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

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

Figure 1

(a) shows the time spectra for all As concentrations at $T=20$ mK. The solid lines denote the theoretical simulation as explained in the main text. (b) outlines the phase diagram of ${\mathrm{YbNi}}_4{\left({\mathrm{P}}_{1-x}{\mathrm{As}}_x\right)}_2$ as determined in Ref. [21] ($\square$) and compared with points derived from our $\mu \mathrm{SR}$ investigations ($◯, ⨂$). (c) depicts the fitted magnetic order parameter ($\mu \mathrm{SR}$ frequencies). The solid line is a fit to the phenomenological function $f_{\mu }$ (see main text).

Figure 2

(a)–(d) represent the muon-spin polarization $P\left(t\right)$ for $x=0$, 0.04, 0.08, and 0.13 at selected temperatures.

Figure 3

Temperature dependence of the dynamic muon-spin relaxation rate $\lambda /T=1/T_{1}T$ for different As concentrations in the paramagnetic state above the ordering temperature. Dotted lines indicate the power-law behavior $1/T_{1}T\propto T^{-n}$. The exponent $n$ slightly decreases from $1.13\pm 0.06$ to $1.01\pm 0.04$ with increasing $x$.

Figure 4

The main panel shows the magnetic field dependence of the dynamic muon-spin relaxation rate ${\lambda }_L$ for $x=0$, 0.04 and 0.08. The inset shows the magnetic field dependence of the muon-spin polarization at $T=150$ mK for $x=0.04$. The top panel shows the magnetic field dependence of the muon-spin polarization at $T=20$ mK for $x=0.08$. The dotted line represents Redfield fits (see main text).