Introduction

Van der Waals (vdW) layered materials such as two-dimensional (2D) transition-metal dichalcogenides (TMDCs) and Bi2Sr2CaCu2O8+x (Bi-2212) have shown various electronic phases that emerge from many-body features such as a charge density wave (CDW) or superconductivity, depending on the temperature and carrier density1,2,3,4,5,6,7,8,9. The vdW interface between dissimilar vdW materials have allowed to investigate the electronic structures of such strongly correlated electron systems. For instance, transport spectroscopy in TMDC/Bi-2212 vdW junctions has revealed gap natures due to the many-body features via the formation of a metal/superconductor proximity junction with a low vdW contact resistance10,11. For tunneling spectroscopy, on the other hand, a graphite/Bi-2212 interface provided a vdW gap tunnel junction, enabling tunneling spectroscopy for the Bi-2212 superconductor11,12. However, it could be hard to apply the graphite to TMDCs for vdW tunneling spectroscopy because graphene/TMDC contacts have been used to form an Ohmic contact13. Thus, for feasible and reliable tunneling spectroscopy for vdW materials, it is crucial to seek a material to form a vdW tunneling gap with any vdW materials. With this purpose, we focus on indium (In) metal in this study.

For evaporated-metal/TMDC contacts, only indium (In) metal has shown vdW contact for TMDCs14,15, owing to its low evaporation temperature of ~ 500 °C. In this case, the In vdW contact provides an Ohmic contact for a few-layer TMDCs. However, the vdW contact at a cryogenic temperature could provide a vacuum tunneling gap with a high contact resistance that makes the flow of current sensitive to the electronic DOS at the interface. Indeed, vdW gap tunneling spectroscopy based on a field-effect transistor (FET) design with carbon nanotubes (CNTs) with In metal contacts was demonstrated, recently16,17. In this previous work, the local conductance peaks observed in the conductance vs bias voltage plot were shown to originate from the van Hove singularities corresponding to the sub-band structures of semiconducting and metallic CNTs.

In the present study, we apply a type of FET with In contacts for various vdW layered materials (i.e., 1T-TaS2, Bi-2212, and 2H-MoS2) to demonstrate that In-contacted vdW gap tunneling spectroscopy is a feasible method to investigate the electrical DOS of vdW layered materials. For the experiments with 1T-TaS2, the zero-bias resistivity showed a sudden increase at T ~ 180 K as the temperature was lowered. At T = 4 K, a plot of the differential conductance for various bias voltages revealed the emergence of an energy gap, i.e., the Mott gap edge, which has been observed in the same material only using scanning tunneling and photoemission spectroscopies18,19,20,21. The In/Bi-2212 junction also showed a superconducting gap of ~ 58 meV at T = 4 K. The gap feature slightly decreased with increasing gate voltage, which indicates that the high-temperature superconductivity could be controlled by electric fields12. Finally, for ~ 10 nm-thick MoS2 FETs, we observed gap features at energy levels of ~ 0.9 eV in differential conductance vs bias voltage curves recorded at T = 4 K, corresponding to a semiconducting bandgap. The formation of the tunneling barrier with a high contact resistance for ~ 10 nm-thick MoS2 is inconsistent with the Ohmic contacts for few-layer (thickness ⪅ 4 nm) MoS2 flakes with In contacts, which might be related to the location of the Fermi level of an In electrode with respect to the bandgap, depending on the thickness of the MoS2 layer. The naturally formed vdW tunnel gap without any artificial insulating barrier is very robust under varying temperature. Our work provides simple and reliable identification of electronic DOS using the simple FET geometry for the vdW materials without sophisticated tools such as scanning tunneling microscope.

