Mott- versus Slater-type Insulating Nature of Two-Dimensional Sn Atom Lattice on SiC(0001)
Semiconductor surfaces with narrow surface bands provide unique playgrounds to search for Mott-insulating state. Recently, a combined experimental and theoretical study [Phys. Rev. Lett. 114, 247602 (2015)] of the two-dimensional (2D) Sn atom lattice on a wide-gap SiC(0001) substrate proposed a Mott-type insulator driven by strong on-site Coulomb repulsion . Our systematic density-functional theory (DFT) study with local, semilocal, and hybrid exchange-correlation functionals shows that the Sn dangling-bond state largely hybridizes with the substrate Si 3 and C 2 states to split into three surface bands due to the crystal field. Such a hybridization gives rise to the stabilization of the antiferromagnetic order via superexchange interactions. The band gap and the density of states predicted by the hybrid DFT calculation agree well with photoemission data. Our findings not only suggest that the Sn/SiC(0001) system can be represented as a Slater-type insulator driven by long-range magnetism, but also have an implication that taking into account long-range interactions beyond the on-site interaction would be of importance for properly describing the insulating nature of Sn/SiC(0001).
pacs:73.20.At, 75.10.Lp, 75.30.Et
Search for Mott-insulating state driven by short-range electron correlations has long been one of the most challenging issues in condensed matter physics ima (); meng (). Since the electrons in two-dimensional (2D) atom lattices can experience strong on-site Coulomb repulsion due to their reduced screening, metal overlayers on semiconductor substrates have attracted much attention for the realization of a Mott-Hubbard insulator nor (); ani (); cor (); mod (); pro (); li (); li2 (), where splits a half-filled band into lower and upper Hubbard bands. For example, the 1/3-monolayer adsorption of Sn atoms on the Si(111) or Ge(111) surface produces the reconstruction in which all the dangling bonds (DBs) of underlying Si or Ge surface atoms are saturated to leave a single DB on each Sn atom cor (); mod (); pro (); li (); li2 (); mor (); car (); bal (); lee1 (); lee2 (). Such Sn-overlayer systems with a half-filled band have been considered as an ideal playground for investigating 2D correlated physics on the triangular lattice cor (); mod (); pro (); li (); li2 (). However, the nature of the insulating ground state in Sn/Si(111) or Sn/Ge(111) has become a controversial issue whether the gap formation is driven by strong Coulomb interactions (Mott-type insulator) cor (); mod (); pro (); li (); li2 () or by long-range magnetic order (Slater-type insulator) lee1 (); lee2 ().
To realize a significantly reduced adatom-substrate hybridization as well as a strongly suppressed screening, Glass . gla () fabricated the phase of Sn overlayer on a wide-gap SiC(0001) substrate (see Fig. 1). In their photoemission experiment on the Sn/SiC(0001) surface system, Glass . observed a large energy gap of 2 eV. To account for the origin of such an insulating phase, Glass . performed the combined density-functional theory and dynamical mean-field theory (DFT + DMFT) calculations for a single-band Hubbard model that includes only the on-site Coulomb repulsion, and reproduced the experimentally observed insulating gap. Meanwhile, their spin-polarized DFT calculation gla () with the local density approximation (LDA) predicted a small energy gap of 0.1 eV for the collinear antiferromagnetic (AFM) ordering. Based on these results, Glass . interpreted the Sn/SiC(0001) surface system as a pronounced Mott-type insulator. However, the theoretical analysis of Glass . gla () leading to the Mott-insulating scenario raises the following questions: (i) Is the single-band Hubbard model employed in the previous DFT + DMFT calculation gla () suitable for describing the insulating nature of the Sn/SiC(0001) system? and (ii) Does the LDA accurately predict the insulating gap formed by the AFM order?
