Atomic disorder of Li0.5Ni0.5O thin films caused by Li doping: estimation from X-ray Debye–Waller factors

The structures of room-temperature epitaxial Li0.5Ni0.5O and NiO thin films are compared. A structural parameter for the atomic disorder is discussed and evaluated using X-ray Debye–Waller factors.


Introduction
Li x Ni 1Àx O has attracted considerable attention recently as a potential material for high-performance electrochromic devices (Moulki et al., 2012), UV detectors (Ohta et al., 2003) and gas sensors (Garduno-Wilches & Alonso, 2013). Recently, cubic type NiO thin films with (111) orientation containing large amounts of Li (up to 50 mol%) were epitaxically grown on ultra-smooth sapphire (0001) substrates by roomtemperature (RT) pulsed laser deposition (Shiraishi et al., 2010;Yang et al., 2014). The ultra-smooth sapphire substrate used in this work has an atomic 'step and terrace' structure on its surface, which consists of atomically flat terraces comprising single oxygen layers separated by periodic atomic steps with 0.22 nm height corresponding to the oxygen layer spacing along the c axis. These films could be grown even though bulk NiO containing over 30 mol% Li undergoes a phase transformation from the cubic to rhombohedral structure (Goodenough et al., 1958).
The Debye temperature is an important characteristic structural parameter of a solid and describes the dynamic motions of atoms in the material. A number of physical parameters such as mean-square atomic displacement (Herbstein, 1961) and elastic constant (Gazzara & Middleton, 1964) depend on the Debye temperature of a solid. Therefore, it is very important to determine the Debye temperature of Li x Ni 1Àx O in order to understand its physical properties, such as the change of heat capacity caused by Li doping, and develop related electronic devices. X-ray diffraction is a valuable method to study thin films and other epitaxial layer materials. The changes of the lattice constant and crystal quality caused by doping are frequently evaluated using this method (Dutta et al., 2010;Palatnikov et al., 2006). However, the effect of atomic disorder caused by doping elements on the X-ray diffraction intensity has seldom been estimated (Sakata & Hashizume, 1997). In this study, we attempted to quantitatively analyze the effect of atomic disorder caused by Li doping on the X-ray diffraction intensity of an Li 0:5 Ni 0:5 O thin film.

History of Debye temperature estimation by X-ray diffraction
Originally, the Debye temperature was determined using X-ray diffraction and was strictly limited to monoatomic cubic crystals. The integrated intensity, I, from a cubic sample can be expressed as (James, 1962) where K is a constant, which is a normalized factor depending on the experimental arrangement, L hkl p is the Lorentz-polarization factor ð1 þ cos 2 2 hkl Þ=ð2 sin 2 hkl cos hkl Þ, h, k and l are the Miller indices of the diffraction plane, is the Bragg diffraction angle, jFj is the modulus of the structure factor, and D is the Debye-Waller factor, expressed as expðÀ2B sin 2 = 2 Þ. Here (James, 1962) and u 2 is the mean-square displacement of atoms from the sites of the average lattice of the solid solution. m is the mass of the atom, k B is the Boltzmann constant, his Planck's constant divided by 2, M is the Debye temperature and T is the experimental temperature. Later, the Debye theory was extended to apply to binary alloys, such as KCl and NaCl, whose specific heats can be fairly well described by the Debye-Waller formula with an appropriate value of the characteristic temperature (James, 1962). The structure factor for binary alloys is formally written as (Murakami, 1953) FðhklÞ Here f is the atomic scattering factor, x, y, z are the atomic positions in the unit cell, and the subscripts A and B refer to the two components of the alloy. r A and r B are the atomic fractions of A and B, respectively. To perform a simple approximation (Kulkarni & Bichile, 1977;Wathore & Kulk-arni, 1980;Pathak & Trivedi, 1973), we let B A = B B = B, and in equation (1) use m = r A m A þ r B m B . This approximation allows a quasi-Debye temperature to be obtained. The experimental results summarized by Lonsdale (1948) are in reasonable agreement with this approximation.

