From 4ec6538aec763b6cee0f79a0ebcd738aab5a3b41 Mon Sep 17 00:00:00 2001
From: Blaise Thompson <blaise@untzag.com>
Date: Mon, 22 Jan 2018 18:37:19 -0600
Subject: MX2 content grab

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-\chapter{MX2}
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+\chapter{MX2}
+
+
+We report the first coherent multidimensional spectroscopy study of a MoS\textsubscript{2} film.  %
+A four-layer sample of MoS\textsubscript{2} was synthesized on a silica substrate by a simplified
+sulfidation reaction and characterized by absorption and Raman spectroscopy, atomic force
+microscopy, and transmission electron microscopy.  %
+State-selective coherent multidimensional spectroscopy (CMDS) on the as-prepared
+MoS\textsubscript{2} film resolved the dynamics of a series of diagonal and cross-peak features
+involving the spin---orbit split A and B excitonic states and continuum states.  %
+The spectra are characterized by striped features that are similar to those observed in CMDS
+studies of quantum wells where the continuum states contribute strongly to the initial excitation of
+both the diagonal and cross-peak features, while the A and B excitonic states contributed strongly
+to the final output signal.  %
+The strong contribution from the continuum states to the initial excitation of both the diagonal
+and cross-peak features, while the A and B excitonic states contributed strongly to the final
+output signal.  %
+The strong contribution from the continuum states to the initial excitation shows that the continuum
+states are coupled to the A and B excitonic states and that fast intraband relaxation is occurring
+on a sub-70 fs time scale.  %
+A comparison of the CMDS excitation signal and the absorption spectrum shows that the relative
+importance of the continuum states is determined primarily by their absorption strength.  %
+Diagonal and cross-peak features decay with a 680 fs time constant characteristic of exciton
+recombination and/or trapping.  %
+The short time dynamics are complicated by coherent and partially coherent pathways that become
+important when the excitation pulses are temporally overlapped.  %
+In this region, the coherent dynamics create diagonal features involving both the excitonic states
+and continuum states, while the partially coherent pathways contribute to cross-peak features.  %
+
+\section{Introduction}  % -------------------------------------------------------------------------
+
+Transition metal dichalcogenides (TMDCs), such as MoS\textsubscript{2}, are layered semiconductors
+with strong spin-orbit coupling, high charge mobility, and an indirect band gap that becomes direct
+for monolayers. \cite{WangQingHua2012a, MakKinFai2010a}  %
+The optical properties are dominated by the A and B excitonic transitions between two HOMO
+spin-orbit split valence bands and the lowest state of the conduction band at the $K$ and $K^\prime$
+valleys of the two-dimensional hexagonal Brillouin zone. \cite{MolinaSanchezAlejandro2013a}  %
+The spin and valley degrees of freedom are coupled in individual TMDC layers as a result of the
+strong spin-orbit coupling and the loss of inversion symmetry.  %
+The coupling suppresses spin and valley relaxation since both spin and valley must change in a
+transition.  %
+These unusual properties have motivated the development of TMDC monolayers for next-generation
+nano/optoelectronic devices as well as model systems for spintronics and valleytronics
+applications. \cite{MakKinFai2010a, XuXiaodong2014a, XiaoDi2012a}  %
+
+Ultrafast dynamics of the MoS\textsubscript{2} A and B electronic states have been measured by
+pump-probe, transient absorption, and transient reflection spectroscopy. \cite{FangHui2014a,
+  KumarNardeep2013a, NieZhaogang2014a, SunDezheng2014a, SimSangwan2013a}  %
+The spectra contain A and B excitonic features that result from ground-state bleaching (GSB),
+stimulated emission (SE), and excited-state absorption (ESA) pathways.  %
+The excitons exhibit biexponential relaxation times of $\approx$10--20 and $\approx$350--650 fs,
+depending on the fluence and temperature.  %
+The dependence on excitation frequency has not been explored in previous ultrafast experiments on
+MoS\textsubscript{2}, but it has played a central role in understanding exciton cooling dynamics
+and exciton-phonon coupling in studies of quantum dots. \cite{KambhampatiPatanjali2011a}  %
+
+Coherent multidimensional spectroscopy (CMDS) is a complementary four wave mixing (FWM) methodology
+that differs from pump-probe, transient absorption, and transient reflection methods.
