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Terahertz Fingerprint of Monolayer Wigner Crystals
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Cite this: Nano Lett. 2022, 22, 3, 1311–1315
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https://doi.org/10.1021/acs.nanolett.1c04620
Published January 20, 2022

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The strong Coulomb interaction in monolayer semiconductors represents a unique opportunity for the realization of Wigner crystals without external magnetic fields. In this work, we predict that the formation of monolayer Wigner crystals can be detected by their terahertz response spectrum, which exhibits a characteristic sequence of internal optical transitions. We apply the density matrix formalism to derive the internal quantum structure and the optical conductivity of the Wigner crystal and to microscopically analyze the multipeak shape of the obtained terahertz spectrum. Moreover, we predict a characteristic shift of the peak position as a function of charge density for different atomically thin materials and show how our results can be generalized to an arbitrary two-dimensional system.

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At low temperatures, electrons arrange in a crystal lattice (1) in order to minimize their repulsive energy, which represents one of the most intriguing quantum phase transitions. Ever since their prediction, the experimental realization of these Wigner crystals (WCs) has remained challenging, considering that the prerequisite of a dominating Coulomb versus kinetic energy requires very low temperatures and charge densities. Therefore, the first realizations of WCs (2−4) have used strong magnetic fields and quasi two-dimensional (2D) quantum well systems in order to quench the kinetic energy via Landau quantization. The advances in the research on transition-metal dichalcogenide (TMD) monolayers (5−7) have delivered new opportunities to realize electron crystallization. Several recent studies (8−12) have reported insulating states at fractional fillings of flat moiré bands, (13−15) which arise in the superlattices of twisted homobilayer and heterobilayer systems. However, similar to the application of magnetic fields, these so-called generalized Wigner crystals heavily rely on the external moiré potential stabilizing the lattice structure and can therefore not be considered as intrinsic electronic phases.
In contrast, TMDs in their monolayer form already represent a unique platform to achieve electron crystallization, as they exhibit large effective masses and a strong Coulomb interaction leading to prominent exciton physics. (16,17) Smoleński et al. have recently demonstrated signatures of a WC formed in hBN-encapsulated MoSe2 monolayers (18) at electron densities of up to n = 3 × 1011 cm–2 without any external fields. As an indicator for the crystallization, the authors have used the emergence of an additional peak in the reflectance spectrum at visible frequencies. This peak stems from excitons with finite center-of-mass momentum that couple to the light cone via Bragg scattering at the WC. Other rather indirect confirmations of the WC, such as a decrease in the conductivity, also have been applied to verify other WC systems. (2−4,8,10−12) However, a direct, more reliable and conclusive experimental proof of the WC, such as a direct spectral signature of the internal structure of the crystal, could not be provided so far.
In this work, we propose to exploit terahertz radiation to directly probe the internal quantum transitions of the WC in TMD monolayers, as illustrated in Figure 1. We have developed a fully quantum mechanical model of the interaction of monolayer WC with low frequency light. Our model is based on the many-particle density matrix formalism, where the Hamiltonian of interacting electrons is transformed to the self-consistent eigenbasis of the WC on a Hartree–Fock level. Consequently, we analyze the linear optical response of the system resulting from transitions between different bands of the electron crystal.

Figure 1

Figure 1. Sketch of the 2D Wigner crystal with a honeycomb lattice and alternating spin polarization. The colored curves underneath the particles illustrate their wave functions. A low-energy light field can probe the internal transition of the Wigner crystal toward excited quantum states to provide a conclusive proof of the phase transition.

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We predict the emergence of a series of terahertz resonances whose position is a function of electron density and its characteristic peak sequence can be explained by the nature of the involved excited quantum states. Consequently, the predicted low frequency response can be used as a direct probe of the Wigner crystallization in arbitrary 2D systems of interacting fermions. We apply our microscopic model to determine the material realistic response of hBN-encapsulated WSe2 monolayers without any open parameters and explain the characteristic response shape with the help of our microscopic model. We predict that the central energy of the internal quantum resonances shifts from ∼4 meV at 109 cm–2 to 24 meV at 8 × 1010 cm–2. Moreover, we demonstrate that an enhanced effective mass and reduced dielectric screening, such as in MoSe2 on SiO2, can push the WC resonances up to 100 meV. Finally, we also present a unitless study for ideal 2D systems, which can be rescaled to arbitrary fermionic systems.

Microscopic Mean Field Theory

Previous numerical studies on WCs based on quantum Monte Carlo methods (19) have demonstrated that the low-density crystal phase is characterized by small correlation energies and that the WCs properties are well-reproduced within a Hartree–Fock description of the Coulomb interaction. (20) To obtain access to the optical response of the WC, we start with the Hamilton operator of interacting electrons,
(1)
which can, e.g., be related to the free charge carriers within n(p)-doped TMD monolayers, such that aσk(†) creates/annihilates electrons (holes) in the lowest conduction band (highest valence band) with spin up at the K point (σ = ↑) or spin down at the −K point (σ = ↓) and momentum k. For low temperature and densities, the electron dispersion εk is determined by an effective mass m*. (21) The Coulomb interaction is treated in a mean field approximation (20,22) via the nonlocal Hartree–Fock potential,
(2)
depending on the Fourier transform of the Coulomb potential Vq, which includes the dielectric environment of the TMD monolayer through the Rytova–Keldysh potential. (23,24) Moreover, the mean field potential also is dependent on the charge distribution entering via the Fourier transform of the Wigner distribution wkσ(q) = ⟨aσk+q†aσk⟩. The Hamiltonian in eq 1 is now diagonalized via the basis transformation Aλσk† = ∑Guσkλ*(G)aσ,k+G†, where we sum over the reciprocal lattice vectors G of the WC. Here, uσkλ(G) is the complete set of Bloch functions fulfilling the periodic Hartree–Fock equation (see the Supporting Information (SI)). The latter is solved iteratively until a self-consistent solution is found, starting with a randomized charge distribution. Hence, in the new basis describing WC electrons the Hamiltonian reads
(3)
where the self-consistent solution requires that the lowest band of the WC (λ = 0) is completely filled (one electron per WC unit cell) and the rest is empty, i.e., fλσk = ⟨Aλσk†Aλσk⟩ = δλ0.
Having determined the energy spectrum of WC electrons Eσkλ and their wave functions uσkλ(G), we can transform the electron-light interaction into the Wigner basis and compute the dynamics of the WC using the Heisenberg’s equation of motion. (25,26) In particular, assuming a weak perturbation through the electromagnetic field, we can derive the linear response of the system, which, e.g., can be specified via the diagonal components of the linear optical conductivity tensor reading:
(4)
where we have introduced the monolayer thickness d, (27) the normalization area A, and a phenomenological dephasing constant Γ. The oscillator strength of transitions between different WC sub-bands with energy ΔEσkλ = Eσkλ – Eσk0 for ê-polarized light is given by the current matrix element Jσkλ=e0ℏ/m*∑Guσkλ*(G)G·êuσk0(G). All details of the derivation and the used parameters are given in the SI. The microscopic model described above is not specific to TMD monolayers and can be applied to arbitrary 2D electron system. In principle, a similar approach can be used to compute the optical response also in generalized WCs within twisted bilayers.

Internal Quantum Structure of WCs

Now, we apply the developed model to the exemplary system of p-doped, hBN-encapsulated WSe2 monolayers at a density of n ≈ 3 × 1010 cm–2 (rs = 50), i.e., deep within the crystalline phase that was predicted to melt at rs = 30–40. (28−30) The temperature is assumed low enough such that only the lowest band of the WC is occupied and the thermal occupation of the excited states can be neglected. Moreover, we focus on the spin-unpolarized (antiferromagnetic) phase, since we assume that the charge density is controlled via an electrostatic gate. As a result, the Wigner crystal is in a chemical equilibrium with an unpolarized charge reservoir. Apart from the WC geometry, the main conclusions qualitatively also hold for spin (valley)-polarized systems. (31)Figure 2 shows the computed band structure for electrons with σ = ↑ (on a cut through the WCs Brillouin zone) as well as the wave functions of the monolayer WC. The band structure is illustrated in Figure 2a, where the background has been color-coded based on the optical activity,
We find a completely flat ground state at ∼17 meV below the edge of the single particle dispersion. We also observe a series of flat excited states, where the first is separated from the ground state by a Wigner gap of ∼10 meV. With the increasing band index, the excited states become more dense and dispersed. In order to interpret this result, we investigate the underlying nature of these quantum states and their respective wave functions.

Figure 2

Figure 2. Internal quantum structure of the WC in p-doped, hBN-encapsulated WSe2 at rs = 50. (a) Band structure of the WC, where only the lowest band is fully occupied. The background color illustrates the oscillator strength for optical transitions from the ground state. (b) The surface plot shows the charge density, whereas the projection contains the spin polarization (↑ is red, ↓ is blue). Also shown are the wave function at k = 0 of the (c) first excited state and (d) the state with the largest oscillator.

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The fully occupied ground state is, in fact, the WC itself, i.e., the corresponding wave functions are eigenstates of the Coulomb potential created by their own collective charge distribution. Here, the charge density (Figure 2b) exhibits strongly localized peaks ordered in a honeycomb lattice (32) with alternating spin polarization (red and blue projections). Moreover, the localized charge distributions exhibit a significant triangular warping, which is usually neglected in variational approaches.
In contrast to the self-consistent ground state, the excited states are unoccupied and therefore do not contribute to the nonlocal Hartree–Fock potential (see the SI). Figure 2c shows the wave function of the first excited state at k = 0, whereas the contours indicate the peaks of the ↑ (red) and ↓ charge density (blue) from Figure 2b. In order to minimize the repulsive Coulomb energy, the excited state is strongly localized in the pockets between the charge peaks. Consequently, the wave function overlap with the ground state is small, which is reflected by the weak optical activity of the first excited state. With the increasing band index, we first find several other localized states with increasing kinetic energy and growing spatial extent. (15) Consequently, the localized orbitals of higher-order bands begin to overlap at higher quantum indices and we find dispersed bands at high energies corresponding to scattering states. Figure 2d shows the excited state with the largest optical activity for transitions from the occupied ground state. As result of its delocalized character and the beneficial exchange interaction between equally spin-polarized electrons, the overlap with the ground state (red contours) is large, resulting in a strong current matrix element. The proposed detection mechanism via low-energy excitations has previously been applied to WCs formed in GaAs quantum wells in strong magnetic fields, where a single peak has been observed at microwave energies, (33,34) the so-called pinning frequency. (35) In contrast, our microscopic model predicts that zero-field monolayer Wigner crystals should exhibit an entire series of resonances at much larger frequencies.
With the knowledge of the internal WC quantum structure, we can now consider the dynamics of the system under external perturbations, e.g., the absorption of light.

