research papers
Unitcell determination of epitaxial thin films based on reciprocalspace vectors by highresolution Xray diffractometry
^{a}Singapore Synchrotron Light Source (SSLS), National University of Singapore (NUS), 5 Research Link, 117603, Singapore, ^{b}Department of Materials Science and Engineering, National University of Singapore (NUS), 9 Engineering Drive 1, 117576, Singapore, ^{c}Department of Materials Science and Engineering and F. Seitz Materials Research Laboratory, University of Illinois UrbanaChampaign, Urbana, IL 61801, USA, and ^{d}South University of Science and Technology of China (SUSTC), Shenzhen, 518055, People's Republic of China
^{*}Correspondence email: slsyangp@nus.edu.sg
Dedicated to Professor Dr Johann Peisl on the occasion of his 80th birthday.
A new approach, based on reciprocalspace vectors (RSVs), is developed to determine Bravais lattice types and accurate lattice parameters of epitaxial thin films by highresolution Xray diffractometry. The lattice parameters of singlecrystal substrates are employed as references to correct the systematic experimental errors of RSVs of thin films. The general procedure is summarized, involving correction of RSVs, derivation of the raw G^{6} space. The estimation of standard error in the lattice parameters derived by this new approach is discussed. The whole approach is illustrated by examples of experimental data. The error of the best result is 0.0006 Å for the lattice parameter of indium tin oxide film. This new RSV method provides a practical and concise route to the crystal structure study of epitaxial thin films and could also be applied to the investigation of surface and interface structures.
and subsequent conversion to the Niggli and the Bravais by matrix calculation. Two methods of this procedure are described: in threedimensional and in sixdimensionalKeywords: crystal ; epitaxial thin films; reciprocalspace vectors; highresolution Xray diffractometry; Nigglireduced cells; Bravais lattice.
1. Introduction
Epitaxial complex oxide thin films have attracted great attention owing to their rich physical phenomena, such as ferroelectric, piezoelectric, ferromagnetic, multiferroic and superconducting, which promise novel functionalities in electronic devices (Zubko et al., 2011). The physical properties of epitaxial complex oxide thin films are very sensitive to distortion of their crystal structures, owing to the strong coupling between charge, spin, orbital and lattice (Schlom et al., 2007; Hwang et al., 2012; Dagotto, 2005). Thus, determination of the crystal structures of epitaxial thin films by highresolution Xray diffractometry (HRXRD), including the crystal symmetry and accurate lattice parameters, as well as domain structures and defects in films, is critical for the understanding and tuning of these physical properties (Cao & Cross, 1993).
Crystal symmetry has a significant effect on the dielectric, ferroelectric and piezoelectric behavior of epitaxial ferroelectric/multiferroic thin films. The ionic displacement in the _{3} (BTO) shows a ferroelectric polarization along the bodydiagonal direction 〈111〉, while for the tetragonal phase it is along the caxis direction 〈001〉, and for the orthorhombic phase it is along the facediagonal direction 〈110〉 (Kay & Vousden, 1949; Jaffe et al., 1971). Piezoelectric properties are also strongly dependent on the crystal symmetry. In the well known Pb(Zr,Ti)O_{3} (PZT) piezoelectric system, the highest piezoelectric response is observed near the morphotropic phase boundary (MPB) between a rhombohedral and a tetragonal phase (Jaffe et al., 1971; Cao & Cross, 1993).
which is determined by the crystal symmetry, gives rise to ferroelectric polarization. For example, the rhombohedral phase of BaTiOLattice parameters of epitaxial thin films, which determine the unitcell distortion and film strain state, are important to know to enable the tuning of such physical properties. For instance, the magnetic _{3} films increases with increasing inplane lattice parameters (Fuchs et al., 2008). Another example is the doubling of the temperature when the inplane lattice parameter of La_{1.9}Sr_{0.1}CuO_{4} decreases slightly from 3.78 to 3.76 Å (Locquet et al., 1998). In SrTiO_{3} (STO) thin films, tuning the lattice parameters by substrate misfit strain at room temperature can drive the paraelectric phase into the ferroelectric phase (Haeni et al., 2004).
temperature of LaCoOIn the determination of crystal structure, the lattice and unitcell dimensions are always the first parameters to be determined, i.e. begins with knowing the Bravais lattice type and lattice parameters. The diffraction pattern of a polycrystalline film can be indexed and the lattice parameters can be obtained with satisfactory precision using powder diffractometry and related computational algorithms (Pecharsky & Zavalij, 2003). Appropriate software includes ab initio indexing programs such as TREOR and DICVOL, employing a trialanderror method, and ITO, which implements zone searching combined with the Delaunay–Ito method (Pecharsky & Zavalij, 2003, p. 399). A simplified diagram for both bulk and film samples is as shown on the lefthand side of Fig. 1, which outlines a general procedure for the determination of crystal structures, for either crystalline or polycrystalline samples.
