ESR investigation of the spin dynamics in (Gd$_{1-x}$Y$_{x}$)$_2$PdSi$_{3}$

We report electron-spin resonance measurements in polycrystalline samples of (Gd$_{1-x}$Y$_{x}$)$_2$PdSi$_{3}$. We observe the onset of a broadening of the linewidth and of a decrease of the resonance field at approximately twice the Nel temperature in the paramagnetic state, this characteristic temperature coincides with the electrical resistivity minimum. The high-temperature behaviour of the linewidth is governed by a strong bottleneck effect.

Electron Spin Resonance ,Gd alloys ,magnetoresistance 76.30.-v ,75.30.Vn ,75.30.Kz

Introduction
============

It is of general believe that due to the close $f$-shell Gd ions do not show exotic magnetic behaviour in solid-state environment. However, recently it has been reported that some Gd alloys exhibit interesting phenomena including large magnetoresistance effects @Eremin01 [@Felser99; @Ahn00; @Mallik98a]. A metal-to-insulator-like transition has been reported for Gd$_5$Ge$_4$ @Levin01. In addition, a large number of the Gd-based intermetallics reveal large negative magnetoresistance (MR) effects arising from peculiar magnetic precursor effects @Mallik98. Recently, the Gd transition-metal silicides and germanides with 2:1:3 stoichiometry (AlB$_2$-type hexagonal structure @Kotsanidis90) came into focus, as some of these compounds even show a Kondo-like minimum in the temperature dependence of the resistivity well above the magnetic ordering temperatures @Mallik98a [@Majumdar00; @Majumdar01]. In polycrystalline Gd$_2$PdSi$_{3}$ ($T_N=21$ K), the compound of interest in this article, a pronounced minimum occurs at about 45 K, which has been confirmed in single crystalline Gd$_2$PdSi$_3$ Saha *et al. *@Saha99. The minimum persists even when diluting the Gd sublattice by Y in (Gd$_{1-x}$Y$_{x}$)$_2$PdSi$_{3}$ @Mallik98a. Chaika *et al. *investigated the electronic structure for $R_2$PdSi$_3$ with $R$=La,Ce,Gd and Tb by photoemission and band-structure calculations @Chaika01 and found that in Gd$_2$PdSi$_3$ the Gd 4*f* level is far below the Fermi level and hence should not significantly hybridize with the band states. In order to shed some light on the unusual magnetic and electronic properties of this compound we systematically investigated the spin dynamics of (Gd$_{1-x}$Y$_{x}$)$_2$PdSi$_{3}$ by electron-spin resonance (ESR).

Experimental details and results
================================

The samples were prepared by arc melting and are taken from the same batch as those used in previous investigations by Mallik *et al. *@Mallik98a. ESR measurements were performed in a Bruker ELEXSYS E500 CW-spectrometer at X-band frequencies ($\nu \approx$ 9.47 GHz) equipped with a continuous He-gas-flow cryostat in the temperature region $4.2<T< 300$ K. The polycrystalline samples were powdered, placed into quartz tubes and fixed with paraffin. ESR detects the power $P$ absorbed by the sample from the transverse magnetic microwave field as a function of the static magnetic field $H$. The signal-to-noise ratio of the spectra is improved by recording the derivative $dP/dH$ with lock-in technique.

ESR spectra and intensity
-------------------------

Typical ESR spectra are presented in Fig. \[spectra\] and illustrate the evolution of the Gd resonance with Y concentration $x$ (left column) and temperature $T$ (right column).

