Abstract
Twodimensional materials can be strongly influenced by their surroundings. A dielectric environment screens and reduces the Coulomb interaction between electrons in the twodimensional material. Since in Mott materials the Coulomb interaction is responsible for the insulating state, manipulating the dielectric screening provides direct control over Mottness. Our manybody calculations reveal the spectroscopic fingerprints of such Coulomb engineering: we demonstrate eVscale changes to the position of the Hubbard bands and show a Coulomb engineered insulatortometal transition. Based on our proofofprinciple calculations, we discuss the (feasible) conditions under which our scenario of Coulomb engineering of Mott materials can be realized experimentally.
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Introduction
Atomically thin twodimensional (2d) materials can be influenced by their environment. This idea is utilized in the Coulomb engineering of semiconductors^{1,2,3,4}, where the dielectric properties of the environment are used to manipulate the optical and electronic properties such as the carrier mobility^{1,5}, the band gap^{2,3,6}, quantum Hall phenomena^{7} and excitons^{3,8,9,10,11,12,13,14}. This tunability is driven by changes in the screening of the Coulomb interaction. Coulomb engineering is noninvasive in the sense that the semiconducting layer is not changed, only its environment. As an application, inhomogeneous dielectric environments can be used to produce semiconductor heterojunctions in homogeneous materials^{2,15}. In traditional semiconducting 2d materials, the screening causes rigid band shifts^{16} and the induced change in the band gap is much smaller than the band gap itself^{3,6} (usually 10–30% of the gap). In moiré correlated electron systems such as twisted bilayer graphene^{17,18,19}, environmental screening turned out to modify superconducting critical temperatures as well as transport gaps. In metallic systems, on the other hand, intrinsic screening can be so large that environmental screening becomes ineffective^{20}. Clearly, a detailed analysis of both internal and external screening is required to determine the feasibility of Coulomb engineering for specific applications.
Here, we simulate the Coulomb engineering of Mott insulators and elucidate its spectroscopic fingerprints, which are experimentally accessible via angular resolved photoemission spectroscopy (ARPES) and scanneling tunneling spectroscopy (STS) experiments. Since correlations induced by the Coulomb interaction open the gap in the electronic excitation spectrum in Mott insulators^{21}, enviromental screening of the Coulomb interaction in this case holds the potential of not only influencing the band gap to a much larger extent than in semiconductors, but even of closing the gap completely.
Results
Modeling Coulomb engineering
Mott insulators are materials that should be metallic according to band theory, but where the electronelectron Coulomb interaction is sufficiently strong to localize the electrons and make the material insulating^{21}. The ratio of the interaction strength and bandwidth determines whether the potential or the kinetic energy dominates, making the system either insulating or conducting. Coulomb engineering works by changing the interaction strength via the dielectric environment, pushing the system towards the conducting phase.
This change from insulator to conductor also dramatically changes the intrinsic dielectric properties of the material itself. In a good conductor, the internal screening is very efficient and because of this the dielectric environment is less important. In the Mott insulator, on the other hand, the electrons are not mobile, and internal screening is inefficient. The quantum manybody physics of correlated electrons is the root cause of the reduced internal screening in Mott systems. In twodimensional systems, such as Mottinsulating ultrathin films, the outofplane environment offers an additional pathway to manipulate the screening. This environmental part of Coulomb engineering can be understood on the level of classical electrostatics when the environment is a simple dielectric with a dielectric constant ϵ_{E}. Considering a monolayer system, some of the field lines connecting charges in the monolayer leave the material and traverse the surrounding dielectric environment, as illustrated in Fig. 1. Thereby, the Coulomb interaction is screened in a peculiar nonlocal manner: the Coulomb interaction (in the momentum representation) of a monolayer of height h encapsulated in a uniform dielectric environment is given^{22,23} by
Here e is the electron charge, q is the absolute value of the momentum transfer, ε_{M} and ε_{E} are the dielectric constants of the material and the environment respectively and x = (1 − ε_{E}/ε_{M})/(1 + ε_{E}/ε_{M}). Since 2πe^{2}/εq is the usual Coulomb interaction in a twodimensional material^{24}, the second part of the formula essentially describes the modification of the dielectric function due to the embedding of the monolayer into the dielectric environment. V is the effective interaction between the lowenergy electrons in the monolayer, so the dielectric constant ε_{M} describes all screening in the monolayer except for the screening by the lowenergy electrons themselves^{25}.
