Spin glass–like transition in the structurally disordered intermetallic compound Sm₂Au_1.1Ge_2.9

Abstract

The novel intermetallic compound Sm₂Au_1.1Ge_2.9 was synthesized using an arc-melting method. A detailed analysis of its crystallographic structure, magnetic and thermal properties has been documented. Powder X-ray diffraction analysis indicated that this compound crystallizes in a disordered variant of the AlB₂-type structure (space group P6/mmm, no. 191) with lattice parameters a = 4.2495(1) Å and c = 4.1300(1) Å. The exact nature of magnetic properties of this intermetallic compound was investigated both by AC and DC magnetization measurements including time evolution of remnant magnetization and magnetic memory effect studies. The investigation of magnetic properties was supplemented by measurements of heat capacity with and without an applied magnetic field. The new Sm₂Au_1.1Ge_2.9 compound is found to exhibit a cluster glass transition at surprisingly high temperature T_f(39 Hz) = 21.9 K. Such a high phase transition temperature makes this material a promising candidate for technical applications e.g. in magnetic memory devices.

Introduction

Spin glasses are defined as materials that exhibit random magnetism as a result of the coexistence of competing antiferromagnetic and ferromagnetic interactions¹. This group of materials has been discovered during studies of diluted magnetic alloys such as Au:Fe3% owing to an employment of the AC susceptibility measurement technique². Thes quickly became an object of interest of solid-state physics scientists due to their unusual properties e.g. a magnetic memory effect and frequency dependent shift of AC magnetization. These features make spin glasses promising candidates for innovative electronic devices, including systems to encode information³. On the other hand, spin glasses are one of the simplest types of disordered materials, which makes them a valued object of metastability studies in glassy systems. Nowadays, diluted magnetic alloys, which reveal glassy behavior are often referred to as canonical spin glasses in order to distinguish them from reported subsequently intermetallic compounds exhibiting spin glass-like features. Alongside canonical spin glasses, there are many groups of materials similar to spin glasses, one of them being cluster glasses, in which magnetic moments occur in clusters rather than as individual spins⁴.

In general, spin glass like materials exhibit slow relaxation of magnetization, and multiple degenerated ground state meaning they can retain information about their magnetic state for long periods. This property can be leveraged in high-density data storage devices. Spin glass devices are expected to be part of neural networks as content-addressable memories (CAMs) or associative memory devices. These memory devices should be capable of learning and recognizing previously learned patterns, should be robust and noise-tolerant. Due to the multiple degenerate energy states, they should also store large amounts of data. Moreover, these materials may serve in applications requiring protection from unwanted magnetic fields, such as in sensitive electronic instruments that need to be shielded from interference².

Rare earth (RE) intermetallic compounds with a nominal stoichiometry RE₂TrX₃ (Tr – transition metal, X—semimetal) are known to present diverse physical properties. For example, La₂NiGe₃⁵ and other analogues with nonmagnetic RE elements^6,7 are usually superconductors, while compounds containing f-block elements are predicted to exhibit intriguing magnetic properties. A complex magnetic behavior was observed in RE₂AgGe₃ compounds⁸ or in Gd₂AgSi₃⁹ and Pr₂NiGe₃¹⁰ with a double magnetic transition, while RE₂NiSi₃ intermetallics tend to display a large magnetocaloric effect^11–13. Recently published RE₂PdGe₃ and RE₂PtGe₃ series^14–19 reveal a glass-like spinning behavior caused by the coexistence of magnetic frustration and structural disorder. In general, RE₂TrX₃ compounds crystalize in AlB₂ derived crystal structures, which are commonly represented by a hexagonal structure (space group P6/mmm, no 191) or a tetragonal α-ThSi₂-type one (space group I4₁/amd, no. 141). Ternary RE₂TrX₃ compounds crystallizing in the hexagonal structure of the AlB₂ type (Fig. 1a, b) are composed of parallel layers of RE ions, which determine their magnetic moments²⁰. These layers are separated by graphite-like nets formed of Tr and X atoms, which are randomly distributed with probability 1:3. This structural disorder causes a variation of the local environment around RE ions, which may lead to the existence of magnetic frustration. Additionally, comparable values of lattice parameters a and c increase random exchange interactions between RE ions, which favor the formation of magnetic clusters¹¹. Moreover, it can be noted that the replacement of La³⁺ ion with other lanthanides cause a shrinkage of the unit cell and generate strains in the crystal structure¹⁹. This behavior is known as a chemical pressure effect and results from the lanthanide contraction. In case of α-ThSi₂ type structure Th positions are occupied by RE ions, while Si are replaced by a quarter of Tr atoms and three quarters of X atoms. This crystal structure can be viewed as a “twisted” AlB₂, which is shown in Fig. 1c. The relationship between different structures of the AlB₂ aristotype can be described using the Bärnighausen tree²¹.

Fig. 1.

