\section{Introduction} \label{sec:intro}
Neutrinos of astrophysical and cosmological origin have been crucial for unraveling neutrino masses and properties. Solar neutrinos provided the first evidence for neutrino oscillations, and hence massive neutrinos. We know that at least two massive neutrinos should exist, as required by the two distinct squared mass differences measured, the atmospheric $\lvert\Delta m^2_{31}\rvert \approx 2.51\cdot 10^{-3}$~eV$^2$ and the solar $\Delta m^2_{21} \approx 7.42\cdot 10^{-5}$~eV$^2$ splittings~\cite{deSalas:2020pgw,Esteban:2020cvm,Capozzi:2021fjo}~\footnote{The current ignorance on the sign of $\lvert\Delta m^2_{31}\rvert$ is translated into two possible mass orderings. In the \emph{normal} ordering (NO), the total neutrino mass is $\sum m_\nu \gtrsim 0.06$~eV, while in the \emph{inverted} ordering (IO) it is $\sum m_\nu \gtrsim 0.10 $~eV.}. However, neutrino oscillation experiments are not sensitive to the absolute neutrino mass scale. On the other hand, cosmological observations provide the most constraining recent upper bound on the total neutrino mass via relic neutrinos, $\sum m_\nu<0.09$~eV at $95\%$~CL~\cite{DiValentino:2021hoh}, where the sum runs over the distinct neutrino mass states. However, this limit is model-dependent, see for example~\cite{DiValentino:2015sam,Palanque-Delabrouille:2019iyz,Lorenz:2021alz,Poulin:2018zxs,Ivanov:2019pdj,Giare:2020vzo,Yang:2017amu,Vagnozzi:2018jhn,Gariazzo:2018meg,Vagnozzi:2017ovm,Choudhury:2018byy,Choudhury:2018adz,Gerbino:2016sgw,Yang:2020uga,Yang:2020ope,Yang:2020tax,Vagnozzi:2018pwo,Lorenz:2017fgo,Capozzi:2017ipn,DiValentino:2021zxy,DAmico:2019fhj,Colas:2019ret}.

The detection of supernova (SN) neutrinos can also provide constraints on the neutrino mass, by exploiting the time of flight delay~\cite{Zatsepin:1968ktq} experienced by a neutrino of mass $m_\nu$ and energy $E_\nu$:

\begin{equation}
    \label{eq:delay}
    \Delta t = \frac{D}{2c}\left(\frac{m_\nu}{E_{\nu}}\right)^2~,
\end{equation}

\noindent where $D$ is the distance travelled by the neutrino. This method probes the same neutrino mass constrained via laboratory-based kinematic measurements of beta-decay electrons~\footnote{The current limit from the tritium beta decay experiment KATRIN (Karlsruhe Tritium Neutrino) is $m_{\beta}<0.8$~eV~\cite{Aker:2021gma} and the expected sensitivity is 0.2~eV~\cite{Drexlin:2013lha}, both at 90\% CL.}. Using neutrinos from SN1987A~\cite{Kamiokande-II:1989hkh,Kamiokande-II:1987idp,Bionta:1987qt,Alekseev:1988gp,Alekseev:1987ej}, a $95\%$ confidence level (CL) current upper limit of $m_\nu<5.8$~eV~\cite{Pagliaroli:2010ik} has ben derived (see also Ref.~\cite{Loredo:2001rx}). Prospects for future SN explosions may reach the sub-eV level~\cite{Pagliaroli:2010ik,Nardi:2003pr,Nardi:2004zg,Lu:2014zma,Hyper-Kamiokande:2018ofw,Hansen:2019giq}. Nevertheless, these forecasted estimates rely on the detection of inverse $\beta$ decay events in water Cherenkov or liquid scintillator detectors, mostly sensitive to $\bar{\nu}_e$ events. An appealing and alternative possibility is the detection of the $\nu_e$ neutronization burst exploiting the liquid argon technology at the DUNE far detector~\cite{DUNE:2020zfm,Rossi-Torres:2015rla}. The large statistics and the very distinctive neutrino signal in time will ensure a unique sensitivity to the neutrino mass signature via time delays.

\section{Supernova electron neutrino events} \label{sec:events}
Core-collapse supernovae emit $99\%$ of their energy ($\simeq 10^{53}$~ergs) in the form of (anti)neutrinos of all flavors with mean energies of $\mathcal{O}(10~\si{\mega\electronvolt})$. The explosion mechanism of a core-collapse SN can be divided into three main phases: the \emph{neutronization burst}, the \emph{accretion phase} and the \emph{cooling phase}. The first phase, which lasts for 25 milliseconds approximately, is due to a fast \emph{neutronization} of the stellar nucleus via electron capture by free protons, causing an emission of electron neutrinos ($e^- + p\rightarrow \nu_e + n$). The flux of $\nu_e$ stays trapped behind the shock wave until it reaches sufficiently low densities for neutrinos to be suddenly released. Unlike subsequent phases, the neutronization burst phase has little dependence on the progenitor star properties. In numerical simulations, there is a second \emph{accretion} phase of $\sim 0.5$~s in which the shock wave leads to a hot accretion mantle around the high density core of the neutron star. High luminosity $\nu_e$ and $\bar{\nu}_e$ fluxes are radiated via the processes $e^- + p\rightarrow \nu_e + n$ and $e^+ + n \rightarrow \bar{\nu}_e + p$ due to the large number of nucleons and the presence of a quasi-thermal $e^+e^-$ plasma. Finally, in the \emph{cooling} phase, a hot neutron star is formed. This phase is characterized by the emission of (anti)neutrino fluxes of all species within tens or hundreds of seconds.

For numerical purposes, we shall make use here of the following quasi-thermal parametrization, representing well detailed numerical simulations~\cite{Keil:2002in,Hudepohl:2009tyy,Tamborra:2012ac,Mirizzi:2015eza}:
\begin{equation}
\label{eq:differential_flux}
\Phi^{0}_{\nu_\beta}(t,E) = \frac{L_{\nu_\beta}(t)}{4 \pi D^2}\frac{\varphi_{\nu_\beta}(t,E)}{\langle E_{\nu_\beta}(t)\rangle}\,,
\end{equation}
and describing the differential flux for each neutrino flavor $\nu_\beta$ at a time $t$ after the SN core bounce, located at a distance $D$. In Eq.~\ref{eq:differential_flux}, $L_{\nu_\beta}(t)$ is the $\nu_\beta$ luminosity, $\langle E_{\nu_\beta}(t)\rangle$ the mean neutrino energy and $\varphi_{\nu_\beta}(t,E)$ is the neutrino energy distribution, defined as:
\begin{equation}
\label{eq:nu_energy_distribution}
\varphi_{\nu_\beta}(t,E) = \xi_\beta(t) \left(\frac{E}{\langle E_{\nu_\beta}(t)\rangle}\right)^{\alpha_\beta(t)} \exp{\left\{\frac{-\left[\alpha_\beta(t) + 1\right] E}{\langle E_{\nu_\beta}(t)\rangle}\right\}}\,,
\end{equation}

\noindent where $\alpha_\beta(t)$ is a \emph{pinching} parameter and $\xi_\beta(t)$ is a unit-area normalization factor.

The input for luminosity, mean energy and pinching parameter values have been obtained from the \texttt{SNOwGLoBES} software \cite{snowglobes}. \texttt{SNOwGLoBES} includes fluxes from the Garching Core-Collapse Modeling Group~\footnote{\url{https://wwwmpa.mpa-garching.mpg.de/ccsnarchive/index.html}}, providing computationally expensive simulation results for a progenitor star of $8.8 M_\odot$~\cite{Hudepohl:2009tyy}.

Neutrinos experience flavor conversion inside the SN as a consequence of their coherent interactions with electrons, protons and neutrons in the medium, being subject to the MSW (Mikheyev-Smirnov-Wolfenstein) resonances associated to the solar and atmospheric neutrino sectors~\cite{Dighe:1999bi}. After the resonance regions, the neutrino mass eigenstates travel incoherently in their way to the Earth, where they are detected as flavor eigenstates.  The neutrino fluxes at the Earth ($\Phi_{\nu_e}$ and $\Phi_{\nu_\mu}=\Phi_{\nu_\tau}=\Phi_{\nu_x}$) can be written as:

\begin{eqnarray}
\label{eq:nue}
   \Phi_{\nu_e}&= &p  \Phi^{0}_{\nu_e} +(1-p) \Phi^{0}_{\nu_x}~;\\
    \Phi_{\nu_\mu}+\Phi_{\nu_\tau} \equiv 2\Phi_{\nu_x} & =& (1-p) \Phi^{0}_{\nu_e} + (1+p) \Phi^{0}_{\nu_x}~,
\end{eqnarray}

\noindent where $\Phi^{0}$ refers to the neutrino flux in the SN interior, and the $\nu_e$ survival probability $p$ is given by $p = |U_{e3}|^2= \sin^2 \theta_{13}$ ($p \simeq |U_{e2}|^2 \simeq \sin^2 \theta_{12}$) for NO (IO), due to adiabatic transitions in the $H$ ($L$) resonance, which refer to flavor conversions associated to the atmospheric $\Delta m^2_{31}$ (solar $ \Delta m^2_{21}$) mass splitting, see e.g.~\cite{Dighe:1999bi}. Here we are neglecting possible non-adiabaticity effects occurring when the resonances occur near the shock wave \cite{Schirato:2002tg,Fogli:2003dw,Fogli:2004ff,Tomas:2004gr,Dasgupta:2005wn,Choubey:2006aq,Kneller:2007kg,Friedland:2020ecy}, as well as the presence of turbulence in the matter density \cite{Fogli:2006xy,Friedland:2006ta,Kneller:2010sc,Lund:2013uta,Loreti:1995ae,Choubey:2007ga,Benatti:2004hn,Kneller:2013ska,Fogli:2006xy}. The presence of non-linear collective effects~\cite{Mirizzi:2015eza,Chakraborty:2016yeg,Horiuchi:2018ofe,Tamborra:2020cul,Capozzi:2022slf} is suppressed by the large flavor asymmetries of the neutronization burst~\cite{Mirizzi:2015eza}.

