x_{2},x_{3}),\quad\forall
x\in{\mathbb{R}}^{3},\;\forall(k_{1},k_{2})\in{\mathcal{R}}.$

$Q$$x_{1}$$x_{2}$$x_{3}$$\Gamma$ Figure 1: Example of a 2D lattice and its
unit cell $\Gamma$.

From now on, we denote by $Q$ the unit cell of
${\mathcal{R}}\subset{\mathbb{R}}^{2}$ and by $\Gamma=Q\times{\mathbb{R}}$ the
unit cell of ${\mathcal{R}}$ seen as a lattice in ${\mathbb{R}}^{3}$. For
$x\in{\mathbb{R}}^{3}$, we denote by
$\underline{x}=(x_{1},x_{2})\in{\mathbb{R}}^{2}$ so that
$x=(\underline{x},x_{3})$.

By means of a thermodynamic limit procedure, it has been shown in [3] that 2D
crystals can be described by a model similar to (1)-(2) posed on the unit cell
$\Gamma$. The main difference is that the 3D Green function
$\displaystyle\frac{1}{\left|x\right|}$ is replaced by the 2D periodic Green
function $G$ solution of

$-\Delta G=4\pi\sum_{k\in{\mathcal{R}}\times\left\\{0\right\\}}\delta_{k}.$

An explicit formula of $G$ is given by [3, equation 11]

$G(x)=-\dfrac{2\pi}{\left|Q\right|}\left|x_{3}\right|+\sum_{\underline{k}\in{\mathcal{R}}}\left(\dfrac{1}{\left|{x}-(\underline{k},0)\right|}-\dfrac{1}{\left|Q\right|}\int_{Q}\dfrac{\mathop{}\\!\mathrm{d}\underline{y}}{\left|{x}-\left(\underline{y}+\underline{k},0\right)\right|}\right).$
(4)

It can be seen as the sum over the lattice of the Coulomb potential created by
a point charge placed at the lattice sites, screened by a uniform background
of negative unit charge. A Fourier decomposition of $G$ can be found in [9].
The 2D crystals energy functional then reads

$\displaystyle{\mathcal{E}}^{m}_{{\rm
per}}(\rho)=\int_{\Gamma}\left|\nabla\sqrt{\rho}\right|^{2}+\int_{\Gamma}\rho^{5/3}+\frac{1}{2}D_{G}(m-\rho,m-\rho),$
(5)

where the Hartree interaction $D_{G}$ is given by

$\displaystyle
D_{G}(f,g):=\int_{\Gamma}\int_{\Gamma}{f(x)g(y)}G(x-y)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y.$

######  Remark 1.1.

In [3], there is an Coulomb correction term in the energy which comes from the
Dirichlet boundary condition at infinity considered in the thermodynamic limit
procedure, so that the energy is

$\displaystyle\widetilde{{\mathcal{E}}^{m}_{{\rm
per}}}(\rho)={\mathcal{E}}^{m}_{{\rm
per}}(\rho)+\dfrac{1}{\left|Q\right|}\int_{\Gamma}\int_{\Gamma}\dfrac{f(x)g(y)}{\left|x-y\right|}.$

In our study, we omit this correction term as it does not affect the problem
from a mathematical point of view.

The ground state is given by the minimization problem

$I^{m}_{{\rm per}}=\inf\left\\{{\mathcal{E}}^{m}_{{\rm
per}}(\rho),\quad\rho\geqslant 0,\quad\sqrt{\rho}\in X_{{\rm
per}},\quad\int_{\Gamma}\rho=\int_{\Gamma}m\right\\}$ (6)

where

$X_{{\rm per}}=\left\\{v\in H^{1}_{{\rm
per}}(\Gamma),\left(1+\left|x_{3}\right|\right)^{1/2}v\in L^{2}_{{\rm
per}}(\Gamma)\right\\}.$

This problem has been studied in [3] along with some basic properties of the
solution, that we summarize in the following Theorem.

######  Theorem 1.2 ([3, Theorem 3.2]).

Let $m\neq 0$ be a smooth non-negative ${\mathcal{R}}$-periodic function with
compact support with respect to $x_{3}$. Then, the minimization problem (6)
has a unique minimizer $\rho$. $u=\sqrt{\rho}$ is the unique solution of the
corresponding Euler-Lagrange system

$\begin{cases}-\Delta u+\frac{5}{3}u^{7/3}+u\Phi=\lambda u,\\\
-\Delta\Phi=4\pi(u^{2}-m),\\\ \end{cases}$ (7)

where $\lambda\in{\mathbb{R}}$. In addition, $u\in L^{\infty}(\mathbb{R}^{3})$
and $\left|u(x)\right|\leqslant\dfrac{C}{1+\left|x_{3}\right|^{3/2}}$, for
$\left|x_{3}\right|>1$, $C>0$ being a constant independent of the density $m$.

######  Remark 1.3.

The fact that, in the inequality
$\textstyle\left|u(x)\right|\leqslant\dfrac{C}{1+\left|x_{3}\right|^{3/2}}$,
for $\left|x_{3}\right|>1$, the constant is independent of the density $m$,
comes from a supersolution method applied to the equation (7) (see [3, Theorem
3.1-Theorem 2.3]).

###  1.3 Homogenization of 2D materials

In the framework of the Thomas-Fermi (TF) model and the reduce Hartree Fock
model (rHF), the recent work [9] studies reduced models for 2D homogeneous
materials. The idea is that if the material is homogeneous, the 3D model is
equivalent to a 1D model; simpler and less costly to simulate. In the present
work, we are interested in the homogenization procedure, which models looking
at a crystal macroscopically, from further and further away. Namely, we put
more and more nuclei in the unit cell, with the right charge normalization,
and we ask the following questions:

  * •

What is the nuclear density at the limit?

  * •

Does the ground state electronic density, potential and energy converge?

  * •

What is the model describing the limit electronic structure?

Let $m$ be a nuclear density satisfying the conditions of Theorem 1.2 and
consider the following sequence of nuclear densities which consists in putting
$N$ small nuclei in the unit cell

$m_{N}(x_{1},x_{2},x_{3}):=m(Nx_{1},Nx_{2},x_{3}),\quad\forall x\in\Gamma.$
(8)

We note that $m_{N}$ is $\frac{1}{N}{\mathcal{R}}$-periodic, which describes a
more homogeneous material than the initial one. The limit when
$N\rightarrow+\infty$ describes a homogeneous material (see Figure 2).

Figure 2: The homogenization process for the nuclear density $m_{N}$,
illustrated at $N=1,4$ and $16$ from left to right.

We will show that this model ”converges”, in some sense that we will precise
later, to the following limit model. For a 1D nuclear density $\mu$, we
introduce the energy functional

${\mathcal{E}}_{1}^{\mu}(\rho)=\int_{\mathbb{R}}\left|\nabla\sqrt{\rho}\right|^{2}+\int_{\mathbb{R}}\rho^{5/3}+\frac{1}{2}D_{1}(\rho-\mu,\rho-\mu)$
(9)

where the 1D Hartree interaction $D_{1}$ is defined for functions decaying
fast enough by

$D_{1}(f,g)=-{2\pi}\int_{\mathbb{R}}\left(\int_{\mathbb{R}}\left|s-t\right|f(s)g(t)\mathop{}\\!\mathrm{d}t\right)\mathop{}\\!\mathrm{d}s$

(more details about the 1D Hartree interaction can be read in [9]). The ground
state of this model is given by the minimization problem

$I^{\mu}_{1}=\min\\{E^{\mu}(\rho),\;\rho\geqslant 0,\;\sqrt{\rho}\in
X_{1},\int_{\mathbb{R}}\rho=\int_{\mathbb{R}}\mu\\},$ (10)

where

$X_{1}=\left\\{v\in H^{1}(\mathbb{R}),\;(1+\left|t\right|)^{1/2}v\in
L^{2}(\mathbb{R})\right\\}.$

The following Theorem, which is a direct consequence of Theorem 1.2 and the
definitions of Hartree interaction terms $D_{G}$ and $D_{1}$, is similar to
[9, Theorem 2.2, Theorem 2.8], which treat Thomas-Fermi and reduced Hartree-
Fock models.

######  Theorem 1.4.

Let $\mu\in{\mathcal{D}}({\mathbb{R}})$. Problem (10) has a unique minimizer.
$u=\sqrt{\rho}$ is the unique solution of the corresponding Euler Lagrange
equations

$\begin{cases}-u^{\prime\prime}+\frac{5}{3}u^{7/3}+u\Phi=\lambda u,\\\
-\Phi^{\prime\prime}=4\pi(u^{2}-\mu),\end{cases}$

where $\lambda\in{\mathbb{R}}$, and there exists $C>0$, independent of $\mu$,
such that for any $\left|t\right|\geqslant 1$,
$\left|u\right|\leqslant\frac{C}{1+\left|t\right|^{3/2}}$.

Our main contribution is the following Theorem.

######  Theorem 1.5.

Let $m$ be a nuclear density satisfying the hypothesis of Theorem 1.2 and for
$N\in{\mathbb{N}}\setminus\left\\{0\right\\}$, let $m_{N}$ be defined as in
(8). We denote by $I_{N}=I^{m_{N}}$ and by $\rho_{N}$ the corresponding ground
state given by Theorem 1.2. Let
$m_{0}(t)=\frac{1}{\left|Q\right|}\int_{Q}m(\underline{x},t)\mathop{}\\!\mathrm{d}\underline{x}$
and denote by $I_{0}=I^{m_{0}}_{1}$ and by $\rho_{0}$ the corresponding ground
state given by Theorem 1.4. The following holds

  1. i.

$\displaystyle\lim_{N\rightarrow\infty}I_{N}=I_{0}$,

  2. ii.

$\rho_{N}$ converges to $\rho_{0}$ in $L^{1}(\Gamma)$, in
$L^{p}_{loc}(\Gamma)$ $\forall 1\leqslant p\leqslant 3$ and almost everywhere,

  3. iii.

$\sqrt{\rho_{N}}$ converges $\sqrt{\rho_{0}}$ weakly in $H^{1}_{\rm
per}(\Gamma)$.

Our result is a zero order approximation of a finite $N$ situation. In
homogenization theory of partial differential equations with periodically
oscillating coefficients, the two scale convergence technique is usually used
to find higher order terms in the approximation [1]. Applying a similar
approach to our problem is a perspective of this work.

##  2 Hartree interaction

We recall in this section some properties of the Green function $G$, the
Hartree interaction $D_{G}$ and prove a uniqueness result on the mean-field
potential $\Phi$. In the subsequent proposition, we introduce a convenient
decomposition for the kernel $G$, that proves to be useful throughout the
paper.

######  Proposition 2.1.

We have

$G(x)=-\dfrac{2\pi}{\left|Q\right|}\left|x_{3}\right|+\dfrac{1}{\left|x\right|}+\psi(x),$

where $\psi\in L^{\infty}(\Gamma)$.

