Materials Selection and Mechanism of Non-linear Conduction in Chalcogenide Selector Devices

The electronic structure and conduction mechanism of chalcogenide-based Ovonic threshold switches (OTS) used as selectors in cross-point memory arrays is derived from density functional calculations and quasi-Fermi level models. The switching mechanism in OTS is primarily electronic. This uses a specific electronic structure, with a wide tail of localized states below the conduction band edge. In amorphous GeSe2−x the conduction band consists of Ge-Se σ*states with a low effective mass, and with a broad tail of localized Ge-Ge σ* states below this band edge. This leads to the OTS behavior. At high fields the electron quasi-EF moves up through these tail states, lowering the conductivity activation energy, and giving the non-linear switching process. The 4:2 coordinated GeSe2−x based alloys are the most favorable OTS material because they have the correct network connectivity to give a high electron mobility and lack of crystallization, a favorable band structure to produce the non-linear conduction, an optimum band gap, and with nitrogen or carbon alloying, a sufficiently low off-current.


Calculations
The early OTS materials were STAG glasses (Si,Te,As,Ge) 6,[12][13][14][15] . These are p-type with a low hole mobility of about 2 × 10 −5 cm 2 /V.s 6,13 , Table 1. More recent OTS materials such as the a-GeSe based semiconductors are n-type with much higher mobility of order 0.7 cm 2 /V.s (and hole mobilities of ~10 −2 cm 2 /V.s) 16,17 . Given the desire for fast switching speeds, this mobility makes GeSe-type materials superior. a-Se consists of molecular units separated by van der Waals bonds, and this gives them a low mobility 16,18 . The difference in mobilities of Ge-and As-based materials arises from their different local structure, as shown in Fig. 1(a). The lower 3-fold coordination of As means that a-As 2 Se 3 and As 2 Te 3 consist of molecular units,  Scientific RepoRts | (2019) 9:1867 | https://doi.org/10.1038/s41598-018-37717-x separated by van der Waals bonds, giving low mobilities 19 because they limited by inter-molecular hopping. This gives their band edge states a high effective mass 20 . On the other hand, the higher 4-fold coordination of Ge or Si in their 4:2 coordinated (silica-like) phases means that the Ge or Si alloys are fully connected networks 21 , with a lower effective mass and higher mobilities 16 . There are three compositional regimes for the Ge or Si chalcogenides, Fig. 1(b). The PCMs use equimolar compositions such as GeTe, which are consistent with their 3:3 coordinated rhombohedral or orthorhombic structures. At the other extreme are the very chalcogen-rich GeSe 2+x or GeS 2+x alloys consist of molecular units, like the a-As 2 Se 3 case. These Se-rich GeSe x alloys were once intensively studied because of interest in the rigidity transition 22 . The OTS materials themselves are 4:2 coordinated but with a higher Ge or Si content so that they form continuous, more highly connected molecular networks.  We now consider the design requirements of OTS devices. It is generally agreed since the work of Adler 6 that the OTS transition is fundamentally of electronic origin. The strongly non-linear change in conductivity corresponds to a negative differential resistance. Ielmini 23,24 modeled this as arising from Poole-Frenkel excitation from midgap trap states to the band edge states, causing a progressive filling of higher energy traps at high applied fields. However, Ielmini's model applies more for high field conduction in wide-gap, weakly screened dielectrics like SiO 2 , rather than for hopping in localized states in moderately screened, disordered semiconductors like GeSe 2 . In this respect, Clima 25 found that the Bohr radius of tail states in a-GeSe x is 1-2 nm, much larger than those of defect states in SiO 2 .
We therefore consider a different interpretation of Ielmini's model, with localized tail states below a mobility edge as in Fig. 2(a) 26,27 . We use the concept of quasi Fermi levels from photoconductivity. At low fields, the electron and hole quasi-Fermi levels lie at the same energy, near midgap. The conductivity is very small, because the activation energy ∆E from E F to the mobility edge is large, half the mobility gap. As the field increases, the occupation of tail states causes the electron and hole quasi Fermi levels to diverge. In particular, the electron E F,n rises up through the broad band of conduction band (CB) tail states, while the hole E F,p remains near midgap because the valence band (VB) tail is quite sharp. The n-type activation energy will decrease sharply, and the conductivity will increase rapidly. This leads to a strongly non-linear dependence of conductivity on applied field. The concept of quasi-Fermi level implies that E F is in equilibrium, so the rate of excitation to these higher states due to 'upwards' hopping equals the rate of thermalization back to lower states.
We now design a chalcogenide band structure to satisfy these requirements. The overall band structure in Fig. 2(b) consists of three types of bands; the bonding (σ) states well below the VB top, the chalcogen lone pair pπ states forming the upper VB, and the antibonding (σ*) states forming the CB [28][29][30] . The pπ band is non-bonding, of high effective mass m*, and it does not have a wide tail state distribution. The GeSe 2 CB consists of σ* states with much lower effective mass. These are Ge sp 3 /Se p states. The band edges of such Ge-Se σ* states are sensitive to disorder and can be localized because they are sp 3 -like as in a-Si, as noted by Hosono 31 . Overall, the band gap has tail states primarily on the CB side, and few on the VB side. This is the configuration required by Fig. 2(a). This same overall band layout holds for a-GeSe, a-SiTe and a-SiSe.
