Research Article

Self-induced spin glass state in elemental and crystalline neodymium

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Science  29 May 2020:
Vol. 368, Issue 6494, eaay6757
DOI: 10.1126/science.aay6757

A structurally ordered spin glass

Spin glasses that form in disordered materials such as magnetic alloys have locally varying magnetic patterns, and their spin relaxation occurs over time scales spanning many orders of magnitude. Kamber et al. used spin-polarized scanning tunneling microscopy to image the magnetism on the (0001) surface of thick, single-crystal films of neodymium as a function of temperature and magnetic field. Despite the lack of structural disorder, they found a spectral distribution of degenerate magnetic wave vectors, or Q states, that exhibited spatiotemporal variation. In this spin-Q glass, pockets of nearly degenerate spin spiral states formed with varying periodicity. Ab initio electronic structure coupled to atomistic spin dynamics calculations suggests that the double hexagonal closed packed crystal structure in neodymium drove strongly frustrated magnetism that created these pockets.

Science, this issue p. eaay6757

Structured Abstract

INTRODUCTION

Spin glasses are one of the more intriguing, but least understood, magnetic states of matter. In stark contrast to ordered magnets, spin glasses form a state characterized by seemingly random and uncorrelated magnetic patterns lacking long-range order. The distinguishing feature of spin glasses is aging: The magnetic state depends on its history, which is driven by multiple time scales. Despite decades of theoretical developments, there is still no clear understanding about when spin glass behavior arises. It is commonly believed that disorder is a key ingredient, in addition to competing magnetic interactions, as exhibited by the textbook example of dilute magnetic alloys.

RATIONALE

Despite more than 50 years of investigation, there is still no consensus on the magnetic ground state of the element neodymium (Nd). Below the Néel temperature, previous experiments reported the onset of static spin spirals with different periodicities, or so-called “multi-Q states,” as well as evidence of additional phase transitions. Although these observations illustrate the complexity in this material, it is not well understood how the multiplicity of these Q states depends on temperature and the exchange landscape of the material and which real-space magnetic interactions cause the multi-Q states.

RESULTS

We show that single-crystalline elemental Nd exhibits unconventional spin glass behavior well described by the recently proposed concept of self-induced spin glassiness. Traditional spin glasses, such as the hallmark metallic alloy Cu-Mn, are based on competing interactions and disorder. By contrast, self-induced spin glasses can be created solely by competing interactions without strong disorder, for example, in systems with long-range magnetic interactions. We studied Nd by growing ultraclean, epitaxial thick islands and films, both of which were representative of bulk Nd. Using spin-polarized scanning tunneling microscopy, magnetic images of the surface revealed strong, local, noncolinear magnetic order at the atomic scale with a multiplicity of Q states and no long-range order (left). We quantified the role of defects on the glassiness, showing that cleaner samples led to more pronounced glassy behavior exemplified by the increased mixing of distinct Q states. The spatially dependent Q states were defined by a spectral distribution of degenerate magnetic wave vectors (right), which varied spatially and with time. Harnessing ab initio and atomic spin dynamics calculations, we quantified the magnetic interactions, which illustrated strongly competing, distance-dependent interactions intricately linked to the crystal structure of Nd. Moreover, the resultant energy landscape illustrates many favorable Q states, showing that glassiness in Nd results from the conditions required for self-induced spin glasses. When probing the response to applied magnetic fields and temperature, we observed the existence of multiple time scales reminiscent of aging in traditional spin glass materials. The calculated autocorrelation function, a mean-field picture of the aging dynamics, also showed aging behavior reminiscent of traditional spin glasses (bottom). Experiments also showed that the aging dynamics were strongly dependent on the underlying value of Q, leading to behavior reminiscent of the concept of dynamic heterogeneity observed in structural glasses. In this way, the energy landscape can be described by a rugged, multiwell potential composed of degenerate Q states. Unlike traditional spin glasses, competing interactions in Nd are driven by its electronic and the structural properties, which lead to self-induced glassiness.

CONCLUSION

Our findings not only suggest that glassy behavior can be found in elements with crystalline order, but also unravel the decades-long debate about the magnetic behavior of Nd. The coexistence of short-range order exhibited by degenerate Q states and aging dynamics manifested by multiple time scales provides the first experimental confirmation of self-induced glassiness. The existence of strong local Q order and the Q-dependent dynamics may imply that multiple but select time scales exist, resulting from a rugged, multiwell landscape, when compared with traditional spin glasses. This provides a new platform with which to explore dynamic heterogeneity in a spin glass material. It remains to be understood what is particularly special about the crystal structure and electronic properties in Nd, and this will necessitate a deeper theoretical understanding of the role of electron correlation effects, as well as the interplay between spin and orbital degrees of freedom. The example here expands our views on magnetic states of matter, inviting further research into aging behavior in magnetic systems.

Spin-Q glass.

(Left) Real-space magnetization image with spin-polarized scanning tunneling microscopy at T = 1.3 K of thick films of Nd(0001). The surface shows multi-Q states but no long-range order. (Right) Sketch of spin-Q glass in both real and reciprocal spaces, with color illustrating the distribution of Q states in real space derived from flat pockets in Q-space. (Bottom) Calculated autocorrelation function for Nd with increasing waiting time (tw) illustrating aging behavior.

