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\section{Introduction}
Let $G$ be a simple undirected graph with the \textit{vertex set} $V(G)$ and the \textit{edge set} $E(G)$. A vertex with degree one is called a \textit{pendant vertex}. The distance between the vertices $u$ and $v$ in graph $G$ is de
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noted by $d_G(u,v)$. A cycle $C$ is called \textit{chordless} if $C$ has no \textit{cycle chord} (that is an edge not in the edge set of $C$ whose endpoints lie on the vertices of $C$).
The \textit{Induced subgraph} on vertex set $S$ is denoted by $\langle
|
S\rangle$. A path that starts in $v$ and ends in $u$ is denoted by $\stackrel\frown{v u}$.
A \textit{traceable} graph is a graph that possesses a Hamiltonian path.
In a graph $G$, we say that a cycle $C$ is \textit{formed by the path} $Q$ if $ | E(C) \set
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minus E(Q) | = 1 $. So every vertex of $C$ belongs to $V(Q)$.
In 2011 the following conjecture was proposed:
\begin{conjecture}(Hoffmann-Ostenhof \cite{hoffman})
Let $G$ be a connected cubic graph. Then $G$ has a decomposition into a spanning tree, a matc
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hing and a family of cycles.
\end{conjecture}
Conjecture \theconjecture$\,$ also appears in Problem 516 \cite{cameron}. There are a few partial results known for Conjecture \theconjecture. Kostochka \cite{kostocha} noticed that the Petersen graph, the pri
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sms over cycles, and many other graphs have a decomposition desired in Conjecture \theconjecture. Ozeki and Ye \cite{ozeki} proved that the conjecture holds for 3-connected cubic plane graphs. Furthermore, it was proved by Bachstein \cite{bachstein} that C
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onjecture \theconjecture$\,$ is true for every 3-connected cubic graph embedded in torus or Klein-bottle. Akbari, Jensen and Siggers \cite[Theorem 9]{akbari} showed that Conjecture \theconjecture$\,$ is true for Hamiltonian cubic graphs.
In this paper, we
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show that Conjecture \theconjecture$\,$ holds for traceable cubic graphs.
\section{Results}
Before proving the main result, we need the following lemma.
\begin{lemma}
\label{lemma:1}
Let $G$ be a cubic graph. Suppose that $V(G)$ can be partitioned into a
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tree $T$ and finitely many cycles such that there is no edge between any pair of cycles (not necessarily distinct cycles), and every pendant vertex of $T$ is adjacent to at least one vertex of a cycle. Then, Conjecture \theconjecture$\,$ holds for $G$.
\en
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d{lemma}
\begin{proof}
By assumption, every vertex of each cycle in the partition is adjacent to exactly one vertex of $T$. Call the set of all edges with one endpoint in a cycle and another endpoint in $T$ by $Q$.
Clearly, the induced subgraph on $E(T) \c
|
up Q$ is a spanning tree of $G$. We call it $T'$. Note that every edge between a pendant vertex of $T$ and the union of cycles in the partition is also contained in $T'$. Thus, every pendant vertex of $T'$ is contained in a cycle of the partition. Now, con
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sider the graph $H = G \setminus E(T')$. For every $v \in V(T)$, $d_H(v) \leq 1$. So Conjecture \theconjecture$\,$ holds for $G$. \vspace{1em}
\end{proof}
\noindent\textbf{Remark 1.}
\label{remark:1}
Let $C$ be a cycle formed by the path $Q$. Then clearl
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y there exists a chordless cycle formed by $Q$.
Now, we are in a position to prove the main result.
\begin{theorem}
Conjecture \theconjecture$\,$ holds for traceable cubic graphs.
\end{theorem}
\begin{proof}
Let $G$ be a traceable cubic graph and $P : v_
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1, \dots, v_n$ be a Hamiltonian path in $G$. By \cite[Theorem 9]{akbari}, Conjecture A holds for $v_1 v_n \in E(G)$. Thus we can assume that $v_1 v_n \notin E(G)$. Let $v_1 v_j, v_1 v_{j'}, v_i v_n, v_{i'} v_n \in E(G)\setminus E(P)$ and $j' < j < n$, $
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1 < i < i'$. Two cases can occur:
\begin{enumerate}[leftmargin=0pt,label=]
\item
\textbf{Case 1.}
Assume that $i < j$. Consider the following graph in Figure \ref{fig:overlapping} in which the thick edges denote the path $P$. Call the three paths between $
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v_j$ and $v_i$, from the left to the right, by $P_1$, $P_2$ and $P_3$, respectively (note that $P_1$ contains the edge $e'$ and $P_3$ contains the edge $e$).
\begin{figure}[H]
\begin{center}
\includegraphics[width=40mm]{engImages/overlapping.pdf}
|
\caption{Paths $P_1$, $P_2$ and $P_3$}
\label{fig:overlapping}
\end{center}
\end{figure}
If $P_2$ has order $2$, then $G$ is Hamiltonian and so by \cite[Theorem 9]{akbari} Conjecture \theconjecture$\,$ holds. Thus we can assume that $P_1$, $P_2$
|
and $P_3$ have order at least $3$. Now, consider the following subcases:\\
\begin{enumerate}[leftmargin=0pt,label=]
\label{case:1}
\item \textbf{Subcase 1.} There is no edge between $V(P_r)$ and $V(P_s)$ for $1 \leq r < s \leq 3$. Since every vertex of $P
|
_i$ has degree 3 for every $i$, by \hyperref[remark:1]{Remark 1}$\,$ there are two chordless cycles $C_1$ and $C_2$ formed by $P_1$ and $P_2$, respectively.
Define a tree $T$ with the edge set
$$ E\Big(\langle V(G) \setminus \big(V(C_1) \cup V(C_2)\big) \r
|
angle\Big) \bigcap \big(\bigcup_{i=1}^3 E(P_i)\big).$$
Now, apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition $\{T, C_1, C_2\}$.\\
\item \textbf{Subcase 2.}
\label{case:edge}
There exists at least one edge between some $P_r$ and $P_s$, $r<s$. With n
|
o loss of generality, assume that $r=1$ and $s=2$. Suppose that $ab \in E(G)$, where $a \in V(P_1)$, $b \in V(P_2)$ and $d_{P_1}(v_j, a) + d_{P_2}(v_j, b)$ is minimum.
\begin{figure}[H]
\begin{center}
\includegraphics[width=40mm]{engImages/ab.pdf}
|
\caption{The edge $ab$ between $P_1$ and $P_2$}
\label{fig:ab}
\end{center}
\end{figure}
Three cases occur: \\
(a) There is no chordless cycle formed by either of the paths $\stackrel\frown{v_j a}$ or $\stackrel\frown{v_j b}$. Let $C$ be the cho
|
rdless cycle $\stackrel\frown{v_j a}\stackrel\frown{ b v_j}$. Define $T$ with the edge set
$$ E\Big(\langle V(G) \setminus V(C)\rangle\Big) \bigcap \big(\bigcup_{i=1}^3 E(P_i)\big).$$
Now, apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition $\{T,C\}$.
