# The nature of the atomic-scale Invar effect in Fe-based bulk metallic glasses

Alexander Firlus, Mihai Stoica, Gavin B.M. Vaughan, Robin E. Schäublin,

Jörg F. Löffler

► To cite this version:

Alexander Firlus, Mihai Stoica, Gavin B.M. Vaughan, Robin E. Schäublin, Jörg F. Löffler. The nature of the atomic-scale Invar effect in Fe-based bulk metallic glasses. Materials Today Nano, 2023, 24, pp.100394. 10.1016/j.mtnano.2023.100394. hal-05080005

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# The nature of the atomic-scale Invar effect in Fe-based bulk metallic glasses

Alexander Firlus<sup>a, *</sup>, Mihai Stoica<sup>a</sup>, Gavin B.M. Vaughan<sup>b</sup>, Robin E. Schäublin<sup>a</sup>,
Jörg F. Löffler<sup>a, **</sup>

## ARTICLE INFO

**Keywords:**

Bulk metallic glasses

Invar effect

Magnetism

Amorphous alloys

Iron

Synchrotron radiation

## ABSTRACT

Fe-based bulk metallic glasses (BMGs) universally show an anomalously low coefficient of thermal expansion below their Curie temperature. This effect is known as the Invar effect, is rarely seen in crystalline materials and also vanishes after the crystallization of BMGs. While it is known that Co and Ni reduce the strength of the Invar effect, the influence of other elements is unknown. Moreover, it is unclear to what extent structural modifications and magnetic interactions of minor alloying elements contribute to the Invar effect. Using synchrotron-based in-situ X-ray diffraction, we show that (Fe<sub>71</sub>B<sub>24</sub>RE<sub>5</sub>)<sub>96</sub>Nb<sub>4</sub> BMGs with the rare earth elements (RE) = Tm, Er, Ho also reveal the Invar effect. This can be seen in the diffraction peaks that shift in accordance with the macroscopically measured thermal expansion. In comparison to a similar FeBYNb BMG, the Invar effect is not influenced by the substitution of Y with RE. This suggests that the minor alloying elements do not contribute to the Invar effect. Neither their influence on the atomic arrangement, the (anti-)ferromagnetic interactions of the 4f electrons of the RE elements nor paramagnetic interactions such as those from Y and Nb have an influence on the Invar effect. These elements only serve to increase the glass-forming ability and the formation of a disordered Fe network but do not contribute themselves to the Invar effect. Additionally, their (anti-)ferromagnetic interaction with Fe does not influence the magneto-structural correlations of the Fe network. Therefore, the Invar effect in these materials not only originates at the atomic scale but can be attributed solely to the magnetic interaction in the disordered Fe network.

## 1. Introduction

Bulk metallic glasses (BMGs) are an alloy class that is characterized by their disordered atomic arrangement and excellent glass-forming ability, which allows them to be cast in sizes of up to several millimeters to centimeter in all dimensions. Their disordered atomic arrangement generates a variety of useful properties such as excellent mechanical properties, e.g. high yield stress and an elastic strain limit of more than 2% [1,2]. Since BMGs vitrify there is no crystallization shrinkage and thus near-net-shape casting is possible. This enables the fabrication of complex geometries in a single production step without the need for further cold working, aging or heat treatment. Consequently, they are interesting candidates for use in high-performance and high-precision tools [3], biomedical applications [4-7], and processing via thermoplastic forming [8] and additive manufacturing techniques [9-11]. Moreover, ferromagnetic Fe-based BMGs show an exceptionally

low coercivity and high saturation magnetization, and are among the best soft-magnetic materials known today [2,12,13].

Another beneficial magnetic effect in Fe-based magnetic glasses is the anomalously low coefficient of thermal expansion (CTE), called the Invar effect. While only very few crystalline alloys show the Invar effect, it is universally observed in all Fe-based BMGs [14-19]. The Invar effect was first discovered in 1896 by Charles Édouard Guillaume [20] in the binary alloy 65Fe35Ni, now called Invar alloy. Below its Curie temperature it has a CTE of $1 \times 10^{-6} \text{ K}^{-1}$, which then suddenly increases to $15 \times 10^{-6} \text{ K}^{-1}$, a value which is typical for metals ($10 - 30 \times 10^{-6} \text{ K}^{-1}$) [21,22]. It is generally agreed that the Invar effect has magnetic origins but its atomic mechanism is not conclusively settled, especially in Fe-based BMGs.

