Lattice dynamics, crystal-field excitations and quadrupolar fluctuations of YbRu$_2$Ge$_2$

TL;DR

This study uses Raman scattering to investigate phononic and crystal-field excitations in YbRu$_2$Ge$_2$, revealing a quasi-quartet ground state, quadrupolar fluctuations, and coupling between phonons and CF transitions, suggesting ferroquadrupolar order.

Contribution

First detailed Raman analysis of YbRu$_2$Ge$_2$ revealing its CF level scheme, quadrupolar fluctuations, and phonon coupling, providing insights into its ferroquadrupolar phase.

Findings

CF ground state is a quasi-quartet with a ~2 cm$^{-1}$ splitting.

Lowest-energy CF transition intensity increases on cooling, indicating coupling to quadrupolar fluctuations.

Phonon intensities increase on cooling, suggesting near-resonant coupling with CF transitions.

Abstract

We report Raman-scattering results of YbRu$_2$Ge$_2$ single crystals to explore the phononic and crystal-field (CF) excitations. This heavy-fermion metal is suggested to enter a ferroquadrupolar (FQ) phase below T$_0$=10 K. The tetragonal CF potential splits the Yb$^{3+}$ $^2F_{7/2}$ ground multiplet into two $\Gamma_6$ and two $\Gamma_7$ Kramers doublets. We establish the following CF level scheme of the ground multiplet: a $\Gamma_6$ ground state, with the two $\Gamma_7$ states at 2 cm$^{-1}$, 95 cm$^{-1}$ and the other $\Gamma_6$ state at 239 cm$^{-1}$. The $\sim$2 cm$^{-1}$ separation between the CF ground and first excited states is in agreement with the previously proposed quasi-quartet CF ground state. The intensity of the lowest-energy CF transition remarkably increases on cooling, indicating a coupling of this CF excitation to the quadrupolar fluctuations above T$_0$. From symmetry analysis, we suggest that the FQ order has B$_{1g}$ symmetry. Moreover, temperature-dependent study of four Raman-active phonon modes shows that the intensities of the A$_{1g}$ and one E$_{g}$ modes increase significantly on cooling, which is explained by a near-resonant coupling between these two phonon modes and CF transitions.

Tables

Table 1. Table 1: The relationship between the scattering geometries and the symmetry channels. For scattering geometry E i E s , E i and E s are the polarizations of incident and scattered light; X, Y, X’, Y’ and Z are the [100], [010], [110], [1 1 ¯ ¯ 1 \overline{1} 0] and [001] crystallographic directions; R and L are right and left circular polarizations. A 1g , A 2g , B 1g , B 2g and E g are the irreducible representations of the D 4h group.

Scattering Geometry	Symmetry Channel
XX	A1g+B1g
XY	A2g+B2g
X’X’	A1g+B2g
X’Y’	A2g+B1g
XZ	Eg
RR	A1g+A2g
RL	B1g+B2g

Table 2. Table 2: Summary of the energy of the phonon and crystal-field (CF) modes. The coupled CF and phonon modes are labeled by “(c)”. Results of this work are compared to inelastic neutron scattering (INS) study  Jeevan ( 2010 ) . Units are cm -1 .

Mode	This work	INS
${\Gamma }_6^{\left(1\right)}\to {\Gamma }_7^{\left(1\right)}$	2	–
${\Gamma }_6^{\left(1\right)}\to {\Gamma }_7^{\left(2\right)}$	95	89
${\Gamma }_6^{\left(1\right)}\to {\Gamma }_6^{\left(2\right)}$ (c)	239	–
A1g (c)	268	260
B1g	168	170
E${}_{}{}^{\left(1\right)}{}_g{}^{}$	138	–
E${}_{}{}^{\left(2\right)}{}_g{}^{}$ (c)	301	–

Table 3. Table 3: The fitting parameters for the energy and FWHM of the four Raman-active optical phonon modes.

Mode	${\omega }_0$	${\omega }_2$	${\Gamma }_0$	${\Gamma }_2$
E${}_{}{}^{\left(1\right)}{}_g{}^{}$	138.4$\pm$0.1	0.70$\pm$0.03	1.7$\pm$0.5	0.3$\pm$0.1
B1g	167.92$\pm$0.01	0.212$\pm$0.002	6.08$\pm$0.02	0.094$\pm$0.005
A1g	270.15$\pm$0.04	2.27$\pm$0.02	3.6$\pm$0.1	1.71$\pm$0.07
E${}_{}{}^{\left(2\right)}{}_g{}^{}$	303.32$\pm$0.02	2.67$\pm$0.01	2.98$\pm$0.05	0.75$\pm$0.03

Equations

$$\omega (T)={\omega }_0-{\omega }_2[1+\frac{2}{e^{\hslash {\omega }_0/2k_{B}T}-1}],$$

$$\Gamma (T)={\Gamma }_0+{\Gamma }_2[1+\frac{2}{e^{\hslash {\omega }_0/2k_{B}T}-1}].$$

$${\chi }_{\mu }(0,T)=\frac{2}{\pi }{\int }_0^{{\omega }_{max}}\frac{{\chi }_{\mu }^{\prime \prime }(\omega ,T)}{\omega }d\omega ,$$

$${\chi }_{\mu }^{\prime \prime }(\omega ,T)\propto \frac{Q_{\mu }^2\omega }{{\omega }^2+{\gamma }_{\mu }^2}$$

$${\chi }_{\mu }(0,T)\propto \frac{Q_{\mu }^2}{T-T_W^{\mu }},$$

$$T_W^{\mu }={\kappa }_{\mu }{Q}_{\mu }^2,$$

$$T_Q^{\mu }=T_W^{\mu }+({\lambda }_{\mu }^2/C_{\mu })Q_{\mu }^2=({\kappa }_{\mu }+{\lambda }_{\mu }^2/C_{\mu })Q_{\mu }^2,$$

