Twist-dependent intra- and interlayer excitons in moir e MoSe 2homobilayers Viviana Villafa ne1Malte Kremser1Ruven H ubner2Marko M. Petri c3Nathan P. Wilson1 Andreas V. Stier1Kai M uller3Matthias Florian4Alexander Steinho2and Jonathan J. Finley1y

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Twist-dependent intra- and interlayer excitons in moir´e MoSe2homobilayers

Viviana Villafa˜ne,1, ∗Malte Kremser,1, ∗Ruven H¨ubner,2Marko M. Petri´c,3Nathan P. Wilson,1

Andreas V. Stier,1Kai M¨uller,3Matthias Florian,4Alexander Steinhoﬀ,2and Jonathan J. Finley1, †

1Walter Schottky Institut and Physik Department,

Technische Universit¨at M¨unchen, Am Coulombwall 4, 85748 Garching, Germany

2Institut f¨ur Theoretische Physik, Universit¨at Bremen, P.O. Box 330 440, 28334 Bremen, Germany

3Walter Schottky Institut and Department of Electrical and Computer Engineering,

Technische Universit¨at M¨unchen, Am Coulombwall 4, 85748 Garching, Germany

4University of Michigan, Dept. of Electrical Engineering and Computer Science, 48109 Ann Arbor, MI, USA

Optoelectronic properties of van der Waals homostructures can be selectively engineered by the

relative twist angle between layers. Here, we study the twist-dependent moir´e coupling in MoSe2

homobilayers. For small angles, we ﬁnd a pronounced redshift of the K-Kand Γ-Kexcitons

accompanied by a transition from K-Kto Γ-Kemission. Both eﬀects can be traced back to the

underlying moir´e pattern in the MoSe2homobilayers, as conﬁrmed by our low-energy continuum

model for diﬀerent moir´e excitons. We identify two distinct intralayer moir´e excitons for R-stacking,

while H-stacking yields two degenerate intralayer excitons due to inversion symmetry. In both cases,

bright interlayer excitons are found at higher energies. The performed calculations are in excellent

agreement with experiment and allow us to characterize the observed exciton resonances, providing

insight about the layer composition and relevant stacking conﬁguration of diﬀerent moir´e exciton

species.

Van der Waals (vdW) homo- and heterostructures

formed from monolayer transition metal dichalcogenides

(TMDs) are unique semiconductor systems in which light

couples to electronic and spin excitations with poten-

tial for novel optoelectronic and valleytronic applications

[1,2]. In angle-aligned TMD heterostructures, correlated

insulating states have been shown to emerge for diﬀerent

fractional charge ﬁllings of moir´e superlattice sites, en-

abling investigation of quantum many-body states [3–8]

with the potential for optical measurement and coher-

ent control of strongly-correlated phases [9–14]. Twisted

homostructures not only allow for controlled tuning of

the underlying moir´e superlattice potential but also en-

hanced formation of hybridized minibands due to the ab-

sence of lattice and energy mismatch in the constitutive

monolayers [15]. Until now, few studies have appeared

on the angle-dependent optical and electronic properties

of twisted MoSe2homobilayers including photolumines-

cence (PL) experiments exhibiting moir´e-trapped trions

for twist angles close to 0°and 60°[16]; and a static elec-

tric dipole moment characterization of diﬀerent excitonic

species on a single MoSe2bilayer with a ﬁxed twist an-

gle of 0°[17]. Recent reports on WS2[18] and MoS2

[19,20] showed that the K-Kexciton transition [18–20]

is insensitive to twist angle.

We combine optical spectroscopy on hBN-encapsulated

MoSe2homobilayers with theory to obtain new informa-

tion about the twist angle dependent optoelectronic re-

sponse. Both, direct K-Kand indirect Γ-Kexcitons,

exhibit an abrupt decrease of the emission energy in the

vicinity of 0°or 60°. This rapid change of exciton en-

ergy is accompanied by the appearance of indirect ex-

citon PL below 8°and above 54°and vanishing direct

exciton PL. Our theoretical predictions based on an ab-

initio-based continuum model are in excellent agreement

with experiment, showing that the transition between dif-

ferent regimes of emission energy can be understood in

terms of increasing exciton localization in moir´e sites at

small twist angles. Furthermore, our model allows us to

characterize the observed exciton resonances, providing

insight into their layer composition and resulting binding

energies.

Samples were prepared using a tear-and-stack tech-

nique [21] combined with a modiﬁed version of the hot-

pick-up method [22,23] to assemble twisted MoSe2ho-

mobilayers with relative stacking angle ∆θ(see the Sup-

plemental Material (SM)). Fig. 1(a) presents optical PL

emission from MoSe2twisted bilayers in the range 0°≤

∆θ≤60°, probed at 10 K using 500 nW of continuous-

wave 532 nm excitation laser focused onto a diﬀraction-

limited spot (100×objective, NA=0.7). PL spectra in

Fig. 1(a) are plotted as a function of neutral K-Kexci-

ton (X0) detuning between homo- and monolayer regions

and normalized with respect to X0intensity measured

in the respective monolayers. We observe that X0and

trion (XT) energies are redshifted relative to the mono-

layer X0. One striking feature of the data presented in

Fig. 1(a) is the complete absence of X0and XTemission

for bilayers having twist angles in the ranges 0°to 8°and

54°to 60°.

Fig. 1(b) presents diﬀerential reﬂectivity measure-

ments for the same series of samples discussed in

Fig. 1(a). We note that the reﬂectivity data reveals clear

signatures of the X0transition from samples having small

twist angles (0°to 8°and 54°to 60°) which could not

be observed in PL experiments due to the low quan-

tum yield of these transitions [24–26]. To analyze the

reﬂectivity spectra, we modeled the refractive index of

arXiv:2210.12076v1 [cond-mat.mes-hall] 21 Oct 2022

-150 -100 -50 0

0.0

0.1

0.2

0.3

0.4

PL intensity (norm. to ML exc.)