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\\
(b) There are two chordless cycles, say $C_1$ and $C_2$, respectively formed by the paths $\stackrel\frown{v_j a}$ and $\stackrel\frown{v_j b}$. Now, consider the partition $C_1$, $C_2$ and the tree induced on the following edges,
$$E\Big(\langle V(G)
|
\setminus \big(V(C_1) \cup V(C_2)\big) \rangle\Big) \; \bigcap \; E\Big(\bigcup_{i=1}^3 P_i\Big),$$
and apply \hyperref[lemma:1]{Lemma 1}.\\
(c) With no loss of generality, there exists a chordless cycle formed by the path $\stackrel\frown{v_j a}$ and the
|
re is no chordless cycle formed by the path $\stackrel\frown{v_j b}$.
First, suppose that for every chordless cycle $C_t$ on $\stackrel\frown{v_j a}$, at least one of the vertices of $C_t$ is adjacent to a vertex in $V(G) \setminus V(P_1)$.
We call one of
|
the edges with one end in $C_t$ and other endpoint in $V(G) \setminus V(P_1)$ by $e_t$. Let $v_j=w_0, w_1, \dots, w_l=a$ be all vertices of the path $\stackrel\frown{v_j a}$ in $P_1$. Choose the shortest path $w_0 w_{i_1} w_{i_2} \dots w_l$ such that $0 <
|
i_1 < i_2 < \dots < l$.
Define a tree $T$ whose edge set is the thin edges in Figure \ref{fig:deltaCycle}.\\
Call the cycle $w_0 w_{i_1} \dots w_l \stackrel\frown{b w_0}$ by $C'$. Now, by removing $C'$, $q$ vertex disjoint paths $Q_1, \dots, Q_q$ which are
|
contained in $\stackrel\frown{v_j a}$ remain. Note that there exists a path of order $2$ in $C'$ which by adding this path to $Q_i$ we find a cycle $C_{t_i}$, for some $i$. Hence there exists an edge $e_{t_i}$ connecting $Q_i$ to $V(G) \setminus V(P_1)$.
|
Now, we define a tree $T$ whose the edge set is,
$$\quad\quad\quad \bigg( E\Big(\langle V(G) \setminus V(C') \rangle \Big)\; \bigcap \; \Big(\bigcup_{i=1}^3 E(P_i)\Big) \bigg) \bigcup \Big(\big\{e_{t_i} \mid 1 \leq i \leq q \big\} \Big).$$
Apply \hyperref[
|
lemma:1]{Lemma 1} $\,$for the partition $\{T,C'\}$.\\
\begin{figure}[H]
\begin{center}
\includegraphics[width=40mm]{engImages/deltaCycle.pdf}
\caption{The cycle $C'$ and the tree $T$}
\label{fig:deltaCycle}
\end{center}
\end{figure}
Ne
|
xt, assume that there exists a cycle $C_1$ formed by $\stackrel\frown{v_j a}$ such that none of the vertices of $C_1$ is adjacent to $V(G) \setminus V(P_1)$. Choose the smallest cycle with this property. Obviously, this cycle is chordless. Now, three cases
|
can be considered:\\
\begin{enumerate}[leftmargin=5pt,label=(\roman*)]
\item There exists a cycle $C_2$ formed by $P_2$ or $P_3$. Define the partition $C_1$, $C_2$ and a tree with the following edge set,
$$E\Big(\langle V(G) \setminus \big(V(C_1) \cup V(
|
C_2)\big)\rangle \Big) \bigcap \Big( \bigcup_{i=1}^3 E(P_i) \Big),$$
and apply \hyperref[lemma:1]{Lemma 1}.\\
\item There is no chordless cycle formed by $P_2$ and by $P_3$, and there is at least one edge between $V(P_2)$ and $V(P_3)$. Let $ab \in E(G)$,
|
$a \in V(P_2)$ and $b \in V(P_3)$ and moreover $d_{P_2}(v_j, a) + d_{P_3}(v_j,b)$ is minimum. Notice that the cycle $\stackrel\frown{v_j a} \stackrel\frown{b v_j}$ is chordless. Let us call this cycle by $C_2$. Now, define the partition $C_2$ and a tree wi
|
th the following edge set,
$$E\Big(\langle V(G) \setminus V(C_2)\rangle \Big) \bigcap \Big( \bigcup_{i=1}^3 E(P_i) \Big),$$
and apply \hyperref[lemma:1]{Lemma 1}.\\
\item There is no chordless cycle formed by $P_2$ and by $P_3$, and there is no edge betwe
|
en $V(P_2)$ and $V(P_3)$. Let $C_2$ be the cycle consisting of two paths $P_2$ and $P_3$. Define the partition $C_2$ and a tree with the following edge set,
$$E\Big(\langle V(G) \setminus V(C_2)\rangle \Big) \bigcap \Big( \bigcup_{i=1}^3 E(P_i) \Big),$$
an
|
d apply \hyperref[lemma:1]{Lemma 1}.
\end{enumerate}
\end{enumerate}
\vspace{5mm}
\item
\textbf{Case 2.}
\label{case:2}
Assume that $j < i$ for all Hamiltonian paths. Among all Hamiltonian paths consider the path such that $i'-j'$ is maximum. Now, thr
|
ee cases can be considered:\\
\begin{enumerate}[leftmargin=0pt,label=]
\item \textbf{Subcase 1.} There is no $s < j'$ and $t > i'$ such that $v_s v_t \in E(G)$. By \hyperref[remark:1]{Remark 1} $\,$ there are two chordless cycles $C_1$ and $C_2$, respecti
|
vely formed by the paths $v_1 v_{j'}$ and $v_{i'} v_n$. By assumption there is no edge $xy$, where $x \in V(C_1)$ and $y \in V(C_2)$.
Define a tree $T$ with the edge set:
$$ E\Big(\langle V(G) \setminus \big(V(C_1) \cup V(C_2)\big) \rangle \Big) \bigcap \B
|
ig( E(P) \cup \{v_{i'}v_n, v_{j'}v_1\} \Big).$$
Now, apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition $\{T, C_1, C_2\}$.\\
\item \textbf{Subcase 2.}
\label{subcase:22} There are at least four indices $s, s' < j$ and $t, t' > i$ such that $v_s v_t,
|
v_{s'} v_{t'} \in E(G)$. Choose four indices $g, h < j$ and $e, f > i$ such that $v_h v_e, v_g v_f \in E(G)$ and $|g-h| + |e-f|$ is minimum.
\begin{figure}[H]
\begin{center}
\includegraphics[width=90mm]{engImages/case2-subcase2.pdf}
\caption{Two
|
edges $v_h v_e$ and $v_g v_f$}
\label{fig:non-overlapping}
\end{center}
\end{figure}
Three cases can be considered:\\
\begin{enumerate}[leftmargin=0pt,label=(\alph*)]
\item There is no chordless cycle formed by $\stackrel\frown{v_g v_h}$ and by $\
|
stackrel\frown{v_e v_f}$.
