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 is the dynamical behavior of the anisotropic order parameter following a quench to the critical point?

Read the following text and answer briefly.

Context 1: \section*{Dynamical Behaviour of $O$ in Lattice Gases}

The dynamical behaviour of the anisotropic order parameter  $m$ [see Eq ~\eqref{eq:def-m} in the Letter] following a quench to the critical point is well described by
the Gaussian theory for all the three lattice gas models studied, $i e ,$ driven lattice gas with either constant (IDLG) or random (RDLG) infinite drive and equilibrium lattice gas (LG)  In other words, in the short-time regime, $m \sim t^{1/2}$ [see Eq  \eqref{eq:mt}] and the Binder cumulant $g$ of the lowest transverse mode [defined in Eq  \eqref{eq:binder}] is zero in this regime  The alternative order parameter $O,$ however, distinguishes between the driven (IDLG, RDLG) and the equilibrium (LG) lattice gases  

In order to understand  this, we first write the phenomenological scaling form for $O$,  analogous to Eq  \eqref{eq:scalingass} in  the Letter,
\begin{eqnarray}
O (t, L_{\parallel} ; S_\Delta) = L_{\parallel}^{-\beta/[\nu(1+\Delta)]} \tilde f_O (t/L_{\parallel}^{z/(1+\Delta)} ; S_\Delta) \quad
\label{eq:Oscalingass}
\end{eqnarray}
We already remarked that, in the LG, this scaling form is not compatible with the prediction $O \sim t^{1/8}  L_{\parallel}^{-1/2}$ of the Gaussian theory   However, following Ref  \cite{AS2002}, it can be argued that, at short times, the only dependence of $O$ on the system size $L_{\parallel}$ is of the form $O \sim L_\parallel^{-1/2}$ which is very well confirmed by numerical simulations  Accordingly,  the generic behaviour of $O$ can be assumed to be
\begin{eqnarray}
O \sim t^{\alpha} L_\parallel^{-1/2}, \label{eq:O}
\end{eqnarray}
where $\alpha$ is a phenomenological exponent to be determined  This, along with Eq  \eqref{eq:Oscalingass}, implies $\tilde f_O(x) \sim x^{\alpha} $ Comparing the finite-size behaviour in Eq ~\eqref{eq:O} with Eq ~\eqref{eq:Oscalingass} one actually infers,
\begin{eqnarray}
\alpha &=& \frac{1+ \Delta -2 \beta/\nu}{2 \, (4- \eta)}  \label{eq:alpha}
\end{eqnarray}
This equation, together with the hyperscaling relation $\Delta - 2 \beta/\nu= - \eta$ in two spatial dimensions, shows that the prediction $\alpha = 1/8$ of the Gaussian theory [see Eq  \eqref{eq:Ot}] can be obtained only when $\eta=0,$ which is the case for the IDLG (exactly) and the RDLG (approximately) but not for the LG  

On the other hand,  Eq ~\eqref{eq:alpha} predicts $\alpha = 1/10$ upon substituting the values of the critical exponents corresponding to the Ising  universality class (LG)  This is consistent with the numerical simulation results presented in the main text, see Fig  \ref{fig:ising}(b) therein 

\begin{figure}[th]
\vspace*{0 2 cm}
 \centering
 \includegraphics[width=10 cm]{ /compare_binder pdf}

\caption{Comparison between the temporal evolution of the Binder cumulants $g$ corresponding to the $12^{th}$ transverse mode, $i e ,$ with $n_\perp =12,$ in the LG (lowest curve), IDLG and RDLG (two upper curves) on a $32 \times 32$ lattice  \label{fig:b}}
 \label{fig:binder}
\end{figure}


The emergence of this new value $1/10$ of the exponent $\alpha$ must be traced back to the non-Gaussian nature of higher fluctuating modes in the LG  In fact, even though the lowest mode behaves identically in all the three models we considered,  characterized by the same behaviour of $m$, higher modes show a significant difference in the non-driven case   


To illustrate this, we measured the Binder cumulants of higher modes which is defined  analogously to Eq ~(11), using transverse modes other than the first, i e , with $\mu=\tilde \sigma(0,2 \pi n_\bot/L_\bot)$ and $n_\bot>1 $  
 Figure \ref{fig:b} compares the same for all the three lattice gases for the mode with $n_\perp =12$ on a $32 \times 32$ lattice  Clearly, the curve corresponding to the LG (lowest, blue) departs from Gaussian behaviour $g=0$ (in practice, $e g ,$ $|g| \lesssim 0 005,$ corresponding to the shaded gray area) much earlier than it does for the IDLG  or RDLG (two upper curves, red and green respectively) 

