# Born and inverse Born series for scattering problems with Kerr
nonlinearities

Nicholas DeFilippis  Department of Mathematics, Drexel University,
Philadelphia, PA, USA  ,  Shari Moskow  Department of Mathematics, Drexel
University, Philadelphia, PA, USA<EMAIL_ADDRESS>and  John C.
Schotland  Department of Mathematics and Department of Physics, Yale
University, New Haven, CT, USA<EMAIL_ADDRESS>
###### Abstract.

We consider the Born and inverse Born series for scalar waves with a cubic
nonlinearity of Kerr type. We find a recursive formula for the operators in
the Born series and prove their boundedness. This result gives conditions
which guarantee convergence of the Born series, and subsequently yields
conditions which guarantee convergence of the inverse Born series. We also use
fixed point theory to give alternate explicit conditions for convergence of
the Born series. We illustrate our results with numerical experiments.

##  1\. Introduction

There has been considerable recent interest in inverse scattering problems for
nonlinear partial differential equations (PDEs) [1, 2, 4, 7, 8, 9, 10, 11, 12,
13, 14]. There are numerous applications in various applied fields ranging
from optical imaging to seismology. In general terms, the problem to be
considered is to reconstruct the coefficients of a nonlinear PDE from boundary
measurements. As in the case of inverse problems for linear PDEs, the
fundamental questions relate to the uniqueness, stability and reconstruction
of the unknown coefficients. In contrast to the linear case (which is very
well studied), the study of inverse problems for nonlinear PDE is still
relatively unexplored. There are uniqueness and stability results for a
variety of semilinear and quasilinear equations. Reconstruction methods for
such problems are just beginning to be developed[4, 5, 12].

In this paper, we consider the inverse problem of recovering the coefficients
of a nonlinear elliptic PDE with cubic nonlinearity. This problem appears in
optical physics, where the cubic term arises in the study of the Kerr effect—a
nonlinear process where self-focusing of light is observed [3]. We show that
it is possible to reconstruct the coefficients of the linear and nonlinear
terms in a Kerr medium from boundary measurements. This result holds under a
smallness condition on the measurements, which also guarantees the stability
of the recovery. The reconstruction is based on inversion of the Born series,
which expresses the solution to the inverse problem as an explicitly
computable functional of the measured data. We note that this method has been
extensively studied for linear PDEs [16]. The extension to the nonlinear
setting involves a substantial reworking of the theory, especially the
combinatorial structure of the Born series itself. We validate our results
with numerical simulations that demonstrate the convergence of the inverse
Born series under the expected smallness conditions.

The remainder of this paper is organized as follows. In section 2, we
introduce the forward problem and state sufficient conditions for its
solvability. The Born series is studied in section 3, the combinatorial
structure of the series is characterized, and sufficient conditions for
convergence are established. We also derive various estimates that are later
used in section 4 to obtain our main result on the convergence of the inverse
Born series. In section 5, we present the results of numerical reconstructions
of a two-dimensional medium. Our conclusions are presented in section 6. The
Appendix contains the proof of Proposition 1.

##  2\. Forward problem

We consider the Kerr effect in a bounded domain $\Omega$ in $\mathbb{R}^{d}$
for $d>2$ with a smooth boundary $\partial\Omega$. The scalar field $u$ obeys
the nonlinear PDE

$\displaystyle\Delta u+k^{2}(1+\alpha(x))u+k^{2}\beta(x)|u|^{2}u$
$\displaystyle=0\quad\text{ in }\quad\Omega\ ,$ (1)
$\displaystyle\frac{\partial u}{\partial\nu}$ $\displaystyle=g\quad\text{ on
}\quad\partial\Omega\ ,$ (2)

where $k$ is the wavenumber and $\nu$ is the unit outward normal to
$\partial\Omega$. The coefficients $\alpha$ and $\beta$ are the linear and
nonlinear susceptibilities, respectively [3] and are taken to be real valued,
as is the boundary source $g$. It follows that $u$ is real valued so that
$|u|^{2}u=u^{3}$. More generally, $u$ is complex valued, in which case our
results carry over with small modifications.

We now consider the solution $u_{0}$ to the linear problem

$\displaystyle\Delta u_{0}+k^{2}u_{0}$ $\displaystyle=0\quad\text{ in
}\quad\Omega\ ,$ (3) $\displaystyle\frac{\partial u_{0}}{\partial\nu}$
$\displaystyle=g\quad\text{ on }\quad\partial\Omega\ .$ (4)

Following standard procedures, we find that the field $u$ obeys the integral
equation

$u(x)=u_{0}(x)-k^{2}\int_{\Omega}G(x,y)\left(\alpha(y)u(y)+\beta(y)u^{3}(y)\right)dy\
.$ (5)

Here the Green’s function $G$ obeys

$\displaystyle\Delta_{x}G(x,y)+k^{2}G(x,y)$
$\displaystyle=\delta(x-y)\quad\text{ in }\quad\Omega\ ,$ (6)
$\displaystyle\frac{\partial G}{\partial\nu_{y}}$ $\displaystyle=0\quad\text{
on }\quad\partial\Omega\ .$ (7)

We define the nonlinear operator $T:C(\overline{\Omega})\rightarrow
C(\overline{\Omega})$ by

$T(u)=u_{0}-k^{2}\int_{\Omega}G(x,y)\left(\alpha(y)u(y)+\beta(y)u^{3}(y)\right)dy.$
(8)

We note that if $u\in C(\overline{\Omega})$ is a fixed point of $T$, that is
$u=T(u)$, then $u$ satisfies equation (5).

The following result provides conditions for existence of a unique solution to
(5).

######  Proposition 1.

Let $T:C(\overline{\Omega})\rightarrow C(\overline{\Omega})$ be defined by (8)
and define $\mu$ by

$\mu=k^{2}\sup_{x\in\Omega}\int_{\Omega}|G(x,y)|dy.$ (9)

If there exists $\gamma>1/2$ such that

$\|\alpha\|_{\infty}<\frac{2\gamma-1}{2\mu(1+\gamma)}$

and

$\|\beta\|_{\infty}<\frac{1}{2\mu\|u_{0}\|^{2}(1+\gamma)^{3}},$

then $T$ has a unique fixed point on the ball of radius
$\gamma\|u_{0}\|_{\infty}$ about $u_{0}$ in $C(\overline{\Omega})$.

The proof is presented in Appendix A.

##  3\. Born series

The forward problem is to compute the field $u$ as measured on
$\partial\Omega$ when $g$ corresponds to a point source on $\partial\Omega$.
The solution to the forward problem is derived by iteration of the integral
equation (5). We thus obtain

$\phi=K_{1}(\zeta)+K_{2}(\zeta,\zeta)+K_{3}(\zeta,\zeta,\zeta)+\cdots\ ,$ (10)

where $\phi=u-u_{0}$ and $\zeta:=(\alpha,\beta)$. The forward operators

$K_{n}:[L^{\infty}(\Omega)]^{2n}\rightarrow
C(\partial\Omega\times\partial\Omega)$

are constructed below. We will refer to (10) as the the Born series. We note
that Proposition 1 guarantees convergence of the Born series.

The forward operator $K_{n}$ is an $n$-linear operator (multilinear of order
$n$) on $[L^{\infty}(\Omega)^{2}]^{n}$. In the following, we do not denote the
dependence of $u_{0}$ on the source explicitly. The first term in the fixed
point iteration is

$u_{1}(x):=T(u_{0})(x)=u_{0}(x)+k^{2}\int_{\Omega}G(x,y)\left[\alpha(y)u_{0}(y)+\beta(y)u_{0}^{3}(y)\right]dy$
(11)

and thus $K_{1}$ is defined by

$K_{1}(\zeta)(x)=k^{2}\int_{\Omega}G(x,y)\left[\alpha(y)u_{0}(y)+\beta(y)u_{0}^{3}(y)\right]dy.$
(12)

Next we observe that

$u_{2}(x):=T(u_{1})(x)=u_{0}(x)+k^{2}\int_{\Omega}G(x,y)\left[\alpha(y)u_{1}(y)+\beta(y)u_{1}^{3}(y)\right]dy.$
(13)

Evidently, expansion of $u_{1}^{3}$ leads to terms which are multilinear in
$\alpha$ and $\beta$. Subsequent iterates become progressively more
complicated. To handle this problem, we introduce the operators:
$a,b:C(\overline{\Omega})\times[L^{\infty}(\Omega)]^{2}\rightarrow
C(\overline{\Omega})$, which are defined by

$a(v,\zeta)=k^{2}\int_{\Omega}G(x,y)\alpha(y)v(y)dy,$ (14)

and

$b(v,\zeta)=k^{2}\int_{\Omega}G(x,y)\beta(y)v(y)dy.$ (15)

The above operators have tensor counterparts which are defined as follows.

######  Definition 1.

Given $T_{l}=T_{l}(\zeta_{1},\cdots,\zeta_{l}),$ a multi-linear operator of
order $l$, define the $l+1$ order multilinear operators $BT_{l}$ and $AT_{l}$
by

$BT_{l}(\zeta_{1},\ldots,\zeta_{l},\zeta_{l+1})=b(T_{l}(\zeta_{1},\ldots,\zeta_{l}),\zeta_{l+1})$

and

$AT_{l}(\zeta_{1},\ldots,\zeta_{l},\zeta_{l+1})=a(T_{l}(\zeta_{1},\ldots,\zeta_{l}),\zeta_{l+1})$

where $b$ and $n$ are given by (15) and (14) respectively.

We will also need a tensor product of multilinear operators.

######  Definition 2.

Given $T_{j}$ and $T_{l}$, multilinear operators of order $j$ and $l$
respectively, define the tensor product $T_{l}\otimes T_{j}$ by

$T_{l}\otimes
T_{j}(\zeta_{1},\ldots,\zeta_{l},\zeta_{l+1},\dots,\zeta_{l+j})=T_{l}(\zeta_{1},\ldots,\zeta_{l})T_{j}(\zeta_{l+1},\dots,\zeta_{l+j}).$

and note that $T_{l}\otimes T_{j}$ is a multilinear operator of order $l+j$.

Note that the tensor product of multilinear operators does not commute. Tensor
products are extended to sums of multilinear operators by bilinearity of the
tensor product, and the tensor product is also associative. In this notation,
we see that if $v$ is a sum of multilinear operators, then

$Tv=u_{0}+Av+Bv\otimes v\otimes v$

yields another sum of multilinear operators ($u_{0}$ is an order zero
operator).

######  Lemma 1.

Viewing the $n$th iterate $u_{n}$ as a sum of multilinear operators, for any
$n$ we have that

$u_{n}=u_{n-1}+\mbox{multilinear operators of order}\ \geq n.$ (16)

###### Proof.

We will prove this by induction. For the base case $n=1$, we have that
$u_{1}=u_{0}+Au_{0}+Bu_{0}\otimes u_{0}\otimes u_{0}$, so the statement holds.
Now assume that the statement holds for $u_{n-1}$. Then

$\displaystyle u_{n}$ $\displaystyle=$ $\displaystyle
u_{0}+Au_{n-1}+Bu_{n-1}\otimes u_{n-1}\otimes u_{n-1}.$ (17)

By inductive hypothesis,

$u_{n-1}=u_{n-2}+w$

where $w$ is a sum of operators of order at least $n-1$. Hence we have that

$u_{n-1}\otimes u_{n-1}\otimes u_{n-1}=u_{n-2}\otimes u_{n-2}\otimes
u_{n-2}+u_{n-2}\otimes u_{n-2}\otimes w\\\ +w\otimes u_{n-2}\otimes
u_{n-2}+u_{n-2}\otimes w\otimes u_{n-2}\\\ +u_{n-2}\otimes w\otimes w+w\otimes
u_{n-2}\otimes w+u_{n-2}\otimes w\otimes w+w\otimes w\otimes w,$ (18)

so that

$u_{n-1}\otimes u_{n-1}\otimes u_{n-1}=u_{n-2}\otimes u_{n-2}\otimes
u_{n-2}+\mbox{multilinear operators of order}\geq n-1.$

Applying $A$ to $u_{n-1}$ and $B$ to $u_{n-1}\otimes u_{n-1}\otimes u_{n-1}$,
we increase the order of each by one. Hence we have that

$\displaystyle u_{n}$ $\displaystyle=$ $\displaystyle
u_{0}+Au_{n-2}+Bu_{n-2}\otimes u_{n-2}\otimes u_{n-2}+\mbox{terms of
degree}\geq n$ (19) $\displaystyle=$ $\displaystyle u_{n-1}+\mbox{terms of
degree}\geq n.$ (20)

The result follows from induction. ∎

Given the previous result, we can now define the forward operators.