Abstract

Spin glasses are a highly complex magnetic state of matter intricately linked to spin frustration and structural disorder. They exhibit no long-range order and exude aging phenomena, distinguishing them from quantum spin liquids. We report a previously unknown type of spin glass state, the spin-Q glass, observable in bulk-like crystalline metallic neodymium thick films. Using spin-polarized scanning tunneling microscopy combined with ab initio calculations and atomistic spin-dynamics simulations, we visualized the variations in atomic-scale noncolinear order and its response to magnetic field and temperature. We quantified the aging phenomena relating the glassiness to crystalline symmetry and the energy landscape. This result not only resolves the long-standing debate of the magnetism of neodymium, but also suggests that glassiness may arise in other magnetic solids lacking extrinsic disorder.

Spin glasses are one of the more intriguing, but least understood, magnetic states of matter (15). Ferromagnets or antiferromagnets form a long-range ordered state when cooled but spin glasses form a state characterized by seemingly random and uncorrelated magnetic patterns. The magnetization pattern in spin glasses can be compared to the amorphous structure of glasses such as silicon dioxide, which exhibit local structural correlations but lack a long-range ordered state. Interest in spin glasses spans many fields, ranging from iron-based superconductors (6) to theoretical machine learning (4, 5, 7), and they have been suggested to be relevant in quantum topological excitations.

Spin glasses are characterized by a glass transition temperature and by aging; i.e., the magnetic state depends on its history, which is driven by a distribution of distinctive spin-relaxation processes with time scales spanning many orders of magnitude (4, 5). Aging also distinguishes spin glasses from so-called quantum spin liquids (8) or spin ices (9), which remain disordered down to zero temperature because of quantum fluctuations and lack of memory. The paradigm of all these types of complex magnets involves magnetic frustration derived from geometry or competing interactions. However, unlike spin liquids, disorder is traditionally also considered necessary to drive non-ergodic behavior in spin glasses.

There is still no clear understanding about when spin glass behavior can arise in magnetic materials. The most commonly debated models (1, 35) describe randomly distributed spins with long-range magnetic interactions of alternating ferromagnetic and antiferromagnetic coupling, such as that seen in prototypical materials such as dilute magnetic alloys (1, 10). In the thermodynamic limit, spin glasses have a hierarchical energy landscape with infinitely many local energy minima separated by energy barriers of multiple heights; therefore, there is a broad distribution of transition times between different minima (35, 11) that results in an absence of local and long-range order. Most models of spin glasses invoke structural disorder, i.e., amorphousness, as a key requirement along with competing magnetic interactions (12). However, self-induced glassiness, a concept introduced initially for the stripe glass behavior in high-temperature superconductors (13, 14), was later developed for magnets (15, 16). Within this framework, competing interactions alone can lead to the glassy state, even in the absence of external disorder, and can give rise to intermediate regimes that exhibit multiwell potentials (17).

We show that single-crystalline elemental neodymium (Nd) exhibits a previously unknown type of spin glass behavior. Spin-polarized scanning tunneling microscopy (SP-STM) on the (0001) surface of thick Nd films revealed that the magnetic state exhibited strong local noncolinear magnetic order but lacked a long-range ordered state. This local order is defined by a spectral distribution of degenerate magnetic wave vectors, or Q states, which varied spatially and with time. We probed the response of this so-called spin-Q glass to applied magnetic fields and variable temperature to quantify its aging behavior and energy landscape. Harnessing ab initio methods and simulations, we quantified the competing long-range magnetic interactions and the favorable Q states, illustrating that this unconventional glassy behavior results from valley-like pockets of degenerate Q states, as has been proposed for self-induced spin glasses (15, 16). Moreover, we performed calculations of the autocorrelation function for pristine Nd and also found multiple relaxation times in its spin dynamics. Our findings not only suggest that glassy behavior and aging can be found in systems with crystalline order, but also unravel an unresolved debate about the magnetic ground state of elemental Nd that has challenged scientists for several decades.

Magnetic ground state of Nd(0001)

Despite more than 50 years of investigations, there is still no consensus on the magnetic ground state of Nd and the origin of the complexity reported in various experimental observations (1826). Below the Néel temperature (TN), neutron diffraction allowed observation of the onset of static multi-Q states with decreasing temperature. However, it is not well understood how the multiplicity of these Q states depends on the exchange landscape of the material and which real-space magnetic interactions cause these various Q states. Moreover, other measurements indicated additional phase transitions below TN, suggesting that the original conclusions from neutron diffraction of a modulated antiferromagnetic structure were oversimplified (23). In addition, there are experimental observations above TN that provide evidence for short-range order as well as a strongly frustrated exchange landscape (26, 27). These open questions illustrate the need for a characterization of the exchange interactions in Nd, as well as a real-space characterization of the local magnetic order.

Atomic-scale visualization of the spin-Q glass state

To characterize the atomic-scale magnetization of the surface of Nd(0001), we used SP-STM and spin-polarized scanning tunneling spectroscopy (SP-STS) (28). We epitaxially grew thick films [up to 100 monolayers (ML)] of Nd(0001)/W(110). We note that lanthanide films prepared in this way on various bcc(110) substrates exhibit high crystallinity and superior surface cleanliness over sputter-annealed bulk single crystals (2937). It has been shown for various lanthanide elements, including Nd, that films grown with bulk lattice parameters above a thickness of 10 ML (31, 35, 3840) exhibit a bulk-like electronic structure >30 ML (32, 33, 41) and start to exhibit bulk-like magnetic behavior in the range of 30 to 50 ML (4245). Superlattice studies showed that 33 to 39 ML–thick Nd films already exhibit the temperature-dependent neutron scattering features known from bulk single crystals (38, 46, 47).