+\cite{XiaoDi2012a, FangHui2014a, KumarNardeep2013a, NieZhaogang2014a, SimSangwan2013a,
+  MakKinFai2012a, SunDezheng2014a}  %
+Rather than measuring the intensity change of a probe beam caused by the state population changes
+induced by a pump beam, CMDS measures the intensity of  a coherent output beam created by
+interactions with three excitation pulses.  %
+The interest in CMDS methods arise from their ability to remove inhomogeneous broadening, define
+interstate coupling, and resolve coherent and incoherent dynamics. \cite{CundiffStevenT2008a,
+  TurnerDanielB2009a, KohlerDanielDavid2014a, YursLenaA2011a, GriffinGrahamB2013a, HarelElad2012a,
+  CundiffStevenT1996a, BirkedaDl1996a, WehnerMU1996a}  %
+CMDS typically requires interferometric phase stability between excitation pulses, so CMDS has been
+limited to materials with electronic states within the excitation-pulse bandwidth.  %
+Multiresonant CMDS is a particularly attractive method for the broader range of complex materials
+because it does not require interferometric stability and is able to use independently tunable
+excitation pulses over wide frequency ranges.  %
+
+The multiresonant CMDS used in this work employs two independently tunable excitation beams with
+frequencies $\omega_1$ and $\omega_2$.  %
+The $\omega_2$ beam is split into two beams, denoted by $\omega_2$ and $\omega_2^\prime$.  %
+These three beams are focused onto the MoS\textsubscript{2} thin film at angles, creating an output
+beam in the phase-matched direction
+$\mathbf{k}_{\mathrm{out}}=\mathbf{k}_1-\mathbf{k}_2+\mathbf{k}_{2^\prime}$ where $\mathbf{k}$ is the wave
+vector for each beam and the subscripts label the excitation frequencies.  %
+Multidimensional spectra result from measuring the output intensity dependence on frequency and
+delay times.  %
+
+\begin{figure}[!htb]
+  \centering
+  \includegraphics[width=0.5\textwidth]{MX2/01}
+  \caption[CMDS tutorial]{(a) Example delays of the $\omega_1$, $\omega_2$, and
+    $\omega_{2^\prime}$ excitation pulses. (b) Dependence of the output intensity on the
+    $\tau_{22^\prime}$ and $\tau_{21}$ time delays for $\omega_1=\omega_2$. The solid lines define
+    the regions for the six different time orderings of the $\omega_1$, $\omega_2$, and
+    $\omega_{2^\prime}$ excitation pulses. We have developed a convention for numbering these time
+    orderings, as shown. (c) Diagram of the band structure of MoS\textsubscript{2} at the $K$
+    point. The A and B exciton transitions are shown. (d) Two dimensional frequency-frequency plot
+    labeling two diagonal and cross-peak features for the A and B excitons.}
+  \label{fig:Czech01}
+\end{figure}
+
+\autoref{fig:Czech01} introduces our conventions for representing multidimensional spectra.  %
+\autoref{fig:Czech01}b,d are simulated data.  %
+\autoref{fig:Czech01}a shows one of the six time orderings of the three excitation pulses where
+$\tau_{22^\prime}\equiv t_2-t_{2^\prime}>0$ and $\tau_{21}\equiv t_2-t_1<0$; that is, the
+$\omega_{2^\prime}$ pulse interacts first and the $\omega_1$ pulse interacts last.  %
+\autoref{fig:Czech01}b illustrates the 2D delay-delay spectrum for all six time orderings when
+$\omega_1$ and $\omega_2$ are both resonant with the same state.  %
+The color denotes the output amplitude.  %
+Along the negative ordinate where $\tau_{22^\prime}=0$, interactions with the $\omega_2$ and
+$\omega_{2^\prime}$ pulses create a population that is probed by $\omega_1$.  %
+Similarly, along the negative absicissa where $\tau_{21}=0$, interactions with the $\omega_2$ and
+$\omega_1$ pulses create a population that is probed by $\omega_{2\prime}$.  %
+The decay along these axes measures the population relaxation dynamics.  %
+Note that these delay representations differ from previous publications by our group.
+\cite{PakoulevAndreiV2006a}  %
+This paper specifically explores the dynamics along the ordinate where $\tau_{22^\prime}$ is zero
+and the $\tau_{21}$ delay is changed.  %
+
+\autoref{fig:Czech01}c depicts the A and B excitonic transitions between the spin-orbit split
+valence bands and the degenerate conduction band states of MoS\textsubscript{2}.  %
+\autoref{fig:Czech01}d illustrates the 2D frequency-frequency spectrum when $\omega_1$ and
+$\omega_2$ are scanned over two narrow resonances.  %
+The spectrum contains diagonal and cross-peaks that we label according to the excitonic resonances
+AA, AB BA, and BB for illustrative purposes.  %
+The dynamics of the individual quantum states are best visualized by 2D frequency-delay plots,
+which combine the features seen in \autoref{fig:Czech01}b,d.  %
+
+This works reports the first multiresonant CMDS spectra of MoS\textsubscript{2}.  %
+It includes the excitation frequency dependence of the A and B excitonic-state dynamics.  %
+These experiments provide a fundamental understanding of the multidimensional MoS\textsubscript{2}
+spectra and a foundation for interpreting CMDS experiments on more complex TMDC
+heterostructures.  %
+The experimental spectra differ from the simple 2D spectrum shown in \autoref{fig:Czech01}d and
+those of earlier CMDS experiments with model systems.  %
+The line shape of the CMDS excitation spectrum closely matches the absorption spectrum, but the
+line shape of the output coherence is dominated by the A and B excitonic features.  %
+The difference arises from fast, $<70$ fs intraband relaxation from the hot A and B excitons of the
+continuum to the band edge.  %
+A longer, 680 fs relaxation occurs because of trapping and/or exciton dynamics.
