Emergent spin-liquid-like state in the bond-disordered breathing pyrochlore

A breathing pyrochlore system is predicted to host a variety of quantum spin liquids. However, perturbations beyond nearest-neighbor Heisenberg interaction are an obstacle to identifying such exotic states. Here, we utilize a bond-alternating disorder to tune a magnetic ground state in the Cr-based breathing pyrochlore. By combining thermodynamic and magnetic resonance techniques, we provide experimental signatures of a spin-liquid-like state in LiGa 1- x In x Cr 4 O 8 ( x =0.2), namely, a nearly T 2 -dependent magnetic specific heat and a persistent spin dynamics by muon spin relaxation ( μ SR). Moreover, 7 Li NMR, ZF- μ SR, and ESR unveil the dichotomic nature of both temporal and thermal spin fluctuations: slowly fluctuating tetramer singlets at high temperatures and a fast fluctuating spin-liquid-like state at low temperatures. Our results suggest that a bond disorder in the breathing pyrochlore offers a new route to achieve an unexplored state of matter.


Introduction
Frustrated quantum magnets provide a fertile ground to discover exotic quantum and topological phenomena, ranging from quantum spin liquids to magnetic monopoles in spin ice to Majorana anyons in the Kitaev honeycomb lattice. 1,2 A prominent instance is a pyrochlore lattice, which is a three-dimensional network of corner-sharing tetrahedra. 3 The pyrochlore Heisenberg antiferromagnet is predicted to harbor a spin-liquid ground state in the absence of further neighbor interactions. [4][5][6] In real materials, the macroscopic ground-state degeneracy can be lifted by a range of perturbations, including spin-lattice couplings, additional magnetic interactions, and quenched disorders. [7][8][9][10] The lifting of the highly degenerate ground states gives rise to emergent phenomena such as zero-energy modes, quantum spin ice, quantum spin liquid, and moment fragmentations. [11][12][13][14][15] By the same token, bond alternation as a perturbation to the pyrochlore system leads to a new structural motif called a breathing pyrochlore lattice. Its spin Hamiltonian is described as ℋ = ∑ , ∈A ⋅ + ′ ∑ , ∈B ⋅ , where J and J' are the nearest-neighbor exchange interactions for the A and B tetrahedra of two alternating sizes, respectively. 16,17 In the breathing pyrochlore, the breathing anisotropy Bf = J'/J gauges a degree of the bond alternation. Remarkably, the breathing pyrochlore can host Weyl magnons in an antiferromagnetically ordered phase, as well as a fracton spin liquid in the presence of Dzyaloshinskii-Moriya interactions not to mention a Coulomb spin liquid. 18,19 On the material side, the Cr-based breathing pyrochlore LiACr4O8 (A = Ga 3+ , In 3+ ; space group 43 ) is a particularly interesting example because Bf can be tuned by varying the composition of Li and A. 17 LiGaCr4O8 , having the breathing anisotropy Bf = 0.6, behaves much like a uniform pyrochlore. On the other hand, LiInCr4O8 has the small Bf = 0.1, in which the A tetrahedra are weakly coupled through inter-tetrahedral interaction J'. Despite the large difference in Bf, both the LiACr4O8 compounds show commonly two-stage symmetry breaking: a magnetostructural phase transition at TS=16 (17) K with subsequent antiferromagnetic ordering at TM=13 (14) K for A=In (Ga). 20 The lasting magnetic orders suggest that the bond alternation alone is not sufficient to repress the effect of spin-lattice coupling that induces a cubic-to-tetragonal transition. In this situation, bond randomness can counteract the detrimental impact of the magnetoelastic transition. This raises the exciting prospect of tracing intrinsic spin dynamics of the breathing pyrochlore lattice to a putative spin-liquid ground state, not being interrupted by long-range magnetic order.
In this work, we investigate spin dynamics and low-energy excitations of the bond-disordered breathing pyrochlore LiGa0.2In0.8Cr4O8 by combining multiple magnetic resonances with thermodynamic techniques. We observe a thermal and temporal dichotomy between a coupled tetramer singlet and a spin-liquid-like state. Spin fluctuations probed on the different time scales give evidence that the emergent spin-liquid state is pertinent to a correlated pyrochlore lattice with dimensional reduction.