Optical Fingerprint of WCs

Now, we evaluate the linear optical conductivity from eq 4 in the energy range of the WCs interband transition. Figure 3 illustrates the real part of the resulting spectrum for two exemplary line widths. For the narrower line width of Γ = 0.1 meV, the spectrum exhibits a distinguished multipeak structure with a characteristic sequence of peak amplitudes. The series of resonances starts at the energy of the Wigner gap (cf. Figure 2a), reflecting the transition to the first excited state, which has a weak oscillator strength, as discussed above. The subsequent series of peaks corresponds to the transition to higher-order excited states, whose oscillator strength increases until a maximum is reached. This is a result of the growing kinetic energy of the excited states with increasing band index. With the increasing orbital size of the excited states, which are mostly localized in the pockets between the WC electrons, their overlap with the ground-state electrons becomes enhanced. After a maximum is reached for the excited state illustrated in Figure 2d, the oscillator strength decreases again. Here, the final states momentum spectrum uσkλ(G) begins to move toward large G vectors, corresponding to scattering states with a large kinetic energy. Consequently, the momentum space overlap determining Jσkλ shrinks.

Figure 3

Figure 3. Terahertz fingerprint of the Wigner crystal. The real part of the optical conductivity exhibits a characteristic series of internal quantum transitions, whose amplitude sequence can be explained by the nature of the involved excited states. Note that the strongly broadened response signal has been scaled by a factor of 2, whereas the Drude signal of free electrons (black line) has been multiplied by 100.

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The calculated conductivity, on the order of 10 Ω–1 cm–1, corresponds to a rather low overall absorbance of the monolayer in the range of α = dRe(σ)/(cϵ0) ≈ 10–4, which is due to the nanometer thickness of the sample. However, comparably weak low-frequency responses stemming from intraexcitonic transitions have already been used to measure time-resolved exciton dynamics in TMD monolayers. (36)
For the larger line width of  Γ = 1.5 meV, the spectrum melts to a single peak centered at ∼15 meV. For comparison, we have also included the Drude response that is expected for an electron gas at the same density. This shows that, even for broad resonances, the terahertz spectrum is undergoing a dramatic change during the formation of the WC. However, we expect a much smaller line width than Γ = 1.5 meV, since the crystallization is occurring at very low temperatures and the phase space for electron–phonon scattering is strongly suppressed within the flat bands of the WC. Moreover, in order to ensure the stability of the WC, ultraclean monolayer samples that are almost free of impurities and disorder are necessary. These samples will, at the same time, also reduce inhomogeneous broadening of WC resonances. Under these conditions, we predict a clear multipeak shape that will allow one to distinguish between different transitions the internal quantum states of the WC.

Density Dependence

Now, we vary the charge density and investigate how the spectral response shifts as the lattice parameter of the WC, , is changed. To this end, we consider the center of the spectral response defined as ωpeak = ∫ωσ(ω)dω/∫σ(ω)dω, where the integrals are only performed over positive frequencies. Figure 4 shows the center energy of the optical response as a function of density for different TMD monolayers. The filled dots represent the values at rs = 30, such that for higher densities quantum melting of the WC is expected. For all three illustrated systems we consider p-doping, since the valence band in TMDs has a significantly larger mass then the conduction band, which enables larger critical densities before quantum melting sets in. Figure 4 illustrates that the optical response shifts significantly to higher energy at enhanced charge densities, which can serve as a clear indicator that the observed peaks indeed stem from transitions of the WC. With increasing density, the charge density peaks are more closely spaced, which enhances the repulsive Coulomb energy for the excited states and therefore increases the transition energies. As a result, we also find that the transition energies for samples on SiO2 are significantly larger, since the Coulomb interaction is much less screened by the substrate. In addition to the spectral shift of the center energy, the amplitude of the optical response also increases with growing densities, since the density of optically active particles is enhanced.

Figure 4

Figure 4. Central energy of the optical response signal as a function of the hole density for different monolayers/substrates. The inset shows the result for an ideal 2D system, where the only system parameter is given by rs and the energy scale is given in units of effective Hartree.

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We have additionally performed calculations for an ideal 2D system with Vq ∝ 1/q, which reproduces the results for TMD monolayers at small densities. For an ideal 2D system, the only relevant system parameter is given by with the effective Bohr radius being defined as aB = 4πϵ0ϵℏ2/(m*e02). Using this density scale, the energies can be given in terms of effective Hartree EH=ℏ2/(m*aB2), such that the results given in the inset of Figure 4 can be rescaled to arbitrary 2D system. The inset furthermore demonstrates that the overall width of the optical response ΔE ≈ Epeak – Egap also grows with increasing density, since the distance between excited states becomes larger.
Finally, we want to note that the susceptibility derived in our work results from an oscillating polarization current as a response to the electromagnetic field. Since the oscillating electrons (holes) are forming the WC crystal itself, the optical resonances found in our work could be interpreted in a classical picture as the excitation of long-range phonon modes of the WC. (32) Moreover, we want to emphasize that the applied model is only valid for weak excitation conditions, i.e., assuming a negligible population of the excited states. A lower bound for the critical power density of the exciting laser can be obtained by estimating a lower bound for the lifetime of excited states. Assuming that the lifetime is shorter than τ = 1 ns, the weak excitation regime can be specified by a power density of I ≪ n(E1 – E0)/τ, which for the systems studied in Figures 2 and 3, yields I ≪ 50 mW/cm2. For stronger excitation powers, we expect optical nonlinearities to occur that could even result in a destruction of the WC.
In conclusion, our study demonstrates that the terahertz response of monolayer Wigner crystals can be used as an unambiguous fingerprint, exhibiting a characteristic sequence of internal quantum transitions. The inherent spectral shift of resonance frequencies with the charge density further represents a strong benchmark for the crystallization. We have predicted the peak positions for different TMD monolayers, as well as other 2D fermion systems. The presented results will guide future experiments toward the detection of Wigner crystallization and the developed approach can be further exploited to theoretically study the interaction dynamics in pure as well as generalized Wigner crystals in twisted bilayers.

Supporting Information

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The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.nanolett.1c04620.

  • Transformation of Hartree–Fock Hamiltonian into Wigner basis; material-specific parameters and interaction matrix elements; derivation of the optical response (PDF)

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  • Corresponding Authors
    • Notes
      The authors declare no competing financial interest.

    Acknowledgments

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    We acknowledge support from Deutsche Forschungsgemeinschaft (DFG) via SFB 1083 (Project B9) and the European Unions Horizon 2020 Research and Innovation Program, under Grant Agreement No. 881603 (Graphene Flagship).