+\cite{SimSangwan2013a}  %
+The intensity of the cross-peaks depends on the importance of state filling and intraband
+relaxation of hot A excitons as well as the presence of interband population trnasfer of the A and
+B exciton states.  %
+
+\section{Methods}  % ------------------------------------------------------------------------------
+
+\begin{figure}[!htb]
+  \centering
+  \includegraphics[width=\textwidth]{MX2/S1}
+  \caption{Schemiatic of the synthetic setup used for Mo thin film sulfidation reactions.}
+  \label{fig:CzechS1}
+\end{figure}
+
+MoS\textsubscript{2} thin films were prepared \textit{via} a Mo film sulfidation reaction, similar
+to methods reported by \textcite{LaskarMasihhurR2013a}.  %
+A 1 nm amount of Mo (Kurt J. Lesker, 99.95\%) metal was electron-beam evaporated onto a fused
+silica substrate at a rate of 0.05 \AA/s.  %
+The prepared Mo thin films were quickly transferred to the center of a 1-inch fused silica tub
+furnace equipped with gas flow controllers (see \autoref{fig:CzechS1}) and purged with Ar.  %
+The temperature of the Mo substrate was increased to 900 $^\circ$C over the course of 15 min, after
+which 200 mg of sulfur was evaporated into the reaction chamber.  %
+Sulfidation was carried out for 30 min, and the furnace was subsequently cooled to room
+temperature; then the reactor tube was returned to atmospheric pressure, and the
+MoS\textsubscript{2} thin film samples were collected.  %
+The MoS\textsubscript{2} samples were characterized and used for CMDS experiments with no further
+preparation.  %
+
+MoS\textsubscript{2} thin film absorption spectra were collected by a Shimadzu 2401PC
+ultraviolet-visible spectraphotometer.  %
+Raman and photoluminescence experiments were carried out in parallel using a Thermo DXR Raman
+microscope with a 100x 0.9 NA focusing objective and a 2.0 mW 532 nm excitation source.  %
+Raman/PL measurements were intentionally performed at an excitation power of $<$8.0 mW to prevent
+sample damage. \cite{CastellanosGomezA2012a}  %
+Contact-mode atomic foce microscopy was performed with an Agilent 5500 AFM.  %
+MoS\textsubscript{2} film thickness was determined by scratching the sample to provide a clean
+step-edge between the MoS\textsubscript{2} film and the fused silica substrate.  %
+TEM samples were prepared following the method outlined by \textcite{ShanmugamMariyappan2012a}
+using concentrated KOH in a 35 $^\circ$C oil bath for 20 minutes.  %
+The delaminated MoS\textsubscript{2} sample was removed from the basic solution, rinsed five times
+with DI water, and transferred to a Cu-mesh TEM grid.  %
+TEM experiments were performed on a FEI Titan aberration corrected (S)TEM under 200 kV accelerating
+voltage.  %
+
+\begin{figure}[!htb]
+  \centering
+  \includegraphics[width=\textwidth]{MX2/10}
+  \caption[Mask and epi vs transmissive.]{(a) Mask. (b) 2D delay spectra at the BB diagonal
+    ($\omega_1=\omega_2\approx1.95$ eV) for transmissive and reflective geometries. Transmissive
+    signal is a mixture of MoS\textsubscript{2} signal and a large amount of driven signal from the
+    substrate that only appears in the pulse overlap region. Reflective signal is representative of
+    the pure MoS\textsubscript{2} response.}
+  \label{fig:Czech10}
+\end{figure}
+
+The coherent multidimensional spectroscopy system used a 35 fs seed pulse, centered at 800 nm and
+generated by a 1 kHz Tsunami Ti-sapphire oscillator.  %
+The seed was amplified by a Spitfire-Pro regenerative amplifier.  %
+The amplified output was split to pump two TOPAS-C collinear optical parametric amplifiers.  %
+OPA signal output was immediately frequency doubled with BBO crystals, providing two $\approx$50 fs
+independently tunable pulses denoted $\omega_1$ and $\omega_2$ with frequencies ranging from 1.62
+to 2.12 eV.  %
+Signal and idler were not filtered out, but played no role due to their low photon energy.  %
+Pulse $\omega_2$ was split into pulses labeled $\omega_2$ and $\omega_{2^\prime}$ to create a total
+of three excitation pulses.  %
+
+\begin{figure}[!htb]
+  \centering
+  \includegraphics[width=\textwidth]{MX2/S4}
+  \caption{OPA outputs at each color explored.}
+  \label{fig:CzechS4}
+\end{figure}
+
+In this experiment we use motorized OPAs which allow us to set the output color in software.  %
+OPA1 and OPA2 were used to create the $\omega_1$ and $\omega_2$ frequencies, respectively.  %
+In \autoref{fig:CzechS4} we compare the spectral envelope generated by the OPA at each set
+color.  %
+Negative detuning values correspond to regions of the envelope lower in energy than the
+corresponding set color.  %
+The colorbar allows for comparison between set color intensities.  %
+The fluence values reported correspond to the brightest set color for each OPA.  %
+A single trace of OPA2 output at set color = 1.95 eV can be found in \autoref{fig:Czech02}.
