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Mehdi Arfaoui, Natalia Zawadzka, Sabrine Ayari, Zhaolong Chen, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Adam Babiński, Maciej Koperski, Sihem Jaziri, Maciej R. Molas

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[Optical properties of orthorhombic germanium sulfide: unveiling the anisotropic nature of Wannier excitons](https://mdr.nims.go.jp/datasets/9e8bb37a-b367-49be-a56f-cd8bc5e1099c)

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Optical properties of orthorhombic germanium sulfide: unveiling the anisotropic nature of Wannier excitonsNanoscalePAPERCite this: Nanoscale, 2023, 15, 17014Received 30th June 2023,Accepted 2nd September 2023DOI: 10.1039/d3nr03168crsc.li/nanoscaleOptical properties of orthorhombic germaniumsulfide: unveiling the anisotropic nature ofWannier excitons†Mehdi Arfaoui, *a Natalia Zawadzka,b Sabrine Ayari,c Zhaolong Chen,d,eKenji Watanabe, f Takashi Taniguchi,g Adam Babiński,b Maciej Koperski,d,eSihem Jaziria and Maciej R. Molas bTo fully explore exciton-based applications and improve their performance, it is essential to understandthe exciton behavior in anisotropic materials. Here, we investigate the optical properties of anisotropicexcitons in GeS encapsulated by h-BN using different approaches that combine polarization- and temp-erature-dependent photoluminescence (PL) measurements, ab initio calculations, and effective massapproximation (EMA). Using the Bethe–Salpeter Equation (BSE) method, we found that the optical absorp-tion spectra in GeS are significantly affected by the Coulomb interaction included in the BSE method,which shows the importance of excitonic effects besides it exhibits a significant dependence on thedirection of polarization, revealing the anisotropic nature of bulk GeS. By combining ab initio calculationsand EMA methods, we investigated the quasi-hydrogenic exciton states and oscillator strength (OS) ofGeS along the zigzag and armchair axes. We found that the anisotropy induces lifting of the degeneracyand mixing of the excitonic states in GeS, which results in highly non-hydrogenic features. A very goodagreement with the experiment is observed.1. IntroductionThe groundbreaking discovery of graphene and its unique pro-perties has sparked further research on the development ofother alternative layered and non-layered materials withvarious optical, electronic, and chemical properties.1,2Recently, transition metal dichalcogenides (TMDs)3,4 andblack phosphorus (BP)5,6 have been extensively researched fortheir high carrier mobility7–9 and strong bound excitons.6,10–12However, the instability of BP under air conditions limits itspractical use.13–15 An alternative class of materials with a puck-ered honeycomb lattice, group-IV monochalcogenides MX(where M = Ge, Sn, or Pb and X = S, Se, or Te),16–19 hasemerged as a promising new class of layered van der Waals(vdW) semiconductors due to their benefits such as low tox-icity,20 high thermal stability,21,22 Earth abundance,19,23 andexcellent absorption energies observed in the visible frequencyrange,24–27 in addition to low thermal conductivities and sig-nificant anisotropic physical properties.19,24,28–31 Amonggroup-IV monochalcogenides, germanium sulfide (GeS) is con-sidered a promising material in optoelectronics applicationsdue to its high optical absorption and optical band gap (BG)in the visible range, which can be effectively tuned by applyingan external strain. This allows for modulation of its emissionwavelength.32–36 GeS also exhibits high photosensitivity, abroad spectral response, and giant piezoelectricity because ofits characteristic “puckered” symmetry.25,27,37 Analogously toBP, the low symmetry orthorhombic crystal structure of GeSresults in unique anisotropic optical, electronic, andvibrational properties along the zigzag (ZZ) and armchair (AC)axes,35,38–41 which makes it suitable for large-scale appli-cations in photovoltaic thermoelectric42 and optoelectronics.19Polarization-resolved photoluminescence (PL), reflectancecontrast (RC), and Raman scattering (RS) measurements†Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d3nr03168caLaboratoire de Physique de la Matière Condensée, Département de Physique,Faculté des Sciences de Tunis, Université Tunis El Manar, Campus Universitaire,1060 Tunis, Tunisia. E-mail: mehdi.arfaoui@fst.utm.tnbInstitute of Experimental Physics, Faculty of Physics, University of Warsaw, Warsaw,Poland. E-mail: maciej.molas@fuw.edu.plcLaboratoire de Physique de l’Ecole normale supérieure, ENS, Université PSL, CNRS,Sorbonne Université, Université de Paris, 24 rue Lhomond, 75005 Paris, FrancedInstitute for Functional Intelligent Material, National University of Singapore,117575, SingaporeeDepartment of Materials Science and Engineering, National University of Singapore,117575, SingaporefResearch Center for Electronic and Optical Materials, National Institute forMaterials Science, 1-1 Namiki, Tsukuba 305-0044, JapangResearch Center for Materials Nanoarchitectonics, National Institute for MaterialsScience, 1-1 Namiki, Tsukuba 305-0044, Japan17014 | Nanoscale, 2023, 15, 17014–17028 This journal is © The Royal Society of Chemistry 2023Open Access Article. Published on 16 October 2023. Downloaded on 12/23/2023 5:53:18 AM.  