# Transition Metal Dichalcogenides as Strategy for High Temperature Electron-Hole Superfluidity

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## Abstract

**:**

## 1. Introduction

_{2}insulating barrier [15]. This signature was observed only at lower densities and is in quantitative agreement with the theoretical predictions [16] of an upper limit of the carrier density for the superfluidity. Above this threshold density, screening kills the superfluidity. The observed transition temperature is low, ${T}_{c}\sim 1.5$ K. Reference [14] had predicted a maximum transition temperature of 17 K in a DBG with a 1.4 nm hBN barrier. Reference [16] pointed out the importance of the strong interband screening from the valence band, a large effect here because of the very small band gap in bilayer graphene [17]. The effect of this additional screening is to reduce the threshold density and the maximum transition temperature.

_{2}, MoSe

_{2}, WS

_{2}, and WSe

_{2}are semiconductors with direct band gaps comparable to that in GaAs [18,19]. The large band gap eliminates the detrimental effects of the interband screening mentioned above for graphene. The electron and hole effective masses in TMDs are larger than for bilayer graphene, and this further increases the coupling strength of the electron-hole pairs [20]. Reference [21] proposed the TMD heterostructure MoSe${}_{2}$-hBN-WSe${}_{2}$ to observe electron-hole superfluidity and to investigate additional novel multicomponent effects resulting from the strong spin-orbit coupling. The splitting of the valence bands ${\lambda}_{v}$ is an order of magnitude larger than the splitting of the conduction bands ${\lambda}_{c}$. The resulting misalignment of the electron and hole bands fundamentally changes the multicomponent nature of the superfluidity. The authors already predicted maximum transition temperatures as high as ${T}_{BKT}\sim 100$ K [21], and very recently enhanced tunneling conductance signatures of Bose-Einstein Condensation (BEC) was reported in this same system with transition temperature ${T}_{c}\sim 100$ K [22] consistent with these predictions.

