Heavy-impurity resonance, hybridization, and phonon spectral functions in ${\text{Fe}}_{1-x}{M}_x\text{Si} (M=\mathrm{Ir}, \mathrm{Os})$

The vibrational behavior of heavy substitutional impurities $(M=\text{Ir},\text{Os})$ in ${\text{Fe}}_{1-x}{M}_x\text{Si}\left(x=0,0.02,0.04,0.1\right)$ was investigated with a combination of inelastic neutron scattering (INS), transport measurements, and first-principles simulations. Our INS measurements on single crystals mapped the four-dimensional dynamical structure factor, $S(\mathbf{Q},E)$, for several compositions and temperatures. Our results show that both Ir and Os impurities lead to the formation of a weakly dispersive resonance vibrational mode, in the energy range of the acoustic phonon dispersions of the FeSi host. We also show that Ir doping, which introduces free carriers, leads to softened interatomic force constants compared to doping with Os, which is isoelectronic to Fe. We analyze the phonon $S(\mathbf{Q},E)$ from INS through a Green's-function model incorporating the phonon self-energy based on first-principles density functional theory simulations, and we study the disorder-induced lifetimes on large supercells. Calculations of the quasiparticle spectral functions in the doped system reveal the hybridization between the resonance and the acoustic phonon modes. Our results demonstrate a strong interaction of the host acoustic dispersions with the resonance mode, likely leading to the large observed suppression in lattice thermal conductivity.

Figure 1

(a) Resistivity $\rho$, (b) Seebeck coefficient $\alpha$, and (c) thermal conductivity $\kappa$ for single-crystalline samples of FeSi, ${\mathrm{Fe}}_{0.98}{\mathrm{Os}}_{0.02}\mathrm{Si}$, and ${\mathrm{Fe}}_{0.98}{\mathrm{Ir}}_{0.02}\mathrm{Si}$ grown by the optical floating-zone technique.

Figure 2

(a) Phonon DOS of FeSi, ${\mathrm{Fe}}_{0.9}{\mathrm{Os}}_{0.1}\mathrm{Si}$, and ${\mathrm{Fe}}_{0.9}{\mathrm{Ir}}_{0.1}\mathrm{Si}$, measured with INS at $T=100\mathrm{K}$ ($E_i=80\mathrm{meV}$). Statistical error bars are comparable to the size of the symbols. (b) Difference in phonon DOS between the doped samples and pure FeSi, normalized to one impurity atom, compared with the energy derivative of FeSi DOS.

Figure 3

Single-crystal scattering function $S(\mathbf{Q},E)$ for (a) FeSi and (b) ${\mathrm{Fe}}_{0.96}{\mathrm{Ir}}_{0.04}\mathrm{Si}$, along the $\Gamma$-X direction $[3,3,L]$, measured with ARCS at $T=100\mathrm{K}$. The TA branch shows a clear splitting at $L=0.37,0.63$ in the doped sample, as indicated by the white arrows. The white areas correspond to missing data.

Figure 4

Single-crystal scattering function $S(\mathbf{Q},E)$ for ${\mathrm{Fe}}_{0.96}{\mathrm{Ir}}_{0.04}\mathrm{Si}$ along $[3,3,L]$ and slightly off the high-symmetry direction at $[2.9,2.9,L]$ and $[3.1,3.1,L]$, measured with ARCS at 100 K. The kink in the TA dispersions is seen around 17 meV in the three panels ($L\sim 1.37$).

Figure 5

Neutron-scattering spectra for single crystals of (a) ${\mathrm{Fe}}_{0.98}{\mathrm{Os}}_{0.02}\mathrm{Si}$ and (b) ${\mathrm{Fe}}_{0.96}{\mathrm{Ir}}_{0.04}\mathrm{Si}$, measured with the HB1 triple-axis spectrometer, at 100 and 300 K. The spectra were measured at the center of the (1,1,3) Brillouin zone ($\Gamma$ point). The green circles show the background from the same Al holder and can be used to mount the samples, measured at 300 K (same collimation and aperture slits). The dashed green curve is a Gaussian fit to the background (centered at 19.6 meV).