The sample morphology can be tuned to either layer-by-layer grown closed films or islands [Stranski–Krastanov (SK) growth], depending on the annealing temperature (39, 42, 4850). We grew two types of samples: SK-grown islands of >50 ML thickness (Fig. 1, A and B) and closed, ~100 ML films (fig. S1B) (50). As discussed above, SK islands should readily represent bulk-like structural, electronic, and magnetic properties. We verified this on thicker (~100 ML) closed films, which showed identical magnetization patterns, as we discuss in the supplementary text, section S2 (50). However, we observed that closed thick films show inferior surface qualities because of the presence of more impurities, as well as screw dislocations that can lead to pinning of magnetic structures (51). Because the latter was not present for our island-grown samples (fig. S1A), we focused on data taken from islands with thickness between 58 and 92 ML and lateral sizes between 58,000 and 200,000 nm2, which may account for differences in previous measurements. Further details of the sample preparation and morphology, as well as a discussion of thin films versus bulk samples, can be found in the supplementary text, section S1 (50).

Fig. 1 Imaging the spin-Q glass state of Nd(0001).

(A) Constant-current STM image of an SK-grown Nd film on W(110) revealing nearly flat-top islands on an Nd wetting layer. The surface defect concentration is <0.01 ML [scale bar, 150 nm; VS = 1 V; tunneling current (It) = 20 pA]. (B) Line profiles along the indicated island edges. The thickness of each island is >50 ML. (C) dI/dV spectrum acquired on the Nd(0001) surface showing the exchange-split surface state [stabilization bias (Vstab) = 1 V; stabilization current (Istab) = 200 pA; modulation bias (Vmod) = 1 mV; T = 40 mK]. (D) Magnetization image illustrating the spatially complex magnetic ground state of a spin-Q glass, which lacks long-range order (T = 1.3 K; B = 0 T; It = 200 pA). The contrast is directly related to variations in the out-of-plane magnetization (Mz) imaged with an out-of-plane sensitive Cr bulk tip (scale bar, 50 nm). (E) Q-space image of the magnetization image in (D) (scale bar, 5 nm−1) illustrating a large distribution of states in Q-space. (F) Line cuts along Γ¯ to M¯ of various Q-space images taken from smaller sections of the image in (D) (compare fig. S5). (G) Close-up views of regions marked in (D) illustrating the local spatial variation of the magnetic order (scale bar, 10 nm). (H) Schematic of the out-of-plane projected magnetization resulting from a superposition of the labeled Q vectors (scale bar, 2 nm). The wave vector amplitudes used are marked with dashed lines in (F).

Using STS at low temperature (0.03 to 7 K), we probed the Nd(0001) surface state, the presence and sharpness of which has been shown to be a probe of the cleanliness of the film (27, 31, 32, 52, 53). The surface state is characterized by an exchange splitting into a majority peak visible below the Fermi energy (EF) [sample bias (VS) = 0] and a minority peak above EF, with an additional narrow peak at EF (Fig. 1C) (49). We saw no difference between spectra taken on the islands and those taken on locally clean areas of the thick film. Furthermore, the magnetic exchange splitting reproduced previous low-temperature data taken on a 30 ML film (27, 54), which is further evidence that our samples are beyond the thin-film limit and fully reflect bulk electronic and magnetic properties.

To get magnetic contrast, we used the spin-polarized nature of the exchange-split surface state and imaged the surface in constant-current mode at two characteristic voltages representing dominant tunneling into the minority state (VS = 200 mV) and out of the majority state (VS = –150 mV), respectively (fig. S2) (50). We used an out-of-plane sensitive antiferromagnetic chromium (Cr) probe, which relates spin-dependent contrast variations directly to the z-projection, the c-axis in the double hexagonally close packed (dhcp) structure, of the magnetization. We consider the subtracted image to remove stronger topographic modulations resulting from buried substrate steps or locally varying sample thickness. We refer to this subtracted image as the magnetization image (Fig. 1, D and G, and fig. S2D). More details on this image-processing procedure can be found in the supplementary text, section S2 (50).

The magnetization images of the surface revealed strong and clear short-range magnetic order with periodicities λ varying from 0.9 to 4.5 nm and oriented along or near the high-symmetry axes. These atomic-scale variations are directly related to local noncolinear magnetic order, with varying periodicities depending on spatial location, which is defined by a superposition of local magnetic wave vectors Qi = 2π/λi. (Fig. 1, G and H, and supplementary text, section S3) (50). Unexpectedly, although clear short-range order could be seen, defined by a local multi-Q state, there was no observable long-range ordered state found for any of the probed experimental conditions.

To better visualize the variations of local multi-Q order, we considered reciprocal-space images obtained through fast Fourier transform (FFT) of the real-space magnetization images, which we refer to as Q-space images. A signature of the lack of long-range order and competing short-range order can be directly visualized by the smeared and broadly distributed spectral weight in various regions in Q-space (Fig. 1E). Q vectors here were derived from real-space SP-STM maps and were not directly measured as in the case of neutron diffraction (see supplementary text, section S3, for further discussion) (50). We only produced and analyzed Q-space images from within a flat terrace on individual islands (typical width, ≈150 nm). Thus, the measured broadening resulted from the spectral distribution of Q, not from averaging over different islands with different orientations. To highlight this, Q-space images were also produced from smaller spatial regions of the same image, which illustrated sharpened and characteristic spectral weight of Q vectors compared with the larger-scale image (Fig. 1G and fig. S4). Various line cuts along the Γ¯ to M¯ direction of multiple Q-space images taken from different spatial regions of Fig. 1D are illustrated in Fig. 1F. The resultant plots revealed a spatial variation in spectral weight in at least three distinct regions, or Q-pockets, with substantial spectral weight along the high-symmetry axes.

