Zero-bias anomaly in nanoscale hole-doped Mott insulators on a triangular silicon surface

Adsorption of 1/3 monolayer of Sn on a heavily doped $p$-type Si(111) substrate results in the formation of a hole-doped Mott insulator, with electronic properties that are remarkably similar to those of the high-$T_c$ copper oxide compounds. In this work, we show that the maximum hole-density of this system increases with decreasing domain size as the area of the Mott insulating domains approaches the nanoscale regime. Concomitantly, scanning tunneling spectroscopy (STS) data at 4.4 K reveal an increasingly prominent zero-bias anomaly (ZBA). We consider two different scenarios as potential mechanisms for this ZBA: chiral $d_{x^2-y^2}+\mathrm{i}{d}_{xy}$ wave superconductivity and a dynamical Coulomb blockade (DCB) effect. The latter arises due to the formation of a resistive depletion layer between the nanodomains and the substrate. Both models fit the tunneling spectra with weaker ZBAs, while the DCB model clearly fits better to spectra recorded at higher temperatures or from the smallest domains with the strongest ZBA. Consistently, STS spectra from the lightly doped substrates display oscillatory behavior that can be attributed to conventional Coulomb staircase behavior, which becomes stronger for smaller sized domains. We conclude that the ZBA is predominantly due to a DCB effect, while a superconducting instability is absent or a minor contributing factor.

Figure 1

(a) Atomic model of the $\surd 3$-Sn structure. (b) STM image of the $\surd 3$-Sn structure. The unit cell is indicated in green.

Figure 2

STM image of a monatomic Sn layer on the $\mathrm{B}\surd 3$ substrate. The image contains a step edge left of center. The terraces mostly consist of the semiconducting $2\surd 3$-Sn phase decorated with disordered islands on top (labeled as “AI”). The $\surd 3$-Sn nanoscale domains form isolated patches near the step edge (left), or inside the $2\surd 3$-Sn domains (right).

Figure 3

STS characterization of the quasiparticle peak (QPP) and the two Hubbard bands at 77 K. (a) dI/dV spectra near the edge of a $16000\mathrm{n}{\mathrm{m}}^2 \surd 3$-Sn domain, and bordering a $2\surd 3$-Sn domain. The domain edge is marked by arrows. The top curve is obtained near the middle of the domain which is out of the range of the STM image. The enhancement of the QPP in the lower dI/dV spectra is apparent from the peak height increase relative to the peak height of the UHB. (b) dI/dV spectra taken from the center of three $\surd 3$-Sn domains, showing the enhancement of the QPP for the smallest domain. (c) Fitting of the $178\mathrm{n}{\mathrm{m}}^2$ spectra in (b) using six Gaussian peaks [23]. The dip feature between −0.04 and +0.04 V is excluded from the fitting. This dip is the ZBA feature, as discussed in the later part of the paper. Its influence on the spectral weight of the QPP is negligible. The single Gaussian peak for the QPP is labeled. The blue tail on the very left accounts for the bulk valence band. The other two Gaussians below the Fermi level account for the LHB while those to the right of the QPP account for the UHB. Curves in (a) and (b) are normalized to their intensity at ∼+1.5 V (outside the plot range), and are shifted vertically for clarity.

Figure 4

STS measurements from different $\surd 3$-Sn nanodomains on the $\mathrm{B}\surd 3$ substrate at 4.4 K. (a) An STM image of a relatively small $\surd 3$-Sn domain near a step edge. (b) Wide-range $dI/dV$ spectra showing the LHB, UHB, and QPP. The sharp spike riding on top of the QPP is the Van Hove singularity (VHS). The bias range displayed in (c) is marked in gray. (c) Small scale $dI/dV$ data measured on different size $\surd 3$-Sn domains. The curves are normalized at their intensity near −100 mV, and are shifted vertically for clarity.

Figure 5

Fitting of the 4.4 K STS data in Fig. 4 with the $d+\mathrm{i}d$ superconducting gap function [Eqs. (2))–(4)]. The simulated spectra (red) are superposed onto the experimental curves (black circles). The best fitting parameters are indicated. The topmost fit to the $3337\mathrm{n}{\mathrm{m}}^2$ spectrum is conducted with fixed ${\mathrm{\Delta }}_0=0$.

