First-principles calculation method for electron transport based on the grid Lippmann-Schwinger equation

We develop a first-principles electron-transport simulator based on the Lippmann-Schwinger (LS) equation within the framework of the real-space finite-difference scheme. In our fully real-space-based LS (grid LS) method, the ratio expression technique for the scattering wave functions and the Green's function elements of the reference system is employed to avoid numerical collapse. Furthermore, we present analytical expressions and/or prominent calculation procedures for the retarded Green's function, which are utilized in the grid LS approach. In order to demonstrate the performance of the grid LS method, we simulate the electron-transport properties of the semiconductor-oxide interfaces sandwiched between semi-infinite jellium electrodes. The results confirm that the leakage current through the $\left(001\right)\mathrm{Si}-{\mathrm{SiO}}_2$ model becomes much larger when the dangling-bond state is induced by a defect in the oxygen layer, while that through the $\left(001\right)\mathrm{Ge}-{\mathrm{GeO}}_2$ model is insensitive to the dangling bond state.

Figure 1

Sketch of the relationship between $z$ and $\zeta$ in the computational model in the case of ${\mathcal{N}}_f=3$. The system consists of the transition region sandwiched between the left and the right semi-infinite crystalline electrodes. In the left electrode, the incident wave and the reflected waves consisting of the propagating and evanescent ones are illustrated by ${\mathrm{\Phi }}^{\mathrm{in}}\left({\zeta }_k\right)$ and ${\mathrm{\Phi }}^{\mathrm{ref}}\left({\zeta }_k\right)$, respectively, while in the right electrode, the transmitted waves composed of the propagating and decaying evanescent ones toward the right side are denoted ${\mathrm{\Phi }}^{\mathrm{tra}}\left({\zeta }_k\right)$. Here, $x\left(y\right)$ and $z$ are coordinates perpendicular and parallel to the nanoscale junction, respectively.

Figure 2

Schematics of unit cells of the (a) Si-${\mathrm{SiO}}_2$ and (b) Ge-${\mathrm{GeO}}_2$ models. Dashed lines indicate the boundaries of the cell. White spheres, H atoms; blue spheres, Si atoms; green spheres, Ge atoms; and red spheres, O atoms.

Figure 3

Schematis of the (a) Si-${\mathrm{SiO}}_2$ and (b) Ge-${\mathrm{GeO}}_2$ models with an oxygen vacancy after geometrical optimization. Dashed lines indicate the boundaries of supercells. Symbols are the same as in Fig. 2.

Figure 4

Conductance of the Si-${\mathrm{SiO}}_2$ and Ge-${\mathrm{GeO}}_2$ models as functions of the incident electron energy measured from the valence band maximum (VBM) of the substrates. The (red) circles and black squares represent conductance spectra of the Si-${\mathrm{SiO}}_2$ and Ge-${\mathrm{GeO}}_2$ models, respectively. The VBM and conduction band minimum (CBM) of Si (Ge) substrate are indicated by dashed (dotted) vertical lines.

Figure 5

Conductance of the Si-${\mathrm{SiO}}_2$ and Ge-${\mathrm{GeO}}_2$ models as functions of the incident electron energy measured from the valence band maximum (VBM) of the substrates. Symbols are the same as in Fig. 4. The VBM and conduction band minimum (CBM) of Si (Ge) substrate are indicated by dashed (dotted) vertical lines.

Figure 6

Contour plot of the charge-density distribution of electrons flowing over the Si-${\mathrm{SiO}}_2$ model with an incident energy of $\mathrm{VBM}+0.41$ eV. Symbols are the same as in Fig. 2. Here the charge density is integrated in the [100] ($x$) direction. Each contour represents twice or half the density of adjacent contours; the lowest contour is $1.97\times {10}^{-4}$ electron/eV/Å${}^2$.

Figure 7

Contour plots of the density of states (DOS) (left panels) and charge density of dangling-bond (DB) states (right panels) for the (a) Si-${\mathrm{SiO}}_2$ and (b) Ge-${\mathrm{GeO}}_2$ models without electrodes. Arrows denote DOS peaks derived from DB states. Energies are measured from the Fermi level $E_F$, and the $z$ coordinate of the atom with a DB is set to 0. In the right panels, the charge density is integrated in the [100] ($x$) direction, and symbolsare the same as in Fig. 2. The coordinate in the $z$ direction corresponds to that in the left panel. Each contour represents twice or half the density of adjacent contours; the lowest contour is $2.56\times {10}^{-2}$ electron/eV/Å ($2.79\times {10}^{-3}$ electron/Å${}^2$) in the left (right) panels.