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      Nature (London, United Kingdom) (2020), 587 (7833), 214-218CODEN: NATUAS; ISSN:0028-0836. (Nature Research)
      Quantum particles on a lattice with competing long-range interactions are ubiquitous in physics; transition metal oxides1,2, layered mol. crystals3 and trapped-ion arrays4 are a few examples. In the strongly interacting regime, these systems often show a rich variety of quantum many-body ground states that challenge theory2. The emergence of transition metal dichalcogenide moire superlattices provides a highly controllable platform in which to study long-range electronic correlations 5-12. Here we report an observation of nearly two dozen correlated insulating states at fractional fillings of tungsten diselenide/tungsten disulfide moire superlattices. This finding is enabled by a new optical sensing technique that is based on the sensitivity to the dielec. environment of the exciton excited states in a single-layer semiconductor of tungsten diselenide. The cascade of insulating states shows an energy ordering that is nearly sym. about a filling factor of half a particle per superlattice site. We propose a series of charge-ordered states at commensurate filling fractions that range from generalized Wigner crystals7 to charge d. waves. Our study lays the groundwork for using moire superlattices to simulate a wealth of quantum many-body problems that are described by the two-dimensional extended Hubbard model3,13,14 or spin models with long-range charge-charge and exchange interactions15,16.
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    9. 9
      Wang, L.; Shih, E.-M.; Ghiotto, A.; Xian, L.; Rhodes, D. A.; Tan, C.; Claassen, M.; Kennes, D. M.; Bai, Y.; Kim, B. Correlated electronic phases in twisted bilayer transition metal dichalcogenides. Nat. Mater. 2020, 19, 861– 866,  DOI: 10.1038/s41563-020-0708-6
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      9
      Correlated electronic phases in twisted bilayer transition metal dichalcogenides
      Wang, Lei; Shih, En-Min; Ghiotto, Augusto; Xian, Lede; Rhodes, Daniel A.; Tan, Cheng; Claassen, Martin; Kennes, Dante M.; Bai, Yusong; Kim, Bumho; Watanabe, Kenji; Taniguchi, Takashi; Zhu, Xiaoyang; Hone, James; Rubio, Angel; Pasupathy, Abhay N.; Dean, Cory R.
      Nature Materials (2020), 19 (8), 861-866CODEN: NMAACR; ISSN:1476-1122. (Nature Research)
      Abstr.: In narrow electron bands in which the Coulomb interaction energy becomes comparable to the bandwidth, interactions can drive new quantum phases. Such flat bands in twisted graphene-based systems result in correlated insulator, superconducting and topol. states. Here we report evidence of low-energy flat bands in twisted bilayer WSe2, with signatures of collective phases obsd. over twist angles that range from 4 to 5.1°. At half-band filling, a correlated insulator appeared that is tunable with both twist angle and displacement field. At a 5.1° twist, zero-resistance pockets were obsd. on doping away from half filling at temps. below 3 K, which indicates a possible transition to a superconducting state. The observation of tunable collective phases in a simple band, which hosts only two holes per unit cell at full filling, establishes twisted bilayer transition metal dichalcogenides as an ideal platform to study correlated physics in two dimensions on a triangular lattice.
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    10. 10
      Zhou, Y.; Sung, J.; Brutschea, E.; Esterlis, I.; Wang, Y.; Scuri, G.; Gelly, R. J.; Heo, H.; Taniguchi, T.; Watanabe, K. Bilayer Wigner crystals in a transition metal dichalcogenide heterostructure. Nature 2021, 595, 48– 52,  DOI: 10.1038/s41586-021-03560-w
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      10
      Bilayer Wigner crystals in a transition metal dichalcogenide heterostructure
      Zhou, You; Sung, Jiho; Brutschea, Elise; Esterlis, Ilya; Wang, Yao; Scuri, Giovanni; Gelly, Ryan J.; Heo, Hoseok; Taniguchi, Takashi; Watanabe, Kenji; Zarand, Gergely; Lukin, Mikhail D.; Kim, Philip; Demler, Eugene; Park, Hongkun
      Nature (London, United Kingdom) (2021), 595 (7865), 48-52CODEN: NATUAS; ISSN:0028-0836. (Nature Portfolio)
      One of the first theor. predicted manifestations of strong interactions in many-electron systems was the Wigner crystal, in which electrons crystallize into a regular lattice. The crystal can melt via either thermal or quantum fluctuations. Quantum melting of the Wigner crystal is predicted to produce exotic intermediate phases5,6 and quantum magnetism because of the intricate interplay of Coulomb interactions and kinetic energy. However, studying two-dimensional Wigner crystals in the quantum regime has often required a strong magnetic field or a more superlattice potential, thus limiting access to the full phase diagram of the interacting electron liq. Here we report the observation of bilayer Wigner crystals without magnetic fields or more potentials in an atomically thin transition metal dichalcogenide heterostructure, which consists of two MoSe2 monolayers sepd. by hexagonal boron nitride. We observe optical signatures of robust correlated insulating states at sym. (1:1) and asym. (3:1, 4:1 and 7:1) electron doping of the two MoSe2 layers at cryogenic temps. We attribute these features to bilayer Wigner crystals composed of two interlocked commensurate triangular electron lattices, stabilized by inter-layer interaction. The Wigner crystal phases are remarkably stable, and undergo quantum and thermal melting transitions at electron densities of up to 6 x 1012 per square centimetre and at temps. of up to about 40 K. Our results demonstrate that an atomically thin heterostructure is a highly tunable platform for realizing many-body electronic states and probing their liq.-solid and magnetic quantum phase transitions.
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    11. 11
      Huang, X.; Wang, T.; Miao, S.; Wang, C.; Li, Z.; Lian, Z.; Taniguchi, T.; Watanabe, K.; Okamoto, S.; Xiao, D. Correlated insulating states at fractional fillings of the WS2/WSe2 moiré lattice. Nat. Phys. 2021, 17, 715– 719,  DOI: 10.1038/s41567-021-01171-w
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      11
      Correlated insulating states at fractional fillings of the WS2/WSe2 moire lattice
      Huang, Xiong; Wang, Tianmeng; Miao, Shengnan; Wang, Chong; Li, Zhipeng; Lian, Zhen; Taniguchi, Takashi; Watanabe, Kenji; Okamoto, Satoshi; Xiao, Di; Shi, Su-Fei; Cui, Yong-Tao
      Nature Physics (2021), 17 (6), 715-719CODEN: NPAHAX; ISSN:1745-2473. (Nature Portfolio)
      The strong electron interactions in the minibands formed in moire´ superlattices of van der Waals materials, such as twisted graphene and transition metal dichalcogenides, make such systems a fascinating platform with which to study strongly correlated states1-19. In most systems, the correlated states appear when the moire´ lattice is filled by an integer no. of electrons per moire´ unit cell. Recently, correlated states at fractional fillings of 1/3 and 2/3 holes per moire´ unit cell have been reported in the WS2/WSe2 hetero-bilayer, hinting at the long-range nature of the electron interaction16. Here we observe a series of correlated insulating states at fractional fillings of the moire´ minibands on both electron- and hole-doped sides in angle-aligned WS2/WSe2 hetero-bilayers, with certain states persisting at temps. up to 120 K. Simulations reveal that these insulating states correspond to ordering of electrons in the moire´ lattice with a periodicity much larger than the moire´ unit cell, indicating a surprisingly strong and long-range interaction beyond the nearest neighbors.
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    12. 12
      Miao, S.; Wang, T.; Huang, X.; Chen, D.; Lian, Z.; Wang, C.; Blei, M.; Taniguchi, T.; Watanabe, K.; Tongay, S. Strong interaction between interlayer excitons and correlated electrons in WSe2/WS2 moiré superlattice. Nat. Commun. 2021, 12, 1– 6,  DOI: 10.1038/s41467-021-23732-6
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      There is no corresponding record for this reference.
    13. 13
      Yu, H.; Liu, G.-B.; Tang, J.; Xu, X.; Yao, W. Moiré excitons: From programmable quantum emitter arrays to spin-orbit–coupled artificial lattices. Sci. Adv. 2017, 3, e1701696  DOI: 10.1126/sciadv.1701696
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      13
      Moir´e excitons: from programmable quantum emitter arrays to spin-orbit - coupled artificial lattices
      Yu, Hongyi; Liu, Gui-Bin; Tang, Jianju; Xu, Xiaodong; Yao, Wang
      Science Advances (2017), 3 (11), e1701696/1-e1701696/7CODEN: SACDAF; ISSN:2375-2548. (American Association for the Advancement of Science)
      Highly uniform and ordered nanodot arrays are crucial for high-performance quantum optoelectronics, including new semiconductor lasers and single-photon emitters, and for synthesizing artificial lattices of interacting quasiparticles toward quantum information processing and simulation of many-body physics. Van der Waals heterostructures of two-dimensional semiconductors are naturally endowed with an ordered nanoscale landscape, i.e., the moir´e pattern that laterally modulates electronic and topog. structures. We find that these moir´e effects realize superstructures of nanodot confinements for long-lived interlayer excitons, which can be either elec. or strain tuned from perfect arrays of quantum emitters to excitonic superlattices with giant spin-orbit coupling (SOC). Besides the wide-range tuning of emission wavelength, the elec. field can also invert the spin optical selection rule of the emitter arrays. This unprecedented control arises from the gauge structure imprinted on exciton wave functions by the moir´e, which underlies the SOC when hopping couples nanodots into superlattices. We show that the moir´e hosts complex hopping honeycomb superlattices, where exciton bands feature a Dirac node and two Weyl nodes, connected by spin-momentum-locked topol. edge modes.
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    14. 14
      Tran, K.; Moody, G.; Wu, F.; Lu, X.; Choi, J.; Kim, K.; Rai, A.; Sanchez, D. A.; Quan, J.; Singh, A. Evidence for moiré excitons in van der Waals heterostructures. Nature 2019, 567, 71– 75,  DOI: 10.1038/s41586-019-0975-z
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      14
      Evidence for moire´ excitons in van der Waals heterostructures
      Tran, Kha; Moody, Galan; Wu, Fengcheng; Lu, Xiaobo; Choi, Junho; Kim, Kyounghwan; Rai, Amritesh; Sanchez, Daniel A.; Quan, Jiamin; Singh, Akshay; Embley, Jacob; Zepeda, Andre; Campbell, Marshall; Autry, Travis; Taniguchi, Takashi; Watanabe, Kenji; Lu, Nanshu; Banerjee, Sanjay K.; Silverman, Kevin L.; Kim, Suenne; Tutuc, Emanuel; Yang, Li; MacDonald, Allan H.; Li, Xiaoqin
      Nature (London, United Kingdom) (2019), 567 (7746), 71-75CODEN: NATUAS; ISSN:0028-0836. (Nature Research)
      Recent advances in the isolation and stacking of monolayers of van der Waals materials have provided approaches for the prepn. of quantum materials in the ultimate two-dimensional limit. In van der Waals heterostructures formed by stacking two monolayer semiconductors, lattice mismatch or rotational misalignment introduces an in-plane moire´ superlattice. It is widely recognized that the moire´ superlattice can modulate the electronic band structure of the material and lead to transport properties such as unconventional supercond. and insulating behavior driven by correlations; however, the influence of the moire´ superlattice on optical properties has not been investigated exptl. Here we report the observation of multiple interlayer exciton resonances with either pos. or neg. circularly polarized emission in a molybdenum diselenide/tungsten diselenide (MoSe2/WSe2) heterobilayer with a small twist angle. We attribute these resonances to excitonic ground and excited states confined within the moire´ potential. This interpretation is supported by recombination dynamics and by the dependence of these interlayer exciton resonances on twist angle and temp. These results suggest the feasibility of engineering artificial excitonic crystals using van der Waals heterostructures for nanophotonics and quantum information applications.
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    15. 15
      Brem, S.; Linderälv, C.; Erhart, P.; Malic, E. Tunable phases of moiré excitons in van der Waals heterostructures. Nano Lett. 2020, 20, 8534– 8540,  DOI: 10.1021/acs.nanolett.0c03019
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      15
      Tunable Phases of Moir´e Excitons in van der Waals Heterostructures
      Brem, Samuel; Linderaelv, Christopher; Erhart, Paul; Malic, Ermin
      Nano Letters (2020), 20 (12), 8534-8540CODEN: NALEFD; ISSN:1530-6984. (American Chemical Society)
