Table Of ContentDipole Analysis of the Dielectric Function of Colour Dispersive Materials:
Application to Monoclinic Ga O
2 3
C. Sturm,1 R. Schmidt-Grund,1 C. Kranert,1 J. Furthmu¨ller,2 F. Bechstedt,2 and M. Grundmann1
1Institut fu¨r Experimentelle Physik II, Universita¨t Leipzig, Linn´estr. 5, 04103 Leipzig, Germany
2Institut fu¨r Festko¨rpertheorie und -optik, Friedrich-Schiller-Universit¨at Jena, Max-Wien-Platz 1, 07743 Jena, Germany
We apply a generalized model for the determination and analysis of the dielectric function of
opticallyanisotropicmaterialswithcolourdispersiontophononmodesandshowthatitcanalsobe
6 generalized toexcitonicpolarizabilities andelectronic band-bandtransitions. Wetakeintoaccount
1 that the tensor components of the dielectric function within the cartesian coordinate system are
0 notindependentfrom eachotherbutareratherprojectionsofthepolarization ofdipolesoscillating
2 along directions defined by the, non-cartesian, crystal symmetry and polarizability. The dielectric
n functionisthencomposedofaseriesofoscillatorspointingindifferentdirections. Theapplicationof
a thismodelisexemplarilydemonstratedformonoclinic(β-phase)Ga2O3 bulksinglecrystals. Using
J thismodel, weare able to relate electronic transitions observed in thedielectric function toatomic
bond directions and orbitals in the real space crystal structure. For thin films revealing rotational
8
2 domains we show that theoptical biaxiality is reduced to uniaxial optical response.
]
i I. INTRODUCTION tation of the dielectric axes depends on the wavelength
c
s and is often called colour dispersion. In the spectral
l- Forthe understanding,designandfabricationofopto- rang with non-vanishing absorption, the situation be-
r electronic devices, the optical properties of the involved comes even more complex. Due to the independent di-
t
m materials have to be known. A well established and agonalizability of the tensor (1) for the real and imag-
. powerful method for the determination of these prop- inary part, the corresponding dielectric axes in general
at erties is spectroscopic ellipsometry1,2. We concentrate do not coincide which each other. Thus, in general, four
m here on the dielectric function (DF), which is usually dielectric axes are present. For these classes of materi-
- obtained by means of numerical model analysis of the als only few reports on the determination of the full di-
d experimentalellipsometry data and then often described electric tensor exists, e.g. for α-PTCDA11, pentacene12,
n by a series of line-shape model dielectric functions in or- BiFeO313,CdWO48,K2Cr2O714,CuSO4·5H2O15andef-
o
derto deducephononproperties,freechargecarriercon- fective anisotropic materials as e.g. slanted columnar
c
[ centrations and the properties of electronic transitions films16. Most of these works are limited to the determi-
(e.g Ref. 2 and 3). For isotropic materials this method nationofthelineshapeofthedielectricfunction,treating
1 is well established. However, in recent years, optically each tensor component of the DF independently of each
v
anisotropic materials, as e.g. Ga O 4–7, CdWO 8 and other. This can, from a technical point of view, result
2 2 3 4
9 lutetium oxyorthosilicate9, went into focus of research in large correlations between the individual tensor ele-
8 since they are promising candidates for optoelectronic ments causing non-physical results. More importantly,
7 applications in the UV spectral range. However, the de- the deeper nature of polarizabilities in the material, like
0 termination of their optical and electronic properties is phonons,excitons,and electronicband-bandtransitions,
1. more challenging compared to isotropic materials since cannot be explored this way. Thus, lineshape model di-
0 they depend on the crystal orientation. The dielectric electicfunctions(MDF)representingtheoscillatorsprop-
6 function is represented by a (frequency-dependent) ten- ertieslike energy,amplitude, broadening,andevenoscil-
1 sor and the determination of its components requires a lationdirectioninameaningfulandphysicalcorrectway
v: series of measurements for various crystal orientations. have to be used.