######  Definition 3.

The $n$th term of the forward series, $K_{n}(\zeta,\ldots,\zeta)$ is defined
to be the sum of all multilinear operators of order exactly $n$ in the $nth$
iterate $u_{n}$.

###  3.1. General formula for the forward operators

Using our tensor notation, the forward series is given by iterations of

$Tv=u_{0}+Av+Bv\otimes v\otimes v.$

Given $u_{0}$, we have

$\displaystyle u_{1}$ $\displaystyle=$ $\displaystyle
Tu_{0}=u_{0}+Au_{0}+Bu_{0}\otimes u_{0}\otimes u_{0},$ $\displaystyle u_{2}$
$\displaystyle=$ $\displaystyle Tu_{1}=u_{0}+Au_{1}+Bu_{1}\otimes u_{1}\otimes
u_{1},$ $\displaystyle u_{n+1}$ $\displaystyle=$ $\displaystyle
Tu_{n}=u_{0}+Au_{n}+Bu_{n}\otimes u_{n}\otimes u_{n}.$

Define $U_{n}$ to be the sum of the first $n$ forward operators, that is,

$\displaystyle U_{n}$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{n}K_{i}(\zeta_{1},\ldots\zeta_{i})$ $\displaystyle=$
$\displaystyle u_{0}+\sum_{i=1}^{n}K_{i}(\zeta_{1},\ldots,\zeta_{i}).$

We know from Lemma 16 that

$u_{n}=U_{n}+w,$

where $w$ is a sum of multilinear operators, all of order $>n$. To find
$U_{n+1}$, we use the iteration

$u_{n+1}=u_{0}+A(U_{n}+w)+B(U_{n}+w)\otimes(U_{n}+w)\otimes(U_{n}+w).$

We know (also from Lemma 16 ) that $K_{n+1}$ will be the sum of all terms here
which are of order $n+1$. Since $w$ contains only terms of order $\geq n+1$,
after applying $A$ or $B$, the result will be of higher order and hence will
not be included in $K_{n+1}$. So any term containing $w$ after expanding out
the tensor product can be dropped, and we have that all terms of $K_{n+1}$
will be contained in the sum

$AU_{n}+BU_{n}\otimes U_{n}\otimes U_{n}.$

Since $A$ and $B$ each add one to the order, $K_{n+1}$ will consist of
$AK_{n}$ and all terms of the form

$BK_{i_{1}}\otimes K_{i_{2}}\otimes K_{i_{3}}$

where the ordered triplets $(i_{1},i_{2},i_{3})$ are such that
$i_{1}+i_{2}+i_{3}=n$. Hence we have derived the following:

$\displaystyle K_{0}$ $\displaystyle=$ $\displaystyle u_{0},$ $\displaystyle
K_{1}$ $\displaystyle=$ $\displaystyle u_{0}+Au_{0}+Bu_{0}\otimes u_{0}\otimes
u_{0},$ $\displaystyle K_{n+1}$ $\displaystyle=$ $\displaystyle
AK_{n}+B\sum_{\begin{subarray}{c}(i_{1},i_{2},i_{3})\\\ i_{1}+i_{2}+i_{3}=n\\\
0\leq i_{1},i_{2},i_{3}\leq n\end{subarray}}K_{i_{1}}\otimes K_{i_{2}}\otimes
K_{i_{3}}.$ (21)

We note that we can count the number of such ordered triples in the above sum
to be

$C(n):=n(n+1)/2+(n+1).$

###  3.2. Bounds on the forward operators.

In order to analyze the inverse Born series, we will need bounds for the
forward operators $K_{i}$. We will see that to apply existing convergence
results about the inverse Born series we need boundedness of the operators as
multilinear forms. We use the notation $|\cdot|_{\infty}$ to denote the bound
on any multilinear operator of order $n$ as follows :

######  Definition 4.

For any multilinear operator $K$ of order $n$ on $[L^{\infty}(\Omega)]^{2n}$,
we define

$|K|_{\infty}=\sup_{\begin{subarray}{c}\zeta_{1},\ldots,\zeta_{n}\end{subarray}}{\|K(\zeta_{1},\dots\zeta_{n})\|\over{\|\zeta_{1}\|\cdots\|\zeta_{n}\|}}.$

Note that, for two multilinear operators $T_{1}$ and $T_{2}$ of the same
order, we have the triangle inequality

$|T_{1}+T_{2}|_{\infty}\leq|T_{1}|_{\infty}+|T_{2}|_{\infty}.$

######  Lemma 2.

The forward operator $K_{n}$ given by (21) is a bounded multilinear operator
from $[L^{\infty}(\Omega)]^{2n}$ to $C(\partial\Omega\times\partial\Omega)$
and

$|K_{n}|_{\infty}\leq\nu_{n}\mu^{n}$ (22)

where

$\mu=k^{2}\sup_{x\in\Omega}\int_{\Omega}|G(x,y)|dy,$ (23)

,

$\nu_{0}=\|u_{0}\|_{C(\overline{\Omega}\times\partial\Omega)},$

and for all $n\geq 1$,

$\nu_{n+1}=\nu_{n}+\sum_{\begin{subarray}{c}(i_{1},i_{2},i_{3})\\\
i_{1}+i_{2}+i_{3}=n\\\ 0\leq i_{1},i_{2},i_{3}\leq
n\end{subarray}}\nu_{i_{1}}\nu_{i_{2}}\nu_{i_{3}}.$ (24)

###### Proof.

We first note that for our product operators in Definitions 1 and 2, we have
that

$|BT_{l}|_{\infty}\leq\mu|T_{l}|_{\infty},$
$|AT_{l}|_{\infty}\leq\mu|T_{l}|_{\infty},$

and

$|T_{l}\otimes T_{j}|_{\infty}\leq|T_{l}|_{\infty}|T_{j}|_{\infty}.$

The proof works then, by induction. The base case clearly holds with trivially
with $|K_{0}|_{\infty}=\nu_{0}$. Assume that for each $i\leq n$, we have

$\|K_{i}\|\leq\nu_{i}\mu^{i}.$

Using (21), we obtain

$\displaystyle|K_{n+1}|_{\infty}$ $\displaystyle\leq$
$\displaystyle|AK_{n}|_{\infty}+|B\sum_{\begin{subarray}{c}(i_{1},i_{2},i_{3})\\\
i_{1}+i_{2}+i_{3}=n\\\ 0\leq i_{1},i_{2},i_{3}\leq
n\end{subarray}}K_{i_{1}}\otimes K_{i_{2}}\otimes K_{i_{3}}|_{\infty}$
$\displaystyle\leq$
$\displaystyle\mu|K_{n}|_{\infty}+\mu\sum_{\begin{subarray}{c}(i_{1},i_{2},i_{3})\\\
i_{1}+i_{2}+i_{3}=n\\\ 0\leq i_{1},i_{2},i_{3}\leq
n\end{subarray}}|K_{i_{1}}|_{\infty}|K_{i_{2}}|_{\infty}|K_{i_{3}}|_{\infty}$

which gives, by the inductive hypothesis

$\displaystyle|K_{n+1}|_{\infty}$ $\displaystyle\leq$
$\displaystyle\nu_{n}\mu^{n+1}+\mu^{n+1}\sum_{\begin{subarray}{c}(i_{1},i_{2},i_{3})\\\
i_{1}+i_{2}+i_{3}=n\\\ 0\leq i_{1},i_{2},i_{3}\leq
n\end{subarray}}\nu_{i_{1}}\nu_{i_{2}}\nu_{i_{3}}$ $\displaystyle=$
$\displaystyle\nu_{n+1}\mu^{n+1}.$

∎

######  Lemma 3.

For the sequence $\\{\nu_{n}\\}$ given by (24) There exist constants $K$ and
$\nu$ (both depending on $\nu_{0}$ but independent of $n$) such that for any
$n\geq 0$,

${\nu_{n}}\leq\nu K^{n}.$

###### Proof.

To prove this, we consider the generating function

$P(x)=\sum_{n=0}^{\infty}\nu_{n}x^{n}.$

We first note that it suffices to prove that this power series has a positive
radius of convergence, since if this is the case, then for some positive $x$
the terms $\nu_{n}x^{n}\rightarrow 0$. In particular, they are bounded by some
$\nu$, which would imply that

$\nu_{n}\leq\nu(1/x)^{n}.$

We now show that $P(x)$ is analytic in some nontrivial interval around zero.
Consider, formally for now, the cube of $P$,

$\displaystyle(P(x))^{3}$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}i_{1},i_{2},i_{3}=0,\ldots,\infty\end{subarray}}x^{i_{1}}x^{i_{2}}x^{i_{3}}\nu_{i_{1}}\nu_{i_{2}}\nu_{i_{3}}$
$\displaystyle=$ $\displaystyle\sum_{n=0}^{\infty}f_{n}x^{n}$

where

$f_{n}=\sum_{\begin{subarray}{c}(i_{1},i_{2},i_{3})\\\ i_{1}+i_{2}+i_{3}=n\\\
0\leq i_{1},i_{2},i_{3}\leq
n\end{subarray}}\nu_{i_{1}}\nu_{i_{2}}\nu_{i_{3}},$

which is exactly as appears in (24). Aow, we multiply (24) by $x^{n}$ and sum
to obtain

$\sum_{n=0}^{\infty}\nu_{n+1}x^{n}=\sum_{n=0}^{\infty}\nu_{n}x^{n}+\sum_{n=0}^{\infty}f_{n}x^{n}.$

One checks that the left hand side is simply $(P(x)-\nu_{0})/x$, and so the
above yields

$(P(x)-\nu_{0})/x=P(x)+(P(x))^{3}.$

So we have

$x(P(x))^{3}+(x-1)P(x)+\nu_{0}=0.$ (25)

This polynomial in $P$ is singular, so it is not clear that it has an analytic
solution at $x=0$. However, if we differentiate with respect to $x$, we obtain

$P^{\prime}(x)=-{(P(x))^{3}+P(x)\over{3x(P(x))^{2}+x-1}}$ (26)

with $P(0)=\nu_{0}$. Since the right hand side is an analytic function of $x$
and $P$ in a neighborhood of $(0,\nu_{0})$, the ode (26) (with initial
condition) has a unique analytic solution in a neighborhood of $x=0$ (see for
example Theorem 4.1 of Teschl [Te]) . Integration of (26) combined with the
initial condition implies that this analytic solution satisfies (25), and
hence its coefficients must satisfy (24). ∎

######  Proposition 2.

The forward operator $K_{n}$ given by (21) is a bounded multilinear operator
from $[L^{\infty}(\Omega)]^{2n}$ to
$C^{0}(\overline{\Omega}\times\partial\Omega)$, and its bound satisfies

$|K_{n}|_{\infty}\leq\nu(K\mu)^{n},$ (27)

where

$\mu=k^{2}\sup_{x\in\Omega}\int_{\Omega}|G(x,y)|dy$ (28)

and $\nu,K,$ both depending on
$\nu_{0}=\|u_{0}\|_{C(\overline{\Omega}\times\partial\Omega)}$, are as in
Lemma 3.

######  Corollary.

The forward Born series

$u=u_{0}+\sum_{n=1}^{\infty}K_{n}(\zeta,\ldots,\zeta)$

where $K_{n}$ are given by (21) converges in $C(\overline{\Omega})$ for

$\|\zeta\|_{\infty}\leq{1\over{K\mu}}$

where

$\mu=k^{2}\sup_{x\in\Omega}\int_{\Omega}|G(x,y)|dy,$ (29)

and $\nu,K,$ both depending on $\nu_{0}=\|u_{0}\|_{C(\overline{\Omega})}$, are
as in Lemma 3.