      Stacking monolayers of transition metal dichalcogenides into a heterostructure with a finite twist-angle gives rise to artificial moir´e superlattices with a tunable periodicity. As a consequence, excitons experience a periodic potential, which can be exploited to tailor optoelectronic properties of these materials. Whereas recent exptl. studies have confirmed twist-angle-dependent optical spectra, the microscopic origin of moir´e exciton resonances has not been fully clarified yet. Here, we combine first-principles calcns. with the excitonic d. matrix formalism to study transitions between different moir´e exciton phases and their impact on optical properties of the twisted MoSe2/WSe2 heterostructure. At angles smaller than 2°, we find flat, moir´e-trapped states for inter- and intralayer excitons. This moir´e exciton phase changes into completely delocalized states at 3°. We predict a linear and quadratic twist-angle dependence of excitonic resonances for the moir´e-trapped and delocalized exciton phases, resp.
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    16. 16
      Ugeda, M. M.; Bradley, A. J.; Shi, S.-F.; da Jornada, F. H.; Zhang, Y.; Qiu, D. Y.; Ruan, W.; Mo, S.-K.; Hussain, Z.; Shen, Z.-X. Giant bandgap renormalization and excitonic effects in a monolayer transition metal dichalcogenide semiconductor. Nat. Mater. 2014, 13, 1091– 1095,  DOI: 10.1038/nmat4061
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      16
      Giant bandgap renormalization and excitonic effects in a monolayer transition metal dichalcogenide semiconductor
      Ugeda, Miguel M.; Bradley, Aaron J.; Shi, Su-Fei; da Jornada, Felipe H.; Zhang, Yi; Qiu, Diana Y.; Ruan, Wei; Mo, Sung-Kwan; Hussain, Zahid; Shen, Zhi-Xun; Wang, Feng; Louie, Steven G.; Crommie, Michael F.
      Nature Materials (2014), 13 (12), 1091-1095CODEN: NMAACR; ISSN:1476-1122. (Nature Publishing Group)
      Two-dimensional (2D) transition metal dichalcogenides (TMDs) are emerging as a new platform for exploring 2-dimensional semiconductor physics. Reduced screening in two dimensions results in markedly enhanced electron-electron interactions, which were predicted to generate giant bandgap renormalization and excitonic effects. Here the authors present a rigorous exptl. observation of extraordinarily large exciton binding energy in a 2-dimensional semiconducting TMD. The authors det. the single-particle electronic bandgap of single-layer MoSe2 by scanning tunnelling spectroscopy (STS), as well as the two-particle exciton transition energy using photoluminescence (PL) spectroscopy. These yield an exciton binding energy of 0.55 eV for monolayer MoSe2 on graphene-orders of magnitude larger than what is seen in conventional 3-dimensional semiconductors and significantly higher than what the authors see for MoSe2 monolayers in more highly screening environments. This finding is corroborated by the authors' ab initio GW and Bethe-Salpeter equation calcns. which include electron correlation effects. The renormalized bandgap and large exciton binding obsd. here will have a profound impact on electronic and optoelectronic device technologies based on single-layer semiconducting TMDs.
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    17. 17
      Chernikov, A.; Berkelbach, T. C.; Hill, H. M.; Rigosi, A.; Li, Y.; Aslan, O. B.; Reichman, D. R.; Hybertsen, M. S.; Heinz, T. F. Exciton binding energy and nonhydrogenic Rydberg series in monolayer WS2. Phys. Rev. Lett. 2014, 113, 076802,  DOI: 10.1103/PhysRevLett.113.076802
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      17
      Exciton binding energy and nonhydrogenic rydberg series in monolayer WS2
      Chernikov, Alexey; Berkelbach, Timothy C.; Hill, Heather M.; Rigosi, Albert; Li, Yilei; Aslan, Ozgur Burak; Reichman, David R.; Hybertsen, Mark S.; Heinz, Tony F.
      Physical Review Letters (2014), 113 (7), 076802CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      We have exptl. detd. the energies of the ground and first four excited excitonic states of the fundamental optical transition in monolayer WS2, a model system for the growing class of atomically thin two-dimensional semiconductor crystals. From the spectra, we establish a large exciton binding energy of 0.32 eV and a pronounced deviation from the usual hydrogenic Rydberg series of energy levels of the excitonic states. We explain both of these results using a microscopic theory in which the nonlocal nature of the effective dielec. screening modifies the functional form of the Coulomb interaction. These strong but unconventional electron-hole interactions are expected to be ubiquitous in atomically thin materials.
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    18. 18
      Smolenski, T.; Dolgirev, P. E.; Kuhlenkamp, C.; Popert, A.; Shimazaki, Y.; Back, P.; Lu, X.; Kroner, M.; Watanabe, K.; Taniguchi, T. Signatures of Wigner crystal of electrons in a monolayer semiconductor. Nature 2021, 595, 53– 57,  DOI: 10.1038/s41586-021-03590-4
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      18
      Signatures of Wigner crystal of electrons in a monolayer semiconductor
      Smolenski, Tomasz; Dolgirev, Pavel E.; Kuhlenkamp, Clemens; Popert, Alexander; Shimazaki, Yuya; Back, Patrick; Lu, Xiaobo; Kroner, Martin; Watanabe, Kenji; Taniguchi, Takashi; Esterlis, Ilya; Demler, Eugene; Imamoglu, Atac
      Nature (London, United Kingdom) (2021), 595 (7865), 53-57CODEN: NATUAS; ISSN:0028-0836. (Nature Portfolio)
      When the Coulomb repulsion between electrons dominates over their kinetic energy, electrons in two-dimensional systems are predicted to spontaneously break continuous-translation symmetry and form a quantum crystal. Efforts to observe this elusive state of matter, termed a Wigner crystal, in two-dimensional extended systems have primarily focused on cond. measurements on electrons confined to a single Landau level at high magnetic fields. Here we use optical spectroscopy to demonstrate that electrons in a monolayer semiconductor with d. lower than 3 x 1011 per cm squared form a Wigner crystal. The combination of a high electron effective mass and reduced dielec. screening enables us to observe electronic charge order even in the absence of a moire potential or an external magnetic field. The interactions between a resonantly injected exciton and electrons arranged in a periodic lattice modify the exciton bandstructure so that an umklapp resonance arises in the optical reflection spectrum, heralding the presence of charge order. Our findings demonstrate that charge-tunable transition metal dichalcogenide monolayer enable the investigation of previously uncharted territory for many-body physics where interaction energy dominates over kinetic energy.
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    19. 19
      Drummond, N.; Radnai, Z.; Trail, J.; Towler, M.; Needs, R. Diffusion quantum Monte Carlo study of three-dimensional Wigner crystals. Phys. Rev. B 2004, 69, 085116,  DOI: 10.1103/PhysRevB.69.085116
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      Diffusion quantum Monte Carlo study of three-dimensional Wigner crystals
      Drummond, N. D.; Radnai, Z.; Trail, J. R.; Towler, M. D.; Needs, R. J.
      Physical Review B: Condensed Matter and Materials Physics (2004), 69 (8), 085116/1-085116/10CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
      We report diffusion quantum Monte Carlo calcns. of three-dimensional Wigner crystals in the d. range rs = 100-150. We have tested different types of orbital for use in the approx. wave functions but none improve upon the simple Gaussian form. The Gaussian exponents are optimized by directly minimizing the diffusion quantum Monte Carlo energy. We have carefully investigated and sought to minimize the potential biases in our Monte Carlo results. We conclude that the uniform electron gas undergoes a transition from a ferromagnetic fluid to a body-centered-cubic Wigner crystal at rs = 106 ± 1. The diffusion quantum Monte Carlo results are compared with those from Hartree-Fock and Hartree theory in order to understand the role played by exchange and correlation in Wigner crystals. We also study "floating" Wigner crystals and give results for their pair-correlation functions.
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    20. 20
      Trail, J.; Towler, M.; Needs, R. Unrestricted Hartree-Fock theory of Wigner crystals. Phys. Rev. B 2003, 68, 045107,  DOI: 10.1103/PhysRevB.68.045107
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      Unrestricted Hartree-Fock theory of Wigner crystals
      Trail, J. R.; Towler, M. D.; Needs, R. J.
      Physical Review B: Condensed Matter and Materials Physics (2003), 68 (4), 045107/1-045107/5CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
      We demonstrate that UHF theory applied to electrons in a uniform potential has stable Wigner crystal solns. for rs≥1.44 in two dimensions and rs≥4.5 in three dimensions. The correlation energies of the Wigner crystal phases are considerably smaller than those of the fluid phases at the same d.
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    21. 21
      Kormányos, A.; Burkard, G.; Gmitra, M.; Fabian, J.; Zólyomi, V.; Drummond, N. D.; Fal’ko, V. k-p theory for two-dimensional transition metal dichalcogenide semiconductors. 2D Materials 2015, 2, 022001,  DOI: 10.1088/2053-1583/2/2/022001
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      Pan, H.; Wu, F.; Das Sarma, S. Quantum phase diagram of a Moiré-Hubbard model. Phys. Rev. B 2020, 102, 201104,  DOI: 10.1103/PhysRevB.102.201104
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      Quantum phase diagram of a Moire-Hubbard model
      Pan, Haining; Wu, Fengcheng; Das Sarma, Sankar
      Physical Review B (2020), 102 (20), 201104CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
      We theor. study a generalized Hubbard model on Moire superlattices of twisted bilayers, and find very rich filling-factor-dependent quantum phase diagrams tuned by interaction strength and twist angle. Strong long-range Coulomb interaction in the Moire-Hubbard model induces Wigner crystals at a series of fractional filling factors. The effective lattice of the Wigner crystal is controlled by the filling factor, and can be triangle, rectangle, honeycomb, kagome, etc., providing a single platform to realize many different spin models on various lattices by simply tuning carrier d. In addn. to Wigner crystals that are topol. trivial, interaction-induced Chern insulators emerge in the phase diagram. This finding paves a way for engineering interaction-induced quantum anomalous Hall effect in Moire-Hubbard systems where the corresponding single-particle Moire band is topol. trivial.
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      Rytova, N. Screened potential of a point charge in a thin film. Moscow Univ. Phys. Bull. 1967, 3, 30
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      Brem, S.; Ekman, A.; Christiansen, D.; Katsch, F.; Selig, M.; Robert, C.; Marie, X.; Urbaszek, B.; Knorr, A.; Malic, E. Phonon-assisted photoluminescence from indirect excitons in monolayers of transition-metal dichalcogenides. Nano Lett. 2020, 20, 2849– 2856,  DOI: 10.1021/acs.nanolett.0c00633
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      Phonon-Assisted Photoluminescence from Indirect Excitons in Monolayers of Transition-Metal Dichalcogenides
      Brem, Samuel; Ekman, August; Christiansen, Dominik; Katsch, Florian; Selig, Malte; Robert, Cedric; Marie, Xavier; Urbaszek, Bernhard; Knorr, Andreas; Malic, Ermin
      Nano Letters (2020), 20 (4), 2849-2856CODEN: NALEFD; ISSN:1530-6984. (American Chemical Society)
      The photoluminescence (PL) spectrum of transition-metal dichalcogenides (TMDs) shows a multitude of emission peaks below the bright exciton line, and not all of them have been explained yet. Here, we study the emission traces of phonon-assisted recombinations of indirect excitons. To this end, we develop a microscopic theory describing simultaneous exciton, phonon, and photon interaction and including consistent many-particle dephasing. We explain the drastically different PL below the bright exciton in tungsten- and molybdenum-based materials as the result of different configurations of bright and momentum-dark states. In good agreement with expts., our calcns. predict that WSe2 exhibits clearly visible low-temp. PL signals stemming from the phonon-assisted recombination of momentum-dark K-K' excitons.
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      Lindberg, M.; Koch, S. W. Effective Bloch equations for semiconductors. Phys. Rev. B 1988, 38, 3342,  DOI: 10.1103/PhysRevB.38.3342
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      Effective Bloch equations for semiconductors
      Lindberg; Koch
      Physical review. B, Condensed matter (1988), 38 (5), 3342-3350 ISSN:0163-1829.
      There is no expanded citation for this reference.
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      Kira, M.; Koch, S. Many-body correlations and excitonic effects in semiconductor spectroscopy. Progress Quantum Electron. 2006, 30, 155– 296,  DOI: 10.1016/j.pquantelec.2006.12.002
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      Laturia, A.; Van de Put, M. L.; Vandenberghe, W. G. Dielectric properties of hexagonal boron nitride and transition metal dichalcogenides: from monolayer to bulk. npj 2D Mater. Appl. 2018, 2, 1– 7,  DOI: 10.1038/s41699-018-0050-x
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    • Abstract

