Towards a Unified Approach to Homography Estimation
Using Image Features and Pixel Intensities

Lucas Nogueira, Ely C. de Paiva

School of Mechanical Engineering

University of Campinas

Campinas, SP, Brazil

Geraldo Silveira

Robotics and Computer Vision research group

Center for Information Technology Renato Archer

Campinas, SP, Brazil

The homography matrix is a key component in various vision-based robotic tasks. Traditionally, homography estimation algorithms are classified into feature- or intensity-based. The main advantages of the latter are their versatility, accuracy, and robustness to arbitrary illumination changes. On the other hand, they have a smaller domain of convergence than the feature-based solutions. Their combination is hence promising, but existing techniques only apply them sequentially. This paper proposes a new hybrid method that unifies both classes into a single nonlinear optimization procedure, applies the same minimization method, and uses the same homography parametrization and warping function. Experimental validation using a classical testing framework shows that the proposed unified approach has improved convergence properties compared to each individual class. These are also demonstrated in a visual tracking application. As a final contribution, our ready-to-use implementation of the algorithm is made publicly available to the research community.

Robot vision; Homography optimization; Hybrid approaches; Vision-based applications.

§ INTRODUCTION

The homography matrix is a key component in computer vision. It relates corresponding pixel coordinates of a planar object in different images, and has been used in a variety of vision-based applications such as image mosaicing
[1], visual servoing [2] and object grasping [3]. The homography estimation task can be formulated as an Image Registration (IR) problem. IR can be defined as a search for the parameters that best define the transformation between corresponding pixels in a pair of images. Solutions to this problem involve the definition of at least four important characteristics [4]: the information space, the transformations models, the similarity measures, and the search strategy.

With respect to the information space, the vast majority of vision-based algorithms use a Feature-Based (FB) approach. In this class, firstly an extraction algorithm searches each image for geometric primitives and selects the best candidates. Then, a matching algorithm establishes correspondences between features in different images. Afterwards, the actual estimation takes place. However, both the extraction and matching steps are error-prone and can produce outliers that affect the quality of the estimation. Additionally, by using only a sparse set of features, these algorithms may discard useful information.

In contrast, Intensity-Based (IB) methods have no extraction and matching steps. These methods are also referred to as direct methods since they exploit the pixel intensity values directly. This allows the estimation algorithm to work with more information than FB methods and does not depend on particular primitives. Thus, it leads to more accurate estimates and is highly versatile. However, an important drawback is that they require a small interframe displacement, i.e., a sufficient overlapping between consecutive images.

The algorithms presented in this work use multidimensional optimization methods as the main search strategy for the image registration problem. When formulated as such, an initial solution is iteratively refined using a nonlinear optimization method. Specifically, the algorithms presented here are derived from the Efficient Second-order Minimization method (ESM) [5]. Its advantages include both a higher convergence rate and a larger convergence domain than standard iterative methods. It allows for a second-order approximation of the Taylor series without computationally expensive Hessian calculations.

The use of the ESM framework has shown remarkable results for IB methods. However, its application within FB methods has been limited so far. As discussed, the two classes of estimation methods have complementary strengths. This work aims to develop a hybrid method that exploits their advantages and reduces their shortcomings. The proposed algorithm is made available as ready-to-use ROS [6] packages and as a C++ library. In particular, a homography-based visual tracking application is also developed. In summary, our contribution is the development of a vision-based algorithm that:

* unifies the intensity- and feature-based approaches to homography estimation into a single nonlinear optimization problem;

* solves that problem using the same efficient minimization method, homography parametrization, and warping function;

* can be applied in real-time settings, such as for homography-based visual tracking as experimentally demonstrated in this paper; and

* its ready-to-use implementation is made publicly available for research purposes as a C++ library and as a ROS package.

The remainder of this article is organized as follows. Section <ref> presents the related works, whereas Section <ref> describes the proposed unified approach.  Section <ref> then reports the benchmarking experiments and the application of the proposed algorithm to visual tracking. Finally, the conclusions are drawn in Section <ref>, and some references are given for further details.

§ RELATED WORKS

The main distinction between IB and FB methods regards their information space. Indeed, on one hand FB requires the extraction and association of geometric primitives in different images before the actual estimation can occur. These primitives can be, e.g., points and lines [1][7]. IB methods simultaneously solves for the estimation problem and pixel correspondences with no intermediate steps [8][9].

The transformation model dictates which parameters are estimated. For example, the original Lucas-Kanade [10] algorithm only estimated translations in the image space. This was later extended to more sophiscated warp functions [11]. Simultaneous Localization And Mapping (SLAM) algorithms commonly use IR to perform the pose and structure estimation [12]. The homography matrix is often used as a transformation model when dealing with predominantly planar regions of interest [13][14][15]. Illumination parameters may also be considered as a component of the transformation model, e.g., in [16].

The quality of the IR is defined by a similarity measure. When an optimization method is applied, this measure is often used as a cost function, such as the Sum of Squared Differences (SSD) [10][17]. Other possibilities include correlation-based metrics [18][19] and mutual information [20].

The last component of IR algorithms is the search strategy. Most real-time applications use a multidimensional optimization approach based on gradient descent. They use the first and second derivatives of the similarity measures with respect to the transformation parameters. The ESM algorithm is such an example, and is applied in the proposed method. Alternative optimization approaches include Gauss-Newton and Levenberg-Marquardt [21]. All of these techniques are most suited to applications with small interframe displacements. Indeed, global techniques are too computationally expensive to be applied in real-time settings. A more thorough review and comparison of image registration algorithms can be found in [22][23].

As for the existing techniques that combine IB and FB methods, their overwhelming majority only applies them sequentially, e.g., [24][25]. In sequential strategies, a FB technique is firstly considered and then its estimated parameters are fed as the initial guess to some IB optimization. This standard combination scheme is thus not optimal and is more time consuming. An exception to that sequential procedure is reported in [26]. However, it aims to estimate the pose parameters, which requires a calibrated camera. The objective of this paper is to estimate the projective homography, i.e., there is no calibrated camera. Furthermore, that existing technique applies a first-order minimization method, and the considered scaling factors do not take into account the convergence properties of the individual approaches, as will be proposed in the sequel.

§ PROPOSED UNIFIED APPROACH

Consider that a reference template has been specified to an estimation algorithm. This is typically a region of interest with predefined resolution inside a larger reference image. Then, a second image, referred to as the current image, is given to that algorithm. The goal is to find the transformation parameters that, when applied to the current image, results in a current template identical to the reference template.

§.§ Transformation Models

The considered transformation models consist of a geometric and a photometric one. The geometric transformation model explains image changes due to variations in the scene structure and/or the camera motion. For a given pixel $\vect{p}^*$ in the reference template that corresponds to pixel $\vect{p}$ in the current image, we model the geometric motion using a homography:

\begin{align}
\mathbf{p} \, &\propto \, \mathbf{H} \, \mathbf{p}^*\\
&= \left[ \,
\frac{h_{11} u^* + h_{12} v^* + h_{13}} {h_{31} u^* + h_{32} v^*
+ h_{33}}, \, \frac{h_{21} u^* + h_{22} v^* + h_{23}} {h_{31}
u^* + h_{32} v^* + h_{33}}, \, 1 \, \right]^\top\\
&= \vect{w}(\vect{H},\vect{p}^*),
\label{equ:homog}
\end{align}

where $\mathbf{p}^* = [u^*, v^*, 1]^\top \in \mathbb{P}^2$ is the homogeneous pixel coordinates in the reference template, $\vect{w}$ is the warping operator, and $\mathbf{H} \in \mathbb{SL}(3)$ is the projective homography matrix with its elements
$\{ h_{ij} \}$. Such matrix has only eight degrees-of-freedom. In general, this situation leads to a reprojection step after each iteration of the minimization algorithm that takes the estimated homography into the Special Linear Group. To avoid this problem, the proposed algorithm parameterizes the homography using its corresponding Lie Algebra [2]. This is accomplished via the matrix exponential function, which maps a region around the identity matrix $\vect{I} \in \mathbb{SL}(3)$ to a region around the origin $\vect{0} \in \mathfrak{sl}(3)$. A matrix $\vect{A}(\vect{v}) \in \mathfrak{sl}(3)$ is the linear combination of eight matrices that form a base of the Lie Algebra. Therefore $\vect{v}$ has eight components. A homography is thus parameterized as

\begin{equation}
\vect{H}(\vect{v}) = \exp(\vect{A}(\vect{v})) . %= \sum_{i=0}^{\infty}\frac{1}{i!} (\vect{A}(\vect{v}))^i.
\label{equ:matrix_exp}
\end{equation}

The homography matrix may be used to extract relative motion and scene structure information [27]. However, this decomposition is out of the scope of this work and is unnecessary for many robotic applications.

The photometric transformation model explains the changes in the image due
to variations in the lighting conditions of the scene. Let us model in this work only global illumination variations, i.e., changes that apply
equally to all pixels in the images. This model is defined as

\begin{equation}\label{eq:I}
\mathcal{I}'(\mathbf{p}) = \alpha \, \mathcal{I}(\mathbf{p}) + \beta,
\end{equation}

where $\mathcal{I}(\mathbf{p}) \geq 0$ is the intensity value of the pixel
$\mathbf{p}$, $\mathcal{I}' (\mathbf{p}) \geq 0$ denotes its transformed
intensity, and the gain $\alpha \in \mathbb{R}$ and the bias $\beta \in \mathbb{R}$ are the
parameters that fully define the transformation. These parameters
can be viewed as the adjustments in the image constrast and brightness, respectively.

§.§ Nonlinear Least Squares Formulation

Consider that the reference template is composed of $m$ pixels. Also, consider that a feature detection and matching algorithm provides $n$ feature correspondences between the reference template and the current image. Ideally, it would be possible to find a vector $\vect{x}^* = \{ \vect{H}^*, \alpha^*, \beta^* \}$ such that:

\begin{align}
\alpha^*\image(\vect{w}(\vect{H}^*,\vect{p}^*_i))+\beta^* &= \image^*(\vect{p}^*_i), \quad &\forall i &= 1, 2, \dots, m, \\
\vect{w}(\vect{H}^*,\vect{q}^*_j) &=\vect{q}_j, \quad &\forall j &= 1, 2, \dots, n,
\end{align}

by substituting (<ref>) in (<ref>), where $\image$ and $\image^*$ are the current and reference images, respectively, $\vect{p}^*_i \in \mathbb{P}^2$ contains the coordinates of the $i$-th pixel of the reference template, and $\vect{q}_j, \vect{q}_j^* \in \mathbb{P}^2$ are the representations of the $j$-th feature correspondence set in the current image and reference template, respectively. The perfect calculation of $\vect{x}^*$ is impossible due to a variety of reasons, including noise in the camera sensor and outliers in the feature matching. This leads to the reformulation of this task as a nonlinear least-squares problem.