Consider the cycle $\stackrel\frown{v_g v_h} \stackrel\frown{v_e v_f}v_g$ and call it $C$. Now, define a tree $T$ with the edge set,
$$\,\,\,E\Big(\langle V(G) \setminus V(C)\rangle \Big) \bigcap \Big( E(P) \cup \{v_1v_{j}, v_{i}
|
v_n\} \Big),$$
apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition $\{T, C\}$.\\
\item With no loss of generality, there exists a chordless cycle formed by $\stackrel\frown{v_e v_f}$ and there is no chordless cycle formed by the path $\stackrel\frown{
|
v_g v_h}$. First suppose that there is a chordless cycle $C_1$ formed by $\stackrel\frown{v_e v_f}$ such that there is no edge between $V(C_1)$ and $\{v_1, \dots, v_j\}$. By \hyperref[remark:1]{Remark 1} $,$ there exists a chordless cycle $C_2$ formed by $
|
\stackrel\frown{v_1 v_j}$. By assumption there is no edge between $V(C_1)$ and $V(C_2)$. Now, define a tree $T$ with the edge set,
$$\quad\quad\quad\quad E\Big(\langle V(G) \setminus \big(V(C_1) \cup V(C_2)\big)\rangle \Big) \bigcap \Big( E(P) \cup \{v_1v
|
_{j}, v_{i}v_n\} \Big),$$
and apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition $\{T, C_1, C_2\}$.
$\;$ Next assume that for every cycle $C_r$ formed by $\stackrel\frown{v_e v_f}$, there are two vertices $x_r \in V(C_r)$ and $y_r \in \{v_1, \dots,
|
v_j\}$ such that $x_r y_r \in E(G)$. Let $v_e=w_0, w_1, \dots, w_l=v_f$ be all vertices of the path $\stackrel\frown{v_e v_f}$ in $P$. Choose the shortest path $w_0 w_{i_1} w_{i_2} \dots w_l$ such that $0 < i_1 < i_2 < \dots < l$. Consider the cycle $w_0 w
|
_{i_1} \dots w_l \stackrel\frown{v_g v_h}$ and call it $C$. Now, by removing $C$, $q$ vertex disjoint paths $Q_1, \dots, Q_q$ which are contained in $\stackrel\frown{v_e v_f}$ remain. Note that there exists a path of order $2$ in $C$ which by adding this p
|
ath to $Q_i$ we find a cycle $C_{r_i}$, for some $i$. Hence there exists an edge $x_{r_i} y_{r_i}$ connecting $Q_i$ to $V(G) \setminus V(\stackrel\frown{v_e v_f})$. We define a tree $T$ whose edge set is the edges,
$$\quad\quad\quad\quad\quad\quad E\Big(\l
|
angle V(G) \setminus V(C)\rangle \Big) \bigcap \Big( E(P) \cup \{v_1v_{j}, v_{i}v_n\} \cup \big\{x_{r_i} y_{r_i} \mid 1 \leq i \leq q\big\} \Big),$$
then apply \hyperref[lemma:1]{Lemma 1} $\,$ on the partition $\{T, C\}$.\\
\begin{figure}[H]
\begin{cente
|
r}
\includegraphics[width=90mm]{engImages/deltaNonOverlapping.pdf}
\caption{The tree $T$ and the shortest path $w_0 w_{i_1}\dots w_l$}
\label{fig:delta-non-overlapping}
\end{center}
\end{figure}
\item There are at least two chordless cycles,
|
say $C_1$ and $C_2$ formed by the paths $\stackrel\frown{v_g v_h}$ and $\stackrel\frown{v_e v_f}$, respectively. Since $|g-h| + |e-f|$ is minimum, there is no edge $xy \in E(G)$ with $x \in V(C_1)$ and $y \in V(C_2)$. Now, define a tree $T$ with the edge
|
set,
$$\quad\quad\quad\quad E\Big( \langle V(G) \setminus \big(V(C_1) \cup V(C_2)\big) \rangle \Big) \bigcap \Big( E(P) \cup \{v_1 v_{j}, v_{i}v_n\} \Big),$$
and apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition $\{T, C_1, C_2\}$.\\
\end{enumerate}
|
\item \textbf{Subcase 3.} There exist exactly two indices $s,t$, $s < j' < i' < t$ such that $v_s v_t \in E(G)$ and there are no two other indices $s', t'$ such that $s' < j < i < t'$ and $v_{s'} v_{t'} \in E(G)$. We can assume that there is no cycle form
|
ed by $\stackrel\frown{v_{s+1} v_j}$ or $\stackrel\frown{v_i v_{t-1}}$, to see this by symmetry consider a cycle $C$ formed by $\stackrel\frown{v_{s+1} v_j}$. By \hyperref[remark:1]{Remark 1} $\,$ there exist chordless cycles $C_1$ formed by $\stackrel\fr
|
own{v_{s+1} v_j}$ and $C_2$ formed by $\stackrel\frown{v_{i} v_n}$. By assumption $v_s v_t$ is the only edge such that $s < j$ and $t > i \;$. Therefore, there is no edge between $V(C_1)$ and $V(C_2)$. Now, let $T$ be a tree defined by the edge set,
$$ E
|
\Big(\langle V(G) \setminus \big(V(C_1) \cup V(C_2)\big)\rangle \Big) \bigcap \Big( E(P) \cup \{v_1v_{j}, v_{i}v_n\} \Big),$$
and apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition \{$T$, $C_1$, $C_2$\}.\\
$\quad$Furthermore, we can also assume that
|
either $s \neq j'-1$ or $t \neq i'+1$, otherwise we have the Hamiltonian cycle $\stackrel\frown{v_1 v_s} \stackrel\frown{v_t v_n} \stackrel\frown{v_{i'} v_{j'}} v_1$ and by \cite[Theorem 9]{akbari} Conjecture \theconjecture$\,$ holds.
$\quad$By symmetry,
|
suppose that $s \neq j'-1$. Let $v_k$ be the vertex adjacent to $v_{j'-1}$, and $k \notin \{j'-2, j'\}$. It can be shown that $k > j'-1$, since otherwise by considering the Hamiltonian path $P': \; \stackrel\frown{ v_{k+1} v_{j'-1}}\stackrel\frown{v_k v_1
|
} \stackrel\frown{v_{j'} v_n}$, the new $i'-j'$ is greater than the old one and this contradicts our assumption about $P$ in the \hyperref[case:2]{Case 2}.
$\quad$We know that $j' < k < i$. Moreover, the fact that $\stackrel\frown{v_{s+1} v_j}$ does not
|
form a cycle contradicts the case that $j' < k \le j$. So $j < k < i$. Consider two cycles $C_1$ and $C_2$, respectively with the vertices $v_1 \stackrel\frown{v_{j'} v_{j}} v_1$ and $v_n \stackrel\frown{v_{i'} v_{i}} v_n$. The cycles $C_1$ and $C_2$ are
|
chordless, otherwise there exist cycles formed by the paths $\stackrel\frown{v_{s+1} v_j}$ or $\stackrel\frown{v_i v_{t-1}}$. Now, define a tree $T$ with the edge set
$$ E\Big(\langle V(G) \setminus \big(V(C_1) \cup V(C_2)\big)\rangle \Big) \bigcap \Big( E
|
(P) \cup \{v_s v_t, v_k v_{j'-1}\} \Big),$$
and apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition \{$T$, $C_1$, $C_2$\}.