We now consider the effects of off-stoichiometry from the stoichiometric GeSe 2 composition. The OTS design should avoid Se-rich compositions, as they create Se-site defects which can mediate slow, structural transitions and which go against the concept of electronic-only transitions. Also, they can introduce Se-site disorder and band tails on the VB side, which go against the desire for unipolar conduction. An excess of Ge from GeSe 2 will introduce Ge-Ge bonds. The σ* state of an isolated Ge-Ge bond will generally lie in the CB. Nevertheless, higher Ge excess will lead to a band tail of localized Ge-Ge σ* states appearing beneath the main CB of Ge-Se σ* states. The width of this Ge-Ge σ* band tail varies with the number of Ge-Ge bonds around each Ge vertex.
Our model is supported by detailed density functional calculations on supercell models. First, note that the band structure of the covalently-bonded chalcogenides differs considerably from that of the crystalline GeTe and GeSbTe compounds. For these, the band gap is narrow and arises from a Peierls distortion 9,10 ( Fig. 3a,b). On the other hand, bonding in a-GeSe 2−x is simple electron-pair covalent bonding, leading to wider gaps (Fig. 3c,d).
We take a-GeSe 2 with divalent Se sites as an example. Figure 4(a) shows a chemically ordered random network model of 4:2 bonded a-GeSe 2 where each Ge site has four Ge-Se bonds. Figure 4(b) shows the partial density of states (PDOS) for this structure. It has a band gap of 1.7 eV in GGA. The calculated DOS here are consistent with the experimental spectra, measured by UPS, XPS and inverse photoemission for the case of the conduction band 21,[32][33][34][35] . The upper valence band has three peaks at −1.4 eV, −3.0 eV and −4.6 eV as seen in UPS. Finally the peak at -9 eV corresponds to the Ge s states. The Se s states lie below the diagram. In the conduction band, the first peak is due to Ge s and Se p states, and the second is due to Ge p and Se p states. Figure 4(c) shows a chemically disordered random network of 4:2 bonded a-GeSe with some Ge-Ge bonds. Figure 4(d) shows its PDOS. The lower CB peak consists mainly of Ge p-like states. Figure 5(a) shows a chemically ordered 4:2 coordinated network of GeSe 2 but with the addition of one Ge-Ge bond. This is formed by removing a Se site from a Ge-Se-Ge bridge and joining the two Ge sites together. While this process would make a gap state in a crystalline model, there are no gap states in the network due to relaxation, as the Ge-Ge σ* state lies just above the CB edge, Fig. 5(b). Figure 5(c) shows a chemical ordered GeSe 2 random network, but including two Ge-Ge bonds with a common vertex. Figure 5(d) shows its PDOS. This network now has gap states corresponding to Ge vertex sites. This behavior occurs generally. As noted earlier, the effective width of a Ge-Ge sub-band varies with the coordination number of the Ge-Ge sub-network. This causes the Ge-Ge bonds to form a distribution of tail states that extend below the conduction band edge. Clima 25 finds that such states have a large Bohr radius and a large localization or inverse participation ratio (IPR) due to their Ge p-like character. Thus overall, our calculations show that a Ge excess within a overall 4:2 coordinated network causes a tailing of localized tail states below the conduction band edge, which will be available to create a non-linear conductivity at high applied fields.   Fig. 6(c,d) shows the network and PDOS of SiTe 2−x with Si-Si bonds. These have the same band layout as a-GeSe 2−x , but with SiTe having a narrower gap and SiSe having a wider gap ( Table 2). The same gap states appear. However, the longer bonds of SiTe cannot carry the high maximum current densities as GeSe, whereas the wider gap SiSe will mean that the non-linearity occurs at too large applied fields. Thus GeSe comes out as having the optimum properties.
We now consider how we can illustrate these trends in terms of phase stability maps. Phase stability maps were first used by Philips and VanVechten 36 for the structural classification of the average valence-4, sp-bonded AB binary compounds. They used the covalent gap and ionic gap parameters derived from optical spectra to separate the covalent zincblende compounds from the ionic rocksalt compounds. The separation was improved by Chelikowsky and Philips 37 who used the quantum defect orbital radii of StJohn and Simons 38 to define two parameters rπ −1 (hybridization) and rσ (ionicity), which acted like the covalent gap and ionic gap, to separate out the zincblende, wurzite, rocksalt and CsCl structures.
The PCM and OTS materials are based on the 5-electron IV-VI binary compounds. Their bonding is primarily between p orbitals not sp orbitals as in 4-electron compounds. Thus, Littlewood 39 defined slightly different parameters, with rπ −1 as before but the rσ parameter involving just p orbitals. These axes led to a structural separation of the rhombohedral, orthorhombic and rocksalt phases of the 5-electron binaries.