+
+After passing through automated delay stages (Newport SMC100 actuators), all three beams were
+focused onto the sample surface by a 1 meter focal length spherical mirror in a distorted BOXCARS
+geometry to form a 630, 580, and 580 $\mu$m FWHM spot sizes for $\omega_1$, $\omega_2$, and
+$\omega_{2^\prime}$, respectively.  %
+
+\begin{figure}[!htb]
+  \centering
+  \includegraphics[width=\textwidth]{MX2/S5}
+  \caption{Spectral delay correction.}
+  \label{fig:CzechS5}
+\end{figure}
+
+\autoref{fig:CzechS5} represents delay corrections applied for each OPA.  %
+The corrections were experimentally determined using driven FWM output from fused silica.  %
+Corrections were approximately linear against photon energy, in agreement with the normal dispersion
+of transmissive optics inside our OPAs and between the OPAs and the sample.  %
+OPA2 required a relatively small correction along $\tau_{22^\prime}$ (middle subplot) to account
+for any dispersion experienced differently between the two split beams.  %
+OPA1 was not split and therefore needed no such correction.  %
+
+\autoref{fig:Czech10}a represents the to-scale mask that defines our distorted BOXCARS
+configuration.  %
+Relative to the center of the BOXCARS mask (small black dot), $\omega_1$, $\omega_2$, and
+$\omega_{2^\prime}$ enter the sample at angles of 5.0, 1.5, and 1.0 degrees.  %
+Each is angled only along the vertical or horizontal dimension, as indicated in
+\autoref{fig:Czech10}a.  %
+This distortion allowed us to remove a large amount of unwanted $\omega_2$ and $\omega_{2^\prime}$
+photons from our signal path (\autoref{fig:Czech10}a red star).  %
+$\omega_1$ photons were less efficiently rejected, as we show below.  %
+The center of the BOXCARS mask was brought into the sample at $\approx$45 degrees.  %
+All three beams had S polarization.  %
+After reflection, the output beam was isolated using a series of apertures, spectrally resolved
+with a monochromator (spectral resolution 9 meV). and detected using a photomultiplier (RCA
+C31034A).  %
+
+Our experimental setup allowed for the collection of both transmissive and reflective
+(epi-directional) FWM signal.  %
+The 2D delay spectra in \autoref{fig:Czech10}b show the presence of a large nonresonant
+contribution at the origin for the transmissive FWM signal and weaker signals from the
+MoS\textsubscript{2} thin film at negative values of $\tau_{21}$ and $\tau_{22^\prime}$.  %
+The nonresonant contribution is much weaker than the signals from the film for the reflective
+signal id is the geometry chosen for this experiment.  %
+This discrimination between a film and the substrate was also seen in reflective and transmissive
+CARS microscopy experiments. \cite{VolkmerAndreas2001a}  %
+
+\begin{figure}[!htb]
+  \centering
+  \includegraphics[width=\textwidth]{MX2/11}
+  \caption[MoS\textsubscript{2} post processing.]{Visualization of data collection and processing.
+    With the exception of (c), each subsequent pane represents an additional processing step on top
+  of previous processing. The color bar of each image is separate. (a) Voltages read by the
+  detector at teach color combination. The large vertical feature is $\omega_1$ scatter; the shape
+  is indicative of the power curve of the OPA. MoS\textsubscript{2} response can be barely seen
+  above this scatter. (b) Data after chopping and active background subtraction at the boxcar (100
+  shots). (c) The portion of chopped signal that is not material response. This portion is
+  extracted by averaging several collections at very positive $\tau_{21}$ values, where no material
+  response is present due to the short coherence times of MoS\textsubscript{2} electronic states.
+  The largest feature is $\omega_2$ scatter. Cross-talk between digital-to-analog channels can also
+  be seen as the negative portion that goes as $\omega_1$ intensity. (d) Signal after (c) is
+  subtracted. (e) Smoothed data. (f) Amplitude level (square root) data. This spectrum corresponds
+  to that at 0 delay in \autoref{fig:Czech03}. Note that the color bar's range is different than in
+  \autoref{fig:Czech03}.}
+  \label{fig:Czech11}
+\end{figure}
+
+Once measured, the FWM signal was sent through a four-stage workup process to create the data set
+shown here.  %
+This workup procedure is visualized in \autoref{fig:Czech11}.  %
+We use a chopper and boxcar in active background subtraction mode (averaging 100 laser shots) to
+extract the FWM signal from $\omega_1$ and $\omega_2$ scatter.  %
+We collect this differential signal (\autoref{fig:Czech11}b) in software with an additional 50
+shots of averaging.  %
+In post-process we subtract $\omega_2$ scatter and smooth the data using a 2D Kaiser window.  %
+Finally, we represent the homodyne collected data as (sig)$^{1/2}$ to make the dynamics and line
+widths comparable to heterodyne-collected techniques like absorbance and pump-probe spectra.  %
+Throughout this work, zero signal on the color bar is set to agree with the average rather than the
+minimum of noise.  %
+Values below zero due to measurement uncertainty underflow the color bar and are plotted in
+white.  %
+This is especially evident in lots such as +120 fs in \autoref{fig:Czech08}, where there is no real
+signal.  %
+IPython \cite{PerezFernando2007a} and matplotlib \cite{HunterJohnD2007a} were important for data
+processing and plotting in this work.
+
+\section{Results and discussion}  % ---------------------------------------------------------------
+
+\begin{figure}[!htb]
+  \centering
+  \includegraphics[width=0.75\textwidth]{MX2/02}
+  \caption[Few-layer MoS\textsubscript{2} thin film characterization.]{Characterization of the
+    few-layer MoS\textsubscript{2} film studied in this work. Optical images of the
+    MoS\textsubscript{2} thin film on fused silica substrate in (a) transmission and (b)
+    reflection. (c) Raman spectrum of the $E_{2g}^1$ and $A_{1g}$ vibrational
+    modes. (d) High-resolution TEM image and its corresponding FFT shown in the inset. (e)
+    Absorption (blue), photoluminescence (green), Gaussian fits to the A and B excitons, along with
+    the residules betwen the fits and absorbance (dotted), A and B exciton centers (dotted) and
+    representative excitation pulse shape (red).}
+  \label{fig:Czech02}
+\end{figure}
+
+The few-layer MoS\textsubscript{2} thin film sample studied in this work was prepared on a
+transparent fused silica substrate by a simple sufidation reaction of a Mo thin film using a
+procedure modified from a recent report. \cite{LaskarMasihhurR2013a}  %
+\autoref{fig:Czech02}a and b show the homogeneous deposition and surface smoothness of the sample
+over the centimeter-sized fused silica substrate, respectively.  %
+The Raman spectrum shows the $E_{2g}^1$ and $A_{1g}$ vibrational modes (\autoref{fig:Czech02}c)
+that are characteristic of MoS\textsubscript{2}. \cite{LiSongLin2012a}  %
+The transmission electron micrograph (TEM) in \autoref{fig:Czech02}d shows the lattice fringes of
+the film with an inset fast Fourier transform (FFT) of the TEM image indicative of the hexagonal
+crystal structure of the film corresponding to the 0001 plane of MoS\textsubscript{2}.