This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.View Article OnlineView Journal  | View Issuehttp://rsc.li/nanoscalehttp://orcid.org/0000-0001-7628-4159http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0002-5516-9415https://doi.org/10.1039/d3nr03168chttps://doi.org/10.1039/d3nr03168chttps://doi.org/10.1039/d3nr03168chttp://crossmark.crossref.org/dialog/?doi=10.1039/d3nr03168c&domain=pdf&date_stamp=2023-11-01http://creativecommons.org/licenses/by/3.0/http://creativecommons.org/licenses/by/3.0/https://doi.org/10.1039/d3nr03168chttps://pubs.rsc.org/en/journals/journal/NRhttps://pubs.rsc.org/en/journals/journal/NR?issueid=NR015042performed over a wide temperature range in GeS confirmed itsanisotropies characteristics.24,29,43 This makes it a highlypromising material for the development of polarization-sensi-tive photodetectors. It is important to examine the many-bodyinteractions present in this material, particularly with regardto the excitonic effect.27,35,40,41,44,45 Despite the tremendousprogress in the study of GeS, many of its fundamental opticalanisotropic properties remain unknown.24 Indeed, our studydelves into the optical properties of the anisotropic nature ofthe Wannier excitons of GeS that remain neglected or incom-pletely understood in previous studies,24,29,43 using differenttheoretical approaches and supported by experimental results.In this work, we investigate the optical response of aniso-tropic excitons in GeS encapsulated with hexagonal BN (h-BN)flakes using a comprehensive approach that combines polariz-ation- and temperature-dependent PL techniques, ab initio cal-culations, and effective mass approximation (EMA). The inno-vative point of our work originates from both the appliedexperimental and theoretical approaches. A high quality of theinvestigated sample was achieved by its preparation under aninert protective gas atmosphere, and GeS was embedded inbetween h-BN flakes to avoid degradation processes.Additionally, to describe the experimental results, weemployed a state-of-the-art approach that utilized density func-tional theory (DFT) and the quasiparticle (QP) correction GWto determine the electronic band structure (BS) of bulk GeS,and the anisotropic effective mass of electrons and holes indifferent crystal directions. Furthermore, using both the inde-pendent particle approximation (IPA) and the GW + Bethe–Salpeter equation (BSE),46,47 we studied the dielectric func-tions for light polarized along the in-plane (x–y) and out-of-plane (z) directions. Indeed, the obtained result from the BSEmodifies the optical absorption spectra in bulk GeS. Thus, itshows the signature of excitonic effects on the opticalresponse. To evaluate the different optical selection rules, wecalculated the direct interband optical transition matrixelements (OME) in different crystal directions, as well as thepercentage of the atomic orbital contribution to the valenceand conduction bands at specific k-points of the Brillouinzone (BZ). We were also able to predict the quasi-hydrogenicexciton states and the oscillator strength (OS) of the excitonsalong both the ZZ and AC directions by integrating ab initiocalculations with the EMA. In fact, the anisotropic effect onthe exciton characteristics in GeS (e.g., exciton binding energy(BE) and the spatial extension of exciton wavefunction) is con-trolled by its reduced mass and dielectric constant. Unlike theisotropic hydrogenic model, the anisotropy lifts the degeneracyof the exciton states, which have the same principal quantumnumber but different radial and angular quantum numbers.This increases in the number of allowed dipole transitions thatcan be probed by terahertz radiation, providing new ways ofcontrolling device emissions.This article is organized as follows: in section 2, we presentthe experimental results obtained for a GeS flake encapsulatedwith h-BN, which include its measured polarization- and temp-erature-dependent PL spectra. In section 3, to gain a deeperunderstanding of the exciton behavior in anisotropicmaterials, we used theoretical methods (DFT and GW + BSE)to calculate the electronic BS of GeS, taking into account theexcitonic effect. In section 4, we analyze the anisotropic behav-ior of GeS using the EMA. By varying the relevant parameters,we control the degree of anisotropy across multiple cases,ranging from isotropic to anisotropic. Our calculations of theexcitonic BE and OS were performed for each case. Overall, ourstudy of GeS as a prototype system for anisotropic layeredmaterials provides an understanding of the