# Opening the black-box of Neighbor Embedding with Hotelling’s $T^{2}$
statistic and $Q$-residuals

Roman Josef Rainer  Michael Mayr  Johannes Himmelbauer  Ramin Nikzad-Langerodi

###### Abstract

In contrast to classical techniques for exploratory analysis of high-
dimensional data sets, such as principal component analysis (PCA), neighbor
embedding (NE) techniques tend to better preserve the local structure/topology
of high-dimensional data. However, the ability to preserve local structure
comes at the expense of interpretability: Techniques such as t-Distributed
Stochastic Neighbor Embedding (t-SNE) or Uniform Manifold Approximation and
Projection (UMAP) do not give insights into which input variables underlie the
topological (cluster) structure seen in the corresponding embedding. We here
propose different ”tricks” from the chemometrics field based on PCA,
$Q$-residuals and Hotelling’s $T^{2}$ contributions in combination with novel
visualization approaches to derive local and global explanations of neighbor
embedding. We show how our approach is capable of identifying discriminatory
features between groups of data points that remain unnoticed when exploring
NEs using standard univariate or multivariate approaches.

###### keywords:

neighbor embedding , t-SNE , UMAP , PCA , chemometrics

††journal: Chemometrics and Intelligent Laboratory Systems

[inst1]organization=Software Competence Center
Hagenberg,addressline=Softwarepark 32a, city=Hagenberg, postcode=4232, state=,
country=Austria

##  1 Introduction

Neighbor embedding techniques have gained increasing popularity in a wide
variety of scientific disciplines such as machine learning [1], biology [2],
physics [3] or engineering [4]. In contrast to classical techniques for
exploratory analysis of high-dimensional data sets, such as principal
component analysis (PCA), these methods tend to better preserve the ”intrinsic
structure” or (local) topology of high-dimensional data when mapped to (i.e.
embedded in) low-dimensional spaces which often provides additional insights.
However, the ability to preserve local structure comes at the expense of
interpretability: Techniques such as t-Distributed Stochastic Neighbor
Embedding (t-SNE) [1] or Uniform Manifold Approximation and Projection (UMAP)
[5] do not give insights into which input variables underlie the topological
(cluster) structure seen in the corresponding embedding. Tools, such as
Embedding Projector (EP) [6] or t-viSNE [7] can help to better understand the
corresponding embedding. EP provides a web-based, interactive environment that
allows different ”views” on high-dimensional data sets and their embedding.
However, while EP is useful to explore embedding in terms of pre-defined
labels, there is no possibility to investigate the relationship between
(semantically) meaningful directions in the embedded space and the input
variables. In general, inspection of how single variables change across an
embedding is straightforward and can be undertaken for low-dimensional data
sets but becomes prohibitive for data sets with hundreds of thousands of
variables. In order to address this issue, t-viSNE comes with a so-called
”dimension correlation tool”, which allows to explore the correlation between
the input variables and a user-defined (e.g. local and/or non-linear)
direction in the embedded space. Bibal et al. ([8]) employ a modified LIME
(local interpretable model-agnostic explanations) approach to explain t-SNE
embedding. Similar to t-viSNE, this approach derives ”explanations” for local
neighborhoods around particular data points in terms of the input variables.
Both methods derive local explanations rather than explaining the (global)
directions of an embedding and require considerable computational resources
for data sampling and local modelling. In addition, both methods require
considerable user-interactions, which might restrict their use to expert users
familiar with the techniques. An alternative approach has been recently
proposed in [9], where the authors use local neighborhood information derived
from UMAP as an additional constraint in PCA. However, for some data sets,
replication of the (non-linear) neighborhood structure with a linear model is
difficult and eventually yields latent variables (LVs) that capture the noise
rather than the systematic variation in the data.

In the current contribution, we propose a simple, yet effective, workflow to
derive explanations of local and global directions of (non-linear) embedding
that is computationally efficient and reduces the overhead of user-
interactions. In brief, we first create the corresponding embedding and fit a
PCA model to the input data. We then employ relative Hotelling’s
$T^{2}$-contributions to derive explanations for the difference between
individual data points or entire clusters. In addition, we propose novel
visualizations that include the PCs scores along with the corresponding
$Q$-residuals in order to understand which data points are well characterized
by individual LVs underlying the data set. The main components of our approach
are summarized in Figure 1.

Figure 1: PCA-derived explanations of neighbor embedding by means of relative
Hotelling’s $T^{2}$ contributions and $Q$-residuals.

We will focus on deriving PCA-based explanations of UMAP embedding, which
build upon t-SNE and now belongs to the most widely used neighbor embedding
methods in the data science community. In general, however, the same workflow
can be applied to any (non-linear/parametric) dimensionality reduction method.
We refer to the excellent review by Wang et al. [10] for an overview of the
current state-of-the-art dimension reduction techniques.

##  2 Theory

###  2.1 Principal Component Analysis (PCA)

Principal component analysis (PCA) is among the most widely used techniques
for exploratory analysis of multivariate datasets [11]. PCA decomposes an
$I\times J$ (e.g. time points $\times$ process variables) matrix $\mathbf{X}$
into a set of _scores_
$\mathbf{T}=[\mathbf{t}_{1},\dots,\mathbf{t}_{A}]^{I\times A}$ and _loadings_
$\mathbf{P}=[\mathbf{p}_{1},\dots,\mathbf{p}_{A}]^{J\times A}$ vectors such
that [12]

$\displaystyle\begin{aligned}
&\mathbf{X}=\sum_{i=1}^{A}\mathbf{t}_{i}\mathbf{p}_{i}^{T}+\mathbf{E}.\\\
&\textbf{s.t.}\;\;\mathbf{t}_{i}^{T}\mathbf{t}_{j}=0\;\text{and}\;\mathbf{p}_{i}^{T}\mathbf{p}_{j}=0\;\;\text{for}\;\;i\neq
j.\end{aligned}$ (1)

Ideally, $A$ in Eq.(1) is chosen such that the systematic variation in
$\mathbf{X}$ is encoded in the mutually orthogonal principal components (PCs)
$\mathbf{t}_{1}\mathbf{p}_{1}^{T},\dots,\mathbf{t}_{A}\mathbf{p}_{A}^{T}$
while the random noise is encoded in the $I\times J$ residual matrix
$\mathbf{E}$. The strength of PCA is rooted in the fact that each PC
represents a separate source of variation underlying the observed data that
can be analyzed at the samples (scores) and variables (loadings) level. While
the former allows to detect (e.g. temporal) trends in (process) data, the
latter allows to investigate the correlations between the original variables
and those trends. In addition, computation of the variance explained by each
PC is straightforward, allowing to quantify the contribution of a trend to the
total variance in a data set. Finally, PCA fits a bilinear model to the data,
which can be useful when aiming at making predictions on new data. This is
exploited e.g. in multivariate statistical process control (MSPC) to detect
departure from normal operating conditions ([13]) or in classification
problems ([14]).

Despite its versatility, several characteristics of PCA are less favorable for
data analysis. First, it is important to note that the objective of PCA is to
find directions in the high-dimensional space of the input variables where the
(explained) variance is largest. Consequently, PCA is good at capturing global
trends while largely neglecting local structures that might be present and
eventually encode important information about the process under investigation.
On the other hand, the latent variables (LVs) that span the PCA subspace are
linear combinations of the original variables. According to the central limit
theorem, linear combinations of random variables converge to normal
distributed LVs [15] as the number of variables increases and scores plots
thus often tend to show spherically shaped clusters where the natural topology
of the data is ”distorted” which is particularly the case for very high-
dimensional data

Over the past decades, several dimension reduction methods have been proposed
that better preserve the (local) structure/topology of high dimensional data,
two of which we find particularly useful. These shall thus be introduced in
the next sections. These will therefore be briefly introduced in the following
sections

###  2.2 Neighbor Embedding

The general idea of Neighbor Embedding (NE) is to preserve (local) distances
between data points in the high-dimensional space when deriving the
corresponding low-dimensional representation.