+\cite{LukowskiMarkA2013a}  %
+The MoS\textsubscript{2} film thickness was determined to be 2.66 nm by atomic force microscopy and
+corresponds to approximately four monolayers.  %
+\autoref{fig:Czech02}3 shows the absorption and fluorescence spectrum of the film along with the A
+and B excitonic line shapes that were extracted from the absorption spectrum. A representative
+excitation pulse profile is also shown in red for comparison.  %
+
+\begin{figure}[!htb]
+  \centering
+  \includegraphics[width=\textwidth]{MX2/S3}
+  \caption[MoS\textsubscript{2} absorbance.]{Extraction of excitonic features from absorbance
+    spectrum. (a) Second derivative spectra of absorbance (black) and fit second derivative
+    spectrum (green). Gaussian fit parameters are shown in the inset table. (b) Absorption curve
+    (black), Gaussian fits (blue and red), and remainder (black dotted).}
+  \label{fig:CzechS3}
+\end{figure}
+
+Extracting the exciton absorbance spectrum is complicated by the large ``rising background'' signal
+from other MoS\textsubscript{2} bands.  %
+With this in mind, we fit the second derivative absorption spectrum to a sum of two second
+derivative Gaussians, as seen in \autoref{fig:CzechS3}.  %
+Conceptually, this method can be thought of as maximizing the smoothness (as opposed to minimizing
+the amplitude) of the remainder between the fit and the absorption spectrum.  %
+The fit parameters can be found in the inset table in \autoref{fig:CzechS3}.  %
+The Gaussians themselves and the remainder can be found in \autoref{fig:CzechS3}.  %
+
+The multiresonant CMDS experiment uses $\approx$70 fs excitation pulses created by two
+independently tunable optical parametric amplifiers (OPAs).  %
+Automated delay stages and neutral density filters set the excitation time delays over all values
+of $\tau_{21}$ with $\tau_{22^\prime}=0$ and the pulse fluence to 90 $\mu$J/cm$^2$ (114
+$\mu$J/cm$^2$) for the $\omega_1$ ($\omega_2$ and $\omega_{2^\prime}$) beam(s).  %
+Each pulse was focused onto the sample using a distorted BOXCARS configuration.
+\cite{EckbrethAlanC1978a}  %
+The FWM signal was spatially isolated and detected with a monochromator that tracks the output
+frequency so $\omega_m = \omega_1$.  %
+In order to compare the FWM spectra with the absorption spectrum, the signal has been defined as
+the square root of the measured FWM signal since FWM depends quadratically on the sample
+concentration and path length.  %
+
+\begin{figure}[!htb]
+  \centering
+  \includegraphics[width=\textwidth]{MX2/03}
+  \caption[MoS\textsubscript{2} frequency-frequency slices.]{2D frequency-frequency spectra of the
+    MoS\textsubscript{2} sample in the epi configuration. In all spectra $\tau_{22^\prime}=0$ fs,
+    while $\tau_{21}$ is designated in the bottom-right corner of each spectral panel. The color
+    bar defines the square root of the intensity normalized to the most intense feature in the
+    series of spectra. The integration of the signal onto the $\hbar\omega_1=\hbar\omega_m$ and
+    $\hbar\omega_2$ axes are represented ans the blue curves in the top and right side plots,
+    respectively. The side plots also contain the absorbance spectrum (black line) to aid
+    intepretation of the dynamics of the integrated 2D signals. The dashed lines mark the centers
+    of the A and B excitons, as designated from the absorption spectrum.}
+  \label{fig:Czech03}
+\end{figure}
+
+The main set of data presented in this work is an $\omega_1\omega_2\tau_{21}$ ``movie'' with
+$\tau_{22\prime}=0$. 
+\autoref{fig:Czec03} shows representative 2D frequency-frequency slices from this movie at
+increasingly negative $\tau_{21}$ times.  %
+Each 2D frequency spectrum contains side plots along both axes that compare the absorbance spectrum
+(black) to the projection of the integrated signal onto the axis (blue).  %
+Along $\omega_1$ (which for negative $\tau_{21}$ times acts as the ``probe'') we observe two peaks
+corresponding to the A and B excitons.  %
+In contrast, we see no well-defined excitonic peaks along the $\omega_2$ ``pump'' axis.  %
+Instead, the signal amplitude increases toward bluer $\omega_2$ values.  %
+The decrease in FWM above 2.05 eV is caused by a drop in the $\omega_2$ OPA power.