## 2. Results

## 3. Discussion

## 4. Materials and Methods

#### 4.1. Materials

#### 4.2. Method

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

2D | Two-dimensional |

DQW | Double Quantum-Wells |

DMG | Double Monolayer Graphene |

hBN | hexagonal Boron Nitride |

DBG | Double Bilayer Graphene |

TMD | Transition Metal Dichalcogenides |

BEC | Bose-Einstein Condensation |

BKT | Berezinskii-Kosterlitz-Thouless |

## References

- Zhu, X.; Littlewood, P.B.; Hybertsen, M.S.; Rice, T.M. Exciton Condensate in Semiconductor Quantum Well Structures. Phys. Rev. Lett.
**1995**, 74, 1633. [Google Scholar] [CrossRef] [Green Version] - Croxall, A.F.; Das Gupta, K.; Nicoll, C.A.; Thangaraj, M.; Beere, H.E.; Farrer, I.; Ritchie, D.A.; Pepper, M. Anomalous Coulomb Drag in Electron-Hole Bilayers. Phys. Rev. Lett.
**2008**, 101, 246801. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Seamons, J.A.; Morath, C.P.; Reno, J.L.; Lilly, M.P. Coulomb Drag in the Exciton Regime in Electron-Hole Bilayers. Phys. Rev. Lett.
**2009**, 102, 026804. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lozovik, Y.E.; Yudson, V.I. Feasibility of superfluidity of paired spatially separated electrons and holes. JETP Lett.
**1975**, 22, 274, (Pis’ma Zh. Eksp. Teor. Fiz.**22**, 556 (1975)). [Google Scholar] - Saberi-Pouya, S.; Conti, S.; Perali, A.; Croxall, A.F.; Hamilton, A.R.; Peeters, F.M.; Neilson, D. Experimental conditions for observation of electron-hole superfluidity in GaAs heterostructures. arXiv
**2019**, arXiv:1910.06631. [Google Scholar] - Neilson, D.; Perali, A.; Hamilton, A.R. Excitonic superfluidity and screening in electron-hole bilayer systems. Phys. Rev. B
**2014**, 89, 060502. [Google Scholar] [CrossRef] [Green Version] - Pieri, P.; Neilson, D.; Strinati, G.C. Effects of density imbalance on the BCS-BEC crossover in semiconductor electron-hole bilayers. Phys. Rev. B
**2007**, 75, 113301. [Google Scholar] [CrossRef] [Green Version] - Novoselov, K.S.; Geim, A.K.; Morozov, S.V.; Jiang, D.; Zhang, Y.; Dubonos, S.V.; Grigorieva, I.V.; Firsov, A.A. Electric field effect in atomically thin carbon films. Science
**2004**, 306, 666. [Google Scholar] [CrossRef] [Green Version] - Britnell, L.; Gorbachev, R.V.; Jalil, R.; Belle, B.D.; Schedin, F.; Katsnelson, M.I.; Eaves, L.; Morozov, S.V.; Mayorov, A.S.; Peres, N.M.; et al. Electron tunneling through ultrathin Boron Nitride crystalline barriers. Nano Lett.
**2012**, 12, 1707. [Google Scholar] [CrossRef] [Green Version] - Min, H.; Bistritzer, R.; Su, J.J.; MacDonald, A.H. Room-temperature superfluidity in graphene bilayers. Phys. Rev. B
**2008**, 78, 121401. [Google Scholar] [CrossRef] [Green Version] - Lozovik, Y.E.; Sokolik, A.A. Coherent phases and collective electron phenomena in graphene. J. Phys. Conf. Ser.
**2008**, 129, 012003. [Google Scholar] [CrossRef] - Gorbachev, R.V.; Geim, A.K.; Katsnelson, M.I.; Novoselov, K.S.; Tudorovskiy, T.; Grigorieva, I.V.; MacDonald, A.H.; Morozov, S.V.; Watanabe, K.; Taniguchi, T.; et al. Strong Coulomb drag and broken symmetry in double-layer graphene. Nat. Phys.
**2012**, 8, 896. [Google Scholar] [CrossRef] - Lozovik, Y.E.; Ogarkov, S.L.; Sokolik, A.A. Condensation of electron-hole pairs in a two-layer graphene system: Correlation effects. Phys. Rev. B
**2012**, 86, 045429. [Google Scholar] [CrossRef] [Green Version] - Perali, A.; Neilson, D.; Hamilton, A.R. High-Temperature Superfluidity in Double-Bilayer Graphene. Phys. Rev. Lett.
**2013**, 110, 146803. [Google Scholar] [CrossRef] - Burg, G.W.; Prasad, N.; Kim, K.; Taniguchi, T.; Watanabe, K.; MacDonald, A.H.; Register, L.F.; Tutuc, E. Strongly Enhanced Tunneling at Total Charge Neutrality in Double-Bilayer Graphene-WSe
_{2}Heterostructures. Phys. Rev. Lett.**2018**, 120, 177702. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Conti, S.; Perali, A.; Peeters, F.M.; Neilson, D. Multicomponent screening and superfluidity in gapped electron-hole double bilayer graphene with realistic bands. Phys. Rev. B
**2019**, 99, 144517. [Google Scholar] [CrossRef] [Green Version] - Zhang, Y.; Tang, T.T.; Girit, C.; Hao, Z.; Martin, M.C.; Zettl, A.; Crommie, M.F.; Shen, Y.R.; Wang, F. Direct observation of a widely tunable bandgap in bilayer graphene. Nature
**2009**, 459, 820. [Google Scholar] [CrossRef] - Mak, K.F.; Lee, C.; Hone, J.; Shan, J.; Heinz, T.F. Atomically thin MoS
_{2}: A new direct-gap semiconductor. Phys. Rev. Lett.**2010**, 105, 136805. [Google Scholar] [CrossRef] [Green Version] - Jiang, H. Electronic band structures of molybdenum and tungsten dichalcogenides by the GW approach. J. Phys. Chem. C
