Introduction

The discovery of spin glass (SG) freezing in the stoichiometric compound URh2Ge21 gave an impetus to the active search and study of SGs without inherent disorder of spin structure although some systems of this type were previously known2. The key point is that the basic physical concepts assume disorder and frustration/alternation as two necessary conditions to the onset of the SG state. It turned out that in URh2Ge2 the crystallographic random-bond disorder originates from the uncontrolled interchange of Rh and Ge ions3 and the enhanced annealing leads the substance to AFM ordering4. The effect of randomness of nonmagnetic ions on the emergence of SG state was found later in some other triple intermetallic compounds5,6,7,8,9. In its turn, SG behavior in perfectly ordered lattices was revealed in two groups of systems and microscopic mechanisms of the in-site self-originated disorder were proposed to explain this phenomenon. In praseodymium intermetallic compounds (PrAu2Si210, PrIr2B211, PrRuSi312, PrRhSn313, Pr3Ir14) the disorder appears due to the ability of Pr ions to form two magnetic states: nonmagnetic singlet and magnetic doublet. The fluctuations between these states are mediated by the exchange coupling and the instability of Pr ion occurs in a curtain critical range of this parameter, which leads to freezing of the magnetic system to SG state15. Pyrochlores A2B2O7 is another group of materials where geometrical frustrations of the lattice favor to the emergence of different exotic states including the SG freezing in well-ordered crystals2,16,17. Recently the microscopic mechanism of the disorder origin in this group was proposed on the example of Y2Mo2O7. The displacements of Mo4+ ions with the corresponding formation of Mo pairs with different angles Mo–O–Mo lead to variation of exchange interactions thus preventing the long-range ordered states18.

In this work we explore the new mechanism of in-site originated disorder and the related SG behavior in the rich borides GdB6 and Gd0.73La0.27B6 with high symmetry cubic lattice (Pm3m − Oh1). The puzzling properties of GdB6 are determined by the mutual shifts of neighbor Gd3+ ions 8S7/2(L = 0, S = 7/2) from the centrally symmetrical positions in the oversized boron lattice19 (Fig. 1). At high temperatures the mean square ion displacement in GdB6 is the highest one in the series of rare earth hexaborides (< δ2 > ≈1 × 10–2 Å2 at T = 300 K)20. It results in strong short range magnetic correlations in the paramagnetic (PM) phase with concomitant shift and broadening of ESR line with decreasing temperature, the deviation from Curie–Weiss behaviour starting already below T ~ 100 K as well as with the large ratio of Curie–Weiss parameter Θ = − 66 K to Neel temperature TN = 15.5 K21. At lower temperatures T < 100 K the Gd3+ ions move in the anharmonic potential caused by their mutual magneto-elastic coupling and the interaction with carriers via the nesting of Fermi surface22. Such dynamic leads to softening of phonon modes22 inducing the growth of magnetic correlations with temperature decrease. The coherent displacement structure of Gd3+ ions and the concomitant AFM order become stabilized below Neel temperature TN = 15.5 K via first order phase transition. The resulting structure is characterized by wavevectors [1/2,0,0] and [1/2,1/2,0] with an additional reflex [1/4,1/4,1/2] developing at temperatures TN2 < 10 K where new AFM2 phase appears23,24,25,26. The onset of the structural phase transition is accompanied by the AFM ordering with the wavevector [1/4,1/4,1/2]27,28. The first order type AFM transition is seen as a jump of resistivity and magnetic susceptibility at TN29,30,31 and also as λ-anomaly in the specific heat32. The onset of AFM2 phase is not always detected in temperature dependencies of physical parameters but is manifested by hysteretic behavior of resistivity below TN233 (Fig. 1b).

Figure 1
figure 1

(a) Structure of GdB6 and schematic shift of Gd3+ ions. The concomitant shifts of boron ions are not shown. (b) Magnetic phase diagram of GdB633.

