Electronic transport in planar atomic-scale structures measured by two-probe scanning tunneling spectroscopy

Miniaturization of electronic circuits into the single-atom level requires novel approaches to characterize transport properties. Due to its unrivaled precision, scanning probe microscopy is regarded as the method of choice for local characterization of atoms and single molecules supported on surfaces. Here we investigate electronic transport along the anisotropic germanium (001) surface with the use of two-probe scanning tunneling spectroscopy and first-principles transport calculations. We introduce a method for the determination of the transconductance in our two-probe experimental setup and demonstrate how it captures energy-resolved information about electronic transport through the unoccupied surface states. The sequential opening of two transport channels within the quasi-one-dimensional Ge dimer rows in the surface gives rise to two distinct resonances in the transconductance spectroscopic signal, consistent with phase-coherence lengths of up to 50 nm and anisotropic electron propagation. Our work paves the way for the electronic transport characterization of quantum circuits engineered on surfaces.

down to about 30 nm.
To establish 2P-STS experiment we perform the following procedure: Two lock-in amplifier setup. AC component (∼30 mV peak to peak, 680 Hz) is added to DC bias voltage on the source probe (tip1).
Tunneling current signals from source (tip1) and drain (tip2) probe are then demodulated at this given frequency, what results in corresponding dI 1 /dV 1 (vertical, differential conductance) and dI 2 /dV 1 (planar, Supplementary Figure 2. Two-probe scanning tunneling spectroscopy (2P-STS). Data were obtained with the application of the protocol with two lock-in amplifiers at the same experiment conditions as discussed in the main text ( Fig. 2c), excluding closer tip1-sample Z 0 distance for blue spectra, which was defined by V sample =-0.5 V and I=20 pA (instead of 10 pA for other data). Note that about two times lower tunneling resistance of tip1-sample junction does not affect the general characteristics of transconductance signal. (b) Set of three 2P-STS transconductance data as a function of tip1 voltage obtained by probes positioned on the same Ge dimer row on c(4×2) reconstructed Ge(001) for different values of tip2-surface low-bias resistance (see legend). tip1-surface distance was defined by V sample =-0.5 V and I=10 pA. The relative distance between probes was 30 nm for black and 37 nm for red and blue spectra. Note that relative intensities of resonances in transconductance dI 2 /dV 1 signals reflect similar trend to calculated spectra presented in Supplementary Figure 11e STM tip1 positions during 2P-STS are marked by different color squares (white ellipse marks points from Fig.5 a,b). Position of tip2 is marked by white circle. The increase of probeto-probe distance during characterization of different relative rows is due to probe geometric constrains. (b) Planar transconductance dI 2 /dV 1 2P-STS signals as a function of tip1 voltage obtained for STM probes located at different reconstruction rows (see a, blue reference spectrum was obtained outside the presented image). The corresponding rows of separation are indicated on the label (0 is the same Ge dimer row). During acquisition of data the sample was grounded and the tunneling contact resistance of the tip2-sample junction was established at ∼50 MΩ and kept constant. tip1-sample distance was established in all cases at Z 0 defined by V sample =-0.5 V and I=20 pA. We observe strong suppression of transconductance dI 2 /dV 1 resonance at 0.35 eV while increasing separation across reconstruction rows between injection and detection of carriers, what confirms its quasi-1D character. The 0.7 eV resonance was in this case still preserved even at 7 rows of separation. For the lock-in detection technique we used additional AC bias applied to tip1. This may induce artificial signal detection on the demodulator of the tip2 current signal due to possible AC coupling between closely spaced STM probes. However, this effect is not affecting general results in the presented experiment, as the direct correspondence of measured dI 2 /dV 1 to differential conductance may be proved by reproduction of the signal from DC current by its numerical differentiation (see Supplementary is obtained from numerical differentiation. Note reproduction of negative differential transconductance regime between peaks, which was observed in some cases (see also Supplementary The fineness of the real space grid was defined using a 250 Ry energy cutoff. Self-consistency was considered to be achieved when the changes of the density matrix elements were less than 10 −5 as well as lower than being the larger amount along the transport direction. The same k-point sampling of the Brillouin zone was used for the Au electrode, which was defined by a unit cell containing 64 atoms (4 layers in the metallic rod).
Since for the later case there are no periodic boundaries in its semi-infinite direction, we used buffer atoms (red box) to help the convergence of the density matrix at the interface between the scattering region and the metal-     (a) Raw one-dimensional FT of the data presented in Fig. 3d. (b) The same FT data with superimposed square, triangular and circle E(k) points obtained by fitting procedure applied to constant energy cross-sections of dI/dV data from Fig. 3d Fig. 3e and Supplementary Figure 14b were obtained by a separate analysis of the spatial pattern for each energy value. First, to reduce noise, the signal from ten adjacent energy points were averaged out. This led to the energy resolution of about 0.018 eV.
