Excitonic instability in transition metal dichalcogenides

When transition-metal dichalcogenide monolayers lack inversion symmetry, their low-energy single particle spectrum can described by tilted massive Dirac Hamiltonians. The so-called Janus materials fall into that category. Inversion symmetry can also be broken by the application of out-of-plane electric fields, or by the mere presence of a substrate. Here we explore the properties of excitons in TMDC monolayers lacking inversion symmetry. We find that exciton binding energies can be larger than the electronic band gap, making such materials promising candidates to host the elusive exciton insulator phase. We also investigate the excitonic contribution to their optical conductivity and discuss the associated optical selection rules.


Introduction
Janus transition metal dichalcogenide (TMD) monolayers are a new type of two-dimensional materials, recently synthesized [1,2] in the form of MoSSe. In this material, one atomic layer of Mo is encapsulated by two different chalcogen layers, namely S and Se. This creates an asymmetry in the direction perpendicular to the plane of the structure, resulting in a dipole moment pointing from the Se to the S layer [3,4]. These first two independent experiments [1,2] triggered a large wave of theoretical studies [5] into the electronic band structure [2], anisotropic elasticity and transport [6], as well as piezoelectric [7], pyroelectric [8], spin [9], and photocarrier [10] properties of Janus structures. Additionally, several potential applications have been studied, such as in water splitting [11], gas sensing [12], and photovoltaics [13].
The band structure and the Berry curvature dipole [14,15,16] of Janus TMD monolayers are well described by a tilted massive Dirac model [17,18,19]. For instance, this two-band effective model captures the pronounced peak in the dipole component near the Fermi level of T − WSTe, seen in DFT calculations [17]. This feature is a consequence of a reduced band gap, which leads to a large Berry curvature [17]. Tilted Dirac cones are also seen in 8 − P mmm borophene [20,21,22], where the low-energy regime near the Dirac points can be accurately described via an effective anisotropic tilted Hamiltonian [20,21,23,24].
Additionally, the importance of spin-orbit coupling in layered organic salts has also been studied via a Hamiltonian with tilted band structure [31,32]. Explicit inversion symmetry breaking and anisotropic terms are also present in other families of materials, such as Weyl semimetals [33] with broken tilt inversion symmetry, where second harmonic generation has been recently studied [34], and type-II semi-Dirac semimetals [35,36,37], where it has been shown that linear and nonlinear anomalous Hall effects can be manipulated via circularly polarized light [38].
A defining feature of massive tilted Dirac cones is the tunability of their electronic band gap, illustrated in Fig. (1). By tuning a single parameter, it is possible to go from a direct gap semiconductor to a metal, passing through an indirect gap semiconductor and a semi-metal. Ab initio calculations [39] suggest that such tunability can be achieved in real materials by the application of an out-of-plane electric field. Thus, materials that host massive tilted Dirac cones are especially attractive as platforms on which new electronic phases can be found. One such phase is the excitonic insulator, first predicted qualitatively by Mott in 1961 [40].
The concept has been subsequently refined and put on firmer grounds by several authors [41,42,43,44]. In brief, when the binding energy of excitons surpasses the value of the electronic band gap, the system becomes unstable against a proliferation of excitons [45]. The true ground state of such a system is an "exciton condensate", akin in many ways to the Cooper pair condensate found in superconductors [43]. This analysis apply equally to small gap semiconductors and semimetals, provided the overlap between conduction and valence bands is small, such that screening of the Coulomb interaction between electrons and holes is negligible. Although predicted more than 60 years ago, the excitonic insulator has eluded conclusive experimental observation until very recently. One of the main difficulties is that the excitonic instability is frequently accompanied by structural instabilities, associated with the softening of phonon modes, which are hard to disentangle from the former.
Here we look into the properties of excitons in 2D massive tilted Dirac electrons. We show that the band gap can be tuned while the exciton binding energy remains constant, moving the system towards an excitonic instability. Importantly, as the magnitude of the gap decreases, the maximum of the valence band and the minimum of the conduction band shift in opposite dirrections in reciprocal space, making the gap indirect. This guarantees that the dielectric function of the material remains finite even as the gap approaches zero, and the exciton binding energies are not strongly affected by the smallness of the gap [41]. It is also noteworthy that the excitonic contribution to the conductivity is insensitive to the tilting parameter; thus, at least in principle, one can expect a sudden change in transport properties as the excitonic instability is reached by tuning of the tilting parameter.
This paper is structured as follows. In Sec. 2, we discuss the tilted Dirac Hamiltonian [18], illustrating the features induced by the tilt parameter on the single particle electronic eigenstates and band structure. In Sec. 3, we introduce the Bethe-Salpeter equation, briefly discussing the electrostatic potential coupling different bands. The screening length of the material is also introduced, and the influence of the tilt parameter on the excitonic states is analyzed. Finally, the optical selection rules and oscillator strengths are discussed, and the excitonic optical conductivity is computed.

