# Fileset [PhysRevX.12.041005.pdf](https://mdr.nims.go.jp/filesets/e770e5a5-2862-4484-a2c6-a65785b4d006/download) ## Creator Jing Liang, Dongyang Yang, Jingda Wu, Jerry I. Dadap, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Ziliang Ye ## Rights [Creative Commons BY Attribution 4.0 International](https://creativecommons.org/licenses/by/4.0/) ## Other metadata [Optically Probing the Asymmetric Interlayer Coupling in Rhombohedral-Stacked

{MoS}_{2}

Bilayer](https://mdr.nims.go.jp/datasets/89f3190f-5624-43f2-add9-4ee00883ac96) ## Fulltext Optically Probing the Asymmetric Interlayer Coupling in Rhombohedral-Stacked MoS2 BilayerOptically Probing the Asymmetric Interlayer Coupling in Rhombohedral-StackedMoS2 BilayerJing Liang ,1,2,* Dongyang Yang ,1,2,* Jingda Wu ,1,2,* Jerry I. Dadap ,1,2 Kenji Watanabe ,3Takashi Taniguchi ,4 and Ziliang Ye 1,2,†1Quantum Matter Institute, The University of British Columbia,Vancouver, British Columbia V6T 1Z4, Canada2Department of Physics and Astronomy, The University of British Columbia,Vancouver, British Columbia V6T 1Z1, Canada3Research Center for Functional Materials, National Institute for Materials Science,1-1 Namiki, Tsukuba 305-0044, Japan4International Center for Materials Nanoarchitectonics, National Institute for Materials Science,1-1 Namiki, Tsukuba 305-0044, Japan(Received 5 May 2022; revised 4 August 2022; accepted 14 September 2022; published 14 October 2022)The interlayer coupling is emerging as a new parameter for tuning the physical properties of two-dimensional (2D) van der Waals materials. When two identical semiconductor monolayers are stacked witha twist angle, the periodic interlayer coupling modulation due to the moiré superlattice may endow exoticphysical phenomena, such as moiré excitons and correlated electronic phases. To gain insight into thesenew phenomena, it is crucial to unveil the underlying coupling between atomic layers. Recently, therhombohedral-stacked transition metal dichalcogenide (TMD) bilayer has attracted significant interestbecause of the emergence of an out-of-plane polarization from nonferroelectric monolayer constituents.However, as a key parameter responsible for the physical properties, the interlayer coupling and itsrelationship with ferroelectricity remain elusive. Here, we probe the asymmetric interlayer couplingbetween the conduction band of one layer and the valence band from the other layer in a 3R-MoS2 bilayer,which can be understood as a result of a layer-dependent Berry phase winding. By performing opticalspectroscopy in a dual-gated device, we show an effective type-II band alignment exists at K points in the3R-MoS2 bilayer. Furthermore, by unraveling various contributions to the band offset, we quantitativelydetermine the asymmetric interlayer coupling and spontaneous polarization in 3R-MoS2. Our results unveilthe physical nature of stacking-induced ferroelectricity in TMD homostructures and have importantimplications for moiré physics in 2D semiconductors.DOI: 10.1103/PhysRevX.12.041005 Subject Areas: Condensed Matter Physics, GrapheneSemiconductor PhysicsInterlayer coupling, a ubiquitous ingredient in van derWaals (vdW) materials, offers unprecedented freedom totailor the electronic band structure of two-dimensional (2D)materials. Through precise control of the layer orientationin an artificial 2D assembly, one can create a periodicallymodulated interlayer coupling in moiré superlattice, whichleads to many intriguing phenomena different from theirmonolayer substituents [1–6]. In semiconducting transitionmetal dichalcogenides (TMDs), such moiré potentials canquench the kinetic energy of electrons and localize them atthe energy local minimum of the moiré superlattice, givingrise to a new platform for creating quantum emitter arraysand simulating various correlated physics [7–17]. In orderto have a full grasp over these emerging phenomena, it istherefore important to understand the underlying interlayercoupling at different localization sites with different layerorientations.Recently, parallel stacked TMD layers have attractedsignificant attention, as they have a broken mirror sym-metry with an out-of-plane spontaneous electrical polari-zation [18–25]. In an artificially stacked TMD bilayer witha marginal twist angle, the crystal structure may sponta-neously relax into multiple rhombohedral domains withalternating polarization [26], which can be electricallyswitched via in-plane sliding motion. As a result, therhombohedrally (R) stacked TMD becomes a new ferro-electric semiconductor with a promising range of electronicand optoelectronic applications [24,25,27]. Here, we*These authors contributed equally to this work.