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 framework capture the reduced-order dynamics?

Read the following text and answer briefly.

Context 1: Paper Info

Title: Interpretable reduced-order modeling with time-scale separation Interpretable reduced-order modeling with time-scale separation
Publish Date: 7 March 2023
Author List: Sebastian Kaltenbach (from CSE-Lab, ETH Zurich, Harvard SEAS), Phaedon-Stelios Koutsourelakis (from CSE-Lab, ETH Zurich, Harvard SEAS), Petros Koumoutsakos (from CSE-Lab, ETH Zurich, Harvard SEAS), Harvard Seas (from CSE-Lab, ETH Zurich, Harvard SEAS)

Figure

FIG  5  Comparison between the phase-space of the reference solution (left) and the phase-space of the predictions
FIG  7  Comparison between predictions and reference solutions for a new initial condition fort = 1 25, 3 75, 7 5, 12 5, 20, 30  (from left to right and top to down) We note that with longer prediction time the uncertainty bounds increases Despite the chaotic nature of the KS equation, the predictive posterior mean is close to the reference solution for t ≤ 12 5

abstract

Partial Differential Equations (PDEs) with high dimensionality are commonly encountered in computational physics and engineering  However, finding solutions for these PDEs can be computationally expensive, making model-order reduction crucial  We propose such a data-driven scheme that automates the identification of the time-scales involved and, can produce stable predictions forward in time as well as under different initial conditions not included in the training data 
To this end, we combine a non-linear autoencoder architecture with a time-continuous model for the latent dynamics in the complex space  It readily allows for the inclusion of sparse and irregularly sampled training data  The learned, latent dynamics are interpretable and reveal the different temporal scales involved 
We show that this data-driven scheme can automatically learn the independent processes that decompose a system of linear ODEs along the eigenvectors of the system's matrix  Apart from this, we demonstrate the applicability of the proposed framework in a hidden Markov Model and the (discretized) Kuramoto-Shivashinsky (KS) equation 
Additionally, we propose a probabilistic version, which captures predictive uncertainties and further improves upon the results of the deterministic framework 

INTRODUCTION

High-fidelity simulations of critical phenomena such as ocean dynamics and epidemics have become essential for decision-making  They are based on physically-motivated PDEs expressing system dynamics that span multiple spatiotemporal scales and which necessitate cumbersome computations   In recent years there is increased attention to the development of data-driven models that can accelerate the solution of these PDEs as well as reveal salient, lower-dimensional features that control the long-term evolution 
In most cases, data-driven reduced-order models are not interpretable  In particular, models based on neural networks despite good predictive capabilities , they offer a black-box description of the system dynamics  A possible remedy is applying a symbolic regression to the learned neural network representation , but this adds additional computational cost due to the two-step procedure 
A number of frameworks such as SINDy allows to learn interpretable dynamics but it relies on the a-priori availability of lower-dimensional descriptors and of time-derivatives which can be very noisy for both simulation and experimental data  Other frameworks are tailored to specific problems such as molecular dynamics  
Here, we present a framework that only needs the value of the observables, and not their derivatives, as training data and is capable of identifying interpretable latent dynamics  The deployment of interpretable latent dynamics ensures that conservation of important properties of that are reflected in the reduced-order model  
The present method is related to approaches based on the Koopman-operator extended Dynamic Mode Decomposition (eDMD) but uses continuous complex-valued latent space dynamics and only requires one scalar variable per latent dimension to describe the latent space dynamics  Therefore we do not have to enforce any parametrizations on the Koopman matrix  
The time-continuous formulation moreover allows to incorporate sparse and irregularly sampled training data and fast generation of predictions after the training phase  By using a complex-valued latent space we can also incorporate harmonic effects and reduce the number of latent variables needed  Linear and non-linear autoencoders are used to map the observed, high-dimensional time-series to the lower-dimensional, latent representation and we identify simultaneously the autoencoder as well as the latent dynamics by optimizing a combined loss function 
Hence the to tasks of dimensionality reduction and discovery of the reduced dynamics are unified while other frameworks treat the two parts separately   Apart from using an architecture based on autoencoders to identify the latent space, projection-based methods could also be employed   We are also proposing a probabilistic version of our algorithm ) that makes use of probabilistic Slow Feature Analysis  
This allows for a latent representation that arart from being time-continuous, can quantify the predictive uncertainty and hierarchically decompose the dynamics into their pertinent scales while promoting the discovery of slow processes that control the system's evolution over long time horizons  The rest of the paper is structured as follows: We introduce the methodological framework as well as algorithmic details in section II 
Particular focus is paid on the interpretability of the inferred lower-dimensional dynamics  In section III we present three numerical illustrations, i e  a system of linear ODEs, a hidden Markov Model and the discretized KS-equation  We then present in section IV the probabilistic extension of the framework and apply it to the KS-equation 
We conclude with a summary and a short discussion about possible next steps  We introduce the autoencoders deployed in this work, followed by the interpretable latent space dynamic and discuss the training process  We consider data from high-dimensional time series x n ∈ R f with n = 1,    , T   We remark that the intervals between the different states do not need to be uniformly spaced 