Explanations of the Invar effect in crystalline 65Fe35Ni are based on the fcc unit cell. Ab-initio calculations have found non-collinear magnetic structures at low volumes that compete with the ferromagnetic

<figure><img src="image_2.png" /></figure>

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<footnote><sup>a</sup> Laboratory of Metal Physics and Technology, Department of Materials, ETH Zurich, 8093 Zurich, Switzerland</footnote>

<footnote><sup>b</sup> European Synchrotron Radiation Facility (ESRF), 71, Avenue des Martyrs, 38043 Grenoble, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>** Corresponding author.</footnote>

<footnote>E-mail addresses: alexander.firlus@mat.ethz.ch (A. Firlus), joerg.loeffler@mat.ethz.ch (J.F. Löffler).</footnote>
state [23]. The low-volume non-collinear spin state can be reached by thermal excitations and this counteracts the normal thermal expansion. However, this mechanism is based on the fcc unit cell and long-range order (LRO), and thus does not hold for metallic glasses.

Despite their short-range order (SRO), lack of long-range order, and absence of any magnetic anisotropy, the ferromagnetic interactions of Fe in BMGs cause a reduced thermal expansion even at the macroscopic scale. These interactions can be so strong that they can even cause a net-negative thermal expansion of the neighboring Fe pairs [17]. It is therefore an important effect that needs to be understood not only from a basic point of view, but also for the design and engineering of magnetic high-performance BMGs.

Most observations of the Invar effect in BMGs are made at the macroscopic scale [14–16]. However, the Invar effect vanishes after crystallization and it is therefore the amorphous SRO (or possibly the lack of LRO) that enables the Invar effect in BMGs. Consequently, it can only be understood via experiments that probe the SRO.

While it is known that Fe plays an important role in the atomic-scale origins of the Invar effect in Fe-based BMGs [17], the role of the other alloying elements is unknown. It is clearly a magnetic effect, so besides structural influences, the minor alloying elements may provide secondary magnetic interactions, among themselves and with Fe. There is no work on the influence of minor alloying elements on the Invar effect. Furthermore, it is unclear how the disordered atomic arrangement favors the Invar effect to make it universal in Fe-based BMGs.

These open questions motivated us to investigate the Invar effect in quaternary (Fe<sub>71</sub>B<sub>24</sub>RE<sub>5</sub>)<sub>96</sub>Nb<sub>4</sub> BMGs with the rare-earth elements RE = Tm, Er, Ho [24], which carry a large magnetic moment in their 4f electrons, and are further denoted as QTm, QEr, and QHo, respectively. Their behavior is compared with an (Fe<sub>71.2</sub>B<sub>24</sub>Y<sub>4.8</sub>)<sub>96</sub>Nb<sub>4</sub> BMG (denoted QY) [25,26]. Because Y is paramagnetic, the two alloy classes differ in the magnetic nature of their minor alloying elements, whereas the atomic sizes in metallic bonding of Y and RE are very similar [27]. This allows us to investigate the influence of minor alloying elements (i.e. Y, Nb, Tm, Er, Ho) on the Invar effect and consequently discover which physical interactions are relevant to explain the universality and strength of the Invar effect in Fe-based BMGs.

Macroscopic magnetometry and dilatometry confirm the presence of the Invar effect in all studied alloys. Through time-resolved X-ray diffraction (XRD) the pair distribution function can be obtained and the

atomic origin of the Invar effect be attributed to Fe-Fe interactions. The substitution of Y with RE does not affect the Invar effect. This suggests that the RE elements and minor alloying elements, such as Y, do not contribute to it via their magnetism. Instead, they only serve to increase the glass-forming ability and enable the formation of a disordered Fe network that causes the Invar effect.

## 2. Results and discussion

### 2.1. Magnetometry

While the origins of the Invar effect are still debated, there is agreement that it results from magnetic interactions. Therefore, the macroscopic magnetic behavior of the BMG samples provides first insights into the peculiarities of the Invar effect.