†Corresponding author.zlye@phas.ubc.caPublished by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI.PHYSICAL REVIEW X 12, 041005 (2022)2160-3308=22=12(4)=041005(10) 041005-1 Published by the American Physical Societyhttps://orcid.org/0000-0001-6348-2068https://orcid.org/0000-0002-0151-3102https://orcid.org/0000-0002-6783-8719https://orcid.org/0000-0001-5100-9396https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0002-1467-3105https://orcid.org/0000-0001-8314-6977https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevX.12.041005&domain=pdf&date_stamp=2022-10-14https://doi.org/10.1103/PhysRevX.12.041005https://doi.org/10.1103/PhysRevX.12.041005https://doi.org/10.1103/PhysRevX.12.041005https://doi.org/10.1103/PhysRevX.12.041005https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/optically probe the spontaneous polarization and interlayercoupling in an R-stacked TMD bilayer. Unlike in thecommon hexagonal polytype, the interlayer coupling inthe rhombohedral polytype is interestingly asymmetric:The conduction band in one layer couples only to thevalence band in the other, but not vice versa, due to thelayer-dependent Berry phase effect. Such an asymmetriccoupling is also the electronic origin of spontaneouspolarization and interlayer potential in TMDs with naturalrhombohedral crystal structure or artificial parallel stackingwith a marginal twist. Here we perform both electric-fieldand doping-dependent optical spectroscopy in a dual-gateddevice made of a homogeneous bilayer MoS2 exfoliatedfrom a chemically synthesized 3R crystal.In a 3R-MoS2 bilayer, the adjacent layers are stacked inthe same direction [Fig. 1(a)]. A relative interlayer dis-placement along the armchair direction gives two inequi-valent stacking configurations: BA or AB stacking order,where the S atom in the top layer lies directly on the top ofthe Mo atom in the bottom layer for BA stacking order[28,29]. The parallel stacking direction maintains thebroken inversion symmetry of the monolayer, while thelateral shift breaks the mirror symmetry between the topand bottom layer, resulting in the C3v point group of the3R-MoS2 bilayer. Under C3 rotational operation, the geo-metric phase of the Bloch states at K points, the high-symmetry points at the Brillion zone edge, has twocontributions: C3ψK ¼ e−ðmþm0Þð2π=3ÞψK , where m is themagnetic quantum number associated with the rotation ofthe atomic orbital andm0 is associated with the Berry phasewhen hopping occurs from one site to the next, which isdependent on the rotation center [Fig. 1(b)] [30–34]. Thesummþm0 is the total azimuthal quantum number (AQN).The dependence of the Berry phase on the rotation centerhas a profound impact on interlayer coupling in 3R-MoS2.In the monolayer, the AQNs between the conduction andvalence bands are different by þ1 or −1, depending on thevalley index, which gives rise to the well-known valleyselection rule [30]. In the 3R-MoS2 bilayer, the lateral shiftleads to a distinct rotational center of two layers, whichcauses a different Berry phase in different layers [Fig. 1(b)].Take the BA-stacked 3R-MoS2 bilayer as an example:(e)(a) (b)(d)(c)FIG. 1. Schematic of asymmetric interlayer coupling in the 3R-MoS2 bilayer. (a) Side view of atomic structures of the 3R-MoS2bilayer with AA (left) and BA (right) stacking configurations, where the yellow and green spheres denote the S and Mo atoms,respectively. (b) Schematic of the phase winding at K points in monolayer MoS2 when the C3 rotational center is located at a Mo site(left) and the hollow center of the hexagon formed by Mo and S (right). The middle is the top view of the atomic structure of the3R-MoS2 bilayer with BA stacking configuration. The small, dashed orange circle denotes the C3 rotational center. (c) Electronic bandstructure at K points of a 3R-MoS2 bilayer with BA stacking configuration. Yellow (green) dashed lines denote the original uncoupledbands. Yellow (green) solid lines denote the bands of the top (bottom) MoS2 layer. Numbers denote the overall azimuthal quantumnumber (AQN) of the conduction or valence band edge at K points. Interlayer coupling t exists only between the conduction band of thetop layer and the valence band of the bottom layer, resulting in a level repulsion t2=Eg between these two bands, where Eg is the directband gap of MoS2. (d) Schematic of spontaneous out-of-plane polarization P pointing from the top layer (yellow) to the bottom layer(green) in a 3R-MoS2 bilayer with BA stacking configuration. (e) Schematic of direct band gap at the K point of the top (yellow) andbottom (green) layer in a 3R-MoS2 bilayer