Measurements and results

Experiments

Single crystals of 1T-TaS2 and Bi-2212 were grown by the usual iodine transport method and solid-state-reaction methods, respectively. A 2H-MoS2 single crystal was commercially purchased (HQ Graphene). We fabricated vdW material-based FETs with In contacts for TaS2, Bi-2212, and MoS2, by carrying out several microfabrication processes. The vdW material flakes on a 500 nm-thick SiO2/Si substrate were prepared via mechanical exfoliation from the vdW materials. We deposited 100 nm-thick In electrodes onto a multilayer flake using traditional electron-beam lithography and thermal deposition processes16. To investigate the quality of the In/vdW material, we collected cross-sectional transmission electron microscopy (TEM) image of an In/few-layered MoS2 junction (Fig. 1a), where the thermally deposited In did not show invasion into the MoS2 layer, resulting in a well-defined vdW gap. Atomic structures of interfaces between In and two vdW materials of 2H-MoS2 and 1T-TaS2 with vdW gaps were prepared by the density functional theory (DFT) calculations as shown in Fig. 1b,c, respectively. The schematic atomic structure for In/Bi-2212 is also plotted in Fig. 1d. Figure 1e shows a schematic of a completed device for the vdW tunneling spectroscopy experiment (Fig. 1f), where the highly doped Si substrate serves as a back-gate electrode. For basic electrical characterizations of the three vdW materials such as carrier density (nH) and Hall mobility (μH), we performed independent electrical measurements for 1T-TaS2 and 2H-MoS2. We measured the resistivity of 1T-TaS2 as a function of T in a four-probe configuration (see Supplementary Fig. 1), where ρ ~ 0.1 Ω cm for 50 < T < 220 K. This is a similar ρ range with a previous report22, thus we expect that nH and μH show a similar trend with temperatures. For 2H-MoS2, we performed the Hall measurement with varying T and back-gate voltage (see Supplementary Fig. 2). For Bi-2212, we estimated nH and μH from literatures showing a similar Tc with a similar hole doping23,24. The list of nH and μH for the three vdW materials was shown in Table 1.

Figure 1
figure 1

In-contacted vdW gap tunneling spectroscopy for vdW layered materials. (a) Cross-sectional TEM image of an In/few-layered MoS2 junction. Scale bar: 2 nm. (b)–(d) Atomic structures at In/2H-MoS2, In/1T-TaS2, and In/Bi-2212 interfaces (schematic), respectively. Here, Atomic structures of In/2H-MoS2 and In/1T-TaS2 were prepared by the DFT calculations. (e) Schematic of the measurement configuration of a device with an FET design. (f) Schematic of the electronic DOS at a vdW material/In interface; this schematic describes the tunneling spectroscopy. CB and VB are the conduction and valence bands of MoS2, respectively. EF is the Fermi energy of the In electrode. Blue and white regions indicate states occupied and unoccupied by electrons, respectively. Under a zero-bias condition, the EF is located at the midgap in the energy-gap region of the vdW material (dashed red line). Under a bias voltage of Vsd, the Fermi level of In becomes aligned with the empty states of the CB, resulting in a tunnel current through the vdW gap. The differential conductance obtained from the tunnel current at a cryogenic temperature reflects the DOS of the vdW material.

Full size image
Table 1 Representative carrier type, carrier density (nH) and Hall mobility (μH) for three vdW materials.
Full size table

vdW gap tunneling spectroscopy for 1T-TaS2

The upper panel of Fig. 2a shows an atomic force microscopy (AFM) image of a fabricated TaS2 device, where the channel length (L) and width (W) are 1.7 μm and 2.2 μm, respectively. The lower panel of Fig. 2a shows the height profile along the dashed white line in the upper panel, indicating that the thickness of TaS2 was ~ 88 nm. Figure 2b shows dI/dVsd as a function of the source–drain voltage (Vsd) and back-gate voltage (Vg) at T = 4 K, which shows a substantial suppression of conductance near zero bias, along with conductance shoulders (indicated by two arrows). In the I–Vsd curve corresponding to Vg = 30 V in Fig. 2c, a relatively flat current region is observed near zero bias, which corresponds to the conductance dip region at the same Vg value in Fig. 2b. The dI/dVsd–Vsd curve corresponding to Vg = 30 V shows a clear gap feature indicated by the bidirectional arrow, which was also observed at Vg =  − 30 V in Fig. 2c. We speculated that the observed gap is related to the Mott gap (ΔMott), exhibiting a gap size of ~ 0.4 eV. The Mott transition in multilayer 1T-TaS2 has been previously shown to be developed in the temperature range 180 ≤ T ≤ 210 K, accompanied by a transition from the nearly-commensurate CDW (NCCDW) phase to the commensurate CDW (CCDW) phase, as revealed by scanning tunneling spectroscopy and photoelectron spectroscopy18,19. A recent study based on resistance measurements as a function of T conducted by using a gate-controlled Li+-ion intercalation method showed that the CCDW phase changed to the NCCDW phase with increasing gate voltage at T = 10 K2. In our case, the width of the conductance dip was nearly constant for Vg values spanning 60 V, possibly because of relatively less change in the carrier density in our gating method with a 500 nm-thick dielectric SiO2 layer. In addition, in Fig. 2d, we displayed already reported dI/dVsd–Vsd curves obtained by conventional scanning tunneling spectroscopy with the same crystal used in this study18. Both of them showed a similar ΔMott size of ~ 0.4 eV, thus we believe that In-contacted vdW gap tunneling spectroscopy provides a credible method to study the electronic states in vdW materials.

