Visualizing the evolution from Mott insulator to Anderson insulator in Ti-doped 1T-TaS2

The electronic evolution of doped Mott insulators has been extensively studied for decades in search of exotic physical phases. The proposed Mott insulator 1T-TaS2 provides an intriguing platform to study the electronic evolution via doping. Here we apply scanning tunneling microscopy (STM) to study the evolution in Ti-doped 1T-TaS2 at different doping levels. The doping Ti atom locally perturbs the electronic and spin state inside the doped star of David and induces a clover-shaped orbital texture at low-doping levels (x < 0.01). The insulator to metal transition occurs around a critical point x = 0.01, in which small metallic and large insulating domains coexist. The clover-shaped orbital texture emerges at a broader energy range, revealing a competition with the electron correlation. It transforms to a disorder-induced Anderson insulating behavior as doping increases. We directly visualize the trapped electrons in dI/dV conductance maps. The comprehensive study of the series of Ti-doped 1T-TaS2 deepens our understanding of the electronic state evolution in a doped strong-correlated system.


INTRODUCTION
The strong electron-electron correlation in Mott insulators is a breeding ground for the emergence of exotic electronic states, most notably in the cuprate high T c superconductors (HTS) 1,2 . Elemental doping can effectively tune a Mott insulator into other electronic phases like a metal, superconductor, etc. 3 . Understanding the evolution from a Mott insulator to a metallic phase, further to a superconducting state has been a central problem in the HTS studies [1][2][3] . Meanwhile, exotic quantum orders like checkerboard charge order, nematic order, etc. emerge during the doping evolution 4,5 . Another intriguing research field is discovering other types of Mott-insulating systems and then studying their exotic electronic states via different tuning methods 6,7 . The quasi-two-dimensional (2D) transition metal dichalcogenides 1T-TaS 2 is such a proposed Mott insulator [8][9][10] .
The bulk 1T-TaS 2 exhibits an abundant phase diagram, which originates in the Mott insulating ground state in the commensurate charge-density-wave (CCDW) phase [11][12][13][14] . The coexistence or competition of electron-electron correlation and electron-phonon interaction has been intensively studied in 1T-TaS 2 [14][15][16] . The insulating state resides on the quasi-2D triangular CCDW super-lattice instead of the atomic lattice. Each element of the super-lattice is called a star of David (SD), including a cluster of 13 Ta atoms (Fig. 1a). The multifolding of SD super-lattice induces a narrow flat band near the Fermi level, which is half-filled with each SD contributing one unpaired orphan electron. A relatively strong correlation then leads to the split Hubbard bands and the Mott insulating gap 10 . Compared with traditional Mott insulators, the super-lattice and Mott insulating state in 1T-TaS 2 are very sensitive to external perturbations, providing complex phenomena for strong-correlation studies. The triangular super-lattice with a geometric frustration also enhances the quantum fluctuation, leading to a possible quantum spin liquid (QSL) state [17][18][19] . Recently, there are revived interests in the insulating nature of 1T-TaS 2 , which may undermine the strong-correlation foundation [20][21][22] . Though numerous experiments have been performed on pristine samples, the origin of the insulating gap remains unclear [23][24][25][26] . The electronic evolution under different tuning methods provides alternative ways to reveal the nature of the insulating gap.
Scanning tunneling microscopy (STM) is a powerful technique to study the electronic evolution of doped Mott insulators at the atomic level. STM can locate the impurity dopants and investigate how a single dopant disturbs the local electronic density of state (DOS) 27,28 . In electronic spatial maps, STM can visualize the doping-dependent evolution of global DOS and the emergence of electronic orders 29,30 . Especially for 1T-TaS 2 , nanostructures always appear in the bulk material [31][32][33][34][35] , and STM is a valuable tool to explore the local electronic state before claiming macroscopic properties.
