Oxide semiconductors (e.g., ZnO, SnO₂, In₂O₃, Ga₂O₃, and InGaZnO) have been widely used for thin film transistors and transparent conducting electrodes for decades due to their high phase stability, easy of synthesis at low temperature, wide bandgap, and high carrier mobility. (Nomura et al., 2003; Fortunato et al., 2012; Park et al., 2012; Troughton and Atkinson, 2019; Lee, 2020; Shi et al., 2021). More recently, oxide semiconductors, especially In₂O₃, have also found their applications as channel materials for back-end-of-line (BEOL) transistors in realizing vertical CMOS technology. (Salahuddin et al., 2018; Hu et al., 2019; Chakraborty et al., 2020; Hu et al., 2020; Si et al., 2020; Si et al., 2021). Oxide semiconductors are typically present in amorphous phase due to their low synthesis temperatures. (Park et al., 2012; Troughton and Atkinson, 2019; Lee, 2020). Understanding the electron transport in amorphous phase oxides would be valuable for tailoring materials properties for the purpose of high-performance electronic device applications.

As the oxide semiconductors are present in thin films, various factors that could influence the electron transport in amorphous oxide semiconductors. It is well known that the oxygen vacancy is ubiquitous in oxides and oxygen vacancies are the sources of electron doping in oxides (Ide et al., 2019; Shi et al., 2021). Film density, which determines the atomic packing density at microscale, is also reported to influence the electron mobility (Ide et al., 2019). Interestingly, Buchholz et al. (Buchholz et al., 2014) have reported a strong dependence of electron mobility on In₂O₃ film crystallinity where the mobility reaches its maximum value at an intermediate degree of nanocrystalline. Other factors, such as interface and surface have been also mentioned influencing the carrier generation and electron transport in the literature (Shi et al., 2021).

Current understanding on the electron transport mechanism in amorphous oxide semiconductors is based on Hosono’s proposed idea which has been widely accepted in the literature (Takagi et al., 2005; Robertson, 2008; Walsh et al., 2011): the electron conduction pathway (ECP) is due to the orbital overlap between extended s orbitals of neighboring metal atoms. This ECP, because of spherical symmetry of s orbital, is immune from local structural distortion present in amorphous phase so that amorphous oxide semiconductors exhibit high electron mobility comparable to the crystalline phase. This concept also explains well the low electron mobility in amorphous Si where ECP is composed of directional sp (Park et al., 2012)-orbital overlaps which can be easily altered by local structure distortion such as bonding angle variation.

The structures of amorphous oxide semiconductors have been investigated mainly from theoretical modeling approaches. Walsh et al. (Walsh et al., 2009) reported that the amorphous oxide phases are defined by the random packing of cation centered polyhedra, where the local metal-oxygen coordination motifs are largely preserved but the long-range packing of these polyhedra is disordered. Later, Buchholz et al. (Buchholz et al., 2014) systematically studied the structural properties of amorphous In₂O₃ (a-In₂O₃) and revealed that long-range order loss is resulted from the InO _x polyhedra random rotation around each other. The random polyhedra packing model is a significant advancement towards the understanding of the structure of amorphous oxide semiconductors.

In this work, we devote to investigating the electron transport mechanisms in In₂O₃ and its implication for amorphous phase electron transport. We base our study on an orbital overlap perspective: the orbital overlap enables electron transport and the continuous orbital overlaps of the involved atomic pair form the ECP. The ECP gives information of how amorphous disorders would influence the electron transport in amorphous phases. Our analysis includes atomic structure characterization, first principles density functional theory (DFT) calculations, and empirically tight binding (ETB) modeling, under which the orbital pair contribution to ECP, i.e., the microscopic electron conduction mechanism, is revealed. Our results provide an important contribution to the current understanding of electron transport mechanism in In₂O₃ and practical guidance on tailoring In₂O₃ properties for high-performance In₂O₃ based electronic devices.

