Experimental visualization of scattering at defects in electronic transport through a single atomic junction

For electronic transport at the nanoscale, coherent scattering at defects plays an important role. Therefore, the capability of visualizing the influence of defects on the conductivity of single atomic junctions may benefit the development of future nano-electronics. Here, we report imaging the coherent scattering from a defect with well-controlled geometry by quantum point contact microscopy recently developed by us. An ∼10$\%$ modulation in transport conductance of a single atomic junction is observed, with a phase shift of nearly π compared to the tunneling conductance. With the well-defined scattering geometry, we performed a theoretical calculation of the conductance and found the result consistent with the experiment.

Figure 1

Setup and basic result. (a) Schematic illustration of conductance measurement for a single atomic junction influenced by coherent scattering from a mono-atomic step edge by QPCM. The curve ρ(r) depicts the local density of states of the surface state electrons, resulting in a spatial modulation of the conductance of the atomic point contact, as is depicted by the curve $G$($x$). Circles depict atoms. (b)–(e) Differential conductance ($\mathrm{d}I$/$\mathrm{d}V$) tunneling images and (f)–(i) $\mathrm{d}I$/$\mathrm{d}V$ QPCM images acquired in constant height mode in the same area close to a step edge on Ag(111) at different bias voltages. (c)–(e) and (g)–(i) are stripes cut from the original data [see Supplemental Fig. S1 (Ref. 15)] as indicated by the dotted rectangles in (b) and (f). The Ag adatom at the top center of (b) is used for performing QPCM. (j) Conductance line profiles of QPCM images (thick lines) and STM images (thin lines) along a straight line orthogonal to the step edge indicated by dashed lines in (b) and (f). Conductance line profiles taken at different bias voltages (from bottom to top $U$ = $-$0.04, 0.00, 0.07, and 0.16 V), are vertically offset by 0.2$G_0$ for QPCM and 3.5 × 10${}^{-4}$$G_0$ for STM. Conductance quantum $G_0$ = 2$e^2$/$h$ = 77.5 μ$\mathrm{S}$.

Figure 2

Differential conductance spectra in tunneling and contact as a function of position (a)–(c) STM topographies show the place where tunneling and contact spectra were acquired. ($U$ = 0.01 V, $I$ = 0.11 nA). Depression in apparent height at the right-top of (b) and (c) is due to the piezo creep. Black at the right-bottom of (b) and (c) is outside the imaging area. (d) Scanning tunneling spectra acquired at the positions approximately indicated by dots in (a), normalized differential conductance is plotted in grayscale (and smoothed) as a function of sample bias voltage and distance from step edge. (e) Point contact spectra acquired at the positions marked in (b), starting from the position of the Ag adatom in the right corner and ending at the step edge in the left. During the measurement, the Ag adatom which forms the contact follows the tip from its position in (b) to that shown in (c). Solid lines in (d) and (e) indicate the positions of local maxima of the standing wave pattern of the Ag(111) surface state according to Eq. (2) [with $m^{*}$ = 0.37$m_e$, $E_{\overline{\Gamma }}$ = $-$0.063 eV, the position of the step edge $x_0$ = $-$3 Å, the phase is ${\phi }_{\mathrm{STM}}$ = $-$1.25π in (d) and ${\phi }_{\mathrm{PCM}}$ = $-$2.05π in (e)].

Figure 3

Conductance modulation as a function of tip-sample distance. (a) Series of $\mathrm{d}I$/$\mathrm{d}V$ spectra from tunneling to point contact. The spectra are taken by approaching the tip by different distances over the center of the Ag adatom shown in Fig. 2b. The wiggles due to coherent scattering of quasiparticles in the surface state are clearly resolved. Vertical lines mark positions of maxima in the conductance trace for comparison. An asterisk to the right marks a spectrum acquired at similar conductance as spectra in Fig. 2d. Panel (b) shows the conductance at $-$0.15 V of spectra in (a) as a function of vertical tip position $d$, the transition from tunneling to point contact can be clearly identified. Each point in (b) corresponds to a single spectrum in (a). (c) Calculated series of differential conductance spectra according to the model discussed in the text for an adatom at a distance $x$ = 129 Å from the step edge [consistent with the distance of the adatom in Fig. 2b]. The differential conductance (solid lines) is plotted together with the imaginary part $-$Im[Σ${}_b$ $+$ Σ${}_s$ ($E$)] (upper dashed line) and the real part Re[Σ${}_b$ $+$ Σ${}_s$ ($E$)] (lower dashed line) of the self-energy in Eq. (3). $-$Im[Σ${}_b$ $+$ Σ${}_s$ ($E$)] is proportional to the LDOS in Fig. 4a. Vertical lines mark positions of maxima in the conductance trace for comparison. An asterisk to the right marks a spectrum calculated with the same parameters (in particular $t$ = 0.8 eV) as in Figs. 4a and 4c. (d) Mean conductance of spectra in (c) is plotted as a function of tip-adatom coupling $t$. Each point in (d) corresponds to a single spectrum in (c).

Figure 4

Theoretical calculation for the scattering pattern in Fig. 2. (a) Model traces for the differential conductance at zero bias voltage in tunneling (empty circle) and in point contact (solid square). The phase shift between the traces in tunneling and point contact is clearly seen. The step edge is assumed to be a black dot scatterer (Ref. 11). (b,c) Bias dependence of the scattering pattern in tunneling [LDOS according to Eq. (1)] and point contact [according to Eq. (3)] plotted in the same way as in Figs. 2d and 2e, respectively. Local maxima are plotted according to Eq. (2) [with $m^{*}$ = 0.37$m_e$, $E_{\overline{\Gamma }}$ = $-$0.063 eV, the phase is ${\phi }_{\mathrm{STM}}$ = $-$1.25π in (b) and ${\phi }_{\mathrm{PCM}}$ = $-$2.1π in (c)].

Figure 5

Model for phase shift and QUIT. (a) Interpretation of the phase shift due to interference of two distinct paths 1 and 2 between tip and substrate. Circles depict atoms; curly lines depict phase shifts from bulk states to adsorbate ${\phi }_b$, between adsorbates ${\phi }_t$, from surface state to adsorbate ${\phi }_{s0}$, and inside surface state φ${}_s$(x). (b) Schematic of QUIT following Refs. 25, 26, 27.