Magnetic order and spin dynamics across a ferromagnetic quantum critical point: $\mu$SR investigations of YbNi$_4$(P$_{1-x}$As$_x$)$_2$

TL;DR

This study uses muon-spin relaxation to investigate how magnetic order and spin dynamics evolve across a ferromagnetic quantum critical point in a heavy-fermion system, revealing persistent slow fluctuations and reduced magnetic moments.

Contribution

It provides detailed experimental insights into the magnetic order and spin dynamics across the ferromagnetic quantum critical point in YbNi$_4$(P$_{1-x}$As$_x$)$_2$, highlighting the homogeneous nature of magnetism and critical slowing down.

Findings

Ordered Yb moments are strongly reduced with As substitution.

Magnetism remains homogeneous without disorder effects.

Critical fluctuations slow down approaching the QCP.

Abstract

In the quasi-1D heavy-fermion system YbNi$_4$(P$_{1-x}$As$_x$)$_2$ the presence of a ferromagnetic (FM) quantum critical point (QCP) at $x_c$ $\approx 0.1$ with unconventional quantum critical exponents in the thermodynamic properties has been recently reported. Here, we present muon-spin relaxation ($\mu$SR) experiments on polycrystals of this series to study the magnetic order and the low energy 4$f$-electronic spin dynamics across the FM QCP. The zero field $\mu$SR measurements on pure YbNi$_4$(P$_{2}$ proved static long range magnetic order and suggested a strongly reduced ordered Yb moment of about 0.04$\mu_B$. With increasing As substitution the ordered moment is reduced by half at $x = 0.04$ and to less than 0.005 $\mu_B$ at $x=0.08$. The dynamic behavior in the $\mu$SR response show that magnetism remains homogeneous upon As substitution, without evidence for disorder effect. In the paramagnetic state across the FM QCP the dynamic muon-spin relaxation rate follows 1/$T_{1}T\propto T^{-n}$ with $1.01 \pm 0.04 \leq n \leq 1.13 \pm 0.06$. The critical fluctuations are very slow and are even becoming slower when approaching the QCP.

Tables

Table 1. Table 1: Comparison of results obtained for YbNi 4 (P 1-x As x ) 2 with different As concentration.

$x$	$n^{\prime }$	fμ(20 mK)	$B_{\mathrm{local}}$(20 mK)	${\mu }_{\mathrm{ord}}$ (20 mK in ${\mu }_B$)
0.00	0.21 $\pm$ 0.02	0.1993	13.87 G	0.04
0.04	0.28 $\pm$ 0.05	0.0989	6.64 G	0.021
0.08	-	-	-

Equations

$$aP(t)$$

$$aP(t)=a_s\left[\frac{1}{3}+\frac{2}{3}(1-{\sigma }^2{t}^2)e^{-\frac{1}{2}{\sigma }^2{t}^2}\right]e^{-\lambda t}+a_{bg}{e}^{-{\lambda }_{bg}t},$$

