Conductance calculations for quantum wires and interfaces: Mode matching and Green’s functions

Landauer’s formula relates the conductance of a quantum wire or interface to transmission probabilities. Total transmission probabilities are frequently calculated using Green’s-function techniques and an expression derived by C. Caroli et al. [J. Phys. C 4, 916 (1971)]. Alternatively, partial transmission probabilities can be calculated from the scattering wave functions that are obtained by matching the wave functions in the scattering region to the Bloch modes of ideal bulk leads. An elegant technique for doing this, formulated by Ando [Phys. Rev. B 44, 8017 (1991)], is here generalized to any Hamiltonian that can be represented in tight-binding form. A more compact expression for the transmission matrix elements is derived, and it is shown how all the Green’s function results can be derived from the mode-matching technique. We illustrate this for a simple model that can be studied analytically, and for an Fe∣vacuum∣Fe tunnel junction that we study using first-principles calculations.

Figure 1

(Color online) Hamiltonian matrix of a quantum wire divided into slices. The left $\left(L\right)$ and right $\left(R\right)$ leads are ideal periodic wires that span the cells $i=-\infty ,\dots ,0$ and $i=S+1,\dots ,\infty$, respectively. The scattering region spans cells $i=1,\dots ,S$.

Figure 2

Parameters of a one-band nearest-neighbor tight-binding model for a single impurity in a one-dimensional chain.

Figure 3

(Color online) Layer density of states (LDOS) at the Fe(001) surface layer. Top panel: majority spin. Top curve (red, dashed): as calculated using the layer Green’s-function method.[35] Bottom curve (blue, solid): as calculated using the mode-matching approach. Both calculations use $\eta =0.0025\mathrm{Ry}$. For clarity, the top curve has been displaced by 10 units along the $y$ axis. Bottom panel: minority spin, the LDOS is shown with a negative sign.

Figure 4

(Color online) The top panels represent the band structures along $\Gamma \text{−}\mathrm{H}$ of bulk Fe for majority and minority spins. Panels (a–h) give the ${\Delta }_1$ contribution to the LDOS ${\rho }_i\left(E\right)$ at $\overline{\Gamma }$ of the Fe∣vacuum interface as a function of the layer position. (a) LDOS of an Fe layer at an infinite distance from the interface; the double-peak structure below $-4\mathrm{eV}$ and single peak (of a double peak structure) above $-1\mathrm{eV}$ correspond to the ${\Delta }_1$ bulk bands. (b) LDOS of the 6th Fe layer (the surface layer is layer 1). (c) LDOS of the 3rd Fe layer; the sharp peak in the bulk band gap corresponds to a surface state. (d) LDOS of the the surface Fe layer; the surface-state peak has maximum amplitude. (e–h) Same for minority spin.

Figure 5

Calculated ${\mathbf{k}}_{\Vert }$-resolved transmission in (a) the majority and (b) the minority spin channels through an Fe∣vacuum∣Fe tunnel junction. The vacuum region has a thickness of $5.733\mathrm{\AA }$ and the Fe electrodes have a parallel magnetization. Only the central part of the Brillouin zone is shown, i.e., $-\pi ∕3a⩽k_x,k_y⩽\pi ∕3a$; the transmission in the other parts is close to zero.