Accordingly, the different dynamical behaviour of $O$, which involves a sum over all modes, can be attributed to the non-Gaussian nature of the higher modes in the LG  
Such a departure is not entirely surprising  In fact, for higher modes, mesoscopic descriptions such as the ones in Eqs  \eqref{eq:L-DLG} or \eqref{eq:g_evol} are not expected to hold, while the anisotropy at the microscopic level could be the mechanism leading to the Gaussianity of higher modes in the driven models 


Context 2: \section{Introduction}

In spite of being an attractive material with excellent electronic properties \cite{ahcn09},  practical applications of graphene as in conventional semiconductor devices are still questionable due to its gapless nature  In particular, the ON/OFF current ratio is low while the saturation of current is poor in pristine graphene transistors \cite{schw10}  Many efforts of bandgap engineering in graphene \cite{yhan07,khar11,lher13,jbai10,zhan09} have been made to solve these issues  The pioneer technique proposed \cite{yhan07} is to cut 2D graphene sheets into 1D narrow nanoribons  In 2D graphene sheets, some options as Bernal-stacking of graphene on hexagonal boron nitride substrate \cite{khar11}, nitrogen-doped graphene \cite{lher13}, graphene nanomesh lattice \cite{jbai10,berr13} and Bernal-stacking bilayer graphene \cite{zhan09} have been explored  However, the possibility to open a sizable bandgap in graphene as large as those of standard semiconductors is still very unlikely  In particular, it requires a very good control of lattice geometry and edge disorder in narrow graphene nanoribbons (GNRs) \cite{quer08} and in graphene nanomesh lattices \cite{hung13}, while the bandgap opening in bilayer graphene by a perpendicular electric field may not be large enough for realistic applications \cite{fior09}  Other methods should be further verified by experiments 

\begin{figure}[!t]
\centering
\includegraphics[width=2 8in]{Fig01 pdf}
\caption{Schematic of unstrained/strained graphene junctions investigated in this work }
\label{fig_sim1}
\end{figure}

On the other hand, graphene was experimentally demonstrated to be able to sustain a much larger strain than conventional semiconductors, making it a promising candidate for flexible electronics (see in a recent review \cite{shar13})  Indeed, strain engineering has been suggested to be an alternative approach to modulating efficiently the electronic properties of graphene nanomaterials  In particular, the bandgap has periodic oscillations in the armchair GNRs \cite{ylu210} while the spin polarization at the ribbon edges (and also the bandgap) can be modulated by the strain in the zigzag cases  In 2D graphene sheets, a finite gap can open under large strains, otherwise, it may remain close to zero but the Dirac points are displaced \cite{cocc10,per209,pere09,huan10}  Many interesting electrical, optical, and magnetic properties induced by strain in graphene have been also explored, e g  see in \cite{bunc07,pere09,kuma12,per010,pell10,guin10,tlow10,zhai11} 

Besides, local strain is a good option to improve the electrical performance of graphene devices \cite{pere09,ylu010,fuji10,juan11,baha13}  For instance, it has been shown to enhance the ON current in a GNR tunneling FET \cite{ylu010} and to fortify the transport gap in GNR strained junctions \cite{baha13}  In a recent work \cite{hung14}, we have investigated the effects of uniaxial strain on the transport in 2D unstrained/strained graphene junctions and found that due to the strain-induced shift of Dirac points, a significant conduction gap of a few hundreds meV can open with a small strain of a few percent  This type of strained junction was then demonstrated to be an excellent candidate to improve the electronic operation of graphene transistors  It hence motivates us to further investigate the properties of this conduction gap so as to optimize the performance of graphene devices  On the one hand, the effects of strain should be, in principle, dependent on its applied direction  On the other hand, because the appearance of conduction gap is a consequence of the shift of Dirac points along the $k_y$-axis, it is predicted that this gap should also depend on the transport direction  Note that here, Oy (Ox) - axis is assumed to be perpendicular (parallel) to the transport direction  The effects of both strain and transport directions will be clarified systematically in the current work 

