dopant-dependent reflectivity and refractive index of microcrystalline h_xwo_3 and li_xwo_3 bronze...

$: Dopant-Dependent Reflectivity and Refractive Index of Microcrystalline H_xWO_3 and Li_xWO_3 Bronze Thin Films$
Dopant-dependent reflectivity and refractive indexof microcrystalline HxWO3 and LixWO3 bronze thin films

Zahid Hussain

Reflectivity spectra of HxWO3 and LixWO3 thin films were measured over the photon energy range from0.4 to 4.2 eV. It was found that microcrystalline tungsten bronzes have reflectances of 8%–30% over thedopant concentration range x �0 � x � 0.25�. Values for the real part of refractive index n were alsodetermined from the refined reflectivity data. The optical data are interpreted by use of a modifiedDrude–Zener model together with a single-oscillator model to differentiate between bound and freeelectronic states. The values of high-frequency dielectric constant εhf of MxWO3 �M � H�, Li�� bronzeswere determined from the refractive-index data for estimation of the effective electronic masses involvedin optical and polaronic transitions. A single-oscillator model showed that oscillator energy Ea anddispersion energy Ed increased and decreased, respectively, with increasing x values, opposite whatoccurs in crystalline tungsten bronzes. These findings support the fact that Bloch electrons are almostabsent; instead, the polaronic species �W5� and W4�� are assumed to control the reflectivity modifications�or variations in the refractive index� that are associated with the microcrystalline tungsten bronzes.© 2002 Optical Society of America

OCIS codes: 310.6860, 240.0310, 160.2100, 160.4760, 120.4530, 230.2090.

1. Introduction

The transition-metal oxides, particularly WO3 andMoO3, either individually or in combination,1–3 con-stitute an interesting class of materials because ofthe various properties that they exhibit. The reasonis that these oxides form a group of predominantlyionic solids that exhibit a wide range of optical andelectrical properties, among them the electrochromiceffect.2–5 The electrochromic effect is the reversiblecoloration induced in both organic and inorganic ma-terials by an applied electric field or current. Themechanism involves the simultaneous electrochemi-cal injection of electrons and monovalent metal ionsinto a film structure to produce the colored state.Electrochromism is of particular interest because ofits potential application in display devices and smartwindows.6–9 The existence of the electro-optical ef-fect has been demonstrated in many transition-metaloxide thin films,5,10–12 and this characteristic has

The author �[email protected]� is with the Department ofElectrical and Electronic Engineering, Imperial College of Science,Technology and Medicine, Exhibition Road, London SW7 2BT,United Kingdom.

Received 29 March 2002; revised manuscript received 24 July2002.

0003-6935�02�316708-17$15.00�0© 2002 Optical Society of America

6708 APPLIED OPTICS � Vol. 41, No. 31 � 1 November 2002

been found to depend strongly on the method of prep-aration of the films. Thin films of tungsten trioxideare electrochromic, inasmuch as electroreduction ofcolorless WO3 forms a colored bronze.

Among the inorganic compounds, WO3 has signif-icant advantages in terms of reversibility, stability,and color efficiency. Because of those qualities, ithas become one of the most promising electrochromic�EC� materials, in that it possesses potential appli-cations in large-area display fabrication,7,12 light-modifying materials,10,11 and gas sensors.13–16

Moreover, it has been shown that porous films of WO3with a high open surface area can be used as a ma-terial for energy conversion applications. Accord-ingly, it is suggested that low-density, highly porous,or disordered tungsten oxide films have a good elec-trochromic property because of the easier diffusion ofprotons in the oxide films.17–20 Thin films of thematerials such as WO3 and MoO3 �amorphous or mi-crocrystalline� offer a number of other favorable prop-erties, e.g., they are angle independent, that exhibit acontinuously variable intensity of coloration, they arecapable of storage of information without energy sup-ply, they exhibit an adequate coloration and bleach-ing rate with UV stability, and they have a largeoperational temperature range.

WO3 microcrystalline thin films doped with hydro-gen gas or lithium metal are transformed into HxWO3and Li WO bronze thin films, which normally oper-
x 3

ate as semiconducting materials at room tempera-ture. These bronzes have modulation of wider rangein transmittance and reflectance and are therefore ofgreat interest in the context of regulating the radi-ated energy for smart windows.5,21–23 Opticallyswitchable electrochromic materials �such as HxWO3and LixWO3 bronzes� are also important in the auto-mobile industry and for automatic control of climaticand day lighting of interior space in buildings andvehicles.6,24,25 In addition, they have significant po-tential for architectural, automotive, and many otherapplications.24–27 It should be noted that the filmdensity could have a strong influence on the switch-ing response time of WO3 thin films.26,28,29

The current interest in gasochromic devices asso-ciated with WO3 thin films arises from applicationsconcerned with gasochromic large-area window ap-plications14,15,30,31 and optically based hydrogen sen-sors.32,33 The development of a hydrogen sensor hasbeen prompted by the potential use of hydrogen fuelin various applications such as monitoring of pipelinecorrosion and for the surveillance of battery recharg-ing.32,33

Lithium-based systems have other superior at-tributes, viz., longer memory and resistance to oxi-dation, compared with their hydrogen counterparts.Nevertheless, the response times of stable electro-chromic cells of HxWO3 and LixWO3 are similar.34,35

Although the EC WO3 thin film can be prepared byvarious methods, such as vacuum evaporation,17,36,37

electron-beam sputtering,16,38,39 solgel activity,39–41

and electrodeposition,42,43 EC so-called tungsten �ormolybdenum� oxide films as used in devices are gen-erally believed to be oxygen deficient20,44,45 and alsoshow a high degree of structural disorder.20,45,46

Further, tungsten oxide films with oxygen-deficientatomic composition possess low packing density andlarge surface area4,47,48 but certainly behave betterthan crystalline films from the point of view of ECresponse.46,49,50 The color change in the amorphous�or microcrystalline� WO3 film is reversible up to con-centration of x � 0.3,26 and, of course, this character-istic strongly depends on the preparatory conditionsof the film.

It is a universal fact that disordered tungsten andmolybdenum oxide films, when ions and electrons areinserted �cathodic coloration�, are transformed froman optically transparent to an absorbing state or to areflecting state, depending on whether the electronsoccupy localized or extended states12,17,18,45; but atpresent there is great interest in the study of the ECeffect shown by amorphous and microcrystalline WO3thin films, preferably when they are in an absorbingstate.17,45

In this paper, considerable reflectivity data aregiven for microcrystalline tungsten bronzes, forwhich the Drude–Zener model has been developed toanalyze the nature of carriers �bound or free� associ-ated with the bronzes. The refractive indices oftungsten bronzes have also been derived from thereflectivity data and have been interpreted by use ofa single effective oscillator model. Some other sci-

entific perspectives have also been taken into accountfor more interpretation of the reported data.

