Phonon softening and metallization of a narrow-gap semiconductor by thermal disorder

Edited* by Mildred Dresselhaus, Massachusetts Institute of Technology, Cambridge, MA, and approved February 9, 2011 (received for review October 4, 2010)
March 7, 2011
108 (12) 4725-4730


The vibrations of ions in solids at finite temperature depend on interatomic force–constants that result from electrostatic interactions between ions, and the response of the electron density to atomic displacements. At high temperatures, vibration amplitudes are substantial, and electronic states are affected, thus modifying the screening properties of the electron density. By combining inelastic neutron scattering measurements of Fe1-xCoxSi as a function of temperature, and finite-temperature first-principles calculations including thermal disorder effects, we show that the coupling between phonons and electronic structure results in an anomalous temperature dependence of phonons. The strong concomitant renormalization of the electronic structure induces the semiconductor-to-metal transition that occurs with increasing temperature in FeSi. Our results show that for systems with rapidly changing electronic densities of states at the Fermi level, there are likely to be significant phonon–electron interactions, resulting in anomalous temperature-dependent properties.
Because many properties of the solid state derive from the electronic structure (1), understanding finite temperature effects on the band structure is crucial to accurately describe materials in realistic operating conditions. The effects of thermal disorder on the electronic structure of materials at high temperature are largely unexplored, however, and the role of the electron–phonon interaction above room temperature has remained controversial (25). We performed detailed investigations of the phonons and electronic structure in Fe1-xCoxSi and found that an adiabatic coupling can lead to pronounced anomalies in the temperature dependence of both phonons and electron states. The mechanism is general and could affect a broad class of materials, including narrow-gap semiconductors, superconductors, heavy-Fermion compounds, and thermoelectrics.
FeSi has attracted a great deal of interest as it exhibits an insulator-metal transition with increasing temperature, and many of its physical properties show anomalous temperature dependences, including the magnetic susceptibility, heat capacity, Seebeck coefficient, thermal expansion, and elastic properties (611). Recently, it has been argued that doping FeSi with Al can lead to a surprising heavy-Fermion metal (12). FeSi has also attracted attention as a possible reaction product at Earth’s core-mantle boundary (1316), and as a candidate thermoelectric material for refrigeration applications (9). The compounds FeSi and CoSi are isostructural, crystallizing in the cubic B20 structure, with similar ion coordinates (17, 18). Although FeSi undergoes a gradual transition from narrow-gap semiconductor (Egap ∼ 0.1 eV) to metal with increasing temperature, the additional d electron in CoSi leads to a metallic state at all temperatures. The anomalous temperature dependences observed in FeSi are absent in CoSi.
Here, we show that the adiabatic electron–phonon coupling has spectacular consequences in FeSi, leading to the metallization of the semiconducting compound as temperature is increased, and causing a large concurrent lowering of phonon frequencies. We conducted comprehensive measurements of the phonons in Fe1-xCoxSi using inelastic neutron scattering (INS) at temperatures 10 ≤ T ≤ 750 K, and detailed ab initio calculations of the full phonon spectra, as well as ab initio molecular dynamics at finite temperature. Both experimental measurements and first-principles calculations show a large suppression of phonon frequencies in FeSi with either increasing temperature or adding charge carriers. Our simulations also reproduce the experimentally observed insulator-metal transition in FeSi with increasing temperature. By contrast, our measurements and calculations show that CoSi exhibits normal phonon behaviors, which is understood from its different electronic structure.
The adiabatic electron–phonon coupling mechanism presented here is general in nature, and similar effects could be occurring in a large number of other systems, in particular in materials where the electronic structure has sharp features near the Fermi level. We present some trends for the anomalous temperature dependence of phonons, depending on the type of features in the electronic structure.