\end{enumerate}
\end{enumerate}
\end{proof}
\noindent\textbf{Re
|
mark 2.}
\label{remark:2}
Indeed, in the proof of the previous theorem we showed a stronger result, that is, for every traceable cubic graph there is a decomposition with at most two cycles.
|
\section{Principle of nano strain-amplifier}
\begin{figure*}[t!]
\centering
\includegraphics[width=5.4in]{Fig1}
\vspace{-0.5em}
\caption{Schematic sketches of nanowire strain sensors. (a)(b) Conventional non-released and released NW structure;
|
(c)(d) The proposed nano strain-amplifier and its simplified physical model.}
\label{fig:fig1}
\vspace{-1em}
\end{figure*}
Figure \ref{fig:fig1}(a) and 1(b) show the concept of the conventional structures of piezoresistive sensors. The piezoresi
|
stive elements are either released from, or kept on, the substrate. The sensitivity ($S$) of the sensors is defined based on the ratio of the relative resistance change ($\Delta R/R$) of the sensing element and the strain applied to the substrate ($\vareps
|
ilon_{sub}$):
\begin{equation}
S = (\Delta R/R)/\varepsilon_{sub}
\label{eq:sensitivity}
\end{equation}
In addition, the relative resistance change $\Delta R/R$ can be calculated from the gauge factor ($GF$) of the material used to make the piezoresis
|
tive elements: $\Delta R/R = GF \varepsilon_{ind}$, where $\varepsilon_{ind}$ is the strain induced into the piezoresistor. In most of the conventional strain gauges as shown in Fig. \ref{fig:fig1} (a,b), the thickness of the sensing layer is typically bel
|
ow a few hundred nanometers, which is much smaller than that of the substrate. Therefore, the strain induced into the piezoresistive elements is approximately the same as that of the substrate ($\varepsilon_{ind} \approx \varepsilon_{sub}$). Consequently,
|
to improve the sensitivity of strain sensors (e.g. enlarging $\Delta R/R$), electrical approaches which can enlarge the gauge factor ($GF$) are required. Nevertheless, as aforementioned, the existence of the large gauge factor in nanowires due to quantum c
|
onfinement or surface state, is still considered as controversial.
It is also evident from Eq. \ref{eq:sensitivity} that the sensitivity of strain sensors can also be improved using a mechanical approach, which enlarges the strain induced into the piezo
|
resistive element. Figure \ref{fig:fig1}(c) shows our proposed nano strain-amplifier structure, in which the piezoresistive nanowires are locally fabricated at the centre of a released bridge. The key idea of this structure is that, under a certain strain
|
applied to the substrate, a large strain will be concentrated at the locally fabricated SiC nanowires. The working principle of the nano strain-amplifier is similar to that of the well-known dogbone structure, which is widely used to characterize the tensi
|
le strength of materials \cite{dogbone1,dogbone2}. That is, when a stress is applied to the dogbone-shape of a certain material, a crack, if generated, will occur at the middle part of the dogbone. The large strain concentrated at the narrow area located a
|
t the centre part with respect to the wider areas located at outer region, causes the crack. Qualitative and quantitative explanations of the nano strain-amplifier are presented as follows.
For the sake of simplicity, the released micro frame and nano
|
wire (single wire or array) of the nano strain-amplifier can be considered as solid springs, Fig. \ref{fig:fig1}(d). The stiffness of these springs are proportional to their width ($w$) and inversely proportional to their length (l): $K \propto w/l$. Conse
|
quently, the model of the released nanowire and micro frames can be simplified as a series of springs, where the springs with higher stiffness correspond to the micro frame, and the single spring with lower stiffness corresponds to the nanowire. It is well
|
-known in classical physics that, for serially connected springs, a larger strain will be concentrated in the low--stiffness string, while a smaller strain will be induced in the high--stiffness string \cite{Springbook}. The following analysis quantitative
|
ly explained the amplification of the strain.
\begin{figure}[b!]
\centering
\includegraphics[width=3in]{Fig2}
\vspace{-1em}
\caption{Finite element analysis of the strain induced in to the nanowire array utilizing nano strain-amplifier.}
\lab
|
el{fig:fig2}
\end{figure}
When a tensile mechanical strain ($\varepsilon_{sub}$) is applied to the substrate, the released structure will also be elongated. Since the stiffness of the released frame is much smaller than that of the substrate, it is safe
|
to assume that the released structure will follows the elongation of the substrate. The displacement of the released structure $\Delta L$ is:
\begin{equation}
\Delta L = \Delta L_m + \Delta L_n = L_m \varepsilon_m + L_n \varepsilon_n
\label{eq:displacem
|
ent}
\end{equation}
where $L_m$, $L_n$ are the length; $\Delta L_m$, $\Delta L_n$ are the displacement; and $\varepsilon_m$, $\varepsilon_n$ are the strains induced into the micro spring and nano spring, respectively. The subscripts m and n stand for th
|
e micro frames and nanowires, respectively. Furthermore, due to the equilibrium of the stressing force ($F$) along the series of springs, the following relationship is established: $F= K_m\Delta L_m = K_n \Delta L_n$, where $K_m$, $K_n$ are the stiffness o
|
f the released micro frames and nanowires, respectively. Consequently the relationship between the displacement of the micro frame (higher stiffness) and nanowires (lower stiffness) is:
\begin{equation}
\frac{\Delta L_m}{\Delta L_n}=\frac{K_n}{K_m}=\frac
|
{L_mw_n}{L_nw_m}
\label{eq:euili}
\end{equation}
Substituting Eqn. \ref{eq:euili} into Eqn. \ref{eq:displacement}, the strain induced into the locally fabricated nanowires is:
\begin{equation}
\varepsilon_n = \frac{\Delta L_n}{L_n} = \frac{1}{1-\frac{
|
w_m-w_n}{w_m}\frac{L_m}{L}}\varepsilon_{sub}
\label{eq:strainamp}
\end{equation}
Equation \ref{eq:strainamp} indicates that increasing the ratio of $w_m/w_n$ and $L_m/L_n$ significantly amplifies the strain induced into the nanowire from the strain a
|
pplied to the substrate. This model is also applicable to the case of nanowire arrays, in which $w_n$ is the total width of all nanowires in the array.
