Table Of ContentAnisotropy, phonon modes, and lattice anharmonicity from dielectric function tensor
analysis of monoclinic cadmium tungstate
A. Mock,1,∗ R. Korlacki,1 S. Knight,1 and M. Schubert1,2,3
1Department of Electrical and Computer Engineering and Center for
Nanohybrid Functional Materials, University of Nebraska-Lincoln, U.S.A.
2Leibniz Institute for Polymer Research, Dresden, Germany
3Department of Physics, Chemistry, and Biology (IFM),
Linko¨ping University, SE 58183, Linko¨ping, Sweden
(Dated:)
WedeterminethefrequencydependenceoffourindependentCdWO Cartesiandielectricfunction
4
tensorelementsbygeneralizedspectroscopicellipsometrywithinmid-infraredandfar-infraredspec-
7
tral regions. Single crystal surfaces cut under different angles from a bulk crystal, (010) and (001),
1
0 areinvestigated. Fromthespectraldependenciesofthedielectricfunctiontensoranditsinversewe
2 determine all long wavelength active transverse and longitudinal optic phonon modes with Au and
Bu symmetry as well as their eigenvectors within the monoclinic lattice. We thereby demonstrate
n
that such information can be obtained completely without physical model line shape analysis in
a
materials with monoclinic symmetry. We then augment the effect of lattice anharmonicity onto
J
our recently described dielectric function tensor model approach for materials with monoclinic and
3
tricliniccrystalsymmetries[Phys. Rev. B,125209(2016)],andweobtainexcellentmatchbetween
all measured and modeled dielectric function tensor elements. All phonon mode frequency and
]
i broadening parameters are determined in our model approach. We also perform density functional
c theory phonon mode calculations, and we compare our results obtained from theory, from direct
s
dielectric function tensor analysis, and from model lineshape analysis, and we find excellent agree-
-
l ment between all approaches. We also discuss and present static and above reststrahlen spectral
r
t rangedielectricconstants. OurdataforCdWO4areinexcellentagreementwitharecentlyproposed
m
generalization of the Lyddane-Sachs-Teller relation for materials with low crystal symmetry [Phys.
. Rev. Lett. 117, 215502 (2016)].
t
a
m PACSnumbers: 61.50.Ah;63.20.-e;63.20.D-;63.20.dk;
-
d
n I. INTRODUCTION
o
c Metal tungstate semiconductor materials (AWO )
4
[
have been extensively studied due to their remarkable
1 optical and luminescent properties. Because of their
v properties, metal tungstates are potential candidates for
3 use in phosphors, in scintillating detectors, and in op-
1 toelectronic devices including lasers.1–3 Tungstates usu-
8
ally crystallize in either the tetragonal scheelite or mon-
0
oclinic wolframite crystal structure for large (A = Ba,
0
. Ca, Eu, Pb, Sr) or small (A = Co, Cd, Fe, Mg, Ni,
1 Zn) cations, respectively.4 The highly anisotropic mono-
0
clinic cadmium tungstate (CdWO ) is of particular in-
7 4
1 terest for scintillator applications, because it is non-
: hygroscopic, has high density (7.99 g/cm3) and there- Figure 1. (a) Unit cell of CdWO4, monoclinic angle β, and
v Cartesian coordinate system (x, y, z) used in this work. (b)
fore high X-ray stopping power,2 its emission centered
i View onto the a - c plane along axis b, which points into
X near 480 nn falls within the sensitive region of typi-
the plane. Indicated is the vector c(cid:63), defined for convenience
r cal silicon-based CCD detectors,5 and its scintillation here. See section IIC9 for details.
a has high light yield (14,000 photons/MeV) with little
afterglow.2 Raman spectra of CdWO have been studied
4
extensively,6–10 and despite its use in detector technolo-
gies, investigation into its fundamental physical proper-
wasidentified,andatentativesymmetryassignmentwas
ties such as optical phonon modes, and static and high-
provided. A broad feature between 260-310 cm−1 re-
frequency dielectric constants is far less exhaustive.