If the monolayer is surrounded by two different materials, a top layer with x_{t} and a bottom layer with x_{b}, where x_{i} = (1 − ε_{i}/ε_{M})/(1 + ε_{i}/ε_{M}) as before, then the effective Coulomb interaction is
An important example is a monolayer with vacuum (ϵ_{vacuum} = 1) on one side. Both onesided^{3} and twosided^{16} setups have been used in experiments and the dielectric model used here accurately describes the experimentally observed Coulomb engineering of conventional semiconductors^{16}.
For demonstrating the possibility and spectroscopic fingerprints of Coulomb engineering in correlated ultrathin films, we choose specific minimal model parameters which are inspired by transition metal dichalcogenides^{26}. We use a triangular lattice with a lattice constant of a ≈ 3.37 Å a monolayer of height h = a and a background dielectric constant ε_{M} = 5 for the monolayer. Except when otherwise noted, all results correspond to twosided screening, Eq. (1).
In addition to the Coulomb interaction, the kinetic energy is the second ingredient required to describe the Mott transition. The simplest model involves a single halffilled band of electrons with hopping between neighboring sites. Combining the potential and kinetic energy results in the extended Hubbard model with Hamiltonian
Here \({c}_{i\sigma }^{{\dagger} },{c}_{i\sigma }^{}\) are the creation and annihilation operators for an electron with spin σ ∈ {↑, ↓} on site i and \({n}_{i}={\sum }_{\sigma }{c}_{i\sigma }^{{\dagger} }{c}_{i\sigma }^{}\) is the electron density on site i. The Coulomb interaction V(r) is the Fourier transform of Eq. (1) and t is the hopping amplitude between neighboring sites \(\left\langle ij\right\rangle\). We use t = 0.3 eV, again inspired by transition metal dichalcogenides.
Mott materials, featuring strong correlations, require an advanced manybody treatment^{27,28}. Here, to understand the Coulomb engineering, we need a consistent treatment of the internal screening in the monolayer, across the insulatormetal transition where the dielectric properties change dramatically. To this end, we use stateoftheart diagrammatic extensions of dynamical meanfield theory^{29}, described below. The results in the main text have been obtained with the Dual Boson method^{30,31} and were crosschecked with GW+DMFT^{29,32,33} calculations for a related model in the Appendix.
Spectral fingerprints of screening
Figure 2 shows how the spectral function evolves from ε_{E} = 1 (the freestanding monolayer) to ε_{E} = 10 (encapsulation in bulk Si would correspond to ε_{E} = 12). The colored curve on the right of each graph shows the local density of states, which is the integral of the spectral function over the Brillouin zone. The screening by the environment leads to substantial changes in the spectral function. Most dramatically, the system changes from an insulator to a metal. A comparison of ε_{E} = 2.0 and ε_{E} = 5.0 shows that the gap at the Fermi level disappears and a quasiparticle band emerges at the Fermi level. This is clearly visible both in the spectral function and the local density of states. Experimentally, the latter can be investigated using scanneling tunneling spectroscopy (STS). Below the Fermi level, the momentumresolved spectral function can be investigated with angular resolved photoemission spectroscopy (ARPES).
