Table Of ContentSelf-consistent iteration procedure in analyzing reflectivity and spectroscopic
ellipsometry data of multilayered materials and their interfaces
T. C. Asmara,1 I. Santoso,1 and A. Rusydi1,a)
NUSNNI-NanoCore, Singapore Synchrotron Light Source, and Department of Physics,
National University of Singapore, Singapore 117576
(Dated: 5 January 2015)
Formultilayeredmaterials,reflectivitydependsonthecomplexdielectricfunctionofalltheconstituentlayers,
and a detailed analysis is required to separate them. Furthermore, for some cases, new quantum states can
occur at the interface which may change the optical properties of the material. In this paper, we discuss
5 various aspects of such analysis, and present a self-consistent iteration procedure, a versatile method to
1
extract and separate the complex dielectric function of each individual layer of a multilayered system. As a
0
case study, we apply this method to LaAlO /SrTiO heterostructure in which we are able to separate the
2 3 3
effects of the interface from the LaAlO film and the SrTiO substrate. Our method can be applied to other
3 3
n
complex multilayered systems with various numbers of layers.
a
J
PACS numbers: 78.20.-e,07.05.Kf, 73.21.Ac
2
]
l I. INTRODUCTION the material in the broad range of 0.5 - 35 eV18,20–22.
e
The broad photon energy range is crucial in order to
-
r Recent technological advances in synthesizing multi- yield a stabilized Kramers-Kronig analysis of the reflec-
t
s layered materials with precise atomic control have made tivitydata(explainedindetailslater),sothatthecorrect
.
t it possible to study the various exotic quantum phe- complex dielectric function ε(ω)= ε1(ω)+iε2(ω) of the
a
m nomena that can occur at the interfaces of dissimilar material can be extracted reliably from the reflectivity.
materials1. The interplay between charge, spin, and or- This technique has been proven to be important in the
d- bital atthese interfaces canlead to variousexciting phe- study of a wide variety of materials, ranging from man-
n nomena such as orbital and spin reconstructions, metal- ganites, graphene, and oxides such as SrTiO318,20–23.
o insulator transitions, magneto-electric coupling, super-
c One important difference between the study of inter-
conductivity, quantum Hall effect, and topological ef-
[ faces with that of bulk materials is the fact that the in-
fects. What makes it even more interesting is that these
terface is buried under one or more layers of parent ma-
1 phenomena can occur even if the parent materials that
terials. Thus, any technique intended to be used in the
v make up the interface are not known to exhibit those
8 study ofinterfaces has to be able to probe the buriedin-
properties. Examples include the ferromagnetic inter-
6 terfacewithoutdisturbingtheparentmaterialssurround-
face(andassociatedcolossalmagnetoresistance)between
3 ingit. Forexample,inLaAlO /SrTiO (LAO/STO)het-
antiferromagnetic insulators LaMnO and SrMnO 2,3, 3 3
0 3 3 erostructure,theLaAlO filmthicknessistypicallyinthe
0 superconducting interface between insulator La2CuO4 orderof1-4nm,whichm3 eansthattheinterfaceisburied
1. andmetallicoverdoped(La,Sr)2CuO44,conductinginter- also at 1 - 4 nm below the surface. In the optical re-
facebetweenbandinsulatorsLaAlO (LAO)andSrTiO
0 3 3 flectivity setup described above, the photon penetration
5 (STO)5,andmanyothers6–17. Anothercaseisgraphene; depth is found to be in the order of 10 - 40 nm, which
1 since it usually needs to be suspended on top of a sub-
is more than sufficient to probe the buried interface of
: strate due to its two-dimensionality, it is also effectively
v LaAlO /SrTiO . In general, this is also applicable to
a multilayered system18,19, and its interaction with the 3 3
i othermultilayeredsystemsaslongasthe depthatwhich
X underlying substrate can be very interesting to study19.
the interface is buried does not exceed the penetration
r In order to study the nature and mechanisms behind
a depth of the photon.
these various interface phenomena, it is very crucial to
have a thorough understanding of the electronic band Inthis paper, we discuss variousaspects of reflectivity
structure at the interface and how it differs from those and dielectric function analysis of both bulk and multi-
of the parent materials. One way to directly probe this layered materials. Especially for the multilayer analysis,
is by measuring the complex dielectric response of the we present a self-consistent iteration procedure, a versa-
material in a broad energy range18,20–22. For example, a tile method to extract and separate the complex dielec-
combination of spectroscopic ellipsometry (0.5 - 5.6 eV) tric function of each individual layer of a multilayered
andultraviolet-vacuumultraviolet(UV-VUV)reflectiv- system. As the case study, we apply this method to
ity (3.7 - 35 eV) can be used to obtain the reflectivity of LaAlO /SrTiO heterostructure5, in which we are able
3 3
to separate the effects of the interface from the LaAlO
3
film and the SrTiO substrate. Our method can be ap-
3
pliedtoothermultilayeredsystemswithvariousnumbers
a)Correspondenceto: [email protected] of layers.