Proposition 2.1 is partially proved in [3]. We detail the proof here for
consistency.

###### Proof.

Using (4), we have

$\displaystyle\psi(x)=-\dfrac{1}{\left|Q\right|}\int_{Q}\dfrac{\mathop{}\\!\mathrm{d}\underline{y}}{\left|{x}-(\underline{y},0)\right|}+\sum_{\underline{k}\in({\mathcal{R}}^{2})^{*}}\left(\dfrac{1}{\left|{x}-(\underline{k},0)\right|}-\dfrac{1}{\left|Q\right|}\int_{Q}\dfrac{\mathop{}\\!\mathrm{d}\underline{y}}{\left|{x}-(\underline{y}+\underline{k},0)\right|}\right).$

Let us show that $\psi\in L^{\infty}(\Gamma)$. The function $\displaystyle
x\mapsto\dfrac{1}{\left|Q\right|}\int_{Q}\dfrac{\mathop{}\\!\mathrm{d}\underline{y}}{\left|{x}-(\underline{y},0)\right|}$
is continuous on $\Gamma$ and goes to zero as
$\left|x_{3}\right|\rightarrow\infty$. Moreover, the sum defining $\psi$
converges normally on a neighborhood of $0_{{\mathbb{R}}^{3}}$ [3, Proposition
3.2]. Therefore it is continuous on that neighborhood, which implies that
$\psi$ is bounded on $Q\times[-\varepsilon,\varepsilon]$, for some
$\varepsilon>0$. From [3, Equation 3.12], there exists $C>0$ such that

$\sum_{\underline{k}\in({\mathcal{R}}^{2})^{*}}\left(\dfrac{1}{\left|{x}-(\underline{k},0)\right|}-\dfrac{1}{\left|Q\right|}\int_{Q}\dfrac{\mathop{}\\!\mathrm{d}\underline{y}}{\left|{x}-(\underline{y}+\underline{k},0)\right|}\right)\leqslant\dfrac{C}{\left|x_{3}\right|^{\alpha}},\quad\forall\alpha<1.$

It follows that the
$x\mapsto\sum_{\underline{k}\in({\mathcal{R}}^{2})^{*}}\left(\dfrac{1}{\left|{x}-(\underline{k},0)\right|}-\dfrac{1}{\left|Q\right|}\int_{Q}\dfrac{\mathop{}\\!\mathrm{d}\underline{y}}{\left|{x}-(\underline{y}+\underline{k},0)\right|}\right)$
is bounded on $x\in Q\times({\mathbb{R}}\setminus[-\varepsilon,\varepsilon])$,
thus $\psi$ is also bounded on the same interval. ∎

As a consequence of the above proposition, we prove useful properties of the
potential $\Phi$. Let us introduce some notations. For some domain
$\Omega\subset{\mathbb{R}}^{d}$ and $1\leqslant p\leqslant{\infty}$, we denote
by

$L^{p}_{\rm unif}(\Omega)=\left\\{f\in L^{p}_{\rm loc}(\Omega),\;\forall
r>0,\;\sup_{x+B_{r}\subset\Omega}\left\|f\right\|_{L^{p}(B(x,r))}<\infty\right\\},$

where $B(x,r)$ is the ball of radius $r$ centered at $x$. For $f\in
L^{p}(\Gamma)$ and $g\in L^{q}(\Gamma)$, the convolution

$(f*_{\Gamma}g)(x)=\int_{\Gamma}f(x-y)g(y)\mathop{}\\!\mathrm{d}y$

is well defined in $L^{r}(\Gamma)$, where
$\frac{1}{p}+\frac{1}{q}=1+\frac{1}{r}$.

######  Proposition 2.2.

Let

$Y_{{\rm per}}=\left\\{f\in L^{1}(\Gamma)\cap
L^{5/3}(\Gamma),\;\int_{\Gamma}f=0,\;\left|x\right|f\in
L^{1}(\Gamma)\right\\}.$

The map

$\begin{array}[]{rll}Y_{\rm per}&\to&L^{\infty}(\Gamma)\\\
f&\mapsto&G*_{\Gamma}f\end{array}$

is well defined and continuous. Moreover, for $f\in Y_{\rm per}$,
$G*_{\Gamma}f$ is the unique solution, up to an additive constant, of the
Poisson equation

$\left\\{\begin{array}[]{rll}\Phi&\in&L^{1}_{\text{\rm unif
}}({\mathbb{R}}^{3})\\\ -\Delta\Phi&=&4\pi f.\end{array}\right.$ (11)

###### Proof.

From Proposition 2.1, we have that
$G(x)=\frac{1}{\left|x\right|}-\dfrac{2\pi}{\left|Q\right|}\left|x_{3}\right|+\psi(x)$,
with $\psi\in L^{\infty}(\Gamma)$. We are thus going to bound the three
functions

$f*_{\Gamma}\left|x_{3}\right|,\;f*_{\Gamma}\frac{1}{\left|x\right|}\;\text{and}\;f*_{\Gamma}\psi$

in $L^{{\infty}}(\Gamma)$. First, since $\psi\in L^{\infty}(\Gamma)$, we have
for any $x\in\Gamma$

$\left|\int_{\Gamma}f(x-y)\psi(y)\mathop{}\\!\mathrm{d}y\right|\leqslant\left\|\psi\right\|_{L^{\infty}(\Gamma)}\left\|f\right\|_{L^{1}(x-\Gamma)}.$

Thus

$\left\|f*_{\Gamma}\psi\right\|_{L^{\infty}(\Gamma)}\leqslant\left\|\psi\right\|_{L^{\infty}(\Gamma)}\left\|f\right\|_{L^{1}(\Gamma)}.$

We move to ${f}*_{\Gamma}\frac{1}{\left|x\right|}$. We have

$\displaystyle\left|f*_{\Gamma}\frac{1}{\left|x\right|}(x)\right|$
$\displaystyle\leqslant\int_{\Gamma}\frac{\left|f(x-y)\right|}{\left|y\right|}\mathop{}\\!\mathrm{d}y\displaystyle\leqslant\int_{\Gamma}\left|f(x-y)\right|\mathop{}\\!\mathrm{d}y+\int_{\Gamma}\frac{\left|f(x-y)\right|}{\left|y\right|}\mathbbm{1}_{\left|y\right|<1}\mathop{}\\!\mathrm{d}y.$

For the second term, we use Hölder inequality with $f\in L^{5/3}(\Gamma)$ and
$\frac{1}{\left|y\right|}\mathbbm{1}_{\left|y\right|<1}\in L^{5/2}(\Gamma)$.
We conclude that

$\begin{array}[]{ll}\left\|\frac{1}{\left|x\right|}*_{\Gamma}f\right\|_{L^{\infty}}&\displaystyle\leqslant\left\|f\right\|_{L^{1}(\Gamma)}+\left\|f\right\|_{L^{5/3}(\Gamma)}\left\|\frac{1}{\left|y\right|}\mathbbm{1}_{\left|y\right|<1}\right\|_{L^{5/2}(\Gamma)}.\end{array}$

Regarding the last term, we use the neutrality assumption to write

$\int_{\Gamma}f(x-y)\left|y_{3}\right|\mathop{}\\!\mathrm{d}y=\int_{\Gamma}f(x-y)(\left|y_{3}\right|-\left|x_{3}\right|)\mathop{}\\!\mathrm{d}y.$

Thus, using the triangle inequality and the ${\mathcal{R}}$-periodicity of
$x\mapsto x_{3}f(x)$, we obtain

$\displaystyle\left|\int_{\Gamma}f(x-y)\left|y_{3}\right|\mathop{}\\!\mathrm{d}y\right|\leqslant\int_{\Gamma}\left|f(x-y)\right|\left|x_{3}-y_{3}\right|\mathop{}\\!\mathrm{d}y=\int_{\Gamma}\left|f(y)\right|\left|y_{3}\right|\mathop{}\\!\mathrm{d}y.$

Finally, we need to prove that $\Phi-G*_{\Gamma}f$ is constant for any
solution $\Phi$ of (11). Notice first that $h=\Phi-G*_{\Gamma}f$ is a harmonic
function over $\mathbb{R}^{3}$. By the mean value Theorem for a harmonic
functions, we have that

$\left(\Phi-h\right)(x)=\int_{B(x,1)}\left(\Phi-h\right)(y)\mathop{}\\!\mathrm{d}y\leqslant\sup_{x\in\Gamma}\int_{B(x,1)}\left(\Phi-h\right)(y)\mathop{}\\!\mathrm{d}y\leqslant\left\|\Phi-h\right\|_{L^{1}_{unif}}.$

Therefore, $\Phi-h$ is harmonic and bounded in $\Gamma.$ By Louisville’s
Theorem, we conclude that $\Phi-h$ is constant. ∎

######  Corollary 2.3.

Let $f\in Y_{per}$, then
$\nabla(G*_{\Gamma}f)\in\left(L^{2}(\Gamma)\right)^{3}$ and there exists
$C>0$, such that

$D_{G}(f,f)=\left\|\nabla(G*_{\Gamma}f)\right\|^{2}_{L^{2}(\Gamma)}\leqslant
C\left\|f\right\|_{L^{1}(\Gamma)}\left(\left\|f\right\|_{L^{1}(\Gamma)}+\left\|f\right\|_{L^{5/3}(\Gamma)}+\left\||x_{3}|f\right\|_{L^{1}(\Gamma)}\right).$

###### Proof.

We have

$\displaystyle D_{G}(f,f)=\int_{\Gamma}(G*_{\Gamma}f)\times
f=\int_{\Gamma}(G*_{\Gamma}f)\times\Delta(G*_{\Gamma}f)=\int_{\Gamma}\left|\nabla(G*_{\Gamma}f)\right|^{2}.$

Besides, by the previous proposition $G*_{\Gamma}f\in L^{\infty}(\Gamma)$ and

$\left\|G*_{\Gamma}f\right\|_{L^{\infty}(\Gamma)}\leqslant(\left\|\psi\right\|_{L^{\infty}(\Gamma)}+1)\left\|f\right\|_{L^{1}(\Gamma)}+\left\|f\right\|_{L^{5/3}(\Gamma)}\left\|\frac{1}{\left|y\right|}\mathbbm{1}_{\left|y\right|<1}\right\|_{L^{5/2}(\Gamma)}+\left\|\left|x\right|f\right\|_{L^{1}(\Gamma)}\leqslant
C\left\|f\right\|_{Y_{{\rm per}}},$

which proves the inequality stated in the corollary.