$$H=H_{CF}+H_{B_{1g}}.$$

$$H_{CF}=B_2^0{\hat{O}}_2^0+B_4^0{\hat{O}}_4^0+B_4^4{\hat{O}}_4^4+B_6^0{\hat{O}}_6^0+B_6^4{\hat{O}}_6^4$$

$$H_{Tetra}=B_2^0{\hat{O}}_2^0+B_4({\hat{O}}_4^0+5{\hat{O}}_4^4)+B_6({\hat{O}}_6^0-21{\hat{O}}_6^4).$$

$$H_{B_{1g}}=\frac{V}{2}({\hat{J}}_x^2-{\hat{J}}_y^2)=\frac{V}{2}({\hat{J}}_{+}^2+{\hat{J}}_{-}^2)$$

$$\Psi \propto (n_{{\Gamma }_5^{(1)}}-n_{{\Gamma }_5^{(2)}})Q_{B_{1g}},$$

$$\Psi \propto \frac{a-b}{a+b},$$

$$I.I.(T)=A{f}_{(1)}(T)[1-f_{(2)}(T)]+B,$$

$$H_{B_{2g}}=\frac{V^{\ast }}{2}({\hat{J}}_x{\hat{J}}_y+{\hat{J}}_y{\hat{J}}_x)=\frac{V^{\ast }}{4i}({\hat{J}}_{+}^2-{\hat{J}}_{-}^2).$$

$$\left(\begin{array}{ccc}0.655& 0& 0\\ 0& -0.346& 0\\ 0& 0& -0.309\end{array}\right)$$

$$\left(\begin{array}{ccc}0.176& -0.748& 0\\ -0.748& 0.176& 0\\ 0& 0& -0.352\end{array}\right)$$

$$H={\omega }_1{\hat{a}}_1^{†}{\hat{a}}_1+{\omega }_2{\hat{a}}_2^{†}{\hat{a}}_2+{\omega }_p{\hat{b}}^{†}\hat{b}+v({\hat{a}}_2^{†}{\hat{a}}_1-{\hat{a}}_1^{†}{\hat{a}}_2)({\hat{b}}^{†}+\hat{b}),$$

$$P(\omega ,T)=\frac{f_1(T)[1-f_2(T)]}{\omega -{\omega }_e+iϵ}-\frac{[1-f_1(T)]f_2(T)}{\omega +{\omega }_e+iϵ}),$$

$$D(\omega ,T)=\frac{1+n({\omega }_p,T)}{\omega -{\omega }_p+iϵ}-\frac{n({\omega }_p,T)}{\omega +{\omega }_p+iϵ},$$

$$I(\omega ,T)\sim \frac{1}{\pi }\Im T^{†}G(\omega ,T)T,$$

$$\begin{gathered} G(\omega ,T)=\frac{1}{1-P(\omega ,T)vD(\omega ,T)v} \\ \left(\begin{array}{cc}D(\omega ,T)& D(\omega ,T)vP(\omega ,T)\\ P(\omega ,T)vD(\omega ,T)& P(\omega ,T)\end{array}\right). \end{gathered}$$

$$\frac{1}{\pi }\Im [T_p^{2}D(\omega ,T)+2T_p{T}_{e}D(\omega ,T)vP(\omega ,T)+T_e^{2}P(\omega ,T)].$$

$$\underset{ϵ\to 0^{+}}{\lim }\frac{1}{\omega \pm {\omega }_0+iϵ}=\text{p.v.}\frac{1}{\omega \pm {\omega }_0}-i\pi \delta (\omega \pm {\omega }_0),$$

$$\begin{gathered} I(\omega ,T)\sim T_p^2[1+n({\omega }_p,T)]\delta (\omega -{\omega }_p) \\ +T_e^2{f}_1(T)[1-f_2(T)]\delta (\omega -{\omega }_e) \\ +2T_p{T}_{e}v[1+n({\omega }_p,T)]f_1(T)[1-f_2(T)] \\ [\frac{\delta (\omega -{\omega }_p)}{{\omega }_p-{\omega }_e}+\frac{\delta (\omega -{\omega }_e)}{{\omega }_e-{\omega }_p}]. \end{gathered}$$

$$\begin{gathered} I_p(\omega ,T)\sim T_p^2[1+n({\omega }_p,T)]\delta (\omega -{\omega }_p) \\ +2T_p{T}_{e}v[1+n({\omega }_p,T)]f_1(T)[1-f_2(T)]\frac{\delta (\omega -{\omega }_p)}{{\omega }_p-{\omega }_e}, \end{gathered}$$

$$T_p^2[1+n({\omega }_p,T)]\{1+2\frac{T_e}{T_p}\frac{v}{{\omega }_p-{\omega }_e}{f}_1(T)[1-f_2(T)]\}\delta (\omega -{\omega }_p).$$

$$\begin{gathered} {\chi }_p^{\prime \prime }(\omega ,T)\sim \\ {T}_p^2\{1+2\frac{T_e}{T_p}\frac{v}{{\omega }_p-{\omega }_e}{f}_1(T)[1-f_2(T)]\}\delta (\omega -{\omega }_p), \end{gathered}$$

$$I.I.(T)\sim T_p^2\{1+2\frac{T_e}{T_p}\frac{v}{{\omega }_p-{\omega }_e}{f}_1(T)[1-f_2(T)]\}.$$

$$I.I.(T)=A{f}_1(T)[1-f_2(T)]+B,$$

Full text

I Introduction

Multipolar interactions and related ordering phenomena have attracted great interest because, unlike commonly-known long-range orders of electric or magnetic dipole moments, multipoles are often related to more exotic phases which are difficult to probe directly by conventional methods Kuramoto et al. (2009); Santini et al. (2009). For systematic investigation of the collective behavior of multipole moments, $f$-electron systems are suitable choices since the strong coupling between spin and orbital degrees of freedom of $f$-electrons facilitates multipole formation. Indeed, the actinide dioxides with $5f$-electrons exhibit a variety of multipolar ordering phenomena Santini et al. (2009). For lanthanides with $4f$-electrons, multipolar, and especially quadrupolar orders, have been discovered for different systems Mignot et al. (2000); Hirota et al. (2000); Paixão et al. (2002); Kuwahara et al. (2006); Cameron et al. (2016).