Full text

††preprint: APS/123-QED

Quantum criticality is a central topic in the field of strongly-correlated electron physics. Canfield (2008); Broun (2008); Sachdev (2008); Gegenwart et al. (2008) In $d$- and $f$-electron based metallic systems close to an antiferromagnetic (AFM) ground state, quantum critical points (QCPs) are well established from both, the experimental and theoretical point of view. On the other hand, the existence of a ferromagnetic (FM) QCP in metals is controversially discussed. Brando et al. (2016) In fact, in clean 2D and 3D itinerant ferromagnets theoretical studies deny the presence of a FM QCP and the second order FM quantum phase transition (QPT) becomes first order before reaching the QCP. Belitz et al. (1999); Chubukov et al. (2004) This has been observed in several $3d$-electron ferromagnets like MnSi Pfleiderer et al. (1997), ZrZn2 Uhlarz et al. (2004) or in U-based systems Aoki et al. (2011). Other theories and experimental studies demonstrate that an inhomogeneous magnetic phase forms in between the paramagnetic and the homogeneous FM phase. Belitz and Kirkpatrick (1997); Conduit et al. (2009); Karahasanovic et al. (2012); Brando et al. (2008); Kotegawa et al. (2013) In other systems the formation of disordered phases, e.g. quantum Griffiths phase or spin-glass-like state, have been observed instead of a FM QCP. Westerkamp et al. (2009); Ubaid-Kassis et al. (2010); Lausberg et al. (2012) In $5f$-electron based metals like, UGe2 Taufour et al. (2010) or UCoGe Huy et al. (2007), it was observed that FM QCPs have been avoided by the formation of superconducting phases. In summary, evidence for the existence of a FM QCP in metallic systems has been lacking so far. However, the recent observation of power-law divergence of the Grüneisen ratio in the Kondo-lattice ferromagnet YbNi4(P1-xAsx)2 at the critical concentration $x_{\mathrm{c}}$ $\approx 0.1$ Steppke et al. (2013), which proves the existence of a QCP, Zhu et al. (2003) has reopened the discussion. This material has quasi-1D crystalline and electronic structures,Krellner et al. (2011) and a strong Kondo interaction.

1 D fluctuations are expected to supress the mechanism which induces the first order transition. Thus, it seems that in such materials FM QCPs could be observable. It became therefore of relevant importance to study the spin dynamics in YbNi4(P1-xAsx)2 across its FM QCP.

The heavy-fermion metal YbNi4P2 possesses a very low Curie temperature $T_{\mathrm{C}}$ = 0.15 K and a small ordered Yb moment due to the strong Kondo effect with a Kondo temperature of 8 K. Krellner et al. (2011); Spehling et al. (2012); Steppke et al. (2013) The quasi-1D crystal structure is due to Yb and Ni-tetrahedra chains along the $c$-axis of the tetragonal unit cell. Pivan et al. (1989) Susceptibility measurements between 50 and 400 K indicate clear Curie -Weiss behaviour with an effective moment close to the value of trivalent Yb. Krellner and Geibel (2012) A recent resonant X-ray emission spectroscopy (RXES) study reveal a low-temperature valence of Yb*+2.988*. Kummer et al. (2016) Non-correlated band-structure calculations found a quasi 1D electronic structure and a non-magnetic Ni4P2 sublattice. Krellner et al. (2011) The absence of Ni-related magnetism in YbNi4P2 is further supported by the fact that LuNi4P2 presents a non-magentic ground state. Kliemt and Cornelius (2016) In YbNi4P2  the presence of strong FM correlations was seen in NMR experiments above 2 K, Sarkar et al. (2012); Baenitz et al. (2013) and in inelastic neutron scattering.Huesges et al. (2015) $\mu$SR experiments were already performed on pure YbNi4P2 and have proved homogeneous static magnetic order with an ordered moment of about 0.04 $\mu_{\mathrm{B}}$. Above $T_{\mathrm{C}}$ the muon spin polarization obeys a time-field scaling relation indicating cooperative critical spin dynamics, with the dynamic spin-spin autocorrelation function following a power law behavior. These results suggest that the NFL behavior above $T_{\mathrm{C}}$ is induced by quasi homogeneous critical spin fluctuations. Spehling et al. (2012)

With a small amount of As substitution for P it is possible to tune $T_{\mathrm{C}}$ continuously to zero. Steppke et al. (2013) In fact, for $x=0.08$, $T_{\mathrm{C}}$ is suppressed below 25 mK and the sample with $x=0.13$ is non-magnetic (see Fig. 1b). For $x=0.08$, both the specific heat $C(T)/T$ and the volume thermal expansion coefficient $\beta(T)/T$ diverge according to power laws with unconventional exponents, namely, $T^{-0.43}$ and $T^{-0.64}$, respectively. Therefore the Grüneisen ratio $\Gamma(T)=\beta(T)/C(T)$ diverges as $T^{-0.22}$ indicating the presence of a nearby FM QCP for $x\approx$ $x_{\mathrm{c}}$.