+
+\begin{figure}[!htb]
+  \centering
+  \includegraphics[width=0.75\textwidth]{MX2/04}
+  \caption[MoS\textsubscript{2} $\omega_1$ Wigner progression.]{Mixed $\omega_1$---$\tau_{21}$
+    time---frequency representations of the 3D data set at five ascending $\omega_2$ excitation
+    frequencies (solid black lines) showing the impact of the $\omega_2$ excitation frequency on
+    the $\omega_1$ spectral line shape as a function of time. The A and B exciton energies are
+    marked as dashed lines within each spectrum.}
+  \label{fig:Czech04}
+\end{figure}
+
+\begin{figure}[!htb]
+  \centering
+  \includegraphics[width=0.75\textwidth]{MX2/05}
+  \caption[MoS\textsubscript{2} $\omega_2$ Wigner progression.]{Mixed $\omega_2$---$\tau_{21}$
+    time---frequency representations of the 3D data set at five ascending $\omega_1$ probe
+    frequencies (solid black lines) showing the impact of the $\omega_1$ excitation frequency on
+    the $\omega_2$ spectral line shape as a function of time. The A and B exciton energies are
+    marked as dashed lines within each spectrum.}
+  \label{fig:Czech05}
+\end{figure}
+
+Figures \ref{fig:Czech04} and \ref{fig:Czech05} show representative 2D frequency-delay slices from
+this movie, where the absicissa is the $\omega_1$ or $\omega_2$ frequency, respectively, the
+ordinate is the $\tau_{21}$ delay time, and the solid bold lines represent five different
+$\omega_2$ or $\omega_1$ frequencies.  %
+The color bar is normalized to the brightest feature in each subplot.  %
+This normalization allows comparison of the time dependence of the line shapes, positions, and
+relative signal amplitudes along the $\omega_1$ or $\omega_2$ axis directly.  %
+
+Each subplot in \autoref{fig:Czech04} is similar to published pump-probe, transient absorption, multidimensionaland
+transient reflection experiments that have measured the electronc dynamics of the A and B excitons.
+\cite{XiaoDi2012a, FangHui2014a, KumarNardeep2013a, NieZhaogang2014a, SunDezheng2014a,
+  SimSangwan2013a, MakKinFai2012a, ThomallaMarkus2006a}  %
+These previous experiments measure relaxation dynamics on the same $\approx$400-600 fs time scale
+that is characteristic of \autoref{fig:Czech04}.  %
+
+Our experiments also show how the spectral features change as a function of the $\omega_2$
+excitation frequency.  %
+The top to subplots of \autoref{fig:Czech04} reflect the changes in the AA and BA features, while
+the bottom two subplots reflect the changes in the AB and BB features.  %
+The figure highlights the changes in the relative amplitude of the A and B features as a function
+of excitation frequency.  %
+Both the line shapes and the dynamics of the spectral features are very similar.  %
+\autoref{fig:Czech05} is an excitation spectrum that shows that the dynamics of the spectral
+features do not depend strongly on the $\omega_1$ frequency.
+
+\begin{figure}[!htb]
+  \centering
+  \includegraphics[width=0.5\textwidth]{MX2/06}
+  \caption[Pathway V, VI liouville pathways.]{Liouville pathways for \autoref{fig:Czech04}. gg and
+    ee designate ground- and excited-state populations, the eg, 2e,e, and e$^\prime$+e,e represent
+    the excitonic and biexcitonic output coherences, and the arrows are labeled with the
+    frequencies or population transfer responsible for the transitions. e and e$^\prime$ represent
+    either A or B excitonic states.}
+  \label{fig:Czech06}
+\end{figure}
+
+The spectral features in Figures \ref{fig:Czech03}, \ref{fig:Czech04} and \ref{fig:Czech05} depend
+on the quantum mechanical interference effects caused by the different pathways.  %
+\autoref{fig:Czech06} shows all of the Liouville pathways required to understand the spectral
+features. \cite{PakoulevAndreiV2010a, PakoulevAndreiV2009a}  %
+These pathways correspond to the time orderings labeled V and VI in \autoref{fig:Czech02}b. The
+letters denote the density matrix elements, $\rho_{ij}$, where g representest the ground state and
+e, e$^\prime$ represent any excitonic state.  %
+Interaction with the temporally overlapped $\omega_2$ and $\omega_{2^\prime}$ pulses excites the ee
+excited-state population and bleaches the ground-state population.  %
+Subsequent interaction with $\omega_1$ creates the output coherences for the diagonal spectral
+features when $\omega_1=\omega_2$ or the cross-peak features when $\omega_1\neq\omega_2$.  %
+The stimulated emission (SE) and ground-state bleaching (GSB) pathways create the eg or e$^\prime$g
+output coherences from the ee and gg populations, respectively, while the excited-state absorption
+pathway creates the 2e,e or e$^\prime$+e,e biexcitonic output coherences.  %
+\autoref{fig:Czech06} also includes a population transfer pathway from the ee excited-state
+population to an e$^\prime$e$^\prime$ population from which similar SE and ESA pathways occur.  %
+Since the ESA pathways destructively interfere with the SE and GSB pathways, the output singal
+depends on the differences between the pathways.  %
+Factors that change the biexcitonic output coherences such as the transition moments, state filling
+(Pauli blocking), frequency shifts, or dephasing rate changes will control the output signal.  %
+State filling and ground-state depletion are important factors for MoS\textsubscript{2} since the
+transitions excite specific electron and hole spin and valley states in individual layers.  %
+
+The state-filling and ground-state bleaching effects on the diagonal and cross-peak features in
+\autoref{fig:Czech03} depend on the spin and valley states in the output coherence.
+\cite{XuXiaodong2014a}  %
+The effects of these spins will disappear as the spin and valley states return to equilibrium.
