prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What are some potential applications of ferromagnetic semiconductors?

Read the following text and answer briefly.

Context 1: \section{introduction}
In the past few years, the synthesis of ferromagnetic semiconductors has become a major challenge for spintronics  Actually, growing a magnetic and semiconducting material could lead to promising advances like spin injection into non magnetic semiconductors, or electrical manipulation of carrier induced magnetism in magnetic semiconductors \cite{ohno00,Bouk02}  Up to now, major efforts have focused on diluted magnetic semiconductors (DMS) in which the host semiconducting matrix is randomly substituted by transition metal (TM) ions such as Mn, Cr, Ni, Fe or Co \cite{Diet02}  However Curie temperatures ($T_{C}$) in DMS remain rather low and TM concentrations must be drastically raised in order to increase $T_{C}$ up to room temperature  That usually leads to phase separation and the formation of secondary phases  It was recently shown that phase separation induced by spinodal decomposition could lead to a significant increase of $T_{C}$ \cite{Diet06,Fuku06}  For semiconductors showing $T_{C}$ higher than room temperature one can foresee the fabrication of nanodevices such as memory nanodots, or nanochannels for spin injection  Therefore, the precise control of inhomogeneities appears as a new challenge which may open a way to industrial applications of ferromagnetism in semiconductors 

The increasing interest in group-IV magnetic semiconductors can also be explained by their potential compatibility with the existing silicon technology  In 2002, carrier mediated ferromagnetism was reported in  MBE grown Ge$_{0 94}$Mn$_{0 06}$ films by Park \textit{et al } \cite{Park02}  The maximum critical temperature was 116 K  Recently many publications indicate a significant increase of $T_{C}$ in Ge$_{1-x}$Mn$_{x}$ material depending on growth conditions \cite{Pint05,Li05,tsui03}  Cho \textit{et al } reported a Curie temperature as high as 285 K \cite{Cho02}  
Taking into account the strong tendency of Mn ions to form intermetallic compounds in germanium, a detailed investigation of the nanoscale structure is required  Up to now, only a few studies have focused on the nanoscale composition in Ge$_{1-x}$Mn$_{x}$ films  Local chemical inhomogeneities have been recently reported by Kang \textit{et al } \cite{Kang05} who evidenced a micrometer scale segregation of manganese in large Mn rich stripes  Ge$_3$Mn$_5$ as well as Ge$_8$Mn$_{11}$ clusters embedded in a germanium matrix have been reported by many authors  However, Curie temperatures never exceed 300 K \cite{Bihl06,Morr06,Pass06,Ahle06}  Ge$_3$Mn$_5$ clusters exhibit a Curie temperature of 296 K \cite{Mass90}  This phase frequently observed in Ge$_{1-x}$Mn$_{x}$ films is the most stable (Ge,Mn) alloy  The other stable compound Ge$_8$Mn$_{11}$ has also been observed in nanocrystallites surrounded with pure germanium \cite{Park01}  Ge$_8$Mn$_{11}$ and Ge$_3$Mn$_5$ phases are ferromagnetic but their metallic character considerably complicates their potential use as spin injectors 
Recently, some new Mn-rich nanostructures have been evidenced in Ge$_{1-x}$Mn$_{x}$ layers  Sugahara \textit{et al } \cite{Sugh05} reported the formation of high Mn content (between 10 \% and 20 \% of Mn) amorphous Ge$_{1-x}$Mn$_x$ precipitates  in a Mn-free germanium matrix  Mn-rich coherent cubic clusters were observed by Ahlers \textit{et al } \cite{Ahle06} which exhibit a Curie temperatures below 200 K  Finally, high-$T_{C}$ ($>$ 400 K) Mn-rich nanocolumns have been evidenced \cite{Jame06} which could lead to silicon compatible room temperature operational devices \\
In the present paper, we investigate the structural and magnetic properties of Ge$_{1-x}$Mn$_x$ thin films for low growth temperatures ($<$ 200$^{\circ}$C) and low Mn concentrations (between 1 \% and 11 \%)  By combining TEM, x-Ray diffraction and SQUID magnetometry, we could identify different magnetic phases  We show that depending on growth conditions, we obtain either Mn-rich nanocolumns or Ge$_{3}$Mn$_{5}$ clusters embedded in a germanium matrix  We discuss the structural and magnetic properties of these nanostructures as a function of manganese concentration and growth temperature  We also discuss the magnetic anisotropy of nanocolumns and  
Ge$_3$Mn$_5$ clusters  

\section{Sample growth}

Growth was performed using solid sources molecular beam epitaxy (MBE) by co-depositing Ge and Mn evaporated from standard Knudsen effusion cells   Deposition rate was low ($\approx$ 0 2 \AA s$^{-1}$)  Germanium substrates were epi-ready Ge(001) wafers with a residual n-type doping and resistivity of 10$^{15}$ cm$^{-3}$ and 5 $\Omega cm$ respectively  After thermal desorption of the surface oxide, a 40 nm thick Ge buffer layer was grown at 250$^{\circ}$C, resulting in a 2 $\times$ 1 surface reconstruction as observed by reflection high energy electron diffraction (RHEED) (see Fig  1a)  Next, 80 nm thick Ge$_{1-x}$Mn$_{x}$ films were subsequently grown  at low substrate temperature (from 80$^{\circ}$C to 200$^{\circ}$C)  Mn content has been determined by x-ray fluorescence measurements performed on thick samples ($\approx$ 1 $\mu m$ thick) and complementary Rutherford Back Scattering (RBS) on thin Ge$_{1-x}$Mn$_{x}$ films grown on silicon  Mn concentrations range from 1 \% to 11\% Mn 

For Ge$_{1-x}$Mn$_{x}$ films grown at substrate temperatures below 180$^{\circ}$C, after the first monolayer (ML) deposition, the  2 $\times$ 1 surface reconstruction almost totally disappears  After depositing few MLs, a slightly diffuse 1 $\times$ 1 streaky RHEED pattern and a very weak 2 $\times$ 1 reconstruction (Fig  1b) indicate a predominantly two-dimensional growth  For growth temperatures above 180$^{\circ}$C additional spots appear in the RHEED pattern during the Ge$_{1-x}$Mn$_{x}$ growth (Fig  1c)  These spots may correspond to the formation of very small secondary phase crystallites  The nature of these crystallites will be discussed below 

Transmission electron microscopy (TEM) observations were performed using a JEOL 4000EX microscope with an acceleration voltage of 400 kV  Energy filtered transmission electron microscopy (EFTEM) was done using a JEOL 3010 microscope equipped with a Gatan Image Filter   Sample preparation was carried out by standard mechanical polishing and argon ion milling for cross-section investigations and plane views were prepared by wet etching with H$_3$PO$_4$-H$_2$O$_2$ solution \cite{Kaga82} 

\begin{figure}[htb]
    \center
    \includegraphics[width= 29\linewidth]{ /fig1a eps}
    \includegraphics[width= 29\linewidth]{ /fig1b eps}
    \includegraphics[width= 29\linewidth]{ /fig1c eps}
    \caption{RHEED patterns recorded during the growth of Ge$_{1-x}$Mn$_{x}$ films: (a) 2 $\times$ 1 surface reconstruction of the germanium buffer layer  (b) 1 $\times$ 1 streaky RHEED pattern obtained at low growth temperatures ($T_g<$180$^{\circ}$C)  (c) RHEED pattern of a sample grown at $T_g=$180$^{\circ}$C  The additional spots reveal the presence of Ge$_3$Mn$_5$ clusters at the surface of the film }
\label{fig1}
\end{figure}

\section{Structural properties \label{structural}}

\begin{figure}[htb]
    \center
	\includegraphics[width= 49\linewidth]{ /fig2a eps}
	\includegraphics[width= 49\linewidth]{ /fig2b eps}
	 \includegraphics[width= 49\linewidth]{ /fig2c eps}
	 \includegraphics[width= 49\linewidth]{ /fig2d eps}
    \caption{Transmission electron micrographs of a Ge$_{1-x}$Mn$_{x}$ film grown at 130$^{\circ}$C and containing 6 \% of manganese  (a) cross-section along the [110] axis : we clearly see the presence of nanocolumns elongated along the growth axis  (b) High resolution image of the interface between the Ge$_{1-x}$Mn$_{x}$ film and the Ge buffer layer  The Ge$_{1-x}$Mn$_{x}$ film exhibits the same diamond structure as pure germanium  No defect can be seen which could be caused by the presence of nanocolumns  (c) Plane view micrograph performed on the same sample confirms the columnar structure and gives the density and size distribution of nanocolumns  (d) Mn chemical map obtained by energy filtered transmission electron microcopy (EFTEM)  The background was carefully substracted from pre-edge images  Bright areas correspond to Mn-rich regions }
\label{fig2}
\end{figure}

In samples grown at 130$^{\circ}$C and containing 6 \% Mn, we can observe vertical elongated nanostructures \textit{i e } nanocolumns as shown in Fig  2a  Nanocolumns extend through the whole Ge$_{1-x}$Mn$_{x}$ film thickness  From the high resolution TEM image shown in Fig  2b, we deduce their average diameter around 3 nm  Moreover in Fig  2b, the interface between the Ge buffer layer and the Ge$_{1-x}$Mn$_{x}$ film is flat and no defect propagates from the interface into the film  The Ge$_{1-x}$Mn$_{x}$ film is a perfect single crystal in epitaxial relationship with the substrate  In Fig  2c is shown a plane view micrograph of the same sample confirming the presence of nanocolumns in the film  From this image, we can deduce the size and density of nanocolumns  The nanocolumns density is 13000 $\rm{\mu m}^{-2}$ with a mean diameter of 3 nm which is coherent with cross-section measurements  In order to estimate the chemical composition of these nanocolumns, we further performed chemical mapping using EFTEM  In Fig  2d we show a cross sectional Mn chemical map of the Ge$_{1-x}$Mn$_{x}$ film  This map shows that the formation of nanocolumns is a consequence of Mn segregation  Nanocolumns are Mn rich and the surrounding matrix is Mn poor  However, it is impossible to deduce the Mn concentration in Ge$_{1-x}$Mn$_{x}$ nanocolumns from this cross section  Indeed, in cross section observations, the columns diameter is much smaller than the probed film thickness and the signal comes from the superposititon of the Ge matrix and Mn-rich nanocolumns  In order to quantify Mn concentration inside the nanocolumns and inside the Ge matrix, EELS measurements (not shown here) have been performed in a plane view geometry \cite{Jame06}  These observations revealed that the matrix Mn content is below 1 \% (detection limit of our instrument)  Measuring the surface occupied by the matrix and the nanocolumns in plane view TEM images, and considering the average Mn concentration in the sample (6 \%), we can estimate the Mn concentration in the nanocolumns  The Mn concentration measured by EELS being between 0\% and 1\%, we can conclude that the Mn content in the nanocolumns is between 30 \% and 38 \% \\
For samples grown between 80$^\circ$C and 150$^\circ$C cross section and plane view TEM observations reveal the presence of Mn rich nanocolumns surrounded with a Mn poor Ge matrix  In order to investigate the influence of Mn concentration on the structural properties of Ge$_{1-x}$Mn$_{x}$ films, ten samples have been grown at 100$^\circ$C and at 150$^\circ$C with Mn concentrations of 1 3 \%, 2 3 \%, 4 \%, 7 \% and 11 3 \%  Their structural properties have been investigated by plane view TEM observations  

\begin{figure}[htb]
    \center
    \includegraphics[width= 98\linewidth]{ /fig3a eps}
	\includegraphics[width= 45\linewidth]{ /fig3b eps}
		\includegraphics[width= 45\linewidth]{ /fig3c eps}
    \caption{Nanocolumns size and density as a function of growth conditions  Samples considered have been grown at 100$^{\circ}$C and 150$^{\circ}$C respectively  (a) Mn concentration dependence of the size distribution  (b) columns density as a function of Mn concentration  (c) Volume fraction of the nanocolumns as a function of Mn concentration }
 \label{fig3}
\end{figure}

For samples grown at 100$^\circ$C with Mn concentrations below 5 \% the nanocolumns mean diameter is 1 8$\pm$0 2 nm  The evolution of columns density as a fonction of Mn concentration is reported in figure 3b  By increasing the Mn concentration from 1 3 \% to 4 \% we observe a significant increase of the columns density from 13000 to 30000 $\rm{\mu m}^{-2}$  For Mn concentrations higher than 5 \% the density seems to reach a plateau corresponding to 35000 $\rm{\mu m}^{-2}$ and their diameter slightly increases from 1 8 nm at 4 \% to 2 8 nm at 11 3 \%  By plotting the volume fraction occupied by the columns in the film as a function of Mn concentration, we observe a linear dependence for Mn contents below 5 \%  The non-linear behavior above 5 \% may indicate that the mechanism of Mn incorporation is different in this concentration range, leading to an increase of Mn concentration in the columns or in the matrix  For samples grown at 100$^\circ$C, nanocolumns are always fully coherent with the surrounding matrix (Fig  4a)  