#### t-SNE

t-SNE models the distance between two data points $\mathbf{x}_{i}$ and
$\mathbf{x}_{j}$ in the high-dimensional space as the (conditional)
probability

$p_{ij}=\frac{p_{j|i}+p_{i|j}}{2N},$ (2)

with

$\displaystyle p_{j|i}=\frac{v_{j|i}}{\sum_{k\neq i}v_{j|i}}$ ; $\displaystyle
v_{j|i}=\text{exp}\left(-\frac{\left\lVert\mathbf{x}_{i}-\mathbf{x}_{j}\right\rVert^{2}_{2}}{2\sigma_{i}^{2}}\right).$
(3)

$N$ denotes the number of data points, and $\sigma_{i}$ (i.e. the Gaussian
kernel bandwidth) controls how fast the probability decreases with increasing
Euclidean distance from the $i$-th data point. Instead of manually choosing
the (optimal) bandwidth for each data point, t-SNE ”automatically” adjusts
$\sigma$ to be small in dense regions and large in sparse regions of the input
space based on the so-called perplexity parameter

$\text{Perplexity}(P_{i})=2^{-\sum p_{j|i}\log_{2}p_{j|i}}$ (4)

that needs to be specified by the user. The perplexity is related to the
number of (close) neighbors around each data point that should be considered
in the embedding. A large perplexity emphasizes global while a small one
focuses on local structure. The distance in the (low-dimensional) embedded
space is modeled using Student’s t-distribution with the corresponding
probability

$\displaystyle q_{ij}=\frac{w_{ij}}{\sum_{k\neq l}w_{kl}}$ ; $\displaystyle
w_{ij}=\frac{1}{1+\left\lVert\mathbf{y}_{i}-\mathbf{y}_{j}\right\rVert^{2}_{2}},$
(5)

where $\mathbf{y}_{i}$ denotes the (two-dimensional) coordinates of the $i$-th
data point in the embedded space. Finally, the so-called _Kullback-Leibler_
(KL) divergence between the high- and low dimensional distributions $P$ and
$Q$ with respect to these coordinates is minimized, i.e.

$\min_{\mathbf{y}_{i},\dots,\mathbf{y}_{N}}\;\text{KL}(P||Q)$ (6)

with $\text{KL}(P||Q)=\sum_{i\neq j}p_{ij}\log\frac{p_{ij}}{q_{ij}}$.
Technically, this is achieved by means of gradient descent using some random
initialization of the $y$-coordinates.

#### UMAP

Conceptually, t-SNE and UMAP share the same basic idea with subtle differences
related to how similarity is modelled. Most importantly, UMAP does not
normalize pair-wise distances, i.e. $\sum_{i,j}p_{ij}\neq 1$ (the same is true
for $q_{ij}$), employs

$p_{j|i}=\text{exp}\left(-\frac{\text{d}(\mathbf{x}_{i},\mathbf{x}_{j})-\rho}{\sigma}\right)$
(7)

as distance metric and minimizes cross-entropy

$\text{CE}(P||Q)=\sum_{i\neq
j}\left(p_{ij}\cdot\log\left(\frac{p_{ij}}{q_{ij}}\right)+(1-p_{ij})\cdot\log\left(\frac{1-p_{ij}}{1-q_{ij}}\right)\right)$
(8)

instead of KL-divergence. The parameter $\rho$ in Eq.(7) represents the
distance from the $i$-th data point to its nearest neighbor and implies a
different distance metric for each data point (that in turn requires a
symetrization that is slightly different from Eq. (2)).
$\text{d}(\mathbf{x}_{i},\mathbf{x}_{j})$ denotes some distance (e.g.
Euclidean distance) between $\mathbf{x}_{i}$ and $\mathbf{x}_{j}$. In
addition, UMAP uses the number of $k$ nearest neighbors

$k=2^{-\sum p_{ij}}$ (9)

when computing $p_{j|i}$ and the family of curves

$q_{ij}=\left(1+a\cdot(\left\lVert\mathbf{y}_{i}-\mathbf{y}_{j}\right\rVert^{2}_{2})^{b}\right)^{-1}$
(10)

instead of perplexity and Student t-distribution, respectively. For the
latter, the parameters $a$ and $b$ are obtained by non-linear least-squares
fitting.

#### t-SNE vs. UMAP

The fact, that UMAP does not normalize the ”probabilities” not only makes it
faster (due to omission of sum computations) and consume less memory
(replacement of gradient descent by stochastic gradient descent) but also
allows one to ”map” new samples to the embedding without changing the position
of the training samples, e.g. by computing

$\hat{\mathbf{y}}_{N+1}=\min_{\mathbf{y}_{N+1}}\;\text{CE}(P_{\mathbf{y}_{1},\dots,\mathbf{y}_{N+1}}||Q_{\mathbf{y}_{1},\dots,\mathbf{y}_{N+1}}).$
(11)

With t-SNE this is not possible, since any new data point changes $p_{j|i}$
and $q_{j|i}$ (through normalization) and thus the (optimal) coordinates of
the training samples. In addition, UMAP better balances global and local
structure preservation. This is because data points that are far apart (small
$p_{ij}$) contribute little to the KL-divergence but contribute to CE through
the second term in Eq. (8).

###  2.3 Interpretability

Unlike in PCA, where each embedded sample can be expressed as linear
transformation of the corresponding inputs, i.e.

$\mathbf{y}_{i}=\mathbf{x}_{i}^{\text{T}}\mathbf{P},$ (12)

the (non-linear) map $f:\mathcal{X}\rightarrow\mathcal{Y}$ remains a black-box
in both t-SNE and UMAP. Since there is no model, interpretability is not
given. To mitigate this issue we propose two complementary workflows to better
understand the embedded space.

###  2.4 Our approach

The idea of using PCA as preprocessing step or to use PCA scores (i.e.
individuals columns of the $\mathbf{T}$ matrix from Eq. (1)) as color code for
NE has been proposed earlier (e.g in [7]) and is a useful tool to get a quick
overview about which variables contribute to the topological structure of the
embedding. We extend this idea by introducing two concepts widely used in
chemometrics: $Q$-residuals and relative $T^{2}$-contributions, and introduce
Voronoi plots for visualization of the corresponding NE.

#### Q-Residuals

are also known as reconstruction loss in the machine learning community. For
an input sample $\mathbf{x}_{i}$

$Q_{i}=\mathbf{x}_{i}(\mathbf{I}-\mathbf{P}_{A}^{\text{T}}\mathbf{P}_{A})\mathbf{x}_{i}^{\text{T}},$
(13)

with $\mathbf{P}_{A}$ denoting the loadings matrix of a PCA model with $A$
components. Samples with low $Q$-residuals are well represented by the model,
i.e. they have small (orthogonal) distances to their low-dimensional
projections (Figure 2). When using the PC scores $\mathbf{t}_{k}$ for
$k\in{1,\dots,A}$ as color code for the embedded data, we here propose to
always also display the corresponding $Q$-residuals in order to see for which
samples the corresponding ”explanations” have strong (small $Q$) and weak
(large $Q$) support. For the latter, a different PC might better explain their
location in the embedding. PC specific $Q$-residuals are calculated by
replacing $\mathbf{P}_{A}$ in Eq.(13) by the $k$-th loadings vector
$\mathbf{p}_{k}$.

Figure 2: $Q$-Residuals and Hotelling’s $T^{2}$-statistic. Figure 3: The X-MAP
tool at a glance. A) Data import, UMAP settings and number of PC’s. B)
Explained variance per PC. C) Selected data points from the UMAP embedding. D)
Voronoi plot of the embedded data (color-coded either by PCA score,
$Q$-residuals or a single input variable) and loadings of the corresponding
PC. E) Relative $T^{2}$ contributions explaining the difference between two
data points or clusters selected in D.