\section{Model and calculations}

In this work, the $\pi$-orbital tight binding model constructed in \cite{per209} is used to investigate the electronic transport through the graphene strained junctions schematized in Fig  1  The Hamiltonian is ${H_{tb}} = \sum\nolimits_{nm} {{t_{nm}}c_n^\dag {c_m}}$ where $t_{nm}$ is the hopping energy between nearest neighbor \emph{n}th and \emph{m}th atoms  The application of a uniaxial strain of angle $\theta$ causes the following changes in the $C-C$ bond vectors:
\begin{eqnarray}
 {{\vec r}_{nm}}\left( \sigma \right) &=& \left\{ {1 + {M_s}\left( \sigma, \theta \right)} \right\}{{\vec r}_{nm}}\left( 0 \right) \\
 {M_s}\left( \sigma, \theta \right) &=& \sigma \left[ {\begin{array}{*{20}{c}}
 {{{\cos }^2}\theta  - \gamma {{\sin }^2}\theta }&{\left( {1 + \gamma } \right)\sin \theta \cos \theta }\\
 {\left( {1 + \gamma } \right)\sin \theta \cos \theta }&{{{\sin }^2}\theta  - \gamma {{\cos }^2}\theta }
 \end{array}} \right] \nonumber
\end{eqnarray}
where $\sigma$ represents the strain and $\gamma \simeq 0 165$ is the Poisson ratio \cite{blak70}  The hopping parameters are defined as $t_{nm} \left( \sigma \right) = t_0 \exp\left[-3 37\left(r_{nm} \left( \sigma \right) /r_0 - 1\right)\right]$, where the hopping energy $t_0 = -2 7$ $eV$ and the bond length $r_{nm} \left( 0 \right) \equiv r_0 = 0 142$ $nm$ in the unstrained case  Therefore, there are three different hoping parameters $t_{1,2,3}$ corresponding to three bond vectors ${\vec r}_{1,2,3}$, respectively, in the strained graphene part of the structure (see Fig  1)  Here, we assume a 1D profile of applied strain, i e , the strain tensor is a function of position along the transport direction Ox while it is constant along the Oy-axis  The transport direction, $\phi$, and strain direction, $\theta$, are determined as schematized in Fig  1  Based on this tight binding model, two methods described below can be used to investigate the conduction gap of the considered strained junctions 

\textbf{Green's function calculations } First, we split the graphene sheet into the smallest possible unit cells periodically repeated along the Ox/Oy directions with the indices $p/q$, respectively (similarly, see the details in \cite{hung12})  The tight-binding Hamiltonian can therefore be expressed in the following form:
\begin{eqnarray}
{H_{tb}} = \sum\limits_{p,q} {\left( {{H_{p,q}} + \sum\limits_{{p_1},{q_1}} {{H_{p,q \to p_1,q_1}}} } \right)}
\end{eqnarray}
where $H_{p,q}$ is the Hamiltonian of cell $\{p,q\}$, and $H_{p,q \to p_1,q_1}$ denotes the coupling of cell $\{p,q\}$ to its nearest neighbor cell $\{p_1,q_1\}$  We then Fourier transform the operators in Eq  (2) as follows:
\begin{eqnarray}
 {c_{p,q}} = \frac{1}{{\sqrt {{M_{cell}}} }}\sum\limits_{{\kappa_y}} {{e^{i{q\kappa_y}}}} {{\hat c}_{p,{\kappa_y}}},
\end{eqnarray}
where $M_{cell}$ is the number of unit cells and $\kappa_y \equiv k_y L_y$ with the size $L_y$ of unit cells along the Oy direction  The Hamiltonian (2) is finally rewritten as a sum of $\kappa_y$-dependent 1D-components:
\begin{eqnarray}
{H_{tb}} &=& \sum\limits_{{\kappa_y}} {\hat H\left( {{\kappa_y}} \right)} \\
\hat H\left( {{\kappa_y}} \right) &=& \sum\limits_p {{{\hat H}_{p \to p - 1}}\left( {{\kappa_y}} \right) + {{\hat H}_p}\left( {{\kappa_y}} \right) + {{\hat H}_{p \to p + 1}}}\left( {{\kappa_y}} \right) \nonumber
\end{eqnarray}
With this Hamiltonian form, the Green's function formalism can be easily applied to compute transport quantities in the graphene strained junction with different transport directions  In particular, the conductance at zero temperature is determined as:
\begin{eqnarray}
 \mathcal{G} \left( \epsilon \right) = \frac{{e^2 W}}{{\pi h L_y}}\int\limits_{BZ} {d{\kappa_y} \mathcal{T}\left( {\epsilon, {\kappa_y}} \right)}
\end{eqnarray}
where $\mathcal{T}\left( {\epsilon,{\kappa_y}} \right)$ is the transmission probability computed from the Green's functions  The integration over $\kappa_y$ is performed in the whole first Brillouin zone  As in ref  \cite{hung13}, the gap of conductance (conduction gap) is then measured from the obtained data of conductance 