mained unexplained. Gabrusenoks et al.8 utilized un-
Infrared (IR) spectra was reported by Nyquist and
polarized far-IR (FIR) reflection measurements from 50-
Kagel,however,noanalysisorsymmetryassignmentwas
5000 cm−1 and identified 7 modes with B symmetry
included.11 Blasse6 investigated IR spectra of HgMoO u
4 but did not provide their frequencies. Jia et al.12 stud-
andHgWO andalsoreportedanalysisofCdWO inthe
4 4 ied CdWO nanoparticles using FT-IR between 400-
spectral range of 200-900 cm−1 and identified 11 IR ac- 4
1400 cm−1, and identified 6 absorption peaks in this
tive modes but without symmetry assignment. Daturi
range without symmetry assignment. Burshtein et al.13
et al.7 performed Fourier transform IR measurements of
utilizedinfraredreflectionspectraandidentified14IRac-
CdWO powder. An incomplete set of IR active modes
4 tive modes along with symmetry assignment. Lacomba-
2
Perales et al.9 studied phase transitions in CdWO at stants. We use generalized ellipsometry for measure-
4
high pressure and provided results of density functional ment of the highly anisotropic dielectric tensor. Fur-
theory calculations for all long-wavelength active modes. thermore, we observe and report in this paper the need
ShevchukandKayun14 reportedontheeffectsoftemper- to augment anharmonic broadening onto our recently
ature on the dielectric permittivity of single crystalline described model for polar vibrations in materials with
(010)CdWO at1kHzyieldingavalueofapproximately monoclinic and triclinic crystal symmetries.15 With the
4
17atroomtemperature. Manyofthesestudieswerecon- augmentation of anharmonic broadening we are able to
ductedonthe(010)cleavageplaneofCdWO ,andthere- achieve a near perfect match between our experimental
4
fore, the complete optical response due to anisotropy in data and our model calculated dielectric function spec-
the monoclinic crystal symmetry was not investigated. tra. In particular, in this work we exploit the inverse of
However, in order to accurately describe the full set of the experimentally determined dielectric function tensor
phononmodesaswellasstaticandhigh-frequencydielec- and directly obtain the frequencies of the longitudinal
tric constants of monoclinic CdWO , a full account for phonon modes. We also demonstrate the validity of a
4
the monoclinic crystal structure must be provided, both recently proposed generalization of the Lyddane-Sachs-
during conductance of the experiments as well as during Teller relation22 to materials with monoclinic and tri-
data analysis. Overall, up to this point, the availability clinic crystal symmetries23 for CdWO . We also demon-
4
ofaccuratephononmodeparametersanddielectricfunc- stratetheusefulnessofthegeneralizationofthedielectric
tion tensor properties at long wavelengths for CdWO functionformonoclinicandtriclinicmaterialsinorderto
4
appears rather incomplete. directlydeterminefrequencyandbroadeningparameters
In this work we provide a long wavelength spec- of all long wavelength active phonon modes regardless
troscopic investigation of the anisotropic properties of their displacement orientations within CdWO . This
4
of CdWO4 by Generalized Spectroscopic Ellipsometry generalization as a coordinate-invariant form of the di-
(GSE). We apply our recently developed model for com- electric response was proposed recently.23 For this anal-
plete analysis of the effects of long wavelength active ysis procedure, we augment the dielectric function form
phonon modes in materials with monoclinic crystal sym- with anharmonic lattice broadening effects proposed by
metry,whichwehavedemonstratedforasimilaranalysis Berreman and Unterwald,24 and Lowndes25 onto the
of β-Ga2O3.15 Our investigation is augmented by den- coordinate-invariantgeneralizationofthedielectricfunc-
sity functional theory calculations. Ellipsometry is an tion proposed by Schubert.23 In contrast to our previous
excellent non-destructive technique, which can be used reportonβ-Ga O ,15wedonotobservetheeffectsoffree
2 3
to resolve the state of polarization of light reflected off chargecarriersinCdWO , andhencetheircontributions
4
or transmitted through samples, both real and imagi- tothedielectricresponse,neededforaccurateanalysisof
nary parts of the complex dielectric function can be de- conductive, monoclinic materials such as β-Ga O , are
2 3
termined at optical wavelengths.16–18 Generalized ellip- ignored in this work. The phonon mode parameters and
sometry extends this concept to arbitrarily anisotropic static and high frequency dielectric constants obtained
materials and, in principle, allows for determination of from our ellipsometry analysis are compared to results
all 9 complex-valued elements of the dielectric function of density functional theory (DFT) calculations. We ob-
tensor.19 Jellison et al. first reported generalized ellip- serve by experiment all DFT predicted modes, and all
sometry analysis of a monoclinic crystal, CdWO4, in the parameters including phonon mode eigenvector orienta-
spectral region of 1.5 – 5.6 eV.20 It was shown that tions are in excellent agreement between theory and ex-
4 complex-valued dielectric tensor elements are required periment.
foreachwavelength,whichweredeterminedspectroscop-
ically, and independently of physical model line shape
functions. Jellison et al. suggested to use 4 indepen-
dentspectroscopicdielectricfunctiontensorelementsin-
stead of the 3 diagonal elements used for materials with
II. THEORY
orthorhombic, hexagonal, tetragonal, trigonal, and cu-
bic crystal symmetries. Recently, we have shown this
approach in addition to a lineshape eigendisplacement A. Symmetry
vector approach applied to β-Ga O .15 We have used
2 3
a physical function lineshape model first described by The cadmium tungstate unit cell contains two cad-
Born and Huang,21 which uses 4 interdependent dielec- mium atoms, two tungsten atoms, and eight oxygen
tric function tensor elements for monoclinic materials. atoms. The lattice constants of wolframite structure
The Born and Huang model permitted determination of CdWO4 are a = 5.026 ˚A, b = 5.078 ˚A, and c = 5.867
all long wavelength active phonon modes, their displace- ˚A, and the monoclinic angle is β = 91.477 (Fig. 1).
ment orientations within the monoclinic lattice, and the CdWO4 possesses33normalmodesofvibrationwiththe
anisotropic static and high-frequency dielectric permit- irreduciblerepresentationforacousticalandopticalzone
tivity parameters. Here, we investigate the dielectric center modes: Γ = 8Ag +10Bg +7Au+8Bu, where Au
tensor of CdWO4 in the FIR and mid-IR (MIR) spec- andBu modesareactiveatmid-infraredandfar-infrared
tral regions. Our goal is the determination of all FIR wavelengths. The phonon displacement of Au modes is
andMIRactivephononmodesandtheireigenvectorori- parallel to the crystal b direction, while the phonon dis-
entations within the monoclinic lattice. In addition, we placement for Bu modes is parallel to the a−c crystal
determine the static and high-frequency dielectric con- plane. All modes split into transverse optical (TO) and
longitudinal optical (LO) phonons.