i For(non-chiral)opticallyanisotropicmaterials,thedi-
X Facing this, Dressel et al.12 proposed an approach as-
electricfunctionisingeneralasymmetrictensorconsist-
r ing of six independent components10, i.e. suming that the dipole moments are aligned to three
a
polarization axis which should coincide with the crys-
ε ε ε tallographic axes. Taking this model into account, the
xx xy xz
ε= ε ε ε . (1) dielectric tensor is fully described by its three indepen-
xy yy yz
ε ε ε dent principal elements and the known angles between
xz yz zz
the crystallographic axis. However, as a consequence of
Due to its symmetry, this tensor can be diagonalized thisapproachtheprincipalaxesoftheindicatrix(related
independently for the real and imaginary part at each to the real part of ε) coincide with those of the conduc-
wavelengthseparately. Inthetransparentspectralrange, tivity tensor (related to the imaginary part of ε) which
i.e. for vanishing imaginary part, the diagonal elements is not generally valid as shown for instance for CdWO 8
4
are the semi-principal axes of the ellipsoid of wave nor- andGa O 7. To overcomethis problem,Ho¨feret al.14,15
2 3
mals and are often called dielectric axes. For materials used for the infrared spectral range a model, developed
with monoclinic or triclinic crystal structure, the orien- earlier by Emslie et al.17, which consists of a sum of
2
damped Lorentz oscillators individually aligned to the symmetry lower than the cubic one. In this case the
axes of their respective dipole moments. For phonons, excitations generally differ between the crystallographic
theseaxesarerelatedtotheatomicelongationsandthus directions in energy, amplitude, broadening, and even in
to some extent to the crystallographic axes. Further, thespatialdirectionoftheirdipolemoment,andthusthe
their dissipative spectral range is usually narrow. Thus DF is a tensor (Eq. (1)).
the question arises if such a model also can be applied Let ε′ being the dielectric response of the ith excita-
i
tospectrallywidespreadexcitationslikeelectronicband- tionandthecoordinatesystemischosen(withoutloosing
band transitions, which consist of numbers of individual generality)in such way that the polarizationdirection is
dipoles whose axes are connected to overlapping atomic along the x-axis. The only non-zero component is then
orbitals of various symmetry and therefore not necessar- given by ε , i.e. ε′ 6= ε′ = 0. However, the po-
xx i,xx i,mn
ily coincide withcrystallographicdirections. Further the larization direction of the excitation and the experimen-
densityofstates(DOS)oftheelectronicbandstructureis tal coordinate system do not coincide with each other in
distributedwithinawideenergyrangeinacomplexman- general and a coordinate transformation has to be per-
ner causing non-symmetric line-shapes of the imaginary formed, independently for each transition. The entire
part of the dielectric function which spectrally overlap dielectric tensor then can be expressed by
for different contributions and directions.
N
Here we demonstrate that the sketched approach is
ε=1+ R(φ ,θ )ε′R−1(φ ,θ ), (3)
generally valid for all kinds of excitations. We demon- i i i i i
strate this exemplarily for monoclinic Ga2O3 (β-phase) Xi=1
single crystals and thin films in the spectral range from with φ and θ being Euler angles, which are in gen-
i i
infraredto vacuumultraviolet. We showthatthis model eral different for each excitation, and R being the ro-
provides a deep insight in electronic properties of the tation matrix. The advantage of this expression is that
materials: Comparingthedirectionsoftheelectronicpo- the components of the resultant dielectric tensor in the
larizabilities obtained by modeling the experimental el- Cartesiancoordinatesystemarenotindependentofeach
lipsometry data using lineshape MDF to the real space other but rather composed of the respective projected
atomic arrangement in the crystal and considering the- part of the excitation’s line-shape function according to
oretical calculated electron density distribution as well the directions of their individual dipole moment. For
asorbital-resolvedDOS,allowsustoassigntheobserved the entire dielectric function it follows that, due to the
transitions to individual orbitals. finite broadening of each excitation and by considering
The paper is organized as follows: In Sec. II, we dis- Kramers-Kronigrelation,the orientationofthe principal
cuss at first the dielectric tensor for all crystal symme- tensor axes of the real and imaginary parts differ from
tries and its composition. After that we demonstrate its each other as it is well known and observed in experi-
applicability to the case of β-Ga2O3 single crystals in ments e.g. for CdWO48 and Ga2O37.