2. Experiment

A. Preparation of Films

Lithium and WO3 were evaporated from separatesources because there is a large difference in theirvapor pressures, and LixWO3 films were prepared bya codeposition technique. WO3 and lithium wereoutgassed for 30 min at 1123 and 353 K, respectively,while the base pressure of the vacuum system wasmaintained at �5 � 10�6 Torr. In the next step,liquid nitrogen was used to drop the partial pressureof water, and eventually the chamber pressure wasmonitored to be �10�7 Torr before evaporation.When the deposition rate of WO3 was �1.5 nm�s, andthe evaporation rate of Li was steady, the shutter wasopened and lithium bronze deposition was carriedout. Controlling the rates of evaporation of the twomaterials yielded a large range of possible values oflithium concentration. For more details on the prep-aration of the samples see Refs. 36 and 51.

Hydrogen tungsten bronzes were prepared by pas-sage of hydrogen gas into WO3 thin films at substratetemperatures of �373 K at pressures close to 6 �10�1 Torr. The hydrogen molecular state was bro-ken into atomic hydrogen with the help of a Microtron200 MKII microwave power generator unit. Theresultant concentration of hydrogen bronze was pro-portional to the exposure time. Details of the appa-ratus and the procedure are given elsewhere.36,52

Each film thickness was measured �an average offour measurements for each film� with a Talysurfprofilometer to a precision of 20 Å �1 Å � 0.1 nm�.Furthermore, every evaporated WO3 thin film wascharacterized as microcrystalline �in nature� of aver-age grain size �50 Å after being examined byTEM36,53 and x-ray diffraction techniques.36,53,54

Shiojiri et al.55 and others37 have shown from TEMstudies that such WO3 thin films have crystallites of�12-Å diameter. Most recent mass spectrometry56

and desorption studies57 have also shown that, dur-ing the evaporation process, the vapor species aremolecular in nature and materialize as bulk of WO2,WO3, W2O5, W2O6, and W3O9, implying that thetungsten is mainly in the �4 and �6 states. Theamount of water contained in the oxide stronglydepends on the residual pressure in the vacuumchamber before evaporation; moreover, the O subs-toichiometry of the film also depends on this pres-sure. Films deposited in high vacuum show poorelectrochromic sensitivity and less water content.However, films deposited at relatively higher pres-sures exhibit better EC sensitivity but more watercontent.

Microcrystalline tungsten thin films as preparedby conventional techniques �evaporation, sputtering,etc.� are normally substoichiometric and have the ra-tio O�W �2.72, as measured by proton backscatter-ing,44 affirmed from Auger electron spectroscopydepth,58,59 and also confirmed by electron spectros-

1 November 2002 � Vol. 41, No. 31 � APPLIED OPTICS 6709

copy for chemical analysis.24 The density of theamorphous or microcrystalline thin films deposited attemperatures of 323–573 K is in the range 5.3–6.6g�cm3, and the density of the polycrystalline filmsdeposited at 673–773 K is in the range 6.6–6.4g�cm3.60 These values are much smaller than thatof 7.3 g�cm3 for a crystalline bulk sample of WO3.17

The main reason for the low values of mass ratioand density is the substantial porosity of these oxidefilms, and the porous nature of the films is explainedas being due to the presence of considerable struc-tural defects such as oxygen ion vacancies, grainboundaries, and other impurities. It should benoted that, even after being annealed in air at 623 Kfor approximately 2–3 h and being transformed into apolycrystalline state, WO3 thin films were still foundto be substoichiometric and to have some degree ofmicroporosity.2,61 Similarly, fully stoichiometricamorphous �or microcrystalline� WO3 thin films �pre-pared in an oxygen environment� are not free fromporosity until physically bound moisture is not ex-pelled from the films after intensive heat treat-ment.2,62 Nonetheless, the reported evaporatedfilms are designated WO3 in what follows.

B. Cell Configuration and Determination of Guest Atoms’x Values

An EC cell was fabricated with a transition-metaloxide, WO3, as a counter electrode and with a pro-pylene carbonate solution of 1-M LiClO4 as an elec-trolyte. The electrochromic cell has the followingconfiguration:

C1, indium tin oxide, MxWO3 bronze �LiClO4�Pt, C2,

where C1 and C2 are the electrodes and Pt is theplatinum wire that is used in the EC cell. A Keithly616 electrometer was used as a constant current�10�5-A� source, and the potential difference betweenthe two electrodes determined with a 1-mol LiClO4solution and a platinum counter was continuouslyrecorded on a chart recorder. Full details of the ap-paratus are given elsewhere.36,51

What we want to know is the x value within thegrains, as the guest atom is taken to reside in thegrains and not along the grain boundaries of micro-crystalline WO3 thin film. This notion is consistentwith the expected ratio of �interior� surface-to-grainbinding energy for a guest atom. It is in accord withthe diffusion behavior, which is rapid in the grain-boundary manifold and slow in the grains. Thevalue of x within the grains �mean grain size, �5 nm�has to be calculated on the basis of 1.0 � 1022 H� �orLi�� ions cm�3 being equivalent to x � 1 when thecoulombmetric titration technique is used. The ex-act amount, x, that was extracted was determinedfrom the numerical model described below.

Determination of concentration x is based on theelectrochemically reversible formation of a tungsten

bronze thin film according to double injection of pos-itive ions and electrons, namely,

�colorless�WO3 � xM� � xe�3MxWO3 �blue color�,

(1)

where 0 � x � 1 and M �H, Li, Na, etc.� is the guestatom inserted into WO3.

Concentration x is defined as the ratio of the num-ber of moles of guest atom to the number of moles ofWO3. If we use the experimentally determined filmdensity f, what we can determine from electrochem-ical extraction is an average x value, �x�, definedas36,52,54

�x� �NT MAf Vf

, (2)

where NT is the total number of guest atoms presentin a volume Vf of a WO3 thin film of density f, M isthe molecular weight, and A is Avogadro’s constant.

If we assume that the density of the WO3 thin filmis the same as the single-crystal density and supposethat the injected amount of hydrogen �or lithium� ina film, at equilibrium, is in the grain as a result of thefilm’s high binding energy, the x value in the grainwill then be given by

x �NT MAs VG

, (3)

where VG is the volume of the film that is grain, asdistinct from the volume ascribed to the grain-boundary regions �VGB�, and s is the WO3 single-crystal density �7.3 g�cm3�.

The total charge removed from the known volumeVf of the film is given by

QT � Itf, (4)

where I is the constant current flowing through thecell and tf is the time for extraction until the end ofthe bleaching process.