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      Figure 1

      Figure 1. Sketch of the 2D Wigner crystal with a honeycomb lattice and alternating spin polarization. The colored curves underneath the particles illustrate their wave functions. A low-energy light field can probe the internal transition of the Wigner crystal toward excited quantum states to provide a conclusive proof of the phase transition.

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      Figure 2

      Figure 2. Internal quantum structure of the WC in p-doped, hBN-encapsulated WSe2 at rs = 50. (a) Band structure of the WC, where only the lowest band is fully occupied. The background color illustrates the oscillator strength for optical transitions from the ground state. (b) The surface plot shows the charge density, whereas the projection contains the spin polarization (↑ is red, ↓ is blue). Also shown are the wave function at k = 0 of the (c) first excited state and (d) the state with the largest oscillator.

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      Figure 3

      Figure 3. Terahertz fingerprint of the Wigner crystal. The real part of the optical conductivity exhibits a characteristic series of internal quantum transitions, whose amplitude sequence can be explained by the nature of the involved excited states. Note that the strongly broadened response signal has been scaled by a factor of 2, whereas the Drude signal of free electrons (black line) has been multiplied by 100.

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      Figure 4

      Figure 4. Central energy of the optical response signal as a function of the hole density for different monolayers/substrates. The inset shows the result for an ideal 2D system, where the only system parameter is given by rs and the energy scale is given in units of effective Hartree.

