Metallic Surface States Probed Within the Microwave Skin Depth of the Putative Topological Insulator YBiPt Compound

Electron Spin Resonance (ESR) experiments of diluted Nd$^{3+}$ ions in the claimed topological insulator (TI) YBiPt are reported. Powdered samples with grain size from $\approx$ 100 $\mu$m to $\approx$ 2,000 $\mu$m were investigated. At low temperatures, 1.6 K $\lesssim$ \emph{T} $\lesssim$ 20 K, the X-band ($9.4$ GHz) ESR spectra show a \emph{g}-value of 2.66(4) and a Dysonian resonance lineshape which shows a remarkably unusual temperature, concentration, microwave power and particle size dependence. These results indicate that metallic and insulating behavior coexist within a skin depth of $\delta \approx$ 15 $\mu$m. Furthermore, the Nd$^{3+}$ spin dynamics in YBiPt are consistent with the existence of a \emph{phonon-bottleneck process} which allows the energy absorbed by the Nd$^{3+}$ ions at resonance to reach the thermal bath via the conduction electrons in the metallic surface states of YBiPt. These results are discussed in terms of the claimed topological semi-metal properties of YBiPt.


I. INTRODUCTION
Topological insulator (TI) materials have recently attracted great attention of the condensed matter scientific community [1][2][3] .Nontrivial topological invariants of the bulk electronic band structure [4][5][6] yield a gapless state on the surface of these materials, which is protected by timereversal symmetry.][16][17] In particular, the series of rare earth (RE) noncentrosymmetric half Heusler ternary semi-metallic compound, REBiPt, has been suggested by first principle calculations to host many three-dimensional topological insulators (3DTIs) owing to the fact that they have a topologically nontrivial band structure with band inversion which leads to a gap-less metallic surface. 18In a 3DTI the bulk behaves as a small gap semiconductor (∆ 10 meV) with robust protected metallic surface states due to their strong spin-orbit (SO) coupling and nontrivial Z 2 topology. 2,19mong the REBiPt compounds, YBiPt has gained distinguished attention due to its unusual transport properties which have been associated with the presence of surface states. 20Moreover, superconductivity has been recently reported in YBiPt with a transition temperature at T c = 0.77 K. 20 Superconductivity in noncentrosymmetric systems is an appropriated framework to study unconventional superconducting phases 21 due to their electronic band structure. 22,23In addition, a su-perconducting phase in connection to non-trivial topology of the electronic bands may create propitious conditions to the investigation of surface states of Majorana fermions. 19,24owever, superconductivity is not restricted to the claimed 3DTI YBiPt (T c = 0.77 K) 20,25 within the RE-BiPt series.In fact, the metallic LuBiPt (T c = 1 K) 26,27 and LaBiPt (T c = 0.9 K) 28 compounds are also superconductor with similar critical temperatures.
Electron spin resonance (ESR) of diluted REs is a powerful local technique that can directly probe the localized magnetic moments and the nature of the interactions with their electronic environment. 29,30Therefore, it may be an useful tool to investigate the dual character, metallic and insulating, of these TI materials.
In this regard, our group has studied the crystalline electric field (CEF) effects of diluted REs (Nd 3+ , Gd 3+ and Er 3+ ) in Y 1−x RE x BiPt about fifteen years ago. 31,32t that time we found an intriguing behavior of the Nd 3+ ESR lineshape in Y 1−x Nd x BiPt which we have chosen not to report so far because this result was disconnected from the study of CEF effects.Now, enlightened by the advent of the field of topological insulators and by the astonishing properties recently discovered in this material, we have revisited those data and performed further experiments in order to elucidate the origin of this unusual lineshape behavior.As such, here we report a detailed investigation of the Nd 3+ ESR lineshape in Y 1−x Nd x BiPt and argue that our results provide important evidence for the existence of surface metallic states in YBiPt.
Our main ESR findings in Y 1−x Nd x BiPt (0.002 x 0.10) are: i ) insulating and metallic behaviors coexisting within a microwave skin depth of δ ≈ 15 µm, which is considered a bulk measurement, even in normal metal-lic systems; 29,30 ii ) the ESR spectra depend on the microwave power, Nd 3+ concentration, temperature and particle size; and iii ) the existence of a phonon-bottleneck relaxation process in this TI.These features are discussed in terms of both phonons and Dirac conduction electrons (ce) contribution to the diffusion of the absorbed microwave energy at resonance by the Nd 3+ ions to the thermal bath.