3
Figure 2. Renderings of TO phonon modes in CdWO with A (b: A (7), g: A (6), h: A (5), i: A (4), k: A (3), m: A (2),
4 u u u u u u u
o: A (1)) and B symmetry (a: B (8), c: B (7), d: B (6), e: B (5), f: B (4), j: B (3), l: B (2), n: B (1)). The respective
u u u u u u u u u u
phonon mode frequency parameters calculated using Quantum Espresso are given in Tab. I.
B. Density Functional Theory ˆe =ˆe . Likewise, a second set of frequencies pertains
l TO,l
to situations when the dielectric loss approaches infin-
Theoretical calculations of long wavelength active Γ- ity for electric fields along directions ˆel. These are the
point phonon frequencies were performed by plane wave so-calledlongitudinaloptical(LO)modeswhoseeigendi-
DFT using Quantum ESPRESSO (QE).26 We used the electric displacement unit vectors are then ˆel = ˆeLO,l.
exchange correlation functional of Perdew and Zunger This can be expressed by the following equations
(PZ).27WeemployOptimizedNorm-ConservingVander-
bilt(ONCV)scalar-relativisticpseudopotentials,28which
|det{ε(ω =ω )}|→∞, (1a)
we generated for the PZ functional using the code TO,l
ONCVPSP29 withtheoptimizedparametersoftheSG15 |det{ε−1(ω =ω )}|→∞, (1b)
LO,l
distribution of pseudopotentials.30 These pseudopoten- ε−1(ω =ω )ˆe =0, (1c)
tials include 20 valence states for cadmium.31 A crystal TO,l TO,l
ε(ω =ω )ˆe =0. (1d)
cellofCdWO4 consistingoftwochemicalunits,withini- LO,l LO,l
tial parameters for the cell and atom coordinates taken
where|ζ|denotestheabsolutevalueofacomplexnumber
from Ref. 32 was first relaxed to force levels less than
ζ. At this point, l is an index which merely addresses
10−5 Ry/Bohr. A regular shifted 4×4×4 Monkhorst-
the occurrence of multiple such frequencies in either or
Pack grid was used for sampling of the Brillouin Zone.33
both of the sets. Note that as a consequence of Eqs. 1,
A convergence threshold of 1×10−12 was used to reach
the eigendisplacement directions of TO and LO modes
selfconsistencywithalargeelectronicwavefunctioncut-
withacommonfrequencymustbeperpendiculartoeach
off of 100 Ry. The fully relaxed structure was then used
other, regardless of crystal symmetry.
for the calculation of phonon modes.
2. The eigendielectric displacement vector summation
C. Dielectric Function Tensor Properties approach
1. Transverse and longitudinal phonon modes It was shown previously that the tensor elements of ε
due to long wavelength active phonon modes in materi-
Fromthefrequencydependenceofageneral, lineardi- als with any crystal symmetry can be obtained from an
electricfunctiontensor,twomutuallyexclusiveandchar- eigendielectricdisplacementvectorsummationapproach.
acteristic sets of eigenmodes can be unambiguously de- In this approach, contributions to the anisotropic di-
fined. One set pertains to frequencies at which dielectric electricpolarizabilityfromindividual,eigendielectricdis-
resonance occurs for electric fields along directions ˆe . placements(dielectricresonances)withunitvectorˆe are
l l
These are the so-called transverse optical (TO) modes addedtoahigh-frequency,frequency-independenttensor,
whose eigendielectric displacement unit vectors are then ε , which is thought to originate from the sum of all di-
∞
4
electric eigendielectric displacement processes at much 5. The Berreman-Unterwald-Lowndes factorized form
shorter wavelengths than all phonon modes15,23
The right side of Eq. 5 is form equivalent to the so-
N called factorized form of the dielectric function for long
(cid:88)
ε=ε + (cid:37) (ˆe ⊗ˆe ), (2) wavelength-active phonon modes described by Berre-
∞ l l l
man and Unterwald,24 and Lowndes.25 The Berreman-
l=1
Unterwald-Lowndes (BUL) factorized form is convenient
where⊗isthedyadicproduct. Functions(cid:37) describethe
l for derivation of TO and LO mode frequencies from
frequency responses of each of the l = 1...N eigendi-
the dielectric function of materials with multiple phonon
electric displacement modes.34 Functions (cid:37) must satisfy
l modes. In the derivation of the BUL factorized form,
causality and energy conservation requirements, i.e., the
however, it was assumed that the displacement direc-
Kramers-Kronig (KK) integral relations and Im{(cid:37) } ≥
l tions of all contributing phonon modes must be parallel.