\textbf{Bandstructure analyses } To determine the conduction gap of strained junctions, we find that another simple way based on the analysis of graphene bandstructures could be efficiently used  It is described as follows  Since the conductance is computed from Eq  (5), the appearance of conduction gap is essentially governed by the gaps of transmission probability, which is determined from the energy gaps in the unstrained and strained graphene sections  These energy gaps can be defined directly from the graphene bandstructures  Therefore, our calculation has two steps, similar to that in \cite{hung14}  From the graphene bandstructures obtained using the tight-binding Hamiltonian above, we first look for the energy gaps $E_{unstrain}^{gap}\left( {{\kappa_y}} \right)$ and $E_{strain}^{gap}\left( {{\kappa_y}} \right)$ for a given $\kappa_y$ of two graphene sections  The maximum of these energy gaps determines the gap $E_{junc}^{gap}\left( {{\kappa_y}} \right)$ of transmission probability through the junction  Finally, the conduction gap $E_{cond gap}$ is obtained by looking for the minimum value of $E_{junc}^{gap}\left( {{\kappa_y}} \right)$ when varying $\kappa_y$ in the whole Brillouin zone 

In particular, the energy bands of strained graphene are given by
\begin{eqnarray}
 E\left( {\vec k} \right) =  \pm \left| {{t_1}{e^{i\vec k{{\vec a}_1}}} + {t_2}{e^{i\vec k{{\vec a}_2}}} + {t_3}} \right|
\end{eqnarray}
where the plus/minus sign corresponds to the conduction/valence bands, respectively  For a given direction $\phi$ of transport, in principle, the vectors $\vec L_{x,y}$ defining the sizes of unit cell along the Ox and Oy directions, respectively, can be always expressed as ${\vec L_x} = {n_1}{\vec a_1} + {n_2}{\vec a_2}$ and ${\vec L_y} = {m_1}{\vec a_1} + {m_2}{\vec a_2}$ with $\cos \phi = \frac{{{{\vec L}_x}\vec L_x^0}}{{{L_x}L_x^0}}$ and $\sin \phi = \frac{{{{\vec L}_x}\vec L_y^0}}{{{L_x}L_y^0}}$ while $\vec L_{x,y}^0 = {\vec a_1} \pm {\vec a_2}$  Note that $n_{1,2}$ and $m_{1,2}$ are integers while $\frac{{{m_1}}}{{{m_2}}} =  - \frac{{{n_1} + 2{n_2}}}{{{n_2} + 2{n_1}}}$, i e , ${\vec L_{x}} {\vec L_{y}} = 0$  In other words, we have the following expressions
\begin{eqnarray}
{{{\vec a}_1} = \frac{{ - {m_2}{{\vec L}_x} + {n_2}{{\vec L}_y}}}{{{n_2}{m_1} - {n_1}{m_2}}};\,\,\,{{\vec a}_2} = \frac{{{m_1}{{\vec L}_x} - {n_1}{{\vec L}_y}}}{{{n_2}{m_1} - {n_1}{m_2}}}}
\end{eqnarray}
On this basis, the energy bands can be rewritten in terms of $\kappa_{x, y} = \vec k \vec L_{x,y} \left( { \equiv {k_{x,y}}{L_{x,y}}} \right)$ by substituting Eqs  (7) into Eq  (6)  This new form of energy bands is finally used to compute the conduction gap of strained junctions 

As a simple example, in the case of $\phi = 0$ (armchair direction), we calculate the conduction gap as follows  First, Eq  (6) is rewritten in the form
\begin{eqnarray}
 E_{\phi = 0}\left( {\vec \kappa} \right) =  \pm \left| {{t_1}{e^{i\kappa_y/2}} + {t_2}{e^{ - i\kappa_y/2}} + {t_3}{e^{ - i\kappa_x/2}}} \right|
\end{eqnarray}
with the vectors $\vec L_{x,y} \equiv \vec L_{x,y}^0$  Using this new form, the energy gap of strained graphene for a given $\kappa_y$ is determined as
\begin{equation}
{E_{strain}^{gap}}\left( {{\kappa_y}} \right) = 2 \left| {\sqrt {{{\left( {{t_1} - {t_2}} \right)}^2} + 4{t_1}{t_2}{{\cos }^2}\frac{{{\kappa_y}}}{2}}  + {t_3}} \right|
\end{equation}
while ${E_{unstrain}^{gap}}\left( {{\kappa_y}} \right)$ is given by the same formula with $t_1$ = $t_2$ = $t_3$ $\equiv$ $t_0$  The gap of transmission probability through the junction is then determined as ${E_{junc}^{gap}}\left( {{\kappa_y}} \right) = \max \left[ {E_{unstrain}^{gap}\left( {{\kappa_y}} \right),E_{strain}^{gap}\left( {{\kappa_y}} \right)} \right]$ and, finally, the conduction gap is given by ${E_{cond gap}} = \min \left[ {E_{junc}^{gap}\left( {{\kappa_y}} \right)} \right]$ for $\kappa_y$ in the whole Brillouin zone 

