Phonon softening and metallization of a narrow-gapsemiconductor by thermal disorderOlivier Delairea,1, Karol Martya, Matthew B. Stonea, Paul R. C. Kenta, Matthew S. Lucasb, Douglas L. Abernathya,David Mandrusa, and Brian C. Salesa

aOak Ridge National Laboratory, 1, Bethel Valley Road, Oak Ridge, TN 37831; and bAir Force Research Laboratory, Wright-Patterson Air Force Base,OH 45433

Edited* by Mildred Dresselhaus, Massachusetts Institute of Technology, Cambridge, MA, and approved February 9, 2011 (received for review October 4, 2010)

The vibrations of ions in solids at finite temperature depend oninteratomic force–constants that result from electrostatic interac-tions between ions, and the response of the electron density toatomic displacements. At high temperatures, vibration amplitudesare substantial, and electronic states are affected, thus modifyingthe screening properties of the electron density. By combininginelastic neutron scattering measurements of Fe1−xCoxSi as a func-tion of temperature, and finite-temperature first-principles calcula-tions including thermal disorder effects, we show that the couplingbetween phonons and electronic structure results in an anomaloustemperature dependence of phonons. The strong concomitantrenormalization of the electronic structure induces the semiconduc-tor-to-metal transition that occurs with increasing temperaturein FeSi. Our results show that for systems with rapidly changingelectronic densities of states at the Fermi level, there are likelyto be significant phonon–electron interactions, resulting in anom-alous temperature-dependent properties.

electron–phonon coupling ∣ metal-insulator transition ∣ thermoelectrics

Because many properties of the solid state derive from the elec-tronic structure (1), understanding finite temperature effects

on the band structure is crucial to accurately describe materialsin realistic operating conditions. The effects of thermal disorderon the electronic structure of materials at high temperature arelargely unexplored, however, and the role of the electron–phononinteraction above room temperature has remained controversial(2–5). We performed detailed investigations of the phonons andelectronic structure in Fe1−xCoxSi and found that an adiabaticcoupling can lead to pronounced anomalies in the temperaturedependence of both phonons and electron states. The mechanismis general and could affect a broad class of materials, includingnarrow-gap semiconductors, superconductors, heavy-Fermioncompounds, and thermoelectrics.

FeSi has attracted a great deal of interest as it exhibits aninsulator-metal transition with increasing temperature, and manyof its physical properties show anomalous temperature depen-dences, including the magnetic susceptibility, heat capacity,Seebeck coefficient, thermal expansion, and elastic properties(6–11). Recently, it has been argued that doping FeSi with Alcan lead to a surprising heavy-Fermion metal (12). FeSi has alsoattracted attention as a possible reaction product at Earth’score-mantle boundary (13–16), and as a candidate thermoelectricmaterial for refrigeration applications (9). The compounds FeSiand CoSi are isostructural, crystallizing in the cubic B20 struc-ture, with similar ion coordinates (17, 18). Although FeSi under-goes a gradual transition from narrow-gap semiconductor(Egap ∼ 0.1 eV) to metal with increasing temperature, the addi-tional d electron in CoSi leads to a metallic state at all tempera-tures. The anomalous temperature dependences observed in FeSiare absent in CoSi.

Here, we show that the adiabatic electron–phonon couplinghas spectacular consequences in FeSi, leading to the metalliza-tion of the semiconducting compound as temperature is in-creased, and causing a large concurrent lowering of phonon

frequencies. We conducted comprehensive measurements of thephonons in Fe1−xCoxSi using inelastic neutron scattering (INS)at temperatures 10 ≤ T ≤ 750 K, and detailed ab initio calcula-tions of the full phonon spectra, as well as ab initio moleculardynamics at finite temperature. Both experimental measure-ments and first-principles calculations show a large suppressionof phonon frequencies in FeSi with either increasing temperatureor adding charge carriers. Our simulations also reproduce theexperimentally observed insulator-metal transition in FeSi withincreasing temperature. By contrast, our measurements andcalculations show that CoSi exhibits normal phonon behaviors,which is understood from its different electronic structure.

