Phonons, $Q$-dependent Kondo spin fluctuations, and $4f$ phonon resonance in $\mathrm{Yb}{\mathrm{Al}}_3$

$4f$ intermediate valence (IV) compounds are canonical hosts of correlated electron physics and can contribute to our understanding of the larger class of correlated electron materials. Here we study the prototype IV compound $\mathrm{Yb}{\mathrm{Al}}_3$ which exhibits a nonintegral valence with a moderately heavy fermion ground state and a large Kondo temperature ($T_{\mathrm{K}}\sim 500–600\mathrm{K}$). To better characterize the correlated physics of $\mathrm{Yb}{\mathrm{Al}}_3$, we have measured the phonon and the magnetic excitation spectra on single crystals of this material by time-of-flight inelastic neutron scattering and inelastic x-ray scattering. We have also performed theoretical calculations of the phonon spectra. We present three findings of these measurements. First, we observe that the measured phonon spectra can be described adequately by a calculation based on standard DFT+$U$ density functional theory. The calculated energies, however, are 10% too low compared to the measured energies. This discrepancy may reflect a hardening of the phonons due to dynamic $4f$ correlations. Second, the low-temperature spin fluctuations on the Kondo energy scale $k_{\mathrm{B}}{T}_{\mathrm{K}}$ have a momentum ($\mathit{Q}$) dependence similar to that seen recently in the IV compound $\mathrm{CeP}{\mathrm{d}}_3$. For that system, the $\mathit{Q}$ dependence has been attributed to particle-hole excitations in a coherent itinerant $4f$ correlated ground state. We suggest a similar origin for the momentum dependence seen in $\mathrm{Yb}{\mathrm{Al}}_3$. This $\mathit{Q}$ dependence disappears as the temperature is raised towards room temperature and the $4f$ electron band states become increasingly incoherent. Such a coherent/incoherent crossover is expected to be generic for correlated electron systems. Third, a low-temperature magnetic peak observed in the neutron scattering near 30 meV shows dispersion identical to a particular optic-phonon branch. This $4f/\mathrm{phonon}$ resonance disappears for $T\ge 150\mathrm{K}$. The phonon spectrum appears to be unaffected by the resonance. We discuss several possibilities for the origin of this unusual excitation, which may be unique to $\mathrm{Yb}{\mathrm{Al}}_3$. We suggest that the excitation may arise from the large amplitude beating of the light Al atoms against the heavy Yb atoms, resulting in a dynamic $4f/3p$ hybridization.

Figure 1

Calculated electronic band structure of $\mathrm{Yb}{\mathrm{Al}}_3$ demonstrating the effects of (a) Coulomb correlation ($U$) and (b) spin-orbit coupling (SOC). Unlike the phonons, the electronic bands are sensitive to $U$ and SOC: larger $U$ values drive the flat $f$ electronic states further from the Fermi level (0 eV), while SOC splits the $4f$ bands.

Figure 2

DFT calculated phonon spectra for $\mathrm{Yb}{\mathrm{Al}}_3$ with varying lattice parameter, $U$ value, and spin-orbit coupling (SOC). Green solid curves correspond to $U=6\mathrm{eV}$ with spin-orbit coupling and the experimental lattice constant ($a_{\exp }$). Red solid and red dashed curves correspond to $U=6\mathrm{eV}$ with no spin-orbit coupling using $a_{\exp }$ and the GGA relaxed lattice constant, respectively. The GGA relaxed lattice constant is larger than the experiment value and thus gives softer phonons. Blue curves correspond to $U=2\mathrm{eV}$ with no spin-orbit coupling using $a_{\exp }$. Note that spin-orbit interactions and varying $U$ in this range have little effect on the phonon dispersions.

Figure 3

The phonon structure factor for $\mathrm{Yb}{\mathrm{Al}}_3$ calculated for two values of $\mathit{Q}$, one near the fcc-diffraction-allowed Γ point (3,3,1) and one near (3,3,2) where fcc diffraction is not allowed. The calculations have been subjected to a Gaussian broadening. The very small structure factor near (3,3,1) reflects the fact that the aluminum atoms within the unit cell have opposing polarizations for optic phonons near Γ.