#### Relative T2-contributions

The Hotelling’s $T^{2}$-statistic is a measure of outlingness w.r.t. the
centroid of the (training) data within the PC space (Figure 2). The
corresponding $T^{2}$-contributions are a measure of influence of each input
variable to the outlingness. For the $i$-th sample

$\mathbf{t}_{\text{cont},i}=\mathbf{t}_{i}\mathbf{\Lambda}^{-1/2}\mathbf{P}_{A},$
(14)

with $\mathbf{\Lambda}$ being a diagonal matrix holding the $A$ leading
Eigenvalues of
$\mathbf{X}^{\text{T}}\mathbf{X}$111https://wiki.eigenvector.com/index.php?title=T-Squared_Q_residuals_and_Contributions.
The corresponding scaling takes care that the same weight is assigned to the
contribution from each subspace dimension to the explanations. Relative
$T^{2}$-contributions on the other hand, can be used to investigate which
variables contribute most to the difference between $\mathbf{x}_{i}$ and
$\mathbf{x}_{j}$ w.r.t. the ($A$-dimensional) PC space:

$\mathbf{t}_{\text{cont},ij}=\mathbf{t}_{\text{cont},i}-\mathbf{t}_{\text{cont},j}$
(15)

The same approach can be used to derive explanations for why entire clusters
within an embedding differ from each other. To do so, the contributions are
computed with respect to the corresponding cluster centroids.

#### Voronoi plots

In order to better explore the non-linear structure of the embedded space, we
propose Voronoi diagrams. A Voronoi diagram is a partitioning of a plane into
regions with each point in that region being closest to a single data point in
terms of Euclidean distance. This allows to see the value of a statistic to be
displayed (e.g. $Q$-Residuals) for individual data points over the whole
embedded space.

#### Implementation

A prototypical implementation of the entire data exploration workflow
including graphical user interface (GUI) was undertaken using the shiny web
application framework for R [16]. We provide open source code at
https://github.com/RNL1/XMAP.git.

Figure 4: Fraction of explained variance for each PC. The number of PCs
included in the model is indicated by the red bar (#PCs = 4).

##  3 X-Map GUI

Figure 3 outlines the functionality of the proposed data exploration tool that
we coin explainable map (X-MAP). First, a .rds file holding the data to be
explored must be provided in long format followed by specifying the UMAP
parameters, i.e. number of nearest neighbors, minimum distance (see Eq. (7),
similarity metric, and number of training epochs (Figure 3A). Subsequently,
the maximum number of PC’s is specified followed by fitting the PCA model and
UMAP embedding. Once model fitting has completed, the percentage of variance
explained by each PC appears in a separate subplot (Figure 4).

In the example shown, more than 94% of the variance is explained by the first
PC. Note that an optimal number of PCs to be retained is proposed and
automatically selected during model fitting but can be manually adjusted after
completion of the training epochs.

In addition to the explained variance plot, a Voronoi diagram of the UMAP
embedding appears with the polygones being by default colored according to the
score of PC 1 side by side with the corresponding loadings (Figure 5).

Figure 5: Voronoi diagram color-coded according to PC1 scores (left) and
corresponding loadings (right).

In the example shown, data points located in the lower right corner of the
embedding exhibit a higher score on PC 1 compared to the points located in the
blue areas. Separation between the two areas is mostly due to lower values of
variables Y8, K4, K11 and A1 for the former as can be seen by inspecting the
corresponding loadings. Figure 6 shows the same embedding with the data points
color-coded according to the value of variable Y8, which is in-line with this
interpretation.

Figure 6: Data points colored according to Y8 variable.

PC 2, on the other hand, encodes information about differences between
clusters in the lower half of the UMAP (Figure 7).

Figure 7: Voronoi diagram color-coded according to PC2 scores (left) and
corresponding loadings (right).

An important criterion for PCA-based interpretation of a non-linear embedding
like UMAP is, how well each data point is represented by the current PC. In
the worst case, a PC that explains a high proportion of variance in the data
(i.e. PC 1 in our example) only explains a small portion of the information
contained in some data points. $Q$-Residuals are a means of diagnosing if some
data points, i.e. outliers, are poorly represented. Figure 10a and 10b show
the same Voronoi diagramm of the embedding with the polygones color-coded
according to the $Q$-residuals with respect to the first and second PC. Note
that the values are min-max normalized and lie between 0 and 1 for samples
with the lowest and highest $Q$-residuum in the data set, respectively.
Notably, a single data point appears as potential outlier (with respect to the
first and second PC) and thus warrants further investigation. For PC 2,
$Q$-residuals indicate that samples located in the lower right corner are
poorly represented. Thus, the corresponding loadings might provide limited
information about these samples.

Figure 8: Principal component 1 Figure 9: Principal component 2

Figure 10: PC-specific $Q$-Residuals. Note that $Q$-values are normalized
between 0 and 1.

In some cases, exploring a non-linear embedding using a single PC at the time
is too complex, which is especially true for data sets with a flat Eigenvalue
distribution. We here propose relative Hotelling’s $T^{2}$ contributions (see
Eq. (15)) in order to include information from all relevant PCs when deriving
explanations for why pairs of data points or clusters differ from each other.
Figure 16 shows the relative $T^{2}$-contributions corresponding to the
difference between the clusters A and B shown in Figure 13. The results
indicate that the two clusters differ mostly by the value of variable Y8,
which does not become evident from the color-coded UMAP shown in Figure 6.
This is because the absolute difference between the two distributions is small
compared to the total variance of the variable across the entire data set
(Figure 15). Altogether, this example underpins the strength of analyzing a
UMAP-type embedding by means of relative $T^{2}$-contributions.

Figure 11: Cluster A Figure 12: Cluster B

Figure 13: Cluster selection for computation of relative
$T^{2}$-contributions.

Figure 14: Relative $T^{2}$-contributions. Figure 15: Histogram of variable Y8

Figure 16: Comparison clusters A and B by means of relative Hotelling’s
$T^{2}$-contributions

##  4 Discussion

We proposed a novel workflow that employs the well-known Hotelling’s $T^{2}$
statistic and $Q$-residuals from PCA to derive explanations for non-linear
neighbor embedding such as UMAP. We integrated these diagnostic tools in a
software prototype using the shiny web application framework for R and
demonstrated how they can be used for exploratory analysis of a multivariate
dataset with UMAP. In particular, we showed how our approach is capable of
identifying discriminatory features between groups of data points that remain
unnoticed when exploring NE using standard univariate (variable-by-variable)
or multivariate (using PC-wise color-coding) approaches which underpins the
strength of the approach.

##  5 Conclusion

Non-linear dimension reduction techniques such as UMAP often provide valuable
insights into high-dimensional data sets beyond those obtained by PCA.
However, interpretation of the former is not as straight forward as with the
latter and thus requires a downstream (confirmatory) data analysis pipeline.
We found previously proposed methodology to ”open the black box of neighbor
embedding” limited in terms of both, user-friendliness and computational
efficiency that would allow non-expert users to ”make sense” out of NE in a
consistent way. Our approach addresses both these shortcomings.

## Acknowledgements

Funding was provided by the Federal Ministry for Climate Action, Environment,
Energy, Mobility, Innovation and Technology (BMK), the Federal Ministry for
Digital and Economic Affairs (BMDW), and the Province of Upper Austria in the
frame of the COMET - Competence Centers for Excellent Technologies programme
managed by the Austrian Research Promotion Agency FFG and the COMET Center
CHASE.

