# Quadriexciton Binding Energy in Electron–Hole Bilayers

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Hamiltonians and Wave Functions

#### 2.1. Polyexciton Wave Function ${\mathsf{\Psi}}_{NX}^{T}$

#### 2.2. Exciton Wave Function

## 3. Results

#### 3.1. Biexciton Binding Energy

#### 3.2. Quadriexciton Binding Energy

#### 3.3. Quadriexciton Pair Correlation Functions

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Ashcroft, N.W.; Mermin, D.N. Solid State Physics; Holt, Rinehart and Winston: New York, NY, USA, 1976. [Google Scholar]
- Lampert, M.A. Mobile and Immobile Effective-Mass-Particle Complexes in Nonmetallic Solids. Phys. Rev. Lett.
**1958**, 1, 450–453. [Google Scholar] [CrossRef] - Moskalenko, S.A. The theory of Mott exciton in AlkaliGallium cristalls. Opt. Spektrosk.
**1958**, 5, 147–155. [Google Scholar] - Ihn, T. Semiconductor Nanostructures; Oxford University Press: Oxford, UK, 2010. [Google Scholar]
- Wang, J.S.Y.; Kittel, C. Excitonic molecules: A possible new form of chemical bonding. Phys. Lett. A
**1972**, 42, 189–190. [Google Scholar] [CrossRef] - Perali, A.; Neilson, D.; Hamilton, A.R. High-Temperature Superfluidity in Double-Bilayer Graphene. Phys. Rev. Lett.
**2013**, 110, 146803. [Google Scholar] [CrossRef] [PubMed] - Li, J.I.A.; Taniguchi, T.; Watanabe, K.; Hone, J.; Levchenko, A.; Dean, C.R. Negative Coulomb Drag in Double Bilayer Graphene. Phys. Rev. Lett.
**2016**, 117, 046802. [Google Scholar] [CrossRef] - Lee, K.; Xue, J.; Dillen, D.C.; Watanabe, K.; Taniguchi, T.; Tutuc, E. Giant Frictional Drag in Double Bilayer Graphene Heterostructures. Phys. Rev. Lett.
**2016**, 117, 046803. [Google Scholar] [CrossRef] - Liu, X.; Watanabe, K.; Taniguchi, T.; Halperin, B.I.; Kim, P. Quantum Hall drag of exciton condensate in graphene. Nat. Phys.
**2017**, 13, 746–750. [Google Scholar] [CrossRef] - Li, J.I.A.; Taniguchi, T.; Watanabe, K.; Hone, J.; Dean, C.R. Excitonic superfluid phase in double bilayer graphene. Nat. Phys.
**2017**, 13, 751–755. [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] - Tan, M.Y.J.; Drummond, N.D.; Needs, R.J. Exciton and biexciton energies in bilayer systems. Phys. Rev. B
**2005**, 71, 033303. [Google Scholar] [CrossRef] - Meyertholen, A.D.; Fogler, M.M. Biexcitons in two-dimensional systems with spatially separated electrons and holes. Phys. Rev. B
**2008**, 78, 235307. [Google Scholar] [CrossRef] - Lee, R.M.; Drummond, N.D.; Needs, R.J. Exciton-exciton interaction and biexciton formation in bilayer systems. Phys. Rev. B
**2009**, 79, 125308. [Google Scholar] [CrossRef] - De Palo, S.; Rapisarda, F.; Senatore, G. Excitonic Condensation in a Symmetric Electron-Hole Bilayer. Phys. Rev. Lett.