Figure 4

The phonon spectra measured by inelastic neutron scattering along the (a) Γ-$X$ [3,3,$L$], (b) Γ-$M$ [$H,H$,2], and (c) Γ-$R$ [$H,H,H$] and equivalent high-symmetry directions, using an initial neutron energy $E_i=60\mathrm{meV}$. The data are shown in different zones in the extended zone scheme, because the phonon structure factor is different in different zones. The solid lines are the result of the DFT phonon calculation for the case that the lattice constant is constrained to the experimental value and for the value $U=6\mathrm{eV}$; the energies obtained from the calculation have been multiplied uniformly by a factor 1.08 to give better agreement with the experimental data. The solid points in panel (a) represent the energy of the maximum in the magnetic resonance near 32 meV as derived in Fig. 13; these clearly track the lower optic-phonon branch at these energies.

Figure 5

Dispersion of acoustic phonons at the Brillouin-zone centers of $\mathrm{Yb}{\mathrm{Al}}_3$ at (a) 300 K and (b) at 10 K. The phonon velocities obtained from fits of linear or sine functions are indicated in the legend. $V_{\mathrm{p}}\left[\mathrm{HK}0\right]$ and $V_{\mathrm{s}}\left[\mathrm{HK}0\right]$ denote the velocities of longitudinal and transverse phonons, respectively, propagating along the [$\mathit{HK}0$] direction of reciprocal space.

Figure 6

Lattice fluctuation spectrum of $\mathrm{Yb}{\mathrm{Al}}_3$ at 300 K as measured by inelastic x-ray scattering (IXS). Individual spectra are shown as color plots. The superimposed markers indicate phonon energies and linewidths determined by fits of the spectra. Blue and green markers denote phonons excited with an overall momentum transfer almost parallel to the [100] and [110] directions of reciprocal space, respectively. The calculated phonon bands of $\mathrm{Yb}{\mathrm{Al}}_3$ are drawn as gray lines. The black lines and markers reproduce the data shown in Fig. 5.

Figure 7

Inelastic neutron spectra for the polycrystalline average of the spectra observed for a single crystal of $\mathrm{Yb}{\mathrm{Al}}_3$. The incident energy is 120 meV. (a) The spectra for three values of momentum transfer $Q$ at $T=5\mathrm{K}$. The solid black line represents the background scattering. The solid colored lines represent fits to the sum of the background scattering, phonons at 14, 22, and 32 meV, and two magnetic peaks, one at 32 meV and one at 48 meV. (b) The magnetic scattering obtained by subtracting the phonons and background as in panel (a) at four temperatures. The solid lines represent quasielastic scattering (with halfwidth Γ) which occurs at elevated temperature. (c),(d) The $Q$ dependence of the scattering at select energies and at $T=5$ and 100 K. The open triangles are the result of subtracting the 100 K data from the 5 K data for $\mathrm{\Delta }E=32.5\mathrm{meV}$. The solid line shows that these subtracted data follow the ${\mathrm{Yb}}^{3+} 4f$ form factor. The solid line for the $\mathrm{\Delta }E=50\mathrm{meV}$ data is the sum of a constant background and a term tracking the $4f$ form factor. [Representative error bars are shown in (a) for the $Q=6\text{−}{\AA }^{-1}$ data; in (b) for the 5 K data; and in (d) for $\mathrm{\Delta }E=32.5\mathrm{meV}$ and $T=5\mathrm{K}$.]

Figure 8

(a) A color $\mathit{Q}$-Δ$E$ slice of the data along $\mathit{Q}=\left(HH,3/2\right)$, showing columns of greater intensity for $H\sim 0.35$ and weak scattering for $H=0$. The same intensity variation can be seen in (b), which shows the scattering in the ($\mathit{HH},L$) plane for $\mathrm{\Delta }E=48\mathrm{meV}$. Bright spots are seen near (0.35,0.35,0.5) and symmetry-equivalent points, while weaker scattering is seen near (0,0,1.5) and yet weaker scattering near (0,0,2). (c) Cuts along ($\mathit{HH}$,3/2) and ($\mathit{HH}$,2) for $\mathrm{\Delta }E=48\mathrm{meV}$ again show strong scattering near (0.35,0.35,1.5), weaker scattering near (0,0,1.5), and yet weaker scattering at (0,0,2).

