Spectroscopic ellipsometry applied to phase transitions in solids: possibilities and limitations

Spectroscopic ellipsometry applied to phase

transitions in solids: possibilities and limitations

Alexandr Dejneka1, Ilze Aulika

2, Vladimir Trepakov

1, Jaromir Krepelka

3, Lubomir

Jastrabik1, Zdenek Hubicka

1, Anna Lynnyk

1

1 Institute of Physics, Academy of Science, Na Slovance 2, 182 21 Prague 8, Czech Republic

2 Institute of Solid State Physics, University of Latvia, Kengaraga 8, LV-1063, Riga, Latvia

3Joint laboratory of UPOL and FZU AS CR, listopadu 50, 772 07 Olomouc, Czech Republic

[email protected]

Abstract: The possibilities of in situ spectroscopic ellipsometry applied to phase transitions investigation in oxide thin films and crystals are examined in this work, along with the use of various parameters calculated from ellipsometric data (band gap energy Eg, refractive index n and surface roughness) together with the directly measured main ellipsometric angles ψ and ∆, for the detection of phase transitions. The efficiency of spectroscopic ellipsometry on “surface” phase transition and its sensitivity to surface defects are also demonstrated.

2009 Optical Society of America

OCIS codes: 240.0240 (Optics at surfaces); 160.4760 (Optical properties); 160.6840 (Thermo-optical materials); 240.0310 (Thin films); 130.2260 (Ferroelectrics); 240.2130 (Ellipsometry and polarimetry)

References and links

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#111953 - $15.00 USD Received 26 May 2009; revised 9 Jul 2009; accepted 13 Jul 2009; published 31 Jul 2009

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1. Introduction

Significant developments in micro and nano-electronics caused a steep rise in interest in the structure, phase state and properties of the surfaces of solids and films. Such fascinating studies are especially important in nanostructure engineering when interfaces and surfaces are playing increasingly important roles. There are only a few existing studies of phase transitions in thin and especially in ultra-thin films (thickness < 10 nm), and the majority of them are destructive methods that demand very specific precise sample preparation. In this situation, it would be very attractive to have nondestructive optical methods of research and diagnostics

Thermo-optical properties of solids have been studied since the 1970′s. The early interest was focused, for example, on amorphous semiconductors (Si, Ge, GaAs, GaP, As2S2, Te, and Se) [1-4], glasses (also disordered SiOx glass systems, where 0 ≤ x ≤ 2), and complex glass systems (such as Pb, Ca, Ge and Ba glasses) [5-7] due to their possible applications as switching elements, optical communication systems, and insulator materials in integrated circuits.

Deviations from the usual linear increase in the refractive index are observed at high temperatures n(T) (for solids without PT) when a change of the phase state occurs in the solids



with PT. For example, Hofmann et al. [8] reported a drastic deviation from the linear behavior of n(T) in BaTiO3 on approaching the cubic-tetragonal PT. It was shown that n(T) is sensitive to the order parameter of PT in transparent crystals [8, 9]. Later, these phenomena were studied systematically in perovskite-type ferroelectrics; for example, Burns, Scott and Dacol observed deviance in n(T) from the standard behavior at temperatures much higher than the temperature Tc of phase transition in Pb(La0.12Zr0.65Ti0.35)O3 and Pb3MgNb2O9 [10], BaTiO3 [12] and Pb(Mg1/3Nb1/3)O3 and Pb(Zn1/3Nb2/3)O3 [13], which was interpreted as the appearance of local disorder [10] and the existence of polar microregions at temperatures higher than Tc [13]; similar deviations were established by Kleemann on KNbO3, SrTiO3 and KTaO3 [12]. Further investigation of ferroelectrics such as SrTiO3 [14, 15], BaTiO3 [15], and dielectric MgZnO thin films [16] showed even more unusual behavior of the refractive index: a decrease of n with increases in T, which is in contradiction with the conventional elasto-optical and thermo-optic contribution to the optical polarizability [14].

Thermo-optical studies in these earlier works were performed by interferometry and linear birefringence, which is a sufficiently sensitive experimental method but at the same time subject to essential restrictions determined by the technique itself. Conventional optical interferometers generally use precise custom-made bulky optics, which are expensive. These interferometers also suffer from the disadvantages of systematic errors due to aberrations of the lenses and surface imperfections of the mirrors in the interferometer. Two-beam interferometers generally have the disadvantage of instability, which comes from vibration and atmospheric disturbances when the interfering beams have different paths in the interferometer. But the main problem of interferometry is its low sensitivity to very thin films and the surfaces.