In this paper, we successfully synthesize the Ti-doped 1T-TaS 2 at a wide range of doping levels. The Ti doping replaces one of the 13 Ta 4+ ([Xe]4f 14 5d 1 ) ions with a nonmagnetic and isovalent Ti 4+ ([Ar]) ion, depleting the unpaired d electron in a SD. We combine STM and density functional theory (DFT) calculation to investigate how the electronic state is locally and globally disturbed. At a lowdoping level x < 0.01, Ti atom locally perturbs the DOS inside SD, while an insulating state remains in undoped SDs. A Ti dopant effectively creates a spin vacancy in doped SD and possibly perturbs the QSL state [17][18][19] regardless of the substitution site. At the same time, the light doping induces a global clover-shaped orbital texture in dI/dV conductance maps. As the doping level increase to x = 0.01, the CCDW phase is completely suppressed, and the low-temperature phase appears as a metallic state. The large and small domains coexist in the STM topography. In the large domain, the DOS of undoped SDs remains insulating as those in pristine samples. While in the small domain, the DOS exhibits a V-shaped metallic behavior near the Fermi level. The small metallic domains act as conducting channels, accompanied by the metallic domain walls. The CDW orbital texture in the metallic domain also expands to a wider energy range with increasing doping, competitive with the weakened correlation. When the doping level reaches 0.08, most SDs are doped by Ti atoms, and those undoped SD are randomly distributed in the topography. The analysis of conductance maps demonstrates that the unpaired electrons are trapped in undoped SDs. The lowtemperature upturn of resistivity is related to the disorder-induced potential fluctuation, showing a typical Anderson insulator behavior.

Perturbation of Ti doping on the ground insulating state
We substitute Ti for Ta atoms to produce 1T-(Ta 1−x Ti x )S 2 , with high-quality single-crystal samples grown by chemical vapor transport method. The similar ionic radii of Ti and Ta make the doping system relatively stable, and the doping has been precisely controlled during sample growth. With the Mott state in 1T-TaS 2 originating from the super-lattice of Ta atoms, a small amount of substitution of Ti for Ta can destroy the Mott insulating state. Figure 1b displays the temperature-dependent resistivity of 1T-(Ta 1−x Ti x )S 2 . For the pristine sample, two clear phase transitions occur within the temperature range [11][12][13] . The transition around 350 K (T I-NC ) is associated with the incommensurate to nearly commensurate (NC) CDW transition. Below 175 K (T NC-C ), the material enters the CCDW phase. The NC-CCDW transition is sensitive to Ti doping. At x = 0.004, T NC-C decreases by about 50 K. From x = 0.01 and on, the transition completely disappears.
To infer the perturbation of 1T-TaS 2 by Ti doping, we first compare the extremely low-doped x = 0.004 sample with an undoped sample. For the undoped sample, STM topography shows a triangular SD super-lattice 16,31 . A typical dI/dV spectrum measured at the center of one SD is displayed in Fig. 1c. The sharp peaks at −190 and 220 meV correspond to the LHB and UHB, respectively. In our previous stacking-order-effect study 32 , we have concluded that this spectrum mainly describes the Mott insulating property of the top single-layer 1T-TaS 2 . For the x = 0.004 sample, the topography maintains a similar triangular SD super-lattice to the undoped one, with each SD appearing as a bright triangle (Fig. 1d). In Fig. 1c, a typical dI/dV spectrum of x = 0.004 sample also largely preserves the nature of the insulating state.
The scattered defective SDs can be identified under a negative bias voltage V b , while difficult to discern under a positive bias voltage. As shown in Fig. 1d, a defective SD exhibits a clover shape with three petals. The defective SD within the red circle retains the three-fold rotational symmetry, while the one within the blue circle is asymmetric. Figure 1e presents the dI/dV spectra measured at the center of these two defective SDs. Compared with the spectrum of undoped SD, the most important changes are a suppression of the LHB (together with the insulating gap); and a transfer of the spectral weight from negative bias to positive bias. For both spectra, some in-gap state exists around the Fermi level, further discussed in the Supplementary Material (Supplementary Note 1). The difference between the two spectra is that the increased DOS at positive bias appears as a plateau for the symmetric defect and a peak for the asymmetric defect. In Fig. 1f, the linecut map suggests that Ti-induced perturbation is localized super-lattice. b Temperature-dependent in-plane resistivity ratio ρ (T)/ρ(360K) for samples with different doping levels. c Typical dI/dV spectra of undoped, x = 0.004, and x = 0.01 samples. d A 9 × 9 nm 2 topography for the x = 0.004 sample (V b = −600 mV, I s = 500 pA). Red and blue circles mark a symmetric and an asymmetric Ti-doped SD, respectively. e The dI/dV spectra measured at the center of an undoped SD, a symmetric, and an asymmetric Ti-doped SDs, respectively. f A series of dI/dV spectra across the symmetric Ti-doped SD, along the red curve labeled in d.
inside the doped SD. All neighboring SDs maintain the Mott insulating state.