Lencer 40 then laid the rπ −1 , rσ coordinates of the high performing PCM compositions onto this phase map. It showed that PCMs were favored where the stable crystal phase has rhombohedral structure which supports   9,10 . We now extend this analysis method to treat selector materials and provide a more detailed understanding of the band structure design that leads to their non-linear conductivity behavior.
To treat the selector materials, we must extend the range of possible structures for the 5-electron semiconductors to higher rπ −1 values by including the covalent 4:2 coordinated SiO x (silica-like) phase. Figure 4(a,b) shows the orthorhombic structure of GeSe containing only heteropolar bonds and Fig. 4(c,d) shows the silica-like phase of GeSe containing also homopolar Ge-Ge bonds 41 .
The phase boundaries are found by calculating the total energies of the different phases in Fig. 7(a,b) using density functional theory at closely spaced compositions across the phase map. Figure 7(a) shows the resulting stability regions of various IV-VI materials. We see that the plot reproduces the previous separation of rhombohedral, orthorhombic and rocksalt structures. A fourth region now appears for the 4:2 bonded silica-like structures, which covers most of the large hybridization parameter. Note that by including closely spaced alloy compositions, we can define accurately where each phase boundary lies. Interestingly, we find that the silica/orthorhombic phase boundary is sloped.
We now lay the experimental data points for OTS compositions of Velea 5 and PCM materials from Lencer 40 onto this map, Fig. 7(b). We see that the PCM region overlaps the rhombohedral and silica-like region, rather than being fully in the rhombohedral region. This indicates a slight offset in the prediction properties of the quantum-defect parameters. We see that the experimental OTS selector compositions lie within the silica-like region of Fig. 7(b). This arises as follows. The maximum current density is a key criteria for selectors. This is limited by electromigration or the breaking of network bonds, just as it is for interconnects 42 . Thus, a large current-carrying capacity requires stronger shorter bonds, and this requires a large hybridization parameter. This means moving to lighter, shorter bond-length elements like Se, S and Si. The larger hybridization parameter moves it away from tellurides and resonant bonding, towards lighter chalcogenides with more covalent bonding. This means better glass formers and slower atomic transition rates. The compounds also remain in the amorphous state to higher working temperatures 6,25 . Given that selectors still require rapid transitions, then they should not use a structural phase transitions as in PCRAMs, but use the intrinsically rapid electronic-only transitions. Thus we arrive at OTS materials. However, we should limit it to well-screened chalcogenides of moderate band-gap to avoid too slow responses due to slow charge trapping as in SiO 2 .
A weakness of the a-GeSe-type alloys as selectors is that their off-current is too high. This can be counteracted by alloying with N or C 43-49 as is well known previously. The addition of N or C both widens the band gap, reducing off-current, and also reduces atomic diffusion and nucleation rates, so raising the crystallization temperature. Thus, the overall selection becomes a-Ge-Se-type systems, not pure a-Ge-Se.
Our results are also consistent with experiment values in more general terms, where these are known. Amorphous Se is found to have a low p-type mobility from its photoconductivity 16 . Amorphous GeSe is different. Thermopower data shows that a-GeSe is n-type for x = 0.50 and for Ge-rich compositions 17 . This is consistent with the defects being mainly on the Ge sites and related to the conduction band. It is also consistent with the photoconductivity data of Govoneanu 4 who found defect states at 0.4-0.5 eV below the CB edge. At low fields, the conductivity activation energy ΔE is found to be roughly half the band gap, showing that conductivity is by activation from E F to the CB mobility edge. At high fields, ΔE is small, consistent with hopping within a narrow, dense range of states near the CB mobility edge. This contrasts with the phase change materials, where in crystalline GeTe and GeSbTe, the Ge vacancy states pin E F close to the valence band edge, making it overall p-type 50 , whereas in the amorphous phase, various not fully identified defects pin E F near midgap 51 .

Conclusions
We have studied the bonding characteristics of Ge-rich Ge x Se 1−x and similar systems as typical chalcogenide OTS materials. It is shown that the network structure of 4:2 coordinated group IV chalcogenide alloys have a low electron effective mass, sizable donor Bohr radii, and a large tail of localized conduction states, leading to both non-linear conduction and a high current carrying capacity as desired for OTS devices. The favored compositions are found to cluster in the region of the hybridization/ ionicity plot where the 4:2 coordinated silica-like phase is most stable. Thus the OTS materials belong to the range of larger hybridization parameter than the PCM materials, explaining their greater structural rigidity which is required for OTS operation. It is found that at the same time as retaining the 4:2 bonded structure, certain Ge-Ge bonding configurations create conduction band tail states that lead to the non-linear conduction. This work promotes a better understanding of the bonding origin of threshold switching in chalcogenides and allows optimization of selector devices.

Methods
To calculate the phase boundaries, we carried out first-principles density functional theory calculations using the plane wave CASTEP code 52 . The generalized gradient approximation (GGA) was used for the exchange-correlation function. Corrections to the GGA description of van der Waals bonding were included using the Tkatchenko and Scheffler scheme 53 . A unit cell of different chalcogenide bonding structures are geometrically optimized with 1000 eV energy cut-off and dense k-point meshes with separation as close as 0.03 Å −1 .