One of the most important and broadly investigated problems is the appearance of phase transitions at the surface of crystals and thin films. The shift in the surface phase transition temperature Tc with respect to the bulk Tc was first predicted for magnetic [17] and later for structural PT [18]. Later, multiple experimental studies on magnetic, polar and structural phase transitions have validated this prediction and shown, that Tc at the surface is substantially higher then Tc in the bulk. There are currently relatively few techniques that can be used to study PT near the surface. Experimentally, enhancements of the surface Tc of more than 20 K were observed for Gd crystals using the spin-polarized low-energy-electron diffraction method [19]. There are reports of the detection of the local structural changes near the surface by conventional nuclear magnetic resonance (NMR) and nuclear quadrupole resonance (NQR), but these techniques have notable exceptions [20], which limit the possibilities of these methods. Closely related methods, such as beta-detected NMR (β-NMR) or NQR (β-NQR), can be more sensitive. Increase of the surface Tc of more than 45 K were observed for single crystals of SrTiO3 using β-NQR [21] and optical second harmonic generation (SHG) methods [22]. The shift in Tc may depend on characteristics of surface quality, such as adsorbents, dislocations, defects, and stress. One of the limitations of β-NQR and SHG is that these methods themselves are very complicated, and moreover, they initially impose intricate requirements on the specific preparation of the sample surface, and afterwards require complex analyses of the experimental data. Finally, the results obtained are often contradictory and require further analysis using other experimental methods.

Relatively recent developments in spectroscopic ellipsometry (SE) have enabled us to investigate the complex refractive index of thin films and bulks in the wide spectral range (thus giving access to fundamental physical parameters such as morphology, crystal quality, chemical composition, or electrical conductivity). SE measures the ratio of the change in the polarization of light reflected from the surface of the sample ρ = (rp/rs)⋅e(δp - δs) = tgψe

i∆ of the perpendicular (s) and parallel (p) polarized reflectivity coefficients rp and rs, which can be written in terms of the main ellipsometric angles ψ and ∆ [23], as in Fresnel’s formulae [24]. SE measurements are not hampered by the need to account for the absolute intensity of the incident light, eliminating the disadvantages of other non-destructive and contactless



optical techniques such as dependence on the intensity of the light source (reflectometry), and vibration and atmospheric disturbances (interferometry). Insomuch as SE measures the ration ρ of two values, and ρ is a complex number that contains phase information, the measurements can be highly accurate, reproducible and sensitive [25]. Typical ellipsometer can measure ψ and ∆ with accuracy higher than 0.02° and 0.1° respectively. Due to this high accuracy the change of refractive index of 10-3 - 10-4, and film thickness changes down to the sub-Å scale can easily be resolved with this technique. Therefore even small changes in the refractive index of solids with the temperature T can be easily detected by SE. The high sensitivity of this technique allows measurements of ultra-thin films of only a couple of nanometers thick, including dynamic (e.g., time- or temperature-dependent) measurements during the modification process of a sample (the growth of a thin film, etching or cleaning of a sample). SE studies offer an opportunity to detect phase transitions at the interface and surface in a non-destructive way, which is important for thin film and crystal studies, without requiring the complicated analyzes of experimental data that are required, for example, in SHG and NQR methods.

The goal of this work is to show the efficiency of thermo-ellipsometric studies of various structural phase transitions in solids. The optical constants of a material are connected with its state and structure and can be evaluated with the necessary precision from direct measurements of the main ellipsometric angles ψ and ∆. The use of such calculated parameters as the band gap energy Eg, refractive index n, and surface roughness, along with ψ and ∆, to evaluate phase transitions, is shown in this work. The use of thermo-ellipsometric investigations for phase transitions detection in perovskites is demonstrated on PbZrxTi1-xO3 [26, 27], Ba0.4Sr0.6TiO3 [28], NaNbO3 [29], thin films and SrTiO3 model single crystals [30].

2. Experimental

Optical measurements were performed by means of a J. A. Woollam variable angle spectroscopic ellipsometer operating in rotating analyzer mode. The main ellipsometric angles ψ and ∆ were measured in the spectral range of 250-1200 nm at the incidence angles of 65, 70 and 75 degrees at room temperature. For the high temperature measurements specially designed sample holder with PC-controlled heating element was used. Low temperature measurements were performed with the Janis liquid helium cryostat at one angle of incidence (70°). The thermocouple was mounted at the surface of the sample. Such configuration allowed us to achieve the accuracy less then 0.5 K. The heating/cooling rate was 1-3 K/min. Temperature dependences were measured in two ways: 1) Angles ψ and ∆ were measured as a function of the wavelength at fixed temperatures 2) dynamic scan of ψ and ∆ at several fixed wavelengths was performed. First type of measurements gave us the temperature dependence of n and k dispersion and optical band gap, and the second one allowed us to obtain the temperature dependence of n and k at fixed wavelengths. The fixed wavelengths were chosen experimentally to increase the sensitivity of SE measurements to any phase transitions and structural changes. In the most cases it is enough to take some wavelength near the absorption edge because any structural changes give a considerable contribution to the interband transitions witch can be seen in the refraction and absorption indices changing [31]. Thickness of the films was calculated from ellipsometric data and also measured by profilometer. Simple model was used to fit room temperature and thermo-ellipsometric data: substrate/film with thickness non-uniformity/roughness. The surface roughness was calculated as an effective medium approximation (EMA) [23]. In a case when anisotropy, inhomogeneity and defects of the films and crystals were not taken into account the calculated values are called as “effective”. Experimental data were analyzed with the software package WVASE32.