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Results and discussion

Magnetic properties of GdB6 and Gd0.73La0.27B6 in PM phase look very similar. At high temperatures magnetization of both compounds obeys to Curie–Weiss law χ ~ μeff2/(T + Θ) (insert in Fig. 2). The effective magnetic moment μeff ≈ 8.2μB derived for GdB6 is in agreement with previously published results34. The determination of μeff for the doped sample strongly depends on the exact actual value of the doping level x. In this case we assume that the effective magnetic moment of Gd3+ ion in the doped crystal is the same as in GdB6 at high temperatures thus obtaining the value x = 0.27 which appear in reasonable correlation with the nominal doping. The doping with La leads to the decrease of the Curie–Weiss parameter Θ from Θ = − 66 K to Θ = − 46 K. Note that the value of Θ in Gd0.73La0.27B6 is not sensitive to the choice of x and remains almost the same if one formally uses the nominal concentration x = 0.22. The decline of Θ to lower value well correlates with the doping level x and is likely resulted from the reduction of the average coordination number z for Gd3+ spins. The deviation of 1/χ(T) from the linear behavior caused by short-range correlations is characteristic of both compounds in the intermediate temperature range, although for GdB6 it begins at higher temperatures (see the inset in Fig. 2). The drastic difference in magnetic behavior of two samples is seen at low temperatures. The first order phase transition to the antiferromagnetic state observed in GdB6 at Neel temperature TN = 15.5 K is consistent with the previously published data29. It appears as a jump on the susceptibility curve χ(T) down on ~ 6% at TN with further gradual decrease of χ (Fig. 2). In contrast to GdB6, the magnetization of Gd0.73La0.27B6 at low temperatures depends on the sample history. The M(T) dependencies show smooth highs at Tf ≈ 10.5 K (Fig. 2) and their traces below Tf are defined by the applied magnetic field value (Fig. 3). This behavior may be a consequence of the spin glass (SG) state formation below Tf (Tf in this case is the temperature of spin ensembles freezing) although in “canonical” SGs the magnetization maximum is believed to be sharp at low magnetic fields35. In our case the peak shape becomes much sharper with the field decrease but remaining slightly round even at H = 20 Oe. However, the magnetization maximum in Gd0.73La0.27B6 can be smeared due to macroscopic inhomogeneity of distribution of La ions in the sample, and further experiments clarifying this question are necessary. The discrepancy between M(T)/H curves begins at one temperature coinciding with the susceptibility maximum thus the further magnetization decrease caused by AFM ordering may be associated with the variety of frozen spin ensembles below Tf. Another crucial criteria characterizing the emergence of SG state is the splitting of zero field cooled (ZFC) and field cooled (FC) magnetization curves. The magnetic field H = 600 Oe was applied to zero-field cooled sample Gd0.73La0.27B6 at T = 2 K, and the sample was heated after that to T > Tf (Fig. 3). The magnetization curve in this case continuously tends to the FC curve and reaches it at T = Tf . Both the above methods unambiguously testify the onset of SG state in Gd0.73La0.27B6 below Tf. It is necessary to note that Gd3+ spins concentration is much higher than the magnetic percolation limit and the long range magnetic order would be expected formally for this composition.

Figure 2
figure 2

Temperature dependencies of magnetic susceptibility of Gd0.73La0.27B6 and GdB6. (inset: inverse susceptibility).

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Figure 3
figure 3

Magnetic susceptibility of Gd0.73La0.27B6 at different fields (inset: FC and ZFC susceptibility).

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Electron spin resonance (ESR) is the method, which is very useful to investigate spin dynamic in the SG state36. It is known from several previous studies that ESR is well observable in the PM phase of GdB6 as a single absorption line, which continuously broadens with temperature decrease and its position shifts to lower fields37,38,39,40. The resonance absorption in the AFM phase was discovered only recently41: the resonance line abruptly transforms to the AFM spectrum with complicated behavior at ν > 39 GHz, while at lower frequencies the line disappears just below TN41. It is remarkable that the line width in PM phase does not diverge down to Neel temperature TN = 15.5 K. In the PM phase of Gd0.73La0.27B6 ESR line temperature behavior is similar to the case of GdB6. However, in the contrast to the parent compound the linewidth diverges at TD ≈ 9.5 K and no resonance absorption is detected below TD (Fig. 4). The analysis of the linewidth temperature dependence ΔH(T) of Gd0.73La0.27B6 demonstrates that it obeys well to the power law: ΔH(T) ~ ((T − TD)/TD)−α with TD = 9.5 K and the exponent α = 1.12 ± 0.5 (Fig. 5). It is necessary to remark that ESR linewidth becomes comparable with the resonance field at temperatures T ≤ 11 K. Thus, the considerable part of the resonance intensity lies out of the range of magnetic field. At this condition the application the line modeling procedure is no more correct. The ΔH parameters for two points in this range evaluated by integration (marked with asterisks in Fig. 4) have underestimated values and were not used in the analysis of ΔH(T) dependence.