Effect of Carrier Doping

Inelastic neutron scattering spectra were measured on powder samples, using the Wide-Angle Range Chopper Spectrometer (ARCS) at the Spallation Neutron Source, Oak Ridge National Laboratory. Details of the experimental procedure and data reduction are given in SI Text.
The neutron-weighted phonon densities of states (DOSs) measured with INS at T = 10 K are shown in Fig. 1A. We observe a large and unexpected shift of the phonon DOSs to lower energies upon increasing Co concentration. This lowering of phonon frequencies is surprising, considering that the specific volume of CoSi is 3% smaller than that of FeSi, which should lead to an increase of phonon frequencies of order 5% (taking an average Grüneisen parameter γ ∼ 1.6 for FeSi and CoSi, as discussed below). Instead, we observe a -13% energy shift for the acoustic peak around 25 meV, and -4% shift for 〈ℏω〉. The small mass increase from Fe to Co can account for only about 1% of this decrease in frequency (ω ∼ M-1/2) and cannot explain the pronounced effect observed experimentally. Thus, the overall decrease of phonon energies upon alloying with Co must originate from changes in interatomic force constants. We show that this is related to the change in electronic structure through the insulator-to-metal transition, which results in increased screening and smaller force constants.
Fig. 1.
(A, Top) Generalized (neutron-weighted) phonon DOS of FeSi, CoSi, and Fe0.5Co0.5Si measured by INS at 10 K. (A, Bottom) Generalized phonon DOS of FeSi and CoSi calculated with DFT at low temperature, weighted with neutron cross-sections, and convolved with the experimental resolution of the instrument. Arrows are discussed in the text. (B) Phonon dispersions calculated from first principles (red lines: FeSi; black dots: CoSi). Labels at the top and bottom indicate standard high-symmetry points and directions of a cubic lattice (defined in SI Text).
The neutron-weighted phonon DOSs of FeSi and CoSi were computed with density functional theory (DFT) for equilibrium structures (details in SI Text). Results are shown in Fig. 1A. The calculations are in excellent agreement with our experimental data at 10 K for both the shape of the DOS, and the energies of the peaks, indicating that DFT is capturing the details of the atomic forces correctly. The DFT results clearly reproduce the energy shift of the phonons from FeSi to CoSi for the acoustic peak (which also involves low-energy optical branches) between 20 and 30 meV, and the highest-energy optical peak between 50 and 60 meV. The computed phonon dispersions (Fig. 1B) show that most branches are shifting down in energy. Two optical branches around 40 and 45 meV have larger energies in CoSi than in FeSi, however. This is also seen in the measured and calculated DOS curves (vertical arrows in Fig. 1A) and is likely associated with a reduced volume and slight changes in ionic positions (SI Text). Nevertheless, the majority of phonon energies decrease significantly with Co doping, which introduces free charge carriers, augmenting screening. Phonon energies previously measured with Raman and infrared spectroscopies (limited to zone-center phonons) are in good agreement with our results (19, 20).