3. Phase transitions in the refractive index and band gap temperature behavior

It is known that in solids, changes in the structural phase transitions are accompanied by changes in band structure. For the cubic crystals with the perovskite structure, the standard formula describing anomalies in the behavior of the refractive index δn(T) in terms of the temperature T is [12]

,)()2)(2/()( 230 ><+−= TPggnTn

F

ij

F

ij δδ (1)

where <δP2> is the autocorrelation function of the fluctuating polarization, F

ijg are the free

electro-optic coefficients and n0 is the unperturbed cubic refractive index. The existence of the phase transitions and the appearance of the order parameter explain the temperature dependence of the polarization and also of the refractive indexes. On the other hand, in the case of thin films any absorption α value in the steepest part of absorption edge can also be used to detect phase transitions [32]. It was shown in [32] that the band gap Eg of proper FE with phase transition of the first order can be presented as

,222

6420 P

cP

bP

aEE gg +++= (2)

where P is the spontaneous polarization and Eg0 is the band gap in the paraelectric phase. From the analyzes of Eq. (2) the phase transition of the first order is accompanied by the jump in the ∆Eg temperature dependence

,2

2P

aEg ≈∆ (3)

and the phase transition of the second order by the jump in the band gap derivation ∆(dEg/dT)

[32], where phase transitions can be seen both in n(T), and Eg(T) dependences. From the experimentally determined main ellipsometric angles ψ and ∆ as functions of the wavelength and temperature T, the refractive index and Eg can be evaluated. The verification of phase transition detection in n(T) and Eg(T) will be shown on self-polarized Pb1.12Zr0.24Ti0.76O3 and Pb1.005Zr0.25Ti0.75O3 made by two methods, Ba0.4Sr0.6TiO3 and NaNbO3 thin films. In some cases, phase transitions can even be seen directly in ψ and ∆ temperature dependence: an example is given on thin films of Ba0.4Sr0.6TiO3.

4. Experimental results

4.1 PbZrxTi1-xO3 films

Recently, the use of ferroelectric thin films of the PbZrxTi1-xO3 family of functional materials in microsystems technology has drawn remarkable interest [33]. PbZrxTi1-xO3 films provide a direct electro-mechanical coupling and are useful for memory, piezoelectric and pyroelectric devices [34]. PbZrxTi1-xO3 thin films have advantage over bulk materials: they can be directly deposited on platinized silicon to allow direct integration with electronics. Due to the applications of PbZrxTi1-xO3 in pyroelectric devices, studies on PZT stability with changes in temperature are very important.

Most of the PbZrxTi1-xO3 films used in modern electronics are in the ferroelectric state at room temperature and become paraelectric at high temperatures. Some of them exhibit self-polarization, which occurs due to the presence of an internal electric field that is at least as large as the coercive field at the Curie temperature. As a consequence, the film will be polarized whenever it is cooled down below the paraelectric-ferroelectric transition temperature. This property is very useful for pyroelectric applications of PbZrxTi1-xO3 films, since the poling procedure, which always requires the application of a strong electric field at elevated temperatures, can be avoided. This is useful since arrays with large numbers of elements would be hard to pole uniformly by conventional techniques. Therefore, any information about phase transition and other temperature perturbations in self-polarized PbZrxTi1-xO3 thin films is extremely important.



0 100 200 300 400 500

2,2

2,3

2,4

2,5

2,6

2,7

IV

III II

I

Opt

ical

ene

rgy

gap

(eV

) Temperature (C

o )

experiment linear fit

a)

0 100 200 300 400 5002.4

2.5

2.6

2.7

2.8

2.9

2.85

2.90

2.95

3.00

3.05

3.10

3.15

3.20

E03 (

eV

)

E0

3 (

eV

)

T (oC)

E03

SP PZT

E03

CSD PZT

b)

Fig. 1. Temperature dependence of band gap energies E02 and E03 of (a) SP PZT and (b) SP and CSD PZT

Self-polarized perovskite Pb1.12Zr0.24Ti0.76O3 made by chemical solution deposition method (CSD PZT) and Pb1.005Zr0.25Ti0.75O3 made by planar multitarget reactive sputtering technique (SP PZT) were grown on Si/SiO2/adhesion layer/(111)Pt substrates [26, 27]. The thickness of the films were ~1000 nm. Because of the inhomogeneity of self-polarized CSD and SP PZT

films, due to their polarization profile [35], we could calculate only the “effective” absorption coefficient using a simple model (substrate)/(one-layer) [26, 27]. The absorption edge analysis itself appears to be a very complicated task in the case of ferroelectric materials. Therefore the band gap Eg02 was taken at a constant absorption coefficient (α = 100 cm-1). This value corresponds to direct optical transitions for many ferroelectric and is often used for Eg estimation in crystals [36].

The temperature dependence of the optical energy gap E02 for SP PZT film is shown in Figure 1a. The value obtained in a linear approximation ∂E02/∂T =7.3⋅10-4 eV/K for phase I is a typical one for the ferroelectric phase of most popular ferroelectric crystals [37]. Transition to phase II can be classified as a second type phase transition. The temperature coefficient of the optical gap in phase II ∂E02/∂T = - 2.6⋅10-3 eV/K is substantially higher than in phase I, which is evidence of a large tetragonal distortion of the PZT film structure. Phase III shows the nonlinear temperature dependence of E02 with a maximum at the lead melting temperature of 327.4°C. This phase starts from the transition point of PbO2, which decomposes to Pb3O4 at about 290°C. Between phase III and phase IV, a band gap jump of first-type phase transitions is seen. Phase IV is a paraelectric one, and the temperature coefficient of its optical gap is ∂E02/∂T = 9.9⋅10-4 eV/K.



