Method to quantify the delocalization of electronic states in amorphous semiconductors and its application to assessing charge carrier mobility of $p$-type amorphous oxide semiconductors

Amorphous semiconductors are usually characterized by a low charge carrier mobility, essentially related to their lack of long-range order. The development of such material with higher charge carrier mobility is hence challenging. Part of the issue comes from the difficulty encountered by first-principles simulations to evaluate concepts such as the electron effective mass for disordered systems since the absence of periodicity induced by the disorder precludes the use of common concepts derived from condensed matter physics. In this paper, we propose a methodology based on first-principles simulations that partially solves this problem, by quantifying the degree of delocalization of a wave function and of the connectivity between the atomic sites within this electronic state. We validate the robustness of the proposed formalism on crystalline and molecular systems and extend the insights gained to disordered/amorphous ${\mathrm{InGaZnO}}_4$ and Si. We also explore the properties of $p$-type oxide semiconductor candidates recently reported to have a low effective mass in their crystalline phases [G. Hautier et al., Nat. Commun. 4, 2292 (2013)]. Although in their amorphous phase none of the candidates present a valence band with delocalization properties matching those found in the conduction band of amorphous ${\mathrm{InGaZnO}}_4$, three of the seven analyzed materials show some potential. The most promising candidate, ${\mathrm{K}}_2{\mathrm{Sn}}_2{\mathrm{O}}_3$, is expected to possess in its amorphous phase a slightly higher hole mobility than the electron mobility in amorphous silicon.

Figure 1

Convergence of the ISWO computed for a hydrogen chain (a), (b) and for crystalline IGZO (c), (d). (a) The evolution of the ISWO value was computed for the first occupied state of a hydrogen chain as a function of the number of atoms present in the unit cell with a DZVP basis. The hydrogen atoms in the linear chain are distant from 1.5 Å. The inset shows a representation of the five-atom hydrogen model indexed with respect to the central atom (referenced to be 0). (b) Convergence with respect to the size of the basis set for an eight-atom model. (c) Error in % on the ISWO value computed for the conduction and valence bands of $c$-IGZO in function of the number of atoms present in the unit cell. (d) Evolution of the deviation induced by the nature of the basis set for a $c$-IGZO model with 189 atoms. All errors are computed with respect to the value obtained for the most converged simulation. For metals, the largest basis set available is a double-zeta valence polarized (DZVP) one. The triple-zeta valence polarized (TZVP), triple-zeta valence twice-polarized (TZV2P), and triple-zeta valence twice-polarized including $f$ orbitals (TZV2PX) labels correspond to simulations where only the basis set of oxygen was increased [26].

Figure 2

Variation of the ISWO values for the lowest-energy valence band of the hydrogen chain (a) as a function of the H-H distance and (b) of the effective mass (${\mathrm{m}}^{*}$). The effective mass is evaluated using a 1 atom/unit cell and the Brillouin zone was sampled with 10 $k$ points along the chain. The effective mass was then obtained by fitting the bottom of the band with a parabola. In the directions perpendicular to the chain, the unit-cell parameters were set to 8 Å.

Figure 3

Representations of an ideal 2D and 3D hydrogen crystal models. The deformations applied to these structures in Figs. 4 and 5 are depicted by arrows. In the 2D case, the deformation is only applied along the $x$ direction. In the 3D one, the deformation is performed along the $x$ and $y$ directions. In both cases, the resulting increases of the bond length span from 1 to 4 Å in the $x$ direction and, in the 3D case, from 1 Å to 2.5 Å in the $y$ one. The atomic structures were rendered with the vesta package [38].

Figure 4

Variation of the ISWO values computed for the lowest-energy valence band for a 2D hydrogen model containing 100 atoms per unit cell forming the square lattice illustrated in Fig. 3. The distance between the atoms is increased along the $x$ direction, and is kept constant at 1.5 Å in the $y$ direction. (a) Variation of the ISWO values as a function of the H-H distance projected along the $x$ (blue circles) and $y$ (green triangles) directions following Eq. (8), as well as the unprotected ISWO analysis (black cross) as given by Eq. (5). (b), (c) Report the same ISWO evolution with respect to the resulting effective mass (${\mathrm{m}}^{*}$). The average effective mass is obtained by the application of a harmonic average along the three Cartesian directions. In the three plots, the blue circles provide the values projected along the $x$ direction, the green triangles along $y$, and the black crosses indicate the unprotected values [Eq. (5)]. For clarity, the unprotected and $y$-projected ISWO curves were multiplied by 10.