with BA stacking configuration. The atomic structure shows the nonequivalent environmentsof the Mo atom in the top and bottom MoS2 layers.JING LIANG et al. PHYS. REV. X 12, 041005 (2022)041005-2The states at K points in the bottom layer obtain an extraquantum number from the Berry phase of m0 ¼ þ1,resulting in the overall AQN ofþ1 and 0 for the conductionand valence bands [Fig. 1(c)]. In contrast, the AQN of theconduction and valence bands in the top layer remains 0and −1. Since electrons can tunnel only between bands ofthe same AQN, the interlayer coupling t exists onlybetween the conduction band of the top layer and thevalence band of the bottom layer at K points. Neither theconduction nor the valence bands at K points are directlycoupled between the two layers, so the band edges can bedistinctively defined for each layer in the 3R-MoS2 bilayerat K points, in contrast to the strongly hybridized valenceband at the Γ point. Such an interesting asymmetricinterlayer coupling induces a level repulsion between thetwo coupled bands and splits the layer degeneracy, resultingin a staggered gap atK points where the valence band edge atthe Brillouin zone corner is localized in the top layer whilethe conduction band edge is localized in the bottom layer,which we refer to as an effective type-II band alignment atKpoints in a 3R-MoS2 homobilayer [Fig. 1(c)] [27,33–35].On the other hand, the asymmetric interlayer coupling alsomixes the Bloch wave functions from different layers, whichcauses the Wannier center of the valence band from thebottom layer to shift toward the top layer while the valenceband of the top layer is unchanged. Consequently, the overallWannier center of all occupied valence band states shiftsto the top layer, resulting in a downward out-of-planepolarization (P) according to modern Berry phase theory[Fig. 1(d)] [33,34,36–38]. The interlayer electrostatic poten-tial associated with the polarization, ϕ0 ¼ Pd0=ε0εm, gen-erates an additional band offset in the 3R-MoS2 bilayer,where ε0 and εm denote the vacuum permittivity and the out-of-plane dielectric constant of MoS2, respectively, and d0 isthe interlayer distance.Last but not least, the local chemical environments of thetwo layers are nonequivalent due to the lateral shift. In theBA stacking, the Mo atom in the bottom layer lies directlybeneath the S atom in the top layer, but the Mo atom in thetop layer does not coincide with the S atom in the bottomlayer. Such an asymmetric atomic configuration is knownfor causing some high-order electronic coupling, whichleads to a larger band gap in the bottom layer than in the top[33,35]. The band-gap energy difference δ is observed inboth artificial and natural TMD bilayers through opticalspectroscopy [Fig. 1(e)]. As a result, the total band offset atthe K points in a 3R-MoS2 bilayer has three contributions:the asymmetric interlayer coupling strength t, the polari-zation-induced intrinsic interlayer potential ϕ0, and theband-gap difference δ, which, according to our effectivefour-band model, can be approximately expressed as(Supplemental Fig. S1 and Note 1 [39])Δc ¼ eϕ0 þt2Edg− δ2; ð1ÞΔv ¼ Δc þ δ ¼ eϕ0 þt2Edgþ δ2: ð2ÞHere, Δc and Δv are the conduction and valence bandoffsets, respectively, and Edg denotes the direct band gap ofthe top layer MoS2. Clearly, if we measure the total bandoffsets and unravel the contributions from each term, thecoupling constant t can be determined.In our experiment, we determine each contribution usingdoping and field-dependent optical spectroscopy. A bilayerMoS2 exfoliated from bulk 3R crystal (HQ graphene Inc.)is employed in this work. In contrast to artificial stacks, thesample is uniformly stacked with a BA stacking order,which is assigned by the interlayer exciton dipole direction.A dual-gated device is fabricated from exfoliated vdWmaterials using a layer-by-layer dry-transfer method [40],which allows independent control of the doping density (n)and the vertical electric field (Ez) between two layers (seeMethods in Supplemental Material [39]). Figure S2 illus-trates the schematic device structure and the optical imageof the dual-gated device employed in this study. Accordingto the thickness ratio of the bottom hexagonal boron nitride(h-BN) gate and the top h-BN gate, one can introduce theexternal doping or electric field by applying a top gate (Vt)in proportion to the bottom gate (Vb) voltage with the sameor opposite polarity (Supplemental Note 2 [39]).We first investigate the optical response of the 3R-MoS2bilayer as a function of carrier density by reflectance contrast(RC) spectroscopy (see Methods in Supplemental Material[39]). All measurements