Table 1. The phase transitions temperatures of PZT thin films made by SP and CSD techniques.

Phase transition SP CSD

Phase I → Phase II 200°C 160°C Phase II → Phase III 290°C 280°C Phase III → Phase IV 400°C 375°C

Thus, the SE allows us to obtain three important temperature points for self-polarized CSD

and SP PZT films in the range from room temperature to 500°C. These points are 200°C, 290°C and 400°C. The last temperature point is the point of the ferroelectric/paraelectric phase transition. The temperature obtained for this phase transition is in good agreement with previous measurements of the lattice parameters c and a of similar films [38]. At 290°C PbO2, which is present in our PZT films at least in a surface layer 80 nm thick [39], decomposes to Pb3O4. At the slightly higher temperature of 327°C, excess metallic lead in the PZT film melts. The point at 200°C is currently unclear, but we speculate that it could be due to an order-disorder transition caused by the appearance of mobile charged lattice defects in the film, probably oxygen vacancies. Just above 200°C, a constriction of the hysteresis loop of Ti-rich PZT films was observed, in combination with a decrease in the internal electric field generated by RIE-induced space charges [40]. Such a deformation of the hysteresis loop is expected for a ferroelectric with a moderate space charge density, which is uniformly distributed [41].

Figure 1b shows the temperature dependence of the optical energy gap E03 (taken at a constant absorption coefficient α = 1000 cm-1) for both CSD and SP PZT samples. In this figure, four linear intervals are seen for both samples, which can be identified with different phases of the PZT material according to Fridkin’s theory [32, 37]. The temperature dependence of E03 in the case of CSD PZT is more linear, and its value is higher, than in the case of SP PZT. The most variation in E03 behavior occurs in the phase II. But the variation of E03(T) at phase II in the CSD PZT is not as large as in the case of SP PZT. The temperature of the phase I → phase II transition is also different for CSD and SP films (Table 1). Moreover, in the CSD PZT a band gap discontinuity appeared at the phase I → phase II transition which is an indication of first-order phase transition. The temperatures of all phase transitions are presented in Table 1. All these temperatures are different for PZT materials prepared with different techniques. This can be explained by the differences in stoichiometry between SP and CSD films and in the space charge accumulation and oxygen vacancies [26].

Use of SE in this case gave a possibility to understand deeply the nature of phase transitions in PZT ceramics prepared with various technologies. Also these data will help to improve the preparation technology and allow the obtainment of PZT films with given properties. For more details about the results and data analyzes of CSD and SP PZT films see Ref. [26, 27, 35, 39].

4.2 Ba0.4Sr0.6TiO3 films

Thin films of ferroelectric BaxSr1–xTiO3 are very promising for a number of applications such as high dielectric constant capacitors, non-volatile memories with low switching voltage, infrared sensors and electro-optic devices such as planar waveguides or optical switches because of their large electro-optical coefficient and low optical losses [42], and tunable microwave applications [43]. The optical, structural, and dielectric properties of BaxSr1–xTiO3 films have been found to depend substantially on the preparation method, composition, film thickness, grain size, and annealing.



50 100 150 200 250 3002.256

2.258

2.260

2.262

2.264

2.266

2.268

n

T (K)

Refractive index of Ba0.4

Sr0.6

TiO3 taken at λλλλ = 400 nm

111 K 216 K 259 K

a)

0 50 100 150 200110

112

114

116

118

120

122

124

∆

∆

∆ ∆ (( ((

οο οο)) ))

T (K)

90 K

b)

Fig. 2. Temperature dependence T of (a) refractive index n and (b) ellipsometric angle ∆ at the wavelength of 400 nm for Ba0.4Sr0.6TiO3thin film.

According to the relation (1), phase transitions is clearly dependent on n(T). Such experimental data are shown in Figure 2a for Ba0.4Sr0.6TiO3 (BST) thin films of thickness ~500 nm deposited on the Si/SiO2/TiO2/Pt(111) by low pressure plasma jet technique. There you can see the characteristic strong changes in the temperature dependence of the refraction index, which are attributed to three characteristic cubic-tetragonal-orthorhombic phase transitions of BST. The temperatures of these transitions appear to be shifted downward in comparison with those of BST ceramics or unstressed films (the phase transitions for this BST composition should be at about 190, 250 and 320 K). The reasons for this shift may be due to the tensile stresses, inhomogeneity and defects within the film and interfaces. The results obtained can be tentatively connected with the dominant compressive stresses of these BST films, which is confirmed by estimation of the lattice constant from x-ray diffraction. The estimated value of the stress in these films was about 0.2%. Unfortunately, for polycrystalline films such measurements can be used only for phase transition detection. In some cases, phase transitions can be directly seen in the main ellipsometric angles Ψ or ∆. An example of such ∆ behavior near the phase transition point in a BST film with low Ba concentration is presented in Figure 2b. The jump at 90 K is attributed to the tetragonal-orthorhombic phase transition. The temperature of this transition is shifted to lower temperatures in comparison to the BST film discussed previously. The difference in stress value between these samples is considered to be one of the reasons for this shift. Further details of BST film preparation and SE data analysis can be found in Ref. [28].