Here, we present muon-spin relaxation ($\mu$SR) studies on polycrystalline YbNi4(P1-xAsx)2 with $x$ = 0, 0.04, 0.08, and 0.13 down to $T=20$ mK. We show that with increasing As substitution the ordered moment decreases continuously to values smaller than 0.005 $\mu_{\mathrm{B}}$ at $x\approx$ $x_{\mathrm{c}}$. In the paramagnetic state across the FM QCP the dynamic muon-spin relaxation rate 1/$T_{1}T\propto T^{-n}$ with an exponent value that decreases across the FM QCP from 1.13 $\pm$ 0.06 for $x\,=\,0$ to 1.01 $\pm$ 0.04 for $x=0.13$. This behavior can not be explained by any existing theory of FM quantum criticality. Finally, from the measured spin autocorrelation time, we find that the spin fluctuations are very slow and becomes even slower for $x\rightarrow$ $x_{\mathrm{c}}$.

The powder samples were prepared by milling single crystals of the respective concentration. The single crystals were grown by a modified Bridgman-method from a Ni-P self-flux in a closed Ta container and are from the same batches as used in Ref. Steppke et al., 2013. The excess flux was removed by centrifuging at high temperatures followed by etching in diluted nitric acid, details are described in Ref. [Kliemt and Krellner, 2016]. The phase purity of the used samples was analyzed by powder x-ray diffraction and magnetization measurements and we found phase pure samples beside small residuals of the flux of order 2%. The positive spin-polarized muons are implanted into the sample in a $\mu$SR experiment, and the time evolution of the muon-spin polarization, $P(t)$, is monitored by detecting the asymmetric spatial distribution of positrons emitted from the muon decay. Schenck (1985); Yaouanc and de Réotier (2011) Zero magnetic field (ZF) and longitudinal applied magnetic fields (LF) with respect to initial muon-spin polarization were used to perform $\mu$SR experiments. All experiments were carried out at the $\pi$M3 beam line at the Swiss Muon Source at the Paul-Scherrer-Institute, Switzerland. For thermal contact, the samples were glued on a Ag plate giving rise to a time and temperature independent background signal due to muons that stopped in the Ag plate (about 20-30 %). The $\mu$SR time spectra were analyzed using the free software package MUSRFIT. Suter and Wojek (2012)

Fig. 1a represents ZF $\mu$SR time spectra for $x=0$, 0.04, 0.08 and 0.13 collected at $T=20$ mK. The plot clearly indicates that the spontaneous frequency of oscillation associated with the static ordered Yb magnetic moment decreases with increasing As concentration. To extract the magnetic order parameters in the ordered state, the muon asymmetry data were modeled by the following function (solid lines in Fig. 1) Spehling et al. (2012); Barsov et al. (1991); Kornilov and Pomjakushin (1991)

[TABLE]

Here, $f_{\mathrm{\mu}}$, $\sigma$ and $\lambda_{bg}$ are the muon-precession frequency, the Gaussian field width of the internal field distribution and the constant background contribution, respectively. The 2/3 oscillating, and the 1/3 non-oscillating terms originate from the spatial averaging in polycrystalline samples, where 2/3 (1/3) of the internal magnetic field components are directed perpendicular (parallel) to the initial muon spin, causing a precession (no precession) of the muon spin. The observation of a 2/3 and 1/3 sample signal fractions, and the Gaussian relaxation of $P(t)$ below $T_{\mathrm{C}}$ proof that 100 % of the sample volume shows static magnetic ordering of the magnetic moments not only for $x\,=\,0.0$ but also for $x\,=\,0.04$. More detailed description of Eq. 1 can be found for example in Refs. Spehling et al., 2012; Barsov et al., 1991; Kornilov and Pomjakushin, 1991.