0,∀ ω ≥0, 1,...l,...N conditions.35,36
Hence, in its original implementation, the application of
the BUL factorized form is limited to materials with or-
thorhombic, hexagonal, tetragonal, trigonal, and cubic
3. The Lorentz oscillator model
crystalsymmetries. SchubertrecentlysuggestedEq.5as
generalization of the BUL form applicable to materials
The energy (frequency) dependent contribution to the
regardless of crystal symmetry.23
long wavelength polarization response of an uncoupled
electric dipole charge oscillation is commonly described
using a KK consistent Lorentz oscillator function with
harmonic broadening37,38
6. The generalized dielectric function with anharmonic
A
(cid:37) (ω)= l , (3) broadening
l ω2 −ω2−iωγ
0,l 0,l
or anharmonic broadening The introduction of broadening by permitting for pa-
rameters γ and Γ in Eqs. 3, and 4 can be shown to
TO,l l
modify Eq. 5 into the following form
A −iΓ ω
(cid:37) (ω)= l l , (4)
l ω2 −ω2−iωγ
0,l 0,l
where A , ω , γ , and Γ denote amplitude, resonance
frequencly,h0a,rlmo0n,licbroadlening,andanharmonicbroad- det{ε(ω)}=det{ε }(cid:89)N ωL2O,l−ω2−iωγLO,l, (6)
ening parameter of mode l, respectively, ω is the fre- ∞ l=1ωT2O,l−ω2−iωγTO,l
quency of the driving electromagnetic field, and i2 =−1
is the imaginary unit. The assumption that functions
(cid:37) can be described by harmonic oscillators renders the where γ is the broadening parameter for the LO fre-
l LO,l
eigendielectric displacement vector summation approach quency ω . A similar augmentation was suggested by
LO,l
of Eq. 2 equivalent to the result of the microscopic de- Gervais and Periou for the BUL factorized form identi-
scription of the long wavelength lattice vibrations given fying γ as independent model parameters to account
LO,l
by Born and Huang in the so-called harmonic approxi- for a life-time broadening mechanisms of LO modes sep-
mation.21Intheharmonicapproximationtheinteratomic arate from that of TO modes.40 Sometimes referred to
forces are considered constant and the equations of mo- as “4-parameter semi quantum” (FPSQ) model, the ap-
tion are determined by harmonic potentials.39 proach by Gervais and Periou allowing for separate TO
From Eqs. 1-4 it follows that ˆe = ˆe , and ω = and LO mode broadening parameters, γ and γ ,
l TO,l 0,l TO,l LO,l
ω . The ad-hoc parameter Γ introduced in Eq. 4 can respectively, provided accurate description of effects of
TO,l l
be shown to be directly related to the LO mode broad- anharmonicphononmodecouplinginanisotropic,multi-
ening parameter γ introduced previously to account ple mode materials with non-cubic crystal symmetry, for
LO,l
for anharmonic phonon coupling in materials with or- example,intetragonal(rutile)TiO ,40,41 hexagonal(cor-
2
thorhombic and higher symmetries, which is discussed rundum) Al O ,42 and orthorhombic (stibnite) Sb S .43
2 3 2 3
below. Inthiswork,wesuggestuseofEq.6toaccuratelymatch
theexperimentallyobservedlineshapesandtodetermine
frequenciesofTOandLOmodes,andtherebytoaccount
4. The coordinate-invariant generalized dielectric function for effects of phonon mode anharmonicity in monoclinic
CdWO .
4
The determinant of the dielectric function ten-
sor can be expressed by the following frequency-
dependent coordinate-invariant form, regardless of crys-
tal symmetry15,23
7. Schubert-Tiwald-Herzinger broadening condition
det{ε(ω)}=det{ε∞}(cid:89)N ωωL22O,l−−ωω22. (5) ThefollowingconditionholdsfortheTOandLOmode
l=1 TO,l broadening parameters within a BUL form42
5
where X = A ,B indicate functions (cid:37)X for long wave-
u u l
0<Im(cid:40)(cid:89)N ωL2O,l−ω2−iωγLO,l(cid:41), (7a) lsepnegctthivealyc.tivTehemoandgeslewαilthdeAnouteasntdheBouriesnytmatmioentroyf, trhee-
ω2 −ω2−iωγ eigendielectric displacement vectors with B symmetry
l=1 TO,l TO,l u
relative to axis a. Note that the eigendielectric displace-
(cid:108) (7b)
ment vectors with A symmetry are all parallel to axis
u
N b, and hence do not appear as variables in Eqs. 9.
(cid:88)
0< (γ −γ ). (7c)
LO,l TO,l
l=1
Thisconditionisvalidforthedielectricfunctionalong
high-symmetry Cartesian axes for orthorhombic, hexag- 10. Phonon mode parameter determination
onal, tetragonal, trigonal, andcubiccrystalsymmetryin
materials with multiple phonon mode bands. For mon- ThespectraldependenceoftheCdWO dielectricfunc-
4
oclinic materials it is valid for the dielectric function for tion tensor, obtained here by generalized ellipsometry
polarizations along crystal axis b. Its validity for Eq. 6 measurements,isperformedintwostages. Thefirststage
has not been shown yet, also not for the conceptual ex- does not involve assumptions about a physical lineshape
pansion for triclinic materials (Eq. 14 in Ref.23). How- model. The second stage applies the eigendielectric dis-
ever,wetesttheconditionforthea-cplaneinthiswork. placement vector summation approach described above.
Stage 1, according to Eqs. 1, the elements of experi-
mentally determined ε and ε−1 are plotted versus wave-
8. Generalized Lyddane-Sachs-Teller relation length,andω andω ,aredeterminedfromextrema
TO,l LO,l
in ε and ε−1, respectively. Eigenvectors eˆ and eˆ
TO,l LO,l
A coordinate-invariant generalization of the Lyddane- can be estimated by solving Eq. 1(c) and 1(d), respec-
Sachs-Teller (LST) relation22 for arbitrary crystal sym- tively.
metrieswasrecentlyderivedbySchubert(S-LST).23 The Stage 2, step (i): Eqs. 9 are used to match simulta-
S-LST relation follows immediately from Eq. 6 setting ω neously all elements of the experimentally determined
to zero tensors ε and ε−1. As a result, ε and eigenvector, am-
∞
plitude,frequency,andbroadeningparametersforallTO
modes are obtained. Step (ii): (B symmetry) The gen-
det{ε(ω =0)} =(cid:89)N (cid:18)ωLO,l(cid:19)2, (8) eralized dielectric function (Eq. 6u) is used to determine
det{ε∞} l=1 ωTO,l the LO mode frequency and broadening parameters. All
other parameters in Eq. 6 are taken from step (i). The
andwhichwasfoundvalidformonoclinicβ-Ga2O3.15We eigenvectorseˆLO,l arecalculatedbysolvingEq.1(d). (Au
investigate the validity of the S-LST relation for mono- symmetry) The BUL form is used to parameterize ε
zz
clinic CdWO4 with our experimental results obtained in and −ε−1 in order to determine the LO mode frequency
zz
this work. and broadening parameters.