4.3 NaNbO3 films

One of the most interesting cases of implementation of the complex sequence of phase transitions of all perovskites is antiferroelectric sodium niobate NaNbO3, which is a promising lead-free piezoelectric material for high temperature applications [44]. NaNbO3 changes its phase from a high-temperature paraelectric to antiferrodistortive and then reaches the antiferroelectric phase at 643 K [45]. The unit cells of phases R and S are the largest reported for a low-symmetry phase of any perovskite.

600 650 700 750 8002.085

2.090

2.095

2.100

2.105

2.110

805 K

793 K

776 K

718 K

Si/SRO (15 nm)/NaNbO3 (230 nm)

Experimental n = n(T ) at 300 nm

Smoothingn

eff

T (K)

643 K

Fig. 3. Temperature dependence of the effective refractive index at the 300 nm of NN thin film [29].

Polycrystalline NaNbO3 (NN) thin films of thickness ~230 nm were deposited on Si/SiO2/Ti/Pt(111)/SrRuO3 substrates by pulsed laser deposition. The details of sample preparation and SP calculations can be found in Ref. [29]. The effective refractive index neff of NN thin films at 300 nm as is presented as a function of temperature in Figure 3. The refractive index has several extremes at 643, 718, 776, 793 and 805 K. Figure 4 presents the band gap energy Eg as a function of temperature, obtained from our thermo-ellipsometric measurements. Significant jumps in Eg can be seen at ~225, 643, 718, 750 and 788 K. The anomalies obtained in n(T) and Eg(T) are interpreted as phase transition temperatures Tc. The slight difference in Tc between n(T) and Eg(T) can be connected with the underestimate number of experimental points in Eg(T), which causes higher errors in Tc detection. On the other hand, it is worth mentioning that the behavior of Eg(T) can be influenced by the discrete “local” interband gap transitions, which can explain the slight difference of Tc observed in n(T) and Eg(T). The minimum of neff and the jump of Eg at 643 K correspond to the P→R phase transition [45-48]. The 1st order phase transition at this T value was also determined by dielectric measurements in our earlier work [49]. The anomalies of neff (or Eg) at 776 (750) and 793 (788) K agree with the values of Tc given for the phase transitions of R→S and S→T1 in [45, 46]. The other two anomalies, at 718 K and 805 K, are explained in other works made by X-ray with coexistence of phases R, S [46], and S, T1 [47, 50].

The low-temperature phase transition was detected at ~225 K (experiment performed during cooling in the low vacuum with Janis cryostat), which is in good agreement with some results on ceramics [51], but is not analogous with any results on single crystals. We did not find any similar data describing this low-temperature phase in the literature. A large thermal hysteresis of ~80 K was reported by Darlington (1973) for the low temperature transition. He found that the single-domain crystals of phase P transformed rapidly to the twinned phase N at the temperature below ~143 K; the reverse transition was again rapid, occurring at ~223 K; and these crystals always returned to the single-domain crystals of phase P. The large hysteresis may be caused by the great structural change required at the transition, which poses



difficulties for the nucleation at the appearing phase. Moreover, the rapid growth of the appearing phase would be the result of a high energy at the interface between the nuclei and bulk. As in the case of “unexpected” phase transitions at 718 and 803 K, where there is possible coexistence between the two phases R and S, and S and T1, here also twinned phases coexist, in this instance N and P [46]. Insomuch as phase N→P appears at different temperatures as reported in many works, there is a reason to consider that it may be caused by different volume fractions of the N/P phases [46], which contribute to the instability of the N→P phase transition. In our room temperature studies of NN thin films [52] we found that these films have pronounced depth profile of the refractive index: the NN film close to the bottom electrode is not identical to that closer to the surface. Such room temperature ellipsometric results are in excellent agreement with evaluated thermo-optical studies, which showed strange anomalies of n at 718 and 803 K, which is connected with this inhomogeneity of the NN film.

0 100 200 300 400 500 600 700 8002,6

2,8

3,0

3,2

3,4

3,6

3,8

TEg ∂∂ /

VIVIVIIIII

Eg (

eV

)

T (K)

Eg(T)

Fit with Varshni

quadratic relation

I

I = - 0.46 meV/K

II = - 1.52 meV/K

III = - 2.57 meV/K

IV = - 2.40 meV/K

V = - 1.34 meV/K

VI = - 1.34 meV/K

Fig. 4. Temperature dependence of the band gap energy of NN thin film. The temperature coefficient of the optical gap ∂Eg/∂T in [meV/K] is given for each phase from I – VI, evaluated by fitting experimental data by Varshni quadratic relation [53].

It is worth mentioning that the structural difference between phases R to T2 of NN is very slight, and therefore a very accurate technique is needed to study them. And as can be seen from these results, elipsometry is a very accurate method to detect even those phase transitions that are followed by small structural differences. Because of the high sensitivity of ellipsometric technique to any global and local structural changes, in the case of thin films we cannot separate the phase transition taking place in the whole film from the surface or interface phase transition. Therefore, we may observe phase transitions at different temperatures from those detected for single crystals, for example, and we can also observe atypical anomalies in n(T) and Eg(T) due to the inhomogeneity of the film. Therefore, the next step of our investigation will be the study of the phase transitions by other methods like specific heat measurements and x-ray, which can not detect phase transition taking place from the interface or surface; and therefore could separate those Tc taking place from the host material of the NN thin film.

