Twisted ZB–CdTe/RS–PbTe (111) heterojunction as a metastable interface structure

The heterostructures of (zinc-blende)–CdTe/(rock-salt)–PbTe are typically found to have their common cubic axes aligned to each other, as in the case of PbTe quantum dots embedded in a CdTe matrix. In this work, we perform both theoretical and experimental studies on the CdTe/PbTe heterostructure in a different geometry: a planar CdTe/PbTe (111) heterostructure. We simulate the epitaxial growth of CdTe (PbTe) on the (111) PbTe (CdTe) substrate, using a density-functional theory. A twisted CdTe/PbTe (111) interface structure has been predicted in the layer-by-layer epitaxial growth on the (111) substrate, in contrast to the non-twisted CdTe/PbTe (111) interface reported in the literature. This predicted structure has been confirmed experimentally in the heterostructure grown by molecular beam epitaxy, using a high-resolution transmission electron microscope. The twisted interface has a lower binding energy than the non-twisted one, indicating that the twisted structure is a metastable phase formed in the non-equilibrium growth process. Additionally, the interface reconstructions of the CdTe/PbTe (111) heterostructure observed by reflection high-energy electron diffraction are explained using the twisted interface model.


Introduction
Narrow gap PbTe is known to have superior thermoelectric and optical properties and great potential for applications in energy and light sources [1][2][3][4]. As one of the few structure-mismatched heterostructures (as opposed to the more familiar 'lattice-mismatched' heterostructures due to the lattice parameter difference), CdTe/PbTe heterojunctions (HJs), which were first realized through epitaxial growth by Koike et al [5] have attracted much attention [6,7] since the synthesis of PbTe quantum dots (QDs) in a CdTe matrix that exhibits intensively enhanced room-temperature mid-infrared luminescence [6][7][8][9][10]. PbTe is in a rocksalt (RS) and CdTe in a zinc-blende (ZB) structure, but they have nearly the same crystal lattice constants: 6.46 and 6.48 Å at 300 K, respectively. Mid-infrared light-emitting diodes with epitaxial PbTe QDs in CdTe have been fabricated [3]. A large Rashba splitting is predicted in CdTe/PbTe/PbSrTe quantum wells (QWs) [11,12], a promising structure for spintronic devices. Recently, an abnormal enhancement of mid-infrared light emission due to coupling with surface plasmons was demonstrated in a polar and structure-mismatched CdTe/PbTe (111) single heterojunction (SHJ) [4]. PbTe QDs are obtained by annealing the [100] oriented CdTe/PbTe/CdTe QWs [8]. The crystalline planes of the (100) facets of the PbTe QDs are found to be parallel to the corresponding (100) planes of CdTe, due to the constrain of the epitaxial growth of the QWs on the (100) substrate, as observed by transmission electron microscopy (TEM) [13]. The structural properties of PbTe/CdTe (100), (110) and (111) interfaces have been theoretically studied by Leitsmann et al [14], where an average interface energy of about 0.2 J m −2 (12.5 meV Å −2 ) was found; no interface reconstruction was reported for these interfaces.
Previous CdTe/PbTe heterostructures are mostly in the form of the QDs of one component embedded in the other; the structure was typically grown on the (100) substrates. In this work, we are interested in the growth of a planar heterostructure on the (111) substrates. The equilibrium structure is more stable than other non-equilibrium structures since it has a lower total energy. However, in a non-equilibrium epitaxial growth, we usually do not get the equilibrium structure, but a metastable structure. A well-known example is the formation of partially ordered III-V alloys during epitaxial growth [15]. There are no reports relating to the growth simulation of CdTe/PbTe (111) HJ. Because the crystal structure of the (111) plane is rather different from that of the (100) and (110) planes for both PbTe and CdTe, the interface crystal structure of the (111) heterostructure could be different from that of the (100) epitaxial structure.
