High-resolution x-ray diffraction of epitaxial bismuth chalcogenide topological insulator layers³

Topological insulators represent a new class of materials exhibiting unique electronic properties. Their bulk electronic structure corresponds to a conventional insulator with an open band gap, while at the surface a gapless two-dimensional (2D) topological surface state (TSS) is formed exhibiting a Dirac-like momentum-energy dispersion and spin-momentum locking [1–3]. In the prototypical bismuth chalcogenide topological insulators Bi₂X₃ (X  =  Te, Se), the surface states are topologically protected by time-reversal symmetry and are immune to surface impurities and backscattering.

The bismuth chalcogenide topological insulators exist in a variety of rhombohedral/ hexagonal phases with different stoichiometric composition Bi_mX_n (denoted (mn) in this work) [4–6]. This creates a homologous series of structures that differ in the atomic layer stacking sequences. In this paper we use the hexagonal 4-index notation of crystallographic planes and vectors in reciprocal space, which follows from the hexagonal elementary unit cell; this representation is used instead of the rhombohedral description. The (mn) phases differ in the sequence of Bi and X basal (0 0 0 1) atomic planes [6–9] along the c-axis of the lattices, however, only the (23) phase exhibits the topologically protected surface states.

Every ideal (mn) phase can be described as a sequence of X-Bi-X-Bi-X quintuple layers (QLs) and Bi–Bi double layers (DLs) [6]. In particular the most prominent (23) phase contains only QLs weakly bonded by van der Waals forces. Other phases like (43), (11), and (45) contain a certain number N of QLs between two subsequent DLs as exemplified by figure 1. Thus, N is related to (mn) by stoichiometry considerations by the simple relation:

$$\begin{eqnarray}&&N=\frac{2}{3m/n-2}=\frac{3}{\delta}-1.\end{eqnarray} \tag{ 1 }$$

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Figure 1 show crystal structures of phases (23), (11), and (43), where the Bi–Bi DLs are highlighted as shaded areas, the elementary unit cells are denoted by black rectangles.

A number of works has been devoted to the investigation of bulk Bi_mX_n crystals, an extensive database of the results can be found in [9–13]. However, little is known about the real structure and crystal quality of Bi_mX_n thin films grown by molecular beam epitaxy (MBE). In our previous papers [14–16] we investigated the crystal structure of MBE grown epitaxial Bi₂Te₃ and Mn-doped Bi₂Te₃ layers using high-resolution x-ray diffraction (HRXRD) and x-ray absorption spectroscopy (EXAFS) and angle-resolved photoelectron spectroscopy (ARPES). We found that the chemical composition, crystal structure as well as the crystal quality substantially depend on the excess flux of Te during the MBE growth. In this paper we present a comparison of HRXRD results on Bi_mTe_n and Bi_mSe_n layers grown with different excess chalcogen (Te, Se) fluxes. We show that the trend of improvement of the crystal structure with the increasing chalcogen flux is common for both investigated systems, but the HRXRD diffraction curves and the dependence of the lattice parameters on the chalcogen flux is substantially different. Our results indicate that the perfection of the stacking order is significantly better in the selenide compared to the telluride system.

The Bi_mX_n epitaxial layers were grown by MBE on BaF₂ (1 1 1) substrates, using compound bismuth telluride, respectively, bismuth selenide effusion sources with nominal Bi₂X₃ composition. Additional pure Te (or Se) source was used to adjust the total chalcogen to bismuth flux ratio and thus, to tune the stoichiometry of the resulting films. For Bi₂Te₃, the additional Te flux varied between 0 and 2.4 Å s⁻¹, while for Bi₂Se₃ layers we used Se fluxes up to 1.4 Å s⁻¹, while the flux rate from the bismuth selenide and bismuth telluride sources was kept constant. The growth temperature was set constant to 350 °C for all samples, the layer thicknesses were between 350 and 500 nm. The details of the growth can be found in [15].