Crystalline In₂O₃ (c-In₂O₃) assumes the cubic (bixbyite type) crystal structure (Ia3, # 206). The lattice constant for bixbyite c-In₂O₃ unit cell is a = 10.1Å. The primitive cell of c-In₂O₃ is shown in Figure 1A. According to Pauling’s rule and other reference works (Pauling, 1929; Walsh et al., 2009; Buchholz et al., 2014), for a typical metal oxide which is an ionic compound, its crystal structure can be viewed as regular packing of MO _x polyhedron (Figure 1B). At short-range order (SRO, defined as the metal-oxygen nearest neighboring bonding), each metal cation is surrounded by coordinated oxygen anion forming polyhedron. The polyhedra are then connected together by sharing common oxygen atoms either at the polyhedral corner (corner sharing), or at the edge (edge sharing), or at the face (face sharing). The medium-range order (MRO) is defined as metal-metal second nearest neighboring bonding. MRO is essentially the metal-metal connectivity and is determined by the common oxygen atom sharing. In In₂O₃, each In atom is coordinated to 6 O atoms forming an octahedron, and different octahedra are linked together by sharing either corner or edge. Each In, in terms of MRO, is coordinated to 12 In atoms, of which 6 are corner sharing (In-In corner) and six are edge sharing (in-In edge). Figure 1C illustrates these two types of In-In coordination: In-In corner where two In atoms share only one common oxygen atom, and In-In edge where two In atoms share two common oxygen atoms.

The band structure of c-In₂O₃ is presented in Figure 2A c-In₂O₃ has a wide band gap of ∼ 3eV due to the large electronegativity difference between oxygen and indium. In terms of band edge features, a strongly dispersive bottom conduction band can be observed with large contribution from indium s-orbitals and lesser contribution from O-2p orbitals. The much flatter top valence band is almost solely dominated by O-2p orbitals. As a result, In₂O₃ has a small electron effective mass and a large hole effective mass. The dispersion polarity between conduction band and valence band can be explained by atomic orbital characters. From tight binding point of view, the band dispersion is due to the atomic orbital overlap and a stronger interatomic orbital overlap leads to a stronger band dispersion. Indium s orbitals are more delocalized and spatially more spreading out with larger orbital radii. As a result, stronger overlap is expected between neighboring metal’s s-orbitals. In comparison, O-2p orbital is more localized and directional with smaller orbital radius leading to weaker p-orbital overlap between neighboring O atoms.

The electron transport is a key property for metals oxides used as transparent conducting oxides and semiconducting channel materials. Drude model states that the carrier mobility depends on the effective mass through μ = eτ/m ^*, where e is the elementary charge and τ is the electron momentum relaxation time. This expression indicates that electron mobility is determined by the band dispersion since the effective mass is simply the band dispersion curvature. Since the band dispersion is due to the interatomic orbital overlap, stronger orbital overlap interaction would therefore lead to smaller effective mass and higher electron mobility. Microscopically, electron transport is achieved by electron delocalization movement facilitated by orbital overlap. The continuous atomic overlap throughout the entire crystal thus forms the ECP. For In₂O₃, the conduction band is dominated by metal’s s-orbital and lesser contribution from O-2p orbital. It is thus expected that two types of orbital overlap, metal-s/metal-s direct orbital overlap and metal-s/O-2p/metal-s indirect orbital overlap, could form the ECP.