Then, for every line we fitted a trigonometric function multiplied by a dying-off exponential function: where A, B, C, L, φ 1 , φ 2 , k 1 and k 2 are parameters. In this formula we generally assume two dominant wave vectors. We also assume here that the interference patterns are independently formed for each band.     with corresponding fits (black and red curves). For lower energies the fits are performed for longer distances (24nm, black curve in d) than for higher energies (15nm, red curves in e and f).
Note that single frequency could not reproduce the data in any of the presented dI/dV distance dependence.
disappear only for distances of ∼25 nm as described in the main text.
Interestingly for energies exceeding 0.7 eV the interference patterns vanish quicker. Here, we argue that the effect is presumably related to more twodimensional character of corresponding bands above 0.7 eV as shown in   Ge atoms forming the dimers and that it is dominated by a π character (4p z orbitals of Ge).
In the Ge(001) surface, the wave function corresponding to the CBE+1 band shows a strong sign oscillation along the dimer wire (Supplementary where the summation goes over all eigenvalues, k x and k y denote wave vector coordinates along the direction perpendicular and parallel to the wires respectively, and η is the broadening parameter (taken in the 0.015-0.02 eV range).
Let us first consider the problem of the energy dependent dI/dV maps as a function of the coordinate perpendicular to the step-egde (y-direction).
For simplicity we consider the step as a perfectly reflecting barrier for electrons propagating along the surface, approximate their wave functions by plane waves and use Tersof-Hamman theory. We easily arrive to where DOS(E, k y ) = dk x ρ(E, k x , k y ) .
Then, for the Fourier transform of the map at a given energy E (omitting an energy dependent constant background) one can get LDOS(E, q) = 1 2π dy LDOS(E, y) e −iqy ∼ DOS(E, q/2) .
In the left panel of Fig. 3e in the main text we plot DOS(E, k y ) and compare it with the Fourier transform of the experimental dI/dV maps.
In agreement with the experiment, for each energy E we obtain higher intensities for two values of k y (particularly in the case of the lower CBE surface band). They corresponds to the higher values of DOS(E, k y ), i.e.
larger density of states, at the lower (k max y ) and higher (k min y ) energy onsets of the surface bands.
We now proceed to analyze the degree of 1-D character of the band structure of Ge(001)-c(4×2) as a function of the energy. Supplementary From 0.37 eV until 0.65 eV, on the other hand, the band gradient exhibits small angles with respect to the k y axis, meaning that the transport is predominantly along the dimer rows. Such behavior is clearly seen at Fig.   4f in the main text, where the incoming states from tip1, evaluated at 0.5 eV and projected into the surface, show a strong localization on a single dimer row.
Starting at CBE+1 (0.67 eV) until 0.75 eV, the transport is again characterized by a 2-dimensional regime. Such behavior may contribute to the observed fast vanishing the interference patterns (∼10 nm) on the step edge experiment for energies ≥ 0.7 eV (∼10 nm) as compared to lower energies (Fig. 3c, main text). Moreover, it may as well explain the dark feature feature observed in the single-point STS stacking plot (Fig. 3d, main text) from 0.7 to 0.8 eV. Finally, our interpretation is in agreement with the simulated eigenchannels on the step-edge setup (Supplementary In the transport simulation setups discussed here so far a twelve-layer Ge(001)-c(4×2) slab was used to represent the sample Ge surface. In order to evaluate the effects that a finite slab might introduce in our calculations and to understand how to align the band structures from slab and bulk, we simulate a semi-infinite Ge(001) substrate being approached by a single metallic tip ( Supplementary Figure 21a-b).
This 2-terminal setup comprises 3934 atoms (31149 orbitals) defined in a supercell of dimensions 32.03×32.03×137.69Å 3 . The same general parameters described in Sec. 5 were adopt here as well. In this case, however, the Ge electrode is defined by a 8-layer bulk Ge (512 atoms) located 28-layers below the surface (blue box in Supplementary Figure 21a). Given the non-periodicity of the the system in the tip-to-bulk direction, more 20-layer Ge were added as buffer (red box in Supplementary Figure 21a). The tip-to-surface distance was fixed at D = 4.5Å and no further geometry optimizations were done, i.e., the Ge surface was fixed by the geometry relaxed without the presence of the tip. This consistent with our previous study, in which we show that relaxations are very small for those large tip-surface separations.
In order to associate the present setup of a full Ge surface with the finite Ge slab calculations, we compared the DOS projected on the surface atoms (blue dashed box in Supplementary Figure 21a Figure 9), we note a very good agreement at higher energies, e.g., the monotonic increment of transmission, and a clear overlap between the onset of the tip-to-bulk transmission with the resonance attributed to CBE+2 in the tip-to-slab transmission.