the infrared and ultraviolet spectral range. Finally, we Equation (3) represents the general case which has to
show by means of a practically relevant β-Ga2O3 thin be used for triclinic crystals and can be simplified de-
filmwhichexhibitrotationdomainsthattheapproachof pending on the crystal symmetry. Crystals with mon-
using directed transitions explains the effective uniaxial oclinic structure exhibit one symmetry axes, represent-
properties ofthe film andenhances the sensitivity to the ing a C rotation axis or the normal of a mirror plane
2
out-of plane component of the dielectric tensor. (or both), which we identify in the following with the y-
direction. The plane perpendicular to y, the x-z-plane,
reveals no symmetry which defines a Cartesian coordi-
II. DIELECTRIC FUNCTION nate system preferentially. Therefore, from symmetry
arguments, considering dipoles polarized either along y
Theopticalresponseofamaterialisdeterminedinfirst or in the x-z-plane, one can simplify Eq. (3) to
order by dipole excitations, e.g. optical phonons, elec-
tronic band-band transitions or excitons which in sum Ny Nxz
are represented by the dielectric function. For isotropic ε=1+ εi,y+ R(φj)ε′j,xzR(φj)−1, (4)
materials, the corresponding dipole moment or polar- i=0 j=0
X X
ization direction of each excitation is macroscopically
withε andε′ beingthecontributionoftherespective
equally distributed in all spatial directions, resulting in i,y j,xz
directions. N and N represent the number of excita-
an isotropic dielectric function, i.e. it is a scalar written y xz
tions with the corresponding polarization directions and
as
as φ we define the angle between the polarization direc-
N tion and the x-axes within the x-z-plane. This leads to
ε=1+ ε , (2) the well known form of the dielectric tensor given by
i
i=1
X ε 0 ε
xx xz
with N being the number of excitations/oscillators. The ε= 0 ε 0 . (5)
yy
situation changes for materials with crystal structure ε 0 ε
xz zz
3
Afurthersimplificationcanbemade fororthorhombic In spectroscopic ellipsometry, the change of the polar-
materials containing three orthogonal twofold rotation izationstateoflightafterinteractionwithasampleisde-
symmetry axis, leading to termined. Inthegeneralcase,thisisexpressedbymeans
ofthe4×4Muellermatrix(MM,M)whichconnectsthe
Nx Ny Nz Stokes vectors of the incident (reflected) light Sin (Sref)
ε=1+ εi,x+ εj,y+ εk,z, (6) by Sref = MSin. In the special case where no energy
i=0 j=0 k=0 transfer between orthogonal polarization eigenmodes of
X X X
the probe light takes place, like for isotropic samples or
a dielectric function tensor which contains only diagonal optically uniaxial samples with the optical axis pointing
elements. In the case of uniaxial materials, e.g. those along the surface normal (as the case for the thin film,
with a hexagonal symmetry, εi,x = εi,y and Nx = Ny cf. Sec. V), the change of the polarization state is ex-
holds. For isotropic material, the numbers of oscillators pressedbytheratioofthecomplexreflectioncoefficients,
in all three directions is the same and therefore the di- i.e. ρ = r˜ /r˜. The index represents the polarization of
p s
electric tensor reduces to the scalar given by Eq. (2). the light polarized parallel (p) or perpendicular (s), re-
ForpracticalapplicationEq.(3)hastobefurthermod- spectively, to the plane of incidence which is spanned by
ified. The real and imaginary parts of the dielectric the surface normal and the light beams propagation di-
function are connected with each other by the Kramers- rection.
Kronig relation. Contributions of excitations at energies
For the determination of the DF, the experimental
higher than the investigated spectral range to the real
dataareanalyzedbytransfer-matrixcalculationsconsid-
part of the DF have to be considered. These contribu-
eringalayerstackmodel. Forthebulksinglecrystals,the
tionsareusuallydescribedbyapolefunction. Inthecase
modelconsistsofasemi-infinitesubstrate(Ga O itself)
2 3
presented here, this means that the identity in Eq. (3)
andasurfacelayeraccountingforsomeroughnessorcon-
has to be replaced by a real valued tensor with the form
taminations. For the infrared spectral range the surface
given by the corresponding crystal structure where each
layer can be neglected. For the thin film the model con-
component is represented by a pole function.
sists of a c-oriented sapphire substrate, the Ga O thin
2 3
film layer and the surface layer. The dielectric function
of sapphire was taken from the literature21. The surface
III. EXPERIMENTAL layer was modelled using an effective medium approxi-
mation (EMA)22 mixing the DF of Ga O and void by
2 3
By using the approach presented in Sec. II and line- 50%:50%for the bulk single crystals.7 For the thin film
shape MDFs, the parametrised dielectric function of β- thisfractionwaschosenasparameterandthebestmatch
Ga O bulksinglecrystalsandthinfilmswasdetermined betweenexperimentandcalculatedspectrawasobtained
2 3
inthemid-infrareduptothevacuum-ultravioletspectral for 80% : 20%. In the following we choose our coordi-
rangebymeansofgeneralizedspectroscopicellipsometry. natesysteminsuchway,thateˆx ka-axis,eˆy kb-axisand
eˆ =eˆ ×eˆ .
