Structural disorder of natural ${\mathrm{Bi}}_m{\mathrm{Se}}_n$ superlattices grown by molecular beam epitaxy

The structure and morphology of ${\mathrm{Bi}}_m{\mathrm{Se}}_n$ epitaxial layers with compositions ranging from ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ to the ${\mathrm{Bi}}_1{\mathrm{Se}}_1$ grown by molecular beam epitaxy with different flux compositions are investigated by transmission electron microscopy, high-resolution x-ray diffraction, and atomic force microscopy. It is shown that the lattice structure changes significantly as a function of the beam flux composition, i.e., Se/BiSe flux ratio that determines the stoichiometry of the layers. A perfect ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ phase is formed only with a sufficiently high additional Se flux, whereas ${\mathrm{Bi}}_1{\mathrm{Se}}_1$ is obtained when only a BiSe compound source without additional Se is used. For intermediate values of the excess Se flux during growth, ${\mathrm{Bi}}_2{\mathrm{Se}}_{3-\delta }$ layers are obtained with the Se deficit $\delta$ varying between 0 and 1. This Se deficit is accommodated by incorporation of additional Bi-Bi double layers into the ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ structure that otherwise exclusively consists of Se-Bi-Se-Bi-Se quintuple layers. While a periodic insertion of such Bi double layers would result in the formation of natural ${\mathrm{Bi}}_m{\mathrm{Se}}_n$ superlattices, we find that this Bi double-layer insertion is rather stochastic with a high degree of disorder depending on the film composition. Therefore, the structure of such epilayers is better described by a one-dimensional paracrystal model, consisting of disordered sequences of quintuple and double layers rather than by strictly periodic natural superlattices. From detailed analysis of the x-ray diffraction data, we determine the dependence of the lattice parameters $a$ and $c$ and distances of the individual (0001) planes $d_j$ as a function of composition, evidencing that only the in-plane lattice parameter $a$ shows a linear dependence on composition. The simulation of the diffraction curves with the random stacking paracrystal model yields an excellent agreement with the experimental data and it brings quantitative information on the randomness of the stacking sequence, which is compared to growth modeling using Monte Carlo simulations. The analysis of transmission electron microscopy data furthermore confirms that the Bi-Bi bilayers contain a large amount of vacancies of up to 25%. Conductivity and Hall data confirm that ${\mathrm{Bi}}_m{\mathrm{Se}}_n$ phases containing Bi-Bi double layers exhibit a rather semimetallic behavior.

Figure 1

Schematic illustration of the ${\mathrm{Bi}}_2{\mathrm{Se}}_3,{\mathrm{Bi}}_1{\mathrm{Se}}_1$, and ${\mathrm{Bi}}_4{\mathrm{Se}}_3$ lattice structures. The ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ phase consists exclusively of Se-Bi-Se-Bi-Se QLs van der Waals bonded to each other, while the ${\mathrm{Bi}}_1{\mathrm{Se}}_1$, and ${\mathrm{Bi}}_4{\mathrm{Se}}_3$ phases contain additional Bi-Bi DL after every $N=2$ and 1 QLs, respectively. The boundaries between the QLs and DLs are denoted by dotted lines, and the crystallographic unit cells are marked by black rectangles.

Figure 2

Atomic force microscopy images (a)–(d) and RHEED patterns (e)–(h) of four different ${\mathrm{Bi}}_2{\mathrm{Se}}_{3-\delta }$ epilayers with different composition grown using different excess Se fluxes ${\mathrm{\Phi }}_{\mathrm{Se}}$ as indicated, while the BiSe flux was kept constant at 0.14 ML/s. The size of the AFM images is $3\times 3\mu {\mathrm{m}}^2$ and the layer thickness 350 nm.

Figure 3

AFM surface images of a ${\mathrm{Bi}}_1{\mathrm{Se}}_1$ (a) and ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ (d) epilayer with 50 nm thickness grown using ${\mathrm{\Phi }}_{\mathrm{Se}}=0$ and 2.13 Å/s, respectively. The corresponding height profiles along the white lines in (a) and (d) are shown in (b) and (e), respectively, shifted vertically for clarity. The profiles in (b) show well-defined steps with the height of either quintuples (QL) or double (DL) layers and only of QL in (d). (c), (f) Show the height distributions calculated over the whole AFM images. The ${\mathrm{Bi}}_1{\mathrm{Se}}_1$ sample consists of both QL and DL blocks, but only one single DL occurs between subsequent QL segments because all the peaks in the height distributions can be ascribed to a single DL followed every time by several QLs. In contrast, the ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ sample consists only of QL layers.

Figure 4

(a) Simulated high-resolution STEM images (left) and the corresponding crystal structure (right) of various $\left(mn\right)$ phases in the $\left(\overline{1}100\right)$ plane. In the sketches of the lattice structure, the blue (red) spheres denote the Bi (Se) atoms, in the simulated STEM images the Bi and Se atomic columns appear as bright white and dark gray spots, respectively. The arrows marked with “c” indicate the crystallographic unit cells with vertical lattice parameter $c$, arrows marked with “a” indicate the in-plane lattice parameters in the $\left[11\overline{2}0\right]$ projection. (b) Averaged intensity profiles of the simulated HRSTEM images in the [0001] $c$ direction. Dashed vertical blue lines indicate the van der Waals (vdW) gaps and orange shaded regions in the cross sections highlight the Bi-Bi double layers (DL).