As expected, no ESR signal could be observed for $x=1$. For $x<1$, within the whole paramagnetic regime the spectrum consists of a broad, exchange narrowed resonance line, which is well fitted by a Dysonian line shape @Feher55. As in the present compounds the linewidth $\Delta H$ is of the same order of magnitude as the resonance field $H_{res}$, both circular components of the exciting linearly polarized microwave field have to be taken into account. Therefore the resonance at the reversed magnetic field $-H_{res}$ has to be included into the fit formula for the ESR signal, given by $$\frac{dP}{dH} \propto \frac{d}{dH}\left\{\frac{\Delta H + \alpha
(H-H_{res})}{(H-H_{res})^2 + \Delta H^2} + \frac{\Delta H + \alpha
(H+H_{res})}{(H+H_{res})^2 + \Delta H^2}\right\} \label{dyson}$$ This is an asymmetric Lorentzian line, which includes both absorption and dispersion, where $\alpha$ denotes the dispersion-to-absorption ratio. Such asymmetric line shapes are usually observed in metals, where the skin effect drives electric and magnetic microwave components out of phase in the sample and therefore leads to an admixture of dispersion into the absorption spectra. For samples small compared to the skin depth one expects a symmetric absorption spectrum ($\alpha$ = 0), whereas for samples large compared to the skin depth absorption and dispersion are of equal strength yielding an asymmetric resonance line ($\alpha$ = 1). At high Y concentrations the spectra are nearly symmetric with respect to the resonance field in accordance with the pure absorption spectra for $\alpha$=0. With decreasing $x$ they become more and more asymmetric corresponding to an increasing parameter $\alpha$. To check, whether the skin effect is the reason for the asymmetric line shape, we have to estimate the skin depth $\delta = (\rho / \mu_0 \omega)^{0.5}$ from the electric resistance $\rho$ and the microwave frequency $\omega = 2
\pi \times 9$ GHz ($\mu_0 = 4 \pi \times 10^{-7}$ Vs/Am). Using the resistivity values determined by Mallik *et al. *@Mallik98a - for example at 60 K for $x = 0$, $\rho$ = 0.25 m$\Omega$cm - we find a skin depth $\delta =$ 5.7 $\mu$m. Therefore the skin depth is comparable to or even smaller than the grain size $(\approx 40 \mu m)$ in accordance with the asymmetry of the ESR spectra. The large increase of the linewidth on decreasing temperatures (see Fig. \[spectra\], right column) results in an increasing uncertainty in determining both, resonance field $H_{\rm res}$ and dispersion-to-absorption ratio $\alpha$ accurately, because both quantities are correlated, while the linewidth remains relatively unaffected. As the resistivity of the present compounds $T \leq 150$ K changes by only 20% @Mallik98a, it is reasonable to fix the parameter $\alpha$ for $T < 50$ K at a value determined for $T>50$ K, where $\alpha$ is found to be nearly temperature independent (e.g. $\alpha = 0.7$ for $x = 0.2$).

The integrated intensity $I_{\mathrm{ESR}}$ of the resonance line measures the spin susceptibility $\chi_{S}$ of the ESR probe, $I_{\mathrm{ESR}}\sim\chi_{S}$. In the case of nonzero dispersion the intensity is determined by $I_{\mathrm{ESR}} = A \cdot \Delta
H^2 (1+\alpha^2)^{0.5}$, where $A$ denotes the amplitude of $dP/dH$ and $(1+\alpha^2)^{0.5}$ takes changes of the dispersion into account. However, one has to be careful with this formula on approaching $\alpha=1$. If the skin depth is small compared to the grain size, the microwave probes only a small fraction of the sample and changes of the resistivity with temperature also change this fraction. Then the ESR intensity is not a sensible measure of the spin susceptibility anymore. Fortunately, the relative changes of the resistivity are small in the temperature regime of interest, and hence the ESR intensity is even useful, when $\alpha$ approaches unity. Figure \[intensity\] shows both $I_{\mathrm{ESR}}$ and the susceptibility data from Mallik *et al.* @Mallik98a for $x=0.2$. Good agreement between the two sets of data can be observed throughout the whole concentration range in the paramagnetic regime. The peaks in the intensity appear at $T=20,15$ and 8K for $x=0,0.2$ and 0.5 respectively.

