Table Of ContentElectron-phonon interaction and spectral weight transfer in Fe Co Si
1−x x
D. Menzel,1,∗ P. Popovich,2,∗ N. N. Kovaleva,2,3 J. Schoenes,1 K. Doll,2 and A. V. Boris2,3
1Institut fu¨r Physik der Kondensierten Materie, Technische Universit¨at Braunschweig,
Mendelssohnstr. 3, D-38106 Braunschweig, Germany
2Max Planck Institute for Solid State Research, Heisenbergstr. 1, D-70569 Stuttgart, Germany
3Department of Physics, Loughborough University, Loughborough, LE11 3TU, United Kingdom
(Dated: January 8, 2009)
AcomprehensiveellipsometricstudywasperformedonFe1−xCoxSisinglecrystalsinthespectral
rangefrom0.01eVto6.2eV.Directandindirectbandgapsof73meVand10meV,respectively,were
9
observedinFeSiat7K.Oneoffourinfrared-activephononsthatisenergetically closetothedirect
0
0 absorption edge is coupled both to the electrons and to the low-energy phonon. This is evident
2 from asymmetry in the phonon line shape and a reduction of its frequency when the absorption
edge shifts across the phonon energy due to the temperature dependence of the direct band gap.
n
As the temperature increases, the indirect gap changes sign, which manifests as a transition from
a
a semiconductor to a semimetal. The corresponding gain of the spectral weight at low energies
J
was recovered within an energy range of several eV. The present findings strongly support the
8
modelindicatingthatFe1−xCoxSicanbewelldescribedinanitinerantpicture,takingintoaccount
self-energy corrections.
]
i
c PACSnumbers: 71.20.Be,78.20.Ci,78.30.Am
s
-
l
r I. INTRODUCTION FeSi can be interpreted in an itinerant picture in which
t
m the magnetic behavior is strongly influenced by a small
. shift in the electronic density of states.10 Doping FeSi
t Due to its unusual properties, the narrow-gap semi-
a with more than 5 at.% Co results in a metallic state
conductor FeSi continues to be the subject of intrigu-
m due to the closing of the band gap. Between 5 and 80
ing debate among theorists and experimentalists with
- regard to its electronic nature. Some groups have de- at.% Co, ferromagnetic order is observed and the sam-
d ple displays a conical spin structure.12 Above 80 at.%
n scribed this material as a correlated system with local
o Fe moments similartothose ofthe compounds knownas Fe1−xCoxSi is paramagnetic, and finally, CoSi is a dia-
magnetic semimetal.13
c Kondo insulators.1 FeSi has an insulating non-magnetic
[ ground state, and its magnetic susceptibility increases Sofar,resultsfromopticalandthermodynamicstudies
1 with rising temperature.2 This behavior has been mod- have not been conclusive. The absolute value of the ex-
v eled by two narrow peaks in the density of states (DOS) perimentally derived band gap varies between 0 and 100
4 onbothsidesofthegap.3 However,localdensityapprox- meV; clearly, the correct size of the intrinsic band gap
1 imation (LDA) calculations do not predictsuch peaks in has still not been reliably determined. Two reasons can
0 the DOS, and this disagreement has been construed as be proposed for this. First, the quality of the samples
1
an indication of correlation effects.4 The fact that a low plays a crucial role in determining the electronic proper-
.
1 temperaturemassenhancementofabout30freeelectron ties. Impurities lead to an only partially developed gap
0 masshasbeenfoundinspecificheatmeasurements5lends with residual states at the Fermi energy, as observed in
9 additional support to the argument for strong electronic earlierhigh-resolutionphotoemissionexperiments.14 The
0
correlations. Results from Raman scattering have been samples usedin the presentstudy havea verylow impu-
:
v interpreted in terms of strong coupling between charge rity concentrationof 0.17 at.% and display a fully devel-
Xi excitations and the lattice.6 opedgapinthedensityofstates.7,8 Second,inpaststud-
ies optical gaps have been determined using reflectivity
r More recent results on high-purity samples indicate
a that FeSi can be well described in an itinerant picture. measurements.5,15,16 However, accurate absolute values
of the dielectric function are difficult to obtain with this
Recent photoemission (PE) investigations of FeSi single
method, especially when the reflectivity is close to unity
crystalshavenotrevealedanyindicationoftheexistence
of a Kondo resonance close to the Fermi energy.7,8 In- in the far-infrared range. Moreover, for absolute values
the comparison with a reference sample is always neces-
stead, good agreement has been found between photoe-
sary and for a correct Kramers-Kronig analysis, values
mission experiments and band structure calculations us-
ing density functional theory;9 this agreement favors an towardzeroandinfinite energymust be extrapolated. A
muchbetter techniqueisthedirectmeasurementofboth
itinerant description rather than a Kondo model. Ra-
the realandthe imaginarypartofthe dielectric function
man spectroscopy on the same FeSi single crystals re-
without the need for a reference sample, which wasdone
veals that the linewidth of an E-phonon increases with
growingtemperature10,11 duetotheshiftofthechemical in this work via spectroscopic ellipsometry.
potential with increasing temperature. Also, magnetic Reflectivity measurements showed that the band-gap
measurementsasafunctionofpressureonCo-substituted disappears at a much lower temperature than expected.