Anomalous Phonon Softening

The temperature dependences of the phonon spectra differ strongly between FeSi and CoSi, as can be seen in Fig. 2. In the case of CoSi, there is a relatively small shift of the phonon DOSs between 10 and 750 K, compatible with the thermal expansion in this range. On the other hand, the phonon DOS of FeSi softens considerably, with all parts of the spectrum shifting to lower energies. In particular, the acoustic peak at approximately 25 meV in the FeSi DOS softens drastically, with a 14% decrease in energy between 10 and 750 K. It is particularly pronounced between 10 and 300 K. This absolute energy shift between 10 and 750 K is about 4 meV, for both the acoustic peak at 25 meV and for the Si peak at 55 meV. We note a clear similarity in the evolution of the phonon DOSs between adding electrons to the system (Co doping) or raising the temperature.
Fig. 2.
(A) Generalized phonon DOS of FeSi and CoSi at 10, 300, 520, and 750 K (measured by INS on powders). Lines below 10 meV are parabolic extrapolations based on a Debye model. (B) Temperature dependence of phonon energies (relative to base temperature) for FeSi and CoSi. Open markers are from position of peaks in the measured DOS. The orange-shaded area covers the range of behaviors for phonons in FeSi. Filled color markers are for single-crystal measurements of specific phonons in FeSi at the Γ, R, X points of the Brillouin zone, obtained from three-axis measurements. The thick curves show the T dependence predicted by the QH model with the Grüneisen parameters from DFT (see text). The dashed-dotted red curve is the QH model using the temperature-dependent Grüneisen parameter reported by Vočadlo et al. (24) The crosses show 〈ℏω〉/〈ℏω0〉 from DFT computations (no disorder) for FeSi at experimental volumes equivalent to temperatures. Red squares and black dots are the results from AIMD calculations at temperatures TMD (top axis) for FeSi and CoSi, respectively.
Generally, one expects that phonon frequencies decrease (soften) with increasing interatomic separations from thermal expansion (21, 22). The Grüneisen parameter γk,j relates the change in the frequency of the mode of wave vector k and polarization j, ωk,j, to the relative change in volume, V: γk,j = -∂ ln ωk,j/∂ ln V. The average thermodynamic Grüneisen parameter, γ = 〈γk,j〉, is given by γ = 3αVBS/CP, with α the linear coefficient of thermal expansion, BS the isentropic bulk modulus, and CP the heat capacity at constant pressure (21). Typically, it takes a positive value, γ ∼ 1.5, and depends weakly on temperature. It provides a measure of anharmonicity, because phonon frequencies do not depend on volume in a perfectly harmonic lattice (21, 23). The widely used quasiharmonic (QH) approximation assumes that the temperature dependence of phonon frequencies arises solely from changes in volume.
The measured change in phonon energies with temperature is shown in Fig. 2B, where it is compared to the QH behavior. The expected phonon softening in the QH approximation was calculated using experimental thermal expansion data (10, 24), and the Grüneisen parameters we obtained from first-principles calculations, γ(FeSi) = 1.61, γ(CoSi) = 1.73. We also calculated the QH softening for FeSi using the experimental Grüneisen parameter reported in ref. 24. The measured phonon softening in FeSi deviates strongly from the QH behavior. At 300 K, the softening of low-energy modes is over four times larger than the QH prediction. On the other hand, phonon energies in CoSi are in good agreement with the QH model. Again, the phonons soften much more in FeSi than in CoSi. These results support previous observations of a large softening of elastic constants in FeSi (8, 10), but show that all phonon modes are affected, not just long-wavelength acoustic modes.
In order to investigate in more detail the anomalous behavior of phonons in FeSi, neutron scattering was performed on a large single crystal of FeSi (m = 8.5 g). Measurements were performed for 30 orientations of the crystal at 10 and 300 K, using the ARCS spectrometer at the Spallation Neutron Source, and combined to fully map the four-dimensional scattering function (details in SI Text). Dispersions were extracted by taking cuts along chosen momentum transfers, , as shown in Fig. 3. Additional measurements were performed at selected points in reciprocal space with the HB-3 triple-axis spectrometer at the High Flux Isotope Reactor. The triple-axis measurements were performed at Γ, X, R, and M points, as well as midway between Γ and the zone-boundary points, at T = 10, 75, 150, 225, and 300 K. Again, there is a significant softening between 10 and 300 K, over the entire Brillouin zone. This is especially clear at the zone boundary (e.g., M point along [011]), and at the zone center (Γ), as pointed out by the arrows in Fig. 3. In Fig. 4, we plot constant wave-vector scans at the X and R points, which show very clearly the large phonon softening between 10 and 300 K. The temperature-dependent changes in phonon frequency for the single-crystal measurements are also plotted in Fig. 2B. The single-crystal data show an anomalously strong softening for all modes measured, including the IR-active modes at 26 and 40 meV, and the phonon DOS also shows a pronounced softening of the high-energy optic modes around 56 meV. This is at odds with reports in ref. 19 that some IR modes do not soften more than predicted by the QH model.
Fig. 3.
Single-crystal phonon dispersions of FeSi, measured by time-of-flight INS (ARCS), illustrating the change in phonon frequencies between 10 K (AC) and 300 K (DF). The abscissa is the varying component of momentum transfer , in reciprocal lattice units. Light blue lines are FeSi dispersions computed with DFT (without thermal disorder). These are reproduced for reference between AC and DF. In C and F, intensities are scaled by a factor of two. Arrows in B, C, E, and F are discussed in the text.
Fig. 4.
Neutron scattering spectra for single-crystal FeSi, at the X and R points of the Brillouin zone, at 10 K and 300 K, from time-of-flight (Left) and three-axis (Right) measurements. The time-of-flight data were summed over multiple symmetry-equivalent points present in the experimental dataset.
The anomaly in FeSi can be explained by the strong coupling between phonons and the electronic structure. Thermal disorder strongly renormalizes the electronic structure, leading to the closing of the narrow gap. This does not affect CoSi, whose Fermi level is far above the gap. We also note that at high temperatures, the slope of 〈ℏω〉/〈ℏω0〉 in FeSi, Fig. 2B, follows the QH model more closely. This indicates that the coupling of electronic structure to thermal disorder is getting weaker at high temperatures, as is expected because the gap is then closed.
A complementary approach is to calculate temperature-dependent, thermal effective Grüneisen parameters, γth, directly from the measured phonon energies and unit cell volume. We calculated such mode-specific parameters, γth,k,j = -(∂ ln ωk,j/∂T)/(∂ ln V/∂T), from our phonon measurements and reported thermal expansivity data (10, 24). We obtain very large values γth = 24 ± 2 for the low-energy peak in the DOS at 100 K, and γth = 16 ± 2 for the zone center 26 meV optic mode at 150 K. These values are an order of magnitude larger than expected in intermetallic compounds (21). For most modes, we observed γth > 4 at temperatures below 300 K. These thermal effective Grüneisen parameters are temperature dependent and tend to become smaller at higher temperatures (SI Text). This trend is in agreement with the decrease in the mode-averaged γ reported in ref. 24, with γ ∼ 3.8 at T < 50 K, decreasing to an asymptotic value approximately 2.1 above room temperature (24), once again showing that the anomalous softening is suppressed with the closing of the gap in the electronic structure. This is further corroborated by performing electronic structure calculations at finite temperatures.