According to our first-principles simulation, the two types of defect patterns can be well reproduced by selecting the central (site A) or peripheral Ta (site B or C) as the Ti substitution sites (Fig.  2c). Two inequivalent peripheral sites (B or C) result in a similar asymmetric pattern with similar doping energy. The case of multiple Ti dopants within one SD is calculated to be energetically unfavored, and is thus not considered for low-doped samples. We further benchmark the defect dI/dV spectrum with first-principles DOS. The DFT calculation with +U correction (DFT + U) on a 1T-TaS 2 single layer provides a good description of the pristine insulating spectrum (Fig. 2a). Extending this calculation to a supercell containing a Ti-doped SD, we can also reproduce the overall spectrum profile and in particular the spectral weight transfer (Fig. 2a). Constraint by the computational sources available, the calculation is performed with respect to a 2 ffiffiffiffiffi 13 p 2 ffiffiffiffiffi 13 p supercell containing 4 SDs (51 Ta + 1 Ti + 104 S). By inspecting the orbital components, the strongly enhanced spectral peak around 300 meV is confirmed to be the empty dorbitals of Ti atoms (Fig. 2b).
The spectral weight transfer has a direct impact on the spin state. While both Ta and Ti have the stable +4 valence state, Ta 4+ is spin half, and Ti 4+ is spin-zero. One occupied state associated with Ta is converted into an unoccupied state associated with Ti. With the LHB originally carrying the spin degree of freedom, the suppression of the LHB is evident in the defect spectrum. The doped SD loses the vital orphan electron, leaving no room for spin degeneracy, and Ti doping effectively tunes a SD into a spin vacancy. Regardless of the substitution site, the DFT + U spin density plot (Fig. 2d) explicitly shows the absence of local moment at defective SDs. The spin state of 1T-TaS 2 is intricate due to strong geometry fluctuation and possible QSL state 17,18 . In the QSL state, the spin vacancy possibly liberates a free moment in neighboring SDs, which increases the local moment density 19 . We do not observe any signals of the liberated free moment with our local probe tool of STM. We perform magnetic susceptibility measurement in three low-doping samples (Supplementary Note 2). The local moment density slightly increases with the doping level. More experiments with other techniques are required to confirm this point [36][37][38] .
We perform dI/dV mapping measurement around a Ti-doped SD at different energies to better understand the Ti doping effect.  Fig. 3f, this map provides further evidence that Ti substitution converts the SD into a spinless site. The high intensity on Ti-doped SD in Fig. 2c, g is also consistent with the spectral weight transfer induced by the Ti doping.
In addition to the local effect of Ti doping, the most surprising result is that unusual orbital textures appear on both Ti-doped and undoped SDs (Fig. 3). In Fig. 3a, the three petals are more separated on Ti-doped SD comparing with those on undoped SDs. Due to LHB suppression at doped SD, the three-petal feature at −380 mV becomes prominent in the negative-voltage-biased topography. The three petal orbital textures have rarely appeared in the pristine sample of 1T-TaS 2 16 , as shown in Supplementary  Fig. 6. In Fig. 3c, d, other intricate orbital textures can also be observed. The map at 360 mV shows inverted triangular patterns.
The map at 560 mV shows a network of triangles with strong intensity distributed on peripheral Ta atoms. Both featured patterns can be roughly reproduced in Fig. 3g, h. Most related with the CDW phase, the orbital textures are described in more detail (Supplementary Note 4). The dilute Ti doping locally impacts the DOS on the Ti-doped SD and globally induces clover-shaped orbital textures in the CDW phase.  randomly scattered in the field of view (FOV). The domains are generally related to the doping-induced subtle modification of crystal structure and a destabilization of the ffiffiffiffiffi 13 p ffiffiffiffiffi 13 p structure 16,33,39,40 . The multiple domain walls provide metallic and conductive transport channels in the NCCDW phase 34,35,41 . The insulating state is still preserved within large domains. The Ti doping effect in the large insulating domain is similar to that in the x = 0.004 sample.

