# A combined inelastic neutron scattering and ab initio lattice dynamics study
of FeSi

Nazir Khan<EMAIL_ADDRESS>Institute for Quantum Materials and
Technologies, Karlsruhe Institute of Technology, D-76021 Karlsruhe, Germany
Sven Krannich  Institute for Solid State Physics, Karlsruhe Institute of
Technology, D-76021 Karlsruhe, Germany  Dominic Boll  Institute for Solid
State Physics, Karlsruhe Institute of Technology, D-76021 Karlsruhe, Germany
Rolf Heid  Institute for Quantum Materials and Technologies, Karlsruhe
Institute of Technology, D-76021 Karlsruhe, Germany  Daniel Lamago  Institute
for Solid State Physics, Karlsruhe Institute of Technology, D-76021 Karlsruhe,
Germany  Laboratoire Léon Brillouin (CEA-CNRS), CEA Saclay, F-91911 Gif-sur-
Yvette, France  A. Ivanov  Institute Laue-Langevin, BP 156, 38042 Grenoble,
France  David Voneshen  ISIS Facility, Rutherford Appleton Laboratory,
Chilton, Didcot, Oxfordshire OX11 0QX, United Kingdom  Department of Physics,
Royal Holloway University of London, TW20 0EX, UK  Frank Weber
<EMAIL_ADDRESS>Institute for Quantum Materials and Technologies,
Karlsruhe Institute of Technology, D-76021 Karlsruhe, Germany

###### Abstract

The phonon renormalization across the semiconductor-to-metal crossover in FeSi
is investigated by inelastic neutron scattering combined with ab-initio
lattice dynamical calculations. A significant part of reciprocal space with a
particular focus on the 110$-$001 scattering plane is mapped by the time-of-
flight inelastic neutron scattering data taken below and above the crossover.
Individual momentum values are investigated in more detail as a function of
temperature. The data reveal that the anomalous phonon softening upon
metallization is not exclusive to the high symmetry $R$ and $\Gamma$ points.
Several other phonon modes around the $R$-point as well as the phonon modes at
the $M$ and $X$ points of the Brillouin zone exhibit anomalous phonon
softening with magnitudes comparable to that observed at the $R$-point. The
momentum dependence of the phonon softening is reproduced by the lattice
dynamical calculation based on the density functional perturbation theory. We
discuss our findings with respect to the nature of the semiconductor-to-metal
crossover in FeSi, for which different microscopic origins have been
prooposed, i.e., lattice thermal disorder and electronic correlation effects.

————————————————————————————————

##  I Introduction

The family of transition-metal mono-silicides (TM-Si) Mn1-xFex-yCoySi
crystallizing in the noncentrosymmetric cubic B20 structure hosts a very rich
phase diagram showcasing a plethora of complex phenomena that are of interest
for fundamental physics and applied science [1, 2, 3, 4, 5, 6]. The phase
diagram [7] as function of chemical substitution ($x$,$y$) shows a wide range
of competing ground states with helimagnetic metallic magnetism (HMM),
paramagnetic metallic (PMM) and paramagnetic insulating regions (PMI) as well
as a magnetic quantum critical point [4]. Exotic phases such as partial order
under hydrostatic pressure [8] and skyrmion lattices in magnetic fields have
been discovered in the end member MnSi [9]. On the other hand, FeSi has the
only insulating ground state and has been the focus of intense research
interests for more than half a century due to its unusual temperature-
dependent electrical, magnetic, and lattice dynamical properties [10, 11, 12].
The $d$-electron compound FeSi is regarded as a correlated narrow-gap
insulator and shows a remarkable similarity to $f$-electron Kondo insulators.
The true nature of the electronic ground state is not yet unambigously
settled. Experiments reveal a small charge gap $\Delta\sim$ 50-70 meV at low
temperatures [10, 13, 14] and a cross-over to a (bad) metal at temperatures
(100 K - 300 K) much smaller than $\Delta$ is observed in transport [15, 16,
17] and optical spectroscopy [13, 18, 14, 17, 19, 20], indicating that the
charge gap is temperature dependent. Recently, size-dependent resistivity
measurements on high quality single crystals of FeSi have provided evidence
for a conducting surface state below 19 K [21] which is reminiscent of a
topological Kondo insulator. The insulator-to-metal crossover at room
temperature is accompanied by a temperature-activated magnetism with a
paramagnetic moment of $\sim$2 $\mu_{B}$ per Fe atom at $T\geq$500 K [10, 12,
22]. Strongly temperature-dependent lattice dynamics in FeSi have been
proposed to originate from metallization [23] and to be magnetically induced
[24].
Theoretical models incorporating strong correlation effects such as local
Coulomb interactions [25], lattice thermal disorder [23, 26], and the orbital
hybridization between Fe 3$d$ and Si 2$p$ bands [27, 28] have been invoked to
explain the anomalous properties of FeSi. The observed phonon softening in
FeSi has been captured by an earlier study based on finite-temperature first-
principles calculations including thermal disorder effects[23]. However, in a
recent study [25] based on realistic many-body calculations argues that
electronic Coulomb correlations alone can quantitatively reproduce the
signatures of the temperature induced cross-over in various observables: the
spectral function, the optical conductivity, the spin susceptibility, and the
Seebeck coefficient.
Here, we address the temperature-induced anomalous phonon renormalization in
FeSi by means of ab initio lattice dynamical calculations and comprehensive
single-crystal inelastic neutron scattering experiments. Going beyond previous
studies, we investigate the momentum dependence of phonon renormalization at
various high-symmetry points as well as in the vicinity of the $R$ point. We
find that our quasi-harmonic calculations for insulating FeSi and quasi-
metallic FeSi resonably-well describe the observed phonon softening over a
large momentum range.
The manuscript is organized as follows. Section II contains theoretical
details. In Sec. III, we present the details of different inelastic neutron
scattering experiments performed on the FeSi single-crystal. Section IV.1
describes the calculated phonon dispersion, electronic and phonon density of
states, and temperature dependence of phonon energies for insulating and
metallic FeSi. Section IV.2 reports the observed lattice dynamics based on
inelastic neutron scattering experiments combined with theoretical
calculations. In Section V we discuss our theoretical and experimental
findings and compare them with earlier studies. The conclusions drawn in
Section VI brings to an end of the manuscript.