Autoencoder

A core assumption of the method is that each high-dimensional state x n can be compressed to a lower-dimensional representation z n ∈ C c with c << f   We identify this lower-dimensional representation by an autoencoder consisiting of a parameterized encoder and decoder  The encoder maps the high-dimensional representation to the latent space as:
The latent space is complex-valued  The decoder reconstructs the high-dimensional representation based on the latent variables as: We denote the parameters of the encoder as well as the decoder by θ  As discussed later in Section II C, both set of parameters are optimized simultaneously during training and therefore there is no need for differentiating them 

Interpretable Latent Space Dynamics

We employ a propagator in the latent space to capture the reduced-order dynamics of the system  In contrast to other time-extended variational autoencoder frameworks, our representation uses complex valued latent variables  In addition the latent variables are treated independently  The latter feature enables us to have an interpretable latent dynamics as well as a model that is especially suitable for being trained in the Small Data regime due to the small number of required parameters 
This is in contrast to temporal propagators such as LSTMs   For each dimension i of the latent variable z we are using the following continuous ODE in the complex plane: By solving this ODE, we can define the operator: Interpretable reduced-order modeling with time-scale separation Here, λ is a vector containing all the individual λ's and ∆t n indicates the time-step between the latent states 
The symbol is used to indicate a component-wise multiplication  We remark that the latent variables and the parameter governing the temporal evolution are complex numbers and their role in describing the system dynamics is similar to that of an eigenvalue  The real part is associated with growth and decay whereas the imaginary part is representing the periodic component 
This approach has similarities with the Koopman-operator based methods and the extended dynamic mode decomposition   In contrast to the methods mentioned before we are using a continuous formulation in the latent space that allows us to incorporate scarce and irregularly sampled training data and directly rely on complex numbers in the latent space 

Training and Predictions

We optimize a loss function that combines both a reconstruction loss as well as a loss associated with the error of our learned propagator in the latent space: (5) We note that we could directly incorporate mini-batch training by only taking the summation over a subset of the N available training data 
For new predictions of unseen states, we use the encoder to generate a latent representations which is then advanced in time by the learned propagator  At a designated time step we are using the decoder to reconstruct the high-dimensional solution  We applied our algorithm to three systems  First, we show that the algorithm is capable of exactly reproducing the solution of a linear ODE and to identify its eigenvalues 
Afterwards we are applying the framework to a high-dimensional process generated by a complex latent dynamics, which is correctly identified  As a final test case, we are applying the algorithm to a Kuramoto Shivashinski (KS) equation  Interpretable reduced-order modeling with time-scale separation