Increasing the Mn content in the samples grown at 150$^\circ$C from 1 3 \% to 11 3 \% leads to a decrease of the columns density (fig 3b)  Moreover, their average diameter increases significantly and size distributions become very broad (see Fig  3a)  For the highest Mn concentration (11 3 \%) we observe the coexistence of very small columns with a diameter of 2 5 nm and very large columns with a diameter of 9 nm  In samples grown at 150$^\circ$C containing 11 3 \% of Mn, the crystalline structure of nanocolumns is also highly modified  In plane view TEM micrographs, one can see columns exhibiting several different crystalline structures  We still observe some columns which are fully coherent with the Ge matrix like in the samples grown at lower temperature  Nevertheless, observations performed on these samples grown at 150$^\circ$C and with 11 3\% Mn reveal some uniaxially \cite{Jame06} or fully relaxed columns exhibiting a misfit of 4 \% between the matrix and the columns and leading to misfit dislocations at the interface between the column and the matrix (see fig  4b)  Thus we can conclude that coherent columns are probably in strong compression and the surrounding matrix in tension  On the same samples (T$_g$=150$^{\circ}$C, 11 3\% Mn), we also observe a large number of highly disordered nanocolumns leading to an amorphous like TEM contrast(fig  4c) 

\begin{figure}[htb]
    \center
   \includegraphics[width= 31\linewidth]{ /fig4a eps}
	\includegraphics[width= 31\linewidth]{ /fig4b eps}
	\includegraphics[width= 31\linewidth]{ /fig4c eps}
    \caption{Plane view high resolution transmission electron micrographs of different types of nanocolumns : (a) typical structure of a column grown at 100$^{\circ}$C  The crystal structure is exactly the same as germanium   (b) Partially relaxed nanocolumn  One can see dislocations at the interface between the columns and the matrix leading to stress relaxation  (c) Amorphous nanocolumn  These columns are typical in samples grown at 150$^{\circ}$C with high Mn contents }
 \label{fig4}
\end{figure}

In conclusion, we have evidenced a complex mechanism of Mn incorporation in Mn doped Ge films grown at low temperature  In particular Mn incorporation is highly inhomogeneous  For very low growth temperatures (below 120$^\circ$C) the diffusion of Mn atoms leads to the formation of Mn rich, vertical nanocolumns  Their density mostly depends on Mn concentration and their mean diameter is about 2 nm  These results can be compared with the theoretical predictions of Fukushima \textit{et al } \cite{Fuku06}: they proposed a model of spinodal decomposition in (Ga,Mn)N and (Zn,Cr)Te based on layer by layer growth conditions and a strong pair attraction between Mn atoms which leads to the formation of nanocolumns  This model may also properly describe the formation of Mn rich nanocolumns in our samples  Layer by layer growth conditions can be deduced from RHEED pattern evolution during growth  For all the samples grown at low temperature, RHEED observations clearly indicate two-dimensional growth  Moreover, Ge/Ge$_{1-x}$Mn$_{x}$/Ge heterostructures have been grown and observed by TEM (see Fig  5)  Ge$_{1-x}$Mn$_{x}$/Ge (as well as Ge/Ge$_{1-x}$Mn$_{x}$) interfaces are very flat and sharp thus confirming a two-dimensional, layer by layer growth mode  Therefore we can assume that the formation of Mn rich nanocolumns is a consequence of 2D-spinodal decomposition 

\begin{figure}[htb]
    \center
	\includegraphics[width= 7\linewidth]{ /fig5 eps}
    \caption{Cross section high resolution micrograph of a Ge/Ge$_{1-x}$Mn$_{x}$/Ge/Ge$_{1-x}$Mn$_{x}$/Ge heterostructure  This sample has been grown at 130 $^{\circ}$C with 6\% Mn  Ge$_{1-x}$Mn$_{x}$ layers are 15 nm thick and Ge spacers 5 nm thick  We clearly see the sharpness of both Ge$_{1-x}$Mn$_{x}$/Ge and Ge/Ge$_{1-x}$Mn$_{x}$ interfaces  Mn segregation leading to the columns formation already takes place in very thin Ge$_{1-x}$Mn$_{x}$ films }
\label{fig5}
\end{figure}

For growth temperatures higher than 160$^\circ$C, cross section TEM and EFTEM observations (not shown here) reveal the coexistence of two Mn-rich phases: nanocolumns and Ge$_{3}$Mn$_{5}$ nanoclusters embedded in the germanium matrix  A typical high resolution TEM image is shown in figure 6  
Ge$_{3}$Mn$_{5}$ clusters are not visible in RHEED patterns for temperatures below 180$^\circ$C  To investigate the nature of these clusters, we performed x-ray diffraction in $\theta-2\theta$ mode  Diffraction scans were acquired on a high resolution diffractometer using the copper K$_\alpha$ radiation and on the GMT station of the BM32 beamline at the European Synchrotron Radiation Facility (ESRF)  Three samples grown at different temperatures and/or annealed at high temperature were investigated  The two first samples are Ge$_{1-x}$Mn$_{x}$ films grown at 130$^\circ$C and 170$^\circ$C respectively  The third one has been grown at 130$^\circ$C and post-growth annealed at 650$^\circ$C  By analysing x-ray diffraction spectra, we can evidence two different crystalline structures  For the sample grown at 130$^\circ$C, the $\theta-2\theta$ scan only reveals the (004) Bragg peak of the germanium crystal, confirming the good epitaxial relationship between the layer and the substrate, and the absence of secondary phases in the film in spite of a high dynamics of the order of 10$^7$  For both samples grown at 170$^\circ$C and annealed at 650$^\circ$C, $\theta-2\theta$ spectra are identical  In addition to the (004) peak of germanium, we observe three additional weak peaks  The first one corresponds to the (002) germanium forbidden peak which probably comes from a small distortion of the germanium crystal, and the two other peaks are respectively attributed to the (002) and (004) Bragg peaks of a secondary phase  The $c$ lattice parameter of Ge$_3$Mn$_5$ hexagonal crystal is 5 053 \AA \ \cite{Fort90} which is in very good agreement with the values obtained from diffraction data for both (002) and (004) lines assuming that the $c$ axis of Ge$_3$Mn$_5$ is along the [001] direction of the Ge substrate 

\begin{figure}[htb]
    \center
	\includegraphics[width= 7\linewidth]{ /fig6 eps}
	\caption{Cross section high resolution transmission electron micrograph of a sample grown at 170$^{\circ}$C  We observe the coexistence of two different Mn-rich phases: Ge$_{1-x}$Mn$_{x}$ nanocolumns and Ge$_3$Mn$_5$ clusters }
\label{fig6}
\end{figure}

In summary, in a wide range of growth temperatures and Mn concentrations, we have evidenced a two-dimensional spinodal decomposition leading to the formation of Mn-rich nanocolumns in Ge$_{1-x}$Mn$_{x}$ films  This decomposition is probably the consequence of: $(i)$ a strong pair attraction between Mn atoms, $(ii)$ a strong surface diffusion of Mn atoms in germanium even at low growth temperatures and $(iii)$ layer by layer growth conditions  We have also investigated the influence of growth parameters on the spinodal decomposition: at low growth temperatures (100$^{\circ}$C), increasing the Mn content leads to higher columns densities while at higher growth temperatures (150$^{\circ}$C), the columns density remains nearly constant whereas their size increases drastically  By plotting the nanocolumns density as a function of Mn content, we have shown that the mechanism of Mn incorporation in Ge changes above 5 \% of Mn  Finally, using TEM observations and x-ray diffraction, we have shown that Ge$_3$Mn$_5$ nanoclusters start to form at growth temperatures higher than 160$^\circ$C 

\section{Magnetic properties \label{magnetic}}

We have thoroughly investigated the magnetic properties of thin Ge$_{1-x}$Mn$_{x}$ films for different growth temperatures and Mn concentrations  In this section, we focus on Mn concentrations between 2 \% and 11 \%  We could clearly identify four different magnetic phases in Ge$_{1-x}$Mn$_{x}$ films : diluted Mn atoms in the germanium matrix, low $T_{C}$ nanocolumns ($T_{C}$ $\leq$ 170 K), high $T_{C}$ nanocolumns ($T_{C}$ $\geq$ 400 K) and Ge$_{3}$Mn$_{5}$ clusters ($T_{C}$ $\thickapprox$ 300 K)  The relative weight of each phase clearly depends on the growth temperature and to a lesser extend on Mn concentration  For low growth temperature ($<$ 120$^{\circ}$C), we show that nanocolumns are actually made of four uncorrelated superparamagnetic nanostructures  Increasing T$_{g}$ above 120$^{\circ}$C, we first obtain continuous columns exhibiting low $T_{C}$ ($<$ 170 K) and high $T_{C}$ ($>$ 400 K) for $T_{g}\approx$130$^{\circ}$C  The larger columns become ferromagnetic \textit{i e } $T_{B}>T_{C}$  Meanwhile Ge$_{3}$Mn$_{5}$ clusters start to form  Finally for higher $T_{g}$, the magnetic contribution from Ge$_{3}$Mn$_{5}$ clusters keeps increasing while the nanocolumns signal progressively disappears 

\begin{figure}[htb]
\center
   \includegraphics[width= 6\linewidth]{ /fig7a eps}
   \includegraphics[width= 3\linewidth]{ /fig7b eps}
\caption{(a) Temperature dependence of the saturation magnetization (in $\mu_{B}$/Mn) of Ge$_{0 93}$Mn$_{0 07}$ samples for different growth temperatures  The magnetic field is applied in the film plane  The inset shows the temperature dependence of a sample grown at 130$^{\circ}$C and annealed at 650$^{\circ}$C for 15 minutes  After annealing, the magnetic signal mostly arises from Ge$_{3}$Mn$_{5}$ clusters  (b) ZFC-FC measurements performed on Ge$_{0 93}$Mn$_{0 07}$ samples for different growth temperatures  The in-plane applied field is 0 015 T  The ZFC peak at low temperature ($\leq$150 K) can be attributed to the superparamagnetic nanocolumns  This peak widens and shifts towards high blocking temperatures when increasing growth temperature  The second peak above 150 K in the ZFC curve which increases with increasing growth temperature is attributed to superparamagnetic Ge$_{3}$Mn$_{5}$ clusters  The increasing ZFC-FC irreversibility at $\approx$ 300 K is due to the increasing contribution from large ferromagnetic Ge$_{3}$Mn$_{5}$ clusters  The nanocolumns signal completely vanishes after annealing at 650$^{\circ}$C for 15 minutes }
\label{fig7}
\end{figure}

In Fig  7a, the saturation magnetization at 2 Tesla in $\mu_{B}$/Mn of Ge$_{1-x}$Mn$_{x}$ films with 7 \% of Mn is plotted as a function of temperature for different growth temperatures ranging from $T_{g}$=90$^{\circ}$C up to 160$^{\circ}$C  The inset shows the temperature dependence of the magnetization at 2 Tesla after annealing at 650$^{\circ}$C during 15 minutes  Figure 7b displays the corresponding Zero Field Cooled - Field Cooled (ZFC-FC) curves recorded at 0 015 Tesla  In the ZFC-FC procedure, the sample is first cooled down to 5 K in zero magnetic field and the susceptibility is subsequently recorded at 0 015 Tesla while increasing the temperature up to 400 K (ZFC curve)  Then, the susceptibility is recorded under the same magnetic field while decreasing the temperature down to 5 K (FC curve)  Three different regimes can be clearly distinguished  \\
For $T_{g}\leq$120$^{\circ}$C, the temperature dependence of the saturation magnetization remains nearly the same while increasing growth temperature  The overall magnetic signal vanishing above 200 K is attributed to the nanocolumns whereas the increasing signal below 50 K originates from diluted Mn atoms in the surrounding matrix  The Mn concentration dependence of the saturation magnetization is displayed in figure 8  For the lowest Mn concentration (4 \%), the contribution from diluted Mn atoms is very high and drops sharply for higher Mn concentrations (7 \%, 9 \% and 11 3 \%)  Therefore the fraction of Mn atoms in the diluted matrix decreases with Mn concentration probably because Mn atoms are more and more incorporated in the nanocolumns  In parallel, the Curie temperature of nanocolumns increases with the Mn concentration reaching 170 K for 11 3 \% of Mn  This behavior may be related to different Mn compositions and to the increasing diameter of nanocolumns (from 1 8 nm to 2 8 nm) as discussed in section \ref{structural} 

\begin{figure}[htb]
\center
   \includegraphics[width= 7\linewidth]{ /fig8 eps}
    \caption{Temperature dependence of the saturation magnetization (in $\mu_{B}$/Mn) of Ge$_{1-x}$Mn$_{x}$ films grown at 100$^{\circ}$C plotted for different Mn concentrations: 4 1 \%; 7 \%; 8 9 \% and 11 3 \% }
\label{fig8}
\end{figure}