Inasmuch as every guest atom removed is singlyionized, the total number of guest ions extracted is

NT � QT�e, (5)

where e is the electronic charge.For the same value of NT, Eqs. �2� and �3� can be

combined as

x � �x�f Vf

s VG(6)

because

Vf � VG � VGB, (7)

i.e., the volume of the film equals the volume of thegrain and the volume of grain boundary. So Eq. �6�can be rewritten as

x � �x�f

s�1 �

VGB

VG� . (8)


It is clear that x depends, apart from the density ofthe film, also on the ratio VGB�VG. Let the meangrain size be L and the thickness of the grain bound-ary be l �where l �� L�; then VG can be expressed as

VG L3,

VGB L2l.

Combining both leads to

VGB

VG�

constant lL

or

VGB

VG�

�

L. (9)

So Eq. �8� can be rewritten as

x � �x�f

s�1 �

�

L� , (10)

where � is the product of grain boundary thickness land some geometrical factor, that depends on theshape of the grains. The value of � was found to be�44 Å 3 Å from the thermodynamic data.54,62 L,the mean grain size of WO3, was found by electrondiffraction studies to be 50 Å 10 Å.36,53

From Eqs. �2�, �4�, �5�, and �10�, the expression forx can be reshaped as

x �Itf M

eVf As�1 �

�

L� . (11)

Using the following values:

I � 10�5 A�cm2,

M � 144 g�mol,

e � 1.6 � 10�19 Coulomb,

A � 6.023 � 1023,

s � 7.3 g�cm3,

� � 44 Å � 3 Å,

L � 50 Å � 10 Å,

and, substituting all these values back into Eq. �11�,we can finally obtain the numerical expression as

x � 6.19 � 10�15 �cm3�s�� tf

Vf� �s�cm3�

or

x � 6.19 � 10�15�tf�Vf�, (12)

where tf is the time for extraction until the end ofbleaching process and Vf indicates the extracted vol-ume of the thin film. Equation �12� was our numer-ical model, which was successfully applied even tohigh values of hydrogen �gas� or lithium �metal� ionsinserted into WO3 thin films.

The values of tf and Vf were both determinedexperimentally for each film under study. The vol-ume of the extracted part of a film required a de-termination of thickness and area, which weremeasured, respectively, by a Taylor–Hobson Taly-surf 4 profilometer and an optical microscope to aprecision of greater than 2%. Taking into accountthe uncertainties in the measurements of extractedvolume and also in the values of � and L yields arelative accuracy of 4%. The overall absolute ac-curacy in the x values with Eq. �12� is correct towithin 10%.

3. Optical Measurements

In an optical experiment it is the transmitted �trans-mittance, T� and the reflected �reflectance, R� inten-sities that are the measured quantities. The ratio ofreflected intensity to incident intensity is called re-flectance, which corresponds to a real Fresnel reflec-tion coefficient. Measuring reflectance at normalincidence is also called reflectivity. Inasmuch as theplane of incidence is undefined for normal incidence,reflectivity �or reflectance� is independent of polariza-tion.

The optical properties of the tungsten bronze thinfilms were determined from the optical transmittance�T� and reflectance �R�, where T and R were bothmeasured �at room temperature� with a Varian 2300double-beam spectrophotometer set with unpolarizedlight near normal incidence �approximately 5° to 10°off normal� over the photon energy range 0.4–4.2 eV.In the transmission mode, silica plates covered withfilms were placed in the path of the sample beam andthe blank silica plates were used in the referencebeam to eliminate the silica absorption cutoff andalso to compensate for any silica impurities. In thereflection mode, a specular reflectance accessory thathad a calibrated front-surface aluminum mirror inthe reference beam was used for recording reflectionspectra. A schematic ray diagram of the spectropho-tometer is shown in Fig. 1.

The observed values of the transmittance �Tobs�and the reflectance �Robs� were recorded by the spec-trophotometer and were later refined to yield trueoptical data. Refinement was required becausethe optical spectra were modulated by interferenceand other various reflection effects. First, I re-moved the effects of the interference by taking thearithmetic mean between successive maxima andminima and drawing a smooth curve through thesepoints. The results gave averaged transmittance�Tobs� and reflectance �Robs� data. These data stillcontained reflection effects. To correct for reflec-tion and to obtain the true final values for trans-mittance and reflectance I adopted the proceduredescribed in what follows. The final quantities,which are the transmittance �Tf � of the thin filmand the reflectance �Rf � from the surface of the thin


film, can generally be expressed by the equa-tions63,64

Tf ��Tobs��1 � Rs�Af��

1 � Rs��1 � �Tobs��(13)

Rf � �Robs� �Tf

2Rs�

1 � Rf�Rs�, (14)

where Rs� and Rf� are the reflectances at the air–silicaand the film–silica interfaces, respectively, and Af� isthe energy absorbed by the film.

For slightly absorbing material, Rs�Af� � 0 in Eq.�13�; also, for silica,

Rs� ��ns � 1�2

�ns � 1�2 � 0.035. (15)

Again, taking Rf� nearly equal to �Robs� yields as thedenominator in Eq. �14�

1 � Rf�Rs� � 1. (16)

After taking all the aforementioned factors into ac-count, we can transform Eqs. �13� and �14� decisivelyinto

Tf � 0.96�Tobs�, (17)

Rf � �Robs� � 0.035Tf2. (18)

Expressions �17� and �18� are the final numericalequations used for refining our optical data. Thetotal error in Tf is �5%, and the maximum error in Rfis �8%.

4. Experimental Results

A. Reflectivity Data and Analysis

The refined reflectivity plots for HxWO3 and LixWO3thin films over the energy range 0.4–4.2 eV areshown in Figs. 2 and 3, respectively. The variationsin reflectance are 8–30% for the reported x values.The broad minimum in reflectance data that occursnear 1.8–2.0 eV causes a broad peak and a broadshoulder in blue bands.36,51,52 This broad minimumin the reflectance appears to be caused �by the ab-sorption of light from the specular beam� by the ex-citation of bound electrons at the film surface.

There are also some other bumps found in some ofthe reported reflectivity data; for example, the bumpin the reflectance spectrum at 3.6 eV is found for thevalues of x � 0.06 and x � 0.10 for LixWO3 bronzes.This bump, which is absent from the other reportedbronzes, is, in fact, related to the history of the prep-aration of these individual films, so the intensity ofimpurity scattering associated with the structural de-fects and the degree of inhomogeneity at the surfacesof these films could play important roles in this dis-tinct bump.

