Introduction
Germanium (Ge), antimony (Sb), and tellurium (Te) ternary alloys (GST), especially those along the GeTe-Sb2Te3 pseudo-binary line, are a special class of materials known as chalcogenides which are phase-change materials (See Figure 1). The unique property combination of these materials lends them naturally to the needs of data storage. In any memory device, there must be sufficient contrast between at least two states in order to represent the 0s and 1s of binary logic. Due to the sufficient contrast in optical properties between the crystalline and amorphous phases, GST has already been widely implemented in optical data storage such as DVDs and Blu-Ray. Since the two phases also exhibit orders of magnitude difference in electronic properties, GST is being heavily researched for use in non-volatile random access memories, which may compete with or replace current FLASH technologies. The transformation between the crystalline and amorphous phases is rapid (~100ns) and reversible with a high stability (>10 years) [1].
Figure 1[6]: Ge-Sb-Te ternary phase diagram
There is still a very primitive understanding of the structure and basic physical properties of GST despite significant experimental and theoretical work. This is partly due to the fact that ternary alloys in general are very difficult to study. There have been many insights into the nature of GST alloys using various characterization techniques (highlighted in the next section). However, this data has not been sufficient in determining the structure of GST. Many groups have turned to ab initio calculations to supplement the experimental data and gain further insight into the structure and properties of GST. These calculations are invaluable because not only do they help characterize the structure, but also provide explanation as to why the system adopts the structure that it does. Once a structure is defined, ab initio calculations will also be able to predict the properties of the system. Successfully characterizing the structure and phase-change mechanism of GST is invaluable for the design of novel materials with optimal property combinations.
This paper will be a review of the efforts in determining the structure of GST via ab initio (density functional theory – DFT) calculations.
Experimental Insight
Much of the previous experimental work has been motivation for various ab initio calculations as well as provide criteria to test the results of said calculations. Therefore, before addressing DFT’s role in the determination of the structure of GST, it is important to highlight some of these experimental findings.
X-ray diffraction (XRD) [8][7]:
The crystal structure is most commonly thought to be a distorted rocksalt structure, that is a structure with two interpenetrating fcc sublattices. The first sublattice is occupied solely by Te atoms. The second sublattice is occupied by Ge, Sb, and vacancies. Vacancy concentrations are relatively high and on the order of 10-25%.
Extended X-ray absorption fine structure (EXAFS) [4]:
There is a splitting of the Ge-Te bonds into shorter and longer bonds. There is a wide distribution of these bond lengths. Furthermore, Ge seems to switch from octahedral to tetrahedral coordination after amorphization.
Coherent phonon spectroscopy (CPS) [2]:
Most GST films exhibit local order with structural subunits of GeTe and Sb2Te3.
X-ray reflectometry (XRR) [9] and atomic force microscopy (AFM) [10]:
Density decrease by about 5-7% upon amorphization.
Differential scanning calorimetry (DSC) [3]:
Crystallization energies in the range of 28-42 meV per atom. This energy is the upper limit for the total energy difference between the two phases with the lower limit given by DFT calculations [11].
Role of Vacancies and Local Distortions
Diffraction experiments have found that GST alloys have a relatively high vacancy concentration on the order of 10-25% [8][7] and also possess a distored rocksalt structure. Studying the electronic structure from DFT calculations has revealed that these vacancies are an intrinsic feature of GST’s structure [12]. It has been suggested that these vacancies may be responsible for many of the interesting properties of GST, such as fast crystallization speed and the phenomena of threshold switching. Therefore, there is great motivation for the study on the role of vacancies in GST.
Wuttig et al performed systematic DFT calculations in order to clarify the origin of these vacancies and the effects of composition, as well as to establish an understanding for the distorted structure. The group began by peforming calculations on a hypothetical alloy, GST-224, and later performed experiments to see if the calculations were able to predict a suitable candidate for a novel alloy.
Total energy calculations were performed on GST-224 in an ideal rocksalt configuration without allowing structural relaxations. Germanium atoms were removed one-by-one and the new total energy recalculated. This procedure was repeated, but instead only removing antimony atoms. All the calculations were repeated, the next time allowing for structural relaxations. The results are shown in Figure 2.
Figure 2: Defect formation energies for Ge (a) and Sb (b) vacancies. The red circles correspond to the ideal rocksalt structure and the blue stars correspond to the distorted (relaxed) rocksalt structure.