Crystal structure of: (a, b) Sm₂Au_1.1Ge_2.9 (AlB₂-type)– big blue spheres are samarium atoms, smaller ones represent gold and germanium, which stochastically occupy hexagonal site with ratio close to 1:3; (c) Gd₂AgSi₃ (α-ThSi₂ type) – large spheres represents gadolinium atoms, while the smaller one stand for silver and silicon randomly distributed with probability 1:3 (all drawings were made employing VESTA software²²).

In this article, we report the successful synthesis of a new intermetallic compound Sm₂Au_1.1Ge_2.9, which is one of the first member of the RE₂AuGe₃ series, next to compound with Yb²³ and Ce, to crystalize in a hexagonal structure²⁴. The magnetic properties investigations established that the studied compound exhibits a cluster glass transition above T = 20 K, which makes it a promising candidate for applications in technology e.g. as magnetic devices components.

Experimental

The polycrystalline sample of Sm₂Au_1.1Ge_2.9 was synthetized under an inert (Zr—gettered pure Ar) atmosphere using an arc melting technique. Constituent amounts of a high purity chemical elements (> 99.99%) was weighted considering 10% molar excess of Sm due to a significant volatility of this metal and then melted together in an arc-furnace MAM-1 GmbH Edmund Bühler. The ingot was rotated several times and remelted to improve the homogeneity of the synthetized material. The sample was weighed after the melting process, and the observed mass losses were comparable to the weight of the excess Sm used, which can suggest that the chemical composition of the ingot is close to the nominal stoichiometry. A sample with a total weight of ~ 0.5 g was synthesized. No thermal treatment was applied to the sample after synthesis.

The sample was characterized by powder X-ray diffraction (pXRD) using a Bruker D2Phaser diffractometer equipped with an XE-T detector (CuK_α radiation). The obtained pXRD pattern was analyzed by the Rietveld profile refinement using the GSAS-II package²⁵.

Scanning electron microscopy (SEM) images and elemental compositions were acquired using an FEI Nova NanoSEM 650 equipped with an energy-dispersive spectroscopy (EDS) attachment. Imaging and excitation of the characteristic radiation of the elements were conducted at an electron energy of 30 keV. Licensed instrument software was used to analyze EDS data using the P/B ZAF (non-standard) method.

The sample density was determined using Archimedes’ method. The measurement was made using isopropyl alcohol [(CH₃)₂CHOH] and an electronic balance Radwag (Radom, Poland) XA 110.4Y.A.

The electronic structure analysis was performed using X-ray photoelectron spectroscopy (XPS) on a Physical Electronics (PHI 5700/660) spectrometer, which operated under ultra-high vacuum (10⁻⁹ Torr) conditions in a UHV cluster, employing a monochromatic Al Kα X-ray source (1486.6 eV). Initially, the sample was placed in a pre-chamber and maintained under vacuum at 10⁻⁸ Torr for a minimum of 12 h. It was then transferred to the preparation chamber and cleaved under UHV conditions (10⁻⁹ Torr) to achieve a fresh surface. The cleaved sample was subsequently moved immediately to the measurement chamber for analysis. Survey spectra were collected at a transition energy of 187.85 eV and core level lines were measured at a transition energy of 23.5 eV. The acquired spectra were processed using XPSPEAK41 software, and the deconvolution of the core level lines was carried out using the Shirley background²⁶ method along with the Gaussian–Lorentzian line shape.

Physical properties of Sm₂Au_1.1Ge_2.9 were investigated by specific heat and magnetization measurements performed on a Quantum Design Physical Property Measurement System (PPMS). A vibrating sample magnetometer (VSM) option was used to perform DC magnetic measurements for different applied magnetic field (up to 9 T) both in Zero Field Cooling mode (ZFC) as well as in Field Cooling mode (FC). The AC magnetization data were collected employing the AC- Measurement System (ACMS) in the frequency range 39 Hz—10 kHz with the magnetic field amplitude μ₀H_ac = 5 mT. Heat capacity investigations were carried out with and without an applied magnetic field by a standard thermal relaxation technique in the temperature range 1.9 K < T < 300 K.

Results and discussion

Similar to reported earlier RE₂TrX₃ compounds e.g. Tb₂Pd_1.25Ge_2.75¹⁴, Tm₂Ni_0.93Si_2.93¹² and Dy₂Pt_1.15Ge_2.85¹⁸, the fully stoichiometric Sm₂AuGe₃ reveals the significant presence of an additional phase, which cannot be removed by a thermal annealing process. In order to obtain a single phase sample, it was necessary to synthetize the intermetallic compound with a defective crystal structure by deliberately changing Au:Ge ratio. The influence of adjusting composition of graphite-like layers on RE₂TrX₃ structure stability was widely discussed in our previous articles^27,28 The best result was achieved for the stoichiometry Sm₂Au_1.1Ge_2.9.