Theoretical analysis65–67 suggests that the domi-nant scattering mechanism in microcrystalline filmsis that of dislocations and is least affected by theinsertion process. So, only random crystallographicshear planes �or dislocations� can exist, which will bea major scattering source responsible for the lowervalues of reflectivity. Also, as a result of dislocationsand substoichiometry �which is difficult to avoid, es-pecially in vacuum-deposited metal oxide films�, theWO3 system adjusts itself to form edge- and evenface-shared WO6 octahedra rather than only corner-shared octahedra �as occurs for stoichiometricWO3�.68,69 In single crystals �or polycrystallinefilms� of MxWO3 �M � H, Li, Na, etc.� there is bothmonopolar dislocation scattering and ionized impu-rity scattering, but ionized impurity scattering dom-inates.65,66

The higher values of reflectivity of the crystalline�fully stoichiometric or substoichiometric� WO3 filmscan be attributed only to an increase in free-electronconcentration on insertion of hydrogen or lithium.For example, the protonated state reflectivity for apolycrystalline film �prepared by rf sputter deposi-tion� exceeds 60% at 2.5 �m wavelength,70 which isconsiderably higher than the value of approximately35% reported for the protonated71 or lithiated72 state

Fig. 1. Schematic ray diagrams for the transmission process: �a�Silica plate covered with film and �b� reference base silica plate.�c� Specular reflection in the final stage, where Io is an incidentlight intensity and Iobs denotes the observed light intensity.


reflectivity for crystallized amorphous films. More-over, WO3 polycrystalline films prepared by rf sput-tering with better stoichiometry �less oxygendeficiency� exhibit reflectivity of 69% at � � 2.5 �mfor LixWO3 �x � 0.5�.6 When WO3 is compared witha reflectance �at the same wavelength� of approxi-mately 82% obtained for a single crystal ofNa0.52WO3,38,65 one might expect that, by further im-provement in stoichiometry �i.e., by attenuation ofthe extended defects and an increase in crystal peri-odicity�, polycrystalline films of WO3 could havehigher lithiated-state reflectivity.6,70 Goldner etal.70 and Svensson and Granqvist73 have describedthe reflective behavior in crystalline HxWO3 or crys-talline LixWO3 in terms of a Drude model modified toaccount for �low enough� free-electron scattering toexhibit relatively high near-IR reflectance modula-tion.

One can view the increase of x in MxWO3 in termsof electrons donated to a matrix conduction bandformed by the overlap of tungsten dt2g orbitals withoxygen p� orbitals74,75 �the ReO3 type �* band ascalculated by Mattheiss76�, leading to delocalizedelectron behavior at x � 0.25.77–79 Moreover, in thishigher range of concentration the bronzes behave ascompletely degenerate semiconductors or metals andshow high electronic conductivities, metallic luster,and also temperature-independent paramagnet-ism.78,80

Now consider the values of x � 0.25; in this rangethe crystalline tungsten bronzes act almost as semi-

conductors.79,81,82 Similarly, microcrystalline �nano-crystalline or amorphous� tungsten bronzes are alsoin the semiconducting mode72,83 for x � 0.32, wheremost of the electrons are localized and the interval-ence band transitions cannot be observed. So, inthis situation, certainly a decided trend away fromthe free-electron behavior should be taken into ac-count.1,82,84

As a matter of fact, the transmittance spectra ofMxWO3 �M � H, Li, Na; x � 0.25� bronzes that havea microcrystalline structure show a typical behaviorof a bell-shaped spectral absorption band with abroader but well-defined absorption peak at 1.2–1.3eV,36,51,52 whereas the reflectivity data of the samebronzes show no phenomenal change in the reflectiv-ity. These results also agree with those of other re-searchers.24,59 The magnitude of the reflectivityover the reported concentration range indicates thenotion that Drude-like processes are almost absent inthese very fine-grained films.

The analysis outlined above supports the fact thatthe charge carriers in amorphous or microcrystallineWO3 or MoO3 thin films �of mean grain size 3–20 nm�remain in bound �or highly localized� states. Conse-quently the reported tungsten bronzes cannot exhibithigh reflectivity or much variation in reflectivity.Instead, they exhibit variable absorptivity and giverise to absorption bands in the near IR.36,51,52

Based on the x-ray photoelectron spectroscopy re-sults, as-deposited WO3 thin films include mainlyW6� and W4� states,85,86 and the number of W4�

Fig. 2. Reflectivity versus photon energy h� of HxWO3 bronze thin films: x � 0.0, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.10, 0.11, 0.13, 0.14.


states increases with increasing oxygen deficiency ofthe WO3 samples. This conclusion has been con-firmed by Gerard et al. with x-ray photoelectron spec-troscopy measurements.87 The process of insertionof lithium or hydrogen monovalent cations �H�, Li��into interstitial lattice sites results first in reducingsome W6� states to W5� states while leaving the W4�

states unaffected. Zhang et al.88 also supported thisnotion, suggesting that some new W5� states are al-ways transformed from the W6� and W4� states dur-ing coloration. Again, from the x-ray photoelectronspectroscopy22,86 and Raman spectroscopic studies89

it is also clear that after EC coloration the number ofW5� states increases with increasing injected charge�hydrogen, lithium, sodium, etc.�.

From the perspective of polaron theory, the boundelectrons can hop from a W5� site to a neighboringW6� by absorption of photon energy, and this processleads to a broad absorption band or to a broaderminimum in reflectivity. This absorptive modula-tion has been attributed to an intervalence transferor to small polaron transitions between the W6� andW4� states and the W5� states.85,86,89 So, the vari-able absorptivity36,51,52 or broadening in the reportedreflectivity minimum has been assigned to polaronichopping transitions between W5� and W6� �or W4��states.85,86,89

When amorphous WO3 is crystallized by heating inair at 773 K for 2 min,19,23 or deposited in a polycrys-

talline form, its absorption peak shifts to 0.95 eV.The coloration density is not so great for this film asfor the amorphous �or microcrystalline� film, and sothis film’s absorption is not so broad, e.g., 1.2 eVinstead of 2.0 eV for its full width at half-maximum.19,23 On crystallization of a WO3 thinfilm, the degree of delocalization of the electrons inthe polycrystalline layer increases, which shifts theabsorption band to lower energies and increases theoscillator strength of the polaron transition �or theoptical intervalence transition� with the polaroncharge transfer.61,90 The oscillator strength is animportant quantity, which is directly related to theabsorption, and its value in both crystalline andamorphous WO3 thin films depends on film prepara-tion and properties.19,20,72 For example, the optical�or polaron� transition for hydrogen, lithium, or so-dium insertion into evaporated WO3 thin films givesan oscillator strength of 0.1.1,72 This value of oscil-lator strength differs from the value 0.2 for sputteredfilms45,61 as well as from the value 0.7 obtained fromthe electron spin resonance data.91 In another in-vestigation the oscillator strength of an optical tran-sition for a color center in a polycrystalline WO3 filmyielded an oscillator strength of 0.35.92 An ex-tremely large effective oscillator strength of 0.51 wasalso obtained for the film deposited by rf reactivemagnetron sputtering at 1000 W.61 Again, thisvalue is five times greater than that of a conventional

