Real-time observation of nanoscale topological transitions in epitaxial PbTe/CdTe heterostructures

Self-organization processes on the nanoscale have become a topic of fundamental relevance for the material sciences and their applications. On the one hand, these bottom-up processes allow for a rather straightforward generation of nanostructures with new functionalities.^1,2 On the other hand, it is important to control the stability of pre-defined nanostructures upon thermal processing.³ Phase separation of heterosystems with a miscibility gap is one of the most widely studied self-organization mechanism. Initially studied during the spinodal decomposition of immiscible liquids⁴ and solid metallic solutions,^5,6 this mechanism is meanwhile being applied to a large variety of material systems for self-assembly down to the nanoscale. For this purpose, phase separation has been exploited both in liquids⁷ and bulk solids⁸ as well as at liquid/solid^9,10 and solid/solid interfaces.^3,11–13 Recently reported applications range from ferromagnetic semiconductors,^14,15 thermoelectric materials,^8,16 and mid-infrared light emitters¹⁷ to nanopattern generation in diblock copolymers.¹⁸ Also, the importance of understanding and controlling phase separation processes has led to intensive efforts to develop models for this type of self-assembly.^19–23 

Most of the aforementioned material systems were characterized either on macroscopic scales, or by higher-resolution snapshots of intermediate or stationary states of the morphological transformation process. Only a few in situ experiments have been reported, which monitored the dynamics of the shape transformations on the nanoscale.^11,12 Systems investigated in this way typically involve either films¹¹ or nanowires^3,12 of polycrystalline noble metals on amorphous substrates. While dewetting phenomena on surfaces are easily accessible by high-resolution imaging techniques, the diffusion properties on a substrate surface with even a small concentration of surface impurities²⁴ as well as the different crystal orientations in polygrains²⁵ are often not well defined.

Such well-defined experimental conditions can be realized in an almost ideal manner in the immiscible²⁶ PbTe/CdTe heterosystem. As we have demonstrated earlier,^13,27,28 thermal annealing of embedded two-dimensional (2D) PbTe layers leads to coherent PbTe quantum dots (QDs)^29,30 in a CdTe host. During all stages of the experiments the two materials remain in the fully coherent solid state, i.e., all processes are exclusively based on bulk diffusion. Such QDs show superior properties in the mid infrared frequency range^17,28 and the PbTe/CdTe material system is also a very promising candidate for thermoelectric applications.^31–33 Besides their application potential, epitaxial PbTe/CdTe layer sequences offer a combination of properties that make them particularly well suited for general transmission electron microscopy (TEM) studies of the phase separation dynamics.

PbTe crystallizes in the rock salt, CdTe in the zinc blende lattice type, which both consist of face-centered cubic (fcc) lattices differing only in the coordinates of their respective two-atomic bases. The different selection rules for electron diffraction at the two lattice types yield high material contrast for dark field (DF) imaging. Moreover, the mass difference of Pb and Cd gives good mass contrast in the bright field (BF) imaging mode. When combined to a PbTe/CdTe heterostructure, the common Te sublattice defines a coherent, single crystalline fcc matrix throughout the entire layer stack (Fig. 1). This and the large miscibility gap lead to coherent, atomically sharp interfaces^27,34 concomitant with correspondingly steep composition gradients. The interface energies of the three facets with the lowest Miller indices, namely {100}, {110}, and {111}, are almost identical in this cubic lattice system.³⁵ This leads in thermal equilibrium to embedded PbTe QDs with the highly symmetric shape of small-rhombicuboctahedrons.¹³ The lattice constants of PbTe and CdTe are almost identical at room temperature.^36,37 Hence, strain effects can be neglected, which otherwise would complicate³⁸ the interpretation of the experiments. Finally, the formation enthalpies of CdTe and PbTe allow for coherent heteroepitaxial growth in a layer-by-layer growth mode at sufficiently low temperatures.³⁹ In this way, 2D PbTe epilayers with low defect densities⁴⁰ can be embedded coherently in a CdTe host crystal.²⁷ Although this layered initial geometry is perfectly phase separated, the interface energy, and thus the total free energy, is not at a global minimum. Hence, the 2D geometry provides ideal starting conditions for monitoring the kinetic pathways that drive the system into its steady state geometry during thermal annealing. In contrast to experimental conditions with a fixed reaction volume, our system can reach its low-energy final state, because the species in the 2D layer can in principle⁴¹ diffuse freely into the third dimension. All these properties of PbTe/CdTe stacks make them an ideal model system to study by in situ TEM in real time the dynamics of thermally induced topological transitions of epitaxial heterolayers.

All samples were grown by molecular beam epitaxy on CdTe/GaAs(001) pseudo-substrates.⁴⁰ The active layer sequence consists of a CdTe buffer, a two-dimensional, 3 or 5 nm thick PbTe epilayer deposited far from thermal equilibrium at a substrate temperature of 220 °C, and finally a 50 nm thick CdTe cap. We found atomically sharp (001) interfaces which are Cd and Te terminated,³⁴ respectively, on opposite faces of the 2D layer. The essential properties of the as-grown samples are depicted in Fig. 1, which shows a high-resolution TEM image of an as-grown CdTe/PbTe/CdTe layer stack and schematic representations of the atomic configurations across the two interfaces.