Fig. 1|
Crystal structure, phase diagram, and magnetic properties. a Schematic structure of the uniform breathing pyrochlore LiACr4O8 (A = Ga, In) and the bond-disordered breathing pyrochlore LiGa0.2In0.8Cr4O8. b T-x phase diagram of LiGa1-xInxCr4O8. The inverse red triangle marks the investigated compound LiGa0.2In0.8Cr4O8. c Temperature dependence of the static magnetic susceptibility χdc(T) of LiGa0.2In0.8Cr4O8. χdc(T) is decomposed into the intrinsic χintr(T) and the orphan spin contribution χorp(T). d Real component of the ac magnetic susceptibility χ'ac(T) as a function of T and B measured with a fixed frequency of ν=100 Hz. e High-field magnetization measured at selected temperatures. The black dashed-dotted and green dashed lines indicate the orphan spin magnetization Morp(T) and the bulk magnetization Mintr(T). A linear extrapolation gives the spin gap of ΔM/kB ~ 20 K. Fig. 1a, the A-site ordered LiACr4O8 (A = Ga, In) spinels are a nearly perfect realization of the breathing pyrochlore lattice due to the significant difference between the Li + and A 3+ valence states. The Ga-for-In substitution disrupts a uniform arrangement of the two alternating tetrahedra and leads to the rich magnetic phase diagram of LiGa1-xInxCr4O8, as shown in Fig. 1b. 21 The magnetic ordering vanishes for x>0.1 or x<0.95. A spin-glass phase occupies a wide range of x=0.1 -0.625. For the Ga-rich compound LiGa0.95In0.05Cr4O8, the 7 Li NMR and neutron experiments revealed the spin nematic transition at Tf = 11 K, driven by the small bond disorder. 22 In the In-rich range of x=0.7 -0.95, LiGa1-xInxCr4O8 exhibits a spin-gap behavior with the lack of magnetic ordering and spin freezing down to 2 K. 21,23 However, little is known about an exact ground state. Here, we have chosen the x=0.8 compound (marked by the inverse triangle in Fig. 1b) to address this issue. Figure 1c presents the T dependence of dc magnetic susceptibility χdc(T) of LiGa0.2In0.8Cr4O8. On cooling, χdc(T) shows a broad hump at around 70 K, and a subsequent upturn below 10 K. Above 100 K, χdc(T) follows the Curie-Weiss law with the Curie-Weiss temperature ΘCW=-386(1) K and the effective magnetic moment μeff =3.873(4) μB (see Supplementary Note 1 and Supplementary Fig. 1) that are consistent with the values in the previous report. 21 The low-T upturn of χdc(T) is reminiscent of orphan spins. We attempt to single out the intrinsic χintr(T) by fitting the T<5 K data to χdc(T) = χintr(T) + χorp(T), where χorp(T)=Corph/(T+ Θorp). We obtain Θorp= -2.0(2) K, confirming the contribution of weakly interacting orphan spins. The orphan spin concentration is estimated to be 0.6 % of the Cr 3+ spins, smaller than the value obtained from the high-field magnetization data (see below). After the subtraction of χorp(T) from χdc(T), we can identify a nearly constant χintr(T<5 K), indicative of the presence of abundant low-energy in-gap states.  Heat capacity To elucidate the nature of low-energy magnetic excitations, we performed heat capacity measurements down to 100 mK in various magnetic fields.

Magnetic susceptibilities and magnetization
In Fig. 2, we plot the heat capacity of LiGa0.2In0.8Cr4O8 as a function of temperature. The total specific heat comprises the sum of lattice, magnetic, and nuclear contributions, Ctot = Clat + Cm + CNS. Here, Clat represents the lattice heat capacity, Cm is the magnetic heat capacity, and CNS is the nuclear Schottky contribution. To isolate the magnetic contribution Cm from Ctot, we estimate the lattice heat capacity Clat using the Debye model and the nuclear Schottky contribution CNS using CNS/T ~ T 3 at low temperatures (see Supplementary Fig. 3a). The Debye model well reproduces the high-T Ctot with the Debye temperature ΘD=669 K, as shown in Fig.   2a. After subtracting the calculated Clat and CNS from Ctot, we obtain the magnetic heat capacity Cm. As evident from Fig. 2a, Cm(T) shows a broad maximum around 70 K, at which both χdc(T) and χ'ac(T,B) indicate the onset of short-range ordering.