Fig. 3. Reflectivity versus photon energy h� of LixWO3 bronze thin films: x � 0.04, 0.06, 0.07, 0.08, 0.10, 0.13, 0.15, 0.16, 0.20, 0.23,0.25.


evaporated WO3 film and 1 order of magnitudegreater than that of the evaporated film annealed at473 K.61 These results suggest that the enhance-ment in oscillator strength is caused by the increasein the degree of extension of the polaron wave func-tion or by some change of the lattice condition in theWO3 thin films.20,90,93 In other words, the oscillatorstrength increases with either increasing crystallin-ity and substiochiometry or from improvement of thestoichiometry and the crystalinity of the thin films atthe same time. However, the oscillator strength fora fully allowed electric-dipole transition involving asingle electron in any kind of absorptive modulationcan have a maximum value of unity.94

B. Real Part of the Refractive Index �n�

The real part of index of refraction n is related towavelength �, angle of incidence �, and the thicknessof the film d by the well-known Fresnel relation63,95

d �12

m�

�n2 � sin2 ��1�2 , (19)

where m is an integer �the order of interference�.The spectral variation of the refractive index can bedetermined by use of the interference method95–97 toverify the reported reflectivity data where fringes canbe easily counted or estimated. The interference os-cillations produced at wavelengths �max and �min oftwo adjacent maxima or minima in the reflectivityspectra can, to a first approximation near normalincidence, be expressed �by splitting of Eq. �19�� intothe following two relations97,98:

d �m2n

�max, (20)

d ��m � 1�2�

2n�min, (21)

where n and d represent the index of refraction andthe thickness of the sample, respectively, and m is theorder of the interference. To avoid uncertainty,measurements were made over to sharp fringeswhere interference order number m could be countedaccurately or where this number could be estimatedwith good precision by insertion of an average valueof the refractive index of WO3 � 2.0 as well as thethickness of the film �d in the above interferencerelations�. The real part of refractive index n wasthen calculated at the respective wavelength extrema�maxima or minima� from the reflectivity spectra.

However, the order of the interference extrema isnot determinable precisely �or is not estimated prop-erly� over the whole range of the reflectivity spectra.When it becomes difficult to evaluate the interferenceorder �m�, then �especially in the visible and thenear-IR regions� the reflectivity values can be used todetermine the values of n at any two wavelengths ��1,

�2� of adjacent maxima or minima by use of the rela-tion96,98

2nd � ��2�1 � �1

�1��1, (22)

where

n � n� � �dnd�

(23)

is the apparent index with the value n � n�, espe-cially near the absorption edge where usually dn�d�is negative. From this point of view it seems that nrather than n� might have been measured with Eq.�22�. In fact, with this technique, only a minor dif-ference was found in the reported values of n com-pared with the values of n�.

Again, when R has only a few �or no� extrema insome parts of the spectra, the practical method ofusing the reflectance maxima or minima does nothelp in estimation of the spectral values of n, specif-ically, in the near-IR region. In this situation, how-ever, both parameters, order m and wavelength �,were eliminated, and the values of n were then cal-culated from the following equation96,98,99:

1 � R1 � R

�2n

1 � n2 . (24)

The values of n �including every aforementioned case�were calculated with an efficient iterative FORTRAN

program, and, from the results, the plots of n versus� of the reported tungsten bronzes are shown in Figs.4–7.

Refractive-index errors, over the wavelength range�UV–near-visible� where the interference fringe tech-nique was used, were 10%. The maximum error inthe refractive index, n, which was due to small un-certainties in the extrema positions ��1, �2� and wasfound by use of Eqs. �22� and �23�, was 8%. Itshould also be noted that R is nearly insensitive tosmall changes in k in the near-IR region of weak

Fig. 4. Real part of refractive index n versus wavelength � ofHxWO3 bronze thin films: x � 0.0, 0.04, 0.05, 0.06, 0.07, and 0.08.


absorption, so the error in n while Eq. �24� was usedwas not more than 5%.

As can be observed from the reported data, there isa broader minimum in the reflectivity as well as inthe refractive-index data over the reported x values,suggesting the presence of two or more non-Drude aswell as feeble Drude components in the reflectivity orrefractive-index data. Full details of this possibilitystill remain obscure. Nevertheless, the broad fea-tures of the experimental curves shown in Figs. 4–7are as follows: �i� in the wavelength range 0.3–0.7�m, i.e., from the UV to the visible spectral range,there is a strong dependence of refractive index n onwavelength �; i.e., n values decrease rapidly withincreasing � for most of the films, and this trend isconsistent with the classic dispersion; �ii� each curvedisplays a minimum �through the visible and in thenear-IR region� with a value of �2.0; �iii� there is arise again after the main minimum, but there is nosharp or broader maximum found in the n � � data;

�iv� the value of n decreases again after wavelength1.3 �m for most of the reported tungsten bronzes.

It is clear from Fig. 4 that in the range of � from 0.3to 2.5 �m the value of n for WO3 obtained is 1.9–2.3.This is in good agreement with those given in theliterature, i.e., 1.9–2.08 for WO3 thin films,17,40,60,100

but less than that of single-crystal WO3, in whichcase the value of n is 2.5.17,101 The real part of therefractive index is low compared with the valuesgiven for bulk WO3. This low value is attributable tothe porous nature of the films. The k value for aWO3 thin film is nearly zero from 0.7 to 2.5 �m andincreases toward the UV range but never exceeds 0.3.This finding is also in agreement with the findings ofother researchers.17,100,102 In addition, the values ofn as calculated with the spectrophotometric tech-nique have been compared with the values as deter-mined by other techniques. For example, index ofrefraction n for a WO3 thin film is determined to be�2.08 near 0.633-�m wavelength, which is close tothe value as determined by ellipsometry.103 It isalso within the range 2.05–2.3, as found by otherresearchers.40,60,104

The real part n of the complex refractive index ofthe reported tungsten bronzes as shown in Figs. 4–7does indicate that the dispersion is strong only in theshort-wavelength region, i.e., from 0.3 to 0.7 �m. Inthe range from 0.7 to 1.3 �m the refractive indexincreases again with the wavelength, and then it de-creases as � goes to 2.5 �m. The dispersion coeffi-cient, dn�d�, for the tungsten bronzes withincreasing wavelength from 300 to 700 nm variesfrom 1.6 � 10�3 to 2.7 � 10�3 nm�1. This high-absorption and high-dispersion region indicates theonset of electronic transitions through the energy gapof WO3 thin film.