are performed at 8 K. We also takethe energy derivative of the RC to highlight small features[Fig. 2(b)]. As shown in Fig. 2(a), the doping-dependent RCspectrum can be divided into three regions. In region II, theFermi level is inside the band gap of both layers. As in thecase of its monolayer counterpart, the 3R-MoS2 bilayerexhibits two prominent exciton peaks, termed A and Bexcitons, associated with the direct optical transition at theK points [41]. In contrast to the monolayer, the intralayer Aexciton is split into two peaks: the low-energy A exciton at1.918 eV (XAt) and the high-energy A exciton at 1.929 eV(XAb). Such a peak splitting (δ ¼ 11.0� 0.5 meV) isobserved in the artificial and natural TMD bilayers and isattributed to the intralayer exciton emission from two layers,which is a powerful tool for probing theoptical response fromdifferent layers. Here, the photon energy difference of theintralayer excitons is directly read from thepeak separation inthe RC spectra, since the gate-dependent RC spectra arecomplex due to the h-BN encapsulation and the presence oftop and bottom gates. The uncertainty of the band-gapdifference is determined by the spectral resolution of0.5 meV. Furthermore, the full width at half maximum ofthe intralayer exciton in the 3R-MoS2 bilayer is about12 meV, broader than that of the monolayer MoS2 [42],which could arise from the phonon scattering-inducedexciton lifetime reduction, as the MoS2 bilayer is anOPTICALLY PROBING THE ASYMMETRIC INTERLAYER … PHYS. REV. X 12, 041005 (2022)041005-3indirect-gap semiconductorwith a valence bandmaximumatthe Γ point.In the electron doping region I, the electron dopingdependences of the two A excitons XAt and XAb are clearlydifferent [Figs. 2(a) and 2(b)]. With increasing electrondoping density, the high-energy XAb blueshifts and transfersits oscillator strength to another emergent peak on the low-energy side of XAt. The observation can be explained by theinteraction between the exciton and degenerate Fermi sea(FS) of excess charge carriers. As shown in Fig. 2(c), theXAbexcitons are dressed by excitations (electron-hole pairs) ofthe FS and split into two branches: a low-energy attractiveexciton polaron (XP−0Ab) and a high-energy repulsive excitonpolaron (XP−Ab) [43–47]. Around the transition area of regionI and region II, we estimate the binding energy of XP−0Ab to be25 meV. With increasing electrostatic gating, the exciton-FSinteraction is enhanced due to the expanding FS, whichenlarges the repulsion between the XP−Ab and XP−0Ab.Consequently, the XP−0Ab redshifts whereas the XP−Ab blue-shifts with increasing electron density, and the high-energyrepulsive exciton-polaron branchXP−Ab transfers its oscillatorstrength to the low-energy attractive exciton-polaron branchXP−0Ab,which is consistentwith previous studies ofmonolayerMoS2 [41]. On the other hand, the intralayer A exciton fromthe top layer XAt remains largely unchanged except for aslight initial redshift, potentially due to theweak screening ora Pauli-blocking-free exciton polaron effect with the chargesin the other layer [Fig. 2(c)]. The different doping depend-ence of the two A excitons suggests that the electrons aredoped into the layer with the larger band gap, indicating atype-II band alignment at the K points of the 3R-MoS2bilayer [48].In contrast to the distinctive response in electron doping,the two MoS2 layers respond similarly to hole doping. Thetwo prominent exciton peaks XAt and XAb are quicklyreplaced by two emergent peaks with lower energy, whichredshift together with the increasing negative voltage[Figs. 2(a) and 2(b)]. Such a doping dependence arisesfrom the valence band maximum at the Γ point in the3R-MoS2 bilayer, where the two layers hybridize strongly.When holes are doped into the bilayer, they become nearlyequally distributed between two layers. As a result, bothXAt and XAb excitons are dressed by electron-hole pairs ofthe FS at the Γ point and become split into low-energy(c)(b)(a)FIG. 2. Doping-dependent reflectance contrast spectrum of intralayer excitons in the 3R-MoS2 bilayer. (a) Contour plot of the doping-dependent reflectance contrast spectra of the 3R-MoS2 bilayer. The gate voltage is applied proportionally on top and bottom gates withrelation Vb ¼ 0.85Vt. The right vertical axis denotes the corresponding doping density (positive n for electron doping and negative n forhole doping). The black dashed lines divide the spectrum into three regions when the Fermi level is in the (I) conduction band, (II) bandgap, and (III) valence band. (b) First energy derivative of the reflectance contrast spectrum in arbitrary units (arb. units) at