HRXRD measurements were carried out on a laboratory x-ray diffractometer equipped by a Cu sealed tube. A multilayer x-ray mirror and a channel-cut Ge monochromator were used to produce a parallel x-ray beam with CuKα wavelength. The scattered radiation was detected by a linear detector. Figure 2 shows the diffraction curves of the Bi_mTe_n (upper panel) and Bi_mSe_n series (lower panel) measured in symmetric reflection geometry, i.e. along the 000L crystal truncation rod. The blue (red) vertical dotted lines represent the theoretical positions of the 000L diffraction maxima in the (23) and (11) phases, respectively (see below). Peaks denoted 'S' are the diffraction maxima of the BaF₂ substrate. The shape of the diffraction curves considerably changes with the chalcogen to bismuth flux ratio. With increasing chalcogen flux, some peaks strongly change their position; for instance the maximum at 000.17 in the (11) phase converts to 000.21 in the (23) phase in both investigated material systems. Some peaks remain almost unchanged—for instance, the 000.12 peak of the (11) phase, which is equivalent to 000.15 in the 23 phase; some pairs of peaks merge creating a strong peak in the (23) phase—for instance the 000.9 and 000.10 peaks of the (11) phase merge to 000.12 in the (23) phase. For Bi_mTe_n, the peaks 000.9 and 000.10 for (11) also merge with increasing Se flux, but eventually completely disappear when reaching the (23) phase.

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Figures 3 and 4 present reciprocal-space maps around asymmetric Bragg points measured in coplanar geometry for Bi_mTe_n (figure 3) and Bi_mSe_n (figure 4) films grown in each case with either a small or large chalcogen fluxes. In the maps we recognize the substrate 313 peak (denoted 'S') and the layer peaks ('L') 1–10.20 of Bi₂Te₃, 1–10.19 of Bi₂Se₃, and 1–10.16 of Bi₁Te₁ and Bi₁Se₁.

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In the following, we present a simple structural model that enables to evaluate the HRXRD diffraction curves and determine the basic structure parameters. We start from a hypothetic hexagonal crystal structure built of identical virtual atoms. The structure consists of an ABCABC... stacking of two-dimensional hexagonal atomic planes with the inter-planar distance d. Such crystal produces a periodic sequence of the diffraction maxima along the symmetric rod 000L at the reciprocal space coordinates Q_p

$$\begin{eqnarray}&&{{Q}_{p}}=\frac{2\pi}{d}p,\end{eqnarray} \tag{ 2 }$$

where p is integer. Now we introduce a periodic chemical modulation replacing the virtual atoms by the Bi and X  =  Se,Te atoms. As we showed above, the structure period of a (mn) phase consists of N QLs (determined using equation (1)) separated by a Bi–Bi DL. In the following we consider only the structures having a single Bi–Bi DL between two subsequent QL blocks; the N values of these structures following from equation (1) must be integer. The distance D between two subsequent DLs in these lattices is

$$\begin{eqnarray}&&D=(5N+2)d,\end{eqnarray} \tag{ 3 }$$

if we assume that the inter-planar distances d are not much affected by the chemical modulation. The vertical hexagonal lattice parameter c of the unit cell must be obviously an integer multiple of D. Since the structures exhibit ABCABC... stacking of the basal planes, the total number of the basal planes in the elementary unit cell must be an integer multiple of 3:

$$\begin{eqnarray}&&c=3td=sD=s(5N+2)d,\end{eqnarray} \tag{ 4 }$$

where s and t are integers. In the (11) phase for instance, N  =  2, D  =  c  =12d (see, e.g. middle panel of figure 1). In the phase (43) with N  =  1, accordingly D  =  7d and c  =  3D  =  21d (see figure 1, right). The (23) phase with the topological surface states contains no Bi-Bi DLs, i.e. D  =  5d and c  =  3D  =  15d.