Crystal-chemical characterization

In Fig. 2 the pXRD patterns for Sm₂Au_1.1Ge_2.9 are shown, which can be well indexed by AlB₂ derived structure with space group P6/mmm. In addition to the peaks associated with the main phase, we also observed peaks originating from trace amounts of the impurity phases identified as Sm (0.1165(7) wt. %), Sm₂Au (0.415(2) wt. %) and Sm₂AuGe₆ (1.9(2) wt. %). More details are presented in Fig. 2b. However, based on the low intensity of the reflections associated with these phases, it can be concluded that they are present only as minor constituents. It should be noted that, unlike in many other RE₂TrX₃ compounds, the (00 l) reflection in Sm₂Au_1.1Ge_2.9 is not significantly broadened compared to other reflections. This indicates the absence of pronounced anisotropic broadening or stacking faults in this material. Such broadening effects have previously been reported in related compounds such as Gd₂Pt_1.1Ge_2.9²⁸, Nd₂PtGe₃¹⁵ and Ce₂PdGe₃²⁹. The calculated values of lattice parameters are equal a = 4.2495(1) Å and c = 4.1300(1) Å (more crystallographic data are gathered in Table 1) implying that a c/a ratio is about 1. Sm₂Au_1.1Ge_2.9 crystallizes in a disordered variant of the AlB₂-type structure as confirmed by a lack of superstructure reflections, which should be observed in case of an ordered variant represented by Ca₂PdGe₃¹⁷. The disordered RE₂TrX₃ structure has a random distribution of Tr and X atoms at honeycomb nets, which occur with probability 1:3. Furthermore, minor differences in lattice parameters cause that strengths of interactions between the nearest neighbors (NN) and the next-nearest neighbors (NNN) are comparable, which promote magnetic frustration³⁰. The coexistence of structural disorder and magnetic frustration in Sm₂Au_1.1Ge_2.9 establish favorable conditions for a spin glass-like state formation, which was investigated by physical properties measurements. In such a scenario, when the degree of disorder is moderate, finite-size regions with correlated spins can form, giving rise to a cluster-glass state rather than a conventional spin glass. Here, magnetic moments within each cluster are aligned ferromagnetically, but the inter-cluster interactions remain random and frustrated, resulting in collective freezing at low temperatures, as observed in Sm₂Au_1.1Ge_2.9.

Fig. 2.

Rietveld refinement of powder XRD data for Sm₂Au_1.1Ge_2.9. Observed data and calculated intensity are represented by red circles and black line respectively. The difference is shown in the lower part by solid blue line. Green vertical ticks correspond to Bragg peaks for space group P6/mmm (no. 191). Inset shows enlarged pXRD patterns in the range 2θ = 28°—34°. Pink circles indicate reflections from Sm, blue squares from Sm₂Au, and orange arrows correspond to Sm₂AuGe₆.

Table 1.

Structural parameters determined for a new intermetallic compound Sm₂Au_1.1Ge_2.9

Nominal composition	Sm2Au1Ge3
Optimal composition	Sm2Au1.1Ge2.9
Experimental composition	Sm2.05(2)Au1.29(2)Ge2.66(3)
Space group	P6/mmm (No. 191)
a (Å)	4.2495(1)
c (Å)	4.1300(1)
V (Å3)	64.589(2)
Molar weight (g mol−3)	728.04
Density (g cm−3)	9.198
Measured density (g cm−3)	9.17(6)
RE (1a)	x = y = z = 0;
	Uiso = 0.0277(4) Å2
Au (2d)	x = 1/3 y = 2/3 z = 0.5;
	Uiso = 0.0266(7) Å2
	Occ. = 0.275
Ge (2d)	x = 1/3 y = 2/3 z = 0.5;
	Uiso = 0.0266(7) Å2
	Occ. = 0.725
Figures of merit:
R (%)	11.4
wR (%)	15.1
GOF	1.34

The EDS technique was employed to measure the actual chemical composition of the resulting sample. The average chemical composition, determined by analyzing multiple locations on the surface (Fig. 3a), is Sm_2.05(2)Au_1.29(2)Ge_2.66(3). This value aligns with both the nominal composition Sm₂AuGe₃ and that derived from pXRD analysis Sm₂Au_1.1Ge_2.9. EDS mapping showed that all elements are mostly uniformly distributed on the sample surface (Fig. 3b–d). However, we observed small Ge-rich precipitates (it can be Sm₂AuGe₆ phase identified in the pXRD pattern). Nonetheless, these are not expected to exert a substantial impact on the physical properties we assessed. EDS mapping revealed that all elements are predominantly uniformly distributed across the sample surface (Fig. 3b–d). Small Ge-rich precipitates were also detected, which may originate from Ge oxides—consistent with the signal observed in the XPS scans. Alternatively, they could be associated with the Sm₂AuGe₆ phase identified in the pXRD patterns. In either case, their presence is not expected to significantly influence the physical properties investigated.