Fig. 1a–c compares the hystereses of the glassy as-cast alloys with their crystallized (at 1000 K) counterparts. Glassy QTm, QEr and QHo start with a saturation magnetization of around 79, 85 and 81 A m<sup>2</sup> kg<sup>-1</sup>, respectively. After crystallization, the saturation magnetization reduces by 46%, 44% and 30% for QTm, QEr and QHo, respectively, which indicates the formation of antiferromagnetic bonds. This is the expected magnetic interaction of Fe-RE alloys, which likely arises during the crystallization event. When alloyed with Fe, the RE elements can be magnetically active even at temperatures above the Curie temperature of their pure form as is illustrated by the comparison of the magnetic hysteresis loops for the glassy alloys of this study. Evidence of the antiferromagnetic activity of RE can be seen in Supplementary Fig. 1, which compares the room-temperature hystereses of QTm, QEr, and QHo to QY, which contains Fe as the only (anti)ferromagnetic element. As antiferromagnetic Fe-RE interactions are independent of the applied field, the initial (static) susceptibility of 644 A m<sup>2</sup> kg<sup>-1</sup> T<sup>-1</sup> (8.1 × 10<sup>-4</sup> m<sup>3</sup> kg<sup>-1</sup>) is the same for all alloys. However, some magnetic moments of Fe are compensated by those of RE such that the saturation magnetization is reduced by 15–20%.

Fig. 1d–f shows the thermomagnetic susceptibility curves for the three alloys. At low temperatures the alloys are ferromagnetic and consequently the susceptibility is constant. At the Curie temperature of 468 K, 476 K and 461 K for QTm, QEr and QHo, respectively, the susceptibility changes to scale like $\chi \sim (T - T_C)^{-1}$. These temperatures coincide with the transition temperatures seen in dilatometry

<figure><img src="image_3.png" /></figure>

**Fig. 1.** Magnetic hysteresis and thermomagnetic curves of QTm, QEr and QHo. The room-temperature hystereses of a QTm, b QEr, and c QHo all show soft-magnetic behavior (green) and a reduction of saturation magnetization and initial susceptibility after crystallization (brown). The thermomagnetic susceptibility curves reveal the Curie temperature of d QTm, e QEr, and f QHo BMG at around 480 K. After crystallization (brown) there is a crystalline phase with a Curie temperature of around 800 K and the remanent glassy matrix shifts its $T_C$ to slightly lower temperatures.
experiments (Fig. 2). During the crystallization process, a phase with a much higher Curie temperature of around 800 K forms in all alloys (see Fig. 1d–f). A very small reduction of the susceptibility at around 820 K can already be seen during the heating of the glass (green curves). It is likely that this results from the growth of quenched-in nuclei at temperatures close to the glass transition temperature. Both, amorphous FeB with around 70 at.% Fe [28] and ternary FeBNb phases [29] are known to have their Curie temperature in that range. Still, after crystallization some amorphous matrix remains that has a Curie temperature of around 440 K, which is slightly lower than the $T_C$ of the initial glassy sample.

## 2.2. Dilatometry

The Invar effect is strongly associated with Fe. For the traditional crystalline FeNi Invar alloy, it was shown via ab-initio calculations on the fcc unit cell and X-ray absorption experiments that only Fe, but not Ni, shows an anomalous correlation between the arrangement of magnetic moments and the average atomic volume [23,30]. For metallic glasses the strength of the Invar effect is tightly linked to a high concentration of Fe, whereas other magnetic elements such as Ni and Co are known to reduce the Invar effect [14]. To date there is no information on the effect of magnetic RE elements on the Invar effect or whether Fe–RE glasses even show the Invar effect.

Fig. 2 shows macroscopic dilatometry experiments of the present Fe–RE BMGs. All glasses exhibit the Invar effect with a similar CTE, which increases from $3 - 4 \times 10^{-6}$ K$^{-1}$ below the Curie temperature to $11 - 12 \times 10^{-6}$ K$^{-1}$ at $T_C$. When heated above their glass transition temperature, $T_g$, the alloys transform from a glass into a supercooled liquid. At around 850 K the CTEs deviate from linearity (i.e. become larger) due to this transition. Immediately above $T_g$, the viscosity decreases by orders of magnitudes [31] and the sample can deform under

<figure><img src="image_4.png" /><figcaption>Fig. 2. Macroscopic coefficient of thermal expansion (CTE) of QTm, QEr and QHo. The CTEs of a QTm, b QEr, and c QHo all show a step-like increase at the Curie temperature. After crystallization they show no signs of the Invar effect anymore. The CTEs of the crystallized samples (brown) are equal to those of the glassy samples (green) above their Curie temperature. For reference, the CTE of QY (black) is also shown.</figcaption></figure>

the applied load (0.25 N) of the dilatometer’s pushing rod. This causes a sharp length reduction just above $T_g$.