II. EXPERIMENTAL DETAILS
Several batches of Y 1−x Nd x BiPt (0.002 x 0.10) were synthesized using a self-flux technique 33 with a starting composition (1 − x)Y:xNd:1Pt:20Bi.The crucible containing the elements was placed in a quartz tube sealed in vacuum and slowly heated up to 1170 0 C.After being kept at this temperature for 2 h, the tube was cooled down to 900 0 C with a rate of 10 0 C/h.The collected crystals have free-growth planes with dimensions up to 4 mm.X-ray powder diffraction was used to verify the cubic crystal structure and F43m space group of YBiPt.The ESR experiments were performed on powdered single crystals of selected particles having sizes greater than 100 µm, corresponding to particles of average size/skin depth ratio, λ = d/δ 6.6.The X-Band (ν ≈ 9.4 GHz) ESR experiments were carried out in a conventional CW Bruker-ELEXSYS 500 spectrometer using a TE 102 cavity coupled to an Oxford helium gas flow system and a quartz/stainless steel cold tail liquid helium dewar.

III. EXPERIMENTAL RESULTS
First of all we will introduce the framework that will be necessary to analyze the ESR lineshape behavior in Y 1−x Nd x BiPt.Due to the high conductivity of metals, the microwave electromagnetic field is attenuated and it only penetrates a small length scale called skin depth (δ).δ is frequently much smaller than the sample dimensions.It leads to a "vertically asymmetric" ESR lineshape named Dysonian (Fig. 1a). 34urthermore, in a metal, a local moment spin system has a very fast relaxation process which allows the resonating spins to transfer the absorbed microwave energy to the lattice very rapidly via exchange interaction with the ce.As such, the ESR signal intensity (doubly integrated ESR spectrum) usually increases linearly as a function of the microwave power at a given temperature, as illustrated in Figure 1b.
In contrast, the microwave goes through the whole volume of an insulating material and the resonating spins present a symmetric ESR lineshape called Lorentzian as shown in Figure 1c.As the insulating materials are free of ce, the relaxation mechanism dominated by phonons is much slower than that of a metal.As a consequence, the ESR signal intensity in an insulator at a given tem-perature can saturate at high microwave power when the population of the spin levels, splitted by the Zeeman effect, tends to be equal.This effect is displayed in Figure 1d.In fact, the overall lineshape of the ESR spectra is analyzed by the general accepted approach where, at resonance, the microwave absorption in a metal is given by the Dyson theory in the diffusionless limit, A/B 2.6 → T D /T 2 1 34,35 .In this limit, the Dyson theory can be approximated to a simple admixture of absorption and dispersion of Lorentzian lineshapes 36 with the A/B ratio changing monotonically from A/B ≈ 1 to A/B ≈ 2.6 for samples size, d, smaller and larger than δ, respectively [34][35][36][37] .However, when A/B exceeds 2.6 the Dyson theory anticipates the presence of diffusive effects, A/B 2.6 → T D /T 2 1, (Ref.34 and Fig. 7) 34,35 .This means that the resonating spins diffuse through the skin-depth with a diffusion time, T D , comparable to the spin-spin relaxation time, T 2 .
Equation 1 gives the derivative of the admixture of absorption (χ ) and dispersion (χ ) of Lorentzian lineshapes.
x = (H 0 − H) γT 2 where H 0 and H are the resonance and the applied magnetic fields respectively, γ is the gyromagnetic ratio, T 2 is the spin-spin relaxation time, α is the admixture of absorption (α = 0) and dispersion (α = 1), and χ 0 is the paramagnetic contribution from the static susceptibility.
In order to understand our results we have introduced the saturation term s = γ 2 H 2 1 T 1 T 2 in our resonance lineshape analysis phenomenologically.H 1 is the strength of the microwave magnetic field and T 1 is the spin-lattice relaxation time. 38igure 2 presents the ESR spectra of Nd 3+ in Y 1−x Nd x BiPt for x = 0.002 as well as natural Gd 3+ impurities at T = 1.6 K and microwave power of P µω ≈ 5 mW for λ 132.By a simple glance at this spectrum one can easily observe the striking and unexpected result that the recorded ESR lineshape for Gd 3+ is typically metallic (Dysonian) corresponding to the diffusionless regime (A/B ≈ 2.6 → T D /T 2 1) while that of the 140 Nd 3+ (I = 0) presents, paradoxically, a completely diffusive lineshape (A/B ≈ 5 → T D /T 2 ≈ 0.4) 34,35 , although both REs are localized magnetic moments diluted in the same material.Also, except for their concentrations, both probes are under the same conditions of T, P µω and particles size much larger than the skin depth.Figure 3a shows the ESR spectra of Y 1−x Nd x BiPt for x = 0.10 at T = 4.2 K and P µω ≈ 8 µW