##  4\. Inverse Born Series

The inverse problem is to reconstruct the coefficients $\alpha$ and $\beta$
from measurements of the scattering data $\phi$. We proceed by recalling that
the Inverse Born series is defined as

$\tilde{\zeta}=\mathcal{K}_{1}\phi+\mathcal{K}_{2}(\phi)+\mathcal{K}_{3}(\phi)+\cdots\
,$ (30)

where the data $\phi\in C(\partial\Omega\times\partial\Omega).$ The inverse
series was analyzed in [15] and later studied in [6]. The inverse operator
$\mathcal{K}_{m}$ can be computed from the formulas

$\displaystyle\mathcal{K}_{1}(\phi)$ $\displaystyle=K_{1}^{+}(\phi),$ (31)
$\displaystyle\mathcal{K}_{2}(\phi)$
$\displaystyle=-\mathcal{K}_{1}\left(K_{2}(\mathcal{K}_{1}(\phi),\mathcal{K}_{1}(\phi))\right),$
(32) $\displaystyle\mathcal{K}_{m}(\phi)$
$\displaystyle=-\sum_{n=2}^{m}\sum_{i_{1}+\cdots+i_{n}=m}\mathcal{K}_{1}{K}_{n}\left(\mathcal{K}_{i_{1}}(\phi),\dots,\mathcal{K}_{i_{n}}(\phi)\right).$
(33)

Here $K_{1}^{+}$ denotes a regularized pseudoinverse of $K_{1}$, and
$\tilde{\zeta}$ is the series sum, an approximation to $\zeta$, when it
exists. Recall that this inverse series requires forward solves for the
background problem only (i.e., applying the forward series operators), and
requires a pseudoinverse and regularization of the first linear operator only;
$\mathcal{K}_{1}=K_{1}^{+}$. The bounds on the forward operators from
Proposition 2 combined with Theorem 2.2 of [6] yield the following result.

######  Theorem 1.

If $\|\mathcal{K}_{1}\phi\|<r$, where where the radius of convergence $r$ is
given by

$r=\frac{1}{2K\mu}\left[\sqrt{16C^{2}+1}-4C\right],$

where $C=\max\\{2,\|\mathcal{K}_{1}\|\nu K\mu\\}.$

$\mu=k^{2}\sup_{x\in\Omega}\int_{\Omega}|G(x,y)|dy,$ (34)

and $\nu,K,$ both depending on
$\nu_{0}=\|u_{0}\|_{C^{0}(\Omega\times\partial\Omega)}$, are as in Lemma 3,
then the inverse Born series converges.

##  5\. Numerical Experiments

In this section we present several numerical experiments using the inverse
Born series (30) to reconstruct $\alpha$ and $\beta$ from synthetic data. In
all cases, used the FEniCS PDE solver library in Python to create the
synthetic data $\phi$. To implement the inverse series we also need the
forward operators; for these we use the recursive formula, implemented as in
Algorithm 1.

Function  _compute-K $(n,\alpha,\beta)$_

if  _$n=0$_ then

return $u_{0}$;

$v_{\alpha}$ := find-K$(n-1,\alpha(n),\beta(n))$;

$v_{\beta}$ := 0;

for  _$i_{1}$ = 0 to $n-1$_ do

for  _$i_{2}$ = 0 to $n-i_{1}-1$_ do

$i_{3}:=n-i_{2}-i_{1}-1$;

$K_{i_{1}}:=$compute-K$(i_{1},\alpha(1:i_{1}),\beta(1:i_{1})$;

$K_{i_{2}}:=$compute-K$(i_{2},\alpha(i_{1}+1:i_{1}+i_{2}),\beta(i_{1}+1:i_{1}+i_{2}))$;

$K_{i_{3}}:=$compute-K$(i_{3},\alpha(i_{1}+i_{2}+1:n-1),\beta(i_{1}+i_{2}+1:n-1))$;

$v_{\beta}:=v_{\beta}+K_{i_{1}}\cdot K_{i_{2}}\cdot K_{i_{3}}$;

return $A(v_{\alpha},\alpha(n))+B(v_{\beta},\beta(n))$;

Algorithm 1 Generation of the terms in the forward series.

We implement the application of the operators $A$ and $B$ by solving a
background PDE (equivalent to integrating against background Green’s function
kernel), again using the FEniCS PDE solver library in Python; taking care to
choose a different FEM mesh from those used for the generation of the
synthetic data. The inverse Born series implementation is the same as in
previous work, see for example , here calling on the above Algorithm 1 to call
the forward operators.

We begin with an experiment in one dimension, on the interval $\Omega=[0,1]$.
Here, we have only two points on the boundary, meaning we can only take
samples at these two points. However, thanks to the nonlinearity, a scaled
source has the potential to yield more information. For example, if we scale
the source and take the right linear combination of the two solutions, we
eliminate $\alpha$. While we don’t implement this explicitly, we do capitalize
on scaling for the one dimensional example, where we found it improved the
reconstruction.

In the following example, we used $12$ scaled sources on each side of the
interval and three different frequencies $k=0.9,1,1.1$, for a total of $72$
sources. We chose the reference functions $\alpha$ and $\beta$ to be

$\alpha,\beta=\begin{cases}\frac{\gamma}{\sqrt{2\pi\epsilon}}e^{\frac{-x^{2}}{2\epsilon}}&\text{if
}0.4\leq x\leq 0.6\\\ 0&\text{otherwise}\end{cases}$

for $\epsilon=0.01$ and $\gamma=0.2$. We see their simultaneous reconstruction
in Figure 1.

Figure 1. One dimensional simultaneous reconstruction of $\alpha$ and $\beta$
using varying source strength and frequency

Next we run several experiments in two dimensions, all with domain $\Omega$
the unit disk. First, we compare the reconstruction of $\alpha$ with $\beta=0$
(traditional linear problem), to the reconstruction of $\beta$ with
$\alpha=0$. In both cases we let the unknown be a piecewise constant. For the
first example, we chose the moderate contrast medium (35), and we see
reconstruction results in Figure 2. Next, we increase the contrast by a factor
of $4$ in (35) and we see results in Figures 4 and 5. Finally, we choose a
very high contrast, the medium (35) multiplied by a factor of $16$. In Figures
6 and 7, we see that the method does not produce good results for $\alpha$,
but is not as bad for $\beta$.

$\beta,\alpha=\begin{cases}1&\text{if }(x-0.3)^{2}+y^{2}\leq 0.2\\\
0&\text{otherwise}\end{cases}$ (35)

(a) $\alpha$ reconstruction with $\beta=0$

(b) $\beta$ reconstruction with $\alpha=0$

Figure 2. Independent reconstruction only for low contrast

(a) $\alpha$ reconstruction with $\beta=0$

(b) $\beta$ reconstruction with $\alpha=0$

Figure 3. Cross-section of independent reconstruction for low contrast

(a) $\alpha$ reconstruction with $\beta=0$

(b) $\beta$ reconstruction with $\alpha=0$

Figure 4. Independent reconstruction for medium contrast

(a) $\alpha$ reconstruction with $\beta=0$

(b) $\beta$ reconstruction with $\alpha=0$

Figure 5. Cross-section of independent reconstruction only for medium contrast

(a) $\alpha$ reconstruction with $\beta=0$

(b) $\beta$ reconstruction with $\alpha=0$

Figure 6. Independent reconstruction for high contrast

(a) $\alpha$ reconstruction with $\beta=0$

(b) $\beta$ reconstruction with $\alpha=0$

Figure 7. Cross-section of independent reconstruction for high contrast

In this next set of examples, we consider the simultaneous reconstruction of
$\beta$ and $\alpha$. In the first example, we take

$\beta=\alpha=\frac{2}{\sqrt{2\pi\epsilon}}\exp{\left(\frac{-|x-x_{0}|^{2}}{2\epsilon}\right)}$
(36)

for $x_{0}=(-.3,3)$ and $\epsilon=0.04$. We see results in Figures 8 and 9. In
the second example, we raise the contrast in (36) by a factor of $4$ ,and we
see that we still get reasonable reconstructions in Figure 10 and Figure 11.

Figure 8. Simultaneous $\alpha$ and $\beta$ reconstruction for high contrast
Figure 9. Cross-section $\alpha$ and $\beta$ reconstruction for high contrast
Figure 10. Simultaneous $\alpha$ and $\beta$ reconstruction for very high
contrast Figure 11. Cross-section $\alpha$ and $\beta$ reconstruction for very
high contrast

##  6\. Discussion

We have considered the Born and inverse Born series for scalar waves with a
cubic nonlinearity of Kerr type. We found a recursive formula for the forward
operators in the Born series. This result gives conditions which guarantee
convergence of the Born series, and also leads to conditions which guarantee
convergence of the inverse Born series. Our results are illustrated results
with numerical experiments.

The ideas developed here provide a framework for studying inverse problems for
a wide class of nonlinear PDEs with polynomial nonlinearities. The formulas
and algorithm for generating the forward operators, the use of the generating
functions, and the resulting reconstruction algorithm are readily
generalizable to this setting and will be explored in future work.

##  7\. Acknowledgments

The authors are indebted to Jonah Blasiak and R. Andrew Hicks for discussions
essential to the proof of Lemma 3. S. Moskow was partially supported by the
NSF grant DMS-2008441. J. Schotland was supported in part by the NSF grant
DMS-1912821 and the AFOSR grant FA9550-19-1-0320.

##  Appendix A Proof of Proposition 1

In this appendix we obtain conditions for existence of a unique solution to
(5) and give alternative conditions on $\alpha$ and $\beta$ that guarantee
convergence of the Born series. Define the linear operator

$G:C^{0}(\overline{\Omega})\rightarrow C^{0}(\overline{\Omega})$

by

$G(v)=-k^{2}\int_{\Omega}G(x,y)v(y)\,dy.$

Then, for $u_{0}$ in $C^{0}(\overline{\Omega})$, we have that $T$ can be
written as

$T(v)=u_{0}+G\left(\alpha v+\beta v^{3}\right).$ (37)

Note that $G$ is compact and bounded. Define $\mu$ by

$\mu=k^{2}\sup_{x\in\Omega}\int_{\Omega}|G(x,y)|dy,$ (38)

Then we have that

$\|G(v)\|\leq\mu\|v\|$

for all $v\in C^{0}(\overline{\Omega})$. We will make use of the following two
lemmas. The first gives conditions to have a contraction.

######  Lemma 4.

For any $f,g\in C^{0}(\overline{\Omega})$ such that $\|f\|,\|g\|\leq R$, we
have

$\|T(f)-T(g)\|\leq q\|f-g\|$

where

$q=\mu(\left\|\alpha\right\|+3R^{2}\left\|\beta\right\|)$

for $\mu$ defined by (38).

###### Proof.

Let $f,g\in B$. Then,

$T(f)-T(g))=k^{2}\int_{\Omega}G(x,y)\left[\alpha(y)\left(f(y)-g(y)\right)+\beta(y)\left(f(y)^{3}-g(y)^{3}\right)\right]\,dy$
(39)

so that

$\|T(f)-T(g)\|\leq\mu\left\|\alpha\left(f-g\right)+\beta\left(f^{3}-g^{3}\right)\right\|\\\
=\mu\left\|\left[\alpha+\beta\left(f^{2}+fg+g^{2}\right)\right](f-g)\right\|\\\
\leq\mu\left(\left\|\alpha\right\|+\left\|\beta\left(f^{2}+fg+g^{2}\right)\right\|\right)\cdot\left\|(f-g)\right\|\\\
\leq\mu\left(\left\|\alpha\right\|+3R^{2}\left\|\beta\right\|\right)\cdot\|f-g\|.$
(40)

∎

The second lemma gives us a ball which $T$ maps into itself.

######  Lemma 5.

Let $r>0$ be given, and let $B=B(u_{0},r)$ be the ball of radius $r$ about
$u_{0}$ in $C^{0}(\Omega)$. Define $R=\|u_{0}\|+r$. Then if

$\mu R\left(\|\alpha\|+R^{2}\|\beta\|\right)<r,$

we have that $T(v)\in B$ for any $v\in B$, and hence $T$ has a fixed point in
$B$.

###### Proof.

Assume that $v\in B$. Then, by the triangle inequality, we have that
$\|v\|\leq R$. Then,

$\displaystyle\|T(v)-u_{0}\|$ $\displaystyle=$ $\displaystyle\|G(\alpha
v+\beta v^{3})\|$ (41) $\displaystyle\leq$ $\displaystyle\mu\left\|\alpha
v+\beta v^{3}\right\|$ (42) $\displaystyle\leq$
$\displaystyle\mu\left(R\|\alpha\|+R^{3}\|\beta\|\right).$ (43)

By hypothesis, this is less than $r$, and hence $T(v)\in B$. ∎

######  Lemma 6.

If the hypotheses of Lemma 5 hold and, additionally,

$\mu\left(\|\alpha\|+3R^{2}\|\beta\|\right)<1,$

then $T$ is a contraction mapping on $B$.