different topgate voltages. Dashed gray lines denote different exciton polarons in region I and region III. (c) The electronic band structure of the3R-MoS2 bilayer with BA stacking configuration at region I (top) or region III (bottom) in (a). The yellow (green) color denotes theelectronic states localized in the top layer (bottom layer). The dashed gray line denotes the Fermi level. In region I, electrons are dopedinto the conduction band at the K point of the bottom layer. XP−0Ab and XP−Ab represent attractive and repulsive exciton polarons,respectively, from the bottom layer when the intralayer excitons XAb are dressed with electron-hole pairs in the FS. In region III, holesare doped into the Γ point. XPþ0Ai and XPþAi represent attractive and repulsive exciton polarons, respectively. The subscript i is t or b,which denotes the optical transitions at the K points from the top or bottom layers.JING LIANG et al. PHYS. REV. X 12, 041005 (2022)041005-4attractive exciton polarons (XPþ0At and XPþ0Ab) and high-energy repulsive exciton polarons (XPþAt and XPþAb), asshown in the first energy derivative of the RC spectra[Figs. 2(b) and 2(c)]. The more abrupt oscillator strengthevolution than that in the electron-doping region mightoriginate from the higher density of states at the Γ pointassociated with the larger effective mass. Since XPþAt isexpected to have similar energy as XPþ0Ab at low doping andshould rapidly diminish with increasing doping, we cannotdistinguish them in the first derivative spectra, but theiropposite doping dependence can be found in the secondderivative spectra (Fig. S3 [39]). As hole doping densityfurther increases, the FS expands, and, consequently, theXPþ0At and XPþ0Ab redshift at the same rate. The bindingenergy of XPþ0At (XPþ0Ab) at a low doping density limit isestimated to be 18 meV (11 meV). The binding energydifference could be caused by the finite layer polarizationof the hole at the Γ point due to the interlayer potential [35].Our doping assignment agrees with the photoluminescence(PL) spectra in Fig. S4 [39]. The doping dependence ofoptical response could also be explained by the trion modelwithout affecting the main conclusion of this work[41,48–50].Next, we utilize the field dependence of the RC spectrain the electron-doping region I to determine the intrinsicinterlayer potential ϕ0. Since the exciton is mainly dressedwith the electron doped in the same layer to form excitonpolarons, as shown in Fig. 2, the oscillator strengths of theexcitons and exciton polarons become efficient probes ofthe doping imbalance between two layers, which isdetermined by the conduction band offset Δc. As a partofΔc arises from the interlayer potential associated with theelectric field between two layers, one can tune Δc byapplying an external electric field with an antisymmetrictop and bottom gate, ΔV ¼ Vt − Vb. When Δc becomeszero, the doped electrons are equally distributed in twolayers, and XAt and XAb should have the same oscillatorstrengths. The specific relationship between Δc and ΔVdepends on the ratio between the averaged gate capacitanceC̄ and the geometric and quantum capacitance of MoS2 (Cmand Cq, respectively), which can be calculated from thefilm thickness and dielectric constants (Supplemental Note2, Fig. S5 [39]). In particular, when the Fermi level is in theconduction band of both layers,ΔcðΔVÞ≈2eCm2CmþCqþ2C̄ϕ0− 2eC̄2CmþCqþ2C̄ΔV; ð3Þwhere e is the electron charge. We note that Δc has adifferent ϕ0 dependence when the Fermi level is inside thegap of one layer or two. Only when the Fermi level is in theconduction bands of both layers can the intrinsic interlayerpotential ϕ0 be obtained by measuring ΔV at Δc ¼ 0condition (Supplemental Note 2 [39]).We now consider the electric field dependence of the RCspectra at a fixed electron doping density (n ¼ 6×1011 cm−2). As Vt is swept between 5.5 and 9.5 V, Vb isalways0.85 times smaller thanVtwith anopposite polarity tocompensate for the h-BN thickness difference [Fig. 3(a)].When the external field is small (Vt < 6.5 V), the electronsare located in the bottom layer. So the exciton in the top layer(XAt) is observed together with the exciton polaron in thebottom layer (XP−0Ab andXP−Ab), similar to region I inFig. 