To resolve the contribution from each type of orbital overlap and to determine the ECP in In₂O₃, we have performed tight binding analysis on the band structure of In₂O₃. In tight binding model, individual orbital overlap interaction can be arbitrarily included or excluded, and their effect on the band dispersion can be examined. This directly provides us a way to determine whether a specific orbital overlap contributes to the band dispersion and eventually electron transport. Figure 2 shows the ETB model of In₂O₃ band structure, with orbital overlap integral empirical parameters fitted to reproduce the DFT band structure. It can be seen that the band dispersion features, especially the bottom conduction band, agree well with the DFT result. Band structures were then calculated based on the following assumptions: (a) all interactions between all atom pairs, (b) In-O orbital overlap interaction excluded, (c) In-In edge interaction excluded, and (d) In-In corner interaction excluded. It is found that when In-O orbital overlap interaction is excluded, the conduction band is identical to that of all interactions included. This suggests that In-O orbital overlap interaction does not contribute to the conduction band dispersion. Therefore, In-O-In indirect orbital overlap behaves as if it is not involved in electron transport and only In-s/In-s direct orbital overlap forms the ECP. The observation that In-O orbital overlap interaction does not contribute to the conduction band dispersion can be explained by the orbital symmetry. In-s orbital presents a spherical symmetry while O-2p orbital has two lobes antisymmetric with respect to the atomic center (Figure 2F). When neighboring atomic orbitals overlap, each O will form a pair of In-s/O-p bonding interaction from one lobe but simultaneously another pair of In-s/O-p antibonding interaction from the other lobe. The bonding and antibonding interaction around O-p completely cancel each other and O atoms always present a zero net orbital interaction with neighboring In atoms. As a result, In-O orbital interaction does not contribute to the E-k dispersion. We thus conclude that in In₂O₃ metal-s/O-p/metal-s indirect overlap does not contribute to the electron transport in metal oxides and only metal-s/metal-s direct orbital overlap forms the ECP.

This above result is in contrast to the traditional understanding that in metal oxides the conduction band dispersion is mainly facilitated by the metal’s s-orbital overlap with lesser contribution from oxygen p-orbital. As mentioned in the Introduction part, electron conduction is due to the orbital overlap between extended s orbitals of neighboring metal atoms. Our results suggest that the electron transport is “entirely” facilitated by metal’s s-orbital “without” oxygen p-orbital involvement. Our results support Hosono’s model that electron conduction is due to the orbital overlap between metal atoms. Our work further develops this model that metal-metal medium range connection is the determining factor to the electron transport. Note that O-p orbitals are still contributing to the band structure including the valence bands. Oxygen atoms also maintain the lattice charge neutrality and their p-orbitals alternate with indium s-orbitals to ensure orbital stability.

It would be instructive to characterize the amorphous phase atomic structure before diving into the electron transport in amorphous oxides. Upon amorphization, InO _x polyhedra will be distorted through O-In-O bond angle and In-O bond length variation. Amorphous phases retain SRO to a large degree in a sense that the local bonding (InO _x polyhedra) is only minorly distorted. SRO is largely maintained in amorphous oxides due to the strong chemical bonding between metal and oxygen atoms. No homopolar bonds are expected for amorphous metal oxides (see radial distribution function of In₂O₃ in Figure 3A). In terms of MRO, a viewpoint on amorphous structure is that MO _x polyhedra in amorphous are connected in a distorted way, contrasting to regular packing in crystalline phase. (Buchholz et al., 2014). Such distortion can be viewed as one MO _x polyhedron being rotated relative another MO _x polyhedron. Figure 3 illustrates this concept using the In₂O₃ structure. The net effect from polyhedra rotation is that In-In medium range connections are distorted. In In₂O₃, the metal-metal medium-range connections are critical to electron transport; therefore, the amorphous disorder influence on electron transport could be revealed from metal-metal MRO. “Polyhedra rotation” in amorphous phases also results in short-range disorder including In-O bonding distance and coordination number change. Figures 3C,D show that when InO _x rotated, two neighboring InO _x polyhedra change from edge sharing to corner sharing which results in In coordination number reduction. This result is consistent with previous work by Buchholz et al. (Buchholz et al., 2014) Figure 3B presents the distribution of the coordination numbers (CN) of In and O atoms. The average CN of In and O are 5.5 and 3.7, respectively, in comparison with six and four in c-In₂O₃. The “polyhedron rotation” in amorphous phase results in certain amount of loss of MRO, which gradually leads to the complete loss of long-range order.