optical response ofanisotropic excitons and can be tailored for other three-dimen-sional (3D) anisotropic materials.2. Experimental resultsTo prevent degradation of the GeS flake, we encapsulated thethick GeS flake with a thickness of about 50 nm in h-BNflakes, see Methods for details. The side-view scheme andoptical image of the investigated GeS are presented in Fig. 1(a)and (b). First, we examined the polarization evolution of thePL spectra measured on the studied sample under 1.88 eV exci-tation, see Fig. 1(c). It is observed that the GeS emission is lin-early polarized along the AC direction, whereas the PL signal isabsent in the ZZ direction. Note that the attribution of the ACand ZZ directions was performed by comparison with previousresults published in ref. 24, 29 and 43, and also confirmed byour theoretical calculations shown below. It is a hallmark ofthe anisotropic optical response of GeS. The upper panel ofFig. 1(c) presents the low-temperature (T = 5 K) PL spectra ofGeS measured in two polarizations corresponding to the ACand ZZ orientations. The PL spectrum consists of three emis-sion lines, denoted X, L1, and L2. In contrast, the corres-ponding polarization-resolved RC spectra, shown in section Aof the ESI,† consist of a single resonance, whose energycoincides with the X emission line and is also polarized alongthe AC direction. We can certainly attribute the X line to thefree neutral exciton,24 while the assignment of the L1 and L2peaks is more questionable, and hence we denoted them aslocalized excitons, see sections A and B of the ESI† for moredetails. The measured shape of the PL spectrum with a ratherlow intensity of L1 and L2 compared to that of X is differentfrom those reported in ref. 24. It may suggest that the hBNencapsulation plays a similar role as for MoS2 MLs, leading tocomplete quenching of the defect-related emission measuredat liquid helium temperature.48 Interestingly, the relativeintensity of the localized and free excitons strongly dependson the excitation energy, see section A of the ESI† for details.Fig. 1(d) presents the temperature evolution of the PL spectrameasured from 5 K to 190 K. Our measurements indicate thatwith increasing temperature, the PL intensity of the excitondecreases until it completely vanishes at 200 K. This resultsuggests that the BE of the exciton, associated with its acti-vation energy, is on the order of several meV. As the L1 and L2intensities are extremely small, we used a different excitationenergy, i.e., 2.41 eV, to investigate their temperature depen-Nanoscale PaperThis journal is © The Royal Society of Chemistry 2023 Nanoscale, 2023, 15, 17014–17028 | 17015Open Access Article. Published on 16 October 2023. Downloaded on 12/23/2023 5:53:18 AM.  This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.View Article Onlinehttp://creativecommons.org/licenses/by/3.0/http://creativecommons.org/licenses/by/3.0/https://doi.org/10.1039/d3nr03168cdence (see section A in the ESI† for details). We found that at atemperature of approximately 70 K, the measured PL spectrumis contributed only by the neutral exciton line. To investigatein detail the temperature evolution of the X line, we deconvo-luted it using Gaussian. The X energy shows a redshift as thetemperature increases from 5 K to 190 K, see Fig. 1(e). Thistype of evolution, characteristic of many semiconductors, canbe expressed by the relationship proposed by O’Donnellet al.,49 which describes the temperature dependence of theBG in terms of an average phonon energy 〈ℏω〉 and reads as:EbgðTÞ ¼ Eg � Shℏωi½cothðhℏωiÞ=2kBT �; ð1Þwhere Eg stands for the BG at absolute zero temperature, S isthe coupling constant, and kB denotes the Boltzmann con-stant. As we can see that the used relationship correctly repro-duces the temperature evolution of the X energy, we concludethat its BE does not depend on the temperature. The deter-mined Eg and S values are of about 1.778 eV and 4.3, respect-ively. The average phonon energy 〈ℏω〉 is found to be around26 meV, which is close to the high density of phonon statesaround 28 meV.24 Panel (f ) of Fig. 1 presents the temperatureevolution of the X linewidth. As the temperature increases, thecarriers have more thermal energy and move more rapidly,leading to a greater distribution of carrier velocities. In semi-conductors, such an evolution can be described by the so-called Rudin’s relationship,50 which is given byγðTÞ ¼ γ0 þ σT þ γ′1expℏω=kT �1; ð2Þwhere γ0 denotes the broadening of a given spectral line at 0 K,the term linear in temperature (σ) quantifies the interaction ofexcitons with acoustic phonons (of negligible meaning for thepresent work), γ′ arises from the interaction of excitons withLO phonons, and ℏω is the LO phonon energy. As can be seenFig. 