For a singlecrystalline epitaxial film, the procedure mentioned above could in principle be used, for example, in some singlecrystal APEX (Bruker, 2009). About 25 reflections (reciprocalspace points) could be used to deduce the lattice parameters, which is the normal case for singlecrystal However, this approach often does not work for a film on a substrate owing to the following facts:
software packages, such as(1) Two sets of diffraction spots (from film and substrate, respectively) can cause an inconsistency in the diffraction pattern and so spoil the procedure, with the result that no lattice parameters can be deduced.
(2) Weaker diffraction intensity from the film yields fewer useful reflections and less precise peak positions. Such diffraction peaks usually have broader width and even the extended Bond methods offer unsatisfactory peak positions (Bond, 1960; Schmidbauer et al., 2012). In HRXRD experiments, the typical accuracy in the θ_{B} is of the order of 0.002° (3.5 × 10^{−5} radians) for a bulk sample. However, for a film peak, the accuracy is much reduced (Schmidbauer et al., 2012). For example, a deviation of about 0.01° in the θ_{B} can lead to a deviation of 0.2° for the angle β for BTO at 409 K (paraelectric to ferroelectric point), as calculated from the 002, 013 and 103 diffraction peaks (Yang, 2012). Since β is very close to 90°, it is hard to judge which crystallographic system, tetragonal or monoclinic, the BTO belongs to. Such an error in the monoclinic angle is critical if the and symmetry at the MPBs of some of the multiferroic films mentioned above are to be recognized (Zeches et al., 2009).
(3) As the Xray beam cannot penetrate through the substrate in the case of the Xray wavelengths typically used in HRXRD setups, backreflection geometry has to be adopted in most experiments. A limited number of diffraction peaks from the film can be collected, again reducing the chance of obtaining accurate lattice parameters.
(4) The existence of
variants results in peaks being wrongly positioned (leading to wrong lattice parameters) or even in failure to locate the peak position, owing to widening and shifting of the peak positions.In these methods, inaccurate θ_{B} angles and fewer diffraction peaks make it hard to obtain accurate lattice parameters, and weaker diffraction intensities make the Bond methods less effective. More seriously, only a small number of d spacings (either accurate or less so) can be obtained, from which it is hard to deduce the Bravais lattice type. A `guess and check' method may often be used in this case, relying mostly on one's luck during the work.
For the procedure after `Bravais lattice' in Fig. 1, we note that a new method to solve and refine the crystal structure (atomic positions inside the crystal cell) has been demonstrated recently for CuMnAs film on a GaAs substrate (Wadley et al., 2013).
In this article, we focus on a reciprocalspace vector (RSV) method using an HRXRD setup, in order to overcome the abovelisted difficulties in lattice parameter determination for singlecrystalline epitaxial film. This method firstly obtains the Niggli basis vectors accurately, with the lattice parameters of the substrate as a reference. The Bravais lattice type together with the lattice parameters are then worked out. In this RSV method, the relationship between the lattices of the film and those of the substrate is clearly revealed, and the lattice parameters can be obtained with high accuracy. The procedure is as shown on the righthand side of Fig. 1.
2. Method 1: and RSVs
In epitaxial film characterization, twodimensional reciprocalspace mapping has been widely used to obtain the necessary RSVs or θ_{B} angles for the determination of the lattice parameters (Catalan et al., 2007; Chu et al., 2009; Daumont et al., 2010; Liu, Yao et al., 2010; Qi et al., 2005; Noheda et al., 2002; Bai et al., 2004; Liu, Yang et al., 2010; Saito et al., 2006). There is always a preset assumption that the crystal axes of the film are in the reciprocalspace map (RSM) plane. In other words, the angle between the crystallographic axes of the film is assumed to be the angle between the RSM planes.
