Experimental observation of pseudogap in a modulation-doped Mott insulator: Sn/Si(111)-($\sqrt{3}\times\sqrt{3}$)R30${}^{\textrm{o}}$

Mott insulators are materials in which constituent electrons cannot move freely because of strong electron repulsions and lie at the heart of strongly correlated electron physics. Upon chemical doping, the electrons delocalize and the Mott insulators can become conductive and even superconductive via a process well-known as Mott transition Mott (1961, 1982); Belitz and Kirkpatrick (1994). Following this transition, many unusual phenomena emerge and have been largely explored in one archetypal example of the perovskite-type copper oxide compounds. These include, for example, the unconventional superconductivity with a record critical temperature ($T_{\textrm{c}}$) at ambient atmosphere Lee et al. (2006), the pseudogap (PG) phase above $T_{\textrm{c}}$ Timusk and Statt (1999) and various different symmetry-broken states Fischer et al. (2007); Damascelli et al. (2003); Timusk and Statt (1999). Among these phenomena, the PG phase stands out and continues to attract attention, because clarifying its microscopic origin from either the performed Cooper pairs Renner et al. (1998); Chen et al. (2005); Geshkenbein et al. (1997) or some competing orders Loret et al. (2019); Caprara et al. (2017); Kampf and Schrieffer (1990) has been considered as an essential prerequisite to understanding the high-$T_{\textrm{c}}$ superconductivity in cuprates. After more than three decades of study, however, the PG phenomenology, which turns out to be more prominent in underdoped regions, remains one of the greatest mysteries in the high-$T_{\textrm{c}}$ cuprate superconductors.

Adsorption of metal atoms (such as Pb, In, and Tl) on semiconductor substrates (e.g. Si(111)) often induces rich surface superstructures and intriguing electronic states at the two-dimensional (2D) limit, including superconductivity Zhang et al. (2010); Uchihashi et al. (2011); Nakamura et al. (2018); Özer et al. (2006), charge density waves Matetskiy et al. (2019); Carpinelli et al. (1996) and unusual magnetism Zhou et al. (2017). Specifically, adsorption of one-third monolayer (ML) Sn on Ge(111) and Si(111) leads to the formation of a ($\sqrt{3}\times\sqrt{3}$)R30${}^{\textrm{o}}$ reconstruction known as $\alpha$ phase Carpinelli et al. (1997); Morikawa et al. (2002); Ottaviano et al. (2000). Such a triangular lattice, which is characteristic of half-filled dangling bond orbitals, forms a conceptually simple and alternative system to explore correlation related Mott physics in the 2D situation Profeta and Tosatti (2007); Modesti et al. (2007); Odobescu et al. (2017). Intriguingly, a superconducting state of $T_{\textrm{c}}$ $\sim$ 4.7 K has been recently observed in Sn/Si(111)-($\sqrt{3}\times\sqrt{3}$)R30${}^{\textrm{o}}$ (hereinafter referred to as $\sqrt{3}$-Sn) on chemical doping by employing heavily boron-doped Si substrates Wu et al. (2020). Such a charge transfer mechanism has been well known as modulation doping in semiconductor heterostructures Hsu and Walukiewicz (1997); Störmer and Tsang (1980); People et al. (1984) and recently evidenced in cuprates Zhong et al. (2016, 2020), thereby calling for further investigation of possible commonalities and distinctions between the doped $\sqrt{3}$-Sn and cuprates.

To date, however, only limited experiments were performed, especially leaving the electronic structure of $\sqrt{3}$-Sn in underdoped regions poorly understood. It remains a puzzle whether the opened energy gap at the Fermi level ($E_{\textrm{F}}$) from the tunneling experiment Ming et al. (2018) relates to superconductivity, and if no why the Coulomb gap increases in magnitude with increasing doping concentration. Here we resolve this issue by combining a molecular beam epitaxy system (MBE) with cryogenic scanning tunneling microscopy (STM). We reveal the PG nature for the opened gap in the underdoped regions, which instead decreases with increasing doping. These behaviors bear a close resemblance to the established electronic phase diagram in cuprate superconductors.