The adiabatic electron–phonon coupling mechanism pre-sented here is general in nature, and similar effects could beoccurring in a large number of other systems, in particular inmaterials where the electronic structure has sharp features nearthe Fermi level. We present some trends for the anomaloustemperature dependence of phonons, depending on the type offeatures in the electronic structure.

Effect of Carrier DopingInelastic neutron scattering spectra were measured on powdersamples, using the Wide-Angle Range Chopper Spectrometer(ARCS) at the Spallation Neutron Source, Oak Ridge NationalLaboratory. Details of the experimental procedure and datareduction are given in SI Text.

The neutron-weighted phonon densities of states (DOSs) mea-sured with INS at T ¼ 10 K are shown in Fig. 1A. We observe alarge and unexpected shift of the phonon DOSs to lower energiesupon increasing Co concentration. This lowering of phononfrequencies is surprising, considering that the specific volumeof CoSi is 3% smaller than that of FeSi, which should lead toan increase of phonon frequencies of order 5% (taking an aver-age Grüneisen parameter γ ∼ 1.6 for FeSi and CoSi, as discussedbelow). Instead, we observe a −13% energy shift for the acousticpeak around 25 meV, and −4% shift for hℏωi. The small massincrease from Fe to Co can account for only about 1% of thisdecrease in frequency (ω ∼M−1∕2) and cannot explain thepronounced effect observed experimentally. Thus, the overalldecrease of phonon energies upon alloying with Co must origi-nate from changes in interatomic force constants. We show thatthis is related to the change in electronic structure through theinsulator-to-metal transition, which results in increased screeningand smaller force constants.

The neutron-weighted phonon DOSs of FeSi and CoSi werecomputed with density functional theory (DFT) for equilibrium

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.

The authors declare no conflict of interest.

*This Direct Submission article had a prearranged editor.1To whom correspondence should be addressed. E-mail: delaireoa@ornl.gov.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1014869108/-/DCSupplemental.

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structures (details in SI Text). Results are shown in Fig. 1A. Thecalculations are in excellent agreement with our experimentaldata at 10 K for both the shape of the DOS, and the energiesof the peaks, indicating that DFT is capturing the details ofthe atomic forces correctly. The DFT results clearly reproducethe energy shift of the phonons from FeSi to CoSi for the acousticpeak (which also involves low-energy optical branches) between20 and 30 meV, and the highest-energy optical peak between 50and 60 meV. The computed phonon dispersions (Fig. 1B) showthat most branches are shifting down in energy. Two opticalbranches around 40 and 45 meV have larger energies in CoSithan in FeSi, however. This is also seen in the measured andcalculated DOS curves (vertical arrows in Fig. 1A) and is likelyassociated with a reduced volume and slight changes in ionicpositions (SI Text). Nevertheless, the majority of phonon ener-gies decrease significantly with Co doping, which introducesfree charge carriers, augmenting screening. Phonon energiespreviously measured with Raman and infrared spectroscopies(limited to zone-center phonons) are in good agreement withour results (19, 20).

Anomalous Phonon SofteningThe temperature dependences of the phonon spectra differstrongly between FeSi and CoSi, as can be seen in Fig. 2. Inthe case of CoSi, there is a relatively small shift of the phononDOSs between 10 and 750 K, compatible with the thermal expan-sion in this range. On the other hand, the phonon DOS of FeSisoftens considerably, with all parts of the spectrum shifting tolower energies. In particular, the acoustic peak at approximately25 meV in the FeSi DOS softens drastically, with a 14% decreasein energy between 10 and 750 K. It is particularly pronouncedbetween 10 and 300 K. This absolute energy shift between 10and 750 K is about 4 meV, for both the acoustic peak at25 meVand for the Si peak at 55 meV. We note a clear similarityin the evolution of the phonon DOSs between adding electrons tothe system (Co doping) or raising the temperature.