Such a perfectly periodic chemical modulation in the c direction gives rise to a periodic sequence of diffraction peaks along the 000L crystal truncation rod, which can be viewed as modulation superlattice satellites to the main peaks in equation (2). These satellite peak positions on the rod are

$$\begin{eqnarray}&&{{Q}_{l}}=\frac{2\pi}{D}l,\end{eqnarray} \tag{ 5 }$$

where l is integer, while the reciprocal-lattice points on the 000L have the positions

$$\begin{eqnarray}&&{{Q}_{L}}=\frac{2\pi}{c}L,\end{eqnarray} \tag{ 6 }$$

with L is integer. For instance, the ideal (23) phase exhibits the maxima at $Q_{l}^{(23)}=2\pi l/(5d)$ and the reciprocal lattice points are at $Q_{L}^{(23)}=2\pi L/(15d)$. Therefore, every third reciprocal lattice point corresponds to an allowed diffraction maximum, i.e. 3$L=l$. This is a result of the structure factor of the (23) unit cell. In the (11) phase all diffraction maxima are allowed, since $Q_{L}^{(11)}\equiv Q_{l}^{(11)}=2\pi l/(12d)$.

In this model we did not yet consider differences in the lattice plane distances between the individual atomic (Bi, X) lattice planes in the crystal structure. The variation of the actual inter-planar distances along the c-axis direction affect the structure factor of the unit cell and consequently, the intensities of the diffraction maxima, but they do not influence the positions of the diffraction peaks, if d in the above relations represents the mean inter-planar distance $\bar{d}$.

For the determination of the mean inter-planar distance $\bar{d}$ it is suitable to use the diffraction peaks of the homogeneous virtual crystal lattice in equation (2). The positions of these peaks depend only on $\bar{d}$ and they are not influenced by the chemical modulation. From the above analysis it follows that these peaks are equivalent to the peaks 000.15⁽²³⁾, 000.30⁽²³⁾,... and 000.12⁽¹¹⁾, 000.24⁽¹¹⁾,... of the (23) and (11) phases and consequently, $\bar{d}$ can be directly determined from their positions. The lateral hexagonal lattice parameters a (but also $\bar{d}$) can be determined from the positions of the layer peaks in the asymmetric reciprocal-space maps, the examples of which are depicted in figures 3 and 4. The modulation period D and consequently the mean number N of QLs between two following DLs and the, consequently, the stoichiometry parameter δ that is related to N by equation (1) can be determined from the distance of nearest diffraction maxima along 000L according to

$$\begin{eqnarray}&&D=\frac{2\pi}{{{Q}_{l}}-{{Q}_{l-1}}}.\end{eqnarray} \tag{ 7 }$$

For this purpose, we used the diffraction peak pairs highlighted by red rectangles in figure 2 and from the figure it follows that in both Bi_mTe_n and Bi_mSe_n systems, the values of N increase with increasing chalcogenide flux so that δ decreases and the structures converge to the ideal (23) phase.

In figure 5 we present the resulting values derived for our sample series. Panels ((a) and (d)) display the δ-dependences of the lateral lattice parameter a and panels ((b) and (d)) that of the mean vertical lattice plane spacing $\bar{d}$. In both material systems, with decreasing δ (i.e. approaching the (23) phase), a decreases, but the change of a is much larger in tellurides than in selenides. Concerning the mean inter-planar distances $\bar{d}$ the tellurides and selenides behave in the opposite way. Approaching the ideal (23) phase, i.e. for $\delta \to 0$, $\bar{d}$ clearly increases in tellurides, while it slightly decreases in selenides (it is noted however, that for the latter system the changes are almost comparable to the error bars).

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The comparison of the values a and $\bar{d}$ obtained from our HRXRD measurement with data reported in literature is not straightforward, since the available databases ([10] for instance) contain data for bulk crystals and our values correspond to epitaxial layer that are strained to some extent. In order to correct our data for the epitaxial strain we use the following formula based on linear elasticity and the fact that the (0 0 0 1) sample surface is stress-free:

$$\begin{eqnarray}&&{{\bar{d}}_{\text{relaxed}}}={{\bar{d}}_{\text{strained}}}\left(1+\frac{2{{C}_{13}}}{{{C}_{33}}}\frac{{{a}_{\text{strained}}}-{{a}_{\text{relaxed}}}}{{{a}_{\text{relaxed}}}}\right),\end{eqnarray} \tag{ 8 }$$

where C_13,33 are the elastic constant, for simplicity assumed independent on the chemical composition. In the telluride systems we used the values of a_relaxed for various ideal (mn) phases from [12] and determined the a_relaxed of our samples from known stoichiometry parameters δ by quadratic interpolation. Then using formula (8) we calculated the corresponding relaxed vales of d and compared with data of ideal periodic phases in [9]. Indeed, our values of d_relaxed correspond well to the values of the reported bulk (mn) phases. Therefore the structure of the epitaxial telluride layers is similar to the bulk structure. A corresponding analysis for our bismuth selenide layers was not possible, since the reported parameters a_relaxed of various bulk (mn) materials tabulated in [7, 10, 12, 17–20] do not exhibit a clear monotonous dependence on δ.