After crystallizing the samples at around 1220 K, the Invar effect vanishes and the CTE remains high even below the Curie temperature. The glass transition and crystallization temperatures are also confirmed via differential scanning calorimetry (DSC), as shown in Supplementary Fig. 2. The CTE of the crystallized samples is identical to that of the glassy but paramagnetic samples. Without the Invar effect, either through heating above $T_C$ or crystallization, the thermal expansion of the BMGs is similar to that of its constituent elements. This is also the case for non-Fe-based BMGs, which are not ferromagnetic and thus cannot show the Invar effect [32]. Therefore, this observation confirms that it is the magnetic interactions in the glassy state that reduce the CTE.

In addition, Fig. 2a also shows the CTE of the QY reference alloy. In general, the reduction of the CTE is very sensitive to slight modifications of the alloy composition [14]. However, here we find that the CTE of QY is equal to that of the Fe–RE glasses both below and above the Curie temperature. The minor element addition of RE elements, which can interact magnetically via their 4f electrons, apparently does not interfere with the magnetic interactions responsible for the Invar effect. It thus follows that only Fe–Fe bonds contribute to the Invar effect while Fe–RE bonds neither support nor oppose it.

## 2.3. X-ray diffraction

The origins of the Invar effect in Fe-based BMGs can only be understood at the atomic scale. In-situ XRD directly probes the interatomic distances rather than the macroscopic thermal expansion and thus can be used to measure changes in atomic bond distances. Michalik et al. reported on the possibility to observe the Curie temperature of Fe-based BMGs via synchrotron-based X-ray diffraction [18,19]. Furthermore, Firlus et al. have recently shown that the Invar effect has clear origins in the magnetic interaction of nearest-neighbor Fe–Fe pairs [17].

All samples were measured via XRD both on heating and cooling and exhibit qualitatively the same behavior. However, the cooling curves are more stable with respect to the geometry of the setup (e.g. influence of thermal expansion of the sample holder). Therefore, the measurements during cooling were used to quantify the atomic-scale Invar effect.

Fig. 3a–c shows typical XRD profiles of the RE-containing BMGs. The contributions from air scattering and boron-nitride layer are already subtracted (see experimental section). As temperature is increased, the position of the diffraction peaks shifts to smaller q values. Only the shift of the main diffraction peak is a measure of thermal expansion as the higher-order diffraction peaks comprise additional information about SRO atomic rearrangements [33]. The diffraction profile within the shaded grey area was fitted with a Lorentzian function plus a linear background. The primary peak position of this fit is shown in Fig. 3d–f as a function of temperature. Over the temperature range of 320–670 K it shifts by roughly 0.5%. Additionally, all samples show a distinct change in slope at the Curie temperature where the thermal expansion abruptly increases. This proves that also in the case of Fe–RE BMGs, the Invar effect has origins in the interatomic distances and is not a purely macroscopic effect.

## 2.4. Pair distribution function

An example of the reduced pair distribution function (rPDF), $G(r)$, is shown in Fig. 4a–c, while the pair distribution function (PDF), $g(r) = \frac{G(r)}{4\pi\rho_{at}r} + 1$, of the first atomic shell is shown in panels d–f. The PDF is a (non-normalized) probability distribution and requires an estimate of the alloy’s atomic density $\rho_{at}$. Here, we set it as $\rho = 0.991 \sum_i f_i \rho_i$, using the weight concentration $f_i$ and density $\rho_i$ of each element i. To account for free volume in the disordered atomic structure the factor of 0.991 is
<figure><img src="image_5.png" /><figcaption>Fig. 3. X-ray diffraction profiles and positions of the first diffraction peaks. The integrated diffraction profiles of a QTm, b QEr, and c QHo show no sign of crystallinity. The area shaded in grey is used for fitting the primary peak positions. For all alloys, d QTm, e QEr, and f QHo, the central peak position diverges from linearity (red line) at the same transition temperature (Curie temperature, shaded in grey) as in the macroscopic dilatometry measurements. The green shaded area marks the ±1 standard deviation of the peak fitting.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4. (Reduced) pair distribution function at around 350 K. It shows five atomic shells for a QTm, b QEr, and c QHo. The first two shells show clear peak splitting, which indicates pronounced SRO and MRO. The pair distribution function of the first atomic shell of d QTm, e QEr, and f QHo shows contributions from multiple atomic pairs. The red bars indicate the scattering factors of the most relevant atomic pairs.</figcaption></figure>

introduced. Thus, the atomic densities are 89.858 nm⁻³, 89.684 nm⁻³, and 88.991 nm⁻³ for QTm, QEr, and QHo, respectively.