ZFC-FC measurements show that the nanocolumns are superparamagnetic  The magnetic signal from the diluted Mn atoms in the matrix is too weak to be detected in susceptibility measurements at low temperature  In samples containing 4 \% of Mn, ZFC and FC curves superimpose down to low temperatures  As we do not observe hysteresis loops at low temperature, we believe that at this Mn concentration nanocolumns are  superparamagnetic in the whole temperature range and the blocking temperature cannot be measured  For higher Mn contents, the ZFC curve exhibits a very narrow peak with a maximum at the blocking temperature of 15 K whatever the Mn concentration and growth temperature (see Fig  7b)  Therefore the anisotropy barrier distribution is narrow and assuming that nanocolumns have the same magnetic anisotropy, this is a consequence of the very narrow size distribution of the nanocolumns as observed by TEM  To probe the anisotropy barrier distribution, we have performed ZFC-FC measurements but instead of warming the sample up to 400 K, we stopped at a lower temperature $T_{0}$  

\begin{figure}[htb]
\center
   \includegraphics[width= 6\linewidth]{ /fig9 eps}
\caption{Schematic drawing of the anisotropy barrier distribution n($E_{B}$) of superparamagnetic nanostructures  If magnetic anisotropy does not depend on the particle size, this distribution exactly reflects their magnetic size distribution  In this drawing the blocking temperature ($T_{B}$) corresponds to the distribution maximum  At a given temperature $T_{0}$ such that 25$k_{B}T_{0}$ falls into the anisotropy barrier distribution, the largest nanostructures with an anisotropy energy larger than 25$k_{B}T_{0}$ are blocked whereas the others are superparamagnetic }
\label{fig9}
\end{figure}

If this temperature falls into the anisotropy barrier distribution as depicted in Fig  9, the FC curve deviates from the ZFC curve  Indeed the smallest nanostructures have become superparamagnetic at $T_{0}$ and when decreasing again the temperature, their magnetization freezes along a direction close to the magnetic field and the FC susceptibility is higher than the ZFC susceptibility  Therefore any irreversibility in this procedure points at the presence of superparamagnetic nanostructures  The results are given in Fig  10a  ZFC and FC curves clearly superimpose up to $T_{0}$=250 K thus the nanocolumns are superparamagnetic up to their Curie temperature and no Ge$_{3}$Mn$_{5}$ clusters could be detected  Moreover for low $T_{0}$ values, a peak appears at low temperature in FC curves which evidences strong antiferromagnetic interactions between the nanocolumns \cite{Chan00} 

\begin{figure}[htb]
\center
    \includegraphics[width= 35\linewidth]{ /fig10a eps}
    \includegraphics[width= 63\linewidth]{ /fig10b eps}
\caption{(a) ZFC-FC measurements performed on a Ge$_{0 887}$Mn$_{0 113}$ sample grown at 115$^{\circ}$C  The in-plane applied field is 0 015 T  Magnetization was recorded up to different T$_{0}$ temperatures: 30 K, 50 K, 100 K, 150 K, 200 K and 250 K  Curves are shifted up for more clarity  (b) ZFC-FC curves for in-plane and out-of-plane applied fields (0 015 T) }
\label{fig10}
\end{figure}

In order to derive the magnetic size and anisotropy of the Mn-rich nanocolumns embedded in the Ge matrix, we have fitted the inverse normalized in-plane (resp  out-of-plane) susceptibility: $\chi_{\parallel}^{-1}$ (resp  $\chi_{\perp}^{-1}$)  The corresponding experimental ZFC-FC curves are reported in Fig  10b  Since susceptibility measurements are performed at low field (0 015 T), the matrix magnetic signal remains negligible  In order to normalize susceptibility data, we need to divide the magnetic moment by the saturated magnetic moment recorded at 5 T  However the matrix magnetic signal becomes very strong at 5 T and low temperature so that we need to subtract it from the saturated magnetic moment using a simple Curie function  From Fig  10b, we can conclude that nanocolumns are isotropic  Therefore to fit experimental data we use the following expression well suited for isotropic systems or cubic anisotropy: $\chi_{\parallel}^{-1}= \chi_{\perp}^{-1}\approx 3k_{B}T/M(T)+\mu_{0}H_{eff}(T)$  $k_{B}$ is the Boltzmann constant, $M=M_{s}v$ is the magnetic moment of a single-domain nanostructure (macrospin approximation) where $M_{s}$ is its magnetization and $v$ its volume  The in-plane magnetic field is applied along $[110]$ or $[-110]$ crystal axes  Since the nanostructures Curie temperature does not exceed 170 K, the temperature dependence of the saturation magnetization is also accounted for by writting $M(T)$  Antiferromagnetic interactions between nanostructures are also considered by adding an effective field estimated in the mean field approximation \cite{Fruc02}: $\mu_{0}H_{eff}(T)$ 
The only fitting parameters are the maximum magnetic moment (\textit{i e } at low temperature) per nanostructure: $M$ (in Bohr magnetons $\mu_{B}$) and the maximum interaction field (\textit{i e } at low temperature): $\mu_{0}H_{eff}$ 

\begin{figure}[htb]
\center
   \includegraphics[width= 7\linewidth]{ /fig11 eps}
\caption{Temperature dependence of the inverse in-plane (open circles) and out-of-plane (open squares) normalized susceptibilities of a Ge$_{0 887}$Mn$_{0 113}$ sample grown at 115$^{\circ}$C  Fits were performed assuming isotropic nanostructures or cubic anisotropy  Dashed line is for in-plane susceptibility and solid line for out-of-plane susceptibility }
\label{fig11}
\end{figure}

In Fig  11, the best fits lead to $M\approx$1250 $\mu_{B}$ and $\mu_{0}H_{eff}\approx$102 mT for in-plane susceptibility and $M\approx$1600 $\mu_{B}$ and $\mu_{0}H_{eff}\approx$98 mT for out-of-plane susceptibility  It gives an average magnetic moment of 1425 $\mu_{B}$ per column and an effective interaction field of 100 mT  Using this magnetic moment and its temperature dependence, magnetization curves could be fitted using a Langevin function and $M(H/T)$ curves superimpose for $T<$100 K  However, from the saturated magnetic moment of the columns and their density (35000 $\rm{\mu m}^{-2}$), we find almost 6000 $\mu_{B}$ per column  Therefore, for low growth temperatures, we need to assume that nanocolumns are actually made of almost four independent elongated magnetic nanostructures  The effective field for antiferromagnetic interactions between nanostructures estimated from the susceptibility fits is at least one order of magnitude larger than what is expected from pure magnetostatic coupling  This difference may be due to either an additional antiferromagnetic coupling through the matrix which origin remains unexplained or to the mean field approximation which is no more valid in this strong coupling regime  As for magnetic anisotropy, the nanostructures behave as isotropic magnetic systems or exhibit a cubic magnetic anisotropy  First we can confirm that nanostructures are not amorphous otherwise shape anisotropy would dominate leading to out-of-plane anisotropy  We can also rule out a random distribution of magnetic easy axes since the nanostructures are clearly crystallized in the diamond structure and would exhibit at least a cubic anisotropy (except if the random distribution of Mn atoms within the nanostructures can yield random easy axes)  Since the nanostructures are in strong in-plane compression (their lattice parameter is larger than the matrix one), the cubic symmetry of the diamond structure is broken and magnetic cubic anisotropy is thus unlikely  We rather believe that out-of-plane shape anisotropy is nearly compensated by in-plane magnetoelastic anisotropy due to compression leading to a \textit{pseudo} cubic anisotropy  From the blocking temperature (15 K) and the magnetic volume of the nanostructures , we can derive their magnetic anisotropy constant using $Kv=25k_{B}T_{B}$: K$\approx$10 kJ m$^{-3}$ which is of the same order of magnitude as shape anisotropy 

\begin{figure}[htb]
\center
    \includegraphics[width= 35\linewidth]{ /fig12a eps}
    \includegraphics[width= 63\linewidth]{ /fig12b eps} 
\caption{(a) ZFC-FC measurements performed on a Ge$_{0 93}$Mn$_{0 07}$ sample grown at 122$^{\circ}$C  The in-plane applied field is 0 015 T  Magnetization was recorded up to different T$_{0}$ temperatures: 50 K, 100 K, 150 K, 200 K and 250 K  Curves are shifted up for more clarity  (b) ZFC-FC curves for in-plane and out-of-plane applied fields  (0 015 T) }
\label{fig12}
\end{figure}

For growth temperatures $T_{g}\geq$120$^{\circ}$C and Mn concentrations $\geq$ 7 \%, samples exhibit a magnetic signal above 200 K corresponding to Ge$_{3}$Mn$_{5}$ clusters (see Fig  7a)  As we can see, SQUID measurements are much more sensitive to the presence of Ge$_{3}$Mn$_{5}$ clusters, even at low concentration, than TEM and x-ray diffraction used in section \ref{structural}  We also observe a sharp transition in the ZFC curve (see Fig  7b, Fig  12a and 12b): the peak becomes very large and is shifted towards high blocking temperatures (the signal is maximum at $T=$23 K)  This can be easily understood as a magnetic percolation of the four independent nanostructures obtained at low growth temperatures into a single magnetic nanocolumn  Therefore the magnetic volume increases sharply as well as blocking temperatures  At the same time, the size distribution widens as observed in TEM  In Fig  12a, we have performed ZFC-FC measurements at different $T_{0}$ temperatures  The ZFC-FC irreversibility is observed up to the Curie temperature of $\approx$120 K meaning that a fraction of nanocolumns is ferromagnetic (\textit{i e } $T_{B}\geq T_{C}$) 
In Fig  12b, in-plane and out-of-plane ZFC curves nearly superimpose for $T\leq$150 K due to the isotropic magnetic behavior of the nanocolumns: in-plane magnetoelastic anisotropy is still compensating out-of-plane shape anisotropy  Moreover the magnetic signal above 150 K corresponding to Ge$_{3}$Mn$_{5}$ clusters that start to form in this growth temperature range is strongly anisotropic  This perpendicular anisotropy confirms the epitaxial relation: (0002) Ge$_{3}$Mn$_{5}$ $\parallel$ (002) Ge discussed in Ref \cite{Bihl06}  The magnetic easy axis of the clusters lies along the hexagonal $c$-axis which is perpendicular to the film plane 

\begin{figure}[ht]
\center
   \includegraphics[width= 35\linewidth]{ /fig13a eps}
   \includegraphics[width= 63\linewidth]{ /fig13b eps} 
\caption{(a) ZFC-FC measurements performed on a Ge$_{0 887}$Mn$_{0 113}$ sample grown at 145$^{\circ}$C  The in-plane applied field is 0 015 T  Magnetization was recorded up to different T$_{0}$ temperatures: 50 K, 100 K, 150 K, 200 K, 250 K and 300 K  Curves are shifted up for more clarity  (b) ZFC-FC curves for in-plane and out-of-plane applied fields (0 015 T) }
\label{fig13}
\end{figure}

For growth temperatures $T_{g}\geq$145$^{\circ}$C the cluster magnetic signal dominates (Fig  13b)  Superparamagnetic nanostructures are investigated performing ZFC-FC measurements at different $T_{0}$ temperatures (Fig  13a)  The first ZFC peak at low temperature \textit{i e } $\leq$ 150 K is attributed to  low-$T_{C}$ nanocolumns ($T_{C}\approx$130 K)  This peak is wider than for lower growth temperatures and its maximum is further shifted up to 30 K  These results are in agreement with TEM observations: increasing $T_{g}$ leads to larger nanocolumns (\textit{i e } higher blocking temperatures) and wider size distributions  ZFC-FC irreversibility is observed up to the Curie temperature due to the presence of ferromagnetic columns  The second peak above 180 K in the ZFC curve is attributed to Ge$_{3}$Mn$_{5}$ clusters and the corresponding ZFC-FC irreversibility persisting up to 300 K means that some clusters are ferromagnetic  We clearly evidence the out-of-plane anisotropy of Ge$_{3}$Mn$_{5}$ clusters and the isotropic magnetic behavior of nanocolumns (Fig  13b)  In this growth temperature range, we have also investigated the Mn concentration dependence of magnetic properties  

\begin{figure}[ht]
\center
    \includegraphics[width= 49\linewidth]{ /fig14a eps}
    \includegraphics[width= 49\linewidth]{ /fig14b eps} 
\caption{Temperature dependence of the saturation magnetization (in $\mu_{B}$/Mn) of Ge$_{1-x}$Mn$_{x}$ films grown at 150$^{\circ}$C plotted for different Mn concentrations: 2 3 \%; 4 \%; 7 \%; 9 \%; 11 3 \%  (b) ZFC-FC measurements performed on Ge$_{1-x}$Mn$_{x}$ films grown at 150$^{\circ}$C  The in-plane applied field is 0 025 T for 2 3 \% and 4 \% and 0 015 T for 8 \% and 11 3 \%  }
\label{fig14}
\end{figure}