All experiments were carried out in a commercial ultrahigh vacuum (UHV) STM apparatus (Unisoku USM 1300), integrated with an MBE chamber for in-situ sample preparation. A magnetic field up to 11 T can be applied perpendicular to the sample surface. The base pressure for both chambers is better than 2.5 $\times$ 10^-10 Torr. Two distinct $p$-type Si(111) wafers (boron-doped) with room temperature resistivities of 0.01 $\Omega\cdot$cm and 0.001 $\Omega\cdot$cm, respectively, were used as substrates for the $\sqrt{3}$-Sn growth. Clean Si(111)-7 $\times$ 7 surfaces were obtained by repeatedly flashing well-degassed substrates at 1200${}^{\textrm{o}}$C. The $\sqrt{3}$-Sn phase was then prepared by evaporating high-purity (99.9999%) Sn from a standard Knudsen cell on Si(111) at approximately 550${}^{\textrm{o}}$C. All STM measurements were conducted at 4.2 K using sharp polycrystalline PtIr tips, which were routinely cleaned by electron beam bombardment in MBE and appropriately calibrated on MBE-prepared Ag/Si(111) films. The STM topographies were acquired in a constant-current mode with the bias $V$ applied to the sample, while the tunneling spectra were measured using a standard lock-in technique with a small bias modulation at 919 Hz.

Figure 1(a) shows a representative STM topography of $\sqrt{3}$-Sn prepared on the 0.001 $\Omega\cdot$cm Si substrates, with the bright spots denoting the Sn adatoms. They occupy the $T_{\textrm{4}}$ sites of the underlying Si (111) surface and are spaced $\sim$ 6.6 Å apart. Individual Sn vacancies occur as dark contrast that change little with the sample bias. Figure 1(b) plots a series of differential conductance dI/dV spectra (from bottom to top) taken along a trajectory (marked by the black arrow) approaching a single Sn vacancy in Fig. 1(a). As anticipated, the Mott Hubbard bands could be readily identified spectroscopically. In line with a previous report Ming et al. (2017), every dI/dV spectrum can be nicely decomposed into six Gaussian peaks (see the best fit of the bottom one to multiple Gaussian function), derived from the slightly appearing valence band (VB), lower Hubbard band (LHB), quasiparticle peak or saying in-gap states (IGS) at around 100 meV, and upper Hubbard band (UHB), respectively. As justified before, the hole doping level of $\sqrt{3}$-Sn can be estimated by calculating the spectra weight of IGS relative to those of UHB and LHB in Fig. 1(c) Ming et al. (2017). Apparently, the IGS become less prominent in the vicinity of Sn vacancy, indicative of a reduced hole doping level there. It is understandable because one missing Sn adatom breaks three Sn-Si bonds, leaving behind three Si dangling bonds. This contributes two extra electrons and reduces the local hole doping. Consequently, the spectral weight of UHB increases as a compensation of the significantly suppressed IGS around the Sn vacancy [Fig. 1(c)]. Such behavior resembles those of both hole- and electron-doped cuprates, in which the IGS also emerge upon doping Zhong et al. (2020); Wang et al. (2020); Hu et al. (2021). A major difference is the occurrence of more prominent IGS in $\sqrt{3}$-Sn, which might originate from the stronger interaction between the $\sqrt{3}$-Sn and underlying Si(111) substrates.

Another noteworthy feature is a small dip that universally occurs near $E_{\textrm{F}}$, circled in Fig. 1(b). To present the feature more clearly, the smaller-energy-scale dI/dV spectra are acquired at the correspondingly identical sites and plotted in Fig. 1(d). A spatially uniform energy gap of approximately 20 meV, irrespective of the existence of Sn vacancy, is identified and symmetric with respect to $E_{\textrm{F}}$. The results are consistent with the previous findings, except that no apparent van Hove singularity is observable below $E_{\textrm{F}}$ in our study Ming et al. (2018, 2017). The robustness of the small energy gap against the Sn vacancy hints at its delocalized nature.

To provide more insights into the observed energy gap, we characterize its dependence on the external magnetic field and temperature in Fig. 2. Firstly, the gap remains essentially unchanged upon the application of a magnetic field up to 10 T [Fig. 2(a)]. This hints that it has nothing to do with superconductivity. Indeed, the gap magnitude of $\sim$ 20 meV turns out to be unrealistically larger than the specific superconducting gap of $\sim$ 1.5 meV identified in this system Wu et al. (2020). Figure 2(b) plots the dependence of the site-specific tunneling dI/dV spectra on temperature ranging from 4.2 K to 77 K. At elevated temperatures, the gap gets gradually suppressed. To elucidate the temperature effect, we calculate the convolution curves of the dI/dV spectrum measured at 4.2 K with a Fermi-Dirac function at varying temperatures and overlay them on the corresponding dI/dV spectra by the red dashed lines. Apparently, they fit nicely to each other especially for the gap depth [Fig. 2(c)], defined as the difference between unity and the normalized zero-bias conductance. This is reminiscent of the PG, which also only smears out at elevated temperatures via thermal broadening, in underdoped cuprates Renner et al. (1998); Fischer et al. (2007). By extrapolating the linear relationship of the PG depth versus temperature to the point where the gap depth is equal to zero, the transition temperature, $T^{*}$, is estimated to be around 99 K. We further compare the spectral weight of IGS relative to LHB and UHB, and therefore obtain the hole doping level of at most 8.5% for our samples investigated. This value is lower than the threshold doping level of 10% required for the emergent superconductivity Wu et al. (2020). This suggests that the $\sqrt{3}$-Sn samples studied here lie in the underdoped regions, which happens to match with the experimental observation of PG.