The annealing experiments were performed with a heatable sample holder inside a TEM instrument. We investigated a temperature range from 240 °C to 350 °C for time spans of up to 3 h. For these experiments five plan-view and one cross-sectional specimens were used. TEM images and video sequences were monitored either in a BF or a DF mode. For consistency, the grayscales of the images and video sequences were chosen such that the CdTe regions appear always darker than the PbTe regions. More details of the material system, instrumentation, and specimen preparation are documented in the supplementary material.⁴¹ 

On two samples with a 5 nm thick 2D PbTe layer the initial breaking-up of the epilayers into a quasi-one-dimensional (1D) percolation network was monitored. Three nanometers thick PbTe layers were then investigated to follow the disintegration of this intermediate state into the steady state geometry with separated PbTe dots. The splitting of the experiments into two batches was necessary, because the QDs from the 5 nm thick PbTe layers reach diameters comparable to the thickness of the CdTe capping layer. A control experiment was conducted on a cross-sectional specimen containing a 5 nm thick PbTe layer to investigate the influence of the thin TEM lamellae.⁴¹ It revealed that under the annealing conditions applied here the influence of the two surfaces of the prepared TEM specimens can be neglected.

Figure 2 shows an overview of the entire topological transformation process of an embedded PbTe layer during annealing. The upper part of Fig. 2 displays a series of 3D renderings revealing the morphological evolution of the 2D PbTe layer into 0D QDs. These snapshots were extracted from a video sequence (provided online with Fig. 2) which covers an in situ TEM experiment on an originally 5 nm thick PbTe layer over a time span of 3 h 10 min and a temperature range from 180 °C to 350 °C. A TEM image of the initial state of the 5 nm thick epilayer is displayed in Figure 2(e). Figures 2(f)–2(h) show related TEM images of 3 nm thick PbTe layers during annealing.

The starting condition (Figs. 2(a) and 2(e), respectively) is defined by an almost closed 2D PbTe epilayer that contains about 130 spots per μm² with diameters <20 nm, where the CdTe host material penetrates the PbTe layer (dark dots in the TEM image of Fig. 2(e)). Such nanoscale disruptions were found with comparable densities in all our specimens and are most likely created from thickness variations during the early stages of PbTe growth. Upon thermal annealing, we observe a rapid lateral expansion of the penetrating CdTe columns (Figs. 2(b) and 2(f)) until they almost touch. However, instead of their predicted coalescence,⁴² the CdTe regions start to expand laterally into more extended PbTe regions, until a percolation network of multiply interconnected, essentially 1D PbTe wires develops. The network arms show rather uniform, octagonal cross sections terminated by the three aforementioned low-energy facets.

Upon further annealing, the percolation network undergoes a second dimensional reduction: The 1D network arms become more and more corrugated and split into separated zero-dimensional (0D) PbTe QDs, which then converge toward their steady-state equilibrium shape, as shown in Figures 2(c) and 2(d) (3D renderings) and Figures 2(g) and 2(h) (experiments). The related topological changes are driven by capillarity forces to minimize the interface free energy. It can be described as a Rayleigh-Plateau instability^43,44 taking place here on the nanoscale in a fully coherent solid-state bulk system. Still, the hierarchy of the observed morphological changes resembles in many respects the breaking-up of a macroscopic water jet into individual droplets.^21,43

In order to study the 1D-0D transition in more detail, we used 3 nm thick embedded PbTe epilayers after the first morphological transition. Figure 3 shows, as an example, the disintegration of a crescent-shaped 1D PbTe structure into individual islands. The disintegration process is induced by three constrictions in the cross section of the curved 1D structure, which cause the detachment of two islands (Figs. 3(b)–3(d)). After the separation of the second QD the remaining constriction in the largest island begins to rebuild and thus does not cause another splitting. Subsequently, all three islands evolve toward their steady state shape.

Simultaneously, bulk diffusion through the coherent host material causes ripening, which occurs, however, on a slower time scale because of the larger distances involved. This leads to the growth of large islands above the time-dependent average volume on the cost of smaller ones, which are shrinking until they dissolve completely. This Ostwald ripening process^45,46 starts in the late stages of the experiment presented in Figs. 2(a)–2(e) with the first appearance of single dots.⁴¹ 

Although the reported experiments resemble in many details the phase separation of a mixture of two immiscible liquids, they occur here in a fully coherent solid-state system with length scales in the nanometer range. To gain further insights into the dominant mechanisms, we implemented the Cahn-Hilliard (CH) model,^19,21 and assessed its scalability into nanoscale dimensions for our model heterosystem. The CH model describes the phase separation of a system into components of extremal composition in the framework of a continuum approach. It is based on a generalized diffusion equation that relates the material flux to the concentration gradient ∇c of the two involved materials. The emerging material fluxes are caused by a decrease of the composition-dependent free energy f(c) of the system. The standard Cahn-Hilliard equation reads

where c is the atomic concentration, t is time, M is the mobility, and f(c) is the composition-dependent free energy. The thickness of the equilibrium interface between two regions of extremal composition increases with |$\sqrt m$|$m$⁠, where m is the coefficient of the gradient term in Eq. (1).