Fig. 3.

SEM image (a) is accompanied by respective EDS maps depicting individual elements of Sm (b), Au (c), and Ge (d) for Sm₂Au_1.1Ge_2.9.

Samarium (Sm), like cerium (Ce), europium (Eu) and ytterbium (Yb), can exist in two valence states, unlike other rare earth elements that typically only exist in the 3 + state. The valence state of a rare earth metal ion significantly influences the physical properties of the chemical compound in which it is found. In some cases, the valence of a rare earth ion may be mixed or fluctuating, leading to unusual behavior in various physical properties. One method to determine the valence state is X-ray photoelectron spectroscopy (XPS). Consequently, we conducted XPS measurements on a sample of the compound Sm₂Au_1.1Ge_2.9 to ascertain the valence state of Sm in this compound. The results of the XPS measurements are collected in Fig. 4. XPS spectrum obtained over a wide range of binding energies (Fig. 4a) revealed peaks corresponding to the elements present in the Sm₂Au_1.1Ge_2.9 compound, along with peaks from carbon and oxygen that stem from atmospheric components in the measurement chamber. All observed peaks were analyzed and described. Near the Fermi level E_F, the XPS spectrum is characterized by a broad feature centered around 5 eV, which includes peaks from Sm 4f., Au 5d, and Ge 4p states. The absence of a distinct peak at E_F suggests that the predominant valence state of Sm is likely 3 +³¹. On the other hand, shoulder structure close to E_F may indicate a small contribution from Sm 2 + states. A similar spectral shape is also observed in other Sm-based compounds³².

Fig. 4.

XPS spectra for Sm₂Au_1.1Ge_2.9: (a) spectrum in a wide binding energy range. All core levels have been identified, (b) valence band spectrum, (c) Sm 3d spectrum, (d) Au 4f. spectrum, (e) Ge 2p spectrum.

The spectra for the Au 4f. states reveal two distinct peaks at binding energies of 84.4(1) eV and 88.1(1) eV. These peaks result from spin–orbit (SO) interaction, which in this case have a strength of of 3.7 eV. Figure 4d illustrates the deconvolution of the Au 4f. peaks. A similar pattern is observed in the Ge 2p states, where the spectrum primarily shows two peaks located at binding energies of 1216.7(1) eV and 1247.9(1) eV. The SO interaction in this instance is 31.2 eV. The spectrum also shows two distinct peaks at 1250.5(1) eV and 1219.4(1) eV, which are likely associated with Ge oxides. Additionally, a broad peak centered at 1232.4(1) eV related to Ge losses is observed between these two pair of peaks. The positions of the main Au and Ge peaks agree very well with literature values for the pure elements³³ confirming that both elements are present predominantly in the Au(0) and Ge(0) states.

Similar for other Sm-based materials, XPS spectrum for 3d Sm states shows two peaks located at energies 1109 eV and 1082 eV formed by SO splitting of Sm³⁺ states (SO splitting is ~ 27 eV). The XPS spectrum does not show peaks attributable to Sm²⁺ states, which should appear below peaks derived from Sm³⁺ states. Based on the XPS measurements, it can be concluded that the valence state of Sm in the Sm₂Au_1.1Ge_2.9 compound is 3 + . The XPS results obtained for Sm₂Au_1.1Ge_2.9 are very similar to those obtained for the Ni sample (Sm₂Ni_0.87Si_2.87), where the authors also observed peaks associated only with Sm³⁺ states³⁴.

Magnetic properties

In order to verify the hypothesis about Sm valence state DC magnetization, measurements were carried out. Figure 5 presents the inverse of molar magnetic susceptibility (1/χ = μ₀H/M) for Sm₂Au_1.1Ge_2.9 as a function of temperature in a range T = 2—300 K recorded for μ₀H = 0.1 T. It is clearly visible that 1/χ(T) curve has a distinctly nonlinear character. This phenomenon is a common feature of samarium containing compounds^19,35–37, which origins from a relatively low energy of the excited state J = 7/2 above the ground state J = 5/2 for Sm³⁺ ion. In consequence the inverse susceptibility curve cannot be well described by a simple Curie–Weiss (CW) law, which should be substituted by the following formula³⁸:

1

where μ_B, N_A, θ_CW, and k_B are the Bohr magneton, the Avogadro number, the paramagnetic CW temperature, and the Boltzmann constant, respectively. The energy difference between the ground and excited multiplet is equal ΔJ = 1494.9 K³⁷ and the Van Vleck constant is represented by α_J. Regarding high temperature region, the above equation can be rewritten as a formula:

2

which has four fitting parameters: θ_CW, α_J, the Curie constant for ⁶H_5/2 state (C_5/2) and ⁶H_7/2 state (C_7/2). The calculated values of these parameters, which are listed in Table 2, may be useful to estimate the effective magnetic moment applying the expression:

3

which leads to μ_eff equal 0.71 μ_B and 2.94 μ_B for ⁶H_5/2 state and ⁶H_7/2 state respectively. These experimental results are close to the theoretical data for Sm³⁺ (μ_theo(⁶H_5/2) = 0.84 μ_B and μ_theo(⁶H_7/2) = 2.58 μ_B) calculated from Hund’s rule³⁹, suggesting a trivalent nature of samarium in Sm₂Au_1.1Ge_2.9. Moreover, the fit to the experimental data yields the positive value of θ_CW suggesting a ferromagnetic ground state.