2
II. GENERAL ANALYSIS OF OPTICS DATA OF BULK Normalized reflectivity of SrTiO
MATERIALS 0.4 Self-normalized reflectivity of Sr3TiO
3
from spectroscopic ellispometry
Unnormalized reflectivity of SrTiO (x1/20)
wishaWicsehelcfif-arnnsotcromdviaseclriuzasinspghthotetecohnspneeinqceutrregotysocroadpneitcgeeremollfiip0ns.e5omt-h5eet.6rcyoemV(Sp.ElSeE)x, eflectivity 00..32 ffCrrooammlc uUolfaVf-t-reVedsU orVen farleencfclteeivc icttiyov niotysf idSmereTraaiOstiuo3rnesm3ents
R
element of dielectric tensor from a single measurement
without the need to perform a Kramers-Kronig trans- 0.1
formation, making it free from any ambiguities that are
related to the normalization of conventional reflectivity 0.0
results24. The raw data measured SE is expressed in 5 10 15 20 25 30 35
terms of Ψ and ∆, which are defined as25 Photon Energy (eV)
r FIG.1. Thenormalized high-energyreflectivity(0.5-35eV)
p
ρ tanΨexp(i∆) , (1)
of SrTiO is compared to the self-normalized reflectivity ob-
≡ ≡ r 3
s
tainedfromspectroscopic ellipsometry (0.5-5.6eV),theun-
normalized UV-VUV reflectivity (3.7 - 35 eV) from the UV-
where ρ is the ratio between r and r , the reflection
p s
VUV reflectivity measurements (scaled down by 20× to fit
coefficientsofp- (paralleltothe planeofincident)ands-
thegraph), and thecalculated reflectivity from off-resonance
(perpendicular to the plane of incident) polarized light,
considerations(>30eV).Toobtainthenormalized reflectiv-
respectively. From the Fresnel equations, r and r are
p s ityinthefullrangeof0.5-35eV,theunnormalizedUV-VUV
defined as
reflectivityis furtherscaled down tomatch thespectroscopic
ellipsometry and the off-resonance data, and then the three
n cosθ n cosθ
rij = j i− i j (2) data are appended together. The raw data are reproduced
p njcosθi+nicosθj withpermission from T.C.Asmaraet al.,Nat. Commun. 5,
3663 (2014)27. Copyright 2014 by NaturePublishing Groups
and (NPG).
n cosθ n cosθ
rij = i i− j j. (3)
s n cosθ +n cosθ part(>30eV)isnormalizedusingcalculationsbasedon
i i j j
off-resonance scattering considerations according to26
Here, n and θ represent the refraction index and angle
r λ
of incident from the surface normal, respectively. The i 0
r =i F(θ)P (2θ), (5)
f
and j indices representthe two materials involvedin the sinθ
photon propagation.
whereristhereflectioncoefficient,r istheclassicalelec-
0
From n, the complex dielectric function ε(ω) can be tron radius (e2/mc2), λ is the photon wavelength, P (θ)
f
obtained using
is the polarization factor (equal to unity for s-polarized
light and equal to cosθ for p-polarized light), and F(θ)
ε(ω)=n(ω), (4) is the structure factor per unit area given by
p
where ω is the photon frequency. The ε(ω) obtained us- 4πzq
F(θ)= n f exp(i sinθ). (6)
q q
ing Eq. 4 can then be converted back to reflection coef- λ
X
q
ficients using Eqs. 2 and 3. The tanΨ and ∆ are essen-
tially ratios of the intensities (for tanΨ) and phases (for The summation is performed over the different types of
∆) of the reflection coefficient of the p- and s-polarized atoms on a particular atomic plane on which the light is
lights(Eq.1),whichmakesthem(andanyquantitiesde- incident, with n denotes the number of atoms of type q
q
rived from them, including reflectivity) self-normalized. inthatparticularplane,f denotesthe tabulatedatomic
q
Thisisanimportantattribute ofSE,sincethe converted form factor corresponding to that atom q, and z de-
q
self-normalized reflectivity can be used to normalize the notes the direction vector normal to the plane in ques-
reflectivity data obtained using other methods, such as tion. From the above step, normalizedreflectivity in the
the UV-VUV reflectivity technique. range of 0.5 - 35 eV can be obtained. As an illustration
Next, we discuss the analysis of the UV-VUV reflec- forthe normalizationprocedure,Figure1 showsthe nor-
tivity data (3.7 - 35 eV) of bulk materials. In optics, a malized high-energy reflectivity (0.5 - 35 eV) of SrTiO
3
material can be considered as a bulk material when its as compared to the self-normalized reflectivity obtained
thickness, d, is morethan five times the photonpenetra- fromSE(0.5-5.6eV),theunnormalizedUV-VUVreflec-
tion depth, D, (see Section IX for detailed discussion of tivity (3.7 - 35 eV), and the off-resonanceconsiderations
D), i.e. d > 5D25. Due to the self-normalized nature of (> 30 eV). (The details of the experimental procedures
SE, the SE-derived reflectivity can be used to normalize used to obtain the data are discussed in Section IV.)