YbRu2Ge2, a heavy-fermion metal with tetragonal structure (space group I4/mmm, No. 139; point group D4h), has been suggested to hold a ferroquadrupolar (FQ) order at low temperature Jeevan et al. (2006); Takimoto and Thalmeier (2008); Thalmeier et al. (2008); Nicklas et al. (2010); Prasad et al. (2010); Jeevan et al. (2011); Rosenberg et al. (2019); Pfau et al. (2019). It undergoes a second-order phase transition at TQ=10 K, before entering into an antiferromagnetic (AFM) phase below TN1=6.5 K Jeevan et al. (2006, 2011). At TN2=5.5 K, there may exist a small change in the magnetic structure Jeevan et al. (2011); Nicklas et al. (2010). Early studies show that the transition at TQ is not magnetic, and TQ increases when magnetic field is applied along the easy direction Jeevan et al. (2006, 2011). This behavior at TQ is similar to that of TmAu2 at its FQ ordering temperature Kosaka et al. (1998), hence suggesting a FQ phase in YbRu2Ge2. The existence of a FQ order is further supported by recent elastoresistivity studies where above TQ the elastoresistivity in the quadrupolar symmetry channels displays a Curie-Weiss behavior Rosenberg et al. (2019). Below TQ, an orthorhombic structural distortion is observed by X-ray diffraction, which confirms that the FQ state breaks B1g ($x^{2}-y^{2}$) symmetry Rosenberg et al. (2019).

The FQ order, namely the ordering of Yb3+ 4f-electron charge distribution at zero wavevector, can be probed indirectly by studying the lattice dynamics and crystal-field (CF) excitations. In a FQ arrangement, aligned charge quadrupoles uniformly distort the lattice via a coupling between the quadrupole moment and the strain field with the same symmetry. The induced distortion reduces the point-group symmetry of the lattice system, splitting degenerate phonon modes; the distortion also modifies the energy and lifetime of the phonon modes of the same symmetry. Such anomalies can be revealed by investigating the phonon spectra. Besides, the quadrupolar moments are carried by the CF ground state of Yb3+. The tetragonal CF potential splits the ${}^{2}F_{7/2}$ ground multiplet into two $\Gamma_{6}$ and two $\Gamma_{7}$ Kramers doublets. The magnetic entropy right above TQ is nearly R ln 4 Jeevan et al. (2006), suggesting that the CF ground state is a quasi-quartet consisting of two quasi-degenerate Kramers doublets. The quasi-quartet ground state was recently confirmed by angle-resolved photo-emission spectroscopy studies Pfau et al. (2019). This quasi-quartet near degeneracy is essential for forming a quadrupolar ground state and deserves a detailed study.

Raman spectroscopy is a conventional tool for studying phonon modes Klein (1982) and CF excitations Schaack (2000). Here we study the lattice dynamics, low-energy quadrupole fluctuations, and CF excitations in YbRu2Ge2. We assign four Raman-active phonon modes, and reveal an anomalous intensity enhancement of two phonon modes on cooling. The three CF transitions within the ${}^{2}F_{7/2}$ ground multiplet are identified and a CF level scheme is in turn established. We demonstrate that low-energy Raman response undergoes remarkable enhancement on cooling towards TQ and that the static electronic Raman susceptibility in the corresponding quadrupole channels follows nearly perfect Curie behavior, signifying that the relatively strong coupling to the lattice in the B1g-symmetry channel enhances by about 10 K the vanishingly small electronic Weiss temperature to the FQ transition temperature TQ.

II Experimental

Single crystals of YbRu2Ge2 were grown by flux method; details of the growth can be found in Ref Rosenberg et al. (2019). Two samples were used in this study: one was cleaved in ambient condition to expose its xy crystallographic plane, the other had a clean as-grown xz crystallographic plane. The xy crystallographic plane was examined under a Nomarski microscope to find about 200$\times$200 $\mu$m2 strain-free area.

Raman scattering measurements were performed in a quasi-back scattering geometry from sample placed in a continuous helium-gas-flow cryostat. We used 476.2, 647.1 and 752.5 nm lines from a Kr+ ion laser for excitation. Incident light with no more than 14 mW power was focused to a 50$\times$100 $\mu$m2 spot. Particularly, for measurements below 10 K, the power of the incident light was reduced to 2 mW. The temperatures reported in this paper were corrected for laser heating, which was estimated to be  0.75 $\pm$ 0.25 K/mW 111Optical absorption coefficient and thermal conductivity data for YbRu2Ge2, which are required for a model estimation of the laser heating Maksimov et al. (1992), are currently unavailable. Hence, we base the estimate of laser heating at low temperatures on comparison with other heavy-fermion metals with comparable electrical conductivity Rosenberg et al. (2019): CeB6 Takase et al. (1980) and URu2Si2 Palstra et al. (1986), for which laser heating for the same experimental set-up was established in the range of 0.5$\sim$1.0 K/mW..

Seven polarization configurations were employed to probe excitations in different symmetry channels. The relationship between the scattering geometries and the symmetry channels Hayes and Loudon (1978) is given in Table 1.