2
When FeSi is cooled down, the band gap is formed obtained by cutting the single crystals into slices; the
and,consequently,spectralweightatlowenergiesislost. cut surfaces were then polished to optical grade using
Someexperimentsindicatethatthisspectralweightdoes a 0.25 µm diamond suspension. The samples were not
not appear just above the gap, as expected in a classical oriented for these experiments which, however, does not
band-like material, but is redistributed over a large en- affect the results of the optical investigations since the
ergyscale,morethan80timesthegapenergy.5,15,17 This crystal structure is cubic.
is also observedin, for example, CePd , which is consid- The IR optical measurements were performed with
3
eredaKondo-insulatortypematerialandwhereasimilar home-built ellipsometers, attached to Fourier transform
shift in the spectral weight from low to high energies is Bruker IFS-66vs spectrometers. As a source we used
found.19 This has led to the conclusion that FeSi is a synchrotron edge radiation at the IR1 beamline of the
strongly correlated insulator. However, based on other ANKA synchrotron for the range of 10−85 meV (80−
reflectivity experiments on FeSi single crystals, it has 680 cm−1), and a SiC glow bar and a mercury arc lamp
been claimed that the missing spectral weight below the for the range from 0.06 to 1.1 eV (500− 9000 cm−1).
gap energy is redistributed around the gap edge, which The angle of incidence was selected to 80.0◦ for all
isacommonpropertyofaconventionalsemiconductor.16 measurements. The infraredspectroscopicinvestigations
We believe thatthis disagreementis due to the difficulty were complemented with measurements in the range
in obtaining absolute values from reflectivity measure- 0.8−6 eV using a commercial Woolam VASE variable
ments. angle rotating-analyzer ellipsometer. In all these setups
Another key issue is the determination of the ionicity the samples were mounted in continuous helium flow
and the charge transfer between the constituents, which cryostats (6−450 K). The base pressure at room tem-
can also be easily derived from the dielectric function. perature was better than 5×10−7 mbar for the IR and
Forthese quantities,accuratedeterminationofthe abso- 5×10−9 mbar for the visible setup.
lute values from ellipsometric data is again mandatory.
In this paper, we report on a decisive examination of
the opticalpropertiesofFe1−xCoxSiby spectroscopicel-
lipsometry from the far-infrared (20 meV) to the UV Wave nu mber (cm-1)
spectral range (6.2 eV). Section II describes the exper- 200 300 40 0 500 600
a)
imental details. In Section IIIA, eigenfrequencies and
2
2500
lineshapes of infrared active modes are determined and 3
4
assigned. In Section IIIB, the determinations of direct 1
2000
and indirect band gaps as functions of Co concentra- )
1
tionandtemperature arepresented. InSectionIIIC,the -m1500
c
electron-phonon coupling is considered as the origin of 1
-Ω
phonon anharmonicity. The self-consistency of the spec- (1000
traisverifiedusingthegeneralizedLyddane-Sachs-Teller σ 1 FeSi
relation. The Born and Szigeti effective charge is calcu- 500 Fe Co Si
0.95 0.05
latedtodeterminethechargetransferbetweenFeandSi. Fe Co Si
0 0.90 0.10
In Section IIID, we address the question of what causes
20 30 40 50 60 70 80
theenhancementofthefar-IRspectralweightduetothe
Photon Energy (meV)
optical gap closing. We report that this additional spec-
b)
trreagliowneiogfh0t.i5s-2tr.5anesVfe.rrFeidnamlloys,tolyurfrocomnctlhuesihoingsh-afrreeqdureanwcny 400 1 2 FFeeSiCo Si
0.95 0.05
in Section IV. 300 Fe Co Si
4 0.90 0.10
200 3
1100
II. EXPERIMENTAL DETAILS ε
0
ForthepreparationoftheFe1−xCoxSisamples99.98% -100
Fe, 99.9+% Co, and Si (ρn =300 Ωcm, ρp =3000Ωcm) -200
were pre-melted in a single arc oven under argon atmo-
20 30 40 50 60 70 80
sphere. Consecutively, single crystals were grown us-
ing the triarc Czochralski technique. The good crys- Photon Energy (meV)
tallinity was evidenced by Laue analysis and by low en-
ergy electron diffraction (LEED) during photoemission
experiments.7,8Averylowconcentrationofmagneticim- Figure1: (Coloronline)Realpartsof(a)opticalconductivity
puritiesinthepureFeSicrystalsisderivedfrommeasure- σ1(ω) and (b) dielectric function ε1(ω) of Fe1−xCoxSi with
ments of the magnetic susceptibility. variousCoconcentrationsxatT =7K.Phononsarelabeled
Fortheopticalmeasurements,individualsampleswere from ’1’ to ’4’.