Earth matter regeneration effects also affect the neutrino propagation in case the SN is shadowed by the Earth for the DUNE detector. The trajectories of the neutrinos depend on the SN location and on the time of the day at which the neutrino burst reaches the Earth. Neutrinos therefore travel a certain distance through the Earth characterized by a zenith angle $\theta$, analogous to the one usually defined for atmospheric neutrino studies. This convention assumes $\cos \theta=-1$ for upward-going events, \emph{i.e.} neutrinos that cross a distance equal to the Earth's diameter, and $\cos \theta\geq 0$ for downward-going neutrinos that are un-shadowed by the Earth. An analytical expression for the electron neutrino fluxes after crossing the Earth~\footnote{In what follows, we shall focus on electron neutrino events, the dominant channel in DUNE.} yields no modifications for NO.
In turn, for IO, an approximate formula for the $\nu_e$ survival probability in Eq.~\ref{eq:nue} and after crossing the Earth, assuming that SN neutrinos have traveled a distance $L(\cos\theta)$ inside the Earth and in a density constant medium, reads as~\cite{Dighe:1999bi,Lunardini:2001pb}:
\begin{widetext}
\begin{eqnarray}
\label{eq:p2e}
p & = & \sin^2\theta_{12} + \sin2\theta^m_{12} \, \label{P2e}
 \sin(2\theta^m_{12}-2\theta_{12})
\sin^2\left(
\frac{\Delta m^2_{21} \sin2\theta_{12}}{4 E \,\sin2\theta^m_{12}}\,L(\cos\theta)
\right)\,,
\end{eqnarray}
\end{widetext}
\noindent
where $\theta^m_{12}$ is the effective value of mixing angle $\theta_{12}$ in matter for neutrinos:

\begin{eqnarray}
\sin2\theta^m_{12} = \frac{\sin^2\theta_{12}}
{\sin^2\theta_{12}+ \left(\cos^2\theta_{12}- \frac{2\sqrt{2}G_F N{e} E}{\Delta m^2_{21}}\right)^2}~.
\end{eqnarray}
In the expression above, $N_e$ refers to the electron number density in the medium, $\sqrt{2}G_F N_e (\textrm{eV})\simeq 7.6 \times 10^{-14} Y_e\rho$, with $Y_e$ and $\rho$ the electron fraction and the Earth's density in g/cm$^3$ respectively.

Our numerical results are obtained calculating $p$ in Eq.~\ref{eq:p2e} in the general case of neutrino propagation in multiple Earth layers, with sharp edge discontinuities between different layers and a mild density dependence within a layer, see \cite{Lisi:1997yc,Fogli:2012ua}. Our method consists in evaluating the evolution operator for the propagation in a single layer using the Magnus expansion \cite{Magnus_exp}, where the evolution operator is written as the exponential of an operator series. In our case, we stop at the second order of the series. With the approximation of the electron density being a fourth order polynomial as a function of the Earth radius, the integrals involved in the Magnus expansion become analytical. The evolution operator over the entire trajectory in the Earth is simply the product of the operators corresponding to each crossed layer.

The neutrino interaction rate per unit time and energy in the DUNE far detector is defined as:
\begin{equation}
\label{eq:rate_DUNE_fun}
R(t,E) = N_\text{target}~\sigma_{\nu_e\text{CC}}(E)~\epsilon(E)~\Phi_{\nu_e}(t,E)~,
\end{equation}
\noindent where $t$ is the neutrino emission time, $E$ is the neutrino energy, $N_\text{target}=\num{6.03e32}$ is the number of argon nuclei for a $40$ kton fiducial mass of liquid argon, $\sigma_{\nu_e\text{CC}}(E)$ is the $\nu_e$ cross-section, $\epsilon(E)$ is the DUNE reconstruction efficiency and $\Phi_{\nu_e}(t,E)$ is the electron neutrino flux reaching the detector per unit time and energy. The total number of expected events is given by $R\equiv \int R(t,E)\mathop{}\!\mathrm{d} t \mathop{}\!\mathrm{d} E$.

As far as cross-sections are concerned, liquid argon detectors are mainly sensitive to electron neutrinos via their charged-current interactions with $^{40}$Ar nuclei, $\nu_e + {^{40} Ar} \rightarrow e^{-} + {^{40} K^{*}}~$, through the observation of the final state electron plus the de-excitation products (gamma rays, ejected nucleons) from $^{40} K^{*}$. We use the MARLEY~\footnote{MARLEY (Model of Argon Reaction Low Energy Yields) is a Monte Carlo event generator for neutrino interactions on argon nuclei at energies of tens-of-MeV and below, see \url{http://www.marleygen.org/} and Ref.~\cite{Gardiner:2021qfr}.} charged-current $\nu_e$ cross-section on $^{40}$Ar, implemented in \texttt{SNOwGLoBES} \cite{snowglobes}. Concerning event reconstruction, we assume the efficiency curve as a function of neutrino energy given in Ref.~\cite{DUNE:2020zfm}, for the most conservative case quoted there of 5~MeV as deposited energy threshold.

\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{IO_eventsperbin_1ms.pdf}
\caption{\label{fig:events}Number of $\nu_e$ events per unit time in the DUNE far detector. The plot zooms into the first 50~ms since core bounce. A SN distance of 10~kpc is assumed. Several histograms are shown: neglecting oscillations (both in the SN and in the Earth), as well as including oscillations for the NO and IO cases. For IO, we show the variation of the Earth matter effects with zenith angle $\theta$.}
\end{center}
\end{figure}

Figure~\ref{fig:events} shows the number of $\nu_e$ events as a function of emission time at the DUNE far detector from a SN explosion at $10$~kpc from Earth, for negligible time delays due to non-zero neutrino masses. We illustrate the case where no oscillations are considered. We also account for oscillations in NO and IO cases, the latter for several possible SN locations with respect to the Earth. The neutronization burst is almost entirely (partially) suppressed in the normal (inverted) mass ordering.

For a SN located at $D=10$~kpc from the Earth and without Earth matter effects, $R$ is found to be 860, 1372 and 1228 for the no oscillations, NO and IO cases, respectively. In other words, the largest total event rate is obtained for the largest swap of electron with muon/tau neutrinos in the SN interior, \emph{i.e.} the smallest value of $p$ in Eq.~\ref{eq:nue}, corresponding to the NO case. This can be understood from the larger average neutrino energy at production of muon/tau neutrinos compared to electron neutrinos, resulting in a higher (on average) neutrino cross-section and reconstruction efficiency.

Finally, as shown in Fig.~\ref{fig:events}, Earth matter effects are expected to have a mild effect on the event rate in all cases. The $\nu_e$ flux is left unchanged in the normal ordering, while Earth matter effects modify slightly the neutronization burst peak in the IO case. The total number of events becomes $R=1206, 1214, 1260, 1200$ for IO and $\cos\theta = -0.3,-0.5,-0.7,-1$, respectively.

\section{Neutrino mass sensitivity} \label{sec:likelihood}
In order to compute the DUNE sensitivity to the neutrino mass, we adopt an 'unbinned' maximum likelihood method similar to the one in \cite{Pagliaroli:2010ik}.

We start by generating many DUNE toy experiment datasets (a few hundreds, typically) for each neutrino oscillation and SN distance scenario, and assuming massless neutrinos. For each dataset, the time/energy information of the $R$ generated events are sampled following the parametrization of Eq.~\ref{eq:rate_DUNE_fun}, and events are sorted in time-ascending order.

Furthermore, we assume a $10\%$ fractional energy resolution in our $\mathcal{O}$(10~MeV) energy range of interest, see~\cite{DUNE:2020zfm}, and smear the neutrino energy of each generated event accordingly. We assume perfect time resolution for our studies. On the one hand, DUNE's photon detection system provides a time resolution better than 1~$\mu$s~\cite{DUNE:2020zfm}, implying a completely negligible smearing effect. On the other hand, even in the more conservative case of non-perfect matching between TPC and optical flash information, the DUNE charge readout alone yields a time resolution of order 1~ms~\cite{DUNE:2020ypp}. While not completely negligible, the time smearing is expected to have a small impact also in this case, considering the typical 25~ms duration of the SN neutronization burst.

Once events are generated for each DUNE dataset, we proceed with our minimization procedure. The two free parameters constrained in our fit are an offset time $t_\text{off}$ between the moment when the earliest SN burst neutrino reaches the Earth and the detection of the first event $i=1$, and the neutrino mass $m_\nu$. The fitted emission times $t_{i,fit}$ for each event $i$ depend on these two fit parameters as follows:
\begin{equation}
\label{eq:emission_t}
t_{i,fit} = \delta t_i  - \Delta t_{i}(m_\nu) + t_\text{off}\,,
\end{equation}
where $\delta t_i $ is the time at which the neutrino interaction $i$ is measured in DUNE (with the convention that $\delta t_1\equiv 0$ for the first detected event), $\Delta t_i(m_\nu)$ is the delay induced by the non-zero neutrino mass (see Eq.~\ref{eq:delay}), and $t_\text{off}$ is the offset time. We do not include any free parameter describing the SN emission model uncertainties in our fit.

By neglecting backgrounds and all the constant (irrelevant) factors, our likelihood $\mathcal{L}$ function \cite{Pagliaroli:2008ur} reads as
\begin{equation}
\label{eq:likelihood_fun}
\mathcal{L}(m_{\nu},t_\text{off}) = \prod_{i=1}^{R}\int R(t_i,E_i)G_i(E)\mathop{}\!\mathrm{d} E~,
\end{equation}
\noindent where $G_i$ is a Gaussian distribution with mean $E_i$ and sigma $0.1E_i$, accounting for energy resolution. The estimation of the $m_\nu$ fit parameter is done by marginalizing over the nuisance parameter $t_\text{off}$. For each fixed $m_\nu$ value, we minimize the following $\chi^2$ function:
\begin{equation}
\label{eq:chi2_fun}
\chi^2(m_{\nu}) = -2 \log(\mathcal{L}(m_{\nu},t_\text{off,best}))~,
\end{equation}
\noindent where $\mathcal{L}(m_{\nu},t_\text{off,best})$ indicates the maximum likelihood at this particular $m_\nu$ value.

The final step in our analysis is the combination of all datasets for the same neutrino oscillation and SN distance scenario, to evaluate the impact of statistical fluctuations. For each $m_\nu$ value, we compute the mean and the standard deviation of all toy dataset $\chi^2$ values. In order to estimate the allowed range in $m_\nu$, the $\Delta\chi^2$ difference between all mean $\chi^2$ values and the global mean $\chi^2$ minimum is computed. The mean 95\% CL sensitivity to $m_\nu$ is then defined as the largest $m_\nu$ value satisfying $\Delta \chi^2<3.84$. The $\pm 1\sigma$ uncertainty on the 95\% CL $m_\nu$ sensitivity can be computed similarly, including into the $\Delta\chi^2$ evaluation also the contribution from the standard deviation of all toy dataset $\chi^2$ values.

