Resonant x-ray scattering method for measuring cation stoichiometry in BaSnO₃ thin films

Recent research has generated interest in the alkaline earth stannates, perovskite oxides with chemical formula ASnO₃, which display high carrier mobility at room temperature $(T=300K)$⁠. High mobility (i.e., strong conductivity with few charge carriers) is linked to better performance in electronic devices such as transistors.¹ Typically, the correlated oxides have relatively large carrier concentration n on the order of 10²¹ cm⁻³ but mobilities less than 10 cm²/(V s). In comparison, the semiconductor silicon has an intrinsic carrier concentration of $1.5×1010cm−3$ with an electron mobility of 1350 cm²/(V s) at room temperature.² 

Research on the alkaline earth stannates has been concentrated on stannates with A = Ba, Sr, and Ca. Of particular interest to us is barium stannate, BaSnO₃. BaSnO₃ (or BSO) is a semiconductor with a wide bandgap, currently estimated at 3.3 eV ≤ E_g ≤ 4.05 eV.³ Upon electron doping with La³⁺, BSO becomes conducting with high carrier mobility at room temperature. La_xBa_1−xSnO₃ (LBSO or BLSO) becomes degenerately doped at 3.7%. Its mobility remains high over a broad doping range with no clear dependence on carrier density. Bulk single crystal LBSO has a mobility of $μ∼100$–320 cm²/(V s),^3–5 which is comparable to the mobility of the semiconductor⁶ ZnO and is a factor of 10 higher than the mobility observed in other perovskites. For example, the common perovskite oxide strontium titanate (SrTiO₃ or STO) has a mobility of $μ∼11cm2/(Vs)$⁠. In addition to its high room-temperature mobility, BSO is a transparent conducting oxide and also possesses excellent thermal stability, maintaining highly stable oxygen stoichiometry and conduction behavior against annealing in various atmospheres at temperatures up to 1000 °C.^4,5 These properties make BSO a promising material for a variety of applications in electronic devices.⁶ 

High mobility materials can be used as nonvolatile switches when combined in epitaxial heterostructures with ferroelectrics. Thermally stable materials have applications as both capacitors and sensors at high temperature⁷ and are also well suited for device fabrication as they resist the formation of oxygen vacancies, which act as scattering centers.⁸ Transparent conducting oxides, which combine high conductivity with reduced optical absorption due to low carrier concentration and the absence of interband transitions, can be used as passive transparent electrodes within flat panel displays and solar cell devices.^7,9 As a thin film, LBSO's mobility is diminished from its bulk mobility due to the introduction of defects. However, while reduced from its bulk value, recent measurements on thin film LBSO grown on STO (001) substrates, as well as various scandate substrates, show that the LBSO mobility remains comparatively high.

With improved epitaxial growth methods, such as the use of evaporated SnO₂ as a cation source,¹⁰ the mobility of LBSO on STO has increased from 70 to 120 cm²/(V s).^4,11–13 Use of SnO₂ addresses issues in the molecular beam epitaxy (MBE) of ternary stannates related to volatile SnO formation and enables the growth of epitaxial, stoichiometric BSO.¹¹ Even higher mobility values have been achieved in LBSO thin films grown on scandate substrates with larger lattice parameters. Although BSO possesses smaller lattice mismatch with these substrates, the scandates are also less widely available as commercial substrates than STO. In 2016, thin film LBSO^14–16 was demonstrated to have a mobility as high as 150 cm²/(V s) on praseodymium scandate PrScO₃ (110).¹⁰ In 2017, a mobility of 183 cm²/(Vs) was achieved on dysprosium scandate DyScO₃ (110).¹⁷ The orthorhombic perovskite DyScO₃ has a pseudocubic lattice parameter $apc=3.943Å$⁠, making it significantly better lattice-matched to BSO $(ϵ=−4.2%)$¹⁷ than the more standard perovskite substrate STO, thereby minimizing the formation of mobility-impeding misfit dislocations in the film. Substrate choice has also since advanced beyond the rare-earth scandates to include Ba₂ScNbO₆.¹⁸ 