In the ensuing discussion, we focus particularly on the three Q-pockets with wave vectors at QA = 1.1 to 2.0 nm−1, QB = 2.7 to 3.5 nm−1, and QC = 4.6 to 5.3 nm−1. From this information, we plotted a schematic of the out-of-plane projected magnetization with respect to the atomic lattice (Fig. 1H) for the two regions shown in Fig. 1G with the defined Q vectors from these Q-pockets. The corresponding Q-space images are a direct visualization of the multi-Q nature acquired over the spatial area of the given images, and they provide a quantitative comparison to previous neutron diffraction studies (1821, 2426). The spectral weight around these pockets, as well as blurring of the intensity of the Q states in larger-scale images (Fig. 1E), was an initial indication of the glassy nature of the magnetic state and reminiscent of spin-based analogs of stripe or checkerboard order in strongly correlated compounds (55).

To illuminate the concept of a spin-Q glass, we qualitatively illustrate the energy landscape in Q-space images for a spin-Q glass compared with a ferromagnet (Fig. 2). A long-range ordered state can be related to a global minimum in Q-space (28, 56, 57), where a single-domain ferromagnetic state is equivalent to a Q = 0 global minimum (Fig. 2A). By contrast, a spin-Q glass is distinguished by the existence of flat valleys defined by a distribution of many local minima, i.e., Q-pockets, at finite Q values (Fig. 2B). Broad Q-pockets led to a lack of a preferentially long-range ordered state. Instead, there were local regions defined by a strong local order derived from a spectral weight of mixed Q vectors within the given pockets, and different regions exhibited random distributions of this spectral weight (see color image in Fig. 2B). The superposition of this spatially varying magnetization led to an overall broadening in the spectral weight. Note that multi-Q states for thin 3d transition metal films show well-ordered domains (58). Within the concept of self-induced glassiness for spins, such pockets may result from a strong competition of magnetic interactions, leading to highly degenerate states (15, 16).

Fig. 2 Energy landscape of a spin-Q glass.

(A) Q-space image of the energy landscape E(Q) of a prototypical ferromagnet, which exhibits a strong global minimum at Q = 0, corresponding to a real-space magnetization pattern M(r) where all spins are aligned. Further minimization leads to the formation of distinct domains (black) separated by domain walls (white), but all domains are defined by a repeating Q-state distribution, where Q = 0. (B) Characteristic Q-space image for a spin-Q glass, which can be distinguished by flat valleys (Q-pockets) at nonzero Q values, which leads to a superposition of a distribution of Q states with different periodicities residing in each pocket. This results in a complex M(r) pattern that lacks long-range order. The spatial distribution of Q states contains regions with local order defined by mixing of Q states (colors) derived from the given Q-pockets.

Theoretical analysis of the magnetic landscape of Nd(0001)

To analyze the origin of the spin-Q glass state in Nd(0001) and its unexpected magnetic patterns (59), we used ab initio calculations to quantify and understand the exchange interactions and the energy landscape in Q-space of bulk Nd, which adopts a dhcp structure that is critical for its magnetic exchange interactions (Fig. 3A). The RSPt code was used for this purpose (see the materials and methods and supplementary text, section S4) (50). The calculated exchange interactions for bulk Nd are shown for both the dhcp and a hypothetical hcp structure. There is a minute energy difference between these crystal structures that results from the large similarities in atomic arrangement.

Fig. 3 Elemental Nd electronic and magnetic landscape.

(A) Calculated Heisenberg magnetic exchange interactions among Nd spin moments with magnitude 2.454 μB in both the dhcp Nd (black) structure and the hypothetical hcp structure (gray). A negative interaction denotes a preference for an antiferromagnetic alignment among the spins, and a positive one denotes a ferromagnetic alignment. Inset, dhcp crystal structure with an ABAC stacking, where the cubic A sites are represented by red spheres and the hexagonal B and C sites by light and dark blue spheres, respectively. (B) Energy landscape for single-Q spin spirals with Q = (Qx, Qy, 2π/c) as evaluated from the calculated exchange interactions for the dhcp structure. (C) Autocorrelation function C(tw, t) = ⟨mi(t + tw) · mi(tw)⟩ for dhcp Nd at T = 1 K. (D) Comparison of Q states for SP-STM (this work, red), neutron diffraction [(21, 25), green], and simulations (this work, black).

The distance dependence of the exchange interactions in the hcp structure illustrates a prototypical behavior with ferromagnetic nearest-neighbor interactions and an oscillating Ruderman-Kittel-Kasuya-Yosida (RKKY)–like interaction at large distances, as seen with other lanthanides such as gadolinium (Gd) (60, 61). By contrast, in the dhcp, at shorter range, the interactions are much weaker and primarily antiferromagnetic, but at larger distances, an RKKY interaction sets in. These strong competing magnetic interactions in Nd create conditions for frustrated magnetism and spin glass formation, as described for self-induced glassiness (1517). Calculated values of the magnetic moment of the surface atoms, and for deeper layers, were similar to the bulk value. The bulk moment was restored within three layers beneath the surface (see supplementary text, section S4) (50).