Mott insulator to metal transition at low-doping level
Within small domains, the dI/dV spectrum appears as a V-shaped metallic one, as shown in Fig. 1c. Small metallic domains provide additional conductive channels besides the domain walls. In Fig. 4b, we show an 8 × 8 nm 2 topography of a metallic domain. A linecut of dI/dV spectrum is measured along the blue line crossing three undoped SDs, with results shown in the waterfall graph in Fig. 4c. Peaks of the V-shaped feature show a periodic variation, with the strongest intensity at the SD center. This lowenergy feature around the Fermi level also shows competition with the CDW peaks at higher energy, with the strongest intensity shifted by half the period. Similar to that in pristine 1T-TaS 2 , the low-energy bands originate from the central Ta orbital, and the CDW bands are from the surrounding Ta orbitals 16 . Besides, some variation exists in different V-shaped metallic spectra. For example, a low-energy in-gap state emerges in the blue spectrum in Fig. 4c. This kind of variation is ubiquitous in metallic domains. In the cuprate superconductor 27,29,30 , the doping triggers a spectral weight transfer from Hubbard bands to low-energy states near the Fermi level, resulting in a V-shaped pseudogap upon doping. For 1T-(Ta 1−x Ti x )S 2 , we observe the appearance of V-shaped (pseudogap) spectra in small domains after Ti doping introduces holes and destroys the electron correlation. The domain wall dI/dV spectrum is also measured and shown in Supplementary Fig. 5, exhibiting a metallic behavior. The metallic transport behavior should be from both the metallic V-shaped spectra within domains and the metallic domain walls, with the former related to the Ti dopinginduced insulator-to-metal transition that we are more interested. Compared with the NCCDW phase in undoped 1T-TaS 2 41 , the lowtemperature NCCDW phase of Ti-doped 1T-TaS 2 (x = 0.01) has some similarities. They all share nano-sized domains and metallic spectra. The detailed spectral differences in two NCCDW phases, however, can not be distinguished by macroscopic transport  measurement. By counting doped SD in both insulating and metallic domains, we think that the doping density does not determine whether the area is insulating or metallic. It is possible that some local material details randomly trigger the appearance of the domain wall network around the critical doping level, and the correlated state in small domains is more prone to be destabilized to a metallic state. For the local Ti doping effect in metallic domains, Fig. 4d shows the dI/dV spectra measured at four Ti-doped SDs in Fig. 4b. The spectral features are different from that of doped SDs in the Mott insulating state. The Ti-induced plateau or peak are both changed to an enhanced peak above the Fermi level. The peak locates at variable energies for different doped SDs. For SD with Ti doping at the B/C site, the peak location in spectrum is relatively closer to the Fermi level. The anomalous enhanced peak at positive bias is still related to the d-orbitals of Ti atom. The in-gap state also exists around the Fermi level, although less apparent against the high peak.
The three-petal orbital texture still exists in dI/dV maps of the metallic domain, even in an energy range several times wider than that in x = 0.004 sample. For the x = 0.004 sample ( Supplementary  Fig. 7), the orbital texture starts to appear around the dip energy between the LHB and the negative-bias CDW peak. For the x = 0.01 sample ( Supplementary Fig. 8), the insulating spectrum changes to a metallic spectrum in the metallic domain. It seems that the negative-bias CDW peak shifts to the lower energy. The energy range of the dip between the LHB and the negative-bias CDW peak is broader, corresponding to the broader range in which the orbital texture exists. The orbital texture with a three-petal feature may be related to a spatially redistributed spectral weight between the central Ta and neighboring Ta atoms. The Mott gap gradually collapses with increasing doping, and the orbital texture's energy range expands, reflecting the competition between electron correlation and the orbital texture structure. When the Ti doping weakens the electron correlation, this competing CDW orbital texture is gradually induced and enhanced. This global change starts to appear even earlier than the V-shaped pseudogap state. In the cuprate superconductor 29 , a checkerboard charge order has emerged in the lightly doped sample. The orbital texture could be a similar emergent phenomenon, indicating a competition and delicate balance between the electron correlation and the orbital texture structure. Another orbital texture has been found to coexist with the Mott insulating state in the MBE-grown single-layer 1T-TaSe 2 42 .

Metal to insulator transition at high doping level
Doping inevitably brings disorder to the system. As the doping amount increases from 0.01 to 0.05, the low-temperature resistivity keeps decreasing (Fig. 1b). However, when the doping rises to 0.08, the low-temperature resistivity shows an upward trend. This reentrance of insulating behavior indicates a competition possibly from the disorder effect under high doping. For the x = 0.08 sample, Fig. 5a displays a 30 × 30 nm 2 topography taken under a bias voltage V b = −600 mV. Ideally, the nominal doping level x = 0.08 corresponds to the doping of one Ti atom per SD. In the negative-voltage-biased topography (Fig. 5a), the three petalshaped SDs are ubiquitous, together with some bright undoped The dI/dV map at the Fermi level. The high intensity area corresponds to the undoped SDs in topography. c The maximum peak map of the dI/dV spectra in the same area. d The averaged dI/dV spectra of different range in the maximum peak map.