4.4 SrTiO3 single crystals: Influence of crystal surface defects on thermo-optical properties

and phase transition behavior

For perovskite crystals, the value of n usually increases as the temperature rises [54]. Fig. 3 illustrates positive thermo-optical effect (TOE) for thin Ba0.4Sr0.6TiO3 films. The opposite behavior of n(T) was observed for NaNbO3 films (Fig. 5). In this case, the negative TOE can be explained by the inhomogeneity of NaNbO3 thin films, since they were found to have depth profile of n (see the room temperature ellipsometry results [52]). But why can both positive and negative TOE be observed in ferroelectric materials?



This has been a substantial thermo-optical problem on perovskites for many years. For brief understanding of this question, it is necessary to analyze the physical contributions taking place in the thermo-optical effect of perovskites in more detail. It was assumed in [31, 55] that the temperature dependence of refraction index for cubic perovskites can be written as

( ) ( ) ,PETOEVETOE dTdndTdndTdn +==+= ηηη (4)

where dn/dT, the thermal coefficient of the refractive index, is considered to be a sum of

contributions from the thermo-optical effect TOEη and from the photoelastic effect PEη ,

connected with the volume expansion (deformation). The proper thermo-optical effect TOEη is

considered of three contributions: 1) the electronic polarizing power EPP

TOEη [56], 2) the

electronic contribution caused by the temperature shift of the absorption edge TAE

TOEη [57, 58],

and 3) the phonon contribution PH

TEOη , controlled by the fundamental phonon frequency 0ω

[59]. Taking into consideration the analyses in [57], the relation (4) can be written as

.PE

PH

TOE

TAE

TOE

EPP

TOEVETOE ηηηηηηη +++=+= (5)

The photoelastic contribution is negative in most cases (dn/dT) < 0, because the thermal expansion decreases the density and so tends to diminish the refractive index with rising temperature. However, in some cases (dn/dT) appears to be positive. The proper effect was believed to be different in magnitude but never positive (dn/dT > 0) [60]. Several works have reported on negative TOE in ferroelectric [14 - 16]. It should be mentioned that the second contribution (relation (5)) is typically considered to be phenomenologically determinable from the behavior of interband Sellmeier’s oscillators. Detailed investigations of the temperature dependence of interband transitions close to the absorption edge showed that the temperature dependence of n and the negative TOE can be connected with a more fine and complicated correlation with the temperature dependence of interband transitions forming the behavior of n in the visible range [30]. In fact, n is related to the difference between the average energies of the conduction and valence band, rather than just a shift of the absorption edge [61]. Certainly, the indirect electron transition band gap ind

gE usually decreases as T rises, but in the

meantime, the bandwidth increases due to the shift of the higher energy bands to the higher energies, for example, the direct electron transition band gap d

gE . If other electron transitions

of the higher energies play a significant role in the TEO, then such phenomena as a decrease in n with increasing temperature can be observed. Thus the analyses of only one part of the absorption as a function of T, for example, d

gE (T), can not give full evidence of the phase

transitions in solids, whereas studies of Eg in terms of T are unambiguous. To generalize this problem, we should be very careful in choosing which Eg will be calculated from the absorption spectra for phase transition detection.

To estimate the SE possibilities for in-depth study of phase transitions, epitaxial monocrystalline films or bulk crystals with well-known properties should be used. For this reason, the SrTiO3 bulk crystal was chosen as a “model” material for further investigations. Surprisingly, up to now the thermo-optics of such popular top models as SrTiO3 were scantily studied (as above, room temperature data in [62, 63], and in more detail within the temperature region 20-900 K in [12]). In our experiment we used two SrTiO3 crystals. The first one was a commercial Verneuil grown [100] oriented substrate from Crystal GmbH and the second one was a TSS grown SrTiO3 specimen from Mateck of a specially cut mono-domain crystal of 18

4hD phase of dimensions ~ 6 mm along [001], 5 mm along [110] and 7 mm

along [110] cubic axes. In our measurements we were mainly focused on the antiferrodistortive phase transition taking place at 105 K.



After the first measurement, we were very surprised by the appearance of a huge jump in the value of n near the phase transition temperature (Fig. 5a) for the Verneuil grown SrTiO3 crystal. Moreover, this jump appeared about 20 degrees higher than the well known phase transition takes place. There are some works describing appearance of surface phase transition in SrTiO3 at T higher than bulk phase transition [21, 22]. In the case of bulk samples, only light reflected from the surface is analyzed. This means that we study the properties of only the surface part of a bulk and high sensitivity of SE can play here a positive role. We suggest that the behavior detected on the surface near the phase transition temperature should be connected with the evaluation of the bulk phase transition, and influenced by various defects on it. To check our “commercial” surface, we performed an x-ray scan and obtained the data presented in Figure 5b. In the case of an ideal monocrystal (single crystal with the absence of structural blocks) the color of each part should be the same. In our case we have 3 different colors, i.e., three blocks where sub-orientations are presented near the surface. Even when the disorientation of these blocks is very small (~0.01°), their coexistence could cause the appearance of a local stress near the surface, which brings distortions connected to the order parameter and can lead to shifts the phase transition temperature [64].