To clarify the impact of the calculated exchange interactions in the dhcp structure of Nd, we evaluated the magnetic energy landscape by means of single-Q spin spirals, magnetic structures that can be parametrized by a single wave vector. For this purpose, we calculated the energy of helical spin spirals. Using the calculated magnetic exchange interactions from Fig. 3A, we parametrized an effective Heisenberg spin Hamiltonian from which the energy E(Q) of the single-Q spirals was then calculated (Fig. 3B). The energy-landscape exploration was performed by fixing the z-component Qz and then sweeping over Qx and Qy in the first Brillouin zone (BZ) (see supplementary text, section S6) (50).

In Fig. 3B, we present the single-Q energy landscape for all possible Qx and Qy combinations in the cell spanned by the dhcp reciprocal lattice vectors for the case when Qz = 2π/c, which corresponded to the configuration with the lowest single-Q energy. This choice of Qz corresponded to a 90° rotation of the moments in adjacent atomic layers within the dhcp unit cell. The Q-dependent energy is color-coded such that red (blue) regions correspond to spin spirals with low (high) energy. This visualization illustrates the complex energy landscape of Nd, as exemplified by the broad and flat, dark-red ring-like structure similar to the deduced landscape in (26).

Instead of a distinct, sixfold degenerate set of strong energy minima that could be expected for a spin-spiral magnet on a hexagonal lattice, the red ring structure showed that the energy barriers between global and local minima in this region of the BZ were very small. In addition, high-energy local minima, or pockets (in red), were distributed with hexagonal symmetry just inside the BZ, and another low-energy valley-like structure formed along the BZ boundary. Magnetic anisotropy was not considered here and we expect that it would bias the ring structure toward the high-symmetry directions, creating pockets akin to those seen in the experiments. Thus, Fig. 3B shows that the energy landscape of Nd has several broad Q-pockets, which supports the formation of the experimentally observed spin-Q glass structure. The presence of strongly competing interactions leading to a glass-like energy landscape in Q-space is a key manifestation of the concept of self-induced glassiness (1315).

Spin-Q glass dynamics: autocorrelation and static correlation function

In addition to the single-Q energy-landscape explorations, we also used Monte Carlo and atomistic spin dynamics (ASD) simulations to find the ground-state magnetic structure. Although not all conventional spin glasses exhibit identical relaxation dynamics, a common trait is aging, i.e., that the relaxation process slows down over time and never settles on a single equilibrium state. Aging is observed in the Edwards–Anderson model (1). To characterize the aging dynamics of dhcp Nd, we studied the two-time autocorrelation function defined as C(tw, t) = 〈mi(t + tw) · mi(tw)〉, where the brackets denote averaging over all sites of the system, mi is the atomic moment, and tw is waiting time. If a system relaxes to a fixed ground state following nonglassy dynamics, then the autocorrelation function should increase with increasing tw. Autocorrelation analysis based on ASD has previously been used to capture the multiple relaxation scales of a conventional spin glass system (10). This approach, which is typically analyzed with a mean-field approach, is well suited to characterizing the multiple time scales indicative of aging (see supplementary text, section S8) (50).

In Fig. 3C, we show the autocorrelation at logarithmically spaced tw for dhcp Nd as it relaxed from a fully disordered state at T = 1 K. The autocorrelation function decayed exponentially toward zero, a typical feature of aging and spin glass behavior (Fig. 3C) (50) and was similar to that of a traditional spin glass system such as Cu-Mn or that of the Edwards–Anderson model (see supplementary text, section S8) (10, 50). In other words, the relaxation process of Nd never arrived at a well-defined energy minimum. We note the autocorrelation analysis often assumes a mean-field picture of aging, whereas a more complete picture should include the Q-dependent contributions to the multiple time scales. These simulations were performed for bulk dhcp Nd, meaning that the observed spin-Q glassy behavior can only be a result of the exchange interactions of the system because neither surface states nor defects were present in the simulations.

In addition to mapping out the aging dynamics of the spin-Q glass state, from these simulations, we obtained real-space spin structures (see supplementary text, section S7) (50) as well as the static correlation function S(Q), which can be compared directly with the experimental Q-space images. In Fig. 3D, we present the simulated peaks of S(Q) in comparison with (i) the observed range of spectral weight from the Q-pockets QA, QB, and QC in the SP-STM experiments [applied magnetic field (Bz) = 0 for both pristine zero-field cooled samples and samples after being exposed to magnetic field] and (ii) reported neutron diffraction data (21, 25). By comparison, the SP-STM and neutron diffraction data agreed well with the simulated results in regions of lower Q values. This agreement indicated that the surface-derived measurements were strongly coupled to the bulk magnetic properties. The simulations also provide distinct local maxima of the correlation function, which so far have not been detected experimentally.