for 6.6 λ 132.Remarkably, it is clear from these data that the observed change of the lineshape, going from A/B ≈ 3 for large particles to A/B ≈ 7 for smaller particles, does not correspond to the A/B values expected from the Dyson theory for diffusionless ESR lineshape (1 A/B 2.6).Instead, the lineshape of the smaller particles presents a strong diffusive shape, A/B ≈ 7 → T D /T 2 ≈ 0.2, despite the fact that the particles size is still larger than the skin depth.Figure 3b, in turn, presents the P µωdependence of the ESR lineshape for the 66 λ 132 sample.These data show that at very low power (≈ 2 µW) the ESR lineshape is closer to the diffusionless limit (A/B ≈ 4 → T D /T 2 ≈ 0.9).However, by increasing P µω up to a power of ≈ 200 µW the lineshape becomes completely diffusive (A/B ≈ 14 → T D /T 2 ≈ 0.02).Yet, up to this power level the integrated ESR spectra grow linearly with [P µω ] 1/2 showing no saturation effects (not shown).Again, these results add up to contrasting behavior displayed by the ESR lineshape of localized magnetic moments in this material.By a further increasing P µω , the lineshape remains diffusive and the double integrated spectra now display saturation effects (see Fig. 6b below).) at low concentration and strong diffusive lineshapes (A/B ≈ 9 -17 → T D /T 2 ≈ 0.10 -0.01) at high concentration, both nearly independent of P µω .Nonetheless, for the intermediate concentration of x = 0.005 (Fig. 4b), the lineshape presents a dramatic and unusual change between these two regimes, similarly to the data of Fig. 3b.This sample at P µω ≈ 3 µW shows a diffusionless ESR lineshape (A/B ≈ 2.3 → T D /T 2 1).But, upon increasing P µω up to an intermediate power of ≈ 80 µW the lineshape becomes noticeably more diffusive (A/B ≈ 3.3 → T D /T 2 ≈ 0.9).Yet, up to these power levels, the double integrated ESR spectra do not show saturation effects (not shown).By a further increasing P µω the lineshape remains diffusive and at higher power levels the double integrated spectra display saturation effects (not shown), similar to the data shown in Figs.6a, 6b.
Figures 5a, 5b and 5c present the T -dependence of the ESR lineshape of Y 1−x Nd x BiPt with 6.6 λ 30 for x = 0.002 at P µω ≈ 5 mW, x = 0.005 at P µω ≈ 5 mW and x = 0.10 at P µω ≈ 0.2 mW, respectively.These results show that at T 10 K the ESR lineshape displays strong diffusive character.The increase of T tends to restore the ESR lineshape into the diffusionless regime (A/B ≈ 2-4) as observed at low-P µω for large particles (see Figs. 2a,  2b).Figures 6a and 6b display the P µω and T -dependence of the double integrated ESR spectra of Y 1−x Nd x BiPt for x = 0.002 and 0.10, respectively, and 6.6 λ 30.Similar results were obtained for the x = 0.005 and x = 0.05 samples (not shown).Strikingly, the saturation effects observed in the ESR spectra confirm that, as far as the relaxation processes are concerned, this system behaves as an insulator regardless the sample concentration.Therefore, we conclude that the ensemble of Nd 3+ ions in the Y 1−x Nd x BiPt (0.002 x 0.10) system saturates as P µω -increases and T -decreases.Notice that at high-T (8 K≤ T ≤ 20 K) the resonance intensity for the x = 0.10 sample saturates at relatively higher microwave power than that of the x = 0.002 sample.This is due to the large diffusive component of the ESR spectra for the x = 0.10 sample (see Fig. 5c). Figure 6c presents the x -dependence of the ESR spectra in Y 1−x Nd x BiPt (0.002 ≤ x ≤ 0.10) at 4.2 K and P µω ≈ 5 mW for 6.6 λ 30.The best fit of the observed spectra to Eq. 1 using 1/T 2 as a fit parameter is shown in red solid line.As expected for inhomogeneous ESR linewidths in insulators, the spin-spin relaxation rate, 1/T 2 , increases as x increases (see inset).Notice that a diffusive-like character of the lineshape is shown, though fortuitously, as x increases (A/B 2.6 → T D /T 2 1). Figure 7 shows the x -dependence of the spin-lattice relaxation rate, 1/T 1 , for Y 1−x Nd x BiPt at T = 4.2 K and 6.6 λ 30.In the analysis of the various investigated samples, T 2 is obtained from the linewidth, ∆H = 1/γT 2 , of the absorption component of the resonance at the lowest microwave power level.For T 1 it is only possible to estimate a lower limit (upper limit of 1/T 1 ) from the saturation factor of the integrated ESR spectra, I sat /I unsat . 39This limitation is due to the lack of an explicit equation which takes into account the ESR in the high microwave power regime for diffusive processes.Figure 7 shows that 1/T 1 slows down as x increases indicating that in this system, as in other insulators, a phonon-bottleneck process dominates the spin-lattice relaxation as the Nd 3+ concentration increases 40 .