##  II Theory

ab initio lattice dynamical calculations based on density functional
perturbation theory (DFPT) were performed in the framework of the mixed basis
pseudopotential method [29]. The exchange-correlation functional was treated
in the local density approximation (LDA) in the Perdew–Wang parameterization
[30]. Norm-conserving pseudopotentials for Fe and Si were constructed
following the schema of Vanderbilt [31] and include the Fe 3$s$ and 3$p$
semicore states in the valence space. The resulting deep potential can be
efficiently treated in the mixed-basis scheme, which combines local functions
together with plane waves for the representation of the valence states. Here,
a combination of plane waves up to a kinetic energy of 22 Ry and local
functions of $s$, $p$, and $d$ symmetry at the Fe sites gave sufficiently
converged results. Brillouin zone integrations were performed with a cubic
8$\times$8$\times$8 $k-$point mesh (24 points in the irreducible Brillouin
zone) in combination with a standard smearing technique using a Gaussian
broadening of $\sigma$=0.05 eV and 0.2 eV. Dynamical matrices were calculated
within DFPT on a cubic 4$\times$4$\times$4 $q$-point mesh, and then a standard
Fourier interpolation technique was applied to get the full phonon dispersion.
Displayed results were obtained for the B20 structure ($P$213) using
experimental lattice constants and optimized internal parameters.

##  III EXPERIMENT

Figure 1: (left) Calculated phonon dispersion in FeSi. The phonons $R$1, $R$2
and $\Gamma M$1 which we investigated also in a previous publication [24] are
marked separately. Phonon dispersions are distinguished for their character.
Solid and dashed lines refer to longitudnal and transverse characters,
respectively. This distinciton was not possible along $\Gamma M$. The inset
shows the high-symmetry points of the Brillouin zone. (right) Calculated
phonon density of states for FeSi (black curve) along with the partial
densities of states for iron (red) and silicon (blue).

A high quality FeSi single crystal weighing $\sim$35 g was used for the
inelastic neutron scattering (INS) experiments. INS measurements on the FeSi
single crystal were performed on the direct-geometry, time-of-flight (TOF),
chopper spectrometer MERLIN at the Rutherford Appleton Laboratory in Harwell,
UK [32]. The Merlin detectors cover scattering angles from $-45{}^{\circ}$ to
+135∘ horizontally and $\pm$30∘ vertically. The sample was mounted on an Al
sample holder using thin Al foil and Al sheet with (110)-(001) plane of the
single crystal as the horizontal scattering plane. The sample was then cooled
by a closed-cycle refrigerator on the instrument. All data [33] were taken
with an incident energy $E_{i}$ = 71.2 meV and covering the same volume of
reciprocal space by rotating the sample over 70∘ with angular steps of 0.25∘.
We used Gd chopper and a frequency of $f$ = 400 Hz was chosen to record data
at $T$ = 10 K and 300 K. The raw data were corrected for detector efficiency
by normalizing the intensities using a standard vanadium sample. Data was
reduced using the Mantid package [34] and analyzed using HORACE [35]. Selected
phonon momenta were investigated in more detail on the thermal triple-axis
spectrometer (TAS) 1T at the ORPHEE reactor at Laboratoire Léon Brillouin, CEA
Saclay. Double-focusing copper (Cu 111) monochromator and graphite (PG 002)
analyser were used. The final energy at the analyzer was set to 14.7 meV
allowing the use of a graphite filter to suppress higher-order scattering. The
single crystal was mounted in a closed-cycle refrigerator allowing
measurements in the temperature range 5 K $\leq$$T$$\leq$ 790 K. The magnetic
field dependence of phonon renormalization for some selected phonon momenta
was investigated on the thermal triple-axis spectrometer IN8 at ILL, Grenoble
using a 10 T vertical cryogenic magnet operating in the temperature range 2 K
$\leq$$T$$\leq$ 300 K [36]. Double-focusing copper (Cu 200) monochromator and
analyser were used in the experiment, and a pyrolytic graphite (PG) was used
as a filter with the final energy at the analyzer fixed to 14.7 meV.
Scattering wave vectors Q = $\tau$ \+ q are expressed in reciprocal lattice
units (r.l.u.) (2$\pi/a$,2$\pi/a$,2$\pi/a$) with the cubic lattice constant
$a$.