In total there are five atomic shells that can be derived from the XRD patterns in Fig. 4a-c. The commonly observed split peaks of the first and second atomic shells are signs of pronounced SRO and medium-range order (MRO) [34,35].

The first atomic shell is located at around 1.85–3.60 Å, which is shaded in Fig. 4d-f. It is split with the primary peak being associated with Fe–Fe pairs at around 2.55 Å and Fe–Nb/RE pairs at larger distances. The red bars represent the X-ray weighting factors [36]

$$w_{ij} = \frac{1}{\left(\sum_k c_k f_k\right)^2} \begin{cases} c_i^2 f_i^2 & i = j \\ 2c_i c_j f_i f_j & i \neq j \end{cases}, \qquad (1)$$

where $c_k$ and $f_k$ are the atomic concentration and X-ray scattering factor of element k, respectively. The X-ray scattering factors were calculated based on the data by Chantler [37] at q = 0 and the synchrotron beam energy as determined by calibration (around 87 keV). The corresponding pair distance is based on the metallic or van-der-Waals radii of the elements of each pair [27]. Of the ten possible pairs, only those that contribute at least 2% are shown. These are only the Fe–X pairs, because of the low scattering factor of B and the low atomic concentrations of Nb and RE.
Firlus et al. have shown that the thermal expansion of Fe–Fe is the atomic origin for the Invar effect in FeBYNb and FeBYMo BMGs [17]. Here the magnetic interaction of Fe–Fe pairs causes a negative contribution to the thermal expansion and can even cause thermal contraction below the Curie temperature of the alloys.

Besides Fe–Fe pairs, there is also a significant number of Fe–Nb and Fe–RE pairs to be found in the first atomic shell of the present BMGs. It is therefore of interest to look at the changes of the complete first atomic shell in the present study.

A good measure for the central position of an atomic shell is the first moment $\langle r \rangle$ of the PDF [17]. It is defined as

$$ \langle r \rangle = \frac{\int_{r_0}^{r_1} rg(r)dr}{\int_{r_0}^{r_1} g(r)dr}, \qquad (2) $$

where $r_0$ and $r_1$ are the boundaries of the atomic shell. For the first atomic shell, the integration limits were kept fixed for all temperatures at 1.85 – 3.6 Å. The shape of $\langle r \rangle$ as a function of temperature is independent of the precise integration limits. More importantly, the transition temperature determined in this way is not affected either. However, there is a (locally) linear relationship between the absolute value of $\langle r \rangle$ and the integration limits such that a variation of the integration limits by 0.05 Å increases $\langle r \rangle$ by 0.07 Å.

Fig. 5a–c shows the temperature evolution of the average Fe–Fe pair distance. It was obtained by fitting a Gaussian peak with a linear background to the rPDF in the range of 2.3 – 2.8 Å. The solid points represent an average of 32 blocks of an equal number of individual data points and the shaded area represents the ±1 standard deviation of the block-averaging. The range of the Curie temperature as derived from the magnetic and macroscopic dilatometry experiments is shaded in grey. In all alloys there is a distinct change in slope at the Curie temperature. When the samples are ferromagnetic, the Fe–Fe pair distances change

very little or even reveal a net-negative thermal expansion. After the magnetization vanishes at the Curie temperature, the Fe–Fe pair distances increase significantly faster. This proves that the Invar effect in FeBNb–RE metallic glasses is also strongly associated with the Fe network and has atomistic origins.

The central position of the first atomic shell, $\langle r \rangle$, is shown in Fig. 5d–f. The Curie temperature, as derived from the magnetic and macroscopic dilatometry experiments, is shaded in grey. Once again, there is a transition from low or even negative thermal expansion to high thermal expansion at the Curie temperature.

## 2.5. Origins of the Invar effect in the disordered atomic arrangement

The macroscopic dilatometry experiments prove that also Fe–RE metallic glasses reveal the Invar effect. This finding further underlines the universality [14] of the Invar effect in magnetic Fe-based BMGs.