Tilted Dirac Hamiltonian
The effective two-band tilted Dirac Hamiltonian has been shown to capture the essential features of both the low-energy band structure and the Berry curvature dipole moment of Janus TMD monolayers, such as WSTe [17]. This Hamiltonian includes an anisotropic term which preserves time-reversal symmetry but explicitly breaks inversion symmetry, tilting the band structure in a specific direction, here considered to be the x-axis. The Hamiltonian can therefore be written as [16,17,18] where (k x , k y ) are the wave vectors, k 2 = k 2 x + k 2 y , (σ x , σ y , σ z ) are the Pauli matrices, σ 0 is the 2 × 2 identity matrix, η = ±1 is a valley-like index, m is the gap, and t tilts the Hamiltonian in the x direction. In Eq. (1), the α parameter is introduced to regulate topological properties as k → ∞ [46]. For clarity, we also write this Hamiltonian in matrix form, where it is given bŷ The band dispersion of this model is given by with λ = ±1 the conduction/valence band index.
Choosing η = −1 as in [17,18], the (non-normalized) eigenvectors are given by where θ = arctan (k y /k x ). By inspection of both Eq. (3) and Eq. (4) it is immediately clear that the tilt parameter t influences neither the difference between the two bands (i.e., E + (k) − E − (k)) nor the eigenstates of the Hamiltonian. Its only effect is to produce a tilt of the dispersion relation along the xaxis. In Fig. 1 we show the dispersion relation for a few representative values of t. The other parameters have been chosen as v = 1 eVÅ, α = 1 eVÅ 2 and m = 0.2 eV [18]. For t = 0 (top left) there is a gap at k = 0, which is continuously suppressed as t → v. Notice also that, for any 0 < t < v, the gap becomes indirect. For t/v ≥ 1 (bottom panels) the model describes a semimetal with zero (t = v) or small t/v 1 carrier density. In Fig. (2), we plot the Fermi rings at a Fermi energy of E F = m/2 for various values of t/v. The top of the valence band crosses the E = m 2 plane at roughly t/v ≈ 1.245, as can be seen from the sudden appearance of the Fermi ring regarding this band. The anisotropy introduced by the tilt of the bands along the k x axis is also clear when compared with the full symmetry along the k y axis.

Excitons in tilted Dirac materials
We now look at the excitons of the massive tilted Dirac fermion model. Our approach is to solve the Bethe-Salpeter equation (BSE). As neither the difference between the two bands nor the eigenvectors depend on t, we do not expect this parameter to play a part in the excitonic properties of the system.

Solving the Bethe-Salpeter equation
The Bethe-Salpeter equation can be written in momentum space as [47,48,49,50] where ψ c;v (k) is the excitonic wave function that we wish to obtain, u v/c k and E v/c k are the single particle electronic wave functions and energies, respectively, and V (k) is an electrostatic potential coupling Taking the thermodynamic limit and rearranging the complex exponentials, the Bethe-Salpeter equation is now given by The process of solving the Bethe-Salpeter equation in monolayer and multilayer systems has been thoroughly discussed recently [51,52], so we will not go into the details of the calculations. We consider the electrostatic interaction to be given by the Rytova-Keldysh potential [53,54], obtained by solving the Poisson equation for a charge embedded in a thin film of vanishing thickness. In momentum space, this potential is given by where α = 1/137 is the fine-structure constant and the mean dielectric constant of the medium above/below the monolayer, here considered to be either hexagonal boron-bitride (hBN) or quartz. The parameter r 0 corresponds to an in-plane screening length related to the 2D polarizability of the material. It can be calculated from the single particle Hamiltonian of the system [55], for t < v, as with m 0 the free electron mass, although ab initio calculations might be necessary for accurate computation of r 0 depending on the material [56].