Ab Initio Molecular Dynamics

In our equilibrium DFT calculations, the positions of the ions and the unit cell volume were optimized to minimize forces on the nuclei and the overall energy, yielding close agreement with experimentally observed structures (SI Text). The electronic DOS for the optimized static structures are shown in Fig. 5 A and B (“0 K”). Measurements in FeSi have reported a range of gap energies Eg ∼ 30–100 meV (6, 7, 9, 10, 19, 25), whereas an indirect gap as small as 10 meV was reported from ellipsometry measurements at low temperature (19). Our calculation for the static low-temperature structure of FeSi does predict a very narrow gap Eg ∼ 120 meV, flanked by two sharp peaks, in agreement with previous calculations (2629), and with photoemission measurements (3033), which observed the very sharp peak at the top of the valence band. The electronic DOS of CoSi is very similar, but shifted to lower energies by 0.6 eV, corresponding to the band filling by the extra d electron of Co, and making CoSi a metal.
Fig. 5.
Electron DOS summed over spin channels for FeSi (A) and CoSi (B), computed with AIMD (T = 300, 600, and 1,200 K), and static DFT calculation on low-T structure (“0 K”). The electronic chemical potential is denoted by μ(T) (= EF at 0 K). At finite temperatures, the electron DOS is obtained from an average over configurations in the corresponding molecular-dynamics simulation. Phonon spectra of FeSi (C) and CoSi (D) computed with AIMD (T = 300, 600, and 1,200 K), and from small-displacement method at the zone-center and zone-boundary points for the static structures (“0 K”). All calculations are for 64-atom supercells. Curves in each panel are offset along the vertical axis for presentation.
We calculated the phonons and the electronic structure at finite temperatures using ab initio molecular dynamics (AIMD) with projector-augmented-wave DFT in the generalized-gradient approximation, as implemented in ref. 34 (details in SI Text). The electronic DOSs for structures at finite temperatures are compared with the 0 K structures in Fig. 5 A and B. In FeSi, thermal disorder leads to the filling of the gap above 600 K. Accompanying the metallization, there is a large increase in N(EF). This agrees with linear muffin-tin orbital calculations of random static disorder (28, 29). In CoSi, the broadening due to disorder is comparable to that in FeSi, but N(EF) is more constant. Because the FeSi electronic gap predicted by DFT is larger than experimental values by about a factor of two, the temperature at which thermal disordering becomes important in AIMD is higher than in experiments. We note that the narrower gap (approximately 90 meV) in CoSi at -0.6 eV is closed above 300 K.
Phonon spectra computed from AIMD (T = 300, 600, and 1,200 K) on a 64-atom 2 × 2 × 2 supercell are shown in Fig. 5 C and D. These spectra capture the zone-boundary and zone-center phonon modes, but not the long-wavelength acoustic modes below 16 meV, owing to the limited size of the simulation cell, and thus constitute a subset of the full DOS (SI Text). Nevertheless, the energies of optical modes up to approximately 55 meV are in good agreement with experiment, and static phonon calculations presented above (Fig. 1A). The AIMD calculations predict a large phonon softening with increasing temperature in FeSi (-8.2% in 〈ω〉 between 300 and 1,200 K), whereas the softening is more limited in CoSi (-2.6%). A large reduction in phonon frequencies is also predicted between FeSi and CoSi at 300 K (-5% in 〈ω〉). These values are in good agreement with our experimental results. The relative change in 〈ω〉 obtained from AIMD is plotted against the MD temperature, TMD, in Fig. 2B. Good agreement is found with the measurements, although the scale for TMD is somewhat larger than for the experimental temperature T, because of the overestimation of the electronic gap.
Because Grüneisen parameters are related to anharmonicity, one may think that the very large thermal Grüneisen parameters (γth) in FeSi are related to strongly anharmonic potentials. To examine this, we performed calculations of frozen-phonon displacement potentials (SI Text). Our calculations indicate a harmonic oscillator potential even for large displacements, ruling out conventional anharmonicity as the cause of the large Grüneisen parameters observed. This conclusion is in agreement with the Raman line shape analysis of ref. 20, which identified the electron–phonon coupling as the source of the temperature dependence, rather than the phonon–phonon interaction associated with anharmonic potentials.
Anomalous temperature dependences of thermodynamic properties in FeSi have been previously interpreted by considering thermal carrier excitations between two thin bands across a narrow gap of fixed width Eg ∼ 80 meV (6, 9, 10). Other interpretations have likened the anomalies to those observed in Kondo insulators (11, 12, 35), but this point of view has recently been challenged by detailed angle-resolved photoemission measurements, which are in very good agreement with the itinerant band structure description from DFT (32, 33). Our AIMD calculations go beyond static DFT calculations and account for the temperature dependence of the gap measured with ellipsometry (19), and the broadening of features with increasing T observed in photoemission measurements (3032, 36) (taking into account the difference in temperature scale mentioned above). We point out that there is a rather striking agreement between the broadening of the electron DOS with T predicted with AIMD (Fig. 5A) and the photoemission data of refs. 30 and 32.
Previous studies have reported the observation of spin fluctuations strengthening with increasing temperature (37). Although our present calculations (without spin fluctuations) reproduce the anomalous softening of phonons in FeSi, we cannot rule out a possible coupling between phonons and spin fluctuations. Jarlborg has shown that the renormalization of the electronic structure by lattice thermal disorder could actually be responsible for the spin fluctuations at high temperature (28).
We also investigated the respective roles of electronic thermal excitations and lattice thermal disorder, by using an elevated temperature for the lattice, Tion, while keeping the electrons at an artificially low temperature, Tel. These calculations show that Tel has a limited effect. Even with the lower Tel = 300 K, the broadening of the electron DOS and filling of the gap occur with increasing Tion as shown in Fig. 5A. The phonon softening is also very similar to the case where Tel = Tion. This agrees with frozen-phonon potentials not depending on Tel (SI Text). Thus, we conclude that the renormalization of the band structure by thermal disorder is mainly responsible for the anomalous temperature dependence in FeSi, whereas electronic excitations across the gap play a more minor role, at least for the phonons. Thermal electronic excitations are of course present, but the occupations of electronic levels do not account for changes in the electron DOS itself (such as the gap closing). The changes in the electron DOS and the large phonon softening are the result of an adiabatic coupling between phonons and electronic structure.