From a log-log plot of Cm/T vs. T in Fig. 2b, we observe a hump at around T = 0.6 K in addition to the high-T broad hump. The lack of a sharp peak excludes long-range magnetic ordering. On the application of an external magnetic field, the T = 0.6 K hump shifts to a higher temperature and is quickly suppressed at µ0H > 3 T. The B-dependence of the low-T hump is attributed to the weakly coupled orphan spins, which become frozen in the small field. In addition, Cm(T)/T displays a moderate B dependence in the T range of T=0.1 -10 K. Overall, the Cm(T)/T data are described by the power-law dependence Cm/T ~ T n . The exponent is determined to be n = 0.68 -0.82 in the T range of T = 2 -10 K and n = 0.67 -0.86 below 1 K.
The observed power-law behavior Cm ~ T 1.68 -1.82 is close to the quadratic T dependence. We recall that in kagome antiferromagnets, the T 2 dependence of Cm is taken as evidence of gapless spinon excitations. 24,25 On the other hand, the coupling between spin and orbital degrees of freedom is evoked in the pyrochlore lattice. 26,27 Obviously, the orbital degrees of freedom are not relevant to the Cr-based breathing pyrochlore. A comparison between the Cm/T data and the calculated curve of a s=3/2 isolated tetrahedron unveils a large discrepancy for temperatures below 10 K (see the inset of Fig. 2b). Thus, we conclude that the observed quadratic dependence of Cm is associated with emergent low-energy excitations of the bond-disordered breathing pyrochlore. We note that the Cm/T data show a steep decrease and deviation from the power-law behavior below 0.1 K. To diagnose the presence of a spin gap, we try to fit the low-T Cm data to Cm ~ exp[-Δlow/kBT]. The extracted Δlow/kB ~0.29(1) K turns out to be tiny (see Supplementary Note 3 and Supplementary Fig. 3b).  Nuclear magnetic resonance We now examine temporal and thermal spin fluctuations by combining multiple magnetic resonance techniques that probe spin correlations on different time scales. The 7 Li NMR spectra and the spin-lattice (spin-spin) relaxation rate 1/T1 (1/T2) are plotted as a function of temperature in Fig. 3a and 3e. At high temperatures, the NMR spectrum exhibits a single sharp line with no quadrupolar splitting, similarly observed in LiInCr4O8. 20,28 As the temperature is lowered, the NMR spectra show a continuous broadening and a shift to higher magnetic fields, emulating the development of magnetic correlations and internal fields down to T = 2.7 K. We observe neither a structured broadening of the NMR spectrum nor a sharp peak in 1/T1, further confirming the absence of long-range magnetic ordering and structural phase transition.
On cooling, 1/T1 displays an exponential-like decrease down to 10 K, and then a subsequent upturn to a power-law growth. The activation behavior of 1/T1 in the T range of 25 -100 K is well fitted with the Arrhenius equation 1/T1 ~ exp(-Δ/kBT) (see the semi-log plot of 1/T1 vs. 1/T in the inset of Fig. 3e). We obtain the spin gap Δ/kB = 35(1) K, somewhat larger than ΔM/kB ~ 20 K extracted from Mintr(H). Generally, the spin gap Δ/kB obtained from 1/T1 is larger than the true spin gap Δmin (q) occurring at a specific point q. [29][30][31] This is because1/T1 maps out the qaverage of the dynamical spin susceptibility, 1/T1 ~ T ΣqA 2 (q)χ''(q,ω0). Here, A(q) is the form factor of hyperfine interactions, and ω0 is the nuclear Larmor frequency. The salient feature is that 1/T1 follows a power law 1/T1 ~ T -n with the exponent of n = -0.46 (3) below T * =10 K, indicative of the development of highly correlated states. The drastically distinct behavior of 1/T1 through T * =10 K suggests the switching of a dominant relaxation mechanism: a high-T activated vs. a low-T critical slowing-down. The spin-spin relaxation rate 1/T2 shows a weak power-law decrease 1/T2 ~ T 0.071(4) as T → 0. A nearly T-independent 1/T2 means that spin dynamics is in the fast fluctuating exchange-narrowed limit.