Ga O crystallizes at ambient conditions in mono- z x y
2 3
clinic crystal structure, the so-called β-phase (Fig. 4).
The angle between the non-orthogonal a- and c-axis is
β = 103.7◦18 resulting in a non-vanishing off-diagonal
element of the dielectric tensor within the Cartesian co- IV. BULK SINGLE CRYSTALS
ordinate system7,19. We investigatedtwo single side pol-
ishedbulksinglecrystalsfromTamuraCorporationwith
A. Infrared spectral range
(010) and (¯201) orientation, allowing access to all com-
ponentsofthedielectrictensor. X-raydiffraction(XRD)
measurements does not revealany hints for the presence The MM in the infrared spectral (250 − 1300cm−1
of rotation domains, twins or in-plane domains. More (31−161meV)) range was measured at angles of inci-
details can be found in Ref. 7. The thin film was de- denceof30◦,50◦ and70◦ fordifferentin-planerotations,
positedonac-planeorientedsapphiresubstratebymeans i.e. rotating the crystal around its surface normal by
ofpulsedlaserdeposition(PLD)atT ≈730◦C. Afterde- 30◦, 60◦ and 90◦. For selected orientations the recorded
position,thesamplewasannealedfor5minatT ≈730◦C spectraareshowninFig.1. Thenon-vanishingblock-off-
and a oxygen partial pressure of p = 800mbar. XRD diagonal elements of the MM demonstrate the optically
measurements confirm the monoclOin2ic crystal structure anisotropic character of the sample.
ofthefilmandthesurfaceorientationwasdeterminedto The dielectric function in the infrared spectral range
be(¯201). Incontrasttothebulksinglecrystals,sixrota- is determined by phonon and free charge carrier os-
tiondomainsareobservedwhicharerotatedagainsteach cillations. The bulk single crystals are not intention-
otherbyanangleof60◦.20Incontrasttobulksinglecrys- ally doped, the latter contribution can be neglected for
tals whichreveala smoothsurfacewithout atomicsteps, the spectral range investigated here. Therefore, only
the surface roughness of the thin film was determined to phonons have to be considered and their contribution is
be R ≈5nm described by Lorentzian oscillators:23
s
4
the mode B(3) only the frequency is given since also the
u
large noise in this spectral range and the probable spec-
tral overlap with B(4) prohibit the determination of its
u
dipole direction.
For comparison we calculated the phonon modes by
ab-initio calculations based on the B3LYP hybrid func-
tionalapproachimplementedintheCRYSTAL14code26.
Thereby we used the basis set of Pandey et al.27 for
gallium and of Valenzano et al.28 for oxygen, which we
slightlymodified,and150k-pointsintheirreducibleBril-
louin zone. The truncation criteria defined by CRYS-
TAL14 code given by five tolerance set to 8,8,8,8, and
16 for our calculations were used for the Coulomb and
exchange infinite sums. Further we used a tolerance of
the energy convergence of 10−11 Hartree. All input pa-
rametersandcalculationconditionscanbe foundin Ref.
26. Thecalculatedlatticeparametersarea=1.2336nm,
b = 0.3078nm and c = 0.5864nm, in reasonable agree-
ment with those reported in the literature18. The corre-
sponding phonon mode energies, oscillator strength and
the direction of the dipoles are also given in Tab. I and
are in excellent agreement with those determined by el-
lipsometry. The excellent agreement is not restricted to
theinfraredactivephononmodesbutisalsoobtainedfor
the Raman active modes26.
FIG. 1. Experimental (symbols) and calculated (lines) spec-
tra of the MM elements of a β-Ga O bulk single crystal for
2 3
◦
anangleofincidenceof70 . Thecorrespondingorientationof
B. Ultraviolet spectral range
thecrystalisgivenbytheEuleranglesontopofeachcolumn
in theyzxnotation.