Impact of impurities on Q-state distribution

Nd films illustrate characteristics of bulk Nd with the dhcp structure, including the expected surface state of Nd(0001) and the expected strain-free lattice constant (50). Although we did not observe bulk screw dislocations and large-scale defects for the island samples, the surface did show the presence of surface impurities. The presence of the interface did not seem to influence the imaged magnetism, e.g., underlying substrate step edges (see supplementary text, section S2) (50). Before discussing the role of the impurities on the magnetic order, we note that the main concentration of impurities in the source material is oxygen and carbon, each about 2000 parts per million by atom (table S1) (50). The impurity density of our surfaces is ~0.01 ML, which is below the detection limit of any surface-averaging spectroscopic technique and among the best reported for lanthanide metal surfaces (53, 62). Of these impurities, we observed the unperturbed intrinsic width of the surface state (54). Because the quality of the surface state (i.e., its intensity and width) in spatially averaging photoemission has been demonstrated to illustrate the structural order as well as cleanliness of the surface (31, 32, 52), we conclude that the surface defects in the present limit have a negligible impact on the overall electronic structure.

In Fig. 4, we show a comparison between two separately prepared samples and their respective FFTs, for which the defect density is sufficiently different. For samples with higher defect densities, imaged at T = 1.3 K, we observed more well-defined Q vectors at higher overall Q values, with a narrower distribution in Q-space compared with samples where the defect density was lower (e.g., Fig. 1). Although local order in this case had a well-defined periodicity, the orientation was not perfectly aligned, leading to a small, angular distribution of Q states. In stark contrast, as the defect density was reduced, the distribution of Q states strongly broadened. Thus, with cleaner samples, there was a stronger manifestation of multiple energy minima as more Q states stabilized. This observation contrasts with typical magnetically ordered systems, where any colinear or noncolinear configuration narrows its distribution in reciprocal space with cleaner samples. In this context, extended line defects (screw and edge dislocations) in lanthanide films were shown to pin magnetic order, whereas local point defects such as atomic adsorbates did not (51). As shown in figs. S1 and S3 (50), thick closed Nd films exhibit line defects as well as higher surface defect densities than the typical islands shown in Fig. 1. The unavoidable presence of dislocations and the higher amount of impurities in bulk single crystals may lead to defect-induced pinning of Q vectors, exemplifying the importance of cleanliness to observe the glassy behavior. However, for the ~100 ML–thick closed films, we observed qualitatively the same magnetic multi-Q structure as the islands (fig. S3) (50). Therefore, despite their seemingly small volume, the islands reflect the magnetic properties of bulk Nd. We conclude that below a critical defect density, a multiwell landscape emerged as a spin-Q glass phase, as illustrated in the defect-free calculations of both the energy landscape and the autocorrelation function (Fig. 3C).

Fig. 4 Effect of defects on the spin-Q glass state.

Magnetization and the corresponding Q-space images of a dirty surface (surface defect concentration of 0.03 ML) (A and B) and a clean surface (<0.01 ML) (C and D) (scale bar, 20 nm, It = 200 pA for magnetization images; scale bar, 5 nm−1 for Q-space images, inverted grayscale). A higher amount of contamination results in pinning of the Q state around the defects, as illustrated in real-space magnetization images. Overall, measurements of several samples showed that the Q-state distribution is essentially unaltered for concentrations ≤0.014 ML. The Q-space image of the dirty sample shows more well-defined Q-pockets caused by the pinning, in contrast to the smeared-out Q-pockets along the high-symmetry axes in the clean sample.

Magnetic field evolution of the spin-Q glass state

We experimentally characterized the response of the magnetization to external magnetic fields. Out-of-plane field dependence is illustrated in Fig. 5 for a few chosen fields up to Bz = 7 T at T = 1.3 K (see supplementary text, section S9) (50). The application of magnetic field should favor states with smaller Q, eventually leading to a preferential Q state near the zone center (56, 63), when the Zeeman energy exceeds the local exchange energy. However, magnetic imaging of a given area at variable magnetic field revealed a different picture. At increasing magnetic fields, no distinct Q state became favorable, with the spectral weight strongly broadening along high-symmetry directions toward higher Q (Fig. 5G). The magnetic order was sensitive to fields on the order of Bz = 0.5 T, illustrating the degeneracy driven by the Q-pockets, whereas at the highest applied fields (Bz = 7 T), there was no distinct Q state, demonstrating the strong local exchange energy.

Fig. 5 Magnetic field evolution of the spin-Q glass state.

Shown are magnetization and the corresponding Q-space images of the same area measured at T = 1.3 K in Bz = 0 T (A and B), in Bz = 4 T (C and D), and in Bz = 7 T (E and F) (scale bar, 30 nm, It = 200 pA for magnetization images; scale bar, 4 nm−1 for Q-space images, inverted grayscale). Increasing magnetic field does not favor a low Q state nor exhibit the favorability of any Q state, as shown by the broadening of the spectral weight in the various pockets. (G) Line cuts along Γ¯ to M¯ of Q-space images (fig. S11) with finer intervals of increasing Bz. The spectral distribution becomes smeared out within the given pockets, leading to no well-defined long-range periodicity at the highest Bz. The surface defect concentration is 0.01 ML.

The application of in-plane fields revealed similar behavior: Favorable Q states collapsed onto an axis related to the direction of the applied field, but with appreciable smearing of the spectral weight along one particular axis that is reminiscent of the so-called archipelago phase seen in neutron scattering (2224) (fig. S13) (50). The field-dependent behavior seen here cannot be attributed to the picture of local domains because there were neither clear domain boundaries that could be traced in the magnetic field nor a repeating or favorable Q structure. In addition, magnetostriction effects should saturate beyond 1 T at the measured temperature, as previously reported (23).