Introduction of a structure parameter
The Debye temperature has been evaluated from X-ray intensities not only at different temperatures for a Bragg peak (Herbstein, 1961) but also from different reflections at a given temperature (Herbstein, 1961;Kulkarni & Bichile, 1977).
Here, a real Debye-Waller factor D 1 for an Li 0:5 Ni 0:5 O thin film was obtained by X-ray diffraction using two different reflections, 111 and 222, with the assumption that the rock-salt lattice is a simple cubic structure composed of one kind of atom and the Li atoms occupy Ni substitutional positions without any atomic disorder. We also assume that the thermal disorder contributes to the ratio of the diffracted intensities between the 111 and 222 reflections for simplicity. Second, a pseudo-Debye-Waller factor D 2 for the Li 0:5 Ni 0:5 O thin film was obtained from the X-ray 111 intensity ratio between Li 0:5 Ni 0:5 O and NiO. Finally, we introduce an atomic disorder parameter exp() ( 0), including the combined effects of thermal vibration, interstitial atoms and defects expressed using D 1 , D 2 and the reported NiO Debye temperature (Freer, 1981). This term is required because the positions of atoms are changed by Li doping. This process is outlined in Fig. 1.

Samples and experiment
Li x Ni 1Àx O thin films were epitaxically grown on ultra-smooth sapphire (0001) substrates (Yoshimoto et al., 1995) by RT pulsed laser deposition. Details of the deposition conditions have been reported elsewhere (Shiraishi et al., 2010;Yang et al., 2014). The compositions of NiO thin films containing large amounts of Li grown at RT were determined using inductively coupled plasma atomic emission spectrometry (Shimadzu ICPS-8100). The lithium contents x in the Li x Ni 1Àx O samples were 0 and 0.5. The periodicities of the intensity oscillations appearing in the X-ray reflectivity curves revealed that the thicknesses t of the NiO and Li 0:5 Ni 0:5 O epitaxial thin films were 400 (21) and 231 (6)   . The cube-onhexagonal epitaxial relationship was the same as that reported previously for similar samples (Sakata et al., 2004;Yang et al., 2014). The ' scan around the nonspecular NiO (111) Bragg positions showed sixfold symmetry (not shown here). This is because two types of terraces formed alternately on the sapphire (0001) face, with 0.2 nm-high steps (Yamauchi et al., 2012). We also recorded the X-ray intensities of six 111 and 222 diffraction peaks.  This suggests that the crystal quality was lowered after Li doping.
The average integrated intensity ratio R 1 between the 111 and 222 Bragg peaks of the Li 0:5 Ni 0:5 O epitaxial thin film was calculated to be 2.5 (2). The contribution of thermal diffuse scattering (TDS) to the measured intensity was neglected in our analysis, because the TDS correction was small (less than 4% even in imperfect single-crystal silicon; Matsumuro et al., 1990;Chipman & Batterman, 1963) in our samples compared with the statistical fluctuation, which was about 7.6% in the NiO thin film. If we assume that all Li atoms are substituted at Ni atom positions, the composition of the Li 0:5 Ni 0:5 O epitaxial thin film is uniform and structural disorder is neglected, the diffracted intensity from the Li 0:5 Ni 0:5 O epitaxial film is The values of S cal 1 are estimated to be 0.331 and 0.328 when we assume that the NiO and Li 0:5 Ni 0:5 O films have the NaCl crystal structure and use the atomic scattering factors (Brown et al., 2006) for their neutral atoms (Li, Ni and O) and ions (Li þ , Ni 2þ and O 2À ), respectively. The calculated results were almost the same. Let us define the observed integrated intensity ratio R 1 : The averaged value of R 1 for the six equivalent reflections was 2.5 (2). Substituting equation (4) into equation (6) allows us to derive the formula for B 1 as follows: The B 1 factor of Li 0:5 Ni 0:5 O was calculated to be 1.8 (4) Å 2 when we used parameters observed from the Li 0:5 Ni 0:5 O films: 111 = 12.19 and 222 = 24.88 , and (111) lattice spacing = 2.368 (4) Å . The corresponding Debye temperature was 281 (39) K according to equations (1) and (2). The error originates from that of B 1 , and it is a little larger than that already determined by X-ray diffraction (Herbstein, 1961), so it may be caused by the large statistical fluctuation of integrated intensity. The corresponding r.m.s. amplitude of atomic vibration was evaluated to be 0.15 Å , as shown in Table 1. The ratio r between the r.m.s. amplitude of atomic vibration and the (111) lattice spacing of the Li 0:5 Ni 0:5 O thin film was 6.3%. This value is larger than that of ideal bulk NiO (4.6%; Freer, 1981), which suggests that Li doping increased the lattice thermal vibration (Borca et al., 2000). D 1 is written as exp½À2B 1 ðsin 1À11 =Þ 2 and is here called the real Debye-Waller factor for the Li 0:5 Ni 0:5 0 thin film.