2(a).With increasing electric field, the offset between two con-duction bands decreases, until the Fermi level reaches theconduction band edge of the top layer (Vt ¼ 6.5 V). Fromthis point on, the electron begins to migrate from the bottomlayer to the top layer [Fig. 3(d)]. The interaction betweenXAtand FS in the top layer forms an attractive and repulsiveexciton polaron, XP−0At and XP−At. As XP−At blueshifts anddiminishes with the expanding FS, the XP−Ab redshifts andgains oscillator strength. Such an oscillator strength transferterminates when the Fermi level is sufficiently away from theconduction band edge of the bottom layer (Vt ¼ 8.2 V).As discussed above, we can quantify the intrinsic inter-layer potential ϕ0 by measuring when the electrons becomeequally distributed in two layers [middle in Fig. 3(d)]. Sincethe oscillator strength of the intralayer exciton is a sensitiveindicator of the electron doping in that layer, we extract thestrength of their peaks from the RC spectrum at fixed photonenergy (1.917 and 1.928 eV). Because the two excitons canhave different intrinsic oscillator strengths and their observedpeak strength can be convoluted with the local field factorand the exciton-polaron formation in the neighboring layer,we normalize the peak strength change with respect to themaximum value within our gate tuning range [Fig. 3(e)].Unsurprisingly, we observe a slope in both excitons wherethe peak strength varies rapidly with the gate voltage. Themiddle point of the slope region coincides with the crossingpoint of the two curves (Vt ¼ 7.35� 0.15 V) indicating theΔc ¼ 0 condition. The uncertainty is estimated from thedifference between twomiddle points, and the crossing pointis found to be largely insensitive to the total doping density n.According to Eq. (3), the intrinsic interlayer potential ϕ0 is58� 1.5 mV, corresponding to an out-of-plane polarizationP ¼ 0.55� 0.02 μC=cm2. Our measured interlayer poten-tial is about 20% larger than the previous report in artificialstacks [24,25], which is expected since our optical meas-urement is performed in a homogeneous sample withoutmixed domains and does not require domain flipping.Additionally, the downward polarization also confirms theBA stacking-order assignment (Supplemental Note 2 [39]).Multiple sets of field-dependent measurements are taken atdifferent locations, and they all show quantitatively similarresults as Fig. 3(a), confirming the homogeneous BA-stacking order of our sample and the intrinsic nature ofour observation. An example of a field-dependent measure-ment at a different location is shown in Fig. S6 [39]. Thedetermination of charge distribution via exciton contrast isOPTICALLY PROBING THE ASYMMETRIC INTERLAYER … PHYS. REV. X 12, 041005 (2022)041005-5reliable, since the low exciton density excited by the broad-bandwhite light source during theRCmeasurement does notaffect the carrier distribution and the finite doped electrons donot cause significant screening of excitons.To quantify the asymmetric interlayer coupling strengtht, we also need to know the total intrinsic band offset. Here,we measure this offset optically by investigating thephotoluminescence (PL) spectra of the interlayer excitons.It is reported that, in the bilayer MoSe2, the interlayerexciton comprises a hole residing in the Γ point and anelectron at the K point [Fig. 4(b)] [35]. Since the electronslocated in the layers with different band edges emit photonsof different wavelengths, the intrinsic conduction bandoffset Δc can be measured by resolving the interlayerexciton PL peaks. In our dual-gated 3R-MoS2 bilayerdevice, the intrinsic interlayer PL spectrum has six peaksaround 1.45 eV. These six peaks can be grouped into threepairs with an energy separation of about 20 meV in eachpair. One prior explanation for such an energy separation isthat each interlayer exciton has a phonon replica, sincephonons are needed for momentum-indirect optical tran-sitions [51]. Available phonons for scattering the electronfrom the K to Γ point have energies of approximately26 meV for the acoustic branch and approximately 47 meVfor the optical branch [51]. The energy splitting in each pairof interlayer excitons is approximately 20 meV, comparableto the energy difference between acoustic and opticalphonons, suggesting the two exciton peaks can be assisted(e)(a)(b) (d)(c)FIG. 3. Electric-field-dependent reflectance contrast spectrum of intralayer excitons in 3R-MoS2 bilayer at a fixed electron dopingdensity. (a) Contour plot of the electric-field-dependent reflectance contrast spectrum of 3R-MoS2 bilayer at a fixed electron dopingdensity. The yellow and green arrows denote the intralayer A exciton, XAt and XAb, from the top and bottom layers, respectively.