Use of anomalous scattering to determine crystalline properties was pioneered in the 1980s by Jerome Karle, who developed single-wavelength anomalous diffraction to solve the phase problem.¹⁹ Advancements in x-ray crystallography have since seen the development of multiwavelength anomalous diffraction,²⁰ a technique that is now widely used in synchrotrons around the world for protein crystallography and has facilitated many drug discoveries. Since this initial development, the tool has been extensively developed and applied in both the hard and the soft x-ray regime.^21–23 In this article, we use the analysis methods developed for diffraction anomalous fine structure by Cross and Sorensen, which is a technique developed to determine an extended x-ray absorption fine structure that is site specific.²⁴ 

We report on a set of three lanthanum-doped BSO thin films grown on DyScO₃ (110) substrates via MBE using a SnO₂ source. Films are lightly doped with $0.57±0.05%La$⁠, with carrier concentration $n=5.44±0.36×1019cm−3$⁠, and display near record room-temperature Hall carrier mobilities, ranging from $μ=84.8$ to 144 cm²/(V s). Film thicknesses, determined from growth rates, and measured Hall mobilities of the three films are recorded in Table I along with growth parameters. All films are grown for 90 min at temperature of 858 ± 6 °C using oxygen plasma with beam equivalent pressure (BEP) of 10⁻⁵ Torr. Ba flux is adjusted via cell temperature to vary the SnO₂/Ba flux ratio, while the SnO₂ cell temperature is held constant for each growth, although some small variation in the SnO₂ flux between growths will occur.¹¹ 

We immediately observe that mobility decreases with film thickness, with the highest mobility sample having the highest SnO₂/Ba flux ratio and thus slowest growth rate. We investigate the Ba/Sn cation stoichiometry of these (La,Ba)SnO₃ samples in order to determine whether there is any additional correlation between stoichiometry and mobility, considering the dependence of these thin films' high mobility on the use of a SnO₂ source of Sn during growth.

To investigate the cation stoichiometry of these films, measurements are taken using resonant x-ray scattering²³ from the Ba L absorption edges. Diffraction measurements are taken at room temperature, and the diffracted intensity is measured using a solid state area detector. Data at the Ba L-I edge (energy $E=5.99keV$⁠) are collected at the NSLS II on beamline 4-ID, while data at the lower Ba L-II and L-III edges (⁠$E=5.62$ and $E=5.25keV$⁠, respectively) are collected at the APS on beamline 33BM. This beamline has a Si(111) monochromator with a 0.8 eV resolution at 5 keV on a four circle diffractometer. This resolution is less than the sharp white line resonance at the edge, which has a width of 4 eV. Rocking curves are measured in the bisecting mode on the LBSO thin films at both high intensity even-order Bragg peaks (Miller indices $H+K+L=even$⁠) and lower intensity odd-order Bragg peaks $(H+K+L=odd)$⁠. Energy scans across the Ba L absorption edges are measured at the same set of Bragg peaks. The range of BSO Bragg peaks accessible at the energies of the Ba L edges include the specular peaks (001) and (002) and nonspecular peaks (202), (112), (101), (102), and (122), with the (103) reflection accessible at the Ba L-I edge only.

Measurements are also taken via conventional x-ray diffraction using an in-lab Rigaku x-ray diffractometer, where the rotating Cu anode emits x-rays of fixed energy $E=8.04keV$⁠. Figures 1(a) and 1(b) show $θ/2θ$ scans measured across the (001) and (002) specular Bragg peaks of both the LBSO film (the broader peak at lower $2θ$⁠) and the DSO substrate (the sharper peak at higher $2θ$⁠) for the three samples. The film lattice parameter c is calculated from the $2θ$ peak positions and confirmed to be approaching 4.116 Å for all three films, with c ranging from $4.104to4.110Å$⁠. Finite thickness oscillations (or fringes), indicating high epitaxial quality and abrupt interfaces, are clearly visible surrounding both the (001) and (002) film peaks. Fitting these oscillations confirm that the film thicknesses are in good agreement with those expected by their respective growth rates.

Finite thickness oscillations are also observed in our synchrotron measurements. Figure 1(c) shows a CCD image of the nonspecular BSO Bragg peak (101), taken during synchrotron measurements, displaying clearly defined finite thickness oscillations along the Q_z-axis (or L-axis). A line scan across the CCD image along the Q_z-axis shows the (101) film peak surrounded by finite thickness oscillations, similar to the oscillations seen in the $θ/2θ$ scans.