1 (a) Side-view scheme and (b) optical image of the investigated GeS encapsulated with h-BN flakes. (c) False-colour map of the low tempera-ture (T = 5 K) polarization-resolved PL spectra measured on GeS flakes under excitation at 1.88 eV. Note that the intensity scale is logarithmic. Thecrystal structures along the ZZ and AC directions are drawn on top of the map. The top panel shows the corresponding PL spectra detected in theAC and ZZ directions. The right panel demonstrates the integrated PL intensity as a function of detection angle. (d) The corresponding temperature-dependent PL spectra measured on GeS flakes with 1.88 eV laser light excitation. The spectra are vertically shifted and are divided by scaling factorsfor clarity. The determined (e) energy and (f ) full width at half maximum (FWHM) of the neutral exciton (X) line. The circles represent the experi-mental results while the curves are fits to the data obtained using eqn (1) and (2).Paper Nanoscale17016 | Nanoscale, 2023, 15, 17014–17028 This journal is © The Royal Society of Chemistry 2023Open Access Article. Published on 16 October 2023. Downloaded on 12/23/2023 5:53:18 AM.  This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.View Article Onlinehttp://creativecommons.org/licenses/by/3.0/http://creativecommons.org/licenses/by/3.0/https://doi.org/10.1039/d3nr03168cin Fig. 1(f ), Rudin’s relationship reproduces the experimentaldata quite well. Due to the anisotropic structure of GeS, weconsider ℏω as a free-fitting parameter in our analysis. Thedetermined values of γ0 and γ′ correspond to about 8.5 meVand 0.13. The fitted ℏω parameter is around 30 meV, which isclose to the energy of the Ag mode (31 meV at T = 5 K).24The main objective of the present work is to construct atheoretical framework to analyze the aforementioned opticalcharacteristics of free neutral excitons in GeS. The free excitonis identified by its distinct spectral peak located at around 1.78eV in the PL spectra of the GeS flake with a thickness of 60 nmat T = 5 K. The low-temperature conditions stabilize andclearly define the excitonic states, yielding an accurate assess-ment of their spectral properties. The reduced thermal exci-tation also enhances the analysis of GeS’s intrinsic opticalproperties.3. First-principles calculations of thequasi-particle and excitonic effects inGeSGeS is a layered material that has an orthorhombic structureand belongs to the Pnma (D2h16) space group.21 This materialcrystallizes in double layers. The unit cell contains eight atomsorganized in two adjacent double layers. The puckered honey-comb lattice of GeS has an anisotropic crystal structure charac-terized by the two orthogonal AC(x) and ZZ (y) directions, asindicated in Fig. 2. The orthorhombic Bravais lattice of GeScan be specified by giving three primitive lattice vectors:a1 = ax̂, a2 = cŷ, and a3 = cẑ, and the reciprocal primitive latticevectors are spanned by: b1 ¼ 2πak̂x, b2 ¼ 2πbk̂y and b3 ¼ 2πck̂z. x̂,ŷ and ẑ are the unit vectors in the directions of the x-axis, they-axis, and the z-axis, respectively.In Table 1, we present the final relaxed lattice parametersand compared them to previously reported experimental andtheoretical data. All calculations were performed with the opti-mized lattice parameters.In Fig. 3(a), we plotted the BS of bulk GeS using DFT with(black dashed curve) and without (red curve) the spin–orbitinteraction. The BS of GeS is not significantly impacted by therelativistic correction effect. Therefore, we disregard this in ourcalculation of many-body simulations. We found that bulk GeSis a semiconductor with a BG of 1.23 eV at the Γ point. Theindirect gap is only 3 meV in energy higher than the directone. Our results are consistent with previous DFTcalculations.51,52 As we can see, the DFT direct BG (red curve)has been significantly underestimated because of the BGproblem with the DFT Kohn–Sham (KS) approach. To over-come this issue, we calculated the relative QP BS using the per-turbative method GW (blue curve). In fact, G and W were con-structed from the KS wavefunctions, and the perturbatively cor-rected KS eigenvalues. The single-shot G0W0 gives an insuffi-ciently small BG when compared to the experimental one. Tosolve this problem, we used the self-consistency of GW onFig. 2 Illustration of the atomic structure of bulk GeS. (a) Top view of the atomic structure of bulk GeS in a 3 × 3 × 1 supercell. (b) Side views of bulkGeS along the AC and ZZ directions, respectively. (c) The bulk first BZ of an orthorhombic Bravais lattice and its projected surfaces. The gray andblack spheres stand for the Ge and S atoms, respectively. The unit cell is indicated by the dashed red rectangle in (a and b).Table 1 Comparison of the lattice parameters estimated at the DFTlevel while accounting for the vdW interaction with the experimentaland recent theoretical resultsReferences a (Å) b (Å) c (Å)G. Ding et al.51 4.74 3.67 10.64S. Hao et al.52 4.44 3.67 10.76T. Grandke et al.53 4.30 3.64 10.47Our work 4.45 3.76 10.77Nanoscale PaperThis journal is © The Royal Society of Chemistry 2023 Nanoscale, 2023, 15, 17014–17028 | 17017Open Access Article. Published on 16 October 2023. Downloaded on 12/23/2023 5:53:18 AM.  