In this section, an RSV method in threedimensional
is introduced, allowing a more accurate determination of crystal lattice parameters than reciprocalspace mapping without any preset assumptions. This method starts with the measurement of RSVs of the film, which is then corrected by rescaling and rotating with respect to the substrate RSVs. The Niggli cell is reduced and the Bravais lattice type with the lattice parameters is then finally worked out.2.1. Obtaining basis vectors in reciprocal and real space
In threedimensional HKL) or
an RSV can generally be represented as (i.e.
where H, K and L are the coordinates based on a selected basis in the along the H, K and L directions, respectively. The coordinate system for the with basis vectors a*, b* and c* is shown in Fig. 2. H, K and L are integers (Miller indices) if a*, b* and c* are chosen from the reciprocals of the basis vectors a, b and c in real space and the RSV falls on the lattice point, i.e. q_{0}; H, K and L may not be integers if another set of basis vectors for a*, b* and c* are chosen, for example, from the substrate lattice parameters, or when the RSV does not fall on the lattice point, i.e. q, in general. Note that there is another definition of the RSV and the scattering vector: Q = 2πq. For simplicity we discuss q here.
By properly selecting three RSVs, e.g. (00L), (H0L) and (0KL), three basis vectors a*, b* and c* can be obtained as follows:
or
The soobtained basis vectors in
may not be primitive and they may also have systematic errors from the measurement and the methodology. They can be rectified by making corrections to the RSVs with the substrate as a reference, as shown in the next section.2.2. Correction of RSVs using the substrate as a reference
As measured RSVs can be different in their lengths and orientations from the correct RSVs in s (scale factor) and a rotational part R (rotation matrix). Such systematic errors may arise from incorrect wavelength and inaccurate angle measurement in the experiment, as discussed above. The scale factor s denotes the length ratio of the correct RSV to its measured RSV. The rotation matrix R can be decomposed into two rotations about the H and K axes, respectively (see Giacovazzo et al., 2002, p. 76), if a righthanded Cartesian coordinate system H, K and L is set up (in Fig. 2). In other cases, a nonCartesian coordinate system could be converted into a Cartesian coordinate system and then a rotation correction applied (see Giacovazzo et al., 2002, p. 74).
the correction should be made and classified as a scaling partAs RSVs of a film are usually in the vicinities of those of the substrate, we have reason to conclude that systematic errors arising from the measurements for the film should be the same, or very close to, those for the substrate. The basic idea is to find a correction for the substrate RSVs first and then make the same correction to the film RSVs.
The corrected RSV (H_{1}K_{1}L_{1}) is related to the measured RSV (h_{1}k_{1}l_{1}) of the substrate (sub) by
As the corrected RSV (H_{1}K_{1}L_{1}) of the substrate is known, s and R can be derived from equation (4). We can then make the same correction to film RSVs:
The scale factor s is the length ratio of the RSV (H_{1}K_{1}L_{1}) to the RSV (h_{1}k_{1}l_{1}) for the substrate as given in equation (4). R can be represented as
where α and β are the rotation angles of (h_{1}k_{1}l_{1}) about the H and K axes, respectively, required to make (h_{1}k_{1}l_{1}) coincident with the corrected (H_{1}K_{1}L_{1}) for the substrate, i.e. a rotation correction. Counterclockwise rotation is positive.
Example for correction of RSV (013)
(−0.0007 0.9996 3.0006) and (−0.0011 1.0065 2.9002) were obtained for an STO substrate and a BiFeO_{3} (BFO) film, respectively. A correction with s = 0.9999, R_{H}(2.333 × 10^{−4}) and R_{K}(−1.800 × 10^{−4}) leads to (013) for the substrate, as it should be, with rotation angles of 0.01337 and −0.01031° about the H and K axes, respectively. The RSV for the BFO film is accordingly corrected as (−0.0004 1.0069 2.8996) using the same scale factor s and rotation matrix R.
2.3. Basis vectors of a film in real space
Using equation (2) or (3) with corrected RSVs from equations (4) and (5), a raw and a primitive with the shortest vectors a, b and c of the film in real space can thus be derived:
where V* and V are the volumes of the raw cell in reciprocal and real space, respectively.
2.4. Niggli cell and Bravais lattice of a film
Raw basis vectors a, b and c of the film can be derived from equation (9). The raw cell can be reduced to a standard Niggli cell, which should satisfy the conditions for two types that
Other main conditions and special conditions can be found in ch. 9.2, p. 750 of Hahn (2006) or on p. 19 of Ma & Shi (1995).
Using Table 4 of Andrews & Bernstein (1988) or Table 1 of Paciorek & Bonin (1992), the raw cell can be transformed into a standard Niggli cell by multiplying the transformation matrix. This transformation is actually a projection of the raw cell into a subspace in G^{6} space (to the standard type of Niggli cell), as described in equation (22) in the next section.