0 50 100 150 200 250 3002.380

2.385

2.390

2.395

2.400

2.405

2.410

cooling

heatingn

T (K) a) b)

Fig. 5. (a) Refractive index at 600 nm and (b) XRD surface omega-scan distribution of (100) surface of Verneuil Crystal GmbH grown SrTiO3 [30].

0 50 100 150 200 250 3002.405

2.410

2.415

2.420

2.425

2.430

n

T (K) a) b)

Fig. 6. (a) Refractive index at 600 nm and (b) XRD surface omega-scan distribution from (110) surface of Mateck TSSG grown SrTiO3 single crystal [30].

A large change of the n in this case can be connected to the mixing of the ellipsometric yields collected from these slightly disoriented blocks that affect the effective n. Another reason for the high value of the jump in n below the antiferrodistortive phase transition is a slight crystal anisotropy, which was not taken into account in our simplistic model. The use of an isotropic model for slightly anisotropic materials is widely accepted in ellipsometry, and in most cases such a calculation is accurate enough. But in our case the mixed signal from



disoriented blocks together with anisotropy neglecting could cause a high “effective” value of the refractive index. Our numerical simulations revealed that using an isotropic approximation for anisotropic materials in some cases can lead to an “effective” refractive index higher than the highest value of the anisotropic one, or lower than the lowest value. For example, if we suppose the surrounding medium with n0 = 1 and a unaxial crystal as an anisotropic medium with refraction indices no = 2.385 for the ordinary beam and ne = 2.380 for the extraordinary beam, the calculated effective refraction index as a function of the anisotropic medium orientation neff (ϕ, ϑ) for the direction of an incidence wave ϕ0 = π/2, ϑ0 = 60° is plotted in the Figure 9 of the appendix. Then it is obvious, in qualitative accordance with experimental results, that for a certain range of anisotropic medium orientations, neff < min (no, ne), and for another range

neff > min (no, ne); the extreme values are minneff = 2.378 and maxneff = 2.387.

For real crystals or structures, such effect could be much higher because of depolarization effects. The details of the simulation of anisotropic media effective refractive index in the approximation of isotropic media are given in the Appendix of this publication.

0 50 100 150 200 250

2.4

2.8

3.2

3.6

4.0

4.4

Experimental

Data smoothing

Ro

ug

hn

ess

(n

m)

T (K) Fig. 7. Surface roughness as a function of the temperature for Mateck TSSG grown SrTiO3 single crystal [30].

To confirm our suggestion about the extremely important role of the surface defects, the same measurements were performed on the second SrTiO3 crystal with a “perfect” surface. The XRD omega scan of this crystal is presented in Figure 6b, where it can be seen that the surface is really perfect without disoriented blocks. In this case, the T dependence of the surface refractive index appeared to be very close to the bulk one (measured by spectrophotometry; Fig. 6a) and did not reveal any jumps or other evidence of phase transition. Nevertheless, the antiferrodistortive phase transition was also detected for this “perfect” crystal. In this case, the “key” parameter was the surface roughness. Its temperature dependence is shown in Figure 7. The physical reason for such changes in the surface roughness is the structural changes of the surface under a second order antiferrodistorsive cubic-tetragonal 18

41

hh DO → structural transition.

4. Conclusions

It was shown in this paper that spectroscopic ellipsometry is a very powerful technique for the study of phase transitions in thin films and crystals. The relative simplicity of the use of ellipsometry enables us to detect phase transitions or other structural perturbations even for ultra-thin films where the use of other techniques is impossible or very complicated. The suitability of use of a proper “key” parameter such as band gap energy, refractive index and surface roughness for phase transition detection was shown on different examples of perovskite oxides (thin films of PbZrxTi1-xO3, Ba0.4Sr0.6TiO3 and NaNbO3 made by various preparation methods and SrTiO3 model single crystals). The high sensitivity of SE allows the



detection of phase transitions even if it takes place only in some part of a film. In this case the calculation of the exact place of such phase transition is rather impossible. This limitation of SE could probably be overcome by the use of shorter wavelengths. It is also important to note that use of SE opens wide possibilities for the study of surface phase transitions, which has become a very popular subject in modern physics. The detection and study of the surface phase transitions by SE is of great significance not only from a fundamental, but also from a technological point of view due to intense developments in micro and nano-electronics for nanostructures engineering, where temperature changes in interfaces and surfaces play an important role.

