General Green's function formalism for layered systems: Wave function approach

The single-particle Green's function (GF) of mesoscopic structures plays a central role in mesoscopic quantum transport. The recursive GF technique is a standard tool to compute this quantity numerically, but it lacks physical transparency and is limited to relatively small systems. Here we present a numerically efficient and physically transparent GF formalism for a general layered structure. In contrast to the recursive GF that directly calculates the GF through the Dyson equations, our approach converts the calculation of the GF to the generation and subsequent propagation of a scattering wave function emanating from a local excitation. This viewpoint not only allows us to reproduce existing results in a concise and physically intuitive manner, but also provides analytical expressions of the GF in terms of a generalized scattering matrix. This identifies the contributions from each individual scattering channel to the GF and hence allows this information to be extracted quantitatively from dual-probe STM experiments. The simplicity and physical transparency of the formalism further allows us to treat the multiple reflection analytically and derive an analytical rule to construct the GF of a general layered system. This could significantly reduce the computational time and enable quantum transport calculations for large samples. We apply this formalism to perform both analytical analysis and numerical simulation for the two-dimensional conductance map of a realistic graphene $p-n$ junction. The results demonstrate the possibility of observing the spatially resolved interference pattern caused by negative refraction and further reveal a few interesting features, such as the distance-independent conductance and its quadratic dependence on the carrier concentration, as opposed to the linear dependence in uniform graphene.

Figure 1

Connection between two widely used approaches to mesoscopic quantum transport: the GF approach and wave function mode matching approach. The former calculates the GF $G\left(E\right)$, while the latter calculates the scattering matrix $S\left(E\right)$. The Fisher-Lee relations allow $S\left(E\right)$ to be constructed from $G\left(E\right)$, while the inverse relation remains absent for a general lattice model.

Figure 2

(a) A layered 2D structure consisting of multiple periodic slices (i.e., leads) and disordered slices (i.e., scatterers). (b) Regarding each slice as a unit cell (filled squares), the structure in (a) becomes a 1D lattice. (c) A semi-infinite lead connected to a scatterer. The unit cell Hamiltonian (filled squares) and nearest-neighbor hopping (double arrows) are $m$-independent inside the lead, but could be disordered inside the scatterer.

Figure 3

Scattering state emanating from a local excitation at $m_0$ in an infinite lead (a), a semi-infinite lead connected to a scatterer on its right (b), and a finite lead sandwiched between two scatterers. The zeroth-order, first-order, and second-order partial waves are denoted by black, blue, and orange arrows, respectively.

Figure 4

Scattering state emanating from a local excitation inside a scatterer (a) or a lead (b). The black arrows denote the zeroth-order partial wave and the blue arrows denote the first-order partial wave due to scattering.

Figure 5

(a) A scatterer $B$ connected to a finite left lead $L$ and a semi-infinite right lead $R$ can be regarded as a composite scatterer $C$ connected to one semi-infinite right lead $R$. (b) Two scatterers sandwiched between three leads $L,M,R$ can be regarded as a com-posite scatterer $C$ connected to two semi-infinite leads $L$ and $R$.

Figure 6

Green's function ${\mathbf{G}}_{m,m_0}$ of an infinite system containing eight scatterers, where $m_0$ and $m$ are both inside the leads or at the surfaces of the scatterers.

Figure 7

(a) Sketch of graphene $p-n$ junction with a smooth interface. Panels (b)–(d) show the choice of primitive vectors and $x,y$ axes when the interface is along the zigzag direction (b), the armchair direction (c), or a more general direction (d). The filled dots mark the Bravais lattice ($A$ sublattice) of graphene. The shaded regions mark the unit cells. In panel (b), the $X$ and $Y$ axes of the Cartesian coordinate system are also shown.

Figure 8

Real energy band (black lines) and complex energy bands (orange lines) of pristine graphene at a fixed $k_y\approx 0.14\times 2\pi /a$.

Figure 9

Transmission and reflection of a right-going eigenmode of energy $E=0$ incident from the $n$ region of a sharp (solid lines and dotted lines) or 5-nm-wide smooth (dashed lines), symmetric (i.e., $-V_L=V_R=V_0$) graphene $p-n$ junction. For $V_0=0.2$, the range of $k_y$ in which the eigenmode is traveling is marked by the double arrow. For $V_0=0.02$, the range of $k_y$ in which the eigenmode is traveling is much narrower.

Figure 10

Contributions of different scattering channels to the Green's function $G({\mathbf{R}}_2,{\mathbf{R}}_1)$ for a smooth linear graphene $p-n$ junction extending from 65 to 135 along the $x$ axis (junction width $\approx 10$ nm). Here ${\mathbf{R}}_1\equiv (X_1,Y_1)$ is chosen to be fixed at $(0,0)$ on the $A$ sublattice, while ${\mathbf{R}}_2\equiv (X_2,Y_2)$ is swept over all the $B$ sublattice sites of the entire structure. (a) Contour plot of $\left|G\right({\mathbf{R}}_2,{\mathbf{R}}_1\left)\right|$ vs ${\mathbf{R}}_2-{\mathbf{R}}_1$. Panels (b) and (c) extract the contributions to $\left|G\right({\mathbf{R}}_2,{\mathbf{R}}_1\left)\right|$ from the incident wave and reflection wave, respectively. Panel (d) shows the contribution to $|G({\mathbf{R}}_2,{\mathbf{R}}_1){|}^2$ due to the interference between the incident wave and the reflection wave.

Figure 11

Spatial map of the scaled conductance (i.e., the transmission coefficient $T_{12}$ times $|{\mathbf{R}}_2-{\mathbf{R}}_1|$) in a sharp graphene $p-n$ junction as a function of ${\mathbf{R}}_2-{\mathbf{R}}_1$. Here ${\mathbf{R}}_1\equiv (X_1,Y_1)$ is fixed at the $A$ sublattice of the unit cell $(0,0)$ and ${\mathbf{R}}_2\equiv (X_2,Y_2)$ is swept over all the lattice sites (including both $A$ and $B$ sublattices) of the entire structure.

Figure 12

Transmission coefficient $T_{12}$ between two STM probes at mirror symmetric locations about the junction interface. (a) $T_{12}$ vs interprobe distance. (b) $T_{12}$ vs $V_0$. Here, the width of the smooth junction is 5 nm for (a) and inset of (b), and the maximum of $T_{12}$ is always rescaled to 1 in each panel.