Generally, one expects that phonon frequencies decrease (soft-en) with increasing interatomic separations from thermal expan-sion (21, 22). The Grüneisen parameter γk;j relates the changein the frequency of the mode of wave vector k and polarizationj, ωk;j, to the relative change in volume, V : γk;j ¼ −∂ lnωk;j∕∂ lnV .The average thermodynamic Grüneisen parameter, γ ¼ hγk;ji, isgiven by γ ¼ 3αVBS∕CP, with α the linear coefficient of thermalexpansion, BS the isentropic bulk modulus, and CP the heatcapacity at constant pressure (21). Typically, it takes a positivevalue, γ ∼ 1.5, and depends weakly on temperature. It providesa measure of anharmonicity, because phonon frequencies donot depend on volume in a perfectly harmonic lattice (21, 23).The widely used quasiharmonic (QH) approximation assumesthat the temperature dependence of phonon frequencies arisessolely from changes in volume.

The measured change in phonon energies with temperature isshown in Fig. 2B, where it is compared to the QH behavior. The

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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 DOSof FeSi and CoSi calculated with DFT at low temperature, weighted with neutron cross-sections, and convolved with the experimental resolution of theinstrument. Arrows are discussed in the text. (B) Phonon dispersions calculated from first principles (red lines: FeSi; black dots: CoSi). Labels at the topand bottom indicate standard high-symmetry points and directions of a cubic lattice (defined in SI Text).

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Fig. 2. (A) Generalized phonon DOS of FeSi and CoSi at 10, 300, 520, and750 K (measured by INS on powders). Lines below 10 meV are parabolicextrapolations based on a Debye model. (B) Temperature dependence ofphonon energies (relative to base temperature) for FeSi and CoSi. Open mar-kers are from position of peaks in the measured DOS. The orange-shadedarea covers the range of behaviors for phonons in FeSi. Filled color markersare for single-crystal measurements of specific phonons in FeSi at the Γ, R, Xpoints of the Brillouin zone, obtained from three-axis measurements. Thethick curves show the T dependence predicted by the QH model with theGrüneisen parameters from DFT (see text). The dashed-dotted red curve isthe QH model using the temperature-dependent Grüneisen parameterreported by Vočadlo et al. (24) The crosses show hℏωi∕hℏω0i from DFTcomputations (no disorder) for FeSi at experimental volumes equivalent totemperatures. Red squares and black dots are the results from AIMD calcula-tions at temperatures TMD (top axis) for FeSi and CoSi, respectively.

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expected phonon softening in the QH approximation was calcu-lated using experimental thermal expansion data (10, 24), and theGrüneisen parameters we obtained from first-principles calcula-tions, γðFeSiÞ ¼ 1.61, γðCoSiÞ ¼ 1.73. We also calculated the QHsoftening for FeSi using the experimental Grüneisen parameterreported in ref. 24. The measured phonon softening in FeSideviates strongly from the QH behavior. At 300 K, the softeningof low-energy modes is over four times larger than the QHprediction. On the other hand, phonon energies in CoSi are ingood agreement with the QH model. Again, the phonons softenmuch more in FeSi than in CoSi. These results support previousobservations of a large softening of elastic constants in FeSi(8, 10), but show that all phonon modes are affected, not justlong-wavelength acoustic modes.