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# Local magnetic and geometric structure in Mn-doped La(Fe,Si)${}_{\text{13}}$

Benedikt Eggert<EMAIL_ADDRESS>Faculty of Physics and Center for
Nanointegration Duisburg-Essen (CENIDE), University of Duisburg-Essen,
Lotharstr.1, 47057 Duisburg, Germany  Johanna Lill  Faculty of Physics and
Center for Nanointegration Duisburg-Essen (CENIDE), University of Duisburg-
Essen, Lotharstr.1, 47057 Duisburg, Germany  Damian Günzing  Faculty of
Physics and Center for Nanointegration Duisburg-Essen (CENIDE), University of
Duisburg-Essen, Lotharstr.1, 47057 Duisburg, Germany  Cynthia Pillich  Faculty
of Physics and Center for Nanointegration Duisburg-Essen (CENIDE), University
of Duisburg-Essen, Lotharstr.1, 47057 Duisburg, Germany  Alexandra Terwey
Faculty of Physics and Center for Nanointegration Duisburg-Essen (CENIDE),
University of Duisburg-Essen, Lotharstr.1, 47057 Duisburg, Germany  Ilyia A.
Radulov  Functional Materials, Institute of Materials Science, Technische
Universität, 64287 Darmstadt, Germany  Fabrice Wilhelm  European Synchrotron
Radiation Facility, 38043 Grenoble cedex, France  Andrei Rogalev  European
Synchrotron Radiation Facility, 38043 Grenoble cedex, France  Mauro Rovezzi
Univ. Grenoble Alpes, CNRS, IRD, Irstea, Météo France, OSUG, FAME, 38000
Grenoble, France  Konstantin Skokov  Functional Materials, Institute of
Materials Science, Technische Universität, 64287 Darmstadt, Germany  Katharina
Ollefs  Faculty of Physics and Center for Nanointegration Duisburg-Essen
(CENIDE), University of Duisburg-Essen, Lotharstr.1, 47057 Duisburg, Germany
Oliver Gutfleisch  Functional Materials, Institute of Materials Science,
Technische Universität, 64287 Darmstadt, Germany  Markus E. Gruner  Faculty of
Physics and Center for Nanointegration Duisburg-Essen (CENIDE), University of
Duisburg-Essen, Lotharstr.1, 47057 Duisburg, Germany  Heiko Wende  Faculty of
Physics and Center for Nanointegration Duisburg-Essen (CENIDE), University of
Duisburg-Essen, Lotharstr.1, 47057 Duisburg, Germany

(August 28, 2024)

###### Abstract

Magnetic cooling has the potential to replace conventional gas compression
refrigeration. Materials such as La(Fe,Si)13 exhibit a sizeable first-order
magnetocaloric effect, and it is possible to tailor the phase transition
towards room temperature by Mn-H-doping, resulting in a large temperature
range for operation. Within this work, we discuss variations of the electronic
and lattice structure in La(Fe,Si)13 with increasing Mn content utilizing
X-ray magnetic circular dichroism (XMCD) and extended X-ray absorption fine
structure spectroscopy (EXAFS). While XMCD shows a decrease of the magnetic
polarization at the Fe K edge, low-temperature EXAFS measurements indicate
increased positional disorder in the La environment that is otherwise absent
for Fe and Mn. First-principles calculations link the positional disorder to
an enlarged Mn-Si distance – explaining the increased positional disorder in
the La surrounding.

##  I Introduction

The discovery of the giant magnetocaloric (MC) effect by Pecharsky and
Gschneider in Gd5Ge2Si2 in 1997 [1] started the research field of magnetic
refrigeration, which aims for an environmentally friendly replacement of gas-
compression refrigeration cycles [2, 3]. The materials of interest exhibit a
significant adiabatic temperature change $\Delta T_{\text{ad}}$ induced by
external applied magnetic fields and isothermal entropy change $\Delta
S_{\text{iso}}$ at their first-order phase transition [4].

Besides the benchmark systems FeRh [5, 6, 7] and Gd[1, 8, 9], current
materials of interest [10, 11, 12, 13], possessing a sizeable magnetocaloric
effect near room temperature, are, e.g. NiMn-based Heusler compounds [14, 15,
16], MnFePSi [17, 18, 19, 20, 21], and H-containing La(Fe,Si)13 [22, 23, 24,
25, 26, 27, 28]. Especially the latter system seems suitable for room
temperature applications due to its small thermal hysteresis width and the
possibility of tailoring the phase transition by adjusting the Si and H
concentration. To further reduce the thermal hysteresis and increase the
cooling efficiency towards the theoretically predicted ones, developing new
mechanisms to tailor the hysteresis of the first-order phase transition of
these magnetocaloric compounds is essential. To implement these new
mechanisms, a complete microscopic understanding of the electronic and
magnetic interactions is of help [29].

The promising magnetocaloric system La(Fe,Si)13H possesses a significant
adiabatic temperature change $\Delta T_{\text{ad}}$ and a drastic volume
decrease $\Delta V/V$ [30] along its isostructural first-order phase
transition, which can be correlated to an itinerant electron metamagnetic
transition (IEM) [31, 32]. The magnetocaloric compound crystallizes in the
NaZn13 (Fm3c) structure (1:13 phase) with cubic shape. Here, two nonequivalent
Fe sites are present, which we will refer to as FeI site (8b Wyckoff position)
and FeII site (96i Wyckoff position). The Si atoms are assumed to occupy
mainly the FeII sites, sharing this site with Fe atoms [33, 34, 35]. The unit
cell can be represented by a cell with fcc primitive vectors containing 28
atoms [30, 36, 37], which simplifies computational studies. The precise ratio
of Si and H in the La(Fe,Si)13 system will determine the phase transition
temperature, which is aimed to be around room temperature. A high Si content
(above x = 1.8 [38]) will turn the phase transition to second order, which
reduces the magnetocaloric effect but avoids dissipative hysteresis losses. In
the literature, many ongoing discussions debate the influence of different
dopings [39] on the size of the magnetocaloric effect and, e.g. phase
transition order or thermal hysteresis width. Examples are Cr-doping, which
first decreases thermal hysteresis width by increasing Cr content to a
concentration of x=0.3. In contrast, for higher doping, the thermal hysteresis
increases again [40]. Besides, Ni-doping changes the first-order phase
transition character to second-order while leading to a rise of the phase
transition temperature T${}_{\text{tr}}$ [41]. While Si is necessary to
stabilize the La(Fe,Si)13 compound, H-doping shifts the phase transition by
about 100 K to RT [42, 33, 43, 44]. A combined nuclear resonant inelastic
X-ray scattering and density functional theory investigation of the
hydrogenated compound could show that the nature of the mechanism is identical
to the parent compound [45, 46]. Partial H-loading makes it possible to tune
the phase transition towards 300 K, while having the disadvantage that partial
H-doping leads to phase separation into a hydrogen-rich and a hydrogen-poor
phase, leading towards the distribution of phase transition temperatures [47].
A complete hydrogen loading can overcome this, but the phase transition occurs
at $T=360\,\mathrm{K}$ for these high H-contents. This makes it necessary to
dope the system further in order to have a system with a phase transition
temperature at 300 K and overcoming the potential risk of phase separation by
the introduction of Mn in the NaZn13structure [44, 48].

The current work intends to resolve the occurring variations in interactions
between the different lattice sites and elements to gain a microscopic picture
of the Mn-doped compound. First, we investigate the La and Fe’s magnetic
moments in the ferromagnetic ground state with increasing Mn concentration by
employing X-ray magnetic circular dichroism (XMCD) in the hard X-ray regime
and compare these results with conventional magnetometry measurements. In a
second step, we discuss changes in the local structure employing extended
X-Ray absorption fine structure spectroscopy (EXAFS) at the Fe, Mn and La
edges, enabling us to compare the local environment and how this is affected
by Mn doping for the different lattice sites. These experimental
investigations are complemented by ab-initio calculations to attribute
structural changes to selected bonds and reveal changes off the respective
magnetic moment. Based on this approach, we can show that by introducing Mn in
the NaZn13-structure, a decrease of the magnetisation and the magnetic moment
of Fe occurs, combined with an increased structural disorder within the first
backscattering shell around the La-site. First-principles calculations and
EXAFS measurements indicate that Mn preferentially occupies the 96i sites,
while first-principles calculations indicate the presence of different local
environments of Mn.

Figure 1: (a) 112 atom cell of LaFe${}_{\text{11.5}}$Si${}_{\text{1.5}}$ in
the NaZn${}_{\text{13}}$ structure. Exemplary representation along the
(111)-direction of (b) La and (c) Fe surrounding occupying the 8a, 8b
(Fe${}_{\text{I}}$), and 96i (Fe${}_{\text{II}}$)-site. Surroundings depict
the local environment within a distance of 3.3 Å (La), 3.3 Å
(Fe@Fe${}_{\text{I}}$), and 4.0 Å (Fe@Fe${}_{\text{II}}$), respectively. The
yellow circle illustrates the central absorbing atom used in the respective
calculations.