##  IV Results

###  IV.1 Density functional perturbation theory

Figure 1 (left) shows calculated phonon dispersions in FeSi along different
high-symmetry directions through the $\Gamma$, $X$, $M$, and $R$ high-symmetry
points of the Brillouin zone as shown in the inset. Different colors are used
to differentiate between different symmetries of the phonon displacement
patterns. The experimental lattice constant at $T$ = 10 K ($a$ = 4.4745Å) was
used for the calculation [37, 38]. The internal parameters were relaxed to
obtain a force-free structure. The resulting values for the optimized
structure are $u$(Fe) = 0.134 and $u$(Si)=0.840 and agree well with
experimental values [39]. The dispersion shown is in good agreement with a
previous report [40]. Dot and circles denote three particular modes ($\Gamma
M$1, $R$1, $R$2) which were investigated by some of us in a previous report
[24]. Figure 1 (right) shows the corresponding phonon density of states
together with the partial density of states for the vibrations of the iron and
silicon atoms. The low-energy part of the phonon spectrum is dominated by the
atomic vibrations of Fe, whereas the vibrations of the significantly lighter
Si atoms dominate the high-energy part of the phonon spectrum. Additional
calculations using optimized lattice constants with LDA and GGA (general
gradient approximation) produced phonon frequencies about 5-10% harder than
those presented in Fig. 1. The origin is that both approximations, i.e., LDA
and GGA, yield smaller lattice constants than experimentally reported.

Figure 2: (a) Electronic density of states (eDOS) in the vicinity of the Fermi
energy $E_{F}$ calculated within density-functional-theory (DFT) using a
Gaussian smearing of $\sigma$ = 0.05 eV (solid line) and 0.2 eV (dashed line).
eDOS calculated in the tetrahedron method (not compatible with the phonon
calculations) is shown as well (dash-dotted line). (b) The calculated
temperature dependence of the energy of the $R$1 phonon in the quasi-harmonic
approximation, where temperature dependent lattice constants were taken from
experimental reports [38, 37]. The red (black) solid lines were calculated for
$\sigma$ = 0.05 eV (0.2 eV). (c, d) Analogous calculations for (c) a high
energy mode at the $R$ point and (d) the phonon mode at q = (0.25,0.25,0),
which is denoted as $\Gamma M$1 in Fig. 1. The dashed lines in (b)-(d)
correspond to the softening based on the reported thermal expansion [37] and a
fixed Grüneisen parameter of 2.

FeSi has an insulating ground state and our calculations of the electronic
structure show indeed a strong suppression of electronic states at $E_{\rm
F}$, reminiscent of a gap in the excitation spectrum [Fig. 2(a)], where a
Gaussian smearing of $\sigma$ = 0.05 eV was used in the calculation. This
value of $\sigma$ corresponds to the smallest value with which numerical
convergence can be safely achieved. A more accurate description of the
electronic density of states can be obtained by integrating over the
electronic states using the tetrahedron method (see blue line in Fig. 2(a)).
In particular, the sharp maxima at the band edges, which were also observed in
electron spectroscopy [41, 42], are well reproduced in the tetrahedron method.
However, lattice dynamical calculations in the used software package are not
compatible with the tetrahedron method and, hence, are based on DFT
calculations employing a Gaussian smearing. For the description of metallic
FeSi at room temperature we used $\sigma$=0.2 eV. The resulting electronic
density of states corresponds to the black curve in Fig. 2(a) and shows
strongly smeared structures. In particular, the density of states in the
region of $E_{F}$ is significantly increased compared to the calculation with
$\sigma$=0.05 eV. Thus, we can use different values of $\sigma$ to simulate
effects on the lattice dynamics related to the closure of the electronic gap.