9. CdWO dielectric tensor model
4
D. Generalized Ellipsometry
We align unit cell axes b and a with −z and x, re-
spectively, andciswithinthe(x-y)plane. Weintroduce
Generalizedellipsometryisaversatileconceptforanal-
vector c(cid:63) parallel to y for convenience, and we obtain a,
ysis of optical properties of generally anisotropic mate-
c(cid:63), −b as a pseudo orthorhombic system (Fig. 1). Seven
rials in bulk as well as in multiple-layer stacks.41–57 A
modes with A symmetry are polarized along b only.
u multiplesample,multipleazimuth,andmultipleangleof
Eight modes with B symmetry are polarized within the
u incidence approach is required for monoclinic CdWO ,
a - c plane. For CdWO , the dielectric tensor elements 4
4 following the same approach used previously for mono-
are then obtained as follows
clinicβ-Ga O .15 Multiple,singlecrystallinesamplescut
2 3
under different angles from the same crystal must be in-
(cid:88)8 vestigated and analyzed simultaneously.
ε =ε + (cid:37)Bucos2α , (9a)
xx ∞,xx l j
l=1
8
(cid:88)
ε =ε + (cid:37)Businα cosα , (9b)
xy ∞,xy l j j 1. Mueller matrix formalism
l=1
8
(cid:88) In generalized ellipsometry, either the Jones or the
ε =ε + (cid:37)Busin2α , (9c)
yy ∞,yy l j Mueller matrix formalism can be used to describe the
l=1
interaction of electromagnetic plane waves with layered
(cid:88)7 samples.37,38,58–60 IntheMuellermatrixformalism,real-
ε =ε + (cid:37)Au, (9d)
zz ∞,zz l valued Mueller matrix elements connect the Stokes pa-
l=1 rameters of the electromagnetic plane waves before and
ε =ε =ε =ε =0, (9e)
xz zx zy yz after sample interaction
6
dielectric tensor in the 4×4 matrix algorithm is
ε ε 0
S M M M M S xx xy
0 11 12 13 14 0 ε= εxy εyy 0 , (11)
S M M M M S
1 = 21 22 23 24 1 . 0 0 εzz
S M M M M S
2 31 32 33 34 2
S3 output M41 M42 M43 M44 S3 input with elements obtained by setting εxx,εxy,εyy, and εzz
as unknown parameters. Then, according to the crys-
(10)
tallographic surface orientation of a given sample, and
with the Stokes vector components defined by S =
0
accordingtoitsazimuthorientationrelativetotheplane
I +I , S = I −I , S = I −I , S = I −I ,
p s 1 p s 2 45 −45 3 σ+ σ−
of incidence, a Euler angle rotation is applied to ε. The
whereI ,I ,I ,I ,I ,andI denotetheintensities
p s 45 −45 σ+ σ−
sample azimuth angle, typically termed ϕ, is defined by
for the p-, s-, +45◦, -45◦, right handed, and left handed
a certain in plane rotation with respect to the sample
circularlypolarizedlightcomponents,respectively.60The
normal. The sample azimuth angle describes the mathe-
Mueller matrix renders the optical sample properties at
maticalrotationthatamodeldielectricfunctiontensorof
a given angle of incidence and sample azimuth, and data
a specific sample must make when comparing calculated
measuredmustbeanalyzed through abestmatchmodel
data with measured data from one or multiple samples
calculation procedure in order to extract relevant physi-
taken at multiple, different azimuth positions.
cal parameters.61,62
Asfirststepindataanalysis,allellipsometrydatawere
analyzed using a wavelength-by-wavelength approach.
Model calculated Mueller matrix data were compared
to experimental Mueller matrix data, and dielectric ten-
2. Model analysis
sor values were varied until best match was obtained.
This is done by minimizing the mean square error (χ2)
The 4×4 matrix formalism is used to calculate the
function which is weighed to estimated experimental er-
Mueller matrix. We apply the half-infinite two-phase
rors (σ) determined by the instrument for each data
model, where ambient (air) and monoclinic CdWO ren-
4 point.19,37,41,42,66Theerrorbarsonthebestmatchmodel
der the two half infinite mediums separated by the plane
calculated tensor parameters then refer to the usual
at the surface of the single crystal. The formalism has
90% confidence interval. All data obtained at the same
beendetailedextensively.37,60,63–65 Theonlyfreeparam-
wavenumberfrommultiplesamples,multipleazimuthan-
eters in this approach are the elements of the dielectric
gles, and multipleangles of incidenceare included(poly-
function tensor of the material with monoclinic crystal
fit)andonesetofcomplexvaluesε ,ε ,ε ,andε is
xx xy yy zz
symmetry, and the angle of incidence. The latter is
obtained. Thisprocedureissimultaneouslyandindepen-
set by the instrumentation. The wavelength only en-
dently performed for all wavelengths. In addition, each
ters this model explicitly when the dielectric function
sample requires one set of 3 independent Euler angle pa-
tensor elements are expressed by wavelength dependent
rameters,eachsetaddressingtheorientationofaxesa,b,
model functions. This fact permits the determination of
c(cid:63)atthefirstazimuthpositionwheredatawereacquired.
the dielectric function tensor elements in the so-called
wavelength-by-wavelength model analysis approach.