The theoretical model is then verified using the finite element analysis (FEA). In the FEA simulation,
|
we compare the strain induced into (i) non released nanowires, (ii) the conventionally released nanowires, and (iii) our nano strain-amplifier structure, using COMSOL Multiphysics \texttrademark. In our nano strain amplifying structure, the width of the r
|
eleased frame was set to be 8 $\mu$m, while the width of each nanowire in the array (3 wires) was set to be 370 nm. The nanowires array structure was selected as it can enhance the electrical conductance of the SiC nanowires resistor which makes the subseq
|
uent experimental demonstration easier. The ratio between the length of nanowires and micro bridge was set to be 1: 20. With this geometrical dimensions, strain induced into nanowires array $\varepsilon_n$ was numerically calculated to be approximately 6 t
|
imes larger than $\varepsilon_{sub}$, Eqn. \ref{eq:strainamp}. The simulation results show that for all structure, the elongation of non-released and released nanowires follow that of the substrate. In addition, strain was almost completely transferred int
|
o conventional released and non-released structures. Furthermore, the ratio of the strain induced in to the locally fabricated nanowires was estimated to be 5.9 times larger than that of the substrate, Fig. \ref{fig:fig2}. These results are in solid agreem
|
ent with the theoretical analysis presented above. For a nanowire array with an average width of 470 nm, the amplified gain of strain was found to be 4.5.
Based on the theoretical analysis, we conducted the following experiments to demonstrate the h
|
igh sensitivity of SiC nanowire strain sensors using the nano strain-amplifier. A thin 3C-SiC film with its thickness of 300 nm was epitaxially grown on a 150 mm diameter Si wafer using low pressure chemical vapour deposition \cite{SiC_growth}. The film wa
|
s \emph{in situ} doped using Al dopants. The carrier concentration of the p-type 3C-SiC was found to be $5 \times 10^{18}$ cm$^{-3}$, using a hot probe technique \cite{philip}. The details of the characteristics of the grown film can be found elsewhere \ci
|
te{Phan_JMC}. Subsequently, I-shape p-type SiC resistors with aluminum electrodes deposited on the surface were patterned using inductive coupled plasma (ICP) etching. As the piezoresistance of p-type 3C-SiC depends on crystallographic orientation, all SiC
|
resistors of the present work were aligned along [110] direction to maximize the piezoresistive effect. Next, the micro scale SiC resistors were then released from the Si substrate using dry etching (XeF$_2$). Finally, SiC nanowire arrays were formed at t
|
he centre of the released bridge using focused ion beam (FIB). Two types of nanowire array were fabricated with three nanowires for each array. The average width of each nanowire in each type were 380 nm and 470 nm, respectively. Figure \ref{fig:fig3} show
|
s the SEM images of the fabricated samples, including the conventional released structure, non-released nanowires, and the nano strain-amplifier.
\begin{figure}[t!]
\centering
\includegraphics[width=3in]{Fig3}
\caption{SEM image of SiC strain se
|
nsors. (a) Released SiC micro bridge used for the subsequent fabrication of the nano strain-amplifier; (b) SEM of a micro SiC resistor where the SiC nanowires array were formed using FIB; (c) SEM of non-released SiC nanowires; (d) SEM of locally fabricated
|
SiC nanowires released from the Si substrate (nano strain-amplifier).}
\label{fig:fig3}
\vspace{-1em}
\end{figure}
The current voltage (I-V) curves of all fabricated samples were characterized using a HP 4145 \texttrademark ~parameter analyzer. The
|
linear relationship between the applied voltage and measured current, indicated that Al made a good Ohmic contact with the highly doped SiC resistance, Fig. \ref{fig:IV}. Additionally, the electrical conductivity of both nanowires and micro frame estimated
|
from the I-V curve and the dimensions of the resistors shows almost the same value. This indicated that the FIB process did not cause a significant surface damage to the fabricated nanowires.
\begin{figure}[b!]
\centering
\includegraphics[widt
|
h=3in]{Fig4}
\vspace{-1.5em}
\caption{Current voltage curves of the fabricated SiC resistors.}
\label{fig:IV}
\end{figure}
The bending experiment was used to characterize the piezoresistive effect in micro size SiC resistors and locally fabrica
|
ted SiC nanowire array. In this experiment one end of the Si cantilever (with a thickness of 625 $\mu$m, and a width of 7 mm) was fixed while the other end was deflected by applying different forces. The distance from the fabricated nanowires to the free e
|
nd of the Si cantilever was approximately 45 mm. The strain induced into the Si substrate is $\varepsilon_\text{sub} = Mt/2EI$, where $M$ is the applied bending moment; and $t$, $E$ and $I$ are the thickness, Young's modulus and the moment of inertia of th
|
e Si cantilever, respectively. The response of the SiC resistance to applied strain was then measured using a multimeter (Agilent \texttrademark 34401 A).
\begin{figure}[h!]
\centering
\includegraphics[width=3in]{Fig5.eps}
\vspace{-1.5em}
\capti
|
on{Experimental results. (a) A comparision between the relative resistance change in the nano strain-amplifiers, non released nanowires and released micro frames; (b) The repeatability of the SiC nanowires strain sensors utilizing the proposed structure.}
|
\label{fig:DRR}
\vspace{-1em}
\end{figure}
The relative resistance change ($\Delta R/R$) of the micro and nano SiC resistors was plotted against the strain induced into the Si substrate $\varepsilon_{sub}$, Fig. \ref{fig:DRR}(a). For all fabricate
|
d samples, the relative resistance change shows a good linear relationship with the applied strain ($\varepsilon_{sub}$). In addition, with the same applied strain to the Si substrate, the resistance change of the SiC nanowires using the nano strain-amplif
|
ier was much larger than that of the the SiC micro resistor and the conventional non-released SiC nanowires. In addition, reducing the width of the SiC nanowires also resulted in the increase of the sensitivity. The magnitude of the piezoresistive effect i
|
n the nano strain-amplifier as well as conventional structures were then quantitatively evaluated based on the effective gauge factor ($GF_{eff}$), which is defined as the ratio of the relative resistance change to the applied strain to the substrate: $GF_
|
{eff} = (\Delta R/R)/\varepsilon_{sub}$. Accordingly, the effective gauge factor of the released micro SiC was found to be 28, while that of the non-released SiC nanowires was 35. From the data shown in Fig. \ref{fig:DRR}, the effective gauge factor of the
|
380 nm and 470 nm SiC nanowires in the nano strain-amplifier were calculated as 150 and 124, respectively. Thus for nanowire arrays with average widths of 380 nm and 470 nm, the sensitivity of the nano strain-amplifier was 5.4 times and 4.6 times larger t
|
han the bulk SiC, respectively. These results were consistent with analytical and numerical models presented above. The relative resistance change of the nano strain-amplifier also showed excellent linearity with the applied strain, with a linear regressio
|
n of above 99\%.
The resistance change of the nano strain-amplifier can also be converted into voltage signals using a Wheatstone bridge, Fig. \ref{fig:DRR}(b). The output voltage of the nano strain-amplifier increases with increasing tensile strains f
|
rom 0 ppm to 180 ppm, and returned to the initial value when the strain was completely removed, confirming a good repeatability after several strain induced cycles. The linearity of the relative resistance change, and the repeatability indicate that the pr
|
oposed structure is promising for strain sensing applications.