Using Table 9.2.5.1 of Hahn (2006, p. 753), the conventional Bravais lattice type and the lattice parameters can be derived from the standard Niggli cell using the transformation matrix M_{N→B}:
For example, the transformation matrix for Bravais lattice type monoclinic mC (No. 10) is
Example of determining a Bravais lattice
As partly shown in the example in §2.2, other RSVs of the same sample are corrected with a similar procedure using equations (4) and (5) to be (002) and (), respectively, from (0.0000 0.0000 1.9992) and (−0.9996 0.0000 2.9993) measured for the substrate; and to be (0.0000 0.0034 1.9382) and (−1.0004 0.0080 2.9009), from (0.0000 0.0034 1.9374) and (−1.0000 0.0080 2.9002) measured for the film. This correction leads to a raw cell with a = 3.898, b = 3.904, c = 4.030 Å, α = 90.36, β = 90.34, γ = 90.19°. This is the case for reduced basis No. 10, monoclinic mC lattice (Hahn, 2006, Table 9.2.5.1). From information given in Table 4 of Andrews & Bernstein (1988) or Table 1 of Paciorek & Bonin (1992), the raw cell can be reduced to a standard Niggli cell with a = b = 3.901, c = 4.030 Å, α = β = 90.35, γ = 90.19°. The Bravais lattice parameters are finally derived as a = 5.507, b = 5.526, c = 4.030 Å, α = 90, β = 90.50, γ = 90°, using the transformation matrix No. 10 (Hahn, 2006, Table 9.2.5.1), which is also represented in equation (15).
Such transformations can be performed using the ACCEL calculation program DeFLaP (determination of film lattice parameters) developed by the authors (available from the authors on request). As the incident beam has poor resolution in one direction, the γ angle has a larger error than the other angles/parameters; this will be discussed in §§3.3 and 4 below.
3. Method 2: vectors in G^{6} space and the unit cell
In this section, a vector in G^{6} space is treated to show how it represents a A similar, but simpler, correction can be made with the substrate as a reference, without separating the rescaling and rotating parts as above. The basis vectors of the Niggli cell are then derived and the Bravais lattice type is determined with the projection method as used in the previous section.
3.1. Representation of a in G^{6} space
A G^{6} space (Andrews & Bernstein, 1988). Actually, from the metric matrix
can be represented as a vector (point) in a sixdimensional Euclidean space, denoted asthere are only six independent components, forming a socalled Niggli matrix (Niggli, 1928):
This forms a vector g in G^{6} space:
Any one such vector in G^{6} space corresponds uniquely to one with the lattice parameters a, b, c, α, β and γ. This onetoone relationship provides us with a convenient method to derive lattice parameters by using this representation in G^{6} space. The correction of the lattice parameters of a becomes simpler as described below.
3.2. Correction of film lattice parameters in G^{6} space
In a real experiment, lattice parameters can be obtained by using, for example, equation (9) to convert them to realspace parameters for both substrate and film. There are, unavoidably, systematic errors in the measurement, as discussed in §2.2. If a correction vector Δg in G^{6} space is needed to obtain the standard lattice parameters for a substrate, the following equations can be used:
where sub indicates substrate. The correction vector Δg is written as
As the corresponding vectors in G^{6} space for substrate and film are close, the systematic error caused in an experiment should apparently be the same or close, as stated in §2.2. Then the same correction should be made to the vector in G^{6} space for the film as for the substrate:
From the calculation above, raw lattice parameters for the film can be derived with the correction Δg from the measured lattice parameters, with the substrate as a reference. This raw cell can be converted to a standard Niggli cell, and the Bravais lattice can finally be determined with equation (14) as shown below.
Using data from Table 4 of Andrews & Bernstein (1988) or Table 1 of Paciorek & Bonin (1992), the raw cell corrected by equation (21) can be transformed (projected) into the Nigglireduced cell by
where the raw cell is projected as a Nigglireduced cell (onto a subspace in G^{6} space) and M_{R→N} is the projection matrix from Table 4 of Andrews & Bernstein (1988) or Table 1 of Paciorek & Bonin (1992). The Bravais lattice will be subsequently determined using the transformation matrix M_{N→B} from equation (14).
3.3. Error estimation
The lattice parameters a, b, c, α, β and γ of the Niggli cell can generally be expressed by the projected vector (Nigglireduced cell) with an error vector δg in G^{6}, i.e.
The error or deviation of the reduced Niggli cell, , can be calculated using the distance between the projected vector [Nigglireduced cell, equation (22)] and its raw vector [raw corrected cell, equation (21)] in G^{6} space:
Alternatively, according to equation (22), we have
where E is a unit matrix.