the UV-VUV reflectivity at the low energy side within From the above step, normalized reflectivity in the
the range of 3.7 - 5.6 eV. Furthermore, the high energy range of 0.5 - 35 eV can be obtained. For isotropic
3
bulk materials, the ε(ω) can be extracted from the III. GENERAL OVERVIEW OF ANALYSIS OF
normalized UV-VUV reflectivity using Kramers-Kronig MULTILAYERED MATERIALS
analysis28–35. The procedure is as following. The reflec-
tioncoefficientandphasedifferencebetweenthereflected If the material is in isotropic bulk form, the Kramers-
and incident light, ϕ, are related through the Kramers- Kroniganalysisisstraightforward,sincethe obtainedre-
Kronig transformation according to flectivity onlydepends onε(ω)ofone material. However
if the material is composed of several layers (i.e. a mul-
r(ω)= R(ω)exp(iϕ(ω)) (7)
tilayer), the analysis becomes more complex due to the
p
and interference between the light reflected from the surface
∞ and those reflected from the interface(s). For example,
ω lnR(x)
ϕ(ω)= P dx+ϕ(0), (8) according to analysis of wave propagation in a strati-
−π Z0 x2−ω2 fied medium, the reflection coefficient of a thin film on a
substrate (i.e. a two-layered material) has the following
where R = r2 is the reflectivity and P is the Cauchy
| | form36,
principal value. From here, the refractive index n and
extinctioncoefficientk canbe extractedfromreflectivity
using r = ramb,film+rfilm,subsei2δfilm , (16)
amb,multi 1+ramb,filmrfilm,subsei2δfilm
1 R
n= − (9)
1+R 2√Rcosϕ where
−
and 2πdi
δ = n cosθ . (17)
i i i
2Rsinϕ λ
k= . (10)
1+R 2√Rcosϕ Here, the subscripts amb, multi, film, and subs repre-
−
sent the ambient, the multilayer, the thin film, and the
Finally,thereal(ε )andimaginary(ε )partsofcomplex
1 2
substrate, respectively, which are the various materials
dielectric function can be obtained from n and k via
involved in the propagationof the photon.
ε =n2 k2 (11) In other words, for a multilayered system the ob-
1
−
tained reflection coefficient, reflectivity (R =
amb,multi
and r 2),and, viaEq.1, ρ, Ψ,and∆dependonthe
amb,multi
| |
ε(ω)ofbothfilmsandsubstrate,alongwiththethickness
ε =2nk. (12)
2 of the film and the angle of incidence37,38,
Like R and ϕ, ε and ε are also related through the
1 2
Kramers-Kronigrelationship according to Ramb,multi =R(εfilm,εsubs,dfilm,θ),
(18)
∞ ρ =ρ(ε ,ε ,d ,θ),
2 xε (x) amb,multi film subs film
2
ε (ω) 1= P dx (13)
1 − π Z x2 ω2
0 − and a detailed analysis is required to separate them. In
and this paper, we discuss various aspects of such analysis,
∞ andpresenttheself-consistentiterationprocedure,aver-
2ω ε (x) 4πσ
ε (ω)= P 1 dx+ DC, (14) satilemethodtoextractandseparatetheε(ω)ofeachin-
2 − π Z x2 ω2 ω
0 − dividual layer of a multilayered system so that they can
where σ is the DC conductivity. be further analyzed separately.
DC
The Kramers-Kronig analysis can be either done di- If the properties of the substrate are not expected to
rectly through function inversion (i.e. by directly using be significantly affected by the presence of the films, the
Eqs. 7-12), numerical approximation, or through fitting. reflectivity of the bare substrate can be separately mea-
In this paper, the analysis is done by fitting35 using sured, andfrom that its ε(ω) canbe separatelyobtained
the Kramers-Kronig-transformable Drude-Lorentz oscil- usingthegeneralproceduredescribedinSectionII.Then,
lators according to if there is only one film layer, the ε(ω) of the film can
be straightforwardly obtained from the total reflectivity
ωp2,k using Eqs. 2, 3, 16, and 17. However, if there are mul-
ε(ω)=ε∞+ . (15)
ω2 ω2 iΓ ω tiple layers of films composed of different materials, and
Xk 0,k− − k the ε(ω) of each material is unknown (or different from
Thehigh-frequencydielectricconstantisdenotedbyε∞; their bulk forms), then the analysis becomes more com-
ω , ω , and Γ are the plasma frequency, the trans- plicated. This isbecause thereareseveralunknowns but
p,k 0,k k
verse frequency (eigenfrequency), and the line width only one equation (Eq. 16)38, which prevents a straight-
(scatteringrate)ofthek-thoscillator,respectively. Since forward mathematical solution. The same problem also
the energy range involved is very broad (covering 0.5 - occurs if the properties of the substrate are affected by
35 eV), the analysis yields a stabilized Kramers-Kronig the presence of the films. For example, if parts of the
transformation. substrate near the interface become modified due to the
4
respectively) are prepared using techniques described
elsewhere27,42. From transport measurements, it is
known that the 2 and 3 uc samples are insulating with
carrier density and conductivity below the measure-
ment limit, while the 4 and 6 uc ones are conducting
with carrier density of 4 6 1013 cm−2 and conduc-
tivity of 4 8 10−5 Ω−−1,×consistent with previous
results39,40,4−2,45(×Figure 2 (b)). Bulk LaAlO and bulk
3
SrTiO samples are also prepared for comparison.
3
The reflectivity is obtained using a combination of
SE (0.5 - 5.6 eV), and UV-VUV reflectivity (3.7 - 35
eV)18,20–22. The details of the optical measurements are
as follow. The SE measurements are performed in the
spectralrangebetween0.5and5.6eVbyusinganSE850
ellipsometeratroomtemperature24. Threedifferentinci-
◦ ◦ ◦
dent angles of 60 , 70 , and 80 from the sample normal
FIG. 2. (a) Crystal structure of LaAlO3/SrTiO3. (b) Trans- areneeded,andtheincidentlightis45◦linearlypolarized
port measurement of LaAlO /SrTiO , showing the LaAlO -
3 3 3 fromtheplaneofincident. Forreflectivitymeasurements
thickness-dependentmetal-insulatortransition. Therawdata
in the high-energy range between 3.7 and 35 eV we use
are reproduced with permission from T. C. Asmara et al.,
Nat. Commun. 5, 3663 (2014)27. Copyright 2014 by Na- the superlumi beamline at the DORIS storage ring of
ture Publishing Groups (NPG). (c) Multilayer consideration HASYLAB (DESY)51. The incoming photon is incident
◦
of LaAlO /SrTiO . at the angle of 17.5 from the sample normalwith linear
3 3
polarization parallel to the sample surface. The sample
chamber is outfitted with a gold mesh to measure the
presence of the films, then in the analysis the interface incidentphotonfluxaftertheslitofthe monochromator.
needs to be treated as an effective additional layer. So, The measurements are performed in ultrahigh vacuum
even if the system is composed of only one layer of thin environment (chamber pressure of 10−9 mbar) at room
film on top of a substrate (an apparent two-layered sys- temperature. The obtained UV-VUV reflectivity data is
tem), it needs to be treated as if it was a three-layered calibrated by comparing it with the luminescence yield
one due to the presence of the interface layer. of sodium salicylate (NaC H O ) and the gold mesh.