SDs scattered in the FOV. The Ti-doped SDs are either symmetric or asymmetric three-petal-shaped. The random sites of Ti doping in SDs result in a periodic potential with disorder-induced fluctuations.
For a detailed analysis of the electronic state, we perform the dI/ dV spectrum measurement on a 200 × 200 grid in this FOV. As shown in Fig. 5b, the zero-bias dI/dV map exhibits a strong spatial fluctuation. The bright triangles with high intensity have a one-toone correspondence with undoped SDs in the topography. We further classify the spectra by seeking the voltage at which the highest peak is located in each spectrum and map the spatialdependent peak voltage. In Fig. 5c, the map of peak voltage shows a random distribution at different SDs but is relatively homogeneous within each SD, which is consistent with that each SD is considered as a super-lattice site in the framework of Mott physics for 1T-TaS 2 .
Guided by the distribution of peak voltage, we calculate the average dI/dV spectra whose peak voltage is within a certain range. Figure 5d shows three representative averaged dI/dV spectra whose peak voltages are within [−100, 100] mV, [100, 210] mV, and [210, 350] mV. For the average spectrum with peak voltage around the Fermi level ([−100, 100] mV), the zero-bias conductance is enhanced, together with a V-shaped feature and two peaks. The spatial distribution (green-yellow spots in Fig. 5c) of these spectra is mainly consistent with the undoped SD positions. The collected information implies that this average spectrum is characteristic of undoped SDs. For the average dI/dV spectra in the range of [100, 210] mV and [210, 350] mV, a strongly enhanced peak exists around 150 and 250 mV, respectively. We attribute the peaks to Ti-induced band features for spectra on doped SDs. For the average spectra with a negative peak voltage, we find in Fig. 5c that some of them correspond to the dark blue spots between SDs. Similar to the periodic variation of pseudogap peaks and CDW peaks, the negative CDW peak can increase to the maximum at spatial positions between SDs.
The x = 0.08 sample has been turned much away from halffilling. For remaining undoped SDs, the central Ta atom contributes an unpaired electron. The DOS of undoped SD maintains the V-shape and two peak features, similar to the V-shaped metallic spectrum in the x = 0.01 sample. The disorderinduced potential fluctuation is likely to be the dominant factor to cause such a consequence. The orphan electrons are trapped and localized at undoped SDs, resulting in an Anderson insulator at this high doping level.

Discussion
The ground state of 1T-TaS 2 is intriguing and puzzling because of the coexistence and competition of different kinds of interactions. The origin of the insulating state of 1T-TaS 2 is still debatable. Recent first-principles calculations and several experiments suggest that a double-layer stacking order leads to a band-gap insulator instead of a Mott insulator [20][21][22][23][24][25][26] . The role of stacking order has been studied in detail in our previous work 32 . In 1T-(Ta 1 −x Ti x )S 2 , the absence of local free moment at doped SD is also slightly against the double-layer band insulator scenario. The large insulating gap is still a correlation-induced Mott insulating gap, the property of the top single-layer 1T-TaS 2 . The possible stackingorder-effect in mosaic domains is also not observed. One intuition is that Ti doping is a strong perturbation within each layer of 1T-TaS 2 and satisfies the doped Hubbard model, which should also be a dominant effect compared with the interlayer stacking perturbation. This point may explain why researchers do not need to consider the stacking-order effect for different doped systems with multiple domains 16 .
In the presence of a strong electron correlation, the triangular SD super-lattice is a possible QSL state due to geometry fluctuation. In the extremely low-doped system, each Ti atom removes one electron in the triangular super-lattice site and perturbs the ground state of 1T-TaS 2 . The Ti doping modifies the local electronic state and alters the spin configuration to a spin vacancy 43 . The spin vacancy possibly liberates a free moment in neighboring sites, and other experimental techniques are required to detect such microscopic spin-channel signals.
When doped away from the half-filling, the Mott insulator generally evolves to a metallic, especially a pseudogap state. With the Mott state residing on the CCDW super-lattice, the insulator to metal transition in 1T-(Ta 1−x Ti x )S 2 starts to appear within small metallic domains, at a critical doping x = 0.01. As the doping level increases to x = 0.08, most SDs include doped Ti atoms and the disorder plays a vital role in the system. The spectrum peak distribution reveals the potential fluctuation, which is crucial to trap and localize unpaired electrons. The reentrance of lowtemperature insulator behavior is compatible with the Anderson insulator phenomenon.