We used a custom triple-grating spectrometer with a liquid-nitrogen-cooled charge-coupled device (CCD) detector for analysis and collection of the scattered light. The data were corrected for the spectral response of the system. The measured secondary-emission intensity $I(\omega,T)$ is related to the Raman response $\chi^{\prime\prime}(\omega,T)$ by $I(\omega,T)=[1+n(\omega,T)]\chi^{\prime\prime}(\omega,T)+L(\omega,T)$, where $n$ is the Bose factor, $\omega$ is energy, $T$ is temperature. $L(\omega,T)$ represents the far tail of photo-luminescence, which in the narrow spectral window of interest was approximated by a linear frequency dependence.

III Results and Discussion

III.1 Lattice Dynamics

The spectra of phonon modes are presented in Fig. 1. By group theory, four Raman-active optical phonon modes are expected for YbRu2Ge2 structure: $A_{1g}\oplus B_{1g}\oplus 2E_{g}$. A1g and B1g modes are accessible in XX geometry and Eg modes in XZ geometry. The phonon energies at 13 K are tabulated in Table 2.

In Fig. 2 we show the temperature dependence of the spectral parameters (energy, FWHM, and integrated intensity) of the phonon modes. The spectral parameters were obtained by fitting the measured spectral peaks with Lorentzian lineshapes.

The temperature dependence of both frequency and FWHM of the phonon modes [Fig. 2(a) and (b)] is in accordance with a simple model assuming anharmonic decay into two phonons with identical frequencies and opposite momenta Klemens (1966); Menéndez and Cardona (1984):

[TABLE]

and

[TABLE]

The fitting results are summarized in Table 3. Small deviations between the measured energy and the fitting curve for the B1g mode could be due to an additional decay channels, for example decay into one acoustic and one optical mode.

The integrated intensity of the A1g and E${}_{g}^{(2)}$ phonon modes has more than 50% increase on cooling, in contrast to the behavior of the B1g phonon mode, whose integrated intensity is nearly temperature-independent [Fig. 2(c)]. The increase of the integrated intensity on cooling suggests a coupling of a CF transition to these two phonon modes Cooper et al. (1987). This coupling is enhanced when the energies of the CF splitting and the phonon modes are close. Indeed, such a CF excitation, 239 cm*-1* at 13 K, exists. The mechanism of this coupling will be discussed in Subsection. III.3.2.

Because the FQ order parameter is of B1g symmetry Rosenberg et al. (2019), the energy and lifetime of the B1g phonon mode are expected to exhibit anomalies across TQ due to electron-phonon coupling. Moreover, breaking of the four-fold rotational symmetry should split the two Eg phonon modes Zhang et al. (2016). However, as shown in the insets of Fig. 2, B1g and Eg phonon modes do not exhibit significant anomaly across TQ. Eg phonon modes do not show notable splitting at 4.5 K, either [Fig. 1]. The splitting of the E${}_{g}^{(1)}$ phonon mode is challenging to observe due to its weak intensity. Because the FWHM of the E${}_{g}^{(2)}$ phonon mode is 4 cm*-1* at 4.5 K, we set the upper limit of the splitting of the Eg phonon modes to be about 4 cm*-1* at 4.5 K.

III.2 Quadrupolar fluctuations

In the tetragonal phase above $T_{Q}$, the four-fold rotational symmetry along the z-axis is preserved and the CF ground state supports no static xy-plane quadrupole moment. However, dynamical quadrupolar fluctuations with zero time average quadrupolar moment are allowed 222Detailed discussion of the symmetry breaking and the wavefunction mixing can be found in Subsection.III.3.

In Fig. 3 we show the spectra of low-energy quadrupolar fluctuations. They are present in RL geometry yet absent in RR geometry [Fig. 3(a)]. By group theory, the absence of A1g and A2g components indicates that the CF ground state is a quasi-quartet composed of one $\Gamma_{6}$ and one $\Gamma_{7}$ doublets.

The static Raman susceptibility $\chi_{\mu}(0,T)$ in the symmetry channel $\mu$ ($\mu$ = B1g or B2g) can be obtained from the Raman response $\chi_{\mu}^{\prime\prime}(\omega,T)$ by virtue of the Kramers-Kronig relation Thorsmølle et al. (2016); Gallais and Paul (2016):

[TABLE]

in which we choose the upper cutoff for the spectra of fluctuations at $\omega_{max}$ = 40 cm*-1*, see Fig. 3.

We use Drude lineshape

[TABLE]

to extrapolate the Raman response below the instrumental cutoff 5 cm*-1*. In Eq. (4), $Q_{\mu}$ is the magnitude of the quadrupolar moment, and $\gamma_{\mu}$ reflects the decay rate. In the Raman scattering process light couples to the system’s charge quadrupole moment.

Theoretically, the low-energy Raman response in the quadrupolar channels contains both the lattice and the electronic contributions Thorsmølle et al. (2016); Gallais and Paul (2016). However, the energy of lattice fluctuations is much lower than the instrumental cutoff (5 cm*-1*), and Eq. (4) only takes into account of the electronic contribution. Thus, only electronic quadrupole fluctuations are included in the derived susceptibility $\chi_{\mu}(0,T)$.

The obtained temperature dependence of the static electronic Raman susceptibilities for both B1g and B2g quadrupole channels are shown in Fig. 4. The static Raman susceptibility $\chi_{\mu}(0,T)$ obeys Curie-Weiss temperature dependence

[TABLE]

where $T^{\mu}_{W}$ is the Weiss temperature:

[TABLE]

in which $\kappa_{\mu}$ measures the strength of the electronic intersite quadrupolar interactions.