BSO Bragg peak rocking curves and energy scans are extracted from the sequences of CCD images captured while at the synchrotron. A comparison of the data measured on even-order and odd-order BSO Bragg peaks is shown in Figs. 2(a) and 2(b). First, we note that the film peak intensity scales with film thickness in the same manner as in the $θ/2θ$ scans shown in Figs. 1(a) and 1(b). Next, we find a significant intensity difference between the even-order Bragg peak (103) and the odd-order Bragg peak (102) along the same rod, with $Ieven(θ)≈20×Iodd(θ)$⁠.

This large intensity difference is due to the similarity in scattering strength between the Ba and Sn cations. Essentially, while the even-order Bragg peak structure factor $Feven$ is dependent on the sum of the cation scattering strengths, the odd-order Bragg peak structure factor $Fodd$ is dependent on the difference of the cation scattering strengths. $Fodd$ is near zero for the Ba and Sn pairing. Because of the near zero structure factor F, the odd-order Bragg peaks of BSO are thus particularly sensitive to the material's cation stoichiometry, i.e., the Ba/Sn ratio. If there is a small change in occupation of either the Ba or Sn, that change will be amplified in the odd-order Bragg peak intensity, allowing measurement of the Ba/Sn ratio with high precision.

Sensitivity to cation stoichiometry is further improved using the technique of resonant x-ray scattering, which changes the scattering strength of a single element. In this experiment, we change the scattering strength of the Ba atom, using scattering from the Ba L absorption edges.

BSO Bragg peak energy scans taken across the three Ba L edges are shown Figs. 2(c)–2(e). Staying fixed on a Bragg peak $(HKL)$⁠, energy is scanned from below the edge to above the edge. Data are measured across the range $ΔE=±0.03keV$ around the Ba L-I edge and across the wider energy range $ΔE=±0.1keV$ around the Ba L-II and L-III edges. As shown in Figs. 2(c)–2(e), the odd-order Bragg peak energy scan has a distinct peak shape, featuring a sharp drop in intensity below the L absorption edge. Oscillations due to fine structure are additionally observed in the scans taken at the L-II and L-III edges taken with a larger energy range. Also in Figs. 2(c)–2(e), we note that the peak has its greatest normalized intensity at the Ba L-III edge, $I(edge)>4$⁠, with decreasing peak intensities at the L-II and L-I edges, where the atomic scattering correction factors $f′$ and $f′′$ have smaller magnitude.

We fit the Bragg peak energy scans, using the real and imaginary atomic scattering factor corrections $f′$ and $f′′$ for the Ba L absorption edges. Owing to the close proximity of the Ba L edges to the lower Sn L edges (at approximately 5 keV), $f′$ and $f′′$ for the Ba L edges could not be determined by measuring fluorescence at the beamline. We therefore self-consistently derive $f′$ and $f′′$ directly from the measured energy scan intensity $I(E)$⁠, using iterative Kramers–Kronig fitting,²⁵ which makes use of the fact that the energy-dependent structure factor $F(E)$ can, for any crystal structure, be decomposed into the sum of its nonresonant (or energy-independent) and resonant (energy-dependent) parts,

where $|a|eiφ$ is the nonresonant part, and the resonant part $f′(E)+if′′(E)$is due wholly to the atomic scattering correction factors at the relevant absorption edge (here, the Ba L edge). The nonresonant part can in turn be decomposed into real and imaginary parts, with $eiφ=cosφ+isinφ$⁠.

Defining three energy-independent coefficients $α=|a|2cos2φ,β=1|a|cosφ$⁠, and$γ=sin⁡φcos⁡φ$⁠, energy scan intensity $I(E)$ can be written as the following simple quadratic equation:

Solving this quadratic equation iteratively, we determine the real correction factor $f′(E)$ as a function of measured intensity $I(E)$⁠, the corresponding imaginary correction factor $f′′(E)$⁠, and the three energy-independent coefficients,

The three energy-independent coefficients $α$⁠, $β$⁠, and $γ$ are taken as fitting parameters; the measured intensity $I(E)$ is known; and $f′′(E)$⁠, being related to $f′(E)$ by the Kramers–Kronig dispersion relation, can be calculated by integrating $f′(E):f′′(E)=−2Eπ∫0∞f′(E′)E2−E′2∂E′$⁠.