Although DFPT is a ground state calculation, changes in the lattice dynamics
because of temperature dependent lattice constants can be studied within the
quasi-harmonic approximation (QH) [43, 44, 45]. For this purpose, the
experimental lattice parameters [37] for temperatures 10 K $\leq$$T$$\leq$ 600
K were used in the calculation of phonon energies. The internal parameters
were relaxed for each lattice constant. The calculation was performed using
both $\sigma$=0.05 eV and $\sigma$=0.2 eV. Calculated temperature dependences
of phonon energies are shown for three selected modes in Fig. 2(b)-(d) (solid
lines). These QH calculations are compared to an estimated softening based on
the reported thermal expansion and using a typical value of the Grüneisen
parameter of 2 [dashed lines in Figs. 2(b)-(d))] [23]. In the QH calculations
(solid lines) with a fixed value of $\sigma$, the softening of the phonons
shown in Figs. 2 (b) and(d) is in good agreement with the estimates from the
Grüneisen model (dashed lines). However, for the high energy mode shown in
Figs. 2(c), the relative softening at high temperatures is overestimated in
the Grüneisen model compared to the QH calculations, which demonstrates that
the QH calculations predict mode-dependent Grüneisen parameters. The QH
calculations reveal additional softening for $\sigma$=0.2 eV [black solid
lines in Figs. 2(b)-(d)], i.e., for quasi- metallic FeSi. The relative
softening due to the metallization is the highest ($\Delta E$=5.6%) for the
low energy phonon mode at q=(0.5,0.5,0.5)[Fig. 2(b)], while the high energy
phonon mode at the same reduced q [Fig. 2(c)] exhibits the lowest softening
($\Delta E$=0.4%). The phonon mode at q=(0.25,0.25,0) [Fig. 2(d)] exhibits an
intermediate value of the relative softening. Therefore, the calculations
suggest that the phonon softening in FeSi due the metallization effect is
strongly dependent on the energy and momentum of a phonon mode, which is
further supported by our experimental results given below.

###  IV.2 Experimental results

Figure 3: Horace plot of the inelastic neutron scattering data at $T$=10 K
with incident energy $E_{i}$=71.2 meV for the FeSi single crystal exhibiting
the coverage in the [HHL] scattering plane for energy transfers binned over
(20$\pm$2) meV. The red cross symbols represent the $R$ points in various
Brillouin zones covered in our experiment.

The coverage of momentum space in our TOF data for the horizontal [110]-[001]
scattering plane is shown in Fig. 3 for energy transfers binned over 20$\pm$2
meV. The [111] direction is of particular interest, since anomalous phonon
renormalization has been observed in FeSi for phonons at the $R$ point, q =
(0.5,0.5,0.5), i.e., the zone boundary along the [111] direction [24]. The
red-cross symbols in Fig. 3 indicate investigated $R$ points in Brillouin
zones within the horizontal scattering plane, which are analyzed in more
detail later in our report.

Figure 4: (a) Calculated phonon intensities overlaid with the corresponding
dispersion lines along Q = [$\zeta$,$\zeta$,$\zeta$], 2.5 $\leq\zeta\leq$ 4,
calculated with the reported lattice constants for $T$= 10 K and $\sigma$=0.05
eV. (b) Observed neutron scattering intensities for the same line in Q space
at $T$=10 K. (c) and (d) show analogous results along the line Q =
(3.5,3.5,3.5)+($\xi$,$-\xi$,0). Dashed rectangles in all panels indicate the
position of the $R$ point at (3.5,3.5,3.5).  Figure 5: Analysis of combined
(a) $R$ point and (b) $X$ point spectra. A linear background was subtracted
from the raw data (squares). The resulting background subtracted data were
approximated by a multi-peak fit (solid line) with the individual peaks shown
as dashed lines. The number of peaks is in reasonable agreement with our
calculations (see text). (c) Relative softening of phonon peaks on heating to
$T$=300 K at the given high-symmetry points plotted as function of the low-
temperature phonon energy. As a reference line, the dashed horizontal line
indicates the softening because of the thermal expansion [37] assuming a fixed
Grüneisen parameter of $\gamma_{\textbf{{k}},j}$ = 2 .

