APS123-QED Excitonic Condensate in Flat Valence and Conduction Bands of Opposite Chirality

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APS/123-QED

Excitonic Condensate in Flat Valence and Conduction Bands of

Opposite Chirality

Gurjyot Sethi1, Martin Cuma2, and Feng Liu1

1Department of Materials Science and Engineering,

University of Utah,

Salt Lake City, Utah 84112, USA

2Center for High Performance Computing,

University of Utah,

Salt Lake City, Utah 84112, USA

(Dated: October 10, 2022)

arXiv:2210.03252v1 [cond-mat.str-el] 6 Oct 2022

Abstract

Excitonic Bose-Einstein condensation (EBEC) has drawn increasing attention recently with the

emergence of 2D materials. A general criterion for EBEC, as expected in an excitonic insulator

(EI) state, is to have negative exciton formation energies in a semiconductor. Here, using exact

diagonalization of multi-exciton Hamiltonian modelled in a diatomic Kagome lattice, we demon-

strate that the negative exciton formation energies are only a prerequisite but insuﬃcient condition

for realizing an EI. By a comparative study between the cases of both a conduction and valence

ﬂat bands (FBs) versus that of a parabolic conduction band, we further show that the presence

and increased FB contribution to exciton formation provide an attractive avenue to stabilize the

EBEC, as conﬁrmed by calculations and analyses of multi-exciton energies, wave functions and re-

duced density matrices. Our results warrant a similar many-exciton analysis for other known/new

candidates of EIs, and demonstrate the FBs of opposite parity as a unique platform for studying

exciton physics, paving the way to material realization of spinor BEC and spin-superﬂuidity.

Excitonic Bose-Einstein condensate (EBEC), ﬁrst proposed in 1960s [1–4], has drawn

recently increasing interest with the emergence of low-dimensional materials where electron

screening is reduced leading to increased exciton binding energy (Eb) [5, 6]. In 1967, Jerome,

et. al. [7], theoretically presented the possibility of an excitonic insulator (EI) phase in a

semi-metal or a narrow gap semiconductor [7–10]. It was shown that the hybridization

gap equation for excitonic condensate order parameter has non-trivial solutions, when Eb

exceeds the semiconductor/semi-metal band gap (Eg). In deep semi-metallic regime, this

gap equation can be solved in analogy to Bardeen-Cooper-Schiﬀer (BCS) superconductor

theory [7, 11]. Due to strong screening of Coulomb potential by the carriers in a semi-metal,

there exists an electron-hole plasma which forms a condensate of weakly paired electrons

and holes at low temperature. On the other hand, in a semiconductor regime, preformed

excitons may condense to form a BEC at low temperatures [7, 11].

This has led to signiﬁcant theoretical [6, 12–19] and experimental [20–32] investigations

into ﬁnding an EI state in real materials. Especially, the EI state in a semiconductor provides

an alternative route to realizing EBEC instead of targeting materials with long-lifetime

excitons, such as optically inactive excitons in bulk Cu2O [33–37] and indirect excitons in

coupled quantum wells [5, 38, 39]. It is worth mentioning that excitonic condensation has

been reported in double layer 2D heterostructures [40–50], where electrons and holes are

separated into two layers with a tunneling barrier in between, and double-layer quantum

Hall systems [51–55] have been shown to exhibit excitonic condensation at low temperature

under a strong magnetic ﬁeld. On the contrary, EIs are intrinsic, i.e., excitonic condensate

stabilizes spontaneously at low temperature without external ﬁelds or perturbations.

However, experimental conﬁrmation of EI state remains controversial [20–32], mainly be-

cause candidate EI materials are very limited. On the other hand, some potential candidate

EIs have been proposed by state-of-the-art computational studies [6, 12–19], based on cal-

culation of single exciton formation energy. It is generally perceived that if single exciton

Ebexceeds the semiconductor Eg, the material could be an EI candidate. But the origi-

nal mean-ﬁeld two-band model studied in Ref. [7] includes inter/intra band interactions,

leading to a non-trivial condensation order parameter, which indicates the importance of

multi-exciton interactions. Hence, in order to ultimately conﬁrm new EI candidates, it

is utmost necessary to analyze and establish the stabilization of multi-exciton condensate

with quantum coherency in the parameter space of multiple bands with inter/intra band

interactions, beyond just negative formation energy for single or multiple excitons.

In this Letter, we perform multi-exciton wave function analyses beyond energetics to

directly assess EBEC for a truly EI state, namely a macroscopic number of excitons (bosons)

condensing into the same single bosonic ground state [56–59]. Especially, we investigate

possible stabilization of EBEC in a unique type of band structure consisting of a pair of

valence and conduction ﬂat bands (FBs) of opposite chirality. These so-called yin-yang FBs

were ﬁrst introduced in a diatomic Kagome lattice [60, 61] and have been studied in the

context of metal-organic frameworks [62] and twisted bilayer graphene [63]. Recently, it

was shown that such FBs, as modelled in a superatomic graphene lattice, can potentially

stabilize a triplet EI state due to reduced screening of Coulomb interaction [6]. However,

similar to other previous computational studies [16–19], the work was limited to illustrating

the spontaneity of only a single exciton formation with a negative formation energy. Here,

using exact diagonalization (ED) of a many-exciton Hamiltonian based on the yin-yang FBs,

in comparison with the case of a parabolic conduction band, we demonstrate that “Eb> Eg”

is actually only a necessary but insuﬃcient condition for realizing an EI state. While both

systems show negative multi-exciton energies, only the former was conﬁrmed with quantum

coherency from the calculation of oﬀ-diagonal long-rang order (ODLRO) of the many-exciton

Hamiltonian. Furthermore, we show that with the increasing FBs contribution to exciton

formation, the excitons, usually viewed as composite bosons made of electron-hole pairs, can

condense like point bosons, as evidenced from the calculated perfect overlaps between the

numerical ED solutions with the analytical form of ideal EBEC wave functions.