Using the Cromer–Liberman theoretical estimates²⁶ as the initial values for atomic scattering correction factors $f′$ and $f′′$⁠, the energy-independent fitting parameters $α$⁠, $β$⁠, and $γ$ converge fairly quickly, typically within five iterations.²⁵ By fitting an odd peak energy scan $I(E)$ taken across a Ba L edge to Eq. (2), we thereby use iterative Kramers–Kronig and derive the $f′$ and $f′′$ spectra self-consistently. Figures 3(a)–3(c) show the iterative Kramers–Kronig fit of odd peak (001) and the corresponding $f′$ and $f′′$ spectra for the Ba L-III edge.

For datasets collected at each Ba L absorption edge, using a single $f′(E)$ and $f′′(E)$ derived from the iterative Kramers–Kronig method, we then fit our complete set of Bragg peak energy scans for all three samples,

where $I0$ is an energy-independent scaling factor, while structure factor F is dependent on both energy E and scattering vector q. Energy scan fits using $f′$ and $f′′$ are sensitive to structural information such as the Debye–Waller factor, oxygen vacancies, and La doping.

For any crystal structure containing Ba, the following equation in the kinematic approximation²³ holds:

For LBSO, we identify $A(q)eiφ(q)$ as the contribution from Sn, La-doping, and oxygen, which have weak energy dependence over the scanned range. All the energy dependence and resonant scattering come from the Ba contribution, namely, from the self-consistently derived real and imaginary corrections $f′(E)$ and $f′′(E)$ to the normal atomic scattering factor $f0,Ba$⁠. Assuming the ideal perovskite structure and the negligible effect from the <1% La doping, we estimate $A(q)$ and $φ(q)$ for our thin films from calculations of the scattering strengths using the Cromer–Mann calculated values.²⁷ 

In Eq. (5), variation of the Ba concentration c with respect to the nonresonant scattering amplitude $A(q)$ reveals the film stoichiometry. Phase $φ(q)$ is the phase difference between the resonant and nonresonant scattering contributions. For $φ(q)=0$⁠, Ba scatters in phase with the Sn, and the structure factor depends on their sum (the ideal case for an even-order Bragg peak). For $φ(q)=π$⁠, Ba scatters out-of-phase with the Sn, and the structure factor depends on their difference (the ideal case for an odd-order Bragg peak).

The measured intensity in Eq. (4) is fit with the parameters $I0$⁠, $φ(q)$⁠, and $c$⁠, holding $f0(q)$ for Ba and $A(q)=Fhkl(f0,Sn,f0,Ox)$ fixed at values derived by Cromer–Mann²⁷ and using the $f′$ and $f′′$ extracted as described above as independent variables. Fitting results are recorded in Table II. These three parameters in Eqs. (4) and (5) are the minimum number of parameters needed to fit the sizes of two complex vectors and the angle between them. The size of the first complex vector is the scattering strength of the barium alone, which is proportional to Ba concentration, and the size of the second vector is the scattering strength of everything else, in this case oxygen and tin. The angle between the two vectors depends on the crystal structure and static and thermal Debye–Waller factors. The c parameter reported in Table II is derived from a ratio of two of the fit parameters to eliminate the unknown scaling factor $I0$⁠,

Equation (6) results in a precise determination of the cation ratio for comparison between samples but with less accuracy. The accuracy depends on measurements and calculations of DWFs and atomic scattering factors. The fits of the data to Eqs. (4)–(6) are shown in Figs. 3(d)–3(g), demonstrating that the $f′$ and $f′′$ spectra derived from iterative Kramers–Kronig fitting of a single peak of a single sample can be superimposed to provide good fits of both even-order and odd-order BSO Bragg peaks for all samples.

Comparing even and odd-order peak intensities in Figs. 3(d)–3(g), we find that the even and odd peak energy scans having distinctly different shapes, with $Iodd(E)>Ieven(E)$ when normalized. Thus, while the odd peak has weaker overall intensity, as seen in the rocking curves in Figs. 2(a) and 2(b), the change in intensity of the odd peak across the L edges is greater than the change of the even peak. The higher intensity odd peak energy scans are expected to have enhanced sensitivity to fitting parameter c, characterizing the cation ratio.