3
The complex reflectance ratio ρ was derived for an
isotropic medium by inverting the Fresnel equations for Wave num ber (cm-1)
the measuredellipsometricparametersΨ and∆, defined 200 300 400 500 600
a)
as
4000 7 K 150 K
r 60 K 200 K
tanΨei∆ = p =ρ. (1) 100 K 300 K
rs 3000
)
Here rp and rs are the Fresnel reflection coefficients for -1m
light polarized parallel and perpendicular to the plane c2000
1
of incidence, respectively. For a semi-infinite isotropic -Ω
(
sample the complex dielectric function ε(ω) follows as σ 11000
2 0
ε(ω)=ε (ω)+iε (ω)=sin2θ 1+ 1−ρ tan2θ 20 30 40 50 60 70 80
1 2
" (cid:18)1+ρ(cid:19) # Photon Energy (meV)
(2) b) 400
where ε , ε , and θ are the real and imaginary parts
1 2 300
of the dielectric function and the angle of incidence, re-
200
spectively.
100
0
1
III. RESULTS AND DISCUSSION ε
-100
-200 7 K 150 K
A. Dielectric response in the phonon energy range 60 K 200 K
-300 100 K 300 K
In our far-infrared spectra of FeSi at T = 11 K, 20 30 40 50 60 70 80
four infrared active phonons are observed at 25.5 meV,
Photon Energy (meV)
40.8 meV, 43.6 meV, and 56.8 meV (206 cm−1, 329
cm−1, 352 cm−1 and 458 cm−1, respectively) (Fig. 1),
in good agreement with previous reports.15,17,18 These
Figure2: (Coloronline)Realpartsof(a)opticalconductivity
modesaresuperimposedonafeaturelesselectronicback-
σ1(ω) and (b) dielectric function ε1(ω) of FeSi measured at
ground. The optical conductivity, σ (ω), of the flat elec-
1 different temperatures.
tronic background increases monotonically with the Co
concentration, making the material more metallic (Fig.
1a). This property is also reflected by the high negative both fully symmetric and can be fitted by classical har-
values of ε1(ω) at low photon energies (Fig. 1b). monic Lorentzian oscillators. In contrast, the two outer-
Theinfluenceofthetemperatureonthedielectricfunc- most phonons, ’1’ and ’4’ (at 25.5 meV and 56.8 meV,
tion of FeSi is displayed in Fig. 2. With increasing tem- respectively), have asymmetric line shapes that cannot
perature,thegapisfilled,leadingtomoremetallicbehav- be described by the Lorentzian equation. In insulating
ior. Therefore, ε1(ω) decreases and becomes negative as systems having more than one infrared-active phonon
σ1(ω)rises,andthe gap,whichisobservableatlowtem- mode at k = 0 these modes are not necessarily inde-
peratures, washes out. This result is in accordance with pendent, but can interact with each other.20 Therefore,
the observation made by Raman spectroscopy.11 From theobservedasymmetrysuggestsacouplingbetweenthe
the evolution of the Raman phonon linewidth with tem- phonon modes. The dielectric function with n phonons
perature, it is shown that the chemical potential, which can be expressed by modified Lorentzian oscillators:21
at zero temperature lies in the middle of the band gap,
is shifted toward lower energy and crosses the top of the
n Ω2−iωQ
valence band near 250 K. j j
In general, it is expected that in metals the phonons ε(ω)=ε1(ω)+iε2(ω)=ε∞+j=1SjΩ2j −ω2−iωΓj (3)
are screened by free electrons and, thus, should have a X
vanishing dipolar moment. However, even in the metal- where Ω and Γ are the resonance frequency and the
j j
lic regime of Fe1−xCoxSi, realized either by Co-doping damping of the jth phonon, respectively, Sj is the os-
or by increasing the temperature, the phonons are still cillator strength, and Q (having the dimensions of fre-
j
observable in our infrared spectra. quency) denotes the coupling of the modes. For Q =0,
j
A detailed analysis of the FeSi phonon response in the jth phonon behaves like an independent classicalos-
the dielectric function shows that the inner phonons ’2’ cillator. In absence of interaction with electronic states
and ’3’ (at 40.8 meV and 43.6 meV, respectively) are the sum of the phonon parameters Q must vanish:
j
4
Figure 4: (Color online) Eigenvectors for the observable IR-
active modes at (a) 25.5 meV, (b) 40.8 meV, (c) 43.6 meV,
and (d) 56.8 meV. Length of arrows represents the scaled
magnitude of thevibrations.
n
Q =0 (4)
j
j=1
X
The four phonons observed can be successfully fitted
by Eq. (3) superimposed on a broad background (Fig.
3). OurfitparametersforthephononsinFeSiat7Kand
150 K are given in Table I. The asymmetry parameters
Q forphonons’2’and’3’areequaltozero,representing
2,3
fully symmetric line-shapes. This implies that these two
phonons are independent classical oscillators. Phonons
’1’ and ’4’ have finite parameters Q 6= 0 with opposite
j
sign, indicating a coupling between these modes. The
sum rule in Eq. (4) is fulfilled at 150 K (Fig. 3b), but
not at 7 K (Fig. 3a). This behavior will be discussed
further in Sec. IIIC.
The factor group analysis for FeSi yields 9 optical
phonons, Γ = 2A + 2E + 5T, which are all identified
byRamanspectroscopy.11 The fiveT-modesarealsoIR-
active.17 Density functional calculations employing the
local density approximation have been performed with
the electronic structure code CRYSTAL06.22 A Gaus-
sianbasissetisused,wheretheatom-centeredbasisfunc-
tions are chosen as in Ref. [9]. The initial step consists
TableI:PhononfitparametersforofFeSiat7K(and150K).