Example 1 to estimate errors
For the Nigglireduced cell of a BFO film, as shown in the examples in §§2.2 and 2.4, we can calculate the deviation as follows:
From the first two components of equation (26), using equation (23) we have 2(a^{2} − 3.901^{2})^{2} = 0.021^{2} + (−0.021)^{2}, a^{2} − 3.901^{2} = and a = 3.901 (3) Å. From the fourth and fifth components of equation (26) and using equation (23) we have (2bc − 2 × 3.901 × 4.030 × )^{2} + (2ac − 2 × 3.901 × 4.030 × )^{2} = 0.008^{2} + (−0.008)^{2}. As the reduced cell has a = b = 3.901, c = 4.030 Å, α = β = 90.35, γ = 90.19°, we get 2 × [2 × 3.901 × 4.030 × ]^{2} = 2 × 0.008^{2}, 2 × 3.901 × 4.030 × = , −2 × 3.901 × 4.030 × × radians = and = . The calculation above is intended only to illustrate the procedure, and the number of significant figures of the values is not always consistent.
Example 2 to estimate errors
To reduce the Niggli cell of a tindoped indium oxide (ITO) film, raw lattice parameters were obtained from RSVs of the film such that a = 5.0683, b = 5.0698, c = 5.0694 Å, α = 89.994, β = 89.958, γ = 89.992°. This is the case for reduced basis No. 3, cubic cP lattice (Hahn, 2006, Table 9.2.5.1). From Table 4 of Andrews & Bernstein (1988) or Table 1 of Paciorek & Bonin (1992), the Nigglireduced cell was projected such that a = b = c = 5.0691 Å, α = β = γ = 90°. The deviation is calculated to be
Using equation (23), the cubic cell can then be expressed as
From the first three components, we have 3(a^{2} −5.0691^{2})^{2} = 0.0090^{2} + (−0.0062)^{2} + (−0.0027)^{2} = (0.0112)^{2}. Hence, a = 5.0691 (6) Å for the cubic ITO film.
4. Experimental conditions and resolution
Using a diffractometer, angles or diffraction positions for a diffraction peak can be measured exactly, from which the corresponding scattering vector q or RSV coordinates can be obtained.
4.1. The scattering vector in fourcircle diffractometer coordinate systems
A standard fourcircle diffractometer is used, as an example, and the coordinate systems are as shown in Fig. 3. The coordinate system convention is as proposed in the SPEC manual (2008 version, p. 163, http://certif.com/ ). Three orthogonal and righthanded coordinate systems are established: (1) the laboratory coordinate system [fixed frame in laboratory, Fig. 3(a)], (2) the diffractometer coordinate system [angular as shown in Fig. 3(a) with Euler circles 2θ, ω, χ and φ] and (3) the sample coordinate system [fixed with sample natural axes, as shown in Fig. 3(b) with a scattering vector q]. Circles 2θ, ω and φ are defined as right handed and χ as left handed. Another definition of the coordinate systems and rotations has been reported by Busing & Levy (1967).
As mentioned in §2.1, the scattering vector Q = K_{2} −K_{1} or Q = 2πq, where
and is oriented with χ and φ angles as shown in Fig. 3(b). K_{1} is the wavevector of the incident Xray beam and K_{2} is the wavevector of the scattered Xray beam (K_{1} = K_{2} = 2π/λ). Correspondingly, k_{1} is the wavenumber vector of the incident Xray beam and k_{2} is the wavenumber vector of the scattered Xray beam, as shown in Fig. 3 (k_{1} = k_{2} = 1/λ). q has component q_{z} along the z direction and q_{φ} in the xy plane. If q lies along the normal direction of the Bragg planes and satisfies the Bragg condition, its magnitude q is then equal to a reciprocalspace vector and diffraction occurs.
In the case shown in Fig. 3(c), the scattering vector q is firstly rotated into the i.e. the xy plane in the coordinate system with angle θ–ω against the q_{z} axis, and then it is rotated to the Bragg condition. For example, BFO (103) can be rotated by χ = −90°. q_{z} is now along the x direction. It is then rotated another 90° in φ such that (103) is in the xy plane at an angle against q_{z}. With a subsequent rotation ω in the one then gets diffraction. The incident Xray beam is at angle α (≡ ω) and the diffracted Xray beam at angle β to the component q_{φ}. The magnitude q is determined as
where k = 1/λ, θ = (2θ)/2, α = ω, β = 2θ−ω and α + β = 2θ.^{1} If ω ≠ θ, i.e. an asymmetrical setting, then α ≠ β and the angle ω does not need to be rotated by θ to satisfy the Bragg condition as shown in Fig. 3(c); if ω = θ, a symmetrical setting, then α = β = θ and ω needs to be rotated by θ to satisfy the Bragg condition.
