\section{Introduction}
Over the last two decades, nanopore technology has developed into a versatile platform for a wide range of applications such as ultrafiltration, bio- and medical sensing, nanofluidics and nano-electronic devices.\cite{Dekker2007,Ahn2014,Zhou2017,Lemay2009,Venkatesan2011,Luan2019} With the advent of solid state nanopore membranes some limitations of biological nanopores can be overcome. Solid state nanopores are more robust and durable under harsh conditions such as elevated temperature, extreme pH environments and high pressure. They are highly tunable in geometry and surface properties and thus enable novel applications. Examples include filtration of water to remove contaminants such as oil\cite{Gao2014} and dyes,\cite{Li2020} nanopore sensors for specific ions,\cite{Liu2015} single molecules,\cite{Howorka2009} proteins,\cite{Houghtaling2018} gases,\cite{Xu2015} biochemicals,\cite{Aramesh2020} blood sugar,\cite{Galenkamp2018} and drugs,\cite{Wang2018} as well as nucleic acid analysis\cite{Tang2019}, DNA\cite{Wang2021a} and protein sequencing.\cite{Yu2019a} The potential of using nanofluidic devices based on nanopore membranes for novel applications such as power generation\cite{Laucirica2021} and electroosmotic pumps\cite{Wu2016} have also been explored in detail.

Solid state nanopore membranes can be manufactured in a wide range of materials such as semiconductors,\cite{Li2001,Storm2003,Kluth2008b,DeGroot2013,Karmi2020} polymers,\cite{ToimilMolares2001,Fink2,Li2004,Kiy2021} alumina\cite{Yuan2004,Nalaparaju2019}, carbon,\cite{Kang2014} and \ch{MoS2}\cite{Ke2019} with only a single\cite{Chtanko2004,Zhang2021} or multiple\cite{Harrell2006,Scopece2006} pores depending on the material and fabrication technique. One major challenge in current nanopore technology is to combine the high throughput rates that nanopores in ultrathin membranes provide with highly asymmetric transport properties characteristic of long, conical nanopores.\cite{PerezSirkin2020,Tang2016,Ulrich2021} Furthermore, translating the performance of single pores to multipore systems without losing performance has proven to be challenging.\cite{Cao2018,Laucirica2021}

Different techniques like e-beam lithography,\cite{Deshmukh1999} ion beam sculpting,\cite{Li2001} e-beam drilling using a transmission electron microscope,\cite{VanDenHout2011} focused ion beam drilling,\cite{Tang2014} dielectric breakdown\cite{Goto2019} or laser-assisted pulling\cite{Cadinu2020} can only be used to fabricate a single or few nanopores and thus lack the scalability that many applications require. Additionally, these techniques do not allow to precisely shape the geometry of the pores. Ion track etching has been shown to create versatile conical nanopores in polymers\cite{Li2004,Wharton2007} and silicon nitride\cite{Vlassiouk2009} in a way that is industrially scalable. In this work, we present a system of conical nanopores in thin amorphous \ch{SiO2} (a-\ch{SiO2}) membranes fabricated using ion track etching. Our membranes feature conical nanopores that exhibit opening angles far greater than those in track etched polymers,\cite{Kaya2016,Hadley2020} while being an order of magnitude thinner. We are able to either manufacture single pores or multipores with adjustable pore density on large areas. As a-\ch{SiO2} is widely used in the semiconductor industry, the fabrication techniques are precise and cost efficient, and can be readily integrated in lab-on-a-chip devices.

Here, we present the fabrication of multipore membranes, and the detailed characterisation of the structure and ionic transport properties. The performance is compared to that of existing nanopore membranes.