Spehling et al. have analyzed a number of potential muon stopping sites, but only considered high symmetry positions. Spehling et al. (2012) Looking at the distribution of atoms within this structure type we suspect the muon to sit on a lower symmetry position near the 4 $c$ site, and therefore calculated the size of the ordered moment from the field expected at this site. From the measured $\mu$SR frequency value $f_{\mu}=0.188$ MHz at $T=20$ mK, a local internal field at the muon site of $B_{\mathrm{local}}=13.87$ G can be determined, using the relation $B_{\mathrm{local}}=2\pi\times$f${}_{\mu}/\gamma_{\mu}$, where $\gamma_{\mu}=2\pi\times$13.55 kHz/G is the muon gyromagnetic ratio. On the other hand the fitted $\mu$SR frequency at 20 mK is 0.09 MHz for $x$ = 0.04, which corresponds to an internal field value of 6.642 G. The values are listed in Table 1. With increasing As substitution the ordered moment is reduced by half in $x=0.04$, and at $x\approx 0.08$, the ordered moment becomes smaller than 0.005 $\mu_{\mathrm{B}}$.

Figure 2 shows the ZF $\mu$SR time spectra for YbNi4(P1-xAsx)2 with $x=0$, 0.04, 0.08, and 0.13 concentrations at representative temperatures. A spontaneous $\mu$SR frequency is observed in the time spectra below $T=140$ mK, and $T=90$ mK for As concentrations $x=0$ and 0.04, respectively (Figs. 2 a,b). This proves the onset of static long-range magnetic ordering of the Yb moments. For $x=0$ and 0.04, above the magnetic ordering the $\mu$SR time spectra reveal a dominant exponential behavior. Here, a stretched exponential function $P(t)$ = $\exp{(-\lambda t)}^{\beta}$ was used to model $P(t)$ indicating that the muon-spin relaxation is due to the dynamics of electronic moments. $\lambda=1/T_{1}$ is the generalized muon-spin relaxation rate, and $\beta$ is the exponent. The exponent $\beta$ is the measure of the homogeneity of the system. Only slight enhancement (value varies 1-1.15) of $\beta$ is observed towards low temperatures suggesting that the system is homogeneous. The same functional form was used to describe $P(t)$ for $x$ = 0.08.  MacLaughlin et al. (2004); Michor et al. (2006)

In comparison to $x$ = 0 and 0.04, a clear development of magnetic order parameter is not observed for $x$ = 0.08 down to $T=19$ mK (see Fig. 2c). However, the $\mu$SR time spectra for $x=0.08$ look very similar to that of $x=0$ and 0.04 above $T>140$ mK and 90 mK, respectively. Given that for the $x=0.08$, $T_{C}$ = 28 mK was observed in $\chi_{AC}$, these $\mu$SR observations suggest that the static ordered magnetic moment is too small to be detected in the $\mu$SR time window which therefore sets an upper limit for the ordered Yb moment of $\mu_{B}$ $\leq$ 0.005 $\mu_{\mathrm{B}}$. Steppke et al. (2013) This is because the frequency of oscillations related to static magnetic order is too small to be captured in the current $\mu$SR time window. Another possibility, the sample pellet did not reach the temperature of the mixing chamber.

In Fig. 1c, the fitted $\mu$SR frequencies, $f_{\mathrm{\mu}}$, are plotted together with a phenomenological fit of the function $f_{\mathrm{\mu}}(T)=f_{\mathrm{\mu}}(0)(1-\frac{T}{T_{\mathrm{C}}})^{n^{\prime}}$ (solid lines) for $T<T_{\mathrm{C}}$, where $n^{\prime}$ is the exponent. From our fit we find that $n^{\prime}$ slightly increases from 0.21(2) to 0.28(5) with increasing As concentration. However the very low data point density for the $T$-range 0.6 $\leq T/T_{\mathrm{C}}\leq$ 1 prevents a precise estimation of exponents.

