Krylov subspace method for evaluating the self-energy matrices in electron transport calculations

We present a Krylov subspace method for evaluating the self-energy matrices used in the Green’s function formulation of electron transport in nanoscale devices. A procedure based on the Arnoldi method is employed to obtain solutions of the quadratic eigenvalue problem associated with the infinite layered systems of the electrodes. One complex and two real shift-and-invert transformations are adopted to select interior eigenpairs with complex eigenvalues on or in the vicinity of the unit circle that correspond to the propagating and evanescent modes of most influence in electron transport calculations. Numerical tests within a density functional theory framework are provided to validate the accuracy and robustness of the proposed method, which in most cases is an order of magnitude faster than conventional methods.

Figure 1

(Color online) Schematic representation of a two-probe device with applied bias $V_b$. The top figure illustrates the Landauer-Büttiker picture of coherent scattering between electron reservoirs kept at chemical potentials ${\mu }_L$ and ${\mu }_R$. The bottom figure shows the device part modeled by two semi-infinite electrodes ($L$ and $R$) and a central region $\left(C\right)$, each divided into principal layers that interact only with nearest-neighbor layers. The layers are described by square Hamiltonian matrices $H_i$ of varying sizes and numbered $i=-\infty ,\dots ,\infty$, as indicated.

Figure 2

(Color online) Positions of the 243 complex eigenvalues [blue (circles)] inside the unit disk for a Au(111) electrode with 27 atoms per unit cell at $E=-2\text{eV}$. The 21 eigenvalues corresponding to propagating modes [red (filled) dots] are located on the unit circle. The modes of most significance in transmission calculations are located within the green (shaded) area given by $0.1\le \left|\lambda \right|\le 1$.

Figure 3

(Color online) Illustration of the complex eigenvalues [blue (circles)] for the Al(100) electrode at $E=3\text{eV}$. The eigenvalues corresponding to the wanted right-going modes [red (filled) dots] can be separated according to their location within three distinct green (shaded) areas of the unit disk and determined efficiently using shift-and-invert spectral transformations to $\pm 1/\sqrt{2}$ and $\widehat{i}/\sqrt{2}$ (crosses).

Figure 4

(Color online) Convergence behavior of the Krylov algorithm for the Al(100) electrode at $E=3\text{eV}$. The figures show the residual norm as a function of iterations for Ritz pairs that satisfy $0.1\le \left|\lambda \right|\le 1+ϵ$, in the case of shift-and-invert transformations to $\pm 1/\sqrt{2}$ and $\widehat{i}/\sqrt{2}$, respectively.

Figure 5

Schematic illustration of the Al(100)-C7-Al(100) two-probe system.

Figure 6

(Color online) Transmission spectrum of the Al(100)-C7-Al(100) system for different bias voltages $V_b$. The self-energy matrices used in the $T\left(E\right)$ calculations have been obtained at the $\Gamma$ point by the proposed Krylov subspace method with parameter ${\lambda }_{\text{min}}$ at several settings: 0.1 [black (full) curve], 0.5 [red (dashed) curve], and 0.99 [blue (dotted) curve]. The bias windows are indicated by the vertical dashed lines.

Figure 7

(Color online) Current as a function of the parameter ${\lambda }_{\text{min}}$ used by the Krylov subspace method for the Al(100)-C7-Al(100) system with applied bias voltages $V_b=\left(\text{a}\right)1$ and (b) $2\text{V}$. The correct currents obtained by conventional methods are $I\approx 38.4$ and $\approx 77.0\mu \text{A}$, respectively, indicated here by the green (dashed) lines.

Figure 8

(Color online) CPU times for computing the left self-energy matrix ${\mathbf{\Sigma }}_L$ plotted as a function of the size $N$ of ${\mathbf{\Sigma }}_L$ for a range of armchair $\left(n,n\right)$ CNT electrodes, where $n=1,\dots ,20$. The dotted and dashed lines indicate $O\left(N^2\right)$ and $O\left(N^3\right)$ computational complexity, respectively.

Figure 9

(Color online) Transmission spectrum of the Au(111)-BDT-Au(111) system for different $k$-point samplings and $V_b=0$. The self-energy matrices used in the $T\left(E\right)$ calculations have been obtained by the generalized Krylov subspace method with parameter ${\delta }_{\text{min}}=0.1$.