In Fig  14a, for low Mn concentrations (2 3 \% and 4 \%) the contribution from diluted Mn atoms in the germanium matrix to the saturation magnetization is very high and nearly vanishes for higher Mn concentrations (7 \%, 9 \% and 13 \%) as observed for low growth temperatures  Above 7 \%, the magnetic signal mainly comes from nanocolumns and Ge$_{3}$Mn$_{5}$ clusters  We can derive more information from ZFC-FC measurements (Fig  14b)  Indeed, for 2 3 \% of Mn, ZFC and FC curves nearly superimpose down to low temperature meaning that nanocolumns are superparamagnetic in the whole temperature range  Moreover the weak irreversibility arising at 300 K  means that some Ge$_{3}$Mn$_{5}$ clusters have already formed in the samples even at very low Mn concentrations  For 4 \% of Mn, we can observe a peak with a maximum at the blocking temperature (12 K) in the ZFC curve  We can also derive the Curie temperature of nanocolumns: $\approx$45 K  The irresversibility arising at 300 K still comes from Ge$_{3}$Mn$_{5}$ clusters  Increasing the Mn concentration above 7 \% leads to: higher blocking temperatures (20 K and 30 K) due to larger nanocolumns and wider ZFC peaks due to wider size distributions in agreement with TEM observations (see Fig  3a)  Curie temperatures also increase (110 K and 130 K) as well as the contribution from Ge$_{3}$Mn$_{5}$ clusters \\
Finally when increasing $T_{g}$ above 160$^{\circ}$C, the nanocolumns magnetic signal vanishes and only Ge$_{3}$Mn$_{5}$ clusters and diluted Mn atoms coexist  The overall magnetic signal becomes comparable to the one measured on annealed samples in which only Ge$_{3}$Mn$_{5}$ clusters are observed by TEM (see Fig  7a) \\
The magnetic properties of high-$T_{C}$ nanocolumns obtained for $T_{g}$ close to 130$^{\circ}$C are discussed in detail in Ref \cite{Jame06} \\
In conclusion, at low growth temperatures ($T_{g}\leq$120$^{\circ}$C), nanocolumns are made of almost 4 independent elongated magnetic nanostructures  For $T_{g}\geq$120$^{\circ}$C, these independent nanostructures percolate into a single nanocolumn sharply leading to higher blocking temperatures  Increasing $T_{g}$ leads to larger columns with a wider size distribution as evidenced by ZFC-FC measurements and given by TEM observations  In parallel, some Ge$_{3}$Mn$_{5}$ clusters start to form and their contribution increases when increasing $T_{g}$  Results on magnetic anisotropy seems counter-intuitive  Indeed Ge$_{3}$Mn$_{5}$ clusters exhibit strong out-of-plane anisotropy whereas nanocolumns which are highly elongated magnetic structures are almost isotropic  This effect is probably due to compensating in-plane magnetoelastic coupling (due to the columns compression) and out-of-plane shape anisotropy  

\section{Conclusion}

In this paper, we have investigated the structural and magnetic properties of thin Ge$_{1-x}$Mn$_{x}$ films grown by low temperature molecular beam epitaxy  A wide range of growth temperatures and Mn concentrations have been explored  All the samples contain Mn-rich nanocolumns as a consequence of 2D-spinodal decomposition  However their size, crystalline structure and magnetic properties depend on growth temperature and Mn concentration  For low growth temperatures, nanocolumns are very small (their diameter ranges between 1 8 nm for 1 3 \% of Mn and 2 8 nm for 11 3 \% of Mn), their Curie temperature is rather low ($<$ 170 K) and they behave as almost four uncorrelated superparamagnetic nanostructures  Increasing Mn concentration leads to higher columns densities while diameters remain nearly unchanged  For higher growth temperatures, the nanocolumns mean diameter increases and their size distribution widens  Moreover the 4 independent magnetic nanostructures percolate into a single magnetic nanocolumn  Some columns are ferromagnetic even if Curie temperatures remain quite low  In this regime, increasing Mn concentration leads to larger columns while their density remains nearly the same  In parallel, Ge$_{3}$Mn$_{5}$ nanoclusters start to form in the film with their $c$-axis perpendicular to the film plane  In both temperature regimes, the Mn incorporation mechanism in the nanocolumns and/or in the matrix changes above 5 \% of Mn and nanocolumns exhibit an isotropic magnetic behaviour due to the competing effects of out-of-plane shape anisotropy and in-plane magnetoelastic coupling  Finally for a narrow range of growth temperatures around 130$^{\circ}$C, nanocolumns exhibit Curie temperatures higher than 400 K  Our goal is now to investigate the crystalline structure inside the nanocolumns, in particular the position of Mn atoms in the distorted diamond structure, which is essential to understand magnetic and future transport properties in Ge$_{1-x}$Mn$_{x}$ films 

\section{Aknowledgements}
The authors would like to thank Dr  F  Rieutord for grazing incidence x-ray diffraction measurements performed on the GMT station of BM32 beamline at the European Synchrotron Radiation Facility 


Context 2: Brooksley Elizabeth Born (born August 27, 1940) is an American attorney and former public official who, from August 26, 1996, to June 1, 1999, was chair of the Commodity Futures Trading Commission (CFTC), the federal agency which oversees the U S  futures and commodity options markets  During her tenure on the CFTC, Born lobbied Congress and the President to give the CFTC oversight of off-exchange markets for derivatives, in addition to its role with respect to exchange-traded derivatives, but her warnings were ignored or dismissed, and her calls for reform resisted by other regulators <ref name="nytimes">Goodman, Peter S  The Reckoning - Taking Hard New Look at a Greenspan Legacy, The New York Times, October 9, 2008 </ref> Born resigned as chairperson on June 1, 1999, shortly after Congress passed legislation prohibiting her agency from regulating derivatives 

Early life and education
Born graduated from Abraham Lincoln High School (San Francisco, California) at the age of 16  She then attended Stanford University, where she majored in English and was graduated with the class of 1961   She initially wanted to become a doctor, but a guidance counsellor at Stanford advised her against medicine, so she majored in English literature instead 

She then attended Stanford Law School, one of only seven women in her class   She was the first female student ever to be named president of the Stanford Law Review  She received the "Outstanding Senior" award and graduated as valedictorian of the class of 1964 

Legal career
Immediately after law school Born was selected as a law clerk to judge Henry Edgerton of the U S  Court of Appeals for the District of Columbia Circuit  It was during this time that she met her first husband, Jacob C  Landau, who was a journalist covering the Federal courts at the time  Following her clerkship, she became an associate at the Washington, D C -based international law firm of Arnold & Porter  Born was attracted to Arnold & Porter because it was one of the few major law firms to have a woman partner at that time, Carolyn Agger, who was the head of the tax practice  Born took a two-year leave of absence from Arnold & Porter to accompany her first husband to Boston, where he had received a fellowship  During that time she worked as a research assistant to law professor Alan Dershowitz 

Born's early career at Arnold & Porter focused on international trade law, in which she represented a number of Swiss industries and the government of Switzerland  She developed a practice representing clients in numerous complex litigation and arbitration cases involving financial market transactions  Among her high-profile cases was the matter of the Hunt Brothers attempt to corner the market in silver in the 1970s  She made partner at Arnold & Porter, after moving to a three-day schedule to help raise her second child, and eventually rose to be the head of the firm's derivatives practice 

Born was among the first female attorneys to systematically address inequities regarding how the laws treated women  Born and another female lawyer, Marna Tucker, taught what is considered to have been the first "Women and the Law" course at Catholic University’s Columbus School of Law  The class exclusively concerned prejudicial treatment of women under the laws of the United States, past and present   Born and Tucker were surprised to discover that there was no textbook on the issue at the time   Born is also one of the co-founders of the National Women's Law Center  Born also helped rewrite the American Bar Association rules to make it possible for more women and minorities to sit on federal bench 

During her long legal career, and into her retirement, Born did much pro bono and other types of volunteer work  She was active in the American Bar Association, the largest professional organization of lawyers in the United States   Initially Born was named a member of the governing council of the ABA's Individual Rights Section, eventually becoming chairperson   Born and Tucker founded the ABA Women's Caucus, the first organization of female lawyers in the ABA   She held several other senior positions in the ABA, including being named the first woman member of the ABA's Standing Committee on the Federal Judiciary   As a member of the Judiciary Committee, Born provided testimony and opinion on persons nominated for federal judgeships  In 1980 she was named chair of the committee   As chair of the committee, Born was invited to address the U S  Congress regarding the nomination of Judge Sandra Day O'Connor to the U S  Supreme Court 

In 1993, Born's name was floated as a possible candidate for Attorney General of the United States, but Janet Reno was nominated 

In July 2009, Nancy Pelosi appointed Brooksley Born as a commissioner to the Financial Crisis Inquiry Commission (FCIC) 

Born and the OTC derivatives market
Born was appointed to the CFTC on April 15, 1994, by President Bill Clinton  Due to litigation against Bankers Trust Company by Procter and Gamble and other corporate clients, Born and her team at the CFTC sought comments on the regulation of over-the-counter derivatives, a first step in the process of writing CFTC regulations to supplement the existing regulations of the Federal Reserve System,  the Options Clearing Corporation, and the National Association of Insurance Commissioners  Born was particularly concerned about swaps, financial instruments that are traded over the counter between banks, insurance companies or other funds or companies, and thus have no transparency except to the two counterparties and the counterparties' regulators, if any   CFTC regulation was strenuously opposed by Federal Reserve chairman Alan Greenspan, and by Treasury Secretaries Robert Rubin and Lawrence Summers  On May 7, 1998, former SEC Chairman Arthur Levitt joined Rubin and Greenspan in objecting to the issuance of the CFTC's concept release  Their response dismissed Born's analysis and focused on the hypothetical possibility that CFTC regulation of swaps and other OTC derivative instruments could create a "legal uncertainty" regarding such financial instruments,  hypothetically reducing the value of the instruments  They argued that the imposition of regulatory costs would "stifle financial innovation" and encourage financial capital to transfer its  transactions offshore  The disagreement between Born and the Executive Office's top economic policy advisors has been described not only as a classic Washington turf war, but also a war of ideologies,  insofar as it is possible to argue that Born's actions were consistent with Keynesian and neoclassical economics while Greenspan, Rubin, Levitt, and Summers consistently espoused neoliberal, and neoconservative policies 

In 1998, a trillion-dollar hedge fund called Long Term Capital Management (LTCM) was near collapse   Using mathematical models to calculate debt risk, LTCM used derivatives to leverage $5 billion into more than $1 trillion, doing business with fifteen of Wall Street's largest financial institutions   The derivative transactions were not regulated, nor were investors able to evaluate LTCM's exposures   Born stated, "I thought that LTCM was exactly what I had been worried about"   In the last weekend of September 1998, the President's working group was told that the entire American economy hung in the balance   After intervention by the Federal Reserve, the crisis was averted   In congressional hearings into the crisis, Greenspan acknowledged that language had been introduced into an agriculture bill that would prevent CFTC from regulating the derivatives which were at the center of the crisis that threatened the US economy   U S  Representative Maurice Hinchey (D-NY) asked "How many more failures do you think we'd have to have before some regulation in this area might be appropriate?"  In response, Greenspan brushed aside the substance of Born's warnings with the simple assertion that "the degree of supervision of regulation of the over-the-counter derivatives market is quite adequate to maintain a degree of stability in the system"  Born's warning was that there wasn't any regulation of them   Born's chief of staff, Michael Greenberger summed up Greenspan's position this way: "Greenspan didn't believe that fraud was something that needed to be enforced, and he assumed she probably did  And of course, she did "  Under heavy pressure from the financial lobby, legislation prohibiting regulation of derivatives by Born's agency was passed by the Congress   Born resigned on June 1, 1999 

The derivatives market continued to grow yearly throughout both terms of George W  Bush's administration  On September 15, 2008, the bankruptcy of Lehman Brothers forced a broad recognition of a financial crisis in both the US and world capital markets   As Lehman Brothers' failure temporarily reduced financial capital's confidence, a number of newspaper articles and television programs suggested that the failure's possible causes included the conflict between the CFTC and the other regulators Faiola, Anthony, Nakashima, Ellen and Drew, Jill  The Crash: Risk and Regulation - What Went Wrong, The Washington Post, October 15, 2008 

Born declined to publicly comment on the unfolding 2008 crisis until March 2009, when she said: "The market grew so enormously, with so little oversight and regulation, that it made the financial crisis much deeper and more pervasive than it otherwise would have been " She also lamented the influence of Wall Street lobbyists on the process and the refusal of regulators to discuss even modest reforms 

An October 2009 Frontline documentary titled "The Warning"  described Born's thwarted efforts to regulate and bring transparency to the derivatives market, and the continuing opposition thereto  The program concluded with an excerpted interview with Born sounding another warning: "I think we will have continuing danger from these markets and that we will have repeats of the financial crisis -- may differ in details but there will be significant financial downturns and disasters attributed to this regulatory gap, over and over, until we learn from experience "