Two separate cost-functions are defined: One for the IB part and another for the FB one. The $i$-th pixel of the reference template contributes to the following row to the IB cost function via the distance

\begin{equation}
a_i(\vect{x}) = \alpha\image(\vect{w}(\vect{H},\vect{p}^*_i))+\beta - \image^*(\vect{p}^*_i),
\end{equation}

and an output vector $\vect{y}_{IB}$ can be constructed as:

\begin{equation}
\vect{y}_{IB} = \begin{bmatrix} a_1 &  a_2 & \cdots & a_m
\end{bmatrix}^\top.
\label{equ:cost_ib}
\end{equation}

The FB cost function is defined using the distance between the features coordinates in each image:

\begin{equation}
\vect{b}_j(\vect{x}) = \vect{w}(\vect{H},\vect{q}^*_j) -\vect{q}_j = \begin{bmatrix} b^u_j & b^v_j & 0
\end{bmatrix},
\end{equation}

where $b^u_j, b^v_j$ are distances between the features in the $u$ and $v$ directions, respectively. The third element is disregarded since it is always zero. Thus, a vector $\vect{y}_{FB}$ can be constructed as:

\begin{equation}
\vect{y}_{FB} = \begin{bmatrix} b^u_1 &  b^v_1&  b^u_2 &  b^v_2 & \cdots & b^u_n & b^v_n
\end{bmatrix}^\top.
\label{equ:cost_fb}
\end{equation}

Using (<ref>) and (<ref>), a unified nonlinear least squares problem can be defined as

\begin{equation}
\min_{\vect{x} = \{\vect{H}, \alpha, \beta\}}  \ \frac{1}{2}
\Bigl(
w_{IB}\norm{ \vect{y}_{IB}(\vect{x}) }^2_2 +
w_{FB}\norm{ \vect{y}_{FB}(\vect{x}) }^2_2
\Bigr),
\label{eq:min_initial_unified}
\end{equation}

where $w_{IB}, w_{FB}$ are carefully chosen weights given to the intensity- and feature-based components of the cost function, respectively, as will be proposed later on. For real-time systems, only local optimization methods can be applied since global ones are too costly. In this case, an initial approximation $\widehat{\vect{x}} = \{ \widehat{\vect{H}}, \widehat{\alpha}, \widehat{\beta} \}$ of the true solution is required. This estimate can be integrated into the least-squares formulation as:

\begin{equation}
\min_{\vect{z} = \{\vect{v}, \alpha, \beta\}}  \ \frac{1}{2}
\Bigl(
w_{IB}\norm{ \vect{y}_{IB}(\vect{x}(\vect{z}) \circ \widehat{\vect{x}}) }^2 +
w_{FB}\norm{ \vect{y}_{FB}(\vect{x}(\vect{z}) \circ \widehat{\vect{x}}) }^2
\Bigr),
\label{eq:min_incremental_unified}
\end{equation}

where the symbol `$\circ$' denotes the composition operation. For the scalars $\alpha$ and $\beta$, it corresponds to the addition, whereas for the homography that operation is the matrix multiplication. Furthermore, to take into account the different number of observations for IB and FB methods, we include normalization factors and define the unified output vector as

\begin{equation}
\vect{y}_{UN} =
\begin{bmatrix}
\sqrt{\frac{w_{IB}}{m}} \vect{y}_{IB} &
\sqrt{\frac{w_{FB}}{2n}} \vect{y}_{FB}
\end{bmatrix}.
\end{equation}

Hence, a more concise unified formulation is achieved:

\begin{equation}
\min_{\vect{z} = \{\vect{v}, \alpha, \beta\}}  \ \frac{1}{2}
\norm{ \vect{y}_{UN}(\vect{x}(\vect{z}) \circ \widehat{\vect{x}})}^2,
\label{eq:min_incremental_unified_concise}
\end{equation}

which can be efficiently solved using [17].

§.§ Weight Choices

The weights $w_{IB}$ and $w_{FB}$ should be carefully selected to ensure the best convergence properties for the algorithm. The following constraints apply to the weights:

\begin{align}\label{eq:convex}
w_{IB} + w_{FB} &= 1,\\ w_{FB}, w_{IB} &> 0.
\end{align}

The idea behind the proposed method for determining the weights is to let the feature-based error be more influential to the optimization when the current solution is far from the true one. As the FB error decreases, then the intensity-based component becomes increasingly more important. This is consistent with the idea that the FB method is better suited to handle large displacements, whereas IB methods have higher accuracy, but only work when the initial guess is sufficiently close to the true solution.

The main measurement used for calculating the weights is the feature-based error associated with the current estimated homography $\widehat{\vect{H}}$. It is calculated using the following root mean squared error (RMSD):

\begin{equation}
RMSD(\vect{y}_{FB}) = \sqrt{ \frac{\sum^n_{j=1} \norm{\vect{w}(\widehat{\vect{H}},\vect{q}^*_j) -\vect{q}_j}^2_2}{n}} = d_{FB}.
\end{equation}

The proposed weights are then defined from

\begin{equation}
w_{FB} = 1 - \exp(-d_{FB})
\end{equation}

and (<ref>). This function allows for a continuous transition where the feature-based weight decreases as its error gets lower, and the intensity-based component becomes increasingly more important in the optimization.

§.§ Local versus Global Search

The processing times may be drastically increased if the feature detection and matching algorithms are allowed to process the entire current image. The proposed method processes only a small region in the current image to obtain good matches whenever possible.

Firstly, a current template is generated by warping the current image with the initial approximation $\widehat{\vect{H}}$. Then, this current template is assigned a score by comparing it with the reference template using the Zero-mean Normalized Cross-Correlated. If this score is higher than a predefined threshold, then the feature detection algorithm searches only within this current template. Otherwise, the current template and $\widehat{\vect{H}}$ are both discarded. In this case, the detection algorithm searches the entire current image for features. The first scenario is referred to as a “local” search, whereas the second one as a “global” search. When the global search is used, it is necessary to recalculate an initial approximation $\widehat{\vect{H}}$. This is done by calculating the homography solely from the features matches between the current image and the reference template.

§ EXPERIMENTAL RESULTS

§.§ Validation Setup

The same testing procedure used in [28] is implemented to validate the algorithm. Firstly, a reference image of size $800 \times 533$ pixels is chosen, and a region of size $100 \times 100$ pixels is selected as the reference template. The coordinates of each corner are independently perturbed in the $\overrightarrow{u}$ and $\overrightarrow{v}$ directions with a zero mean Gaussian noise and standard deviation of $\sigma$ pixels (see Figure <ref>). The relation between the original corner points and the perturbed ones defines a test homography. The reference image is then transformed by this test homography. The algorithm receives the reference template and the transformed image with the identity element as the initial guess for the photogeometric transformation. From this input, the algorithm produces an estimated homography. In turn, this homography is used to transform each reference corner point. If the average residual error between the actual perturbed corner points and the estimated perturbed ones is less than 1 pixel, the result is declared to have converged. 1,000 test cases are randomly generated for each value of the perturbation $\sigma \in [0, 20]$ and used as input for each evaluated algorithm. In all tests, 3 levels of a multiresolution pyramid are used. In each level, a maximum of 3 iterations of the algorithm are allowed to execute.

This setup is used to compare different algorithms. Three criteria are analyzed: Convergence domain, convergence rate and timing analysis. The methods differ on whether they use only the IB or the FB component (SURF is here applied for feature detection and description) in the cost function, or both for the Unified case. Another difference is the use of a ZNCC predictor to improve the initialization in some methods. Finally, some algorithms do not consider the photometric part of the transformation space. These algorithms along with their characteristics are summarized in Table <ref>.

Homography estimation algorithms used for comparisons.

Method     IB     FB     Predictor     Photometric

§.§ Convergence Domain

Figure <ref> shows that the proposed Unified algorithms have a larger convergence domain than all pure FB or IB versions. It also shows that the use of the ZNCC predictor in the unified version does not affect its frequence of convergence, as well as that the IBG (i.e., IB with robustness to Global illumination changes) and ESM algorithms have a very similar performance. The latter is expected because there are no lighting changes in this validation setup.

Another interesting observation is that the results of the algorithms in the FB class (FB_ESM and the algorithm available in OpenCV) were significantly worse than the ones in the IB class, although it was expected that they would have a higher convergence domain. This suggests that there is still room for improving the FB components of the estimation, which would in turn lead to a further improvement in the unified method as well.

§.§ Convergence Rate

Figure <ref> compares the convergence rate of the homography estimation algorithms under a perturbation of magnitude $\sigma=10$. This rate is displayed as the progression of the root mean squared (RMS) error between the coordinates of the 4 corners of the reference template and the estimated transformation of the current template. Out of the 1,000 test cases, only those where the estimation converged are considered here. Note that the results from the OpenCV algorithm is omitted because it was used as a black-box, and therefore the sequence of
homographies at each iteration cannot be accessed. The $x$-axis of Figure <ref> contains each important step in the optimization. The first step, which is labeled “predictor”, is the result of the ZNCC prediction step. The second step, which is labeled “global”, is the step where the algorithm decides to search for features in the entire current image, as described in Section <ref>. Of course, these two steps are not performed by every algorithm. Afterwards, steps from the iterative optimization method follow. They are separated by pyramids level, such that the notation “X-Y” represents pyramid level X at iteration Y.

Figure <ref> allows for several observations. Firstly, the FB_ESM performance is very dependent on the “global” step. After this step, it is the algorithm with the best RMS value. However, it is not capable to improve this value too much in the subsequent optimization steps. When the other algorithms reach the third level of the pyramid, they all outperform its RMS. The behaviour of ESM, IBG and IBG_P is very similar as  they share the same framework. A small difference between them is that IBG_P is able to converge even for cases with a slightly higher initial RMS error, due to the prediction step. After that step, however, all these three algorithms perform quite similarly.