###### Proof.

Since $\mu\left\|\alpha\right\|+3R^{2}\left\|\beta\right\|<1$ by assumption,
by Lemma 4 we have that $d(T(f),T(g))\leq q\cdot d(f,g)$ where $q<1$. Since
$T$ also maps $B$ into itself, $T$ is a contraction mapping on $B$. ∎

Clearly if Lemma 6 holds, by the Banach fixed point theorem $T$ will have a
unique fixed point on $B$. Furthermore, if we start with initial function
$u_{0}$ in $B$, then fixed-point iteration will converge to the unique fixed
point, which in this case is the solution to the integral of the partial
differential equation defined by (5). The iterates of the fixed point
iteration will generate the (forward) Born series.

######  Remark.

We note that If $\beta$ and $\alpha$ satisfy the (more restrictive) condition

$\mu(\|\alpha\|+3R^{2}\|\beta\|)<\frac{r}{R},$

then both Lemma 5 and Lemma 6 are also satisfied.

The following is well known for the linear case, see for example [CoKr].

######  Proposition 3.

If $\beta=0$ and $\|\alpha\|<\frac{1}{\mu}$, then $T$ has a unique fixed point
on all of $C^{0}(\Omega)$.

###### Proof.

In this case, Lemma 5 and Lemma 6 are both satisfied if

$\mu\|\alpha\|<\frac{r}{r+\|u_{0}\|}.$

Since by assumption $\mu\|\alpha\|<1$, there exists $r_{0}$ such that for any
$r>r_{0}$

$\mu\|\alpha\|<\frac{r}{r+\|u_{0}\|}<1.$

Therefore $T$ has a unique fixed point on $B(u_{0},r)$ for any $r>r_{0}$, so
the fixed point must be unique on all of $C^{0}(\Omega)$. ∎

######  Proposition 4.

If $\alpha=0$ and $\beta<\frac{4}{27\mu\|u_{0}\|^{2}}$, then $T$ has a unique
fixed point in the ball $B(u_{0},\|u_{0}\|/2)$.

###### Proof.

Let $r=\|u_{0}\|/2$. This means that $R=3\|u_{0}\|/2$. Then,

$\displaystyle\mu R\left(\|\alpha\|+R^{2}\|\beta\|\right)$ $\displaystyle=\mu
R^{3}\|\beta\|$ $\displaystyle=\mu(3\|u_{0}\|/2)^{3}\|\beta\|$
$\displaystyle<\frac{27\mu\|u_{0}\|^{3}}{8}\cdot\frac{4}{27\mu\|u_{0}\|^{2}}$
$\displaystyle=\frac{\|u_{0}\|}{2}=r$

Thus, the hypothesis needed for Lemma 5 is satisfied. Additionally, we have
that

$\displaystyle\mu\left(\|\alpha\|+3R^{2}\|\beta\|\right)$
$\displaystyle=3R^{2}\mu\|\beta\|$
$\displaystyle=3(3\|u_{0}\|/2)^{2}\mu\|\beta\|$
$\displaystyle<\frac{27\mu\|u_{0}\|^{2}}{4}\cdot\frac{4}{27\mu\|u_{0}\|^{2}}$
$\displaystyle=1,$

so that the condition for Lemma 6 to hold is also satisfied, and hence $T$ has
a unique fixed point on $B(u_{0},r)$. ∎

######  Proposition 5.

If there exists some $\gamma>1/2$ such that

$\|\alpha\|<\frac{2\gamma-1}{2\mu(1+\gamma)}$

and

$\|\beta\|<\frac{1}{2\mu\|u_{0}\|^{2}(1+\gamma)^{3}},$

then $T$ has a unique fixed point in the ball $B(u_{0},\gamma\|u_{0}\|)$.

###### Proof.

Let $r=\gamma\|u_{0}\|$, which that $R=(1+\gamma)\|u_{0}\|$. The hypotheses
then imply that Lemma 5 holds, because

$\displaystyle\mu R(\|\alpha\|+R^{2}\|\beta\|)$
$\displaystyle=\mu(1+\gamma)\|u_{0}\|(\|\alpha\|+(1+\gamma)^{2}\|u_{0}\|^{2}\|\beta\|)$
$\displaystyle=\mu(1+\gamma)\|u_{0}\|\left(\frac{2\gamma-1}{2\mu(1+\gamma)}+\frac{1}{2\mu(1+\gamma)}\right)$
$\displaystyle=\|u_{0}\|\left(\frac{2\gamma-1}{2}+\frac{1}{2}\right)$
$\displaystyle=\gamma\|u_{0}\|=r.$

For Lemma 6, we have that

$\displaystyle\mu(\|\alpha\|+3R^{2}\|\beta\|)$
$\displaystyle=\mu(\|\alpha\|+3(1+\gamma)^{2}\|u_{0}\|^{2}\|\beta\|)$
$\displaystyle<\mu\left(\frac{2\gamma-1}{2\mu(1+\gamma)}+(1+\gamma)^{2}\|u_{0}\|^{2}\frac{1}{2\mu\|u_{0}\|^{2}(1+\gamma)^{3}}\right)$
$\displaystyle=\frac{2\gamma-1}{2(1+\gamma)}+\frac{1}{2(1+\gamma)}$
$\displaystyle=\frac{\gamma}{1+\gamma}<1.$

Thus $T$ has a unique fixed point on the ball $B(u_{0},\gamma\|u_{0}\|)$. ∎

## References

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11institutetext:  Department of Physics, Osaka University, 560-0043, Toyonaka,
Osaka, Japan 22institutetext: Yukawa Institute for Theoretical Physics, Kyoto
University, 606-8502, Kyoto, Japan 33institutetext: J-PARC Branch, KEK Theory
Center, Institute of Particle and Nuclear Studies, KEK, 203-1, 319-1106,
Shirakata, Tokai, Ibaraki, Japan

#  YITP-22-105, J-PARC-TH-0276
Anomalous enhancement of dilepton production due to soft modes in dense quark
matter

Toru Nishimura 11 2 2<EMAIL_ADDRESS>Masakiyo Kitazawa
22 3 3 1 1 Teiji Kunihiro 22

###### Abstract

We explore how the dilepton production rate is modified near the critical
temperature of color superconductivity and QCD critical point by the soft
modes inherently associated with the phase transitions of second-order. It is
shown that the soft modes affect the photon self-energy significantly through
so called the Aslamasov-Larkin, Maki-Thompson and density of states terms,
which are known responsible for the paraconductivity in the metalic
superconductivity, and cause an anomalous enhancement of the production rate
in the low energy/momentum region.

##  1 Introduction

Rich physics is expected to exist in high baryon-density matter at finite
temperature, and may be hopefully revealed by experimental programs in
relativistic heavy-ion collisions (HIC) such as the beam energy scan program
at RHIC, HADES and NA61/SHINE, as well as the future plans at FAIR, NICA and
J-PARC-HI. In this report, we show a promising observability of the color
superconducting (CSC) Alford:2007xm  phase transition and QCD critical point
(CP) in these experiments through an anomalous enhancement of the dilepton
production rate (DPR) caused by the soft modes inherently associated with
these phase transitions of second-order; the CSC phase transion is assumed to
be second-order Nishimura:2022mku . We calculate the effects of the soft modes
on the DPR by extending the theory of the paraconductivity in metals
book_Larkin , and show that the soft modes lead to an anomalous enhancement of
the DPR at low energy region near each phase transition.

##  2 Formalism

To explore the temperature and density region around the 2-flavor color
superconductivity (2SC) and QCD CP, which is expected to be realized at
relatively low densities, we employ the 2-flavor NJL model

$\displaystyle\mathcal{L}$
$\displaystyle=\bar{\psi}i(\gamma^{\mu}\partial_{\mu}-m)\psi+\ G_{\rm
S}[(\bar{\psi}\psi)^{2}+(\bar{\psi}i\gamma_{5}\vec{\tau}\psi)^{2}]+\ G_{\rm
C}(\bar{\psi}i\gamma_{5}\tau_{2}\lambda_{A}\psi^{C})(\bar{\psi}^{C}i\gamma_{5}\tau_{2}\lambda_{A}\psi),$
(1)

where $\tau_{2}$ ($\gamma_{A=2,5,7}$) is the antisymmetric component of the
Pauli (Gell-Mann) matrices for the flavor $SU(2)_{f}$ (color $SU(3)_{c}$), and
$\psi^{C}(x)\equiv C\bar{\psi}^{T}(x)$ with $C=i\gamma_{2}\gamma_{0}$. We
consider the chiral limit for the analysis of the DPR near the critical
temperature ($T_{c}$) of the 2SC, while we set to $m=4$ MeV for investigating
the QCD CP. The scalar coupling constant $G_{S}=5.01\leavevmode\nobreak\
\rm{MeV^{-2}}$ and the three-momentum cutoff $\Lambda=650$ MeV are used, which
leads to the pion decay constant $f_{\pi}=93\leavevmode\nobreak\ \rm{MeV}$ and
the chiral condensate $\langle\bar{\psi}\psi\rangle=(-250\leavevmode\nobreak\
\rm{MeV})^{3}$, while the diquark coupling $G_{C}$ is treated as a free
parameter.

The DPR is related to the retarded photon self-energy $\Pi^{R\mu\nu}(k)$ as

$\displaystyle\frac{d^{4}\Gamma}{d^{4}k}=-\frac{\alpha}{12\pi^{4}}\frac{1}{k^{2}}\frac{1}{e^{\omega/T}-1}g_{\mu\nu}{\rm
Im}\Pi^{R\mu\nu}(k),$ (2)

where $k=(\mathbf{k},\omega)$ is the four momentum of the photon and $\alpha$
is the fine structure constant. In this study, we calculate the modification
of $\tilde{\Pi}^{\mu\nu}(k)$ due to the soft modes of the 2SC and QCD CP. In
this section, we first calculate $\tilde{\Pi}^{\mu\nu}(k)$ including the
diquark soft modes of 2SC Nishimura:2022mku  in Sec. 2.1, and then deal with
the soft modes of the QCD CP that are the mesonic modes in Sec. 2.2.

###  2.1 Modification of $\Pi^{\mu\nu}$ by diquark soft modes

It is known that the diquark fluctuations form a collective mode with a
significant strength above but near $T_{c}$ of the 2SC Kitazawa:2005vr ;
Kunihiro:2007bx . The collective mode associated with the second-order nature
of phase transition is called the soft mode. The propagator of the diquark
soft modes in the random-phase approximation (RPA) in the imaginary-time
formalism is given by

$\displaystyle\tilde{\Xi}_{C}(k)=\frac{1}{{G_{C}}^{-1}+\mathcal{Q}_{C}(k)},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{Q}_{C}(k)=\begin{minipage}[h]{52.03227pt}
\centering\includegraphics[keepaspectratio,scale={0.05}]{qq-corr-crop.pdf}
\@add@centering\end{minipage}=-8\int_{p}{\rm
Tr}[\mathcal{G}_{0}(k-p)\mathcal{G}_{0}(p)],$ (3)

where $\mathcal{Q}_{C}(k)=\mathcal{Q}_{C}(\mathbf{k},i\nu_{n})$ is the one-
loop $qq$ correlation function with
$\mathcal{G}_{0}(p)=\mathcal{G}_{0}(\mathbf{p},i\omega_{m})=1/[(i\omega_{m}+\mu)\gamma_{0}-\mathbf{p}\cdot\mathbf{\gamma}]$
the free quark propagator, $\omega_{m}$ ($\nu_{n}$) the Matsubara frequency
for fermions (bosons), and ${\rm Tr}$ the trace over the Dirac indices. The
retarded propagator is obtained by the analytic continuation
$\Xi^{R}_{C}(\mathbf{k},\omega)=\tilde{\Xi}_{C}(\mathbf{k},i\nu_{n}\rightarrow\omega+i\eta)$.