\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{chi2_10kpc.pdf}
\caption{\label{fig:chi2}$\Delta\chi^2(m_\nu)$ profiles as a function of neutrino mass $m_\nu$, for DUNE generated samples assuming massless neutrinos and a SN distance of 10~kpc. We show the no oscillations' case together with the results for NO and IO. The mean sensitivities and their $\pm 1\sigma$ uncertainties are shown with solid lines and filled bands, respectively. The horizontal dotted line depicts the $95\%$~CL.}
\end{center}
\end{figure}

\begin{table}
\centering
\caption{Mean and standard deviation of the $95\%$~CL sensitivity on neutrino mass from a sample of DUNE SN datasets at $D=10$~kpc, for different neutrino oscillation scenarios. For the IO case, we give sensitivities for different zenith angles $\theta$.}
\label{tab:m_nu_mass_bounds}
\begin{tabular}{@{\extracolsep{0.5cm}}ccc@{\extracolsep{0cm}}}
\toprule
Neutrino mass ordering  & $\cos\theta$ & $m_\nu$(eV) \\
\midrule
No oscillations                    & $0$ & $0.51^{+0.20}_{-0.20}$ \\
\midrule
Normal Ordering                    & $0$ & $2.01^{+0.69}_{-0.55}$ \\
\midrule
\multirow{5}*{Inverted Ordering}   & $0$ & $0.91^{+0.31}_{-0.33}$ \\
                                   & $-0.3$ & $0.85^{+0.33}_{-0.30}$ \\
                                   & $-0.5$ & $0.88^{+0.29}_{-0.33}$ \\
                                   & $-0.7$ & $0.91^{+0.30}_{-0.32}$ \\
                                   & $-1$ & $0.87^{+0.32}_{-0.28}$ \\
\bottomrule
\end{tabular}
\end{table}

Our statistical procedure, and its results for a SN distance of $D=10$~kpc, can be seen in Fig.~\ref{fig:chi2}. The $\Delta\chi^2$ profiles as a function of neutrino mass are shown for no oscillations, and oscillations in SN environment assuming either NO or IO. Earth matter effects have been neglected in all cases. After including Earth matter effects as previously described, only the IO expectation is affected. Table~\ref{tab:m_nu_mass_bounds} reports our results on the mean and standard deviation of the $m_{\nu}$ sensitivity values for different $\cos\theta$ values, that is, for different angular locations of the SN with respect to the Earth.

As can be seen from Fig.~\ref{fig:chi2} and Tab.~\ref{tab:m_nu_mass_bounds}, 95\% CL sensitivities in the 0.5--2.0~eV range are expected. The best, sub-eV reach, results are expected for the no oscillations and IO scenarios. Despite the largest overall event statistics, $R=1372$, the NO reach is the worst among the three cases, of order 2.0~eV. This result clearly indicates the importance of the shape information, in particular of the sharp neutronization burst time structure visible in Fig.~\ref{fig:events} only for the no oscillations and IO cases. Table~\ref{tab:m_nu_mass_bounds} also shows that oscillations in the Earth's interior barely affect the neutrino mass sensitivity.

\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{mass_sensitivity_comparison_errbar.pdf}
\caption{\label{fig:distance}Dependence of the $95\%$~CL neutrino mass sensitivity with the distance $D$ from Earth at which the SN explodes. The mean and standard deviation of the expected sensitivity values are shown with solid lines and filled bands, respectively.}
\end{center}
\end{figure}

Figure ~\ref{fig:distance} shows how the $95\%~$CL sensitivity on the neutrino mass varies with the SN distance $D$. Both the mean and the standard deviation of the expected sensitivity values are shown. In all scenarios, the sensitivities to $m_\nu$ worsen by about a factor of 2 as the SN distance increases from 5 to 25~kpc. As is well known, as the distance $D$ increases, the reduced event rate ($R\propto 1/D^2$) tends to be compensated by the increased time delays for a given $m_\nu$ ($\Delta t_i(m_\nu)\propto D$). Our analysis shows that this compensation is only partial, and better sensitivities are obtained for nearby SNe.

\section{Conclusions} \label{sec:conclusions}

The capability to detect the electron neutrino flux component from a core-collapse SN in our galactic neighborhood makes large liquid argon detectors powerful observatories to obtain constraints on the absolute value of neutrino mass via time of flight measurements.
Exploiting the signal coming from charged-current interactions of $\nu_e$ with argon nuclei, a 0.9~eV sensitivity on the absolute value of neutrino mass has been obtained in DUNE for the inverted ordering (IO) of neutrino masses, a SN distance of 10~kpc and at 95\% CL. The sensitivity is expected to be significantly worse in the normal ordering (NO) scenario, 2.0~eV for the same SN distance and confidence level. The sensitivity difference between the two orderings demonstrates the benefit of detecting the $\nu_e$ neutronization burst, whose sharp time structure would be almost entirely suppressed in NO while it should be clearly observable in DUNE if the mass ordering is IO. The mild effects of oscillations induced by the Earth matter, affecting only the inverted mass ordering, and of the SN distance from Earth, have been studied. The DUNE sensitivity reach appears to be competitive with both laboratory-based direct neutrino mass experiments (such as KATRIN) and next-generation SN observatories primarily sensitive to the $\bar{\nu}_e$ flux component (such as Hyper-Kamiokande and JUNO).

\begin{acknowledgments}
This work has been supported by the Spanish grants FPA2017-85985-P, PROMETEO/2019/083 and PROMETEO/2021/087, and by the European ITN project HIDDeN (H2020-MSCA-ITN-2019/860881-HIDDeN). The  work  of  FC  is  supported  by  GVA  Grant No. CDEIGENT/2020/003.
\end{acknowledgments}