**2012**, 116, 7664. [Google Scholar] [CrossRef] - Fogler, M.M.; Butov, L.V.; Novoselov, K.S. High-temperature superfluidity with indirect excitons in van der Waals heterostructures. Nat. Commun.
**2014**, 5, 4555. [Google Scholar] [CrossRef] [Green Version] - Conti, S.; der Donck, M.V.; Perali, A.; Peeters, F.M.; Neilson, D. A doping-dependent switch from one- to two-component superfluidity at temperature above 100K in coupled electron-hole van der Waals heterostructures. arXiv
**2019**, arXiv:1909.03411. [Google Scholar] - Wang, Z.; Rhodes, D.A.; Watanabe, K.; Taniguchi, T.; Hone, J.C.; Shan, J.; Mak, K.F. Evidence of high-temperature exciton condensation in two-dimensional atomic double layers. Nature
**2019**, 574, 76. [Google Scholar] [CrossRef] [PubMed] - Strinati, G. A Survey on the Crossover from BCS Superconductivity to Bose-Einstein Condensation. Phys. Essays
**2000**, 13, 427. [Google Scholar] [CrossRef] - Randeria, M.; Duan, J.M.; Shieh, L.Y. Superconductivity in a two-dimensional Fermi gas: Evolution from Cooper pairing to Bose condensation. Phys. Rev. B
**1990**, 41, 327. [Google Scholar] [CrossRef] [PubMed] - Pistolesi, F.; Strinati, G.C. Evolution from BCS superconductivity to Bose condensation: Role of the parameter k
_{F}ξ. Phys. Rev. B**1994**, 49, 6356. [Google Scholar] [CrossRef] [PubMed] - Singh, G.; Jouan, A.; Herranz, G.; Scigaj, M.; Sánchez, F.; Benfatto, L.; Caprara, S.; Grilli, M.; Saiz, G.; Couëdo, F.; et al. Gap suppression at a Lifshitz transition in a multi-condensate superconductor. Nat. Mater.
**2019**, 18, 948. [Google Scholar] [CrossRef] [Green Version] - Kosterlitz, J.M.; Thouless, D.J. Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C: Solid State
**1973**, 6, 1181. [Google Scholar] [CrossRef] - Saberi-Pouya, S.; Zarenia, M.; Perali, A.; Vazifehshenas, T.; Peeters, F.M. High-temperature electron-hole superfluidity with strong anisotropic gaps in double phosphorene monolayers. Phys. Rev. B
**2018**, 97, 174503. [Google Scholar] [CrossRef] [Green Version] - Conti, S.; Perali, A.; Peeters, F.M.; Neilson, D. Multicomponent electron-hole superfluidity and the BCS-BEC crossover in double bilayer graphene. Phys. Rev. Lett.
**2017**, 119, 257002. [Google Scholar] [CrossRef] [Green Version] - Xiao, D.; Liu, G.B.; Feng, W.; Xu, X.; Yao, W. Coupled Spin and Valley Physics in Monolayers of MoS
_{2}and Other Group-VI Dichalcogenides. Phys. Rev. Lett.**2012**, 108, 196802. [Google Scholar] [CrossRef] [Green Version] - Zhu, Z.Y.; Cheng, Y.C.; Schwingenschlögl, U. Giant spin-orbit-induced spin splitting in two-dimensional transition-metal dichalcogenide semiconductors. Phys. Rev. B
**2011**, 84, 153402. [Google Scholar] [CrossRef] [Green Version] - Kośmider, K.; González, J.W.; Fernández-Rossier, J. Large spin splitting in the conduction band of transition metal dichalcogenide monolayers. Phys. Rev. B
**2013**, 88, 245436. [Google Scholar] [CrossRef] [Green Version] - Van der Donck, M.; Peeters, F.M. Interlayer excitons in transition metal dichalcogenide heterostructures. Phys. Rev. B
**2018**, 98, 115104. [Google Scholar] [CrossRef] [Green Version] - Kumar, P.; Chauhan, Y.S.; Agarwal, A.; Bhowmick, S. Thickness and Stacking Dependent Polarizability and Dielectric Constant of Graphene–Hexagonal Boron Nitride Composite Stacks. J. Phys. Chem. C
**2016**, 120, 17620. [Google Scholar] [CrossRef] - Yu, F.F.; Ke, S.S.; Guan, S.S.; Deng, H.X.; Guo, Y.; Lü, H.F. Effects of Se substitution and transition metal doping on the electronic and magnetic properties of a MoS
_{x}Se_{2-x}/h-BN heterostructure. Phys. Chem. Chem. Phys.**2019**, 21, 20073. [Google Scholar] [CrossRef] [PubMed] - Gerber, I.C.; Marie, X. Dependence of band structure and exciton properties of encapsulated WSe
_{2}monolayers on the hBN-layer thickness. Phys. Rev. B**2018**, 98, 245126. [Google Scholar] [CrossRef] [Green Version] - Shanenko, A.A.; Aguiar, J.A.; Vagov, A.; Croitoru, M.D.; Milošević, M.V. Atomically flat superconducting nanofilms: Multiband properties and mean-field theory. Supercond. Sci. Tech.
**2015**, 28, 054001. [Google Scholar] [CrossRef] - Vargas-Paredes, A.A.; Shanenko, A.A.; Vagov, A.; Milošević, M.V.; Perali, A. Cross-band versus intra-band pairing in superconductors: Signatures and consequences of the interplay. arXiv
**2019**, arXiv:1906.06528. [Google Scholar] - Lozovik, Y.E.; Sokolik, A.A. Multi-band pairing of ultrarelativistic electrons and holes in graphene bilayer. Phys. Rev. A
**2009**, 374, 326. [Google Scholar] [CrossRef] - Kochorbe, F.G.; Palistrant, M.E. Superconductivity in a two-band system with low carrier density. J. Exp. Theor. Phys.
**1993**, 77, 442. [Google Scholar] - Giorgini, S.; Pitaevskii, L.; Stringari, S. Condensate fraction and critical temperature of a trapped interacting Bose gas. Phys. Rev. A
**1996**, 54, 4633. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Salasnich, L.; Manini, N.; Parola, A. Condensate fraction of a Fermi gas in the BCS-BEC crossover. Phys. Rev. A
**2005**, 72, 023621. [Google Scholar] [CrossRef] [Green Version] - Benfatto, L.; Capone, M.; Caprara, S.; Castellani, C.; Di Castro, C. Multiple gaps and superfluid density from interband pairing in a four-band model of the iron oxypnictides. Phys. Rev. B
**2008**, 78, 140502. [Google Scholar] [CrossRef] [Green Version] - Botelho, S.S.; Sá de Melo, C.A.R. Vortex-Antivortex Lattice in Ultracold Fermionic Gases. Phys. Rev. Lett.
**2006**, 96, 040404. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 1.**Low-energy multiband structure resulting from spin-orbit coupling in n–WSe${}_{2}$/hBN/MoSe${}_{2}$–p. The MoSe${}_{2}$ bands are the valence bands after a standard particle-hole transformation to positive energies. Because of the large band gap in TMDs, we are able to neglect the remote bands. ${\lambda}_{c}$ ( ${\lambda}_{v}$) is the spin-orbit splitting of the conduction (valence) bands. The dashed arrows labelled $\left\{bb\right\}$ and $\left\{tt\right\}$ indicate schematically the pairing channels.