All the profiles described above are clearly thecharacteristics of classic dispersion associated withbound electrons. The above analysis also indicatesthat the concentration of bound electrons is dominant

Fig. 5. Real part of refractive index n versus wavelength � ofHxWO3 bronze thin films: x � 0.09, 0.10, 0.11, 0.13, 0.14.

Fig. 6. Real part of refractive index n versus wavelength � ofLixWO3 bronze thin films: x � 0.04, 0.06, 0.07, 0.08, 0.10, 0.13.

Fig. 7. Real part of refractive index n versus wavelength � ofLixWO3 bronze thin films: x � 0.15, 0.16, 0.20, 0.23, 0.25.


in the strong dispersive region of the spectral rangestudied.

The experimental refractive-index data on microc-rystalline MxWO3 bronzes could lead to the conclu-sion that, whenever the WO3 thin film is in thecolored state and has a broader minimum in therefractive-index data in a narrow wavelength inter-val, then, at that stage, the dopant-dependent pol-aron transition between the W4� �or W6�� and W5�

species always reaches a maximum value �with theoscillator strength not more than 0.1 for the evapo-rated WO3 thin films�,1,72 particularly in that narrow-wavelength interval. Moreover, the shift in thebroader minimum in the n representation at differentlevels of coloration matches the observed shift of thepeak of the polaron absorption band of the microcrys-talline tungsten oxide bronzes36,51,52 toward the bluewith increasing x value. This kind of interpretationis consistent with the model proposed by Zhang etal.88 and also accords with the results of other re-search teams.39,44,105 So, with the polaron concept,the broader minimum in refractive index n and itsshift toward higher photon energy as x increases canbe ascribed to a hopping transition of small polaronsfrom one set of levels of localized W5� states to asecond set of levels of localized W6� �or W4�� stateswithin the energy gap of the host material, WO3.

When the film is amorphous or microcrystalline,the polarons are tightly localized at the W site as aresult of Anderson localization.87,106,107 With the de-velopment of film crystallinity, a delocalization of thepolaron wave function can occur, and the oscillatorstrength of the charge involved can be enhanced toany value less than 1.

5. Discussion

The minimum in reflectivity as a function of concen-tration x, as shown in Figs. 2 and 3, is a direct con-sequence of the increase in optical absorption andoptical density and is greatly associated with boundelectrons. The reflective behavior of a crystallinesemiconductor �fully stoichiometric or substoichio-metric� is attributed to an increase in free-electronconcentration following insertion of H or Li, but onecannot apply the free-electron scattering theory tointerpret the reflectivity data associated with the mi-crocrystalline tungsten bronzes. Instead, theDrude–Zener model is developed in Subsection 5.Afor analysis of the reflectivity data and also to unravelthe nature of the charge carriers involved. With aview to technical applications, the dopant-dependentrefractive-index data are interpreted in Subsection5.B by use of the single-effective-oscillator model.

A. Modified Drude–Zener model

The total dielectric constant of Mx WO3 �M � H�, Li�,Na�� microcrystalline bronze thin films can be ex-pressed by the Drude-Zener model108,109:

εtotal � 1 �ne2x

4�2εo me*�p2 � εhf, (25)

where n is the number of conduction electrons in aunit cell; x is the concentration of guest atoms; me* isthe effective electronic mass; �p is the screenedplasma frequency �not the real metallic plasma fre-quency�, and εhf is the high-frequency dielectric con-stant. Near the minimum reflectivity εtotal � 0, sorelation �25� can be transformed into

�p2 � Zx, (26)

where

Z �e2

4�εo

n�me*�1 � εhf�

. (27)

Using the equivalent of x � 1 to the number of con-duction electrons n � 1.1 � 1028 m�3 and also sub-stituting e2�4�εo � 2.3 � 10�28 �kg m�3 s�2�transforms Eq. �27� into

Z � 0.884 � 1030� 1�*�1 � εhf�

��s�2�, (28)

where �* � me*�me.From Eqs. �26� and �28� we have

�p � 0.94 � 1015� x�*�1 � εhf�

�1�2

�Hz�, (29)

which yields

�p � 31.4 � 103� x�*�1 � εhf�

�1�2

�cm�1�. (30)

Equation �30� gives a modified form of the Drude–Zener reflectivity model based on quasi-free-electrontheory. Before we calculate the values of screenedplasma frequency �p over the reported x range, theexperimental values of εhf and �* were determined byuse of the following steps:

In the first step, I determined the values ofscreened plasma energy h�p from the reported data,using the well-known optical relation110

h�p � h�rmin�εhf � 1εhf

�1�2

, (31)

where h�rmin is the minimum reflectivity photon en-ergy.

Experimental values of h�rmin were directly deter-mined from Figs. 2 and 3. The values of high-frequency dielectric constant εhf were obtained fromthe plots of �n2 � 1��1 versus �h��2 shown in Figs.8–11.

In the second step, I obtained the expression forelectron mass ratio, �*, by reshaping Eq. �29� as

h2�p2 � 15.13� x

�*�1 � εhf�� eV�2,

or

�* � 15.13� xh2�p

2�1 � εhf�� . (32)


The values of electron mass ratio �* were then cal-culated from Eq. 32. Finally, the values of �p werethen generated from Eq. �30�.

All the optical parameters obtained are listed inTables 1 and 2, whereas Fig. 12 shows �p versus x1�2

for the HxWO3 and LixWO3 bronzes.Let us compare this numerical analysis with the

other findings in the literature with regard to crys-talline and microcrystalline tungsten bronzes. Incrystalline tungsten bronzes, different samples �withdifferent concentrations of hydrogen or lithium� ex-hibit different levels of reflective modulation over abroad wavelength range in the near IR and at highhydrogen or lithium insertion levels �x � 0.5� in thevisible as well.70,111,112 These samples, which havedifferent plasma frequencies, also reflect differentcrystallographic phases and lattice structures andsubsequently show higher effective masses.66,70,73,112

Furthermore, the absorption coefficient for crystal-line HxWO3 �or LixWO3� always increases linearly

with x1�2,65,67,70 suggesting that the absorptive mod-ulation also arises from free-carrier scattering.Thus both the reflective and the absorptive modula-tions associated with the crystallized bronzes are con-trolled mainly by free carriers rather than by boundcarriers or small polarons.65,67,70,112

In the reported reflectivity data �Figs. 2 and 3�,however, there is no sharp minimum and also there isno indication of a steep rise after this minimum.The frequency plots of �p versus x1�2 are not found tobe linear, indicating that free-carrier scattering doesnot dominate the reflective and the absorptive behav-ior of reported microcrystalline MxWO3 bronzes.The high-frequency dielectric constant for crystallinetungsten oxide film is �4.8,17,113 whereas the �aver-age� high-frequency dielectric constant for the micro-crystalline tungsten oxide thin film is found near4.04, which is obviously less than its crystalline coun-

Fig. 8. Plot of �n2 � 1��1 versus �h��2 of HxWO3 bronze thin films:x � 0.04, 0.05, 0.06, 0.07, 0.08. Dotted curves, least-squares fits.