Muon spin relaxation (μSR)
To discriminate between static and dynamic magnetism, we performed μSR experiments for LiGa0.2In0.8Cr4O8. In Fig. 3b, we present ZF-μSR spectra of LiGa0.2In0.8Cr4O8 at selected temperatures. At high temperatures, the ZF-μSR spectra show a Gaussian-like shape and gradually change to an exponential-like form. With decreasing temperature, the muon spin polarization rapidly relaxes in the initial time interval (t = 0 -2 μs).
We observe neither oscillating muon signal down to T = 25 mK nor a recovery to 1/3 of the initial polarization, evidencing the formation of a dynamically fluctuating state. 32 For quantitative analysis, we fitted the ZF-μSR spectra to the sum of a stretched exponential function and an exponential function, Pz(t) = afastexp[-(λft) β ] + aslowexp [-λst]. Here, afast (aslow) is the fraction of the fast (slow) relaxation component, and λf (λs) is the muon spin relaxation rate for the fast (slow) relaxing component. We summarize the obtained parameters in Fig. 3f and Supplementary Information (see Supplementary Fig. 4). As the temperature decreases, the slow relaxation component suddenly appears below T * =10 K, at which the 7 Li 1/T1 shows an upturn to the power law. Furthermore, the slow muon spin relaxation rate λs(T) shows a Tindependent behavior (λs ~ 0.042 μs -1 ) below 1 K. Figure 3f is a log-log plot of the fast relaxation rate vs. T. As the temperature is lowered, λf(T) exhibits a weakly increasing T-dependence down to 10 K, and then a power-law increase with the exponent of n=-0.71 (6). We note that the steep increment of λf(T) concurs with the levelingoff of χint(T) in Fig. 1c and the upturn of 1/T1 in Fig. 3e. Upon further cooling toward T = 25 mK, λf(T) flattens out below 2 K, entering a persistent-spin-dynamical regime. Such a levelingoff of the muon spin relaxation rate is often observed in an assortment of quantum spin liquid candidates. [33][34][35] With decreasing temperature, the stretched exponent β gradually decreases from β=2 (a Gaussian-like decay) to β=1 (an exponential decay) below T * =10 K (see Supplementary Fig. 4b). The low-T exponential relaxation suggests the persisting spin fluctuations to T = 25 mK without forming frozen moments. Fig. 3c and 3g are the LF-μSR spectra and the LF dependence of λf(HLF) measured at T = 25 mK. Remarkably, the LF-µSR spectra exhibit substantial relaxation even at HLF =1 T, lending further support to the dynamic ground state. In the Redfield model LF ( ) = 2 2 〈 loc 2 〉 /( 2 + 2 2 ), the LF dependence of λf(HLF) is related to the fluctuation frequency ν, and the fluctuating time-averaged local field 〈 loc 2 〉. 36 Here, is the muon gyromagnetic ratio. Fitting to the λf(HLF) data yields ν = 1.6(1) GHz and Hloc=296(11) G (see Fig. 3g). We note that the fluctuation frequency is faster than the value reported thus far for spin-liquid materials while the magnitude of the local field is rather large, possibly due to a large spin number s=3/2. 33,35 Taken the μSR data together, we conclude that the emergent correlated spin state below T * =10 K has a dynamic nature of a GHz fluctuation frequency.

Electron spin resonance
To take a close look at spin fluctuations on the GHz time scale, we turn to high-frequency ESR. As shown in Fig. 3d, with decreasing temperature, the ESR spectra broaden continuously down to 2.3 K and shift to a lower field. We observe no impurity signals. All of the ESR spectra are fitted with a single Lorentzian profile, suggesting that they originate solely from the correlated Cr spins. The extracted parameters are plotted as a function of temperature in Fig. 3h and Supplementary Fig. 5. The peak-to-peak linewidth ΔHpp is associated with the development of spin-spin correlations, while the resonance field Hres reflects the build-up of internal magnetic fields.
As the temperature is lowered, ΔHpp(T) exhibits a critical increase, which is described by a power-law ΔHpp(T) ~ T -n . We can identify a small change of the exponent from n=-0.37(1) to 0.30(2) through T * =10 K. This weak anomaly in the exponent is in sharp contrast to the drastically changing character of the magnetic correlations on the MHz time scale.