Figure 5

Measured HRSTEM images of the sample grown at ${\mathrm{\Phi }}_{\mathrm{Se}}=0$ (a) and ${\mathrm{\Phi }}_{\mathrm{Se}}=17$ Å/s (b) with composition corresponding to ${\mathrm{Bi}}_1{\mathrm{Se}}_1$ and ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, respectively. In (c), (d) the zoomed-in HRSTEM images from the white rectangular areas in (a) and (b) are depicted, corresponding to regions of perfect ordering as expected for the ideal (1:1) and (2:3) phases. Underneath, the corresponding line profiles are shown. The dashed blue lines and light orange stripes denote the van der Waals gaps and the Bi-Bi double layers, respectively.

Figure 6

(a) Measured diffraction curves of the ${\mathrm{Bi}}_2{\mathrm{Se}}_{3-\delta }$ sample series along the symmetric $000Lc$ direction. The curves are displaced vertically and their parameter is the excess Se flux ${\mathrm{\Phi }}_{\mathrm{Se}}$ in Å/s used for growth listed as numbers on the right-hand side for each curve, which decreases from 17 Å/s to 0 from top to bottom. Red (blue) numbers denote the diffraction indices $L_{\left(11\right),\left(23\right)}$ of the ideal (11) and (23) phases, respectively. The blue diffraction maxima (000.15) and (000.30) highlighted by the green shaded areas are the “true” maxima of the virtual homogeneous lattice as explained in the text. From their positions $Q_{1,2}$ the mean interplanar distance $d$ of the 2D hexagonal lattice planes was derived as shown in Fig. 9. When the excess Se flux ${\mathrm{\Phi }}_{\mathrm{Se}}$ decreases, the diffraction maxima $\left(000.L\right)\left(L=3,12,18,27\right)$ split into pairs; the splitting is highlighted by orange shaded areas. The distance $\mathrm{\Delta }Q$ of these pairs is used to determine the average modulation period $D$ of the natural superlattice structures as shown in Fig. 8. The substrate peaks are denoted by “S.” (b) Shows the expected simulated diffraction curves of ideally periodic ${\mathrm{Bi}}_m{\mathrm{Se}}_n$ phases of natural superlattices with the structures exemplified in Fig. 1 and structure parameters taken from the literature.

Figure 7

Mean number $N$ of QLs between two Bi-Bi DLs (left axis) and corresponding ${\mathrm{Bi}}_2{\mathrm{Se}}_{3-\delta }$ composition $\delta$ (right axis) calculated using Eq. (5) plotted as a function of the excess Se flux ${\mathrm{\Phi }}_{\mathrm{Se}}$ used for growth. The values $\delta =0$, i.e., $N\to \infty$ and $\delta =1(N=2)$ correspond to the (23) and (11) phases as indicated by the blue and red lines, respectively. The vertical arrow indicates the critical Se flux needed to obtain pure ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. The solid line represents the expected composition calculated for an Se incorporation coefficient ${\gamma }_{\mathrm{Se}}=0.5$ as described in the text.

Figure 8

Measured mean interplanar distance $d$ (a) and in-plane lattice parameter $a$ (b) of the ${\mathrm{Bi}}_2{\mathrm{Se}}_{3-\delta }$ epilayers as functions of composition $\delta$ (black points). The open circles and dashed lines denote the relaxed values $d^{\left(\mathrm{rel}\right)},a^{\left(\mathrm{rel}\right)}$ calculated using the procedure described in the text, taking into account the biaxial strain induced by the thermal expansion mismatch, assuming that the layers are fully relaxed at the growth temperature. The tabulated lattice parameters and interplanar distances for the bulk phases of (23) [9], (67) [41], (11) [10, 12], and (43) [11] are denoted by the colored dots.

Figure 9

Measured (points) and fitted (blue to red lines) diffraction curves of all samples using the disordered random stacking paracrystal model as described in the text. The order parameters of the curves are values of the excess Se flux ${\mathrm{\Phi }}_{\mathrm{Se}}$ in Å/s (upper numbers) and the derived film composition $\delta$ (lower numbers) deduced from the average number $N$ of the QLs between the Bi-DL obtained from the fits. The colors of the simulated diffraction curves (from red to blue) highlight the transition from the (11) to (23) phase.

Figure 10

(a) Interplanar distances $d_j,j=1,...,7$, as functions of the stoichiometry parameter $\delta$ derived from the fit of the diffraction curves of Fig. 9. The locations of the $d_j$ within the crystal lattice structures are indicated on the left in (b). The distances $d_{4,...,7}$ denoted by empty symbols only occur in ${\mathrm{Bi}}_2{\mathrm{Se}}_{3-\delta }$ samples with sufficiently large $\delta$. The blue ticks on the left axis denote the values for the (23) phase taken from [9] [(a)] and from [51] [(b)]. In the inset, we plotted the $\delta$ dependence of the occupation $c_{\mathrm{Bi}}$ of the Bi sites in the DLs. (b) Shows the sketches of the crystal structures of the (23) and (11) phases; the interplanar distances $d_j$ are indicated by numbers.

Figure 11

Relative deviation ${\sigma }_N/N$ of the mean number of QLs as function of $\delta$ derived using the random stacking paracrystal model for the fit of the experimental diffraction curves. The full lines represent the results of the kinetic Monte Carlo simulation with various values of the “memory” parameter $\xi$ [(a)] and parameter $\chi$ [(b)] (see text).