\section{Experimental}
\subsection{a-\ch{SiO2} membrane fabrication}
Figure~\ref{f:fab} shows the workflow involved in the fabrication of free-standing silicon dioxide membranes. Double side polished <100> Silicon wafers of \SI{100}{\mm} diameter with \SI{1}{\um} wet thermal silicon dioxide were purchased from WaferPro, LLC (USA) (i). The wafers were then RCA cleaned before being spin-coated with a thin coating of TI prime (Microchemicals, GmbH) for 25 seconds at \SI{3000}{\text{rpm}} which acts as an adhesion promoter. On the backside of the wafer, negative photoresist maN-1420 (Microchemicals, GmbH) was spin-coated for \SI{30}{\second} at \SI{3000}{\text{rpm}} (ii). After that, UV lithography was used to pattern custom windows measuring $475\times\SI{475}{\um\tothe{2}}$ (iii). The silicon was then exposed in the customised window region by removing the a-\ch{SiO2} layer using reactive ion etching from the backside of the wafer (iv). The photoresist was removed and the exposed \ch{Si} was then anisotropically etched using a wet etching solution containing \SI{5}{\%} tetramethylammonium hydroxide (TMAH) (Sigma-Aldrich, 331635) which leads to the formation of free-standing membranes (v). TMAH has a higher selectivity of a-\ch{SiO2}/\ch{Si} than other etchants like potassium hydroxide, and the exposed \ch{Si} can be completely etched without the need for extra silicon nitride layers. The membranes were then RCA cleaned to eliminate any contaminants they may have acquired during the process.  On a \SI{100}{\mm} Si wafer, this method results in the creation of 220 free-standing a-\ch{SiO2} membranes with a frame size of $5.6\times\SI{5.6}{\mm\tothe{2}}$ and a window size of $55\times\SI{66}{\um\tothe{2}}$.
\begin{figure}
	\centering
	\includegraphics{fab.png}
	\caption{Fabrication process of free standing a-\ch{SiO2} membranes.}
	\label{f:fab}
\end{figure}

\subsection{Ion transport measurements}
All electrolyte solutions were prepared using demonised water (\SI{18.2}{\mega\ohm\cm}, arium\textsuperscript{\textregistered} mini Sartorius ultrapure water system). Potassium chloride (\ch{KCl}) was obtained from Thermo Fisher Scientific ($\ge\SI{99.8}{\percent}$, catalog number: AJA383-500). The pH of the solutions was adjusted using potassium hydroxide ($\ge\SI{85.0}{\percent}$, ChemSupply Australia, product code: PA161-500G) and hydrochloric acid (\SI{36}{\percent}, Thermo Fisher Scientific, catalogue number: R18601E). The pH and conductivity of the electrolyte were measured using an Orion Star\texttrademark A215 pH/Conductivity Benchtop Multiparameter Meter from Thermo Fisher Scientific.
\begin{figure}
	\centering
	\includegraphics[width=8.3cm]{setup_picture.png}
	\caption{The custom-built setup used for conductometric experiments. The membrane is sealed between two fluid compartments. Using electrodes on either side of the membrane a bias voltage can be applied and the ionic current measured. The entire setup is enclosed in a stainless steel Faraday cage.}
	\label{f:setup_picture}
\end{figure}

The current measurements were performed in a custom-built setup that features two Teflon cells that hold a liquid volume of \SI{730}{\ul} each (see Figure~\ref{f:setup_picture}). The nanopore membrane is sealed between two silicone gaskets and the Teflon cells, which are filled with the electrolyte. \ch{Ag}/\ch{AgCl} electrodes were fabricated by the reaction of silver wires ($\varnothing$\SI{1}{\mm}) and sodium hypochlorite (\SI{10}{\percent}, Sigma-Aldrich) and placed in the liquid on either side of the membrane. The Teflon cells are enclosed in a stainless steel Faraday cage. The current-voltage ($I-V$) curves were recorded using a Moku:Lab (Liquid Instruments) connected to a custom-built voltage amplifier with a gain of \SI{e6}{\V\V\tothe{-1}}. The voltage sweep was started at \SI{0}{\V}, ramped up to \SI{1}{\V}, reduced to \SI{-1}{\V} and returned to \SI{0}{\V} with a sweep rate of \SI{200}{\mV\s\tothe{-1}}. This cycle was repeated until a stable current was reached, as the membrane has to adjust to the electrolyte. Each measurement was repeated twice, and the membrane was rinsed with demonised water in between.