+\cite{ZengHualing2012a, ZhuBairen2014a, MaiCong2013a}
+If we assume spin relaxation is negligible, the FWM transitions that create the diagonal features
+involve either the A or B ESA transitions, so the resulting 2e,e state includes two spin-aligned
+conduction band electrons and valence band holes.  %
+Similarly, the cross-peak regions denoted by AB or BA in \autoref{fig:Czech01}d will have two
+transitions involving an A exciton, so the initial e$^\prime$+e,e state will include antialigned
+spins.  %
+A quantitative treatment of the cancellation effects between the GSB, SE, and ESA pathways requries
+knowledge of the transition moments and state degeneracies and is beyond the scope of this paper.
+\cite{WongCathyY2011a}  %
+
+The most important characteristic of the experimental spectra is the contrast between the absence
+of well-resolved excitonic features that depend on $\omega_2$ in Figures \ref{fig:Czech03} and
+\ref{fig:Czech05} and the well-defined excitonic features that depend on $\omega_1$.  %
+It is also important to note that the projections of the signal amplitude onto the $\omega_2$ axis
+in \autoref{fig:Czech03} match closely with the continuum features in the absorption spectrum and
+that the line shapes of the features along the $\omega_2$ axis in \autoref{fig:Czech05} do not
+change appreciably for different delay times or $\omega_1$ values. %
+It shoudl be noted that the excitation pulse bandwidth (see \autoref{fig:Czech02}e) contributes to
+the absence of well-resolved A and B excitonic features along $\omega_2$.  %
+The similarity to the continuum states in the absorption spectrum and the absence of a strong
+dependence on $\omega_1$ show that the continuum states observed at higher $\omega_2$ frequencies
+participate directly in creating the final output coherences and that their increasing importance
+reflects the increasing absorption strength of higher energy continuum states.  %
+In contrast, the features dependent on $\omega_1$ in Figures \ref{fig:Czech03} and
+\ref{fig:Czech04} match the line shapes of the A and B excitonic resonances.  %
+Although the relative amplitudes of the spectral features depend on the $\omega_2$ frequency, they
+do not depend strongly on the delay times.  %
+These characteristics show that hot A and B excitonic states undergo rapid intraband population
+relaxation over a $<$70 fs time scale set by excitation pulses to the A and B excitonic states
+excited by $\omega_1$.
+
+A central feature of Figures \ref{fig:Czech03} and \ref{fig:Czech04} is that the AB region is much
+brighter than the BA region.  %
+This difference is suprising because simple models predict cross-peaks of equal amplitude, as
+depicted in \autoref{fig:Czech01}d.  %
+The symmetry in simple models arises because the AB and BA cross-peaks involve the same four
+transitions.  %
+The symmetry between AB and BA may be broken by material processes such as population relaxation
+and transfer, the output coherence dephasing rates, and the bleaching and state-filling effects of
+the valence and conduction band states involved in the transitions.  %
+
+We believe that ultrafast intraband population transfer breaks the symmetry of AB and BA
+cross-peaks.  %
+For the BA peak, the interactions with $\omega_2$ and $\omega_{2^\prime}$ generate only A excitons
+that do not relax on $<$70 fs time scales.  %
+For the AB peak, $\omega_2$ and $\omega_{2^\prime}$ generate two kinds of excitons: (1) B excitons
+and (2) hot excitons in the A band.  %
+Relaxation to A may occur by interband transitions of B excitons or intraband transitions of hot A
+excitons.  %
+Either will lead to the GSB, SE, and ESA shown in the ee $\rightarrow$ e$^\prime$e$^\prime$
+population transfer pathways of \autoref{fig:Czech06}.  %
+This relaxation must occur on the time scale of our pulse-width since the cross-peak asymmetry is
+observed even during temporal overlap.  %
+We believe that intraband relaxation of hot A excitons is the main factor in breaking the symmetry
+between AB and BA cross-peaks.  %
+\autoref{fig:Czech04} shows that B $\rightarrow$ A interband relaxation occurs on a longer time
+scale.  %
+The B/A ratio is higher when $\omega_2$ is resonant with the B excitonic transition than when
+$\omega_2$ is lower than the A exciton frequency (the top subplot).  %
+If population transfer of holes from the B to A valence bands occurred during temporal overlap, the
+B/A ratio would be independent of pump frequency at $\tau_21<0$.