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    • References

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        Understanding and controlling disorder is key to nanotechnol. and materials science. Traditionally, disorder is attributed to local fluctuations of inherent material properties such as chem. and structural compn., doping or strain. Here, we present a fundamentally new source of disorder in nanoscale systems that is based entirely on the local changes of the Coulomb interaction due to fluctuations of the external dielec. environment. Using two-dimensional semiconductors as prototypes, we exptl. monitor dielec. disorder by probing the statistics and correlations of the exciton resonances, and theor. analyze the influence of external screening and phonon scattering. Even moderate fluctuations of the dielec. environment are shown to induce large variations of the bandgap and exciton binding energies up to the 100 meV range, often making it a dominant source of inhomogeneities. As a consequence, dielec. disorder has strong implications for both the optical and transport properties of nanoscale materials and their heterostructures.
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      8. 8
        Xu, Y.; Liu, S.; Rhodes, D. A.; Watanabe, K.; Taniguchi, T.; Hone, J.; Elser, V.; Mak, K. F.; Shan, J. Correlated insulating states at fractional fillings of moiré superlattices. Nature 2020, 587, 214– 218,  DOI: 10.1038/s41586-020-2868-6
        8
        Correlated insulating states at fractional fillings of moire superlattices
        Xu, Yang; Liu, Song; Rhodes, Daniel A.; Watanabe, Kenji; Taniguchi, Takashi; Hone, James; Elser, Veit; Mak, Kin Fai; Shan, Jie
        Nature (London, United Kingdom) (2020), 587 (7833), 214-218CODEN: NATUAS; ISSN:0028-0836. (Nature Research)
        Quantum particles on a lattice with competing long-range interactions are ubiquitous in physics; transition metal oxides1,2, layered mol. crystals3 and trapped-ion arrays4 are a few examples. In the strongly interacting regime, these systems often show a rich variety of quantum many-body ground states that challenge theory2. The emergence of transition metal dichalcogenide moire superlattices provides a highly controllable platform in which to study long-range electronic correlations 5-12. Here we report an observation of nearly two dozen correlated insulating states at fractional fillings of tungsten diselenide/tungsten disulfide moire superlattices. This finding is enabled by a new optical sensing technique that is based on the sensitivity to the dielec. environment of the exciton excited states in a single-layer semiconductor of tungsten diselenide. The cascade of insulating states shows an energy ordering that is nearly sym. about a filling factor of half a particle per superlattice site. We propose a series of charge-ordered states at commensurate filling fractions that range from generalized Wigner crystals7 to charge d. waves. Our study lays the groundwork for using moire superlattices to simulate a wealth of quantum many-body problems that are described by the two-dimensional extended Hubbard model3,13,14 or spin models with long-range charge-charge and exchange interactions15,16.
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      9. 9
        Wang, L.; Shih, E.-M.; Ghiotto, A.; Xian, L.; Rhodes, D. A.; Tan, C.; Claassen, M.; Kennes, D. M.; Bai, Y.; Kim, B. Correlated electronic phases in twisted bilayer transition metal dichalcogenides. Nat. Mater. 2020, 19, 861– 866,  DOI: 10.1038/s41563-020-0708-6
        9
        Correlated electronic phases in twisted bilayer transition metal dichalcogenides
        Wang, Lei; Shih, En-Min; Ghiotto, Augusto; Xian, Lede; Rhodes, Daniel A.; Tan, Cheng; Claassen, Martin; Kennes, Dante M.; Bai, Yusong; Kim, Bumho; Watanabe, Kenji; Taniguchi, Takashi; Zhu, Xiaoyang; Hone, James; Rubio, Angel; Pasupathy, Abhay N.; Dean, Cory R.
        Nature Materials (2020), 19 (8), 861-866CODEN: NMAACR; ISSN:1476-1122. (Nature Research)
        Abstr.: In narrow electron bands in which the Coulomb interaction energy becomes comparable to the bandwidth, interactions can drive new quantum phases. Such flat bands in twisted graphene-based systems result in correlated insulator, superconducting and topol. states. Here we report evidence of low-energy flat bands in twisted bilayer WSe2, with signatures of collective phases obsd. over twist angles that range from 4 to 5.1°. At half-band filling, a correlated insulator appeared that is tunable with both twist angle and displacement field. At a 5.1° twist, zero-resistance pockets were obsd. on doping away from half filling at temps. below 3 K, which indicates a possible transition to a superconducting state. The observation of tunable collective phases in a simple band, which hosts only two holes per unit cell at full filling, establishes twisted bilayer transition metal dichalcogenides as an ideal platform to study correlated physics in two dimensions on a triangular lattice.
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      10. 10
        Zhou, Y.; Sung, J.; Brutschea, E.; Esterlis, I.; Wang, Y.; Scuri, G.; Gelly, R. J.; Heo, H.; Taniguchi, T.; Watanabe, K. Bilayer Wigner crystals in a transition metal dichalcogenide heterostructure. Nature 2021, 595, 48– 52,  DOI: 10.1038/s41586-021-03560-w
        10
        Bilayer Wigner crystals in a transition metal dichalcogenide heterostructure
        Zhou, You; Sung, Jiho; Brutschea, Elise; Esterlis, Ilya; Wang, Yao; Scuri, Giovanni; Gelly, Ryan J.; Heo, Hoseok; Taniguchi, Takashi; Watanabe, Kenji; Zarand, Gergely; Lukin, Mikhail D.; Kim, Philip; Demler, Eugene; Park, Hongkun
        Nature (London, United Kingdom) (2021), 595 (7865), 48-52CODEN: NATUAS; ISSN:0028-0836. (Nature Portfolio)
        One of the first theor. predicted manifestations of strong interactions in many-electron systems was the Wigner crystal, in which electrons crystallize into a regular lattice. The crystal can melt via either thermal or quantum fluctuations. Quantum melting of the Wigner crystal is predicted to produce exotic intermediate phases5,6 and quantum magnetism because of the intricate interplay of Coulomb interactions and kinetic energy. However, studying two-dimensional Wigner crystals in the quantum regime has often required a strong magnetic field or a more superlattice potential, thus limiting access to the full phase diagram of the interacting electron liq. Here we report the observation of bilayer Wigner crystals without magnetic fields or more potentials in an atomically thin transition metal dichalcogenide heterostructure, which consists of two MoSe2 monolayers sepd. by hexagonal boron nitride. We observe optical signatures of robust correlated insulating states at sym. (1:1) and asym. (3:1, 4:1 and 7:1) electron doping of the two MoSe2 layers at cryogenic temps. We attribute these features to bilayer Wigner crystals composed of two interlocked commensurate triangular electron lattices, stabilized by inter-layer interaction. The Wigner crystal phases are remarkably stable, and undergo quantum and thermal melting transitions at electron densities of up to 6 x 1012 per square centimetre and at temps. of up to about 40 K. Our results demonstrate that an atomically thin heterostructure is a highly tunable platform for realizing many-body electronic states and probing their liq.-solid and magnetic quantum phase transitions.
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      11. 11
        Huang, X.; Wang, T.; Miao, S.; Wang, C.; Li, Z.; Lian, Z.; Taniguchi, T.; Watanabe, K.; Okamoto, S.; Xiao, D. Correlated insulating states at fractional fillings of the WS2/WSe2 moiré lattice. Nat. Phys. 2021, 17, 715– 719,  DOI: 10.1038/s41567-021-01171-w
        11
        Correlated insulating states at fractional fillings of the WS2/WSe2 moire lattice
        Huang, Xiong; Wang, Tianmeng; Miao, Shengnan; Wang, Chong; Li, Zhipeng; Lian, Zhen; Taniguchi, Takashi; Watanabe, Kenji; Okamoto, Satoshi; Xiao, Di; Shi, Su-Fei; Cui, Yong-Tao
        Nature Physics (2021), 17 (6), 715-719CODEN: NPAHAX; ISSN:1745-2473. (Nature Portfolio)
        The strong electron interactions in the minibands formed in moire´ superlattices of van der Waals materials, such as twisted graphene and transition metal dichalcogenides, make such systems a fascinating platform with which to study strongly correlated states1-19. In most systems, the correlated states appear when the moire´ lattice is filled by an integer no. of electrons per moire´ unit cell. Recently, correlated states at fractional fillings of 1/3 and 2/3 holes per moire´ unit cell have been reported in the WS2/WSe2 hetero-bilayer, hinting at the long-range nature of the electron interaction16. Here we observe a series of correlated insulating states at fractional fillings of the moire´ minibands on both electron- and hole-doped sides in angle-aligned WS2/WSe2 hetero-bilayers, with certain states persisting at temps. up to 120 K. Simulations reveal that these insulating states correspond to ordering of electrons in the moire´ lattice with a periodicity much larger than the moire´ unit cell, indicating a surprisingly strong and long-range interaction beyond the nearest neighbors.
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      12. 12
        Miao, S.; Wang, T.; Huang, X.; Chen, D.; Lian, Z.; Wang, C.; Blei, M.; Taniguchi, T.; Watanabe, K.; Tongay, S. Strong interaction between interlayer excitons and correlated electrons in WSe2/WS2 moiré superlattice. Nat. Commun. 2021, 12, 1– 6,  DOI: 10.1038/s41467-021-23732-6
        There is no corresponding record for this reference.
      13. 13
        Yu, H.; Liu, G.-B.; Tang, J.; Xu, X.; Yao, W. Moiré excitons: From programmable quantum emitter arrays to spin-orbit–coupled artificial lattices. Sci. Adv. 2017, 3, e1701696  DOI: 10.1126/sciadv.1701696
        13
        Moir´e excitons: from programmable quantum emitter arrays to spin-orbit - coupled artificial lattices
        Yu, Hongyi; Liu, Gui-Bin; Tang, Jianju; Xu, Xiaodong; Yao, Wang
        Science Advances (2017), 3 (11), e1701696/1-e1701696/7CODEN: SACDAF; ISSN:2375-2548. (American Association for the Advancement of Science)
        Highly uniform and ordered nanodot arrays are crucial for high-performance quantum optoelectronics, including new semiconductor lasers and single-photon emitters, and for synthesizing artificial lattices of interacting quasiparticles toward quantum information processing and simulation of many-body physics. Van der Waals heterostructures of two-dimensional semiconductors are naturally endowed with an ordered nanoscale landscape, i.e., the moir´e pattern that laterally modulates electronic and topog. structures. We find that these moir´e effects realize superstructures of nanodot confinements for long-lived interlayer excitons, which can be either elec. or strain tuned from perfect arrays of quantum emitters to excitonic superlattices with giant spin-orbit coupling (SOC). Besides the wide-range tuning of emission wavelength, the elec. field can also invert the spin optical selection rule of the emitter arrays. This unprecedented control arises from the gauge structure imprinted on exciton wave functions by the moir´e, which underlies the SOC when hopping couples nanodots into superlattices. We show that the moir´e hosts complex hopping honeycomb superlattices, where exciton bands feature a Dirac node and two Weyl nodes, connected by spin-momentum-locked topol. edge modes.
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      14. 14
        Tran, K.; Moody, G.; Wu, F.; Lu, X.; Choi, J.; Kim, K.; Rai, A.; Sanchez, D. A.; Quan, J.; Singh, A. Evidence for moiré excitons in van der Waals heterostructures. Nature 2019, 567, 71– 75,  DOI: 10.1038/s41586-019-0975-z
        14
        Evidence for moire´ excitons in van der Waals heterostructures
        Tran, Kha; Moody, Galan; Wu, Fengcheng; Lu, Xiaobo; Choi, Junho; Kim, Kyounghwan; Rai, Amritesh; Sanchez, Daniel A.; Quan, Jiamin; Singh, Akshay; Embley, Jacob; Zepeda, Andre; Campbell, Marshall; Autry, Travis; Taniguchi, Takashi; Watanabe, Kenji; Lu, Nanshu; Banerjee, Sanjay K.; Silverman, Kevin L.; Kim, Suenne; Tutuc, Emanuel; Yang, Li; MacDonald, Allan H.; Li, Xiaoqin
        Nature (London, United Kingdom) (2019), 567 (7746), 71-75CODEN: NATUAS; ISSN:0028-0836. (Nature Research)
