Fully resolved currents from quantum transport calculations

We extract local current distributions from interatomic currents calculated using a fully relativistic quantum mechanical scattering formalism by interpolation onto a three-dimensional grid. The method is illustrated with calculations for $\mathrm{Pt}|\mathrm{Ir}$ and $\mathrm{Pt}|\mathrm{Au}$ multilayers as well as for thin films of Pt and Au that include temperature-dependent lattice disorder. The current flow is studied in the “classical” and “Knudsen” limits determined by the sample thickness relative to the mean free path $\lambda$, introducing current streamlines to visualize the results. For periodic multilayers, our results in the classical limit reveal that transport inside a metal can be described using a single value of resistivity $\rho$ combined with a linear variation of $\rho$ at the interface while the Knudsen limit indicates a strong spatial dependence of $\rho$ inside a metal and an anomalous dip of the current density at the interface which is accentuated in a region where shunting is incomplete.

Figure 1

Schematics of (a) electron scattering in a thin film geometry of finite thickness $d$ illustrating the role of surface scattering. An electron wave-packet (depicted by the black arrows) undergoing only specular scattering at the surface (blue arrow) results in a uniform current distribution $j$ flowing in the $z$ direction whereas diffuse scattering from a rough surface (red arrow) results in a nonuniform current profile $j\left(x\right)$. (b) Scattering at an interface (thick black horizontal line) between two slabs of finite thicknesses $d_1$ and $d_2$ where the horizontal dashed lines represent a surface or the next interface in a bilayer/multilayer geometry. At an interface, a part of the incident electron wave ($I$) is transmitted ($T$) and the rest is reflected ($R$) specularly or diffusely eventually leading to current equilibration along $y$. A parallel resistivity model describes $x$-independent resistivities and, corresponding to these, uniform current distributions. A realistic current distribution $j_1\left(x\right)$ and $j_2\left(x\right)$ resulting from a resistivity gradient across the interface and from details of the scattering (specular and diffuse) is sketched on the right. Note that the spatial dependence is only expected to be significant over a lengthscale determined by the mean free path of the materials. Coordinate axes are shown for reference.

Figure 2

A scattering region ($\mathcal{S}$) is sandwiched between lattice-matched ballistic left ($\mathcal{L}$) and right ($\mathcal{R}$) leads which are semi-infinite in the $\pm z$-direction, respectively. Superlattice periodicity is imposed in the $xy$ plane by means of an $N\times M$ supercell in the $x$ and $y$ directions, respectively. This construction makes it possible to simulate a wide range of disordered systems. Coordinate axes are shown for reference for an fcc lattice with $N=M=10$ and $x=\left[110\right]$, $y=\left[1\overline{1}0\right]$, $z=\left[001\right]$.

Figure 3

(a) An arbitrary lattice described by the translation vectors ${\mathbf{T}}_1,{\mathbf{T}}_2$ (and ${\mathbf{T}}_3$; not shown) is affine transformed into (b) an equivalent orthogonal lattice described by ${\mathbf{T}}_1^{\prime },{\mathbf{T}}_2^{\prime }$ (and ${\mathbf{T}}_3^{\prime }$; not shown). (c) The lattice is then discretized into $n_x{n}_y{n}_z$ boxes equal to the number of atoms. Here $n_i$ with $i=x,y,z$ is the number of atomic layers in the $i\mathrm{th}$ direction. (d) The final grid where a single box $b_{ijk}\equiv b(x_i,y_j,z_k)$ is explicitly shown in three dimensions to help visualize the grid. The example shown is for a $4\times 4$ lateral supercell in the $xy$ plane with only two consecutive layers in the $z$ direction shown for clarity. Cartesian coordinate axes are shown for reference.

Figure 4

Illustration of the discrete current scheme where the grid is zoomed to show how the current $j^{PQ}$ between atoms $P$ and $Q$ is interpolated into the box $b$. The current contribution from “wire” $PQ$ to the box $b$ comes from the segment $UV$ (colored red) that lies inside the box.

Figure 5

Sketch of an $N\times M$ lateral supercell used to model transport in a lattice-matched $AB$ multilayer with $A=\text{Au}$ and $B=\text{Pt}$. For clarity, only six layers in the $z$ direction are explicitly shown, the separation of the layers is exaggerated and the leads sandwiching this geometry in the $\pm z$ directions are not shown. The $x$ direction is the crystal [110] direction. Part of an fcc layer perpendicular to the [001] direction with in-plane crystallographic directions is shown on the left. The conventional cubic axes of an fcc lattice are $X,Y$ and $Z(\equiv z)$.

Figure 6

Current distribution in a thin film of Au at 335 K [59]. (a) Top curve, black symbols: current density ${\overline{j}}_z\left(x\right)$ obtained by averaging over $y$ and $z$. Color symbols: ${\overline{j}}_z(x,z_0)$ obtained by averaging over $y$ and $z=z_0\pm 5$ layers in the $z$ direction for four different values of $z_0$ that are offset from the black curve in steps of $\mathrm{\Delta }j=0.002$ for clarity. The error bars, which are smaller than the symbol size, indicate the average deviation over ten random configurations of disorder. (b) Streamlines of the current vector $\mathbf{j}(x,z)=j_x\mathbf{i}+j_z\mathbf{k}$ in the $xz$ plane. The color contour in the background corresponds to the magnitude of the current.