7 5 3
Forthenearly-isotropicbulkLaAlO andbulkSrTiO ,
3 3
the reflectivity normalization along with the extraction
IV. CASE STUDY: LAO/STO HETEROSTRUCTURE ofε(ω)fromthenormalizedreflectivitycanbeperformed
usingthegeneralproceduredescribedinSectionII.(The
One example is the LaAlO /SrTiO heterostructure normalized reflectivity and extracted ε(ω) of both are
3 3
(Figure 2 (a)), which is also the system that will be shown later in Figure 6.) On the other hand, the analy-
used as the case study throughout this paper. When sisofLaAlO3/SrTiO3isnotasstraightforward,duetoits
thin film of LaAlO is deposited on SrTiO , a conduct- heterostructurenatureaswellasthepresenceofthecon-
3 3
ingquasitwo-dimensionalelectrongas(2DEG)withhigh ductinglayeratitsinterface. Forthisreason,amultilayer
mobility and carrier density emerges at the interface5. consideration based on a boundary analysis of Fresnel
Interestingly, this quasi-2DEG only emerges when the equationneeds to be takeninto accountinanalyzingthe
LaAlO3 film reaches a certain critical thickness, usually SEdataandtheUV-VUVreflectivityofLaAlO3/SrTiO3.
4 unit cells (uc) or more39. Below the critical thick- Inthismultilayeranalysis,LaAlO3/SrTiO3 (particularly
ness, the system remains insulating. Furthermore, the the conducting samples) consists of three layers: the
interface is also found to exhibit magnetism40–44 and LaAlO3filmlayeratthetop,thebulkSrTiO3layeratthe
superconductivity45–47, andtwo-dimensionalcoexistence bottom and an interface layer sandwiched between the
of both has evenbeen observed48–50. These observations two at the middle, representing the quasi-2DEG at the
are very remarkableconsidering LaAlO and SrTiO are interface (Figure 2 (c)), consistent with previous obser-
3 3
non-magnetic insulators in their bulk state27. Thus, vation using cross-sectional conducting tip atomic force
the understanding of electronic band structure at the microscopy52.
LaAlO /SrTiO interfaceisverycrucialtorevealthena- According to analysis of wave propagation in a strat-
3 3
ture and mechanisms of these interesting phenomena. ified medium, the reflection coefficient of a three-layer
For the case study, four samples of LaAlO /SrTiO system like LaAlO /SrTiO can be expressed through
3 3 3 3
with varying thicknesses of LaAlO (2, 3, 4, and 6 uc, Fresnel equations as36
3
ramb,fLAO+rfLAO,intei2δfLAO +ramb,fLAOrfLAO,intrint,STOei2δint +rint,STOei2(δfLAO+δint)
r = , (19)
amb,multi 1+ramb,fLAOrfLAO,intei2δfLAO +rfLAO,intrint,STOei2δint +ramb,fLAOrint,STOei2(δfLAO+δint)
5
where the subscripts fLAO, int, and STO representthe
LaAlO film, the interface layer, and the SrTiO sub-
3 3
strate,respectively. Thus,thereflectivity(and,byexten-
sion via Eq. 1, ρ, Ψ, and ∆) of LaAlO /SrTiO contains
3 3
mixed information from all three constituent layers37,38,
R =R(ε ,ε ,ε ,d ,d ,θ),
amb,multi fLAO int STO fLAO int
(20)
ρ =ρ(ε ,ε ,ε ,d ,d ,θ).
amb,multi fLAO int STO fLAO int
This makes the extraction of the ε(ω) of individual
layer non-trivial, because there are too many unknown
factorsinvolved. Sinceε(ω)ofthebulkSrTiO substrate,
3
ε , can be measured independently and LaAlO film
STO 3
thickness,d ,ofeachsampleareknown,Eqs.1-4,17,
fLAO
and19leftuswith3unknownvariables: ε(ω)ofLaAlO
3
film, ε , (which might be different from that of bulk
fLAO
LaAlO ), ε(ω) of interface layer, ε , and the thickness
3 int
of the interface layer, dint, FIG.3. Diagramdepictingtheangle-dependentiterationpro-
cess.