\section{Introduction}
Spin-orbit interaction (SOI) plays an important role in the widely
studied spin-related effects and spintronic devices. In the latter
it can be either directly utilized to create spatial separation of
the spin-polarized charge carries or indirectly influence the device
performance through spin-decoherence time. In 2D structures two
kinds of SOI are known to be of the most importance, namely Rashba
and Dresselhaus mechanisms. The first one characterized by parameter
$\alpha$ is due to the structure inversion asymmetry (SIA) while the
second one characterized by $\beta$ is due to the bulk inversion
asymmetry (BIA). Most brightly both of the contributions reveal
themselves when the values of $\alpha$ and $\beta$ are comparable.
In this case a number of interesting effects occur: the electron
energy spectrum becomes strongly anisotropic \cite{AnisotrSpectrum},
the electron spin relaxation rate becomes dependent on the spin
orientation in the plane of the quantum well
\cite{AverkievObserved}, a magnetic break-down should be observed in
the Shubnilov de Haas effect\cite{magn}. The energy spectra
splitting due to SOI can be observed in rather well-developed
experiments as that based on Shubnikov--de Haas effect. However,
these experiments can hardly tell about the partial contributions of
the two mechanisms leaving the determination of the relation between
$\alpha$ and $\beta$ to be a more challenging task. At the same
time, in some important cases spin relaxation time $\tau_s$ and spin
polarization strongly depend on the $\frac{\alpha}{\beta}$ ratio. In
this paper we consider the tunneling between 2D electron layers,
which turns out to be sensitive to the relation between Rashba and
Dresselhaus contributions. The specific feature of the tunneling in
the system under consideration is that the energy and in-plane
momentum conservation put tight restrictions on the tunneling.
Without SOI the tunneling conductance exhibits delta function-like
maximum at zero bias broadened by elastic scattering in the layers
\cite{MacDonald}, and fluctuations of the layers width
\cite{VaksoFluctuations}. Such a behavior was indeed observed in a
number of experiments \cite{Eisenstein,Turner,Dubrovski}. Spin-orbit
interaction splits the electron spectra into two subbands in each
layer. At that energy and momentum conservation can be fulfilled for
the tunneling between opposite subbands of the layers at a finite
voltage corresponding to the subbands splitting. However, if the
parameters of SOI are equal for left and right layers, the tunneling
remains prohibited due to orthogonality of the appropriate spinor
eigenstates. In \cite{Raichev} it was pointed out that this
restriction can also be eliminated if Rashba parameters are
different for the two layers. A structure design was proposed
\cite{Raikh} where exactly opposite values of the Rashba parameters
result from the built-in electric field in the left layer being
opposite to that in the right layer. Because the SOI of Rashba type
is proportional to the electric field, this would result in
$\alpha^R=-\alpha^L$, where $\alpha^L$ and $\alpha^R$ are the Rashba
parameters for the left and right layers respectively. In this case
the
 peak of the conductance should occur at the voltage $U_0$ corresponding
to the energy of SOI: $eU_0=\pm2\alpha k_F$, where $k_F$ is Fermi
wavevector. In this paper we consider arbitrary Rashba and
Dresselhaus contributions and show how qualitatively different
situations can be realized depending on their partial impact. In
some cases the structure of the electrons eigenstates suppresses
tunneling at ever voltage. At that the scattering is important as it
restores the features of voltage-current characteristic containing
information about SOI parameters. Finally the parameters $\alpha$
and $\beta$ can be obtained in the tunneling experiment which unlike
other spin-related experiments requires neither magnetic field nor
polarized light.
\section{Calculations}
We consider two 2D electron layers separated by potential barrier at
zero temperature (see Fig.\ref{fig:layers}). We shall consider only
one level of size quantization and not too narrow barrier so that
the electrons wavefunctions in the left and right layers overlap
weakly.
The system can be described by the phenomenological tunneling Hamiltonian \cite{MacDonald,MacDonald2,VaksoFluctuations}
\begin{figure}[h]
\leavevmode
 \centering\epsfxsize=180pt \epsfbox[30 530 500 760]{fig1.eps}
\caption{\label{fig:layers} Energy diagram of two 2D electron
layers.}
\end{figure}
\begin{equation}
\label{HT0} H=H_{0}^L+H_{0}^R+H_T,
\end{equation}
where $H_{0}^L,H_{0}^R$ are the partial Hamiltonians for the left
and right layers respectively, $H_T$ is the tunneling term. With
account for the elastic scattering and SOI in the layers the partial
Hamiltonians and the tunneling term have the the following form in
representation of secondary quantization:
\begin{equation}
\label{eqH}
\begin{array}{l}
H_{0}^l = \sum\limits_{k,\sigma} {\varepsilon^l_{k} c^{l+}_{k\sigma}
 c^l_{k\sigma } } +  \sum\limits_{k,k',\sigma}  {V^l_{kk'} c^{l+}_{k\sigma}c^l_{k'\sigma }} + H^l_{SO}  \\
H_T  = \sum\limits_{k,k',\sigma,\sigma'} {T_{kk'\sigma\sigma'}\left( {c^{L+}_{k\sigma} c^{R}_{k'\sigma'}  + c^{R+}_{k'\sigma'}  c^L_{k\sigma} } \right)}, \\
 \end{array}
\end{equation}
Here index $l$ is used for the layer designation and can take the
values $l=R$ for the right layer, $l=L$ for the left layer. By $k$
here and further throughout the paper we denote the wavevector
aligned parallel to the layers planes, $\sigma$ denotes spin
polarization and can take the values $\sigma=\pm 1/2$.
$\varepsilon_k^l$ is the energy of an electron in the layer $l$
having in-plane wavevector $k$. It can be expressed as:
\begin{equation}
\label{spectrum}
 \varepsilon _k^l  = \varepsilon+\varepsilon_0^l+\Delta^l,
 \end{equation}
where $\varepsilon=\frac{\hbar^2k^2}{2m}$, $m$ being electron's
effective mass, $\varepsilon_0^l$ and $\Delta^l$ are the size
quantization energy and the energy shift due to external voltage for
the layer $l$ . We shall also use the value $\Delta^{ll'}$ defined
as
$\Delta^{ll'}=(\Delta^l-\Delta^{l'})+(\varepsilon_0^l-\varepsilon_0^{l'})$.
Similar
 notation will be used for spin polarization denoted by indices $\sigma$, $\sigma'$.
 The second term in  the Hamiltonian (\ref{eqH}) $V_{kk'}^l$ is the matrix element of the scattering operator.
  We consider only elastic scattering. The tunneling
term $H_T$ in (\ref{eqH}) is described by the tunneling constant
$T_{kk'\sigma\sigma'}$, which
 has the meaning of size quantization levels splitting due to
the wavefunctions overlap. By lowercase $t$ we shall denote the
overlap integral itself. Our consideration is valid only for the
case of weak overlapping, i.e. $t\ll1$. Parametrically $T\sim
t\varepsilon_F$, where $\varepsilon_F$ is the electrons Fermi
energy.  The term $H^{l}_{SO}$ describes the spin-orbit part of the
Hamiltonian:
\begin{equation}
\label{eqSOH}
 \hat{H}^l_{SO}=\alpha^l \left( \bm{\sigma}  \times \bm{k}
\right)_z + \beta^{l} \left( {\sigma _x k_x  - \sigma _y k_y }
\right),
\end{equation}
where $\sigma_i$ are the Pauli matrices, $\alpha^l,\beta^l$ are
respectively the parameters of Rashba and Dresselhaus interactions
for the layer $l$. In the secondary quantization representation:
\begin{eqnarray}
 \hat {H}_{SO}^l =\alpha^l \sum\limits_k {\left( {k_y
-ik_x } \right)c_{k\sigma }^{l+} c_{k\sigma '}^l +} \left( {k_y
+ik_x }
\right)c_{k\sigma '}^{l+} c_{k,\sigma }^l \nonumber \\
 +\beta^l \sum\limits_k
{\left( {k_x -ik_y } \right)c_{k\sigma }^{l+} c_{k\sigma '}^l +}
\left( {k_x +ik_y } \right)c_{k\sigma '}^{l+} c_{k\sigma }^l
\label{eqSOHc}
\end{eqnarray}
The operator of the tunneling current can be expressed as
\cite{MacDonald}:
\begin{equation}
\label{current0}
 \hat{I} = \frac{{ie}}{\hbar
}\sum\limits_{k,k',\sigma,\sigma'} T_{kk'\sigma\sigma'}
\left(\hat\rho_{kk'\sigma\sigma'}^{RL}-\hat\rho_{k'k\sigma'\sigma}^{LR}
\right),
\end{equation}
where
$\hat\rho_{kk'\sigma\sigma'}^{ll'}=c_{k,\sigma}^{l+}c_{k',\sigma'}^{l'}$
 We shall assume the case of in-plane momentum and the spin
projection being conserved in the tunneling event so the tunneling
constant $T_{kk'\sigma\sigma'}$ has the form
$T_{kk'\sigma\sigma'}=T\delta_{kk'}\delta_{\sigma\sigma'}$, where
$\delta$ is the Cronecker symbol. The tunneling current is then
given by
 \begin{equation}
 \label{current}
 I = \frac{ie}{\hbar}
T \int dk\: \mathrm{Tr} \left( \left<\hat\rho^{RL}_{k\sigma}\right>
-\left<\hat\rho^{LR}_{k\sigma}\right>\right),
\end{equation}
where $<>$ denotes the expectation value in quantum-mechanical
sense. For further calculations it is convenient to introduce vector
operator
$\bm{\hat{S}}^{ll'}_{kk'}=\left\{\hat{S_0},\bm{\hat{s}}\right\}=\left\{\mathrm{Tr}\left(\hat\rho^{ll'}_{kk'\sigma\sigma'}\right),\mathrm{Tr}\left({\bm
\sigma}\hat\rho^{ll'}_{kk'\sigma\sigma'}\right) \right\}$. This
vector fully determines the current because the latter can be
expressed through the difference
$\hat{S}^{RL}_{0k}-\hat{S}^{LR}_{0k}$. The time evolution of
$\bm{\hat{S}}^{ll'}_{kk'}$ is governed by:
\begin{equation}
\label{drodt}
\frac{d\bm{\hat{S}}_{kk'}^{ll'}}{dt}=\frac{i}{\hbar}[H,\bm{\hat{S}}_{kk'}^{ll'}]
\end{equation}
In the standard way of reasoning \cite{Luttinger} we assume
adiabatic onset of the interaction with characteristic time
$w^{-1}$. We will set $w=0$ in the final expression. With this
(\ref{drodt}) turns into:
\begin{equation}
\label{drodt0}
(\bm{\hat{S}}_{kk'}^{ll'}-\bm{\hat{S}}_{kk'}^{(0)ll'})w=\frac{i}{\hbar}[H,\bm{\hat{S}}_{kk'}^{ll'}]
\end{equation}
Here $\bm{\hat{S}}_{kk'}^{(0)ll'}$ represents the stationary
solution of (\ref{drodt}) without interaction. By interaction here
we mean the tunneling and the elastic scattering by impurities but
not the external voltage. The role of the latter is merely shifting
the layers by $eU$ on the energy scale. From such defined
interaction it immediately follows that the only non-zero elements
of $\bm{\hat{S}}_{kk'}^{(0)ll'}$ are that with $l=l'$ and $k=k'$. In
further abbreviations we will avoid duplication of the indices i.e.
write single $l$ instead of $ll$ and $k$ instead of $kk$:
\begin{equation}
\label{Sdiag}
\bm{\hat{S}}_{kk'}^{(0)ll'}=\bm{\hat{S}}_{k}^{(0)l}\delta_{kk'}\delta_{ll'}
\end{equation}
 With use of fermion commutation rules
 \begin{eqnarray*}
 \left\{ {c_i c_k } \right\} = \left\{ {c_i^ +  c_k^ +  } \right\} = 0 \\
 \left\{ {c_i c_k^ +  } \right\} = \delta _{ik}
 \end{eqnarray*}
the calculations performed in a way similar to \cite{Luttinger}
bring us to the following system of equations
 with respect to
$\bm{\hat{S}}_{k}^{ll'}$:
\begin{eqnarray}
 0= \left( {\Delta^{ll'}+i\hbar w } \right){\bf{\hat
S}}_k^{ll'} + T\left( {{\bf{\hat S}}_k^{l'}  - {\bf{\hat S}}_k^l }