IV. ANALYSIS AND DISCUSSION
The Dysonian ESR lineshape of diluted Gd 3+ and Nd 3+ in YBiPt (Fig. 2) combined with the change of the Nd 3+ ESR lineshape with the size of the powdered particles (Fig. 3a), assure us that these REs are probing the presence of ce within the skin depth of δ ≈ 15 µm.On the other hand, in our previous study on crystal field effects of diluted REs (Nd 3+ , Gd 3+ and Er 3+ ) in Y 1−x RE x BiPt we have demonstrated that the exchange coupling between the RE localized magnetic moments and ce is very weak in this system. 31,32This conclusion was drawn from the very small thermal broadening of the ESR linewidth (Korringa relaxation rates), the negligible g-shifts (Knight shifts) measured for these REs 41 and also corroborated by the small Sommerfeld coefficient (γ 0.1 mJ/molK) found for this material. 32,42][33] Therefore, as in any insulator in the diluted limit, the relaxation of the localized magnetic moments to the thermal bath should mainly happen through the lattice phonons via SO coupling (λ 4f L N d .S N d ) 40 and not by the exchange interaction with ce ((g j -1)J f s J N d .sce ) 41 .As such, in spite of the observed metallic ESR lineshapes of dysonian-like 34,35 (see Figs 3, 4, 5 and 6), the low-T ESR linewidth of diluted REs in YBiPt is expected to be inhomogeneous (see Fig. 6c) and the spin-spin relaxation time much shorter than the spin-lattice relaxation time (T 2 T 1 ) as in any insulator.Moreover, Figures 6a  and 6b show that the double integrated Nd 3+ ESR spectra saturates for T 20 K and P µω 5 mW, confirming the slow spin-lattice relaxation and indicating that the relaxation process is driven by phonons via SO coupling.This result combined with the dependence of the Nd 3+ (Γ 6 Kramer doublet; S ef f = 1/2) ESR lineshape on the particle size (Fig. 2a), microwave power (Figs.3b, 4a,  4b, 4c), temperature (Figs.5a,b,c) and concentration (Fig. 6c) indubitably assure us that the insulating and metallic characters coexist in the YBiPt system.
Then, for the analysis of the ESR spectra we shall use Eq. 1 above.It is worth noting that one has to take our phenomenological lineshape analysis that uses the simple incorporation of the saturation term s = γ 2 H 2 1 T 2 T 1 into the admixture of absorption and dispersion with care.This may not completely describe the complex phenomenon involving the process of resonant microwave absorption diffusing to the thermal bath in the presence of a phonon-bottleneck process (see below).
Figure 8a shows the simulations of the data shown in Figs.3b and 4b using Eq. 1.The spin-spin, 1/T 2 , and spin-lattice, 1/T 1 , relaxation rates were kept constant in these simulations.The simulated spectra show that the ESR lineshape changes as P µω increases, going from a diffusionless to a broad diffusive-like regime.Despite the broadening of the spectra, these simulations reproduce reasonably well the general lineshape features presented in Figs.3b and 4b.Then, to avoid this broadening, not observed experimentally, we have forced the linewidth, ∆H = 1/γT 2 , to narrow as P µω increases, holding 1/T 1 constant.Figure 8b displays the simulated spectra for the ESR data of Fig. 4b.The inset shows the extracted phenomenological P µω -dependence of 1/γT 2 .Notice that the narrowing of the linewidth begins around P µω ≈ 5 mW where the resonance starts to saturate (non-thermal equilibrium) (see Figs. 6a, 6b).From this point on, an exponential behavior is obtained for the spin-spin relaxation rate, 1/T 2 ∼ e −aPµω , where a is a fitting parameter.The decrease of ∆H with P µω can be ascribed as a reduction of 1/T 2 due to an evanescent local fluctuating field (secular and non-secular broadenings) 43 , caused by the saturation of the ensemble of Nd 3+ ions.Nonetheless, we believe that our most important and striking experimental result of the present work is the dramatic change of the Nd 3+ ESR lineshape between the diffusionless (A/B ≈ 2.6; T D /T 2 1) and diffusive regimes (A/B 2.6 → T D /T 2 1) observed for P µω 200 µW and T 10 K.This change is clearly seen in the size (Fig. 3a), P µω (Figs.3b and 4b) and T -dependence (Figs.5a,b,c) of the ESR spectra.