This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.View Article Onlinehttp://creativecommons.org/licenses/by/3.0/http://creativecommons.org/licenses/by/3.0/https://doi.org/10.1039/d3nr03168ceigenvalues only (evGW), see section D in the ESI† for details.At this theoretical level, the BG increases to 1.78 eV with theQP correction of 0.55 eV at Γ. Our results are in excellent agree-ment with the experimental bulk GeS BG energies extrapolatedto 0 K given in ref. 54 and 55, as well as with previous theore-tical calculations with GW in ref. 56 and 57. In addition, theQP corrections are slightly dispersed with respect to the KSeigenenergies. In fact, we found that by fitting the QP correc-tions data to a linear curve, the conduction and valence bandsare slightly stretched by 7% and 1%, respectively. The bandprofiles for the lowest conduction band and the highestvalence band were also extracted from DFT and are shown inFig. 3(b) and (c), respectively. The valence band has threemaximum locations in the AC direction: K4, K3, and Γ, withthe highest maximum located at the Γ-point. The conductionband has a minimum located at the Γ-point.In Fig. 3(d–f ), we calculated the normalized squared OMEMc;vi (k) = |Pc;vi |2 = |uv,k|pi|uc,k|2 between the top valence bandand the bottom conduction band along a specific direction i.Here, uv(c),k is the single particle Bloch function of the valence(v) and conduction bands (c) obtained by the DFT-KS calcu-lation for the wavevector k. pi is the momentum operator alongthe i direction. Mc;vi (k) measures the k-dependent opticalstrength of the c–v interband transition. Furthermore, theOME contain all the symmetry-imposed selection rules.Fig. 3(e) and (f) show that the k-resolved OME for light polar-ized in the AC and ZZ directions exhibit distinct responses,which reveals the anisotropic nature of GeS. Of particularinterest are the valleys located in K1, K2, K3, K4, and Γ in theΓX and ΓY directions, where the allowed OME are significantlyhigher in ΓX for light polarized in the AC direction.Additionally, the OME are not allowed in the vicinity of Γ-pointfor light polarized in the ZZ direction. By calculating the directinterband OME, we calculate the linear optical characteristicsthat can be derived from the complex dielectric tensor εi,j(ω,q)= ε1i,j(ω,q) + iε2i,j(ω,q) for q → 0, where ε1i,j(ω,q) and ε2i,j(ω,q) arethe real and imaginary parts of the dielectric tensor, see sec-tions C and D of the ESI.† ℏω is the photon energy, q is thephoton wavevector and i, j = x, y, or z are the subscripts thatcorrespond to the Cartesian directions.In Fig. 4, we plotted the three components of the imaginarypart of the dielectric function for linear light polarized alongthe axis of the AC (x), ZZ (y) and perpendicular (z) directions tothe atomic planes for bulk GeS. In fact, Fig. 4 shows that ε2(ω)Fig. 3 Electronic properties of GeS from ab initio calculations. (a) Electronic BS of bulk GeS calculated by generalized gradient approximation(GGA) without spin–orbit coupling (red), with spin–orbit coupling (black dashed) and evGW (blue) methods. (b) and (c) 2D BSs of the topmostvalence band and the bottom most conduction band, respectively. The global maximum (minimum) of the VBM (CBM) is located at the AC directionΓ X path of the BZ. (d) Interband transitions of OME between v and c bands. (e) Color contour of the dipole OS distribution in the first BZ for lightpolarized in AC (e) and ZZ (f ) directions derived using DFT. The darker color indicates a larger dipole strength.Paper Nanoscale17018 | Nanoscale, 2023, 15, 17014–17028 This journal is © The Royal Society of Chemistry 2023Open Access Article. Published on 16 October 2023. Downloaded on 12/23/2023 5:53:18 AM.  This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.View Article Onlinehttp://creativecommons.org/licenses/by/3.0/http://creativecommons.org/licenses/by/3.0/https://doi.org/10.1039/d3nr03168cstrongly depends on the direction of polarization. This behav-ior reveals the anisotropic character of bulk GeS. The lightabsorption of GeS has a wide range from near-infrared to near-ultraviolet light (1–9 eV). However, compared to the experi-mental results, the exciton peak does not appear in Fig. 4 withthe IPA method (blue line). This result is expected since theIPA method does not include the electron–hole (e–h) inter-action. To account for the e–h interaction, we plotted, in Fig. 4(black line), the dielectric function using the GW + BSEmethod. A detailed description of the GW + BSE method canbe found in section D of the ESI.