Büttiker probes and the recursive Green's function: An efficient approach to include dissipation in general configurations

An efficient and compact approach to the inclusion of dissipative effects in nonequilibrium Green's function (NEGF) simulations of electronic systems is introduced. The algorithm is based on two well-known methods in the literature, first that of the so-called recursive Green's function (RGF) and second that of Büttiker probes. Numerical methods for exact evaluation of the Jacobian are presented by a direct extension to RGF which can be modularly included in any codebase that uses it presently. Then using both physical observations and numerical methods, the computation time of the Büttiker probe Jacobian is improved significantly. An improvement to existing phonon models within Büttiker probes is then demonstrated in the simulation of fully atomistic graphene nanoribbon based field effect transistors in $n-i-n$ and $p-i-n$ operation.

Figure 1

Unit cells of armchair nanoribbons, each unit cell consists of $2\times n_y$ carbon atoms. Even $n_y$ configurations have screw symmetry, whereas odd $n_y$ configurations have mirror symmetry. Strength and pigment of colors demonstrate equivalent atoms under (left) screw and (right) mirror operations. Note that while the screw symmetry of even $n_y$ nanoribbons is broken by differing chemical potentials, the mirror symmetry of odd $n_y$ nanoribbons persists.

Figure 2

(Left) Electronic band structure and (right) electronic density of states (DOS) for the $n_y=16$ armchair nanoribbon. Only positive $k_x$ are shown as the band structure is symmetric about $k_x=0$. The band gap between the valence and conduction bands is approximately 680 meV and the separation between the first and second bands at $k_x=0$ is approximately 138 meV.

Figure 3

A schematic demonstrating the design of the graphene FETS. Three gates, $V_{\text{S}},V_{\text{G}},V_{\text{D}}$, are applied to the device which rigidly shift the position of the bands (Fig. 2). $V_{\text{S}}$ and $V_{\text{D}}$ are placed on the semi-infinite contacts with chemical potentials ${\mu }_{\text{S}},{\mu }_{\text{D}}$ corresponding to the source and drain, respectively, and $V_{\text{G}}$ shifts the energy of the central third of the atoms in the device. Intermediate regions then equilibrate by electron-electron interactions [2]. The number of atoms in the transverse direction is to scale, whereas the longitudinal direction has been reduced to approximately one eighth for ease of viewing. For our simulations we mimic the smooth interpolation between regions by replacing the step functions that describe the gates by error functions.

Figure 4

Electronic response of a biased NINFET using the RGF method, incorporating both acoustic and optical phonon scattering as a function of position within the device. (Left) Spectral function $A$. (Middle) Electron Green's function $G^n$. (Right) Nonequilibrium filling function $f_{\text{neq}}=G^n/A$. Green dashed line corresponds to the conduction band edge and the red dash-dotted line corresponds to the valence band edge. Both the $A$ and $G^n$ matrices are in units of ${\mathrm{eV}}^{-1}$, and $f_{\text{neq}}$ is by definition unitless. Data are averaged over unit cells for presentation.

Figure 5

NINFET energy resolved partial $G^n$ matrices in units of ${\mathrm{eV}}^{-1}$. (a) Optical phonon contribution. (b) Acoustic phonon contribution. (c) Lead contributions (including dissipation). (d) All components summed. Green dashed line corresponds to the conduction band edge and the red dash-dotted line corresponds to the valence band edge. Data are averaged over unit cells and presented using a ${log}_{10}$ color axis.

Figure 6

Electronic response of a biased PINFET using the RGF method, incorporating both acoustic and optical phonon scattering as a function of position within the device. (Left) Spectral function $A$. (Middle) Electron Green's function $G^n$. (Right) Nonequilibrium filling function $f_{\text{neq}}=G^n/A$. Green dashed line corresponds to the conduction band edge and the red dash-dotted line corresponds to the valence band edge. Both the $A$ and $G^n$ matrices are in units of ${\mathrm{eV}}^{-1}$, and $f_{\text{neq}}$ is by definition unitless. Data are averaged over unit cells for presentation.

Figure 7

PINFET energy resolved partial $G^n$ matrices in units of ${\mathrm{eV}}^{-1}$. (a) Optical phonon contribution. (b) Acoustic phonon contribution. (c) Lead contributions (including dissipation). (d) All components summed. Green dashed line corresponds to the conduction band edge and the red dash-dotted line corresponds to the valence band edge. Data are averaged over unit cells and presented using a ${log}_{10}$ color axis.