Phonon j Ω Γ S Q
j j j j
(meV) (meV) (meV)
Figure 3: (Color online) Real (open circles) and imaginary
(open squares) parts of the measured complex dielectric re- ’1’ 25.5 (25.2) 0.67 (0.99) 9.40 (10.40) -5.05 (-2.33)
sponseε(ω)ofFeSi. Thesolidcurvespresenttheresultofthe ’2’ 40.8 (40.6) 1.66 (1.80) 13.9 (13.7) 0 (0)
fitting with Eqn. (3) at (a) 7 K and (b) 150 K (Q values in ’3’ 43.6 (43.3) 0.39 (0.39) 0.25 (0.19) 0 (0)
j
meV). ’4’ 56.8 (56.5) 1.61 (2.07) 1.93 (2.36) 9.65 (2.38)
5
of a geometry optimization. In the next step, the sec- close for 15 at.% (see inset of Fig. 5b). As a function of
ond derivatives of the energy with respect to the atom temperature, the direct band gap follows the empirical
positions are obtained using analytical first derivatives. Varshni equation26 (Fig. 6):
Finally,thesecondderivativesarecalculatednumerically
bydisplacingatoms. Inthisway,themass-weightedHes-
αT2
sian matrix is obtained, and diagonalization gives the E (T)=E (0)− (8)
dir dir
eigenvalues and eigenvectors. T +β
Of the five computed IR-active T-modes, predicted at
The parameters obtained from the fit using Eq. (8)
energies of 27.8 meV, 31.6 meV, 41.2 meV, 47.6, and
are summarized in Tab. II.
57.5 meV, four modes are observed in our IR-spectra at
Indirect gap. For parabolic bands, ε follows from the
25.5meV,40.8meV,43.6meV,and56.8meV.Thetran- 2
equation24
sition matrix element of the fifth phonon mode, which is
eoxbpseercvtaedbleatin31th.6eminefrVa,reisdtsopoecstmraa.llTfohretehigeenmvoedcteortos fboer ε2 = Cjajb′sω−2(~ω+~ωph−∆Eind)+ (9)
theobservedIR-activephononsareshowninFig. 4. The + Cjejm′ω−2(~ω−~ωph−∆Eind)
47.6 meV and 57.5 meV modes are mainly relatedto sil-
icon motion.
a)
25
B. Band gaps of Fe1−xCoxSi
7 K
20
Substituting CoforFeinFeSileadstoaparamagnetic 4) 100 K
-s
semiconductorforx<0.05andaferromagneticmetalfor 0 150 K
0.05 < x < 0.8.23 However, the size of the band gap in 610 15 200 K
(
the semiconductingregimehasnotbeenreportedconsis- 2
2) 10
tently. From optical reflectivity measurements, a band ω
2
ε
gap of ≈80 meV has been reported.5 In contrast, pho- (
5
toemission data reveal a considerably smaller value for
thegapof≈30meV.7 Hereweresolvethisapparentcon- FeSi
tradiction by giving evidence for the existence of both a b) 20 80
direct(∆E )andanindirectbandgap(∆E )inFeSi.
Direct gadpir. In the case of a direct opticalintdransition, 4) eV)60
-s 15 m40
tbheecaimlcauglainteadryaspart of the dielectric function ε2(ω) can 6010 ∆E(dir 20
(
22) 10 00 5 10 15 20
ω2 Co conc . (at.%)
~ω −1 ε(
ε =A ∆Edir (5) 5
2 q( ~ω )2 Fe Co Si
∆Edir 0.95 0.05
0
with 0 20 40 60 80 100 120
Photon Energy (meV)
A= 2e2(2µ)3/2|P |2∆E−3/2 (6)
m2~ cv dir
e
where µ= m−1+m−1 −1 is the reduced mass of the Fofigtuhreep5h:o(nCoonlorresopnolninsee)aPtldotiffoefre(nεt2(tωe)mωp2e)r2ataufrteesrfsourb(tara)cFtieoSni
c v
effective masses in the valence and the conduction band and (b) Fe0.95Co0.05Si as a function of photon energy. The
and|P |isth(cid:0)emomentum(cid:1)matrixelement.25 Fromthese intercept of the dashed line with the energy axis defines the
cv
direct energy gap. Inset: Dependence of the direct gap of
equations one can derive
Fe1−xCoxSi on the Co concentration x. The dashed line is a
(ε ω2)2 ∝~ω−∆E (7) guide tothe eye.