\right)+{\bf{M(}}k{\bf{)\hat S}}_k^{ll'} \nonumber \\
- \sum\limits_{k'} {\left( {\frac{{A_{kk'} {\bf{\hat S}}_k^{ll'}  -
B_{kk'} {\bf{\hat S}}_{k'}^{ll'} }}{{ {\varepsilon'  - \varepsilon
-\Delta^{ll'} }  + i\hbar w}} + \frac{{B_{kk'} {\bf{\hat S}}_k^{ll'}
- A_{kk'} {\bf{\hat S}}_{k'}^{ll'} }}{{ {\varepsilon   -
\varepsilon' -\Delta^{ll'} }  + i\hbar w}}} \right)}
 \label{system1}
 \end{eqnarray}
\begin{eqnarray}
i\hbar w\left( {{\bf{\hat S}}_k^{\left( 0 \right)l} - {\bf{\hat
S}}_k^l } \right) = T\left( {{\bf{\hat S}}_k^{l'l}  - {\bf{\hat
S}}_k^{ll'} } \right) + {\bf{M}}(k){\bf{\hat S}}_k^l \nonumber \\ +
\sum\limits_{k'} { {\frac{{2i\hbar wA_{kk'} \left( {{\bf{\hat
S}}_k^l  - {\bf{\hat S}}_{k'}^{l'} } \right)}}{{\left( {\varepsilon'
- \varepsilon } \right)^2  + \left( {\hbar w} \right)^2 }}} },
\label{system2}
\end{eqnarray}
where $\bm{M}$ is a known matrix, depending on $k$ and parameters of
spin-orbit interaction in the layers. Here we also introduced the
quadratic forms of the impurities potential matrix elements:
\begin{eqnarray}
A_{kk'}  \equiv \left| {V_{k'k}^{l} } \right|^2 \nonumber \\
B_{kk'} \equiv V_{k'k}^{l} V_{kk'}^{l'}
\label{correlators}
\end{eqnarray}
As (\ref{system1}) and (\ref{system2}) comprise a system of linear
integral equations these quantities enter the expression
(\ref{current}) for the current linearly and can be themselves
averaged over spatial distribution of the impurities. In order to
perform this averaging we assume the short range potential of
impurities:
\begin{equation}
\label{ImpuritiesPotential} V\left( r \right) = \sum\limits_a
{V_0^{} \delta \left( {r - r_a } \right)}
\end{equation}
The averaging immediately shows that the correlators
$\left<A_{kk'}\right>\equiv A$ and $\left<B_{kk'}\right>\equiv B$
have different parametrical dependence on the tunneling transparency
$t$, namely
\begin{equation}
\label{T2}
 \frac{B}{A}\sim t^{2}\sim T^2
 \end{equation}
We emphasize that this result holds for non-correlated distribution
of the impurities as well as for their strongly correlated
arrangement such as a thin layer of impurities placed in the middle
of the barrier. The corresponding expressions for these two cases
are given below. Index 'rand' stands for uniform impurities
distribution and 'cor' for their correlated arrangement in the
middle of the barrier $(z=0)$:
\begin{eqnarray}
 {B^{rand} }  = \frac{{V_0^2 n}}{W}\int {dz}
f_l ^2 (z)f_{l'} ^2 (z)\sim\frac{{V_0^2
n}}{W}\frac{{t^2 }}{d} \nonumber \\
  {A^{rand} }
  = \frac{{V_0^2 n}}{W}\int {dz} f_l^4\left(z\right)
\sim\frac{{V_0^2 n}}{W}\frac{1}{d}
\nonumber \\
{B^{cor} }   = \frac{{V_0^2 n_s }}{W}f_l ^2 (0)f_{l'} ^2
(0)\sim\frac{{V_0^2 n_s}}{W}\frac{{t^2 }}{d}
\nonumber \\
 {A^{cor} }   = \frac{{V_0^2 n_s
}}{W}f_l ^4 \left( 0 \right)\sim\frac{{V_0^2 n_s}}{W}\frac{1}{d},
\label{correlators1}
\end{eqnarray}
where $n$ and $n_s$ are bulk and surface concentrations of the
impurities, $W$ is the lateral area of the layers, $d$ is the width
of the barrier and $f(z)$ is the eigenfunction corresponding to the
size quantization level, $z$ is coordinate in the direction normal
to the layers planes, $z=0$ corresponding to the middle of the
barrier\cite{Raikh}.
 Unlike \cite{Raikh} and according to (\ref{T2}) we
conclude that the correlator $\left<B_{kk'}\right>$ has to be
neglected as soon as we shall be interested in calculating the
current within the order of $T^2$. In the hereused method of
calculation this result appears quite naturally, however, it can be
similarly traced in the technique used in \cite{Raikh} (see
Appendix). For the same reason the tunneling term should be dropped
from (\ref{system2}) as it would give second order in $T$ if
(\ref{system2}) substituted into (\ref{system1}). According to
(\ref{correlators}) $A$ can be expressed in terms of electrons
scattering time:
\begin{equation}
\label{tau} \frac{1}{\tau } = \frac{{2\pi }}{\hbar }\nu\left\langle
{\left| {V_{kk'} } \right|^2 } \right\rangle   = \frac{{2\pi
}}{\hbar }\nu A ,
\end{equation}
where $\nu$ is the 2D density of states $\nu=\frac{m}{2\pi\hbar^2}$.
By means of Fourier transformation on energy variable the system
(\ref{system1}),(\ref{system2}) can be reduced to the system of
linear algebraic equations. Finally ${{\bf{\hat S}}_k^{ll'} }$ can
be expressed as a function of ${{\bf{\hat S}}_k^{\left( 0 \right)l}
}$. Consequently the current (\ref{current}) becomes a function of
$\left<\hat{\rho}_{k\sigma}^{(0)R}\right>$,
$\left<\hat{\rho}_{k\sigma}^{(0)L}\right>$. For the considered case
of zero temperature:
    \[
\left<\rho _{k\sigma}^{(0)l}\right>  = \frac{1}{2W}  \theta \left(
{\varepsilon _F^l  + \Delta ^l - \varepsilon - \varepsilon _\sigma }
\right),
\]
where
\[
\varepsilon _\sigma   =  \pm \left| {\alpha ^l \left( {k_x  - ik_y }
\right) - \beta ^l \left( {ik_x  - k_y } \right)} \right|,
\]. Without loss of generality we shall consider the
case of identical layers and external voltage applied as shown in
Fig.\ref{fig:layers}:
\begin{eqnarray*}
\varepsilon_0^R=\varepsilon_0^L\\
\Delta^L=-\frac{eU}{2}, \Delta^R=+\frac{eU}{2}\\
 \Delta^{RL}=-\Delta^{LR}=eU
\end{eqnarray*}
 The calculations can be simplified with account for
two small parameters:
\begin{eqnarray}
 \xi=\frac{\hbar}{\varepsilon_F\tau}\ll1  \nonumber \\
\eta=\frac{eU}{\varepsilon_F}\ll1 \label{deltaef}
\end{eqnarray}
With (\ref{eftau}) calculation yields the following expression for
the current:
\begin{equation}
\label{currentfinal0}  I = \frac{{ie}}{{2\pi \hbar }}T^2 \nu
\int\limits_0^\infty {\int\limits_0^{2\pi } {\left( {\zeta ^L  +
\zeta ^R } \right)\mathrm{Tr}\left( {\rho _\sigma ^{\left( 0
\right)R}  - \rho _\sigma ^{\left( 0 \right)L} } \right)d\varepsilon
d\varphi } },
\end{equation}
where
\[
\label{constants} \zeta ^l = \frac{{C^l \left[ {\left( {C^l }
\right)^2 - 2bk^2 \sin2\varphi  - gk^2 } \right]}}{{\mathop {\left(
{f + 2d\sin2\varphi } \right)}\nolimits^2 k^4  - 2\left( {C^l }
\right)^2 \left( {c + 2a\sin2\varphi } \right)k^2  + \left( {C^l }
\right)^4 }}, \]
 \[ C^l\left(U\right)  = \Delta ^l  + i\frac{\hbar
}{\tau },
\]
\begin{eqnarray}
 a  = \alpha ^L \beta ^L  + \alpha ^R \beta ^R \nonumber  \\
 b  = \left( {\beta ^L  + \beta ^R } \right)\left( {\alpha ^L  + \alpha ^R } \right)\nonumber \\
 c  = \left( {\beta ^L } \right)^2  + \left( {\beta ^R } \right)^2  + \left( {\alpha ^L } \right)^2  + \left( {\alpha ^R } \right)^2 \nonumber \\
 d  = \alpha ^L \beta ^L  - \alpha ^R \beta ^R \nonumber \\
 f  = \left( {\beta ^L } \right)^2  - \left( {\beta ^R } \right)^2  + \left( {\alpha ^L } \right)^2  - \left( {\alpha ^R } \right)^2 \nonumber \\
 g  = \mathop {\left( {\beta ^L  + \beta ^R } \right)}\nolimits^2  + \mathop {\left( {\alpha ^L  + \alpha ^R } \right)}\nolimits^2  \nonumber \\
\label{constants}
 \end{eqnarray}
Parameters $a$-$g$ are various combinations of the Rashba and
Dresselhaus parameters of SOI in the layers. Both types of SOI are
known to be small in real structures so that:
\begin{equation}
\alpha k_F\ll\varepsilon_F, \; \beta k_F\ll\varepsilon_F
\end{equation}
This additional assumption together with (\ref{deltaef}) reduces
(\ref{currentfinal0}) to
\begin{equation}
\label{currentfinal}  I =  \frac{{ie^2 }}{{2\pi \hbar }}T^2 \nu
WU\int\limits_0^{2\pi } {\left[ {\zeta ^L \left( {\varepsilon_F }
\right) + \zeta ^R \left( {\varepsilon_F } \right)} \right]d\varphi
}
\end{equation}
The integral over $\varphi$ in (\ref{currentfinal}) can be
calculated analytically by means of complex variable integration.
However, the final result for arbitrary $\alpha^l,\beta^l$  is not
given here for it is rather cumbersome. In the next section some
particular cases are discussed.
\section{Results and Discussion}
The obtained general expression (\ref{currentfinal}) can be
simplified for a few particular important relations between Rashba
and Dresselhaus contributions. These calculations reveal
qualitatively different dependencies of the d.c. tunneling current
on the applied voltage.
\begin{figure}[h]
 \leavevmode
 \centering\epsfxsize=210pt \epsfbox[130 350 700 800]{fig2.eps}
\caption{\label{fig:tunnelingmain}Tunneling conductance, a:
$\varepsilon_F=10$ meV, $\alpha=\beta=0$, $\tau=2*10^{-11}$ s; b:
same as a, but $\alpha k_F=0.6$ meV; c: same as b, but
$\beta=\alpha$; d: same as c, but $\tau=2*10^{-12}$ s.}
\end{figure}
The results of the calculations shown below were obtained using the
following parameters: Fermi energy $\varepsilon_F=10$ meV,
spin-orbit splitting was taken to resemble $GaAs$ structures:
$\alpha k_F=0.6$ meV.
\subsection{No Spin-Orbit Interaction}
 In the absence of SOI ($\alpha^R=\alpha^L=0$, $\beta^R=\beta^L=0$) the
energy spectrum for each of the layers forms a paraboloid:
\begin{equation}
E^l(k)=\varepsilon_0+\frac{\hbar^2k^2}{2m}\pm \frac{eU}{2}.
\end{equation}
  According to our assumptions (\ref{current0}),(\ref{current}), the tunneling takes place at:
\begin{eqnarray}
   E^R=E^L\nonumber \\
   k^R=k^L
   \label{conservation}
\end{eqnarray}
Both conditions are satisfied
  only at $U=0$ so that a nonzero external voltage does not produce any current
  despite it produces empty states in one layers aligned to the filled states in the other layer
  (Fig.\ref{fig:layers}). The momentum conservation restriction in (\ref{conservation}) is weakened if the electrons scatter at the impurities.
   Accordingly, one should expect a nonzero tunneling current
  within a finite voltage range in vicinity of zero.
 For the considered case the general formula (\ref{currentfinal}) is simplified radically as all the parameters (\ref{constants})
 have zero values. Finally we get the well-known
 result\cite{MacDonald}:
\begin{equation}
\label{currentMacDonald}
 I = 2e^2 T^2 \nu
WU\frac{{\frac{1}{\tau }}}{{\left( {eU} \right)^2 + \left(
{\frac{\hbar }{\tau }} \right)^2 }}. \end{equation}
 The conductance defined as $G(U)=I/U$ has  Lorentz-shaped peak at $U=0$
 turning into delta function at $\tau\rightarrow\infty$.
 This case is shown in (Fig.\ref{fig:tunnelingmain},a).
  All the curves in Fig.\ref{fig:tunnelingmain} show the results of the
calculations for very weak scattering. The corresponding scattering
time is taken $\tau=2*10^{-11}s$.
\subsection{Spin-Orbit Interaction of Rashba type}
 The spin-orbit interaction gives
qualitatively new option for the d.c. conductance to be finite at
non-zero voltage. SOI splits the spectra into two subbands. Now an
electron from the first subband of the left layer can tunnel to a
state in a second subband of the right layer. Let us consider a
particular case when only Rashba type of SOI interaction exists in
the system, its magnitude being the same in both layers, i.e.
$|\alpha^R|=|\alpha^L|\equiv \alpha$, $\beta^R=\beta^L=0$. In this
case the spectra splits into two paraboloid-like subbands "inserted"
into each other. Fig.\ref{fig:spectraRashba} shows their
cross-sections for both layers,