A tight-binding model based on diatomic Kagome lattice is considered for the kinetic

energy part of the Hamiltonian, as shown in Fig. 1(a). Our focus will be on comparing the

many-excitonic ground states of superatomic graphene lattice (labelled as EISG), which is

already known to have a negative single exciton formation energy [6], and the ground states

of a model system (labelled as EIP B) with a parabolic conduction band edge, in order to

reveal the role of FBs in promoting an EI state. The interatomic hopping parameters for

the two systems are: t1= 0.532 eV; t2= 0.0258 eV; t3= 0.0261eV for EISG, benchmarked

with density-functional-theory (DFT) results [6, 64], and t1= 0.62 eV; t2= 0.288 eV;

t3= 0.0 eV for EIP B. An interesting point to note here is that for EISG,t2< t3. This

is an essential condition to realize yin-yang FBs in a single-orbital tight-binding model as

has been discussed before, which can be satisﬁed in several materials [60–62]. The insets in

Fig. 1(c) and 1(d) show the band structures for EISG and EIP B , respectively. Coulomb

repulsion between electrons is treated using an extended Hubbard model as

H=Hkin +Hint =X

<r,r0>n

tnc†

rcr0+

<r,r0>n

Vnc†

rcrc†

r0cr0,(1)

where tnis the nth nearest-neighbor (NN) hopping parameter, and Vnis nth NN Hubbard pa-

rameter. Each of the Vnis calculated using the Coulomb potential, U(r > ro) = e2/(4πor),

with a very low dielectric constant (∼1.02) due to the presence of FBs in a 2D lattice

[6] and a cutoﬀ (ro) for onsite interactions. The Hubbard interaction terms are projected

onto all three conduction and valence bands. Spin indices in the Hamiltonian are omitted.

We distinguish triplet and singlet excitonic states by the absence and presence of excitonic

exchange interaction, respectively [64, 65]. The Hamiltonian is exactly diagonalized for a

ﬁnite system size (2 ×3) for converged results [64], which includes 36 lattice sites (equiv-

alent to a 6 ×6 trigonal lattice) with 18 electrons for a half-ﬁlled intrinsic semiconductor.

With Neh number of electrons (holes) in conduction (valence) bands, exciton population

(EP) is deﬁned as Neh divided by the total number of allowed reciprocal lattice points (i.e.,

2×3 = 6). Throughout this work we focus on the ground state of Eqn. 1 with varying

EPs.

(a)

−0.25 0.00 0.25 0.50 0.75

Tri plet Ef(eV)

a. u.

(c)

𝐄𝐈𝐒𝐆

−0.20.00.20.4

Tri pl et Ef(eV)

a. u.

𝐄𝐈𝐏𝐁

−1

E(eV)

MΓKM

0.0

0.5

Triplet Singlet

Ef(eV)

BSE-GW

TB-ED

(d)

(b)

1.0

−0.2

0.0

MΓKM

E(eV)

E (eV)

FIG. 1. (a) Schematic of diatomic Kagome lattice with ﬁrst three NN hopping integrals labelled as

t1,t2, and t3, respectively. (b) Single exciton Efcalculated using ED (blue bars) compared with

GW-BSE results [6] (red bars) for EISG. (c), and (d) Triplet excitonic density of states for EISG,

and EIP B respectively. Excitonic states with negative and positive formation energies are shown

in yellow and orange, respectively. Inset shows the band excitation contributions to the ﬁrst triplet

level, indicated by the width of bands in red for EISG ((c)) and green for EIP B ((d)), respectively.

We ﬁrst calculate the energies and wavefunctions for a single exciton, i.e., Neh = 1

(EP=1/6), to benchmark the single-exciton results of EISG with those obtained using ﬁrst-

principles GW-BSE method for this lattice [6]. Importantly, our model calculation results,

especially the trends of exciton levels, match very well with GW-BSE (Fig. 1(b), Fig.

S2 [64]). One clearly sees in Fig. 1(b) for EISG that the formation of triplet exciton is

spontaneous with a negative formation energy (Ef), while that of singlet is positive. These

key agreements validate our model for further analysis. In Fig. 1(c) and 1(d), we plot

triplet excitonic density of states for EISG and EIP B , respectively. Both systems have a

negative lowest triplet Ef, indicative of the possibility that both systems can be a triplet

EI. The insets of Fig. 1(c) and 1(d) show the band excitation contribution to the lowest

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APS/123-QEDExcitonicCondensateinFlatValenceandConductionBandsofOppositeChiralityGurjyotSethi1,MartinCuma2,andFengLiu11DepartmentofMaterialsScienceandEngineering,UniversityofUtah,SaltLakeCity,Utah84112,USA2CenterforHighPerformanceComputing,UniversityofUtah,SaltLakeCity,Utah84112,USA(Dated:October10,2...

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