Fig. 5.

The temperature dependence of the inverse magnetic susceptibility for Sm₂Au_1.1Ge_2.9. The red line is fit to modified Curie – Weiss law (see text for details). Inset (a): Temperature dependence of DC magnetic susceptibility (χ = M/μ₀H) for different magnetic fields in zero-field-cooled (ZFC) and field-cooled (FC) mode. Inset (b): The dependence of the irreversibility temperature (T_irr) for Sm₂Au_1.1Ge_2.9 as a function of applied field μ₀H.

Table 2.

Selected physical parameters value for Sm₂Au_1.1Ge_2.9.

Parameter	Sm2Au1.1Ge2.9
Tirr(0) (K)	21.4
Tf(39 Hz) (K)	21.9
δTf (-)	0.019
Ea/kB (K)	58.4(4)
T0 (K)	18.5(3)
zυ’ (-)	10.2(4)
S (emu/g)	2.32(4) × 10–4
β (-)	0.66(2)
ξ(2500 s, 10 K) (Å)	8733(6)
θCW (K)	21.89(2)
μeff (μB)	0.71 (C5/2) / 2.94 (C7/2)
ϕ	1.1

The exact nature of magnetism in the described intermetallic compound was investigated by the χ(T) measurements in a low temperature region both in ZFC and FC mode for different applied magnetic fields. The gathered results are depicted in the inset (a) of Fig. 5, which shows an irreversibility between ZFC and FC curves near T = 20 K. The ZFC branch has evidence maximum at the thermal magnetic irreversibility temperature (T_irr), and below this point the magnetic susceptibility continuously decreases with a temperature drop. Meanwhile the FC line is increasing with a decreasing temperature, which resembles rather ferromagnetic behavior instead reaching almost constant value expected for canonical spin glasses⁴⁰. Moreover, the maximum value of χ_ZFC is about one order of magnitude larger than observed in a simple cluster glass Sm₂PdGe₃¹⁹, which may be a result of ferromagnetic clusters formation. These phenomena are characteristic features of a class of materials known as ferromagnetic spin glasses, which is represented by some members of RE₂TrX₃ family e.g. U₂IrSi₃⁴¹_, Eu₂AuGe₃⁴² and Ho₂PtSi₃⁴³. Furthermore, the mentioned bifurcation tends to shift to lower temperatures with an increasing value of an applied magnetic field, which is clearly visible in the inset (b) of Fig. 6. The irreversibility temperature for each of applied magnetic fields was determined as a dχ_ZFC (T)/dT = 0 and for μ₀H = 0.01 T its value is equal T_irr = 21.3 K. Cluster glasses usually obey de Almeida–Thouless dependence, which describes the dependence between T_irr and an applied magnetic field H and may be expressed by an equation⁴⁴:

4

where T_irr(0) ≈ 21.4 K is an irreversibility temperature in an absence of the external magnetic field and A is a constant. Following a mean field approximation for spin glasses κ = 2/3, while cluster glasses are characterized by a lower value of this parameter⁴⁵. In the studied case the T_irr(μ₀H) plot is almost linear, which is another evidence of ferromagnetic clusters formation². The occurrence of a glassy-like state is usually in line with an existence of a magnetic frustration, which may be described by an empirical criterion proposed by Ramirez: ϕ =|θ_cw|/T_irr³⁰. For Sm₂Au_1.1Ge_2.9 this parameter is close to 1, which is rather expected for compounds with long range ordering than for spin glass—like materials. However, this phenomenon was earlier observed for other ferromagnetic cluster glasses e.g. PrRhSn₃⁴⁶ and probably suggests a tendency to a magnetic ordering at the ground state. The further investigation of magnetic properties of Sm₂Au_1.1Ge_2.9 included the isothermal magnetization measurements carried out at different temperatures.

Fig. 6.

The isothermal magnetization as a function of an applied magnetic field at different temperatures for Sm₂Au_1.1Ge_2.9.