In this Letter, we investigate the nature of the insulating ground state of Sn/SiC(0001) by using the systematic DFT calculations with the LDA, semilocal (GGA), and hybrid exchange-correlation functionals as well as the LDA + DMFT calculation. All of the DFT calculations predict the AFM ground state, but the calculated band gap largely depends on the employed exchange-correlation functionals. Specifically, the hybrid DFT results for the band gap and the density of states (DOS) agree well with photoemission data. It is revealed that the Sn 5, 5, and 5 orbitals largely hybridize with the substrate Si 3 and C 2 orbitals, leading to three surface bands due to the crystal-field splitting. Such an unexpectedly large hybridization between the Sn DB state and the substrate states not only facilitates the superexchange interactions between neighboring Sn atoms to stabilize the AFM order, but also implies that long-range interactions beyond the on-site interaction should be taken into account for properly describing the insulating nature of Sn/SiC(0001). The present results suggest that the Sn/SiC(0001) surface system can be more represented as a Slater-type insulator via long-range magnetism rather than the previously gla () proposed Mott-type insulator via strong on-site Coulomb repulsion.
We begin to optimize the atomic structure of the NM structure using the LDA, GGA, and hybrid DFT calculations method (). The optimized NM structure obtained using LDA is displayed in Fig. 1. We find that the LDA height difference between the Sn atom and its bonding Si atoms is = 2.09 Å and that between the first C-layer atoms is = 0.19 Å, in good agreement with those ( = 2.03 Å and = 0.21 Å) of a previous LDA calculation gla (); note1 (). The values of and slightly change by less than 0.05 Å, depending on the employed exchange-correlation functionals. Figures 2(a) and 2(b) show the LDA band structure and partial density of states (PDOS) projected onto the Sn 5 and substrate Si 3 and C 2 orbitals, respectively. Interestingly, we find that Sn DB electrons form three surface bands designated as , , and [see Fig. 2(a)]. The bands projected onto the Sn 5, 5, and 5 orbitals obviously indicate that originates from the 5 orbital while and have mixed 5 and 5 characters (see Fig. 1S of the Supplemental Material supp ()). The higher energy of the state relative to the almost degenerate and states can be attributed to the effect of crystal-field splitting: i.e., the electrostatic repulsion between the Sn 5 and Si 3 orbitals is likely larger than that between the Sn 5 (or 5) and Si 3 orbitals. It is noted that, for the state, the Sn 5 PDOS is nearly equal in magnitude to the sum of the PDOS of Si 3 and C 2 orbitals, while, for the and states, the Sn 5 + 5 PDOS only amounts to 65% of the sum of the PDOS of Si 3 and C 2 orbitals [see Fig. 2(b) and Fig. 2S of the Supplemental Material supp ()]. Such a large hybridization between the Sn DB state and the substrate Si and C states is well reflected by the conspicuously mixed charge character of the localized Sn-DB and the delocalized SiC(0001)-substrate electrons [see the inset of Fig. 2(a)].
As shown in Fig. 2(a), the state crosses the Fermi level , producing a half-filled band. Despite its delocalized charge character as mentioned above, the state has a small band width of 0.31, 0.33, and 0.55 eV, obtained using the LDA, GGA, and hybrid DFT calculations, respectively. This flat-band-like feature is likely to be attributed to a large separation of 5.3 Å between Sn atoms within the unit cell. Because of such a narrow band width of the state, the electronic instabilities such as a charge or spin density wave (CDW/SDW) may be expected. For the CDW instability, we find that the 33 structure containing three Sn atoms (i.e., U, U, and D atoms in Fig. 3S of the Supplemental Material supp ()) of different heights is more stable than the NM structure by 13.4, 19.0, and 182.8 meV per unit cell for LDA, GGA, and hybrid DFT, respectively (see Table I).