Fig. 9. Plot of �n2 � 1��1 versus �h��2 of HxWO3 bronze thin films:x � 0.09, 0.10, 0.11, 0.13, 0.14. Dotted curves, least-squares fits.

Fig. 10. Plot of �n2 � 1��1 versus �h��2 of LixWO3 bronze thinfilms: x � 0.04, 0.06, 0.07, 0.08, 0.10, 0.13. Dotted curves, least-squares fits.

Fig. 11. Plot of �n2 � 1��1 versus �h��2 of LixWO3 bronze thinfilms: x � 0.15, 0.16, 0.20, 0.23, 0.25. Dotted curves, least-squares fits.


terpart. Moreover, the variations found in mass ra-tio �* lie from 0.03 to 0.10 �Tables 1 and 2�, which areobviously lower than the density-of-states effectivemass �of value approximately 1–3 me� as found forcrystalline NaxWO3 by magnetic susceptibility,114

transport, and specific heat.115 The investigatedvalues of the effective masses are also not comparableto an optical mass of 0.86 for ReO3 or to 0.5 forNa0.5WO3,108 as found by optical measurements.The above analysis demonstrates clearly that the ef-fective electronic masses of the reported tungstenbronzes are admittedly of bound electrons, and mostlikely the sources of scattering associated with thereported tungsten bronzes are only the extended de-fects such as crystallographic shear planes and ion-ized impurities.117,118

So, based on an electron–phonon interaction, weinterpret broader reflectivity minima and the varioushumps in the reported reflectivity data as the signa-ture of trapped electrons �polarons, and more pre-cisely small polarons� at specific W6� and W4� sites,and different coloration in the bronzes is then formedwith increasing numbers of W5� states within thetungsten oxide matrices. Following the polaronmodel, the reflective behavior of microcrystallinetungsten bronzes is related solely to the small pol-aron transition between the W5� and W4� �or W6��states,57,88,119 and the dopant-dependent reflectivityminimum, and its continuous shift toward higher fre-quencies, as x increases, is then just the polaron op-tical transfer from lower-energy �W5�� states tohigher-energy �W6� or W4�� states within the opticalbandgap of WO3.

B. Single-Effective-Oscillator Model

The refractive index is closely related to the electronicpolarizability of ions and to the local field inside thematerial, and therefore its values are interpreted interms of single-oscillator dispersion of the electronicdielectric constant. So, in terms of dispersion en-ergy Ed, single-oscillator energy Ea, and lattice-oscillator energy E1, refractive index n over the rangeof photon energy E can be expressed in the form120–122

n2�E� � 1 � Ed Ea��Ea2 � E2� � �E1�E�2. (33)

However, over the reported energy range, latticeterm E1 can be neglected and Eq. �33� can then berewritten as

n2�E� � 1 �Ea Ed

�Ea2 � E2�

, (34)

which again can equivalently be reshaped as

�n2�� 1��1 � �Ea�Ed� � ��h��2�Ea Ed�, (35)

where dispersion energy Ed is related to the inter-band transition strength and is normally indepen-dent of the absorption threshold, i.e., the bandgap.The value of Ed does not depend on the lattice con-stant but is definitely related to the charge distribu-tion within each unit cell. Furthermore, Ed is also

Fig. 12. Plot of screened plasma frequency �p versus concentra-tion x1�2 of HxWO3 and LixWO3 bronze thin films.

Table 1. Modified Drude–Zener Model Parameters for HxWO3

Thin Filmsa

Film xThickness

��m� εhf

h�rmin

�eV�h�p

�eV� �*�p �cm�1

� 103�

HW1 0.04 0.285 5.42 2.10 1.90 0.026 15.37HW2 0.05 0.29 6.26 1.94 1.78 0.033 14.34HW3 0.06 0.30 7.45 1.75 1.63 0.040 13.23HW4 0.07 0.28 7.25 1.91 1.773 0.041 14.28HW5 0.08 0.28 8.14 1.85 1.733 0.044 14.00HW6 0.09 0.28 5.76 1.82 1.656 0.073 13.41HW7 0.10 0.295 6.49 2.07 1.904 0.056 15.32HW8 0.11 0.29 8.63 2.00 1.88 0.049 15.16HW9 0.13 0.28 7.33 2.25 2.09 0.054 16.88HW10 0.14 0.31 6.88 1.88 1.74 0.089 14.03

ax, value �in HxWO3 grains�, thickness, high-frequency dielectricconstant εhf, minimum reflectivity energy h�rmin, plasma energyh�p; electron mass ratio �*; and screened plasma frequency �p �cf.Eqs. �30�–�32��.

Table 2. Modified Drude–Zener Model Parameters for LixWO3

Thin Filmsa

Film xThickness

��m� εhf

h�rmin

�eV�h�p

�eV� �*�p �cm�1

� 103�

LW1 0.04 0.42 7.90 1.61 1.510 0.030 12.15LW2 0.06 0.325 5.52 1.72 1.502 0.062 12.10LW3 0.07 0.29 6.62 1.70 1.570 0.056 12.72LW4 0.08 0.32 6.90 1.80 1.660 0.055 13.47LW5 0.10 0.27 6.55 1.85 1.703 0.069 13.76LW6 0.13 0.25 13.19 1.89 1.830 0.041 14.84LW7 0.15 0.29 9.20 1.90 1.701 0.077 13.72LW8 0.16 0.27 10.80 1.88 1.784 0.064 14.45LW9 0.20 0.25 11.00 1.91 1.820 0.076 14.70LW10 0.23 0.285 9.33 1.94 1.830 0.100 14.82LW11 0.25 0.30 12.11 2.00 1.920 0.078 15.53

ax value �in LixWO3 grains�, thickness, high-frequency dielectricconstant εhf, minimum reflectivity energy h�rmin, plasma energyh�p, electron mass ratio �*; screened plasma frequency, �p �cf. Eqs.�30�–�32��.


found to be closely related to the chemical bonding.Single-oscillator energy Ea, however, is related to thebandgap and also depends on the lattice constant.

When the refractive-index data were fitted to Eq.�35�, the resultant plots of �n2 � 1��1 versus �h��2

with least-squares fitting were as shown in Figs.8–11. Intercepts Ea�Ed and slopes �1�EaEd weredetermined, and the required parameters, Ea and Ed,were evaluated. These are listed in Tables 3 and 4for HxWO3 and LixWO3 bronzes, respectively. Themaximum error in Ea and Ed values �including un-certainty in film-thickness measurement, concentra-tion determination, and analysis of the reflectancedata� is estimated to be of the order of 9%.