As the In-In medium range connection forms the ECP, it is therefore essential to reveal the In-In medium-range connection to shed light on the electron transport in a-In₂O₃. The metal-metal coordination behavior is the focus for MRO analysis. Based on tight binding analysis, electrons transport through neighboring In-In orbital overlap and In-In edge sharing contributes more to the ECP than In-In corner sharing. The amorphous disorder modifies the In-In connection and thus affects the electron transport. The “polyhedron rotation” in a-In₂O₃ effectively leads to the polyhedral edge sharing “degrading” to corner sharing (Figure 3D). Figure 4 presents the statistical result of In-In connectivity resolving the In-In edge connection and In-In corner connection in a typical a-In₂O₃ structure. In c-In₂O₃, each In is coordinated to 12 In atoms, with six of them bridged by edge sharing O atom and six of them by corner sharing O atom. In contrast, in a-In₂O₃, the overall In to In CN has reduced from 12 to an average of 11.5. Since the In-In connection is the ECP, the decreased CN of In-In pair will reduce the possible conduction pathways. Another important feature shown in Figure 4 is that upon amorphization, the number of In-In corner sharing increases while the number of In-In edge sharing decreases. The In-In edge sharing transforms to In-In corner sharing due to “InO₆ polyhedra” rotation. Since In-In edge is more efficient in In-In orbital overlapping and facilities more to the electron transport, this transition further reduces electron conduction pathways in amorphous In₂O₃ and limits the amorphous phase mobility. Supplementary Figure S1 also shows that In-In distance also influences the In-In orbital pair overlap interaction strength.

The above results indicate that in a-In₂O₃ the local In-s/In-s orbital overlapping and In-In medium-range connection determine the electron transport effectiveness. The average In-In orbital overlapping integral quantifies the electron mobility in a-In₂O₃. It is well-known that in covalent bonded systems like Si and Ge, the amorphous phase mobilities typically degrade by 10–100 times compared to the crystalline phase mobility. (Clark, 1967; Marshall et al., 1986). In contrast, ionically bonded oxides can retain their high electron mobility in amorphous phases. (Nomura et al., 2004; Robertson, 2008). This is due to the spherical symmetry of s orbital which is immune from local structural distortion present in amorphous phase, while sp ³-orbital overlaps are easily altered by local structure distortion. In our work, we found that the In-In coordination in a-In₂O₃, which is the structural origin for s-orbital overlap, does not disrupt significantly compared to c-In₂O₃. This is manifested by the slight change of In-In bond distance (3.39Å→3.44Å) and In CN of In (reducing from 12 in a-In₂O₃ to 11.5 in c-In₂O₃). The insignificant In-In connection change from crystalline to amorphous phase is the underlying reason for undisrupted s-orbital overlap in a-In₂O₃. The theoretical finding presented in this work can be validated by neutron diffraction experiment where the relationship between the In-In position correlation and electron mobility can be mapped. The above result implies that enhancing the In-In connection could improve the amorphous phase electron transport. For example, incorporating higher oxygen content during In₂O₃ thin film growth could tune In-In corner sharing into In-In edge sharing so that s-orbital overlap is strengthened. Densifying the film might decrease In-In bond distance and increase In-In CN, both of which are beneficial to In-In s-orbital overlap. These results could provide useful guidance for device engineers for better optimizing a-In₂O₃ based electronic devices.

In conclusion, we have investigated the electron conduction mechanism in In₂O₃ and its implication for amorphous phase transport from an orbital overlap facilitated electron transport perspective. Combined DFT and ETB calculations show that the electron transport is facilitated by the neighboring metal atomic s orbital overlap without oxygen’s p-orbital involvement. The ECP in In₂O₃ relies only on the metal-metal medium range connection. This electron conduction mechanism is extended to amorphous phase where the amorphous disorder influences the electron transport through impacting the metal-metal MRO including metal-metal CN and metal-metal separation. Our results contribute to the current theoretical understanding of electron transport in oxide semiconductors and to the practical advancement of oxides in technological applications.