R =R(ε ,ε ,d ,θ),
amb,multi fLAO int int
(21)
ρ =ρ(ε ,ε ,d ,θ).
amb,multi fLAO int int
following (Figure 3). According to previous studies, the
(Note that even though ε(ω) has real and imaginary thickness of the conducting interface might be around
parts, they are connected through the Kramer-Kronig 2 - 10 nm45,52,57–61, so the initial guess for dint, dint[0],
relationship,seeEqs.13and14.) Thisposesachallenge, can be reasonably set as 5 nm. In general, there are two
because (assuming there is no change in ε across boundaryconditionsthatcanbeappliedwhensettingthe
STO
the samples) there are 3 unknowns but only 1 equation initialguess,andalsoforconfirmingthephysicalvalidity
(Eq.19),whichpreventsastraightforwardmathematical of the convergedvalue after iteration, of the thickness of
solution38. To overcome this problem, it can be noted films and interfaces:
that the light phase, δ, in Eq. 17 depends mainly on two 1. The thickness should not be lower than the thickness
parameters: theincidentangle,θ,(angle-dependent)53–55 of 1 uc of the materials. For films, this is the thinnest
andthelayerthickness,d,(thickness-dependent)54. This physical limit for the layer-by-layerdeposition, while for
means Eq. 17 can be used to diversify Eq. 19 by varying interfacesthisistotakeintoaccountanyinterfacerough-
either of these two parameters, so that the number of ness effects.
equations can match the number of unknown variables 2. Thethicknessshouldnotbehigherthanfivetimesthe
(in this case, three). (Note that Eq. 19 can also be di- photon penetration depth D (see Section IX for detailed
versified by varying the ambient within which the mea- discussion of D), since beyond this limit the material is
surementis performed54,56, e.g. by purgingthe measure- optically considered to be bulk25.
ment chamber with different ambient gas or immersing Meanwhile, the initial guess for εfLAO can be set as
thesetupinsidedifferentliquids,howeverconcernsabout the same as ε(ω) of bulk LaAlO3, εbLAO, which can be
surface contamination on surface-sensitive samples may obtainedindependentlybymeasuringitseparately. With
make this method less versatile.) This enables us to per- thesetwovariablesfixed,Eq.1isfittedintoexperimental
◦
form a self-consistent iteration procedure on the reflec- value of Ψ and ∆ measured at θ = 60 using Eq. 15
tivity(andthusalsoρ,Ψ,and∆viaEq.1)data,sothat by appropriatelyadjusting the Drude-Lorentzoscillators
each unknown variable can be extracted separately. that make up the ε(ω) of interface layer35. At this first
step,byexpandingEq.21tothefirstorder,thetentative
ε(ω) of interface after fitting, ε , can be expressed
int[1]
V. ANGLE-DEPENDENT ITERATION PROCEDURE as37,38,
∂ρ /∂d
In SE (0.5 - 5.6 eV), the Ψ and ∆ measurements are ε =ε +(d d ) amb,multi int
done atthree differentincidentangles: 60◦, 70◦,and80◦ int[1] int int− int[0] ∂ρamb,multi/∂εint
(22)
from the sample normal, which results in three sets of Ψ ∂ρ /∂ε
amb,multi fLAO
+(ε ε ) .
and ∆ data. Since Eq. 1 can be diversified via Eqs. 17 fLAO− bLAO ∂ρ /∂ε
amb,multi int
and19byvaryingθ53–55,thisgivesusthethreeequations
necessary to perform an angle-dependent iteration pro- It can be seen that at this step, ε deviates from
int[1]
cedure to extract the three unknown variables: ε , theactualvalueofε duetothestill-impropervaluesof
fLAO int
ε , and d . d andε . Tosimplifythenotation,generalizedad-
int int int fLAO
As a representative, the iteration for the SE data of dition and subtraction operators, and , respectively,
⊕ ⊖
the 4 uc LaAlO /SrTiO sample can be performed as can be introduced to represent the correlation effects of
3 3
6
d and ε on ε such that,
int fLAO int[1]
ε =ε (εd εd ) (ε ε ). (23)
int[1] int⊕ int⊖ int[0] ⊕ fLAO⊖ bLAO
Here, εd andεd areintroducedto representthe cor-
int int[0]
relation effects that d and d have on ε , re-
int int[0] int[1]
spectively. For example, to the first order, εd can be
int
expressed according to Eq. 22 as,
∂ρ /∂d
εd = amb,multi intd . (24)
int ∂ρ /∂ε int
amb,multi int
Another advantage of these generalized operators nota-
tions is that they also allows the higher orders of Eq. 21
expansion to be implicitly included in Eq. 23, mean-
ing that Eq. 23 can be taken as the generalized form of
Eq. 22. Thus, due to their convenience, these operators
shall be used throughout this paper.
After the first step described above, the newly-fitted
ε is in turn fixed, and d is appropriately adjusted
int[1] int
so that Eq. 1 can now be fitted into experimental value
◦
of Ψ and ∆ measured at θ = 70 . Here, the tentative
value of d , d , can be expressed as,
int int[1]
d =d (dε dε ) (dε dε ). (25)
int[1] int⊕ fLAO⊖ bLAO ⊕ int⊖ int[1]
Again, dε , dε , dε , and dε are introduced to
fLAO bLAO int int[1]
representthe correlationeffects that ε , ε , ε ,
fLAO bLAO int
and ε have on d , respectively. As a representa-
int[1] int[1]
tive, dε can be expressed to the first order as,
fLAO
∂ρ /∂ε
dε = amb,multi fLAOε . (26)
fLAO ∂ρ /∂d fLAO
amb,multi int
FIG.4. Comparisonbetweentheexperimentally-measuredΨ
Then, the newly-adjusted dint[1] is also fixed (along and ∆ of 4 uc LaAlO3/SrTiO3 and their fitted va◦lues after
with the previously-fittedε ), andEq.1 is fitted into iteration for all three incident angles. (a) For 60 incident
int[1] ◦ ◦
◦ angle. (b) For 70 incident angle. (c) For 80 incident angle.
experimental value of Ψ and ∆ measured at θ = 80 us-
The fitted values match the measured Ψ and ∆ very well for
ingEq.15byappropriatelyadjustingthe Drude-Lorentz
all three incident angles, confirming the stability of the iter-
oscillators that make up the ε(ω) of LaAlO film layer.