In 2009 Born, along with Sheila Bair of the FDIC, was awarded the John F  Kennedy Profiles in Courage Award in recognition of the "political courage she demonstrated in sounding early warnings about conditions that contributed" to the 2007-08 financial crisis   According to Caroline Kennedy, "Brooksley Born recognized that the financial security of all Americans was being put at risk by the greed, negligence and opposition of  powerful and well connected interests     The catastrophic financial events of recent months have  proved them [Born and Sheila Bair] right "  One member of the President's working group had a change of heart about Brooksley Born   SEC Chairman Arthur Levitt stated "I've come to know her as one of the most capable, dedicated, intelligent and committed public servants that I have ever come to know", adding that "I could have done much better  I could have made a difference" in response to her warnings 

In 2010, a documentary film Inside Job further alleged that derivatives regulation was ineffective from the Clinton administration on  Along with fellow whistleblower, former IMF Chief Economist Raghuram Rajan, who was also scorned by the economic establishment, Brooksley Born was cited as one of the authorities arguing that financial derivatives increase economic risk 

 Personal life 
Born is married to Alexander E  Bennett (also retired from Arnold & Porter)   She has five adult children - two from a previous marriage to Jacob Landau and three stepchildren  Notably, Born was named a partner at Arnold & Porter while working part-time so she could raise her two young children   When both of her children were school-age, Born returned to practice full-time 

References

External links
Attorney profile at Arnold & Porter
Brooksley Born (2009 Winner) of the Profiles in Courage Award, with acceptance speech transcript and NECN video

Profile at MarketsWiki
Speeches and statements
"Testimony Of Brooksley Born Chairperson of the CFTC Concerning The Over-The-Counter Derivatives Market", before the House Committee On Banking And Financial Services, July 24, 1998 
"The Lessons of Long Term Capital Management L P ", Remarks of Brooksley Born, Chairperson of the CFTC, Chicago-Kent-IIT Commodities Law Institute, Chicago, Illinois, October 15, 1998 
 Interview: Brooksley Born for "PBS Frontline: The Warning", PBS, (streaming VIDEO 1 hour), October 20, 2009 
Articles
Manuel Roig-Franzia  "Credit Crisis Cassandra:Brooksley Born's Unheeded Warning Is a Rueful Echo 10 Years On", The Washington Post, May 26, 2009
 Taibbi, Matt  "The Great American Bubble Machine", Rolling Stone'', July 9–23, 2009

1940 births
American women lawyers
Arnold & Porter people
Clinton administration personnel
Columbus School of Law faculty
Commodity Futures Trading Commission personnel
Heads of United States federal agencies
Lawyers from San Francisco
Living people
Stanford Law School alumni
21st-century American women
Stanford University alumni



ground_truth: Spin injection into non magnetic semiconductors, or electrical manipulation of carrier induced magnetism in magnetic semiconductors.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What award did Brooksley Born receive in 2009?

Read the following text and answer briefly.

Context 4: \section{Introduction}\label{S1}

The multiple access interferences (MAI) is the root of user
limitation in CDMA systems \cite{R1,R3}  The parallel least mean
square-partial parallel interference cancelation (PLMS-PPIC) method
is a multiuser detector for code division multiple access (CDMA)
receivers which reduces the effect of MAI in bit detection  In this
method and similar to its former versions like LMS-PPIC \cite{R5}
(see also \cite{RR5}), a weighted value of the MAI of other users is
subtracted before making the decision for a specific user in
different stages \cite{cohpaper}  In both of these methods, the
normalized least mean square (NLMS) algorithm is engaged
\cite{Haykin96}  The $m^{\rm th}$ element of the weight vector in
each stage is the true transmitted binary value of the $m^{\rm th}$
user divided by its hard estimate value from the previous stage  The
magnitude of all weight elements in all stages are equal to unity 
Unlike the LMS-PPIC, the PLMS-PPIC method tries to keep this
property in each iteration by using a set of NLMS algorithms with
different step-sizes instead of one NLMS algorithm used in LMS-PPIC 
In each iteration, the parameter estimate of the NLMS algorithm is
chosen whose element magnitudes of cancelation weight estimate have
the best match with unity  In PLMS-PPIC implementation it is assumed
that the receiver knows the phases of all user channels  However in
practice, these phases are not known and should be estimated  In
this paper we improve the PLMS-PPIC procedure \cite{cohpaper} in
such a way that when there is only a partial information of the
channel phases, this modified version simultaneously estimates the
phases and the cancelation weights  The partial information is the
quarter of each channel phase in $(0,2\pi)$ 

The rest of the paper is organized as follows: In section \ref{S4}
the modified version of PLMS-PPIC with capability of channel phase
estimation is introduced  In section \ref{S5} some simulation
examples illustrate the results of the proposed method  Finally the
paper is concluded in section \ref{S6} 

\section{Multistage Parallel Interference Cancelation: Modified PLMS-PPIC Method}\label{S4}

We assume $M$ users synchronously send their symbols
$\alpha_1,\alpha_2,\cdots,\alpha_M$ via a base-band CDMA
transmission system where $\alpha_m\in\{-1,1\}$  The $m^{th}$ user
has its own code $p_m( )$ of length $N$, where $p_m(n)\in \{-1,1\}$,
for all $n$  It means that for each symbol $N$ bits are transmitted
by each user and the processing gain is equal to $N$  At the
receiver we assume that perfect power control scheme is applied 
Without loss of generality, we also assume that the power gains of
all channels are equal to unity and users' channels do not change
during each symbol transmission (it can change from one symbol
transmission to the next one) and the channel phase $\phi_m$ of
$m^{th}$ user is unknown for all $m=1,2,\cdots,M$ (see
\cite{cohpaper} for coherent transmission)  According to the above
assumptions the received signal is
\begin{equation}
\label{e1} r(n)=\sum\limits_{m=1}^{M}\alpha_m
e^{j\phi_m}p_m(n)+v(n),~~~~n=1,2,\cdots,N,
\end{equation}
where $v(n)$ is the additive white Gaussian noise with zero mean and
variance $\sigma^2$  Multistage parallel interference cancelation
method uses $\alpha^{s-1}_1,\alpha^{s-1}_2,\cdots,\alpha^{s-1}_M$,
the bit estimates outputs of the previous stage, $s-1$, to estimate
the related MAI of each user  It then subtracts it from the received
signal $r(n)$ and makes a new decision on each user variable
individually to make a new variable set
$\alpha^{s}_1,\alpha^{s}_2,\cdots,\alpha^{s}_M$ for the current
stage $s$  Usually the variable set of the first stage (stage $0$)
is the output of a conventional detector  The output of the last
stage is considered as the final estimate of transmitted bits  In
the following we explain the structure of a modified version of the
PLMS-PIC method \cite{cohpaper} with simultaneous capability of
estimating the cancelation weights and the channel phases 