By fitting the data, the ratio of $Q_{B_{1g}}$ to $Q_{B_{2g}}$ is determined to be nearly 1.4. The derived Weiss temperatures, T${}^{B_{1g}}_{W}$ $\sim$ $-2$ and T${}^{B_{2g}}_{W}$ $\sim$ $+2$ K 333We note that in the elastoresistivity studies, the Weiss temperature obtained from a fit to the static quadrupole-strain susceptibility is also a measure of an extrapolated bare electronic quadrupolar transition temperature because of clamped lattice Rosenberg et al. (2019). This is the reason why the Weiss temperatures extrapolated from the Raman susceptibility and from quadrupole-strain susceptibility agree.. The nearly exact Curie law is not surprising because both direct-exchange and super-exchange between local quadrupolar moments are expected to be vanishingly weak due to compactness of the $f$-orbitals. Although itinerant electrons provide effective coupling between the local magnetic dipole moments at Yb3+ sites, and the resulting RKKY interaction 444See Ref. Van Vleck (1962) and the references therein. leads to AFM order below TN1=6.5 K, these itinerant electrons do not provide a significant effective coupling between the local electric quadrupole moments at Yb3+ sites.

The true B1g-symmetry FQ order develops at TQ = 10 K Rosenberg et al. (2019), about 10 K above the Weiss temperature T${}^{B_{1g}}_{W}$. Because YRu2Ge2, the non-magnetic analog of the same structure, has no orthorhombic transition Bouvier et al. (1996); Rosenberg et al. (2019), the quadrupolar fluctuations of YbRu2Ge2 lattice themselves should have little tendency towards a structural instability. Nevertheless, coupling between the lattice strain fields and the local electronic quadrupole moments of the same symmetry enhances the transition temperature Böhmer et al. (2015); Böhmer and Meingast (2016); Thorsmølle et al. (2016); Gallais and Paul (2016):

[TABLE]

where $\lambda_{\mu}$ measures the coupling between the local charge quadrupole moments on Yb3+ sites and the lattice strain fields, and $C_{\mu}$ is the symmetrized elastic constant, which is (C11-C12)/2 for the B1g channel or C66 for the B2g channel Mitsumoto et al. (2013). The true quadrupolar transition temperature TQ equals to the largest of two T${}^{\mu}_{Q}$. Because the FQ order in YbRu2Ge2 has B1g symmetry, the T${}^{B_{1g}}_{Q}$ = TQ and non-realized T${}^{B_{2g}}_{Q}$ < TQ.

Tuning an additional parameter (magnetic field, pressure or doping, for instance) may affect the electron-lattice coupling and induce a transition from B1g FQ ordering to B2g FQ ordering. Indeed, although TQ stays constant up to application 9 GPa pressure with zero magnetic field Nicklas et al. (2010) and increases with in-plane magnetic field at ambient pressure Jeevan et al. (2006), experimental results do show suppression of TQ by Si Prasad et al. (2010) or Rh Pfau et al. (2019) doping, and by applying magnetic field under 1.23 GPa pressure Nicklas et al. (2010). These results suggest a competition between B1g- and B2g-symmetry FQ order.

III.3 Crystal-Field Excitations

Within the ${}^{2}F_{7/2}$ multiplet, there are three CF excitations corresponding to transitions from the CF ground state to the three CF excited states. From group theoretical considerations Koster (1963), the CF transitions between levels of the same symmetry (i.e. $\Gamma_{6}\rightarrow\Gamma_{6}$ or $\Gamma_{7}\rightarrow\Gamma_{7}$) contain A1g, A2g and Eg representations, whereas those between levels of different symmetry (i.e. $\Gamma_{6}\rightarrow\Gamma_{7}$ or $\Gamma_{7}\rightarrow\Gamma_{6}$) contain B1g, B2g and Eg symmetry representations. The Raman intensities in different symmetry channels may vary due to matrix element effect.

The lowest-energy CF transition, namely the transition between the two quasi-degenerate Kramers doublet does not clearly exhibit itself in the low-energy Raman spectra [Fig. 3]. The CF excitations from the ground state to the remaining two higher energy states are shown in Fig. 5 at 95 cm*-1* and 239 cm*-1*. These two transitions are expected to appear in all Raman-active symmetry channels, because the two low-lying doublets within the quasi-quartet have roughly the same population at 11 K. With the 476.2 nm excitation, the 95 cm*-1* transition indeed appears as a weak peak for four linear polarizations, while the 239 cm*-1* transition overlaps with the strong A1g phonon mode. With the 647.1 nm excitation, instead, the 95 cm*-1* transition becomes weaker, but the 239 cm*-1* transition is identifiable, manifesting itself as a peak in the RL spectrum and a shoulder in the RR spectrum.

The symmetry of the CF levels are assigned by the following argument: because YbRh2Si2 and YbIr2Si2, which have the same lattice structure as YbRu2Ge2, both have a $\Gamma_{6}$ CF ground state Kutuzov and Skvortsova (2011); Leushin and Ivanshin (2008), we suggest that the CF ground state of YbRu2Ge2 is also of $\Gamma_{6}$ symmetry (denoted as $\Gamma_{6}^{(1)}$). The other Kramers doublet within the quasi-quartet, in turn, is of $\Gamma_{7}$ symmetry (denoted as $\Gamma_{7}^{(1)}$).

The small difference of the excitation energy measured in RR and RL geometries near 239 cm*-1* [Fig. 5(b)] serves as an estimation of the quasi-quartet splitting. Using the Lorentzian fits, we find that the excitation energy measured in RR geometry is higher by 2 $\pm$ 1 cm*-1* than that in RL geometry at 25 K and 15 K. Therefore, the symmetry of the CF state at 239 cm*-1* is defined to be $\Gamma_{6}$ (denoted as $\Gamma_{6}^{(2)}$), and the quasi-quartet splitting is estimated to be 2 $\pm$ 1 cm*-1*. Because there are only two $\Gamma_{6}$ and two $\Gamma_{7}$ states within the ${}^{2}F_{7/2}$ multiplet, the CF state at 95 cm*-1* can only be of $\Gamma_{7}$ symmetry (denoted as $\Gamma_{7}^{(2)}$).