We remark that $\Xi^{R}_{C}(\mathbf{k},\omega)$ satisfies the Thouless
criterion, that is $[\Xi^{R}_{C}(\mathbf{0},0)]^{-1}=0$ at $T=T_{c}$ for the
second-order phase transition. The criterion shows that
$\Xi^{R}_{C}(\mathbf{k},\omega)$ has a pole at the origin at $T=T_{c}$, and
hence the diquark fluctuations become soft near $T_{c}$ Kitazawa:2005vr . This
fact also allows us to adopt the time-dependent Ginzburg-Landau (TDGL)
approximation

$\displaystyle[\Xi^{R}_{C}(\mathbf{k},\omega)]^{-1}=A(\mathbf{k})+C(\mathbf{k})\omega,$
(4)

near but above $T_{c}$, where
$A(\mathbf{k})=G_{C}^{-1}+Q^{R}_{C}(\mathbf{k},0)$ and $C(\mathbf{k})=\partial
Q^{R}_{C}(\mathbf{k},\omega)/\partial\omega\leavevmode\nobreak\ |_{\omega=0}$.
It is known that Eq. (4) reproduces $\Xi^{R}_{C}(\mathbf{k},\omega)$ over wide
ranges of $\omega$, $\mathbf{k}^{2}$ and $T(>T_{c})$ Kitazawa:2005vr .

To construct $\tilde{\Pi}^{\mu\nu}(k)$ that inolves the soft mode, we start
from the one-loop diagram of $\tilde{\Xi}_{C}(\mathbf{k},i\nu_{n})$, which is
the lowest contribution of the soft modes to the thermodynamic potential
Nishimura:2022mku . The photon self-energy is then constructed by attaching
electromagnetic vertices at two points of quark lines in the thermodynamic
potential. One then obtains four types of diagrams shown in Fig. 1. These
diagrams are called the Aslamasov-Larkin (AL) (Fig. 1(a)), Maki-Thompson (MT)
(Fig. 1(b)) and density of states (DOS) (Fig. 1(c, d)) book_Larkin  terms,
respectively, in the theory of metallic superconductivity. Each contribution
to the photon self-energy, $\tilde{\Pi}_{\rm AL}^{\mu\nu}(k)$,
$\tilde{\Pi}_{\rm MT}^{\mu\nu}(k)$ and $\tilde{\Pi}_{\rm DOS}^{\mu\nu}(k)$,
respectively, is given by

$\displaystyle\tilde{\Pi}_{\rm AL}^{\mu\nu}(k)$
$\displaystyle=3\int_{q}\tilde{\Gamma}^{\mu}(q,q+k)\tilde{\Xi}_{C}(q+k)\tilde{\Gamma}^{\nu}(q+k,q)\tilde{\Xi}_{C}(q),$
(5) $\displaystyle\tilde{\Pi}_{\rm MT(DOS)}^{\mu\nu}(k)$
$\displaystyle=3\int_{q}\tilde{\Xi}_{C}(q)\mathcal{R}_{\rm
MT(DOS)}^{\mu\nu}(q,k),$ (6)

where $\tilde{\Gamma}^{\mu}(q,q+k)$ and
$\mathcal{R}^{\mu\nu}(q,k)=\mathcal{R}_{\rm MT}^{\mu\nu}(q,k)+\mathcal{R}_{\rm
DOS}^{\mu\nu}(q,k)$ satisfy the following Ward-Takahashi (WT) identities

$\displaystyle k_{\mu}\tilde{\Gamma}^{\mu}(q,q+k)$
$\displaystyle=(e_{u}+e_{d})[\mathcal{Q}_{C}(q+k)-\mathcal{Q}_{C}(q)],$ (7)
$\displaystyle k_{\mu}\mathcal{R}^{\mu\nu}(q,k)$
$\displaystyle=(e_{u}+e_{d})[\tilde{\Gamma}^{\nu}(q-k,q)-\tilde{\Gamma}^{\nu}(q,q+k)],$
(8)

where $e_{u}=2|e|/3$ ($e_{d}=-|e|/3$) is the electric charge of up (down)
quarks. The total photon self-energy is then given by
$\tilde{\Pi}^{\mu\nu}(k)=\tilde{\Pi}_{\rm free}^{\mu\nu}(k)+\tilde{\Pi}_{\rm
AL}^{\mu\nu}(k)+\tilde{\Pi}_{\rm MT}^{\mu\nu}(k)+\tilde{\Pi}_{\rm
DOS}^{\mu\nu}(k)$. One can explicitly check that this photon self-energy
satisfies the WT identity using Eqs. (7) and (8).

To obtain the DPR with Eq. (2), the calculation of the spatial components of
$\Pi^{R\mu\nu}(k)$ is sufficient since $\Pi^{R00}(k)$ is expressed in terms of
the longitudinal part from the WT identity as
$\Pi^{R00}(k)=\mathbf{k}^{2}\Pi^{R11}(k)/k_{0}^{2}$ with
$k=(k_{0},|\mathbf{k}|,0,0)$. For the vertex functions (7) and (8), we
approximate $\tilde{\Gamma}^{\mu}(q,q+k)$ and $\mathcal{R}^{\mu\nu}(q,k)$
using Eq. (4). For $\tilde{\Gamma}^{i}(q,q+k)$, we employ an ansatz

$\displaystyle\tilde{\Gamma}^{i}(q,q+k)=-(e_{u}+e_{d})\frac{\mathcal{Q}_{C}(\mathbf{q}+\mathbf{k},0)-\mathcal{Q}_{C}(\mathbf{q},0)}{|\mathbf{q}+\mathbf{k}|^{2}-|\mathbf{q}|^{2}}(2q+k)^{i},$
(9)

which is a real number and satisifes the WT identity (7) with Eq. (4). One can
also obtain $\mathcal{R}^{ij}(q,k)$ in the same manner, and finds that it is a
real number as well. Using this vertex and Eq. (4), the imaginary part of
$\Pi^{Rij}_{\rm MT}(q)+\Pi^{Rij}_{\rm DOS}(q)$ vanishes. Therefore, the MT and
DOS terms do not contribute to the calculation of the DPR. This result is in
accordance with the case of the metallic superconductivity book_Larkin .

(a) |  (b) |  (c) |  (d)
---|---|---|---
Figure 1: Diagrammatic representations of the Aslamasov-Larkin (a), Maki-
Thompson (b) and density of states (c,d) terms with the diquark soft mode. The
wavy and double lines are photons and the diquark soft modes, respectively.

###  2.2 Modification of $\Pi^{\mu\nu}$ by mesonic soft modes near QCD CP

The photon self-energies including the mesonic soft modes of the QCD CP can
also be calculated with a similar manner. The propagator of the mesonic mode
is evaluated by the RPA as

$\displaystyle\tilde{\Xi}_{S}(k)=\frac{1}{{G_{S}}^{-1}+\mathcal{Q}_{S}(k)},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{Q}_{S}(k)=\begin{minipage}[h]{52.03227pt}
\centering\includegraphics[keepaspectratio,scale={0.05}]{qqbar-corr-crop.pdf}
\@add@centering\end{minipage}=12\int_{p}{\rm
Tr}[\mathcal{G}_{0}(p-k)\mathcal{G}_{0}(p)],$ (10)

where $\mathcal{Q}_{S}(k)$ is the one-loop $q\bar{q}$ correlation function. As
described in Ref. Fujii:2004jt , $\Xi^{R}_{S}(k)$ has a discontinuity at the
light cone, and its strength is limited to the space-like region. This
property is contrasted with the diquark soft mode (3), which does not have the
discontinuity at the light cone, while the diquark soft mode also has a
significant strength in the space-like region. The photon self-energy with the
mesonic modes is given by

$\displaystyle\tilde{\Pi}_{\rm AL}^{\mu\nu}(k)$
$\displaystyle=\sum_{f}\int_{q}\tilde{\Gamma}^{\mu}_{f}(q,q+k)\tilde{\Xi}_{S}(q+k)\tilde{\Gamma}^{\nu}_{f}(q+k,q)\tilde{\Xi}_{S}(q),$
(11) $\displaystyle\tilde{\Pi}_{\rm MT(DOS)}^{\mu\nu}(k)$
$\displaystyle=\sum_{f}\int_{q}\tilde{\Xi}_{S}(q)\mathcal{R}_{f\leavevmode\nobreak\
\rm MT(DOS)}^{\mu\nu}(q,k),$ (12)

where $f=u,d$ are the indices of the flavors.
$\tilde{\Gamma}^{\mu}_{f}(q,q+k)$ and
$\mathcal{R}^{\mu\nu}_{f}(q,k)=\mathcal{R}_{f\leavevmode\nobreak\ {\rm
MT}}^{\mu\nu}(q,k)+\mathcal{R}_{f\leavevmode\nobreak\ {\rm
DOS}}^{\mu\nu}(q,k)$ satisfy the WT identity, similarly to Eqs. (7) and (8).

##  3 Numerical results and summary

|
---|---
Figure 2:  DPRs per unit energy $\omega$ and momentum $\mathbf{k}$ above
$T_{c}$ of the 2SC at $\mu=350\leavevmode\nobreak\ {\rm MeV}$ (left)
Nishimura:2022mku  and of the QCD CP at $\mu=\mu_{\rm CP}$ (right). The thick
lines are the contribution of the soft modes, and the thin lines are the
results for the free quark gas.

In the left panel of Fig. 2, we show the DPR per unit energy and momentum
$d^{4}\Gamma/d^{4}k$ near the 2SC phase transition for $T=1.01T_{c}$,
$1.1T_{c}$ and $1.5T_{c}$ at $\mu=350\leavevmode\nobreak\ {\rm MeV}$
($T_{c}=42.94\leavevmode\nobreak\ {\rm MeV}$) Nishimura:2022mku . The thick
lines are the contribution from the soft modes, while the thin lines are the
ones of the free quark gas. One sees that the DPR from the soft modes is
anomalously enhanced in the small $\omega$ and $\mathbf{k}$ region in
comparison with the free quark gas for $T\lesssim 1.5T_{c}$, and this
enhancement becomes more pronounced as $T$ approaches $T_{c}$. This result is
expected through the properties of the soft modes. Shown in the right panel of
Fig. 2 is the DPR near the QCD CP for $T=1.01T_{\rm CP}$, $1.05T_{\rm CP}$ and
$1.2T_{\rm CP}$ at $\mu=\mu_{\rm CP}$, where $(T_{\rm CP},\mu_{\rm
CP})=(35.07,322.20)\leavevmode\nobreak\ {\rm MeV}$ is the location of the QCD
CP. One finds that the anomalous enhancement of the DPR is also observed
similarly to the case of the 2SC.

The low-energy part of the DPR is related to the electrical conductivity
$\sigma$ and relaxation time $\tau$, which are obtained from differentiations
of $\rho_{L}(\mathbf{0},\omega)=\sum_{i}{\rm
Im}{\Pi_{AL}^{R}}^{ii}(\mathbf{0},\omega)/\pi$ at $\omega=0$. The analysis of
$\sigma$ and $\tau$ thus allows us to understand behavior of the DPR near
$T_{c}$ more quantitatively. We find that these quantities diverge at $T_{c}$
with $\sigma\propto\epsilon^{-1/2}$ and $\tau\propto\epsilon^{-3/2}$ for the
2SC, while for the case of the QCD CP, $\sigma\propto\epsilon^{-2/3}$ and
$\tau\propto\epsilon^{-1}$ with $\epsilon=|T-T_{c}|/T_{c}$, as will be
discussed in the forthcoming publication. The origin of the difference of two
cases is due to the $T$-dependence of $A(\mathbf{k})$ and the
$\mathbf{k}$-dependence of $C(\mathbf{k})$ in Eq. (4).

In this report, we studied how the soft modes of the 2SC and QCD CP affect the
DPR near but above $T_{c}$. The modification of the photon self-energy due to
the soft modes are investigated through the analysis of the Aslamasov-Larkin,
Maki-Thompson and density of states terms with the TDGL approximation for the
soft mode propagator and vertices. The enhancement of the DPR found in this
study would allow us to detect these signals of the 2SC and QCD CP in the
future HIC experiments.