We studied the dispersion of phonons going away from the $R$ point along the
[111] and [1$\bar{1}$0] directions. Since previous data show strong
renormalization of longitudinal phonons, we particularly investigated the
region around Q = (3.5,3.5,3.5). There, we looked for anomalous phonon
renormalization on heating from 10 K to 300 K and compared our results to ab-
initio lattice dynamical calculations. We point out that the data obtained on
MERLIN cover a large range of wave vectors. For instance, large phonon momenta
near zone centers $\tau$ = (3,3,3) and (4,4,0) [see Fig. 3] can be accessed on
triple-axis spectrometers in expense of the resolution or scattering
intensity. Figure 4(a) and (b) exhibit the calculated and experimentally
observed phonon intensities at $T$=10 K, respectively, overlaid with the
corresponding dispersion lines in the [$\zeta$,$\zeta$,$\zeta$] - Energy plane
for 2.5 $\leq\zeta\leq$ 4 including $R$ points at half-integer values. The
DFPT calculation shown in Fig. 4(a) was performed using $\sigma$=0.05 eV and
the experimental lattice parameter at 10 K [38]. Figure 4(c) and (d) are
analogous plots, except that they exhibit dispersion along the line Q =
(3.5,3.5,3.5)+($\xi$,$-\xi$,0). Overall calculation and experiment are in good
agreement. We now look into the phonon renormalization on heating to room
temperature in two ways: (1) combining data obtained at symmetry-equivalent
wave vectors to improve statistics and (2) investigating phonons close to the
$R$ point along the high symmetry directions shown in Fig. 4.

####  IV.2.1 Softening at high-symmetry points

Despite the large size of the investigated FeSi single crystal, scattering
intensities at individual absolute momenta Q are often not good enough to
unambiguously approximate peak functions to reveal the temperature dependence
of phonon energies. Therefore, we employed the following strategy for selected
high-symmetry points: First, we obtained constant-Q scans for individual wave
vectors from the $S(\textbf{{Q}},\omega)$ datasets of FeSi collected at 10 K
and 300 K using the Phonon Explorer software [46] with a binning size ($\Delta
H$=$\Delta K$=$\Delta L$) of $\pm$0.075 r.l.u. Then, data from symmetry-
equivalent wave vectors was combined to improve the statistics. The combined
energy scans at 10 K for the $R$ and $X$ points are shown in Fig. 5(a) and
Fig. 5(b), respectively. To determine the peak positions of the spectra, a
linear sloping background was subtracted from the raw data sets [squares in
Figs. 5(a) and (b)]. Gaussian functions were approximated to the background
subtracted data [spheres in Fig. 5(a) and Fig. 5(b)].

Figure 6: (a)-(c) Phonon dispersion in the vicinity of the $R$ point along
three different directions, i.e., (a) $\delta\textbf{{q}}\parallel$ [111], (b)
$\delta\textbf{{q}}\parallel$ [1$\bar{1}$0], and (c)
$\delta\textbf{{q}}\parallel$ [11$\bar{2}$]. Solid and dashed lines denote
DFPT calculations for insulating ($a_{T=10K}$ = 4.55 Å, $\sigma$ = 0.05 eV)
and metallic ($a_{T=300K}$ = 4.57Å, $\sigma$ = 0.2 eV) FeSi, respectively, and
the symbols are the corresponding experimental results. (d)-(f) Relative
softening of the observed phonons as a function of phonon momentum transfer q.
As a reference line, the dashed lines indicate the softening due to the
thermal expansion upon heating up to 300 K using a fixed Grüneisen parameter
of 2.

Starting with the $R$ point, we note that the high symmetry results in 4-fold
degenerate phonon energies. Hence, there are only 6 different phonon energies.
Our calculations show that for specific $R$ points with q $\parallel$ $\tau$,
only four of these modes with well-separated energies have non-zero structure
factors. We first identified these four modes from the spectra obtained by
combining the energy scans at Q=(1.5,1.5,1.5), Q=(2.5,2.5,2.5), and
Q=(3.5,3.5,3.5) as shown in the inset of Fig. 5(a). As phonon energies remain
the same for the same reduced q=(0.5,0.5,0.5), the energies of the other two
phonon modes in the combined energy scan [Fig. 5(a)] can now be determined
following the prescription of DFPT. DFPT calculations suggest that the
energies of these two modes lie close to the energies of the $R1$ and $R2$
modes identified from the spectrum of the phonons with longitudinal symmetry
along the [111] direction [see Fig. 1 and inset of Fig. 5(a)]. Thus, a
cumulative peak fitting to the combined energy scan [Fig. 5(a)] enabled the
determination of all the six phonon energies for the $R$ point phonons. For
the spectrum at other high symmetry points ($M$, $X$, and $\Gamma$) the
situation is less favorable. Therefore, we considered only those peaks that
are clearly visible in the spectra with significant intensities at both 10 K
and 300 K for the cumulative peak fitting, e.g., the $X$ point shown in Fig.
5(b). The relative softening, $\Delta E$(%), of different phonon modes at the
$R$, $M$, $X$, and $\Gamma$ points of the Brillouin zone due to the crossover
from the insulating state ($T$=10 K) to the metallic state ($T$=300 K) is
shown in Fig. 5(c) together with the softening expected based on thermal
expansion and a fixed $\gamma_{\textbf{{k}},j}$ = 2 as a reference line. Most
of the phonon modes shown in Fig. 5(c) exhibit relative softening beyond this
reference value ($\Delta E$=1.5%) due to the thermal expansion upon heating up
to 300 K. The columns in Fig. 5(c) demonstrate that several phonon modes at
the $M$, $X$, and $\Gamma$ points exhibit strong softening comparable to the
anomalous softening of the $R$ point phonon modes reported previously [24]. We
also see that the relative softening is stronger for lower energy phonons
entailing larger involvement of the heavier Fe atoms in lower-frequency
vibrations (see the partial phonon DOS plotted in Fig. 1). Note that the
relative softening obtained from the cumulative peak fitting for some phonons
with very low intensities (and therefore low reliability) are not shown in
Fig. 5(c).