Finally, let us note that the Unified algorithms have a behaviour that combines the FB and IB methods, as desired. The UNIF_P uses both the “predictor” and “global” steps. Interestingly, the global search is less applied in that version than the UNIF one because of the prediction step. This explains its smaller initial reduction in RMS value. On the other hand, less usage of the global step leads to a improvement in the processing times, as shown in the next section. After these steps, both the Unified algorithms behave similarly to IB ones, with the advantage of having a better initialization procedure.

§.§ Timing analysis

Figure <ref> shows how the average time needed to run the estimation algorithms varies depending on the magnitude of perturbation. This time is measured in a Intel i7-6700HQ processor, and is averaged over the subset of the 1,000 cases only when the estimation has converged. The most noticeable aspect of this graph is that pure IB algorithms have nearly constant time, regardless of the perturbation level. In constrast, the algorithms that have a feature-based component need more time to process images with higher perturbation levels. This phenomenon can be explained by considering the effect of the global versus local feature search. As the perturbation level increases, the number of occasions where the algorithm applies the global search also increases. This step, however, is very computationally expensive. The UNIF_P manages to have a lower processing time because the prediction step increases the probability that the local search is used. Therefore, the UNIF_P can be seen as a compromise between having the advantage of being capable of performing global search, without taking a big penalty in the processing times.

[thick, scale=1.0]
axis lines=middle,
ylabel near ticks,
xlabel near ticks,
xlabel= Magnitude of Perturbation,
ylabel= Processing Time (s),
enlargelimits = false,
ytick=0,0.02, 0.04, 0.06, 0.08, 0.1, 0.12, 0.14, 0.16,
yticklabel style=
/pgf/number format/fixed,
/pgf/number format/precision=2

legend style=at=(axis cs:15.0,0.08),anchor=north west, nodes=scale=0.5, transform shape,]
table [y=ib_basic,x=Sigma]data/timing_all.dat;
table [y=ibg_regular,x=Sigma]data/timing_all.dat;
table [y=ibg_predictor,x=Sigma]data/timing_all.dat;

table [y=robust_fb,x=Sigma]data/timing_all.dat;
table [y=unified,x=Sigma]data/timing_all.dat;
[color=green, mark=triangle*] table [y=unified_predictor,x=Sigma]data/timing_all.dat;

Processing times for different perturbation levels.

However, these results also show that more research is needed to develop a method that is able to reliably perform in real-time settings for large perturbations. The IB methods are already capable of that when they converge, requiring less than 0.02s/image. The FB and Unified methods may need up to 0.12s, which may be unacceptable for some applications.

§.§ Use Case: Visual Tracking

The proposed algorithm is publicly available for research purposes as a C++ library and as a ROS package [29], along with its technical report [30]. This section shows its application to homography-based visual tracking. Results are available at [31]. The prediction step is applied, as recommended for real-time tracking applications. Figure <ref> shows some excerpts of this tracking experiment. An interesting result is that the proposed unified visual tracker can recover from full occlusions. Even after completely removing the tracked region from the current image, the tracker can recover given its feature-based ability to perform the “global” search. Additionally, it can be seen that the algorithm is robust to large global illumination changes, and that in some cases it can recover from complete failure even under severe lighting variations.

§ CONCLUSIONS

This paper proposes a first step towards a truly unified optimal approach to homography estimation. The results show that improved convergence properties are indeed obtained when combining both classes of feature- and intensity-based methods into a single optimization procedure. This can help vision-based applications to handle faster robot motions. Future work will focus on reducing the processing time of the unified algorithm, specially when very large interframe displacements lead to a global search for features.

§ ACKNOWLEDGMENT

This work was supported in part by the CAPES under Grant 88887.136349/2017-00, in part by the FAPESP under Grant 2017/22603-0, and in part by the InSAC project (CNPq under Grant 465755/2014-3, FAPESP under Grant 2014/50851-0).

[1]
O. Faugeras, Q.-T. Luong, and T. Papadopoulo, The geometry of multiple
images.1em plus 0.5em minus 0.4emThe MIT Press, 2001.

[2]
S. Benhimane and E. Malis, “Homography-based 2D visual tracking and
servoing,” The International Journal of Robotics Research, vol. 26, no. 7,
2007, pp. 661–676.

[3]
B. Neuberger, G. Silveira, M. Postolov, and M. Vincze, “Object grasping in
non-metric space using decoupled direct visual servoing,” in Proc. Austrian
Robotics Workshop & OAGM Workshop, 2019, pp. 99–104.

[4]
L. Brown, “A survey of image registration techniques,” ACM computing surveys,
vol. 24, no. 4, 1992, pp. 325–376.

[5]
S. Benhimane and E. Malis, “Real-time image-based tracking of planes using
efficient second-order minimization,” in Proc. IEEE/RSJ IROS, 2004, pp.

[6]
M. Quigley, K. Conley, B. Gerkey, J. Faust, T. Foote, J. Leibs, R. Wheeler, and
A. Y. Ng, “ROS: An open-source robot operating system,” in Proc. ICRA
workshop on open source software, 2009.

[7]
R. Szeliski, “Image alignment and stitching: A tutorial,” Foundations and
Trends in Computer Graphics and Vision, vol. 2, no. 1, 2007.

[8]
M. Irani and P. Anandan, “All about direct methods,” in Proc. Workshop on
Vision Algorithms: Theory and practice, 1999.

[9]
G. Silveira, “Contributions to direct methods of estimation and control from
visual data,” Ph.D. dissertation, Ecole des Mines de Paris, 2008.

[10]
B. Lucas and T. Kanade, “An iterative image registration technique with an
application to stereo vision,” 1981.

[11]
J. R. Bergen, P. Anandan, K. J. Hanna, and R. Hingorani, “Hierarchical
model-based motion estimation,” in Proc. ECCV, 1992, pp. 237–252.

[12]
J. Zhang and S. Singh, “Visual-lidar odometry and mapping: Low-drift, robust,
and fast,” in Proc. IEEE ICRA, 2015, pp. 2174–2181.

[13]
G. Silveira, E. Malis, and P. Rives, “An efficient direct approach to visual
SLAM,” IEEE transactions on robotics, vol. 24, no. 5, 2008, pp. 969–979.

[14]
R. Mur-Artal, J. M. M. Montiel, and J. D. Tardos, “ORB-SLAM: a versatile and
accurate monocular SLAM system,” IEEE Transactions on Robotics, vol. 31,
no. 5, 2015, pp. 1147–1163.

[15]
D. DeTone, T. Malisiewicz, and A. Rabinovich, “Deep Image Homography
Estimation,” 2016.

[16]
G. Silveira, “Photogeometric direct visual tracking for central
omnidirectional cameras,” Journal of Mathematical Imaging and Vision,
vol. 48, no. 1, 2014, pp. 72–82.

[17]
G. Silveira and E. Malis, “Unified direct visual tracking of rigid and
deformable surfaces under generic illumination changes in grayscale and color
images,” International Journal of Computer Vision, vol. 89, 2010, pp.

[18]
G. D. Evangelidis and E. Z. Psarakis, “Parametric image alignment using
enhanced correlation coefficient maximization,” IEEE Transactions on Pattern
Analysis and Machine Intelligence, vol. 30, no. 10, 2008, pp. 1858–1865.

[19]
L. M. Fonseca and B. Manjunath, “Registration techniques for multisensor
remotely sensed imagery,” 1996.

[20]
P. Viola and W. M. Wells III, “Alignment by maximization of mutual
information,” International journal of computer vision, vol. 24, no. 2,
1997, pp. 137–154.

[21]
S. Baker and I. Matthews, “Lucas-kanade 20 years on: A unifying framework,”
International journal of computer vision, vol. 56, 2004.

[22]
B. Zitova and J. Flusser, “Image registration methods: a survey,” Image and
Vision Computing, vol. 21, 2003, pp. 977–1000.

[23]
A. K. Singh, “Modular tracking framework: A unified approach to registration
based tracking,” Master's thesis, University of Alberta, 2017.

[24]
Y. Jianchao, “Image registration based on both feature and intensity
matching,” in Proc. IEEE International Conference on Acoustics, Speech, and
Signal Processing, 2001, pp. 1693–1696.

[25]
A. Ladikos, S. Benhimane, and N. Navab, “A real-time tracking system combining
template-based and feature-based approaches,” in Proc. VISAPP, 2007, pp.

[26]
P. F. Georgel, S. Benhimane, and N. Navab, “A unified approach combining
photometric and geometric information for pose estimation,” in Proc. BMVC,

[27]
E. Malis and M. Vargas, “Deeper understanding of the homography decomposition
for vision-based control,” INRIA, Tech. Rep., 2007.

[28]
S. Baker and I. Matthews, “Equivalence and efficiency of image alignment
algorithms,” in Proc. IEEE CVPR, 2001.

[29]
L. Nogueira and G. Silveira, “englishGitHub -
visiotec/vtec_ros: ROS packages from the VisioTec group,” 2020, URL:
<https://github.com/visiotec/vtec_ros> [Accessed 22 August 2020].

[30]
L. Nogueira, E. de Paiva, and G. Silveira, “VTEC robust intensity-based
homography optimization software,” no. CTI-VTEC-TR-01-19, Brazil, 2019.

[31]
L. Nogueira, “englishUnified Intensity- And
Feature-Based Homography Estimation Applied To Visual Tracking,” 2020,
URL: <https://www.youtube.com/watch?v=oArw449qp1E> [Accessed 22 August


# Weyl Semimetallic Phase in High Pressure CrSb2 and Structural Compression
Studies of its High Pressure Polymorphs

Carl Jonas Linnemann  Emma Ehrenreich-Petersen  Davide Ceresoli  Timofey
Fedotenko  Innokenty Kantor  Mads Ry Vogel Jørgensen  Martin Bremholm
<EMAIL_ADDRESS>
###### Abstract

In this study, high pressure synchrotron powder X-ray diffraction is used to
investigate the compression of two high pressure polymorphs of CrSb2. The
first is the CuAl2-type polymorph with an eight-fold coordinated Cr, which can
be quenched to ambient conditions from high-pressure high-temperature
conditions. The second is the recently discovered MoP2-type polymorph, which
is induced by compression at room temperature, with a seven-fold coordinated
Cr. Here, the assigned structure is unambiguously confirmed by solving it from
single-crystal X-ray diffraction. Furthermore, the electrical properties of
the MoP2-type polymorph were investigated theoretically and the resistance
calculations under pressure were accompanied by resistance measurements under
high pressure on a single crystal of CrSb2. The calculated electronic band
structure for the MoP2-type phase is discussed and we show that the polymorph
is semimetallic and possesses type-I Weyl points. No further phase transitions
were observed for the CuAl2-type structure up to 50 GPa and 40 GPa for the
MoP2-type structure. Even though the CuAl2-phase has the highest coordination
number of Cr, it was found to be less compressible than the MoP2-phase having
a seven-fold coordinated Cr, which was attributed to the longer Cr-Sb distance
in the CuAl2-type phase. The discovery of a type-I Weyl semimetallic phase
redin CrSb2 opens up for discovering other Weyl semimetals in the transition
metal di-pnictides under high pressure.