For the sample with $x=0.13$, all $\mu$SR time spectra reveal a pronounced Gaussian functional form instead of a stretched exponential (see Fig. 2d) as expected for a non-magnetic ground state in which the muon-spin polarization, $P(t)$, is due to the interaction with nuclear dipole moments only. $P(t)$ is best modeled by the following functional form:

[TABLE]

where $\sigma$ is the Gaussian field width and $\lambda$ is the muon-spin relaxation rate. In a simple paramagnet, $P(t)$ is due to static randomly oriented nuclear dipole moments with a $T$-independent Gaussian relaxation rate $\sigma$, theoretically described by the Kubo-Toyabe function,Hayano et al. (1979) which is the bracketed term in Eq. 2. An additional exponential term, $\exp(-\lambda t)$, is introduced to account for the extra muon relaxation due to electronic moments, where $\lambda$ is the dynamic muon-spin relaxation rate due to fluctuations of the electronic moments. At all temperatures, the data were analyzed using a fixed $\sigma=0.1316$ MHz determined at high temperatures in the PM state where $P(t)$ is due to nuclear moments only.

Figure 3 shows the fitted $\lambda/T$ ($=1/T_{1}T$) values as a function of $T$ for $x$ = 0, 0.04, 0.08 and 0.13. An increase of $\lambda/T$ is observed with lowering temperature following a power law behavior $1/T_{1}T\propto T^{-n}$. The exponent value decreases slightly with increasing $x$ from 1.13 $\pm$ 0.06 for $x=0$ to 1.01 $\pm$  for $x=0.13$. It should be noted that for the compound $x=0.13$, a different functional form was used to describe the experimental data. The fact that $1/T_{1}T\propto T^{-n}$ with a nearly same exponent for all concentrations is consistent with the observation that the temperature dependencies of the thermodynamic quantities, $C(T)/T$ and $\beta(T)/T$, show the same power-law exponent in the PM region of the phase diagram across the FM QCP. This indicates that disorder does not play a relevant role. For the sample with $x=0.08$, located closest to $x_{\mathrm{c}}$, $1/T_{1}T\propto T^{-1.07\pm 0.03}$ indicating that 1/$T_{1}$ is approximately $T$ independent .

Itinerant ferromagnetism is often discussed on the basis of spin fluctuation theories. Moriya (1985) Within the self-consistent renormalization (SCR) theory, the spin-lattice relaxation rate $1/T_{1}T$ in itinerant ferromagnets scales with the uniform magnetic susceptibility $\chi$ in 3D and $1/T_{1}T\propto\chi^{3/2}$ in 2D. Moriya (1985); Hatatani and Moriya (1995) The exponents have been calculated at the FM QCP and the expected value is $n=4/3$ in 3D, while in 2D there is no long range FM order and the SCR theory predicts $1/T_{1}T\propto T^{-3/2}(-\ln T)^{-3/2}\approx T^{-1.4}$ for $T\rightarrow 0$. Stewart (2001); Hatatani and Moriya (1995) Our exponents are therefore not close to these predictions, they are smaller. In addition, the unusual power law exponent in the thermodynamic quantities support our present findings, and suggests the inadequacy of the itinerant spin-fluctuations theory for the FM fluctuation in the close proximity of the QCP. Steppke et al. (2013)

Previous nuclear magnetic resonance (NMR) experiments have detected a spin-lattice relaxation rate $1/T_{1}T$ which follows a $\sim T^{-n}$ function also with a small $n=0.75$ in a broad temperature range above 2 K. Sarkar et al. (2012) Although NMR experiments were performed in an external magnetic field and crystalline electric field effects have to be considered in this temperature range, the exponent is still small compared to other Yb-based FM systems studied at the FM QPT in the same $T$ range. Carretta et al. (2009)

To understand the nature of magnetic correlations, in particular whether the magnetism is static or dynamic, $\mu$SR experiments in different longitudinal fields (LFs) up to $B_{\mathrm{LF}}=300$ G were performed. We found that above $T_{\mathrm{C}}$ and with small applied fields the muon spin-relaxation is completely suppressed in the samples with $x=0.0$ and 0.04 (see Ref. Spehling et al., 2012 and Fig. 4c). This indicates that the zero-field relaxation in the magnetic ordered state originates from static internal fields on the microsecond timescale.