3 ation. Therawdataarereproduced with permission from T.
Here,thetentativeε(ω)ofLaAlO3 film, εfLAO[1],canbe C.Asmaraetal.,Nat. Commun. 5,3663(2014)27. Copyright
expressed as,
2014 by NaturePublishing Groups (NPG).
ε =ε (ε ε ) (εd εd ). (27)
fLAO[1] fLAO⊕ int⊖ int[1] ⊕ int⊖ int[1]
Afterthat,theprocessisrepeatedbygoingbacktoΨand and dfLAO, these quantities can be converted to reflec-
∆ values measured at θ = 60◦ and subsequently cycling tivity in the 0.5 - 5.6 eV range using Eqs. 2 - 4, 17, and
through the incident angles, fitting only one variable at 19, which then can be used to normalize the UV-VUV
each step while keeping the other two fixed. reflectivity.
Convergence is reached when, at a certain step N, Figure5(a)illustratestheiterationprocessofthe4uc
εfLAO[N], εint[N], and dint[N] can satisfy Eq. 1 for all LaAlO3/SrTiO3byshowingtheevolutionofdintthrough
threeincidentangles,asshowninFigure4. Atthispoint, eachiterationstep. Astheiterationprogresses,thevalue
εfLAO[N] εfLAO, εint[N] εint, dint[N] dint, and ofdintslowlyapproachesadistinctasymptoticvalue,and
thecorrela→tioneffectsbetwee→nthesethreepar→ametersare at step 5 it finally converges into 5.2 nm.
∼
minimized (see SectionVI for a more rigoroustreatment For 6 uc LaAlO /SrTiO , the iterationprocess can be
3 3
of this convergence condition). In other words, the iter- performed similarly, since the only difference is d ,
fLAO
ation results form a universal fitting that can match the which is known and can be appropriately adjusted using
data from all incident angles. The iteration thus results Eq. 17. Figure 6 shows the fitted Ψ and fitted ∆ after
intheconvergedvaluesofε ,ε ,andd ,asshown convergence that match the measured Ψ and measured
fLAO int int
in Figure 14 later. Along with the already-known ε ∆ of 6 uc LaAlO /SrTiO . The iteration progress of
STO 3 3
7
FIG.6. Comparisonbetweentheexperimentally-measuredΨ
and ∆ of 6 uc LaAlO /SrTiO and their fitted values after
3 3
FIG. 5. (a) Iteration progress of the interface layer thick- iteration for all three incident angles. (a) For 60◦ incident
ness for 4 uc LaAlO3/SrTiO3. (b) Iteration progress of the angle. (b) For 70◦ incident angle. (c) For 80◦ incident angle.
interface layer thickness for 6 uc LaAlO3/SrTiO3, showing The fitted values match the measured Ψ and ∆ very well for
thecomparisonbetweenthearbitraryandpredefinedstarting all three incident angles, confirming the stability of the iter-
pointsof6nmand5.2nm,respectively. (c)Iterationprogress ation. Therawdataarereproduced with permission from T.
oftheinterfacelayerthicknessfor6ucLaAlO3/SrTiO3,show- C.Asmaraetal.,Nat. Commun. 5,3663(2014)27. Copyright
ingthecomparison betweenthearbitrarystartingpointsof2 2014 by NaturePublishing Groups (NPG).
nm and 10 nm, respectively.
canbe achievedwith fewer steps, while still reachingthe
dint for 6 uc LaAlO3/SrTiO3 is shown in Figure 5 (b). same converged value of dint ≈ 5.3 nm. Further tests
In this case, the initial guess for d is set to be 6 nm, of the stability of the iteration process are also done by
int
and the final converged value is found to be 5.3 nm, setting the initial guess for dint to be 2 nm and 10 nm,
very close to the 4 uc value of 5.2 nm. This∼indicates respectively. From Figure 5 (c), it canbe seenthat both
that the properties of 4 and 6∼uc LaAlO3/SrTiO3 are iterations are indeed able to converge to the same dint
very similar, and any apparent differences in Ψ, ∆, and value of 5.3nm, althoughthey need considerablymore
∼
reflectivity values between the two samples are mainly stepstoconvergebecausetheinitialguessesdeviatealot
caused by the difference in d . In fact, because of more from the converged value. These results confirm
fLAO
this, since from 4 uc iteration the converged values for theself-consistencyoftheiterationprocess,showingthat
ε ,ε ,andd of4ucLaAlO /SrTiO arealready evenifitstartswithdifferentinitialguesses,theiteration
fLAO int int 3 3
obtained,thosevaluescanalsobeusedastheinitialguess does eventually converge into the same final results.
for the 6 uc LaAlO /SrTiO iteration. It can be seen For the insulating samples (2 and 3 uc
3 3
from Figure 5 (b) that with those better starting points, LaAlO /SrTiO ), the iteration-based analysis is
3 3
the iteration process can be simplified and convergence alsoperformedsimilarly. The fitted Ψ andfitted ∆ after
8
FIG.7. Comparison betweentheexperimentally-measuredΨ FIG.8. Comparisonbetweentheexperimentally-measuredΨ
and ∆ of 2 uc LaAlO3/SrTiO3 and their fitted values after and ∆ of 3 uc LaAlO3/SrTiO3 and their fitted values (after
iteration for all three incident angles. (a) For 60◦ incident iteration) for all three incident angles. (a) For 60◦ incident
angle. (b) For 70◦ incident angle. (c) For 80◦ incident angle. angle. (b) For 70◦ incident angle. (c) For 80◦ incident angle.