From the present results and previous measurements on superconducting compounds, we can identify general trends in the temperature dependence of phonons in relation to the electronic structure. When electron–phonon coupling and anharmonicity are weak, temperature effects are well captured by the quasiharmonic volume mechanism alone. The resulting temperature dependence of phonon energies is illustrated in Fig. 6A, in the case of a material with a smooth electron DOS. On the other hand, in materials where the electronic DOS exhibits sharp features at the Fermi level, the electron–phonon coupling can lead to nonstandard behaviors. In the case of a narrow-gap semiconductor, such as FeSi, thermal disorder can lead to the closing of the gap and an increase in the density at the Fermi level, N(EF). This increases the screening of atomic forces and leads to an extra phonon softening, compared to the QH volume dependence (Fig. 6B). Conversely, in the case of a sharp peak at the Fermi level, thermal disorder can lead to a broadening of the peak, suppressing N(EF) and the electronic screening of force constants. In this case, the adiabatic electron–phonon coupling would induce a phonon stiffening, as illustrated in Fig. 6C, that competes with the softening from increased volume. We have recently reported such a phonon stiffening (up to T ∼ 1,000 K) in superconducting A15 compounds and body-centered cubic alloys, which do exhibit a peak at the Fermi level (38, 39). The same mechanism of renormalization of the electron DOS by phonon excitations (thermal disorder) is thus capable of explaining the anomalous temperature dependence of phonons in rather different classes of materials.
Fig. 6.
Trends in the temperature dependence of phonon energies for different electronic DOS, N(E). The Fermi energy is denoted by EF, and ω represents an average phonon energy, as a function of temperature T. The dotted line representing the quasiharmonic (QH) behavior in A is reproduced in B and C for comparison.
These results illustrate the importance of the coupling between phonons and electron states when the electronic band structure exhibits sharp features around the Fermi level. Similar effects arising from adiabatic electron–phonon couplings at high temperature are likely to be occurring in a number of other materials. In this regard, narrow-gap semiconductors, heavy-Fermion compounds, superconductors with sharp peaks at the Fermi level, or thermoelectric materials with large slopes at N(EF) could potentially all be affected by this type of coupling. Such effects should not be dismissed a priori in electronic structure calculations, if one wants to predict the high-temperature behavior of materials accurately.