∎

##  3 Homogenization procedure: proof of Theorem 1.5

This section is devoted to the proof of Theorem 1.5. The strategy of the proof
is as follows. We start by proving the convergence of the nuclear densities
$(m_{N})$. Indeed, when $N$ increases, the nuclear density $m_{N}$ becomes
more homogeneous and the sequence $(m_{N})$ converges to the 2D homogeneous
density
$m_{0}(x_{3})=\frac{1}{\left|Q\right|}\int_{Q}m(x)\mathop{}\\!\mathrm{d}\underline{x}$
(see Lemma 3.3). The electronic densities $\rho_{N}$ are uniformly bounded
with respect to $N$. We can thus extract convergent subsequences (see
Proposition 3.5). Both convergences give the upper bound $I\leqslant\liminf
I_{N}$. To prove the lower bound $I_{0}\geqslant\limsup I_{N}$, we use
$\rho_{0}$ as a test function. The proof is divided into four steps.

#### Step 1: Properties of the sequence $(m_{N})$

We state in this section two properties of the sequence $(m_{N})$.

######  Lemma 3.1.

For $p\geqslant 1$ and $f\in L^{1}_{\rm loc}({\mathbb{R}})$ such that
$x\mapsto f(x_{3})m^{p}(x)\in L^{1}(\Gamma)$, we have

$\displaystyle\int_{Q}m_{N}^{p}(\underline{x},x_{3})\mathop{}\\!\mathrm{d}\underline{x}=\int_{Q}m^{p}(\underline{x},x_{3})\mathop{}\\!\mathrm{d}\underline{x},$

and

$\displaystyle\int_{\Gamma}f(x_{3})m_{N}^{p}(x)\mathop{}\\!\mathrm{d}x=\int_{\Gamma}f(x_{3})m^{p}(x)\mathop{}\\!\mathrm{d}x.$

###### Proof.

We have

$\int_{Q}m_{N}^{p}(x_{1},x_{2},x_{3})\mathop{}\\!\mathrm{d}x_{1}\mathop{}\\!\mathrm{d}x_{2}=\int_{Q}m^{p}(Nx_{1},Nx_{2},x_{3})\mathop{}\\!\mathrm{d}x_{1}\mathop{}\\!\mathrm{d}x_{2}=\frac{1}{N^{2}}\int_{NQ}m^{p}(y_{1},y_{2},x_{3})\mathop{}\\!\mathrm{d}y_{1}\mathop{}\\!\mathrm{d}y_{2}.$

As $m$ is $Q$-periodic then,
$\displaystyle\int_{NQ}m^{p}(y_{1},y_{2},x_{3})\mathop{}\\!\mathrm{d}y_{1}\mathop{}\\!\mathrm{d}y_{2}=N^{2}\int_{Q}m^{p}(y_{1},y_{2},x_{3})\mathop{}\\!\mathrm{d}y_{1}\mathop{}\\!\mathrm{d}y_{2}$.
Hence, the first claim is proved. The second claim easily follows. ∎

######  Remark 3.2.

By the second point of Lemma 3.1, we have that
$\int_{\Gamma}m_{N}=\int_{\Gamma}m=\int_{\Gamma}m_{0}$, so that any admissible
state for $I_{N}$ is also an admissible state for $I_{0}$, and vice versa.

######  Lemma 3.3.

The sequence $(m_{N})$ converges to $m_{0}$ weakly in $L^{p}(\Gamma)$ $\forall
1\leqslant p<+{\infty}$.

###### Proof.

From the $Q$-periodicity of $m$, we have for any $x_{3}\in{\mathbb{R}}$ (see
for instance [13])

$m_{N}(\cdot,\cdot,x_{3})\rightharpoonup m_{0}(\cdot,\cdot,x_{3})\quad\mbox{
in }L^{p}(Q)\quad\forall 1\leqslant p<{\infty}.$ (12)

For $\varphi\in L^{q}(Q)$ and $\psi\in L^{q}({\mathbb{R}})$, with
$\frac{1}{q}+\frac{1}{p}=1$, we have
$f_{N}(x_{3}):=\displaystyle\int_{Q}m_{N}(x)\varphi(\underline{x})\psi(x_{3})\mathop{}\\!\mathrm{d}\underline{x}\to
f(x_{3}):=m_{0}(x_{3})\psi(x_{3})\int_{Q}\varphi(\underline{x})\mathop{}\\!\mathrm{d}\underline{x}$
a.e. and

$\displaystyle\left|f_{N}(x_{3})\right|$
$\displaystyle\leqslant\left\|\varphi\right\|_{L^{q}(Q)}\left|\psi(x_{3})\right|\left(\int_{Q}\left|m_{N}(\underline{x},x_{3})\right|^{p}\mathop{}\\!\mathrm{d}\underline{x}\right)^{1/p}=\left\|\varphi\right\|_{L^{q}(Q)}\left|\psi(x_{3})\right|\left(\int_{Q}\left|m(\underline{x},x_{3})\right|^{p}\mathop{}\\!\mathrm{d}\underline{x}\right)^{1/p}.$
(13)

Thus, by dominated convergence Theorem,

$\int_{\mathbb{R}}f_{N}(x_{3})\mathop{}\\!\mathrm{d}x_{3}=\int_{\Gamma}m_{N}(x)\varphi(\underline{x})\psi(x_{3})\mathop{}\\!\mathrm{d}x\to\int_{\mathbb{R}}f(x_{3})\mathop{}\\!\mathrm{d}x_{3}=\int_{\Gamma}m_{0}(x_{3})\varphi(\underline{x})\psi(x_{3})\mathop{}\\!\mathrm{d}x.$

By the density of $L^{q}(Q)\otimes L^{q}({\mathbb{R}})$ in $L^{q}(\Gamma)$,
the proof is complete. ∎

#### Step 2: Convergence of electronic densities

The following proposition gives uniform bounds on quantities of interest with
respect to $N$.

######  Proposition 3.4.

There exist various constants $C$ independent of $N$ such that the following
bounds hold.

  1. I.

$\displaystyle I_{N}\leqslant C$,

  2. II.

$\displaystyle\left\|\sqrt{\rho_{N}}\right\|_{H^{1}(\Gamma)}\leqslant C$,

  3. III.

$\displaystyle\left\|\rho_{N}\right\|_{L^{p}({\Gamma})}\leqslant C$ for all
$1\leqslant p\leqslant 3$,

  4. IV.

$\displaystyle D_{G}(\rho_{N}-m_{N},\rho_{N}-m_{N})\leqslant C$,

###### Proof.

As $\rho_{N}$ is the minimizer of $I_{N}$ and since $m$ is an admissible test
function for ${\mathcal{E}}^{m_{N}}_{\rm per}$, we have

$I_{N}={\mathcal{E}}_{per}^{m_{N}}(\rho_{N})\leqslant{\mathcal{E}}_{per}^{m_{N}}(m)=\int_{\Gamma}\left|\nabla\sqrt{m}\right|^{2}+\int_{\Gamma}m^{5/3}+\frac{1}{2}D_{G}(m-m_{N},m-m_{N}).$

By corollary 2.3 applied to $f=m-m_{N}$ and Lemma 3.1, we have

$\begin{array}[]{lll}D_{G}(m-m_{N},m-m_{N})&\leqslant&C\left\|m-m_{N}\right\|_{L^{1}(\Gamma)}\left(\left\|m-m_{N}\right\|_{L^{1}(\Gamma)}+\left\|m-m_{N}\right\|_{L^{5/3}(\Gamma)}\right.\\\
&&\left.+\left\||x_{3}|(m-m_{N})\right\|_{L^{1}(\Gamma)}\right)\\\
&\leqslant&C\left\|m\right\|_{L^{1}(\Gamma)}\left(\left\|m\right\|_{L^{1}(\Gamma)}+\left\|m\right\|_{L^{5/3}(\Gamma)}+\left\||x_{3}|m\right\|_{L^{1}(\Gamma)}\right).\end{array}$

Points II-IV are easily deduces from point point I. ∎

######  Proposition 3.5.

There exists a non negative function $\rho_{0}\in L^{1}(\Gamma)\cap
L^{5/3}(\Gamma)$ such that, up to a subsequence,

  * •

$(\rho_{N})$ converges to $\rho_{0}$ strongly in $L^{1}(\Gamma)$ and
$L^{p}_{loc}(\Gamma)$ for all $1\leqslant p\leqslant 3$, weakly in
$L^{p}(\Gamma)$ and almost everywhere on $\mathbb{R}^{3}$,

  * •

$(\nabla\sqrt{\rho_{N}})$ converges to $\nabla\sqrt{\rho_{0}}$ weakly in
$(L^{2}(\Gamma))^{3}$

  * •

$\rho_{0}-m_{0}\in Y_{\rm per}$

  * •

The sequence $\nabla\Phi_{N}=\nabla\left(G*_{\Gamma}(\rho_{N}-m_{N})\right)$
converges to $\nabla\Phi_{0}=\nabla\left(G*_{\Gamma}(\rho_{0}-m_{0})\right)$
weakly in $(L^{2}(\Gamma))^{3}$.

As a consequence

${\mathcal{E}}_{\rm per}^{m_{0}}(\rho_{0})\leqslant\liminf I_{N}.$

###### Proof.

The sequence $(\sqrt{\rho_{N}})_{N}$ is uniformly bounded with respect to $N$
in $H^{1}(\Gamma)$ by Proposition 3.4. Then, up to a subsequence, it converges
to some non negative function $u_{0}$ weakly in $H^{1}(\Gamma)$, strongly in
$L^{p}_{loc}(\Gamma)$ for all $2\leqslant p\leqslant 6$ and almost everywhere
on $\mathbb{R}^{3}$. Besides, by Theorem 1.2, there exists $C\geqslant 0$ such
that for any $N$ and any $x\in{\mathbb{R}}^{3}$ such that
$\left|x_{3}\right|\geqslant 1$, it holds

$\left|u_{N}(x)\right|\leqslant\frac{C}{1+\left|x_{3}\right|^{3/2}}.$

By the almost everywhere convergence, $u_{0}$ satisfies the same estimate, and
we have for $\rho_{0}:=u_{0}^{2}$ the following estimate

$\left|x_{3}\right|\rho_{0}(x)\leqslant\rho_{0}(x)1_{\left|x_{3}\right|\leqslant
1}+\frac{C}{1+\left|x_{3}\right|^{3}}\in L^{1}(\Gamma).$

Thus $\rho_{0}\in L^{1}(\Gamma)\cap L^{5/3}(\Gamma)$ and
$\left|x_{3}\right|\rho_{0}\in L^{1}(\Gamma)$.