2 dir
When (ε ω2)2 is plotted versus the photon energy,
2 Table II:Parameters of theVarshnifit for 0%, 5% and 10%
the direct band gap is obtained by the intersection of
Co-doped FeSi
the fitted lines and the energy axis (e.g., for FeSi and
Fe0.95Co0.05Si in Fig. 5). For FeSi, a direct band gap of Co conc. Edir.(0) (meV) α β (K)
0 at.% 73 0.00047 490
73meVat7Kisfound,whichisinreasonableaccordance
with the results from earlier optical experiments.5 This 5 at.% 63 0.00043 480
10 at.% 43 0.00015 470
direct gap still exists for 5 at.% Co and does not even
6
(a)
1.6
7 K
FeSi ) 1.4
1
Ω (meV)’4’ 60 FFee00..995CCoo0.01.S05iSi 20.515-ω) (10 s 0011....6802 112050000 KKK G1
E, dir ε(20.4 G2
∆ 40 0.2
0.0
-0.04 -0.02 0.00 0.02 0.04 0.06 0.08
Energy (meV)
0 50 100 150 200 250 300 (b) 10
Temperature (K) 5 V)15
e
m
) 0 E (ph10
V -5
Figure 6: (Color online) Temperature dependence of the di- e 5
sroeclitdglainpe∆s rEedpirreosefnFte1t−hxeCfiotxsSui,sifnogr xEq=n.0,(08.)0.5Tahnedl0in.1e. wTihthe (mnd--1150 0 50 T1 0(K0) 150 200
crosses (−×−) denotes the frequency of phonon ’4’ in FeSi. ∆Ei-20 FeSi
Variation of Ω′4′ for different x is smaller than 1 meV. -25 Fe0.98Co0.02Si
Fe Co Si
-30 0.95 0.05
which includes the matrix elements Cjj′ for the absorp- 0 50 100 150 200 250
tionandtheemissionofaphonon. Forindirectinterband Temperature (K)
transitions, a plot of (K ·~ω)1/2 vs. the photon energy
(where K is the absorption coefficient) may show linear
intercepts that can be attributed to the absorption of
chargecarriersacrossthe gapinvolvingphononemission Figure7: (Coloronline)(a)Plotof(ε2(ω)ω2)1/2ofFeSiafter
and/or absorption, from which the size of the indirect subtractionofthephononresponseatdifferenttemperatures.
gap can be derived.27 The straight lines represent the linear relation of Eq. (10).
Because (b)Temperaturedependenceoftheindirectbandgapenergy.
Inset: Energy of the transition-assisting phonon vs. temper-
k
ε =2nk and K ∝ ∝kω ⇒ ε ω2 ∝Kω, (10) ature.
2 2
λ
we plot (ε ω2)1/2 as function of photon energy in Fig.
2
7. For analysis of the indirect gap, the modeled phonon Because the ellipsometric measurements were per-
modes have been subtracted from the imaginary part of formed using a Fourier spectrometer, the sample was il-
the dielectric function, ε (ω). For each temperature, luminatedbysynchrotronradiationinthespectralrange
2
two linear intercepts (G and G ) may be attributed between 0.4 meV and 1.2 eV. Even if it is assumed that
1 2
to phonon emission and absorption, respectively. How- only 1% of the synchrotron beam flux reaches the sam-
ever, the indirect absorption is temperature dependent ple, the surface is irradiated with ≈ 1011 photons per
due to the phonon assistance. The absorption of a second. This leads to a continuous generation of optical
phonon is proportionalto the Bose distribution function phonons, which may also decay into low-energy acoustic
e(Eph/kBT)−1 −1,whichhasavalueofonly≈10−9 for phonons. Therefore, a sufficient number of optically ex-
T = 7 K and a phonon with an expected energy in the cited phonons may allow for phonon absorption, even
(cid:0)order of 10 meV(cid:1). Therefore, at low temperatures below at low temperatures. If so, one is able to derive the
200 K, only absorption branches due to the emission of absolute value of the indirect gap from the arithmetic
a phononcanbe observed,andthese canonly takeplace mean of the intersects of G1 and G2 with the energy
at photon energies ~ω > ∆E + E . For ∆E ≈ axis. At 7 K, the indirect gap for FeSi is 10 meV and
ind ph ind
30meV,asderivedinourpreviousphotoemission7,28 and decreases with increasing temperature (Fig. 7b). The
Ramanexperiments,11 andE ≈10meV,~ω shouldex- indirect gap changes sign and becomes negative above
ph
ceed 40 meV. Thus, only the linear slopes G can be 100K.Thismanifestsasacrossoverfromsemiconducting
1
attributed to an indirect transition with the emission to semimetallic behavior,where the valence and conduc-
of a phonon; the lines G cannot be assigned to inter- tivitybandsapproacheachotherandoverlapatdifferent
2
bandtransitions,whichareassistedbythermallyexcited points of the Brillouin zone.
phonons. The indirect gap is also reduced in Fe1−xCoxSi with
7
increasing Co concentration and closes at low tempera- there is no need to infer strongly correlated electrons in
ture in Fe Co Si. The phonon that assists the indi- this compound.