It is well known that substitution of Fe by Co or Ni strongly reduces the strength of the Invar effect [14]. In particular, it is the CTE in the ferromagnetic state that increases while the CTE in the paramagnetic state is largely unaffected. This also applies to reducing the Fe concentration in favor of non-magnetic elements as demonstrated for Fe$_{75}$Mo$_5$P$_{10}$C$_{7.5}$B$_{2.5}$. A reduction of the Fe concentration by 4 at.% causes an increase in the CTE of the ferromagnetic state by about 60% [14]. Therefore, the strength of the Invar effect is very sensitive to the Fe concentration regardless of the magnetic properties of the substituting element. This indicates that the substitution of Fe influences the Invar effect via the atomic arrangement rather than through magnetic interactions.

The alloy systems in this study, (Fe$_{71}$B$_{24}$RE$_5$)$_{96}$Nb$_4$ with RE = Tm, Er, Ho, are derived from the well-studied (Fe$_{71.2}$B$_{24}$Y$_{4.8}$)$_{96}$Nb$_4$ alloy [15–17] by substituting Y, i.e. a minor alloying element, with RE. This leaves the Fe concentration unchanged but introduces a second magnetic atomic species (interacting via 4f electrons) to the alloy. Interestingly, the strength of the Invar effect, i.e. the difference in CTE of the paramagnetic and ferromagnetic state, is unaffected by this substitution.

<figure><img src="image_7.png" /><figcaption>Fig. 5. Average Fe–Fe pair distance and central position of the first atomic shell. The Curie temperature from the dilatometry and magnetometry experiments is shaded in grey. The Fe–Fe pair distances (green dots) of a Tm, b QEr, and c QHo all clearly show an anomalously low thermal expansion below the Curie temperature. The standard deviation is shaded in light-green. The central positions of the first atomic shell of d Tm, e QEr, and f QHo also show a clearly reduced thermal expansion below the Curie temperature of the respective alloy. The thermal expansion can be even negative, as seen for the alloys QEr and QHo.</figcaption></figure>
Two aspects have to be considered to develop the conclusions that can be derived from this observation: (1) the influence on the atomic arrangement during vitrification, and (2) the magnetic interactions within the Fe network.

Small variations of minor alloying elements can drastically change the glassy atomic arrangement and thus the glass-forming ability [25, 26]. Since the Curie temperature is much lower than the glass transition temperature, there are no magnetic interactions during vitrification.

While the bonding preference plays a role in forming a densely packed glass [38], most models of the SRO and MRO of metallic glasses are topological [34]. Tm, Er, and Ho have atomic radii of 1.74 Å, 1.75 Å, and 1.76 Å, respectively, which are close to that of Y at 1.80 Å, and they are all the largest atomic species in their respective alloy ($r_{Nb} = 1.46$ Å, $r_{Fe} = 1.26$ Å, $r_{B} = 0.98$ Å) [27]. Thus, the RE elements serve the same function in building the glassy SRO and MRO as yttrium, but it can be seen in Fig. 4a-c that the first atomic shell of QTm, QEr, and QHo has a much stronger shoulder than the one of (Fe<sub>71.2</sub>B<sub>24</sub>Y<sub>4.8</sub>)<sub>96</sub>Nb<sub>4</sub> [17]. For this reason the average pair distance of Fe–Nb and Fe–RE cannot be extracted from the PDF through peak fitting. Only the Fe–Fe peak is sufficiently pronounced and separated from the other atomic pairs. The absence of a pronounced shoulder in the SRO of FeBYNb may be explained by the similar chemistry of Fe–Y and Fe–Nb [39], whereas the RE elements have a different bonding behavior with Fe than Nb does. It is known that neither Y nor Nb affects the Fe–Fe bond distances but that they rather dilute the number of such bonds [39]. Since there is no influence on the ferromagnetic CTE, which is very sensitive to variations of the Fe network, it can be concluded that also the RE elements do not affect the formation of the Fe network during vitrification.