\subsection{Theoretical background}
The total conductance of a membrane $G\ind{m}$ with $n$ conical nanopores is the sum of the conductance of each individual nanopore $G\ind{p}$, given by $G\ind{m}=\sum_n G\ind{p}=n G\ind{p}$. As the average distance between pores is on the order of \SI{10}{\um}, inter-pore effects can be neglected.\cite{Cao2018} The pore conductance consists of the bulk conductance $G\ind{b}$, the surface conductance $G\ind{s}$, and the access resistance at the cone tip $R\ind{a,t}$ and base $R\ind{a,b}$:\cite{Hall1975,Lee2012}
\begin{equation}
    G\ind{m}=n\left(\frac{1}{G\ind{b}+G\ind{s}}+R\ind{a,t}+R\ind{a,b}\right)^{-1}\label{eq:cond}
\end{equation}

As charged conical nanopores often exhibit asymmetric transport properties between positive and negative applied voltage, we analysed the conductance $G_0$ in the vicinity of \SI{0}{\V} where it is largely linear in a small bias interval. We determined the conductance in the linear regime of the $I-V$ curve around \SI{0}{\V} between \SIlist{-.1;.1}{\V}. The different contributions are:\cite{Frament2012}
\begin{align}
    G_{\text{b},0}&=\kappa\ind{b}\frac{\pi r\ind{t} r\ind{b}}{L}\\
    G_{\text{s},0}&=\kappa\ind{b}\frac{r\ind{b}-r\ind{t}}{\ln\left(\frac{r\ind{b}}{r\ind{t}}\right)}\frac{\pi}{L}\frac{\left|\sigma\right|}{e c}
\end{align}
where $\kappa\ind{b}$ is the bulk conductivity, $e$ is the elementary charge, $c$ is the salt concentration, $r\ind{t}$ is the radius at the cone tip, $r\ind{b}$ is the radius at the cone base, $L$ is the pore length, $\sigma$ is the surface charge density, and $\text{Du}_i=\left|\sigma\right|/(e c r_i)$ is the Dukhin number at the pore orifices.

The access resistance according to Hall is given by\cite{Hall1975,Lee2012}
\begin{equation}
    R_{\text{a},i}=\frac{1}{\kappa\ind{b} r_i}\frac{1}{4+\text{Du}_i}
\end{equation}
and describes the rate at which the ions pass into and out of the nanopore at the cone tip and base.

In a-\ch{SiO2} nanopores, the surface charge is regulated by protonation and deprotonation reactions of silanol (\ch{SiOH}) groups at the pore surface that can render the surface negatively or positively charged:\cite{Yang2020}
\begin{align*}
    \ch{SiOH &<=>[$K\ind{a}$] SiO- + H+}\\
    \ch{SiOH + H+ &<=>[$K\ind{b}$] SiOH2+}
\end{align*}
with the acid and base dissociation constants $K\ind{a}$ and $K\ind{b}$ that are related to the $\text{p}K_i$ of the surface $K_i=10^{-\text{p}K_i}$. The surface charge density $\sigma$ is controlled by the local $\zeta$-potential:\cite{Smeets2006,Frament2012}
\begin{align}
    \zeta(\sigma)=-\frac{k\ind{B}T}{e}\ln\left(\frac{\sigma(\zeta)}{e\Gamma+\sigma(\zeta)}\right)+\frac{k\ind{B}T\ln10}{e}(\text{pH}-\text{p}K\ind{a})-\frac{\sigma(\zeta)}{C}\label{eq:sig1}
\end{align}
with the Boltzmann constant $k\ind{B}$ and the temperature $T=\SI{22}{\degree C}$, the surface density of chargeable sites $\Gamma=\SI{8}{\nm\tothe{-2}}$,\cite{Iler2,Hiemstra1989,Behrens2001} and the Stern layer capacitance $C=\SI{2.9}{\F\m\tothe{-2}}$.\cite{Hiemstra1989,Behrens2001} The terms $-\ln\left(\sigma/(e\Gamma+\sigma)\right)$ and $(\text{pH}-\text{p}K\ind{a})$ are only valid for negatively charged surfaces and are replaced by $\ln\left((e\Gamma-\sigma)/\sigma\right)$ and $(\text{p}K\ind{b}-\text{pH})$ if the surface charge becomes positive, i.e. below the isoelectric point pI at which the surface carries no net charge.\cite{Frament2013}