        Recent advances in the isolation and stacking of monolayers of van der Waals materials have provided approaches for the prepn. of quantum materials in the ultimate two-dimensional limit. In van der Waals heterostructures formed by stacking two monolayer semiconductors, lattice mismatch or rotational misalignment introduces an in-plane moire´ superlattice. It is widely recognized that the moire´ superlattice can modulate the electronic band structure of the material and lead to transport properties such as unconventional supercond. and insulating behavior driven by correlations; however, the influence of the moire´ superlattice on optical properties has not been investigated exptl. Here we report the observation of multiple interlayer exciton resonances with either pos. or neg. circularly polarized emission in a molybdenum diselenide/tungsten diselenide (MoSe2/WSe2) heterobilayer with a small twist angle. We attribute these resonances to excitonic ground and excited states confined within the moire´ potential. This interpretation is supported by recombination dynamics and by the dependence of these interlayer exciton resonances on twist angle and temp. These results suggest the feasibility of engineering artificial excitonic crystals using van der Waals heterostructures for nanophotonics and quantum information applications.
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      15. 15
        Brem, S.; Linderälv, C.; Erhart, P.; Malic, E. Tunable phases of moiré excitons in van der Waals heterostructures. Nano Lett. 2020, 20, 8534– 8540,  DOI: 10.1021/acs.nanolett.0c03019
        15
        Tunable Phases of Moir´e Excitons in van der Waals Heterostructures
        Brem, Samuel; Linderaelv, Christopher; Erhart, Paul; Malic, Ermin
        Nano Letters (2020), 20 (12), 8534-8540CODEN: NALEFD; ISSN:1530-6984. (American Chemical Society)
        Stacking monolayers of transition metal dichalcogenides into a heterostructure with a finite twist-angle gives rise to artificial moir´e superlattices with a tunable periodicity. As a consequence, excitons experience a periodic potential, which can be exploited to tailor optoelectronic properties of these materials. Whereas recent exptl. studies have confirmed twist-angle-dependent optical spectra, the microscopic origin of moir´e exciton resonances has not been fully clarified yet. Here, we combine first-principles calcns. with the excitonic d. matrix formalism to study transitions between different moir´e exciton phases and their impact on optical properties of the twisted MoSe2/WSe2 heterostructure. At angles smaller than 2°, we find flat, moir´e-trapped states for inter- and intralayer excitons. This moir´e exciton phase changes into completely delocalized states at 3°. We predict a linear and quadratic twist-angle dependence of excitonic resonances for the moir´e-trapped and delocalized exciton phases, resp.
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      16. 16
        Ugeda, M. M.; Bradley, A. J.; Shi, S.-F.; da Jornada, F. H.; Zhang, Y.; Qiu, D. Y.; Ruan, W.; Mo, S.-K.; Hussain, Z.; Shen, Z.-X. Giant bandgap renormalization and excitonic effects in a monolayer transition metal dichalcogenide semiconductor. Nat. Mater. 2014, 13, 1091– 1095,  DOI: 10.1038/nmat4061
        16
        Giant bandgap renormalization and excitonic effects in a monolayer transition metal dichalcogenide semiconductor
        Ugeda, Miguel M.; Bradley, Aaron J.; Shi, Su-Fei; da Jornada, Felipe H.; Zhang, Yi; Qiu, Diana Y.; Ruan, Wei; Mo, Sung-Kwan; Hussain, Zahid; Shen, Zhi-Xun; Wang, Feng; Louie, Steven G.; Crommie, Michael F.
        Nature Materials (2014), 13 (12), 1091-1095CODEN: NMAACR; ISSN:1476-1122. (Nature Publishing Group)
        Two-dimensional (2D) transition metal dichalcogenides (TMDs) are emerging as a new platform for exploring 2-dimensional semiconductor physics. Reduced screening in two dimensions results in markedly enhanced electron-electron interactions, which were predicted to generate giant bandgap renormalization and excitonic effects. Here the authors present a rigorous exptl. observation of extraordinarily large exciton binding energy in a 2-dimensional semiconducting TMD. The authors det. the single-particle electronic bandgap of single-layer MoSe2 by scanning tunnelling spectroscopy (STS), as well as the two-particle exciton transition energy using photoluminescence (PL) spectroscopy. These yield an exciton binding energy of 0.55 eV for monolayer MoSe2 on graphene-orders of magnitude larger than what is seen in conventional 3-dimensional semiconductors and significantly higher than what the authors see for MoSe2 monolayers in more highly screening environments. This finding is corroborated by the authors' ab initio GW and Bethe-Salpeter equation calcns. which include electron correlation effects. The renormalized bandgap and large exciton binding obsd. here will have a profound impact on electronic and optoelectronic device technologies based on single-layer semiconducting TMDs.
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      17. 17
        Chernikov, A.; Berkelbach, T. C.; Hill, H. M.; Rigosi, A.; Li, Y.; Aslan, O. B.; Reichman, D. R.; Hybertsen, M. S.; Heinz, T. F. Exciton binding energy and nonhydrogenic Rydberg series in monolayer WS2. Phys. Rev. Lett. 2014, 113, 076802,  DOI: 10.1103/PhysRevLett.113.076802
        17
        Exciton binding energy and nonhydrogenic rydberg series in monolayer WS2
        Chernikov, Alexey; Berkelbach, Timothy C.; Hill, Heather M.; Rigosi, Albert; Li, Yilei; Aslan, Ozgur Burak; Reichman, David R.; Hybertsen, Mark S.; Heinz, Tony F.
        Physical Review Letters (2014), 113 (7), 076802CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
        We have exptl. detd. the energies of the ground and first four excited excitonic states of the fundamental optical transition in monolayer WS2, a model system for the growing class of atomically thin two-dimensional semiconductor crystals. From the spectra, we establish a large exciton binding energy of 0.32 eV and a pronounced deviation from the usual hydrogenic Rydberg series of energy levels of the excitonic states. We explain both of these results using a microscopic theory in which the nonlocal nature of the effective dielec. screening modifies the functional form of the Coulomb interaction. These strong but unconventional electron-hole interactions are expected to be ubiquitous in atomically thin materials.
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      18. 18
        Smolenski, T.; Dolgirev, P. E.; Kuhlenkamp, C.; Popert, A.; Shimazaki, Y.; Back, P.; Lu, X.; Kroner, M.; Watanabe, K.; Taniguchi, T. Signatures of Wigner crystal of electrons in a monolayer semiconductor. Nature 2021, 595, 53– 57,  DOI: 10.1038/s41586-021-03590-4
        18
        Signatures of Wigner crystal of electrons in a monolayer semiconductor
        Smolenski, Tomasz; Dolgirev, Pavel E.; Kuhlenkamp, Clemens; Popert, Alexander; Shimazaki, Yuya; Back, Patrick; Lu, Xiaobo; Kroner, Martin; Watanabe, Kenji; Taniguchi, Takashi; Esterlis, Ilya; Demler, Eugene; Imamoglu, Atac
        Nature (London, United Kingdom) (2021), 595 (7865), 53-57CODEN: NATUAS; ISSN:0028-0836. (Nature Portfolio)
        When the Coulomb repulsion between electrons dominates over their kinetic energy, electrons in two-dimensional systems are predicted to spontaneously break continuous-translation symmetry and form a quantum crystal. Efforts to observe this elusive state of matter, termed a Wigner crystal, in two-dimensional extended systems have primarily focused on cond. measurements on electrons confined to a single Landau level at high magnetic fields. Here we use optical spectroscopy to demonstrate that electrons in a monolayer semiconductor with d. lower than 3 x 1011 per cm squared form a Wigner crystal. The combination of a high electron effective mass and reduced dielec. screening enables us to observe electronic charge order even in the absence of a moire potential or an external magnetic field. The interactions between a resonantly injected exciton and electrons arranged in a periodic lattice modify the exciton bandstructure so that an umklapp resonance arises in the optical reflection spectrum, heralding the presence of charge order. Our findings demonstrate that charge-tunable transition metal dichalcogenide monolayer enable the investigation of previously uncharted territory for many-body physics where interaction energy dominates over kinetic energy.
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      19. 19
        Drummond, N.; Radnai, Z.; Trail, J.; Towler, M.; Needs, R. Diffusion quantum Monte Carlo study of three-dimensional Wigner crystals. Phys. Rev. B 2004, 69, 085116,  DOI: 10.1103/PhysRevB.69.085116
        19
        Diffusion quantum Monte Carlo study of three-dimensional Wigner crystals
        Drummond, N. D.; Radnai, Z.; Trail, J. R.; Towler, M. D.; Needs, R. J.
        Physical Review B: Condensed Matter and Materials Physics (2004), 69 (8), 085116/1-085116/10CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
        We report diffusion quantum Monte Carlo calcns. of three-dimensional Wigner crystals in the d. range rs = 100-150. We have tested different types of orbital for use in the approx. wave functions but none improve upon the simple Gaussian form. The Gaussian exponents are optimized by directly minimizing the diffusion quantum Monte Carlo energy. We have carefully investigated and sought to minimize the potential biases in our Monte Carlo results. We conclude that the uniform electron gas undergoes a transition from a ferromagnetic fluid to a body-centered-cubic Wigner crystal at rs = 106 ± 1. The diffusion quantum Monte Carlo results are compared with those from Hartree-Fock and Hartree theory in order to understand the role played by exchange and correlation in Wigner crystals. We also study "floating" Wigner crystals and give results for their pair-correlation functions.
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      20. 20
        Trail, J.; Towler, M.; Needs, R. Unrestricted Hartree-Fock theory of Wigner crystals. Phys. Rev. B 2003, 68, 045107,  DOI: 10.1103/PhysRevB.68.045107
        20
        Unrestricted Hartree-Fock theory of Wigner crystals
        Trail, J. R.; Towler, M. D.; Needs, R. J.
        Physical Review B: Condensed Matter and Materials Physics (2003), 68 (4), 045107/1-045107/5CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
        We demonstrate that UHF theory applied to electrons in a uniform potential has stable Wigner crystal solns. for rs≥1.44 in two dimensions and rs≥4.5 in three dimensions. The correlation energies of the Wigner crystal phases are considerably smaller than those of the fluid phases at the same d.
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      21. 21
        Kormányos, A.; Burkard, G.; Gmitra, M.; Fabian, J.; Zólyomi, V.; Drummond, N. D.; Fal’ko, V. k-p theory for two-dimensional transition metal dichalcogenide semiconductors. 2D Materials 2015, 2, 022001,  DOI: 10.1088/2053-1583/2/2/022001
        There is no corresponding record for this reference.
      22. 22
        Pan, H.; Wu, F.; Das Sarma, S. Quantum phase diagram of a Moiré-Hubbard model. Phys. Rev. B 2020, 102, 201104,  DOI: 10.1103/PhysRevB.102.201104
        22
        Quantum phase diagram of a Moire-Hubbard model
        Pan, Haining; Wu, Fengcheng; Das Sarma, Sankar
        Physical Review B (2020), 102 (20), 201104CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
        We theor. study a generalized Hubbard model on Moire superlattices of twisted bilayers, and find very rich filling-factor-dependent quantum phase diagrams tuned by interaction strength and twist angle. Strong long-range Coulomb interaction in the Moire-Hubbard model induces Wigner crystals at a series of fractional filling factors. The effective lattice of the Wigner crystal is controlled by the filling factor, and can be triangle, rectangle, honeycomb, kagome, etc., providing a single platform to realize many different spin models on various lattices by simply tuning carrier d. In addn. to Wigner crystals that are topol. trivial, interaction-induced Chern insulators emerge in the phase diagram. This finding paves a way for engineering interaction-induced quantum anomalous Hall effect in Moire-Hubbard systems where the corresponding single-particle Moire band is topol. trivial.
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      23. 23
        Rytova, N. Screened potential of a point charge in a thin film. Moscow Univ. Phys. Bull. 1967, 3, 30
        There is no corresponding record for this reference.
      24. 24
        Brem, S.; Ekman, A.; Christiansen, D.; Katsch, F.; Selig, M.; Robert, C.; Marie, X.; Urbaszek, B.; Knorr, A.; Malic, E. Phonon-assisted photoluminescence from indirect excitons in monolayers of transition-metal dichalcogenides. Nano Lett. 2020, 20, 2849– 2856,  DOI: 10.1021/acs.nanolett.0c00633
        24
        Phonon-Assisted Photoluminescence from Indirect Excitons in Monolayers of Transition-Metal Dichalcogenides