(b) Reflectance contrast spectrum at different top gate voltages in steps of 0.5 V. The bottom gate voltage is Vb ¼ −0.85Vt þ 1.65. Theyellow and green dashed lines denote XAt and XAb from the top and bottom layers, respectively. (c) From low energy to high energy,electric-field dependence of XP−0At (attractive exciton polaron in the top layer), XP−0Ab (attractive exciton polaron in the bottom layer), XAtor XP−At (intralayer exciton or repulsive exciton polaron, respectively, in the top layer), and XAb or XP−Ab (intralayer exciton or repulsiveexciton polaron, respectively, in the bottom layer) energies extracted from (a). (d) Schematics of band alignment and existing opticaltransitions of a BA-stacked 3R-MoS2 bilayer at a fixed electron doping density when (I) the Fermi level reaches the conduction bandedge of the top layer with Δc ¼ n=Cq, (II) the Fermi level is in the conduction band of both layers and electrons are equally doped intotwo layers with Δc ¼ 0, and (III) the Fermi level is at the conduction band edge of the bottom layer with Δc ¼ −n=Cq. (e) Normalizedintensity changes of XAt and XAb extracted from (a) along the fixed photon energy indicated by yellow and green arrows. The crossingpoint denotes the same oscillator strength of XP−At and XP−Ab when the conduction band offset is zero as shown in the middle panel of (d).JING LIANG et al. PHYS. REV. X 12, 041005 (2022)041005-6by different phonon branches. More studies are required todetermine the specific phonon modes responsible for theoptical transition. In the following, we focus on the low-energy branch of each pair.The origin of the three interlayer exciton pairs is clarifiedby measuring their doping and electric field dependence.Since the two high-energy pairs disappear in the dopedregimes [Fig. 4(a)], we attribute them to the intrinsicinterlayer exciton, XIb and XIt. The highest-energy pairis emitted from the top layer, as the top layer has a higherconduction band than the bottom one at zero field. Thebroad peaks of the lowest energy are likely to result fromtrions emitted from the bottom layer (TIt). They areobservable in region II, since the bilayer is always dopedby bond charges induced by the spontaneous polarization.Such an assignment is confirmed by the electric fielddependence of the emission energy [Fig. 4(c)]. With Ezvarying from negative to positive, XIt redshifts and XIb(d)(b)(c)(a)FIG. 4. Doping-dependent and electric-field-dependent photoluminescence spectrum of Γ-K interlayer excitons in the 3R-MoS2bilayer. (a) Contour plot of the doping-dependent PL spectrum of momentum-indirect interlayer excitons in the 3R-MoS2 bilayerwithout an external electric field. The gate voltage is applied proportionally on top and bottom gates with relation Vb ¼ 0.85Vt. At zerodoping, from low energy to high energy, three pairs of peaks indicate TIb (interlayer trion), XIb (interlayer Γ-K exciton), and XIt(interlayer Γ-K exciton), respectively; XIt is more evident in (c) as indicated by the yellow arrow. The top is an average PL spectrum inregion II with the black arrow denoting the XIt (interlayer Γ-K exciton). cps: counts per second. (b) Type-II band alignment of BA-stacked MoS2 bilayer at K points of the Brillouin zone and momentum-indirect Γ-K transitions. Δc and Δv denote the conduction bandoffset and valence band offset at K points, respectively. (c) Contour plot of the electric-field-dependent PL spectrum in the 3R-MoS2bilayer with zero doping. The green (yellow) arrow denotes the interlayer exciton XIb (XIt). The gate voltage is applied proportionally ontop and bottom gates with relation Vb ¼ −0.85Vt. (d) Electric-field dependence of interlayer excitons extracted from (c). Solid linesdenote the linear fitting results. The dashed gray line denotes the same energy of XIb and XIt when Δc ¼ 0.OPTICALLY PROBING THE ASYMMETRIC INTERLAYER … PHYS. REV. X 12, 041005 (2022)041005-7blueshifts, respectively, and they cross each other atVt ¼ 6.5 V. Similar to the picture in Fig. 3, such aStark shift is due to the tuning of the conduction bandoffset at the K point, while the valence band maximum atthe Γ point remains largely unchanged, since the two layershybridize strongly. When the conduction band offset is zero(Δc ¼ 0), XIt and XIb have the same photon energycorresponding to the crossing point in Fig. 4(c). The dipolemoment of the interlayer exciton can be extracted from theslope of the Stark shift to be 0.31 e · nm, which agrees withthe picture that the electron is localized in one layer whilethe hole is shared between the two, so their out-of-planedistance is about half of the interlayer distance(d ∼ 0.65 nm). The measured dipole moment agrees withthe calculated value of the Γ-K interlayer exciton, which ismuch larger than that of the Γ-Q interlayer exciton, thusfurther confirming our assignment [35]. Interestingly, wefind that the interlayer trion peaks have a similar field-dependent Stark shift as the interlayer