Figure 7

Distribution of the charge current ${\overline{j}}_z$ in a $\mathrm{Pt}|\mathrm{Au}$ multilayer. For ${\overline{j}}_z\left(x\right)$ in (a) $j_z(x,y,z)$ is averaged over $y$ and over $z$. The horizontal black lines indicate the mean values obtained by averaging separately over $x\in$(Pt,Au). For ${\overline{j}}_z\left(z\right)$ in (b) $j_z(x,y,z)$ is averaged over $x$ and $y$ with $x\in \mathrm{Pt}$ and $x\in \mathrm{Au}$ separately [61]. (c) Streamlines of the current vector $\mathbf{j}(x,z)=j_x\mathbf{i}+j_z\mathbf{k}$ averaged over $y$ and superimposed on a color contour of $j=\sqrt{j_x^2+j_z^2}$ in the $xz$ plane. Error bars represent the mean deviation over ten configurations of thermal disorder.

Figure 8

Average resistivity of Pt films calculated as a function of the effective thickness $d/\lambda$ where $d$ is the thickness of the film in the $x$ direction for transport in the $z$(001) direction and $\lambda$ is the mean-free path of bulk Pt at RT. The separation of periodically repeated thin films by vacuum (“vacuum thickness”) is modelled using five layers of “empty” spheres in the $x$ direction. Calculations for each thickness are done for two cases with five (blue) and three (red) atomic layers repeated periodically in the $y$ direction. The dashed lines are calculated using the Fuchs-Sondheimer model [4, 15] for three different values of the specularity coefficient $p$ that describes the amount of completely diffusive surface scattering: completely specular ($p=1$), partially specular ($p=0.5$), and completely diffusive ($p=0$). According to Eq. (20), choosing $p=1$ yields the bulk resistivity ${\rho }_b$ irrespective of $d/\lambda$.

Figure 9

(a) Top curve, black symbols: current density ${\overline{j}}_z\left(x\right)$ obtained by averaging over $y$ and $z$. Color symbols: ${\overline{j}}_z(x,z_0)$ obtained by averaging over $y$ and $z=z_0\pm 5$ layers in the $z$ direction for four different values of $z_0$ that are offset from the black curve in steps of $\mathit{\Delta }j=0.002$ for clarity. The error bars, which are smaller than the symbol size, indicate the average deviation over ten random configurations of disorder. (b) Streamlines of the current vector $\mathbf{j}(x,z)=j_x\mathbf{i}+j_z\mathbf{k}$ in the $xz$ plane. The color contour in the background corresponds to the magnitude of $\mathbf{j}(x,z)$.

Figure 10

Distribution of the charge current ${\overline{j}}_z$ in a $\mathrm{Pt}|\mathrm{Ir}$ multilayer. For ${\overline{j}}_z\left(x\right)$ in (a) $j_z(x,y,z)$ is averaged over $y$ and over $z$. The horizontal black lines indicate the asymptotic values obtained by averaging separately over $x\in$(Pt,Ir) omitting the region of rapid variation close to the $\mathrm{Pt}|\mathrm{Ir}$ interface in either layer. For ${\overline{j}}_z\left(z\right)$ in (b) $j_z(x,y,z)$ is averaged over $x$ and $y$ with $x\in \mathrm{Pt}$ and $x\in \mathrm{Ir}$ separately. (c) Streamlines of the current vector $\mathbf{j}(x,z)=j_x\mathbf{i}+j_z\mathbf{k}$ obtained from averaging $j_z(x,y,z)$ over $y$ and superimposed on a color contour of $j=\sqrt{j_x^2+j_z^2}$ in the $xz$ plane. The scarcely visible error bars that are smaller than the symbol sizes represent the mean deviation over ten configurations of thermal disorder.

Figure 11

Distribution of charge current ${\overline{j}}_z$ injected from ballistic Pt leads into a 300 K $\mathrm{Pt}|\mathrm{Au}$ multilayer (top) and into a 800 K $\mathrm{Pt}|\mathrm{Ir}$ multilayer (bottom). To obtain ${\overline{j}}_z\left(x\right)$ in (a) and (d), $j_z(x,y,z)$ is averaged over $y$ and over the ten central layers in the $z$ direction furthest from the leads. The horizontal black lines in (d) indicate the asymptotic value of ${\overline{j}}_z$ obtained by averaging $j_z(x,y,z)$ over $y$ and $z$ and $x\in \mathrm{Pt}$ or $x\in \mathrm{Ir}$ omitting atomic layers near the $\mathrm{Pt}|\mathrm{Ir}$ interface where the current varies continuously. In (b) and (e), ${\overline{j}}_z\left(z\right)$ is obtained by averaging $j_z(x,y,z)$ over $x$ and $y$ with $x\in \mathrm{Pt}$ and $x\in$(Au, Ir) separately. The grey curves indicate the corresponding profiles on injecting from ballistic Au (Fig. 7) and Ir (Fig. 10) leads into $\mathrm{Pt}|\mathrm{Au}$ and $\mathrm{Pt}|\mathrm{Ir}$ multilayers, respectively. (c) and (f) show streamlines of $\mathbf{j}(x,z)=j_x\mathbf{i}+j_z\mathbf{k}$ averaged over $y$ and superimposed on a color contour of $j(x,z)=\sqrt{j_x^2+j_z^2}$ in the $xz$ plane. Error bars represent the mean deviation over ten configurations of thermal disorder.