† In fact, compared with theIPA results, the Coulomb interaction included in the BSEmethod modifies the optical absorption spectra in bulk GeS aswell as a redistribution of OS, implying that excitonic effectshave an important influence on the optical properties. Thedetected low-energy peak in the BSE spectrum is attributed tothe presence of a bound exciton. In comparison to the experi-mental results, the calculated energy of the first bright excitonpeak, 1.77 eV, differs by only about 10 meV from the corres-ponding experimental value from Fig. 1, ∼1.78 eV. In addition,our results show that GeS has remarkable optical anisotropy,as evidenced by the presence of an excitonic peak only for lightpolarized along the AC direction. This intriguing observationaligns very well with our experimental results in Fig. 1. The firstbright exciton BE is ∼10 meV. Our experimental measurements oftemperature-dependent PL spectra also support this result. Insection A of the ESI,† we estimated the exciton BE to be ∼11 meVunder 2.41 eV excitation, and this value can slightly increase(∼16 meV) under resonant excitation at 1.88 eV. This BE is alsocomparable to those of bulk materials such as BP (30 meV),58MoS2 (25 meV),59 wurtzite GaN (21 meV)60 and GaAs quantumwells (∼10 meV).61 This is in contrast to layered group-VII TMDrhenium dichalcogenides (ReX2, where X = S, Se), which exhibit asignificantly higher binding energy of 120 meV.62 Moreover, theresults in the upper panel of Fig. 4 show the contributions of theoccupied and unoccupied bands to ε2, where each band islabeled with a distinct index, i.e. VB1 → VB4 for the occupiedbands and CB1 → CB4 for the unoccupied bands. In particular,the size of each green circle is proportional to the value of ε2(ω).Indeed, the analysis reveals that the first peak in ε2(ω) for the ACdirection is predominantly formed by the direct optical transitionbetween the occupied band VB1 and the unoccupied band CB1.Note that we neglect the phonon-assisted related optical absorp-tion, which requires more computational resources and is beyondthe scope of this work.The red shades in Fig. 5(a) show the first bright excitonweight originating from the vertical interband transition,which contributes to exciton formation and, as a result, toabsorption spectra. The inset characterizes the transitions inthe first reciprocal BZ. In GeS, it is mainly contributed by tran-sitions in the nearby Γ valley along ΓX, ΓY and ΓZ. The valenceband minimum (VBM) along these directions is occupiedmainly by the p- and s-orbitals of the Ge and S atoms. The con-duction band maximum (CBM) is populated by p- and s-orbi-tals of the Ge atom and the S atom, see section E of the ESI.†In order to explain the anisotropic behavior shown in Fig. 4and 5, we evaluated the different optical selection rules. Infact, in one-photon spectroscopy, the transition dipole selec-tion rules have to satisfy two conditions: (i) the change inangular momentum between the valence and conductionstates should satisfy (Δℓ = ℓ − ℓ′ = ±1) and (ii) since the parityof the momentum operator is odd, the conduction andvalence bands should have opposite parity in the i directionand the same parity in other directions. For incident lightpolarized along the AC direction, the dipole transitions areallowed for px ↔ s, px ↔ dx2 − y2. px ↔ dz2, and py(z) ↔ dxy(zx).Furthermore, the allowed transitions for the light polarizedalong the ZZ direction are py ↔ s, py ↔ dx2 − y2, py ↔ dz2, andpxz ↔ dxy(zy). For bulk GeS, we calculated the percentage of theFig. 4 Optical properties of GeS obtained from ab initio calculations.Imaginary part of the dielectric functions calculated using IPA and GW +BSE methods, with light polarized in the x, y, and z directions, respect-ively. The red vertical bars represent the normalized OS (arb. units). TheDFT and GW BGs are denoted by the black and orange dashed verticallines, respectively. The size of each green circle in the upper panelcorresponds to the value of ε2(ω). The theoretical value of the first brightexciton energy, EX = 1.77 eV, is marked by the dashed vertical violet line,while the dashed vertical orange line points to the correspondingexperimental value of about 1.78 eV.Nanoscale PaperThis journal is © The Royal Society of Chemistry 2023 Nanoscale, 2023, 15, 17014–17028 | 17019Open Access Article. Published on 16 October 2023. Downloaded on 12/23/2023 5:53:18 AM.  This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.View Article Onlinehttp://creativecommons.org/licenses/by/3.0/http://creativecommons.org/licenses/by/3.0/https://doi.org/10.1039/d3nr03168ccontribution of the atomic orbital to the valence and the con-duction wavefunctions at a particular k-point, as shown inTable 2. We can clearly see that the d-orbitals do not contrib-ute to the interband transition between the CBM and theVBM. Unlike GeS, ReX2 bands near the gap are mainly shapedby Re metal d-orbitals, leading to stronger spin–orbit coup-ling.63 The optical absorption for polarization in the AC (ZZ)direction, which is related to the interband transition from theVBM to the CBM along the ΓX (ΓY) direction, occurs only frompx(y) ↔ s transitions. The projected OME in the AC and ZZdirections (see Fig. 3(d)) show that the allowed interband tran-sition in the Γ–X direction is more significant than the tran-sition in the Γ–Y direction. Indeed, in the vicinity of the high-symmetry Γ-point, Mc;vy (k) vanishes and only Mc;vx (k) contrib-utes to this interband transition, which may explain the strongsignal of the exciton peak in the AC direction. Thus, bulk GeSstrongly absorbs AC-polarized light with an energy EX, and it isalmost transparent along the ZZ direction for the same energy.This phenomenon is the result of selection rules associatedwith the symmetries of this anisotropic material.To investigate the exciton spatial extension, we plotted therelative e–h wavefunction in real space, see Fig. 5(b–d). Thisshows how these excitonic wavefunctions unfold over the real-space lattice. Indeed, we fixed the position of the hole on thetop of the S atom (which contributes mainly to the top of thevalence band, see section E of the ESI† for more informationon the PDOS) separated by about 1 Å within the unit cell.Because the exciton wavefunction spreads over many unit lat-tices (more delocalized), they are more like Wannier-type exci-tons. In contrast to GeS, other anisotropic materials such asReX2 due to their weak interlayer interactions, even for thebulk case, the exciton is mainly confined to a single layer(∼68%), leading to a relatively high binding energy of 120 meVand a small Bohr radius of 0.95 nm.62In the following section, we used a semi-analytical theore-tical model based on the EMA. In Table 3, we determined theeffective masses (mν;ieff ) of electrons (ν = e) and holes (ν = h) aswell as the static dielectric constant (εi) in GeS for differentcrystal directions (i). These parameters can be inserted directlyFig. 5 (a) Excitonic weights showing the most important electronictransitions along the high symmetry points of the BZ of GeS. Electronictransitions representing exciton weights are projected onto the ground-state electronic dispersion, which are depicted by the red shade. Thecolored inset shows the 2D projected exciton wavefunction distributionin k-space. Both the excitonic weights and k-space wavefunctions showthat the first bright exciton originates from direct optical transitions inthe vicinity of the Γ valley along the Γ X, Γ Y and Γ Z directions. (b, c andd) Normalized squared exciton wavefunction of first bright exciton inGeS using the BSE for incident light polarized along the AC direction.We fix the position of the hole near the S atom. Side view on the bottomand top view on the top.Table 2 PDOS of the VBM and CBM band wavefunctions at the special k-points. The percentage contributions from each atomic orbital to thesewavefunctions are listedDirection k StateS atom Ge atoms pz px py s pz px py dz2 dxz dyz dx2−y2 dxyΓ CBM 7 0.9 8.4 0 7.9 64 6.8 0 0 0 0 0 0VBM 0.9 59.1 5 0 20.9 12.6 0.3 0 0 0 0 0 0AC K1 CBM 6.2 3 7.5 0.0 8.4 60.2 9.2 0 0 0 0 0 0VBM 0.97 60 4.7 0.0 20 11.8 0.86 0.0 0 0 0 0 0ZZ K2 CBM 6.7 1.65 8 0.0 8.1 62 7.8 0.0 0 0 0 0 0VBM 0.9 59.8 4.8 0 20 12 0.58 0.0 0 0 0 0 0AC K3 CBM 4.8 8 6.2 0.0 9 54.7 12 0.0 0 0 0 0 0VBM 1.2 57.3 5.1 0.0 20.6 13 1.6 0.0 0 0 0 0 0AC K4 CBM 15.5 7 9.1 1.3 3.7 11.5 3 45.5 0 0 0 0 0VBM 2.2 13 26.4 2 31.4 12.2 6.14 6.1 0 0 0 0 0Table 3 Values of effective me;ieff (mh;ieff) and reduced mass μi in the unitof free electron mass (m0), and dielectric constant εi along the ΓX, ΓYand ΓZ directions. The subscripts i refer to the x, y, z crystallographydirectionme;ieff (m0) mh;ieff (m0) μi (m0) εiΓ–X 0.845 1.011 0.46 10.94Γ–Y 1.519 1.599 0.78 11.29Γ–Z 0.019 0.101 0.016 10.72Paper Nanoscale17020 | Nanoscale, 2023, 15, 17014–17028 This journal is © The Royal Society of Chemistry 2023Open Access Article. Published on 16 October 2023. Downloaded on 12/23/2023 5:53:18 AM.  This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.View Article Onlinehttp://creativecommons.org/licenses/by/3.0/http://creativecommons.org/licenses/by/3.0/https://doi.org/10.1039/d3nr03168cinto the Schrödinger equation describing the interactionbetween e–h pairs. me;ieff and mh;ieff were calculated in terms ofeffective free electron mass units (m0) by means of a parabolicfitting of the valence and conduction band curvatures near theΓ k-point, see Fig. 3(a). mν;ieff and εi are strongly dependent onthe crystal direction, revealing the influence of the crystal an-isotropy on the optical properties of the excitonic states. Theexciton reduced mass (μi = me;ieffmh;ieff /(me;ieff + me;ieff )) in the in-plane direction is larger compared to that in the out-of-planedirection, leading to a strong compression of the exciton Bohrradius (aib ∝ εiℏ2/e2μi) in the in-plane direction. As a result,our models will consider the exciton as an unconfined e–hpair, since the confinement potential is negligible comparedto the Coulomb potential.4. Anisotropic Wannier excitontheory within the EMABSE demonstrates that the dominant contribution to the firstbright exciton in GeS originates from the band states that liein the vicinity of the Γ valley along the ΓX, ΓY, and ΓZ direc-tions (see Fig. 5). We used this result to analyze theSchrödinger equation that describes the unconfined exciton ina pristine sample, taking into account the interaction betweenthe e–h pair with anisotropic mν;ieff and εi. The Hamiltonian formodeling excitons in bulk semiconductors, along with thetransformation to exciton coordinates, is described in sectionF of the ESI.