We show now that $\rho_{N}$ converges to $\rho_{0}$ strongly in
$L^{1}(\Gamma)$. Let $\varepsilon>0$ and $R$ large enough such that for any
$M,N\in{\mathbb{N}}\setminus\\{0\\}$

$\displaystyle\int_{\left|x_{3}\right|>R}\left|\rho_{N}-\rho_{M}\right|\leqslant\frac{1}{R}\int_{\left|x_{3}\right|>R}\left|x_{3}\right|\left|\rho_{N}-\rho_{M}\right|\leqslant\frac{C}{R}\leqslant\frac{\varepsilon}{2}.$

By the strong convergence of $\rho_{N}$ in $L^{1}_{\rm loc}(\Gamma)$, for
$M,N$ large enough, we have

$\int_{\left|x_{3}\right|<R}\left|\rho_{N}-\rho_{M}\right|\leqslant\frac{\varepsilon}{2},$

which proves that $(\rho_{N})$ is a Cauchy sequence. Besides,
$(\nabla\Phi_{N})$ is uniformly bounded in $L^{2}(\Gamma)$. Thus, it weakly
converges, up to a subsequence, to $W\in\left(L^{2}(\Gamma)\right)^{3}$. Since
$-\Delta\Phi_{N}=4\pi\left(\rho_{N}-m_{N}\right)$, then,
$\forall\varphi\in{\mathcal{D}}(\Gamma)$, we have

$\int_{\Gamma}\nabla\Phi_{N}\nabla\varphi=4\pi\int_{\Gamma}\left(\rho_{N}-m_{N}\right)\varphi.$
(14)

By definition of the weak limit, the LHS of (14) converges to
$\int_{\Gamma}W\nabla\varphi$, and since $\rho_{N}-m_{N}$ weakly converges to
$\rho_{0}-m_{0}$ in $L^{1}(\Gamma)$, then

$\forall\varphi\in{\mathcal{D}}(\Gamma)\quad\int_{\Gamma}W\nabla\varphi=4\pi\int_{\Gamma}\left(\rho_{0}-m_{0}\right)\varphi.$

Therefore ${\rm div\;}W=4\pi(\rho_{0}-m_{0})$. By taking the mixed Fourier
transform defined in [3, Eq. 3.13] , we get for $\underline{R}$ in the dual
lattice ${\mathcal{R}}^{*}$ and $\eta\in{\mathbb{R}}$

$2i\pi\left(\begin{array}[]{c}R_{1}\\\ R_{2}\\\
\eta\end{array}\right)\cdot\widehat{W}(R,\eta)=(\widehat{\rho_{0}-m_{0}})(R,\eta).$

For $\underline{R}=0$ and $\eta=0$, we obtain
$\displaystyle\int_{\Gamma}\rho_{0}-m_{0}=0$. It follows that
$\rho_{0}-m_{0}\in Y_{\rm per}$. Thus by Proposition 2.2,
$\Phi_{0}:=G*_{\Gamma}(\rho_{0}-m_{0})$ is well defined. Finally, we show that
$\nabla\Phi_{0}=W$. Let
$\varphi=(\varphi_{1},\varphi_{2},\varphi_{3})\in{\mathcal{D}}(\Gamma)^{3}$.
We have ${\rm div\;}\varphi\in Y_{\rm per}$, thus $G*_{\Gamma}({\rm
div\;}\varphi)\in L^{\infty}(\Gamma)$. Therefore

$\displaystyle\langle\nabla G*_{\Gamma}(\rho_{N}-m_{N}),\varphi\rangle$
$\displaystyle=\langle\rho_{N}-m_{N},G*_{\Gamma}{\rm div\;}\varphi\rangle$
$\displaystyle\to\langle\rho_{0}-m_{0},G*_{\Gamma}{\rm
div\;}\varphi\rangle=\langle\nabla
G*_{\Gamma}(\rho_{0}-m_{0}),\varphi\rangle.$

∎

#### Step 3: Identification of the limit

We show in this section that $\rho_{0}$ is invariant with respect to
$(x_{1},x_{2})$ and it is indeed the unique minimizer of
${\mathcal{E}}_{1}^{m_{0}}$. This will prove that the convergences in
Proposition 3.5 hold for the whole sequence and prove points ii-iii of Theorem
1.5.

We start by showing that $\rho_{0}$ is the unique minimizer of
${\mathcal{E}}^{m_{0}}_{\rm per}$.

######  Proposition 3.6.

$\rho_{0}$ is the unique minimizer of ${\mathcal{E}}^{m_{0}}_{\rm per}$.

###### Proof.

Let $\rho\in\left\\{\rho\geqslant 0,\;\sqrt{\rho}\in X_{\rm
per},\;\int_{\Gamma}\rho=\int_{\Gamma}m_{0}\right\\}$ an admissible test
function for ${\mathcal{E}}^{m_{0}}_{\rm per}$. Let us show that

${\mathcal{E}}_{\rm per}^{m_{0}}(\rho_{0})\leqslant{\mathcal{E}}_{\rm
per}^{m_{0}}(\rho).$

As $\int_{\Gamma}m_{0}=\int_{\Gamma}m_{N}$, $\rho$ is also an admissible test
function for ${\mathcal{E}}_{\rm per}^{m_{N}}$. Thus

$I_{N}={\mathcal{E}}_{\rm per}^{m_{N}}(\rho_{N})\leqslant{\mathcal{E}}_{\rm
per}^{m_{N}}(\rho).$

We know from Proposition 3.5 that

${\mathcal{E}}^{m_{0}}_{\rm per}(\rho_{0})\leqslant\liminf
I_{N}\leqslant\liminf{\mathcal{E}}_{\rm per}^{m_{N}}(\rho).$

It thus remains to show that ${\mathcal{E}}_{\rm
per}^{m_{N}}(\rho)\to{\mathcal{E}}_{\rm per}^{m_{0}}(\rho)$, which boils down
to showing that

$D_{G}(\rho-m_{N},\rho-m_{N})\to D_{G}(\rho-m_{0},\rho-m_{0}).$ (15)

We recall that

$D_{G}(f,g)=\int_{\Gamma\times\Gamma}f(x)f(y)G(x-y)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y,$

with $G(x)=-2\pi\left|x_{3}\right|+\frac{1}{\left|x\right|}+\psi(x)$, $\psi\in
L^{\infty}(\Gamma)$. We denote by $h_{N}=\rho-m_{N}$ and $h=\rho-m_{0}$. We
recall as well that for any $x_{3}\in{\mathbb{R}}$ $h_{N}(\cdot,x_{3})$
converges to $h(\cdot,x_{3})$ weakly in $L^{p}(Q)$, $\forall 1\leqslant
p<{\infty}$. Therefore, for any $x_{3},y_{3}\in{\mathbb{R}}$,
$h_{N}(\cdot,x_{3})h_{N}(\cdot,y_{3})$ converges to
$h(\cdot,x_{3})h(\cdot,y_{3})$ weakly in $L^{p}(Q\times Q)$. We have
$-2\pi\left|x_{3}-y_{3}\right|+\psi(x-y)\in L^{\infty}(Q\times Q)$, and
$\frac{1}{\left|x-y\right|}\in L^{p}(Q\times Q)$ for any $1\leqslant p<2$.
Therefore, for any $x_{3},y_{3}\in{\mathbb{R}}$

$\int_{Q\times
Q}h_{N}(x)h_{N}(y)G(x-y)\mathop{}\\!\mathrm{d}\underline{x}\mathop{}\\!\mathrm{d}\underline{y}\to\int_{Q\times
Q}h(x)h(y)G(x-y)\mathop{}\\!\mathrm{d}\underline{x}\mathop{}\\!\mathrm{d}\underline{y}.$
(16)

To use a dominated convergence argument, we need to bound the LHS of (16)
uniformly with respect to $N$ by an $L^{1}$ function with respect to
$(x_{3},y_{3})$. We start with the term with $\psi$. We have

$\left|\int_{Q\times
Q}h_{N}(x)h_{N}(y)\psi(x-y)\mathop{}\\!\mathrm{d}\underline{x}\mathop{}\\!\mathrm{d}\underline{y}\right|\leqslant\left\|\psi\right\|_{L^{\infty}(\Gamma)}\int_{Q}\left|h_{N}(x)\right|\mathop{}\\!\mathrm{d}\underline{x}\int_{Q}\left|h_{N}(y)\right|\mathop{}\\!\mathrm{d}\underline{y},$

where

$\int_{Q}\left|h_{N}\right|\mathop{}\\!\mathrm{d}\underline{x}\leqslant\int_{Q}{\rho}\mathop{}\\!\mathrm{d}\underline{x}+\int_{Q}{m_{N}}\mathop{}\\!\mathrm{d}\underline{x}=\int_{Q}\rho\mathop{}\\!\mathrm{d}\underline{x}+\int_{Q}{m}\mathop{}\\!\mathrm{d}\underline{x}.$
(17)

The RHS of (17) is indeed an $L^{1}$ function wrt to $x_{3}$ and $y_{3}$. We
move to the term with $\left|x_{3}\right|$. We have

$\displaystyle\left|\int_{Q\times
Q}h_{N}(x)h_{N}(y)\left|x_{3}-y_{3}\right|\mathop{}\\!\mathrm{d}\underline{x}\mathop{}\\!\mathrm{d}\underline{y}\right|$
$\displaystyle\leqslant\int_{Q}\left|x_{3}\,h_{N}(x)\right|\mathop{}\\!\mathrm{d}\underline{x}\int_{Q}\left|h_{N}(y)\right|\mathop{}\\!\mathrm{d}\underline{y}+\int_{Q}\left|y_{3}\,h_{N}(y)\right|\mathop{}\\!\mathrm{d}\underline{y}\int_{Q}\left|h_{N}(x)\right|\mathop{}\\!\mathrm{d}\underline{x}$
$\displaystyle\leqslant\left(\int_{Q}\left|x_{3}\right|\left(\rho(x)+m(x)\right)\mathop{}\\!\mathrm{d}\underline{x}\right)\left(\int_{Q}\left(\rho(y)+m(y)\right)\mathop{}\\!\mathrm{d}\underline{y}\right)$
$\displaystyle\quad+\left(\int_{Q}\left|y_{3}\right|\left(\rho(y)+m(y)\right)\mathop{}\\!\mathrm{d}\underline{y}\right)\left(\int_{Q}\left(\rho(x)+m(x)\right)\mathop{}\\!\mathrm{d}\underline{x}\right).$
(18)

Again, the RHS of (3) is an $L^{1}$ function wrt $x_{3}$ and $y_{3}$. Finally,
for the term with $\frac{1}{\left|x\right|}$, we split the term as follows

$\displaystyle\int_{Q\times
Q}h_{N}(x)h_{N}(y)\frac{1}{\left|x-y\right|}\mathop{}\\!\mathrm{d}\underline{x}\mathop{}\\!\mathrm{d}\underline{y}$
$\displaystyle=\int_{Q\times
Q}\rho(x)\left(\rho(y)-2m_{N}(y)\right)\frac{1}{\left|x-y\right|}\mathop{}\\!\mathrm{d}\underline{x}\mathop{}\\!\mathrm{d}\underline{y}$
$\displaystyle+\int_{Q\times
Q}m_{N}(x)m_{N}(y)\frac{1}{\left|x-y\right|}\mathop{}\\!\mathrm{d}\underline{x}\mathop{}\\!\mathrm{d}\underline{y}.$