In conclusion, this work presents a novel mechanical approach to obtain highly sensitive piezoresistance in nanowires based on a nano strain-amplifier. The key factor of the nano strain-
|
amplifier lies on nanowires locally fabricated on a released micro structure. Experimental studies were conducted on SiC nanowires, confirming that by utilizing our nano strain-amplifier, the sensitivity of SiC nanowires was 5.4 times larger than that of c
|
onventional structures. This result indicated that the nano strain-amplifier is
|
an excellent platform for ultra sensitive strain sensing applications.
|
\section{Introduction}\label{intro}
Gas has a fundamental role in shaping the evolution of galaxies,
through its accretion on to massive haloes, cooling and subsequent
fuelling of star formation, to the triggering of extreme luminous
activity around super
|
massive black holes. Determining how the
physical state of gas in galaxies changes as a function of redshift is
therefore crucial to understanding how these processes evolve over
cosmological time. The standard model of the gaseous interstellar
medium (IS
|
M) in galaxies comprises a thermally bistable medium
(\citealt*{Field:1969}) of dense ($n \sim 100$\,cm$^{-3}$) cold
neutral medium (CNM) structures, with kinetic temperatures of $T_{\rm
k} \sim 100$\,K, embedded within a lower-density ($n \sim
1$\,cm$^{
|
-3}$) warm neutral medium (WNM) with $T_{\rm k} \sim
10^{4}$\,K. The WNM shields the cold gas and is in turn ionized by
background cosmic rays and soft X-rays (e.g. \citealt{Wolfire:1995,
Wolfire:2003}). A further hot ($T_{\rm k} \sim 10^{6}$\,K) ionized
|
component was introduced into the model by \cite{McKee:1977}, to
account for heating by supernova-driven shocks within the inter-cloud
medium. In the local Universe, this paradigm has successfully
withstood decades of observational scrutiny, although ther
|
e is some
evidence (e.g. \citealt{Heiles:2003b}; \citealt*{Roy:2013b};
\citealt{Murray:2015}) that a significant fraction of the WNM may
exist at temperatures lower than expected for global conditions of
stability, requiring additional dynamical processes
|
to maintain local
thermodynamic equilibrium.
Since atomic hydrogen (\mbox{H\,{\sc i}}) is one of the most abundant
components of the neutral ISM and readily detectable through either
the 21\,cm or Lyman $\alpha$ lines, it is often used as a tracer of
the
|
large-scale distribution and physical state of neutral gas in
galaxies. The 21\,cm line has successfully been employed in surveying
the neutral ISM in the Milky Way
(e.g. \citealt{McClure-Griffiths:2009,Murray:2015}), the Local Group
(e.g. \citealt{Kim:200
|
3,Bruns:2005,Braun:2009,Gratier:2010}) and
low-redshift Universe (see \citealt{Giovanelli:2016} for a
review). However, beyond $z \sim 0.4$ (\citealt{Fernandez:2016})
\mbox{H\,{\sc i}} emission from individual galaxies becomes too faint
to be detectable by
|
current 21\,cm surveys and so we must rely on
absorption against suitably bright background radio (21\,cm) or UV
(Lyman-$\alpha$) continuum sources to probe the cosmological evolution
of \mbox{H\,{\sc i}}. The bulk of neutral gas is contained in
high-colu
|
mn-density damped Lyman-$\alpha$ absorbers (DLAs, $N_{\rm HI}
\geq 2 \times 10^{20}$\,cm$^{-2}$; see \citealt*{Wolfe:2005} for a
review), which at $z \gtrsim 1.7$ are detectable in the optical
spectra of quasars. Studies of DLAs provide evidence that the a
|
tomic
gas in the distant Universe appears to be consistent with a
multi-phase neutral ISM similar to that seen in the Local Group
(e.g. \citealt*{Lane:2000}; \citealt*{Kanekar:2001c};
\citealt*{Wolfe:2003b}). However, there is some variation in the cold
an
|
d warm fractions measured throughout the DLA population
(e.g. \citealt*{Howk:2005}; \citealt{Srianand:2005, Lehner:2008};
\citealt*{Jorgenson:2010}; \citealt{Carswell:2011, Carswell:2012,
Kanekar:2014a}; \citealt*{Cooke:2015}; \citealt*{Neeleman:2015}).
|
The 21-cm spin temperature affords us an important line-of-enquiry in
unraveling the physical state of high-redshift atomic gas. This
quantity is sensitive to the processes that excite the ground-state of
\mbox{H\,{\sc i}} in the ISM
(\citealt{Purcell:195
|
6,Field:1958,Field:1959b,Bahcall:1969}) and
therefore dictates the detectability of the 21\,cm line in absorption.
In the CNM the spin temperature is governed by collisional excitation
and so is driven to the kinetic temperature, while the lower densities
|
in the WNM mean that the 21\,cm transition is not thermalized by
collisions between the hydrogen atoms, and so photo-excitation by the
background Ly $\alpha$ radiation field becomes important. Consequently
the spin temperature in the WNM is lower than the
|
kinetic temperature,
in the range $\sim$1000 -- 5000\,K depending on the column density and
number of multi-phase components (\citealt{Liszt:2001}). Importantly,
the spin temperature measured from a single detection of extragalactic
absorption is equal to
|
the harmonic mean of the spin temperature in
individual gas components, weighted by their column densities, thereby
providing a method of inferring the CNM fraction in high-redshift
systems.
Surveys for 21\,cm absorption in known redshifted DLAs have been
|
used
to simultaneously measure the column density and spin temperature of
\mbox{H\,{\sc i}} (see \citealt{Kanekar:2014a} and references
therein). There is some evidence for an increase (at $4\,\sigma$
significance) in the spin temperature of DLAs at redsh
|
ifts above $z =
2.4$, and a difference (at $6\,\sigma$ significance) between the
distribution of spin temperatures in DLAs and the Milky Way
(\citealt{Kanekar:2014a}). The implication that at least 90\,per\,cent
of high-redshift DLAs may have CNM fractions
|
significantly less than
that measured for the Milky Way has important consequences for the
heating and cooling of neutral gas in the early Universe and star
formation (e.g. \citealt*{Wolfe:2003a}). However, these targeted
observations rely on the limited
|
availability of simultaneous 21\,cm
and optical/UV data for the DLAs and assumes commonality between the
column density probed by the optical and radio sight-lines. The first
issue can be overcome by improving the sample statistics through
larger 21\,cm li
|
ne surveys of high-redshift DLAs, but the latter
requires improvements to our methodology and understanding of the gas
distribution in these systems. There are also concerns about the
accuracy to which the fraction of the source structure subtended by
the
|
absorber can be measured in each system, which can only be
resolved through spectroscopic very long baseline interferometry
(VLBI). It has been suggested that the observed evolution in spin
temperature could be biased by assumptions about the radio-source
|
covering factor (\citealt{Curran:2005}) and its behaviour as a
function of redshift (\citealt{Curran:2006b, Curran:2012b}).
In this paper we consider an approach using the statistical constraint
on the average spin temperature achievable with future large
|
21\,cm
surveys using precursor telescopes to the Square Kilometre Array
(SKA). This will enable independent verification of the evolution in
spin temperature at high redshift and provide a method of studying the
global properties of neutral gas below $z \
|
approx 1.7$, where the
Lyman\,$\alpha$ line is inaccessible using ground-based observatories.