**2002**, 88, 206401. [Google Scholar] [CrossRef] [PubMed] - Maezono, R.; López Ríos, P.; Ogawa, T.; Needs, R.J. Excitons and biexcitons in symmetric electron-hole bilayers. Phys. Rev. Lett.
**2013**, 110, 216407. [Google Scholar] [CrossRef] [PubMed] - De Palo, S.; Tramonto, F.; Moroni, S.; Senatore, G. Quadriexcitons and excitonic condensate in a symmetric electron-hole bilayer with valley degeneracy. Phys. Rev. B
**2023**, 107, L041409. [Google Scholar] [CrossRef] - Reynolds, P.J.; Ceperley, D.M.; Alder, B.J.; Lester, W.A. Fixed-node quantum Monte Carlo for moleculesa) b). J. Chem. Phys.
**1982**, 77, 5593–5603. [Google Scholar] [CrossRef] - Umrigar, C.J.; Nightingale, M.P.; Runge, K.J. A diffusion Monte Carlo algorithm with very small time-step errors. J. Chem. Phys.
**1993**, 99, 2865–2890. [Google Scholar] [CrossRef] - Foulkes, W.M.C.; Mitas, L.; Needs, R.J.; Rajagopal, G. Quantum Monte Carlo simulations of solids. Rev. Mod. Phys.
**2001**, 73, 33–83. [Google Scholar] [CrossRef] - Toulouse, J.; Umrigar, C.J. Optimization of quantum Monte Carlo wave functions by energy minimization. J. Chem. Phys.
**2007**, 126, 084102. [Google Scholar] [CrossRef] - Umrigar, C.J.; Toulouse, J.; Filippi, C.; Sorella, S.; Hennig, R.G. Alleviation of the Fermion-Sign Problem by Optimization of Many-Body Wave Functions. Phys. Rev. Lett.
**2007**, 98, 110201. [Google Scholar] [CrossRef] - Sorella, S.; Casula, M.; Rocca, D. Weak binding between two aromatic rings: Feeling the van der Waals attraction by quantum Monte Carlo methods. J. Chem. Phys.
**2007**, 127, 014105. [Google Scholar] [CrossRef] [PubMed] - Kato, T. On the eigenfunctions of many-particle systems in quantum mechanics. Commun. Pure Appl. Math.
**1957**, 10, 151–177. [Google Scholar] [CrossRef] - Mahan, G.D.; Subbaswamy, K. Local Density Theory of POLARIZABILIY; Plenum Press: New York, NY, USA, 1990; sect. 3.3. [Google Scholar]
- Available online: https://en.wikipedia.org/wiki/Numerov%27s_method (accessed on 8 May 2023).
- Toulouse, J.; Assaraf, R.; Umrigar, C.J. Zero-variance zero-bias quantum Monte Carlo estimators of the spherically and system-averaged pair density. J. Chem. Phys.
**2007**, 126, 244112. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Binding energy of the biexciton ${E}_{B}(2X,d)$ (see Equation (16)) as a function of the interlayer separation d. Together with our results from DMC (solid blue dots), we also report the binding energies from Ref. [13] (open red dots). The line joining the DMC data is only a guide to the eye.