The first term is bounded by

$\displaystyle\left|\int_{Q\times
Q}\rho(x)\left(\rho(y)-2m_{N}(y)\right)\frac{1}{\left|x-y\right|}\mathop{}\\!\mathrm{d}\underline{x}\mathop{}\\!\mathrm{d}\underline{y}\right|$
$\displaystyle\leqslant\left\|\rho*\frac{1}{\left|x\right|}\right\|_{L^{\infty}(\Gamma)}\left\|\rho+2m(\cdot,x_{3})\right\|_{L^{1}(Q)}$

which is an $L^{1}$ function with respect to $x_{3}$. For the second term we
use Young inequality with $p=q=r=3/2$. We obtain

$\displaystyle\left|\int_{Q\times
Q}m_{N}(x)m_{N}(y)\frac{1}{\left|x-y\right|}\mathop{}\\!\mathrm{d}\underline{x}\mathop{}\\!\mathrm{d}\underline{y}\right|$
$\displaystyle\leqslant\left\|m_{N}(\cdot,x_{3})\right\|_{L^{3/2}(Q)}\left\|m_{N}(\cdot,y_{3})\right\|_{L^{3/2}(Q)}\left\|\frac{1}{\left|x\right|}\right\|_{L^{3/2}(Q)}$
$\displaystyle\leqslant\left\|m(\cdot,x_{3})\right\|_{L^{3/2}(Q)}\left\|m(\cdot,y_{3})\right\|_{L^{3/2}(Q)}\times\left\|\frac{1}{\left|\underline{x}\right|}\right\|_{L^{3/2}(Q)}.$
(19)

As $m$ is compactly supported in $x_{3}$,
$x_{3}\mapsto\left\|m(\cdot,x_{3})\right\|_{L^{3/2}(Q)}$ is also compactly
supported and it is an $L^{3/2}$ function, thus, it is an $L^{1}$; which
concludes the proof. ∎

######  Proposition 3.7.

The density $\rho_{0}$ is invariant wrt $(x_{1},x_{2})$, it is indeed the
unique minimizer of the 1D model ${\mathcal{E}}_{1}^{m_{0}}$ and

$I_{0}={\mathcal{E}}_{\rm
per}^{m_{0}}(\rho_{0})={\mathcal{E}}_{1}^{m_{0}}(\rho_{0}).$

###### Proof.

For any $\underline{R}\in{\mathbb{R}}^{2}$, $\tau_{\underline{R}}\rho_{0}$ is
also a minimizer of ${\mathcal{E}}_{\rm per}^{m_{0}}$, where $\tau$ is the
translation operator. Thus, by the convexity of the functional
${\mathcal{E}}_{\rm per}^{m_{0}}$ and the uniqueness of its minimizer (see
Theorem 1.2), we deduce that $\rho_{0}$ is invariant with respect to
$(x_{1},x_{2})$. To conclude the proof of the proposition, we notice that for
any ${\mathbb{R}}^{2}$-invariant function $f$

$D_{G}(f,f)=D_{1}(f,f),$

as shown in [9, Propo. 3.1].

∎

#### Step 4: Convergence of the energy

We prove in this section point i of Theorem 1.5, which consists of the
convergence of the energy.

######  Proposition 3.8.

We have

$I_{0}=\lim I_{N}.$

###### Proof.

By Propositions 3.5 and 3.7, we have

$I_{0}\leqslant\liminf I_{N}.$

To prove the lower bound on $I_{0}$, we use $\rho_{0}$ as a test function for
${\mathcal{E}}_{\rm per}^{m_{N}}$ and obtain

$\begin{split}I_{N}\leqslant\int_{\Gamma}\left|\nabla\sqrt{\rho_{0}}\right|^{2}+\int_{\Gamma}\rho_{0}^{5/3}+D_{G}(\rho_{0}-m_{N},\rho_{0}-m_{N})\to{\mathcal{E}}_{\rm
per}^{m_{0}}(\rho_{0})=I_{0},\end{split}$

where the last convergence is obtained using (15). Thus

$\limsup I_{N}\leqslant I_{0},$

which concludes the proof of the proposition.

∎

##  4 Numerical illustration

In this section, we illustrate the convergence result of Theorem 1.5. We
recall that the minimization problem (6) has a unique minimizer $\rho$, and
$u=\sqrt{\rho}$ is the unique solution of the corresponding Euler-Lagrange
equations

$\left\\{\begin{array}[]{rll}\Delta
u+\frac{5}{3}u^{\frac{7}{3}}+u\Phi&=&-\lambda u,\\\
-\Delta\Phi&=&4\pi(u^{2}-\mu).\end{array}\right.$ (20)

We solve the 3D periodic Euler-Lagrange system in (20) by coupling a spectral
approach and a fixed point iterative scheme111A python implementation of the
solution procedure can be found at https://github.com/sa3dben/TFW_model. We
adopt the following iterative fixed point scheme : starting form an arbitrary
initial guess $u_{0}$, at each fixed point iteration, a linear eigenvalue
problem is solved

$\left(\Delta+\frac{5}{3}u_{n}^{4/3}+\Phi_{n}\right)u_{n+1}=-\lambda_{1}u_{n+1},$
(21)

with $\Phi_{n}$ the solution to the Poisson equation
$-\Delta\Phi_{n}=4\pi(u_{n}^{2}-\mu)$. The fixed point iterations are
continued until convergence, i.e.
$\left\|u_{n}-u_{n+1}\right\|_{2}\leqslant\varepsilon$ for some tolerance
$\varepsilon>0$. We take $\varepsilon=10^{-6}$ in our simulations.

We adopt a periodic setting, with the unit cell
$\textstyle\Gamma=Q\times[-\frac{L}{2},\frac{L}{2}]$, with
$Q=\left[-\frac{1}{2},\frac{1}{2}\right]^{2}$. We use a Fourier series
decomposition approach to solve the linearized equation (21). The solution is
expected to be ${\mathcal{R}}-$periodic with respect to $x_{1}$ and $x_{2}$
and decaying at infinity with respect to $|x_{3}|$. Taking a large value for
$L$, it is reasonable to impose period conditions in the third direction as
well. Therefore, we consider the Fourier basis

$f_{k}(x)=\frac{1}{\sqrt{L}}e^{i2\pi\left(\underline{x}\cdot\underline{k}+\frac{k_{3}x_{3}}{L}\right)}.$

We illustrate the convergence results in Theorem 1.5 using the series of
nuclei densities $m_{N}$ defined as

$m_{N}(x_{1},x_{2},x_{3})=\mu(N\cdot\,x_{1})m_{0}(x_{3})=\frac{\pi}{2}\left|\cos(N\pi
x_{1})\right|\times 5\exp\left(-\frac{x_{3}^{2}}{8}\right).$

We note that $\int_{Q}\mu(N\cdot
x_{1})\,\rm{d}x_{1}\,\rm{d}x_{2}=\frac{\pi}{2}\int_{Q}\left|\cos(N\cdot\pi
x_{1})\right|=1$. We take $L=2\pi$, which proves effective as the function
$m_{0}(x_{3})=5\exp\left(-\frac{x_{3}^{2}}{8}\right)$ rapidly decreases in the
$x_{3}$ direction. The same behaviour is assumed for the solutions $u_{N}$,
and is validated with 1D simulations for $(u_{0},\rho_{0})$, the 1D solution
corresponding to $m_{0}$.

The different functions are sampled on a grid of dimensions $(200\times
N,4,300)$. We keep the discretization constant along the $y$ and $z$-axis as
the homogenization is only applied in the $x_{3}$ dimension. As we will only
compute the zero Fourier mode along $x_{2}$, four points are largely enough.
We compute the positive Fourier modes up to indices
$K=(K_{1},K_{2},K_{3})=(4\times N,0,6)$. On the $x_{3}$-axis, we use the first
$7$ Fourier modes, found to be sufficient for accurate reconstruction of
$m_{N}$ and $m_{0}$ along that axis. Simulations in 1D prove that the solution
$(u_{0}(x_{3}),\rho_{0}(x_{3}))$, is also accurately represented by these $7$
modes. We assume that this is also the case in 3D for all values of $N$. The
choice of $(4N+1)$ Fourier modes in the $x_{1}$ direction, is motivated by the
need for increased Fourier modes as $N$ grows, to capture the changing
characteristics and new harmonics for $m_{N}$ and ’eventually’ for the
solution $(\rho_{N},u_{N})$. In fact, we know that $(\rho_{N},u_{N})$ will
rather converge to a constant along the axis $x_{1}$ by Theorem 1.5, however
we enforce a hard check for this by computing also large modes for $\rho_{N}$
and $u_{N}$ .

Different $L^{p}$-norms of the error $e_{N}=\rho_{N}-\rho_{0}$ are presented
in Figure 3 for values of $N=1,\ldots,5$. We, indeed, observe the convergence
stated in Theorem 1.5. The same figure shows some convergence rates estimates.
We conjecture that we have a theoretical convergence rate of $1/N^{r}$ with
$\frac{10}{3}<r<4$ for the different norms of $e_{n}$ and the gradient $L^{2}$
norm $\left\|\nabla{u_{n}}-\nabla{u_{0}}\right\|_{2}$. The discrepancies in
Figure 3 at $N=4,5$ are probably due to numerical errors.

Figure 3:  Convergence analysis for $e_{n}$ norms (left) and convergence rates
estimation for $e_{n}$ norms and the gradient $L^{2}$ norm
$\left\|\nabla{u_{n}}-\nabla{u_{0}}\right\|_{2}$ (right).

Figure 4 illustrates the convergence of the ground state energy
$I_{N}={\mathcal{E}}^{m_{N}}_{per}(\rho_{N})$ to the 1D energy
$I_{0}={\mathcal{E}}^{m_{0}}_{1}(\rho_{0})$, proved in Theorem 1.5. The
estimated convergence rate for the energy is of the order
$\approx\dfrac{1}{N^{3/2}}$.

Figure 4: Convergence analysis for the energies $I_{N}$ (left) and convergence
rate estimation (right).

######  Remark 4.1.

It is noteworthy to mention that, for $N\geqslant 6$, and using higher order
Fourier modes, the presence of numerical errors hinders a clear observation of
further convergence for $e_{N}$.