![\[spectra04\] ESR spectra of (Gd$_{0.8}$Y$_{0.2})_2$PdSi$_3$ for $T\leq T_{\mathrm N}$ (4-15 K), which have to be multiplied by the given amplification factors in order to get the original amplitude.](figure3.eps){width="80mm"}

Below the Néel-Temperature $T_{\rm N}$ we still observe a distorted resonance signal for $x\leq 0.5$ as shown in Fig. \[spectra04\] for $x=0.2$, but due to the reasons mentionend above deviations between ESR intensity and the magnetic susceptibility appear. The inverse quantities $1/I_{\mathrm{ESR}}$ and $1/\chi$, which are sketched for $x=0.2$ in the inset of Fig. \[intensity\], exhibit a linear temperature dependence in the paramagnetic regime ($T>T_{\rm N}$). The samples with $x=0,0.2$ and 0.5 follow a ($T-\Theta_{\mathrm CW})^{-1}$ Curie-Weiss (CW) law, with positive Curie-Weiss temperatures $\Theta _{\mathrm CW}$ of 25,17 and 6 K, respectively. For $x>0.5$ the reciprocal ESR intensities $1/I_{\mathrm{ESR}}$ show no deviations from a Curie law ($\Theta _{\mathrm CW}=0$). The resonance lines below $T_{\rm N}$ (Fig. \[spectra04\]) exhibit a splitting into a broad absorption band with decreasing temperature, which can be interpreted in terms of antiferromagnetic powder spectra with a small antiferromagnetic gap @Smirnov02. However, the observation of resonance lines below $T_{\rm N}$ is a rather unusual feature of intermetallics @Taylor74.

Temperature dependence of the ESR linewidth and the resonance field
-------------------------------------------------------------------

Generally, the ESR linewidth $\Delta H$ measures the spin-spin relaxation rate $1/T_2$ of the Gd-spins, whereas the resonance field $H_{\mathrm{res}}$ or effective g-value ($\bar
\omega_{\mathrm L}=g\mu_B H_{\mathrm{res}}$) gives information about the local static magnetic field at the Gd site. The linewidth is plotted versus temperature for all investigated samples in Fig. \[linewidth\]. As the temperature is lowered, the linewidth shows a monotonous behaviour that can be fitted very well with a linear function $\Delta H = c\cdot T$. Then the linewidth passes a mininum at $T_{min}\approx 2 T_N$ and increases drastically while approaching the transition temperature to the magnetically ordered state. This feature is clearly seen for $x\leq 0.5$, whereas for higher Y concentrations we can only detect the onset of a line broadening in the temperature range investigated.

![\[linewidth\] Temperature dependence of the ESR linewidth $\Delta H$ in (Gd$_{\mathrm{1-x}}$Y$_{\mathrm{x}})_2$PdSi$_3$. The solid lines represent a linear fit for $T>T_{min}$, yielding a concentration dependence (inset) of the slope according to eq. (\[korringaslope\]).](figure4.eps){width="80mm"}

In Fig. \[resonancefield\] we show the temperature dependence of the resonance field $H_{\mathrm{res}}$ for all samples under investigation. Again, we have to distinguish between the Gd-rich concentration range $x\leq 0.5$ and the samples with $x\geq
0.8$:\
At high temperatures $H_{\mathrm{res}}$ is nearly constant for all concentrations and yields a g-value $g \approx 1.99$ slightly below the free-electron value. For $T< 100$ K and $x\leq
0.5$ we observe a dramatic shift of the resonance to lower fields, which coincides with the onset of the line broadening mentioned above. Finally, a distinct kink and an upturn to higher fields follows. The peak-like minima for $x=0,0.2,0.5$ are found at 20, 15 and 6 K respectively. These temperatures are consistent with the ESR intensity and the magnetic-ordering temperatures obtained from measurements of the heat capacity and the magnetic susceptibility @Mallik98a.

Discussion
==========

The temperature dependencies of the $g$-value and the Gd-ESR linewidth bear important information on the interaction between the localized Gd-4f spins and the conduction electrons (for a review on the theory of ESR in metals, see e.g. Barnes @Barnes81, for experimental reviews see Taylor @Taylor75 and Elschner and Loidl @Elschner97). Due to its half-filled 4f shell ($4f^7$, spin $J=S=7/2$) the orbital momentum of Gd$^{3+}$ vanishes ($L=0$), and therefore the Gd spin does not relax directly to the lattice. But the energy is transferred via the exchange interaction to the conduction electrons, which themselves couple to the phonons of the lattice. This process is usually known as Korringa relaxation and yields a linear increase of the resonance linewidth $\Delta H_{\rm K}
\propto bT$ with increasing temperature $T$, where the slope $b$ is proportional to the squared electronic density of states $N^2(E_{\rm F})$ at the Fermi energy.