We would like to notice that the Green's function calculations and the banstructure analyses give the same results of conduction gap in the junctions where the transition region between unstrained and strained graphene sections is long enough, i e , larger than about 5 to 6 nm  In the case of short length, as discussed in \cite{baha13,hung14}, this transition zone can have significant effects on the transmission between propagating states beyond the energy gaps and hence can slightly enlarge the gap of conductance, compared to the results obtained from the bandstructure calculations 

\section{Results and discussion}

\begin{figure}[!t]
\centering
\includegraphics[width=3 0in]{Fig02 pdf}
\caption{Dependence of graphene bandgap (in the unit of eV) on the applied strain and its direction: tensile (a) and compressive (b)  The radius from the central point indicates the strain strength ranging from 0 (center) to 30 $\%$ (edge of maps) while the graphene lattice is superimposed to show visibly the strain direction  The orange circle corresponds to the strains of $\sigma = 23 \%$ }
\label{fig_sim2}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=3 4in]{Fig03 pdf}
\caption{Conductance ($G_0 = e^2W/hL_y$) as a function of energy in graphene strained junctions for $\sigma = 4 \%$ with different strain directions  The transport along the armchair direction ($\phi = 0$) is considered  The data obtained in a uniformly strained graphene is displayed for the comparison }
\label{fig_sim6}
\end{figure}
\begin{figure*}[!t]
\centering
\includegraphics[width=5 8in]{Fig04 pdf}
\caption{Local density of states (left panels) and corresponding transmission coefficient (right panels) for three different wave-vectors $k_y$ obtained in an unstrained/strained graphene junction of $\sigma = 4 \%$, and $\theta \equiv \phi = 0$  On the top is a schematic of graphene bandedges illustrating the strain-induced shift of Dirac points along the $k_y$-direction }
\label{fig_sim4}
\end{figure*}
\begin{figure*}[!t]
\centering
\includegraphics[width=5 6in]{Fig05 pdf}
\caption{Maps of conduction gap in unstrained/strained graphene junctions: tensile (a,c) and compressive cases (b,d)  The transport is along the armchair $\phi = 0$ (a,b) and zigzag $\phi = 30^\circ$ directions (c,d)  The strain strength ranges from 0 (center) to 6 $\%$ (edge of maps) in all cases }
\label{fig_sim4}
\end{figure*}
First, we re-examine the formation of the bandgap of graphene under a uniaxial strain  From Eq  (9), it is shown that a strain-induced finite-bandgap appears only if ${E_{strain}^{gap}}\left( {{\kappa_y}} \right) > 0$ for all $k_y$ in the first Brillouin zone, i e , ${k _y} \in \left[ { - \frac{\pi}{L_y}, \frac{\pi}{L_y}} \right]$, otherwise, the bandgap remains zero  Hence, the condition for the bandgap to be finite is either
\begin{equation*}
 \left| {{t_1} - {t_2}} \right| > \left| {{t_3}} \right|\,\,\,\,\,{\rm{OR}}\,\,\,\,\,\left| {{t_3}} \right| > \left| {{t_1} + {t_2}} \right|
\end{equation*}
and the corresponding values of bandgap are
\begin{equation*}
 {E_{gap}} = 2\left( {\left| {{t_1} - {t_2}} \right| - \left| {{t_3}} \right|} \right)\,\,\,\,\,{\rm{OR}}\,\,\,\,\,2\left( {\left| {{t_3}} \right| - \left| {{t_1} + {t_2}} \right|} \right)
\end{equation*}
This result was actually reported in \cite{per209,hase06}  We remind as displayed in Fig  2(a) that a finite bandgap opens only for strain larger than $\sim 23 \%$ and the zigzag (not armchair) is the preferred direction for bandgap opening under a tensile strain \cite{per209}  We extend our investigation to the case of compressive strain and find  (see in Fig  2(b)) that (i) the same gap threshold of $\sigma \simeq 23 \%$ is observed but (ii) the preferred direction to open the gap under a compressive strain is the armchair, not the zigzag as the case of tensile strain  This implies that the properties of graphene bandstructure at low energy should be qualitatively the same when applying strains of $\left\{ {\sigma ,\theta } \right\}$ and of $\left\{ {-\sigma ,\theta + 90^\circ} \right\}$  This feature can be understood by considering, for example, strains of $\left\{ {\sigma , \theta = 0} \right\}$ and of $\left\{ {-\sigma , \theta = 90^\circ} \right\}$  Indeed, these strains result in the same qualitative changes on the bond-lengths, i e , an increased bond-length $r_3$ and reduced bond-lengths $r_{1,2}$
Context 3: 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



ground_truth: It is well described by the Gaussian theory.