Assume $\alpha_m^{(s-1)}\in\{-1,1\}$ is a given estimate of
$\alpha_m$ from stage $s-1$  Define
\begin{equation}
\label{e6} w^s_{m}=\frac{\alpha_m}{\alpha_m^{(s-1)}}e^{j\phi_m} 
\end{equation}
From (\ref{e1}) and (\ref{e6}) we have
\begin{equation}
\label{e7} r(n)=\sum\limits_{m=1}^{M}w^s_m\alpha^{(s-1)}_m
p_m(n)+v(n) 
\end{equation}
Define
\begin{subequations}
\begin{eqnarray}
\label{e8} W^s&=&[w^s_{1},w^s_{2},\cdots,w^s_{M}]^T,\\
\label{e9}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!X^{s}(n)\!\!\!&=&\!\!\![\alpha^{(s-1)}_1p_1(n),\alpha^{(s-1)}_2p_2(n),\cdots,\alpha^{(s-1)}_Mp_M(n)]^T 
\end{eqnarray}
\end{subequations}
where $T$ stands for transposition  From equations (\ref{e7}),
(\ref{e8}) and (\ref{e9}), we have
\begin{equation}
\label{e10} r(n)=W^{s^T}X^{s}(n)+v(n) 
\end{equation}
Given the observations $\{r(n),X^{s}(n)\}^{N}_{n=1}$, in modified
PLMS-PPIC, like the PLMS-PPIC \cite{cohpaper}, a set of NLMS
adaptive algorithm are used to compute
\begin{equation}
\label{te1} W^{s}(N)=[w^{s}_1(N),w^{s}_2(N),\cdots,w^{s}_M(N)]^T,
\end{equation}
which is an estimate of $W^s$ after iteration $N$  To do so, from
(\ref{e6}), we have
\begin{equation}
\label{e13} |w^s_{m}|=1 ~~~m=1,2,\cdots,M,
\end{equation}
which is equivalent to
\begin{equation}
\label{e14} \sum\limits_{m=1}^{M}||w^s_{m}|-1|=0 
\end{equation}
We divide $\Psi=\left(0,1-\sqrt{\frac{M-1}{M}}\right]$, a sharp
range for $\mu$ (the step-size of the NLMS algorithm) given in
\cite{sg2005}, into $L$ subintervals and consider $L$ individual
step-sizes $\Theta=\{\mu_1,\mu_2,\cdots,\mu_L\}$, where
$\mu_1=\frac{1-\sqrt{\frac{M-1}{M}}}{L}, \mu_2=2\mu_1,\cdots$, and
$\mu_L=L\mu_1$  In each stage, $L$ individual NLMS algorithms are
executed ($\mu_l$ is the step-size of the $l^{th}$ algorithm)  In
stage $s$ and at iteration $n$, if
$W^{s}_k(n)=[w^s_{1,k},\cdots,w^s_{M,k}]^T$, the parameter estimate
of the $k^{\rm th}$ algorithm, minimizes our criteria, then it is
considered as the parameter estimate at time iteration $n$  In other
words if the next equation holds
\begin{equation}
\label{e17} W^s_k(n)=\arg\min\limits_{W^s_l(n)\in I_{W^s}
}\left\{\sum\limits_{m=1}^{M}||w^s_{m,l}(n)|-1|\right\},
\end{equation}
where $W^{s}_l(n)=W^{s}(n-1)+\mu_l \frac{X^s(n)}{\|X^s(n)\|^2}e(n),
~~~ l=1,2,\cdots,k,\cdots,L-1,L$ and
$I_{W^s}=\{W^s_1(n),\cdots,W^s_L(n)\}$, then we have
$W^s(n)=W^s_k(n)$, and therefore all other algorithms replace their
weight estimate by $W^{s}_k(n)$  At time instant $n=N$, this
procedure gives $W^s(N)$, the final estimate of $W^s$, as the true
parameter of stage $s$ 

Now consider $R=(0,2\pi)$ and divide it into four equal parts
$R_1=(0,\frac{\pi}{2})$, $R_2=(\frac{\pi}{2},\pi)$,
$R_3=(\pi,\frac{3\pi}{2})$ and $R_4=(\frac{3\pi}{2},2\pi)$  The
partial information of channel phases (given by the receiver) is in
a way that it shows each $\phi_m$ ($m=1,2,\cdots,M$) belongs to
which one of the four quarters $R_i,~i=1,2,3,4$  Assume
$W^{s}(N)=[w^{s}_1(N),w^{s}_2(N),\cdots,w^{s}_M(N)]^T$ is the weight
estimate of the modified algorithm PLMS-PPIC at time instant $N$ of
the stage $s$  From equation (\ref{e6}) we have
\begin{equation}
\label{tt3}
\phi_m=\angle({\frac{\alpha^{(s-1)}_m}{\alpha_m}w^s_m}) 
\end{equation}
We estimate $\phi_m$ by $\hat{\phi}^s_m$, where
\begin{equation}
\label{ee3}
\hat{\phi}^s_m=\angle{(\frac{\alpha^{(s-1)}_m}{\alpha_m}w^s_m(N))} 
\end{equation}
Because $\frac{\alpha^{(s-1)}_m}{\alpha_m}=1$ or $-1$, we have
\begin{eqnarray}
\hat{\phi}^s_m=\left\{\begin{array}{ll} \angle{w^s_m(N)} &
\mbox{if}~
\frac{\alpha^{(s-1)}_m}{\alpha_m}=1\\
\pm\pi+\angle{w^s_m(N)} & \mbox{if}~
\frac{\alpha^{(s-1)}_m}{\alpha_m}=-1\end{array}\right 
\end{eqnarray}
Hence $\hat{\phi}^s_m\in P^s=\{\angle{w^s_m(N)},
\angle{w^s_m(N)+\pi, \angle{w^s_m(N)}-\pi}\}$  If $w^s_m(N)$
sufficiently converges to its true value $w^s_m$, the same region
for $\hat{\phi}^s_m$ and $\phi_m$ is expected  In this case only one
of the three members of $P^s$ has the same region as $\phi_m$  For
example if $\phi_m \in (0,\frac{\pi}{2})$, then $\hat{\phi}^s_m \in
(0,\frac{\pi}{2})$ and therefore only $\angle{w^s_m(N)}$ or
$\angle{w^s_m(N)}+\pi$ or $\angle{w^s_m(N)}-\pi$ belongs to
$(0,\frac{\pi}{2})$  If, for example, $\angle{w^s_m(N)}+\pi$ is such
a member between all three members of $P^s$, it is the best
candidate for phase estimation  In other words,
\[\phi_m\approx\hat{\phi}^s_m=\angle{w^s_m(N)}+\pi \]
We admit that when there is a member of $P^s$ in the quarter of
$\phi_m$, then $w^s_m(N)$ converges  What would happen when non of
the members of $P^s$ has the same quarter as $\phi_m$? This
situation will happen when the absolute difference between $\angle
w^s_m(N)$ and $\phi_m$ is greater than $\pi$  It means that
$w^s_m(N)$ has not converged yet  In this case where we can not
count on $w^s_m(N)$, the expected value is the optimum choice for
the channel phase estimation, e g  if $\phi_m \in (0,\frac{\pi}{2})$
then $\frac{\pi}{4}$ is the estimation of the channel phase
$\phi_m$, or if $\phi_m \in (\frac{\pi}{2},\pi)$ then
$\frac{3\pi}{4}$ is the estimation of the channel phase $\phi_m$ 
The results of the above discussion are summarized in the next
equation
\begin{eqnarray}
\nonumber \hat{\phi}^s_m = \left\{\begin{array}{llll} \angle
{w^s_m(N)} & \mbox{if}~
\angle{w^s_m(N)}, \phi_m\in R_i,~~i=1,2,3,4\\
\angle{w^s_m(N)}+\pi & \mbox{if}~ \angle{w^s_m(N)}+\pi, \phi_m\in
R_i,~~i=1,2,3,4\\
\angle{w^n_m(N)}-\pi & \mbox{if}~ \angle{w^s_m(N)}-\pi, \phi_m\in
R_i,~~i=1,2,3,4\\
\frac{(i-1)\pi+i\pi}{4} & \mbox{if}~ \phi_m\in
R_i,~~\angle{w^s_m(N)},\angle
{w^s_m(N)}\pm\pi\notin R_i,~~i=1,2,3,4 \\
\end{array}\right 
\end{eqnarray}
Having an estimation of the channel phases, the rest of the proposed
method is given by estimating $\alpha^{s}_m$ as follows:
\begin{equation}
\label{tt4}
\alpha^{s}_m=\mbox{sign}\left\{\mbox{real}\left\{\sum\limits_{n=1}^{N}
q^s_m(n)e^{-j\hat{\phi}^s_m}p_m(n)\right\}\right\},
\end{equation}
where
\begin{equation} \label{tt5}
q^{s}_{m}(n)=r(n)-\sum\limits_{m^{'}=1,m^{'}\ne
m}^{M}w^{s}_{m^{'}}(N)\alpha^{(s-1)}_{m^{'}} p_{m^{'}}(n) 
\end{equation}
The inputs of the first stage $\{\alpha^{0}_m\}_{m=1}^M$ (needed for
computing $X^1(n)$) are given by
\begin{equation}
\label{qte5}
\alpha^{0}_m=\mbox{sign}\left\{\mbox{real}\left\{\sum\limits_{n=1}^{N}
r(n)e^{-j\hat{\phi}^0_m}p_m(n)\right\}\right\}



ground_truth: In 2009, Brooksley Born received the John F. Kennedy Profiles in Courage Award.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What kind of ultracold neutral plasmas does this study focus on?

Read the following text and answer briefly.

Context 3: \section{Introduction}

Ultracold neutral plasmas studied in the laboratory offer access to a regime of plasma physics that scales to describe thermodynamic aspects of important high-energy-density systems, including strongly coupled astrophysical plasmas \cite{VanHorn,Burrows}, as well as terrestrial sources of neutrons \cite{Hinton,Ichimaru_fusion,Atzeni,Boozer} and x-ray radiation \cite{Rousse,Esarey}   Yet, under certain conditions, low-temperature laboratory plasmas evolve with dynamics that are governed by the quantum mechanical properties of their constituent particles, and in some cases by coherence with an external electromagnetic field    

The relevance of ultracold plasmas to such a broad scope of problems in classical and quantum many-body physics has given rise to a great deal of experimental and theoretical research on these systems since their discovery in the late 90s   A series of reviews affords a good overview of progress in the last twenty years \cite{Gallagher,Killian_Science,PhysRept,Lyon}   Here, we focus on the subset of ultracold neutral plasmas that form via kinetic rate processes from state-selected Rydberg gases, and emphasize in particular the distinctive dynamics found in the evolution of molecular ultracold plasmas   

While molecular beam investigations of threshold photoionization spectroscopy had uncovered relevant effects a few years earlier \cite{Scherzer,Alt}, the field of ultracold plasma physics began in earnest with the 1999 experiment of Rolston and coworkers on metastable xenon atoms cooled in a magneto optical trap (MOT) \cite{Killian}   

This work and many subsequent efforts tuned the photoionization energy as a means to form a plasma of very low electron temperature built on a strongly coupled cloud of ultracold ions   Experiment and theory soon established that fast processes associated with disorder-induced heating and longer-time electron-ion collisional rate processes act to elevate the ion temperatures to around one degree Kelvin, and constrain the effective initial electron temperature to a range above 30 K \cite{Kuzmin,Hanson,Laha}   

This apparent limit on the thermal energy of the electrons can be more universally expressed for an expanding plasma by saying that the electron correlation parameter, $\Gamma_e$, does not exceed 0 25, where, 
\begin{equation}
\Gamma_e = \frac{e^2}{4\pi \epsilon_0 a_{ws}}\frac{1}{k_B T_e}
\label{eqn:gamma_e}
\end{equation}
defines the ratio of the average unscreened electron-electron potential energy to the electron kinetic energy   $a_{ws}$ is the Wigner-Seitz radius, related to the electron density by, $\rho_e = 1/(\frac{4}{3} \pi a_{ws}^3)$   These plasmas of weakly coupled electrons and strongly coupled ions have provided an important testing ground for ion transport theory and the study of electron-ion collision physics \cite{Strickler} 

Soon after the initial reports of ultracold plasmas formed by direct photoionization, a parallel effort began with emphasis on the plasma that forms spontaneously by Penning ionization and electron-impact avalanche in a dense ultracold Rydberg gas \cite{Mourachko}   This process affords less apparent control of the initial electron temperature   But, pulsed field-ionization measurements soon established that the photoionized plasma and that formed by the avalanche of a Rydberg gas both evolve to quasi-equilibria of electrons, ions and high-Rydberg neutrals \cite{Rolston_expand,Gallagher}   

Early efforts to understand plasmas formed by Rydberg gas avalanche paid particular attention to the process of initiation   Evolution to plasma in effusive atomic beams was long known for high-Rydberg gases of caesium and well explained by coupled rate equations \cite{Vitrant}   But, low densities and ultracold velocity distributions were thought to exclude Rydberg-Rydberg collisional mechanisms in a MOT   

In work on ultracold Rydberg gases of Rb and Cs, Gallagher, Pillet and coworkers describe the initial growth of electron signal by a model that includes ionization by blackbody radiation and collisions with a background of uncooled Rydberg atoms \cite{Mourachko,Gallagher,Li,Comparat,Tanner}  This picture was subsequently refined to include many-body excitation and autoionization, as well as attractive dipole-dipole interactions \cite{Viteau,Pillet}, later confirmed by experiments at Rice \cite{Mcquillen}   

The Orsay group also studied the effect of adding Rydberg atoms to an established ultracold plasma   They found that electron collisions in this environment completely ionize added atoms, even when selected to have deep binding energies \cite{Vanhaecke}   They concluded from estimates of electron trapping efficiency that the addition of Rydberg atoms does not significantly alter the electron temperature of the plasma   

Tuning pair distributions by varying the wavelength of the excitation laser, Weidem\"uller and coworkers confirmed the mechanical effects of van der Waals interactions on the rates of Penning ionization in ultracold $^{87}$Rb Rydberg gases \cite{Amthor_mech}   They recognized blackbody radiation as a possible means of final-state redistribution, and extended this mechanical picture to include long-range repulsive interactions \cite{Amthor_model}   This group later studied the effects of spatial correlations in the spontaneous avalanche of Rydberg gases in a regime of strong blockade, suggesting a persistence of initial spatial correlations \cite{RobertdeSaintVincent}   

Robicheaux and coworkers have recently investigated the question of prompt many-body ionization from the point of view of Monte Carlo classical trajectory calculations \cite{Goforth}   For atoms on a regular or random grid driven classically by an electromagnetic field, they find that many-body excitation enhances prompt ionization by about twenty percent for densities greater than $5 6 \times 10^{-3}/(n_0^2 a_0)^3$, where $n_0$ is the principal quantum number of the Rydberg gas and $a_0$ is the Bohr radius   They observed that density fluctuations (sampled from the distribution of nearest neighbour distances) have a greater effect, and point to the possible additional influence of secondary electron-Rydberg collisions and the Penning production of fast atoms not considered by the model, but already observed by Raithel and coworkers \cite{Knuffman}   

The Raithel group also found direct evidence for electron collisional $\ell$-mixing in a Rb MOT \cite{Dutta}, and used selective field ionization to monitor evolution to plasma on a microsecond timescale in ultracold $^{85}$Rb $65d$ Rydberg gases with densities as low as $10^8$ cm$^{-3}$ \cite{WalzFlannigan}   Research by our group at UBC has observed very much the same dynamics in the relaxation of Xe Rydberg gases of similar density prepared in a molecular beam \cite{Hung2014}   In both cases, the time evolution to avalanche is well-described by coupled rate equations (see below), assuming an initializing density of Penning electrons determined by Robicheaux's criterion \cite{Robicheaux05}, applied to an Erlang distribution of Rydberg-Rydberg nearest neighbours   

Theoretical investigations of ultracold plasma physics have focused for the most part on the long- and short-time dynamics of plasmas formed by direct photoionization \cite{PhysRept,Lyon}   In addition to studies mentioned above, key insights on the evolution dynamics of Rydberg gases have been provided by studies of Pohl and coworkers exploring the effects of ion correlations and recombination-reionization on the hydrodynamics of plasma expansion \cite{Pohl:2003,PPR}   Further research has drawn upon molecular dynamics (MD) simulations to reformulate rate coefficients for the transitions driven by electron impact between highly excited Rydberg states \cite{PVS}, and describe an effect of strong coupling as it suppresses three-body recombination \cite{Bannasch:2011}   MD simulations confirm the accuracy of coupled rate equation descriptions for systems with $\Gamma$ as large as 0 3   Newer calculations suggest a strong connection between the order created by dipole blockade in Rydberg gases and the most favourable correlated distribution of ions in a corresponding strongly coupled ultracold plasma \cite{Bannasch:2013}   