0.95 0.05
rect transition in FeSi has an energy of ~ω ≈17 meV,
ph
which is temperature independent within the accuracy
of the measurement. The phonon energy is smaller than C. Electron-phonon interaction
thatofallopticalphononsknownfromfactorgroupanal-
ysis and Raman spectroscopy,11 which therefore implies
AswehavediscussedinSectionIIIAtheinnerphonons
that an acoustic phonon assists the optical transition.
areclassicaloscillators,whereasphonons’1’and’4’show
In the band structure calculation, using density func- strong anharmonicity. The origin of the phonon anhar-
tional theory with the gradient corrected functional of monicity is due to phonon-phonon and electron-phonon
PerdewandWang(PW91),threeindirecttransitionsbe- interactions. The parameter Q is introduced in Eq. (3)
tween Γ–M, Γ–R, and Γ–X can be identified (Fig. 8). to describe the asymmetric lineshape of the anharmonic
Theminimumdirectgap(0.18eV)occursalongtheΓ–M phonons. For insulating materials with strong phonon-
direction and the minimum indirect gap corresponds to phonon interaction in absence of electron-phonon inter-
transitions from the valence band in the Γ–R direction action, the Eq. (4) is satisfied. We found that this is not
to the conduction band in the Γ–M direction (0.12 eV). the caseinFeSi,wherephonon’1’hasamuchlessasym-
It is to be noted here that the calculated energy val- metriclineshapethanphonon’4’,|Q |<|Q |. Thisindi-
1 4
ues areoverestimatedcomparedwiththe experimentally cates that the additional lineshape distortion of phonon
measured values. This is evidenced in high-resolution ’4’ comes from the interaction of this phonon with elec-
photoemission experiments, which reveal a renormaliza- tronic states, forming the absorption edge.
tionofthe Fe bandsclosetothe Fermienergydue tothe The shape of the broad electronic continuum in the
self-energy of the Fe-3d electrons.8 Because the transi- optical conductivity in Fig. 2a has a steep positive slope
tion matrix elements are not accessible from this calcu- nearthe absorptionedge. Becausephonon’4’isenerget-
lation, it is not clear which of the three possible indirect ically close to the direct absorption edge one would ex-
transitions is dominant in the infrared spectrum. The pect that the damping of the longitudinal LO vibration
minimum gap energies are indicated in Fig. 8. The gap ismuchhigherthanthatofthe correspondingtransverse
observed in photoemission (PE) experiments7 (28 meV) TO vibration, causing the strong lineshape asymmetry.
is slightly larger than the indirect gap derived from the Moreover,bothtypesofdamping(LOandTO)havedif-
ellipsometric measurements. However, because the data ferent temperature dependencies, because the electronic
inthePEspectraarerestrictedtoalimitedsectionofthe background is getting flatter with increasing tempera-
Brillouin zone, the gap observed in a certain crystalline ture; both parameters Q and Q and their sum tends
1 4
direction is not necessarily the smallest one. to zero. The direct absorption edge shifts towards lower
Contrary to the suggestions in Refs. 15 and 17, the energiesandcrossesthe energyofthe fourthTO phonon
closing of the gap in FeSi at 200 K seems to be due to at ≈150 K in FeSi and ≈100 K in Fe Co Si (Fig.
0.95 0.05
indirect transitions, rather than to the direct gapthat is 6). Above these temperatures the damping of the LO
notexpectedtocloseuntil700K;fromthispointofview, and TO vibrations for phonon ’4’ becomes equal (Fig.
9). Additionally, the indirect gap is filled at higher tem-
peraturesandthe chargecarriersscreenoutthe phonon-
phononinteraction. Thereforeasymmetric lineshape for
all four phonons is observed.
Increasingthe Co-dopingenlarges the electronic back-
ground, causing the outermost phonons become more
and more symmetric. In case of higher doping (more
than 10% Co), the phonon asymmetry vanishes as the
enhancedelectronicbackgroundscreensoutthe coupling
between the phonons.
There are two contributions to the change of phonon
frequency with temperature: the shift arising from the
volumechangeduetothermalexpansionandthephonon-
phononinteraction. Thephononfrequencyshift,δω,due
to anharmonicity effects is given as:32
δω =ω0 e−γR0TαVdT −1 , (11)
(cid:16) (cid:17)
Figure8: CalculatedbandstructureofFeSiusingthedensity whereω0,γ,αV arethephononfrequencyatzerotem-
functionaltheory. Bothindirect(I)anddirect(D)transitions perature, the mode Gru¨neisen parameter, and the volu-
are shown in theband structure. metric thermal expansion coefficient, respectively. We
8
used one Gru¨neisen parameter for all modes, calculated
by the relation γ = α V /(C χ ), taking the molar
V molar p S a)
volume V = 13.6 cm3/mol.33 The values of the spe- 1.04
molar
cific heat, C , and the adiabatic compressibility, χ , of FeSi
p S
FeSi are taken from [31] and [3], respectively. Using the 1.03
thermal expansion values34 for our FeSi samples and ne- K)
0
glecting the phonon-phononinteraction,we estimate the 01.02
3
Gru¨neisenparameterat3.4fortemperaturesupto300K. Ω(j
Theexpectedfrequencychangesduetothetemperature- )/1.01
T
driven lattice expansion for FeSi and Fe0.95Co0.05Si are Ω(j 25.5 meV (A)
calculated using this Gru¨neisen parameter and the ther- 1.00 56.8 meV (A)
mal expansion data and are shown in Fig. 10. 40.8 meV (L)
The fact that phonon ’4’ is coupled more strongly to
b)
the electronic states can be seen in Fig.10. Phonons ’1’ 1.04
to ’3’ follow the expected temperature dependence due Fe Co Si
to anharmonicity effects, whereas an effective softening 1.03 0.95 0.05
can be observed for phonon ’4’. This effective softening
)
vanishes when the absorption edge crosses the phonon K1.02
0
energy (Fig. 6). 0
3
Despite the existence of a strong electron-phonon in- Ω(j1.01
teraction, in the metallic regime the IR-active phonons T)/ 25.5 meV (A)
are not fully screened by the conduction electrons. Be- Ω(j1.00 55.0 meV (A)
cause an electric dipole moment is necessary to excite a 40.2 meV (L)
phononoptically,furtherinsightintothis peculiaritycan 0.99
beobtainedbyananalysisofthechargetransferbetween 0 100 200 300
Temperature (K)
Figure10: (Coloronline)Relativechangesofphononfrequen-
a)
ciesΩ (T)forthetwoasymmetric(A)andoneLorentzian(L)
10 25.5 meV modesJin(a)FeSiand(b)Fe0.95Co0.05Siwithdecreasingtem-
56.8 meV
peraturebelow 300 K.Thesolid lines represent theexpected
frequencychangeduetothermalexpansion. Thedashedlines
V) 5 are guides to theeyes (see text).