In order to investigate in more detail the anomalous behaviorof phonons in FeSi, neutron scattering was performed on a largesingle crystal of FeSi (m ¼ 8.5 g). Measurements were performedfor 30 orientations of the crystal at 10 and 300 K, using the ARCSspectrometer at the Spallation Neutron Source, and combinedto fully map the four-dimensional scattering function Sð ~Q;EÞ(details in SI Text). Dispersions were extracted by taking cutsalong chosen momentum transfers, ~Q, as shown in Fig. 3. Addi-tional measurements were performed at selected points in reci-procal space with the HB-3 triple-axis spectrometer at the HighFlux Isotope Reactor. The triple-axis measurements were per-formed at Γ, X, R, and M points, as well as midway betweenΓ and the zone-boundary points, at T ¼ 10, 75, 150, 225, and300 K. Again, there is a significant softening between 10 and300 K, over the entire Brillouin zone. This is especially clearat the zone boundary (e.g., M point along [011]), and at the zonecenter (Γ), as pointed out by the arrows in Fig. 3. In Fig. 4, we plotconstant wave-vector scans at the X and R points, which showvery clearly the large phonon softening between 10 and 300 K.The temperature-dependent changes in phonon frequency forthe 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 26and 40 meV, and the phonon DOS also shows a pronounced soft-ening of the high-energy optic modes around 56 meV. This is atodds with reports in ref. 19 that some IR modes do not softenmore than predicted by the QH model.

The anomaly in FeSi can be explained by the strong couplingbetween phonons and the electronic structure. Thermal disorderstrongly renormalizes the electronic structure, leading to the clos-ing of the narrow gap. This does not affect CoSi, whose Fermilevel is far above the gap. We also note that at high temperatures,the slope of hℏωi∕hℏω0i in FeSi, Fig. 2B, follows the QH modelmore closely. This indicates that the coupling of electronic struc-ture 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, directlyfrom the measured phonon energies and unit cell volume. Wecalculated such mode-specific parameters, γth;k;j ¼ −ð∂ lnωk;j∕∂TÞ∕ð∂ lnV∕∂TÞ, from our phonon measurements and reportedthermal expansivity data (10, 24). We obtain very large valuesγth ¼ 24� 2 for the low-energy peak in the DOS at 100 K,and γth ¼ 16� 2 for the zone center 26 meV optic mode at150 K. These values are an order of magnitude larger thanexpected in intermetallic compounds (21). For most modes, weobserved γth > 4 at temperatures below 300 K. These thermaleffective Grüneisen parameters are temperature dependentand tend to become smaller at higher temperatures (SI Text). Thistrend is in agreement with the decrease in the mode-averaged γreported in ref. 24, with γ ∼ 3.8 at T < 50 K, decreasing to anasymptotic value approximately 2.1 above room temperature(24), once again showing that the anomalous softening is sup-pressed with the closing of the gap in the electronic structure.This is further corroborated by performing electronic structurecalculations at finite temperatures.

Ab Initio Molecular DynamicsIn our equilibrium DFTcalculations, the positions of the ions andthe unit cell volume were optimized to minimize forces on thenuclei and the overall energy, yielding close agreement withexperimentally observed structures (SI Text). The electronic DOSfor the optimized static structures are shown in Fig. 5 A and B(“0 K”). Measurements in FeSi have reported a range of gapenergies Eg ∼ 30–100 meV (6, 7, 9, 10, 19, 25), whereas anindirect gap as small as 10 meV was reported from ellipsometrymeasurements at low temperature (19). Our calculation for thestatic low-temperature structure of FeSi does predict a verynarrow gap Eg ∼ 120 meV, flanked by two sharp peaks, in agree-ment with previous calculations (26–29), and with photoemission

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measurements (30–33), which observed the very sharp peak atthe top of the valence band. The electronic DOS of CoSi is verysimilar, but shifted to lower energies by 0.6 eV, correspondingto the band filling by the extra d electron of Co, and making CoSia metal.