Figure 2
figure 2

vdW gap tunneling spectroscopy for 1T-TaS2. (a) Upper panel: AFM image of a 1T-TaS2 FET. Scale bar: 1 µm. Lower panel: height profile along the dashed white line in the upper panel. (b) dI/dVsd as a function of Vsd and Vg at T = 4 K. The two arrows indicate the conductance shoulders corresponding to a gap feature. (c) dI/dVsd–Vsd curves at Vg =  ± 30 V and I–Vsd curve at Vg = 30 V. These curves clearly show an energy-gap feature, indicated by the Mott gap (ΔMott) with a magnitude of ~ 0.4 eV. (d) Conventional scanning tunneling spectra obtained from the same 1T-TaS2 single crystal at T = 77 K, which shows a similar ΔMott size with that in (c). (d) is reproduced with permission from18, 1994 AMERICAN PHYSICAL SOCIETY.

Full size image

To investigate the phase transition, we obtained dI/dVsd–Vsd curves at Vg = 30 V over the temperature range 4 ≤ T ≤ 210 K, as shown in Fig. 3a, where the curves are vertically shifted as much as 20 nS for clarity. The pronounced gap feature at T = 4.2 K was smeared with increasing T up to T ~ 140 K and became featureless at T ≥ 180 K, whereas the conductance dip near zero bias was still observed. Figure 3b shows the zero-bias resistivity as a function of T, as extracted from Fig. 3a at Vsd = 0 V. A sudden increase is observed at T ~ 180 K (indicated by an arrow) with decreasing T. This behavior was found to be consistent with previous observations of the Mott transition accompanied by the phase transition from the NCCDW phase to the CCDW phase near this temperature, TCCDW&Mott19,20. In that Mott transition, a gap was not fully opened in the investigated T region, resulting in a so-called a pseudogap structure19,20. In our case, such behavior was observed in the region indicated by a dashed bidirectional arrow in Fig. 3b, where the zero-bias resistivity monotonically increases with decreasing T. The dI/dVsd–Vsd curves corresponding to 80 ≤ T ≤ 180 K in Fig. 3a show pseudogap-hump structures at Vsd ~  ± 0.3 V with a finite zero-bias conductance, representing a small but finite density of state at the Fermi energy (EF). Importantly, at T < 60 K, the resistivity increases substantially faster with decreasing T (see Fig. 3b) and the zero-bias conductance finally decreases to nearly zero at T < 20 K (Fig. 3a), where conductance peaks corresponding to the gap edges are clearly observed at Vsd ~  ± 0.2 V, as shown in Fig. 3a (see also Fig. 2c). This result indicates that the Mott gap was fully developed at T < 20 K21. To test the reproducibility of these results, we also fabricated two additional 1T-TaS2 devices with a thickness similar to that of the first 1T-TaS2 device, and the curves for each device showed a clear conductance peak at one polar Vsd of − 0.20 V and 0.18 V at T = 4 K, respectively, as shown in Fig. 3c (vertical arrows). Asymmetric gap features have been frequently observed in tunneling spectroscopy results for two electrodes (tip and metal contacts) with substantially different contact resistance levels (see Fig. 2d)18.

Figure 3
figure 3

Temperature dependence of Mott gap. (a) dI/dVsd as a function of Vsd at Vg = 30 V for various temperatures, where each curve is vertically shifted for clarity. (b) Zero-bias resistivity at Vsd = 0 V as a function of T, as obtained from (a). Near T ~ 180 K, the resistivity suddenly increased with decreasing T, as indicated by the arrow. The region indicated by a dashed bidirectional arrow indicates a pseudogap region. (c) dI/dVsd as a function of Vsd for two other 1T-TaS2 devices, each of which showed a gap feature, as indicated by the two arrows.