A critical line broadening in the paramagnetic state is due to the development of local spin correlations below the Curie-Weiss temperature ΘCW and is generic to frustrated magnets subject to critical spin fluctuations. We note that the observed critical exponent of n = 0.30 -0.37 is much smaller than n = 0.6 -0.7 for the uniform pyrochlore MCr2O4 (for M = Zn, Mg, Cd) 37 , as shown in Fig. 3h. Compared to the uniform pyrochlore, the reduced exponent in LiGa0.2In0.8Cr4O8 is interpreted in terms of the dimensional reduction as discussed below. As such, the emergent low-T state bears a resemblance to the ground state inherent to a pyrochlore lattice.

Fig. 4| Schematics of the emergent magnetic subsystems in a bond-disordered breathing
pyrochlore. Each square represents the tetrahedra formed by the Cr spins. The Cr spins are omitted for clarity. The orange and green squares are the alternating A and B tetrahedra for LiInCr4O8. The grey square denotes the B' tetrahedra introduced by the Ga-for-In substitution. Due to an energy hierarchy of the tetrahedral (EA~EB'>EB), the breathing pyrochlore is effectively decomposed into two subsystems: (i) coupled tetramer singlets (red shaded circle) and (ii) diluted pyrochlore lattices (blue dashed region).

Discussion
With the aid of various thermodynamic and magnetic resonance techniques (0 Hz -1 THz), we are able to disclose the temporal and thermal structures of magnetic correlations in the bond-disordered breathing pyrochlore LiGa0.2In0.8Cr4O8.
We identify the characteristic temperature T * =10 K, below which a highly correlated spin state emerges from high-T tetramer singlet fluctuations. The emergent low-energy state features nearly T-squared Cm as well as critical spin fluctuations over a wide time scale of 10 -12 -10 -4 sec, as evident from a power-law 1/T1 ~ T -0.46 (3) , λf(T) ~ T -0.71 (6) , and ΔHpp(T) ~ T -0. 30 . We recall that the energy scale of T * =10 K is smaller than the spin gap of Δ/kB=20 -35 K, deduced from M(H) and 1/T1. Yet, it is bigger than a few kelvins of weakly correlated orphan spins. The latter contribution is quenched by applying the external field of µ0H > 3 T. As such, the low-T correlated state does not stem from extrinsic defect spins. Rather, it pertains to the in-gap magnetic state created by the bond randomness in a breathing pyrochlore. Here, we stress that there is no hint for random singlets whose fingerprints are scalings of thermodynamic quantities, namely, χ(T)~ T 1-α , M(H) ~ T α , and Cm(T) ~ T α with 0< α <1. 38 Therefore, the low-T correlated state is distinct from a bond-disorder induced random singlet.
We next turn to the temporal spin dynamics. The thermodynamic data reveal the spin gap decorated with the in-gap state. On the slow time scale (1 -10 4 Hz), the NMR data demonstrate a dichotomic spin dynamics, namely, high-T singlet correlations and low-T critical correlations. On the fast time scale (10 4 -10 12 Hz), the µSR and ESR data give no signature of the tetramer singlet correlations. Instead, highly correlated spins prevail over the whole measured temperature range and show a spin-liquid behavior. This behavior means that the singlet fluctuations average out beyond the time scale of the NMR technique at low temperatures. Another perspective is that LiGa0.2In0.8Cr4O8 comprises two spin subsystems whose temporal magnetic correlations differ from each other. As the tetramer singlet forms an entangled state of four spins, its spin fluctuations are slower than the spin fluctuations of the rest subsystem that are governed by two-spin correlations. In the GHz time window, the high-T critical spin dynamics persists to the low-T state. More specifically, the power-law dependence of ΔHpp(T), which is an ESR signature of the pyrochlore-like spin dynamics, is observed down to T=2.3 K. This gives indirect evidence that the low-T in-gap state is linked to a pyrochlore-like subsystem.
Lastly, we discuss the origin of the dichotomic magnetic correlations emerging from the bondalternating disorder in LiGa0.2In0.8Cr4O8. As sketched in Fig. 4, the Ga-for-In substitution introduces the B' tetrahedra randomly to the alternating A and B tetrahedra. Consequently, the perturbed system entails two different networks consisting of (i) the alternating A and B tetrahedra with Bf=0.1 and (ii) the alternating A and B' tetrahedra with Bf=0. 6. Neglecting a small fraction of the alternating B and B' tetrahedra with Bf=0.17, the large difference in Bf brings about two subsystems: (i) coupled tetramer singlets and (ii) diluted pyrochlore lattices.