Spectral fingerprints of the Coulomb engineering are visible even without crossing the insulatormetal transition: a comparison of ε_{E} = 1 and ε_{E} = 2 shows that the size of the gap within the insulating phase is reduced by the screening. On the metallic side of the transition, the Hubbard side bands are still visible in the spectra and we can trace how their energy changes due to Coulomb engineering. To get a better view, Fig. 3a shows the Energy Distribution Curves (EDCs, the crosssections of the spectral function A(k, E) at fixed momentum) at k = M. Both the lower and upper Hubbard band move towards the Fermi level as ε_{E} increases, and for ε_{E} ≥ 5 a quasiparticle peak close to the Fermi level is visible. In addition to the changing position of the peaks, the figure also clearly shows the spectral weight transfer to the quasiparticle peak at the expense of the Hubbard bands.
The Coulomb engineering of Mott insulators leads to eVsized changes in peak positions, as shown in Fig. 3b, c. There, the peak position of the upper and lower Hubbard bands is shown. This energy has been extracted from the EDCs at k = Γ and k = K, corresponding to the bottom and top of the band. The environmental screening changes the position of the Hubbard bands by as much as 0.8 eV. The largest changes occur for small ε_{E}.
Inside the metallic phase at large ε_{E}, the effects of environmental screening are also visible in the quasiparticle band, close to the Fermi energy. The spectral weight in this band increases, as was visible in the local spectral function. In addition, the effective electron mass is reduced from m^{*}/m = 3.6 at ε_{E} = 5 to m^{*}/m = 2.0 at ε_{E} = 20. Screening has an effect on both the dynamical and static contributions to the effective mass^{34,35}. The overall reduction of m^{*} with ε_{E} shows that the reduction of the dynamical contribution \({(1\frac{\partial {{\mathrm{Re}}}\,\Sigma ({{{\bf{k}}}},\omega )}{\partial \omega })}^{1}\) due to the reduced Coulomb interaction is more important than the increase in the static contribution (see “Methods”).
Role of temperature
The spectral functions of Fig. 2 correspond to an electronic temperature of T = 0.1 eV (1160 K). This temperature is obviously rather high for experiments, however we should stress that we are only simulating the electronic problem, where the energy scale is set by the bandwidth 9t = 2.7 eV. The electronic temperature is small compared to this bandwidth. On the other hand, the typical energy scale for magnetic phenomena is much lower. Our calculations become substantially more difficult and expensive for lower temperatures, limiting the range where we can perform meaningful calculations. For ε_{E} = 20, the screening is already quite effective, making the system less correlated, and this reduces the computational cost and allows us to reach lower temperatures. The spectra at reduced temperatures of T = 0.05 eV (580 K) and T = 0.025 eV (290 K, room temperature) are shown in Fig. 4. The overall shape of the spectral function does not change substantially in this temperature range, the main difference is that spectral features become sharper at low temperature. The bandwidth sets the scale for the temperatures, for a system like magicangle twisted bilayer graphene, where the bandwidth is two orders of magnitude smaller, our calculations would correspond to meV temperatures (10 K) and all gaps and changes in gaps are also on the meVscale.
Substrate
So far, we have considered the setup of Fig. 1, a monolayer surrounded on both sides by a dielectric environment. Another experimentally relevant scenario is a Mott monolayer on a dielectric substrate with vacuum on the other side. In that case, the screened Coulomb interaction is given by Eq. (2). Figure 5 shows the resulting density of states (filled curves) in comparison with twosided screening (lines). The starting point ε_{E} = 1 is the same, it corresponds to vacuum on both sides. As ε_{E} increases, the onesided screening is less efficient and the resulting spectra retain more Mottish features (smaller quasiparticle peak, Hubbard bands). Still, the overall physics remains the same and Coulomb engineering across the insulatormetal transition is possible. This shows that experiments can use substrate screening instead of both sided screening when it is more convenient.
Not just about U
Internal screening in (quasi)twodimensional materials is frequently incomplete, leading to substantial intersite Coulomb interactions^{36,37}, as shown in Table 1 for the present model. Environmental screening changes the relative magnitudes of the onsite and intersite Coulomb matrix elements. The influence of intersite interactions on the electronic properties is determined by the spatial extent of charge excitations^{38}: only shortranged (onsite and nearestneighbor) interactions are relevant sufficiently deep in the insulating phase, whereas larger interaction lengths are important for the delocalized electrons in a metal. The crux here is that Coulomb engineering pushes the system across the boundary between the two distinct screening regimes, insulator and metal. Thus, screening in both phases has to be captured appropriately and consistently in a single theoretical description, necessitating advanced theoretical techniques such as the diagrammatic extensions of DMFT used here.