Above the ordering temperature, we could fit all data for all As concentrations with a simple exponential function $P(t)=P(0)e^{-\lambda_{L}t}$ as shown in Fig. 4 (solid lines). The single exponential fit in the LF suggests that the system is free from magnetic disorder and is intrinsically homogeneous. The spin dynamics is characterized by a narrow distribution of correlation times for $x=0$ as well for $x=0.08$ near the FM QCP. With increasing field, the relaxation rates $\lambda_{L}$ decreases. The field dependence of $\lambda_{L}$ is presented in the main panel of Fig. 4b for all $x\leq 0.1$. They show very similar values. From the magnetic field dependence of $\lambda_{L}$, the spin autocorrelation time, $\tau_{c}$ can be estimated using $\lambda_{L}(B_{LF})=(2\gamma^{2}_{\mu}\left\langle B^{2}_{\mathrm{loc}}\right\rangle\tau_{c})/\left[1+(\gamma^{2}_{\mu}B^{2}_{LF}\tau^{2}_{c})\right]$. This is so called the Redfield formalism where $\tau_{c}$ is independent of the applied magnetic field $B_{LF}$. Here, $B_{\mathrm{loc}}(t)$ describes the time-varying local magnetic field at the muon site due to fluctuations of neighboring Yb $4f$ moments, with a local time averaged second moment $\Delta^{2}=\gamma^{2}_{\mu}\left\langle B^{2}_{\mathrm{loc}}\right\rangle$, and a single fluctuation time $\tau_{c}$, meaning a single exponential auto-correlation function characterized by a single time constant. For $\hbar\omega$ $<<$ $k_{B}T$ ($\omega$ giving the spin fluctuation frequency), the fluctuation-dissipation theorem Toll (1956) relates $\tau_{c}$ to the imaginary component of the local $q$-independent $f$-electron dynamic susceptibility, i.e., $\tau_{c}(B)=(k_{B}T)[\chi^{\prime\prime}/\omega]$. Dotted curves in the main panel of Fig. 4b are fits to the experimental data. The results of the fits give $\Delta^{2}\approx 0.1$, 0.1, and 0.108 (MHz) and $\tau_{c}\approx 6.00\times 10^{-7}$, 9.11$\times 10^{-7}$, and 14.11$\times 10^{-7}$ s for $x=0.00$, 0.04, and 0.08, respectively. With increasing $x$, $\tau_{c}$ becomes larger and it becomes even larger for $x\rightarrow$ $x_{\mathrm{c}}$. The estimated $\tau_{c}$ value for $x=0.08$ is much larger than that found in YbRh2Si2 Ishida et al. (2003) and in CeFePO Lausberg et al. (2012), respectively. These fluctuations can therefore be considered as very slow critical spin fluctuations. On the other hand similar $\Delta^{2}$ values for all the concentration suggest rather homogeneous system because $\Delta^{2}$ is the second moment of the field distribution at the muon stopping sites. If the system is inhomogeneous then one would expect rather different field distribution for different As concentration.

In conclusion, the present $\mu$SR experiments show that by introducing a small amount of As at the P site in the system YbNi4(P1-xAsx)2  the magnetic order remains homogeneous and the static ordered Yb moment is suppressed continuously from about 0.04 $\mu_{\mathrm{B}}$ to values lower than 0.005 $\mu_{\mathrm{B}}$ for $x\rightarrow$ $x_{\mathrm{c}}$ = 0.1. There is no evidence for disorder effect in the $\mu$SR samples. Notably, in the paramagnetic region of the phase diagram and across the FM QCP the dynamic muon-spin relaxation rate 1/$T_{1}T\propto T^{-n}$ with $n\approx 1$ indicating 1/$T_{1}$ is approximately $T$ independent. The critical fluctuations are very slow in the pure compound and become even slower when approaching the FM QCP. All these findings support the presence of a clean FM QCP in this system. However, the power laws observed in the $T$-dependence of thermodynamic quantities and in $1/T_{1}T$ remain to be understood.

We thank Z. Huesges for having prepared the samples for $\mu$SR experiments and C. Geibel for fruitful discussions. We are thankful to DFG for the financial assistance through the grant no. SA 2426/1-1, KR 3831/4-1, BR 4110/1-1, and SFB 1143 for the project C02.