###### keywords:

Crystal binding and equation of state, Electronic band structure, Phase
transitions , High pressure, X-ray diffraction

[1]organization=Department of Chemistry and iNANO, Aarhus
University,addressline=Langelandsgade 140,postcode = 8000,city=Aarhus C,
country = Denmark [2]organization = Deutsches Elektronen-Synchrotron DESY,
addressline = Notkestr. 85, postcode = 22607, city = Hamburg, country =
Germany [3]organization = Consiglio Nazionale delle Ricerche - Istituto di
Scienze e Tecnologie Chimiche “G. Natta”, addressline = via Golgi 19, postcode
= 20133, city = Milano, country = Italy

[4]organization = MAX IV Laboratory, Lund University,addressline = Fotongatan
2,city = Lund, postcode = 225 94,country = Sweden

##  1 Introduction

Many redcompounds exist in different polymorphs with different properties,
such as WP2, which exists in the low temperature modification $\alpha$-WP2 [1]
and the high temperature modification $\beta$-WP2 crystallizing in the
$Cmc2_{1}$ space group [2]. Both polymorphs are semimetals where $\beta$-WP2
is a type-II Weyl semimetal and $\alpha$-WP2 is a trivial semimetal [3, 4].
Pressure can also induce phase transitions such as in CrSb2 where, recently, a
new high pressure polymorph was discovered independently by Li et al. as well
as Ehrenreich-Petersen et al., extending the number of known polymorphs of
CrSb2 to three [5, 6]. At ambient pressure CrSb2 crystallizes in the marcasite
structure (Fig. 1 a)) and has an antiferromagnetic (AFM) ordering below
$T_{N}\approx$273\text{\,}\mathrm{K}$$, with a 1x2x2 supercell with the
magnetic moments perpendicular to the (101) plane [7]. The marcasite phase is
a $n$-type semiconductor with a narrow bandgap of
$0.1\text{\,}\mathrm{e}\mathrm{V}$ redand has been shown to possess large
anisotropic magnetoresistance in its surface states [8, 9]. The first high-
pressure polymorph of CrSb2 (herein named HP-I) was discovered by Takizawa et
al. by high pressure synthesis above
$5.5\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ and
$600\text{\,}\mathrm{\SIUnitSymbolCelsius}$ [10]. This polymorph crystallizes
in a CuAl2-type structure (Fig. 1 b)), redin which TiSb2 and VSb2 also
crystallize [11]. Here, TiSb2 has been shown to be a three-dimensional Dirac-
semimetal [12]. redThe HP-I phase of CrSb2 exhibits metallic behaviour and is
an itinerant electron ferromagnet with a Curie temperature of approximately
$160\text{\,}\mathrm{K}$ [13]. At approximately $90\text{\,}\mathrm{K}$ it has
an AFM spin density wave transition, which can be suppressed under the
application of magnetic fields. It has been shown that at pressures above
$0.5\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ the two states merge together to
a single AFM state, where the AFM order is suppressed to lower temperatures by
application of pressure and vanishes completely at
$P_{c}\approx$9\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$$, indicating an AFM
quantum critical point [14]. The newly discovered high pressure polymorph
(herein named HP-II) has been shown to appear at room temperature and
pressures above $10\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$. It is meta-stable
and cannot be recovered to ambient pressure [5, 6]. Furthermore laser-heating
experiments under high pressure have shown that the phase transforms into the
high-pressure CuAl2-type phase, indicating a high activation barrier for the
phase transition to the HP-I phase and that it is more stable than the HP-II
phase [6]. Li et al. showed that the polymorph exhibits n-p conduction
switching as well as that the high pressure phase exhibits metallic conduction
[5]. There have been disagreements regarding the structure, where Li et al.
assigned it to the arsenopyrite structure-type, while Ehrenreich-Petersen et
al. assigned it to have a MoP2-type structure using a combination of DFT
calculations and powder diffraction [5, 6]. The MoP2-type structure has the
orthorombic $Cmc2_{1}$ space-group, where Mo is coordinated to seven P atoms.
This structure is furthermore isostructural with $\beta$-WP2, and both MoP2
and $\beta$-WP2 have been studied immensely as they are robust type-II Weyl-
semimetals [2, 3]. Both materials have been shown to possess extremely high
magnetoresistance and high conductivity[15]. redFurthermore, high harmonic
generation up to the 10th order has been shown for $\beta$-WP2 [16] and due to
their high magnetoresistance both materials are potential candidates for high-
gain high-frequency amplifiers with low power dissipation, which could replace
High-electron-mobility transistors in quantum computers [17]. redFor both
compounds no structural phase transitions under the application of pressure
have been shown, however $\beta$-WP2 has been shown to possess a pressure
induced Lifshitz transition [18, 19]. Both MoP2 and WP2 have also shown to
crystallize in the monoclinic OsGe2-type structure. For WP2 it is a low
temperature modification, while it for MoP2 has been shown to be a recoverable
high-pressure high-temperature polymorph [2, 18].

redIn this study the high pressure structural properties of both high pressure
polymorphs of CrSb2 are investigated experimentally through powder diffraction
at high pressures as well as theoretically through density functional theory
(DFT) calculations. Furthermore, the MoP2-type structure is shown to be the
correct structure for the HP-II polymorph of CrSb2 through single crystal
X-ray diffraction at 12 GPa. The electronic properties of this polymorph are
furthermore investigated through resistance measurements under high pressure
at ambient temperature accompanied by theoretical calculations regarding the
resistivity and Hall coefficients. The band structure is calculated at 12 GPa
and the topology of the band crossings is determined to examine if the HP-II
phase possesses Weyl points. The verification of the MoP2-type structure, and
furthermore the evidence of Weyl-points opens up for discovering other high
pressure transition-metal di-pnictides, which potentially host Weyl points.

Figure 1: Crystal structure and Cr coordination of the different high-pressure
polymorphs of CrSb2: a) Marcasite, b) CuAl2-type (HP-I) and c) MoP2-type (HP-
II).

##  2 Materials and methods

###  2.1 Powder synthesis of CrSb2

Stoichiometric amounts of Cr (99.95%) and Sb (99.95%) were ground in a mortar,
pressed to a pellet and sealed in an evacuated quartz tube, which was heated
at a rate of $50\text{\,}\mathrm{\SIUnitSymbolDegree}\mathrm{C}$/min to
$600\text{\,}\mathrm{\SIUnitSymbolDegree}\mathrm{C}$, kept at that temperature
for $48\text{\,}\mathrm{h}$ and then quenched in air.

###  2.2 Single crystal growth of CrSb2

Single crystals of CrSb2 were grown using a flux method adapted from Ref. [8].
Cr (99.95%) and Sb (99.99%) with a molar ratio of 6:94 were ground in a mortar
and sealed in an evacuated quartz tube. The tube was heated to
$1000\text{\,}\mathrm{\SIUnitSymbolDegree}\mathrm{C}$ at
$50\text{\,}\mathrm{\SIUnitSymbolDegree}\mathrm{C}$/min, kept at that
temperature for $36\text{\,}\mathrm{h}$ followed by cooling to
$640\text{\,}\mathrm{\SIUnitSymbolDegree}\mathrm{C}$ at
$2\text{\,}\mathrm{\SIUnitSymbolDegree}\mathrm{C}$/h and then cooled to room
temperature. The crystals were isolated from the Sb-flux through
centrifugation. The phase purity of the crystals was verified by determining
the unit cell of a single crystal using a STOE Stadivari single crystal
diffractometer with Mo K-$\alpha$ radiation (Table S1).

###  2.3 High pressure synthesis of CuAl2-type CrSb2

A pellet of marcasite CrSb2 powder was loaded in a large volume press in a
14/8 COMPRES octahedral assembly [20], pressurized to
$7\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ at $1\text{\,}{GPa}\mathrm{/}{h}$
and heated to $900\text{\,}\mathrm{\SIUnitSymbolDegree}\mathrm{C}$ for
$1\text{\,}\mathrm{h}$ followed by quenching. The phase purity of the sample
was verified using powder X-ray diffraction using a Rigaku Smartlab
diffractometer with Co K-$\alpha_{1,2}$ radiation (Fig. S1).

###  2.4 High pressure diffraction measurements

Single-crystal X-ray diffraction (SC-XRD) measurements under high pressure
were conducted at the Extreme Conditions Beamline (ECB, P02.2)[21], PETRA III,
DESY ($\lambda$=0.2899 Å, beam size $\approx 2\times 2.7$
$\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}$, Perkin Elmer XRD 1621 detector).
Three single crystals of CrSb2 were loaded in a plate-diamond anvil cell (DAC)
(Almax EasyLab) with $600\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ culet
diamonds. Additionally, a small ruby sphere was loaded that served as the
pressure calibrant during the experiment by the ruby fluorescence method [22].
A 16:3:1 mixture of methanol-ethanol-water was used as the pressure
transmitting medium (PTM). A stainless-steel gasket pre-indented to
$72\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ micron with a
$300\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ hole drilled by electric
discharge machining (EDM), served as the sample chamber. The pressure was
increased to 12 GPa, i.e. above the structural phase transition of CrSb2 and
the DAC was heated to $100\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for one hour
to relieve strain. The pressure was allowed to equilibrate for several hours
before the measurement and was measured right before and after the diffraction
experiment with no observed pressure deviation. As most of the single crystals
had split into separate crystal domains during the phase transition a suitable
spot for diffraction was chosen and diffractograms were obtained with a
rotation from $-37\degree$ to $37\degree$ with $0.5\degree$ steps with an
acquisition time of 0.5 s and a
$50\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ Pt absorber inserted in the
beam. Reduction and integration of the data was performed using CrysAlisPro
[23] and the structure was solved using SHELXT in Olex2 and refined using
SHELXL [24, 25, 26]. During integration and reduction of the data, frames with
a $R_{\mathrm{int}}$ above 0.3 were rejected.