The coupled tetramer singlets are responsible for the high-T and slow spin fluctuations. The diluted pyrochlore subsystem weakly bonded to the tetramer singlets contributes to the emergence of the low-T spin-liquid-like state. LiInCr4O8 is in the vicinity of a critical point and is proximate to a Coulomb phase. 28,39 However, the magnetoelastic coupling does not allow for exploring this exotic phase. Nonetheless, the bond disorder offers an efficient way to overcome restrictions imposed by the magnetostructural transition. As a by-product of quenching symmetry breaking, we obtain a finite-size breathing pyrochlore lattice that dictates a low-energy and fast spin behavior. As such, the low-T state of LiGa0.2In0.8Cr4O8 may realize a diluted Coulomb spin liquid, deserving further theoretical and experimental investigations.
To conclude, we have investigated the temporal and thermal spin dynamics of a bonddisordered breathing pyrochlore compound. We find that a dichotomy of magnetic correlations is linked to two emergent subsystems. The observed low-T spin-liquid-like state bears pertinent spin fluctuations to a diluted breathing pyrochlore lattice, which opens up a venue for a fundamentally new state.

Methods
Sample synthesis Polycrystalline samples of LiGa0.2In0.8Cr4O8 were synthesized by a solidstate reaction method. Stoichiometric amounts of Li2CO3, Cr2O3, Ga2O3, and In2O3 were mixed in 1:4:0.2:0.8 molar ratio and thoroughly ground in a mortar. The mixture was pelletized and sintered at 800 ℃ in the air for 12 h. The substance was ground, pressured into pellets, and finally sintered at 1000 ℃ for 24 h and 1100 ℃ for 72 h. We checked the quality of the samples by X-ray diffraction measurements.
Magnetic properties characterization dc and ac magnetic susceptibilities were measured using a superconducting quantum interference device magnetometer and vibrating sample magnetometer (Quantum Design MPMS and VSM). High-field magnetization experiments were performed at the Dresden High Magnetic Field Laboratory using a pulsed-field magnet (20 ms duration). The magnetic moment was detected by a standard inductive method with a pick-up coil system in the field range of μ0H = 0 -60 T.
Heat capacity Heat capacity measurements were carried out in the temperature and field range of T=0.03 -300 K and B=0 -9 T using a commercial set-up of Quantum Design PPMS with a thermal relaxation method. The raw heat capacity data Ctot/T follow a T -3 power-law behavior at extremely low temperatures, corresponding to a nuclear Schottky contribution (see Supplementary Fig. 3a). After subtracting the nuclear Schottky contribution CNS/T = AT -3 from Ctot/T, we present the magnetic and lattice contributions to the specific heat in the main text.
Nuclear magnetic resonance 7 Li (I = 3/2, γN = 16.5471 MHz/T) NMR measurements were conducted at National High Magnetic Field Laboratory (Tallahassee, USA) by using a locally developed NMR spectrometer equipped with a high-homogeneity 17 T sweepable magnet. 7 Li NMR spectra were recorded by a fast Fourier transform of spin-echo signals while sweeping the field at a fixed frequency ν = 198.317 MHz. The nuclear spin-lattice (spin-spin) relaxation time T1 (T2) was measured by a modified inversion recovery (Hahn pulse) method width π/2 = Muon spin relaxation (μSR) μSR measurements were carried out on the DR spectrometer ( 3 He/ 4 He dilution refrigerator, 25 mK ≤ T ≤ 10 K) at M15 beamline and the LAMPF spectrometer ( 4 He cryostat, 2 < T < 300 K) at M20 beamline in TRIUMF (Vancouver, Canada). The polycrystalline samples of LiGa0.2In0.8Cr4O8 were wrapped with a silver foil and then attached to the sample holder. After the mounted samples were inserted into the cryostat, μSR spectra were measured in zero-field, longitudinal field (parallel to muon spin direction), and transverse field (perpendicular to muon spin direction).

Electron spin resonance (ESR)
High-frequency ESR experiments were performed at National High Magnetic Field Laboratory (Tallahassee, USA). The ESR spectra were recorded using a quasioptical heterodyne spectrometer in the temperature range of T = 2 -290 K, 40 enabling the detection of a magnetic field derivative of a microwave absorption signal. An external magnetic field was swept with a 12.5 T sweepable superconducting magnet at the fixed frequency ν = 240 GHz.