However, as shown in Figs. 4(a)–4(d), no immediately visible difference in energy scan measured intensity is observed between the three films. For both even-order and odd-order BSO Bragg peaks, measured at all three Ba L absorption edges, normalized energy scan intensities of the three films overlap. This consistency in $I(q,E)$ for the three samples is reflected in our fitting results (see Table II), where we find minimal change of cation ratio c. Across the three samples, we calculate a standard deviation of cation ratio $σc≈1%$⁠. No systematic change can thus account for the large range of measured mobilities displayed by these films. Instead, our results attest to a consistency in cation stoichiometry across the three samples, despite their differing Hall mobilities, cation flux ratios, and thicknesses. This consistency is due to the thermodynamic control of stoichiometry that is typical for absorption-controlled growth, a classic example of which is thin film GaAs. Ideally, an analysis of the phase difference in Table II may reveal a systematic change in the concentration of extended defects, such as misfit and threading dislocations, that may have a large effect on mobility and may be consistent with the thickness and growth rate dependence.

To quantify the precision with which this resonant scattering method can be used to determine cation stoichiometry, we plot the energy scan intensity difference between each sample and the averaged intensity of all three samples at each Bragg peak. This noisy data $Δintensity$⁠, indicating the error or resolution of our measurement is plotted in Figs. 4(e)–4(g). Fitting these data to Eqs. (4) and (5) with parameters $I0,A(q)$⁠, and $φ(q)$ held constant yields a change of cation ratio of $Δc<1%$⁠. Comparing the magnitude of the noise with the calculated intensity differences, we determine the precision of $c≈1%$⁠, with the highest intensity odd-order peak (001) appearing to have marginally greater resolution than the other peaks.

This method of determining stoichiometry can be generalized to many thin films and crystal structures. Its precision does not depend on a comparison with standards but on the magnitude of the changes in $f′$ and $f′′$ compared to the amplitude $a(q)$ in Eq. (1). For BaSnO₃, the conditions are ideal in that the scattering strengths for Ba and Sn are almost the same. The precision of the measurement decreases as the scattering strengths become different but may be applied to other structures when, in a binary structure, the cations diffract out of phase for a particular direction in reciprocal space. We also note that the technique is complementary to and more precise than Rutherford backscattering (RBS), which works best when the elements in the film and the substrate have widely differing masses. For the case of BaSnO₃ on DyScO₃, the Ba, Sn, and Dy signals all overlap in an RBS spectrum, making a stoichiometry determination with a precision <1% difficult. The anomalous diffraction technique is also insensitive to the chemical composition of the substrate compared to, for example, wavelength dispersive x-ray fluorescence for thin films grown on substrates with the same cations. One drawback is that anomalous diffraction cannot determine an overall film cation ratio if some of the cations are in different phases that have Bragg reflections at different locations in reciprocal space. For such a case, x-ray fluorescence or Rutherford backscattering is better suited. This feature of the technique is also a strength in that it is free of background from the substrate.

In summary, LBSO on DSO thin films, possessing a range of high room-temperature carrier mobilities, were investigated for their cation stoichiometry using resonant synchrotron scattering. Resonant scattering from the three Ba L absorption edges was used to increase sensitivity to cation stoichiometry, allowing measurements of greater precision. The analysis of energy scans measured across both even-order and odd-order BSO Bragg peaks determined that there was <1% change in stoichiometry between the three films. No systemic change in cation stoichiometry could be correlated with the films' large range of high mobilities and cation arrival rates, which instead appear to scale with film thickness and growth rate. This method of determining cation stoichiometry through resonant x-ray scattering has a precision comparable with and is complementary to Rutherford backscattering, and unlike XPS, it is not a surface-sensitive technique. However, this method is also highly dependent on perovskite composition, with the near-zero structure factor of the odd-order BSO Bragg peaks increasing sensitivity to the cation stoichiometry. Further work would include measuring control samples using MBE grown films where we systematically vary stoichiometry by varying the arrival rate at lower temperatures where the growth is not absorption controlled.

This work was supported by the Office of Naval Research under Grant No. N00014-18-1-2704. This research used resources of the Advanced Photon Source, a U.S. Department of Energy (DOE) Office of Science User Facility, operated for the DOE Office of Science by Argonne National Laboratory under Contract No. DE-AC02-06CH11357. Extraordinary facility operations were supported in part by the DOE Office of Science through the National Virtual Biotechnology Laboratory, a consortium of DOE national laboratories focused on the response to COVID-19, with funding provided by the Coronavirus CARES Act. This research used sector 4ID of the National Synchrotron Light Source II, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Brookhaven National Laboratory under Contract No. DE-SC0012704. We also acknowledge a calculation of the RBS spectra by H. Hijazi, Rutgers University and the Rutgers RBS laboratory of surface modification, LSM.

The data that support the findings of this study are available from the corresponding author upon reasonable request.