The energies of the CF excitations at 15 K are summarized in Table 2 555We note that would the CF ground state be $\Gamma_{7}$ symmetry, the symmetry of the CF states at 2 cm*-1*, 95 cm*-1* and 239 cm*-1* should instead be $\Gamma_{6}$, $\Gamma_{6}$ and $\Gamma_{7}$, respectively..

In an inelastic neutron scattering study of YbRu2Ge2, excitations at 89 cm*-1*, 170 cm*-1* and 260 cm*-1* are resolved at 5 K with the magnitude of momentum transfer being $\sim$1.9Å*-1* (Ref. Jeevan (2010)). Their data well match our assignments; the comparison is shown in Table 2. This consistency not only supports our assignments, but also suggests that the CF excitations and optical phonon modes have little dispersion.

III.3.1 Angular Electron-Cloud Distribution of the Crystal-Field States

We use the following single-ion Hamiltonian to calculate the angular electron-cloud distribution at Yb sites:

[TABLE]

The first term in Eq. (8)

[TABLE]

is the general expression for a CF potential of tetragonal site symmetry Fischer and Herr (1987). The $\hat{O}^{0}_{2}$, $\hat{O}^{0}_{4}$, $\hat{O}^{4}_{4}$, $\hat{O}^{0}_{6}$, and $\hat{O}^{4}_{6}$ operators are Stevens operators Stevens (1952). The five $B$’s are the CF coefficients.

From the CF level diagram, we cannot uniquely determine the CF Hamiltonian and wavefunctions if we assume tetragonal site symmetry. Hence, we approximate the real tetragonal CF potential with a dominating cubic CF potential Lea et al. (1962) plus a small $\hat{O}^{0}_{2}$ axial term:

[TABLE]

A cubic CF potential would split the ${}^{2}F_{7/2}$ multiplet into one quartet $\Gamma_{8}$, one doublet $\Gamma_{7}$, and one doublet $\Gamma_{6}$ states of Oh group. Reducing the cubic symmetry to the tetragonal symmetry, the quartet $\Gamma_{8}$ state of Oh group would be split into one $\Gamma_{7}$ and one $\Gamma_{6}$ states of D4h group. Because YbRu2Ge2 has a quasi-quartet CF ground state, it is possible that this quasi-quartet is induced by a small tetragonal perturbation to a large cubic CF potential. This small perturbation is represented by the first term in Eq. (10). We cannot rule out an alternative scenario that the quasi-quartet CF ground state of YbRu2Ge2 is of accidental degeneracy, rather than derived from the quartet $\Gamma_{8}$ state of cubic symmetry. Nevertheless, the Hamiltonian $H_{Tetra}$ preserves the 4-fold rotational symmetry along z-axis, and is sufficient to provide qualitative insights. In Appendix A we show that based on our assumption, the ratio of $Q_{B_{1g}}$ to $Q_{B_{2g}}$ is calculated to be 1.34, close to the experimentally determined ratio of 1.4 [Subsection.III.2]. This consistency supports our choice of Eq. (10). Experimentally, the wavefunction of the CF ground state could be determined by core-level non-resonant inelastic X-ray scattering, which has been used for Ce-based heavy fermion systems Sundermann et al. .

The second term in Eq. (8)

[TABLE]

represents the effective quadrupole-field (QF) potential of B1g symmetry. $V$ measures the strength of the QF potential.

Above TQ, there is no static B1g QF potential and we define $H=H_{Tetra}$. We diagonalize $H_{Tetra}$ in the basis of $|J,m_{J}\rangle$, where $J=7/2$ and $m_{J}$ are the quantum numbers of $\hat{\mathbf{J}}$ and $\hat{J}_{z}$, respectively. After diagonalization, the CF transition energies can be expressed in terms of $B_{2}^{0}$, $B_{4}$, and $B_{6}$. We fit the experimentally determined CF level diagram by these three adjustable parameters. There are four sets of parameters which reproduce the level diagram, and we choose the set with the smallest $B_{2}^{0}$ value. The fitting results thus are $B_{2}^{0}$ = -0.164 cm*-1*, $B_{4}$=0.0518 cm*-1*, and $B_{6}$=-0.00442 cm*-1*. The corresponding angular electron-cloud distribution of the CF states is plotted in Fig. 6.

Below TQ, there is a finite static B1g QF potential, here we define $H=H_{Tetra}+H_{B_{1g}}$. We assume that the values of $B_{2}^{0}$, $B_{4}$, and $B_{6}$ do not change. We diagonalize $H$ in the basis of $|J,m_{J}\rangle$, and after diagonalization, the CF transition energies can be expressed in terms of $V$. We find that $V$=0.523 cm*-1* renders an additional 2 cm*-1* splitting of the ground quartet. In Fig. 7, we plot the angular electron-cloud distribution of the ground quartet for $V$=0.523 cm*-1*. The charge distribution looks different from [100] and [010] directions because the $\Gamma_{5}^{(1)}$ and $\Gamma_{5}^{(2)}$ doublets carry B1g quadrupole moment. Furthermore, the quadrupole moment carried by $\Gamma_{5}^{(1)}$ state and that carried by $\Gamma_{5}^{(2)}$ state have approximately same magnitude but an opposite sign 666Strictly speaking, the quadrupole moments have the same magnitude only if we ignore the contribution by higher energy CF states..