Finally, we address the structure quality of the layers. In case of a perfectly periodic sequence of QLs and DLs corresponding to a given (mn) phase, the 000L-diffraction curve would exhibit a periodic sequence of diffraction maxima in positions according to equation (5). A simple inspection of the measured data reveals that this is not the case except for the samples with zero additional chalcogen flux (corresponding to the (11)) phase and the sample for the (23) phase obtained at the highest flux values that does not exhibit any inserted Bi-DLs. In the intermediate cases, only some diffraction maxima survive. In order to describe this effect we developed a statistical model of a Bi_mX_n lattice assuming that the number of QLs between two following Bi–Bi DLs is random with the mean value N. In reference [15] we presented the details of the model and a formula for the diffracted intensity averaged over all random sequences of QLs and DLs. The mean result of the model connects the full-widths at half maxima (FWHMs) δQ of the modulation satellites with the root-mean square (rms) deviation σ_N of the number of QLs between two DLs according to

$$\begin{eqnarray}&&{{\sigma}_{N}}\approx \frac{\delta Q\langle {{D}^{2}}\rangle}{10\pi d}.\end{eqnarray} \tag{ 9 }$$

This approximate relation was derived assuming that the broadening of the diffraction maxima due to the finite film thickness is negligible with respect to the broadening caused by disorder. Figures 5(c) and (e) show the resulting dependencies of the relative root mean square deviations σ_N/N on δ as obtained for our Bi_mTe_n and Bi_mSe_n epilayers. From the figures it follows that both tellurides and selenides exhibit a similar behavior. The (11) phase (δ  =  1) grown under zero additional chalcogen flux is well ordered; in this phase with $N=2$ σ_N/N is very small. Increasing the flux, the mean number N of the QLs between the Bi DL increases. This results in a strong increase of σ_N/N, meaning that the insertion of the Bi–Bi DL in the lattice structure is not very well correlated. For the samples grown with high chalcogen flux, $N\to \infty$ and ${{\sigma}_{N}}\to 0$ from definition. In these curves, the FWHMs of the diffraction peaks indeed correspond to the full layer thickness.

High-resolution x-ray diffraction was performed for investigation of the lattice structure of bismuth chalogenide topological insulator epilayers of rhombohedral lattices Bi_mX_n  ≡  Bi₂X_3−δ (X  =  Te, Se) grown by molecular beam epitaxy using different fluxes of the chalcogen atoms. It was demonstrated that for both tellurides and selenides with increasing chalcogen flux the structure evolves from Bi₁X₁ (zero additional flux) towards Bi₂X₃ (large additional flux), where the stoichiometry parameter δ of the investigated structures was determined from the distance of the modulation satellites of the diffraction curves. This transition is connected with a change of the lattice parameters, i.e. with increasing flux the lateral lattice parameter a decreases both in selenides and tellurides, whereas the mean hexagonal (0 0 0 1) lattice plane distances increases with increasing flux in the tellurides, but slightly decreases in selenides. For the evaluation of the structure quality we used our previously developed stochastic structure model to determine the mean-square deviation of the number of X-Bi-X-Bi-X quintuple layers present between consecutive Bi–Bi double layers. From our analysis we show that the Bi₁X₁ Bi₂X₃ phases are well ordered in both material systems, whereas for the intermediate compositions a high degree of disorder in the sequence of QL and DL layers exists that should significantly affect the topological and electronic properties of these materials.

The work was supported by Czech Science Foundation (project 14-08124S) and the Austrian Science Funds (project SFB 025, IRON).