+
+\begin{figure}[!htb]
+  \centering
+  \includegraphics[width=\textwidth]{MX2/07}
+  \caption[MoS\textsubscript{2} transients.]{Transients taken at the different $\omega_1$ and
+    $\omega_2$ frequencies indicated by the colored markers on the 2D spectrum. The dynamics are
+    assigned to a 680 fs fast time constant (black solid line) and a slow time constant represented
+    as an unchanging offset over this timescale (black dashed line).}
+  \label{fig:Czech07}
+\end{figure}
+
+\autoref{fig:Czech07} shows the delay transients at the different frequencies shown in the 2D
+spectrum.  %
+The colors of the dots on the 2D frequency-frequency spectrum match the colors of the
+transients.  %
+The transients were taken with a smaller step size and a longer time scale than the delay space
+explored in the 3D data set.  %
+The transients are quite similar.  %
+Our data are consistent with both monomolecular biexponential and bimolecular kinetic models and
+cannot discriminate between them.  %
+We have fit the decay kinetics to a single exponential model with a time constant of 680 fs and an
+offset that represents the long time decay.  %
+The 680 fs decay is similar to previously published pump-probe and transient absorption
+experiments. \cite{NieZhaogang2014a, SunDezheng2014a, DochertyCallumJ2014a}  %
+
+\begin{figure}[!htb]
+  \centering
+  \includegraphics[width=\textwidth]{MX2/08}
+  \caption[MoS\textsubscript{2} frequency-frequency slices near pulse overlap.]{2D
+    frequency-frquency spectra near zero $\tau_{21}$ delay times. The signal amplitude is
+    normalized to the brightest features in each spectrum.}
+  \label{fig:Czech08}
+\end{figure}
+
+The spectral features change quantitatively for delay times near temporal overlap.  %
+\autoref{fig:Czech08} shows a series of 2D spectra for both positive and negative $\tau_{21}$ delay
+times with $\tau_{22^\prime}=0$.  %
+Each spectrum is normalized to its brightest feature.  %
+The spectra at positive $\tau_{21}$ delay times become rapidly weaker as the delay times become
+more positive until the features vanish into the noise at +120 fs.  %
+The spectra also develop more diagonal character as the delay time moves from negative to positive
+values.  %
+The AB cross-peak is also a strong feature in the spectrum at early times.  %
+
+\begin{figure}[!htb]
+  \centering
+  \includegraphics[width=0.5\textwidth]{MX2/09}
+  \caption[Pathways I, III Liouville pathways.]{Liouville pathways for the $\omega_1$, $\omega_2$,
+    and $\omega_{2^\prime}$ time ordering of pulse interactions. e and e$^\prime$ represent either
+    A or B excitonic states.}
+  \label{fig:Czech09}
+\end{figure}
+
+The pulse overlap region is complicated by the multiple Liouville pathways that must be
+considered.  %
+Additionally, interference between scattered light from the $\omega_1$ excitation beam and the
+output signal becomes a larger factor as the FWM signal decreases.  %
+\autoref{fig:Czech09} shows the $\omega_1$, $\omega_2$, and $\omega_{2^\prime}$ time ordered
+pathway that becomes and important consideration for positive $\tau_{21}$ delay times.  %
+Since $\tau_{22^\prime}=0$, the initial $\omega_1$ pulse creates an excited-state coherence, while
+the subsequent $\omega_2$ and $\omega_{2^\prime}$ pulses create the output coherence.  %
+The output signal is only important at short $\tau_{21}>0$ values because the initially excited
+coherence dephases very rapidly.  %
+When $\omega_1\neq\omega_2$, the first two interactions create an e$^\prime$e zero quantum
+coherence that also dephases rapidly.  %
+However, when $\omega_1=\omega_2$, the first two interactions create an ee, gg population
+difference that relaxes on linger time scales.  %
+The resulting signal will therefore appear as the diagonal feature in \autoref{fig:Czech08}
+(\textit{e.g.}, see the +40 fs 2D spectrum).  %
+In addition to the diagonal feature in \autoref{fig:Czech08}, there is also a vertical feature when
+$\omega_1$ is resonant with the A excitonic transition as well as the AB cross-peak.  %
+These features are attributed to the pathways in \autoref{fig:Czech06}.  %
+Although these pathways are depressed when $\tau_{21}>0$, there is sufficient temporal overlap
+between the $\omega_2$, $\omega_{2^\prime}$, and $\omega_1$ pulses to make their contribution
+comparable to those in \autoref{fig:Czech09}.  %
+More positie values of $\tau_{21}$ emphasize the \autoref{fig:Czech09} pathways over the
+\autoref{fig:Czech06} pathways, accounting for the increasing percentage of diagonal character at
+increasingly positive delays.  %
+
+\section{Conclusions}  % --------------------------------------------------------------------------
+
+This paper presents the first coherent multidimensional spectroscopy of MoS\textsubscript{2} thin
+films.  %
+CMDS methods are related to the earlier ultrafast pump-probe and transient absorption methods since
+they all share bleaching, stimulated emission, Pauli blocking, and excited-state absorption
+pathways, but they differ in how these pathways define the spectra.  %
+In addition, CMDS methods have many additional pathways that become important when the coherence
+dephasing times are longer than the excitation pulse widths.  %
+In this work, the dephasing times are short so the pathways are identical to transient
+absorption.  %
+This work reports the first frequency-frequency-delay spectra of MX\textsubscript{2} samples.  %
+These spectra are complementary to previous work because they allow a direct comparison between the
+initially excited excitonic states and the states creating the final output coherence.  %
+The spectra show that the same hot A and B exciton continuum states that are observed in the
+absorption spectrum also dominate the CMDS excitation spectra.  %
+They also show that rapid, <70 fs intraband relaxation occurs to create the band-edge A and B
+excitonic features observed in the CMSD spectrum.  %
+The relative intensity of the diagonal peak features depends on the relative absorption strength of
+the A and B excitons.  %
+The relative intensity of cross-peak features in the 2D spectra depends on the excitation
+frequency.  %
+Excitation at or above the B exciton feature creates strong cross-peaks associated with hot A and B
+excitons that undergo ultrafast intraband population transfer.  %
+Excitation below the B excitonic feature creates a weak cross-peak indicating A-induced B-state
+bleaching but at a lower signal level corresponding to the lower optical density at this energy.  %
+Population relaxation occurs over $\approx$680 fs, either by transfer to traps or by bimolecular
+charge recombination.  %
+
+These experiments provide the understanding of MoS\textsubscript{2} coherent multidimensional
+spectra that will form the foundation required to measure the dynamical processes occurring in more
+complex MoS\textsubscript{2} and other TMDC heterostructures with quantum-state resolution.  %
+The frequency domain based multiresonant CMDS methods described in this paper will play a central
+role in these measurements.  %
+They use longer, independently tunable pulses that provide state-selective excitation over a wide
+spectral range without the requirement for interferometric stability.  %
+
-- 
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