The FQ phase transition reflects the competition between the entropy and energy terms in the Helmholtz free energy of the system. Above TQ, the entropy term dominates and the system prefers a quasi-degenerate CF ground state. Below TQ, instead, the system pursues lowest possible energy, and an orthorhombic quadrupolar field fulfills the goal: this field mixes the wavefunctions of the quasi-degenerate $\Gamma_{6}^{(1)}$ and $\Gamma_{7}^{(1)}$ states, increasing their separation and in turn reducing the ground state energy. In view of group-theoretical considerations, the $\Gamma_{6}$ and $\Gamma_{7}$ irreducible representations of the D4h group become the $\Gamma_{5}$ representation of the D2h group. Correspondingly, the $\Gamma_{6}^{(1)}$ and $\Gamma_{7}^{(1)}$ states of the D4h tetragonal phase are mixed by the Hamiltonian $H_{B_{1g}}$, and become the $\Gamma_{5}^{(1)}$ and $\Gamma_{5}^{(2)}$ states of the D2h orthorhombic phase.

There are two obvious choices for the macroscopic order parameter of the B1g-symmetry FQ phase. First is the quadrupole moment per unit volume:

[TABLE]

where $n_{\Gamma_{5}^{(1)}}$ and $n_{\Gamma_{5}^{(2)}}$ are the occupancy for the $\Gamma_{5}^{(1)}$ and $\Gamma_{5}^{(2)}$ states, respectively. The second choice is the lattice orthorhombicity, which couples to the quadrupolar order:

[TABLE]

where $a$ and $b$ are the in-plane lattice constant. The orthorhombicity as a function of temperature has been measured by X-ray diffraction Rosenberg et al. (2019).

III.3.2 Coupling between the Crystal-Field Transition and the Phonon Modes

The coupling between the $\Gamma_{6}^{(1)}\rightarrow\Gamma_{6}^{(2)}$ CF transition and the A1g and E${}_{g}^{(2)}$ phonon modes originates from the modulation of the electron-cloud distribution of CF states by lattice vibration [Fig. 6]. Such coupling is allowed by group theory because $\Gamma_{6}\otimes\Gamma_{6}\,=\,A_{1g}\oplus A_{2g}\oplus E_{g}$. We note that the phonon energy and linewidth can be well accounted for by the anharmonic decay model [Fig. 2(a) and (b)], suggesting that renormalization due to electron-phonon coupling is small. In the Appendix B we show that for small coupling strength, the temperature dependence of the integrated intensity of the phonon modes, $I.I.(T)$, has the following phenomenological expression:

[TABLE]

where $A$ and $B$ are two constants; $f_{(1)}(T)=2/Z(T)$ measures the occupancy of the $\Gamma_{6}^{(1)}$ CF state, and $f_{(2)}(T)=2e^{-E_{4}/k_{B}T}/Z(T)$ measures the occupancy of the $\Gamma_{6}^{(2)}$ CF state. $Z=2\sum_{i=1}^{4}e^{-E_{i}/k_{B}T}$ is the partition function; $E_{1}$=0 cm*-1*, $E_{2}$=2 cm*-1*, $E_{3}$=95 cm*-1* and $E_{4}$=239 cm*-1* are the energies of the CF levels [Table 2].

In Eq. (14), the constant $B$ represents the temperature-independent spectral weight of the phonon mode. Without the interaction $v$ and in the absence of a phase transition, the integrated intensity of the phonon modes is expected to be temperature-independent. The first term, which is temperature-dependent, can be interpreted as the spectral weight transferred from the CF mode to the phonon mode. This transferred spectral weight is proportional to the occupancy of the ground CF state $\Gamma_{6}^{(1)}$, and the un-occupancy of the excited CF state $\Gamma_{6}^{(2)}$. The constant $A$ is a measure of the transferred spectral weigh at zero-temperature 777We note that there is no general sum rule for Raman spectroscopy..

Because of the phase transition at TQ=10 K, Eq. (14) is only valid above 10 K. In addition, group theory allows the $\Gamma_{7}^{(1)}\rightarrow\Gamma_{6}^{(2)}$ CF mode to couple to the E${}_{g}^{(2)}$ phonon mode, which is not considered by simplified Eq. (14). Because the splitting between the $\Gamma_{6}^{(1)}$ and $\Gamma_{7}^{(1)}$ states is only 2 cm*-1*, including the contribution from the $\Gamma_{7}^{(1)}\rightarrow\Gamma_{6}^{(2)}$ CF mode will only influences the fitting curve at temperature much lower than TQ, a temperature range where Eq. (14) is invalid.

We use Eq. (14) to fit the phonon intensity data above 10 K in Fig. 2(c). For the A1g phonon mode, $A=3.14\pm 0.08$ and $B=0.06\pm 0.03$; for the E${}_{g}^{(2)}$ phonon mode, $A=2.08\pm 0.05$ and $B=0.35\pm 0.02$. These values show that at low-temperature, the integrated intensity of the A1g and E${}_{g}^{(2)}$ modes is mainly contributed by the transferred spectral weight. The fitting curves match the data well, which further supports our CF level scheme.

IV Conclusion

In summary, the Raman scattering study of YbRu2Ge2 focuses on the origin of the ferroquadrupolar transition, as well as on the spectroscopy of phonons and CF excitations within the ${}^{2}F_{7/2}$ ground multiplet of Yb3+ ion.

The deduced CF level scheme verifies the proposed quasi-quartet ground state, and we estimate that the splitting between two quasi-degenerate Kramers doublets is about 2 cm*-1*. The static electronic Raman susceptibilities in both B1g and B2g quadrupole channels essentially exhibit Curie law, signifying relatively strong coupling to the lattice in the B1g-symmetry channel that enhances the vanishingly small electronic Weiss temperature to the temperature of quadrupole phase transition at 10 K.

The temperature dependence of the energy and FWHM of the observed phonon modes are described by anharmonic decay model. The integrated intensities of the A1g and E${}_{g}^{(2)}$ phonon modes show more than 50% enhancement on cooling, which is explained by strong coupling between these phonons and the CF transitions with similar energies.

Acknowledgements.