Besides the topological influence of RE there is also the magnetic interaction with the Fe network. Ab-initio calculations on the classical FeNi Invar alloy have shown that it is only Fe but not Ni that can contribute to the Invar effect [23], but a particular concentration of 35 wt% of Ni is required to achieve the Fe environment that is responsible for the Invar effect [20,40–42]. In Fe-based BMGs this effect is universal. Supplementary Fig. 1 compares the hysteresis loops of the Fe–RE glasses and the FeBYNb glass and proves that the RE elements are (anti)ferromagnetically active even at room temperature. Nevertheless, the Invar effect is not affected by the Fe–RE interactions. It can therefore be concluded that the Invar effect is solely a consequence of the magnetic interactions of Fe–Fe. Firlus et al. [17] have also observed an anomalous thermal expansion with a change in slope of Fe–Y, Nb and Fe–Y, Mo bonds at the Curie temperature, albeit a weak one. It was at that time unclear whether the paramagnetic elements play a causal role in the atomic-scale Invar effect. With the information about the irrelevance of (anti)ferromagnetic Fe–RE interactions, it can now be concluded that it is unlikely that paramagnetic or diamagnetic elements would contribute to the Invar effect via their magnetic interactions, which are orders of magnitude weaker than the (anti) ferromagnetic ones. Therefore, the role of these atomic species is assumed to be purely topological in the sense that they are required to form the glassy Fe network and improve glass-forming ability. Any signs of the Invar effect in Fe–X bonds are just a secondary consequence of the anomalous thermal expansion of the Fe network in which the minor alloying elements are embedded. This means that they are only affected “mechanically” by their (contracting) Fe-rich environment.

### 3. Conclusions

The Invar effect is universally seen in magnetic Fe-based bulk metallic glasses. A reduced CTE in the ferromagnetic state characterizes it. After crystallization, the Invar effect vanishes and the CTE is similar to the one in the paramagnetic, glassy state. The BMGs (Fe<sub>71</sub>B<sub>24</sub>RE<sub>5</sub>)<sub>96</sub>Nb<sub>4</sub> with RE = Tm, Er, Ho were investigated for their Invar effect via macroscopic dilatometry and magnetometry. These experiments constitute the first confirmation of the Invar effect in rare-

earth containing Fe-based BMGs. Additionally, the atomic origins of the Invar effect were studied by synchrotron-based in-situ XRD. It reveals that the Invar effect originates at the atomic scale. In particular, Fe–Fe bonds show the same anomalous thermal expansion as is seen at the macroscopic scale.

The ferromagnetic elements Co and Ni are known to reduce the Invar effect in Fe-based BMGs, but here it is found that the RE elements, which couple antiferromagnetically with Fe, do not have any influence on it. This likely stems from their different role in the atomic arrangement. Ni and Co substitute Fe within the Fe-network while the bigger RE atoms tend to be located in the backbone that surrounds it [43]. A similar BMG is (Fe<sub>71.2</sub>B<sub>24</sub>Y<sub>4.8</sub>)<sub>96</sub>Nb<sub>4</sub> from which the alloys in this study were derived by substituting Y with RE. Y and the RE elements have very similar atomic radii. Thus, the absence of any influence on the Invar effect suggests that the role of those elements is purely topological. They enable good glass-forming ability and the formation of a disordered Fe network but do not actively contribute to the Invar effect. This leaves Fe as the only atomic species in those alloys that can enable and control the Invar effect.

This further implies that the minor alloying elements can be varied to optimize other material properties while maintaining the Invar effect and thus almost zero thermal expansion. For example, it may be possible to increase the RE concentration (without reducing glass-forming ability) to exchange the ferromagnetic Fe–Fe interactions with antiferromagnetic Fe–RE interactions. In this way a macroscopically non-magnetic glassy alloy with zero thermal expansion could be created.

### 4. Materials and methods

#### 4.1. Sample preparation

All master alloys were prepared from pure elements with purities of 99.5% for B, 99.97% for Fe, and 99.99% for Tm, Er, and Ho. Nb was included in the master alloy via an 25Fe75Nb pre-alloy made from 99.95% pure Nb (and 99.97% pure Fe). The elements were alloyed through arc melting in a Ti-gettered argon (99.999% pure) atmosphere. The resulting button was flipped over and remelted several times to ensure homogeneity. The casting into rods of 3 mm diameter was done by arc melting a piece from the master alloy and pulling the melt into a water-cooled copper mold by suction casting.