Figure 4
figure 4

Experimental ESR spectra for Gd0.73La0.27B6 at ν = 60 GHz (open symbols) and the corresponding fits (red solid lines).

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Figure 5
figure 5

Temperature dependencies of the linewidth of GdB6 (closed circles) Gd0.73La0.27B6 (open rhombus and asterisks). For clarity power low fits are shown only for x = 0.27 by dashed lines.

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The finding of the power law of ΔH(T) dependence in Gd0.73La0.27B6 stimulates the looking for the same behavior in GdB6 by adjusting the temperature of the width divergence TD < TN. It is seen that such an analysis results in the coincidence of ΔH(T) dependencies of both compounds (Fig. 5). The parameters for the ΔH(T) power law dependence in this case are TD = 12 K and α = 1.09 ± 0.05. Critical dependencies in both systems persist in very wide temperature range: they last up to T* ~ 75 K in GdB6 and up to T* ~ 55 K in Gd0.73La0.27B6, where they go into high temperature asymptotic. It is remarkable that the tendency of the line broadening with temperature decrease takes place in both also in the range T > T* although the growth is much weaker then at lower temperatures. The ΔH(T) dependencies can be described by power law in high temperature interval as well with exponents α = 0.14 for x = 0 and α = 0.22 for x = 0.27. However it should be born in mind that such an analysis is questionable due to small temperature range of the observation and large temperature separation from TD.

Before discussing the results, it is necessary to examine some important problems regarding the interpretation of ESR experiment. First, the total linewidth may contain, apart from the “critical” term, also additional contributions, which become substantial at temperatures much higher then TD. In the pioneer work of Huber, where the power law was suggested for ESR in SGs, this contribution to the linewidth was described as a temperature independent term B42. Indeed, this type of high temperature behavior takes place in some of SGs43. However, many systems demonstrate either increase or decrease of the linewidth with temperature growth. In the first case the corresponding dependence obeys to Korringa law a + bT44 and it is often observed in metallic alloys21,45. The second dependence type is more usual for diluted semiconductors and some frustrated antiferromagnets46,47. The analysis of high temperature behavior in this case have shown that ΔH(T) dependence can be described by semi-empirical function B(1 + Θ/T) where Θ is Curie–Weiss parameter47,48. Note, that in both latter cases the magnitude of ΔH variation at high temperatures may be comparable with its change in the “critical” region. Thus the separation of these two contributions introduces additional uncertainty to the final result.

Another problem is the analysis of ESR line parameters when approaching TD. Due to strong broadening of the line and its shift to lower fields, the considerable part of spectral intensity goes beyond the measuring field range and the data can’t be correctly analyzed by a numerical modeling. The analysis of line shape near TD can be additionally disturbed due to the deformation of the line shape at the condition when the applied field becomes weaker compared with the random local fields. Then the shape becomes no longer simple, neither Lorentzian nor Gaussian49. In this regard, it is necessary to emphasize the importance of increasing the measurement frequency, which allows to expand proportionally the range of observation of the critical dependence.

In the current study the area of critical behavior begins already at temperatures T*/TD ~ 6–7 where the changeover of the dominant spin relaxation process takes place (Fig. 4). Results of inelastic X-ray scattering experiment show the appearance and further grows of the anharmonicity of free energy potential of Gd3+ ions with a decrease in temperature happening somewhere in the range 40–300 K50, and the switching between the two types of dependencies of the linewidth at T* occurs apparently for this reason. The use of the high frequency ν = 60 GHz in the experiment allowed us to analyze the resonance lines with widths up to ΔH ≈ 20 kOe. Due to the above circumstances the range of observation of the critical behavior with α ≈ − 1 reached almost two tens, which to our knowledge, considerably exceeds all previous measurements.