The surface charge density can be calculated using the planar Graham equation, if the electrical double layer (EDL) is not overlapping.\cite{Behrens2001,VanDerHeyden2005,Israelachvili2} This is valid if the Debye length is smaller than the pore tip radius, which is the case for every electrolyte concentration except two, as shown in Table~\ref{t:debye}. The surface charge density is then given as:
\begin{equation}
    \sigma(\zeta) = \frac{2\varepsilon\varepsilon_0}{\lambda\ind{D}}\frac{k\ind{B}T}{e}\sinh{\left(\frac{e\zeta(\sigma)}{2k\ind{B}T}\right)}\label{eq:sig2}
\end{equation}
with the permittivity of the liquid $\varepsilon=80.2$, the vacuum permittivity $\varepsilon_0$ and the Debye length $\lambda\ind{D}=\sqrt{\sum_i\varepsilon\varepsilon_0k\ind{B}T/e^2c_iz_i}$. $c_i$ and $z_i$ are the concentration and charge of the ionic species \ch{K+}, \ch{Cl-}, \ch{H+} and \ch{OH-}, respectively. Solving Equation~\ref{eq:sig1} and \ref{eq:sig2} self consistently yields the $\zeta$-potential as well as the surface charge density as a function of the pH and the electrolyte concentration. Hence, in the non-linear least squares fitting of the conductance as a function of pH and concentration using Equation~\ref{eq:cond} the $\text{p}K_a$, $\text{p}K_b$ and pI are fitting parameters and thus directly obtained.

\begin{figure*}
 \centering
 \includegraphics{etching_charact.png}
 \caption{(a) Nanopore fabrication process using the track etch technique. (i) The a-\ch{SiO2} membrane is irradiated with swift heavy ions (Au ions), creating damage in the material. (ii) The damage is etched using hydrofluoric acid, forming conical nanopores. (b-f) Structural characterisation using scanning electron microscopy (SEM), atomic force microscopy (AFM) and small angle X-ray scattering (SAXS). (b) SEM: Cross-section of a nanopore from a different membrane, fabricated under the same conditions. (c) AFM: High resolution image of a nanopore. (d) SEM: Plan-view of a nanopore. (e-f) SAXS: (e) 2D scattering image and (f) fit to the data of a different membrane, fabricated under identical conditions.}
 \label{f:charact}
\end{figure*}

\section{Results \& discussion}
\subsection{Nanopore fabrication}
The nanopores were created in \SI{1}{\um} thick, $55\times\SI{66}{\um\tothe{2}}$ a-\ch{SiO2} windows using the track etch technique,\cite{Hadley2018} which is schematically shown in Figure~\ref{f:charact}(a). First, the a-\ch{SiO2} windows were irradiated with \SI{2.2}{\GeV} \ch{^{197}Au} ions at the Universal Linear Accelerator UNILAC (GSI Helmholtz Centre for Heavy Ion Research Darmstadt, Germany) with a fluence of \SI{e6}{\text{ions }\cm\tothe{-2}} (i). When passing through the material, the ions generate a narrow cylindrical damage region along their path.\cite{Kluth2008} These so-called ion tracks are more susceptible to chemical etching than the undamaged material. After irradiation, the membrane was taped to a silicon wafer using Kapton tape and immersed in \SI{2.5}{\percent} hydroflouric acid at room temperature, which removes material along the track thus at a higher rate than the surrounding matrix (ii).\cite{Hadley2020} This process creates highly uniform conical nanopores in the thin a-\ch{SiO2} windows. To stop the etching, the membranes were rinsed with and stored in demonised water until they were characterised.

\begin{figure*}
 \centering
 \includegraphics{setup.png}
 \caption{Conductometric nanopore experiments: Scheme of the setup with a negatively charged nanopore membrane. Due to the negative surface charge, there are more cations than anions transported through the pore. In the positive case (left), the pore is ion depleted, representing the low conductance state. For negative voltages (right), the concentration inside the pore is increased, corresponding to the high conductance state.}
 \label{f:setup}
\end{figure*}

With this fabrication technique, we are able to manufacture nanopore membranes with a wide range of different morphologies. By varying the thickness of the a-\ch{SiO2} layer, nanopores with lengths between \SIrange{200}{1000}{\nm} can be created. By combining a variable irradiation fluence between \SIrange{e6}{e9}{\text{ions }\cm\tothe{-2}} and window sizes ranging from $30\times30$ to $700\times\SI{700}{\um\tothe{2}}$, membranes containing between 10 to 5,000,000 nanopores can be achieved. Single ion irradiation leading to a single pore is also possible.\cite{Guo2008} The half-cone angle can be tuned between \SIrange{10}{18}{\degree} by using different irradiation energies, and adjusting the etching conditions.\cite{Hadley2019}

Here, we are presenting a detailed structural characterisation and the ionic transport properties of conical nanopores in an a-\ch{SiO2} membrane. Further descriptions of the experiments are given in the SI.