Membrane driven Mao-Bell type DACs with culet sizes of
$300\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ (For the HP-I phase) or
$200\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ (For the HP-II phase) were
used for the high pressure powder X-ray diffraction measurements (HP-PXRD). Re
gaskets indented from approx.
$250\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ to approx.
$40\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ were used with a
$120\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ hole drilled by EDM for
the HP-I phase and a laser-drilled
$100\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ hole for the HP-II phase.
Silicone oil was used as the PTM for the HP-I phase and for the HP-II phase Ne
was gas-loaded as the PTM using the gas-loading system at Sector 13, APS [27].
To obtain small grain sizes, the samples were suspended in ethanol and the
supernatant was removed and dried. Thin sheets of samples were loaded together
with thin sheets of Cu (for the HP-I phase) or Au (for the HP-II phase) for
pressure determination [28, 29]. A small ruby sphere was also added, which
served to determine the pressure during the loading and closing of the cell.
For the HP-I phase the HP-PXRD measurements were conducted at the DanMAX
beamline at the MAX IV synchrotron ($\lambda$=0.3542 Å, beam size $\approx
7.4\times 7.2$ $\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}$, Dectris Pilatus3X
2M CdTe detector). The diffractograms for the HP-II phase have been shown, but
not refined, in Ref. [6], and the HP-PXRD measurements for the HP-II phase of
CrSb2 were conducted at the Advanced Photon Source at the GSECARS beamline
13-ID-D ($\lambda$=0.3344 Å, beam size $\approx 4\times
3\,$\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}$$, Dectris Pilatus CdTe 1M
detector). All 2D powder diffractograms were integrated using Dioptas [30]. Le
Bail refinements of the unit cell parameters were performed using the FullProf
suite [31] and the unit cell parameters were fitted to the third order Birch-
Murnaghan equation of state (EoS) using the EoSFit7 GUI [32]. Crystal
structure models were drawn using the VESTA software [33].

###  2.5 Resistance measurement of CrSb2 under pressure

The resistance measurements were performed using an iDiamond (Almax Easylab)
containing eight tungsten microprobes partially embedded in an epitaxially
grown diamond [34]. The culet size of the anvil was
$250\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ with bevels. A BX-90 DAC
was used with a stainless-steel gasket indented from
$250\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ to
$50\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ using two standard diamond
anvils redin order not to damage the electrodes on the iDiamond. A hole with a
diameter of $120\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ was drilled
using EDM. Steatite was used as the pressure transmitting medium. A suitable
single crystal of CrSb2 was mounted on the designer anvil together with a ruby
for pressure determination [22] and the resistance was measured in a two-wire
reddirect-current geometry with a current of 1 mA using a Keithley 2450
sourcemeter.

###  2.6 DFT modeling

DFT calculations were performed with the plane wave pseudopotential code
Quantum Espresso [35, 36] with ultrasoft pseudopotentials and the PBEsol
exchange-correlation functional [37]. The plane wave and density cutoffs were
45 and 450 Ry, respectively. In all calculations we sampled the reciprocal
space with a nearly constant density of k-points of $\simeq$0.2 Å-1. We used a
Gaussian smearing of 0.005 Ry for the electronic temperature. Within our DFT
setup, the band gap of the AMFe marcasite structure is 0.06 eV, which slightly
underestimates the experimental value of 0.07 eV [38]. A Hubbard U correction
of 2.3 eV as in Ref. [38] results in a worse agreement with the experimental
AFMe lattice parameters, and, contrary to Ref. [38], the AFMe marcasite
structure becomes metallic, with Sb-5$p$ orbital bands crossing the Fermi
level. As a function of pressure, variable-cell geometry optimizations of
CrSb2 were performed in three magnetic ground states: non-magnetic (NM),
ferromagnetic (FM), and antiferromagnetic (AFM). For the marcasite structure
we used the AFMe ordering reported in Ref. [38]. For the HP-II structure we
generated all AFM orderings compatible with a 2$\times$2$\times$2 supercell
and selected the lowest energy one (which we named AFM1, Fig. S2).

The electrical conductivity was calculated with the BoltzTraP2 code [39] at
300 K in the Constant Relaxation Time Approximation (CRTA), with a
20$\times$20$\times$20 k-point mesh. The search for Weyl points requires the
evaluation of the DFT Hamiltonian on an extremely dense k-point grid. For this
purpose, we used PAOFLOW2 [40] to construct a tight-binding Hamiltonian from
the self-consistent plane wave Hamiltonian. The BFGS algorithm is used to
locate band crossings in reciprocal space. Finally, the chirality (i.e. the
$\mathbb{Z}_{2}$ invariant) of each band crossing was calculated using Z2PACK
[41].

##  3 Results and discussion

###  3.1 The crystal structure of the HP-II phase obtained from single-crystal
X-ray diffraction

Table 1: Results from the single-crystal diffraction experiment compared to
the data from DFT and powder diffraction from ref. [6]

| Single crystal diffraction | Ref. [6]a
---|---|---
Pressure (GPa) | 11.9(2) | 12.0(7)
Space group | $Cmc2_{1}$ | $Cmc2_{1}$
$a$ (Å) | 3.0752(2) | 3.06723(6)
$b$ (Å) | 12.4476(19) | 12.4495(5)
$c$ (Å) | 5.8741(2) | 5.85682(9)
$\alpha,\beta,\gamma$ ($\degree$) | 90 | 90
$V$ (Å3) | 224.85(4) | 223.645(11)
Atomic coordinates |
Cr | (0, 0.3942(5), 0.1490(6)) | (0, 0.39431, 0.15191)
Sb1 | (0, 0.20702(17), 0.3545(2)) | (0, 0.20806, 0.36102)
Sb2 | (0, 0.05627(18), 0) | (0, 0.05501, 0)
Refinement statistics |
Completeness | 55.9% |
$I/\sigma(I)$ | 95.7 |
$R_{1}$(%) | 5.82 |
$wR_{2}$ (%) | 16.10 |
$R_{\mathrm{int}}$ (%) | 0.62 |
GooF | 1.150 |
$\lambda$ (Å) | 0.2899 |

  * a

The unit cell was standardized to match the setting from the single-crystal
diffraction experiment.

The results from the SC-XRD refinement are shown in Table 1 together with the
results from the DFT-calculations and powder diffraction from Ehrenreich-
Petersen et al. [6]. Here, the structural solution from the single crystals of
CrSb2 at 12 GPa shows that the HP-II structure is of the MoP2-type in
agreement with our previous paper [6].

###  3.2 Electrical properties of the HP-II phase

The electrical resistance of the HP-II phase was measured for pressures up to
$21\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ and is shown in Fig. 2a) together
with the calculated resistivity. Between 5 and 10 GPa a small linear decrease
of the resistance is observed, which was also observed by Li et al. indicating
a decrease of the band-gap of the marcasite phase with pressure, which also
has been shown for some pressure intervals in FeSb2 [42, 43]. After the onset
of the phase transition at 10 GPa [6], a sharp decrease of the resistance is
observed, indicating a metallization of the system. redThe residual resistance
of $130\text{\,}\mathrm{\SIUnitSymbolOhm}$ after the phase transition is high
for a metal, which is attributed to the resistance in the electrodes, which
for a similar anvil has been reported to be
$110\text{\,}\mathrm{\SIUnitSymbolOhm}$ [34].

Figure 2: a) Resistance of CrSb2 under high pressure: The experimentally
measured resistance of CrSb2 under the phase transition to the MoP2-structure
is shown together with the calculated resistivity for the marcasite phase and
the MoP2-type phase. The dashed line marks the onset of the phase transition
from the marcasite to the HP-II phase [6]. b) Calculated Hall coefficient. The
dashed line marks the phase transition pressure from DFT [6] and the line at
$R_{\mathrm{H}}=0$ serves as a guide to the eye. c) The carrier concentration
calculated from the Hall coefficient. Figure 3: Band structure of the HP-II
AFM1 structure at 12 GPa around the Fermi level, with and without Spin Orbit
Coupling (SOC).

The evolution of the electronic density of states as a function of pressure is
shown in Fig. S3 for the marcasite- and the HP-II polymorph. As reported in
Ref. [38], in the marcasite structure the magnetic order has a very strong
impact on the electronic structure and the experimental AFMe order opens a
$\sim$0.2 eV gap between the Cr-projected valence and conduction states. GGA
calculations (as in the present case) predict a vanishing band gap, with a low
density of states and low electron velocity. On the contrary GGA+U
calculations [38] produce a narrow band of 0.35 eV, which overestimates the
experimental band gap of $\sim$0.07 eV. Upon increasing pressure, the density
of states reveals that a pseudogap is generated by the splitting of $3d$ Cr
orbitals. In the marcasite structure, the band edges near the Fermi level have
a large Cr-$3d$ character, while the Sb atomic orbitals contribute largely to
the electronic properties below $-$2 eV and above $+$3 eV from the Fermi level
(not shown in the figure). Overall, this results in a semi-metallic or semi-
conducting behaviour. The situation is strikingly different for the HP-II
polymorph. First, the system is metallic with a large density of states at the
Fermi level, and the magnetic order has a modest impact on the electronic
structure. The density of states of the HP-II structure at $E_{\mathrm{F}}$
decreases as a function of pressure. The results thus confirm that the
marcasite to HP-II transition is accompanied by a sudden metallization of the
system. This leads to the lower resistivity of the HP-II phase compared to the
marcasite phase. Furthermore, the drop of the resistance between 5 and 10 GPa
for the marcasite phase is also reproduced by the calculations, however with a
pressure offset of 4 GPa. The calculated Hall-coefficient (Fig. 2 b)) is in
accordance with the measurements by Li et al. [5] with a change to a positive
Hall coefficient after the phase transition. The carrier concentration
obtained from the Hall coefficient (Fig. 2 c)) is on the same order of
magnitude as in Li et al., however our calculated maximum is 4 GPa lower than
the observed maximum by Li et al.[5] which is attributed to computational
effects as well as effects of non-hydrostaticity in the measurements. The fact
that the calculations on the MoP2-type model for the high-pressure phase are
consistent with the results by Li et al. provides further evidence that the
MoP2-type structure is correct. The band structure of the AFM1 structure at 12
GPa is shown in Fig. 3 both with and without Spin Orbit Coupling (SOC).