Despite the long history of studies of ESR in SGs and the large amount of experimental data obtained, the relationship between the behavior of the resonance line and the properties of the SG state remains an unresolved issue. Generally, SGs can be attributed to the group of systems with short-range magnetic correlations, such as frustrated and low-dimensional magnets, in which the linewidth dependence ΔH(T) is characterized by strong growth when the temperature decreases toward some critical value TD. According to the most common scenario, ΔH(T) shows the critical divergence ((T − TD)/TD)−α , which was observed experimentally and was justified by theoretical models42. As for SGs, first observation of the power law of the linewidth with the power exponent α = − 1 was reported for MnCu system51. It should be noted that this value of α is quite common for various systems and is observed both in other SGs52 and in frustrated and one-dimensional magnets (TD equals zero in the latter case)53. The experimental and theoretical results supports the opinion that the exponent in SGs should be α = − 136. However, in many SGs the value of α differs markedly from α = − 147,54 and in some semiconductors α even varies by almost an order of magnitude depending on the doping level47. Thus, numerous experimental results provide a deep basis for considering ΔH(T) dependence in SGs as power law, however it is not clear how basic line broadening mechanisms affect on the power exponent α.

Mention should be made of another approach used to describe the linewidth behavior at low temperatures, which includes the Arrhenius-type dependence of ΔH(T): Aexp(− T/T0)55,56,57. Since this model, in contrast to critical behavior, predicts the finite value of ΔH at any temperature, the type of dependence can be uniquely determined by careful measurement and analysis of the resonance line near TD.

The observation of the exponent close to α = − 1 in GdB6 and Gd0.73La0.27B6 is a rather remarkable fact, since it adds one more system with a unique mechanism of the occurrence of short-range correlations to the group of diverse magnets, that demonstrate this type of behavior. This suggests the existence of a common dynamical mechanism for broadening of the resonance line in these systems. In this matter, GdB6 is not only one more compound in this set but it can serve as a model system due to the simple crystal structure, the absence of static disorder and the definiteness of the magnetic moment of each cell. One more question posed by current research is the possible difference between the divergence temperature of ESR TD and the SG temperature Tf observed in the dependence of magnetization in Gd0.73La0.27B6 (TD = 9.5 K and Tf = 10.5 K). As far as we know, these temperatures were implicitly considered equal in all previous works and the present observation of their discrepancy is the first direct detection of the problem.

It is interesting to consider the features of low-temperature phases in GdB6 and Gd0.73La0.27B6 from the point of view of their behavior in the PM phase. The coincidence of ΔH(T) dependencies testifies the identity of the origin of short-range magnetic correlations in the PM phases of both compounds. Apparently this phenomenon is caused by mutual shifts of Gd3+ ions which form dynamical configurations with random distribution of inter-ionic distances and as a consequence of the exchange energy. The ESR in this case can be considered as a sum of resonances from individual Gd3+ ions moving in random effective magnetic fields (static and microwave), which cause the line broadening and its shift. Such disordered spin structures freeze in the case of Gd0.73La0.27B6 when the temperature decreases below Tf. However, the transition to the AFM state with a coherent ion shift structure in GdB6 occurs at higher temperature TN = 15.5 K thus masking the expected SG temperature Tf = 13.2 K which can be estimated using the relation Tf/TD in Gd0.73La0.27B6. This fact indicates another competitive physical mechanism responsible for the onset of the ordered phase below TN. Indeed, ESR confirms this assumption. According to the experiment, the gyromagnetic ratio γ obtained from ESR changes abruptly at TN and the value γ persists at all temperatures in the AFM phase58, which signify the onset of new magnetic state of Gd pairs. This effect is similar to the formation of dimers although the magnetic ground state of Gd pairs is different from singlet and its genesis requires a separate study. In its turn, ESR in Gd0.73La0.27B6 doesn’t show any signature of new state of Gd3+ emergence. Apparently, the preference of the long-rang order in this compound is destroyed by La doping and the system freeze as a configuration of mutually shifted individual Gd3+ ions with various interionic distances. Due to the inhomogenity of La in the sample, one can raise the question of the influence of its distribution on the formation of SG state. According to the above discussion SG transition is qualitatively determined by dynamic of Gd ions matrix, although the doping features can affect as the freezing temperature and the observed difference between Tf and TD.