##  II Experimental details

Mn-containing LaFe${}_{\text{13-x-y}}$Mn${}_{\text{x}}$Si${}_{\text{y}}$ with
a fixed Si content of y=1.4 and a Mn content ranging from x=0 to x=0.5 were
prepared by induction melting in an argon atmosphere from pure elements.
Afterwards, the as-cast samples were homogenised by annealing at 1323 K for
seven days in an Ar-filled and sealed quartz ampoule, followed by quenching in
water. The obtained ingots were then crushed into a powder for further
measurements. We refer the reader to Refs. [25, 44] for additional details on
the sample preparation. Field-dependent magnetometry was performed using a
Quantum Design PPMS DynaCool with the vibrating sample magnetometer option
applying an external magnetic field up to 9 T with a sample temperature range
between 1.8 K up to 400 K. Element-specific extended X-ray absorption fine
structure (EXAFS) measurements were conducted at beamline BM30B at the ESRF
[49]. Spectra have been recorded at the corresponding K or L3 absorption edge
in partial fluorescence yield detection mode at 45° incident geometry using an
energy dispersive detector to collect the signal originating from the
respective characteristic Kα or Lα fluorescence line. In addition, potential
saturation effects due to self-absorption have been corrected using the Booth
algorithm [50] within the DEMETER package [51]. X-ray magnetic circular
dichroism (XMCD) measurements were conducted at beamline ID12 at the ESRF at
low temperatures (T=3 K) with an external magnetic field of $\pm$2 T applied
parallel to the incoming X-ray beam. The obtained XMCD spectra arise as the
difference between X-ray absorption spectra measured with opposite helicities
of the incoming photons. The measurements are performed in opposite directions
of the magnetic field to avoid potential artefacts in the dichroic signal.

##  III Computational details

Parameter-free first-principles calculations were carried out in the framework
of the density functional theory (DFT) with the help of the Vienna Ab-initio
Simulation Package (VASP) [52]. A plane wave basis set with an energy-cutoff
$E_{\mathrm{cut}}=400\,$eV was used with exchange and correlation in the
generalised gradient approximation with the functional of Perdew, Burke and
Ernzerhof [53] and scalar-relativistic approximation with collinear spin-
polarisation. For the accurate description of the core electrons, we took
advantage of the projector augmented wave (PAW) approach [54], employing
potentials to a valence electron configuration of 5s25p65d16s2 for La,
5p63d74s1 for Fe, 5p63d64s1 for Mn and 3s23p2 for Si. For the structural
optimisations (cell parameters, angles and atomic positions) in the 28-atom
cell as used in our previous work [45] were carried out with of
7$\times$7$\times$7 Monkhorst-Pack $k$-mesh which yields 172 $k$-points in the
irreducible Brillouin zone (IBZ) and a finite temperature smearing according
to Methfessel and Paxton [55] with a broadening of
$\sigma=\,0.1\mathrm{\,eV}$. The self-consistency cycle was stopped when the
difference in energy between two consecutive cycles fell below $10^{-7}$ eV,
structures were considered relaxed below a threshold of
$10^{-7}\mathrm{\,eV}$. Total energy and magnetic moments were determined in a
further step with a finer 11$\times$11$\times$11 k-mesh (666 points in the
IBZ), and the tetrahedron method with Bloechl corrections [56] for the
Brillouin zone integration.

The program package FEFF9 [57, 58, 59] was used for the calculation of the
oscillatory fine structure of the X-ray absorption spectra in the
ferromagnetic phase ($T=$20 K) at the La L3, Mn K and Fe K edges. FEFF
calculations have been performed on the 112 atom cell representation of
LaFe${}_{\text{11.5}}$Si${}_{\text{1.5}}$ (see Fig. 1) in the
NaZn${}_{\text{13}}$ lattice structure, by averaging over all different
occurring lattice sites. To model the fine structure of Mn, we assumed the
absorber to be Mn, while the other atoms had not been adjusted. Due to the
similar backscattering amplitude and phase shift of Mn and Fe, this assumption
is justified. For the description of occurring dynamic disorder (e.g. phonons
& zero-point vibrations), we used a Debye model with a Debye temperature of
380 K [60] and a measurement temperature of 20 K. Besides, an additional
static mean square relative displacement $\sigma^{2}_{\text{stat}}$ of
0.001$\,\text{\AA}^{2}$ has been assumed to model the presence of static
disorder (grain boundaries or lattice distortions). Additional descriptions of
the first scattering shell of the La, Mn and Fe atoms have been performed with
the ARTEMIS package [51] and Larch [61, 62].

##  IV Results & Discussion

###  IV.1 The magnetic structure

In order to track the effect of Mn-doping on the magnetic structure, we
performed field-dependent magnetometry measurements. Figure 2 depicts
isothermal magnetisation measurements for different Mn-concentration between
$\pm$9 T at low temperatures. As shown in the inset of Figure 2, the
saturation magnetisation reduces by approximately 10 % when Mn replaces 2.5 %
of the Fe atoms while a Mn-concentration of approximately 4.5 % leads to a
reduction of almost 30 % of the saturation magnetisation.

Figure 2: Field-dependent magnetometry performed at T = 5 K for La(Fe,Si)13
with varying Mn content. The obtained magnetisation values have been corrected
with respect to occurring secondary bcc-Fe phases, as discussed in Ref. [63].
The inset shows the obtained saturation magnetisation as a function of Mn
concentration.

To resolve and attribute the changes in the magnetisation to the specific
elements, we performed element-specific X-ray magnetic circular dichroism
(XMCD) measurements in the hard X-ray regime at the beamline ID12 of the ESRF.
For the Fe magnetic moment, we performed measurements at the Fe K edge in
order to probe the 1s$\xrightarrow{}$4p transition (E1 transition), making it
possible to obtain the magnetic dichroism arising from the orbital moment of
the 4p states and the magnetic moment of the 3d-states through the quadrupolar
transition (E2-transition). Figure 3(a) depicts the obtained Fe K edge XMCD
spectra for different Mn concentrations. Focusing on the XMCD spectra (inset),
we can see that with increasing Mn content, a reduction of the XMCD intensity
occurs first at photons energy of 7111 eV – corresponding to the quadrupolar
transitions into localised 3d states. Only the largest Mn concentration
possesses an intensity reduction at higher photon energies (above 7112 eV),
indicating a decreased spin polarisation of delocalised 4p states. In
addition, the XAS at photon energies of 7113 eV shows a broadening close to
the Fermi energy. The reduced Fe moment might originate from a redistribution
in the density of states due to the hybridisation of Fe-Mn 3d states [64] and
can also be assigned as a feature of the reduced magnetisation [65].

Figure 3: (a) X-ray absorption near edge spectra and X-ray magnetic circular
dichroism signals for all
LaFe${}_{\text{11.6-x}}$Mn${}_{\text{x}}$Si${}_{\text{1.4}}$ samples at the Fe
K edge, measured at 3 K in an external field of 2 T. Inset shows the near edge
region of the Fe K edge XMCD signal. (b) X-ray absorption near edge spectra
and X-ray magnetic circular dichroism signals for all except the maximum doped
LaFe${}_{\text{11.6-x}}$Mn${}_{\text{x}}$Si${}_{\text{1.4}}$ at the La L2,3
edges.

Besides, we performed measurements to unravel the magnetic dichroism of La 5d
states by probing the 2p$\xrightarrow{}$5d transition at the La L2,3 edges
(see Figure 3(b)). Usually, it is possible to extract the orbital and spin
moment by the so-called sum rule analysis [66, 67]. A quantitative discussion
of the La 5d moments goes beyond the scope of this work, and the reasoning
will be explained in the following. The La spectrum depicts a well-developed
dichroic oscillating spectral fine structure – leading to a significant error
in the determination of the integrated XMCD signal. In addition, the
quadrupolar transitions contributing to the isotropic absorption result in
underestimating the extracted 5d moment. The obtained XMCD spectra reveal two
aspects. First, the La 5d moments align antiparallel to the applied field (and
the Fe-moment). At the same time, it seems that the magnetic dichroism and the
isotropic absorption (number of unoccupied states) do not react to the Mn
doping in the range of the investigated doping concentration. Based on the
spectra, we can assume that the induced La moment is constant within
experimental error. Concentrating on the XMCD shape, we observe that the XMCD
peaks lay within the rising edge of the absorption edge – indicating a
derivative-like signal. The majority of the XMCD signal in the near edge
region can be modelled (not shown here) by applying an energy shift of 50 and
120 meV to the L3 and L2 edge, respectively. This energy shift within the La-
partial density of states is caused by the 3d-5d hybridization leading to a
slight exchange splitting in the spin-resolved density of states [45].
Differences in the determined energy shifts reflect the varying lifetimes of
the excited states at the respective absorption edges [68].