 arrows show spin orientation. By applying a certain external
voltage $U_0=\frac{2\alpha k_F}{e}$,
$k_F=\frac{\sqrt{2m\varepsilon_F}}{\hbar}$ the layers can be shifted
on the energy scale in such a way that the cross-section of the
"outer" subband of the right layer coincides with the "inner"
subband of the left layer (see solid circles in
Fig.\ref{fig:spectraRashba}). At that both conditions
(\ref{conservation}) are satisfied. However, if the spin is taken
into account, the interlayer transition can still remain forbidden.
It happens if the appropriate spinor eigenstates involved in the
transition are orthogonal. This very case occurs if
$\alpha^R=\alpha^L$, consequently the conductance behavior remains
the same as that without SOI. Contrary, if the Rashba terms are of
the opposite signs, i.e. $\alpha^R=-\alpha^L$ the spin orientations
in the "outer" subband of the right layer and the "inner" subband of
the left layer are the same and the tunneling is allowed at a finite
voltage but forbidden at $U=0$ . This situation, pointed out in
\cite{Raichev,Raikh} should reveal itself in sharp maxima of the
conductance at $U=\pm U_0$ as shown in
Fig.\ref{fig:tunnelingmain},b. From this dependence the value of
$\alpha$ can be immediately extracted from the position of the peak.
Evaluating (\ref{constants}) for this case and further the
expression (\ref{currentfinal}) we obtain the following result for
the current:
\begin{equation}
\label{currentRaikh}  I = \frac{{2e^2T^2 W\nu U\frac{\hbar }{\tau
}\left[ {\delta^2 + e^2 U^2  + \left( {\frac{\hbar }{\tau }}
\right)^2 } \right]}}{{\left[ {\left( {eU - \delta } \right)^2 +
\left( {\frac{\hbar }{\tau }} \right)^2 } \right]\left[ {\left( {eU
+ \delta } \right)^2  + \left( {\frac{\hbar }{\tau }} \right)^2 }
\right]}},
\end{equation}
where $\delta=2\alpha k_F$. The result is in agreement with that
derived in\cite{Raikh}, taken for uncorrelated spatial arrangement
of the impurities. As we have already noted we do not take into
account interlayer correlator $\left<B_{kk'}\right>$
($\ref{correlators}$) because parametrically it has higher order of
tunneling overlap integral $t$ than the intralayer correlator
$\left<A_{kk'}\right>$. Therefore the result (\ref{currentRaikh}) is
valid for arbitrary degree of correlation in spatial distribution of
the impurities in the system.
\begin{figure}[h]
 \leavevmode
 \centering\epsfxsize=220pt \epsfbox[130 500 700 800]{fig3.eps}
 \caption{\label{fig:spectraRashba}Cross-section of electron energy spectra in the left(a) and right (b) layer for
 the case
$\alpha^{L}=-\alpha^{R}, \beta^{L}=\beta^{R}=0$.}
\end{figure}
It is worth noting that the opposite case when only Dresselhaus type
of SOI exists in the system leads to the same results. However, it
is rather non-practical to study the case of the different
Dresselhaus parameters in the layers because this type of SOI
originates from the crystallographic asymmetry and therefore cannot
be varied if the structure composition is fixed. For this case to be
realized one needs no make the two layers of different materials.
\subsection{Both Rashba and Dresselhaus contributions}
The presence of Dresselhaus term in addition to the Rashba
interaction can further modify the tunneling conductance in a
non-trivial way. A special case occurs if the magnitude of the
Dresselhaus term is comparable to that of the Rashba term. We shall
always assume the Dresselhaus contribution being the same in both
layers: $\beta^{L}=\beta^{R}\equiv\beta$. Let us add the Dresselhaus
contribution to the previously discussed case so that
$\alpha^{L}=-\alpha^{R}\equiv\alpha,\;\alpha=\beta$. The
corresponding energy spectra and spin orientations are shown in
Fig.\ref{fig:spectraRD}. Note that while the spin orientations in
the initial and final states are orthogonal for any transition
between the layers, the spinor eigenstates are not, so that the
transitions are allowed whenever the momentum and energy
conservation requirement (\ref{conservation}) is fulfilled. It can
be also clearly seen from Fig.\ref{fig:spectraRD} that the condition
(\ref{conservation}), meaning overlap of the cross-sections a. and
b. occurs only at few points. This is unlike the previously
discussed case where the overlapping occurred within the whole
circular cross-section shown by solid lines in
Fig.\ref{fig:spectraRashba}. One should naturally expect the
conductance for the case presently discussed to be substantially
lower. Using (\ref{currentfinal}) we arrive at a rather cumbersome
expression for the current:
\begin{figure}[h]
 \leavevmode
 \centering\epsfxsize=220pt \epsfbox[130 500 700 810]{fig4.eps}
\caption{\label{fig:spectraRD}Cross-section of electron energy
spectra in the left(a) and right (b) layer for
 the case
$\alpha^{R}=-\alpha^L=\beta$.}
\end{figure}
\begin{eqnarray}
   I = eT^2 W\nu U\left[ {\frac{{G_ - \left(
{G_ - ^2  - \delta ^2 } \right)}}{{\sqrt {F_ -  \left( {\delta ^4  +
F_ -  } \right)} }} - \frac{{G_ +  \left( {G_ + ^2  - \delta ^2 }
\right)}}{{\sqrt {F_ +  \left( {\delta ^4  + F_ +  } \right)} }}}
\right], \label{CurrentSpecial}
\end{eqnarray}
where \begin{eqnarray*}
G_ \pm   = eU \pm i\frac{\hbar }{\tau } \\
F_ \pm   = G_ \pm ^2 \left( {G_ \pm ^2  - 2\delta^2 } \right).
\end{eqnarray*}
Alternatively, for the case of no interaction with impurities a
precise formula for the transition rate between the layers can be
obtained by means of Fermi's golden rule. We obtained the following
expression for the current:
\begin{equation}
\label{CurrentPrecise} I = \frac{{2\pi eT^2 W}}{{\hbar \alpha ^2
}}\left( {\sqrt {K + \frac{{8m\alpha ^2 eU}}{{\hbar ^2 }}}  - \sqrt
{K - \frac{{8m\alpha ^2 eU}}{{\hbar ^2 }}} } \right),
\end{equation} where
\[
K = 2\delta^2  - e^2 U^2  + \frac{{16m^2 \alpha ^4 }}{{\hbar ^4 }}
\]
Comparing the results obtained from (\ref{CurrentSpecial}) and
(\ref{CurrentPrecise}) is an additional test for the correctness of
(\ref{CurrentSpecial}). Both dependencies are presented in
Fig.\ref{fig:goldenRule} and show a good match. The same dependence
of conductance on voltage is shown in Fig.\ref{fig:tunnelingmain},c.
As can be clearly seen in the figure the conductance is indeed
substantially suppressed in the whole voltage range. This is
qualitatively different from all previously mentioned cases.
Furthermore, the role of the scattering at impurities appears to be
different as well. For the considered above cases characterized by
resonance behavior of the conductance, the scattering broadens the
resonances into Lorentz-shape peaks with the characteristic width
$\delta=\hbar/(e\tau)$. Contrary, for the last case the weakening of
momentum conservation, caused by the scattering, increases the
conductivity and restores the manifestation of SOI in its dependence
on voltage. Fig.\ref{fig:tunnelingmain},d shows this dependence for
a shorter scattering time $\tau=2*10^{-12}$. The reason for that is
the weakening of the momentum conservation requirement due to the
elastic scattering. One should now consider the overlap of the
spectra cross-sections the circles in Fig.\ref{fig:spectraRD} having
a certain thickness proportional to $\tau^{-1}$. This increases the
number of points at which the overlap occurs and, consequently, the
value of the tunneling current. As the calculations show, for
arbitrary $\alpha$ and $\beta$ the dependence of conductance on
voltage can exhibit various complicated shapes with a number of
maxima, being very sensitive to the relation between the two
contributions. The origin of such a sensitivity is the interference
of the angular dependencies of the spinor eigenstates in the layers.
A few examples of such interference are shown in
Fig.\ref{fig:variousRD}, a--c. All the dependencies shown were
calculated for the scattering time $\tau=2*10^{-12}$ s.
Fig.\ref{fig:variousRD},a summarizes the results for all previously
discussed cases of SOI parameters, i.e. no SOI (curve 1), the case
$\alpha_R=-\alpha_L, \beta=0$ (curve 2) and
$\alpha_R=-\alpha_L=\beta$ (curve 3). Following the magnitude of
$\tau$ all the reasonances are broadenered compared to that shown in
Fig.\ref{fig:tunnelingmain}. Fig.\ref{fig:variousRD},b (curve 2)
demonstrates the conductiance calculated for the case
$\alpha_L=-\frac{1}{2}\alpha_R=\beta$, Fig.\ref{fig:variousRD},c
(curve 2) -- for the case $\alpha_L=\frac{1}{2}\alpha_R=\beta$. The
curve 1 corresponding to the case of no SOI is also shown in all the
figures for reference. Despite of a significant scattering parameter
all the patterns shown in Fig.\ref{fig:variousRD} remain very
distinctive. That means that in principle the relation between the
Rashba and Dresselhaus contributions to SOI can be extracted merely
from the I-V curve measured in a proper tunneling experiment.
\begin{figure}[h]
 \leavevmode
 \centering\epsfxsize=190pt \epsfbox[130 350 700 800]{fig5.eps}
\caption{\label{fig:goldenRule}Tunneling conductance calculated for
the case $\alpha^R=-\alpha^L=\beta$ and very weak scattering
compared to the precise result obtained through Fermi's golden rule
calculation.}
\end{figure}
\begin{figure}[h]
  \hfill
  \begin{minipage}[t]{0.5\textwidth}
    \begin{center}
      \centering\epsfxsize=170pt \epsfbox[70 650 266 801]{fig6a.eps}
    \nonumber
    \end{center}
  \end{minipage}
  \hfill
  \begin{minipage}[t]{0.5\textwidth}
    \begin{center}
      \epsfxsize=170pt \epsfbox[70 650 266 801]{fig6b.eps}
      \nonumber
    \end{center}
  \end{minipage}
  \hfill
  \begin{minipage}[t]{0.5\textwidth}
    \begin{center}
      \epsfxsize=170pt \epsfbox[70 650 266 801]{fig6c.eps}
    \end{center}
  \end{minipage}
  \caption{\label{fig:variousRD}Tunneling conductance calculated for various parameters of
SOI}
\end{figure}
\section{Summary}
As we have shown, in the system of two 2D electron layers separated
by a potential barrier SOI can reveal itself in the tunneling
current. The difference in spin structure of eigenstates in the
layers results in a sort of interference in the tunneling
conductance. The dependence of tunneling conductance on voltage
appears to be very sensitive to the parameters of SOI. Thus, we
propose a way to extract the parameters of SOI and, in particular,
the relation between Rashba and Dresselhaus contributions in the
tunneling experiment. We emphasize that unlike many other
spin-related experiments the manifestation of SOI studied in this
paper should be observed without external magnetic field. Our
calculations show that the interference picture may be well resolved
for GaAs samples with the scattering times down to $\sim 10^{-12}$
s, in some special cases the scattering even restores the traces of
SOI otherwise not seen due to destructive interference.
\section*{ACKNOWLEDGEMENTS}
 This work has been supported in part by RFBR, President
of RF support (grant MK-8224.2006.2) and Scientific Programs of RAS.