For the P µω -dependent case shown in Fig 4, the ESR spectra display a drastic lineshape change well below ≈ 5 mW, the microwave power limit where saturation effects begin to be observed (Figs.6a, 6b).Besides, this drastic lineshape change is revealed by our simulated spectra only at P µω ≥ 20 mW.For the samples of Figs.3b and 4b this lineshape change occurs below ≈ 200 µW and ≈ 80 µW, respectively, (also for the sample with x = 0.05, not shown).Therefore, we conclude that the lineshape change for P µω 200 µW has nothing to do with the saturation phenomenon.Then, in Eq. 1 we again force the lineshape to change in the region of low-P µω adjusting the α parameter phenomenologically.Figure 8c presents, following the same procedure adopted for Figure 8b, the fits to the data of Figure 5b.It means that the admixture of absorption/dispersion (α in Eq. 1) was adjusted to fit the observed lineshape change to a pure diffusive regime (T D /T 2 1).The extracted P µω -dependence of α is shown in the inset of Figure 8c.Notice that the fits can not exactly reproduce the two minimum lateral of the resonances, although they show a change in the lineshape the same P µω (≈ 100 µW) as observed experimentally.This P µω is well below the regime where the saturation effects start to set in (≈ 5 mW).We believe that the inability of our fits to reproduce exactly the experimental spectra exactly may be due to the simplified phenomenological approach employed.
Regarding to the T -dependent ESR lineshape of Figure 5a, Figure 9 presents the fits of the spectra to Eq. 1.For these fits the spin-lattice relaxation rates, 1/T 1 , were estimated from the saturation factors 39 of Figure 6a (see inset of Fig. 9).Moreover, the T 2 and α parameters have been forced to assume the values that best fit the observed spectra.The obtained T -dependence of 1/T 2 , 1/T 1 and α parameter are shown in the inset of Fig. 9.These results show that as T increases both 1/T 1 and 1/T 2 increase, restoring the local fluctuating field as the ensemble of Nd 3+ ions reach their thermal equilibrium (unsaturated state).In other words, the phonon relaxation, via SO coupling, restore the local fluctuating field.Now, based on the strong evidence displayed in Fig. 7 that the spin lattice relaxation of the Nd 3+ ions is controlled by the phonon-bottleneck phenomenon in the high Nd 3+ concentration limit, we conclude that the phononthermal bath contact must be poor and the Kapitza resistance 44 should be relatively high in YBiPt.Therefore, we suggest that the abrupt change of the lineshape, from diffusionless (α ≈ 0.5 → T D /T 2 1) to diffusive (α ≈ 1 → T D /T 2 1), between ≈ 2 µW and ≈ 100 µW (Figs.3b and 4b) and for T 10 K (Figs. 5a, 5b, 5c), is associated to with the formation of a long life time phonons reservoir due to the poor phonon-thermal bath contact within the skin depth of δ ≈ 15 µm.Since the SO interaction (λ ce l ce .sce ) is an important ingredient to form a TI [4][5][6] we argue that these long life time phonons could excite the ce in the metallic surface state of this TI compound via SO coupling.Which, in turn, deliver the microwave energy absorbed at resonance by the Nd 3+ ions to the thermal bath.Then, this combination of long life time phonons and metallic surface states plays the role of ce diffusing across the skin depth in the usual Dyson theory for normal metals 35 .Figure 10 presents an illustrative route diagram indicating a plausible path (thick solid blue arrows) for the net flow of microwave energy absorbed at resonance reaching the thermal bath.
The intriguing results of Fig. 3a for x = 0.10 at low-P µω clearly show that a diffusionless lineshape occurs for the larger particles while for smaller ones a diffusive lineshape does.Such results can be also easily simulated (not shown) adjusting the α parameter in Eq. 1 as done for Figures 8c and 9. Notice that even for the smallest particles their size are several times larger than the skin depth.We believe that this remarkable lineshape change is due to the increase in the surface/volume ratio for the smaller particles which turns the long life time phonons interaction with the Dirac ce more effective.Therefore, smaller particles would favor the diffusion of the microwave energy absorbed at resonance by the Nd 3+ ions to the thermal bath.It is also worth mentioning that our microwave photons with energy hν 0.5 K may promote electrons across the gap of the Dirac cones, increasing the density of the ce on the surface as the microwave power increases.Thus, such an increase would favor the diffusion of the microwave energy absorbed at resonance by the Nd 3+ ions to reach the thermal bath.This ce activation can be induced by electric dipolar transitions associated with the electric component of the applied microwave 45 .Nevertheless, this electronic activation may not be so relevant here because a TE 102 ESR resonator was used in the ESR experiments.