The fitted values match the measured Ψ and ∆ very well for The fitted values match the measured Ψ and ∆ very well for
all three incident angles, confirming the stability of the iter- all three incident angles, confirming the stability of the iter-
ation. Therawdataare reproducedwith permission from T. ation. Therawdataarereproduced with permission from T.
C.Asmaraetal.,Nat. Commun. 5,3663(2014)27. Copyright C.Asmaraetal.,Nat. Commun. 5,3663(2014)27. Copyright
2014 by NaturePublishing Groups (NPG). 2014 by NaturePublishing Groups (NPG).
VI. THICKNESS-DEPENDENT ITERATION
PROCEDURE
convergence that match the measured Ψ and measured
∆ of 2 and 3 uc LaAlO /SrTiO are shown in Figures 7 Fromthe iteration-basedanalysisofSE data,the ε(ω)
3 3
and 8, respectively. For the sake of consistency and to of each individual constituent layer of LaAlO /SrTiO ,
3 3
make layer-by-layer comparison between insulating and along with their thicknesses, can be extracted. These
conducting LaAlO /SrTiO more readily apparent, the quantities can then be converted into reflectivity within
3 3
interface layer is still initially retained in the iteration the 0.5 - 5.6eV rangeusing Eqs. 2 - 4, 17, and19. From
process. However, as shown later in Figure 14, after here,thenormalizationprocedureoftheUV-VUVreflec-
analyzing the normalized reflectivity in the full range of tivity data (3.7 - 35 eV) is similar to that of bulk mate-
0.5 - 35 eV, the ε(ω) of the (artificial) interface layer is rials: using the SE-derived reflectivity to normalize the
found to be very similar to that of bulk SrTiO , making low-energyside (3.7- 5.6eV) andthe off-resonancescat-
3
insulating LaAlO /SrTiO effectively a two-layer struc- teringconsiderationsaccordingtoEqs.5and6tonormal-
3 3
ture instead. This can be explained by the absence of ize the high-energy side (> 30 eV). The normalized re-
the conducting interface in insulating LaAlO /SrTiO . flectivityoftheLaAlO /SrTiO samplesalongwiththat
3 3 3 3
9
0.4
Bulk SrTiO
3
Bulk LaAlO
3
2 uc LaAlO/SrTiO
0.3 3 3
3 uc LaAlO/SrTiO
y 3 3
vit 4 uc LaAlO3/SrTiO3
ecti 0.2 6 uc LaAlO3/SrTiO3
efl (a)
R
0.1
0.0
10
of bulk LaAlO
8 (cid:1)1of bulk LaAlO3
(cid:1)2 3
of bulk SrTiO
6 (cid:1)1of bulk SrTiO3
(cid:1)2 3
2
1, 4 (b)
2
0
-2
5 10 15 20 25 30 35
Photon Energy (eV)
FIG.9. (a)ReflectivityofLaAlO /SrTiO atdifferentthick-
3 3
nesses of LaAlO film as compared tobulkLaAlO and bulk
3 3
SrTiO . (b) Complex dielectric functions, ε(ω) = ε (ω)+
3 1
iε (ω), of bulk LaAlO and bulk SrTiO . The raw data are
2 3 3
reproduced with permission from T. C. Asmara et al., Nat.
Commun. 5, 3663 (2014)27. Copyright 2014 by Nature Pub-
lishing Groups (NPG).
FIG.10. Diagramdepictingtheiterationprocessofthe2and
3 uc LaAlO /SrTiO samples at (a) the initial step (step 0),
3 3
(b) step 1, and (c) step 2.
of bulk LaAlO and bulk SrTiO is shown in Figure 9
3 3
(a).
The same challenge present in the analysis of SE data
initialstep(step0),the3ucLaAlO /SrTiO isregarded
3 3
of LaAlO /SrTiO due to its multilayered structure is
3 3 asonlyhavingtwolayers: the bulkSrTiO substratebe-
3
also present in analyzing the high-energy reflectivity
lowandacompositelayer(comp)ontop(Figure10(a)).
data. Even though d is already known to be 5.3
int This compositelayerrepresentsthe mixture betweenthe
∼
nm from the SE angle-dependent iteration analysis, it
unknown LaAlO film and the unknown interface layer,
3
still leaves us with two unknowns (high photon energy
ε and high photon energy ε ), but only one equa- ε =ε ε . (28)
fLAO int comp[0] fLAO int
⊕
tion (Eq. 19), which prevents a straight-forward math-
Similar to Eq. 23, Eq. 28 can be explicitly expressed to
ematical solution. Furthermore, due to a fixed incident
◦ the first order as,
angle of 17.5 from the sample normal, similar angle-
dependentiterationasthe onedonefortheSEdatacan-
∂R /∂ε
amb,multi int
not be performed. To address this, we note that Eq. 19 εcomp[0] =εfLAO+εint∂R /∂ε . (29)
canalsobediversifiedthroughEq.17byvaryingthelayer amb,multi fLAO
thickness54,inparticulardfLAO. Itisforthisreasonthat Because the separately-measuredεSTO is known (Fig-
wehaveintentionallyfabricatedapairofinsulatingsam- ure9(b)), ε canbe obtainedbyfitting Eq.16into
comp[0]
ples (2 and 3 uc of LaAlO3) and a pair of conducting reflectivity of 3 uc LaAlO3/SrTiO3 (R3uc) using Eq. 15
samples (4 and 6 uc of LaAlO3). Each pair has similar by appropriatelyadjusting the Drude-Lorentzoscillators
respective physics with only difference in dfLAO, which that make up εcomp[0] (similar to the procedure used in
can be rectified by appropriately adjusting Eq. 17. This the angle-dependent iteration, see Sec. V). After that,
means for each case (insulating and conducting), there forthe 2 ucLaAlO /SrTiO ,the interfacelayerisadded
3 3
are two unknowns and two equations for Ramb,multi, so between the SrTiO3 substrate and the composite layer
a self-consistent iteration can be used to extract ε(ω) of so that it now is regarded as having three layers (Fig-
each individual layer. ure 10 (a)). This interface layer is added for the sake of
Let us first discuss the iteration procedure for the in- consistency and to make layer-by-layer comparison be-
sulatingsamples(2and3ucofLaAlO /SrTiO ). Asthe tween insulating and conducting LaAlO /SrTiO more
3 3 3 3
10
readily apparent. The newly-fitted ε along with
comp[0]
the already-known ε is now taken as the input (i.e.