It is known that not all IV-VI materials have stable RS structures and that some of them are distorted by the strong pseudo-Jahn-Teller (PJT) coupling [16,17]. Recently, the distorted PbTe RS local structure, which behaves like that in ferroelectrically distorted lone-pair-active Pb 2+ compounds, such as PbTiO 3 [18], was found by an atomic pair distribution function analysis at room temperature [19]. This finding implies that the RS structure PbTe may easily become unstable under certain circumstances. Nevertheless, a bulk CdTe crystal is typically an insulator with a wide band gap (1.6 eV) in the ZB structure formed by the sp 3 hybridization [20,21]. The ZB structure has alternating interplanar spacings along the [111] direction, whereas the RS structure has equally spaced (111) planes. Modeling the structure-mismatched CdTe/PbTe (111) interfaces is not a straightforward task, because the (111) surfaces of PbTe and CdTe are polar and the atomic relaxation calculation is hard to converge due to the presence of intrinsic as well as artificial electric fields induced by the polar surfaces or interfaces. As experimentally demonstrated by Cai et al [4], the interfacial effect plays an important role in the strong luminescence enhancement. Therefore, it is of fundamental interest as well as pivotal for understanding the experimental observations to model the crystal structure and corresponding electronic structure of the interface for the SHJ of CdTe/PbTe.
We firstly simulate the epitaxial growth of CdTe (PbTe) on a PbTe (CdTe) (111) substrate using a density-functional theory (DFT) approach. A 'double external field correction' method is adopted to remove the artifacts due to the use of the supercell approach. A metastable interface structure is predicted to form in the layer-by-layer epitaxial growth. Specifically, the (111) epilayer is rotated with respect to the substrate about the [111] axis by 180 • , termed a 'twisted' structure that has a slightly higher total energy than the 'normal' (non-twisted) structure. The formation of the twisted interface structure is confirmed in the materials grown by molecular beam epitaxy (MBE) using a high-resolution transmission electron microscope (HR-TEM). Interface reconstruction during the growth of the CdTe/PbTe SHJ was characterized by reflection high-energy electron diffraction (RHEED). The predicted twisted structure is expected to yield a very different electronic structure near the interface; this work lays the foundation for the future exploration of this type of unusual semiconductor heterostructure.

Theoretical method
A DFT approach within a generalized gradient approximation [22], implemented in a Vienna ab initio simulation package [23,24], is applied to study the PbTe/CdTe interface. Spin-orbit coupling is taken into account since this effect plays an important role in materials with heavy elements, such as Pb and Te [25]. The interaction of the valence electrons with the remaining ions was modeled by pseudopotentials generated within the projector augmented wave method [26]. The Cd 4d and Pb 5d electrons are treated as valence electrons since the outermost occupied d states give rise to shallow semicore bands, which contribute to chemical bonding in IV-VI materials [27]. The heterostructure is modeled by a slab with a vacuum space of approximately 4 nm. The Brillouin zone integration is performed by a summation over Monkhorst-Pack special k-points [28]. 5 × 5 × 1 k-point meshes are used for the vacuum slab modeling.
In the real material growth, for instance, the CdTe epitaxial layer is grown on a PbTe buffer layer that is first grown on a BaF 2 (111) substrate. However, in the modeling, we consider only a single heterostructure such as CdTe on PbTe, without having to include the BaF 2 substrate, because in the real growth the buffer layer is rather thick (of the order of 1 µm) so the buffer layer can be viewed as the substrate.
To simulate the layer-by-layer growth process, e.g. CdTe growing on a Te terminated PbTe (111) surface, we use a 24 bi-layer PbTe as the substrate. We start the calculation with only one Cd atomic layer of CdTe on the PbTe surface, allowing the atoms to relax; we then repeat the calculation by adding a Te layer and relaxing the structure, and so on until all the desired CdTe layers are included.
Since the [111] orientations in both RS-PbTe and ZB-CdTe are polar, the combination of the two slabs leads to a supercell with chemically, structurally and electrostatically different interfaces. In the following, the surfaces (interfaces) with the directions of [111] and [111] of CdTe are labeled as A (Cd-terminated) and B (Te-terminated), respectively. To correct the artificial electric field caused by the use of the supercell approach, we propose a double external field (DEF) correction method. Taking the growth of CdTe on a PbTe (111) buffer layer as an example, when the slab contains a small number of atomic layers of PbTe and CdTe, there are three electrostatic potential (ESP) differences: CdTe , which is induced by the CdTe surface and the CdTe/PbTe interface; vacuum 1,2 induced by the PbTe surface and CdTe surface; and PbTe induced by the PbTe surface and the CdTe/PbTe interface. These ESP differences could be obtained by the calculation of the plane-averaged ESP along the growth orientation, as shown in figure 1(a). As a result, three different electric fields are induced along the [111] direction in the CdTe film, the vacuum layer and the PbTe layer, respectively. It is known that the ESP differences do not depend on the slab size [14,29], which we have verified by changing the thickness of the PbTe layer in the slab model. Therefore, the electric field in the PbTe layer tends to zero when the thickness of the PbTe layer reaches infinity. As the PbTe layer would be much thicker than the CdTe film, the electric field with real physical meaning is only that in the CdTe film, as shown in figure 1(b).