\subsection{Structural characterisation}
Scanning electron microscopy (SEM) is a useful tool to estimate the pore base radius, number of pores and pore distribution across the nanopore membrane. Figure~\ref{f:charact}(b) and (d) show a cross-section and a plan-view of a conical nanopore. To acquire a cross-section, the membrane is broken in half. As this is a destructive technique, Figure~\ref{f:charact}(b) is from a different membrane fabricated under identical conditions. From the SEM data, we conclude that the membrane contains 16 nanopores with an average minimum distance to the neighbouring pore of \SI{7.8(46)}{\um} (see SI, Figure~S1 and S2) and a pore base radius of \SI{169(7)}{\nm}. The spacing of the nanopores is sufficient to rule out any meaningful contributions of inter-pore interactions.\cite{Cao2018}

The base radius of the pores can also be determined using atomic force microscopy (AFM). As presented in Figure~\ref{f:charact}(c), AFM provides a high resolution image of the nanopore, from which a base radius of \SI{158.3(26)}{\nm} is obtained. The difference in base radius compared to the SEM measurements might be due to the difficulty in defining the edge of the pore in either measurement technique and limited resolution. For more information, see the SI.

Using ellipsometry and surface profiling, the membrane thickness and hence the pore length was determined to \SI{710(5)}{\nm}, indicating a thinning down from the original thickness of \SI{1}{\um} by almost \SI{300}{\nm} during the etching process. Due to the interference of the ellipsometry laser with the nanopores, the thickness of the membrane could not determined directly by ellipsometry only. Thus, the step between an un-etched area of the membrane was compared to the etched area using surface profiling.

To gain precise measurements of the cone angle of the nanopores, synchrotron-based small angle X-ray scattering (SAXS) was performed at the SAXS/WAXS beamline at the Australian Synchrotron (Melbourne, Australia) with an X-ray energy of \SI{12}{\keV}. By tilting the membrane with respect to the X-ray beam and analysing the change in scattering pattern, the cone angle of the pores can be determined. This measurement has been performed using a different membrane that was fabricated under identical conditions irradiated with a fluence of \SI{e8}{\text{ions }\cm\tothe{-2}} to obtain a sufficiently high SAXS signal. It can be assumed that the etching process and the SAXS measurements are unaffected by the difference in fluence, as in both cases the pore overlap is negligible.\cite{Kluth2008b} Figure~\ref{f:charact}(e) shows the 2D scattering image with the characteristic x-wing pattern and oscillations in intensity for a tilt angle of \SI{41}{\degree}. By performing a fit of the 2D scattering data (Figure~\ref{f:charact}(f)) to our theoretical model we find that the half-cone angle of the nanopores is \SI{12.6(1)}{\degree} (more SAXS measurements are presented in the SI, Figure~S3). A detailed explanation of the measurement and data analysis technique is given by Hadley \textit{et al}.\cite{Hadley2018}

Utilising the structural characterisation using the different measurement techniques, we can predict the tip radius of the nanopore $r\ind{t}$ using the relation
\begin{align}
    r\ind{t}=r\ind{b}-L\tan\vartheta
\end{align}
where $r\ind{b}$ is the base radius, $L$ is the length, and $\vartheta$ is the half-cone angle of the nanopore. The tip radius calculated by this method varies between \SIrange{0}{11}{\nm} due to the uncertainties determining the base radius using AFM and SEM (a more comprehensive analysis is given in the SI). It is important to note that the cone angle determined by SAXS has an uncertainty of only \SI{.1}{\degree} and thus does not contribute significantly to this. Conductometry measurements however will provide a better measure for the tip radius, which is presented in the following section.