Figure 6 presents the magnetization dependence M_DC(μ₀H) measured across a temperature range of 15 K to 300 K. At higher temperatures, the M_DC(μ₀H) curves exhibit an approximately linear behavior, consistent with the expected response of paramagnetic materials in the Curie–Weiss regime. Notably, the magnetization curve at T = 30 K does not show saturation, even under high magnetic fields, and displays an “S”-shaped profile characteristic of a Brillouin-like response. Such behavior has been previously reported for other members of the RE₂TrX₃ family^11,14,18,19, and is indicative of spin glass–like magnetic behavior. At T = 15 K, the emergence of a small hysteresis loop was observed, which may be attributed to the presence of a secondary magnetic impurity phase. However, similar phenomena have also been reported for other compounds within the RE₂TrGe₃ series^27,47, suggesting this feature may be intrinsic to the structural family.

The appropriate classification of the magnetic transition at T_irr requires also AC magnetization studies, which results are depicted in Fig. 7. The clearly visible peak at M’(T) appears close to T_irr and strongly depends on the applied AC frequency, which is a one of characteristic features of spin glass-like materials¹. The increase of υ cause a shift of the observed maximum to higher temperatures and its value at the lowest measured frequency is usually employed to determine a freezing temperature T_f = 21.9 K. The relative shift of this parameter may be described by a criterion proposed by Mydosh⁴:

5

Fig. 7.

Temperature dependence of the real part of the AC magnetization M’(T) for Sm₂Au_1.1Ge_2.9. The inset (I) shows ln(τ) plotted as a function of ln(t_red) with the solid red line, which represents the fit to the power-law divergence. The inset (II) shows plot of the freezing temperature (T_f) versus 1/ln(τ₀υ) with a Vogel-Fulcher law fit (blue solid line).

The calculated value of δT_f = 0.019 is an order of magnitude larger than expected for canonical spin glasses (~ 10^–3), although it is in range typical for cluster glass materials for example CeCu₄Mn_0.9Al_0.1—δT_f = 0.0177(1)⁴⁸, Er₂NiSi₃—δT_f = 0.02¹¹, and Sm₂PdGe₃—δT_f = 0.016¹⁹. The AC measurements data can be further analyzed applying a dynamical scaling theory, which may be expressed by a power law dependence⁴⁹:

6

where τ ~ 1/υ is the relaxation time associated with a measurement frequency υ and zυ’ is a dynamic critical exponent, which is related to a correlation length (ξ = (T_f /T_SG − 1)^{− υ’}, τ ∼ ξ^z)¹¹. The parameter τ₀ is a characteristic flipping time of the magnetic entities and vary from τ ₀ = 10^–7 s for cluster glasses to τ ₀ = 10^–13 s for canonical spin glasses¹. The spin glass temperature (T_SG) is defined as a temperature in the static limit (υ → 0), however for simplicity is usually equated with T_irr. The fit of the above formula to depicted in the inset I of Fig. 7 log(τ) versus log(t_red) plot (t_red is a reduced temperature and may be established as t_red = (T_f—T_SG)/T_SG) lead to values of zυ’ and τ₀ estimated as 10.2(4) and 10^–9 s respectively, which indicate the cluster glass transition in Sm₂Au_1.1Ge_2.9. Another possibility to explain the spin glass-like phenomenon is regarding T_f as a temperature of a thermal activation of interactions between magnetic spins, which may be denoted by an empirical Vogel-Fulcher (VF) law⁵⁰:

7

where E_a is an activation energy and T₀ means the measure of interaction strength between magnetic spins, which is known as a VF temperature. This equation can be also referred as a dependence between T_f and υ:

8

which is useful to determine E_a and T₀ from a linear fit to T_f vs. − 1/ln(τ₀υ) plot (the inset II of Fig. 7). Obtained values of these parameters are gathered in Table 2. The E_ak_B/T₀ ratio is much greater than one, which suggests a weak coupling between magnetic entities and confirm the hypothesis about the cluster glass formation⁴⁹. Finally, this assumption is in accordance with a Tholence criterion⁵⁰: δT_Th = (T_f – T₀)/ T_f ≈ 0.16, which gives a similar result to previously reported RE₂TrX₃ compounds e.g. δT_Th = 0.12 for Nd₂PtGe₃¹⁵, δT_Th = 0.32 for Ho₂Pd_1.3Ge_2.7²⁹ and δT_Th = 0.10(2) for Ce₂PdGe₃⁵¹.