This holds as long as the back-scattering rate $\delta_{ei}$ of the conduction electrons to the Gd ions is small compared to the scattering rate $\delta_{el}$ of the conduction electrons to the lattice. If however the former becomes comparable or even larger than the latter, the so called bottleneck effect occurs and the Korringa law is modified by the ratio $B =
\delta_{el}/\delta_{ei}$ of both rates. This results in a reduced linear increase following @Barnes81: $$\label{korringaslope} \Delta H(T)=\Delta H_K(T)\frac{B}{1+B}$$ As the back-scattering rate $\delta_{ei}$ is proportional to the concentration of the localized spins (here $\delta_{ei}\sim$ Gd concentration $(1-x)$), it is possible to prove the influence of the bottleneck effect in a given compound by the dependence of the linewidth on the concentration of magnetic ions.

The inset of Fig. \[linewidth\] shows the slope $c$ of the linear high-temperature increase of the linewidth $\Delta
H(T)=c\cdot T$ in (Gd$_{1-x}$Y$_x$)$_2$PdSi$_3$ as a function of the Gd-concentration (1-x). It is well fitted by eq. (2), assuming a constant electron-lattice relaxation rate $\delta_{el}$, where $B$ can be expressed as $B=A/(1-x)$ with a constant parameter $A$. The resulting Korringa rate $b=10$ Oe/K is typical for usual metals and related intermetallic compounds (e.g. LaCu$_2$Si$_2$ @Schlott88). Hence, at high temperatures the system behaves like a regular metal under the strong influence of the bottleneck effect for all Gd concentrations. Usually in unbottlenecked metallic hosts the $g$-value of Gd differs from the value in non-metallic hosts by $\Delta g =N(E_F)J_{ie}(0)$ with the coupling constant $J_{ie}$ between the Gd-spins and the band states @Barnes81. The fact that at high temperatures the $g$ value is not strongly shifted from $g=1.997$, which is usually observed for Gd$^{3+}$ in insulators @Elschner97, is in accordance with the strong bottleneck. The 122-type compounds have been intensively investigated by ESR by Kaczmarska and coworkers during the last 15 years @Kaczmarska96. They found that the ESR behaviour of dense Gd systems depends on the transition metal with the thermal broadening parameter $c$ decreasing with increasing number of $d$ electrons from Co to Cu. For GdPd$_2$Ge$_2$ a value of 2.0 Oe/K was found,@Kaczmarska89 which is enhanced in comparison to 1.2 Oe/K for Gd$_2$PdSi$_3$ (see Fig. 3). This result is in agreement with regard to the relative Gd concentration, which is higher in the latter compound.

The increase of the resonance linewidth and the shift of the resonance field towards lower temperatures on approaching magnetic order is a common feature of intermetallic compounds containing rare-earth ions like Gd as reported in the experimental review by Taylor and Coles @Taylor74. Generally, for antiferromagnetic compounds they find no ESR signal below $T_{\mathrm N}$, but a broadening of the line at about $1.5T_{\mathrm N}<T<10T_{\mathrm
N}$. Some systems like Pd$_3$Gd and EuAl$_4$, however, show an ESR signal also below $T_{\mathrm N}$, which has been attributed this as common to materials which undergo metamagnetic transitions in magnetic fields comparable to the resonance field @Taylor74. Exactly such a behaviour has been reported for Gd$_2$PdSi$_3$. At 2 K Saha *et al.* observed two metamagnetic transitions in fields of 3 and 9 kOe @Saha99. On approaching $T_{\mathrm
N}$ from below both transitions coincide at about 3 kOe, the value of the resonance field. Together with the spectra we observed below $T_{\mathrm N}$ (see Fig. \[spectra04\]) for $x\leq 0.5$, we conclude that our system belongs to the same peculiar class of intermetallics as Pd$_3$Gd and EuAl$_4$.\
Apart from the existence of these signals and the suggested connection to metamagnetism, we found another very interesting feature:\
To analyze the linewidth in (Gd$_{1-x}$Y$_x$)$_2$PdSi$_3$ at low temperatures in more detail, we remind the magneto-resistance effects, which indicate the importance of the scattering of the conduction electrons on the localized Gd spins. Indeed, the comparison of the resistivity data reported by Mallik et al. @Mallik98a with the EPR linewidth shows that the onset of the line broadening and the resonance shift (Fig. \[resonancefield\]) coincides with the increase of resistivity for $x\leq 0.5$. In Fig. \[dHresist\] we show the ESR linewidth and resistivity for the pure compound using a scaling that reveals the similarity of the increase in resistivity and linewidth.