Figure 6

Phonon dispersions of B20 FeSi computed from first principles using density functional theory. The horizontal red-dashed line symbolizes the position of the resonance mode for the Os or Ir impurities in the doped material.

Figure 7

Dynamical structure factor, $S(\mathbf{Q},E)$, of FeSi for $\mathbf{Q}$ along $[-3,-3,L]$ ($-1\le L\le 1$). (a) Experimental data measured with ARCS at 300 K. (b) First-principles calculation, with phonon thermal occupation factor set to $T=300\mathrm{K}$. Integration ranges are $[-3.1,-2.9]$ along $[H,H,0]$ and $[-0.1,0.1]$ along $[K,-K,0]$, for both experimental and simulated data. Note that the strong horizontal bar of intensity for $E<1.5\mathrm{meV}$ in (a) corresponds to experimental background.

Figure 8

(a) Total phonon density of states for FeSi, ${\mathrm{Fe}}_{31}{\mathrm{OsSi}}_{32}$, and ${\mathrm{Fe}}_{31}{\mathrm{IrSi}}_{32}$. (b) Partial phonon density of states for Ir and Os impurities in ${\mathrm{Fe}}_{31}{\mathrm{OsSi}}_{32}$ and ${\mathrm{Fe}}_{31}{\mathrm{IrSi}}_{32}$. All first-principles calculations were performed at the theoretical equilibrium lattice parameters $a=4.4492$, $4.4612$, and $4.4613\AA$, respectively, for FeSi, ${\mathrm{Fe}}_{31}{\mathrm{OsSi}}_{32}$, and ${\mathrm{Fe}}_{31}{\mathrm{IrSi}}_{32}$. All spectra are normalized to unity.

Figure 9

Local potential-energy curves for Os and Ir impurities in FeSi. The calculations were performed on 64-atom cells of ${\mathrm{Fe}}_{31}{\mathrm{OsSi}}_{32}$ and ${\mathrm{Fe}}_{31}{\mathrm{IrSi}}_{32}$, with the impurity displaced along [100].

Figure 10

Electronic DOS, $N\left(E\right)$, for FeSi, ${\mathrm{Fe}}_{31}{\mathrm{OsSi}}_{32}$, and ${\mathrm{Fe}}_{31}{\mathrm{IrSi}}_{32}$, computed on the relaxed cells. The Fermi level is taken as the origin of energy (at the top of the valence band in the case of FeSi and ${\mathrm{Fe}}_{31}{\mathrm{OsSi}}_{32}$).

Figure 11

Partial phonon DOS of (a) Os and (b) Ir atoms computed from first principles for the supercells ${\mathrm{Fe}}_{31}X{\mathrm{Si}}_{32}$ (solid curves) and ${\mathrm{Fe}}_{24}{X}_8{\mathrm{Si}}_{32}$ (dashed curves), with $X=\text{Os},\text{Ir}$. (c) The isolated-impurity partial DOS for Os and Ir derived from the simplified Green's-function model described in the text.

Figure 12

Dynamical structure factor, $S(\mathbf{Q},E)$, computed using the Green's-function model described in the text, for an impurity resonance mode of energy $E_R=17\mathrm{meV}$ and a sine-wave acoustic dispersion branch.

Figure 13

Atomically resolved configuration-averaged phonon spectral function $\langle A_n(\mathbf{q},E)\rangle$ of ${\mathrm{Fe}}_{0.96}{\mathrm{Os}}_{0.04}\mathrm{Si}$ obtained from first-principles calculations of 10 supercells with 320 atoms on average. The vertical lines correspond to fixed momenta curves in Fig. 14.

Figure 14

Band-resolved configuration-averaged phonon spectral function $\langle A_j({\mathbf{q}}_0,E)\rangle$ of ${\mathrm{Fe}}_{0.96}{\mathrm{Os}}_{0.04}\mathrm{Si}$ at (a) ${\mathbf{q}}_0=0.35\Gamma X$, (b) ${\mathbf{q}}_0=0.45\Gamma X$, and (c) ${\mathbf{q}}_0=0.55\Gamma X$, corresponding to the vertical lines in Fig. 13. Bands $j=1,2$ correspond to transverse acoustic modes and $j=3$ to longitudinal acoustic modes.