\begin{figure}
    \begin{minipage}{.5\textwidth}
        \includegraphics{iv.png}
    \end{minipage
    \begin{minipage}{.5\textwidth}
        \includegraphics{G_ph.png}
        \includegraphics{sc_ph.png}
    \end{minipage}
  \caption{(a) $I-V$ curves in \SI{100}{\mymmol} \ch{KCl} for the pH series in acidic and basic conditions. (b) Experimental membrane conductance (symbols) including fit to Equation~\ref{eq:cond} (lines) for the pH series at a concentration of \SI{100}{\mymmol}. The conductance is normalised by the electrolyte conductivity, as the conductivity slightly varies with pH (inset). (c) Calculated surface charge densities as a function of pH for different concentrations for our nanopore system. The surface charge densities were calculated using the experimentally obtained $\text{p}K_a$, $\text{p}K_b$ and pI values and Equation~\ref{eq:sig1} and \ref{eq:sig2}.}
  \label{f:ph}
\end{figure}

\subsection{Ion transport and surface properties}
The ionic transport characteristics of the nanopores were analysed as a function of electrolyte concentration and pH conditions. The measurements were conducted by inserting the nanopore membrane between two compartments, which are filled with a potassium chloride (\ch{KCl}) solution at concentrations ranging from \SIrange{.1}{1000}{\mymmol} and pH values between \numlist{1.8;10.1}{}. The pH was adjusted by adding \ch{HCl} or \ch{KOH}. Using \ch{Ag}/\ch{AgCl} electrodes on either side of the membrane, we measured the current-voltage ($I-V$) characteristics of the pores between \SIlist{-1;1}{\V} to determine the membrane conductance and ion current rectification (ICR). The conductance was determined in the linear regime of the $I-V$ curve around \SI{0}{\V} between \SI{-.1}{\V} and \SI{.1}{\V}. The convention for conical nanopore experiments is that the polarity of the voltage is determined by the electrode at the cone base, as shown in Figure~\ref{f:setup}.

In a charged nanopore, the concentration of counter-ions to the surface charge is greatly increased and thus the majority of the ionic current is carried by them. As the electric field in the tip region of a conical nanopore is greatly increased, transport of majority charge carriers from tip to base is promoted and a large ionic current is observed. This case is called the high conductance state. If the polarity of the electric field is reversed, transport is hindered, which is referred to as the low conductance state.

Figure~\ref{f:ph}(a) shows the $I-V$ curves for the pH series. In acidic conditions, the high conductance state of the membrane occurs at positive voltages, indicating a positive surface charge which is the opposite in basic conditions, indicating a negative surface charge. In either case, the high conductance state is strongly dependent on the pH, while the low conductance state is almost unaffected. In the high conductance state, the ionic current is dominated by surface effects, indicating a large change in surface charge with varying pH. As in the low conductance state the current is dominated by bulk behaviour, the ionic current is almost unaffected by changes in pH.

Figure~\ref{f:ph}(b) shows the membrane conductance as a function of pH. The symbols are our measurements and the solid line represents a fit to Equation~\ref{eq:cond}. The membrane conductance is normalised by the electrolyte conductivity as it changes slightly with pH (inset). The minimum conductance occurs at a pH of 4.5, at which the surface charge density reaches \SI{0}{\V\m\tothe{-2}} as plotted in Figure~\ref{f:ph}(c). Figure~\ref{f:ph}(c) shows the surface charge densities that are calculated using the experimentally obtained $\text{p}K_a$, $\text{p}K_b$ and pI values and Equation~\ref{eq:sig1} and \ref{eq:sig2}. At the point of zero surface charge density, which is the isoelectric point pI, the pores only exhibit bulk current behaviour. In more acidic conditions, the surface becomes positively charged and in more basic conditions the pore becomes negatively charged.

\begin{figure}
    \begin{minipage}{.5\textwidth}
        \includegraphics{G_c.png}
    \end{minipage
    \begin{minipage}{.5\textwidth}
        \includegraphics{sc_c.png}
    \end{minipage}  
  \caption{(a) Experimental membrane conductance (symbols) including fit to Equation~\ref{eq:cond} (lines) for the concentration series for three different pH values, and pH as a function of concentration (inset). (b) Calculated surface charge densities as a function of concentration for different pH values for our nanopore system. The surface charge densities were calculated using the experimentally obtained $\text{p}K_a$, $\text{p}K_b$ and pI values and Equation~\ref{eq:sig1} and \ref{eq:sig2}.}
  \label{f:concentration}
\end{figure}