Figure 6

Fitting of the 4.4 K STS data in Fig. 4 with the DCB model. The tunneling process is modeled by a double junction system, corresponding to the tip-$\surd 3$-Sn domain junction (T junction) and the $\surd 3$-Sn domain-Si bulk junction (S junction), as sketched on top. The lower panel shows the fittings (red) overlapping the data (black circle). The top curve is the DOS used for the fitting. The fitting parameters are: $T=4.4\mathrm{K}$; $C=180\mathrm{aF}, R=0.06R_Q$ ($3337\mathrm{n}{\mathrm{m}}^2$); $C=72.5\mathrm{aF}, R=0.22R_Q$ ($1215\mathrm{n}{\mathrm{m}}^2$); $C=32.5\mathrm{aF}, R=0.56R_Q$ ($412\mathrm{n}{\mathrm{m}}^2$); $C=22.5\mathrm{aF}, R=0.96R_Q$ ($232\mathrm{n}{\mathrm{m}}^2$); and $C=4.5\mathrm{aF}, R=4.9R_Q$ ($44\mathrm{n}{\mathrm{m}}^2$). The curves are shifted vertically for clarity.

Figure 7

Plot of the parameters obtained from fitting a series of STS spectra to the DCB model as a function of the $\surd 3$-Sn domain size. Each data point represents the average of the fitting parameters, $C$ (a) and $R$ (b), estimated from multiple spectra taken from the same domain. The error bars represent the range of the best fit parameters. The red lines represent linear fits to the data on the double logarithmic scale.

Figure 8

Temperature-dependent dI/dV spectra of three $\surd 3$-Sn domains, with sizes of (a) 12 370, [(b) and (d)] 1148, and [(c) and (e)] $243\mathrm{n}{\mathrm{m}}^2$. Temperatures indicated in (a) also apply to the other data sets following the same sequence. [(b) and (c)] Experimental data (black circles) on the two smaller domains are fitted (red) to the DCB model. The simulated spectra in each data set are computed using the capacitance and resistance values obtained from the DCB fit to the 4.4 K data. [(d) and (e)] Fitting the same set of data as in (b) and (c) but using the Dynes function [Eqs. (12) and (13)]. The resulting fit parameters are indicated.

Figure 9

Comparison of the $dI/dV$ data from $\surd 3$-Sn domains grown on p-type Si substrates with three different doping levels. (a) An STM image of the $\surd 3$-Sn domains grown on the p-0.008 Si substrate. The surface contains a mix of the $\surd 3$-Sn and $2\surd 3$-Sn phases. The $2\surd 3$-Sn domains are indicated with dashed lines. On the same terraces, neighboring $\surd 3$-Sn domains are separated by domain boundaries (labelled as “DB”), appearing as bright lines. Two neighboring $\surd 3$-Sn domains are labelled “L” and “S”, representing large and small domains. (b) STS measured on relatively large $\surd 3$-Sn domains (3337, 6764, and $6000\mathrm{n}{\mathrm{m}}^2$ for the $\mathrm{B}\surd 3$, p-0.004, and p-0.008 substrates, respectively). (c) STS measured on relatively small $\surd 3$-Sn domains (232, 356, and $234\mathrm{n}{\mathrm{m}}^2$ for the $\mathrm{B}\surd 3$, p-0.004, and p-0.008 substrates, respectively). The curves in (b) and (c) are shifted vertically for clarity.

Figure 10

Tip-sample-separation dependent STS measurements of a large $\surd 3$-Sn domain ($6000\mathrm{n}{\mathrm{m}}^2$) grown on the p-0.008 Si substrate, recorded at 4.4 K. (a) With decreasing tip-sample separation (or increasing set point current), the spectral features (including the QPP) shift to higher energy, implying a tip induced band bending effect in the Si substrate. (b) At smaller energy scale, the fine spectral features shift inward with decreasing tip-sample separation, implying a change in T-junction capacitance, $C_T$. In each plot, the curves are shifted vertically for clarity; the shifting features are tracked with dashed lines. Set point voltages and currents are indicated for the farthest and closest tip-sample distance. The tip-sample separation is also indicated on the right and is measured relative to the farthest tunneling set point.