####  IV.2.2 Softening around $R$ point

After having focused on high-symmetry points in reciprocal space, we explore
the phonon softening in FeSi in more detail as function of wave vector close
to the $R$ point at Q = (3.5,3.5,3.5) [some data shown in Figs. 4(b)(d)]. In
Fig. 6(a)-(c) we plot the observed dispersion going away from this $R$ point
along [111], [1$\bar{1}$0] and [11$\bar{2}$] directions, respectively, at
$T$=10 K (solid symbols) and $T$=300 K (open symbols). The experimental
results are compared to the corresponding calculated dispersion line with
$\sigma$ = 0.05 eV and 10 K lattice constants (solid lines) and $\sigma$ = 0.2
eV and room temperature lattice constants (dashed lines). There, we only show
the dispersion lines of modes which have non-zero structure factor, and
therefore, this number is different in panels (a-b-c) of Fig. 6. The
experimental phonon energies are determined from the Gaussian fits to constant
momentum lines extracted from the TOF data set, e.g., the data shown in Fig.
4(b,d). The assignment of the observed peaks to the calculated phonon
dispersion is comparatively straightforward for q = [$h$,$h$,$h$] since this
is the highest symmetry direction in FeSi and only longitudinal phonon modes
have non-zero intensities. The increasing number of phonon lines having non-
vanishing structure factor along q=[0.5+$h$,0.5-$h$,0.5] and
[0.5+$h$,0.5+$h$,0.5-2$h$] allow only for a rough assignment of observed peaks
to the calculated dispersion lines. Fig. 6(d)-(f) exhibits the relative
softening of the low temperature phonon frequencies in FeSi as a function of
the phonon momentum transfer, q, upon metallization at $T$=300 K. All the
observed low-lying phonons ($E<$ 35 meV) with fixed $\gamma_{k,j}$ = 2
[horizontal line in Figs. 6(d)-(f)] exhibit softening beyond the estimate from
thermal expansion and also comparable to that observed at the $R$ point
($\Delta E$=7%) [grey shaded wave vectors in Figs. 6(a)-(c)]. On average the
softening of the high energy branches (purple) is significantly smaller than
that for intermediate energies (green) and the softening is largest for the
lowest energy phonons (orange). These observations manifest that the anomalous
phonon renormalization is not exclusive to the phonons at the high symmetry
points of the Brillouin zone.

Figure 7: Temperature-dependent energies of selected phonon modes with
energies up to 40 meV for temperatures 10 K $\leq T\leq$ 790 K. The phonon
energies were normalized to the value at $T$ = 10 K. Red and black lines
correspond to quasi-harmonic calculations with $\sigma$ = 0.05 eV and
$\sigma$= 0.2 eV, respectively (see text).

A detailed temperature dependence of softening for some selected phonon modes
is investigated in the temperature range 10 K$\leq T\leq$790 K ($E\leq$40
meV). In Figs. 7(a)-(e) we plot the variation of normalized phonon energies
(symbols) obtained from the data collected at the thermal TAS 1T across the
insulator-to-metal crossover temperature for five different phonon modes at a
constant momentum transfer, Q. The energies were normalized to the value at
$T$ = 10 K. Red and black solid lines show the calculated phonon energies
based on QH approximation for $\sigma$= 0.05 eV and $\sigma$= 0.2 eV
corresponding to an insulating- and metallic-type electronic DOS, respectively
[see Fig. 2(a)]. The data in Figs. 7(a)-(e) reveal that all the five phonon
modes exhibit anomalous softening limited to the temperature range 100 K$\leq
T\leq$300 K. The slope of softening below 100 K and above 300 K agrees
reasonably well with the QH calculations. Where our calculations show various
degrees of agreement with the experimental observations, the results clearly
demonstrate a difference in softening of up to a factor of 2 at room
temperature, even for very similar phonon energies [see Figs. 7(a) and (e)].

####  IV.2.3 Phonon line brodening

Figure 8: Phonon spectra collected at $T$=100 K and 690 K in the vicinity of
(a,b) the zone center $\tau$ = (2,2,0) and (c,d) the $R$ point at
Q=(1.5,1.5,1.5). For each spectrum a linear background was subtracted before
peak fitting. Spectra at $T$= 690 K were fitted with Lorentz curves convoluted
with the Gaussian resolution determined from the corresponding spectra at $T$=
100 K. (e) Q-dependence of the approximated Lorentzian linewidth at the wave
vectors Q=(2,2,$h$).