\begin{table}
  \caption{Debye length $\lambda\ind{D}$ (calculated using Equation~\ref{eq:sig2}) for different electrolyte concentrations. Extension of the EDL through the nanopore at the pore tip indicated by the ratio of Debye length to tip radius $\lambda\ind{D}/r\ind{t}$.}
  \label{t:debye}
  \begin{tabular}{lll}
    \hline
    Concentration, $c$ [\SI{}{\mymmol}]  & Debye length, $\lambda\ind{D}$ [\SI{}{\nm}]  & $\lambda\ind{D}/r\ind{t}$ [$\%$]  \\
    \hline
    0.1 & 30.8 & 540 \\
    1 & 9.72 & 171 \\
    10 & 3.07 & 54 \\
    100 & 0.97 & 17 \\
    1000 & 0.31 & 5.4 \\
    \hline
  \end{tabular}
\end{table}

Figure~\ref{f:concentration}(a) shows the membrane conductance as a function of electrolyte concentration for three pH values. The symbols are our measurements and the solid line the fit to Equation~\ref{eq:cond}. At a pH of 3.6, the surface is positively charged, at a pH of 5.7, the pores are slightly negatively charged and at a pH of 10.1, the pores are strongly negatively charged, which is reflected in Figure~\ref{f:concentration}(b). At high concentrations, the overall conductance is dominated by bulk behaviour, as the Debye lengths becomes smaller (see Table~\ref{t:debye}). The Debye length describes the screening length of the surface charge; a reduced Debye length is correlated to a thinner electrical double layer (EDL). At lower concentrations, where the slope of the conductance changes, surface effects start to dominate over bulk behaviour. 

By parallel analysis and non-linear least squares fitting of the pH and concentration dependent membrane conductance with Equation~\ref{eq:cond}, the $\text{p}K_a$, $\text{p}K_b$ and pI values as well as pore size can be determined. We obtain a pore size of \SI{5.7(1)}{\nm}. This is well within the range of the pore size determined by the structural characterisation. The $\text{p}K_a$, $\text{p}K_b$ and pI are \num{7.6(1)}, \num{1.5(2)} and \num{4.5(1)} respectively, demonstrating that nanopores in track etched a-\ch{SiO2} exhibit similar properties to a-\ch{SiO2} nanoparticles and nanochannels, for which $\text{p}K_a$, $\text{p}K_b$ and pI values of 6.6~--~8, 0~--~2 and <~5 respectively have been reported.\cite{VanDerHeyden2005,Wang2010,Andersen2011,Yeh2013,Barisik2014}

\subsection{Ionic current rectification}
Ionic current rectification (ICR) is an important characteristic of (asymmetric) nanopores in which the membrane exhibits a directional preference for transport of charged species. ICR can be achieved by different means, such as by a varying surface charge density along the nanopore,\cite{Daiguji2005} asymmetries in the electrolyte,\cite{Hsu2018} or geometry of the pores such as our conical nanopores. ICR is defined as the ratio between the current in the high conductance state to the low conductance state:
\begin{equation}
    \text{ICR} = 
    \begin{cases}
    \left|\frac{I\left(\SI[retain-explicit-plus]{+1}{\V}\right)}{I\left(\SI{-1}{\V}\right)}\right|&\text{if pH}<\text{pI}\\
    \left|\frac{I\left(\SI{-1}{\V}\right)}{I\left(\SI[retain-explicit-plus]{+1}{\V}\right)}\right|&\text{if pH}>\text{pI}
    \end{cases}
\end{equation}

\begin{figure}
    \begin{minipage}{.5\textwidth}
        \includegraphics{rect_ph.png}
    \end{minipage
    \begin{minipage}{.5\textwidth}
        \includegraphics{rect_c.png}
    \end{minipage}
  \caption{Ionic current rectification (ICR) as a function of (a) pH at a concentration of \SI{100}{\mymmol} and (b) concentration for three different pH values. The dashed line indicates a rectification ratio of 1, i.e. no rectification and a symmetric current.}
  \label{f:icr}
\end{figure}

