Efficient wave-function matching approach for quantum transport calculations

The wave-function matching (WFM) technique has recently been developed for the calculation of electronic transport in quantum two-probe systems. In terms of efficiency it is comparable to the widely used Green’s function approach. The WFM formalism presented so far requires the evaluation of all the propagating and evanescent bulk modes of the left and right electrodes in order to obtain the correct coupling between device and electrode regions. In this paper we will describe a modified WFM approach that allows for the exclusion of the vast majority of the evanescent modes in all parts of the calculation. This approach makes it feasible to apply iterative techniques to efficiently determine the few required bulk modes, which allows for a significant reduction of the computational expense of the WFM method. We illustrate the efficiency of the method on a carbon nanotube field-effect-transistor device displaying band-to-band tunneling and modeled within the semiempirical extended Hückel theory framework.

Figure 1

(Color online) Schematic illustration of a nanoscale two-probe system in which a device is sandwiched between two semi-infinite bulk electrodes.

Figure 2

Schematic representation of WFM applied to layered two-probe systems, where the central device region, consisting of layers $i=1,\dots ,n$, is attached to left and right semi-infinite electrodes. The incoming propagating mode from the left electrode is scattered in the central region and ends up as reflected and transmitted superpositions of propagating and evanescent modes.

Figure 3

(Color online) (a) Positions of the Bloch factors ${\lambda }_k\left(\left|{\lambda }_k\right|\le 1\right)$ obtained for a bulk Au(111) electrode with 27 atoms per unit cell at $E=-1.5\text{eV}$. (b) Amplitudes of the corresponding normalized electrode modes ${\mathbf{\varphi }}_k$ moving through ten layers of the ideal bulk electrode. A total of 243 modes are shown of which three are propagating (colored/dashed) and the rest are evanescent (circles/black).

Figure 4

Two-probe system in which the $C$ region boundaries are expanded by $l$ extra electrode layers.

Figure 5

(Color online) Error (absolute) in the calculated total transmission (circles/solid red lines) and reflection (squares/solid blue lines) coefficients $T^{\prime }$ and $R^{\prime }$ as a function of $l$. The panels show the cases of ${\lambda }_{\text{min}}$ set to 0.5, 0.3, and 0.1, which corresponds to 3, 14, and 31 Au bulk modes (out of 243, see Fig. 3) taken into account, respectively. Dashed line indicates the first-order error estimate ${\lambda }_{\text{min}}^l$. The upward-pointing and downward-pointing triangles (green and yellow lines) show error estimates obtained from Eq. (23).

Figure 6

(Color online) Schematic illustration of a carbon nanotube (8,4) band-to-band tunneling device. The carbon nanotube is positioned on Li surfaces next to an arrangement of three gates.

Figure 7

(Color online) (Left panel) Representation of the electrostatic induced shift of the valence- and conduction-band edges along the length of the device for gate potentials $V_{\text{gate}\text{B}}=-2.0$, 1.0, 2.0, and 4.0 eV. (Right panel) The corresponding transmission spectrum. Dotted line shows the position of the Fermi level and the solid line shows the transmission coefficient for an ideal CNT(8,4).

Figure 8

(Color online) Conduction in units of the conductance quantum $G_0$ as a function of the gate B potential. In the calculations we use a dielectric constant of 4, $V_{\text{gate}\text{A}}=-2.0\text{eV}$, and vary $V_{\text{gate}\text{B}}$ from $-2.0$ to 4.0 eV as indicated.

Figure 9

(Color online) Transmission coefficients $T$ and $\tilde{T}$ calculated with the standard WFM method (black solid) and the method of this work (red dashed), respectively, and the difference $\left|\tilde{T}-T\right|$ (blue line) as a function of energy $E$ in the $V_{\text{gate}\text{B}}=2\text{V}$ case.