Relative energy (meV)

0 5 55 60

1.38

1.40

1.42

1.44

1.46

1.48

1.50

Energy (eV)

Twist angle (�)

0°

54°

60°

Energy shift (meV)

×5

×5 ×0.5

×20

×0.2

ML X0ML X0

0°

12°

20°

30°

40°

54°

8°

60°

0°

12°

20°

30°

40°

54°

60°

8°

DR X0

PL X0

PL XT

�

PL Xind

×5

×10

(a) (b) (c)

(e)

(d)

×0.1

-75 -50 -25 0 25

-1

Differential reflectivity

Relative energy (meV)

FIG. 1: Photoluminescence (a) and reﬂectivity (b) spectra for MoSe2homobilayers with diﬀerent twist angles. The energies

are given as the detuning with respect to the neutral exciton in the monolayer region of each sample, labeled by dotted lines.

Black spectra in both panels show an exemplary monolayer signal. (c) Energy shift between monolayer and bilayer regions

for excitons and trions extracted from the data presented in panels (a) and (b). Green (blue) data points represent the

neutral exciton shift extracted from reﬂectivity (PL) measurements. Red data points represent trion shifts extracted from PL

measurements. Error bands for the extracted spectral positions are respectively color coded with the reﬂectivity measurements

error band being extended across all panels for comparison. Negative shifts correspond to an energy reduction in the bilayer

region with respect to the monolayer region. Indirect exciton (Xind) PL spectra (d) and extracted energies (e) as a function

of twist angle of the MoSe2homobilayers.

the MoSe2monolayer as a single Lorentz oscillator, ex-

cept in the case of 0°where we used two oscillators, and

calculated the diﬀerential reﬂectivity using the transfer-

matrix method (see the SM). The simulated reﬂectivity

curves are shown as diﬀuse color lines in Fig. 1(b) with

good agreement between experiment and theory. The

reﬂectivity of 0°, 8°and 20°show a double-peak struc-

ture predicted by our theoretical model steaming from

diﬀerent hybridized K-Kexcitons, as we will see be-

low. We evaluated the average energies of X0and XT,

as obtained from PL and reﬂectivity measurements. In

Fig. 1(c), we present the energy shift as a function of

the stacking angle, relative to the energy of the corre-

sponding excitonic transition (X0or XT) in the mono-

layer. Green data points within the green error band in

the uppermost panel in Fig. 1(c) denote relative X0en-

ergies measured via diﬀerential reﬂectivity. Blue and red

data points in the middle and bottommost panel show

the energy shift of X0and XTrespectively, determined

from PL measurements. At large twist angles (away from

0°and 60°), X0and XTexhibit similar behavior: Both

are redshifted by ∼20 meV with respect to the monolayer

emission with only a minor twist angle dependence. The

observed energetic reduction for X0and XTtransitions

in the homobilayer region can be explained considering

two fundamentally diﬀerent eﬀects: (i) hybridization of

electronic states between layers [27,28] and (ii) static di-

electric screening induced by hBN encapsulation and the

mutual proximity of the second MoSe2monolayer [29–

32]. In fact, X0energy has a strong sensitivity to its

dielectric environment enhanced by the bilayers reduced

dimensionality and dielectric constant mismatch between

hBN and MoSe2[33]. Consequently, for stacking angles

around ∆θ'30°, static screening is the dominant eﬀect

inducing a constant redshift in the X0and XTenergies

[31,32,34,35]. Remarkably, as twist angles of 0°or

60°are approached the redshift increases to ∼40 meV.

In this case, hybridization eﬀects between the layers are

dominant and twist-angle dependent. Figs. 1(d) and (e)

present measured PL spectra and energies for indirect

Γ-Kexcitons (Xind), appearing 200 meV below the ob-

served X0transition. We observe a strong energy depen-

dence as a function of stacking angle, similarly as the

one found for X0and XT. For twist angles in the ranges

0°to 8°and 54°to 60°, the decrease in the intensity of

the X0PL emission is accompanied by an increase on

PL emission into these Xind momentum indirect states

at lower energies [17]. As we now continue to show, the

experimentally-observed X0and XTenergetic shifts and

Xind PL emission increase for low stacking angles orig-

inate from moir´e physics, a behaviour predicted by our

low-energy continuum model.

For small stacking angles, the moir´e pattern can be

seen as a smooth variation of the stacking conﬁgurations

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Twist-dependentintra-andinterlayerexcitonsinmoireMoSe2homobilayersVivianaVillafa~ne,1,MalteKremser,1,RuvenHubner,2MarkoM.Petric,3NathanP.Wilson,1AndreasV.Stier,1KaiMuller,3MatthiasFlorian,4AlexanderSteinho,2andJonathanJ.Finley1,y1WalterSchottkyInstitutandPhysikDepartment,TechnischeUniversita...

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Twist-dependent intra- and interlayer excitons in moir e MoSe 2homobilayers Viviana Villafa ne1Malte Kremser1Ruven H ubner2Marko M. Petri c3Nathan P. Wilson1 Andreas V. Stier1Kai M uller3Matthias Florian4Alexander Steinho2and Jonathan J. Finley1y.pdf

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Twist-dependent intra- and interlayer excitons in moir e MoSe 2homobilayers Viviana Villafa ne1Malte Kremser1Ruven H ubner2Marko M. Petri c3Nathan P. Wilson1 Andreas V. Stier1Kai M uller3Matthias Florian4Alexander Steinho2and Jonathan J. Finley1y

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