4. Model dielectric function analysis
A second analysis step is performed by minimizing
3. Wavelength-by-wavelength analysis thedifferencebetweenthewavelength-by-wavelengthex-
tractedε ,ε ,ε ,andε spectraandthosecalculated
xx xy yy zz
Twocoordinatesystemsmustbeestablishedsuchthat by Eqs. (9). All model parameters were varied until cal-
one that is tied to the instrument and another is tied culatedandexperimentaldatamatchedascloseaspossi-
to the crystallographic sample description. The system ble(bestmatchmodel). Forthesecondanalysisstep,the
tied to the instrument is the system in which the di- numericaluncertaintylimitsofthe90%confidenceinter-
electric function tensor must be cast into for the 4×4 val from the first regression were used as “experimental”
matrix algorithm. We chose both coordinate systems to errors σ for the wavelength-by-wavelength determined
be Cartesian. The sample normal defines the laboratory ε ,ε ,ε ,andε spectra. Asimilarapproachwasde-
xx xy yy zz
coordinate system’s zˆaxis, which points into the surface scribed, for example, in Refs. 15, 37, 41, 42, and 67. All
of the sample.15,63 The sample surface then defines the best match model calculations were performed using the
laboratory coordinate system’s xˆ - yˆ plane. The sam- software package WVASE32 (J. A. Woollam Co., Inc.).
plesurfaceisattheoriginofthecoordinatesystem. The
planeofincidenceisthexˆ-zˆplane. Notethatthesystem
(xˆ, yˆ, zˆ) is defined by the ellipsometer instrumentation III. EXPERIMENT
through the plane of incidence and the sample holder.
One may refer to this system as the laboratory coordi- Two single crystal samples of CdWO were fab-
4
nate system. The system (x, y, z) is fixed by our choice ricated by slicing from a bulk crystal with (001)
to the specific orientation of the CdWO crystal axes, a, and (010) surface orientation. Both samples were
4
b, and c as shown in Fig. 1 with vector c(cid:63) defined for then double side polished. The substrate dimensions
convenience perpendicular to a-b plane. One may refer are 10mm×10mm×0.5mm for the (001) crystal and
tosystem(x,y,z)asourCdWO system. Then,thefull 10mm×10mm×0.2mm for the (010) crystal.
4
7
Mid-infrared (IR) and far-infrared (FIR) generalized agonal parts of the Mueller matrix, are plotted within
spectroscopic ellipsometry (GSE) were performed at the same panels. Therefore, the panels represent the up-
room temperature on both samples. The IR-GSE mea- perpartofa4×4matrixarrangement. Becausealldata
surementswereperformedonarotatingcompensatorin- obtained are normalized to element M , and because
11
fraredellipsometer(J.A.WoollamCo.,Inc.) inthespec- M = M , the first column does not appear in this
1j j1
tral range from 333 – 1500 cm−1 with a spectral resolu- arrangement. The only missing element is M , which
44
tion of 2 cm−1. The FIR-GSE measurements were per- cannot be obtained in our current instrument configu-
formed on an in-house built rotating polarizer rotating ration due to the lack of a second compensator. Data
analyzer far-infrared ellipsometer in the spectral range are shown for wavenumbers (frequencies) from 80 cm−1
from 50 – 500 cm−1 with an average spectral resolution – 1100 cm−1, except for column M = M which only
4j j4
of 1 cm−1.68 All GSE measurements were performed at containsdatafromapproximately250cm−1–1100cm−1.
50◦, 60◦, and 70◦ angles of incidence. All measurements AllotherpanelsshowdataobtainedwithintheFIRrange
are reported in terms of Mueller matrix elements, which (80 cm−1 – 500 cm−1) using our FIR instrumentation
are normalized to element M . The IR instrument de- anddataobtainedwithintheIRrange(500cm−1 –1100
11
termines the normalized Mueller matrix elements except cm−1) using our IR instrumentation. Data from the re-
for those in the forth row. Note that due to the lack of a maining5azimuthorientationsforeachsampleatwhich
compensator for the FIR range in this work, neither ele- measurements were also taken are not shown for brevity.
ment in the fourth row nor fourth column of the Mueller The most notable observation from the experimental
matrixisobtainedwithourFIRellipsometer. Datawere Mueller matrix data behavior is the strong anisotropy,
acquired at 8 in-plane azimuth rotations for each sam- whichisreflectedbythenonvanishingoffdiagonalblock
ple. Theazimuthpositionswereadjustedbyprogressive, elementsM13,M23,M14,andM24,andthestrongdepen-
counterclockwise steps of 45◦. dence on sample azimuth in all elements. A noticeable
observationisthattheoffdiagonalblockelementsinpo-
sition P1 for the (001) surface in Fig. 3 are close to zero.