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: How does the conduction gap depend on the strain direction?

Read the following text and answer briefly.

Context 2:  However, for the same strain strength, because of the exponential dependence of hoping energies on the bond-lengths, the compressive strain generally induces a larger bandgap than the tensile one, as can be seen when comparing the data displayed in Figs  2(a) and 2(b)  To conclude, we would like to emphasize that a large strain is necessary to open a bandgap in graphene  This could be an issue for practical applications, compared to the use of graphene strained junctions explored in \cite{hung14} 

We now go to explore the properties of conduction gap in the graphene strained junctions  In Fig  3, we display the conductance as a function of energy computed from Eq  (5) using the Green's function technique  As discussed above, a small strain of a few percent (e g , 4 $\%$ here) can not change the gapless character of graphene, i e , there is no gap of conductance in the case of uniformly strained graphene  However, similar to that reported in \cite{hung14}, a significant conduction-gap of a few hundreds meV can open in the unstrained/strained graphene junctions  The appearance of this conduction gap, as mentioned previously, is due to the strain-induced shift of Dirac points and is explained as follows  Actually, the strain causes the lattice deformation and can result in the deformation of graphene bandstructure  Therefore, the bandedges as a function of wave-vector $k_y$ in unstrained and strained graphene can be illustrated schematically as in the top panel of Fig  4  As one can see, the shift of Dirac points leads to the situation where there is no value of $\kappa_y$, for which the energy gaps $E_{unstrain}^{gap}\left( {{\kappa_y}} \right)$ and $E_{strain}^{gap}\left( {{\kappa_y}} \right)$ are simultaneously equal to zero  This means that the transmission probability always shows a finite gap for any $\kappa_y$  For instance, the energy gap is zero (or small) in the unstrained (resp  strained) graphene section but finite in the strained (resp  unstrained) one in the vicinity of Dirac point $k_y = K_{unstrain}$ (resp  $K_{strain}$)  Accordingly, as illustrated in the pictures of LDOS in the left panels of Fig  4 and confirmed in the corresponding transmissions in the right panels, clear gaps of transmission are still obtained  Far from these values of $k_y$, $E_{unstrain}^{gap}\left( {{\kappa_y}} \right)$ and $E_{strain}^{gap}\left( {{\kappa_y}} \right)$ are both finite (e g , see the LDOS plotted for $k_y = K_{gap}$) and hence a finite gap of transmission also occurs  On this basis, a finite gap of conductance is achieved  More important, Fig  3 shows that besides the strength of strain, the strain effect is also strongly dependent on the applied direction  For instance, the conduction gap takes the values of $\sim$ 295, 172 and 323 meV for $\theta = 0$, $30^\circ$ and $90^\circ$, respectively 

Below, we will discuss the properties of the conduction gap with respect to the strain, its applied direction, and the direction of transport  Note that due to the lattice symmetry, the transport directions $\phi$ and $\phi + 60^\circ$ are equivalent while the applied strain of angle $\theta$ is identical to that of $\theta + 180^\circ$  Hence, the data obtained for $\phi$ ranging from $-30^\circ$ to $30^\circ$ and $\theta  \in \left[ {0^\circ ,180^\circ } \right]$ covers the properties of conduction gap in all possible cases 