We calculated the phonons and the electronic structure atfinite temperatures using ab initio molecular dynamics (AIMD)with projector-augmented-wave DFT in the generalized-gradientapproximation, as implemented in ref. 34 (details in SI Text). Theelectronic DOSs for structures at finite temperatures are com-pared with the 0 K structures in Fig. 5 A and B. In FeSi, thermaldisorder leads to the filling of the gap above 600 K. Accompany-ing the metallization, there is a large increase in NðEFÞ. Thisagrees with linear muffin-tin orbital calculations of random staticdisorder (28, 29). In CoSi, the broadening due to disorder iscomparable to that in FeSi, but NðEFÞ is more constant. Becausethe FeSi electronic gap predicted by DFT is larger than experi-mental values by about a factor of two, the temperature at whichthermal disordering becomes important in AIMD is higher thanin experiments. We note that the narrower gap (approximately90 meV) in CoSi at −0.6 eV is closed above 300 K.

Phonon spectra computed from AIMD (T ¼ 300, 600, and1,200 K) on a 64-atom 2 × 2 × 2 supercell are shown in Fig. 5 CandD. These spectra capture the zone-boundary and zone-centerphonon modes, but not the long-wavelength acoustic modesbelow 16 meV, owing to the limited size of the simulation cell,and thus constitute a subset of the full DOS (SI Text). Neverthe-less, the energies of optical modes up to approximately 55 meVare in good agreement with experiment, and static phonon cal-culations presented above (Fig. 1A). The AIMD calculations pre-dict a large phonon softening with increasing temperature in FeSi

(−8.2% in hωi between 300 and 1,200 K), whereas the softeningis more limited in CoSi (−2.6%). A large reduction in phononfrequencies is also predicted between FeSi and CoSi at 300 K(−5% in hωi). These values are in good agreement with ourexperimental results. The relative change in hωi obtained fromAIMD is plotted against the MD temperature, TMD, in Fig. 2B.Good agreement is found with the measurements, although thescale for TMD is somewhat larger than for the experimental tem-perature 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. Toexamine this, we performed calculations of frozen-phonon dis-placement potentials (SI Text). Our calculations indicate a harmo-nic oscillator potential even for large displacements, ruling outconventional anharmonicity as the cause of the large Grüneisenparameters observed. This conclusion is in agreement with theRaman line shape analysis of ref. 20, which identified the elec-tron–phonon coupling as the source of the temperature depen-dence, rather than the phonon–phonon interaction associatedwith anharmonic potentials.

Anomalous temperature dependences of thermodynamicproperties in FeSi have been previously interpreted by consider-ing thermal carrier excitations between two thin bands across anarrow gap of fixed width Eg ∼ 80 meV (6, 9, 10). Other interpre-tations have likened the anomalies to those observed in Kondoinsulators (11, 12, 35), but this point of view has recently beenchallenged by detailed angle-resolved photoemission measure-ments, which are in very good agreement with the itinerant bandstructure description from DFT (32, 33). Our AIMD calculationsgo beyond static DFT calculations and account for the tempera-ture dependence of the gap measured with ellipsometry (19),and the broadening of features with increasing T observed inphotoemission measurements (30–32, 36) (taking into accountthe difference in temperature scale mentioned above). We pointout that there is a rather striking agreement between the broad-ening 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 fluctua-tions strengthening with increasing temperature (37). Althoughour present calculations (without spin fluctuations) reproducethe anomalous softening of phonons in FeSi, we cannot ruleout a possible coupling between phonons and spin fluctuations.Jarlborg has shown that the renormalization of the electronicstructure by lattice thermal disorder could actually be responsiblefor the spin fluctuations at high temperature (28).

We also investigated the respective roles of electronic thermalexcitations and lattice thermal disorder, by using an elevatedtemperature for the lattice, T ion, while keeping the electrons atan artificially low temperature, Tel. These calculations show thatTel has a limited effect. Even with the lower Tel ¼ 300 K, thebroadening of the electron DOS and filling of the gap occurwith increasing T ion as shown in Fig. 5A. The phonon softeningis also very similar to the case where Tel ¼ T ion. This agrees withfrozen-phonon potentials not depending on Tel (SI Text). Thus,we conclude that the renormalization of the band structure bythermal disorder is mainly responsible for the anomalous tem-perature dependence in FeSi, whereas electronic excitationsacross the gap play a more minor role, at least for the phonons.Thermal electronic excitations are of course present, but theoccupations of electronic levels do not account for changes inthe electron DOS itself (such as the gap closing). The changesin the electron DOS and the large phonon softening are the resultof an adiabatic coupling between phonons and electronicstructure.