## References

  * [1] G. Allaire, Homogenization and two-scale convergence, SIAM Journal on Mathematical Analysis, 23 (1992), pp. 1482–1518.
  * [2] R. Benguria, H. Brezis, and E. Lieb, The Thomas-Fermi-von weizsäcker theory of atoms and molecules, Communications in Mathematical Physics, 79 (1981), pp. 167–180.
  * [3] X. Blanc and C. Le Bris, Thomas-fermi type theories for polymers and thin films, Advances in Differential Equations, 5 (2000), pp. 977–1032.
  * [4] X. Blanc and R. Monneau, Screening of an applied electric field inside a metallic layer described by the Thomas-Fermi-von Weizsäcker model, Advances in Differential Equations, 7 (2002), pp. 847 – 876.
  * [5] A. Carvalho, P. E. Trevisanutto, S. Taioli, and A. H. C. Neto, Computational methods for 2d materials modelling, Reports on Progress in Physics, 84 (2021), p. 106501.
  * [6] I. Catto, C. Le Bris, and P. L. Lions, Mathematical theory of thermodynamic limits : Thomas- Fermi type models, Oxford University Press, 1998.
  * [7] V. Chaudhary, P. Neugebauer, O. Mounkachi, S. Lahbabi, and A. E. Fatimy, Phosphorene—an emerging two-dimensional material: recent advances in synthesis, functionalization, and applications, 2D Materials, 9 (2022), p. 032001.
  * [8] A. Geim and I. Grigorieva, Van der Waals heterostructures, Nature, 499 (2013), pp. 419–425.
  * [9] D. Gontier, S. Lahbabi, and A. Maichine, Density functional theory for two-dimensional homogeneous materials, Communications in Mathematical Physics, 388 (2021), pp. 1475–1505.
  * [10] D. Gupta, V. Chauhan, and R. Kumar, A comprehensive review on synthesis and applications of molybdenum disulfide MoS2 material: Past and recent developments, Inorganic Chemistry Communications, 121 (2020), p. 108200.
  * [11] F. Hussain, M. Imran, and H. Ullah, Density Functional Theory (DFT) Study of Novel 2D and 3D Materials, Springer Singapore, Singapore, 2017, pp. 269–284.
  * [12] E. H. Lieb and B. Simon, The Thomas-Fermi theory of atoms, molecules and solids, Advances in Mathematics, 23 (1977), pp. 22–116.
  * [13] D. Lukkassen and P. Wall, On weak convergence of locally periodic functions, Journal of Nonlinear Mathematical Physics, 9 (2002), pp. 42–57.
  * [14] A. Patra, S. Jana, P. Samal, F. Tran, L. Kalantari, J. Doumont, and P. Blaha, Efficient band structure calculation of two-dimensional materials from semilocal density functionals, The Journal of Physical Chemistry C, 125 (2021), p. 11206–11215.
  * [15] K. Ren, M. Sun, Y. Luo, S. Wang, J. Yu, and W. Tang, First-principle study of electronic and optical properties of two-dimensional materials-based heterostructures based on transition metal dichalcogenides and boron phosphide, Applied Surface Science, 476 (2019), pp. 70–75.
  * [16] J. Saha and A. Dutta, A review of graphene: Material synthesis from biomass sources, Waste and Biomass Valorization, 13 (2022), p. 1385–1429.
  * [17] Q. Tang, Z. Zhou, and Z. Chen, Innovation and discovery of graphene-like materials via density-functional theory computations, WIREs Comput Mol Sci, 5 (2015), pp. 360–379.
  * [18] S. Tawfik, O. Isayev, C. Stampfl, J. Shapter, D. Winkler, and M. J. Ford, Efficient prediction of structural and electronic properties of hybrid 2d materials using complementary DFT and machine learning approaches, ChemRxiv, (2018).
  * [19] C. F. v. Weizsäcker, Zur theorie der kernmassen, Zeitschrift für Physik, 96 (1935), pp. 431–458.


[affiliation=1]NanZhou [affiliation=1]YouhaiJiang [affiliation=1]JialinTan
[affiliation=1]ChongminQi

# PLDNet: PLD-Guided Lightweight Deep Network Boosted by Efficient Attention
for Handheld Dual-Microphone Speech Enhancement

###### Abstract

Low-complexity speech enhancement on mobile phones is crucial in the era of
5G. Thus, focusing on handheld mobile phone communication scenario, based on
power level difference (PLD) algorithm and lightweight U-Net, we propose PLD-
guided lightweight deep network (PLDNet), an extremely lightweight dual-
microphone speech enhancement method that integrates the guidance of signal
processing algorithm and lightweight attention-augmented U-Net. For the
guidance information, we employ PLD algorithm to pre-process dual-microphone
spectrum, and feed the output into subsequent deep neural network, which
utilizes a lightweight U-Net with our proposed gated convolution augmented
frequency attention (GCAFA) module to extract desired clean speech.
Experimental results demonstrate that our proposed method achieves competitive
performance with recent top-performing models while reducing computational
cost by over 90%, highlighting the potential for low-complexity speech
enhancement on mobile phones.

###### keywords:

low complexity, power level difference, attention, dual microphone, speech
enhancement

##  1 Introduction

Noise reduction represents a thoroughly explored domain within signal
processing, characterized by a rich tapestry of prior research. Numerous
conventional monaural methods have been developed over the years, utilizing
robust noise estimation models [1, 2, 3]. However, dual-microphone speech
enhancement methods demonstrate greater robustness in addressing transient
noise and interference concurrently with stationary noise. Thus, they have
attracted significant research attention. For example, researchers have
proposed and developed power level difference (PLD) algorithms, which leverage
the power variance of received dual-microphone signals to mitigate both
stationary and non-stationary noise [4, 5, 6, 7]. Additionally, coherence
function based approaches have proven popular for suppressing noise and
interference [8, 9]. Other widely used techniques in dual-microphone speech
enhancement include beamforming-based methods such as minimum variance
distortionless response (MVDR) [10, 11] and generalized sidelobe canceler
(GSC) [12].

Over the past decade, there has been a proliferation of deep learning based
monaural speech enhancement algorithms [13, 14, 15, 16, 17] which have
achieved state-of-the-art performance. For dual-microphone speech enhancement,
Tan et al. explored a real-time approach using densely-connected convolutional
recurrent network (DC-CRN) and obtained notable results [18]. Subsequently, Xu
et al. further advanced the field by utilizing multi-head cross-attention
(MHCA) mechanism [19]. Additionally, we observe that the multi-scale temporal
frequency convolutional network with axial attention (MTFAA-Net) demonstrated
remarkable performance in the L3DAS22 3D speech enhancement challenge [17].

Recently, there has been a trend to combine signal processing and deep
learning approaches for speech enhancement. For example, an adaptive linear
filter is often placed before a deep neural network (DNN) to perform acoustic
echo cancellation [17]. Additionally, a hybrid approach integrating a Kalman
filter and a recurrent neural network (RNN) was proposed by [20] and
demonstrated greatly improved echo cancellation performance with low
computational cost. Furthermore, an iterative refinement approach combining a
DNN and a multi-frame multi-channel Wiener filter (mfMCWF) achieved first
place in the L3DAS22 3D speech enhancement challenge [21]. Compared to pure
deep learning approaches [17, 22], these hybrid techniques can promise better
or competitive performance with reduced computational complexity, owing to the
integration of signal processing priors.

In this study, for the handheld mobile phone communication scenario, we
propose a low-complexity dual-microphone speech enhancement method by
integrating signal processing and deep learning approaches. For this
application scenario, the vocal component captured by the primary microphone
is substantially stronger than that of the secondary microphone. For the
signal processing guidance, we employ a PLD-based algorithm to pre-process the
dual-microphone spectrum. We then utilize a lightweight U-Net with our
proposed gated convolution augmented frequency attention (GCAFA) module to
extract the desired clean speech. Experimental results demonstrate that our
proposed method can achieve competitive performance with recent top-performing
models while reducing computational cost by over 90%. This highlights the
potential of PLDNet for low-complexity speech enhancement on mobile phones.

##  2 PLDNet

Figure 1: Diagram of PLDNet.

###  2.1 Signal Model

We define the signal model in the time-frequency (T-F) domain as:

$\begin{aligned} Y_{m}(k,\ell)&=X_{m}(k,\ell)+V_{m}(k,\ell)+U_{m}(k,\ell)\\\
&=X_{m}(k,\ell)+N_{m}(k,\ell),(m=1,2)\end{aligned}$

(1)

where $Y_{m}(k,\ell)$, $X_{m}(k,\ell)$ and $N_{m}(k,\ell)$ are spectral
coefficients of received, desired and undesired signals, respectively.
$V_{m}(k,\ell)$ and $U_{m}(k,\ell)$ are stationary noise signal and transient
interference signal. Furthermore, $k$ and $\ell$ are the frequency and time
indices, whereas $m=1$ corresponds to the primary microphone and $m=2$ to the
secondary microphone.

###  2.2 PLD-based Pre-processing

Drawing conceptual inspiration from GSC postfiltering algorithm [23], our PLD-
based pre-processing algorithm consists of two main components: a stationary
noise estimator [3] and a dual-channel collaborative noise reduction module.
After obtaining an estimation for the stationary noise power spectrum
$\lambda_{m}(k,\ell)$ using the noise estimation algorithm [3], we calculate
the posterior SNR $\gamma_{m}(k,\ell)$ of each microphone as:

$\gamma_{m}(k,\ell)=\frac{\left|Y_{m}(k,\ell)\right|^{2}}{\lambda_{m}(k,\ell)}$
(2)

Then we compute the ratio between the energy of the voice signals obtained
from two microphones as:

$\kappa(k,\ell)=\frac{{\left|Y_{1}(k,\ell)\right|^{2}}-{\lambda_{1}(k,\ell)}}{{\left|Y_{2}(k,\ell)\right|^{2}}-{\lambda_{2}(k,\ell)}}$
(3)

The speech presence probability (SPP) is defined as:

$\begin{aligned} \psi(k,\ell)=\begin{cases}1,&\text{if}\
\gamma_{1}(k,\ell)>\gamma_{\text{thresh}}\ \text{and}\\\ &\
\kappa(k,\ell)>\kappa_{\text{high}}\\\
\dfrac{\kappa(k,\ell)-\kappa_{\text{low}}}{\kappa_{\text{high}}-\kappa_{\text{low}}},&\text{if}\
\gamma_{1}(k,\ell)>\gamma_{\text{thresh}}\ \text{and}\\\ &\
\kappa_{\text{low}}<\kappa(k,\ell)<\kappa_{\text{high}},\\\
0,&\text{otherwise}\end{cases}\end{aligned}$

(4)

Table 1: Values of the constants used in the PLD-based pre-processing
algorithm.

$k_{low}=8$ | $k_{high}=113$ | $\gamma_{thresh}=1.69$ | $\tilde{\psi}_{thresh}=0.25$
---|---|---|---
$\kappa_{low}=1.5$ | $\kappa_{high}=3$ | $\gamma_{low}=1$ | $\gamma_{high}=4.6$

Global SPP $\tilde{\psi}(\ell)$ is defined as:

$\tilde{\psi}(\ell)=\dfrac{1}{k_{high}-k_{low}+1}\sum_{k=k_{low}}^{k_{high}}\psi(k,\ell)$
(5)

Then $\gamma_{1}(k,\ell)$, $\psi(k,\ell)$ and $\tilde{\psi}(\ell)$ are used to
compute the a posterior signal absence probability $\hat{q}(k,\ell)$ defined
as:

$\hat{q}(k,\ell)=\left\\{\begin{array}[]{l}1,\ \ \ \ \text{if}\
\gamma_{1}(k,\ell)\leqslant\gamma_{\text{low }}\text{or}\ \
\tilde{\psi}(\ell)\leqslant\tilde{\psi}_{thresh},\\\
\max(\dfrac{\gamma_{\text{high }}-\gamma_{1}(k,\ell)}{\gamma_{\text{high
}}-\gamma_{\text{low }}},1-\psi(k,\ell)),\text{ otherwise }\end{array}\right.$

(6)

where $\tilde{\psi}_{thresh}$ is a predetermined threshold.