If the reverse rotations of the sample for ω, χ and φ are performed, the components of the scattering vector q in the original coordinate system (i.e. before it was rotated into the Bragg condition) can be traced back as
Similar results can be found on p. 154 of Aslanov et al. (1998) and p. 284 of Bennett (2010). For an RSV in an orthorhombic system, we have
where a*, b* and c* are the lattice parameters in For other crystal systems lower than the orthorhombic system, the B matrix should be used to convert from equation (31) to equation (32) (Aslanov et al., 1998; Bennett, 2010; Busing & Levy, 1967). If we set k = 1 and ω = θ in equation (31), the result is just the coordinates obtained for the case of the symmetrical setting as commonly used in fourcircle diffractometers.
4.2. Xray beam conditions and resolution in RSV measurement
Typical experimental conditions are given in Table 1 for the diffraction station on beamline BL14B1 at the Shanghai Synchrotron Radiation Facility (SSRF). A substrate of LaSrAlO_{4} (LSAO) was tested for this purpose. Δα and Δχ were estimated from the full width at halfmaximum (FWHM) of rocking scans measured for vertical and horizontal divergence, respectively, of the incident beam. Δβ was obtained from a 2θ scan. These are combined widths, as a result of convolution of the instrumental widths with the crystal diffraction widths of the LSAO substrate. As the intrinsic diffraction width of LSAO is much smaller (of the order of arcseconds), these combined widths can be used to represent the divergences of the incident Xray beam.

The worst errors obtained when measuring peak positions for general RSVs, denoted as δα, δχ and δβ, are estimated as δα ≤ ±Δα/2, δχ ≤ ±δχ/2 and δβ ≤ ±Δβ/2, respectively, which serve as a kind of accuracy of the peak positions. Furthermore, we adopt half of δα, δχ and δβ as the standard uncertainties to describe the margins of error for the measurements, which are much larger than the angular precisions of the circles in the diffractometer.
Differentiating equation (31), the deviation of the scattering vector or RSV can be derived. For RSV (200), symmetrical setting, χ = 0°, φ = 0° and ω = θ, it is worked out as
Apparently direct rocking angles α (or φ now) and χ measure the divergences Δα and Δχ, respectively, with δφ = δα in this case.
For RSV (013), asymmetrical setting, χ = −90° and keeping φ unrotated at φ = 0°, χ_{0} = 90°−(θ−ω), we have a kind of fixedφ case (mode 3 in the SPEC manual). The deviation is worked out as
The deviations in H, K and L are expressed as
In the equations above, q is as shown in equation (30), δq = (δθ), δθ = (δα + δβ)/2, δω = δα and δφ = δχ in the asymmetric case. For STO (a = 3.9053 Å), the worst errors for measuring (013) in mode (3) as described in §4.1 can be calculated to be, on average, less than 0.001 for H, K and L, using equation (35). We choose its halfvalue, i.e. 0.0005, to calculate the standard uncertainties for the lattice parameters. For the STO {200} type, the uncertainty is even less, owing to the simpler and symmetrical diffraction condition. In a real measurement, it is below 0.0005 in H, K and L from our experience in this work. This estimation should be reasonable and reliable.
In the example above of cubic ITO film, the lattice parameter can be determined with an error of 0.0006 Å, which is estimated from the projection error [equation (27)] of the raw cell to the Nigglireduced cell. The lattice parameters are even better determined than for other films, owing to the ITO film's sharp and strong diffraction peaks.
5. Structural study of ferroelectric films
In this section, the determination of crystal lattice types and lattice parameters for two ferroelectric films is demonstrated. One film is single crystalline and the other is twinned.
5.1. and lattice parameter determination for a PbZr_{0.52}Ti_{0.48}O_{3} (PZT 52/48) film
The RSVs of a PZT 52/48 film on an STO substrate were measured at the SSLS and at the SSRF, both showing very close measured RSVs as below. An SrRuO_{3} (SRO) layer as a bottom electrode was grown between the film and substrate. As such, the PZT film is near the MPB composition; its structure is puzzlingly between monoclinic and tetragonal symmetry.
The RSMs in Fig. 4 show that a single centered spot is formed for each RSV, indicating that no twin exists in the film or the substrate. For the STO substrate, (002), (03) and (013) were measured in threedimensional at the SSRF as (0.0004 −0.0011 2.0001), (−1.0005 0.0011 3.0001) and (0.0034 0.9986 3.0004).