To compensate the artificial electric fields in the vacuum layer and the PbTe layer, a DEF correction scheme is suggested: one external field is used to compensate the artificial electric field in the vacuum layer, which is similar to the dipole correction introduced by Neugebauer and Scheffler [30], with the additional external ESP given as where [z V min , z V max ] is the interval of the vacuum region along the [111] direction and z 0 is the position with the lowest plane-averaged charge density in the vacuum region that has been set to zero in figure 1. The other external field is used to compensate the artificial electric field in the PbTe region to mimic the limit of an infinitely thick PbTe substrate. The corresponding external ESP can be written as here z i and z sub s are the interface position and the bottom surface position of the PbTe along the [111] direction, respectively. In the correction procedure, the ESP differences ( vacuum 1,2 , PbTe ) vanish self-consistently, i.e., after the self-consistent field calculation is completed the  (1) and (2); the total free energy is corrected by equation (3) [31]: where Z I is the valence and R I is the position of the Ith ion, which leads to a new electron density n(r). The ESP should be calculated again using the new electron distribution. This procedure continues until the ESP differences vacuum 1,2 and PbTe are smaller than a tolerant value.

Experimental setup
We grew the PbTe(CdTe)/CdTe(PbTe) HJ sample by solid-source MBE using a different growth process (i.e., CdTe on PbTe or PbTe on CdTe). The PbTe/CdTe HJs were grown on CdTe (111) single-crystal substrates, which were cut along the [111] direction into 800 µm thick slices. Because of the big difference between the thermal expansion coefficients of CdTe and PbTe (α CdTe = 4.9 × 10 -6 K −1 , α PbTe = 19.8 × 10 -6 K −1 ), the 1 µm thick PbTe film was grown at 648 K and slowly cooled down to room temperature, which can effectively reduce the interface defects caused by the thermal stress relaxation. To observe the interface structure of PbTe/CdTe HJ we performed HR-TEM characterization on a cross-sectional specimen along the 110 zone axis. For the CdTe/PbTe HJ, a 1 µm PbTe buffer layer was first grown on BaF 2 (111) substrates at 620 K and then a CdTe layer was grown on the PbTe buffer at 520 K. The interface reconstruction during the growth process was observed by the RHEED patterns.

The growth simulation of CdTe/PbTe(111) heterojunctions
To determine the favorable interface structures for different growth processes (i.e., CdTe on PbTe or PbTe on CdTe), we perform the relaxation calculation using the DEF correction method. The [111]-oriented layer (either PbTe or CdTe) is simulated by passivating the dangling bonds of its bottom surface using pseudo-hydrogen atoms to remove the influence of the bottom surface states. Twenty-four bi-layers are used for the buffer material in the slab. The atomic geometries are obtained by atomic position relaxation until the Hellmann-Feynman forces are smaller than 10 meV Å −1 .
In contrast to interfaces A and B of the PbTe QDs embedded in the CdTe matrix [8], in which the PbTe crystal planes are parallel to corresponding planes of CdTe (e.g. When CdTe grows on the PbTe (111) layer, which is simulated by sequentially adding Cd and Te layers on the PbTe slab, the calculation shows that it forms the twisted interface A structure, no matter which layer (Pb or Te) terminates the PbTe (111) surface. It is understandable that the Pb terminated PbTe surface is first followed by a Te layer, because Pb and Cd atoms do not bond directly. The Te atoms form ionic bonds with the Pb atoms on the surface, which continues the RS structure. Since the interplanar spacings of PbTe are equal, the tellurided Pb terminated PbTe (111) surface is equivalent to a Te terminated PbTe surface. However, when PbTe grows on a CdTe (111) layer, there are two possibilities (i.e. surfaces A and B) because the (111) interplanar spacings of the Cd layers and Te layers are not equal to each other, which results in formation of two different PbTe/CdTe interface structures when the PbTe layer grows on the A and B surfaces of the CdTe. Our simulation shows that it forms the structure of a twisted interface A (B) when the PbTe grows on the CdTe surface A (B). Compared to the structure in figures 2(a) and (c), the twisted structures keep the RS structure, PbTe, and the ZB structure, CdTe, as close as possible to their bonding situations in the bulks near the interface. For example, the PbTe (CdTe) structure consists of periodic rectangles (hexagons), but the shapes of the rectangles (hexagons) at the interface (one side of which is formed by the Cd-Te (Pb-Te) bonds) are quite different (see figures 2(a) and (c)). The twisted structure in figures 2(b) and (d) mostly keeps the bulk shape of the rectangles (hexagons), whereas the non-twisted structure in figures 2(a) and (c) evidently changes the bulk shape of the rectangles (hexagons) as highlighted in figure 2. At this point, the twisted structure forms more easily in the epitaxial growth than the non-twisted one at the interface.