We thank K. Haule and G. Khanal for discussions. The spectroscopic work at Rutgers (M.Y. and G.B.) was supported by NSF Grant No. DMR-1709161. The work at Stanford (E.W.R. and I.R.F.), including the crystal growth and characterization, was supported by the Gordon and Betty Moore Foundation Emergent Phenomena in Quantum Systems Initiative through Grant No. GBMF4414. G.B. also acknowledges the QuantEmX grant from ICAM and the Gordon and Betty Moore Foundation through Grant No. GBMF5305 allowing to make G.B. a collaborative visit to Stanford. Work at NICBP was supported by IUT23-3 grant.

Appendix A The effect of the B2g quadrupole-field potential

For completeness, we analyze here the effect of the B2g QF potential on the ground quasi-quartet. Following the treatment in Subsection. (III.3.1), we take $H=H_{Tetra}+H_{B_{2g}}$, where 888Asterisks are used to distinguish the symbols used for $H_{B_{2g}}$ from those used for $H_{B_{1g}}$.

[TABLE]

We find that for B2g potential, $V^{*}$ = 0.668 cm*-1* renders a 2 cm*-1* additional splitting of the ground quartet. On the contrary, for B1g potential $V$=0.523 cm*-1* renders a 2 cm*-1* additional splitting of the ground quartet. Hence smaller B1g QF potential is needed to induce the same additional splitting of the quasi-quartet. This result is consistent with the conclusion that the coupling between the local quadrupole moments and the lattice strain field is stronger in the B1g channel than in the B2g channel.

In Fig. 8, we plot the angular electron-cloud distribution of the ground quartet for $V^{*}$ = 0.668 cm*-1*. The charge distribution looks different from [110] and [1$\overline{1}$0] directions because the $\Gamma_{5}^{(1)*}$ and $\Gamma_{5}^{(2)*}$ doublets carry B2g quadrupole moment.

The traceless tensor of the electric quadrupole moments Jackson (1999), written in Cartesian coordinate with arbitrary units, for the $\Gamma_{5}^{(1)}$ wavefunction generated by the B1g field, Eq. (11), has the following values:

[TABLE]

Hence the magnitude of the B1g-symmetry electric quadrupole moment $Q_{B_{1g}}$ of the charge distribution of the $\Gamma_{5}^{(1)}$ wavefunction has a value of 1.00 when $V$=0.523 cm*-1*.

The same tensor for the $\Gamma_{5}^{(1)*}$ wavefunction generated by the B2g field, Eq. (15) has the following values:

[TABLE]

The magnitude of the B2g-symmetry electric quadrupole moment $Q_{B_{2g}}$ of the charge distribution of the $\Gamma_{5}^{(1)*}$ wavefunction is 0.748 when $V^{*}$ = 0.668 cm*-1*.

Therefore, for the same additional splitting of the ground quasi-quartet, the calculated ratio of $Q_{B_{1g}}$ to $Q_{B_{2g}}$ is 1.34. We recall that the experimentally determined ratio of $Q_{B_{1g}}$ to $Q_{B_{2g}}$ is 1.4. This consistency supports the assumptions made in Eq. (10), and shows that the wavefunctions we use are close to the real wavefunctions.

Appendix B Derivation of Eq. (14)

The Hamiltonian of the coupled CF transition and phonon mode can be written in second-quantization form as

[TABLE]

where $\hat{a}^{{\dagger}}$ and $\hat{a}$ are fermionic creation and destruction operators; $\hat{b}^{{\dagger}}$ and $\hat{b}$ are bosonic creation and destruction operators. The first and second terms describe respectively the energy of the lower and upper CF level; the third term is the phonon energy; and the last term is the coupling between the CF transition and phonon mode. Coefficient $v$ measures the strength of the coupling, which we take as a real number.

The CF transition corresponds to a bubble-shape Feynman diagram of electron-hole pair. Neglecting self-energy, the propagator has the following form:

[TABLE]

where $f_{1}(T)$ and $f_{2}(T)$ are respectively the temperature-dependent occupancy of the lower and upper CF level; $\omega_{e}=\omega_{2}-\omega_{1}$ is the energy of the CF transition; and $\epsilon$ is an infinitesimal positive value.

The phonon propagator is

[TABLE]

where $n(\omega_{p},T)$ is the Bose distribution function.

The experimentally-measured scattering rate, $I(\omega,T)$, has the form

[TABLE]

where T^{{\dagger}}=\left(\begin{array}[]{cc}T_{p}&T_{e}\end{array}\right) is the vertex of the light scattering, and $G(\omega,T)$ is the Green’s function of the Hamiltonian in Eq. (18). $G$ can be obtained by solving Dyson equation:

[TABLE]

In the following derivation, we assume $v$ is small so that the prefactor in the above expression can be replaced by unity. Then the approximated form of $I(\omega,T)$ is

[TABLE]

By virtue of Sokhotsky’s formula

[TABLE]

where p.v. stands for principle value, we derive from Eq. (23) the Stokes part of the scattering rate:

[TABLE]

Therefore, the phonon scattering rate, $I_{p}(\omega,T)$, is

[TABLE]

which can be arranged into

[TABLE]

The phonon response function, $\chi^{\prime\prime}_{p}(\omega,T)$, in turn, is

[TABLE]

and integration of $\chi^{\prime\prime}_{p}(\omega,T)$ yields the integrated intensity of the phonon mode $I.I.(T)$:

[TABLE]

Eq. (29) can be cast in a phenomenological form:

[TABLE]

where $A\sim\frac{T_{e}T_{p}v}{\omega_{p}-\omega_{e}}$ and $B\sim T_{p}^{2}$ are two constants. Eq. (30) is the same as Eq. (14) used in the Main Text to fit the experimentally-measured temperature-dependence of the integrated intensity of the A1g and E${}_{g}^{(2)}$ phonon modes.