ThenumericDFintheUVspectralrangewasrecently
reportedbyus,obtainedbyusingaKramers-Kronigcon-
sistentnumericalanalysis7. Inordertoextracttheprop-
erties of the contributing electronic transitions, e.g. en-
AγE
0 ergyandelectronicorbitalsinvolved,andtodemonstrate
ε(E)= , (7)
E2−E2−iγE the universal applicability of Eq. (3) for electronic tran-
0
sitions we analysed the contribution of each transition
with A, E0 and γ being the amplitude, energy and to the entire DF by using line-shape model dielectric
broadeningofthephononmode,respectively. Thecalcu- functions. Symmetry consideration and band structure
lated MM spectra are shown in Fig. 1 as red solid lines properties7 yieldthatthetransitionsarepolarizedeither
yielding good agreement with the experimental ones. along the y-axes or within the x-z-plane. Thus the DF
Note, that a similarly good match is obtained by us- can be written as in Eq. (4) with a set of excitonic tran-
ing a Kramers-Kronig consistent numerical analysis and sitions and Gaußian oscillators. We have been shown
consider the four components of the DF (Eq. (5)) to be by density functional theory calculations combined with
independent from each other. many-body perturbation theory including quasiparticle
In the investigated spectral range, 9 of the totally and excitonic effects7, that the DF in the spectral range
12 optical infrared active phonon modes are observable. from the fundamental absorption edge on up to some
Their properties are summarized in Tab. I. For the eV higher are dominated by excitonic correlation ef-
modeswhichhaveadipolemomentinthea-cplane(Bu- fects. Thus, several excitonic contributions have been
symmetry)the polarizationdirectionwith respectto the considered in modeling and were described by a model
a-axis is given by the angle φ, which was found to dif- dielectric function developed by C. Tanguy for Wan-
fer for each phonon mode. This is also in agreement nier excitons taking into account bound and unbound
with the results recently reported by Schubert et al.19 states.29–31Thecontributionofweaklypronouncedband-
The phonon mode B(4) was not observablein our exper- band-transitionswhere summarizedby using a Gaussian
u
iment. This can be attributed to the weak sensitivity oscillator. A further Gaussian oscillator was included to
to this mode caused by its low amplitude which is pre- consider contributions of transitions at energies higher
dicted by ab-initio calculations (see below) and to the thantheinvestigatedspectralrangeduetotheirspectral
pronouncednoisecausedbythe lowsensitivityofthe de- broadening. These contributions together with the pole
tector of our setup in this spectral range. Further, for function were considered for each dielectric tensor com-
5
A fexp/f0,e fcalc/f0,c γ φexp φtheo E0,exp E0,theo E0,exp E0,theo
(cm−1) (◦) (◦) (cm−1) (cm−1) (cm−1) (cm−1) (cm−1) (cm−1) (cm−1)
A(1) - - 0.01 - - - - 160.7 155 154.8 155 155.7 141.6
u
Bu(1) - - 0.41 - - 101 - 224.3 250 213.7 216 202.4 187.5
Bu(2) 92 - 0.33 8 - 176 253 267.3 290 262.3 300 260.4 251.6
B(3) - - 0.03 - - 39 - 281.9 310 279.1 337 289.7 265.3
u
A(u2) 51 - 0.50 21 - - 295 300.5 n.o. 296.6 352 327.5 296.2
Bu(4) 86 0.20 0.16 4 166 173 357 361.0 375 356.8 374 365.8 343.6
B(5) 82 0.96 0.96 17 46 47 430 434.2 455 432.5 500 446.8 410.5
u
A(u3) 83 0.77 0.78 13 - 447 447.0 525 448.6 526 475.7 383.5
Bu(6) 73 1.00 1.00 15 128 130 572 560.8 640 572.5 626 589.9 574.3
A(u4) 73 0.39 0.43 5 - 662 665.8 668 663.2 656 678.4 647.9
B(7) 32 0.25 0.23 7 28 0 691 692.5 692 692.4 720 705.8 672.6
u
Bu(8) 10 0.13 0.13 11 74 76 743 742.5 731 743.5 760 753.8 741.6
this work Ref. 24 Ref. 19 Ref. 24 Ref. 19 Ref. 25
TABLEI.Amplitudes(A),oscillator strength(fexp),dampingparameters(γ)andenergies(E0,exp)ofthephononmodes. The
angle φ represents the determined angle between the a-axes and the direction of the dipole moment in the x-z-plane. The
exp
phonon energy, oscillator strength and direction of the dipole from ab-initio calculations are given by E f and φ ,
0,calc clc calc
respectively. For comparison the experimentally determined as well as calculated oscillator strength was normalized to those
of the Bu(6) mode.
ponent independently because they may originate from label direction Type A E γ φ
different transitions. (eV) (meV) (◦)
The experimentally recorded and the calculated spec- X a-c Exciton 15.0 4.88 70 110
1
tra of the MM elements are shown for selected orienta- X2 a-c Exciton 18.0 5.10 800 17
tions in Fig. 2, yielding good agreement. The difference X3 a-c Exciton 14.9 6.41 210 41
between the experimental and the calculated spectra for X4 a-c Exciton 28.0 6.89 190 121
G a-c Gauss 0.27 6.14 1,343 124
energies E > 7eV was also observed by using the above 1
Xb b Exciton 8.3 5.41 75
mentioned numerical Kramers-Kronig consistent analy- 1
Xb b Exciton 20.1 5.75 139
sis and might be caused by the limitation of the used 2
approach for the description of the surface layer.7 This X3b b Exciton 7.0 6.93 253
can be attributed to the fact that the sensitivity to this
TABLE II. Parameters of the UV model dielectric functions
layer is strongly enhanced in this spectral range due to
for the observed transitions within the investigated spectral
the enhanced absorption and therefore reduced penetra-
range. The angle φ represents the orientation of the dipole
tion depth.
moment in thea-c-planewith respect to thex-axis.