Discussion
After this computational demonstration of Coulomb engineering and its spectral fingerprints, we analyse qualitatively how the relevant parameters and scales enter, to guide experimental realizations. The effective height h of the monolayer determines the length and momentum scale where the environment becomes important. For the limit of large qh ≫ 1, Equation (1) reduces to a twodimensional Coulomb interaction with dielectric constant ε_{M}. In other words, on short length scales the interaction is exclusively determined by the monolayer. The relevant dimensionless parameter is h/a, the ratio between the height of the monolayer and the inplane lattice constant. For the Coulomb engineering to be efficient, h/a should not be large. This is confirmed by GW+DMFT calculations, as described in the “Methods” section. We should add that throughout this paper, we assume that the charge density of the electronic orbitals is homogeneous on the scale a of the unit cell. If, instead, the correlated Wannier orbitals have a radius r_{WF} ≪ a, then more detailed modeling of the screening on the atomic scale is necessary.
The control parameter in our study is ε_{E}, the dielectric constant of the environment. This means that the environment should not influence the monolayer in other ways such as hybridization^{39} or (pseudo)doping^{40}, or the bandwidth widening that occurs in, e.g., strontium iridate superlattices^{41}. A spacer layer between the Mott monolayer and the substrate can help reduce the coupling^{39}. This strategy has the disadvantage that it increases the distance between the substrate and the monolayer, making the screening somewhat less efficient. We should also note that our model of the environment assumes that ϵ_{E} is independent of frequency and momentum. This is a reasonable approximation for dielectric substrate with a wide band gap, where dynamical screening effects only come into play at higher energies.
The internal dielectric constant ε_{M} of the monolayer sets the scale for ε_{E}, as the expression for x shows. Typical values of ε_{E} range from 1 for vacuum, via 3.9 for SiO_{2}^{42} to 12 for Si^{43}. If ε_{M} is large, i.e., if screening inside the monolayer is already very effective, then much larger changes in the dielectric environment are needed to change V(q). For the possibility to turn an insulator into a metal, the monolayer material itself should be an insulator and not too far way from the metallic state. If the goal is only to change the size of the Mott gap then the original material can be deeper inside the Mott phase.
The Hubbard interaction U between electrons on the same site can serve as a useful indicator for the screening possibilities, when we compare a freestanding monolayer (ε_{E} = 1) and the corresponding bulk material (ε_{E} = ε_{M}). For the (supposed) Mott insulator CrI_{3}, the Hubbard interaction is reduced from U = 2.9 eV for the monolayer to U = 2.0 eV for the bulk^{44}. For the strongly correlated metal SrVO_{3} there is a reduction from U = 3.7–3.95 eV in the monolayer to U = 3.3 eV in the bulk^{45}. These substantial reductions of the Coulomb interaction are achieved by replacing vacuum with other Mott layers, meaning that a more effective dielectric environment will reduce the Coulomb interaction even more.
Meanwhile, the interlayer spacing and inplane lattice constants in CrI_{3} are both ~7 Å so that h/a ≈ 1^{46}. Layered cuprate materials, another material family known for strong correlation physics, is worse in this respect, with typical values^{47} of a ≈ 4 Å and h ≈ 12 Å, so h/a = 4. While it is possible to exfoliate cuprate single layers^{48}, their thickness compared to the lattice constant renders Coulomb engineering more challenging. For transition metal dichalcogenides, h/a < 2 can be achieved^{26}, which is better than the cuprates but not as good as CrI_{3}.