In addition, values of the high-frequency dielectricconstants εhf for the tungsten bronzes were also ob-tained from the intercepts of the plots of �n2 � 1��1 at�h��2 � 0. These are also listed in Tables 1 and 2.The value of high-frequency dielectric constant εhfobtained was found to be �4.08 for microcrystallineWO3, which is close to the value as reported byDeb.17,113

It can easily be observed from Figs. 8–11 that thereare no abrupt changes in the slopes of the plots of�n2 � 1��1 versus �h��2, so there is no indication of

x-dependent crystallographic phases with respect tothe reported tungsten bronzes. However, thechanges in the slopes do show some surface modifi-cations, depending upon the amount of colorationdensity.

With reference to crystalline WO3, the value of Edis equal to 20.74 eV,123 but, for the reported microcrystalline WO3, the value of Ed �Table 3� was foundto be equal to 18.02 eV. The values of Ed for thetungsten bronzes �Tables 3 and 4� were also found tolie in the range 9.01–13.51 eV. This finding doessupport the fact that the samples investigated aredeficient in density and therefore have smaller chem-ical bondings than crystalline MxWO3. Further-more, the values of Ea �Tables 3 and 4� were found tobe in the range from 1.08 to 5.06 eV, a bit higher thanthose of crystalline materials,123 indicating biggerbandgaps associated with the microcrystalline tung-sten bronzes than with their crystalline counterparts.

6. Summary

Reflectivity spectra of HxWO3 and LixWO3 thin filmsin the concentration range from x � 0.0 to x � 0.25were measured for photon energies from 0.4 to 4.2 eV.It was found that, after the reflectivity data are re-fined, microcrystalline tungsten bronzes have reflec-tances of 8–30%. The accuracy of the reflectancedata is estimated to be 8%.

A detailed set of values for the real part of refrac-tive index n was also determined from the refinedreflectivity data by use of various numerical tech-niques that depend on the nature of the dispersionover various spectral regions in the data. The max-imum uncertainty in calculating the values of n isless than 10% over the reported spectral range.

The refractive-index data of the WO3 thin film in-dicate that dispersion is large in the short-wavelength region, i.e., from 0.3 to 0.7 �m, butchanges slowly over the range 0.7–2.5 �m. Thevalue of high-frequency dielectric constant εhf wasfound to be 4.08 for the microcrystalline WO3 thinfilm, less than that of crystalline WO3.

For Mx WO3 �M � H�, Li�� bronze thin films, mostof the refractive-index data were found to be highlydispersive in the range 0.3–0.7 �m, but in the range0.7–1.3 �m the refractive index increased again withthe wavelength, and then it decreased as � tended to2.5 �m. All these profiles in the n � � data areclearly characteristics of classical dispersion associ-ated with the bound electrons.

I measured the values of the high-frequency dielec-tric constant εhf of MxWO3 bronzes from therefractive-index data to gauge the effective electronicmasses associated with the optical and with the pol-aron transitions. The calculated values of the elec-tronic effective mass were found to be much lowerthan the values of the density-of-states effectivemass. This again provides a convincing argumentthat the charge carriers involved in the optical tran-sitions are bound carriers.

Using the modified Drude–Zener model, I foundthat for the reported reflectivity data the screened

Table 3. Wemple–Didomenico Single-Oscillator Model Parameters forHxWO3 Thin Filmsa

Film x Thickness ��m� Ea �eV� Ed �eV�

W 0.00 0.31 5.06 18.02HW1 0.04 0.285 2.23 9.04HW2 0.05 0.29 1.71 9.24HW3 0.06 0.30 1.42 9.46HW4 0.07 0.28 1.61 10.16HW5 0.08 0.28 1.43 10.13HW6 0.09 0.28 2.03 9.66HW7 0.10 0.295 1.43 10.16HW8 0.11 0.29 1.42 10.87HW9 0.13 0.28 1.80 11.63HW10 0.14 0.31 1.53 9.01

ax value �in HxWO3 grains�, thickness, single-effective oscillatorenergy Ea, dispersion energy Ed �cf. Eq. �35��.

Table 4. Wemple–Didomenico Single-Oscillator Model Parameters forLixWO3 Thin Filmsa

Film x Thickness ��m� Ea �eV� Ed �eV�

W 0.00 0.31 5.06 18.02LW1 0.04 0.42 1.44 9.90LW2 0.06 0.325 2.47 11.24LW3 0.07 0.29 1.86 10.53LW4 0.08 0.32 1.92 11.36LW5 0.10 0.27 1.93 10.75LW6 0.13 0.25 1.08 13.51LW7 0.15 0.29 1.40 11.68LW8 0.16 0.27 1.21 11.49LW9 0.20 0.25 1.18 10.42LW10 0.23 0.285 1.141 10.98LW11 0.25 0.30 1.53 11.30

ax value �in LixWO3 grains�, thickness, single-effective oscillatorenergy Ea, dispersion energy Ed �cf. Eq. �35��.


plasma frequency �p does not show any linear rela-tion to x1�2 as demanded by free-electron theory.

Using a single-effective-oscillator model, I foundthe values of dispersion energy Ed for the reportedtungsten bronzes to be in the range 9.01–18.02 eV, abit smaller than those of crystalline bronzes. Thevalues of single-oscillator energy Ea were found tovary from 1.08 to 5.06 eV, a bit higher than those ofcrystalline bronzes, indicating bigger bandgaps asso-ciated with the microcrystalline tungsten bronzes.These findings do indicate that the samples investi-gated have smaller density and therefore havesmaller chemical bondings than crystalline Mx WO3bronzes.

All the experimental results and interpretationssupport the ideas that the injected electrons arestrongly localized in optically absorbing bound statesand that the Drude free electrons are almost absentin the reported microcrystalline tungsten bronzes.They imply that the usual electron-scattering theo-ries cannot apply to these bronzes. Instead, the po-laron species �W5� and W4� species� are assumed tocontrol all the reflectivity developments or changes inrefractive index associated with the tungstenbronzes.

Based on an electron–phonon interaction, thesmall reflectivity crests should correspond to the mul-titude of small polaron absorption processes causedby the trapped electrons on tungsten sites, and theobserved shift of the minimum reflectivity �or broaderminimum refractive index� toward higher frequencywith increasing x value in Hx WO3 �or Lix WO3� thinfilms can then be ascribed to a hopping transition ofpolarons from lower-energy localized states �W5�� tohigher-energy localized states �W6� or W4�� withinthe energy gap of the host material, WO3.

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dopant-dependent reflectivity and refractive index of microcrystalline h_xwo_3 and li_xwo_3 bronze...

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