5. Appendix

The permittivity εd and permeability µd tensors of (anisotropic) medium in the default Cartesian coordinate system can be transformed to the new coordinate system via the orthogonal matrix A determined using the unit vector (cosϕsinϑ, sinϕsinϑ, cosϑ) normal to the plane of the material:

,0

/1

A

−−−

=

zyx

yz

zxyx

nnn

nn

nnnn

γγγγγ

where ϕ, ϑ are spherical coordinates and ( ) .12/12 −

−= xnγ The transformation relations may

then be read as ε = ATεdA, µ = ATµdA. The Maxwell equations can be solved by approximation of plane electromagnetic waves

with the tangential components kx = cxk0, ky = cyk0 of a wave vector, where k0 = ω/c is a vacuum wave number with frequencyω, c is the speed of light in the vacuum, and cx = n0cosϕ0sinϑ0, cy = n0sinϕ0sinϑ0, and ϕ0, ϑ0 are the spherical coordinates of the unit wave vector of the incident wave measured in the (isotropic) surrounding medium with refractivity index n0. In accordance with Berreman [65], for the tangential field components

−

=

x

y

y

x

H

E

H

E

z

0

0

0

0

)(

µεµε

ψ

we have the Maxwell equations

),(Did

)(dz

cz

z ψωψ−=

where the components of the matrix D are generally defined from the relations

,D11zz

yzy

zz

zxxcc

µµ

εε

−−= ,D00

012 µµ

µµµµ

εε

zz

zyyzyy

zz

xc−+−=

,D13zz

yzx

zz

zyx cc

µµ

εε

+−= ,D00

014 µµ

µµµµ

εε

zz

zxyzyx

zz

yxcc+−−=

,D00

02

21 εεεε

εε

µµ

zz

zxxzxx

zz

yc−+−= ,D

022 µ

µεε zyy

zz

xzxcc

−−=

,D00

023 εε

εεεε

µµ

zz

zyxzxy

zz

yxcc−+−= ,D24

zz

zxy

zz

xzy cc

µµ

εε

+−=

,D31zz

xzy

zz

zxy cc

µµ

εε

+−= ,D00

032 µµ

µµµµ

εε

zz

xzzyxy

zz

yxcc+−−=



,D33zz

xzx

zz

zyy cc

µµ

εε

−−= ,D00

02

34 µµµµ

µµ

εε

zz

zxxzxx

zz

yc−+−=

,D00

041 εε

εεεε

µµ

zz

zxyzyx

zz

yxcc

−+= ,D42zz

zyx

zz

yzxcc

µµ

εε

+−=

,D00

02

43 εεεε

εε

µµ

zz

zyyzyy

zz

xc−+−= ,D44

zz

zxx

zz

yzy cc

µµ

εε

−−=

where ε0, µ0 are the permittivity and permeability of vacuum. Denoting by D0 and D1 the matrices of an isotropic surrounding medium and anisotropic medium, respectively, we can factorize them using their eigenvectors arranged in the columns of the admittance matrices Yj and their eigenvalues in the diagonal matrices Λj, i.e., Dj = YjΛjYj

-1, j = 1, 2 [66]. When the four field modes are ordered in such a way that 1 and 2 represent the incident and reflected s-polarized waves and modes 3 and 4 the incident and reflected p-polarized waves in the surrounding isotropic medium (and likewise for the anisotropic medium), then it is possible to define the matrix S = Y0

-1Y1 transferring the tangential mode components of contra-propagating waves. Utilizing the boundary conditions, the four amplitude reflectivities using the elements of the transfer matrix S may be defined as follows: r21 = (S21S33 – S23S31)/d, r23 = (S11S23 – S21S13)/d, r41 = (S41S33 – S43S31)/d, r43 = (S11S43 – S41S13)/d, where d = S11S33 – S13S31 and the index ij means that the incident is i-mode and the reflected j-mode. We can define the effective refraction index neff of a fictive isotropic medium by averaging the same ellipsometric parameters ψ, ∆ obtained experimentally:

.)iexp(tan4341

2321

rr

rr

++

=∆ψ

a) b)

Fig. 9. Effective refraction index neff as a function of anisotropic medium orientation ϕ, ϑ, n0 = 1, no = 2.385, ne = 2.380, ϕ0 = π/2, ϑ0 = 60° in (a) three and (b) two dimensions.

In more detail, we have [67] neff = η - ik and

),costan1/(tansin 20000 ∆+= ψϑϑnf ,tansin2),tan1( 02

201 ψψ ∆=−= fxfx

( ) ( )( ) ,221

2/12/1

40

22

21

20

222

21

20

22

21

+−++++−= fxxfxxfxxη ./21 ηκ xx=

As an example, let us suppose a surrounding medium with n0 = 1 and SrTiO3 as an anisotropic medium with refraction indices no = 2.385 for an ordinary beam and ne = 2.380 for an extraordinary beam. The effective refraction index is shown in Figure 9 as a function of the anisotropic medium orientation neff (ϕ, ϑ) for the direction of an incidence wave ϕ0 = π/2, ϑ0 =



60°. Then it is obvious, in qualitative accordance with experimental results, that for a certain range of anisotropic medium orientations, neff < min (no, ne), and for another range,

neff > min

(no, ne); the extreme values are minneff = 2.378 and maxneff = 2.387.

Acknowledgments

This research was supported by the grants 202/09/J017, KJB100100703 and AV0Z10100522, projects KAN301370701 of the ASCR, 1M06002 of the MSMT CR, European Social Fund, the 6th Framework Program of the Multifunctional & Integrated Piezoelectric Devices (MIND) and UNESCO LÓREAL Latvian National Fellowship for Woman in Science.



Spectroscopic ellipsometry applied to phase transitions in solids: possibilities and limitations

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Transcript of Spectroscopic ellipsometry applied to phase transitions in solids: possibilities and limitations