Based on the results obtained by magnetometry, the Fe K, and La L2,3 XMCD
spectra, the decrease in the magnetisation can be attributed to a reduction of
the Fe magnetic moment.

###  IV.2 First-principles calculations

Table 1:  Total energy $\Delta E_{\mathrm{tot}}$, total moment $\mu_{\mathrm{tot}}$ per formula unit (f.u.) and averaged element-resolved magnetic moments $\mu_{i}$ of LaFe11.5Si1.5 (first row) and LaFe11Si1.5Mn0.5 given per atom as obtained from our DFT calculations. $n_{\mathrm{Si}}$ specifies the number of Si atoms in the closest neighbourhoods of Mn (below $3\,{\mathrm{\AA}}$) in the specific configuration. The total energy $\Delta E_{\mathrm{tot}}$ is specified per 28-atom primitive cell relative to the average total energy across all inequivalent configurations with Mn placed on the 96i sites. The first column specifies the sites at which the Mn is placed. This refers either to a specific position indicated by the letter in parentheses or the average (avg.).  |  | $\Delta E_{\mathrm{tot}}$ | $n_{\mathrm{Si}}$ | $\mu_{\mathrm{tot}}$ | $\mu_{\mathrm{La}}$ | $\mu_{\mathrm{Fe}}$ | $\mu_{\mathrm{Si}}$ | $\mu_{\mathrm{Mn}}$
---|---|---|---|---|---|---|---|---
| Mn@ | (meV) |  | ($\mu_{\mathrm{B}}$/f.u.) | ($\mu_{\mathrm{B}}$/atom)
LaFe11.5Si1.5 | - | - | - | 24.47 | -0.200 | 2.219 | -0.101 | -
LaFe11Si1.5Mn0.5 | 96i avg. | 0.0 | 1.3 | 21.31 | -0.181 | 2.142 | -0.087 | -2.598
| 96i(a) | -81.2 | 0 | 21.20 | -0.181 | 2.131 | -0.092 | -2.540
| 96i(b) | -29.1 | 1 | 20.97 | -0.183 | 2.125 | -0.087 | -2.851
| 96i(c) | 17.7 | 2 | 21.36 | -0.177 | 2.146 | -0.088 | -2.587
| 96i(d) | -81.0 | 2 | 21.28 | -0.185 | 2.139 | -0.089 | -2.549
| 96i(e) | 43.7 | 1 | 21.36 | -0.179 | 2.143 | -0.087 | -2.534
| 96i(f) | 112.8 | 1 | 21.56 | -0.179 | 2.158 | -0.085 | -2.483
| 96i(g) | 17.0 | 2 | 21.41 | -0.185 | 2.153 | -0.080 | -2.641
LaFe11Si1.5Mn0.5 | 8b avg. | 128.6 | 1.5 | 23.53 | -0.186 | 2.196 | -0.093 | 0.658
| 8b(a) | 114.4 | 3 | 23.39 | -0.190 | 2.197 | -0.091 | 0.420
| 8b(b) | 142.9 | 0 | 23.67 | -0.181 | 2.195 | -0.094 | 0.895
Table 2:  Structural parameters of the LaFe11.5Si1.5 (first row) and LaFe11Si1.5Mn0.5 (all other rows) configurations obtained from our DFT calculations, in terms of the (Average) volume per atom $V_{\mathrm{at}}$ and the averaged interatomic distances $d$ within the closest neighbour shells (below $4\,$Å for La and $3\,$Å for Fe and Mn) between specified pairs of elements and all atoms (all).  | $V_{\mathrm{at}}$ |  $d$(Mn-Si) |  $d$(FeII-Si) |  $d$(FeI-Si) |  $d$(La-Si) |  $d$(La-all) |  $d$(FeI-all) |  $d$(FeII-all) |  $d$(Mn-all)
---|---|---|---|---|---|---|---|---|---
Mn@ | (Å3/at.) | (Å) | (Å)
- | 13.281 | - | 2.4948 | 2.4034 | 3.3781 | 3.3328 | 2.4505 | 2.5099 | -
96i avg. | 13.236 | 2.5566 | 2.4952 | 2.4086 | 3.3691 | 3.3296 | 2.4459 | 2.5071 | 2.5082
96i(a) | 13.214 | - | 2.4919 | 2.4043 | 3.3713 | 3.3281 | 2.4454 | 2.5071 | 2.4848
96i(b) | 13.226 | 2.6015 | 2.4926 | 2.3958 | 3.3719 | 3.3283 | 2.4462 | 2.5065 | 2.5280
96i(c) | 13.244 | 2.5626 | 2.4901 | 2.4218 | 3.3752 | 3.3296 | 2.4482 | 2.5076 | 2.4928
96i(d) | 13.225 | 2.6261 | 2.4866 | 2.4229 | 3.3504 | 3.3279 | 2.4454 | 2.5052 | 2.5124
96i(e) | 13.228 | 2.4998 | 2.5103 | 2.4058 | 3.3587 | 3.3282 | 2.4444 | 2.5074 | 2.4988
96i(f) | 13.262 | 2.5375 | 2.4990 | 2.4114 | 3.3672 | 3.3332 | 2.4456 | 2.5075 | 2.5082
96i(g) | 13.256 | 2.5119 | 2.4956 | 2.3983 | 3.3889 | 3.3321 | 2.4462 | 2.5087 | 2.5327
8b avg. | 13.247 | 2.4210 | 2.4940 | 2.4083 | 3.3683 | 3.3308 | 2.4505 | 2.5066 | 2.4422
8b(a) | 13.243 | 2.4210 | 2.4921 | - | 3.3657 | 3.3302 | 2.4653 | 2.5064 | 2.4270
8b(b) | 13.250 | - | 2.4958 | 2.4083 | 3.3708 | 3.3313 | 2.4357 | 2.5068 | 2.4573

To obtain a microscopic understanding of the site-preference of Mn and its
impact on the magnetic and structural properties, we carried out DFT
calculations of LaFe11Si1.5Mn0.5 in its 28-atom primitive cell containing two
formula units. The primitive cell is spanned by face-centered cubic lattice
vectors and represents $1/8$ of the conventional 112 atom cell shown in Fig.
1. The Si atoms were distributed across the 96i in an arrangement which
preserves rhombohedral long-range symmetry. This distribution allowed us
previously to provide computationally efficient modelling of the vibrational
density of states of ferro- and paramagnetic LaFe11.5Si1.5 yielding an
excellent agreement with experiment [45, 65, 60, 46]. The remaining
rhombohedral symmetry in the cell leaves 7 inequivalent three-fold degenerate
FeII on the 96i positions and two inequivalent FeI sites on the 8b positions
of the 28-atom prototype cell with cubic symmetry. In the course of our
calculations, we systematically replaced one Fe-atom on these nine
inequivalent sites with one Mn and evaluated the changes in magnetic
properties and lattice structure.