Linear ODE

We are considering a two-dimensional ODE system for x = y 1 y 2 : Based on the obtained training data we run our algorithm using a linear encoder and decoder structure as well as two latent variables z  The loss function was optimized using the Adam algorithm   As we consider a linear ODE we can analytically compute the eigenvalues involved and compare it with the parameters λ identified by our algorithm 
We observe in Figure that the algorithm was able to recover the correct values, i e  the eigenvalues 7 and 3 of the given linear ODE  The system does not have a periodic component and the two imaginary parts correctly go to zero, whereas the real parts converge to the reference value  Moreover we are also able to identify for the linear mapping between our latent variables z and the training data a matrix consisting of a multiple of the eigenvectors (1,1) and (1,-1) and thus the correct solution 
This example was chosen to show that the algorithm is able to quickly identify the exact solution of a linear ODE in terms of its linearly independent components 

Hidden multiscale dynamics

We consider eight-dimensional synthetic time series data produced by an underlying twodimensional complex valued process  In particular, the data points x are generated by first solving for the temporal evolution for the two complex-valued processes p 1 and p 2 and than mapping to the eight-dimensional space by using a randomly sampled linear mapping W  
One of the two processes used to generate the data is chosen to be much slower than the other one and both processes have a periodic component  dp 2 dt = (−0 9 + 1 5i)p 2 (8) As training data we consider 40 time series with 150 data points each, obtained by simulating the described processes for a maximum of t = 15 s and then sampling from the obtained data points 
Hence the training data consists of: • 40 time-series • with each consisting 150 observations of the x at a uniform time-step ∆t = 0 0025 The autoencoder obtained consists of one linear layer for both the decoder as well as the encoder  The model is trained for 5000 iterations using the Adam optimizer and a learning rate of 10 −3  
The results for the convergence of the parameters λ 1 and λ 2 can be found in Figure   We note that the process which is slower decaying and thus more responsible for the long-term evolution of the system has a higher convergence rate than the faster process  With the obtained parameters λ as well as the trained autoencoder, we compute predictions based on the last time step used for training, i e  we apply the encoder to obtain our latent representation and than use the latent dynamics to advance the latent representation in time 
Afterwards, we employ the decoder to reconstruct the full high-dimensional system  The results can be found in Figure and show very good agreement between predictions and reference data  This example shows that our model is successfully able to carry out dimensionality reduction and moreover indicates that the convergence rate between latent processes can be different 
The latter is relevant when training models as for accurate predictions all latent processes and their dynamics should be converged 

Kuramoto-Sivashinsky

Finally, we applied our algorithm to the KS equation and aim to identify a reduced-order model for the solution u(y, t): We employed periodic boundary conditions, µ = 1 and a domain size y ∈ [0, 22]  For this domain-size, the KS-equation exhibits a structurally stable chaotic attractor as discussed in The black lines divides the area for which training data was given from the area without raining data 
;   The equation is discretized in space using a discretization step of 22 64 resulting in a state vector x of dimension 64 and a nonlinear system of coupled ODEs  This is solved using a stiff fourth-order solver  We employed a non-linear encoder and decoder with four fully-connected layers each and ReLU-activation functions as well as Dropout Layers between the fully-connected layers 
We trained the model for 200000 iterations using Adam and a learning rate of 5 • 10 4 and assuming a five-dimensional latent space  We obtained the λ's in Figure   Four latent variables have λ's close to zero and thus a slow temporal dynamic that is responsible for the long-term evolution whereas one latent variable is quickly decaying 
Based on the obtained parameters, we do predictions based on an unseen initial condition not contained in the training data  We are able to reconstruct the correct phase space based on our predictions despite only using a very limited amount of training data  The results for the phase space can be seen in Figure  
Although the small-scale fluctuations in the temporal dynamics are not well captured, the model identifies the correct manifold which has a good accuracy compared to the reference solution  All phase-spaces were obtained by using a finite-difference operator on the data or predictions  These results are in accordance Interpretable reduced-order modeling with time-scale separation with whose LSTM-based temporal dynamic model was also able to find the correct phase space but not to track the actual dynamics for long-term predictions 
Our model is not able to account for noise in the temporal evolution and thus dealing with chaotic, small-scale fluctuations is challenging  We believe that a probabilistic version of our algorithm could be advantageous here  This section contains a fully probabilistic formulation for the deterministic model discussed before 
We replace the Autoencoder with a Variational Autoencoder and the ODE in the latent space with a SDE  The loss function which we optimize is the Evidence-Lower Bound (ELBO) 