        Brem, Samuel; Ekman, August; Christiansen, Dominik; Katsch, Florian; Selig, Malte; Robert, Cedric; Marie, Xavier; Urbaszek, Bernhard; Knorr, Andreas; Malic, Ermin
        Nano Letters (2020), 20 (4), 2849-2856CODEN: NALEFD; ISSN:1530-6984. (American Chemical Society)
        The photoluminescence (PL) spectrum of transition-metal dichalcogenides (TMDs) shows a multitude of emission peaks below the bright exciton line, and not all of them have been explained yet. Here, we study the emission traces of phonon-assisted recombinations of indirect excitons. To this end, we develop a microscopic theory describing simultaneous exciton, phonon, and photon interaction and including consistent many-particle dephasing. We explain the drastically different PL below the bright exciton in tungsten- and molybdenum-based materials as the result of different configurations of bright and momentum-dark states. In good agreement with expts., our calcns. predict that WSe2 exhibits clearly visible low-temp. PL signals stemming from the phonon-assisted recombination of momentum-dark K-K' excitons.
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      25. 25
        Lindberg, M.; Koch, S. W. Effective Bloch equations for semiconductors. Phys. Rev. B 1988, 38, 3342,  DOI: 10.1103/PhysRevB.38.3342
        25
        Effective Bloch equations for semiconductors
        Lindberg; Koch
        Physical review. B, Condensed matter (1988), 38 (5), 3342-3350 ISSN:0163-1829.
        There is no expanded citation for this reference.
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      26. 26
        Kira, M.; Koch, S. Many-body correlations and excitonic effects in semiconductor spectroscopy. Progress Quantum Electron. 2006, 30, 155– 296,  DOI: 10.1016/j.pquantelec.2006.12.002
        There is no corresponding record for this reference.
      27. 27
        Laturia, A.; Van de Put, M. L.; Vandenberghe, W. G. Dielectric properties of hexagonal boron nitride and transition metal dichalcogenides: from monolayer to bulk. npj 2D Mater. Appl. 2018, 2, 1– 7,  DOI: 10.1038/s41699-018-0050-x
        27
        Dielectric properties of hexagonal boron nitride and transition metal dichalcogenides: from monolayer to bulk
        Laturia, Akash; Van de Put, Maarten L.; Vandenberghe, William G.
        npj 2D Materials and Applications (2018), 2 (1), 1-7CODEN: DMAAAH; ISSN:2397-7132. (Nature Research)
        Hexagonal boron nitride (h-BN) and semiconducting transition metal dichalcogenides (TMDs) promise greatly improved electrostatic control in future scaled electronic devices. To quantify the prospects of these materials in devices, we calc. the out-of-plane and in-plane dielec. const. from first principles for TMDs in trigonal prismatic and octahedral coordination, as well as for h-BN, with a thickness ranging from monolayer and bilayer to bulk. Both the ionic and electronic contribution to the dielec. response are computed. Our calcns. show that the out-of-plane dielec. response for the transition-metal dichalcogenides is dominated by its electronic component and that the dielec. const. increases with increasing chalcogen at. no. Overall, the out-of-plane dielec. const. of the TMDs and h-BN increases by around 15% as the no. of layers is increased from monolayer to bulk, while the in-plane component remains unchanged. Our computations also reveal that for octahedrally coordinated TMDs the ionic (static) contribution to the dielec. response is very high (4.5 times the electronic contribution) in the in-plane direction. This indicates that semiconducting TMDs in the tetragonal phase will suffer from excessive polar-optical scattering thereby deteriorating their electronic transport properties.
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      28. 28
        Yoon, J.; Li, C.; Shahar, D.; Tsui, D.; Shayegan, M. Wigner crystallization and metal-insulator transition of two-dimensional holes in GaAs at B = 0. Phys. Rev. Lett. 1999, 82, 1744,  DOI: 10.1103/PhysRevLett.82.1744
        28
        Wigner Crystallization and Metal-Insulator Transition of Two-Dimensional Holes in GaAs at B =0
        Yoon, Jongsoo; Li, C. C.; Shahar, D.; Tsui, D. C.; Shayegan, M.
        Physical Review Letters (1999), 82 (8), 1744-1747CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
        The authors report the transport properties of a low disorder two-dimensional hole system (2DHS) in the GaAs/AlGaAs heterostructure, which has an unprecedentedly high peak mobility of 7 × 105 cm2/V s, with a hole d. of 4.8 × 109<p<3.72 × 1010 cm-2 in the temp. range of 50 mK<T<1.3 K. From their T, p, and elec. field dependences, the metal-insulator transition in zero magnetic field in this exceptionally clean 2DHS occurs at rs = 35.1 ± 0.9, which is in good agreement with the crit. rs for Wigner crystn. rcs = 37 ± 5, predicted by Tanatar and Ceperley for an ideally clean 2-dimensional system.
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      29. 29
        Drummond, N.; Needs, R. Phase diagram of the low-density two-dimensional homogeneous electron gas. Phys. Rev. Lett. 2009, 102, 126402,  DOI: 10.1103/PhysRevLett.102.126402
        29
        Phase Diagram of the Low-Density Two-Dimensional Homogeneous Electron Gas
        Drummond, N. D.; Needs, R. J.
        Physical Review Letters (2009), 102 (12), 126402/1-126402/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
        We use quantum Monte Carlo methods to calc. the zero-temp. phase diagram of the two-dimensional homogeneous electron gas. We find a transition from a paramagnetic fluid to an antiferromagnetic triangular Wigner crystal at d. parameter rs=31(1) a.u. and a transition to a ferromagnetic crystal at rs=38(5) a.u. The fully spin-polarized fluid is never stable. We search for, but do not find, the ferromagnetic "hybrid" phase proposed by H. Falakshahi and X. Waintal.
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      30. 30
        Zarenia, M.; Neilson, D.; Partoens, B.; Peeters, F. Wigner crystallization in transition metal dichalcogenides: A new approach to correlation energy. Phys. Rev. B 2017, 95, 115438,  DOI: 10.1103/PhysRevB.95.115438
        30
        Wigner crystallization in transition metal dichalcogenides: a new approach to correlation energy
        Zarenia, M.; Neilson, D.; Partoens, B.; Peeters, F. M.
        Physical Review B (2017), 95 (11), 115438/1-115438/5CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
        A review. We introduce a new approach for the correlation energy of one- and two-valley two-dimensional electron gas (2DEG) systems. Our approach is based on an interpolation between two limits, a RPA at high densities and a classical approach at low densities which gives excellent agreement with available Quantum Monte Carlo (QMC) calcns. The two-valley 2DEG model is introduced to describe the electron correlations in monolayer transition metal dichalcogenides (TMDs). We study the zero-temp. transition from a Fermi liq. to a quantum Wigner crystal phase in monolayer TMDs. Consistent with QMC, we find that electrons crystallize at rs = 31 in one-valley 2DEG. For two valleys, we predictWigner crystn. at rs = 30, implying that valley degeneracy has little effect on the crit. rs, in contrast to an earlier claim.
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      31. 31
        Tanatar, B.; Ceperley, D. M. Ground state of the two-dimensional electron gas. Phys. Rev. B 1989, 39, 5005,  DOI: 10.1103/PhysRevB.39.5005
        31
        Ground state of the two-dimensional electron gas
        Tanatar; Ceperley
        Physical review. B, Condensed matter (1989), 39 (8), 5005-5016 ISSN:0163-1829.
        There is no expanded citation for this reference.
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      32. 32
        Bonsall, L.; Maradudin, A. Some static and dynamical properties of a two-dimensional Wigner crystal. Phys. Rev. B 1977, 15, 1959,  DOI: 10.1103/PhysRevB.15.1959
        32
        Some static and dynamical properties of a two-dimensional Wigner crystal
        Bonsall, Lynn; Maradudin, A. A.
        Physical Review B: Solid State (1977), 15 (4), 1959-73CODEN: PLRBAQ; ISSN:0556-2805.
        The static ground-state energy of a 2-dimensional Wigner crystal was obtained for each of the 5 2-dimensional Bravais lattices. At const. electron no. d., the hexagonal lattice has the lowest energy. Phonon dispersion curves were calcd. for wave vectors along the sym. directions in the 1st Brillouin zone for the hexagonal lattice. In the long-wavelength limit, 1 of the 2 branches of the dispersion relation vanishes with vanishing 2-dimensional wave vector .vector.q as q, the 2nd as q1/2. The coeff. of q in the former branch is purely imaginary for certain directions of propagation in the square point energy and low-temp. thermodn. functions were obtained for the hexagonal lattice. The dielec. susceptibility tensor of a 2-dimensional Wigner crystal was detd. in the long-wavelength limit, in the presence of a static magnetic field perpendicular to the crystal; the result was used to obtain the dispersion relation for plasma oscillations in the electron crystal.
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      33. 33
        Li, C.-C.; Yoon, J.; Engel, L.; Shahar, D.; Tsui, D.; Shayegan, M. Microwave resonance and weak pinning in two-dimensional hole systems at high magnetic fields. Phys. Rev. B 2000, 61, 10905,  DOI: 10.1103/PhysRevB.61.10905
        33
        Microwave resonance and weak pinning in two-dimensional hole systems at high magnetic fields
        Li, C.-C.; Yoon, J.; Engel, L. W.; Shahar, D.; Tsui, D. C.; Shayegan, M.
        Physical Review B: Condensed Matter and Materials Physics (2000), 61 (16), 10905-10909CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
        Microwave frequency cond. Re(σxx) of high quality two-dimensional hole systems (2DHS) in a large perpendicular magnetic field (B) is measured with the carrier d. (ns) of the 2DHS controlled by a backgate bias. The high-B insulating phase of the 2DHS exhibits a microwave resonance that remains well defined, but shifts to higher peak frequency (fpk) as ns is reduced. In different regimes, fpk vs ns can be fit to fpk.varies.ns-1/2 or to fpk.varies.ns-3/2. The data clearly indicate that both carrier-carrier interactions and disorder are indispensable in detg. the dynamics of the insulator. The ns dependence of fpk is consistent with a weakly pinned Wigner crystal in which domain size increases with ns, due to larger carrier-carrier interaction.
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      34. 34
        Chen, Y.; Lewis, R.; Engel, L.; Tsui, D.; Ye, P.; Pfeiffer, L.; West, K. Microwave resonance of the 2D Wigner crystal around integer Landau fillings. Phys. Rev. Lett. 2003, 91, 016801,  DOI: 10.1103/PhysRevLett.91.016801
        34
        Microwave resonance of the 2D Wigner crystal around integer Landau fillings
        Chen, Yong; Lewis, R. M.; Engel, L. W.; Tsui, D. C.; Ye, P. D.; Pfeiffer, L. N.; West, K. W.
        Physical Review Letters (2003), 91 (1), 016801/1-016801/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
        The authors have obsd. a resonance in the real part of the finite frequency diagonal cond. using microwave absorption measurements in high-quality 2D electron systems near integer fillings. The resonance exists in some neighborhood of filling factor around corresponding integers and is qual. similar to the previously obsd. resonance of weakly pinned Wigner crystals in the high B and very small filling factor regimes. Data measured around both ν = 1 and ν = 2 are presented. The authors interpret the resonance as the signature of the Wigner crystal state around integer Landau levels.
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      35. 35
        Chitra, R.; Giamarchi, T.; Le Doussal, P. Pinned Wigner crystals. Phys. Rev. B 2001, 65, 035312,  DOI: 10.1103/PhysRevB.65.035312
        There is no corresponding record for this reference.
      36. 36
        Pöllmann, C.; Steinleitner, P.; Leierseder, U.; Nagler, P.; Plechinger, G.; Porer, M.; Bratschitsch, R.; Schüller, C.; Korn, T.; Huber, R. Resonant internal quantum transitions and femtosecond radiative decay of excitons in monolayer WSe2. Nat. Mater. 2015, 14, 889– 893,  DOI: 10.1038/nmat4356
        There is no corresponding record for this reference.

    • Supporting Information

      Supporting Information

      The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.nanolett.1c04620.

      • Transformation of Hartree–Fock Hamiltonian into Wigner basis; material-specific parameters and interaction matrix elements; derivation of the optical response (PDF)


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