From the above set of RSVs, the raw unitcell parameters of the substrate can be derived with very good precision such that a = 3.9005, b = 3.9043, c = 3.9048 Å, α = 89.984, β = 90.015, γ = 90.007°.
For the PZT film, corresponding RSVs were measured also at the SSRF as (0.0095 −0.0011 1.9010), (−0.9422 0.0001 2.8544) and (0.0154 0.9578 2.8504).
The raw cell was calculated for the PZT film such that a = 4.0873, b = 4.0710, c = 4.1085 Å, α = 89.917, β = 90.097, γ = 89.963°, with the same correction made as for the substrate lattice parameters, i.e. cubic a = 3.9053 Å. A tetragonal cell was finally deduced such that a = 4.079 (8) , c = 4.109 (2) Å, where the error in a is calculated using the difference from the projected cell and that in c is estimated from the accuracy in measuring RSVs as discussed in the sections above.
To confirm the tetragonal system and lattice parameters, a symmetry test has been conducted. The diffraction data and derived structure factors are given in Table 2. 2θ_{B} has very close values for the four diffraction vectors of the {103} family. Although the intensity correction made for the sample (irregular film shape) was not perfect, it can be seen that the deduced structure factors are also close for the family. So the crystal lattice of the PZT film has tetragonal symmetry with the parameters as shown above.

5.2. and latticeparameter determination for a BFO film
As shown for the PZT film above, the RSV method can be used to derive the shows the RSMs of a BFO film on an LSAO substrate, grown by pulsedlaser deposition (Chen, Luo et al., 2011; Chen, Luo et al., 2010; Chen, Prosandeev et al., 2011). There are four phases coexisting in the film, as shown in Fig. 5(a), i.e. a bulk rhombohedrallike phase (marked as Rlike), a tetragonallike monoclinic phase (Tlike, M_{C}), a tilted rhombohedrallike phase (Tri1, 1 and 2) and a tilted tetragonallike phase (Tri2, I and II). All phases are also indicated in the atomic force microscopy (AFM) topograph in Fig. 5(d).
and lattice parameters for a singlecrystalline epitaxial film. For a twinned film, it is also possible to obtain the crystal structure parameters if the variants can be sorted out from each other to form one consistent set of RSVs. Fig. 5The RSVs for the substrate (denoted as LSAO in the mappings) were measured to be (−0.0003 0.00034 1.9999), (−1.0001 0.00041 3.0000) and (−0.00027 0.99983 2.9999).
From the above set of RSVs, raw unitcell parameters of the substrate can be obtained with very good precision such that a = 3.7577, b = 3.7590, c = 12.637 Å, α = 90.00, β = 89.99, γ = 90.02°. These lattice parameters should be corrected to the standard tetragonal ones so that a = 3.7564, c = 12.636 Å. To work out the lattice parameters for the Tri1 phase in the BFO film, we select only one set of variants out of the eight variants; the selected variants are marked as 1 in Fig. 5(a), 1 in Fig. 5(b) and 1a in Fig. 5(c). The corresponding RSVs were measured to be (−0.0881 −0.0084 2.0170), (−1.0872 −0.0061 2.9680) and (−0.1371 0.9710 3.0463).
Careful identification was made to confirm and measure the above RSVs, as the spots of this set of the rhombohedrallike phase are not located in any coordinate plane of the (a) and 5(b), marked as 1 or 2. The same is true of the spots of the Tri2 phase, marked as I or II.
(in the substrate coordinate system). Only a tracelike spot can be seen in Figs. 5The raw cell for the Tri1 phase was calculated such that a = 3.9265, b = 3.8163, c = 4.1717 Å, α = 90.817, β = 90.297, γ = 89.359°, with the same correction made as for the substrate lattice parameters, to convert to the standard tetragonal parameters. A triclinic cell was finally obtained, such that a = 3.927 (3), b = 3.816 (3), c = 4.172 (1) Å, α = 89.18 (4), β = 89.70 (4), γ = 89.36 (6)°, where the errors in the parentheses were estimated from the deviations in measuring the RSVs as discussed above.
The Tri2 phase can be calculated to be triclinic, the Rlike phase to be monoclinic M_{A} and the Tlike phase to be monoclinic M_{C}.
A similar example for a BFO film on an LAO substrate has been investigated (Chen, Prosandeev et al., 2011), from which triclinic cells for the Tri1 and Tri2 phases, a monoclinic M_{A} cell for the Rlike phase and a monoclinic M_{C} cell for the Tlike phase have been concluded.