Full size image

vdW gap tunneling spectroscopy for Bi-2212

The as-grown Bi-2212 crystal used in the present study exhibits a slightly hole-overdoped character; the superconducting transition temperature (Tc) measured from other crystals obtained from the same batch was ~ 88 K25. The upper panel of Fig. 4a shows an AFM image of a Bi-2212 device with L ~ 0.4 μm and W ~ 0.4 μm. The lower panel of Fig. 4a shows a height profile along the dotted line depicted in the upper panel of Fig. 4a, indicating that the thickness of the Bi-2212 is ~ 70 nm. Figure 4b shows dI/dVsd as a function of Vsd at Vg =  − 40 V for various temperatures, where the curves are vertically shifted as much as 50 nS. At T = 4 K, the gap feature was observed at Vsd ~  − 58 and 50 mV, which are assigned as Vp− and Vp+, respectively, as indicated by two red arrows in the figure. The observed gap sizes are consistent with previous observations of the superconducting gap energy, \(\Delta\)26,27. The peak signature at Vp+ is relatively weak compared with that at Vp−, similar to the TaS2 case. The Vp− value corresponding to the conductance peak decreases with increasing T, as indicated by the dashed green line, and is smeared out near the Tc at T = 80 K. Figure 4d shows |Vp−| as a function of T at Vg =  − 40 V, where |Vp−| decreases with decreasing T. For comparison, we added a dashed curve representing \(\Delta \left( T \right)\)/e based on the expression28

$$\Delta \left( T \right) = \Delta_{0} \tanh \left( {\frac{\pi }{a}\sqrt {b\left( {\frac{{T_{{\text{c}}} }}{T} - 1} \right)} } \right)$$
(1)

where e is the elementary charge; \(\Delta_{0}\) = 58 meV, a = 2.14, and b = 4/3 for a weak-coupling 2D d-wave superconductor; and Tc = 88 K. Although we lack exact information about the Tc at Vg =  − 40 V, the data qualitatively follows Eq. (1). Thus, we conclude that the gap feature originates from the superconducting gap. We also measured dI/dVsd as a function of Vsd at Vg =  − 30 V as T was varied (Fig. 4c). As depicted by the dashed green line, the superconducting gap energy decreases with increasing T. However, we note that the gap feature relatively weakens at Vg =  − 30 V compared with that at Vg =  − 40 V. For instance, at Vg =  − 30 V, Vp−  decreases to approximately − 46 mV at T = 4 K, accompanied by a reduction of the conductance peak height. The conductance peak corresponding to the superconducting gap edge at the positive Vsd region (Vp+) even disappears at T = 4 K. For comparison, we added |Vp−| as a function of T obtained at Vg =  − 30 V in Fig. 4d. Recently, Liao et al. reported that the superconductor–insulator transition in Bi-2212 could be achieved using a back-gate method with a solid ionic conductor, where the carrier density was modulated by intercalated Li+ ions12. This result may indicate that the superconductivity of Bi-2212 can also be manipulated by a back-gate field with a SiO2/Si substrate design as our case, although further study is needed to quantitatively demonstrate the feasibility of this concept.

Figure 4
figure 4

vdW gap tunneling spectroscopy for Bi-2212. (a) Upper panel: AFM image of a Bi-2212 device. Scale bar: 1 μm. Lower panel: height profile along the dashed black line in the upper panel. (b), (c) dI/dVsd as a function of Vsd for various temperatures, as obtained at Vg =  − 40 and − 30 V, respectively. Dashed green lines follow the gap edges as T is varied. (d) Closed squares and circles: |Vp−| as a function of T for Vg =  − 40 and − 30 V, respectively. Dashed curve: a plot based on Eq. (1) for comparison.

Full size image

vdW gap tunneling spectroscopy for 2H-MoS2

The inset of Fig. 5a shows an AFM image of the MoS2 FET (MS1) with a thickness of ~ 10 nm. The MS1 device showed a traditional n-type transfer curve for Vsd = ± 0.5 V at T = 4 K (Fig. 5a), where the I–Vg curves show an asymmetric behavior depending on the polarity of Vsd. Figure 5b shows a dI/dVsd map as a function of Vsd and Vg. In Schottky FETs, the zero-conductance region observed when Vsd is swept in a depletion state decreases with positively increasing Vg for an n-type device because the width of the Schottky barrier decreases with positively increasing Vg. Although the conductance map in Fig. 5b appears to show such behavior for Vg < 30 V (region i), a robust zero-conductance region that is independent of the change in Vg was observed at − 0.45 V ≤ Vsd ≤ 0.45 V (see dashed yellow lines) in the Vg region labeled as region ii. Figure 5c shows I and dI/dVsd as functions of Vsd at Vg = 40 V in region ii. Conductance shoulders, indicated by the two vertical dashed lines, are separated from each other by an energy scale of ~ 0.96 eV, which is close to the interval bandgap of ~ 1.2 eV expected for multilayer MoS229.