The parameters of the best-match MDF are summa-
rized in Tab. II and III. We extracted a exciton bind-
ing energy of about Eb =270meV for all contributions. Gauss Pole ε∞
X A E γ E A
Notethatweconsideredthesameexcitonbindingenergy
(eV) (eV) (eV) A
forallexcitonictransitionsbecauseofthestrongcorrela-
tionbetweenenergyofthe fundamentalboundstate and εxx 2.64 9.69 2.7 200.8 15.5 0.907
the corresponding binding energy. εyy 1.81 9.78 3.8 52.7 10.5 1.392
εzz 1.84 8.99 1.7 91.5 11.9 1.126
The dispersion of the tensor elements for the entire
εxz 0.26 8.49 0.5 -0.086
investigated spectral range is shown in Fig. 3. The con-
tributions of excitonic transitions to ε are shown as red
2 TABLE III. Parameters of the UV model dielectric function
solid lines. The orientation of the corresponding dipole describing the contributions of the high energy transition to
moments in the x-z-plane is indicated by the arrows in thedielectric function in theinvestigated spectral range.
the inset. In agreementwith our theoretical calculations
andthenumericMDF,7thetwoenergeticallylowesttran-
sitions (labeled as X1 and X2) are strongly polarized late the directions of the dipole moments of all four
alongthex-andz-direction,respectively. Athigherener- pronounced excitonic excitations within this plane (X
1
gies,therearetransitionsalongy-axis(b-axis)andwithin ...X ), as obtained from the ellipsometry model, to
4
the x-z-plane (a-c-plane). atomic bonds inthe crystalstructure asshowninFig. 4.
Based on calculated charge distribution33 and atomic For transitions along y, no direct assignment to individ-
arrangement within the x-z-plane (a-c-plane), we re- ual orbitals was possible because of the complex distri-
6
the dipole directions to the atomic bonds in Fig. 4. It
turns out that the transition X , almost directed along
2
x (a) involves O and Ga(II) and also reveals a high
amplitude in the DF. Ga(II) is located between O(II)
and O(III). But the dipole direction only fits to the
bondGa(II)-O(III),soitseemsthattransitionstoGa(II)
statesintheconductionbandonlyappearwhenO(III)is
involvedandarenotpossibleinvolvingO(II).Thiscanbe
understoodconsideringthe coordinationofthe O-atoms,
which is higher (6 bonds) for O(II), suggesting the or-
bitals to be more s-like compared to O(III) (4 bonds)
which dominate the DOS near the valence band maxi-
mum. The transitions X and X are assigned to take
3 4
placebetweenGa(I) andO(III). The directionsobtained
frommodelanalysisofthe DFdoesnotfitasgoodasfor
transition X , maybe caused in correlation effects due
2
to spectral overlap of different contributions to the DF.
Finally, transition X , directed almost along c, was as-
1
signed to take place either between O(I) and O(III) or
between two O(II) atoms, or both. While the first pos-
sibility involvesdifferently coordinatedatoms suggesting
dipole allowedtransitions between p- and s-like orbitals,
the second possibility involves only highly coordinated
atoms (s-like character) and thus should be dipole for-
bidden. The relatively high amplitude of this transition
is not clear at first place, because following Ref. 33, the
FIG. 2. Experimental (symbols) and calculated (lines) spec- charge density between the involved atoms and also the
tra of the MM elements of a β-Ga2O3 bulk single crystal for DOS of the oxygen orbitals in the conduction band is
◦
anangleofincidenceof70 . Thecorrespondingorientationof
predicted to be relatively weak.
thecrystalisgivenbytheEuleranglesontopofeachcolumn
These results nicely demonstrate the potential of the
in theyzxnotation.
usedmodelapproachforthedielectrictensortogaindeep
insight into electronic properties of highly anisotropic
materials.