Since such a buckled NM 33 structure accompanies a charge transfer from the D to the U (or U) atoms, it is most likely to reduce Coulomb repulsions between Sn DB electrons compared to the NM structure. We note that the calculated band structure of the NM 33 structure exhibits the presence of occupied surface states at (see Fig. 4S of the Supplemental Material supp ()), indicating a metallic feature. To find the possibility of SDW, we perform the spin-polarized LDA, GGA, and hybrid DFT calculations for the ferromagnetic (FM) and AFM 2 structures, which were considered in the previous LDA calculation gla (). We find that all of the employed exchange-correlation functionals favor the FM and AFM structures over the NM and 33 structures (see Table I). Here, the AFM structure is more stable than the FM structure, consistent with the previous LDA calculation gla (). It is noted that the stabilities of the two magnetic structures relative to the NM structure increase in the order of LDA GGA hybrid DFT calculations (see Table I). In the optimized AFM structure, two Sn atoms within the 2 unit cell are at the same height, indicating a structural symmetry as observed by low-energy electron diffraction and scanning tunneling microscopy gla ().
Figures 3(a), 3(b), and 3(c) show the LDA, GGA, and hybrid-DFT band structures of the AFM structure, which give the band gap of 0.12, 0.30, and 1.97 eV, respectively. The band gap obtained using hybrid DFT is found to be closer to that (2 eV) measured by photoemission spectroscopy gla (). As shown in Fig. 3(c), the DOS obtained using hybrid DFT exhibits the three peaks located at 1.09, 1.83, and 2.17 eV below , which are associated with the , , and states, respectively. On the other hand, photoemission spectra gla () showed the presence of two peaks at 1.0 and 2.4 eV, which were interpreted to originate from the Sn DB state and the SiC bulk states, respectively. Based on the present DOS results, we however interpret the upper and lower photoemission peaks in terms of the and (or ) surface states, respectively.
To understand the underlying mechanism for the gap opening of the AFM spin ordering, we plot in Fig. 3(d) the spin-polarized local DOS projected onto the two Sn atoms at A and B sites, together with their spin characters. It is seen that the occupied (unoccupied) spin-up and spin-down states are localized at the A(B) and B(A) sites, respectively. Here, the hybridization between the occupied spin-up (spin-down) state at the A(B) site and the unoccupied spin-up (spin-down) state at the B(A) site gives rise to a gap opening sato (). Such superexchange interaction super1 (); super2 () between the occupied and unoccupied electronic states can be facilitated due to a large hybridization of the Sn 5 orbitals with the Si 3 and C 2 orbitals [see Fig. 2(b) and Fig. 2S of the Supplemental Material supp ()]. This superexchange interaction is well represented by a large spin delocalization [see Fig. 3(d)] with the spin moments of 0.33, 0.12, and 0.10 for Sn, Si (outermost-layer) and C (outermost-layer) atoms, respectively (see the hybrid DFT results in Table IS of the Supplemental Material supp ()). We note that the calculated spin moments of Sn, Si, and C atoms increase in the order of LDA GGA hybrid DFT calculations (see Table IS), corresponding to that of the stabilization energy of the AFM structure (see Table I). On the basis of our DFT calculations, we can say that the magnetically driven insulating state of Sn/SiC(0001) with a large spin delocalization can be characterized as a Slater-type insulator.
The existence of the long-range AFM order due to the sizable hybridization between the Sn DB state and the substrate states raises questions about the reliability of the previous LDA + DMFT study gla () in which a single-band Hubbard model including only the on-site Coulomb interaction was employed. Here, the single band representing the DB state dominantly localized at Sn atoms invokes strong on-site Coulomb repulsion with suppressed electron hoping, driving the gap formation. Despite the fact that such a model Hamiltonian does not incorporate long-range interactions due to the largely hybridized state, we solve it within the LDA + DMFT scheme DMFT (); dmftref (); qmc (). Figure 4(a) shows the calculated DOS for the AFM and paramagnetic phases obtained at = 100 and 300 K, respectively. The observed insulating gap of 2 eV is found to be well reproduced with = 1.8 eV, similar to the previous gla () LDA + DMFT calculation. As shown in Fig. 4(b), the paramagnetic phase is transformed into the AFM phase below 100 K. Note that such a phase transition little changes the insulating gap [see Fig. 4(a)]. Therefore, the LDA + DMFT results indicate that the gap formation is not driven by the AFM order but attributed to the on-site interaction, representing a Mott-type insulator. Accordingly, the spin magnetic moment obtained using LDA + DMFT is 1 for Sn atom [see Fig. 4(b)]. Such a localized magnetic moment inherent in the Mott phase drastically contrasts with the large spin delocalization over Sn atoms and Si substrate atoms obtained using the hybrid DFT calculation [see Fig. 3(d) and Table IS]. Future experiments are anticipated to resolve such different features of spin magnetic moment between the Mott-type and Slater-type insulators by measuring the surface magnetic moments at Sn/SiC(0001).