Then gain function $G(k,\ell)$ is obtained using the optimally modified log-
spectral amplitude (OMLSA) [2] method. It is subsequently applied to the
primary microphone spectrum as:

$\hat{X}_{\mathrm{PLD}}(k,\ell)=G(k,\ell)Y_{1}(k,\ell)$ (7)

The resulting predicted target speech spectrum
$\hat{X}_{\mathrm{PLD}}(k,\ell)$ is then utilized as a guidance feature for
subsequent speech enhancement model. The constants employed in the PLD-based
pre-processing algorithm are presented in Table 1.

###  2.3 GCAFA-Boosted Lightweight U-Net

####  2.3.1 Main structure

In this study, we leverage the speech enhancement capability of multi-scale
multi-resolution U-Net architecture [15, 17, 18] along with our proposed GCAFA
module. Specifically, we utilize the same main structure as MTFAA-Net [17],
which represents the state-of-the-art real-time monaural speech enhancement
model.

As depicted in Figure 1, the Phase Encoder (PE) initiates the process by
extracting complex information from three inputs, producing real-valued
feature maps. This is enhanced by the inclusion of spectrum from the PLD pre-
processing technique, which assists in learning directional cues. The
architecture further comprises layers for downsampling convolution (DC) and
upsampling convolution (UC), Temporal Frequency Convolution Modules (TFCMs),
the novel GCAFA modules, and a deep filter-based Mask Estimating and Applying
(MEA) module, interconnected by skip connections for feature integration. Each
encoder block starts with a DC layer, followed by a TFCM and a GCAFA module
for comprehensive temporal and frequency analysis. Following the last encoder
block, two backbone blocks incorporate TFCMs and GCAFA modules. Each Decoder
blcok contains a UC layer, a TFCM, a GCAFA module, and a skip connection. The
MEA module employs a 2-stage approach to refine speech masks, initially
leveraging a deep filter to harness adjacent frequency and temporal
information for magnitude mask estimation, subsequently enhancing phase mask
precision to uplift speech quality. Further insights into the PE, DC, UC,
TFCM, and MEA components are available in [17], with our specific parameters
outlined in Section 3.1. The GCAFA module is elaborated in the subsequent
section.

####  2.3.2 GCAFA

Figure 2: Flow diagrams of projection block (a), gated convolution block (b),
and GCAFA (c).

GCAFA module integrates a low-complexity frequency attention mechanism,
specifically designed to augment the model's capacity for speech processing. A
pivotal observation underpinning our design is that while multi-scale dilated
convolution blocks proficiently handle extended temporal contexts, they are
inherently limited in addressing frequency-related nuances due to their
constrained frequency kernel sizes.

To mitigate this limitation, the GCAFA module is engineered to adeptly capture
and interpret global frequency dependencies. As illustrated in Figure 2, it
incorporates a single-head self-attention mechanism that operates specifically
along the frequency axis, ensuring a focused and effective analysis of
frequency components. Additionally, pointwise convolution layers are utilized
in both the input and output stages to encourage information exchange between
channels. Finally, a gating mechanism is strategically implemented to regulate
the flow of information.

The operational workflow of the GCAFA module begins with the input feature
map, denoted as $D\in\mathbb{R}^{B\times C\times F\times T}$. This feature map
undergoes initial processing in a gated convolution block, which projects it
onto a dimension of $3\times C^{\prime}$ channels. The resultant feature map
is then partitioned into three segments along the channel dimension. At this
juncture, the single-head self-attention mechanism comes into play, operating
along the frequency axis to refine the frequency-specific features. Following
the attention mechanism, the feature map's channel number is restored to $C$
through a projection block. Finally, The process concludes with another gated
convolution block. Notably, two residual connections are incorporated into the
GCAFA module to facilitate training process and ensure stable convergence.

####  2.3.3 Loss function

We use hybrid loss function combing waveform and spectral loss functions. The
waveform loss function is defined as:

$L_{\mathrm{wave}}=\frac{\sum_{i=1}^{N}\left|\hat{x}_{1}(i)-x_{1}(i)\right|}{\sum_{i=1}^{N}\left|x_{1}(i)\right|}$
(8)

where $N$ is the number of samples in the waveform, $\hat{x}_{1}(i)$ denotes
the predicted waveform and $x_{1}(i)$ denotes the ground truth waveform. The
spectral loss function is defined as:

$\begin{aligned} L_{\mathrm{spec}}=\sum_{K\in
2^{6},\ldots,2^{11}}(&\frac{\sum_{k}\sum_{\ell}||\hat{X}_{1}(k,\ell)-X_{1}(k,\ell)||^{2}}{\sum_{k}\sum_{\ell}||X_{1}(k,\ell)||^{2}}+\\\
&\sum_{k}\sum_{\ell}\frac{||\hat{X}_{1}(k,\ell)-X_{1}(k,\ell)||^{2}}{||X_{1}(k,\ell)||^{2}})\end{aligned}$

(9)

where $\hat{X}_{1}(k,\ell)$ and $X_{1}(k,\ell)$ denotes the predicted and
ground truth spectra, $K$ is the frequency resolution.

The final loss function is defined as:

$L=L_{\mathrm{wave}}+\alpha L_{\mathrm{spec}}$ (10)

where $\alpha$ is the weight of $L_{\mathrm{spec}}$.

##  3 Experiments

###  3.1 Experimental Setup

In this study, we utilize three datasets for our experimental analysis,
including VCTK [24] for source speakers, LibriSpeech [25] for interference
speakers and WHAM! [26] for noise signals. From the VCTK dataset, we use 99,
5, and 5 speakers for training, validation, and testing, respectively. For the
LibriSpeech dataset, we use the train-clean-100, dev-clean and test-clean
subsets for training, validation, and testing, respectively. For the WHAM!
dataset, we employ 58, 10, and 9 hours of data for training, validation, and
testing, respectively.

To generate simulated signals, similar to [18], we simulate a rectangular room
of dimension $10\times 7\times 3\ \mathrm{m}^{3}$, with the target speech
source located at the room's center. The primary microphone is placed on the
same horizontal plane with the source, with a distance ranging from 2 to 5 cm.
The secondary microphone is placed above the primary microphone, with a
distance of 15 cm between the two microphones and a zenith angle ranging from
$0^{\circ}$ to $15^{\circ}$. Furthermore, we generate all room impulse
responses (RIRs) using image method, with RT60 values ranging from 0.2 to 0.5
s.

To simulate babble interference, similar to [18], we randomly select 72 speech
clips form 72 different speakers in LibriSpeech dataset, and place them on a
horizontal circle centered at the primary microphone, with a radius of 3 m. To
simulate diffuse noise, we choose 2 noise clips from WHAM! dataset and mix
them up using the method detailed in [27]. Subsequently, we create all
mixtures with SNR and SIR ranging from 0 to 20 dB and volume levels ranging
from -40 to -10 dB with a sampling rate of 16 kHz.

Table 2: Performance comparison of different speech enhancement models and
ablation study of PLDNet on simulated test dataset. The best results of all
models are highlighted in bold. The best results of PLDNet are highlighted in
bold italics.

Model | #Params (M) | FLOPs/s (G) | SI-SDR | NB-PESQ | WB-PESQ | STOI | SIG | BAK | OVRL
---|---|---|---|---|---|---|---|---|---
Unprocessed | - | - | 7.325 | 2.364 | 1.333 | 0.880 | 2.735 | 2.082 | 1.962
PLD | - | 0.002 | 12.643 | 2.743 | 1.805 | 0.885 | 3.130 | 3.203 | 2.529
MTFAA-Net (mic 1) [17] | 2.149 | 4.542 | 16.384 | 3.080 | 2.305 | 0.941 | 3.149 | 4.038 | 2.898
MTFAA-Net (2 mics) [17] | 2.149 | 4.549 | 19.107 | 3.262 | 2.603 | 0.963 | 3.186 | 4.034 | 2.926
DC-CRN [18] | 10.810 | 24.054 | 20.525 | 3.323 | 2.764 | 0.967 | 3.270 | 4.042 | 2.999
MHCA-CRN [19] | 50.553 | 38.351 | 20.571 | 3.361 | 2.788 | 0.969 | 3.306 | 3.993 | 3.009
DC-CRN (scale-down) | 0.228 | 0.349 | 16.472 | 3.008 | 2.204 | 0.940 | 2.997 | 4.013 | 2.747
MHCA-CRN (scale-down) | 0.585 | 0.497 | 16.860 | 3.118 | 2.398 | 0.946 | 3.051 | 3.987 | 2.786
PLDNet | 0.155 | 0.312 | 19.219 | 3.294 | 2.653 | 0.962 | 3.203 | 4.004 | 2.925
\- w/o GCAFA | 0.115 | 0.199 | 17.949 | 3.274 | 2.658 | 0.959 | 3.148 | 4.018 | 2.885
\- w/o PLD spectrum | 0.115 | 0.194 | 17.879 | 3.192 | 2.501 | 0.958 | 3.144 | 4.014 | 2.881
\- w/o secondary microphone | 0.115 | 0.191 | 15.880 | 2.992 | 2.208 | 0.933 | 3.093 | 4.030 | 2.841

For the model settings, the PE's complex convolutional layer has 4 output
channels, followed by 3 encoder blocks each with a DC layer (kernel: (7, 1),
stride: (4, 1)), a TFCM with depth of 6, and a GCAFA module. In each GCAFA
module, $C^{\prime}$ is half of $C$. The output channels of 3 DC layers are
16, 24, and 40 across blocks. Two backbone blocks include dual 6-layer TFCMs
and a GCAFA module each. The decoder inversely mirrors the encoder, with MEA's
deep filter size set at (1, 3). Loss weight $\alpha$ is set at 1.0. STFT is
conducted wiht a 32 ms window and 16 ms hop. Training involves 60 epochs,
NovoGrad optimizer, cosine annealing scheduler, a 3e-3 initial learning rate,
and batch size of 16. Notably, PLDNet's unoptimized version exceeds 3x real-
time on an MT6789 ARM CPU, underscoring its real-time viability.

For comparison with recent SOTA dual-microphone speech enhancement models, we
reproduce MTFAA-Net [17], DC-CRN [18] and MHCA-CRN [19] on the same dataset.
To compare with lightweight models, we reduce the model sizes of DC-CRN and
MHCA-CRN to a fair level and train the models on the same dataset.