e
m
(
Q
0
Fe and Si.
The generalized multimode Lyddane-Sachs-Teller
FeSi relation35,36
-5
b) 4 ωL2O,i = εstat (12)
10 2556..50 mmeeVV iY=1ωT2O,i εopt
which compares the values of the dielectric function
V) 5 with the phonon frequencies, is fulfilled to within 20%
e
m and therefore represents a self-consistency check of our
Q ( measurements (see Tab. III). With the derived trans-
0
verse and longitudinal phonon frequencies, the Born ef-
fective charge e∗ can be expressed by36
i
Fe Co Si
-5 0.95 0.05
0 50 100 150 200 250 300 e∗
N i =ε ω2 −ω2 (13)
Temperature (K) Mi opt LO,j TO,j
i j
X X(cid:0) (cid:1)
∗
yielding e = 2.4±0.6 for all investigated Co concen-
i
Figure 9: (Color online) Temperature dependence of the trations 0 ≤ x ≤ 0.05 (Tab. IV). The consideration of
asymmetry parameter Q for phonons ’1’ and ’4’ in (a) FeSi polarization effects results in the Szigeti effective charge
and (b) Fe0.95Co0.05Si. e∗, which includes the screening of the phonons due to
s
9
free carriers37 by the expression of Clausius-Mosotti
Table IV:Born and Szigeti effective charge of Fe1−xCoxSi.
∗ ∗ 3 Born effective Szigeti effective
e =e . (14)
s iε +2 charge e∗ charge e∗
opt i S
FeSi 1.81 0.03
The Szigeti effective charge is lower than 0.1e for all Fe0.98Co0.02Si 2.46 0.05
Co concentrations. Despite the nearly complete screen- Fe0.95Co0.05Si 3.00 0.08
ing, the phonon response is still visible in the infrared
spectra. From the density functional calculations, it is
known that the additional electron charge from each Co
dopand essentially resides on the Co site.11 Therefore,
the oscillations between Fe and Si may not be affected 2mV ω
N (ω)= σ (ω)dω, (15)
by the charge density localized on the Co and, in addi- eff πe2 1
Z0
tion, the ratio of Co with respect to Fe is small. A simi-
larineffectivescreeningofthephononsisobservedinthe where Neff, m, e, andV are the effective number of elec-
semiconductor U Ru Sn, where the low carrier concen- trons,the free-electronmassandcharge,andthe volume
2 2
trationof3.5×1020cm−3isresponsiblefortheineffective of one FeSi formula unit. In the limit ω → ∞ the effec-
screening.38InFeSithecarrierconcentrationisontheor- tive number of electrons is equal to the total number of
der of several times 1018 cm−3 at low temperature, even electrons in one formula unit. The lattice parameter for
lower than in U2Ru2Sn. FeSiwastakenfrom[33]andusedtocalculateNeff forall
In conclusion, two coupling effects can be observed. Co-doping. The estimated error of Neff due to thermal
First, a coupling between phonons ’1’ and ’4’ exists as expansion and doping is less than 1%.
indicated by an asymmetry parameter of opposite sign Figure 11 shows the real parts of the optical conduc-
and, second, the phonons are affected by an electron- tivity,σ1(ω),andofthedielectricfunction,ε1(ω),ofFeSi
phonon coupling that changes as a function of temper- in the range 40 meV to 6.5 eV for 7 K and 200 K (fitted
ature because electronic decay channels appear at lower phononsaresubtracted). The leftpanels ofFig.11show
excitation energies when the gap is filled up. This re- that the optical response is strongly temperature depen-
sult is corroborated by Raman experiments, in which dentwithinseveralgapenergies,∆Edir ≈73meV.Above
the electron-phonon coupling is observedin terms of the 1 eV, the optical conductivity is nearly temperature in-
broadening of the phonon line-shapes. The linewidth in- dependent. From Neff(ω) integrated up to 1 eV (Fig.11,
creases as function of temperature, and the broadening insert), we estimate that 0.2 electrons per Fe atom par-
saturates at about 250 K as a consequence of the chemi- ticipate in the optical transitions up to this energy.