It should be noted that such PG feature has been alternatively explained as a dynamical Coulomb blockade effect Ming et al. (2018). Given that several different Si(111) substrates with varying doping levels have been explored and the resistive depletion layer between the $\sqrt{3}$-Sn domains and the substrates has been largely modified, the PG feature should profoundly change with the varying substrates in the Coulomb blockade model. This contradicts with the universal occurrence of PG feature in $\sqrt{3}$-Sn, even when the underlying Si(111) substrates are more lightly doped with a resistivity of 0.01 $\Omega\cdot$cm (not shown). More importantly, if only the Coulomb blockade comes into play, the accompanying Coulomb staircase behavior commonly occurs as discrete peaks beside the Coulomb gap, which has never been observed. Thus, it seems more tempting to attribute the dip-like feature to the PG opening at $E_{\textrm{F}}$, as predicated from a quantum Monte Carlo simulation of the 2D Hubbard model Moukouri et al. (2000).

In order to corroborate this opinion, we have explored the doping evolution of PG by taking advantage of the $\sqrt{3}$-Sn nanodomain size-dependent hole doping level, albeit unknown cause Ming et al. (2018). Here the $\sqrt{3}$-Sn nanodomains are isolated and laterally encircled by the neighboring Sn/Si(111)-($2\sqrt{3}\times 2\sqrt{3}$)R30${}^{\textrm{o}}$ (shorten as $2\sqrt{3}$-Sn) phase, as illustrated in Fig. 3(a). Figure 3(b) represents the characteristic dI/dV spectra, disclosing well-defined PG on various $\sqrt{3}$-Sn nanodomains with varying areas. The PG shrinks in magnitude with increasing $\sqrt{3}$-Sn nanodomain area and saturates to about 20 meV above 600 nm² [Fig. 3(c)]. In extremely small $\sqrt{3}$-Sn nanodomains, a U-shaped gap develops with an apparent electron-hole asymmetry, analogous to the previous study Ming et al. (2018). A sharp and significant distinction is that the hole doping level, estimated from the relative spectral weight analysis just as Fig. 1(b), decreases as the area of $\sqrt{3}$-Sn nanodomains is reduced [Fig. 3(d)]. This has been unambiguously confirmed as well by exploring the spatial-dependent dI/dV spectra perpendicular to the boundary between the $\sqrt{3}$-Sn and $2\sqrt{3}$-Sn phases in Figs. 3(e) and 3(f). Similar to those near a single Sn vacancy, the tunneling conductance dI/dV spectra near the phase boundary are characteristic of significantly weaken IGS. This means a reduced hole doping level along the phase boundary and thus for the small $\sqrt{3}$-Sn nanodomain as a whole. The large discrepancy from the previous study might arise from either the distinct sample preparation details or the different measurement temperatures. In the previous study, the large- and small-energy scale dI/dV spectra are separately measured at 77 K and 4.4 K, which renders a direct comparison of them probably questionable. On the other hand, we have performed all the spectroscopic dI/dV measurements at the identical temperature of 4.2 K. This makes the present findings more convincing.

Figure 4 summarizes the doping-dependent magnitude of PG (purple stars) and the superconducting energy gap (blue star) in $\sqrt{3}$-Sn. All PG data are readily extracted from various nanodomains of $\sqrt{3}$-Sn supported by the heavily-doped Si(111) substrates (0.001 $\Omega\cdot$cm). It should be emphasized that we have also measured the $\sqrt{3}$-Sn on a lightly doped Si(111) substrate with the resistivity of 0.01 $\Omega\cdot$cm, which reveals a similar PG except for significantly suppressed gap edge peaks. These behaviors resemble those of cuprate PG in a prominent manner, and the nanodomain size-dependent PG behavior is also expected for a doped 2D Mott insulator Moukouri et al. (2000).

In summary, we have spectroscopically observed an unusual PG and established the electronic phase diagram in a modulation-doped Mott insulator $\sqrt{3}$-Sn on the Si(111) substrates. The PG, electronic phase diagram, and chemical doping scheme are found to be analogous to those of the high-$T_{\textrm{c}}$ cuprate superconductors. Further chemical doping control and experimental investigation of $\sqrt{3}$-Sn might find the microscopic mechanism of PG and superconductivity in this doped Mott insulator, by implications in cuprates.