It may clearly be seen that creating vacancies is energetically favorable and that there is an ideal composition. Allowing for relaxations further decreases the energy of the system. Compare these findings to that of other semiconductors (e.g. Si and GaAs) which posess very high vacancy formation energies. These results may also be surprising considering the Ge-Te is the strongest bond in this system, yet removing Ge still causes a decrease in energy. The results are not trivial and require explanation.
To analyze these results, Wuttig et all employed an energy-partitioning method known as crystal orbital Hamilton population (COHP) after using DFT to calculate the electronic structure. The curves generated from this analysis provide information about the bonding and antibonding states of the system. It was concluded that creating vacancies annihilated the energetically unfavorable antibonding in Ge-Te and Sb-Te. However, it must be pointed out that removing Te is actually energetically unfavorable since it weakens low-lying bonding states.
Analyzing the pair correlation functions for Ge, Sb, and Te for the different alloys studied, it was found that there are significant local distortions for the nearest-neighbour Ge-Te bonds, resulting in a distribution of shorter and longer Ge-Te bonds. The same effect happens to Sb-Te bonds but not to the same degree. Experimental results from EAXHS has shown the same effect in other GST alloys [4]. The splitting into shorter and longer bonds is known as the Peierls effect.
The structural distortions allow for the optimization of the Te electronic structure. This may be described considering the nature of the p-orbitals of the components in the system. One p-orbital of Te can be described as a non-bonding orbital filled with two electrons. Germanium on the other hand has a completely unfilled p-orbital. Thus, when adding Ge to an Sb-Te alloy, it reduces the number of non-bonding states as Ge-Te bonds are formed between the two p-orbitals just recently mentioned. Sb is insufficient compared to Ge since Sb has more p-electrons.
Thus, Figure 2 may most completely be described by the balance between the need to annihilate antibonding states (creating Ge or Sb vacancies) and also to account for the Peierls unstable Te atom (introducing Ge to removed non-bonding states).
To supplement their calculations, Wuttig et all fabricated thin films of the alloys they studied using DFT to see if they had the stability and favorable properties compared to the more common/popular GST alloys. Their ideal alloys were indeed stable and possesed very similar properties compared to the more traditionally studied alloys on the GeTe-Sb2Te3 pseudo-binary line. In fact, they also discovered that one of their alloys had superior optical properties compared to what has been previously developed.
Characterizing the amorphous phase
While the crystal structure of GST is more or less agreed upon, the amorphous phase has still not been able to be unambiguously defined. Lang et al. performed DFT calculations for a vacancy in GST with an ideal rocksalt structure [5] and found their results contradictory to a previously proposed model of the crystalline phase and the mechanism of phase transformation. Upon further investigation, they proposed a model of the amorphous phase of GST.
Kolobov et al. performed extended x-ray absorption fine structure (EXAFS) analysis to find a wide distribution of shorter and longer Ge-Te bonds in GST. To account for this phenomena, his group suggested the building block for the crystalline phase of GST was a planar ring of Ge, Sb, and Te atoms surrounding a vacancy, with the Te atoms relaxing away from the vacancy and the Ge and Sb atoms relaxing toward the vacancy. They also suggested a mechanism of amorphization where the Ge atoms switches from sixfold coordination to being coordinated by four Te atoms. The relaxation behavior in Kolobov’s model (KB model) [4] was in direct contradiction to the previously mentioned DFT calculations performed by Lang et al. However, their calculations still agreed with the experimental results of Kolobov.
Lang performed DFT calculations to determine the energy barrier required for the Ge flipping mechanism proposed in the KB model. Figure 3 shows two cases that were calculated in DFT. In both cases the Ge atom starts coordinated with six Te atoms. In case 1, the Ge atom flips to be coordinated by two Sb atoms and four Te atoms (leaving behind a vacancy). In case 2, the Ge atoms flips to be coordinated only by four Te atoms (system having three vacancies). The energy was calculated with DFT for various systems being a snapshot of a path of the Ge atom from the starting configuration to end configuration of each case. The rigorous way to find a barrier height would be to consider multiple paths, but since only one path was taken in these calculations, the results at best will provide an upper limit for the barrier height or show that there is no barrier. It was found that case 1 had no barrier while a barrier existed for case 2. They increased their confidence by allowing for structural relaxation of the ending configurations and doing some molecular dynamics simulations. After structural relaxations, the Ge atoms returned to its starting position in case 1 (which is reasonable since there is no barrier) and in case 2 the Ge-Te bond was lengthened.
Figure 3: Atomic configurations before and after the flip. Case 1 [(a) and (b)] and Case 2 [(c) and (d)]. The red arrows show the Ge atom that flips, and the blue and yellow arrows show the Sb and Te nearest neighbors, respectively.