\section{Introduction}
Nano-manufacturing by polymer self-assembly is attracting interests in recent
decades due to its wide applications~\cite{FINK:1998}. The numerical simulation
of this process can be used to research the mechanisms of phase separation of
polymer blends and predict the unobservable process states and unmeasurable
material properties. The mathematical principles and numerical simulation of
self-assembly via phase separation has been extensively
studied~\cite{SCOTT:1949,HSU:1973,CHEN:1994,HUANG:1995,ALTENA:1982,ZHOU:2006,TONG:2002,HE:1997,MUTHUKUMAR:1997,KARIM:1998}. But few specific software
toolkit have been developed to efficiently investigate this phenomenon. \par

A computer program is developed in MATLAB for the numerical simulation of the
polymer blends phase separation.
With this software, the mechanisms of the phase separation are investigated.
Also the mobility, gradient energy coefficient energy, and the surface energy
in the experiment are estimated with the numerical model. The software can
evaluate the physical parameters in the numerical model by implementing the
real experimental parameters and materials properties. The numerical simulation
results can be analyzed with the software and the results from the simulation
software can be validated with the experimental results. \par

\begin{figure}[!ht]
	\centering
	\includegraphics[width=\textwidth]{fg_gui_screen_shot.eps}
	\caption{Screenshot of the simulation program graphical user interface.
	\label{fg_gui_screenshot}}
\clearpage
\end{figure}

\section{Fundamentals}
The numerical model for phase separation of polymer blends are established and
validated with experimental results work~\cite{SHANG:2010}. The free energy profile during the phase separation in a inhomogeneous mixture
is described by Cahn-Hilliard
Equation~\cite{CAHN:1958, CAHN:1959, CAHN:1961, CAHN:1965}, as shown below,

\begin{equation}
	F(C_1,C_2,C_3)=\int_{V} \left\{ f(C_1,C_2,C_3)+\displaystyle\sum_{i=1,2,3} [\kappa_i (\nabla C_i)^2] \right\} dV \label{cahn_hilliard_intro}
\end{equation}

where $f$ is the local free energy density of homogeneous material, $\phi _i$
is the lattice volume fraction of component $i$, and $\kappa_i$ is the gradient
energy coefficient for the component $i$. The total free energy of the system
is composed by two items as shown in Equation~\ref{cahn_hilliard_intro}. The
first item is the local free energy and the second is the composition gradient
contribution to the free energy. \par

In our study, the local free energy is in the form of Flory-Huggins equation,
which is well know and studied for polymer blends~\cite{HUANG:1999}
The ternary Flory-Huggins Equation is shown as follows,

\begin{equation}
	\begin{split}
		f(C_1,C_2,C_3)
			 &=  \frac{RT}{v_{site}}\bigg( \frac{C_1}{m_1}\ln{C_1}+\frac{C_2}{m_2}\ln{C_2} + C_3\ln{C_3} \\
			& \chi_{12}C_1C_2+\chi_{13}C_1C_3+\chi_{23}C_2C_3\bigg)
\label{eq_flory_huggins_intro}
	\end{split}
\end{equation}

where $R$ is the ideal gas constant, $T$ is the absolute temperature,
$v_{site}$ is the lattice site volume in the Flory-Huggins model, $m_i$ is the
degree of polymerization of component $i$, and $C_i$ is the composition for the
component $i$. \par

There are some parameters in the numerical model which can not be measured
directly, such as the gradient energy coefficient and the mobility. These
parameters have to be estimated from the experimental
parameters.The gradient energy coefficient, $\kappa$, determines the influence
of the composition gradient to the total free energy of the domain.
The value of $\kappa$ is difficult to measure experimentally. Though efforts
have been made by Saxena and Caneba~\cite{SAXENA:2002} to estimate the
gradient energy coefficient in a ternary polymer system from experimental
methods, few experimental results are published for our conditions. Initially,
the value of $\kappa$ can be estimated by the interaction distance between
molecules~\cite{WISE_THESIS:2003},

\begin{equation}
	\kappa=\frac{RTa^2}{3v_{site}}\label{eq_gradient_energy_coefficient}
\end{equation}

where $a$ is the monomer size. A modified equation to calculate $\kappa$
considering the effects of the composition is reported by Gennes,
et al.~\cite{GENNES:1980}.

\begin{equation}
	\kappa_i=\frac{RTa^2}{36v_{site}C_i}
\end{equation}

where the subscript, $i$, represents component $i$.  \par

The mobility is estimated from the diffusivity of the components. The mobility
of the polymer blends with long chains can be estimated by the equation as
follows~\cite{GENNES:1980},

\begin{equation}
M_i=\frac{C_i}{m_i}\frac{D_mN_ev_{site}}{RT}
\end{equation}

where $m_i$ is the degree of polymerization as stated before, $D_m$ is the
diffusivity of the monomer, and $N_e$ is the effective number of monomers per
entanglement length. Because of the scarce experimental data for $N_e$, a more
generalized form is employed for our study,

\begin{equation}
M=\frac{Dv_{site}}{RT}\label{eq_mobility}
\end{equation}

The time evolution of the composition of component $i$ can be represented
as~\cite{HUANG:1995,BATTACHARYYA:2003,GENNES:1980,SHANG:2009},\par

\begin{equation}
	\begin{split}
		\frac{\partial C_i}{\partial t}
		&= M_{ii}\left[ \frac{\partial f}{\partial C_i}-\frac{\partial f}{\partial C_3}-2\kappa_{ii}\nabla^2C_i-2\kappa_{ij}\nabla^2C_j\right] \\
		& +M_{ij}\left[ \frac{\partial f}{\partial C_j}-\frac{\partial f}{\partial C_3}-2\kappa_{ji}\nabla^2C_i-2\kappa_{jj}\nabla^2C_j \right]
	\end{split}\label{eq6_paper2}
\end{equation}

where the subscripts $i$ and $j$ represent components 1 and 2, and\par

\begin{equation}
	\begin{aligned}
		M_{ii}=&(1-\overline{C}_i)^2M_i+\overline{C}_i^2\displaystyle\sum_{j\neq i}M_j\qquad i=1,2;j=1,2,3\\
		M_{ij}=&-\displaystyle\sum_{i\neq j}\left[(1-\overline{C}_i)\overline{C}_j\right]M_i+\overline{C}_i\overline{C}_jM_3\qquad i=1,2;j=1,2
	\end{aligned}
\end{equation}

where $\overline{C}_i$ is the average composition of component $i$. To simplify
the solution of Equation \ref{eq6_paper2}, $\kappa_{ii}=\kappa_i+\kappa_3$, and
$\kappa_{12}=\kappa_{21}=\kappa_3$, where $\kappa_i$ is the gradient energy
coefficient in Equation~\ref{eq_gradient_energy_coefficient}. \par

For detailed discussion and practical scientific cases with this software can
be found in our previous
works~\cite{SHANG:2008,SHANG:2009,SHANG:2009THESIS}.\par

\section{The MATLAB Program for Simulation of Polymer Phase Separation}

\subsection{Design Principles}
The program is developed in MATLAB m-code. A graphical user interface (GUI) is
implemented in the program created with MATLAB GUI editor. MATLAB is widely
used in scientific computation and has many toolkits and commonly used
mathematical functionalities. But implementing the software in MATLAB the
efficiency of development is greatly improved. Also, by developing the program
in MATLAB, the program is cross platform. \par

The software is designed for daily usage of simulation and experiment
scientists. The program is light weighted and programmed with high computation
efficiency so that it can produce significant science results in a common PC.
It also extensible to a parallel version or implement code to use the high
computation performance of GPU. The GUI is implemented so that the users can
conveniently input the experiment parameters. The results as well as the user
settings can be saved and revisited by the program. Also, for better assistance
to a real productive environment, the simulation model is carefully designed,
so that the users provide the real processing and material parameters and the
program will produce quantitative results comparable to experimental results.
Analytical tools are also provided with the program for post-processing of the
results. \par

\subsection{Numerical Methods}
To solve the partial differential equation, the discrete cosine transform
spectral method is employed. The discrete cosine transform (DCT) is applied
to the right hand side and left hand side of Equation~\ref{eq6_paper2}. The
partial differential equation in the ordinary space then transformed into an
ordinary differential equation in frequency space. When the ODE in frequency
space then is solved, the results are transformed back to the ordinary
space. \par