**Figure 2.**(

**a**) The maximum of the superfluid gaps ${\Delta}_{bb}$ and ${\Delta}_{tt}$ as a function of the density n. For reference, the upper horizontal axis shows the density of the top bands ${n}_{t}$. Both densities have units of ${10}^{12}$ cm${}^{-2}$. (

**b**) Chemical potential as a function of the density n. The zero energy is fixed at the minimum of the bottom bands ${\epsilon}_{b}\left(0\right)$. The ${E}_{B}^{b}/2$ and ${E}_{B}^{t}/2$ are the two-body bound state energies.

**Figure 3.**Condensate fraction ${C}_{bb}$ and ${C}_{tt}$ as a function of n (${10}^{12}$ cm${}^{-2}$). The upper horizontal axis shows the density of the top bands ${n}_{t}$ (${10}^{12}$ cm${}^{-2}$). The blue shaded area is the BEC regime.

**Table 1.**Experimental (E) and theoretical (T) properties and results of the different systems. ${m}_{e}^{*}$ and ${m}_{h}^{*}$: electron and hole effective masses; ${E}_{g}$: energy band gap; d: layer separation; ${n}_{0}$: superfluid threshold density (${10}^{11}$ cm${}^{-2}$); ${T}_{c}$: transition temperature for superfluidity.

${\mathit{m}}_{\mathit{e}}^{*}$ (${\mathit{m}}_{\mathit{e}}$) | ${\mathit{m}}_{\mathit{h}}^{*}$ (${\mathit{m}}_{\mathit{e}}$) | ${\mathit{E}}_{\mathit{g}}$ (eV) | ${\mathit{d}}^{\mathit{T}}$ (nm) | ${\mathit{d}}^{\mathit{E}}$ (nm) | ${\mathit{n}}_{0}^{\mathit{T}}$ | ${\mathit{n}}_{0}^{\mathit{E}}$ | ${\mathit{T}}_{\mathit{c}}^{\mathit{T}}$ (K) | ${\mathit{T}}_{\mathit{c}}^{\mathit{E}}$ (K) | |
---|---|---|---|---|---|---|---|---|---|