Tate and coworkers have studied ultracold plasma avalanche and expansion theoretically as well as experimentally   Modelling observed expansion rates, they recently found that $^{85}$Rb atoms in a MOT form plasmas with effective initial electron temperatures determined by initial Rydberg density and the selected initial binding energy, to the extent that these parameters determine the fraction of the excited atoms that ionize by electron impact in the avalanche to plasma \cite{Forest}   This group also returned to the question of added Rydberg atoms, and managed to identify a crossover in $n_0$, depending on the initial electron temperature, that determines whether added Rydberg atoms of a particular initial binding energy act to heat or cool the electron temperature \cite{Crockett}    

Our group has focused on the plasma that evolves from a Rydberg gas under the low-temperature conditions of a skimmed, seeded supersonic molecular beam   In work on nitric oxide starting in 2008 \cite{Morrison2008,Plasma_expan,Morrison_shock,PCCP}, we established an initial kinetics of electron impact avalanche ionization that conforms with coupled rate equation models \cite{Saquet2011,Saquet2012,Scaling,haenelCP} and agrees at early times with the properties of ultracold plasmas that evolve from ultracold atoms in a MOT   We have also observed unique properties of the NO ultracold plasma owing to the fact that its Rydberg states dissociate \cite{Haenel2017}, and identified relaxation pathways that may give rise to quantum effects \cite{SousMBL,SousNJP}   The remainder of this review focuses on the nitric oxide ultracold plasma and the unique characteristics conferred by its evolution from a Rydberg gas in a laser-crossed molecular beam   


\section{Avalanche to strong coupling in a molecular Rydberg gas}

\subsection{The molecular beam ultracold plasma compared with a MOT}

When formed with sufficient density, a Rydberg gas of principal quantum number $n_0>30$ undergoes a spontaneous avalanche to form an ultracold plasma \cite{Li,Morrison2008,RobertdeSaintVincent}   Collisional rate processes combine with ambipolar hydrodynamics to govern the properties of the evolving plasma   For a molecular Rydberg gas, neutral fragmentation, occurs in concert with electron-impact ionization, three-body recombination and electron-Rydberg inelastic scattering   Neutral dissociation combined with radial expansion in a shaped distribution of charged particles, can give rise to striking effects of self-assembly and spatial correlation \cite{Schulz-Weiling2016,Haenel2017}    

The formation of a molecular ultracold plasma requires the conditions of local temperature and density afforded by a high mach-number skimmed supersonic molecular beam   Such a beam propagates at high velocity in the laboratory, with exceedingly well-defined hydrodynamic properties, including a propagation-distance-dependent density and sub-Kelvin temperature in the moving frame \cite{MSW_tutorial}   The low-temperature gas in a supersonic molecular beam differs in three important ways from the atomic gas laser-cooled in a magneto-optical trap (MOT) 

The milli-Kelvin temperature of the gas of ground-state NO molecules entrained in a beam substantially exceeds the sub-100 micro-Kelvin temperature of laser-cooled atoms in a MOT   However, the evolution to plasma tends to erase this distinction, and the two further characteristics that distinguish a beam offer important advantages for ultracold plasma physics:  Charged-particle densities in a molecular beam can exceed those attainable in a MOT by orders of magnitude   A great many different chemical substances can be seeded in a free-jet expansion, and the possibility this affords to form other molecular ultracold plasmas, introduces interesting and potentially important new degrees of freedom governing the dynamics of their evolution 


\subsection{Supersonic molecular beam temperature and particle density}

Seeded in a skimmed supersonic molecular beam, nitric oxide forms different phase-space distributions in the longitudinal (propagation) and transverse coordinate dimensions   As it propagates in $z$, the NO molecules reach a terminal laboratory velocity, $u_{\parallel}$, of about 1400 ${\rm ms^{-1}}$, which varies with the precise seeding ratio   

The distribution of $v_{\parallel}$, narrows to define a local temperature, $T_{\parallel}$, of approximately 0 5 K   The beam forms a Gaussian spatial distribution in the transverse coordinates, $x$ and $y$   In this plane, the local velocity, $v_{\perp}(r)$ is defined for any radial distance almost entirely by the divergence velocity of the beam, $u_{\perp}(r)$   Phase-space sorting cools the temperature in the transverse coordinates, $T_{\perp}$ to a value as low as $\sim 5$ mK \cite{MSW_tutorial}   

The stagnation pressure and seeding ratio determine the local density distribution as a function of $z$   For example, expanding from a stagnation pressure of 500 kPa with a 1:10 seeding ratio, a molecular beam propagates 2 5 cm to a skimmer and then 7 5 cm to a point of laser interaction, where it contains NO at a peak density of $1 6 \times 10^{14}$ cm$^{-3}$   

Here, crossing the molecular beam with a laser beam tuned to the transition sequence, ${\rm X} ~^2 \Pi_{1/2} ~N'' = 1 \xrightarrow{\omega_1} {\rm A} ~^2\Sigma^+ ~N'=0  \xrightarrow{\omega_2} n_0 f(2)$ forms a Gaussian ellipsoidal volume of Rydberg gas in a single selected principal quantum number, $n_0$, orbital angular momentum, $\ell = 3$, NO$^+$ core rotational quantum number, $N^+ = 2$ and total angular momentum neglecting spin, $N=1$   

A typical $\omega_1$ pulse energy of 2 $\mu$J and a Gaussian width of 0 2 mm serves to drive the first step of this sequence in a regime of linear absorption   Overlapping this volume by an $\omega_2$ pulse with sufficient fluence to saturate the second step forms a Rydberg gas ellipsoid with a nominal peak density of $5 \times 10^{12}$ cm$^{-3}$  \cite{Morrison2008,MSW_tutorial}   Fluctuations in the pulse energy and longitudinal mode of $\omega_1$ cause the real density to vary   For certain experiments, we find it convenient to saturate the $\omega_1$ transition, and vary the density of Rydberg gas by delaying $\omega_2$   An $\omega_1$-$\omega_2$ delay, $\Delta t$, reduces the Rydberg gas density by a precise factor, $e^{-\Delta t/\tau}$, where $\tau$ is the 200 ns radiative lifetime of NO ${\rm A} ~^2\Sigma^+ ~N'=0$ \cite{Carter,Hancock} 


\subsection{Penning ionization}

The density distribution of a Rydberg gas defines a local mean nearest neighbour distance, or Wigner-Seitz radius of $ a_{ws} =  \left(3/4 \pi \rho \right)^{1/3} $, where $\rho$ refers to the local Rydberg gas density   For example, a Rydberg gas with a density of $ \rho_0=0 5 \times 10^{12}$ cm$^{-3} $ forms an Erlang distribution \cite{Torquato 1990} of nearest neighbour separations with a mean value of $ 2 a_{ws}=1 6$  $\mu$m   

A semi-classical model \cite{Robicheaux05} suggests that 90 percent of Rydberg molecule pairs separated by a critical distance, $ r_c = 1 8 \cdot 2 n_0^2 a_0 $ or less undergo Penning ionization within 800 Rydberg periods   We can integrate the Erlang distribution from $ r=0 $ to the critical distance $r = r_c$ for a Rydberg gas of given $n_0$, to define the local density of Penning electrons ($ \rho_e$ at $t=0$) produced by this prompt interaction, for any given initial local density, $\rho_0$ by the expression:
\begin{equation}
\rho_e(\rho_0,n_0) = \frac{0 9}{2} \cdot 4 \pi \rho_0 ^2\int_0^{r_{c}} r^2 \mathrm{e}^{-\frac{4\pi}{3}\rho_0 r^3}\mathrm{d}r \quad 
\label{eqn:Erlang}
\end{equation}

Evaluating this definite integral yields an equation in closed form that predicts the Penning electron density for any particular initial Rydberg density and principal quantum number 
\begin{equation}
\rho_e(\rho_0,n_0) =\frac{0 9 \rho_0}{2}(1-\mathrm{e}^{-\frac{4\pi}{3}\rho_0 r_c^3}) \quad 
\label{Eq:PenDens}
\end{equation}
\begin{figure}[h!]
\centering
\includegraphics[scale=0 33]{Penning_Latice pdf}
\caption{Distributions of ion-ion nearest neighbours following Penning ionization and electron-impact avalanche simulated for a predissociating molecular Rydberg gas of initial principal quantum number, $n_0$, from 30 to 80, and density of 10$^{12}$ cm$^{-3}$   Dashed lines mark corresponding values of $a_{ws}$  Calculated by counting ion distances after relaxation to plasma in 10$^6$-particle stochastic simulations  Integrated areas proportional to populations surviving neutral dissociation }
\label{fig:PL}
\end{figure}

Prompt Penning ionization acts on the portion of the initial nearest-neighbour distribution in the Rydberg gas that lies within $r_c$   When a molecule ionizes, its collision partner relaxes to a lower principal quantum number, $n'<n_0/\sqrt{2}$   This close-coupled interaction disrupts the separability of Rydberg orbital configurations in the Penning partner   This causes mixing with core penetrating states that are strongly dissociative   Penning partners are thus very likely to dissociate, leaving a spatially isolated distribution of ions   We refer to the spatial correlation that results as a Penning lattice \cite{Sadeghi:2014}   The extent of this effect varies depending on the local density and the selected initial principal quantum number   Figure \ref{fig:PL} shows the degree to which Rydberg gases with initial principal quantum numbers from 30 to 80 form a Penning lattice for an initial density of $1 \times 10^{12} ~{\rm cm}^{-3}$   

\subsection{Spontaneous electron-impact avalanche}

The electrons produced by prompt Penning ionization start an electron impact avalanche   The kinetics of this process are well described by a set of coupled rate equations that account for state-to-state electron-Rydberg inelastic scattering, electron-impact ionization and three-body ion-electron recombination \cite{PPR,Saquet2011,Saquet2012,Scaling} using detailed rate coefficients,  $k_{ij}$, $k_{i,ion}$ and $k_{i,tbr}$ validated by MD simulations \cite{PVS}   
\begin{eqnarray}
-\frac{d\rho_i}{dt}&=&\sum_{j}{k_{ij}\rho_e\rho_i}-\sum_{j}{k_{ji}\rho_e\rho_j} \nonumber\\
&& +k_{i,ion}\rho_e\rho_i-k_{i,tbr}\rho^3_e 
  \label{level_i}
\end{eqnarray}
\noindent and,
\begin{equation}
\frac{d\rho_e}{dt}=\sum_{i}{k_{i,ion}\rho_e^2}-\sum_{i}{k_{i,tbr}\rho^3_e}
  \label{electron}
\end{equation}

The relaxation of Rydberg molecules balances with collisional ionization to determine an evolving temperature of avalanche electrons to conserve total energy per unit volume  
\begin{equation}
E_{tot}=\frac{3}{2}k_BT_e(t)\rho_e(t)-R\sum_i{\frac{\rho_i(t)}{n_i^2}},
  \label{energy}
\end{equation}
Here, for simplicity, we neglect the longer-time effects of Rydberg predissociation and electron-ion dissociative recombination \cite{Saquet2012} 

Such calculations show that the conversion from Rydberg gas to plasma occurs on a timescale determined largely by the local Penning electron density, or Penning fraction, $P_f = \rho_e/\rho_0$, which depends on the local density of Rydberg molecules and their initial principal quantum number   

Avalanche times predicted by coupled rate equation calculations range widely   For example, in a model developed for experiments on xenon, simulations predict that a Rydberg gas with $n_0 = 42$ at a density of $8 8 \times 10^8 ~{\rm cm}^{-3}$ ($P_f = 6 \times 10^{-5}$) avalanches with a half time of  40 $\mu$s \cite{Hung2014}   At an opposite extreme, rate equations estimate that a Rydberg gas of NO with $n_0=60$ at a density of $1 \times 10^{12} ~{\rm cm}^{-3}$ ($P_f = 0 3$) rises to plasma in about 2 ns \cite{Saquet2012}   

\begin{figure}[h!]
\centering
\includegraphics[width=  49 \textwidth]{SFI_n=49 pdf}
   \caption{Contour plots showing SFI signal as a function the applied field for an $nf(2)$ Rydberg gas with an initial principal quantum number, $n_0=49$   Each frame represents 4,000 SFI traces, sorted by initial Rydberg gas density   Ramp field beginning at 0 and 150 ns (top, left to right), and  300 and 450 ns (bottom) after the $\omega_2$ laser pulse   The two bars of signal most evident at early ramp field delay times represent the field ionization of the $49f(2)$ Rydberg state respectively to NO$^+$ X $^1\Sigma^+$ cation rotational states, $N^+=0$ and 2   The signal waveform extracted near zero applied field represents the growing population of plasma electrons   
   }
\label{fig:SFI}
\end{figure}

Selective field ionization (SFI) probes the spectrum of binding energies in a Rydberg gas   Applied as a function of time after photoexcitation, SFI maps the evolution from a state of selected initial principal quantum number, $n_0$, to plasma \cite{Haenel2017}   Figure \ref{fig:SFI} shows SFI spectra taken at a sequence of delays after the formation of $49f(2)$ Rydberg gases of varying density     

Here, we can see that a $49f(2)$ Rydberg gas with an estimated initial density $\rho_0 = 3 \times 10^{11} ~{\rm cm}^{-3}$ relaxes to plasma on a timescale of about 500 ns   Observations such as these agree well with the predictions of coupled rate-equation calculations   We can understand this variation in relaxation dynamics with $\rho_0$ and $n_0$ quite simply in terms of the corresponding density of prompt Penning electrons these conditions afford to initiate the avalanche to plasma   

Figure \ref{fig:scaled_rise} illustrates this, showing how rise times predicted by coupled rate-equation simulations for a large range of initial densities and principal quantum number match when plotted as a function of time scaled by the ultimate plasma frequency and fraction of prompt Penning electrons   The dashed line gives an approximate account of the scaled rate of avalanche under all conditions of Rydberg gas density and initial principal quantum number in terms of the simple sigmoidal function:

\begin{equation}
\frac{\rho_e}{\rho_0} = \frac{a}{b+e^{-c\tau}},
  \label{scaledEq1}
\end{equation}
where,
\begin{equation}
\tau = t \omega_e P_f^{3/4},
  \label{scaledEq2}
\end{equation}
in which $\omega_e$ is the plasma frequency after avalanche, $P_f$ is the fraction of prompt Penning electrons, and $a = 0 00062$,  $b =   0 00082$ and $c =     0 075$ are empirical coefficients   

\begin{figure}[h!]
\centering
\includegraphics[width=  4 \textwidth]{sim_analytical_density pdf}