STO
thecompositelayerbecomesthe ”fixedlayer”),andthen
Eq. 19 is fitted into reflectivity of 2 uc LaAlO /SrTiO
3 3
(R ) using Eq. 15 to extractthe newly-addedinterface
2uc
layer (the ”variable layer”) ε .
int[0]
For the next step (step 1), the 3 uc sample is also re-
garded as having three layers (Figure 10 (b)), and this
timetheinterfacelayerisdesignatedtobetheinputfixed
layer in order to extract ε(ω) of the now-variable com-
posite layer from R . After the extraction, the ε
3uc comp
for this step 1 becomes
εcomp[1] =εfLAO εint εint[0]. (30) FIG.11. Diagramdepictingtheiterationprocessofthe2and
⊕ ⊖
3 ucLaAlO /SrTiO samples at a certain general step n.
3 3
Then, the focus is again shifted to R . The newly-
2uc
adjusted composite layer (ε ) is designated as the
comp[1]
input fixed layer to extract ε(ω) of the variable interface and ε converges to ε ,
comp[N+1] fLAO
layer,
N
ε =ε δε , (31)
int[1] int[0]⊕ int[1] εcomp[N+1] =εfLAO εint (εint[0] δεint[i])
⊕ ⊖ ⊕
X
from R . This ε is slightly different than ε , i=1 (40)
2uc int[1] int[0]
by an amount of δεint[1]. In step 2 (Figure 10 (c)), this →εfLAO⊕εint⊖εint
procedure is repeated again, and by the end of that step ε .
fLAO
→
ε(ω) of the layers becomes
In other words, when this point is reached, the ε(ω) of
εcomp[2] =εfLAO εint (εint[0] δεint[1]) (32) eachindividuallayerisabletobeisolatedseparately,and
⊕ ⊖ ⊕
the iteration procedure converges successfully.
and
In order to give a better presentation of how this it-
εint[2] =εint[0] δεint[1] δεint[2]. (33) eration procedure is applied to the LaAlO3/SrTiO3 case
⊕ ⊕ study,ε(ω)ofthecompositeandinterfacelayersof2and
The iteration procedure is repeated until it eventually 3 uc LaAlO /SrTiO at various iteration steps is shown
3 3
achievesconvergence(seediscussionbelow),andatacer- in Figure 12. Fromthere, it canbe seen that as the iter-
taingeneralstepn(Figure11),ε(ω)ofthecompositeand ationprogresses,thedifferencebetweenε(ω)ofeachcon-
interface layers can be expressed as, secutive step becomes progressively smaller. Eventually,
the ε(ω) at step 6 becomes virtually indistinguishable to
n−1
that of step 5, which means that at step 6 the iteration
ε =ε ε (ε δε ) (34)
comp[n] fLAO⊕ int⊖ int[0]⊕ int[i] converges. The ε(ω) of the interface layer is successfully
X
i=1
separated from the composite layer, and the ε(ω) of the
and compositelayerbecomesequaltotheε(ω)oftheLaAlO
3
n film layer.
ε =ε δε . (35) Moreover, to further ensure the validity of the result-
int[n] int[0] int[i]
⊕
X ing ε and ε , a consistency check can be per-
i=1 int fLAO
formed by inserting ε and ε along with
int[N] comp[N+1]
Eventually, at a certain step N, ε(ω) of the layers be-
ε into Eq. 19 via Eqs. 2 - 4, 17, and 19 and confirm-
STO
come such that,
ing that the resulting R can indeed reproduce
amb,multi
the experimentally-measuredreflectivity of both 2 and 3
δε 0, (36)
int[N]
→ uc LaAlO /SrTiO simultaneously by appropriately ad-
3 3
justing the LaAlO thickness factor in Eq. 17 (see Fig-
3
εint[N] εint[N−1], (37) ure 13). Thus, at the point of convergence the extracted
≈
ε(ω) of eachindividual layeris able to universallyfit the
and
reflectivity of both 2 and 3 uc LaAlO /SrTiO .
3 3
εcomp[N] εcomp[N−1]. (38) For the 4 and 6 uc LaAlO3/SrTiO3 samples (the con-
≈
ducting case), the iteration process should essentially be
This is our point of convergence. At this point, ε
int[N] the same as the iteration procedure for the insulating
converges to ε ,
int case. However, in practice it can be more complex than
that. This is because in the conducting sample the in-
N
ε =ε δε ε , (39) terface layer is conducting, so its optical properties can
int[N] int[0] int[i] int
⊕Xi=1 → be very different than that of bulk SrTiO3. This makes