Based on the above consideration, it is worth paying attention to the low temperature range (T < 10–12 K) inside the AFM phase of GdB6 where a new state of AFM2 develops. Note that in spite of several X-ray studies23,24,25,26 the displacement structure in this area is not recognized26. The anomalous features of AFM2 phase are the hysteretic behavior of some physical parameters as well as the sample dependence of this effect. So, the hysteresis was observed in the resistivity and magnetization at TN2 in some experiments29,59 although it is absent in some other samples33. The dependence, which is known to exhibit hysteresis in all studied samples, is magnetoresistance33,59. Moreover, the value TN2 varies in the range 5–12 K in different experiments23,24,25,26,29,33,59. In the recent density functional calculations of GdB6 the basic structural and electronic properties as well as the stability of different AFM structures were determined60. It turns out that two types of magnetic orders, E-AFM and C-AFM (illustrated in Fig. 6), lay energetically close to each other in the wide range of the Coulomb repulsion parameter U with energy difference 0.6–1 meV (6–12 K) between them60. In its turn, the hysteretic properties of ESR absorption suggest the coexistence in AFM2 phase of domains with different types of magnetic structure. Four lines resonance structure develops below T ≤ 12 K41 and the resonance “sweep up” spectra become different from that “sweep down” ones in the range 4.2 K ≤ T < 10 K . (Fig. 6). This is reflected in the redistribution of the sum absorption intensity between different resonance lines, while the total intensity of integral spectrum remains constant. It was assumed in the previous study that ESR spectrum in AFM phase is determined by low-symmetrical crystal field arising due to the shift of Gd3+ ions41. However, the change in line intensity can hardly be explained within the framework of this hypothesis. Indeed, any change of crystal field parameters would rather affect lines positions but not exclusively the distribution of intensities. On the other hand, this observation is consistent with the assumption that the AFM phase consists of domains with different magnetic orders. Moreover, proportional change of the lines A and B as well as C and D (Fig. 6) suggests that each pair of lines belongs to different domains. At the same time, the arrangement of pairs in each structure is clearly defined, as evidenced by the small width of the resonance line, and also slightly different from each other. Note, that the magnetic structure one of two lowest states E-AFM is consistent with neutron experiment results27,28,60. The absence of the satellites of the second structure may be caused by small volume of the corresponding domain as well as the complexity of the neutron experiment in GdB627,28. The considerable difference in domains volumes can be seen from the difference of intensities of the corresponding lines, which takes place at all frequencies41.

Figure 6
figure 6

Hysteretic behavior of ESR in AFM2 phase of GdB6. Two types of possible magnetic structures are illustrated. The inset shows the hysteresis in magnetoresistance33.

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The competition of two magnetic structures can explain the puzzling feature of AFM2 phase which lies in the fact that hysteresis is observed in magnetoresistance and ESR (Fig. 6)33, while is not visible in magnetization field dependence. Apparently, the magnetization at low temperatures is determined by the exchange energy in Gd pairs characterized by the exchange field HE ≈ 245 kOe61, and one can assume that this parameter in the magnetic structures of close energy is also close. Then the volume redistribution between domains of different type will not noticeably affect the sum magnetization. In its turn, the position of resonance line is determined by the parameter (HAHE)1/2, which include the anisotropy field HA as well. Thus, possible variations in HA in different positions of pairs can change the position of the lines without a detectable effect on the magnetization. The redistribution of Gd pairs between different positions in this case affects not only on ESR mode intensities but also on resistivity via the change of the configuration of scattering array consisting of nonequivalent Gd pairs thus causing its hysteresis.