Next we compare the total energies or the binding energies of the two types of interface structures. The equation below is used to calculate the binding energy where the three terms are respectively the total energy of the PbTe, CdTe, and CdTe/PbTe slab. PbTe or CdTe slabs each consist of 24 (111) bi-layers. The twisted and non-twisted heterostructures are calculated in the dipole corrected stoichiometric slab approximation [29]. To keep the system stoichiometric, it is necessary to have both interface A and B in the same slab. It turns out that the binding energy for the non-twisted interface (24.0 meV Å −2 ) is larger than that of the twisted one (21 meV Å −2 ), which indicates that the non-twisted interface structure is the energetically favorable equilibrium phase and the twisted one is a metastable phase formed in the non-equilibrium growth process. As a matter of fact, a metastable phase is often observed in the non-equilibrium epitaxial growth by MBE and MOCVD; for instance, a well-known phenomenon, spontaneous ordering, has been observed in the epitaxial growth of a semiconductor alloy, GaInP on GaAs in particular [32]. The twisted interface structure is observed by our HR-TEM measurement in a PbTe/CdTe SHJ grown by MBE, as shown in figure 3. The inset of figure 3 illustrates the atomic arrangements near the interface. The inclined angle between the (100) planes of PbTe and CdTe is 70.6 • (2 arccos( √ 2/3)). We note that because of the twisted interface structure, it is not feasible to form rhombocubo-octahedral PbTe QDs by annealing the [111] oriented CdTe/PbTe/CdTe QWs.

Interface reconstructions during growth processes
The PJT coupling in PbTe can be sensitively dependent on the tensile strain [33]. According to our relaxation calculation for the bulk PbTe and CdTe, the lattice constant of these two materials is 6.422 and 6.647 Å (0 K), respectively, and the lattice mismatch ε of the PbTe(CdTe)/CdTe(PbTe) HJs is 0.7%. Therefore, PbTes have a small tensile strain and the RS structure of the PbTe may be distorted by PJT coupling at the interface. The vacuum slab with 2 × 2 lateral unit cells is used to test the interface reconstruction while the relaxation is only allowed along the growth direction. Simulation shows that bulk PbTe has equal interplanar spacings (1.85 Å at 0 K) along the [111] direction. However, at the interface of PbTe/CdTe HJ, the PJT coupling drives the Pb and Te fcc sub-lattices of the RS structure deviated slightly from each other along the [111] direction by 0.3 Å due to the tensile strain. As a result, the interplanar spacings of PbTe (111) at the interface region become alternately wide (2.15 Å) and narrow (1.55 Å), which is called rhombohedral distortion [33].
The non-twisted (111) interface of the PbTe QDs in the CdTe matrix gives no evidence for an interface reconstruction [14]. However, the twisted structure in the CdTe/PbTe (111) HJ induces reconstruction due to the competition between the RS structure and ZB structure at the interface. We notice that one of four atoms from different lateral unit cells in the same atomic layer has a vertical displacement compared to the other three atoms when CdTe is grown on the PbTe (111), as shown in figures 4(a) and (c). The vertical displacement in the first Cd layer is 0.01 Å, it gradually diminishes within a few bi-layers, which means that a 2 × 2 reconstruction is formed at interface A. For interface B, we obtain a 2 × 1 reconstruction, as shown in figure 4(d), but the maximum vertical displacement is extremely small, ∼6 × 10 -3 Å.