butionofatomicbonds. Pleasenotethattheuncertainty
in the experimentally determined dipole moment direc-
tions amounts to up to 10◦, caused by the simplification
V. THIN FILM
due to the used model functions, which summarize spec-
trally over different individual transitions. As all these
transitions reveal no contribution to the dielectric ten- As mentioned above, the PLD grown β-Ga2O3 thin
sor component ε , only bonds located solely within the film exhibit (¯201) surface orientation with 6 in-plane ro-
yy
sub-planes of the x-z-plane (a-c-plane) are considered tation domains, rotated by multiples of 60◦. As their
(cf. Fig 4). It is found that all excitonic transitions size is much smaller than the optically probed sample
but the first one, which appears to take place between area of about 5×8mm2, the measured optical response
oxygen atoms, are between differently coordinated gal- isdeterminedbyanaverageoverthesedomains. Foruni-
lium and oxygen. In the following discussion we will use formdistributionofthese rotationdomains,the effective
thenomenclaturegivenbyGeller18andlabelthetetrahe- dielectric function is given by
drally and octahedrally coordinated Ga atoms as Ga(I)
andGa(II),respectively,whilethe three differentsitesof 6
ε= R(φ )εmonoR−1(φ )
the oxygen atoms are labeled as O(I), O(II) and (OIII) i i
(cf. Fig. 4). Xi=1
Bandstructurecalculationsrevealthatthe uppermost 0.5(ε′xx+εyy) 0 0
valence bands are dominated by oxygenp-orbitals,while = 0 0.5(εxx+εyy) 0 , (8)
the DOS of the lowest conduction bands is composed 0 0 ε′zz
of almost equal contributions from Ga-s, O-s, and O-p
orbitals.5,33,34 Thus, dipole allowed transitions can take with φ = (i−1)π/3 the rotation angle of the ith ro-
place from O-p orbitals to Ga-s and O-s orbitals. It tation domain (i = 1...6) and R(φ) being the rotation
turnsoutthatthestatesneartheconductionbandmini- matrixaroundthesurfacenormal. Equation(8)issimilar
mumarepreferentiallydeterminedby octahedrallycoor- to those of a uniaxial material with ε = 0.5(ε′ +ε )
⊥ xx yy
dinated Ga(II).33 This is reflected by the assignment of andε =ε′ (⊥andk: perpendicularandparalleltothe
k zz
7
FIG. 3. Dielectric function (black solid line) of a β-Ga O bulk single crystal in the infrared and UV spectral range. The red
2 3
solid lines represent the excitonic contribution in the investigated UV spectral range whereas the red dashed lines represent
the contribution of the high-energy contributions. The arrows in the insets depict the orientation of the corresponding dipole
moment and their relative amplituderatio.
opticalaxis) with orientationof the effective opticalaxis in Fig. 5 in terms of the pseudo dielectric function2
along the surface normal. Note that ε′ and ε′ are the
xx zz
tensor components for the coordinate system with the <ε>=<ε1 >+i<ε2 >
x- and z axis parallel and perpendicular to the sample 2
1−ρ
surface, respectively. =sin2Φ 1+tan2Φ (9)
1+ρ
" (cid:18) (cid:19) #
For such samples with the sensitivity to ε is usually
k with angle of incidence Φ. Below E ≈ 4.8eV oscilla-
limited due to the high index of refraction of the inves-
tions due to multiple reflection interferences caused by
tigated material resulting in a propagation direction of
the interfaceswithin the sample areobservedwhichvan-
the wave within the sample with only very small angles
ish with the onset of the absorption at higher energies.
offtheopticalaxis. Butthereisafiniteprojectionofthe
For the parametric model of the dielectric function of
electro-magnetic field strength onto the optical axis and
the thin film we used the same set of model dielectric
thus the optical response is determined by ε and ε in
⊥ k functions as for the bulk single crystal. The calculated
any case, which have to be considered in order to obtain
spectra are shown as red solid lines in Fig. 5 and a good
a physical meaningful dielectric function35. However, in
agreementbetweentheexperimentalandcalculateddata
contrasttoahomogeneousuniaxialmaterial,thoseeffec-
is apparent. The tensor components of the dielectric
tive ε and ε are not independent from each other. As
⊥ k function of the thin film are shown in Fig. 6. For com-
shown in Sec. II and demonstrated in Sec. IV the com-
parison, the components calculated from DF of the bulk
ponents ε′ and ε′ reflect the same transitions and are
xx zz singlecrystalbyusingEq.(8)areshownasdashedlines.