We also investigated the wave vector dependence of the broadening in zero
field on the TAS 1T using a Cu (111) monochromator to improve the experimental
resolution. Such an investigation is much more demanding than a mere
determination of phonon energies. The approximated line width depends
sensitively on the reliability of the estimated experimental background.
Furthermore, phonons modes have to be well-separated in energy to allow an
unambiguous analysis of the peak widths. Hence, we could only investigate the
line width of phonon modes along the line Q = (2,2,$h$) with $h$ = 0-0.5
r.l.u. We approximated the data at $T$ = 100 K with Gaussians and used them as
resolution function for further analysis. The high temperature data were
analyzed by a Lorentzian convoluted with the Gaussian resolution determined at
low temperature. Thus, we deduced the linewidth $\Gamma_{L}$ of the intrinsic
phonon broadening at $T$ = 690 K which is shown in Fig. 8(e). The observed
phonon broadening along $\Gamma-X$ even increases for the mode around 25 meV
but it practically vanishes for the mode above 30 meV. The broadening of the
phonon modes dispersing along the $\Gamma-R$, but away from $R$, cannot be
determined unambiguously because the asymmetric lineshapes at $T$ = 690 K
indicate strong overlap of multiple phonon peaks, e.g., at Q=(1.75,1.75,1.75)
[Fig. 8(d)].

####  IV.2.4 Magnetic field dependence

Figure 9: Phonon spectra at the zone center $\tau$ = (2,2,0) with and without
applied magnetic field at (a) room temperature and (b) $T$=100 K. Horizontal
bars indicate the width of the peak observed at low temperature. Dashed lines
indicate the background taken from an empty can measurement for the same
experimental configuration.

One hallmark of FeSi is the temperature-activated magnetism [10] where a large
magnetic moment at high temperatures of up to 2 $\mu_{B}$ per Fe ion is
suppressed at low temperatures $T\leq$ 100 K. Previously, we observed that
phonons at the $R$ point are broadened with temperature following closely this
temperature-activated behavior. Therefore, we performed INS measurements in an
applied magnetic field of 10 T on the TAS IN8 to investigate a possible
magnetic field dependence. We focused on the $R$ point at Q = (1.5,1.5,1.5) as
well as the zone center at Q = (2,2,0) (Fig. 9). While a structured background
due to the complex sample environment prevents a detailed quantitative
analysis, the raw data allow two conclusions: (1) A vertical magnetic field of
10 T has not detectable effect on the phonon peaks at either wave vector. (2)
The previously reported broadening is clearly visible, e.g., at Q = (2,2,0)
for the main peak around 25 meV (Fig. 9).

####  IV.2.5 Results-overview

In summary, our detailed INS investigation of the lattice dynamics of FeSi
reveals: (1) The anomalous softening in FeSi is limited in the temperature
range 100 K$\leq T\leq$300 K. The softening below 100 K and above 300 K can be
approximated by the calculation of volume effect due to thermal expansion. (2)
The anomalous softening is widely distributed in Q though not all modes show
it. (3) Overall, the softening is stronger for phonons below 35 meV, which
(according to PDOS) are Fe dominated. (4) Broadening occurs also at various
wave vectors - although only a few wave vectors could be checked due to the
low symmetry and unclear selection rules in FeSi. (5) Application of magnetic
fields up to 10 T has no effect on the phonon properties of FeSi.