Exotic orders emerge in the electronic evolution of the doped Mott system. For example, the checkerboard charge order appears in the lightly doped cuprate superconductor 29 . The competition between near-degenerate ground states is prominent in the theoretical calculation of the doped Hubbard model [44][45][46] . In 1T-(Ta 1−x Ti x )S 2 , a clover-shaped orbital texture starts to appear in the low-doping-level sample and exists within a broader energy range as the doping increases. The energy-dependent charge order in STM dI/dV maps relates the charge order with corresponding electronic bands. The energy-dependent orbital texture reveals both the charge-order-band relation and the orbital distribution pattern of such order. Our results present the competition between the much complex charge order behavior and the electronic correlation.
In summary, we performed detailed STM measurements to study the electronic evolution of 1T-(Ta 1−x Ti x )S 2 at a wide range of doping level. The precise dopant positioning enables the detection of Ti-induced local disturbance. Combining with the first-principles calculation, we demonstrate a slight Ti doping (x < 0.01) locally modifies the DOS of doped SD and creates a spin vacancy on site. When the doping x > 0.01, small metallic domains and domain walls lead to the insulator-metal transition. In the evolution process, some orbital textures emerge globally and compete with the suppressed electronic correlation. As the doping increases, the disorder drives the system to an Anderson insulator. The comprehensive study of 1T-(Ta 1−x Ti x )S 2 deepens our understanding of the strongly correlated physics and deserves more theoretical exploration.

Sample preparation
The high-quality 1T-(Ta 1−x Ti x )S 2 single-crystal samples were grown by the chemical vapor transport (CVT) method with iodine as the transport agent. Ta (99.99%, Aladdin), Ti (99.99%, Aladdin), and S (99.99%, Aladdin) powders with a nominal mole ratio of 1 − x:x:2 were weighted, mixed with 0.2 g of I 2 , and placed into silicon quartz tubes. The quartz tubes were sealed under high vacuum and heated in the two-zone furnace for 10 days, in which the temperature of source and growth zones were fixed at 900 and 800°C. Then the tubes were removed and quenched in an ice water mixture, after which single crystals were collected for measurements. The doping has been uniformly and precisely controlled in high-quality samples. The average stoichiometry was determined by examination of multiple points using X-ray energy dispersive spectroscopy (EDS). Since the minimum resolution of EDS is about 0.01 for Ti content, the doping content smaller than 0.01 is calibrated by the relative phase transition temperatures.

STM measurements
The STM measurements were performed in two different commercial STM systems showing no obvious difference. The samples were gradually cooled to around 80 K, in situ cleaved and transferred to the STM head for an immediate measurement. Electrochemically etched tungsten tips were used after calibration on single crystalline with an Au (111) surface. Data were collected at either 4.5 K or 77 K, both of which are far below the transition temperature. Other than a small thermal broadening, the data taken at 77 K is very similar to that taken at 4.5 K, and does not affect the physics we discussed in this work. The dI/dV spectra were taken in a standard lock-in technique with a frequency of 1213.7 Hz and an amplitude of 10 mV 31 .

First-principles calculation
The first-principles calculations were performed using the Vienna ab initio Simulation Package [47][48][49][50] , with the projector augmented wave method 51 and the Perdew-Burke-Ernzerhof exchange-correlation functional 52 . The +U correction was employed to capture the Coulomb interaction of Ta 5d orbitals on the Hartree-Fock level, following the simplified (rotational invariant) approach introduced by Dudarev 53 . We applied an effective U = 2.27 eV as previously derived from the linear-response calculation 54 . All our simulation cells contain a single layer of 4 SDs and with a 15 Å vacuum to prevent interlayer coupling between different layers. The plane-wave cutoff was set to 300 eV. For integration over the Brillouin zone, a Γ-centered 3 × 3 × 1 k-point mesh for the computation of the structure relaxations and the self-consistent electronic densities, and a denser, 7 × 7 × 1 mesh for the computation of the densities of states. Convergence is reached if the consecutive energy difference is within 10 −5 eV for electronic iterations and the forces are less than 0.002 eV Å −1 . We relaxed the structures after substituting Ti for Ta. Substitution of the central Ta site (site1) is found to have the lowest energy. The substitution energies of the two peripheral Ta sites are 193 meV per Ti (site 2) and 178 meV per Ti (site 3) higher.

DATA AVAILABILITY
The source data and related supporting information are available from the corresponding author upon request.