In Fig  5, we present the maps of conduction gap with respect to the strain and its applied direction in two particular cases: the transport is either along the armchair ($\phi = 0$) or the zigzag ($\phi = 30^\circ$) directions  Both tensile and compressive strains are considered  Let us first discuss the results obtained in the armchair case  Figs  5(a,b) show that (i) a large conduction gap up to about 500 meV can open with a strain of 6 $\%$ and (ii) again the conduction gap is strongly $\theta$-dependent, in particular, its peaks occur at $\theta = 0$ or $90^\circ$ while the gap is zero at $\theta \approx 47^\circ$ and $133^\circ$ for tensile strain and at $\theta \approx 43^\circ$ and $137^\circ$ for compressive strain  In principle, the conduction gap is larger if the shift of Dirac points in the $\kappa_y$-axis is larger, as discussed above about Figs  3-4  We notice that the strain-induced shifts can be different for the six Dirac points of graphene \cite{kitt12} and the gap is zero when there is any Dirac point observed at the same $\kappa_y$ in the two graphene sections  From Eq  (9), we find that the Dirac points are determined by the following equations:
\begin{eqnarray*}
  {\cos}\frac{\kappa_y}{2} &=& \pm \frac{1}{2}\sqrt{\frac{{t_3^2 - {{\left( {{t_1} - {t_2}} \right)}^2}}}{{{t_1}{t_2}}}}, \\
  \cos \frac{{\kappa_x}}{2} &=& \frac{{{t_1} + {t_2}}}{{\left| {{t_3}} \right|}}\cos \frac{{\kappa_y}}{2},\,\,\,\sin \frac{{\kappa_x}}{2} = \frac{{{t_2} - {t_1}}}{{\left| {{t_3}} \right|}}\sin \frac{{\kappa_y}}{2},
\end{eqnarray*}
which simplify into ${\cos}\frac{\kappa_y}{2} = \pm \frac{1}{2}$ and, respectively, $\cos \left( {\frac{{{\kappa _x}}}{2}} \right) = \mp 1$ in the unstrained case  Hence, the zero conduction gap is obtained if
\begin{equation*}
  \frac{{t_3^2 - {{\left( {{t_1} - {t_2}} \right)}^2}}}{{4{t_1}{t_2}}} = \frac{1}{4}
\end{equation*}
Additionally, it is observed that the effects of a strain $\{\sigma,\theta\}$ are qualitatively similar to those of a strain $\{-\sigma,\theta+90^\circ\}$, i e , the peaks and zero values of conduction gap are obtained at the same $\theta$ in these two situations  To understand this, we analyze the strain matrix $M_s \left(\sigma,\theta\right)$ and find that in the case of small strains studied here, there is an approximate relationship between the bond lengths under these two strains, given by \[{r \left( \sigma, \theta \right)} - {r \left( -\sigma, \theta + 90^\circ\right)} \simeq \sigma \left( {1 - \gamma } \right) r_0,\] which is $\theta$-independent for all \emph{C-C} bond vectors  It implies that there is a fixed ratio between the hopping energies $t_i \left( \sigma, \theta \right)$ and $t_i \left( -\sigma, \theta + 90^\circ\right)$ and hence there is the similar shift of Dirac points in these two cases 
\begin{figure}[!t]
\centering
\includegraphics[width=3 4in]{Fig06 pdf}
\caption{Map showing the dependence of conduction gap on the directions ($\theta,\phi$) for $\sigma = 4 \%$  The top is a diagram illustrating the rotation of Dirac points in the \emph{k}-space with the change in the transport direction $\phi$ }
\label{fig_sim6}
\end{figure}
\begin{figure*}[!t]
\centering
\includegraphics[width=5 5in]{Fig07 pdf}
\caption{Maps of conduction gap obtained in tensile/compressive strained junctions  The transport along the armchair/zigzag directions is considered in (a,b)/(c,d), respectively  The strains $\sigma_c = -2 \%$ and $\sigma_t = 2 \%$ are applied in (a,c) while $\sigma_c = -1 \%$ and $\sigma_t = 3 \%$ in (b,d) }
\label{fig_sim4}
\end{figure*}

We now go to analyze the properties of conduction gap shown in Figs  5(c,d) where the transport is along the zigzag direction $\phi = 30^\circ$  In fact, the conduction gap in this case can reach a value as high as that of the case of $\phi = 0$ but has different $\theta$-dependence  In particular, the conduction gap has peaks at $\theta \approx 47^\circ$ and $133^\circ$ for tensile strain and at $\theta \approx 43^\circ$ and $137^\circ$ for compressive strain, where it is zero in the case of $\phi = 0$  It is also equal to zero at $\theta = 0$ and $\theta = 90^\circ$ where the peaks of conduction gap occur in the latter case of $\phi = 0$  The relationship between these two transport directions can be explained as follows  On the one hand, based on the analyses above for $\phi = 0$, we find that for a given strength of strain, a maximum shift of Dirac points along the $k_y$-axis corresponds to a minimum along the $k_x$-one and vice versa when varying the strain direction $\theta$  On the other hand, as schematized in the top of Fig  6 below, the change in the transport direction results in the rotation of the first Brillouin zone, i e , the $k_x$ (resp  $k_y$) axis in the case of $\phi = 30^\circ$ is identical to the $k_y$ (resp  $k_x$) axis in the case of $\phi = 0$  These two features explain essentially the opposite $\theta$-dependence of conduction gap for $\phi = 30^\circ$, compared to the case of $\phi = 0$ as mentioned  Again, we found the same qualitative behavior of conduction gap when applying the strains of $\{\sigma,\theta\}$ and $\{-\sigma,\theta+90^\circ\}$ 