###  3.2 Results and Analysis

For performance evaluation, we adopt seven evaluation metrics, namely SI-SDR
[28], NB-PESQ, WB-PESQ [29], STOI [30], and SIG, BAK, OVRL using DNSMOS P.835
[31]. Additionally, we compute the number of parameters and FLOPs per second
for all models using DeepSpeed toolkit [32]. We compare the performance of
various recent models against PLDNet, and present an ablation study of PLDNet.
All models are causal versions for fair comparison.

The analysis summarized in Table 2 shows that evaluated speech enhancement
models outperform the PLD method in simulated environments, highlighting the
superiority of modern deep learning over traditional approaches. A comparison
between the monaural and binaural MTFAA-Net versions emphasizes the importance
of multi-microphone inputs, especially in complex auditory settings involving
multiple interfering speakers. An evaluation of MTFAA-Net, DC-CRN, and MHCA-
CRN reveals a link between model complexity and performance improvements, up
to a certain limit.

Moreover, PLDNet marginally exceeds MTFAA-Net's dual-microphone setup in
performance, utilizing less than 10% of its parameters and computational
resources. It also significantly surpasses scaled-down versions of DC-CRN and
MHCA-CRN, with its metrics closely approaching those of DC-CRN, despite DC-CRN
having roughly 70 times more parameters and FLOPs. MHCA-CRN, the largest model
compared, leads in performance but requires over 100 times the parameters and
FLOPs compared to PLDNet. This highlights the efficacy of combining signal
processing principles with a lightweight, attention-enhanced U-Net framework,
challenging much larger models. An ablation study further reveals the GCAFA
module's substantial improvement in metrics like SI-SDR, underlining its role
in enhancing model performance. Moreover, PLD spectrum guidance boosts NB-PESQ
and WB-PESQ scores, indicating superior speech quality. This suggests
significant performance gains from signal processing-based guidance without
heavy computational demands. A noticeable gap exists between dual-microphone
and monaural models, as seen in the final row. And it's pertinent to highlight
that it represents a version of monaural MTFAA-Net with 20 $\times$ fewer
FLOPs and devoid of the attention mechanism, which performs marginally
inferior to the original monaural MTFAA-Net. This underlines the challenges
monaural models face in dual-channel enhancement, particularly due to the lack
of spatial cues, leading to minimal differentiation in performance among them.

Figure 3: Spectrograms from ablation study of PLDNet on a noisy sample, the
corresponding NB-PESQ scores are displayed in parentheses.

Figure 3 presents spectrograms from an ablation study on PLDNet, applied to a
noisy speech sample, alongside their corresponding NB-PESQ scores in
parentheses. These results underscore PLDNet's effectiveness in minimizing
background noise and maintaining essential speech elements. A noticeable
difference in NB-PESQ scores between model versions with and without PLD
spectrum guidance highlights its significance. The monaural model variant,
however, exhibits considerable speech distortion, pointing out its
deficiencies in safeguarding speech clarity.

##  4 Conclusion

In this study, we introduce PLDNet, an innovative, low-complexity speech
enhancement approach that combines a PLD-based pre-processing technique with a
streamlined U-Net architecture, enhanced by our GCAFA module. Targeting noise
reduction in mobile phone communications, our method has shown to rival the
performance of larger models with significantly lower computational demands.
Future research will delve into merging signal processing with deep learning
and the feasibility of real-time mobile deployment.

## References

  * [1] R. Martin, ``Spectral subtraction based on minimum statistics,'' _power_ , vol. 6, no. 8, pp. 1182–1185, 1994.
  * [2] I. Cohen and B. Berdugo, ``Speech enhancement for non-stationary noise environments,'' _Signal processing_ , vol. 81, no. 11, pp. 2403–2418, 2001\.
  * [3] I. Cohen, ``Noise spectrum estimation in adverse environments: Improved minima controlled recursive averaging,'' _IEEE Transactions on speech and audio processing_ , vol. 11, no. 5, pp. 466–475, 2003.
  * [4] N. Yousefian, A. Akbari, and M. Rahmani, ``Using power level difference for near field dual-microphone speech enhancement,'' _Applied Acoustics_ , vol. 70, no. 11-12, pp. 1412–1421, 2009.
  * [5] M. Jeub, C. Herglotz, C. Nelke, C. Beaugeant, and P. Vary, ``Noise reduction for dual-microphone mobile phones exploiting power level differences,'' in _2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2012, pp. 1693–1696.
  * [6] J. Zhang, R. Xia, Z. Fu, J. Li, and Y. Yan, ``A fast two-microphone noise reduction algorithm based on power level ratio for mobile phone,'' in _2012 8th International Symposium on Chinese Spoken Language Processing_. IEEE, 2012, pp. 206–209.
  * [7] Z. Fu, F. Fan, and J. Huang, ``Dual-microphone noise reduction for mobile phone application,'' in _2013 IEEE International Conference on Acoustics, Speech and Signal Processing_. IEEE, 2013, pp. 7239–7243.
  * [8] N. Yousefian and P. C. Loizou, ``A dual-microphone speech enhancement algorithm based on the coherence function,'' _IEEE Transactions on Audio, Speech, and Language Processing_ , vol. 20, no. 2, pp. 599–609, 2011.
  * [9] N. Yousefian and P. C. Loizou, ``A dual-microphone algorithm that can cope with competing-talker scenarios,'' _IEEE transactions on audio, speech, and language processing_ , vol. 21, no. 1, pp. 145–155, 2012.
  * [10] O. L. Frost, ``An algorithm for linearly constrained adaptive array processing,'' _Proceedings of the IEEE_ , vol. 60, no. 8, pp. 926–935, 1972\.
  * [11] T. Higuchi, N. Ito, T. Yoshioka, and T. Nakatani, ``Robust mvdr beamforming using time-frequency masks for online/offline asr in noise,'' in _2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2016, pp. 5210–5214.
  * [12] K. Buckley and L. Griffiths, ``An adaptive generalized sidelobe canceller with derivative constraints,'' _IEEE Transactions on antennas and propagation_ , vol. 34, no. 3, pp. 311–319, 1986.
  * [13] Y. Xu, J. Du, L.-R. Dai, and C.-H. Lee, ``A regression approach to speech enhancement based on deep neural networks,'' _IEEE/ACM Transactions on Audio, Speech, and Language Processing_ , vol. 23, no. 1, pp. 7–19, 2014.
  * [14] H.-S. Choi, J.-H. Kim, J. Huh, A. Kim, J.-W. Ha, and K. Lee, ``Phase-aware speech enhancement with deep complex u-net,'' in _International Conference on Learning Representations_ , 2019.
  * [15] Y. Hu, Y. Liu, S. Lv, M. Xing, S. Zhang, Y. Fu, J. Wu, B. Zhang, and L. Xie, ``Dccrn: Deep complex convolution recurrent network for phase-aware speech enhancement,'' _arXiv preprint arXiv:2008.00264_ , 2020.
  * [16] X. Hao, X. Su, R. Horaud, and X. Li, ``Fullsubnet: A full-band and sub-band fusion model for real-time single-channel speech enhancement,'' in _ICASSP 2021-2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2021, pp. 6633–6637.
  * [17] G. Zhang, L. Yu, C. Wang, and J. Wei, ``Multi-scale temporal frequency convolutional network with axial attention for speech enhancement,'' in _ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2022, pp. 9122–9126.
  * [18] K. Tan, X. Zhang, and D. Wang, ``Deep learning based real-time speech enhancement for dual-microphone mobile phones,'' _IEEE/ACM transactions on audio, speech, and language processing_ , vol. 29, pp. 1853–1863, 2021.
  * [19] X. Xu, R. Gu, and Y. Zou, ``Improving dual-microphone speech enhancement by learning cross-channel features with multi-head attention,'' in _ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2022, pp. 6492–6496.
  * [20] D. Yang, F. Jiang, W. Wu, X. Fang, and M. Cao, ``Low-complexity acoustic echo cancellation with neural kalman filtering,'' in _ICASSP 2023-2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2023, pp. 1–5.
  * [21] Y. Lu, S. Cornell, X. Chang, W. Zhang, C. Li, Z. Ni, Z. Wang, and S. Watanabe, ``Towards low-distortion multi-channel speech enhancement: The espnet-se submission to the l3das22 challenge,'' in _ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2022, pp. 9201–9205.
  * [22] A. Li, W. Liu, C. Zheng, and X. Li, ``Embedding and beamforming: All-neural causal beamformer for multichannel speech enhancement,'' in _ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2022, pp. 6487–6491.
  * [23] I. Cohen, ``Analysis of two-channel generalized sidelobe canceller (GSC) with post-filtering,'' _IEEE Transactions on Speech and Audio Processing_ , vol. 11, no. 6, pp. 684–699, 2003.
  * [24] C. Veaux, J. Yamagishi, K. MacDonald _et al._ , ``Cstr vctk corpus: English multi-speaker corpus for cstr voice cloning toolkit,'' _University of Edinburgh. The Centre for Speech Technology Research (CSTR)_ , 2017.
  * [25] V. Panayotov, G. Chen, D. Povey, and S. Khudanpur, ``Librispeech: an asr corpus based on public domain audio books,'' in _2015 IEEE international conference on acoustics, speech and signal processing (ICASSP)_. IEEE, 2015, pp. 5206–5210.
  * [26] G. Wichern, J. Antognini, M. Flynn, L. R. Zhu, E. McQuinn, D. Crow, E. Manilow, and J. L. Roux, ``Wham!: Extending speech separation to noisy environments,'' _arXiv preprint arXiv:1907.01160_ , 2019.
  * [27] E. A. Habets, I. Cohen, and S. Gannot, ``Generating nonstationary multisensor signals under a spatial coherence constraint,'' _The Journal of the Acoustical Society of America_ , vol. 124, no. 5, pp. 2911–2917, 2008.
  * [28] J. Le Roux, S. Wisdom, H. Erdogan, and J. R. Hershey, ``SDR–half-baked or well done?'' in _ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2019, pp. 626–630.
  * [29] A. W. Rix, J. G. Beerends, M. P. Hollier, and A. P. Hekstra, ``Perceptual evaluation of speech quality (PESQ)-a new method for speech quality assessment of telephone networks and codecs,'' in _2001 IEEE international conference on acoustics, speech, and signal processing. Proceedings (Cat. No. 01CH37221)_ , vol. 2. IEEE, 2001, pp. 749–752.
  * [30] C. H. Taal, R. C. Hendriks, R. Heusdens, and J. Jensen, ``A short-time objective intelligibility measure for time-frequency weighted noisy speech,'' in _2010 IEEE international conference on acoustics, speech and signal processing_. IEEE, 2010, pp. 4214–4217.
  * [31] C. K. Reddy, V. Gopal, and R. Cutler, ``DNSMOS P. 835: A non-intrusive perceptual objective speech quality metric to evaluate noise suppressors,'' in _ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2022, pp. 886–890.
  * [32] J. Rasley, S. Rajbhandari, O. Ruwase, and Y. He, ``Deepspeed: System optimizations enable training deep learning models with over 100 billion parameters,'' in _Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_, 2020, pp. 3505–3506.