Influence of the tilt parameter
When computing the momentum matrix element present in Eq. (9), a dependence on t is only present on its x component as (10) However, the diagonal terms proportional to t are canceled by the orthogonality relation of the two eigenvectors when u c k |P x | u v k is computed. Explicitly, this term reads with a ± and b ± the normalized spinor components of the eigenvectors in Eq. (4), written generically as This cancellation, together with the fact that neither the difference of energy between the two bands nor the eigenvectors themselves depend on t, implies that the t parameter will not change the results obtained from solving the Bethe-Salpeter equation. As such, the obtained excitonic states will be independent of the tilt parameter t.
First considering the TMD encapsulated in hBN, the energies of first and second s-series states are, respectively, E 1s = 134 meV and E 2s = 176 meV. When t ≈ 1.1v, the top of the valence band crosses the excitonic level, as shown in Fig. (3-a). This marks the onset of the instability against the spontaneous formation of excitons. Although the system is a semimetal for this value of the ratio t/v, the carrier density is still very small, indicating that screening is still weak and the long-range character of the electron-hole interaction should be preserved.
By changing the material by which the TMD is encapsulated, it is possible to tune the exciton binding energy. For instance, by replacing hBN by quartz (whose relative dielectric constant is 3.8 [57]), the energy of the 1s exciton is E 1s = 56.6 meV. In this case, the onset of the excitonic instability happens for t ≈ 0.83v, well into the semiconducting regime, as shown in Fig. (3-b). The fact that, for all t = 0, the gap is indirect, guarantees that the renormalization of the exciton binding energy and the dielectric function are small even for an arbitrarily small gap [41].
In Fig. (4) we plot the absolute value squared of the first two s-series excitonic wave functions where the TMD has been encapsulated in quartz. These plots are centered at k = 0 for a square region of side 20 Å −1 .

Excitonic Conductivity
In the dipole approximation, and considering normal incidence, the optical conductivity is given by [48] σ (1) where the sum over n represents the sum over excitonic states with energy E n and wave function ψ n , and Γ n is a phenomenological broadening parameter considered to be n-dependent in a similar fashion as [51]. In Eq. (13), Ω n,α is defined as with u v k |r α | u c k the interband dipole operator matrix element in the α direction, obtained using the relation Inserting this relation into Eq. (13), we then write the excitonic xx-conductivity as The optical selection rules are directly obtained from the phase factors of the single particle states in Eq. (12) when the commutator u v k |[H, r α ]| u c k is explicitly expanded. Recalling Eq. (10), as well as the discussion regarding eigenvector orthogonality that followed, the allowed transitions are associated with states with angular momentum = 0 (s-series states) and = ±2 (d-series states). Explicitly, the commutator reads with A (k) and B (k) the radial dependence of both the spinor components and the numerical parameters in the momentum matrix of Eq. (10). These two functions are then given by To compare the oscillator strength of the two possible types of transitions, we compute the oscillator strength for d-series transitions as where f n;d (k) is the excitonic radial wave function for d-series states. On the other hand, the oscillator strength for s-series transitions is given by where f n;s (k) is the excitonic radial wave function for s-series states. The first resonance for the = 2 angular momentum series occurs at around 156 meV, and we obtain an oscillator strength around 2 orders of magnitude smaller than that of the s-series transitions closest to it. In Fig. 5, we plot the real part of the excitonic xx-conductivity with a broadening parameter of Γ = 3 meV. In its inset, the contribution from d-series states is also plotted, magnified by a factor of 100 as to improve comparison of the oscillator strengths of both types of transitions.

Conclusions
We studied the properties of excitons in Janus TMD monolayers modeled by a tilted massive Dirac Hamiltonian.
We have shown that, as the tilt parameter increases, the band gap is continuously suppressed, whereas the maximum of the valence band and the minimum of the conduction band shift in opposite directions along the tilting axis, making the gap indirect. Notably, the exciton binding energies remain unchanged as the tilting is enhanced. This means that the (indirect) gap can be made smaller than the exciton binding, a situation that has been predicted to lead to an excitonic instability, and possibly the formation of an excitonic insulator phase.
Finally, we also considered the excitonic linear conductivity, discussing the optical selection rules for the system. With the model Hamiltonian considered, only states with angular momentum = 0 or | | = 2 can be excited, with the resonances associated with = 0 transitions more than two orders of magnitude greater than those associated with = ±2 transitions.
As expected from the solutions of the Bethe-Salpeter equation, the excitonic linear conductivity was also fully independent of the tilt parameter, even when comparing light polarized in . Real part of the excitonic xx-conductivity for a material described by a tilted Dirac Hamiltonian encapsulated in quartz with broadening parameter Γ = 3 meV, and a N = 450 point Gauss-Legendre quadrature. First ten states of each excitonic series were considered for the total conductivity. Vertical dashed lines represent the bandgap of the system. The conductivity is given in units of the conductivity of monolayer graphene σ 0 = e 2 /4 . In the inset, we plot the contribution from only d-series states scaled by two orders of magnitude to improve readability and comparison of the relative intensity. The vertical dashed line representing the bandgap has been aligned by the same value in both the main plot and the inset.
either the parallel or the perpendicular direction of the tilt axis. Importantly, the excitonic contribution to the conductivity is insensitive to the value of the tilting parameter t. As the system is pushed towards the excitonic instability by tuning t, one can expect σ xx (ω) to remain unchanged until the system reaches the instability point, where σ xx (ω) is expected to change abruptly, marking the onset of the phase change.