It is even possible to reach h < a when a is the size of an emergent superlattice which can be much larger than the interatomic distance of the underlying lattice. One example is twisted bilayer graphene^{49,50,51} with a moiré superlattice hundreds of times larger than the graphene unit cell so that h/a can become genuinely small. We should note that ε_{M} is quite large in twisted bilayer graphene^{52} and that the insulating state is not of the idealized MottHubbard type but likely involves spontaneous symmetry breaking in the valley and spin degrees of freedom^{53,54,55,56}. Correspondingly, the interpretation of first possible experimental reports^{17,18,19} of Coulomb engineering related effects in twisted bilayer graphene is intricate^{52,57}.
A clearer situation appears for certain charge density wave phases in 2d materials: 1TTaS_{2}^{58,59,60}, NbSe_{2} and TaSe_{2} have socalled commensurate charge density waves with a starofDavid lattice reconstruction, the emergent scale is \(\sqrt{13}\) times the original lattice constant. The starofDavid reconstructed TaSe_{2} monolayer has a Mott gap of ~0.1 eV^{61}, and lends itself as a natural candidate for the exploration of Coulomb engineering of Mott materials.
We have demonstrated that the dielectric environment can be used to control Mott insulating layered materials and that Coulomb engineering across the insulatormetal transition is possible. Our calculations show the spectroscopic fingerprints of Coulomb engineering, namely eVscale movement of the Hubbard bands and the appearance of a quasiparticle band as the system turns metallic. Based on our modeling, we can identify necessary conditions for Coulomb engineering. To be effective, the dielectric environment should be close to the monolayer, while also avoiding other monolayerenvironment couplings like pseudodoping and hybridization. Furthermore, the monolayer itself should not screen too strongly. A good indicator is if the Hubbard parameter U is much larger for the monolayer than for the corresponding bulk material. Our results open a perspective for the fabrication of heterostructures by the application of dielectric covering on parts of a Mott monolayer. In this way, the environment can create local phase transitions in correlated materials: metallic paths in an otherwise insulating layer or the control of unconventional superconducting phases via the reduction of the effective interaction. The sharpness of these dielectrically controlled heterostructures is controlled by the electronic Green’s function^{2}, so in Mott insulators where the electrons are localized, heterostructures can be atomically sharp.
Methods
Dual Boson
The system is studied in the grand canonical ensemble with density fixed to halffilling. For our computations, we use the Dual Boson method^{30} with the implementation described in ref. ^{31}. It consists of a selfconsistency cycle to determine the optimal Anderson Impurity Model (AIM). In this cycle, we include vertex corrections to the susceptibility in the ladder approach to ensure that the susceptibility satisfies charge conservation at small momenta. This is important when dealing with longranged V(q). When selfconsistency has been reached, we calculate the spatial selfenergy \(\tilde{\Sigma }(k,\nu )\) in the last iteration using the secondorder diagram of Fig. 2b from ref. ^{31}, which together with the AIM selfenergy is used to determine the Green’s function G and finally the spectral function A(k, E). The diagrammatic calculations are done on a 64 × 64 periodic lattice, sufficient to rule out electronic finitesize effects, especially since the electrons are rather localized in the regime studied here. We study the nonmagnetic insulatormetal transition, we do not allow for magnetic order in our calculations. The Anderson Impurity Model is solved using the ALPS/w2dynamics (for GW+DMFT, see below) CTHYB solvers^{62,63,64,65}. All our calculations are done in Matsubara space, the analytical continuation to real energies is done to determine A(k, E). For this, we use OmegaMaxEnt^{66}. To verify the reliability, we have also performed stochastic continuation using Spektra (https://spektra.app)^{67,68} for a representative subset of our calculations.
Effective mass
In the metallic phase, the effective electronic mass is a useful indicator of the strength of correlations. The effective mass renormalization consists of two parts^{34,35}, a dynamical contribution \({(1\frac{\partial {{\mathrm{Re}}}\,\Sigma ({{{\bf{k}}}},\omega )}{\partial \omega })}^{1}\) and a static contribution. The dynamical contribution decreases as the Coulomb interaction is reduced by screening, but the spatial contribution actually increases due to Focklike band widening. The latter effect turns out to be smaller, leading to an overall decrease of the effective mass upon increased screening. The values of the effective mass given in the Results section were determined on the ΓM high symmetry line, at the Fermi surface crossing. In our calculations, the effective mass depends only weakly on momentum.