The interface reconstruction is a result of the competition between the RS structure and ZB structure at the interface. As illustrated in figure 4(c), when CdTe grows on the surface of PbTe (111), 3/4 of the Cd atoms (α atoms) of the A-C1 layer come up from the A-C1 plane trying to form the RS lattice with the Te atoms in the P1 layer, while leaving other Cd atoms almost keeping the ZB lattice with the Te atoms. Shown in figure 4(d), when the PbTe grows on the surface B of CdTe (111), one half of the Pb atoms (β atoms) of the B-P1 layer go down slightly from the plane, trying to form the ZB lattice with the Te atoms in the B-C1 layer. It is well known that Pb and Te atoms form ionic bonds in bulk PbTe, whereas Cd and Te atoms form mostly covalent bonds in bulk CdTe. Compared to the bulk PbTe and CdTe crystals, the bonds around the CdTe/PbTe interface cannot be described simply by either an ionic or covalent character since both the Pb-Te and Cd-Te bonds are covalent bonds with a partial ionic property. Figure 5 shows the calculated electron density around the interface, showing that the Cd atoms, which come up from the C1 layer, form Cd-Te bonds in more of an ionic nature than the other Cd atoms at interface A (the atom α in figure 5(a)), and the lower Pb atoms form Pb-Te bonds in more of a covalent nature than the other Pb atoms at interface B (the atom β in figure 5(b)), which explains the competition between RS and ZB structures at the interface. In order to enhance the color contrast, the Cd 4d and Pb 5d electron densities, which are highly localized around the Cd and Pb ion cores, are not displayed (under the gray atom balls).
Our simulation predicts that the surface of a CdTe (111) B layer has a weak 2 × 1 surface reconstruction. The reconstruction is caused by the distortion of the sp 3 dangling bonds on the CdTe (111) surface, which is analogous to the 2 × 1 surface reconstruction on a Si (111) surface. Figure 6(a) shows the RHEED pattern measured from the CdTe (111) surface B along the [110] direction. The narrow faint fringes are the reconstruction lines, which disappear in the measured RHEED pattern along the [112] direction. The reconstruction fringes fade away as soon as PbTe grows on it, as shown in figure 6(b), because the vertical displacement in the PbTe layers is too small to be discriminated at the interface B. The surface of a PbTe (111) does not show any reconstructions, as illustrated in figure 6(c). However, bright reconstruction fringes appear as soon as CdTe grows on it, as observed by RHEED patterns shown in figure 6(d) along both the [110] and [112] directions, which implies a 2 × 2 reconstruction at the twisted interface A. It is also noted that the reconstruction fringes are very bright at the growth temperature (∼520 K), but they gradually become weaker and weaker as the temperature decreases. According to our calculation, for the PbTe grown on the CdTe (111) B surface at 0 K, the RS structure of PbTe is distorted by both the tensile strain and the rhombohedral distortion due to the PJT coupling in PbTe. However, for the CdTe grown on the PbTe (111) surface at 0 K, the ZB structure of CdTe is only slightly distorted by the compressive strain. As a result, the vertical atom displacement at the PbTe/CdTe HJ is smaller (∼6 × 10 -3 Å) than that at the CdTe/PbTe HJ (0.01 Å) at 0 K, as shown in figure 4, which implies a stronger distortion of the RS or ZB structure will restrain the competition between the two lattice structures at the interface. When the CdTe grows on the PbTe (111) surface, at the growth temperature (∼520 K), the CdTe and PbTe are perfectly lattice matched (ε < 0.05%). Thus the distortion caused by the strain vanishes and the vertical displacements become much larger since the competition between RS and ZB structures becomes stronger. Instead of partial ionic bonds, the α atoms of the A-C1 layer in figure 4(c) may form an RS structure with fully ionic bonds. Furthermore, when the PbTe grows on the CdTe (111) B surface at 620 K, the lattice mismatch becomes greater (ε > 0.16%) and the competition between the two lattice structures is restrained again, which explains why we cannot see the reconstruction fringes of the RHEED pattern in figure 6(b). Based on both the theoretical calculation for 0 K and the RHEED patterns at a high temperature, we can conclude that the vertical atom displacement is quite small (∼0.01 Å) at 0 K, but it becomes larger with the increasing temperature and reaches a maximum value around 500 K.

Conclusion
The structural properties of twisted CdTe/PbTe (111) polar interfaces are investigated both theoretically and experimentally. The epitaxial growth of CdTe (PbTe) on a polar PbTe (CdTe) surface is simulated by DFT calculations and a preferred twisted CdTe/PbTe (111) interface structure is predicted and verified by HR-TEM observation in the MBE grown PbTe/CdTe SHJ. Interface reconstructions of CdTe/PbTe (111) induced by the twisted structure are observed by RHEED and interpreted by theoretical modeling. The twisted interface structure is a metastable phase, as compared to the non-twisted structure. The formation of this metastable structure is the result of the non-equilibrium growth process, similar to the observation of spontaneous ordering in various III-V semiconductor alloys that is also driven by the surface effect. We expect that the twisted interface will yield interesting physics to be explored in comparison with the normal interface.