![\[dHresist\] Temperature dependencies of ESR linewidth ($\diamond$) and resistivity ($\bullet$, taken from @Mallik98a) in Gd$_2$PdSi$_{3}$.](figure6.eps){width="100mm"}

Unfortunately, only limited resistivity data has been reported for EuAl$_4$ @Taylor74 and Pd$_3$Gd making it difficult to judge unambiguously, whether a correlation between linewidth and resistivity could be another common property of those materials. To date, the origin of such a behaviour cannot easily be explained, but due to the fact that the Gd 4*f* level in Gd$_2$PdSi$_3$ is about 2 eV below the Fermi level the Kondo effect cannot be responsible for the observed minima @Chaika01, which is in agreement with the fact that the Korringa rate and therefore the density of states at the Fermi level is independent on $x$. A comparison with other metallic systems reveals that also in many ordered and disordered magnetic compounds a minimum of the ESR linewidth can be found at twice the critical temperature @Elschner97. The increase of the linewidth is attributed to spin fluctuations, which could also give rise to the resistivity behaviour. Another system where a correlation between ESR linewidth and resistivity has been observed is Fe doped La$_{2-x}$Sr$_{x}$CuO$_{4+ \delta}$ @Kruschel93, where localization effects were considered to explain that feature. Indeed, a localization scenario has also been proposed by Mallik *et al. *in order to explain the resistivity minimum in (Gd$_{1-x}$Y$_x$)$_2$PdSi$_3$ @Mallik98a. Very recently, Eremin *et al.* @Eremin01 gave an explanation for the magneto-resistance effects of another Gd alloy, namely GdI$_2$, which orders ferromagnetically close to room temperature @Ahn00 [@Felser99]. The concomitant anomalous peak in the resistivity has been attributed to scattering of the 5$d$ conduction electrons by localized 4$f^7$ electrons as a result of the specific topology of the Fermi surface. A similar mechanism could also account for the correlation in resistivity and linewidth in (Gd$_{1-x}$Y$_{x}$)$_2$PdSi$_{3}$.

Conclusion
==========

In conclusion we performed electron-spin resonance measurements in (Gd$_{1-x}$Y$_x$)$_2$PdSi$_3$. On approaching magnetic order, both the ESR linewidth and the resonance field are correlated with the resistivity minimum for $x\leq 0.5$. The high-temperature behaviour of the linewidth can be well described by a strong bottleneck effect, yielding a constant density of states at the Fermi level. The observation of antiferromagnetic-resonance-like absorption lines below the Néel temperature can be attributed to a metamagnetic transition in a magnetic field comparable to the resonance field. High-Field ESR measurements in single crystals should be performed in order to investigate the frequency and field dependence of the resonance lines. Thus, it should be possible to separate the influences of metamagnetic transitions, spin fluctuations and localization scenarios. Hopefully, such investigations will also trigger theoretical studies on localization effects in compounds with stable localized moments, which do not show a Kondo effect.

Acknowledgements
================

This work was supported in part by the BMBF under contract no. 13N6917 (EKM) and by the Deutsche Forschungsgemeinschaft (DFG) via the SFB 484.

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