6. Discussion and summary
There is often coexistence of multiple martensitelike twin variants in metallic alloys, intermetallics or oxide films. The distortion of the cell might be very large and there are possibly overlapping peaks. In order to get one set of correct RSVs, one should carefully recognize and separate the peaks, with a highresolution diffractometry setup using an intense synchrotron source. The example in Fig. 5 shows that one set of RSVs for every morphotropic phase in the BFO film has been separated and the lattice parameters can finally be determined. With separated peak intensities, one could obtain the fractional ratio of twin variants, e.g. for the abundance of adomains and cdomains in PZT films (Lee & Baik, 1999; Nagarajan et al., 1999). Otherwise, a kind of averaged structure is obtained if the peaks cannot be resolved. Such a crystal structure could still be solved and refined using some of the suitable twin laws as in a normal crystal structure analysis procedure (Sheldrick, 2008), from which the variant ratio could also be worked out.
In summary, we have developed a procedure, based on RSVs, to determine the Bravais lattice type and the lattice parameters for an epitaxial film. Three independent (noncoplanar) reciprocal vectors (00L), (H0L) and (0KL) are firstly obtained and corrected using the substrate as a reference. The three shortest vectors are then deduced to form the Nigglireduced cell. The Bravais lattice type is finally determined and the lattice parameters are calculated accordingly. Such a procedure could be performed by converting or selecting the corresponding vectors in real space as well. An error of 0.001 Å or better could be achieved. Some structures of multiferroic films have been successfully determined using the RSV method (Chukka et al., 2011; Chen, Ren et al., 2011; Chen, Luo et al., 2011, 2010; Chen, Prosandeev et al., 2011; Chen, You et al., 2010; Chen et al., 2012; Kumar et al., 2013; Liu et al., 2012, 2011, 2013; Liu, Yao et al., 2010; Liu, Yang et al., 2010; Saito et al., 2006).
The RSV method has the following advantages:
(1) It is a concise and direct method for calculating the lattice parameters from the three reciprocalspace vectors or a sixdimensional vector in G^{6} space, without any prior knowledge or assumption of the crystal structure. In other methods, more diffraction data and iteration are needed for indexing and leastsquares as described in Fig. 1. Hence, the experiment duration for the RSV method is shortened, thus saving beam time.
(2) It is an accurate method using the substrate as a reference, independent of Xray wavelength and accounting for instrument misalignment. In other methods, the wavelength should necessarily be calibrated if synchrotron Xrays are used. Although the d values can be more accurately obtained using the Bragg equation with more accurate θ_{B} angles, for example, in the Bond methods, there is still a difficulty in determination of the lattice type: a `guess and check' method has to be used. Moreover, such methods could cause inaccuracy in determination of the lattice parameters, especially in monoclinic angles when they are close to 90°, and particularly when the film diffraction intensity is low, as discussed in the Introduction. For further discussion refer to the articles by De Caro & Tapfer (1998), Shilo et al. (2001), Karmazin & James (1972) and Fatemi (2005).
(3) It offers not only lattice parameters but also basis vectors a, b and c in real space in the framework of the substrate coordinate system, i.e. the orientations of the crystal basis vectors for both the film and the substrate. Analyses on the length and orientation of the basis vectors of the film relative to that of the substrate will yield information on lattice mismatch (strain status), crystallographic tilt and step bunching in the surface terrace (Kim et al., 2011). It is worth mentioning that such basisvector orientation analyses may lead to an understanding of the interface structure formation between film and substrate. For example, a rotation of the lattice network about the normal of the surface will result in a twist boundary between film and substrate, while rotation about an inplane axis will result in a smallangle boundary at the interface. Further exploration of the RSV method as applied to the study of interfaces is proposed.
Footnotes
^{1}Note that these angles of α and β are unrelated to those defined in the rotation matrix of equations (6), (7) and (8) or those in the conventional symbols for crystal lattice parameters.
Acknowledgements
The authors are grateful for the technical support received from beamline BL14B1 of the SSRF for the data collection under project Nos. 11sr0395 and j10sr0092. PY is supported by the SSLS via NUS Core Support C380003003001. PY acknowledges support from the Alexander von Humboldt Foundation under the ID 1031847. JW acknowledges the grant support of MOE, Singapore Ministry of Education Academic Research Fund Tier 1 (grant No. T110702P06, R284000054112). We thank Professor Tom Wu and Dr Rami Chukka for the samples.
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