In an early attempt at a genuinely blind 21\,cm absorption survey,
\cite{Darling:2011} used pilot data from the Arecibo Legacy Fast
Arecibo L-band Feed Array (ALF
|
ALFA) survey to obtain upper limits on
the column density frequency distribution from 21\,cm absorption at
low redshift ($z \lesssim 0.06$). However, they also noted that the
number of detections could be used to make inferences about the ratio
of the spin
|
temperature to covering factor. Building upon this work,
\cite{Wu:2015} found that their upper limits on the frequency
distribution function measured from the 40\,per\,cent ALFALFA survey
({$\alpha$}.40; \citealt{Haynes:2011}) could only be reconciled wit
|
h
measurements from other low-redshift 21\,cm surveys if the typical
spin temperature to covering factor ratio was greater than 500\,K. At
higher redshifts, \cite{Gupta:2009} found that the number density of
21\,cm absorbers in known \mbox{Mg\,{\sc ii}} ab
|
sorbers appeared to
decrease with redshift above $z \sim 1$, consistent with a reduction
in the CNM fraction. We pursue this idea further by investigating
whether future wide-field 21\,cm surveys can be used to measure the
average spin temperature in dista
|
nt galaxies that are rich in atomic
gas.
\section{The expected number of intervening \mbox{H\,{\sc i}}
absorbers}\label{section:expected_number}
We estimate the expected number of intervening \mbox{H\,{\sc i}}
systems towards a sample of background rad
|
io sources by evaluating the
following integral over all sight-lines
\begin{equation}\label{equation:expected_number}
\mu = \iint{f(N_{\rm HI},X)\,\mathrm{d}X\,\mathrm{d}N_{\rm HI}},
\end{equation}
where $f(N_{\rm HI}, X)$ is the frequency distribution
|
as a function
of column density ($N_{\rm HI}$) and comoving path length ($X$). We
use the results of recent surveys for 21\,cm emission in nearby
galaxies (e.g. \citealt{Zwaan:2005}) and high-redshift Lyman-$\alpha$
absorption in the Sloan Digitial Sky Su
|
rvey (SDSS;
e.g. \citealt*{Prochaska:2005}; \citealt{Noterdaeme:2009}), which show
that $f(N_{\rm HI}, X)$ can be parametrized by a gamma function of the
form
\begin{equation}
f(N_{\rm HI}, X) = \left({f_{\ast} \over N_{\ast}}\right)\left({N_{\rm HI} \ov
|
er N_{\ast}}\right)^{-\beta}\exp{\left(-{N_{\rm HI} \over N_{\ast}}\right)}\,\mathrm{cm}^{2},
\end{equation}
where $f_{\ast} = 0.0193$, $\log_{10}(N_{\ast}) = 21.2$ and $\beta =
1.24$ at $z = 0$ (\citealt{Zwaan:2005}), and $f_{\ast} = 0.0324$,
$\log_{10}(N
|
_{\ast}) = 21.26$ and $\beta = 1.27$ at $z \approx 3$
(\citealt{Noterdaeme:2009}). While the observational data do not yet
constrain models for evolution of the \mbox{H\,{\sc i}} distribution
at intermediate redshifts between $z \sim 0.1$ and
$3$\footnote{
|
Measurements of $f(N_{\rm HI},X)$ at intermediate
redshifts come from targeted ultra-violet surveys of DLAs using the
\emph{Hubble Space Telescope} (\citealt*{Rao:2006};
\citealt{Neeleman:2016}). However, due to the limited sample sizes
these are c
|
urrently an order-of-magnitude less sensitive than the
nearby 21-cm and high-redshift optical Lyman-$\alpha$ surveys.}, it
is known to be much weaker than the significant decline seen in the
global star-formation rate and molecular gas over the same epoc
|
h
(e.g. \citealt{Lagos:2014}). We therefore carry out a simple linear
interpolation between the low and high redshift epochs to estimate
$f(N_{\rm HI},X)$ as a function of redshift.
\begin{figure}
\centering
\includegraphics[width=0.475\textwidth]{width_d
|
ist.pdf}
\caption{The distribution of 21\,cm line widths based on existing
detections of intervening absorption at $z > 0.1$ (see the text for
details of this sample). The sample size in each bin is denoted by
the number above and errorbars denote th
|
e standard deviation. The
solid red line is a log-normal fit to the data, from which we draw
random samples for our analysis.}\label{figure:width_dist}
\end{figure}
The probability of detecting an absorbing system of given column
density depends on t
|
he sensitivity of the survey, the flux density and
structure of the background source and the fraction of \mbox{H\,{\sc
i}} in the lower spin state, given by the spin temperature. We
express the column density ($N_{\rm HI}$; in atoms\,cm$^{-2}$) in
ter
|
ms of the optical depth ($\tau$) and spin temperature ($T_{\rm
spin}$; in K) by
\begin{equation}\label{equation:column_density}
N_{\rm HI} = 1.823\times10^{18}\,T_{\rm spin} \int{\tau(v)\mathrm{d}v},
\end{equation}
where the integral is performed acros
|
s the spectral line in the system
rest-frame velocity $v$ (in km\,s$^{-1}$). We then express the optical
depth in terms of the observables as
\begin{equation}
\tau = -\ln\left[1 + {\Delta{S}\over c_{\rm f}S_{\rm cont}}\right],
\end{equation}
where $\Delt
|
a{S}$ is the observed change in flux density due to
absorption, $S_{\rm cont}$ is the background continuum flux density
and $c_{\rm f}$ is the (often unknown) fraction of background flux
density subtended by the intervening gas.
\begin{figure}
\centering
|
\includegraphics[width=0.465\textwidth]{covfact_dist.pdf}
\caption{The distribution of \mbox{H\,{\sc i}} covering factors from
the main sample of \citet{Kanekar:2014a}, which were estimated using
the fraction of total continuum flux density in the quas
|
ar core. The
sample size in each bin is denoted by the number above and errorbars
denote the standard deviation. The solid red line is the uniform
distribution, from which we draw random samples for our
analysis.}\label{figure:covfact_dist}
\end{fi
|
gure}
We assume that a single intervening system can be described by a
Gaussian velocity distribution of full width at half maximum (FWHM)
dispersion ($\Delta{v_{\rm 50}}$) and peak optical depth ($\tau_{\rm
peak}$), so that \autoref{equation:column_den
|
sity} can be re-written
as
\begin{equation}\label{equation:column_density_gaussian}
N_{\rm HI} = 1.941\times10^{18}\,T_{\rm spin}\,\tau_{\rm peak}\,\Delta{v_{\rm 50}}.