In order to avoid any ambiguity in interpretation of the spin-glass phenomena, the time dependent DC magnetization for Sm₂Au_1.1Ge_2.9 was measured in ZFC as well as in FC mode, which were described in details in previously published articles^11,43,52. The collected data are presented in Fig. 8, which appears that the studied compound exhibits a time evolution of magnetization known as an aging effect^1,2. This occurrence may be explained by a nonequilibrium dynamics below T_f which arise from a magnetization reorientation. The magnetic relaxation behavior in spin glass—like compounds obeys a stretched exponential law expressed by a function⁴⁹:

9

where M₀ corresponds to an initial magnetization defined as magnetization at t = 0, M_g describes a glassy component of the magnetization and τ is a characteristic relaxation time. The last of fitting parameters – β signifies a stretching exponent, which for a different glassy systems lies in the range 0 < β < 1, where β = 0 imply a lack of relaxation⁵³. The alternative method to describe the relaxation phenomenon is using a logarithmic dependence⁵²:

10

where S is the magnetic viscosity. The reference time t₀ has only limited physical relevance and is usually orders of magnitude larger than τ₀⁵⁰. All of these parameters depend on measuring conditions, especially from a temperature and an applied magnetic field. The values of fitting parameters estimated using both of mentioned above equations are listed in Table 2 and correspond to results reported for other cluster glasses from RE₂TrX₃series^{11,14,15,52,54},. The mismatch between the initial magnetization calculated from the first equation (M₀ = 0.01479(9) emu/g) and the second one (M₀ = 0.0028(2) emu/g) is a consequence of a magnetic memory effect, which is further discussed. The inset of Fig. 8 shows a time dependence of magnetization at different temperatures. It is obvious that the ageing effect occurs only below T_f and disappears at higher temperatures. This phenomenon confirms our assumption about spin glass—like transition in Sm₂Au_1.1Ge_2.9.

Fig. 8.

Time dependent remnant magnetization for Sm₂Au_1.1Ge_2.9 measured in ZFC and FC mode at T = 10 K. Red line represents a fit to the equation M(t) = M(0) + Sln(t/t₀ + 1) and a blue one exhibits a fit of the stretched exponential law: M = M₀ – M_gexp(-(t/τ)^β). The inset exhibit time dependent remnant magnetization behavior for Sm₂Au_1.1Ge_2.9 measured at ZFC mode at different temperatures.

The non-equilibrium dynamics in spin glass—like materials can be also observed in a time evolution of the magnetic viscosity⁵⁵, which is shown in Fig. 9. The time dependent magnetic viscosity can be described by the response function proposed by Joh⁵⁶:

11

where M_FC means the thermoremanent magnetization in a function of time after switching off an external magnetic field H. The curve S(t) for T = 10 K presented in Fig. 9 exhibits a maximum, which can be employed to estimate the effective waiting time . This parameter is usually larger than t_w resulting from the cooling protocol⁵⁵. The average cluster size is usually described by a quantity called the coherence length (ξ), which can be calculated by knowing the parameters and t_w. For this purpose, should be used the formula⁵⁶:

12

where E_z is a Zeeman energy, which can be interpreted as a magnetic energy associated with a change in magnetic field. The Zeeman energy is directly correlated with a volume within which the spins (N_s) are effectively locked together during barrier hopping by an equation⁵⁷:

13

where χ_FC is the spin glass susceptibility per spin measured in the FC conditions. The parameter N_s stands for the number of correlated spin units that flip together within a volume ξ³, thus the correlation length can be described as:

14

Fig. 9.

The relaxation rate S(t) for Sm₂Au_1.1Ge_2.9 (discussed in the text) at T = 10 K and in a magnetic field of 0.01 T after waiting for t_w = 2500 s in zero applied magnetic field.

In case of Sm₂Au_1.1Ge_2.9 the calculated value of ξ was equal 8733(6) Å, which is about an order of magnitude larger than typical values for spin glasses^57,58. This result is in a good agreement with the hypothesis about a cluster glass transition in the investigated sample.

Another intriguing property of spin glass—like materials is a magnetic memory effect, which is also a result of non-equilibrium dynamics occurring at different glassy systems. This phenomenon may be observed employing a procedure proposed by Sun et al.⁵⁹. The sample of Sm₂Au_1.1Ge_2.9 was cooled down from a paramagnetic region to the lowest attainable temperature (T = 2 K for PPMS device) in the presence of an external magnetic field μ₀H = 0.01 T. The cooling process was stopped at chosen temperatures below T_f (T_stop = 15 K, 10 K and 5 K) in which the sample was maintained at zero field conditions for a waiting time t_w = 1 h. Then the same magnetic field was reapplied, and the cooling was resumed. After attaining T = 2 K, the M(T) dependence was measured during heating with the same applied magnetic field. Figure 10 presents data collected employing the mentioned above procedure. This plot clearly indicates that the magnetic memory effect can be found in the sample investigated. The observed peculiarities of M_mem curve are similar to those revealed for various spin glass-like materials^11,43,52,60 and are in line with a presumption about the spin glass—like state formation.

Fig. 10.

(a) Memory effect of Sm₂Au_1.1Ge_2.9 measured in FC protocol The M_stop denotes the results obtained in the cooling process, the M_mem denotes the results obtained in the heating process and the M_ref denotes the reference results obtained in the standard FC process. (b) The FC protocol used to obtain memory effect.