Aging of the spin-Q glass state through the magnetic field

Although field-dependent imaging illustrated the degeneracy within the Q-pockets, the distinguishing property of a spin glass is the observation of aging. Aging, in a mean-field picture, can be described by the existence of multiple relaxation time scales, as exemplified in Fig. 3C, leading to a magnetization state that never fully relaxes (10). To monitor the evolution of the ground state, we repeatedly (i = 1, … , n) applied the following procedure, starting from the pristine (i = 0) state: (i) sweep up the magnetic field to Bz,i (maximum, 7 T) and stay at that value for a finite time τi, (ii) reduce the magnetic field to zero, and (iii) image the ith zero-field state of the same area as the pristine state. Comparing several field-dependent cycles, the zero-field magnetic state at T = 1.3 K illustrates sequential redistributions of the spectral weight within and between the various Q-pockets (Fig. 6).

Fig. 6 Aging and glassy behavior of the spin-Q glass state.

(A and B) Magnetization and the corresponding Q-space images of the surface at T = 1.3 K in its pristine state at B = 0 T. The same measurements were performed at the exact same area in B = 0 T after subsequent magnetic field sweeps and intermittent probing, leading to the magnetization and Q-space images after Bz,1 = +4 T, τ1 = 105 s (C and D), after Bz,2 = –4 T, τ2 = 105 s (E and F), and after Bz,3 = +7 T, τ3 = 105 s (G and H). Typical sweep rates were on the order of 225 mT/min. The sequence shows that the system never reverts to the initial zero-field cooled state (scale bar, 30 nm, It = 200 pA for magnetization images; scale bar, 4 nm−1 for Q-space images, inverted grayscale). The surface defect concentration is 0.01 ML.

Consistent with aging, the magnetic state did not relax to a given ground state after perturbation, nor was there a clear tendency toward a different but particular ground state distribution. Instead, the overall trend was a redistribution between all pockets toward a broader overall distribution along the high-symmetry directions, with the angular distribution sharpening along the high-symmetry axes. Moreover, the system never reverted to the initial zero-field cooled state. These findings ruled out that field-dependent cycling can be understood as the evolution of a favorable domain. To rule out hysteretic effects, we also performed field sweeps at positive and negative fields, but there was no clear correlation between such subsequent conditions, nor was there any correlation with local defects on the surface. Similar aging effects, and magnetic field evolution, were seen at different temperatures (fig. S15) (50), as we detail below, as well as with the application of an in-plane field and near island edges (fig. S14 and supplementary text, sections S10 and S11) (50).

Multiple and Q-dependent relaxation times at elevated temperature

The evolution of intermittent spectral weight distributions, which were frozen after subsequent field-sweep cycles, is a signature of slow aging dynamics and a hierarchy of states separated by small energies. By contrast, the application of temperature illustrated the presence of fast dynamics concomitant with slow relaxation dynamics. To illustrate this effect, we show the effect of temperature by imaging the same area at different temperatures (Fig. 7). Compared with T = 1.3 K (Fig. 7B), the spectral weight within the QA pockets “melted” away at T = 4.2 K (Fig. 7D). In addition, in the real-space image, the associated long-wavelength pattern was no longer visible (Fig. 7C) when compared with images of the identical area at T = 1.3 K (Fig. 7A). Our time resolution was limited to ~1 ms so that we were imaging the time-averaged magnetization. Thus, the loss of spectral intensity in the QA pockets was most likely caused by increased fluctuations of the magnetization states within the QA pockets, which were thermally activated at this temperature. Thus, the fluctuations of the magnetization led to an overall reduction of the measured time-averaged spin polarization. Nevertheless, we observed similar slow aging behavior after application of magnetic field at T = 4.2 K for the other Q-pockets as seen at lower temperatures for exactly the same sample area (Fig. 7F). This observation illustrated that there were at least two different and Q-dependent time scales present at sufficient temperature.

Fig. 7 Temperature dependence of the spin-Q glass state.

Magnetization and the corresponding Q-space images of the exact same area at T = 1.3 K (A and B) and at T = 4.2 K (C and D) in B = 0 T. Warming up the surface from 1.3 to 4.2 K results in depopulation of the QA pocket. (E and F) The same measurements were performed after a +4 T magnetic field sweep in the same area at T = 4.2 K in B = 0 T, illustrating similar out-of-plane magnetic field aging behavior (scale bar, 30 nm, It = 200 pA for magnetization images; scale bar, 3 nm−1 for Q-space images, inverted grayscale). The surface defect concentration is 0.01 ML.

Although it is unclear how to describe the Q-dependent dynamical behavior of the magnetization, the presence of multiple relaxation times is an indication of glassy behavior in a mean-field picture, analogous to the aging effects seen in metallic alloys (3, 10). There may be some ruggedness to the energy barriers, symbolic of a multiwell landscape, resulting from strong local order in this system. In structural glasses, local order leads to a more complex picture of the dynamics, referred to as dynamic heterogeneity (64). In this picture, there can be different spatial regions exhibiting substantially different dynamics related to the local correlations. The existence of different and Q-dependent time scales may be evidence for dynamic heterogeneity in Nd. Therefore, it is not entirely clear how many various time scales can be seen in this system and at what temperatures compared with conventional spin glasses, which have more flat energy landscapes. Combined with the observations that lower defect densities led to a smearing of the Q states, these results exemplify the complex aging behavior related to the multiwell energy landscape illustrated by theory in Fig. 3. The application of higher temperature led to more well-defined Q states (e.g., for T = 7 K); by comparison, the same area imaged at T = 40 mK showed glassy distributions similar to the previously shown data taken at T = 1.3 K (see supplementary text, section S11) (50). This result rules out transitions into various long-range ordered multi-Q states (21, 25) with decreasing temperature and instead exemplifies the emergence of a spin-Q glass state.