Estimation of the effect of atomic disorder caused by Li doping on the X-ray diffraction intensity
In order to obtain the structure parameter that can be used to estimate the effect of atomic disorder [exp()] caused by Li doping on the X-ray diffraction intensity of an Li 0:5 Ni 0:5 O thin film, let us propose a factor D 2 from the X-ray 111 intensity ratio between Li 0:5 Ni 0:5 O and NiO as shown in Fig. 1. If we assume that all of the Li atoms are substituted for Ni atoms and the composition of the Li 0:5 Ni 0:5 O epitaxial thin film is uniform (Fig. 3a), then structural disorder can be neglected.
Let us define the measured average intensity ratio R 2 between the 111 Bragg peaks of the Li 0:5 Ni 0:5 O and NiO films: The average value of R 2 for the six equivalent reflections was 0.029 (4). We also define S cal 2 jFð111Þ cal Li 0:5 Ni 0:5 O j 2 jFð111Þ cal NiO j 2 : The value of S cal 2 is 0.148 when we assume that the NiO and Li 0:5 Ni 0:5 O films have the NaCl crystal structure and use the atomic scattering factors for their neutral atoms. The formula for our proposed parameter B 2 is obtained after we substitute equation (4) into equation (8): B 2 was evaluated to be 6.5 (15) Å 2 using the B NiO value of 0.939 Å 2 obtained from M (317.4 K at RT;Freer, 1981) of NiO according to equations (1) and (2). We also used parameters ( NiO 111 = 12.05 ) observed from the NiO film. These obtained parameters, the r.m.s. amplitude of atomic vibration and its ratio compared with the lattice spacing are listed in Table 2. D 2 is written as exp½À2B 2 ðsin 111 =Þ 2 and is here called the pseudo-Debye-Waller factor for the Li 0:5 Ni 0:5 0 thin film. B 2 is much greater than B 1 . We do not consider the atomic disorder when obtaining the B 1 value. To consider the difference between B 1 and B 2 , let us assume the relation of D 2 and D 1 using the atomic disorder parameter expressed as D 2 ¼ expðÞD 1 . expðÞ was evaluated to be 0.66 using NiO as reference. This value is much smaller than unity, which suggests that the range in which the atoms are located might be ca 0.14 Å wider than that for B 1 . In general, Debye-Waller factors may include not only the influence of the thermal disorder but also that of the atomic disorder. The structural disorder that we evaluated looks to have a very small influence on B 1 . Furthermore, let us assume another extreme case where all of the Li atoms are located on interstitial sites. Interstitial Li atoms are found not only in cubic Li-doped NiO (Guo et al., 2013) but also in many other crystal structures like ZnO (Chawla et al., 2009), ZnSe (Sasaki et al., 1993) and Li 3 NbO 4 (McLaren et al., 2004). The positions of the Li atoms are (1/4, 1/4, 1/4), (3/4, 1/4, 1/4), (3/4, 3/4, 1/4), (1/4, 3/4, 1/4), (1/4, 1/4, 1/2), (3/4, 1/4, 1/2), (3/4, 3/4, 1/2), (1/4, 3/4, 1/2), as shown in Fig. 3(b). Therefore, the structure factor of Li 0:5 Let us introduce B 3 for all the Li atoms sitting in interstitial sites. The value is 3.1 (15) Å 2 using equations (8) and (11), as shown in Table 2. The difference between B 3 for the Li interstitial model and B 1 is smaller than that between B 2 for the Li substitute model and B 1 . This implies that some Li atoms are probably located in the interstitial sites in a heavily doped Li 0:5 Ni 0:5 O thin film. These results help us to understand the physical properties of Li x Ni 1Àx O to develop Li x Ni 1Àx O-based devices.

Concluding remarks
In conclusion, the conventional B 1 factor of an Li 0:5 Ni 0:5 O thin film was estimated to be 1.8 (4) Å 2 using X-ray diffraction by considering the 111 and 222 reflections. The corresponding Debye temperature was 281 (39) K. Furthermore, the pseudo-Debye-Waller factor of the Li 0:5 Ni 0:5 O thin film was obtained using the intensity ratio between the 111 Bragg peaks of Li 0:5 Ni 0:5 O and NiO thin films. The atomic disorder parameter that we proposed was evaluated to be 0.66. The disorder may include combined effects of thermal vibration, interstitial atoms and defects caused by Li doping. Interstitial Li atoms could be present in heavily Li-doped Li 0:5 Ni 0:5 O thin films because the proposed B 3 factor was smaller than B 2 determined for a film with Li ions only in Ni sites.

Figure 3
Crystal structure models of Li 0:5 Ni 0:5 O with (a) all of the Li atoms substituted for Ni atoms and (b) all of the Li atoms located in interstitial sites.