Comparing the total energies, we find a clear preference for Mn to be located
on the 96i sites, see Table 1, the average across all 7 inequivalent choices
being almost 130 meV below the average over Mn placed on the 8b sites, which
is a considerable difference. Thus, while it cannot be excluded that during
heat treatment at elevated temperatures some Mn atoms will replace FeI, the
vast majority of the Mn atoms can be expected to reside on the positions of
FeII. Among the Mn placed on the 96i sites, there is a large spread regarding
the total energy of the respective configurations. While for two
configurations, 96i(a) and 96i(d), their energy is sizeable 80 meV below the
average, we also find several configurations with $17$, $44$ and even
$113\,$meV above it. This indicates that there might be specific local
environments which are particularly favourable for the placement of an Mn atom
and may thus lead to a kind of short-range order. These are, however, not
straight-forward to identify from our calculations in the 28-atom cell, with
its pseudo-disordered placement of Si, and thus a systematic comparison in
larger cells is necessary, which is beyond the scope of the present work.
However, we may conclude that one obvious parameter, the number of Si atoms in
the closest neighbour shells of Mn, $n_{\mathrm{Si}}$, which is also given in
Table 1 does not correlate with the energy of the configuration.

Substitutional Mn on the 96i sites enters a rather stable high moment state
with an average magnitude of $2.6\,\mu_{\mathrm{B}}$, which is oriented
oppositely to the ferromagnetically aligned Fe-moments. This is roughly half
of a Bohr magneton larger than the average moment of Fe and corresponds to the
smaller valence electron number e/a of Mn. However, the magnetic moments of Mn
are considerably smaller compared to other magnetocaloric TM compounds such as
Ni-Mn-based Heusler alloys [69]. Ferromagnetic configurations with Mn on 96i
positions aligned parallel to Fe turned out to have significantly larger
energy and were not considered further. The antiparallel orientation of Mn
leads as a side effect to a slight reduction of the induced moments of La by
0.02 $\mu_{\mathrm{B}}$ and Si by roughly 0.013 $\mu_{\mathrm{B}}$, which is
in agreement with the measurements presented in the preceding subsection. The
Fe moments, in turn, reduce by 0.066 $\mu_{\mathrm{B}}$ on average.

If Mn is placed on the 8b sites, a high moment state can not be stabilized in
the calculations. Instead, a low moment state with a magnitude below 1
$\mu_{\mathrm{B}}$ is realized, which was previously also observed for Fe on
the 8b sites [45, 65]. Our data cannot give a final conclusion about the
orientation of the Mn-moment; thus antiparallel or non-collinear alignments
may be possible as well.

The low spin configuration of Mn on the FeI sites or the antiparallel
orientation of Mn on the FeII sites results in a decrease in the average
ground state volume of $0.25\,$% and $0.33\,$%, respectively, relative to the
volume of LaFe11.5Si1.5, see Table 2. The volume change depends substantially
on the position of the Mn on the inequivalent Fe-sites. Furthermore, the
decrease in average volume does, however, not result in a homogeneous
reduction of the nearest neighbour interatomic spacings, which are also listed
in Table 2. For the case that Mn is placed on the FeII (96i) sites, the
average distance of La to all neighbours closer than $4\,$Å reduces by
$0.096\,$% on average, for FeI to $0.19\,$% and FeII to $0.11\,$% all
neighbours closer than $3\,$Å. For Mn atoms on the 96i sites, which introduce
the antiparallel moments, the average distance to all nearest neighbors
reduces only by $0.067\,$% compared to the nearest neighbor distances of FeII
in LaFe11.5Si1.5. The origin of this reduced trend may be deduced to the
significantly larger distances of Mn to the closest Si atoms, which expand by
a considerable $2.5\,$% compared to their FeII counterparts, which remain
nearly constant in the presence of Mn despite the reduction of the total cell
volume. This points out the paramount role of the Fe-Si bonding in stabilizing
this alloy while Mn attempts to establish a larger distance to the surrounding
Si. In fact, in Table 1 those configurations of Mn on 96i sites reveal the
lowest energy, which have no nearest Si neighbours or a particularly large
separation. In turn, the distances between La and Si decrease by $0.27\,$% on
average. A comparison of the electronic density of states (not shown) does not
exhibit signs of an increased hybridization between the partial contributions
of the two elements. Therefore, we tentatively exclude stronger La-Si bonding
as the origin. Instead, we rather ascribe this to the large spacing between La
and its neighbours, which offers more space for adapting to structural
modification arising from Mn-Si interactions as compared to Fe.

When Mn is placed on the FeI 8b sites, the La-Si distances decrease to a
similar extent, while the FeII-Si distances remain largely constant on
average, while Mn-Si distances are again significantly larger. Here, the
configuration with no nearest Mn-Si neighbours offers the lowest energy.

###  IV.3 The microscopic geometric structure

Extended X-Ray absorption fine structure spectroscopy (EXAFS) measurements
were performed at the Fe K, Mn K, and La L3 edges for all Mn-concentrations at
the beamline BM30B (ESRF). By performing EXAFS measurements of the different
Mn-doped compounds, it is possible to follow the influence of the Mn
concentration on the short-range ordering around the probed atom, especially
unravelling the site occupation of the Mn atoms. Besides, it is possible to
determine upcoming changes in the local environment of the probed element. The
oscillatory fine structure following the element-specific absorption edge can
be understood in terms of the EXAFS equation:

$\begin{split}\chi(k)=&\sum_{j}^{\,}S_{0}^{2}(k)N_{j}f_{j}(k)\exp\left(-2\frac{R_{j}}{\lambda(k)}\right)\exp\left(-2\sigma^{2}_{j}k^{2}\right)\\\
&\cdot\frac{\sin\left(2kR_{j}-\dfrac{4}{3}C_{3,j}(T)k^{3}+\delta_{j}(k)\right)}{2kR^{2}_{j}}\end{split}$
(1)

Hereby, the oscillatory part is given through the last term, where $k$ is the
wave number of the photoelectron, $R_{j}$ is the distance to neighbour atoms
in shell $j$ and $\delta_{j}(k)$ originates from the finite size of the
scattering potential giving rise to a phase shift of the scattered wave. The
oscillation strength mainly depends on the number of identical atoms per shell
$N_{j}$. Depending on the scattering element, the shape of the scattering
potential changes in an energy-dependent manner, considered through the
backscattering amplitude $f_{j}(k)$. The finite lifetime of the photoelectron
leads to a damping of the fine structure at larger wavenumbers $k$. It can be
described by an energy-dependent mean free path $\lambda(k)$ of the
photoelectron. In addition, structural disorder leads to additional damping of
the fine structure in the form of the Debye-Waller factor and is characterised
through the mean square relative displacement $\sigma^{2}_{j}$ which can be
further divided in the sum of the static contribution
$\sigma^{2}_{\text{static}}$ and the temperature-dependent dynamic
contribution $\sigma^{2}_{\text{dynamic}}(T)$ arising from lattice vibrations.
In order to ensure minimal contributions of dynamic disorder, the measurements
were performed at $T=20$ K, so we assume to be in the regime of zero-point
vibrations that can be well described within a Debye model. The third cumulant
$C_{3}(T)$ reflects anharmonic lattice vibrations [70] that occur at elevated
temperatures. Due to the low measurement temperature, anharmonic contributions
can be neglected within the scope of this study [71]. As equation (1)
indicates, EXAFS is a technique averaging all absorber atoms. Thus, one
spectrum arises from the sum of the probed element’s lattice sites and the
respective local environment.

Figure 4: EXAFS oscillations $k\chi\left(k\right)$ for
LaFe${}_{\text{11.6-x}}$Mn${}_{\text{x}}$Si${}_{\text{1.4}}$. at the Fe K edge
(a), Mn K edge (b), and La L3 edge (c) measured at 20 K in the absence of a
magnetic field. (d)-(f) show the corresponding Fourier transformations
$\left|FT\left(k\chi\left(k\right)\right)\right|\left(R\right)$ that was
performed with the indicated window (grey line in subfigures (a)-(e)). The
black lines depict ab-initio FEFF-calculations with an assumed Debye
temperature $\theta_{\text{D}}$ of 380 K and a measurement temperature of 20 K
for the lowest Mn-concentration. The inset in subfigure (e) depicts the
Fourier transform from FEFF calculations, where Mn occupies the 8b site (black
line) or 96i site (red line).