There,axisbisalignedalmostperpendiculartotheplane
IV. RESULTS AND DISCUSSION
of incidence. Hence, the monoclinic plane with a and c
is nearly parallel to the plane of incidence, and as a re-
A. DFT Phonon Calculations
sult almost no conversion of p to s polarized light occurs
and vice versa. As a result, the off diagonal block ele-
The phonon frequencies and transition dipole compo-
ments of the Mueller matrix are near zero. The reflected
nentswerecomputedattheΓ-pointoftheBrillouinzone
light for s polarization is determined by ε alone, while
using density functional perturbation theory.69 The re- zz
the p polarization receives contribution from ε , ε ,
sults of the phonon mode calculations for all long wave- xx xy
and ε , which then vary with the angle of incidence.
lengthactivemodeswithA andB symmetryarelisted yy
u u A similar observation was made previously for a (¯201)
in Tab. I. Data listed include the TO resonance frequen-
surface of monoclinic β=Ga O .15 While every data set
cies, and for modes with B symmetry the angles of 2 3
u (sample,position,azimuth,angleofincidence)isunique,
the transition dipoles relative to axis a within the a−c
alldatasetssharecharacteristicfeaturesatcertainwave-
plane. Renderingsofatomicdisplacementsforeachmode
lengths. Verticallinesindicatefrequencies,whichfurther
were prepared using XCrysDen70 running under Silicon
below we will identify with the frequencies of all antici-
Graphics Irix 6.5, and are shown in Fig. 2. Frequencies
pated TO and LO phonon mode mode frequencies with
of TO modes calculated by Lacomba-Perales et al. (Ref.
A and B symmetries. All Mueller matrix data were
9) using GGA-DFT are included in Tab. I for reference. u u
analyzed simultaneously during the polyfit, wavelength-
We note that data from Ref. 9 are considerably shifted
by-wavelength best match model procedure. For every
withrespecttoours,whileourcalculateddataagreevery
wavelength, up to 528 independent data points were in-
closely with our experimental results as discussed below.
cludedfromthedifferentsamples,azimuthpositions,and
angle of incidence measurements, while only 8 indepen-
dentparametersforrealandimaginarypartsofε ,ε ,
xx xy
B. Mueller matrix analysis ε , and ε were searched for. In addition, two sets of
yy zz
3 wavelength independent Euler angle parameters were
Figures3and4depictrepresentativeexperimentaland looked for. The results of polyfit calculation are shown
bestmatchmodelcalculatedMuellermatrixdataforthe in Figs. 3 and 4 as solid lines for the Mueller matrix ele-
(001) and (010) surfaces investigated in this work. In- ments. We note in Figs. 3 and 4 the excellent agreement
sets in Figures 3 and 4 show schematically axis b within between measured and model calculated Mueller matrix
the sample surface and perpendicular to the surface, re- data. Furthermore, the Euler angle parameters, given
spectively, and the plane of incidence is also indicated. in captions of Figs. 3 and 4, are in excellent agreement
Graphsdepictselecteddata,obtainedat3differentsam- with the anticipated orientations of the crystallographic
ple azimuth orientations each 45◦ apart. Panels with in- sample axes.
dividual Mueller matrix elements are shown separately,
and individual panels are arranged according to the in-
dicesoftheMuellermatrixelement. Itisobservedbyex-
perimentaswellasbymodelcalculationsthatallMueller
matrix elements are symmetric, i.e., M = M . Hence,
ij ji
elements with M =M , i.e., from upper and lower di-
ij ji
8
TableI.PhononmodeparametersforA andB modesofCdWO obtainedfromDFTcalculationsusingQuantumEspresso.
u u 4
Renderings of displacements are shown in Fig. 2.
X =B X =A
u u
Parameter k=1 2 3 4 5 6 7 8 k=1 2 3 4 5 6 7
AX [(eB)2/2]this work 2.61 3.31 0.29 1.38 0.12 0.18 0.27 0.10 0.52 1.47 0.65 0.28 0.43 0.03 0.15
k
ω [cm−1] this work 786.47 565.46458.33285.00264.05225.70156.97108.50 863.40 669.13510.16407.97329.74285.88138.11
TO,k
α [◦] this work 22.9 111.5 8.3 69.9 59.8 126.8 157.2 28.3 - - - - - - -
TO,k
ω [cm−1] Ref. 9 743.6 524.2 420.9 255.2 252.9 225.9 145.0 105.6 839.1 626.8 471.4 379.4 322.1 270.1 121.5
TO,k
Figure 3. Experimental (dotted, green lines) and best match model calculated (solid, red lines) Mueller matrix data obtained
from a (001) surface at three representative sample azimuth orientations. (P1: ϕ = −1.3(1)◦, P2: ϕ = 43.7(1)◦, P3: ϕ =
88.7(1)◦). Data were taken at three angles of incidence (Φ = 50◦,60◦,70◦). Equal Mueller matrix data, symmetric in their
a
indices, are plotted within the same panels for convenience. Vertical lines indicate wavenumbers of TO (solid lines) and LO
(dotted lines) modes with B symmetry (blue) and A symmetry (brown). Fourth column elements are only available from
u u
the IR instrument limited to approximately 333 cm−1. Note that all elements are normalized to M . The remaining Euler
11
angle parameters are θ = 88.7(1) and ψ = −1.3(1) consistent with the crystallographic orientation of the (001) surface. The
inset depicts schematically the sample surface, the plane of incidence, and the orientation of axis b in P3.