Discussion

We have demonstrated that the magnetic ground state of elemental Nd is a spin-Q glass. We have observed aging, i.e., glassiness, in an elemental magnetic solid with minimal amounts of chemical or extrinsic disorder, which is direct experimental confirmation of self-induced glassiness (15). Moreover, the coexistence of short-range order exhibited by multi-Q pockets along with aging behavior in a material without extrinsic disorder cannot be captured by traditional mean-field theoretical descriptions of spin glasses (1, 2). The self-induced glassy behavior observed here is a departure from the traditional magnetic alloys, where disorder drives the glassy dynamics and, likewise, many intermediate time scales can be observed. The energy landscape of Nd may have some amount of ruggedness resulting from the strong local Q order (17), providing a new material system with which to study dynamic heterogeneity in a spin glass material (64). With the advent of techniques that can probe picosecond dynamics with STM (65), it may be interesting to develop magnetically sensitive, time-resolved methods that can resolve the picosecond dynamics in Nd, which can then be compared directly with ASD simulations. It remains to be understood what is particularly special about the interplay between crystal structure and electronic properties in Nd that leads to this exotic behavior, necessitating a deeper theoretical understanding of the role of electron correlation effects, the possible influence of the surface, as well as the interplay between spin and orbital degrees of freedom.

The conclusions drawn here resolve the long-standing debate about the magnetic state of Nd in that the various reported multi-Q transitions as a function of temperature can be understood within a picture of multiple minima in Q-space with different depths that can be thermally activated. Likewise, the establishment of spin-Q glass order raises the question of the dynamical behavior of spins through the variation in local relaxation times and may provide a material platform with which to explore exotic topological quasiparticles similar to fractons (6668). The example here expands our views on magnetic states of matter and invites numerous further experimental and theoretical investigations to understand the emergence of aging behavior in magnetic systems.

Materials and methods

The experimental studies were performed in two home-built, ultrahigh vacuum systems that allow for cleaning the W(110) single-crystal substrates, molecular beam epitaxy, and annealing of Nd (see supplementary text, section S1), as well as transfer into the cryogenic STM, all in situ. The first system operates at a base temperature of 1.2 K with magnetic fields up to 9 T perpendicular to the sample. The second system operates at a base temperature of 30 mK with magnetic fields up to 9 T perpendicular and 4 T parallel to the sample (69). The SP-STM measurements were performed using an antiferromagnetic Cr tip with out-of-plane spin polarization (see supplementary text, section S2). Apart from a global plane subtraction, all STM topography images are unprocessed raw data. Magnetization images were produced by subtraction of the majority and the minority SP-STM images, and corresponding Q-space images were produced by computing the FFT using MATLAB. For the latter, we again did not apply any postprocessing steps (see supplementary text, section S2).

The theoretical studies included ab initio calculations of bulk Nd as well as cubic- and hexagonal-terminated slabs with up to 13 layers. The electronic structure was calculated within density functional theory using the RSPt software (70). We used the local spin-density approximation for the exchange and correlation functional. In these calculations, we made use of the standard model of the lanthanides and treated the 4f electrons as localized, unhybridized particles with a magnetic moment according to Russel–Saunders coupling. The dispersive states were expanded by means of 6s, 6p, and 5d orbitals in a multiple-basis fashion, as described in (70). In addition, we included pseudocore 5s and 5p states to hybridize with the rest of the valence states. No approximation was made concerning the shape of the charge density or potential in a so-called full-potential description. The simulations were done using 32,768 and 1152 k-points of the full BZ for bulk dhcp and slabs, respectively (see supplementary text, section S4). The Monte Carlo and ASD calculations were performed using The Uppsala Atomic Spin Dynamics (UppASD) software (71, 72) (see supplementary text, section S5).

Supplementary Materials

science.sciencemag.org/content/368/6494/eaay6757/suppl/DC1

Supplementary Text

Figs. S1 to S16

Tables S1 and S2

References (73100)

References and Notes

  1. See the supplementary materials.
Acknowledgments: We thank P. Thunström for assistance with the spin-polarized core calculations. Funding: This work was supported by the Swedish National Infrastructure for Computing (SNIC), NWO, and VIDI project no. 680-47-534, “Manipulating the interplay between superconductivity and chiral magnetism at the single-atom level.” This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (SPINAPSE: grant no. 818399). We also acknowledge support from the Swedish Research Council (VR), the Knut and Alice Wallenberg Foundation (KAW), the Foundation for Strategic Research (SSF), Energimyndigheten, eSSENCE, and StandUP. Author contributions: U.K., A.E., and M.S. conducted the experiments. U.K., A.E., M.S., N.H, D.W., and A.A.K. analyzed the experimental data. A.A.K. and D.W. designed the experiments. D.I. performed ab initio calculations. A.B. performed the spin-dynamics simulations. M.I.K., L.N., and O.E. provided additional theoretical support on all calculations. All authors contributed to the writing of the manuscript. Competing interests: The authors declare no competing financial interests. Data and materials availability: All data needed to evaluate the conclusions of this study are available in the main text or the supplementary materials. Any additional data pertaining to this study can be made available, pending approval, by contacting the corresponding author.

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