Figure 5
figure 5

vdW gap tunneling spectroscopy for 2H-MoS2. (a), (d) Current vs. back-gate voltage (I–Vg) curves at Vsd =  ± 0.5 V and T = 4 K for MS1 and MS2, respectively. Insets of (a), (d): AFM images of MS1 (scale bar: 1 μm) and MS2 (scale bar: 2 μm), respectively. The thickness of the MoS2 flakes in both cases was ~10 nm, corresponding to ~ 14 MoS2 layers. (b), (e) Differential conductance (dI/dVsd) as a function of Vsd and Vg of MS1 and MS2, respectively. The zero-conductance region on the Vsd axis became smaller with increasing Vg in region “i” and did not change in region “ii”. The two dashed lines indicate the edges of the bandgap. (c), (f) dI/dVsd and I as a function of Vsd at Vg = 40 and 64 V for MS1 and MS2, respectively. The left and right vertical dashed lines located at the conductance shoulders correspond to the edges of the conduction and valence bands, respectively.

Full size image

To confirm the consistency of the vdW gap tunneling spectroscopy, we fabricated another ~10 nm-thick MoS2 device (MS2; inset of Fig. 5d). The overall behaviors of the electrical properties of the MS2 device, such as the n-type behavior and the robust zero-conductance region (region ii) in Fig. 5d,e, respectively, show similar trends as those of the MS1 device. The conductance shoulders in Fig. 5f with the two vertical dashed lines provide an energy scale of ~ 0.92 eV at Vg = 64 V in region ii, which is also similar to that of the MS1 device.

To determine the origin of the robust zero-conductance region, we considered the possibility of tunnel barriers with a high contact resistance due to the vdW gap between the MoS2 and In electrodes. Figure 6a,b show the band structures of MoS2 and In with vdW gap tunnel barriers at the MoS2/In interfaces (vertical gray bars) for representative Vg regions labeled as i and ii in Fig. 5b, respectively. For simplicity, we only considered the MS1 device in Fig. 5a–c. The upper and lower solid black curves correspond to the conduction-band (CB) and valence-band (VB) edges of MoS2, respectively. The light-blue region under the VB edge indicates the states occupied by electrons. For Vg ~  − 20 V, the MoS2 band was found to be shifted upward, whereas the band edges were fixed at the junction interfaces, where the left electrode was grounded. EF(In) was located within the bandgap (Eg) without Vsd, as shown in the left panel of Fig. 6a (horizontal red line); thus, a sufficiently high Vsd is needed to overcome the Eg region. The middle and right panels of Fig. 6a show Vsd conditions in which the EF(In) of the right electrode reaches the CB and VB edges, respectively. Nevertheless, electrons do not flow to the edges of the MoS2 because they experience a large Schottky barrier width. Vsd values greater than those corresponding to the band edges are thus needed to make a narrower Schottky barrier for the flow of electrons. With increasing Vg, the bands for the MoS2 bend downward, leading to a relatively narrow Schottky barrier. Thus, the ± Vsd that allow the current to flow decrease with increasing Vg, corresponding to region i in Fig. 5b. In region ii in Fig. 5b, the interval between ± Vsd locations for the finite conductance edges were found to change only slightly as Vg was varied. The band diagrams corresponding to region ii are plotted in Fig. 6b. In this region, the MoS2 electronic bands were observed to be substantially bent downward for Vg ~ 40 V. Here, the alignment of EF(In) with the two band edges with proper Vsd values of − Vc and + Vv enabled the electrons to tunnel between the electrode and MoS2 under the assumption that the Schottky barrier widths were sufficiently narrow to allow the tunnel event, as indicated by horizontal arrows in the middle and right panels in Fig. 6b. The alignment resulted in no variation of the interval between − Vc and + Vv values, as indicated by the two dashed parallel lines in Fig. 5b. We note that the electrostatic band bending also occurs because of the vdW tunnel junction with a finite Vsd, which leads to an additional band bending in a lower direction with the positive Vsd.

