Imaging the transition from diffusive to Landauer resistivity dipoles

TL;DR

This work investigates how resistivity dipoles formed around defects transition from diffusive to Landauer (ballistic) behavior in ultra-thin Bi films. Using scanning tunneling potentiometry, it images dipoles around holes of varying sizes in Bi/Si(111) and analyzes their amplitudes to reveal a crossover from linear to constant scaling with defect size, signaling the shift from $p_{\rm D} = E_0 a^2$ (diffusive) to $p_{\rm L} = \frac{16}{3}\frac{\hbar j}{k_{\rm F} e^2} a_\perp$ (Landauer). From the transition point and diffusive fits, the authors extract the Fermi wave vector $k_{\rm F}$ and the carrier mean free path $\lambda$, and show consistency with Drude conductivity via $\sigma = ne^2\lambda/(\hbar k_{\rm F})$. The results provide nanoscale real-space evidence for Landauer’s transport limit and offer a robust method to benchmark theories of the diffusion-ballistic crossover in two-dimensional electron systems.

Abstract

A point-like defect in a uniform current-carrying conductor induces a dipole in the electrochemical potential, which counteracts the original transport field. If the mean free path of the carriers is much smaller than the size of the defect, the dipole results from the purely diffusive motion of the carriers around the defect. In the opposite limit, ballistic carriers scatter from the defect$-$for this situation, Rolf Landauer postulated the emergence of residual resistivity dipoles that are independent of the defect size and thus impose a fundamental limit on the resistance of the parent conductor. Here, we study resistivity dipoles around holes of different sizes in two-dimensional Bi films on Si(111). Using scanning tunneling potentiometry to image the dipoles, we find a crossover from linear to constant scaling behavior of their amplitudes with defect size, manifesting the transition from diffusive to Landauer dipoles. The extracted parameters of the transition allow us to estimate the Fermi wave vector and the carrier mean free path in our Bi films.

Figures (4)

• Figure 1: Resistivity dipole at a point-like defect in a thin film. (a) Schematic of a circular hole with radius $a$ in a two-dimensional conductor with sheet conductivity $\sigma$. Injecting and draining a homogeneous current density $j$ in lateral (or $x$) direction results in a uniform electric transport field $E_0$ far away from the hole. (b) Schematic resistor network, with a black circle in the center representing the hole shown in panel (a). White resistors connect nodes in the film, with resistance according to the film's sheet conductivity $\sigma$, while dark resistors, which connect to nodes inside the insulating hole, block current from entering the hole. (c) Dipole potential around the hole (outline indicated by circle) calculated from a $201\times201$ nodes$^2$ resistor network upon injecting a current in $x$ direction. The linear background corresponding to $E_0=20\rm\, kV/m$ has been subtracted. (d) Cross section of the potential along the dashed line in panel (c) (dashed red curve). The hole size is indicated by the shaded region. The dotted teal curve shows the analytical solution in the diffusive limit [Eqs. (\ref{['eq1:potentialDipole']}) and (\ref{['eq2:dipoleMoment_diffusive']})]. The solid teal curve shows the total potential before the subtraction of the linear background due to the driving field $E_0$ (scaled by a factor 0.05).
• Figure 2: Scanning tunneling potentiometry on thin Bi films. (a) Scanning tunneling micrograph of 4 ML Bi deposited on a Si(111)-$7\times 7$ sample. (b) Histogram of panel (a), showing that the majority of the Bi film has a thickness of 4 ML (inset) with holes extending down to the Si substrate [dark areas in panel (a)] and a small number of additional islands on top (bright areas). (c) SEM image of the tips during the potentiometry experiments. The distance between the current-injecting tips is $d=25\,$µ m. (d) Schematic of the sample cross section and potentiometry setup implemented into a multi-tip STM. The two outer tips (in point contact) inject a lateral current density $j$ into the film, while the central tip (in tunneling mode) maps the electrochemical potential. The tips are positioned under scanning electron microscope (SEM) observation.
• Figure 3: Potentials of exemplary resistivity dipoles. (a) Cross sections of the potential around a large hole with $2a\simeq {16.5}\,\mathrm{nm}$, along the dotted lines in the insets, measured at current density $j= (7.90 \pm 0.56)\rm\,A/m$. Insets: Experimental topography (top left) and potentiometry map (bottom left), hole map resulting from threshold detection (top right), and potential map resulting from resistor network calculation (bottom right). All maps in the panel cover the same $x$ range as the main plot. The shaded region in the main panel corresponds to the cross section of the hole in the upper right inset. (b) Same as panel (a), but for a smaller hole with $2a\simeq 7\,\mathrm{nm}$ at $j= (7.41 \pm 0.23) \rm\,A/m$. For the large hole, the measured dipole potential (black squares) and fits of Eq. (\ref{['eq1:potentialDipole']}) to the experimental data [solid black curves, fitting the two sides of the dipoles separately, using Eq. (S1)] agree well with the calculated analytical dipole potential $V_\mathrm{D}$ in the diffusive limit [Eq. (\ref{['eq2:dipoleMoment_diffusive']}), dotted teal curve], and with the diffusive dipole potential resulting from the resistor network calculation $\tilde{V}_\mathrm{D}$ (dashed red curve). Around the small hole, $V_\mathrm{dipole}$ is significantly larger than both $V_\mathrm{D}$ and $\tilde{V}_\mathrm{D}$.
• Figure 4: Transition from diffusive to Landauer resistivity dipoles. Experimental resistivity dipole $\rho_{\rm dipole} = |V_{\rm dipole}(a)|/j$ as function of the effective hole size $a^* \equiv |\tilde{V}_\mathrm{D}(a)|/E_0$ for data set A (squares) and data set B (circles). Solid symbols are the result of point-symmetric fits of the dipole potential with Eq. (\ref{['eq1:potentialDipole']}), while empty symbols for data set B result from separate fits to both sides of the dipole potential [Eq. (S1)]. The solid red line corresponds to the diffusive limit and the solid green line to the Landauer limit. Inset: Zoom into the transition region.