exciton.When Ez ¼ 0, the energy difference between XIt and XIbof 58.0� 0.5 meV is equal to the intrinsic conduction bandoffset Δc. The uncertainty of Δc is also determined by thespectral resolution of 0.5 meV as that of δ. The spinconfiguration of conduction bands should not affect thedetermination as long as the two sets of spin bands have thesame band offset. Considering the optically determined Δc,δ, and ϕ0, we extract the asymmetric interlayer coupling ofthe 3R-MoS2 bilayer to be t ¼ 100� 25 meV. This con-clusion demonstrates that the asymmetric interlayer cou-pling is not negligible, in contrast to some estimation in theliterature [38]. Instead, our measured asymmetric couplingstrength in the 3R bilayer is of the same order as thepredicted interlayer coupling strength between valencebands in the 2H bilayer [38], in agreement with first-principles calculations [34]. Our four-band Hamiltonianshould be regarded as an effective model which describesthe essential physics involving states near the conductionand valence band edges of the 3R-MoS2 bilayer. As such,the interlayer coupling parameter t should also be regardedas an effective parameter which includes, in principle,contributions from all interlayer tunneling processesbetween the conduction band in one layer and the valenceband in the other. Besides the band-gap difference δ, thehigher-order corrections of remote bands to the band offsetare qualitatively discussed in Supplemental Note 1 [39].As a summary, the polarization-induced intrinsic inter-layer potential ϕ0, the interlayer coupling t, and the directband-gap difference δ correspondingly contribute58� 1.5, 6� 2.5, and −5.5� 0.25 meV to the intrinsicconduction band offset Δc of 58 meV at K points in3R-MoS2 (Table I). The experimental results agree withprevious self-consistent calculations which relate theintrinsic interlayer coupling and interlayer potential[27]. Since ϕ0 is determined only by the ratio of theaveraged gate capacitance C̄ and the geometric capaci-tance of bilayer MoS2 (Cm), neither the size of the spin-orbit coupling-induced conduction band splitting nor thespin configuration of the lowest band affect our meas-urement conclusions [52]. The contribution from the Qpoints is not expected to have a qualitative impact on ourconclusions either for the same reason.In conclusion, we leverage the distinctive sensitivity ofthe Fermi polaron to charges doped in different layers toprobe the intrinsic interlayer potential and, thus, thespontaneous polarization in a homogeneous 3R-MoS2bilayer. Compared to the graphene electrical sensingapproach to determining spontaneous polarization, ouroptical technique does not rely on polarization switchingvia interlayer sliding, rendering it suitable for quantifyingspontaneous polarization in a wide range of materials. Incombination with the optically measured band gaps andband offsets, we quantitatively determine the strength of anasymmetric interlayer coupling at K points, which isfundamental to the ferroelectric and optoelectronic appli-cations of semiconducting TMDs where sliding ferroelec-tricity is observed [24]. Last but not least, our results lay animportant foundation for understanding the moiré super-lattice formed by hetero- or twisted homostructures.Besides the exotic physics predicted from the position-dependent interlayer coupling [53], the hopping betweenlayers is known for determining the moiré potential depthas well as the bandwidth of the moiré bands, whosecompetition with the Coulomb interaction has led to arange of correlated insulating states observed in variousexperiments [54]. Therefore, the full understanding ofinterlayer coupling in different stacking orders is crucialfor exploring these new semiconducting moiré materials.We acknowledge support from the Natural Sciences andEngineering Research Council of Canada, CanadaFoundation for Innovation, New Frontiers in ResearchFund, Canada First Research Excellence Fund, and MaxPlanck–UBC–UTokyo Centre for Quantum Materials [MaxPlanck Society (Germany), University of British Columbia(Canada), and the University of Tokyo (Japan)]. Z. Y. is alsosupported by the Canada Research Chairs Program. Theauthors thank Benjamin T. Zhou for the helpful discussion.TABLE I. The measured conduction or valence band offset at theK points, the asymmetric interlayer coupling, thedirect band-gap difference, and the polarization-induced interlayer potential in a 3R-MoS2 bilayer.Δc Δv t δ ϕ058� 0.5 meV 69� 1 meV 100� 25 meV 11� 0.5 meV 58� 1.5 mVJING LIANG et al. PHYS. REV. X 12, 041005 (2022)041005-8[1] E. Suárez Morell, J. D. 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