\end{equation}
If we further assume that the rms spectral noise is Gaussian, with a
st
|
andard deviation $\sigma_{\rm chan}$ per independent channel
$\Delta{v_{\rm chan}}$, then the 5$\sigma$ column density detection
limit is given by
\begin{equation}
N_{5\sigma} \approx 1.941\times10^{18}\,T_{\rm spin}\,\tau_{\rm 5\sigma}\,\Delta{v_{\rm co
|
nv}},
\end{equation}
where
\begin{equation}\label{equation:optical_depth_limit}
\tau_{5\sigma} \approx -\ln\left[1 - {5\,\sigma_{\rm chan}\over c_{\rm f}\,S_{\rm cont}}\sqrt{\Delta{v}_{\rm chan}\over \Delta{v_{\rm conv}}}\right],
\end{equation}
and $\De
|
lta{v_{\rm conv}} \approx \sqrt{\Delta{v}_{\rm chan}^{2} +
\Delta{v}_{50}^{2}}$, which is the observed width of the line, given
by the convolution of the physical velocity distribution and the
spectral resolution of the telescope. We now redefine $\mu$ a
|
s the
expected number of intervening \mbox{H\,{\sc i}} detections in our
survey as a function of the column density sensitivity along each
sight-line where each comoving path element
$\delta{X}(z)$\footnote{For the purposes of this work we adopt a flat
$
|
\Lambda$ cold dark matter cosmology with $H_{0}$ =
70\,km\,s$^{-1}$, $\Omega_\mathrm{M}$ = 0.3 and $\Omega_{\Lambda}$ =
0.7. } in the integral defined by
\autoref{equation:expected_number} is given by
\begin{equation}
\delta{X}(z)=
\begin{cases}
|
{\delta{z}\,(1+z)^{2}\over \sqrt{(1+z)^{2}(1+z\Omega_{\rm
M})-z(z+2)\Omega_{\rm \Lambda}}}, & \text{if}\ N_{\rm HI} \geq N_{5\sigma}, \\
0, & \text{otherwise}.
\end{cases}
\end{equation}
To calculate the column density sensitivity for each comov
|
ing element
we draw random samples for $\Delta{v}_{50}$ and $c_{\rm f}$ from
continuous prior distributions based on existing evidence. In the case
of $\Delta{v}_{50}$ we use a log-normal distribution obtained from a
simple least-squares fit to the sample
|
distribution from previous
21-cm absorption surveys reported in the literature (see
\autoref{figure:width_dist})\footnote{References for the literature
sample of line widths shown in \autoref{figure:width_dist}:
\citet*{Briggs:2001}; \citet*{Carilli:19
|
93};
\citet*{Chengalur:1999}; \citet{Chengalur:2000, Curran:2007b,
Davis:1978, Ellison:2012, Gupta:2009, Gupta:2012, Gupta:2013};
\citet{Kanekar:2001b,Kanekar:2003b}; \citet{Kanekar:2001c,
Kanekar:2006, Kanekar:2009a, Kanekar:2013, Kanekar:2014
|
a};
\citet{Kanekar:2003a}, \citet*{Kanekar:2007};
\citet*{Kanekar:2014b}; \citet{ Lane:2001, Lovell:1996, York:2007,
Zwaan:2015}.}, assuming that this correctly describes the true
distribution for the population of DLAs. However, direct measurement
|
of the \mbox{H\,{\sc i}} covering factor is significantly more
difficult and so for the purposes of this work we draw random samples
assuming a uniform distribution between 0 and 1. In
\autoref{figure:covfact_dist}, we show a comparison between this
assum
|
ption and the sample distribution estimated by
\cite{Kanekar:2014a} from their main sample of 37 quasars. Kanekar et
al. used VLBI synthesis imaging to measure the fraction of total
quasar flux density contained within the core, which was then used as
a pr
|
oxy for the covering factor. By carrying out a two-tailed
Kolmogorov-Smirnov (KS) test of the hypothesis that the Kanekar et
al. data are consistent with our assumed uniform distribution, we find
that this hypothesis is rejected at the 0.05 level, but not
|
at the
0.01 level (this outcome is dominated by the paucity of quasars in the
sample with $c_{\rm f} \lesssim 0.2$). It is therefore possible that
the population distribution of \mbox{H\,{\sc i}} covering factors may
deviate somewhat from the uniform distr
|
ibution assumed in this
work. We discuss the implications of this further in
\autoref{section:covering_factor}.
\section{A 21\,cm absorption survey with
ASKAP}\label{section:all_sky_survey}
We use the Australian Square Kilometre Array Pathfinder (ASKAP
|
;
\citealt{Johnston:2007}) as a case study to demonstrate the expected
results from planned wide-field surveys for 21\,cm absorption
(e.g. the ASKAP First Large Absorption Survey in \mbox{H\,{\sc i}} --
Sadler et al., the MeerKAT Absorption Line Survey --
|
Gupta et al., and
the Search for HI absorption with AperTIF -- Morganti et al.). ASKAP
is currently undergoing commissioning. Proof-of-concept observations
with the Boolardy Engineering Test Array (\citealt{Hotan:2014}) have
already been used to successful
|
ly detect a new \mbox{H\,{\sc i}}
absorber associated with a probable young radio galaxy at $z = 0.44$
(\citealt{Allison:2015a}). Here we predict the outcome of a future
2\,h-per-pointing survey of the entire southern sky ($\delta \leq
+10\degr$) using the
|
full 36-antenna ASKAP in a single 304\,MHz band
between 711.5 and 1015.5\,MHz, equivalent to \mbox{H\,{\sc i}}
redshifts between $z = 0.4$ and 1.0.
\begin{figure}
\centering
\includegraphics[width=0.475\textwidth]{nsources_flux.pdf}
\caption{The number o
|
f radio sources in our simulated southern sky
survey ($\delta \leq +10\degr$) estimated from existing catalogues
at $L$-band frequencies (see the text for details). The grey region
encloses the expected number across the 711.5 - 1015.5\,MHz ASKAP
f
|
requency band, assuming a canonical spectral index of $\alpha =
-0.7$.}\label{figure:nsources_flux}
\end{figure}
Our expectations of the ASKAP performance are based on preliminary
measurements by \cite{Chippendale:2015} using the prototype Mark {\sc
I
|
I} phase array feed. We estimate the noise per spectral channel
using the radiometer equation
\begin{equation}
\sigma_{\rm chan} = {S_{\rm system} \over \sqrt{n_{\rm pol}\,n_{\rm ant}\,(n_{\rm ant} - 1)\,\Delta{t}_{\rm in}\,\Delta{\nu}_{\rm chan}}},
\e
|
nd{equation}
where $S_{\rm system}$ is the system equivalent flux density, $n_{\rm
pol}$ is the number of polarizations, $n_{\rm ant}$ is the number of
antennas, $\Delta{t}_{\rm in}$ is the on-source integration time and
$\Delta{\nu}_{\rm chan}$ is the s
|
pectral resolution in frequency. The
sensitivity of the telescope in the 711.5 - 1015.5\,MHz band is
expected to vary between $S_{\rm system} \approx 3200$ and $2000$\,Jy,
with the largest change in sensitivity between 700 and 800\,MHz. ASKAP
has dual line
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ar polarization feeds, 36 antennas and a fine filter bank
that produces 16\,416 independent channels across the full 304\,MHz
bandwidth, so the expected noise per 18.5\,kHz channel in a 2\,h
observation is approximately 5.5 - 3.5\,mJy\,beam$^{-1}$ across t
|
he
band. In the case of an actual survey, the true sensitivity will of
course be recorded in the spectral data as a function of redshift (see
e.g. \citealt{Allison:2015a}), but for the purposes of the simulated
survey presented in this work we split the ba
|
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