Phonons reservoir
Thermal Bath This means that the sample is located at the minimum of the microwave electric field in the cavity.Additionally, we should mention that we have carried out similar studies of Gd 3+ in Y 1−x Gd x BiPt for 0.01 x 0.05.Saturation effects at high microwave power, change of the ESR lineshape below ≈ 100 mW and phonon-bottleneck effects, similar to those of Nd 3+ were also observed.However, describing the details of the Gd 3+ ESR in YBiPt is beyond the scope of this work.
Finally, we believe that the results presented here will help to shed new light on the experimental characterization of the surface metallic states in TI materials.In particular, this work should motivate further ESR work in other TI materials.However, it is possible that the effects of the surface states would be favored to be observable by ESR in the presence of a phonon-bottleneck regime.

V. CONCLUSIONS
The systematic ESR study of the Nd 3+ Γ 6 Kramer doublet (S ef f = 1/2) CEF ground state in the cubic noncentrosymmetric half Heusler semiconduc-tor/semimetallic compound Y 1−x Nd x BiPt revealed that this system presents, simultaneously, metallic and insulating features.This dual character was verified by the saturation effects and relaxation processes observed on the Dysonian (metallic lineshape) of the Nd 3+ ESR spectra.Also, our phenomenological approach to analyze the ESR lineshape suggests that saturation effects do affect the local fluctuation field and contribute to slow down the effective spin-spin relaxation rate, 1/T 2 , i.e., narrowing down the inhomogeneous ESR linewidth.
Moreover, the dramatic evolution of the lineshape between diffusionless (A/B 2.6 → T D /T 2 1) and diffusive regimes (A/B 2.6 → T D /T 2 1) at microwave powers below ≈ 200 µW, where no saturation effects are observed, strongly suggests that this peculiar behavior is caused by a subtle combination between the increasing presence of the phonon-bottleneck process (as the Nd 3+ concentration increases) and the highly conducting metallic surface of this TI material.
Finally, we conclude that the phonon-bottleneck process allows the observation of several striking features on the Nd 3+ ESR metallic linshape behavior within a skin depth of δ ≈ 15 µm in YBiPt, supporting the 3DTI character of this compound.We argue that the phonon-bottleneck process allowed the formation of long life time phonons reservoir where these phonons, via SO interaction, couple to highly mobile Dirac ce at the Fermi level that, finally, deliver the microwave energy absorbed at resonance by the Nd 3+ ions to the thermal bath.We have also mentioned that electromagnetically excited electrons across the Fermi level at the Dirac cones by our low energy microwave photons may contribute to facilitate the observation of the diffusive effect of the Nd 3+ ESR lineshape in Y 1−x Nd x BiPt, although this contribution may not be so relevant due to our experimental configuration.