Figure 9

Magnetic spectra at three temperatures obtained from the total intensity by subtracting the background and dividing by the $4f$ form factor to reduce the data to the first Brillouin zone. At 5 K (a) the intensity in the Kondo scale (∼48 meV) peak decreases in the sequence $I$(1/2,1/2,1/2) > $I$(1/2,1/2,0) > $I$(1/2,0,0) > $I$(0,0,0) while the intensity in the peak near 32 meV follows the opposite sequence. At 100 K (b), the Kondo-scale scattering—albeit weaker—maintains this $\mathit{Q}$ dependence while the 32 meV scattering is strongly reduced. At 300 K (c) the scattering is $\mathit{Q}$ independent and quasielastic. Representative error bars are shown in (a) for the (0,0,0) data. In most cases the error bars are close to the symbol size.

Figure 10

The measured total scattering at four values of $\mathit{Q}$ and at 5 and 100 K using an incident neutron energy $E_i=60\mathrm{meV}$. The solid lines represent the background scattering; the dashed lines are the sum of the background and magnetic scattering at 100 K. The 32 meV phonon is essentially absent at (1,1,1) and (1,1,3/2), so that the magnetic scattering dominates, while the phonons dominate the 30–35 meV spectra at (3/2,3/2,0) and (1/2,1/2,1/2). [Representative error bars are shown in (a) for the 5 K data.]

Figure 11

The behavior of the 32 meV magnetic resonance. (a) The result of subtracting the background and the 30–35 meV optic phonons to obtain the magnetic spectra at 5 K. The resonance is strong at Γ and $X$, weaker at $M$, and vanishingly small at $R$ where the Kondo scale scattering dominates. (b) The temperature dependence of the (1,1,0) magnetic scattering. The 32 meV resonance vanishes rapidly as the temperature is raised. [Representative error bars are shown in (a) for the data at the Γ point and in (b) for the 5 K data.]

Figure 12

The top panel compares the resonant scattering along Γ-$X$ at 5 K to the calculated phonons (with lattice constant constrained to the experimental value and with energies multiplied by an additional factor 1.08). The phonon structure factor vanishes at (1,1,1), so the scattering there is entirely magnetic. The leading edge of the resonance tracks the optic-phonon dispersion along this direction. The lower panel shows that the resonance has disappeared by 150 K.

Figure 13

A comparison of the magnetic-resonance-dominated scattering at 5 K along (1,1,1 + $x$) to the phonon-dominated scattering at (3,3,2 + $x$). The initial neutron energy is $E_i=60\mathrm{meV}$. The magnetic resonance peak clearly tracks the increasing energy of the lower optic mode in the 30–35 meV range. [Representative error bars are shown in (a) for the (1,1,1) data.]

Figure 14

(a) The scattering minus background at (1,1,0) at 5 and 150 K measured with an initial neutron energy $E_i=60\mathrm{meV}$. The phonon peaks at 150 K have the same amplitude as at 5 K. The solid line is the sum of these phonon peaks and a magnetic line simulated with a Fano line shape (blue line) with the resonance factor $q=2.5$ and the bare linewidth 4.5 meV. (b) The scattering at (3,3,0), is in an equivalent zone as (a). The magnetic contribution at 5 K is reduced by the $4f$ form factor. (c) The scattering at (3,3,2) is dominated by the phonons, with a small magnetic contribution at 5 K. Note that the 31- and 35 meV peaks do not change position or amplitude with temperature. [Representative error bars are shown in (a) for the 5 K data.]

Figure 15

Comparison of IXS spectra at the $X$ points of reciprocal space at 10 and 300 K, in settings where $\mathit{Q}$ is parallel (bottom panel) and not parallel (top panel) to ${\mathit{q}}_{\mathrm{ph}}$. The spectra have been corrected for the Bose population factor and an arbitrary scale factor has been applied.

Figure 16

Two optic-phonon oscillation modes near 32 meV. Red spheres: Yb; blue spheres: Al. Upper panel: mode 9 at the Γ point. In this mode, the Al atoms at $(1/2,1/2,0)$ and $(0,1/2,1/2)$ move with equal amplitude but in opposite directions along the $y$ direction, while the other Al atom and the Yb atom remain fixed. Lower panel: mode 7 at the $M$ point. In this mode, the Al atoms at (1/2,0,1/2) move in the $x$ direction while the (0,1/2,1/2) atoms move in the $-y$ direction with equal amplitude while the other atoms remain fixed. Mode 9 at Γ is associated with the resonance while mode 7 at $M$ mode resonates weakly in comparison.