Figure 4: a) Brillouin zone of the HP-II AFM1 structure at 12 GPa with Weyl
points $w_{1}$ and $w_{2}$. Red points have positive chirality, blue points
have negative chirality. The $w_{2}$ points are very close to the b3=0
($k_{z}$=0) plane. b) Band structure passing through the $w_{1}$ points. c)
Band structure passing through the $w_{2}$ points.

The effect of SOC is minor in this system and it results in narrow splittings
of the bands in the vicinity of the Fermi level. At 12 GPa the system is a
semimetal with a pseudogap and low density of states at the Fermi level (Fig.
S3). The AFM1 magnetic order removes the _C_ centering from the $Cmc2_{1}$
space group. Even though the space group is non-symmorphic, we verified that
the system is not an _altermagnet_ [44, 45], i.e. that no spin-splitting
occurs at any k-point of the Brillouin zone. The large number of band
crossings or near-crossings around the Fermi level, could reveal the presence
of Weyl points. Similarly to the parent structure compounds, MoP2 and WP2 [3],
we found two symmetry independent type-I Weyl points. The results are reported
in Table 2. From Fig. 4 a) $w_{1}$ points are located near to the top and
bottom face of the Brillouin zone. Due to the orthorhombic symmetry, $w_{1}$
points with opposite chirality sit aligned along the $b_{2}$ and $b_{3}$
reciprocal lattice vectors. The $w_{2}$ points are instead located near the
$k_{z}=0$ ($b_{3}=0$) plane and are composed by very close pairs, as in MoP2.
As the $w_{2}$ points have the same chirality they are robust with respect to
strain and small deformations. The band structure passing through the $w_{1}$
and $w_{2}$ points reveals tilted cones (Fig. 4 b-c)), with just one point
crossing the constant energy plane (type-I Weyl points).

Table 2: Coordinates, chirality and energy of Weyl points in the HP-II AFM1 structure at 12 GPa. For each point listed, the remaining 7 ones are generated by applying the mirror symmetries. Weyl point | kx (Å-1) | ky (Å-1) | kz (Å-1) | Chirality | E$-$EF (eV)
---|---|---|---|---|---
$w_{1}$ | 0.2141 | 0.0792 | 0.4625 |  $+$1 |  $-$0.1271
$w_{2}$ | 0.3729 | 0.1096 | 0.0055 |  $+$1 |  $-$0.1129

###  3.3 Equation of state of the high pressure polymorphs of CrSb2

The diffractograms for the HP-II structure are shown in Fig. 5 a) and the
refined unit cell parameters are shown together with the values from the DFT
calculations in Fig. 5 b-e) together with the Birch-Murnaghan EoS fits (Table
3 for the volume and redTable S3 in the SI for the linear moduli).

Figure 5: a) Background subtracted waterfall plot of the diffractograms for
the HP-II phase. b-e) Experimental and calculated unit cell parameters for the
HP-II structure.

The volume reddecreased with pressure in a scattered way, which is attributed
to strain observed for the Au pressure standard as well as a change in the
measured sample position at 19.3 GPa and change in the measured Au position at
21.7 GPa. From the linear moduli the $b$-axis was seen to be the least
compressible while the $c$-axis was the most compressible. As the angle
between the bond of the Sb-Sb dimer and the $b$-axis and $c$-axis is approx.
$45\degree$, this bond stabilizes both axes equally. Therefore, the lower
compressibility of the $b$-axis is attributed to the compression of the Cr-Sb
bonds, as they are oriented more along the $b$\- than the $c$-axis. Along the
$a$-axis both the Cr-Cr and Sb-Sb bond distances are equal to the lattice
constant, and here the intermediate compressibility is attributed to the Cr-Cr
interactions. The parent compounds, MoP2 and redWP2, have a higher bulk
modulus, redhowever the same trend for the linear moduli with
$M_{0,b}>M_{0,a}>M_{0,c}$ is also seen for MoP2 redand WP2 [19, 46].

For the HP-I structure the diffractograms are shown in Fig. 6 a). The refined
unit cell parameters and the unit cell parameters calculated by DFT are shown
in Fig. 6c) together with the Birch-Murnaghan EoS fits (Table 3 for the volume
and redTable S4 in the SI for the linear moduli). The $a$\- and $c$-axis were
seen to decrease relatively smoothly with pressure, while for the volume some
points around 5-$15\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ were scattered
around the fit. The obtained ambient pressure unit cell values (Table S4 and
Table 3 for the volume) are in good agreement with the values obtained by
Takizawa et al. and Jiao et al. [10, 13, 14].

Table 3: Equation of state parameters for both high pressure polymorphs of
CrSb2 as well as for related structures.

|  $V_{0}$ (Å3) |  $K_{0}$ (GPa) | $K^{\prime}_{0}$ | Comments
---|---|---|---|---
CuAl2-type structures: |  |  |
HP-I CrSb2 | 241.68(15) | 77(2) | 6.2(3) | This study
TiSb2 |  | 91a |  | Tavassoli et al. [47]
V_0.97Sb2 |  | 85a |  | Failamani et al. [48]
CuAl2 | 179.5(6) | 117(13) | 4 (fixed) | Grin et al. [49]
MoP2-type structures: |  |  |
HP-II CrSb2 | 251.8(14) | 91(3) | 4 (fixed) | This study
MoP2 |  | 238(20) |  | Soto et al. [46]
redWP2 | red174.8(2) | red234.0(34) | red4(fixed) | redChi et al. [19]
Marcasite-type: |  |  |
CrSb2 | 137.2(2) | 33(3) | 13.6(19) | Ehrenreich-Petersen et al. [6]
| 130.7(2) | 74.2(2) | 4 (fixed) | Li et al. [5]

  * a

Measured by resonant ultrasound spectroscopy

Figure 6: a) Background subtracted waterfall plot of the diffractograms for
the HP-I phase. b-d) Experimental and calculated unit cell parameters for the
HP-I structure

From the obtained linear moduli the $c$-axis was seen to be more compressible
redthan the $a$-axis, which also has been reported for TiSb2 by Armbrüster et
al. which they attributed to weaker Ti-Ti bonds compared to the Sb-Sb bonds
[11]. By comparing the HP-I phase to its isostructural compounds TiSb2 and
V_1-xSb2 a decreasing trend in bulk modulus (Table 3) is seen going from Ti-V-
Cr. As the bulk modulus of the elements increases from Ti-V-Cr [50, 51, 52],
and the $M$-Sb distance ($M$=Ti, V, Cr) decreases when going from Ti-V-Cr an
increasing trend in the bulk modulus would be expected. However, the Sb-Sb
distance decreases when going in this direction, and as the Sb-Sb bond had
been shown to be the least compressible, this increase in bond distance
explains the decreasing bulk modulus and furthermore verifies that the Sb-Sb
bond is the least compressible [10, 11].

###  3.4 Discussion of bulk moduli among the polymorphs of CrSb2

In Table 4 selected bond distances for the three polymorphs are shown. To
facilitate a better comparison among the polymorphs, the atomic coordinates
from the SC-XRD experiment were used for the unit cell extrapolated to 0 GPa
for the HP-II phase. At 0 GPa the Cr-Sb distance in the HP-II polymorph is
similar to the Cr-Sb distance in the marcasite phase, where the coordination
number is six. Thereby the higher bulk-modulus of the HP-II phase can be
explained by its higher coordination number of seven. In the HP-I phase Cr is
coordinated to eight Sb atoms, however the bulk modulus for this polymorph is
lower than for the HP-II phase, which is attributed to the longer Cr-Sb
distance in the HP-I polymorph of 2.834 Å [13]. As the HP-II structure is not
stable at 0 GPa, the bulk moduli are also compared at 12 GPa, where the bulk
modulus for the HP-II phase is 136(3) GPa and for the HP-I phase 145(1) GPa.
Here, the Cr-Sb bond length is still shorter for the HP-II phase, and the
slightly higher bulk modulus for the HP-I phase can be explained by its
shorter Cr-Cr bond distance.

Table 4: Bond distances of the different polymorphs of CrSb2

| Cr-Cr (Å) | Cr-Sb (Å) | Sb-Sb (Å) | Reference
---|---|---|---|---
0 GPa |  |  |  |
Marcasite-type | 3.2715(7) | 2.689(9) ($\times$4) | 2.84(2) | [53]
| 2.746(8) ($\times$2) |  |
HP-I (CuAl2-type) | 2.8720(2) | 2.8339(14) | 2.914(9) | [13]
HP-II (MoP2-type)a | 3.172(13) | 2.714(11) | 2.907(7) | This study
| 2.729(7) ($\times$2) |  |
| 2.772(9) ($\times$2) |  |
| 2.743(6) ($\times$2) |  |
12 GPa |  |  |  |
HP-I (CuAl2-type) | 2.7413(17) | 2.7298(7) | 2.8431(7) | This study
HP-II (MoP2-type) | 3.0751(2) | 2.624(7) | 2.803(3) | This study
| 2.645(4) ($\times$2) |  |
| 2.635(4) ($\times$2) |  |
| 2.683(6) ($\times$2) |  |

  * a

The values at 0 GPa were obtained using the $L_{0}$ values from table S3 and
atomic coordinates obtained through SC-XRD.

##  4 Conclusion

Using single crystal X-ray diffraction at 12 GPa it was shown that the high
pressure phase of CrSb2 crystallizes in the orthorombic $Cmc2_{1}$ space
group, with a MoP2-type structure with unit cell constants of $a=3.0752(2)$ Å,
$b=12.4476(19)$ Å and $c=5.8741(2)$ Å well in accordance with the obtained
unit cell from DFT calculations and Le Bail refinements of powder
diffractograms. Furthermore, electrical resistance measurements under pressure
had shown that the phase transition is accompanied by a metallization of the
system, and the experimentally measured resistance was verified by resistivity
calculations. Furthermore, the calculations showed that the Hall coefficient
changes from positive to negative during the phase transition from the
marcasite to the MoP2-type phase.