## References

  * (1) M.G. Alford, A. Schmitt, K. Rajagopal and T. Schäfer, Rev. Mod. Phys. 80 (2008) 1455.
  * (2) M. Kitazawa, T. Koide, T. Kunihiro and Y. Nemoto, Phys. Rev. D 65 (2002) 091504; ibid., Phys. Rev. D 70 (2004) 056003; ibid., Prog. Theor. Phys. 114 (2005) 117.
  * (3) T. Kunihiro, M. Kitazawa and Y. Nemoto, PoS CPOD07 (2007) 041.
  * (4) T. Nishimura, M. Kitazawa and T.Kunihiro, PTEP 2022 (2022) 093D02.
  * (5) A. I. Larkin and A. A. Varlamov, Springer Berlin Heidelberg (2008).
  * (6) H. Fujii and M. Ohtani, Phys. Rev. D 70 (2004) 014016.


# Tailored Molybdenum Carbide Properties and Graphitic Nano Layer Formation by
Plasma and Ion Energy Control during Plasma Enhanced ALD

Eldad Grady  Marcel Verheijen  Tahsin Faraz  Saurabh Karwal  W.M.M. Kessels
Ageeth A. Bol  Department of Applied Physics, Eindhoven University of
Technology, Den Dolech 2, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

###### Abstract

We demonstrate the extensive study on how film density and crystallinity of
molybdenum carbide ($MoC_{x}$) can be tailored during plasma-enhanced ALD
(PEALD) by controlling either the plasma exposure time or the ion energy. We
investigated $MoC_{x}$ films grown using $Mo(^{t}BuN)_{2}(NMe_{2})_{2}$ as the
precursor and $H_{2}/Ar$ plasma as the co-reactant at temperatures between
150°C and 300°C. We discover a threshold for graphitic layer formation at high
mean ion energies during the PEALD cycle. The supplied high energy dose allows
for hybridised $sp^{2}$ carbon bonds formation, similar to high temperature
annealing. The graphitisation of the $MoC_{x}$ surface takes place at
temperature of 300${}^{\circ}C$. The graphitic film show a (101) plane
diffraction peak with dominant intensity in XRD, and a typical $sp^{2}$ C1s
peak along with carbidic metal in XPS measurements. Surface roughness of the
film lowers significantly at the graphitisation regime of deposition. This low
temperature graphitisation by high energy plasma ions during PEALD shows a
great promise to advancing graphene and graphite composites at low temperature
by PEALD for future applications.

##  I Introduction

Graphene and graphitic based composites are of great interest in research of
next generation electronics, protective layer coatings, and flexible
electronics [1, 2]. Transition metal of group IV-VI have shown to be highly
suitable for the high temperature CVD graphene growth process due to its
thermal stability, and low thermal coefficient matching Si substrates [3]
These characteristics result in lower graphene film stress, as the mismatch
between graphene and underlying catalytic substrate is significantly reduced.
molybdenum carbide ($MoC_{x}$) is a refractory metal carbide compound that
combines the physical properties of ceramics and the electrical properties of
metals, with hardness and mechanical strength (300 - 535 GPa), high thermal
stability, chemical inertness and a metallic like electrical conductivity [4,
5]. The atomic radii ratio of carbon to molybdenum of 0.556 makes it highly
suitable to form interstitial carbides with a mixture of covalent, ionic and
metallic bonds between the metal and the carbon atoms which are responsible
for its unique set of properties [4]. These characteristics are of great
interest for diffusion barriers in ICs, superconductors and various MEMS
applications. Furthermore, molybdenum carbide has made a new addition to the
2D material family named MXenes [6]. As an IC diffusion barrier $MoC_{x}$ is
ideally amorphous and dense, while for superconductivity the cubic $\delta-
MoC_{0.75}$ shows the highest transition temperature (14.3 K) [7] for
$MoC_{x}$ films.

In order to accommodate the broad spectrum of usage, the ability to separately
control the crystallinity and density of $MoC_{x}$ would be an asset for
tailoring film properties to the specific application needs. Thus far, various
techniques have been implemented to synthesize cubic $MoC_{x}$, such as high
energy ion bombardment of amorphous MoC, CVD, PVD and ultra high pressure
synthesis, at high temperatures [8, 9, 10]. While some techniques were
successful in the synthesis, the high temperatures required make it unsuitable
for integration in temperature sensitive devices.

Atomic layer deposition (ALD) offers a cyclic soft deposition technique with a
broad and low temperature window, submonolayer thickness control due to its
self limiting nature, along with high uniformity and conformality. These
merits allow for deposition on temperature sensitive applications as well as
for deposition of layers sensitive for the kinetic ion impact during
sputtering.

While substantial work has been done on various molybdenum compounds such as
molybdenum oxide [11] and nitride , so far, very little work has been done on
ALD of $MoC_{x}$. Recently, a PEALD process using
$Mo(^{t}BuN)_{2}(NMe_{2})_{2}$ precursor with a $H_{2}/N_{2}$ plasma as co-
reactant was developed with the aim of achieving low resistivity films [12].
Bertuch et al. demonstrated the deposition of a range of
$MoC_{x}-MoC_{x}N_{y}$ compounds depending on the $H_{2}/N_{2}$ ratios used. A
predominantly molybdenum carbide material with nitrogen impurities was
produced with pure $H_{2}$ plasma at 150 ∘C, which showed the lowest (170
$\mu{\Omega-cm}$) resistivity. Here, we present the first extensive study on
how film density and crystallinity of $MoC_{x}$ can be tailored independently
during plasma-enhanced ALD (PEALD) by controlling either the plasma exposure
time or the ion energy [13, 14, 15]. We investigated $MoC_{x}$ films grown
using $(^{t}BuN)_{2}(NMe_{2})_{2}Mo$ as the precursor and $H_{2}/Ar$ plasma as
the co-reactant at temperatures between 150∘C and 300∘C. Additionally, we
present the effects of enhancing the impinging ion energy on the physical and
chemical properties of $MoC_{x}$ thin films, controlled via RF biasing of the
substrate table during the plasma step of the PEALD. We discover a threshold
for graphitic layer formation at high mean ion energies during the PEALD
cycle. While plasma assisted graphene fabrication has attracted significant
interest due to the low temperature synthesis [16], to the best of our
knowledge this is the first report of nano graphitic layers formation at low
temperature using PEALD.

##  II Experimental methods

$MoC_{x}$ thin films have been deposited by plasma enhanced atomic layer
deposition (PEALD) at various temperatures and plasma conditions. In this part
the deposition process is explained and the analysis techniques described.

A. Film depositions

Plasma enhanced atomic layer deposition was performed on 100 mm Si (100)
wafers coated with 450 nm of thermally grown $SiO_{2}$. The depositions were
performed in an Oxford instruments FlexAL2 ALD reactor, which is equipped with
an inductively coupled remote RF plasma (ICP) source (13.56 MHz) with alumina
dielectric tube. The reactor was pumped down to a base pressure of $1\cdot
10^{-6}Torr$ with a turbomolecular pump. The samples were loaded and unloaded
from a low pressure loadlock chamber, allowing for a cooldown after deposition
in vacuum conditions. The reactor’s deposition table was set to temperatures
between 150∘C and 300∘C, and the wall temperature to 150∘C.

One ALD cycle consists of subsequently a precursor dose step, a purge step, a
plasma exposure step and again a purge step. The
($Mo(^{t}BuN)_{2}(NMe_{2})_{2}$ precursor ($98\%$, Strem Chemicals) was stored
in a stainless steel container, which was kept at 50 ∘C. The precursor
container was bubbled by an argon flow of 50 sccm during the precursor dose
step, while flowing argon at the same rate from the ICP chamber to avoid
deposition in the ICP tube. The total pressure in the reactor during the 6
seconds long precursor injection was set to 200 mTorr. During the plasma
exposure step, the ICP RF power was set to 100W. $H_{2}/Ar$ with 4:1 ratio and
total flow rate of 50 sccm was fed from the top ICP tube and the plasma was
ignited for the desired exposure time, typically 20 to 80 seconds. Automatic
pressure control (APC) valves were fully opened during plasma exposure to
reduce the pressure to 7 mTorr. Purge steps (4 to 5 seconds) were set after
precursor and plasma half cycles to evacuate residual precursor gas, reaction
byproducts and plasma species with 100 sccm argon flow and open APC valves, at
pressure 25 mTorr.

B. Film analyses

Film thickness and optical properties of the deposited films have been studied
with a J.A. Woollam UV-spectroscopic ellipsometer (SE) using a model
comprising one Drude and two Tauc-Lorenz oscillators. Data was obtained within
the range of 190 nm – 990 nm, and refractive index (n) and extinction
coefficient (k) were determined. Growth per cycle (GPC) was calculated using
in-situ J.A. Woollam IR-SE every 10 deposition cycles. Optical resistivity was
derived from the first term of the Drude oscillator, corresponding to the
imaginary part $\epsilon{{}_{2}}$.

Sheet resistance values of the $MoC_{x}$ films deposited on 450 nm
$SiO_{2}$/Si were determined using a Keithley 2400 SourceMeter and a Signatron
probe. Film thickness was evaluated using ex-situ UV-SE, to calculate the
electrical resistivity at room temperature.

The film composition was analysed by X-ray photoelectron spectroscopy (XPS)
with a Thermo Scientific KA1066 spectrometer, using monochromatic Al $K\alpha$
x-rays with an energy of 1486.6 eV. The films have been sputtered with $Ar+$
ion gun prior to scans, in order to remove surface oxide and adventitious
carbon. A continuous electron flood gun was employed during measurements to
compensate for charging. XPS studies have been performed to evaluate the film
composition as function of plasma conditions. Three main components were
evaluated by their corresponding peaks, namely molybdenum by the mo3d peak,
oxygen by the O1s peak and carbon by the C1s peak. The ratio between each peak
area to the sum of all peaks area gave the partial atomic percentage of the
element. The precursor molecule used for deposition contains 4 nitrogen atoms
bonded to the molybdenum atom, therefore nitrogen is expected to be found in
the film. However, due to overlapping N1s and Mo3p peaks, deconvolution of the
N1s peak is not reliable for low N1s peak intensity.

Rutherford backscattering spectroscopy (RBS) and elastic recoil detection
(ERD) were performed by AccTec BV, Eindhoven, The Netherlands, using a
Singletron with a 2 MeV He+ beam to determine the chemical composition of the
films. ERD was performed with the detector at a recoil angle of 30∘. RBS is
performed at scattering angles of 150∘ and 105∘. The mass density was
calculated using the measured mass density as obtained from RBS/ERD
measurements and the film thickness as obtained from SE. RBS measurement
complement the XPS estimation of film composition, and specifically provide
further information on nitrogen content in the film.

Film crystallinity and preferred crystal orientation was studied by Gonio
x-ray diffraction. Experiments were conducted with PanAnlytical X’pert PROMRD
diffractometer operated using $CuK\alpha(\lambda=1.54A)$. A JEOL ARM 200
transmission electron microscope (TEM) at 200 kV was used to analyse the
microstructure of films deposited on planar TEM windows. These windows
consisted of $\sim 15$ nm $Si_{3}N_{4}$ membranes coated with 5 nm of
$SiO_{2}$ grown using ALD. This ensured a $SiO_{2}$ starting surface while
maintaining transparency to the electron beam. High-angle-annular-dark-field
TEM (HAADF-TEM) modes was employed to characterise the samples.

##  III Results and discussion

Atomic layer film deposition has been performed with
$Mo(^{t}BuN)_{2}(NMe_{2})_{2}$ precursor and 40/10 sccm of $H_{2}/Ar$ plasma.
The growth per cycle (GPC) was measured with an in-situ SE by measuring the
film’s thickness every 10 cycles. Precursor saturation occurs after 6 seconds
of dosing, and plasma time saturation after 20 seconds of exposure. The
saturation curve for the precursor and the plasma time performed at 300∘C is
presented in figure 1. Typical saturated GPC value is 0.365Å. The effect of
temperature on the GPC in saturated conditions has been studied as well for
150∘C and 250∘C yielding values of about 0.25Å(see figure 2). The significant
increase in GPC at 300∘C has been explained elsewhere [11] as precursor
decomposition. However, we see no evidence to that effect, as the precursor
saturation curve confirms no increase in GPC after 6 seconds dosing.
Furthermore, film thickness as measured across a wafer with ex-situ SE, show
high uniformity typical for ALD. Additionally, previous work in our group
using the same precursor for $MoO_{x}$ deposition showed no decomposition at
the same table temperature [12].The characterisation of the $MoC_{x}$
properties with varying temperature provides an insight to observed rise in
GPC.