   \caption{Rise in fractional electron density as a function of time scaled by the plasma frequency, $\omega_e$ and fraction, $\rho_e(t=0)/\rho_0 = P_f$, of prompt Penning electrons   Simulation results shown for $n_0 = 30$, 50 and 70 with initial densities, $\rho_0 = 10^9,~10^{10},~10^{11},~{\rm and}~10^{12}~{\rm cm}^{-3}$   
   }
\label{fig:scaled_rise}
\end{figure}


\subsection{Evolution to plasma in a Rydberg gas Gaussian ellipsoid}

As outlined above, the local density and principal quantum number together determine the rate at which a Rydberg gas avalanches to plasma   Our experiment crosses a 2 mm wide cylindrically Gaussian molecular beam with a 1 mm diameter TEM$_{00}$ $\omega_1$ laser beam to produce a Gaussian ellipsoidal distribution of molecules excited to the A $^2\Sigma^+$ $v=0, ~N'=0$ intermediate state   A larger diameter $\omega_2$ pulse then drives a second step that forms a Rydberg gas in a single $n_0f(2)$ state with the spatial distribution of the intermediate state   

We model this shaped Rydberg gas as a system of 100 concentric ellipsoidal shells of varying density \cite{haenelCP}   Coupled rate equations within each shell describe the avalanche to plasma   This rate process proceeds from shell to shell with successively longer induction periods, determined by the local density as detailed above   The rising conversion of Rydberg molecules to ions plus neutral dissociation products conserves the particle number in each shell   We assume that local space charge confines electrons to shells, conserving quasi-neutrality   Electrons exchange kinetic energy at the boundaries of each shell, which determines a single plasma electron temperature   

\begin{figure}[h!]
\centering
\includegraphics[width=  5 \textwidth]{shell_model_100}
   \caption{(top frame) Cross-sectional contour diagram in the $x,y$ plane for $z=0$ describing the distribution of ion plus electron density over 100 shells of Gaussian ellipsoid with initial dimensions, $\sigma_x= 0 75$ mm and $\sigma_y= \sigma_z = 0 42$ mm and an initial $n_0 = 50$ Rydberg gas density, $\rho_0 = 2 \times 10^{11}$ cm$^{3}$ after an evolution time of 100 ns   (bottom frame) Curves describing the (dashed) ascending ion and (solid) descending Rydberg gas densities of each shell as functions of evolution time, for $t=20$, 40, 60, 80 and 100 ns   
   }
\label{fig:shell}
\end{figure}

The upper frame of Figure \ref{fig:shell} shows contours of NO$^+$ ion density after 100 ns obtained from a shell-model coupled rate-equation simulation of the avalanche of a Gaussian ellipsoidal Rydberg gas of nitric oxide with a selected initial state, $50f(2)$ and a density of $2 \times 10^{11}$ cm$^{-3}$   Here, we simulate a relaxation that includes channels of predissociation at every Rydberg level and redistributes the energy released to electrons, which determines a uniform rising electron temperature for all shells   

For comparison, the lower frame plots curves describing the ion density of each shell as a function of time from 20 to 100 ns, as determined by applying Eq \ref{scaledEq1} for the local conditions of initial Rydberg gas density   This numerical approximation contains no provision for predissociation   Coupled rate-equation simulations for uniform volumes show that predissociation depresses yield to some degree, but has less effect on the avalanche kinetics \cite{Saquet2012}   Therefore, we can expect sets of numerically estimated shell densities, scaled to agree with the simulated ion density at the elapsed time of 100 ns to provide a reasonable account of the earlier NO$^+$ density profiles as a function of time    


\begin{figure}[h!]
\centering
\includegraphics[width=  4 \textwidth]{Shell_pop}
   \caption{Global population fractions of particles as they evolve in the avalanche of a shell-model ellipsoidal Rydberg gas with the initial principal quantum number and density distribution of Figure \ref{fig:shell}
   }
\label{fig:shell_yields}
\end{figure}

For each time step, the difference, $\rho_0 - \rho_e$ defines the neutral population of each shell   We assign a fraction of this population to surviving Rydberg molecules, such that the total population of NO$^*$ as a function of time agrees with the prediction of the shell-model simulation, as shown in Figure \ref{fig:shell_yields}   We consider the balance of this neutral population to reflect NO$^*$ molecules that have dissociated to form N($^4$S) $+$ O($^3$P)   Figure \ref{fig:shell} plots these surviving Rydberg densities as a function of radial distance for each evolution time   At the initial density of this simulation, note at each time step that a higher density of Rydberg molecules encloses the tail of the ion density distribution in $x$    

\subsection{Plasma expansion and NO$^+$ - NO$^*$ charge exchange as an avenue of quench}

We regard the ions as initially stationary   The release of electrons creates a radial electric potential gradient, which gives rise to a force, $-e\nabla \phi_{k,j}(t)$, that accelerates the ions in shell $j$ in direction $k$ according to \cite{Sadeghi 2012}:

\begin{align}
\frac{-e}{m'}\nabla \phi_{k,j}(t) = & \frac{\partial u_{k,j}(t)}{\partial t} \notag  \\
= & \frac{k_BT_e(t)}{m'\rho_j(t)} \frac{\rho_{j+1}(t) - \rho_j(t)}{r_{k,j+1}(t) - r_{k,j}(t)},
  \label{dr_dt}
\end{align}

\noindent where $\rho_j(t)$ represents the density of ions in shell $j$   

The instantaneous velocity, $u_{k,j}(t)$ determines the change in the radial coordinates of each shell, $r_{k,j}(t)$, 
\begin{equation}
\frac{\partial r_{k,j}(t)}{\partial t}=u_{k,j}(t) = \gamma_{k,j}(t) r_{k,j}(t),
  \label{dr_dt}
\end{equation}
\noindent which in turn determines shell volume and thus its density, $ \rho_j(t)$    
The electron temperature supplies the thermal energy that drives this ambipolar expansion   Ions accelerate and $T_e$ falls according to: 

\begin{equation}
\frac{3k_B}{2}\frac{\partial T_e(t)}{\partial t}= -\frac{m'}{\sum_{j}{N_j}}\sum_{k,j}{N_j u_{k,j}(t)\frac{\partial u_{k,j}(t)}{\partial t}},
  \label{dr_dt}
\end{equation}
\noindent where we define an effective ion mass, $m'$, that recognizes the redistribution of the electron expansion force over all the NO$^+$ charge centres by resonant ion-Rydberg charge exchange, which occurs with a very large cross section \cite{PPR}   
\begin{equation}
m' =\left (1+ \frac{\rho^*_{j}(t)}{ \rho_j(t)}\right) m ,
  \label{dr_dt}
\end{equation}
\noindent in which $\rho^*_{j}(t)$ represents the instantaneous Rydberg density in shell $j$ 

The initial avalanche in the high-density core of the ellipsoid leaves few Rydberg molecules, so this term has little initial effect   Rydberg molecules predominate in the lower-density wings   There, momentum sharing by charge exchange assumes a greater importance   

We see this most directly in the $\omega_2$ absorption spectrum of transitions to states in the $n_0 f(2)$ Rydberg series, detected as the long-lived signal that survives a flight time of 400 $\mu$s to reach the imaging detector   The balance between the rising density of ions and the falling density of Rydberg molecules depends on the initial density of electrons produced by prompt Penning ionization   As clear from Eq \ref{Eq:PenDens}, this Penning fraction depends sensitively on the principal quantum number, and for all principal quantum numbers, on the initial Rydberg gas density   

\begin{figure}[h!]
\centering
\includegraphics[width=  4 \textwidth]{w2_spectra}
   \caption{Double-resonant spectra of nitric oxide Rydberg states in the $nf$ series converging to NO$^+$ $v=0$, $N^+=2$ (designated, $nf(2)$), derived from the late-peak signal obtained after a flight time of 400 $\mu$s by scanning $\omega_2$ for $\omega_1$ tuned to NO A $^2\Sigma^+$ $v=0$, $N'=0$ for initial $nf(2)$ densities from top to bottom of 0 07, 0 10, 0 13, 0 19, 0 27, 0 30, 0 32 and $3 \times 10^{12}$ cm$^{3}$   
  }
\label{fig:w2_spectra}
\end{figure}

Figure \ref{fig:w2_spectra} shows a series of $\omega_2$ late-signal excitation spectra for a set of initial densities   Here, we see a clear consequence of the higher-order dependence of Penning fraction - and thus the NO$^+$ ion - NO$^*$ Rydberg molecule balance - on $n_0$, the $\omega_2$-selected Rydberg gas initial principal quantum number   This Penning-regulated NO$^+$ ion - NO$^*$ Rydberg molecule balance appears necessary as a critical factor in achieving the long ultracold plasma lifetime required to produce this signal   We are progressing in theoretical work that explains the stability apparently conferred by this balance   


\subsection{Bifurcation and arrested relaxation}

Ambipolar expansion quenches electron kinetic energy as the initially formed plasma expands   Core ions follow electrons into the wings of the Rydberg gas   There, recurring charge exchange between NO$^+$ ions and NO$^*$ Rydberg molecules redistributes the ambipolar force of the expanding electron gas, equalizing ion and Rydberg velocities   This momentum matching effectively channels electron energy through ion motion into the overall $\pm x$ motion of gas volumes in the laboratory   The internal kinetic energy of the plasma, which at this point is defined almost entirely by the ion-Rydberg relative motion, falls   Spatial correlation develops, and over a period of 500 ns, the system forms the plasma/high-Rydberg quasi-equilibrium dramatically evidenced by the SFI results in Figure \ref{fig:SFI}   

\begin{figure}[h!]
\centering
\includegraphics[width=  4 \textwidth]{Bifurcation pdf}
   \caption{$x,y$ detector images of ultracold plasma volumes produced by 2:1 aspect ratio ellipsoidal Rydberg gases with selected initial state, $40f(2)$ after a flight time of 402 $\mu$s over a distance of 575 mm   Lower frame displays the distribution in $x$ of the charge integrated in $y$ and $z$   Both images represent the unadjusted raw signal acquired in each case after 250 shots   
   }
\label{fig:bifurcation}
\end{figure}

In the wings, momentum redistribution owing to cycles of ion-Rydberg charge transfer retards radial expansion \cite{Pohl2003,PPR}   By redirecting electron energy from ambipolar acceleration to $\pm x$ plasma motion, NO$^+$ to NO$^*$ charge exchange dissipates electron thermal energy   This redistribution of energy released in the avalanche of the Rydberg gas to plasma, causes the ellipsoidal Rydberg gas to bifurcate \cite{Schulz-Weiling2016,Haenel2017}, forming very long-lived, separating charged-particle distributions   We capture the electron signal from these recoiling volumes on an imaging detector as pictured in Figure \ref{fig:bifurcation}   Here, momentum matching preserves density and enables ions and Rydberg molecules to relax to positions that minimize potential energy, building spatial correlation   

The semi-classical description of avalanche and relaxation outlined above forms an important point of reference from which to interpret our experimental observations   The laser crossed molecular beam illumination geometry creates a Rydberg gas with a distinctively shaped high-density spatial distribution   This initial condition has an evident effect on the evolution dynamics   We have developed semi-classical models that explicitly consider the coupled rate and hydrodynamic processes governing the evolution from Rydberg gas to plasma using a realistic, ellipsoidal representation of the ion/electron and Rydberg densities \cite{haenelCP}   No combination of initial conditions can produce a simulation that conforms classically with the state of arrested relaxation we observe experimentally  


\subsection{A molecular ultracold plasma state of arrested relaxation}

Thus, we find that spontaneous avalanche to plasma splits the core of an ellipsoidal Rydberg gas of nitric oxide  As ambipolar expansion quenches the electron temperature of this core plasma, long-range, resonant charge transfer from ballistic ions to frozen Rydberg molecules in the wings of the ellipsoid quenches the ion-Rydberg molecule relative velocity distribution  This sequence of steps gives rise to a remarkable mechanics of self-assembly, in which the kinetic energy of initially formed hot electrons and ions drives an observed separation of plasma volumes  These dynamics redistribute ion momentum, efficiently channeling electron energy into a reservoir of mass-transport  This starts a process that evidently anneals separating volumes to a state of cold, correlated ions, electrons and Rydberg molecules  

We have devised a three-dimensional spin model to describe this arrested state of the ultracold plasma in terms of two, three and four-level dipole-dipole energy transfer interactions (spin flip-flops), together with Ising interactions that arise from the concerted pairwise coupling of resonant pairs of dipoles \cite{SousMBL,SousNJP}   

The Hamiltonian includes the effects of onsite disorder owing to the broad spectrum of states populated in the ensemble and the unique electrostatic environment of every dipole   Extending ideas developed for simpler systems \cite{Burin1,Sondhi}, one can make a case for slow dynamics, including an arrest in the relaxation of NO Rydberg molecules to predissociating states of lower principal quantum number 

Systems of higher dimension ought to thermalize by energy transfer that spreads from rare but inevitable ergodic volumes (Griffiths regions) \cite{Sarang2, Roeck_griffith, RareRegions_rev, Thermal_inclusions}   However, a feature in the self-assembly of the molecular ultracold plasma may preclude destabilization by rare thermal domains:  Whenever the quenched plasma develops a delocalizing Griffiths region, the local predissociation of relaxing NO molecules promptly proceeds to deplete that region to a void of no consequence 

In summary, the classical dynamics of avalanche and bifurcation appear to create a quenched condition of low temperature and high disorder in which dipole-dipole interactions drive self-assembly to a localized state purified by the predissociation of thermal regions   We suggest that this state of the quenched ultracold plasma offers an experimental platform for studying quantum many-body physics of disordered systems  \\

\section{Acknowledgments}
This work was supported by the US Air Force Office of Scientific Research (Grant No  FA9550-17-1-0343), together with the Natural Sciences and Engineering research Council of Canada (NSERC), the Canada Foundation for Innovation (CFI) and the British Columbia Knowledge Development Fund (BCKDF)  





ground_truth: A subset that form via kinetic rate processes from state-selected Rydberg gases.