Figure 5

Variation of the ISWO analysis computed for the lowest-energy valence band state for a cubic bcc stack of hydrogen containing 125 atoms per unit cell (illustrated in Fig. 3). The distance between the hydrogen atomic sites is increased asymmetrically in the $x$ direction by steps of 0.1 Å and by steps of 0.05 Å in the $y$ direction. The distance along $z$ is kept constant at 1.5 Å. (a) Indicates the variation of the ISWO projected in the $x$ and $y$ directions as a function of the H-H bond length in their respective directions. (b) Reports the evolution of the unprotected ISWO values versus effective mass. (c), (d) Provide the same information projected along the $z$ direction and for the unprotected case.

Figure 6

Variation of the ISWO values computed for the conduction band of crystalline ${\mathrm{InGaZnO}}_4$ as a function of a change of the electronic exchange correlation. This modulation is driven by an increase of the Hubbard term applied to the $d$ orbital of the metals and to the $p$ orbital of the oxygen (green dots) or to the inclusion of a part of Hartree-Fock exchange (blue crosses).

Figure 7

ISWO and IPR values of the lowest-energy state of graphite, bilayer (2L), monolayer graphene and of some organic molecules (blue triangles) and diamond, bulk silicon, germanium, and tin (green circles). The inset provides the representation of the wave functions of diamond, benzene, and polyethylene (${\mathrm{C}}_{24}{\mathrm{H}}_{26}$) with and without fluorine and finally of two interacting molecules of fluorinated polyethylene (${\mathrm{C}}_{12}{\mathrm{H}}_{12}\mathrm{F}$) close to each other (rendered with the vesta [38] package). The wave functions of silicon, germanium, and tin are visually similar to the one of diamond.

Figure 8

Comparison of the ISWO and IPR profiles computed for 13 different atomistic models of $a$-IGZO containing from 84 to 980 atoms. The distribution of the conduction (green) and valence bands (blue) is obtained by averaging the ISWO and IPR values over an energy range of 0.3 eV with respect to the first conduction/valence states to account for the energy window accessible to the charge carriers in $a$-IGZO transistor [42]. The red dashed lines represent the value of the conduction/valence ISWO averaged for all models.

Figure 9

Evolution of the ISWO as a function of the band energy for an $a$-IGZO model containing 980 atoms (a) and a model of $a$-Si (b) with 120 atoms. The dashed vertical lines indicate the position of the Fermi level within the band gap. The band-gap value is highly underestimated due to the use of a semilocal exchange correlation functional (PBE functional [29]). Note that the band gap is further reduced compared to the crystalline phase due to the amorphization process [30].

Figure 10

Comparison of the ISWO values computed for (a) the valence and (b) the conduction bands of crystalline and amorphous phases of a selection of promising $p$-type oxides candidates (ZrOS, ${\mathrm{Sr}}_4{\mathrm{As}}_2\mathrm{O}, {\mathrm{NaNbO}}_2, {\mathrm{Ca}}_4{\mathrm{P}}_2\mathrm{O}, {\mathrm{Sb}}_4{\mathrm{Cl}}_2{\mathrm{O}}_5, {\mathrm{Tl}}_4{\mathrm{V}}_2{\mathrm{O}}_7, {\mathrm{K}}_2{\mathrm{Sn}}_2{\mathrm{O}}_3$). For each oxide, three different amorphous models containing a different number of atoms have been built and the averaged ISWO is provided. The error bars provide the lowest and highest values obtained for each material. For comparison purpose, the results obtained for three models of amorphous silicon and 13 models of $a$-IGZO (Fig. 8) are also provided. The red dashed lines corresponds to the lowest ISWO value obtained for the conduction band of the different $a$-IGZO models.

Figure 11

Evolution of the ISWO in function of the energy for (a) ${\mathrm{K}}_2{\mathrm{Sn}}_2{\mathrm{O}}_3$, (b) ${\mathrm{Tl}}_4{\mathrm{Cl}}_2{\mathrm{O}}_5$, and (c) ${\mathrm{Sb}}_4{\mathrm{Cl}}_2{\mathrm{O}}_5$. The vertical lines set the Fermi levels in the structures.