G W+DMFT
To illustrate that our findings are representative of Mott monolayers in general, we have also performed GW+DMFT^{32} calculations for a square lattice system, based on the implementation of ref. ^{33,35}. The nonlocal contributions of the selfenergy, calculated in a singleshot GW calculation, are added to the local DMFT selfenergy^{65}.
The GW+DMFT treatment of the interaction in this model was restricted to local (U) and nearestneighbor (V_{nn}) terms. To make a comparison with the dielectric model of Eq. (1), we map the (ε_{E}, ε_{M}, h/a) model onto the twoparameter effective interaction \(\hat{V}({{{\bf{q}}}})=U+2{V}_{{{\mathrm{nn}}}}[\cos ({q}_{x})+\cos ({q}_{y})]\) via two constraints:
The first constraint ensures that the overall magnitude of the interaction is the same in the effective model, which is important for the Mott physics, the second constraint that the potential energy cost of checkerboard chargedensity waves is equal for both interactions. The latter is important to ensure that the effective model is not biased towards the CDW phase of the square lattice extended Hubbard model^{69}. For the material parameters, we used a = 2.27 Å, ε_{M} = 2.9 and both h/a = 0.6 and h/a = 1. The hopping was set to t = 0.5 eV, the temperature to T = 0.025 eV (290 K, room temperature).
The numerical advantage of the GW+DMFT approach compared to the Dual Boson method is that it does not involve socalled vertex corrections, which makes it substantially cheaper and allows us to explore larger parts of phase space even at low temperatures. Here, we use it to study the effect of h/a. At the same time, the neglect of vertex corrections is an approximation, which is potentially problematic for collective excitations on longer length scales. This is why we cut off the interaction after the nearestneighbors in this model. The square lattice at halffilling has particlehole symmetry, for the present investigation this, for example, means that the lower and upper Hubbard band should appear at the same (absolute) energy, which is used to estimate the error bars in Fig. 6, coming from uncertainty in the analytical continuation.
The results are shown in Figs. 6, 7 and 8. As in the Dual Boson results, the ε_{E} induced insulatormetal transition is clearly visible. Here, this transition occurs around ε_{E} = 5 for h/a = 0.6 and ε_{E} = 10 for h/a = 1. As ε_{E} increases, the Hubbard bands move towards the Fermi surface.
Data availability
The datasets generated during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
The authors thank P. Hofmann for useful discussions and K. Ghanem for the help with the stochastic continuation. The authors acknowledge the NorthGerman Supercomputing Alliance (HLRN) for providing computing resources via project number hbp00047 that have contributed to the research results reported in this paper. D.S. and J.M.T. acknowledge support by the Austrian Science Fund (FWF) through project ‘LinReTraCe’ P 30213N36. Some calculations were performed on the Vienna Scientific Cluster (VSC). G.S. acknowledges financial support from the DFG through the WürzburgDresden Cluster of Excellence on Complexity and Topology in Quantum Matter—ct.qmat (EXC 2147, projectid 390858490). T.O.W. acknowledges support from the DFG through QUAST (FOR 5249, No. 449872909) and via the Cluster of Excellence “CUI: Advanced Imaging of Matter” (EXC 2056, No. 390715994).
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E.G.C.P.v.L. performed the Dual Boson calculations, M.S. performed the analytical continuations, D.S. and J.M.T. performed the GW+DMFT calculations. All authors contributed to the interpretation of the results and to the writing of the manuscript.
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van Loon, E.G.C.P., Schüler, M., Springer, D. et al. Coulomb engineering of twodimensional Mott materials. npj 2D Mater Appl 7, 47 (2023). https://doi.org/10.1038/s4169902300408x
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DOI: https://doi.org/10.1038/s4169902300408x
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