determined by the projection A′ /A′ = sin2φ′/cos2φ′
zz xx For the thin film, we needed to adjust energies and am-
of their amplitudes A (φ′ is the angle of the oscillation
plitudes of the transitions and even the dipoles’ orien-
direction of the individual dipoles with respect to the
tation angles φ within the x-z-plane (a-c-plane). Com-
sample surface). This offers in the present case more
pared to the DF of the single crystal a blue-shift of the
sensitivity for determination of the tensor component ε
k transition energies up to 100meV and a lowering of the
as compared to homogeneous uniaxial materials.
oscillator strengths is observed for the thin film. The
reduced oscillator strength in the investigated spectral
The uniaxial behaviour of the film with the optical range cannot explain the lowering of the real part of the
axis parallelto the surface normalis reflectedby vanish- dielectric constant and therewith of the index of refrac-
ingoff-diagonalelementsoftheMM.Therefore,standard tion in the visible spectral range alone. Therefore, the
ellipsometryissufficientformeasuringthefullopticalre- reduced refractive index indicates also a reduced oscilla-
sponse (cf. Sec. III). The experimental data are shown tor strength of the high energy transitions compared to
8
FIG. 4. (a-c): Schematic representation of projections of thecrystal structureof β-Ga O intotheb-a-plane(a), b-c-plane(b)
2 3
and c-a-plane(c). The unit cell is indicated by theblack framed boxes. Bonds are indicated by lines between the atoms. The
respectiveCartesian coordinatesxandz areindicated,y pointsalongb. ThetetrahedrallycoordinatedGa(I)atomsareshown
inblueandtheoctahedrallycoordinatedGa(II)areshowningreen. Theoxygenatomsaremarkedinred. (d)Thedirectionsof
thedipolemomentswithinthec-a-/x-z-planeareindicated forthetransitions X ...X at theleft sideofthemiddlerow. (e)
1 4
Sub-layersofthec-a-plane(left: layer1,right: layer2)asindicatedalsointheupperrow. Here,theoxygenatomsatdifferent
lattice sites are highlighted by colours as O(I) red, O(II) orange and O(III) yellow (see also text). The dashed coloured lines
relatethedipoledirectionsofthetransitionsX ...X toatomicbondswithinthecrystalstructure. Pleasenotethatonlyone
1 4
example is shown for each different transition. (Images created by VESTA (Ref. 32).)
the bulk single crystal. We relate these changes of the
DFpropertiescomparedtothebulksinglecrystalonthe
one hand to crystal imperfections typically lowering the
oscillatorstrengthofelectronictransitionsby dissipative
processes. On the other hand also strain will be possi-
bly presentin the thin film, causing changesin the bond
length and maybe also torsion of the unit cell causing
different dipole moment orientations.
VI. SUMMARY
FIG. 5. Real (a) and imaginary (b) part of the thin film’s
pseudodielectric function for angle of incidence60◦ and 70◦. Wehavedeterminedthedielectricfunctionofβ-Ga2O3
The experimental and calculated data are shown as symbols by using a generalized oscillator model taking into ac-
and red solid lines, respectively. count the direction of the dipole moments for each tran-
sition. Within this model, the components of the dielec-
9
By means of the determined direction of the dipoles we
assign the involved orbitals for the observed transitions.
Forthethinfilmweshowedthatthepresenceofrotation
domains leads to the formation of an effective uniaxial
material. The sensitivity to the out-of-plane component
of the dielectric tensor is enhanced compared to pure
uniaxial materials since it is connected to the in-plane
component. This allows a precise determination of this
component even if the optical axis is perpendicular to
the surface, which is relevant for applications in opto-
electronics.
FIG. 6. Real (a) and imaginary (b) part of the tensor com-
ponentsoftheGa2O3 thinfilm(solidlines). Forcomparison, ACKNOWLEDGMENTS
thecomponentscalculated byEq.(8)usingthesinglecrystal
values are shown as dashed lines.
We thank Hannes Krauß, Vitaly Zviagin and Stef-
fen Richter for the support of the ellipsometry mea-
surements. This work was supported by the Deutsche
Forschungsgemeinschaft within Sonderforschungsbereich
tric tensor within the cartesian coordinate system are 762 - ”Functionality of Oxide Interfaces”. We also
not independent from each other but are determined by acknowledge financial support of the Austrian Fond
the projection of the corresponding dipole direction. In zur F¨orderung der Wissenschaftlichen Forschung in the
doing so, we could determine the tensor components of framework of SFB25 - ”Infrared Optical Nanostruc-
the DF of β-Ga O bulk single crystals and thin films. tures”.
2 3
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