DQW | 0.067 | 0.3 | 1.5 | 15 [5] | 25 [2,3] | <0.7 [5] | − | ∼1 [5] | − |

DMG | 0 | 0 | 0 | 1.0 [11] | 1.0 [12] | − | − | − | − |

DBG | 0.04 | 0.04 | ≤0.25 | 1.4 [14,16] | 1.4 [15] | 7.0 [16] | 8.0 [15] | 17 [14] | 1.5 [15] |

TMD | 0.3–0.5 | 0.4–0.6 | 1.5–2.0 | 1.0 [21] | 1.0 [22] | 150 [21] | 10 [22] | ≳100 [21] | 100 [22] |

**Table 2.**(

**a**) $({E}_{B}^{b}-\delta \lambda )$ in meV for different combinations of TMD monolayers. The ${E}_{B}^{b}$ are calculated for double TMD monolayers with separation $d=1$ nm. $\delta \lambda $ is determined from Table 3. Multicomponent superfluidity is possible only when $({E}_{B}^{b}-\delta \lambda )>0$ (marked in bold). (

**b**) Corresponding two-body binding energy for the $\left\{bb\right\}$ bands: ${E}_{B}^{b}$ in meV.

(a) | p-MoS${}_{2}$ | p-MoSe${}_{2}$ | p-WS${}_{2}$ | p-WSe${}_{2}$ | (b) | p-MoS${}_{2}$ | p-MoSe${}_{2}$ | p-WS${}_{2}$ | p-WSe${}_{2}$ |
---|---|---|---|---|---|---|---|---|---|

n-MoS${}_{\mathbf{2}}$ | 249 | 228 | −37 | −69 | n-MoS${}_{\mathbf{2}}$ | 396 | 405 | 390 | 388 |

n-MoSe${}_{\mathbf{2}}$ | 276 | 253 | −9 | −49 | n-MoSe${}_{\mathbf{2}}$ | 405 | 412 | 400 | 390 |

n-WS${}_{\mathbf{2}}$ | 252 | 232 | −28 | −65 | n-WS${}_{\mathbf{2}}$ | 375 | 385 | 375 | 368 |

n-WSe${}_{\mathbf{2}}$ | 260 | 233 | −24 | −57 | n-WSe${}_{\mathbf{2}}$ | 372 | 375 | 368 | 365 |

${\mathit{m}}_{\mathit{e}}^{*}$ (${\mathit{m}}_{\mathit{e}}$) | ${\mathit{m}}_{\mathit{h}}^{*}$ (${\mathit{m}}_{\mathit{e}}$) | ${\mathit{E}}_{\mathit{g}}$ (eV) | ${\mathit{\lambda}}_{\mathit{c}}$ (eV) | ${\mathit{\lambda}}_{\mathit{v}}$ (eV) | |
---|---|---|---|---|---|

MoS${}_{\mathbf{2}}$ | 0.40 | 0.48 | 1.66 | −0.003 | 0.15 |

MoSe${}_{\mathbf{2}}$ | 0.43 | 0.50 | 1.47 | −0.021 | 0.18 |

WS${}_{\mathbf{2}}$ | 0.33 | 0.30 | 1.79 | 0.027 | 0.43 |

WSe${}_{\mathbf{2}}$ | 0.36 | 0.30 | 1.60 | 0.038 | 0.46 |

Masses Ratio | [0pt]Effective Masses | [0pt]Valence Band Screening | [0pt]Multicomponent Superfluidity | |
---|---|---|---|---|

DQW | ${m}_{h}^{*}\sim $10 ${m}_{e}^{*}$ | 0.07 − 0.3 | no | no |

DMG | ${m}_{h}^{*}={m}_{e}^{*}$ | 0 | yes | no superfluidity |

DBG | ${m}_{h}^{*}={m}_{e}^{*}$ | 0.04 | yes | no |

TMD | ${m}_{h}^{*}\sim $1.3 ${m}_{e}^{*}$ | 0.4 − 0.5 | no | yes |

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**MDPI and ACS Style**

Conti, S.; Neilson, D.; Peeters, F.M.; Perali, A.
Transition Metal Dichalcogenides as Strategy for High Temperature Electron-Hole Superfluidity. *Condens. Matter* **2020**, *5*, 22.
https://doi.org/10.3390/condmat5010022

**AMA Style**

Conti S, Neilson D, Peeters FM, Perali A.
Transition Metal Dichalcogenides as Strategy for High Temperature Electron-Hole Superfluidity. *Condensed Matter*. 2020; 5(1):22.
https://doi.org/10.3390/condmat5010022

**Chicago/Turabian Style**

Conti, Sara, David Neilson, François M. Peeters, and Andrea Perali.
2020. "Transition Metal Dichalcogenides as Strategy for High Temperature Electron-Hole Superfluidity" *Condensed Matter* 5, no. 1: 22.
https://doi.org/10.3390/condmat5010022