The calculated band-structure at 12 GPa has shown that this polymorph is a
robust type-I Weyl semimetal. Furthermore, the compressibility of both high
pressure polymorphs of CrSb2 were investigated, showing no further phase
transitions for both polymorphs, and Le Bail refinements of the unit cell
parameters showed that the MoP2 phase was less compressible than the
CuAl2-type phase which was attributed to the longer Cr-Sb distance in the
CuAl2-type polymorph.

##  5 Acknowledgements

This research was funded by the Independent Research Fund Denmark (7027-00077B
and 1026-00409B) and we furthermore acknowledge the Danish Agency for Science,
Technology, and Innovation for funding the instrument center DanScatt. We
acknowledge MAX IV Laboratory for beamtime on the DanMAX beamline (Proposal
20211063). Research conducted at MAX IV, a Swedish national user facility, is
supported by the Swedish Research council under contract 2018-07152, the
Swedish Governmental Agency for Innovation Systems under contract 2018-04969,
and Formas under contract 2019-02496. DanMAX is funded by the NUFI grant no.
4059-00009B. Portions of this work were performed at GeoSoilEnviroCARS (The
University of Chicago, Sector 13), Advanced Photon Source (APS), Argonne
National Laboratory. GeoSoilEnviroCARS is supported by the National Science
Foundation – Earth Sciences (EAR – 1634415). This research used resources of
the Advanced Photon Source, a U.S. Department of Energy (DOE) Office of
Science User Facility operated for the DOE Office of Science by Argonne
National Laboratory under Contract No. DE-AC02-06CH11357. Use of the COMPRES-
GSECARS gas loading system was supported by COMPRES under NSF Cooperative
Agreement EAR -1606856 and by GSECARS through NSF grant EAR-1634415 and DOE
grant DE-FG02-94ER14466. This research used resources of the Advanced Photon
Source, a U.S. Department of Energy (DOE) Office of Science User Facility
operated for the DOE Office of Science by Argonne National Laboratory under
Contract No. DE-AC02-06CH11357. We acknowledge DESY (Hamburg, Germany), a
member of the Helmholtz Association HGF, for the provision of experimental
facilities. Parts of this research were carried out at PETRA III using the
P02.2 beamline. Beamtime was allocated for proposal H-20010016. DC
acknowledges support from ICSC – Centro Nazionale di Ricerca in High
Performance Computing, Big Data and Quantum Computing, funded by European
Union – NextGenerationEU (Grant number CN00000013).

## References

  * [1] F. Hulliger, New representatives of the NbAs2 and ZrAs2 structures, Nature 204 (Nov 1964).  doi:10.1038/204775a0.
  * [2] S. Rundqvist, T. Lundström, X-ray studies of molybdenum and tungsten phosphides, Acta Chemica Scandinavica 17 (1963) 37–46.  doi:10.3891/acta.chem.scand.17-0037.
  * [3] G. Autès, D. Gresch, M. Troyer, A. A. Soluyanov, O. V. Yazyev, Robust type-II weyl semimetal phase in transition metal diphosphides ${X}$P2 (${X}=$Mo, W), Physical Review Letters 117 (2016) 066402.  doi:10.1103/PhysRevLett.117.066402.
  * [4] J. Du, Z. Lou, S. Zhang, Y. Zhou, B. Xu, Q. Chen, Y. Tang, S. Chen, H. Chen, Q. Zhu, H. Wang, J. Yang, Q. Wu, O. V. Yazyev, M. Fang, Extremely large magnetoresistance in the topologically trivial semimetal $\alpha-$WP2, Physical Review B 97 (2018) 245101\.  doi:10.1103/PhysRevB.97.245101.
  * [5] C. Li, K. Liu, S. Peng, Q. Feng, D. Jiang, T. Wen, H. Xiao, B. Yue, Y. Wang, Rewritable pressure-driven n–p conduction switching in marcasite-type CrSb2, Chemistry of Materials 35 (3) (2023) 1449–1457.  doi:10.1021/acs.chemmater.2c03673.
  * [6] E. Ehrenreich-Petersen, M. F. Hansen, J. Jeanneau, D. Ceresoli, F. Menescardi, M. Ottesen, V. Prakapenka, S. N. Tkachev, M. Bremholm, Seven-coordinated high-pressure phase of CrSb2 and experimental equation of state of MSb2 (M =Cr, Fe, Ru, Os), Inorganic Chemistry 62 (31) (2023) 12203–12212.  doi:10.1021/acs.inorgchem.3c00227.
  * [7] H. Holseth, A. Kjekshus, A. F. Andresen, Compounds with the marcasite type crystal structure. vi. neutron diffraction studies of CrSb2 and FeSb2, Acta Chemica Scandinavica 24 (1970) 3309–3316.  doi:10.3891/acta.chem.scand.24-3309.
  * [8] B. C. Sales, A. F. May, M. A. McGuire, M. B. Stone, D. J. Singh, D. Mandrus, Transport, thermal, and magnetic properties of the narrow-gap semiconductor CrSb2, Physical Review B 86 (2012) 235136.  doi:10.1103/PhysRevB.86.235136.
  * [9] K. Nakagawa, M. Kimata, T. Yokouchi, Y. Shiomi, Surface anisotropic magnetoresistance in the antiferromagnetic semiconductor CrSb2, Phys. Rev. B 107 (2023) L180405.  doi:10.1103/PhysRevB.107.L180405.
URL https://link.aps.org/doi/10.1103/PhysRevB.107.L180405

  * [10] H. Takizawa, K. Uheda, T. Endo, M. Shimada, High pressure phase transition of CrSb2, Review of High Pressure Science and Technology/Koatsuryoku No Kagaku To Gijutsu 7 (1998) 1043–1045.  doi:10.4131/jshpreview.7.1043.
  * [11] M. Armbrüster, W. Schnelle, U. Schwarz, Y. Grin, Chemical bonding in TiSb2 and VSb2: A quantum chemical and experimental study, Inorganic Chemistry 46 (16) (2007) 6319–6328.  doi:10.1021/ic070284p.
  * [12] W. Xia, X. Shi, Y. Wang, W. Ge, H. Su, Q. Wang, X. Wang, N. Yu, Z. Zou, Y. Hao, W. Zhao, Y. Guo, The de Haas-van Alphen quantum oscillations in a three-dimensional Dirac semimetal TiSb2, Applied Physics Letters 116 (14) (2020) 142103.  doi:10.1063/5.0001566.
URL https://doi.org/10.1063/5.0001566

  * [13] H. Takizawa, K. Uheda, T. Endo, A new ferromagnetic polymorph of CrSb2 synthesized under high pressure, Journal of Alloys and Compounds 287 (1) (1999) 145–149.  doi:https://doi.org/10.1016/S0925-8388(99)00056-0.
  * [14] Y. Y. Jiao, Z. Y. Liu, M. A. McGuire, S. Calder, J.-Q. Yan, B. C. Sales, J. P. Sun, Q. Cui, N. N. Wang, Y. Sui, Y. Uwatoko, B. S. Wang, X. L. Dong, J.-G. Cheng, High-pressure phase of CrSb2: A new quasi-one-dimensional itinerant magnet with competing interactions, Physical Review Materials 3 (2019) 074404.  doi:10.1103/PhysRevMaterials.3.074404.
  * [15] N. Kumar, Y. Sun, N. Xu, K. Manna, M. Yao, V. Süss, I. Leermakers, O. Young, T. Förster, M. Schmidt, H. Borrmann, B. Yan, U. Zeitler, M. Shi, C. Felser, C. Shekhar, Extremely high magnetoresistance and conductivity in the type-II weyl semimetals WP2 and MoP2, Nature Communications 8 (Nov 2017).  doi:10.1038/s41467-017-01758-z.
  * [16] Y.-Y. Lv, J. Xu, S. Han, C. Zhang, Y. Han, J. Zhou, S.-H. Yao, X.-P. Liu, M.-H. Lu, H. Weng, Z. Xie, Y. B. Chen, J. Hu, Y.-F. Chen, S. Zhu, High-harmonic generation in weyl semimetal $\beta$-WP2 crystals, Nature Communications 12 (2021) 6437.  doi:https://doi.org/10.1038/s41467-021-26766-y.
  * [17] A. Toniato, B. Gotsmann, E. Lind, C. B. Zota, Weyl semi-metal-based high-frequency amplifiers, in: 2019 IEEE International Electron Devices Meeting (IEDM), 2019, pp. 9.4.1–9.4.4.  doi:10.1109/IEDM19573.2019.8993575.
  * [18] X. Liu, Z. Yu, J. Li, Z. Xu, C. Zhou, Z. Dong, L. Zhang, X. Wang, N. Yu, Z. Zou, X. Wang, Y. Guo, A new transition metal diphosphide $\alpha$-MoP2 synthesized by a high-temperature and high-pressure technique, Chinese Physics B 32 (1) (2023) 018102.  doi:10.1088/1674-1056/ac633d.
URL https://dx.doi.org/10.1088/1674-1056/ac633d

  * [19] Z. Chi, J. Zhang, Z. Gong, F. Peng, X. Wang, G. Dong, Y. Li, Y. Shi, Y. Ge, X. Yang, Z. Zhang, G. Xu, N. Hao, C. Zhou, J. Qin, Pressure-induced Lifshitz transition in the type-II Weyl semimetal WP2, Materials Today Physics 42 (2024) 101372.  doi:https://doi.org/10.1016/j.mtphys.2024.101372.
URL https://www.sciencedirect.com/science/article/pii/S2542529324000488