For f(c) we used the standard mean-field free-energy functional²¹ 

In a heterosystem with a miscibility gap this fourth order functional has two minima, which determine the extremal compositions as a function of the two parameters a and b. These can easily be mapped to the physical concentration range [c_min, c_max] defined by the solubility limits at the given temperature. More details on the CH model and the employed numerical procedure are described in the supplementary material.⁴¹ 

Snapshots of the CH results are shown in Fig. 4 in an arrangement that can be directly compared to the experiments shown in Figure 2. Simulated video sequences capturing the complete time evolution are provided with the online version of Fig. 4. Initially, some rectangular pinholes were placed randomly in the PbTe layer to mimic the experimental conditions. The pinholes grow rapidly due to capillarity effects, a process that decreases the total interface energy (Fig. 4(a)). The expanding pinholes then lead to a spatially disordered percolation network of interconnected, quasi-1D wires (Fig. 4(b)), in close agreement with the experiments. Once the process has reached this stage, both in the experiments and in the simulations, the dynamics clearly slows down and seems to remain stable over extended time scales.

In the next phase, the percolation network breaks up into elongated, quasi-1D segments (Fig. 4(c)) which then evolve into nano-spheres (Fig. 4(d)). The spherical shape reflects the isotropic nature of the CH model. In fact, the rhombicuboctahedral shape observed in the experiments is the closest approximation to a sphere a cubic crystal can reach with the aforementioned iso-energy facets {100}, {110}, and {111}.

While approaching their steady-state shape, the QDs simultaneously undergo Ostwald ripening. Indeed, we found in both the experiments and the simulations the shrinking and finally disappearance of smaller dots. As mentioned, this process occurs on a slower time scale compared to the disruption of the 2D layer, because of the larger distances to be overcome by bulk diffusion. The second, accelerated video sequence⁴¹ accompanying the online version of Fig. 4 extends into the Ostwald-ripening phase and reveals the dynamics of the different processes.

The observed morphological dynamics in the CH model covers all relevant morphological and dynamic features observed in the experiments. Evidently, in an immiscible system the morphological transformation of an unstrained, embedded 2D layer into separated objects is dominated by capillary effects even in a single-crystalline environment and on the nanoscale. This finding allows us to estimate the expected dimensions of the topological phases that evolve before the onset of Ostwald ripening from an initial 2D layer of thickness w. Assuming abrupt interfaces, we considered the interface areas of highly symmetric object types in the three dimensionality classes D observed in both the experiments and the CH simulations. These are: for D = 2 a layer of thickness w corresponding to the initial state of the experiments; for D = 1 cylinders of diameter d_c, which were inspired by the CH results in Figure 4(c); and for D = 0 spheres of diameter d_d, which approximate the experimental equilibrium shape (Fig. 5). We then determined the respective critical diameters for which a transition from one object class to the next reduces the total interface area, and thus the interface energy. The total volume is preserved at all times.

These purely geometrical arguments lead to a surprisingly simple relation between the characteristic sizes z of the nano-objects and their dimensionality class D, namely

Since these critical sizes are only the lower bounds determined by the energetic threshold arguments, the real, maximally unstable growth wavelengths are larger, and our analytic approximation therefore underestimates the dot sizes. For the second transition, i.e., the decomposition of cylinders into isolated dots via the Rayleigh-Plateau instability,⁴³ the maximally unstable wavelength is known to be⁴⁷ λ_max = 4.5·d_c. With this correction we arrive at a preferential dot diameter of d_d = 1.89·d_c = 3.78·w. Table I compares these values with the average dot sizes |$d_d^{\exp } (av)$|$ddexp(av)$ from Ref. 28 and the results from the annealing experiments presented here. The agreement is within a factor of two, even though our model does not contain a correction for the maximal unstable growth wavelengths of the first transition. Moreover, the experimental average values contain some Ostwald ripening, which becomes more pronounced for smaller w. This is the case because the resulting dot spacings decrease with the dot size, which leads to more pronounced inter-dot diffusion.

In summary, we performed under almost ideal experimental conditions real-time TEM experiments to monitor in situ the dynamics of the thermally induced topological transformations of an immiscible heterosystem in the coherent solid state. Commencing with a well-defined 2D epilayer, we follow on the nanoscale the evolution into a quasi 1D percolation phase, which subsequently breaks up into 0D quantum dots. Cahn-Hilliard simulations reproduce all relevant morphological transformation steps of the experiments. As the CH approach does not depend on the atomistic details of the studied system, we can conclude that the observed topological phase separation is dominated by bulk diffusion processes driven by the minimization of the free energy. Moreover, the scaling of the CH equation down to nanometer dimensions is a general feature of the model. Finally, we derived a parameter-free geometrical model for the observed topological transitions, which is in good qualitative agreement with the experimental findings.

This work was financially supported by FWF (Vienna) via SFB 025: IRoN. Valuable discussions with W. Heiss, R. Leitsmann, and F. Bechstedt are gratefully acknowledged.