###  III.1 Plasma Time and Temperature effects on Films

$MoC_{x}$ depositions at 150∘C with 20 seconds plasma have produced films with
C/Mo ratio of 0.7 as estimated by XPS (see table 1). The oxygen content was
estimated to be 12.4% at 150∘C with 20 seconds plasma. Increasing the plasma
exposure time to 80 seconds shows a slight decrease in C/Mo ratio to 0.68 with
a sharp decrease of oxygen content to 5.7%, as demonstrated in figure 3. The
growth per cycle (GPC) for both cases remains unchanged at 0.25Å. All films
deposited at 150∘C show no diffraction peaks in XRD measurements and are
deemed to be completely amorphous. Deposition at 250∘C with 20 seconds plasma
produced a film with 5% hydrogen, 5% nitrogen and C/Mo ratio of 0.63, however
no [O] was detected in the bulk. Hydrogen content could originate both from
the plasma coreactant gas, and the [H] rich precursor molecule, while nitrogen
content originates solely from the precursor molecule, $C_{12}H_{30}N_{4}Mo$,
that contributes 4 [N] atoms. Each of the [N] atoms is directly bonded to the
[Mo], and to CH ligands. The $H_{2}/Ar$ plasma is able to break most N-Mo
bonds with 20s plasma, explaining the relatively low rate of [N] in the film.
The high content of [C] in the film on the other hand, suggests that the
removed ligands are redeposited during the plasma step, ultimately producing a
$MoC_{x}$ film. RBS measurements reveal a film mass density of 8.9
$g/{cm^{3}}$, and [Mo] GPC of 1.33 $\frac{atom}{nm^{2}\times{cycle}}$ and [C]
of 0.83 $\frac{atom}{nm^{2}\times{cycle}}$. The XRD pattern show diffraction
peaks corresponding to the (111) and (220) lattice planes of the delta-cubic
$MoC_{x}$. The (111) peak width appears broad and distorted, along with
several sharper smaller peaks between 28-33${}^{\circ}\,2\theta$. With 80
seconds plasma at 250∘C the C/Mo ratio reduced from 0.63 to 0.56 and the
nitrogen and hydrogen content from 5% to below detection limits ($\pm{1}\%$
and $\pm{5}\%$ respectively), as table 2 shows. The removal of [N] content
supports the role of the $H_{2}/Ar$ plasma in removal of N-Mo bonds, and
redeposition of [C] from the ligands, allbeit [C] removal effects are revealed
with the prolonged plasma exposure. As no [O] was present in the bulk, XPS
shows no significant change in O1s peak (see figure 3) with prolonged plasma,
as oppose to film deposited at 150∘C. Moreover, the film’s mass density has
increased with the longer plasma exposure from 8.9 to 9.2 $g/{cm^{3}}$. Film
deposition at 250∘C and 80 seconds plasma exhibits a single diffraction peak
of the (111) lattice plane, while other diffraction peaks are completely
suppressed (see figure 4). The broad peak width indicates small order
crystalline domains, embedded in mostly amorphous domains. GPC for both
conditions at 250∘C is at 0.25Å, similar to deposition at 150∘C within error
margins. The growth of atoms per $nm^{2}cycle$ show a relatively unchanged
growth in Mo atoms (1.33 and 1.35 for 20s and 80s respectively), while the
growth of C was reduced from 0.83 $\frac{atom}{nm^{2}\times{cycle}}$ to 0.76
$\frac{atom}{nm^{2}\times{cycle}}$ which correlates to the slight increase in
density. The removal of [C] content indicates more efficient removal of
ligands during deposition with prolonged plasma time. Film deposition at 300∘C
with 20 seconds plasma shows a significant increase in GPC from $0.25\AA\,$
for lower temperatures to $0.37\AA$. RBS profiling reveal an increase in C/Mo
ratio from 0.63 to 0.92 upon increasing the temperature from 250∘C to 300∘C.
Atomic growth per cycle a slight increase in Mo (1.35
$\frac{atom}{nm^{2}\times{cycle}}$ at 250∘C to
1.45$\frac{atom}{nm^{2}\times{cycle}}$ at 300∘C), while [C] content increased
by 60% from 0.83 to 1.34 $\frac{atom}{nm^{2}\times{cycle}}$. Moreover, 4% [N]
content was detected in the film by RBS and 8% [H] by ERD measurements. The
increase in film impurities content reflected on the mass density, which
sharply declined from 8.9 $g/{cm^{3}}$ to 7.0 $g/{cm^{3}}$ with the increase
in temperature. Prolonging the plasma exposure time to 80 seconds results in
decreased C/Mo from 0.92 to 0.78, a sharper decrease than that observed at
250∘C. Correspondently, [Mo] atoms deposition rate increases from 1.45 to 1.63
$\frac{atom}{nm^{2}\times{cycle}}$, and [C] decreases from 1.34 to 1.28
$\frac{atom}{nm^{2}\times{cycle}}$. The increase of [Mo] growth is a factor of
10 higher at 300∘C than for the same plasma time at 250∘C
($\Delta{[Mo]}_{250}=0.02\,at./{nm^{2}cycle}$ and
$\Delta{[Mo]}_{300}=0.2\,at./{nm^{2}cycle}$). Additionally, [N] and [H]
content decreased below detection level for 80 seconds plasma. The lower C/Mo
ratio and removal of additional impurities reflected in a higher mass density
of 8.0 $g/{cm^{3}}$. The film deposited with 80 seconds plasma show the
dominant diffraction peak at 36.6∘ $\,2\theta$ of the (111) plane. Comparison
of film deposited with 80 seconds plasma exposure at different temperatures,
shows C/Mo ratio of 0.7 at 150∘C which is reduced to 0.56 at 250∘C and then
increased to 0.78 at 300∘C. By prolonging the plasma exposure time, the total
ion energy dose to the deposition substrate is increased linearly with time
($eV_{ion}/s\times{Plasma_{seconds}}$). The increased energy dose resulted in
higher mass density and correspondently lower C/Mo ratio and mitigation of
film impurities below detection levels. This reflected on the lowered film
resistivity, with two fold decrease from 272 $\mu{\Omega-cm}$ to 143
$\mu{\Omega-cm}$ at 300∘C for 20s and 80s plasma respectively. However, the
effect on film crystallinity is not significant.

Generally, the films deposited at 300∘C show higher peak intensities and
narrower peak widths than films at 250∘C which are mostly amorphous. Hence,
more crystalline material is present with larger crystal grains at higher
temperature, whose effect on crystallinity is more significant than plasma
exposure time.

###  III.2 Biased Substrate effects on ALD Films

The total energy dose to the deposition substrate can be alternatively altered
by increasing the ion energy. By applying radio-frequency (RF) bias voltage to
the substrate we can control the energy of ions impinging on the surface. An
RF bias with time average voltage ($V_{bias}$) between -100V to -200V was
applied to study the effects on the film properties. For these experiments we
use 20 seconds of total plasma time, for a comparison with the saturated non
biased plasma time. The plasma during biased ALD is comprised of 10s non
biased followed by 10s biased plasma.
The mean energy of the ion energy is calculated from the following
equation:$E_{ion}=e\cdot{V_{sheath}}=e\cdot{(V_{plasma}-V_{surface})}$
Where $V_{plasma}$ is the ion energy for 0V bias, measured to be 25eV for 100W
plasma power at 7mTorr pressure, and $V_{surface}$ is the time averaged bias
voltage. Thus for an applied time averaged $V_{b}ias$ ( $<V_{bias}>$) of
-100V, the total mean ion energy is 125 eV. All bias experiments performed
were done at 300∘C, as the largest fluctuation in film density and content
occurs at this temperature. Figure 5 shows the effect of applied bias on the
GPC. When $-100V_{bias}$ was applied the GPC declined sharply from 0.37Å at
300∘C with $<0V_{bias}>$ to 0.22Å. RBS measurements reveal the cause for this
significant decline in a lowered [Mo] and [C] GPC upon applying bias voltage.
[Mo] GPC declined from 1.45 to 1.02 $\frac{atom}{nm^{2}\times{cycle}}$ and [C]
with over 60% decline from 1.34 to 0.81 $\frac{atom}{nm^{2}\times{cycle}}$.
The corresponding C/Mo ratio reduced from 0.92 for the non biased case to 0.79
with $-100V_{bias}$. An effect of $MoC_{x}$ densification is noted, with mass
density measured at 8.2 $g/{cm^{3}}$ for $-100V_{bias}$ compared to a 7.0
$g/{cm^{3}}$ for $<0V_{bias}>$.

Physical alterations are also first seen with the application of surface bias
voltage, giving the typical cubic $\delta-MoC_{0.75}$ structure with
diffraction peaks at 35.75∘ and 41.42∘ $\,2\theta$ (see figure 7). The
dominant peak is the (111) plane, as the (200) plane peak almost completely
suppressed. The peak intensity increased by 2 orders of magnitude upon
applying $-100V_{bias}$, and peak FWHM decreased by half
($0V_{bias}=1.03\,^{\circ}2\theta$ and $-100V_{bias}=0.49\,^{\circ}2\theta$).
Crystallite size has been calculated by fitting the (111) and analysing the
peak broadening using Scherrer equation:

$\tau=K\cdot\lambda/(\beta\cdot\cos{\theta})$

The XRD data points to a highly crystalline material with crystallite size
doubling in comparison for non biased deposition. Crystallite size for 20s pl
without bias is calculated to be 9.0 nm while extending the plasma time yields
a crystallite size of 7.6 nm, however this slight decrease could be attributed
to increased strain in the denser film with 80s pl. When $-100V_{bias}$ bias
is applied, crystallite doubles in size to 19.4 nm. As can be seen in figure
8, the plan view high angle annular dark-field (HAADF) TEM images support
these findings. The size of the visible structured material is significantly
increased and voids previously seen for non biased deposition not detectable
with $-100V_{bias}$.

###  III.3 Graphitisation during PEALD

With increased $V_{bias}$, further physical alterations and formation of
graphitic nano layers are revealed in the $MoC_{x}$ film. With
$<-135V_{bias}>$ applied, the (111) lattice peak of the cubic $\delta-
MoC_{0.75}$ is diminished, and new diffraction peak appears corresponding to
crystalline graphite. The emerging peak at 44.5$\,2\theta$∘ match the (101)
graphite lattice plane and is the dominating peak intensity, indicating strong
graphitization with higher ion energies ($E_{ion}=155eV$). The observed
emerging phase is consistent with increasing ion energy further
($-187V_{bias},\,E_{ion}=212eV$), however the (101) peak intensity is somewhat
diminished in comparison to the previous bias step at $-130V_{bias}$. The GPC
drops continuesly with increasing $V_{bias}$, from 0.22Å for -100V to 0.15Å
and 0.14Å for -135V and -187V respectively. While C/Mo ratio dropped with the
application of $-100V_{bias}$ from 0.92 for non biased ALD to 0.79, this ratio
has stagnated for higher applied $V_{bias}$, and the C/Mo ratio remained at
$\sim$0.80.

###  III.4 Plasma Time and Bias Effects on Film Resistivity

The effect of plasma time on $MoC_{x}$ film resistivity has been studied using
four-point probe (4PP) measurements. Film of comparable thickness of 30 nm
have been deposited at 300∘C with plasma time ranging from 20 seconds (begin
of saturation) up to 80 seconds. Difference in film thickness is within SE
error margin. As shown in figure 10, a significant decrease in resistivity was
measured with increase of plasma time. For 20 seconds plasma resistivity was
272 $\mu{\Omega-cm}$ and a decrease to 242 $\mu{\Omega-cm}$ is seen with
double the plasma time (40 seconds). Further increase of plasma time to 60
seconds decreased resistivity to 181 $\mu{\Omega-cm}$, and at the final plasma
time of 80 seconds 143 $\mu{\Omega-cm}$ was measured. By increasing plasma
time from 20 s to 40 s, XPS measurements revealed a reduction of C/Mo ratio
from 0.96 to 0.85. With a total 80 s plasma time, the C/Mo ratio decreased
further to 0.79. As was demonstrated above, the reduction in C/Mo ratio
increased mass density of the $MoC_{x}$ film, and the higher ratio of [Mo] to
[C] improves its electrical conductivity.