C. Dielectric tensor analysis are shown as dotted lines in Fig. 5 for ε , ε , ε , and
xx xy yy
ε , and in Fig. 6 as dotted lines for ε−1, ε−1, ε−1, and
zz xx xy yy
The wavelength-by-wavelength best match model di- ε−1. A detailed preview into the phonon mode proper-
zz
electric function tensor data obtained during the polyfit
9
Figure 4. Same as Fig. 4 for the (010) sample at azimuth orientation P1: ϕ = 0.5(1)◦, P2: ϕ = 45.4(1)◦, P3: ϕ = 90.4(1)◦.
θ=0.03(1) and ψ =0(1), consistent with the crystallographic orientation of the (010) surface. Note that in position P3, axis
b which is parallel to the sample surface in this crystal cut, is aligned almost perpendicular to the plane of incidence. Hence,
the monoclinic plane with a and c is nearly parallel to the plane of incidence, and as a result almost no conversion of p to s
polarized light occurs and vice versa. As a result, the off diagonal block elements of the Mueller matrix are near zero. The
inset depicts schematically the sample surface, the plane of incidence, and the orientation of axis b, shown approximately for
position P3.
ties of CdWO is obtained here without physical line- the tensor elements ε−1, ε−1, ε−1 when magnitudes ap-
4 xx xy yy
shape analysis. In Fig. 5, a set of frequencies can be proachlargevalues. Thesefrequenciesareagaincommon
identifiedamongthetensorelementsε ,ε ,ε ,where to all elements ε−1, ε−1, ε−1, and thereby reveal the fre-
xx xy yy xx xy yy
their magnitudes approach large values. In particular, quencies of 8 LO modes with B symmetry. The same
u
the imaginary parts reach large values.71 These frequen- considerationholdsforε revealing7LOmodeswithA
zz u
ciesarecommontoallelementsε ,ε ,ε ,andthereby symmetry. Theimaginarypartofε−1 attainspositiveas
xx xy yy xy
reveal the frequencies of 8 TO modes with B symme- well as negative extrema at these frequencies, and which
u
try. The same consideration holds for ε revealing 7 is due to the respective LO eigen dielectric displacement
zz
LO modes with A symmetry. The imaginary part of unit vector orientation relative to axis a. We note that
u
ε attains positive as well as negative extrema at these depicting the imaginary parts of ε and ε−1 alone would
xy
frequencies, and which is due to the respective eigen di- sufficetoidentifythephononmodeinformationdiscussed
electric displacement unit vector orientation relative to above. We further note that the inverse tensor does not
axis a. As can be inferred from Eq. 9(b), the imagi- contain new information, however, in this presentation
nary part of ε takes negative (positive) values when the properties of the two sets of phonon modes are most
xy
α is within {0··· − π} ({0...π}). Hence, B TO convenientlyvisible. Wefinallynotethatuptothispoint
TO,l u
modes labeled 2, 6, and 7 are oriented with negative an- no physical model lineshape model was applied.
gletowardsaxisa. Asimilarobservationcanbemadein
Fig.6,whereasetoffrequenciescanbeidentifiedamong
10
Figure 5. Dielectric function tensor element ε (a), ε (b), ε (c), and ε (d). Dotted lines (green) indicate results from
xx xy yy zz
wavelength by wavelength best match model regression analysis matching the experimental Mueller matrix data shown in
Figs. 4 and 3. Solid lines are obtained from best match model lineshape analysis using Eqs. 9 with Eq. 4. Vertical lines in
panel group [(a), (b), (c)], and in panel (d) indicate TO frequencies with B and A symmetry, respectively. Vertical bars in
u u
(a), (c), and (d) indicate DFT calculated long wavelength transition dipole moments in atomic units projected onto axis x, y,
and z, respectively.
D. Phonon mode analysis plane. In particular, modes B -2, B -6, and B -7 reveal
u u u
eigenvectorswithintheinterval{0···−π},andcauseneg-
a. TO modes: Figs. 5 and 6 depict solid lines ob- ative imaginary resonance features in εxy. Accordingly,
tained from the best match mode calculations using their unit eigen displacement vectors in Tab. II reflect
Eqs. 9 and the anharmonic broadened Lorentz oscilla- values larger than 90◦. The remaining mode unit vec-
tor functions in Eq. 4. We find excellent match between tors possess values between {0...π} and their resonance
all spectra of both tensors ε and ε−1.72 The best match features in the imaginary part of εxy are positive.
model parameters are summarized in Tab. II. As a re- PreviousreportshavebeenmadeofCdWO4 TOmode
sult, we obtain amplitude, broadening, frequency, and frequencies and their symmetry assignments for,6–8,13
eigenvector parameters for all TO modes with A and however,noneprovideacompletesetofIRactivemodes.
u
B symmetries. Wefind8TOmodefrequencieswithB Due to biaxial anisotropy from the monoclinic crystal,
u u
symmetry and 7 with A symmetry. Their frequencies reflectivity measurements do not provide enough infor-
u
are indicated by vertical lines in panel group [(a), (b), mation to determine directions of the TO eigenvectors.
(c)] and panel (d) of Fig. 5, respectively, and which are Therefore, no previously determined TO mode frequen-
identical to those observed by the extrema in the imagi- cies could be accurately compared here.
b. TO displacement unit vectors: A schematic pre-
nary parts of the dielectric tensor components discussed
sentationoftheoscillatorfunctionamplitudeparameters
above. As discussed in Sect. IIC, element ε provides
xy ABu and the mode vibration orientations according to
insight into the relative orientation of the unit eigen dis- k
anglesα fromTab.IIwithinthea-cplaneisshown
placement vectors for each TO mode within the a - c TO,k
inFig.7(a). InFig.7(b)wedepicttheprojectionsofthe