FIG. 1 :
FIG. 1: (color online) a) and b) illustrate the microwave penetration and ESR lineshape; c) and d) the microwave power dependence of the ESR signal intensity in metallic and insulating samples, respectively.

FIG. 6 :
FIG. 6: (color online) Pµω and T -dependence of the integrated ESR spectra for Y1−xNdxBiPt and 6.6 λ 30 for: a) x = 0.002 and b) x = 0.10.The solid lines are guide to the eyes.c) x -dependence, 0.002 x 0.10, of the ESR spectra for Y1−xNdxBiPt at T = 4.2 K, Pµω ≈ 5 mW and 6.6 λ 30.Red solid lines are the fits of the ESR spectra to Eq. 1 using 1/T 2 as a fitting parameter.The inset shows the increase of 1/T 2 as x increases.The solid line is a guide to the eyes.

1 XFIG. 7 :
FIG.7:x -dependence of 1/T 1 for Y1−xNdxBiPt at T = 4.2 K and 6.6 λ 30.The data was obtained from the analysis of the saturation factors39 of the Nd 3+ double integrated ESR spectra of various samples reported in this work.The dashed line is a guide to the eyes.

FIG. 8 :
FIG.8:(color online) Theoretical Pµω-dependence (Eq. 1) of the ESR lineshape for Y1−xNdxBiPt (x = 0.005) at T = 4.2 K and 6.6 λ 30; a) simulations with constant relaxation rates, 1/T 2 and 1/T 1; b) simulations with 1/T 1 constant and 1/T 2 following the Pµω-dependence (logarithmic scale) shown in the inset and; c) the red lines are fits of the data of Fig.4bwith Eq.1 using constant 1/T 1.The extracted parameters of absorption/dispersion admixture, α, and 1/T 2 are shown in the inset.The solid lines are guide to the eyes.

1 )FIG. 9 :
FIG.9:(color online) The red lines are fittings to Eq. 1 of the T -dependence ESR lineshape for Y1−xNdxBiPt (x = 0.002) at Pµω≈ 5 mW and 6.6 λ 30.For these fittings the spin-lattice relaxation rate, 1/T 1, was obtained from the saturation data of Fig.5a, the spin-spin relaxation rate, 1/T 2, and α were free parameters that best fit the experimental spectra.Their T -dependence are shown in the inset.The dashed lines are guide to the eyes.

Nd 3+ ensemble Diffusion of microwave energy absorbed at resonance assisted by a "phonon bottleneck process"
Nd(g j -1)J.J Nd .sce λ ce l ce .sceti hν λ f L Nd .S