##  V Discussion

The lattice dynamics in FeSi have been investigated before based on inelastic
neutron scattering [23, 24], Raman and infrared spectroscopies [20, 47]. The
INS study addressed the phonon softening in FeSi based on the phonon DOS
measured on a powder sample at a few selected temperatures and the single-
crystal phonon dispersion taken at 10 K and 300 K. The Raman studies in FeSi
mainly measured the energies of some zone-center phonons at 5 K and the
temperature dependence of the phonon line width for a particular phonon mode
($E$ mode). Their experimental observations are in good agreement with our
detailed study performed on a high-quality single crystal. From the
experimental point of view, our study provides a deeper look into the
anomalous phonon softening in FeSi. Unlike previous studies, our detailed
temperature dependence of phonon energies (see Fig. 7) reveals clearly that
the anomalous softening is only limited to the temperature window 100 K$\leq
T\leq$300 K. Additionally, based on the quantitative assessment we show that
the relative softening (Fig. 5 and Figs. 6) and broadening (Fig. 8) are not
exclusive to the high symmetry points$-$ rather it is widely distributed in Q.
Generally, we find that the magnitude of relative softening is strongest for
phonons below 35 meV. The calculation of partial phonon density of states
[Fig. 1] implies that the low energy phonons entail larger involvement of Fe
atoms. As the conduction electrons result mainly from the Fe state, increased
screening and, thus, reduced interatomic force-constant due to metallization
could explain the stronger softening for the lower energy modes.
Based on finite temperature molecular-dynamics (MD) calculations, anomalous
temperature dependence of phonon in FeSi has been previously interpreted by
considering the renormalization of the electronic structure by thermal
disorder, leading to the closing of the narrow gap [23]. Though the simulated
phonon density of states at different temperatures capture the phonon
softening upon metallization, the MD calculations could not reproduce
correctly the temperature at which thermal disorder becomes important [23].
Similarly, the closure of the electronic gap was predicted to happen at
significantly higher temperatures [23].
A later study, based on a combination of density functional theory and
dynamical mean-field theory (DMFT), proposed a new scenario in which FeSi is a
band insulator at low temperatures and is metalized with increasing
temperature through electronic correlation induced incoherence due to the
unlocking of iron fluctuating moments [25]. According to their theoretical
picture, the cross-over to bad metallic behavior is not caused by a narrowing
of the excitation gap, rather it is filled with incoherent weight that emerges
with increasing temperature. This picture is validated by quantitative
agreement with optical spectroscopy experiments which show spectral weight
transfers over several electronvolts$-$ a common hallmark of correlation
effects [48].
Thus, whether the thermal disorder effects are essential to simulate the
lattice dynamics and spin fluctuations in FeSi at high temperatures, needs
further investigation. In fact, our calculations for insulating FeSi and
quasi-metallic FeSi, without considering any thermal disorder effects, appear
to capture the observed phonon dispersion (Fig. 4 and Fig. 6) and anomalous
softening due to the metallization (Fig. 7) quite well. In molecular dynamics
simulations [23] which considers the thermal disorder effects, the gap
$\Delta_{DFT}$ was shown to vanish abruptly for temperatures of the order of
$\Delta_{DFT}/2$. This seems to be inconsistent with our experimental
observation (Fig. 7) where we see that the anomalous phonon softening starts
at around 100 K and ends at around 300 K, suggesting that the narrow gap in
FeSi is gradually filled by electronic states and the filling is complete
above 300 K. Such a gap closure in FeSi is consistent with the optical
spectroscopy experiments [19, 20] indicating the role of correlation effects
in the electronic structure renormalization in FeSi. Thus, our study promotes
that like other anomalous physical properties of FeSi, the microscopic origin
behind the anomalous lattice dynamics is also electronic in nature.
We also observed that broadening of the phonon linewidth is not limited only
to the phonons at the high symmetry points [Fig. 8(e)]. It appears that both
phonon energy and linewidth exhibit strong renormalization over the entire
Brillouin zone. The broadening at high temperatures could be attributed to a
strong electron-phonon coupling as FeSi behaves like a metal as high
temperatures. However, in our previous study [24] we observed that the phonon
broadening increases with increasing temperature beyond 300 K, and mimic the
$T$-dependence of magnetization due to the development of a magnetic moment at
the Fe site. This implies that the strength of electron-phonon coupling
increases with growing magnetism in FeSi at high temperatures, however, the
presence of other interaction paths such as spin-phonon coupling can-not be
ruled out.
Additionally, we showed that a magnetic field of 10 T has no effect on the
lattice dynamics even in the metallic phase where Fe develops a temperature
induced local magnetic moment [Fig. 9]. This is likely due to the mismatch in
energy scales between the 10 T field and the spin fluctuations which are
significant at very high temperatures. Thus, it could be very interesting to
see how the lattice dynamics in the isostructural metallic helimagnet MnSi
respond to an external magnetic field. There, a magnetic field as low as 0.5 T
can fully spin-polarize the system through an intermediate conical magnetic
phase [9].

##  VI Conclusion

We have reported a comprehensive inelastic neutron scattering and ab initio
theoretical investigation of the lattice dynamics of FeSi, focusing on the
phonon renormalization across the temperature induced metal-to-insulator
cross-over. The anomalous softening is found to be limited to the temperature
range where FeSi gradually crosses over from a small-gap semiconductor to a
bad metal. The softening is widely distributed in Q and appears stronger for
phonons having energies below 35 meV i.e., vibrations dominated by Fe motions.
Phonon broadening is also distributed in Q space. Overall, our theoretical
calculations are in reasonable agreement with the experimental observations,
which demonstrates that the anomalous phonon renormalization can also be
reproduced without considering lattice thermal disorder effects. Apparently,
magnetic field has no effects on the phonons in FeSi, however, we anticipate
that lattice dynamics in the isostructural but magnetically ordered MnSi may
reveal some interesting field-dependent behavior.

###### Acknowledgements.

This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Project No. 419331252. Experiments at the ISIS
Pulsed Neutron and Muon Source were supported by a beamtime allocation from
the Science and Technology Facilities Council.

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