Lang’s findings suggested that the KB model was incapable of describing the phase transition and the group proposed a different building block for the amorphouse phase. The building block is derived from a plane of atoms surrounding a vacancy in the crystal structure. It is a planar ring consisting of eight atoms: two Sb, two Ge, and four Te atoms. The ring was placed in a supercell large enough such that it would be considered isolated and then allowed to relax in a DFT calculation. The resulting, distorted ring was proposed as the building block of the amorphous phase and is shown in Figure 4. The bulk would be made up of many of these rings, misaligned and distorted, with the addition of Te atoms for linking between rings and maintaining stoichiometry.
Figure 4: Building blocks of the amorphous phase. (a) the planar ring after relaxing the atoms around a vacancy in an isolated system and (c) a layer of the distorted ring model before (strong colors) and after (weak colors) refinement.
Lang et al. performed independent experimental measurements to create RDFs using electron diffraction. Their experimental data was deemed accurate and in acceptable agreement compared to other experimental measurements (e.g. Kolobov’s EXAFS data). They compared the RDF data of their computer model to that of experiment and found close agreement. Reverse Monte Carlo (RMC) was used to further refine their model. RMC is similar to other Monte Carlo algorithms but instead of minimizing the energy, the criteria for valid atomic movements is to not deviate strongly from experimental data. The resulting refined structure had an even closer fit to the experimental data.
The phase transformation mechanism was then proposed for the distorted, planar ring model. During quenching, not given time to sufficiently diffuse, the atoms may adopt the ring structure as to minimize nearest-neighbor energy. Given that the model is a derivative of the actual crystal structure, it justifies the rapid transformation characteristic of these alloys.
Experimental data suggests that GST alloys along the Ge-Te and Sb2Te3 pseudo-binary line possess some local order containing units of GeTe and Sb2Te3 [2]. Eom et al. independently proposed an alternative to the KB model since it did not incorporate the structures of GeTe and Sb2Te3. The group constructed three-dimensional building blocks by creating all combinations of a GeTe subunit attached to a Sb2Te3 subunit along a single direction (Figures 5 and 6) [1].
Figure 5: Local geometries of (a) Sb2Te3 and (b) GeTe. Legend: Ge-Blue, Sb-Red, Te-Black, Vacancy-White.
Figure 6: Double layer building blocks constructed from the subunits in Figure 5. Legend: Ge-Blue, Sb-Red, Te-Black, Vacancy-White.
DFT total energy calculations were performed on supercells filled with each of these building blocks simultaneously allowing for relaxations. For comparison purposes, they also calculated the total energy of the planar ring. The results are summarized in Table 1. It should be pointed out that the planar ring was intermediate in energy compared to the three-dimensional building blocks. Given that the energy difference was small between various building blocks, the authors proposed that there may be multiple choices for the structure of GST.
Table 1: Computed total energy for various local structures as seen in Figure 6.
Another result of Kolobov’s EXAFS data that was not rigorously considered in the previous models is the fact that the germanium atom flips from octahedral coordination in the crystalline phase to tetrahedral, at least partially, coordination in the amorphous phase. This motivated Welnic et al. to perform DFT calculations on structures that explicitly accounted for the tetrahedral coordination [11]. These structures were: the rocksalt structure with all octahedrally coordinated atoms; the chalcopyrite structure (similar to zincblende but with three different sublattices) that has all atoms tetrahedrally coordinated; and the spinel structure where only Ge atoms are tetrahedrally coordinated and the Sb and Te atoms are octahedrally coordinated. The calculations were repeated a second time allowing for relaxations. The results are provided in Figure 7.
Figure 7: Electronic ground-state energy per atom plotted versus lattice parameter. All energies are relative to the ideal rocksalt structure (minimum set to zero).
Chalcopyrite was ruled out as a candidate for the amorphous structure due to a density change upon amorphization much greater than experimental data. The spinel structure was deemed a likely candidate for the local order in the amorphous phase since its density change, difference in energy compared to the distorted rocksalt structure, and its bulk modulus all corresponded well with experimental data. Furthermore, the local order was also in agreement with Kolobov’s EXAFS data.
Conclusions
Despite significant experimental and theoretical efforts, there is yet to be an unambiguously defined structure for GST. However, as was outlined in this review, thanks to DFT there has been great progress in this regard and many new models have been proposed. The structural details that are known in GST were justified with DFT via electronic structure arguments. In some cases, DFT has even been able to aid in the design of novel GST alloys.