Results for Mn-concentrations Mn0.00, Mn0.15, Mn0.25, Mn0.30 and Mn0.50 of the
measurements are shown in Figure 4. The fine structure oscillations at the Mn
and Fe K edge indicate that Mn occupies both the 96i and 8b sites.
Furthermore, FEFF calculations describe the experimentally obtained EXAFS
spectra at the La L3 and Fe K for the initial sample and the Mn K edge with a
slightly adapted unit cell to model the Mn0.50 doped sample. The FEFF
calculations are depicted as black dotted lines, while the measurements are
coloured, depending on the Mn concentration. For the FEFF calculations, a
correlated Debye model [59] with a Debye-Temperature $\Theta_{\text{D}}$ of
380K̇ [60] was used to take dynamical disorder effects into account, arising
from finite temperatures. Additionally, a static disorder of
$\sigma^{2}_{\text{static}}=0.001\,$Å was taken into account, representing
slight lattice disorder in an expected range. Subfigures 4 (a) - (c) show the
oscillatory fine structure of all samples at the mentioned three edges. With
these measurements and calculations, we aim to resolve the occupation sites of
the Mn atoms in the NaZn13 structure and then continue to discuss the
influence of Mn doping on the short-range order.

The experimental fine structures of the Mn K edges and Fe K edges exhibit
similarities in frequency and fine structure. Site-resolved calculations of
the 8b and 96i sites for the Fe K edge and Mn K edge indicate that Mn atoms
cannot be solely found on the 8b sites. The site-resolved Fourier transforms
of 8b and 96i sites (see inset of Figure 4(e)) reveal the first backscattering
shell at R=2.1 Å corresponding to a bond length of approximately 2.4 Å.
Assuming that Mn occupies the 8b site, the Fourier transform of the
corresponding FEFF calculation depicts a clear contribution at 3.5 Å that is
not present in our experimental results or FEFF calculations for Mn occupying
the 96i site. The missing contribution at R=3.5 Å indicates that the majority
of the Fe and Mn fine structure originates from 96i-surroundings. Due to the
small spectral contribution of the 8b sites, it is not possible to state
whether the Mn atoms occupy both sites, the 8b and 96i sites or only the 96i
sites. The derived occupation is consistent with previously published
Mossbauer results [63] and our first-principles calculation (discussed above).
In contrast, a recent investigation of Cr-doped La(Fe,Si)${}_{\text{13}}$ [40]
reveals that Cr only occupies the 8b site. At the same time, the occupation of
the 96i-site is energetically unfavoured by 0.31 eV per Cr atom compared to
the 8b-occupation [40]. This indicates a clear difference of the site
occupation between Mn and Cr.

By comparing the different Mn-concentrations for each edge, it can be seen
that while for the Fe K edge (Subfigure 4 (a)) and the Mn K edge (Subfigure 4
(b)), there is no change in the oscillatory fine structure visible beyond the
error of measurement, the La L3 edge shows a clear decreasing envelope of its
fine structure with increasing Mn-concentration. The oscillatory frequency
remains the same, which denotes no change in the distance $R_{j}$. The
decrease in the envelope indicates increasing structural disorder at a
constant temperature, an effect of rising static disorder. This effect is
again visible by looking at the Fourier transform of the EXAFS signal for the
respective edges (see subfigures 4 (d) - (f)). The Fourier transform of the
EXAFS signal is connected to the pair distribution function, and therefore,
each contribution represents atoms in a neighbouring shell. While the Fourier
transforms of the Fe K and Mn K edge lay on top of each other (within
experimental accuracy), the La L3 edge shows an intensity decrease with
increasing Mn concentration.

Figure 5: Extracted mean square relative displacements $\sigma^{2}$ for the
first backscattering shell of La, Fe, and Mn. Fits performed for measurements
taken at T = 20 K.

To further analyze the effect of structural disorder, a modelling of the first
backscattering shell has been employed with single scattering paths within the
ARTEMIS and Larch package [51, 62]. For this model, we assume a uniform mean
square relative displacement (MSRD) $\sigma^{2}$. Based on these calculations,
we can extract variations of the structural disorder for the different Mn
concentrations (see Fig. 5). The overall mean square relative displacements
(static and dynamic disorder) of the Fe and Mn K edges are constant within the
error for all Mn concentrations. However, the La L3 edge presents a strong
increase of the MSRD with increasing Mn concentration. These results depict
that the Fe and Mn environment presents no observable response to the
increasing Mn concentration, representing no drastic change in the local
structure or increasing disorder in the local environment around the Fe and Mn
absorbers, respectively.

In contrast, the drastic increase of the mean square relative displacement
seen for La L3 edge needs to be discussed further. We assume that phonons and
zero-point vibrations of Fe and Mn do not differ substantially due to their
similar atomic weight. Therefore, a La-Fe or a La-Mn scattering path’s mean
square relative displacement should be identical and not depend on the Mn
concentration. On the other hand, a La-Si path possesses a different MSRD due
to the small weight of Si compared to La, Fe, or Mn and would show larger
variations. As discussed in section IV.2, Mn-Si bonds possess increased
interatomic distances compared to other FeII-Si ones (see Table 2). Besides,
our DFT calculations predict a distribution of Mn-Si and La-Si distances,
leading to the increased mean square relative displacement $\sigma^{2}$ of Mn
and La. However, due to the high amount of degenerated single scattering paths
in the first backscattering shell of La (22 paths) and similar backscattering
amplitudes for La-Si, La-Fe and La-Mn, it is impossible to experimentally
distinguish these spectral contributions further. Since EXAFS averages all
local environments of the probed element, the increased structural disorder is
only reflected at the La edge and not at the TM K edges (e.g. through a TM-Si
scattering path) since Fe or Mn, in principle, occupy 93 (85 Fe II and 8 Fe I)
out of 112 positions per unit cell. In contrast, Si only occupies 11 of these,
with La having eight individual positions. Due to the investigated compound’s
small Si content, each Fe atom possesses only one Si as a nearest-neighbour on
average (see Table 1). In contrast, all La-atoms have a higher probability for
Si atoms in their closest surroundings. Therefore the La surrounding is more
sensitive to the structural disorder caused by the Mn-Si bonds.

##  V Summary

In summary, we combined element-specific XMCD, EXAFS, and first-principles
calculations to follow the evolution with element-specific contrast of the
magnetic moments and the lattice structure in La(Fe,Si)${}_{\text{13}}$ with
increasing Mn concentration. By employing XMCD at the Fe K and La L2,3 edges,
we could reveal that the 3d Fe moment reduces – leading to a decreased
magnetisation. Furthermore, DFT calculations indicate that Mn predominately
occupies the 96i site with an average moment of 2.6$\,\mu_{\mathrm{B}}$ that
couples antiferromagnetically to the otherwise ferromagnetically aligned Fe
moments [64, 63] and is an additional source for a decreased magnetisation
that we observe in conventional magnetometry. Besides, DFT calculations in our
28-atom cell indicate the presence of specific local environments that are
favourable for Mn and might lead to a kind of short-range order, which does
not only depend on the amount of Si nearest-neighbours.

Looking at the structural response, we can see that the introduction of Mn on
96i-sites reduces the volume and, in addition, leads to an inhomogeneous
reduction of nearest-neighbour interatomic spacings. Here, we want to
highlight the occurrence of relatively large Mn-Si distances that are 2.5 %
larger compared to other FeII-Si bonds (see Table 2). These large Mn-Si bond
lengths lead to a variation of La-Si spacings, resulting in structural
disorder that we observe in the La environment through EXAFS but not in the Fe
and Mn surroundings. The drastic increase of structural disorder with
increasing Mn content might be a reason for the formation of additional
secondary phases besides the NaZn13 structure observed in previous
experimental studies [44].

##  VI Acknowledgements

This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) within TRR 270 (subprojects A03, B01, B05, and B06),
Project-ID 405553726. We thank the ESRF for allocating beamtime at ID12 and
BM30 and the financial support through projects HC-3255 and MA-4020.
Calculations were performed at the MagnitUDE high performance computing system
of the Center of Computational Sciences and Simulation (CCSS) at the
University of Duisburg-Essen (DFG INST 20876/209-1 and 20876/243-1 FUGG).

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