Model Structure

We postulate the following relations for our probabilistic model using an Ornstein-Uhlenbeck (OU) for each dimension i of the latent space and a Wiener process W t in the latent space: We again assume that the latent variables z t are complex-valued and a priori independent  Complex variables were chosen as their evolution includes a harmonic components which are observed in many physical systems 
We assume an initial conditions z 0,i ∼ CN (0, σ 2 0,i )  The total parameters associated with the latent space dynamics of our model are thus {σ 2 0,i , σ 2 i , λ i } c i=1 and will be denoted by θ together with all parameters responsible for the decoder mapping G (see next section)  These parameters along with the state variables z t have to be inferred from the data x t  
Based on probabilistic Slow Feature Analysis (SFA) , we set σ 2 i = 2; (λ j ) and σ 2 0,i = 1  As a consequence, a priori, the latent dynamics are stationary  A derivation and reasoning for this choice can be found in Appendix A  Hence the only independent parameters are the λ i , the imaginary part of which can account for periodic effects in the latent dynamics 

Variational Autoencoder

We employ a variational autoencoder to account for a probabilistic mappings from the lower-dimensional representation z n to the high-dimensional system x n   In particular we are employing a probabilistic decoder The encoder is used to infer the state variables z based on the given data and thus defined in the inference and learning section 

Inference and Learning

Given the probabilistic relations , our goal is to infer the latent variables z 0:T as well as all model parameters θ  We follow a hybrid Bayesian approach in which the posterior of the state variables is approximated using amortized Variational Inference and Maximum-A-Posteriori (MAP) point-estimates for θ are computed 
The application of Bayes' rule for each data sequence x 0:T leads to the following posterior: where p(θ) denotes the prior on the model parameters  In the context of variational inference, we use the following factorization of the approximate posterior i e  we infer only the mean µ and variance σ for each state variable based on the given data points 
This conditional density used for inference is the encoder-counterpart to the probabilistic decoder defined in the section before  It can be readily shown that the optimal parameter values are found by maximizing the Evidence Lower Bound (ELBO) F(q φ (z 0:T ), θ) which is derived in Appendix B  We compute Monte Carlo estimates of the gradient of the ELBO with respect to φ and θ with the help of the reparametrization trick and carry out stochastic optimization with the ADAM algorithm  

Results for the probabilistic extension

We applied our probabilistic version to the KS-equation  We used the same settings as for the deterministic approach but considered up to 10 complex latent variables  The obtained λ's are in Figure   The probabilistic model allows us to quantify the uncertainty in predictions  In Figure predictions for various time-steps and the respective uncertainty bounds are shown for an unseen initial condition 
Due to the chaotic nature of the KS-equation and the small amount of training data, the underlying linear dynamic of our model is only able to capture the full dynamics for a limited time horizon  Fortunately, due to the probabilistic approach the model is capable of capturing chaotic fluctuations with increasingly wide uncertainty bounds 
We also computed the phase space representation for the KS-equation based on the predictions obtained by our model and compare it with the reference solution  The probabilistic model identifies the correct manifold with a better accuracy than the deterministic model  As some of the small-scale fluctuations are accounted as noise, the resulting manifold is more concentrated at the origin and the obtained values are slightly smaller than the reference manifold although their shape is very similar 
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: \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



ground_truth: By using a propagator in the latent space.

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:  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



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 is the average magnetic moment per column in Ge$_{1-x}$Mn$_{x}$ films?

Read the following text and answer briefly.

Context 3: 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 





ground_truth: 1425 $\mu_{B}$.