Within the hypothesis of the coexistence of domains with two magnetic structures the transition between AFM2 and AFM1 phases inside AFM phase looks like gradual “mixing” of stable domains when temperature increases. So, the AFM1 phase consists from random dynamical complexes of Gd ions. However, in contrast to the PM phase, the structural element of it is Gd pair and possible configurations are restricted by certain combinations of E-AFM and C-AFM clusters. The structure of AFM1 phase can be described within the framework of a single state as shift wave of Gd ions26. Disturbance of positions and arrangements of Gd pairs lead to strong broadening of ESR line in this phase. However, the resonance line exactly cover the position of the spectrum of A,B,C,D lines thus confirming the lack of magnetic structures except those existing in AFM2 phase. Therefore it is possible to assume that multi-line ESR spectrum structure is caused by presence of several types of domains. In its turn, the domains formation may be expected to be very sensitive to the minor intrinsic defects and impurities of the crystal structure which would explain the large spreading of TN2 value found in the experiment.

In the connection of the present study, it is necessary to mention the rich borides, where the possibility of the glass behavior caused by the displacement of RE ions from their position in the lattice is discussed in literature. The signs of SG behavior were found in PrB662,63 and the presence of the “cage-glass” state was claimed taking place in dodecaborides LuB12 and ZrB1264,65. However, as for magnetic and nonmagnetic RE ions the origin of the assumed glass effects is inherent defects in the boron lattice which induce the ions shift. Moreover the estimated defect concentration in these compounds is rather small (< 5%)66 and the existence of the volume glass states remains the disputable question67. In this respect GdB6 and Gd0.73La0.27B6 are first systems where the origin of SG behavior is not caused by the inherent disorder in the magnetic Gd3+ ion system but is induced by ions shift with the formation of random spin configurations. It leads to short range spin correlations in the PM phase and then SG freezing in Gd0.73La0.27B6 or SG effects in AFM2 phase of GdB6 with temperature lowering. It should be emphasized that doping with La is not a source of the SG state in the system but rather leads to suppression of competitive coherent ordering with simultaneous decrease in Tf.

In conclusion, the magnetization measurements of Gd0.73La0.27B6 have shown the onset of SG state below Tf = 10.5 K. It manifests itself by the maximum on M(T) at Tf and by the dependence of M on the sample history below Tf. The identity of the temperature dependencies of ESR linewidth in the PM phase of Gd0.73La0.27B6 and GdB6, which follow the power law ((T − TD)/TD)−α with α ≈ − 1, clearly demonstrate the SG origin of short range magnetic correlations underlying the line broadening. In the case of Gd0.73La0.27B6 it leads to the width divergence at TD = 9.5 K while in GdB6 the coherent AFM phase transition takes place at TN = 15.5 K hiding the related SG temperature TD = 12 K. The observed behavior is caused by the shift of Gd3+ ions from the centrally symmetrical positions in the rigid boron lattice. In Gd0.73La0.27B6 dynamical displacement complexes get frozen at Tf resulting to the SG phase. The coherent displacement of Gd ions compete in GdB6 with random configurations leading first to first order phase transition and then at T < TD to the onset of complicated low temperature phase with peculiar hysteretic behavior.

Methods

The single crystals of GdB6 and of Gd1−xLaxB6 (with nominal composition x = 0.22) were grown by the induction zone melting in argon atmosphere. The sample of GdB6 is identical to those ones studied previously in transport (Fig. 1b) and ESR measurements33,41. The quality of both crystals is verified by the X-ray diffraction technique, microprobe analysis and SEM. The latter method did not allow the exact actual concentration of La to be measured due to inhomogeneous distribution of the dopant with a spread of x of several percent. The ESR measurements have been done using the setup based on Agilent PNA network analyzer68. Method for cavity measurements of strongly correlated metals, where samples are fixed as a part of the bottom plate of the cylindrical cavity69,70,71,72 was applied. Experiments were carried out in cylindrical cavity operating on TE011 mode at the frequency ν = 60 GHz. The magnetic field was applied along [100] crystallographic direction in both cases. The experimental resonance curves were analyzed as the sum of real χ1(H) and imaginary χ2(H) parts of microwave magnetic susceptibility where χ1(H) and χ2(H) were taken as Lorentz functions, except in the case of very wide lines that is discussed in the text. Magnetic measurements have been carried out with the help of SQUID magnetometer MPMS-5 (Quantum Design) at fields up to 5 T.