Specific heat

The heat capacity measurements were carried out in order to confirm the volume character of observed in Sm₂Au_1.1Ge_2.9 magnetic transitions. Figure 11a shows the temperature dependence of the specific heat gathered in the absence of an external magnetic field. It should be noted that the C_p(T) plot reaches saturation at room temperature slightly above the value calculated from the Dulong-Petit limit: 3nR ≈ 150 J mol⁻¹ K⁻¹, where n is the number of atoms per formula unit (n = 6) and R is the gas constant (R = 8.314 J mol⁻¹ K⁻¹). This behavior may originate from an electron excitation of Sm ions or be related to the anharmonicity effect resulting from structural disorder⁶¹.

Fig. 11.

Specific heat (C_p) of Sm₂Au_1.1Ge_2.9. (a) Temperature dependence of (C_p) vs. T. Dashed line represents Dulong-Petit limit. The insets shore presents T in the low temperature range measured in zero and finite magnetic field μ₀H. T_f—freezing temperature from magnetic measurements, T_max—maximum in temperature dependence of specific heat. (b) Plot of C_p/T vs. T² at low temperatures. The solid line represents fit to Debye model (Eq. 15).

The C_p(T) dependence does not have a sharp maximum typical for materials with long-range magnetic ordering. The absence of such an anomaly is typical for spin glass-type systems¹. At temperatures higher than T_f determined from magnetic measurements, a broad shoulder is seen. In addition, the magnetic field has little effect on the observed anomaly. Furthermore, the inset of Fig. 11a reveals a small kink around 30 K, which is most likely associated with trace amounts of secondary phases, such as Sm₂Au or Sm₂AuGe₆.

At T ≈ 4 K, a small anomaly was observed, which most likely comes from an impurity phase. Taking into account the results of pXRD and SEM/EDS measurements, it can be assumed that this phase transition is related to the presence of Sm₂AuGe₆.

The range of lowest temperatures (Fig. 11b) can be described by the simplified Debye equation⁶²:

15

where γ is electronic specific heat coefficient and Θ_D is the Debye temperature. Fitting carried out in the range from 5 to 12 K yielded the following values of the parameters γ = 192(1) mJ mol⁻¹ K⁻¹ and Θ_D = 159.0(1) K. The value of the γ is significantly higher (by two orders of magnitude) than for simple metals and their alloys. An increase in γ value can be related to various effects such as heavy-fermion-type behavior, spin fluctuations, crystal electric field, disorder, etc. The values obtained are typical for this type of material^51,63. Using the formula⁶²:

16

the electronic density of states N(E_F) at the Fermi level was also determined and is equal to 81 states/(eV atom). Measurements of specific heat and their analysis confirm that there are no signs of long-range magnetic ordering. The absence of a clear anomaly, along with the material’s low sensitivity to changes in the magnetic field and a large γ coefficient, suggests that this material behaves like a spin-glass.

Conclusions

The intermetallic compound Sm₂Au_1.1Ge_2.9 is the member of the RE₂AuGe₃ family which crystallizes in hexagonal structure (space group P6/mmm). This material was synthetized using an arc melting technique and can be obtained in single phase only by deliberately tweaking the nominal stoichiometry. The pXRD investigations indicated that the compound studied crystallize in a disordered variant of the AlB₂ derived structure, which promotes a spin glass-like state formation. Lattice parameters, which were calculated using the Rietveld refinement are equal a = 4.2495(1) Å and c = 4.1300(1) Å. Magnetic susceptibility and specific heat measurements reveal that the sample of Sm₂Au_1.1Ge_2.9 exhibits a spin glass—like ordering. It can be observed at T_f (39 Hz) = 21.9 K the transition from paramagnet to ferromagnetic cluster glass, which is confirmed both by the analysis of AC and DC susceptibility measurements. The compound exhibits the magnetic memory effect. The presented results pave the way for future research in RE₂AuGe₃ family, which is a promising candidate for investigations of novel materials with application potential in magnetic memory devices. Further research on this intermetallic compound may include measurements of its electrical properties and magnetocaloric studies.

Author contributions

L.S. Litzbarski: Conceptualization, Methodology, Validation, Investigation, Data curation, Writing—original draft, Writing—review & editing, Visualization, Supervision, Project administration, Resources, Funding acquisition. M.J. Winiarski: Conceptualization, Methodology, Validation, Investigation, Data curation, Writing—review & editing, Visualization. K. Synoradzki: Conceptualization, Methodology, Validation, Investigation, Data curation, Writing—original draft, Writing—review & editing, Visualization. P. Skokowski : Investigation, Writing—review & editing. M. Matczak: Investigation. T. Klimczuk: Writing—review & editing, Resources. A. Bajorek: investigation, Data curation. B. Andrzejewski: Writing – review & editing, Supervision.

All authors have given approval to the final version of the manuscript.

Funding

This work was supported by the National Centre of Science (Poland) under the project 2023/07/X/ST11/00345 (“MINIATURA”).

Data availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.