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ConclusionsFrom the present results and previous measurements on super-conducting compounds, we can identify general trends in the tem-perature dependence of phonons in relation to the electronicstructure. When electron–phonon coupling and anharmonicityare weak, temperature effects are well captured by the quasihar-monic volume mechanism alone. The resulting temperaturedependence of phonon energies is illustrated in Fig. 6A, in thecase of a material with a smooth electron DOS. On the otherhand, in materials where the electronic DOS exhibits sharp fea-tures at the Fermi level, the electron–phonon coupling can leadto nonstandard behaviors. In the case of a narrow-gap semicon-ductor, such as FeSi, thermal disorder can lead to the closing of

the gap and an increase in the density at the Fermi level, NðEFÞ.This increases the screening of atomic forces and leads to an extraphonon softening, compared to the QH volume dependence(Fig. 6B). Conversely, in the case of a sharp peak at the Fermilevel, thermal disorder can lead to a broadening of the peak, sup-pressingNðEFÞ and the electronic screening of force constants. Inthis case, the adiabatic electron–phonon coupling would induce aphonon stiffening, as illustrated in Fig. 6C, that competes with thesoftening from increased volume. We have recently reported sucha phonon stiffening (up to T ∼ 1;000 K) in superconducting A15compounds and body-centered cubic alloys, which do exhibit apeak at the Fermi level (38, 39). The same mechanism of renor-malization of the electron DOS by phonon excitations (thermaldisorder) is thus capable of explaining the anomalous tempera-ture dependence of phonons in rather different classes of ma-terials.

These results illustrate the importance of the coupling betweenphonons and electron states when the electronic band structureexhibits sharp features around the Fermi level. Similar effectsarising from adiabatic electron–phonon couplings at high tem-perature are likely to be occurring in a number of other materials.In this regard, narrow-gap semiconductors, heavy-Fermion com-pounds, superconductors with sharp peaks at the Fermi level, orthermoelectric materials with large slopes at NðEFÞ could poten-tially all be affected by this type of coupling. Such effects shouldnot be dismissed a priori in electronic structure calculations, ifone wants to predict the high-temperature behavior of materialsaccurately.

ACKNOWLEDGMENTS. We thank D. J. Singh and S. E. Nagler for helpfuldiscussions. The Research at Oak Ridge National Laboratory’s SpallationNeutron Source, High Flux Isotope Reactor, and Center for NanophaseMaterials Sciences was sponsored by the Scientific User Facilities Division,Office of Basic Energy Sciences, US Department of Energy (DOE). ARCS datareduction benefited from DANSE software developed under National ScienceFoundation Grant DMR-0520547. This research used resources of theNational Energy Research Scientific Computing Center, which is supportedby the Office of Science of the US DOE. O.D. was partially supported bythe US DOE, Office of Basic Energy Sciences, as part of an Energy FrontierResearch Center, DOE Grant DE-SC0001299. D.M. and B.S. acknowledgefunding from DOE Materials Sciences and Technology Division.

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Electron DOS

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Phonons

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N(E)

N(E)

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low Thigh T

EF

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anomalous

anomalous

B

A

C

Fig. 6. Trends in the temperature dependence of phonon energies fordifferent electronic DOS, NðEÞ. The Fermi energy is denoted by EF, and ωrepresents an average phonon energy, as a function of temperature T .The dotted line representing the quasiharmonic (QH) behavior in A is repro-duced in B and C for comparison.

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