We thank D. J. Singh and S. E. Nagler for helpful discussions. The Research at Oak Ridge National Laboratory’s Spallation Neutron Source, High Flux Isotope Reactor, and Center for Nanophase Materials Sciences was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, US Department of Energy (DOE). ARCS data reduction benefited from DANSE software developed under National Science Foundation Grant DMR-0520547. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the US DOE. O.D. was partially supported by the US DOE, Office of Basic Energy Sciences, as part of an Energy Frontier Research Center, DOE Grant DE-SC0001299. D.M. and B.S. acknowledge funding from DOE Materials Sciences and Technology Division.

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Supporting Information


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Proceedings of the National Academy of Sciences
Vol. 108 | No. 12
March 22, 2011


Submission history

Published online: March 7, 2011
Published in issue: March 22, 2011


  1. electron–phonon coupling
  2. metal-insulator transition
  3. thermoelectrics


We thank D. J. Singh and S. E. Nagler for helpful discussions. The Research at Oak Ridge National Laboratory’s Spallation Neutron Source, High Flux Isotope Reactor, and Center for Nanophase Materials Sciences was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, US Department of Energy (DOE). ARCS data reduction benefited from DANSE software developed under National Science Foundation Grant DMR-0520547. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the US DOE. O.D. was partially supported by the US DOE, Office of Basic Energy Sciences, as part of an Energy Frontier Research Center, DOE Grant DE-SC0001299. D.M. and B.S. acknowledge funding from DOE Materials Sciences and Technology Division.


*This Direct Submission article had a prearranged editor.



Olivier Delaire1 [email protected]
Oak Ridge National Laboratory, 1, Bethel Valley Road, Oak Ridge, TN 37831; and
Karol Marty
Oak Ridge National Laboratory, 1, Bethel Valley Road, Oak Ridge, TN 37831; and
Matthew B. Stone
Oak Ridge National Laboratory, 1, Bethel Valley Road, Oak Ridge, TN 37831; and
Paul R. C. Kent
Oak Ridge National Laboratory, 1, Bethel Valley Road, Oak Ridge, TN 37831; and
Matthew S. Lucas
Air Force Research Laboratory, Wright-Patterson Air Force Base, OH 45433
Douglas L. Abernathy
Oak Ridge National Laboratory, 1, Bethel Valley Road, Oak Ridge, TN 37831; and
David Mandrus
Oak Ridge National Laboratory, 1, Bethel Valley Road, Oak Ridge, TN 37831; and
Brian C. Sales
Oak Ridge National Laboratory, 1, Bethel Valley Road, Oak Ridge, TN 37831; and


To whom correspondence should be addressed. E-mail: [email protected].
Author contributions: O.D. designed research; O.D., K.M., M.B.S., P.R.C.K., M.S.L., D.L.A., and B.C.S. performed research; O.D., K.M., M.B.S., and P.R.C.K. analyzed data; and O.D., M.B.S., P.R.C.K., D.M., and B.C.S. wrote the paper.

Competing Interests

The authors declare no conflict of interest.

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    Phonon softening and metallization of a narrow-gap semiconductor by thermal disorder
    Proceedings of the National Academy of Sciences
    • Vol. 108
    • No. 12
    • pp. 4697-5139







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