Figure~\ref{f:icr} shows the ICR of our membrane as a function of pH and concentration. The rectification ratio peaks at around 10 in acidic conditions at a pH between 2 and 3 at concentrations between \SIrange{10}{100}{\mymmol}, corresponding to surface charge densities of \SIrange{.03}{.04}{\coulomb\m\tothe{-2}}. This behaviour was theoretically predicted by Tseng \textit{et al}.;\cite{Tseng2016} In the high conductance state, when the surface charge is low, an increase in surface charge will cause a large increase in the concentration of ions within the pore and thus the current. Conversely, in the low conductance state, the concentration of ions barely increases by increasing the surface charge and thus the current remains almost constant. This effect can be observed in our measurement. Figure~\ref{f:ph}(a) shows a large change in the high conductance state between pH values of 4.5 to 2.3, whereas the low conductance state is almost stable, thus ICR increases. However, when the surface charge is increased further, the concentration and thus current saturates in the high conductance state. In the low conductance state, the concentration and thus current will increase significantly. We observe this effect when adjusting the pH from 2.3 to 1.8, and hence ICR decreases. Thus for any given pore geometry, ICR will have a maximum value at a certain surface charge density, which is \SIrange{.03}{.04}{\coulomb\m\tothe{-2}} in our nanopore system.
  
\section{Conclusions}
We presented a detailed characterisation of conical nanopores in a-\ch{SiO2} that were fabricated using the track etch technique. The geometrical structure of the nanopore membrane was characterised using SAXS, AFM, SEM, ellipsometry, and surface profiling. The multipore membrane used in this research contains 16 nanopores with tip radii of \SI{5.7(1)}{\nm}. We determined the important surface parameters $\text{p}K_a$, $\text{p}K_b$ and pI, which are \num{7.6(1)}, \num{1.5(2)} and \num{4.5(1)} respectively, which is in good agreement to those from other \ch{SiO2} systems. The maximum ICR was determined to be around 10 at pH 2.3. It is important to note that these measurements have been performed with a multipore system. Frequently, characterisations are performed using single nanopores, and scaling those to multipores can lead to a significant loss in performance.\cite{Cao2018,Wang2021} ICR is an extremely important property of nanopore systems, as ICR enables the development of innovative applications, as well as improving the performance of non-rectifying membranes. ICR has been utilised to create new biosensors,\cite{Cai2015a,Perez-Mitta2018} as well as demonstrate superior performance in power generation.\cite{Li2019,Laucirica2021} Electroosmotic pumps (EOP) that operate in AC mode have also been developed.\cite{Wu2016} As the performance of such a pump is proportional to the opening angle of the pore, we estimate that our system has the potential to outperform the best AC-EOP by an order of magnitude.\cite{Wu2016}

a-\ch{SiO2} offers a platform that is much thinner with larger cone angles and thus higher throughput than polymers, while being more resistant to temperature and chemicals. It also provides the capabilities of readily integration in lab-on-a-chip devices. Additionally, a-\ch{SiO2} is well suited for chemical surface functionalisation through simple silane chemistry,\cite{Abdelsamad2018,Sanz-Perez2018,Cashin2018,Karnati2019} allowing further tuning of the transport characteristics, which can be well utilised due to the large surface area of the pore walls. It also opens up new surface modification techniques that are readily available for a-\ch{SiO2} but are lacking for other materials,\cite{Graf2012,Tanaka2013,Eggenberger2019b} potentially enabling applications with significantly enhanced performance to currently established techniques. This nanopore system thus provides a new versatile platform for the development of advanced integratable sensor and separation systems.
\begin{acknowledgement}
P.K. acknowledges financial support from the Australian Government through the Australian Research Council under the ARC Discovery Project Scheme (DP180100068). Part of this research was undertaken at the SAXS/WAXS beamline at the Australian Synchrotron, part of ANSTO. The irradiated a-\ch{SiO2} membranes are part of the experiment UMAT, which was performed at the beam line X0 at the GSI Helmhotzzentrum für Schwerionenforschung, Darmstadt (Germany) in the frame of FAIR Phase-0. This work used the ACT node of the NCRIS-enabled Australian National Fabrication Facility (ANFF-ACT).
\end{acknowledgement}

\begin{suppinfo}
The following file is available free of charge: "Supporting Information: Number of pores and average minimum distance; Small angle X-ray scattering analysis; Atomic force microscopy analysis; Pore tip radius calculation (PDF)".
\end{suppinfo}

\section*{Conflicts of interest}
There are no conflicts of interest to declare.