Figure 6
figure 6

Band diagrams in MoS2 device with various Vg and Vsd values. (a), (b) Band diagrams with zero and finite Vsd conditions in Vg regions corresponding to regions “i” and “ii” depicted in Fig. 5b,e, respectively. EF denotes the Fermi energy of In electrodes. Dark- and light-blue regions indicate bands filled by electrons of In and MoS2, respectively. The upper and lower solid lines in the MoS2 region represent the edges of the conduction and valence bands, respectively. Eg is the energy gap of MoS2. Solid red lines in the In region indicate Fermi levels depending on Vsd conditions. Vertical red arrows depict the direction of the changing Fermi levels with finite Vsd conditions from the zero-bias condition (dashed red lines). Horizontal red arrows represent the current flow directions via tunneling.

Full size image

The observed tunnel behavior with high contact resistances at In-contacted devices with 10 nm-thick (n ~ 15, where n is the number of MoS2 layers) MoS2 layers appears to be inconsistent with the behavior of In/MoS2 devices with MoS2 thicknesses ≤ 4 nm (n ≤ 6), which exhibit Ohmic behavior13. For this reason, we considered the location of EF(In). Figure 5c,f show that the location of zero bias is nearly midway between the two vertical dashed lines, which implies that EF(In) for n ~ 15 MoS2 is located near the midgap, indicating non-Ohmic contact. However, in the case of few-layer MoS2, EF(In) is located just below the CB edge, resulting in Ohmic contact under the tunneling (or field-emission) mechanism through the vdW gap13. In this sense, performing vdW gap tunneling spectroscopy for n ≤ 6 is not possible, although the bandgap drastically increases from ~ 1.4 to ~ 1.9 eV when n is changed from 6 to 12. Additional experimental and theoretical studies are needed to understand the current flowing between In and various-thickness MoS2 to know the limit of In-contacted vdW gap tunneling spectroscopy for MoS2.

Conclusions

We carried out In-contacted vdW gap tunneling spectroscopy for 1T-TaS2, Bi-2212, and 2H-MoS2 using an FET geometry. We clearly observed the Mott gap (~ 0.4 eV), superconducting gap (~ 58 meV), and semiconducting bandgap (~ 0.9 eV) of 1T-TaS2, Bi-2212, and MoS2, respectively, by analyzing conductance curves as a function of Vsd at T = 4 K. Thus, we propose that vdW gap tunneling spectroscopy provides a feasible method to reveal the electronic band structure of inert vdW layered 2D materials. For semiconductor vdW materials of MoS2, we found that In-contacted vdW gap tunneling spectroscopy is applicable for only bulk MoS2 (n ~ 15), which could be related to the location of the Fermi level of In with respect to the midgap of MoS2. This reflects that the relative location of the Fermi level of In with respect to the midgap of vdW material may reveal the limitations of In-contacted vdW gap tunneling spectroscopy. For non-semiconductor vdW materials of 1T-TaS2 and Bi-2212 with thickness of tens of nanometers, we confirmed the In-contacted vdW gap tunneling spectroscopy is applicable while we need further study for the validity for a few layers.

Methods

Samples

We fabricated electrical devices with In contacts for TaS2, Bi-2212, and MoS2 on 500 nm-thick SiO2/Si substrates by conventional microfabrication processes. We deposited 100 nm-thick In electrodes onto a multilayer flake in a thermal evaporator at a vacuum of 3 × 10−6 Torr. The substrate was attached to a stage, which was cooled to 100 K using liquid N2 during the In deposition process. The low temperature of the sample stage ensured a homogeneously deposited In layer with a uniform thickness and without grain boundaries16.

Measurements

We performed electrical measurements using a two-probe configuration in a closed-cycle refrigerator with a base temperature of 4 K. Bias voltages were applied by Keithley 213 quad voltage source and current was measured by a current amplifier (Ithaco 1211, DL).

Computations

DFT calculations for Fig. 1b,c are carried out within generalized gradient approximation (GGA) for exchange–correlation (xc) functionals30,31, implemented in the Vienna ab initio simulation package (VASP)32.