**Figure 2.**Logarithmic plot of the biexciton binding energy ${E}_{B}(2X,d)$ as a function of the interlayer separation d. Data are fitted to the function $1/\mathrm{log}[{E}_{0}(2X,d)/{E}_{B}(2X,d)]=({d}_{c}-d)/D+{({d}_{c}-d)}^{2}/{D}_{1}$ as suggested in Ref. [13]. We compare results from our DMC simulation and results from Ref. [13].

**Figure 4.**Logarithmic plot of the quadriexciton binding energy ${E}_{B}(4X,d)$ as a function of interlayer separation d. $-1/log\left({E}_{B}(4X,d)\right)$ (solid blue dots) is compared with $-1/log\left[{E}_{0}(4X,d)\right]$ (solid dark-red line, see Equation (21)). The DMC data in the range from $d=0.65$ have been fitted to the functions $f\left(d\right)=(A/{d}^{4}){e}^{-D/({d}_{c}-d)}$ (solid blue line) and $q\left(d\right)=A{e}^{-b/({d}_{c}-d)}$ (solid dark green line). In the box, we report the values of the critical distance for unbinding according to the two fitting functions, together with the reduced ${\chi}^{2}$.

**Figure 5.**Electron–hole pair correlation functions for the quadriexciton at several distances d. In panel (

**a**), extrapolated DMC ${g}_{eh}\left(r\right)$ values are shown for distances $d=0.0,0.2,0.4,0.5$ and $0.65\left({a}_{B}^{*}\right)$ with black, blue, orange, red and dark red solid points, respectively. Lines joining the DMC data are only a guide to the eye. In panel (

**b**), we show the quantity $2\pi {\int}_{0}^{r}dttg\left(t\right)$ that sums up to 1 in all cases for the r-ranges considered here.

**Figure 6.**Electron–electron pair correlation functions for the quadriexciton at several distances d. In panel (

**a**), extrapolated DMC ${g}_{ee}\left(r\right)$ values are shown for distances $d=0.0,0.2,0.4,0.5$ and $0.65\left({a}_{B}^{*}\right)$ with black, blue, orange red and dark red solid points, respectively. Lines joining the DMC data are only a guide to the eye. In panel (

**b**), we show the quantity $2\pi {\int}_{0}^{r}dttg\left(t\right)$ that sums up to 1 in all cases for the r-ranges considered here.

**Table 1.**Energy per particle ${E}_{X}/2$ (exciton), ${E}_{2X}/4$ (biexciton) and ${E}_{4X}/8$ (quadriexciton) for various distances d. Excitonic energies were estimated using the Numerov algorithm [25,26]. The energies for the excitonic complexes were obtained from DMC simulations with ${N}_{w}=1760$ walkers and the time-step bias removed by an extrapolation to the zero time step.

d | ${\mathit{E}}_{\mathit{X}}/2$ | ${\mathit{E}}_{2\mathit{X}}/4$ | ${\mathit{E}}_{4\mathit{X}}/8$ |
---|---|---|---|

0.000 | −1.00000000 | −1.096435(6) | −1.32793(5) |

0.100 | −0.76594269 | −0.80450(4) | −0.89400(5) |

0.200 | −0.64517991 | −0.66372(1) | −0.70988(3) |

0.300 | −0.56511027 | −0.57420(1) | −0.59997(3) |

0.400 | −0.50651906 | −0.510687(6) | −0.52533(2) |

0.500 | −0.46112073 | −0.46273(1) | −0.47050(1) |

0.550 | −0.44192983 | −0.442815(6) | −0.448179(6) |

0.600 | −0.42457683 | −0.425001(5) | −0.42844(1) |

0.650 | −0.40878665 | −0.408919(7) | −0.410877(6) |

0.675 | −0.40140710 | −0.401466(8) | −0.402841(9) |

0.700 | −0.39433924 | −0.3943676(9) | −0.39498(8) |

0.710 | −0.39159457 | −0.391622(8) | −0.39202(2) |

0.720 | −0.38889512 | −0.3889012(5) | −0.38909(1) |

0.740 | −0.38362702 | −0.3836290(4) | - |

0.750 | −0.38105610 | −0.3810579(5) | - |

0.760 | −0.37852580 | −0.3785260(1) | - |

0.780 | −0.37358293 | −0.37358300(2) | - |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Malosso, C.; Senatore, G.; De Palo, S.
Quadriexciton Binding Energy in Electron–Hole Bilayers. *Condens. Matter* **2023**, *8*, 44.
https://doi.org/10.3390/condmat8020044

**AMA Style**

Malosso C, Senatore G, De Palo S.
Quadriexciton Binding Energy in Electron–Hole Bilayers. *Condensed Matter*. 2023; 8(2):44.
https://doi.org/10.3390/condmat8020044

**Chicago/Turabian Style**

Malosso, Cesare, Gaetano Senatore, and Stefania De Palo.
2023. "Quadriexciton Binding Energy in Electron–Hole Bilayers" *Condensed Matter* 8, no. 2: 44.
https://doi.org/10.3390/condmat8020044