  * [20] K. D. Leinenweber, J. A. Tyburczy, T. G. Sharp, E. Soignard, T. Diedrich, W. B. Petuskey, Y. Wang, J. L. Mosenfelder, Cell assemblies for reproducible multi-anvil experiments (the COMPRES assemblies), American Mineralogist 97 (2-3) (2012) 353–368.  doi:10.2138/am.2012.3844.
  * [21] H.-P. Liermann, Z. Konôpková, W. Morgenroth, K. Glazyrin, J. Bednarčik, E. E. McBride, S. Petitgirard, J. T. Delitz, M. Wendt, Y. Bican, A. Ehnes, I. Schwark, A. Rothkirch, M. Tischer, J. Heuer, H. Schulte-Schrepping, T. Kracht, H. Franz, The Extreme Conditions Beamline P02.2 and the Extreme Conditions Science Infrastructure at PETRAIII, Journal of Synchrotron Radiation 22 (4) (2015) 908–924.  doi:10.1107/S1600577515005937.
  * [22] K. Syassen, Ruby under pressure, High Pressure Research 28 (2) (2008) 75–126.  doi:10.1080/08957950802235640.
  * [23] CrysAlisPRO, Oxford Diffraction /Agilent Technologies UK Ltd, Yarnton, England.
  * [24] O. V. Dolomanov, L. J. Bourhis, R. J. Gildea, J. A. K. Howard, H. Puschmann, OLEX2: a complete structure solution, refinement and analysis program, Journal of Applied Crystallography 42 (2) (2009) 339–341.  doi:10.1107/S0021889808042726.
  * [25] G. M. Sheldrick, Crystal structure refinement with SHELXL, Acta Crystallographica Section C 71 (1) (2015) 3–8.  doi:10.1107/S2053229614024218.
  * [26] G. M. Sheldrick, SHELXT – Integrated space-group and crystal-structure determination, Acta Crystallographica Section A 71 (1) (2015) 3–8.  doi:10.1107/S2053273314026370.
  * [27] M. Rivers, V. B. Prakapenka, A. Kubo, C. Pullins, C. M. Holl, S. D. Jacobsen, The COMPRES/GSECARS gas-loading system for diamond anvil cells at the advanced photon source, High Pressure Research 28 (3) (2008) 273–292.  doi:10.1080/08957950802333593.
  * [28] A. Dewaele, P. Loubeyre, M. Mezouar, Equations of state of six metals above 94 GPa, Physical Review B 70 (2004) 094112.  doi:10.1103/PhysRevB.70.094112.
  * [29] Y. Fei, A. Ricolleau, M. Frank, K. Mibe, G. Shen, V. Prakapenka, Toward an internally consistent pressure scale, Proceedings of the National Academy of Sciences 104 (22) (2007) 9182–9186.  doi:10.1073/pnas.0609013104.
  * [30] C. Prescher, V. B. Prakapenka, DIOPTAS: a program for reduction of two-dimensional x-ray diffraction data and data exploration, High Pressure Research 35 (3) (2015) 223–230.  doi:10.1080/08957959.2015.1059835.
  * [31] J. Rodríguez-Carvajal, Recent advances in magnetic structure determination by neutron powder diffraction, Physica B: Condensed Matter 192 (1) (1993) 55–69.  doi:https://doi.org/10.1016/0921-4526(93)90108-I.
  * [32] J. Gonzalez-Platas, M. Alvaro, F. Nestola, R. Angel, EosFit7-GUI: a new graphical user interface for equation of state calculations, analyses and teaching, Journal of Applied Crystallography 49 (4) (2016) 1377–1382.  doi:10.1107/S1600576716008050.
  * [33] K. Momma, F. Izumi, VESTA3 for three-dimensional visualization of crystal, volumetric and morphology data, Journal of Applied Crystallography 44 (6) (2011) 1272–1276.  doi:10.1107/S0021889811038970.
  * [34] S. T. Weir, J. Akella, C. Aracne-Ruddle, Y. K. Vohra, S. A. Catledge, Epitaxial diamond encapsulation of metal microprobes for high pressure experiments, Applied Physics Letters 77 (21) (2000) 3400–3402.  doi:10.1063/1.1326838.
  * [35] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. D. Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P. Seitsonen, A. Smogunov, P. Umari, R. M. Wentzcovitch, QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials, Journal of Physics: Condensed Matter 21 (39) (2009) 395502.  doi:10.1088/0953-8984/21/39/395502.
  * [36] P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. B. Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carnimeo, A. D. Corso, S. de Gironcoli, P. Delugas, R. A. DiStasio, A. Ferretti, A. Floris, G. Fratesi, G. Fugallo, R. Gebauer, U. Gerstmann, F. Giustino, T. Gorni, J. Jia, M. Kawamura, H.-Y. Ko, A. Kokalj, E. Küçükbenli, M. Lazzeri, M. Marsili, N. Marzari, F. Mauri, N. L. Nguyen, H.-V. Nguyen, A. O. de-la Roza, L. Paulatto, S. Poncé, D. Rocca, R. Sabatini, B. Santra, M. Schlipf, A. P. Seitsonen, A. Smogunov, I. Timrov, T. Thonhauser, P. Umari, N. Vast, X. Wu, S. Baroni, Advanced capabilities for materials modelling with QUANTUM ESPRESSO, Journal of Physics: Condensed Matter 29 (46) (2017) 465901.  doi:10.1088/1361-648x/aa8f79.
  * [37] J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E. Scuseria, L. A. Constantin, X. Zhou, K. Burke, Restoring the density-gradient expansion for exchange in solids and surfaces, Physical Review Letters 100 (2008) 136406.  doi:10.1103/PhysRevLett.100.136406.
  * [38] G. Kuhn, S. Mankovsky, H. Ebert, M. Regus, W. Bensch, Electronic structure and magnetic properties of CrSb2 and FeSb2 investigated via ab initio calculations, Physical Review B 87 (8) (feb 2013).  doi:10.1103/physrevb.87.085113.
  * [39] G. K. Madsen, J. Carrete, M. J. Verstraete, BoltzTraP2, a program for interpolating band structures and calculating semi-classical transport coefficients, Computer Physics Communications 231 (2018) 140–145.  doi:10.1016/j.cpc.2018.05.010.
  * [40] F. T. Cerasoli, A. R. Supka, A. Jayaraj, M. Costa, I. Siloi, J. Sławińska, S. Curtarolo, M. Fornari, D. Ceresoli, M. B. Nardelli, Advanced modeling of materials with PAOFLOW 2.0: New features and software design, Computational Materials Science 200 (2021) 110828.  doi:10.1016/j.commatsci.2021.110828.
  * [41] D. Gresch, G. Autès, O. V. Yazyev, M. Troyer, D. Vanderbilt, B. A. Bernevig, A. A. Soluyanov, Z2pack: Numerical implementation of hybrid wannier centers for identifying topological materials, Physical Review B 95 (7) (feb 2017).  doi:10.1103/physrevb.95.075146.
  * [42] H. R. Aliabad, S. Rabbanifar, M. Khalid, Structural, optoelectronic and thermoelectric properties of FeSb2 under pressure: Bulk and monolayer, Physica B: Condensed Matter 570 (2019) 100–109.  doi:https://doi.org/10.1016/j.physb.2019.06.001.
  * [43] A. Mani, J. Janaki, A. T. Satya, T. G. Kumary, A. Bharathi, The pressure induced insulator to metal transition in FeSb2, Journal of Physics: Condensed Matter 24 (7) (2012) 075601.  doi:10.1088/0953-8984/24/7/075601.
  * [44] L.-D. Yuan, Z. Wang, J.-W. Luo, A. Zunger, Prediction of low-Z collinear and noncollinear antiferromagnetic compounds having momentum-dependent spin splitting even without spin-orbit coupling, Physical Review Materials 5 (1) (jan 2021).  doi:10.1103/physrevmaterials.5.014409.
  * [45] I. I. Mazin, K. Koepernik, M. D. Johannes, R. González-Hernández, L. Šmejkal, Prediction of unconventional magnetism in doped FeSb2, Proceedings of the National Academy of Sciences 118 (42) (oct 2021).  doi:10.1073/pnas.2108924118.
  * [46] V. Soto, K. Knorr, L. Ehm, C. Baehtz, B. Winkler, M. Avalos-Borja, High-pressure and high-temperature powder diffraction on molybdenum diphosphide, MoP2, Zeitschrift für Kristallographie - Crystalline Materials 219 (6) (2004) 309–313.  doi:doi:10.1524/zkri.219.6.309.34642.
  * [47] A. Tavassoli, A. Grytsiv, F. Failamani, G. Rogl, S. Puchegger, H. Müller, P. Broz, F. Zelenka, D. Macciò, A. Saccone, G. Giester, E. Bauer, M. Zehetbauer, P. Rogl, Constitution of the binary m-sb systems (m = Ti, Zr, Hf) and physical properties of MSb2, Intermetallics 94 (2018) 119–132.  doi:https://doi.org/10.1016/j.intermet.2017.12.014.
  * [48] F. Failamani, P. Broz, D. Macciò, S. Puchegger, H. Müller, L. Salamakha, H. Michor, A. Grytsiv, A. Saccone, E. Bauer, G. Giester, P. Rogl, Constitution of the systems {V,Nb,Ta}-Sb and physical properties of di-antimonides {V,Nb,Ta}Sb2, Intermetallics 65 (2015) 94–110.  doi:https://doi.org/10.1016/j.intermet.2015.05.006.
  * [49] Y. Grin, F. R. Wagner, M. Armbrüster, M. Kohout, A. Leithe-Jasper, U. Schwarz, U. Wedig, H. Georg von Schnering, CuAl2 revisited: Composition, crystal structure, chemical bonding, compressibility and raman spectroscopy, Journal of Solid State Chemistry 179 (6) (2006) 1707–1719.  doi:https://doi.org/10.1016/j.jssc.2006.03.006.
  * [50] Y. K. Vohra, P. T. Spencer, Novel $\gamma$-phase of titanium metal at megabar pressures, Physical Review Letters 86 (2001) 3068–3071.  doi:10.1103/PhysRevLett.86.3068.
  * [51] W. A. Crichton, J. Guignard, E. Bailey, D. P. Dobson, S. A. Hunt, A. R. Thomson, High-temperature equation of state of vanadium, High Pressure Research 36 (1) (2016) 16–22.  doi:10.1080/08957959.2015.1123256.
  * [52] A. Marizy, G. Geneste, P. Loubeyre, B. Guigue, G. Garbarino, Synthesis of bulk chromium hydrides under pressure of up to 120 GPa, Physical Review B 97 (2018) 184103.  doi:10.1103/PhysRevB.97.184103.
  * [53] H. Holseth, A. Kjekshus, Compounds with the marcasite type crystal structure. II. on the crystal structures of the binary pnictides, Acta Chemica Scandinavica 22 (1968) 3284–3292.  doi:10.3891/acta.chem.scand.22-3284.