Next, we investigate the conduction gap with respect to different transport directions $\phi$  We display a ($\theta,\phi$)-map of conduction gap for $\sigma = 4 \%$ in Fig  6 and, in the top, an additional diagram illustrating the rotation of Dirac points in the $k-$space with the change in the transport direction  It is clearly shown that (i) a similar scale of conduction gap is obtained for all different transport directions, (ii) there is a smooth and continuous shift of $E_{cond gap}-\theta$ behavior when varying $\phi$, and (iii) the same behavior of $E_{cond gap}$ is also observed when comparing the two transport directions of $\phi$ and $\phi+30^\circ$, similarly to the comparison above between $\phi = 0^\circ$ and $30^\circ$  The data plotted in Fig  6 additionally shows that $E_{cond gap}$ takes the same value in both cases of $\{\phi,\theta\}$ and $\{-\phi,-\theta\}$ with a remark that the strains of $-\theta$ and $180^\circ-\theta$ are identical  Moreover, the values of $\theta$ and $\phi$, for which the conduction gap has a peak or is equal to zero, have an almost linear relationship  In particular, the relationship for conduction gap peaks is approximately given by $\theta = \theta_A - \eta_s \phi$  For tensile strains, $\eta_s$ takes the values of $\sim 1 5667$ and $1 4333$ for $\theta_A = 0$ and $90^\circ$, respectively  On the opposite, it is about $1 4333$ and $1 5667$ for $\theta_A = 0$ and $90^\circ$, respectively, for compressive strain cases  All these features are consequences of the rotation of Dirac points in the $k$-space with respect to the transport direction $\phi$ as illustrated in the diagram on the top and the lattice symmetry of graphene 

Finally, we investigate other junctions based on compressive and tensile strained graphene sections  The idea is that in this type of strained junction, the shifts of Dirac points are different in two graphene sections of different strains, which offers the possibilities to use smaller strains to achieve a similar conduction gap, compared to the case of unstrained/strained junction  In Fig  7, we display the maps of conduction gap with respect to the directions of compressive ($\theta_c$) and tensile ($\theta_t$) strains in two cases of transport direction $\phi = 0$ (armchair) and $30^\circ$ (zigzag) for given strain strengths  Indeed, as seen in Fig  7(a,b), with smaller strains $\left\{ {{\sigma _c},{\sigma _t}} \right\} = \left\{ { - 2\% ,2\% } \right\}$ or $\left\{ { - 1\% ,3\% } \right\}$, similar conduction gap of about 310 meV can be achieved (see Figs  7(a,b)) while it requires a strain of 4 $\%$ in the unstrained/strained junctions discussed above  However, since the shift of Dirac points is strongly dependent on the direction of applied strains and the transport direction, the properties of conduction gap are more complicated than in the latter case  In particular, our calculations show that the preferred transport directions to achieve a large conduction gap are close to the armchair one  Otherwise, the conduction gap is generally smaller, similarly to the data for $\phi = 30^\circ$ compared to $\phi = 0$, as shown in Fig  7  Additionally, it is shown that the preferred directions of applied strains in the case of $\phi = 0$ are close to ${\theta _c} \equiv {\theta _t} = 0$ or $90^\circ$ 

\section{Conclusion}

Based on the tight binding calculations, we have investigated the effects of uniaxial strain on the transport properties of graphene strained junctions and discuss systematically the possibilities of achieving a large conduction gap with respect to the strain, its applied direction and the transport direction  It has been shown that due to the strain-induced deformation of graphene lattice and hence of graphene bandstructure, a finite conduction gap higher than 500 meV can be achieved for a strain of only 6 $\%$  Moreover, as a consequence of the shift of Dirac points along the $k_y$-axis, the conduction gap is strongly dependent not only on the strain strength but also on the direction of applied strain and the transport direction  A full picture of these properties of conduction gap has been presented and explained  The study hence could be a good guide for the use of this type of unstrained/strained graphene junction in electronic applications 

\textbf{\textit{Acknowledgment }} This research in Hanoi is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103 02-1012 42  We also acknowledges the French ANR for financial support under the projects NANOSIM-GRAPHENE (Grant no  ANR-09-NANO-016) and MIGRAQUEL (Grant no  ANR-10-BLAN-0304) 


Context 3:  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



ground_truth: Peaks occur at certain strain directions, while the gap is zero at others.

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