cal potential crossing the top of the valence band.11 In order to analyze the temperature dependent redis-
tribution of the spectral weight, in Fig. 12 we plot the
difference between the normalizedeffective chargedensi-
D. High-energy spectral weight transfer ties at 200 K and 7 K for FeSi, calculated as
In a conventional semiconductor, thermally excited
∆N (ω)=N200K(ω)−N7K(ω). (16)
charge carriers increase the far-IR conductivity at high eff eff eff
temperatures. On the other hand, because the elec-
A full recovery of the spectral weight happens when
trons partially occupy the conduction band and holes
∆N (ω) approaches zero. In FeSi, ∆N (ω) increases
are present at the top of the valence band, absorption eff eff
withinthegapenergyrangebelow0.1eV,thendecreases
at energies just above the gap is reduced. For a con-
upto2.5eV,withasmallbumpat0.35eV,andremains
ventionalsemiconductor, it is expected that the spectral
positive throughout. One can identify three distinct ab-
weight lost below the gap is recovered, with increasing
sorption bands at 0.15 eV (α), 0.65 eV (β), and 2.0 eV
temperature, in an energy range on the order of a few
(γ), whose spectral weight is transferred to the far-IR
times the gap. The integrated optical conductivity can
range due to closing of the band gap. The following in-
bequantitativelyanalyzedbydefiningtherestrictedsum
crease of ∆N above 2.5 eV can be attributed to the
rule function: eff
broadening of a strong transition above 4 eV with in-
creasingtemperature. This preventsthe complete recov-
ery of the spectral weight near 3 eV (Fig. 12), which is
Table III: Self-consistency check using the generalized
almost 40 times the direct optical gap energy.
Lyddane-Sachs-Tellerrelation
The optical spectral weight transfer due to dynamic
Q4i=1ωL2O,i/ωT2O,i εstat/εopt renormalization effects is widely seen in strongly corre-
FeSi 1.03 1.22
latedelectronmaterials,whichreflectsthe redistribution
Fe0.98Co0.02Si 1.01 1.15 of the spectral weight between Hubbard bands. In this
Fe0.95Co0.05Si 1.03 1.23 scenario, the optical spectral weight is recovered on the
energyscaleofon-siteCoulombinteractions. Supporting
10
8.0
)
-1m) 2.0 β -1m
c
c 1
-1σΩ (m1 1.0 α el per Fe) 00..1105 4.0-σΩ (m1
N (eff 00..0005 γ
0.0 0.2 0.4 0.6 0.8 1.0
hν (eV)
7 K
5.0
200 K
100
1
ε
ε1
0.0
0
-5.0
0.2 0.4 0.6 1.0 2.0 3.0 4.0 5.0 6.0
Photon energy (eV) Photon energy (eV)
Figure 11: (Color online) Real parts of optical conductivity σ1(ω) and dielectric function ε1(ω) for FeSi at 7 K (black) and
200 K (red). Phonons are subtracted. Inset: Normalized effective charge density at 7 K.
this scenario,ithasbeenreportedthatinFeSi, the spec- behavior arises primarily from changes in the valence
tralweightis notfully recoveredupto atleast6 eV.15,18 and conduction band occupancies at different points of
Contrarytothepreviousreports,wefindthatthespec- theBrillouinzoneduetophonon-assistedindirecttransi-
tral weight is recovered within a 3 eV spectral range. tions. Phononsarenotavailableforvibronicprocessesat
Moreover, we find that FeSi is as an indirect gap semi- low temperature. As temperature increases, the phonon
conductor, exhibiting the temperature-driven semicon- assistance is dramatically enhanced, both by the ther-
ductortosemimetalcrossover. Insemimetallicmaterials, mal excitation of phonons and by concomitant complete
significanttemperature-dependentchangesintheoptical closing of the indirect gap. Our findings imply that the
spectrumareexpectedovera broadspectralrange. This electron-phonon interaction and semimetallic character
of FeSi play the dominant role in the observed broad-
band spectral weight transfer, although one should do
not completely exclude the relevance of the alternative
scenario.
α
0.009
β
IV. CONCLUSION
e)
F
er 0.006 γ We have presented spectroscopic ellipsometry studies
p
N (el eff 0.003 coaflFpeh1−onxoCnoxsSpiec(txra=sh0ow−f0o.u2r) IsRin-galceticvreymstaoldse.s,Oinuragorpetei--
∆ ment with previous reports. We suggest the assignment
of all phonon modes observed by comparing them with
0.000 the results of density functional calculations. A detailed
0.0 1.0 2.0 3.0 4.0 analysisofthephononresponseinthedielectricfunction
Photon ene rgy (eV) of FeSi shows that the inner phonons at 40.8 meV (329
cm−1) and 43.6 meV (352 cm−1) are fully symmetric,
whereasthe outermostphonons at25.5meV (206cm−1)
Figure 12: Effective charge density difference ∆Neff(ω) be- and 56.8 meV (458 cm−1) have asymmetric line shapes,
tween200Kand7KinFeSi. Absorptionbandsat0.15, 0.65 which cannot be described by the Lorentzian equation.
and 2.0 eV are shown with arrows. The asymmetry parameters introduced for these modes