#### 4.2. Magnetometry

The magnetic properties were studied by vibrating sample magnetometry at 15.9 Hz and AC susceptibility measurements in an MPMS3 device from QuantumDesign. The AC susceptibility was measured with an excitation of 80 A m<sup>-1</sup> at 113 Hz without any bias field applied. The error in the magnetic moment measurements was confirmed with a Pd reference sample to be less than 1%. A temperature error of less than 5 K was confirmed by measuring the Curie temperature of FeNi and pure Ni reference samples.

#### 4.3. Dilatometry

16–19 mm long BMG rods of 3 mm diameter were cut after casting and their surfaces were polished to ensure parallel faces. The dilatometry was performed with a Netzsch DIL 402 C dilatometer applying a heating/cooling rate of 5 K min<sup>-1</sup> in a 99.999% pure argon atmosphere with 80 ml/min flow rate. The temperature profile consisted of three sections: heating to 950 °C, cooling to 25 °C, and heating to 950 °C. The heating and cooling segments were separated by isothermal sections of at least 60 min, which also served to provide enough time for complete crystallization of the sample. The temperature calibration was done by melting pure In, Sn, Pb, Zn, eutectic AgCu, Ag, and Cu. The coefficient of thermal expansion (CTE) was obtained from the relative length change by a Savitzky-Golay differentiation filter using a second-order
polynomial and 20-point window size (corresponding to about 1 K).

## 4.4. In-situ X-ray diffraction

The in-situ XRD experiments were performed in transmission at the ID15A beamline [44] at ESRF Grenoble. A 0.5 mm × 0.5 mm monochromatic beam of 87 keV photons was used to illuminate the samples for 0.5 s per frame. The diffraction pattern was recorded with a Debye Pilatus3 X CdTe 2 M detector. The samples were thin disks of 3 mm diameter and covered in boron nitride to avoid oxidation. Heating and cooling was performed at 5 K min⁻¹ in a Linkam THMS600 heating stage. Calibration with CeO₂ and data integration was done with the pyFAI software package [45]. Pair distribution functions were derived from the integrated diffraction profiles with the pdfgetx3 software [46]. Integration limits of 1.24 – 16.50 Å⁻¹, 1.20 – 16.00 Å⁻¹, and 1.20–15.90 Å⁻¹ were chosen for QTm, QEr, and QHo, respectively.

## Author contributions

J. F. L., M. S. and A. F. designed the study, and J. F. L. supervised the work. A. F. prepared the samples and performed the dilatometry and magnetometry measurements. A. F., M. S. and G. B.M. V. planned the XRD experiments and carried them out. A. F. analyzed and correlated the data. All authors discussed and validated the results, and wrote the manuscript.

**AF:** Conceptualization(E), Data curation(L), Formal Analysis(L), Investigation(L), Methodology(L), Project administration(E), Validation (E), Visualization(L), Writing – original draft(L), Writing – review & editing(E), MS: Conceptualization(E), Formal Analysis(S), Investigation (E), Methodology(E), Project administration(S), Validation(E), Writing – original draft(S), Writing – review & editing(E), GBMV: Data curation (S), Formal Analysis(S), Investigation(E), Methodology(E), Validation (E), Writing – review & editing(E), RES: Validation(E), Writing – review & editing(E), JFL: Conceptualization(E), Funding acquisition(L), Project administration(E), Supervision(L), Validation(E), Writing – review & editing(E). L = Leading, E = Equal, S = Supporting.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Data availability

The raw X-ray diffraction datasets of this study are made publically available by ESRF: "Time-resolved evolution of SRO/MRO/LRO of Fe-based bulk metallic glasses with Invar effect", DOI 10.15151/ESRF-ES-428373998. The calibrated datasets generated and analyzed during the current study are available in the ETH Research collection: Datasets to "The nature of the atomic-scale Invar effect in Fe-based bulk metallic glasses", DOI 10.3929/ethzb-000577538.

## Acknowledgements

The authors thank Sebastian Reitz and Christian Wegmann for technical help and Stefan Stanko for fruitful discussions. They also express special thanks to the ‘Seminar für Statistik’ for advice on the statistical data analysis. This work was carried out with the support of ESRF Grenoble, proposal HC-4361. The authors also gratefully acknowledge the support by an ETH Research Grant (ETH-47 17-1; J. F. L.) and the ETH + initiative within the framework of SynMatLab (Laboratory for Multiscale Materials Synthesis and Hands-On Education; J. F. L.).

# Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.mtnano.2023.100394.

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