RF bias of the substrate table of average $-100V_{bias}$ has been applied for
the duration of 10 seconds, after 10 seconds of non-biased plasma exposure.
Film resistivity was measured for films of comparable thickness of 30 nm. We
correlate the change in resistivity to GPC and density rates to gain further
indications of chemical and physical effects to the film. Figure 5 depicts GPC
for biased ALD, and figure 12 the resistivity for the corresponding bias
voltages. With $-100V_{bias}$, film resistivity was decreased to 144 and GPC
ratio was decreased from 0.36 (for $0V_{bias})$) to 0.22.

and 146 $\mu{\Omega-cm}$ respectively. The GPC ratio was decreased from 0.22 Å
for -100V to 0.16Å for -135V $<V_{bias}>$. As shown above, the C/Mo reduced
upon application of bias voltage from 0.92 to $\sim$ 0.8, which reflected in
higher mass density. The produced denser and more metallic film translated to
better electrical conductivity. Increasing the bias voltage to -187V gives a
slight rise in resistivity to 156 $\mu{\Omega-cm}$ and further decrease in GPC
to 0.15Å. At -210V $<V_{bias}>$ a significant increase to 200 $\mu{\Omega-cm}$
is shown with 0.14Å GPC.

##  IV Conclusions

We presented here the ability to tailor $MoC_{x}$ film by controlling the ion
energy and the total energy dose during PEALD. Additionally, we discover
graphitic layer formation during the PEALD cycle with mean ion energies
between 150eV to 220eV. $sp^{2}$ hybridised carbon bonds appear to form during
the high energy dose to the film surface with average bias voltage is
increased to -135$V_{b}ias$. Furthermore, the effects of plasma time exposure,
and the ion energy in the plasma during PEALD modification of the $MoC_{x}$
film electrical, chemical and physical properties were investigated. Both
prolonged plasma and biasing are successful in mitigating [N] impurities in
the film, and reducing C/Mo ratio. Consequently, a significantly lowered film
resistivity was attained of 143 $\mu{\Omega-cm}$. Increasing the total energy
dose, which is a product of the mean $E_{i}$ and the total plasma exposure
time. With grounded substrate table ($<0V_{bias}>$) we can increase the total
energy dose by prolonging the plasma time, with mean ion energy fixed at 25eV.
By increasing the total plasma time from 20s to 80s, we increased film density
from 7 to 8, while retaining the film physical properties. With a fixed plasma
exposure time of 20 seconds, we increased the mean ion energy by substrate
biasing, thus increasing the total energy dose as well. We demonstrated an
increase in mass density with increased total energy dose, while no
significant change to crystallinity for the same $E_{i}$. With substrate bias
we elevated the impinging ion energies, and demonstrated physical effects to
the film. With mean ion energy of 125eV, a highly crystalline and dense film
was fabricated corresponding the cubic $\delta-{MoC_{0.79}}$. We can achieve
the same high density, low resistivity film in 20s plasma exposure time by
biasing, as with long plasma exposure time of 80s without biasing. However, by
increasing the mean ion energy, we see physical alterations similar to
annealing effects on film, substituting the thermal energy with plasma ion
energy. With mean ion energy of $\sim{200}eV$ we see the most crystalline
graphitic layer formation, indicating an optimal energy region for graphene
formation. The diminishing graphite crystallinity beyond mean ion energy of
200 eV indicates deterioration of the graphite top layer by high energy
$ArH^{+}$ ions. The increase in resistivity with further decrease in GPC for
$<V_{bias}>$ beyond -135V suggest microstructural changes to the film, more so
for the higher -210V. These changes could be in form of voids due to increased
ion energy bombardment. The bi-modal distribution of ion energy shows that
part of the incident ion have energies higher than -210 eV. The diminished
intensity and increased resistivity suggest that these high energy ions at the
higher end of the spectrum, could have a sputtering effect on the graphitic
top layer when impinging on the surface. Further work needs to be done in
optimising the graphene layer by exploring ideal RF bias plasma exposure time,
and mean ion energy, as the two determine the total energy dose to the film
surface and quality of graphitisation. We postulate, that the high energy ions
are able to break Mo-C bonds and achieve a dominantly graphitic film in the
[C] rich atmosphere. A trade-off between ion energy and the plasma exposure
time needs to be explored, to fine tune the graphitisation.

The applied RF bias voltage effect on film crystallinity and composition is
shown to be critical. By controlling the mean ion energy of the impinging ions
on the surface of the deposited film, we gain a powerful mean of influence on
crystallinity and density alike. These coupled physical and chemical effects
are derived from the composition of the coreactant $H_{2}/Ar$ gas in the
plasma. [H] interaction with the surface is mainly chemical in nature, while
the heavier Ar atoms contribute to the physical effects with their kinetic
energy. The produced $ArH^{+}$ ions in the plasma combine both these features,
hence the coupled chemical and physical effects. By varying the ion energy, we
alter predominantly the kinetic impact effect on film, therefore the sharp
changes to the $MoC_{x}$ crystallinity.

In conclusion, we demonstrated the control of mass density independent from
film crystallinity, and a method to control both features by PEALD, without
post deposition thermal annealing treatments, or high temperature deposition.
Furthermore, this work paves the way to fabrication of graphene layers with
PEALD, crucial for the production of stacked layers, with atomic precision and
high film purity. Moreover, the fabrication can be performed in temperatures
as low as 300${}^{\circ}C$. This effect could play a significant role in
various applications that require low thermal budget. The ability of control
in PEALD of $MoC_{x}$ opens a window for the integration of tailor-made thin
$MoC_{x}$, as well as low temperature graphene films in a wide range of
applications previously unavailable.

### Acknowledgements

E. Grady would like to acknowledge the financial support of the Dutch
Technology Foundation STW (project number 140930), which is part of the
Netherlands Organisation for Scientific Research (NWO). E. Grady thanks
Cristian van Helvoirt for XRD measurements, Jeroen Gerwen and Janneke
Zeebregts for their technical support.

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| XPS | 4PP
---|---|---
Temperature (∘C) |  Plasma time (s) |  Oxygen (atomic%) |  $\frac{C}{Mo}$ ratio |  Resistivity $\mu{\Omega-cm}$
150 |  20 |  12.4 |  0.70 |  154
150 |  80 |  5.6 |  0.72 |  145
Table 1: Atomic % of elements by XPS and electrical resistivity measurements

| ERD | XPS | RBS
---|---|---|---
Temperature (∘C) |  Plasma time(s) |  [H] (atomic%) |  $\frac{C}{Mo}$ ratio |  $\frac{C}{Mo}$ ratio |  [N] (atomic%) |  [Mo] ($\frac{atom}{nm^{2}\times{cycle}}$) |  [C] ($\frac{atom}{nm^{2}\times{cycle}}$) |  Density ($\frac{g}{cm^{3}}$)
150 |  20 |  – |  0.70 |  – |  – |  – |  – |  –
150 |  80 |  – |  0.72 |  – |  – |  – |  – |  –
250 |  20 |  5 $\pm{5}$ |  0.64 |  0.63 |  5 $\pm{1}$ |  1.33 |  0.83 |  8.86
250 |  80 |  $<d.l.$ |  0.75 |  0.56 |  1 |  1.35 |  0.76 |  9.2
300 |  20 |  8 |  0.96 |  0.92 |  3.5 |  1.45 |  1.34 |  7
300 |  80 |  $<d.l.$ |  0.91 |  0.78 |  $<d.l.$ |  1.63 |  1.28 |  8
$<d.l.$: below detection limits of the element. |
Table 2: Atomic % of elements and film density by RBS measurements of non
Biased $MoC_{x}$ at various temperatures and plasma times.

| ERD | RBS
---|---|---
Temperature (∘C) |  Plasma time(s) |  [H] (atomic%) |  [N] (atomic%) |  $\frac{C}{Mo}$ ratio |  [Mo] ($\frac{atom}{nm^{2}\times{cycle}}$) |  [C] ($\frac{atom}{nm^{2}\times{cycle}}$) |  Density ($\frac{g}{cm^{3}}$)
0 |  20 |  8 |  4 |  0.92 |  1.45 |  1.34 |  7.0
0 |  80 |  $<d.l.$ |  $<d.l.$ |  0.78 |  1.63 |  1.28 |  8.0
-100 |  20 |  $<d.l.$ |  $<d.l.$ |  0.79 |  1.02 |  0.81 |  8.2
$<d.l.$: below detection limits of the elements.
Table 3: Atomic % of elements and film density by RBS measurements of non
Biased $MoC_{x}$ at various temperatures and plasma times.

Figure 1: Growth per cycle (GPC) of $MoC_{x}$ deposited at 300∘C determined
using a in-situ SE, as a function of (a) precursor dose time (b) plasma
exposure time. The lines serve as a guide to the eye Figure 2: Growth per
cycle (GPC) during initial 500 cycles. $MoC_{x}$ films deposited using
($Mo(^{t}BuN)_{2}(NMe_{2})_{2}$ precursor and$H_{2}/Ar$ plasma (without
substrate biasing) as a function of deposition temperature

Figure 3: XPS core level spectra of the O1s peak for $\sim$22 nm $MoC_{x}$
films deposited at (a) 150 °C and (b) 250 °C. The red and blue lines represent
films deposited using $H_{2}/Ar$ plasma exposure times of 20s and 80s,
respectively Figure 4: $\theta$-$2\theta$ x-ray diffractograms for 30 nm MoC
films deposited using different temperatures and $H_{2}/Ar$ plasma exposure
conditions (different plasma exposure times without biasing) Figure 5: Growth
per cycle (left axis) and C/Mo ratio (right axis) of $MoC_{x}$ films deposited
at 300∘C as a function of the time-averaged bias voltage applied during the
$H_{2}/Ar$ plasma exposure step. Figure 6: $\theta$-$2\theta$ x-ray
diffractograms for 30 nm $MoC_{x}$ films deposited at 300∘C without any
substrate biasing with 20, and with a time-averaged bias voltage ranging from
-100V to -187V applied during the last half (10 s) of the 20 s $H_{2}/Ar$
plasma exposure step. Figure 7: $\theta$-$2\theta$ x-ray diffractograms for 30
nm MoCs films deposited at 300∘C without any substrate biasing with 20 and 80
s plasma, and with a time-averaged bias voltage of -100V applied during the
last half (10 s) of the 20 second $H_{2}/Ar$ plasma exposure step. Figure 8:
Plan-view high angle annular dark-field (HAADF) TEM images for molybdenum
carbide films deposited at 300C with (a) 0 V or no bias, 20 s plasma and (b) 0
V 80 s plasma, and (c) -100 V (d) -135 V and (e) -210 V bias applied during
the last half (10 s) of the 20 s $H_{2}/Ar$ plasma exposure step. Figure 9:
Plan-view high angle annular dark-field (HAADF) TEM images for molybdenum
carbide films deposited at 300C with (a) 0 V or no bias, 20 s plasma and (b) 0
V 80 s plasma, and (c) -100 V (d) -135 V and (e) -210 V bias applied during
the last half (10 s) of the 20 s $H_{2}/Ar$ plasma exposure step. Figure 10:
Resistivity of $\sim{30}$ nm $MoC_{x}$ films deposited at 300∘C expressed as a
function of the$H_{2}/Ar$ plasma exposure time. The resistivities were
measured using four point probe. Figure 11: Resistivity of $\sim{30}$ nm
$MoC_{x}$ films deposited at 300∘C with time-averaged bias voltages ranging
from 0 V (i.e., no biasing) to -210 V applied during the last 10 seconds of
the 20 seconds $H_{2}/Ar$ plasma exposure step. The resistivities measured
using four point probe are denoted by the stars while those derived using
spectroscopic ellipsometry are denoted using squares. Centre: resistivity of
$\sim{30}$ nm $MoC_{x}$ films deposited at 300∘C expressed as a function of
the $H_{2}/Ar$ plasma exposure conditions. Figure 12: Properties of $\sim{30}$
nm $MoC_{x}$ films deposited at 300∘C with time-averaged bias voltages ranging
from 0 V (i.e., no biasing) to 210 V applied during the last 10 seconds of the
20 seconds $H_{2}/Ar$ plasma exposure step. Graphitisation regime begins with
mean ion energies higher than 125eV (-100$V_{b}ias$), and shows a drastic
decrease in surface roughness, and a peak in mass density at -135$V_{b}ias$.