It is noteworthy that the charge character of the state exhibits a large delocalization up to the third deeper Si and C substrate layers, which in turn gives some lateral overlap between neighboring Sn atoms [see the inset of Fig. 2(a)]. Such an extension of the half-filled surface state calls for the importance of long-range interactions which were not considered in the previous gla () and present LDA + DMFT calculations. Indeed, a recent fully self-consistent GW + DMFT study hans () for the analogous X/Si(111) systems (with X = C, Si, Sn, and Pb) reported that taking into account long-range Coulomb interactions is mandatory because of their comparable magnitude with that of the on-site Coulomb interaction. It was shown that the inclusion of long-range interactions within the extended Hubbard model changes the ground-state character of the X/Si(111) systems hans (): i.e., without long-range interactions, all the X/Si(111) systems are in the Mott phase, but, as long-range interactions are added, Sn/Si(111) and Pb/Si(111) become closer to a metallic phase. Compared to the Sn/Si(111) system, Sn/SiC(0001) has the 20% smaller nearest-neighbor distance of Sn atoms as well as the relatively lower dielectric screening of the SiC substrate, thereby leading to an increase in the intersite interactions. It is thus expected that the nonlocal interaction effects in Sn/SiC(0001) might significantly influence the stability of the Mott phase obtained by using only the on-site interaction. For more accurate simulation of the present system, the extended Hubbard model including long-range interaction terms will be demanded in future theoretical work. There still remains an interesting challenge of how to equally consider all of the on-site interaction, long-range interactions, and magnetic response in the Sn/SiC(0001) system.
To conclude, we have presented two different pictures for the insulating nature of the Sn overlayer on a wide-gap SiC(0001) substrate using the LDA, GGA, and hybrid DFT calculations and the LDA + DMFT calculation. The DFT calculations drew the Slater-type picture with a long-range AFM order, while the LDA + DMFT calculation supported the Mott-type picture driven by strong on-site Coulomb repulsion. Unexpectedly, the Sn DB state was found to largely hybridize with the substrate Si and C states, thereby facilitating the stabilization of the AFM spin ordering via superexchange interactions. This intriguing electronic structure of the present system raises an important issue of how long-range interactions beyond the on-site interaction should be taken into account to diminish the Mott phase. Our findings will not only caution against the realization of the Mott-insulating phase in metal overlayers on semiconductor substrates, but also stimulate further experimental studies for the exploration of the magnetic phases of Sn/SiC(0001).
Acknowledgement. This work was supported by National Research Foundation of Korea (NRF) grant funded by the Korea Government (MSIP) (2015R1A2A2A01003248). The calculations were performed by KISTI supercomputing center through the strategic support program (KSC-2013-C3-043) for the supercomputing application research.
Corresponding author: email@example.com
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1. Sn -orbitals projected band structure of the NM structure
To figure out the electronic characters of the , , and states, we plot in Fig. 1S the bands projected onto the Sn 5, 5, and 5 orbitals. It is shown that originates from the 5 orbital while and have mixed 5 and 5 characters.
2. Partial density of states of the NM structure
Figure 2S shows the partial density of states (PDOS) of the NM structure, projected onto the Si 3, 3, and 3 components as well as the C 2, 2, and 2 ones.
3. Geometry of the NM 33 structure
4. Band structure of the NM 33 structure
Figure 4S shows the calculated band structure of the NM 33 structure. It is seen that there is a half-filled band crossing the Fermi level, indicating a metallic feature.
5. Spin moments of Sn, Si, and C atoms in the AFM structure