Comparing to conventional finite element method, the spectral method is more
efficient and accurate. This method enabled the program to solve the equation
in a reasonable computation time to investigate the changes of the phase
separation during an real time span long enough to observe the phase evolution.
The spectral method is only
applied to the spatial coordinates since the time length of the evolution is
not predictable. Actually the real time for phase evolution is usually one of
the major concerns as the result of the simulation. \par

The DCT takes a considerable portion of the computation time. Especially in a
3-dimensional numerical model, the 3-dimensional DCT function with conventional
approach has a complexity of $O(n^3)$, which can not be practical for real
application on a PC. To overcome this computational difficulty, the code can
either be translated to C code embedded in MATLAB m scripts, or a different
mathematical approach can be implemented as well. In this program, the DCT is
calculated from the fast Fourier transform (FFT) which is optimized in
MATLAB. \par

\subsection{Quantitative Simulation with Real Experimental Parameters}
Many of previous numerical simulations in the self-assembly with polymer blends
phase separation are qualitative other than quantitative. The results can only
be used to provide non-quantitative suggestions to the experiments. While this
program implemented a numerical model which quantitatively simulates the
experimental results with the real processing and material parameters. Most of
inputs in to this program can be directly measured and read from the instrument
or material labels. For some of the physical parameters such as $\kappa$ and
the mobility, the program can provide a start value from the calculation with
the theoretical model. The user may need to validate the value by comparing
the simulation results to the experimental results. Eventually, a more accurate
estimation can be find with optimization methods by setting the difference
between the simulation and experiment results as the cost function. \par

Besides the parameters in Cahn-Hilliard equation, other effects such as the
evaporation, substrate functionalization, and the degree of polymerization are
also implemented with the real conditions. The final results are saved and
summarized. The characteristic length of result pattern from simulation and its
compatibility with the substrate functionalization are calculated. These
numbers can be used to compare with the experimental results. \par

\subsection{Data Visualization and Results Analysis}
When running the program, the message from the software will be output to the
working console of MATLAB. The messages will show the current state and real
time results of the simulation. Also, when the simulation is started, the phase
pattern will be plotted in a real time plot window. Users can set the frequency
of real time plot and the scale factor on the domain of the contour plot in
the GUI. The results of the simulation will be saved to a folder designated by
the user. The real time plot will be saved to the result folder. The
quantitative results will be saved as several comma separated values (CSV)
text files. The result folder can be loaded into the analysis toolkit of the
program and the user can view the assessment values such as the characteristic
length, the compatibility parameters, and the composition profile wave in depth
direction with convenient plotting tools. Usually these results such as the
composition profile in each direction in the domain are difficult to observe
in experiment results. \par

\begin{figure}[!ht]
	\centering
	\includegraphics[width=\textwidth]{fg_gui_running.eps}
	\caption{The simulation is running with the real time plot of the
	current ternary phase morphology.
	\label{fg_gui_running}}
\clearpage
\end{figure}

\section{Examples}
To demonstrate the capability of this program, example simulation cases are
shown in this paper. The results of numerical simulation have been validated
with the experimental results in our previous work~\cite{SHANG:2010}. To
compare the simulated results with a real experimental system, we directed
the morphologies of polystyrene (PS) / polyacrylic acid (PAA) blends using
chemically heterogeneous patterns.  More specifically, alkanethiols with
different chemical functionalities were patterned by electron beam
lithography, which were then used to direct the assembly of PS/PAA blends
during the spin coating from their mutual solvent~\cite{MING:2009}. The
experimental conditions are implemented into the numerical simulation. The
effects such as the substrate functionalization and the solvent evaporation
are involved in the numerical modeling.
The parameters difficult to measure are acquired with the optimization methods
~\cite{SHANG:2009}.
\par

Sophisticated techniques are required to investigate the composition profile in
the depth of the polymer film~\cite{GEOGHEGAN:2003}. While the numerical
simulation results can provide the composition profile in each position of the
file, the composition profile change in depth direction can be easily accessed.
To investigate the composition wave allow the direction perpendicular to the
film surface, a thick film is implemented to the numerical simulation. This kind
of film is not only difficult to fabricate and characterize in experiments, however
in the numerical modeling, the user only needs to change the mesh grid domain size.
The depth profiles with different substrate functionalization are shown in
Figure~\ref{fg_thick_film}, where $|f_s|$ denotes the surface energy term from the
substrate functionalization. This term will be added to the total free energy
on the interface of the polymer film and the substrate. The initial thickness
of the film is 1 mm and decreases to 8 $\mu m$ due to the evaporation of the
solvent. The thickness results are scaled by 0.5 to fit in the figures. It can
be seen that a higher surface interaction force can result in a faster substrate
directed phase separation in the film. A stronger substrate interface attraction
force can direct the phase separation morphology near the substrate surface. While
with a lower surface energy, the phase separation dynamics in the bulk of the
film overcomes the substrate attraction force. It can be seen that at 30 seconds,
the substrate functionalization has little effects on the morphology on the
substrate surface. Also, the checker board structure can be seen near the
substrate surface with a higher surface energy~\cite{KARIM:1998}. \par

\begin{figure}[!ht]
    \centering
    \includegraphics[width=\textwidth]{fg_thick_film.eps}
    \caption{The phase separation in a thick film. \label{fg_thick_film}}
\clearpage
\end{figure}

\iffalse
  \qquad
(Figure~\ref{fg_thick_film} The phase separation in a thick film.)\par
  \qquad
\fi

To investigate the effects of a more complicated pattern, a larger domain is
simulated. The pattern on the substrate applied on the substrate surface is
shown in Figure~\ref{fg_chn_pattern}. The substrate pattern is designed to
investigate the effects of various shapes and contains components such as
squares, circles, and dead end lines in different sizes. The initial surface
dimensions of the model are changed to 12$\mu m\times$12$\mu m$. The initial
thickness of the film is 1mm and shrinks during the solvent evaporation. The
elements in the modelling is 384$\times$384$\times$16. The average composition
ratio of PS/PAA is changed to 38/62 to match the pattern. The result patterns
from the simulation can be seen in Figure~\ref{fg_complicated_patterns}. \par

\begin{figure}[!ht]
	\centering
	\includegraphics[width=\textwidth]{fg_chn_pattern.eps}
	\caption{The substrate pattern with complicated features.
	\label{fg_chn_pattern}}
\clearpage
\end{figure}
\iffalse
  \qquad
(Figure~\ref{fg_chn_pattern}The substrate pattern with complicated features.)\par
  \qquad
\fi

\begin{figure}[!ht]
	\centering
	\includegraphics[width=\textwidth]{fg_complicated_patterns.eps}
	\caption{The effects of complicated substrate patterns.
	\label{fg_complicated_patterns}}
\clearpage
\end{figure}
\iffalse
  \qquad
(Figure~\ref{fg_complicated_patterns}The effects of complicated substrate
patterns.)\par
  \qquad
\fi

It can be seen that in a larger domain with complicated substrate patterns, the
attraction factor has to be increased to obtain a better replication. In
general, the increase of the attraction factor will increase the refinement of
the pattern according to the substrate pattern. But since the substrate pattern
has geometrical features in different sizes, the attraction factor has to be
strong enough to force the intrinsic phase separation with unified
characteristic length to match the substrate pattern in different sizes. This
would be the main challenge to the replication of complicated patterns. It has
been reported by Ming et. al.~\cite{MING:2009} that the addition of the
copolymer can improve the refinement of the final patterns in experiments. The
reason is that the PAA-b-PS block copolymer will concentrate in the interface
of the PS and PAA domains in the phase separation, therefore decreasing the
mixing free energy. Fundamentally, the addition of the block copolymer
increased the miscibility of the two polymers. To simulate these phenomena, the
Flory-Huggins interaction parameter is decreased from 0.22 to 0.1 to increase
the miscibility of PS/PAA in the modelling. The result pattern is also shown in
Figure~\ref{fg_complicated_patterns}, in comparison to the cases without the
addition of block copolymers. It can be seen that the refinement of the phase
separated pattern is improved by the addition of the block copolymer. The $C_s$
values of the phase separation with complicated patten are measured and plotted
in Figure~\ref{fg_cs_complicated_patterns}. \par

\begin{figure}[!ht]
	\centering
	\includegraphics[width=\textwidth]{fg_cs_complicated_patterns.eps}
	\caption{The effects of complicated substrate patterns. \label{fg_cs_complicated_patterns}}
\clearpage
\end{figure}
\iffalse
  \qquad
(Figure~\ref{fg_cs_complicated_patterns}The effects of complicated substrate patterns.)\par
  \qquad
\fi
A assessment parameter, $C_s$, the compatibility parameter is introduced to
evaluate the replication of the morphology to the substrate pattern, where
a higher $C_s$ value denotes a better
replication of the polymer film morphology according to the substrate pattern
It can be seen in Figure~\ref{fg_cs_complicated_patterns} that the $C_s$ value
for the system with block copolymer is 7.69E-01, which is higher than the
system without the block copolymer when attraction forces are the same. The
decrease of the Flory-Huggins interaction parameter increases the miscibility
of the polymers, which will decrease the miscibility gap of the polymers, as
can be seen in Equation~\ref{eq_flory_huggins_intro}. The two phase at
equilibrium will be less concentrated in different types of polymer. This is an
issue may need to be concerned when the interaction parameter of the two
polymers is changed. \par

\section{Conclusion}

A computer program for simulation of polymer self-assembly with phase separation
is introduced. The program is developed in MATLAB m code and designed to
assist the scientists in real working environments. The program is able to
simulate the experiment results quantitatively with real experimental
parameters. The unmeasurable physical parameters such as the gradient energy
coefficient and the mobility can be estimated with the program. The program
provides a graphical user interface and analytical toolkits. This program
can help the scientists in research in polymer phase separation mechanisms and
dynamics with high efficiency, convenience of usage, quantitative results
analysis, and validated reliability.

\section{Acknowledgement}
The authors would thank the efforts of Liang Fang and Ming Wei for providing
help in the experimental procedurals. The authors also appreciate the
valuable suggestions and comments from other users and testers of this program.
This project is a part of the research in Center of High-rate Nanomanufacturing,
sponsored by National Science Foundation (grant number NSF-0425826).

\bibliographystyle{unsrt}