† Using the relative r = (x, y, z) and center-of-mass(COM) RCM = (XCM, YCM, ZCM) coordinates, the resultingSchrödinger equation can be written as:64–66�ℏ221MX;x@2@XCM2 þ1MX;y@2@YCM2 þ1MX;z@2@ZCM2� ��� ℏ221μx@2@x2þ 1μy@2@y2þ 1μz@2@z2 !� e2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiεyεzx2 þ εxεzy2 þ εxεyz2p �ϒXj ðRCM; rÞ ¼ EXj ϒXj ðRCM; rÞ;ð3Þwhere the first term describes the kinetic energy of the freeCOM motion of the exciton in direction i, with exciton massesMX,i = me;ieff + mh;ieff . The last two terms describe the relativemotion of the e–h pair, bound via the attractive Coulomb inter-action. EXj ðKÞ ¼ EGWg þ E relj þ Pi¼x;y;zℏ2Ki2=ð2MX;iÞ andϒXj ðRCM; rÞ ¼ffiffiffiffiffiffiffiffiffi1=VpexpiRCMKΨ relj ðrÞ are the eigenenergy and eigen-function solutions of the Schrödinger equation, respectively.Here, Erelj and Ψrelj (r) represent the eigenenergy and the eigen-vector of the relative motion of the e–h pair. The COM wavevector is denoted by K = (Kx, Ky, Kz). The volume of a bulk semi-conductor, V = NΩ, depends on the number of primitive cells(N) and the volume of the unit cell (Ω). To obtain the equationthat describes Wannier excitons in an anisotropic medium, itis more convenient to transform from anisotropic masses toanisotropic potentials via the change of variables,ξ ¼ ffiffiffiffiffiffiffiffiffiffiμx=μ̄px; , η ¼ffiffiffiffiffiffiffiffiffiffiμy=μ̄qy; , ζ ¼ ffiffiffiffiffiffiffiffiffiffiμz=μ̄pz. Thus, the equationfor the relative motion of the e–h pair with K = 0 readsHrelX ¼ � ℏ22μ̄@2@ξ2þ @2@η2þ @2@ζ2� �� e2ε̄ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAξ2 þ Bη2 þ Cζ2p ; ð4Þwhere the kinetic term retains its conventional form but with amodified mass. The anisotropic parameters A ¼ μ̄μxεyεzε̄2, B ¼μ̄μyεxεzε̄2and C ¼ μ̄μzdεxεyε̄2, assumed to be real and positive, esti-mate the degree of anisotropy. μ̄ ¼ε̄31εxμxþ 1εyμyþ 1εzμz ! !�1and ε̄ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiεxεyεz3p represent theaverage reduced mass and the dielectric constant, respectively.The problem is treated in spherical coordinates, whereρ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiξ2 þ η2 þ ζ2p. This transformation integrates the an-isotropy into the potential while maintaining the conventionalform of an isotropic COM system with a reduced mass equal toμ̄. We utilized the new coordinate system for our calculationsand only switch back to Cartesian coordinates x, y, and z whenpresenting the contour plots of the density wavefunction forclarity and interpretation. Expanding the relative wavefunctionin a basis of 3D-hydrogenic wavefunction, Φn,ℓ,m(ρ, θ, φ), isconvenient to solve eqn (4). We express the relative wavefunc-tion Ψ relj¼ðñ;‘̃;m̃Þðρ; θ;ϕÞ ¼Pn;‘;mCn;‘;mΦn;‘;mðρ; θ;ϕÞ. Here, Cn,ℓ,m areexpansion coefficients and Φn,ℓ,m(ρ,θ,φ) are the basis wavefunc-tions, where n, ℓ, and m represent the primary, azimuthal, andmagnetic quantum numbers, respectively. The numerical diag-onalization method was adopted for the resolution of eqn (4).In fact, the indices n, ℓ, and m refer to the dominant contri-butions of the coefficients Cn,ℓ,m to the relative excitonic wave-function Ψ relj¼ðñ;‘̃;m̃Þðρ; θ;ϕÞ, corresponding to the coefficient ofthe highest weight. The matrix elements of the relativeHamiltonian can be written as:Φn;‘;m� ��HrelX Φn′;‘′;m′�� � ¼ � R̄yn2δn;n′δ‘;‘′δm;m′ þ Φn;‘;m� ��Hper Φn′;‘′;m′�� �ð5Þwith R̄y = e2/(2ɛ̄āb) is the 3D-effective Rydberg energy. āb = ɛ̄ℏ2/(μ̄e2) is the 3D-exciton effective Bohr radius and the perturbedHamiltonianHper ¼ e2ε̄ρ1� 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðA cosðϕÞ2 þ B sinðϕÞ2Þ sinðθÞ2 þ C cosðθÞ2q0B@1CA:ð6ÞThe perturbation reduces the symmetry of the system and,therefore, breaks the degeneracy of the excitonic state com-pared to the isotropic cases. For further information on theNanoscale PaperThis journal is © The Royal Society of Chemistry 2023 Nanoscale, 2023, 15, 17014–17028 | 17021Open Access Article. Published on 16 October 2023. Downloaded on 12/23/2023 5:53:18 AM.  This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.View Article Onlinehttp://creativecommons.org/licenses/by/3.0/http://creativecommons.org/licenses/by/3.0/https://doi.org/10.1039/d3nr03168cmatrix elements and the excitonic Hamiltonian, see section Fof the ESI.†By the variation of the appropriate parameters (μi, εi and A,B, C), we can control the degree of anisotropy. To visualize theeffects of anisotropy, Fig. 6(a) shows the first low-lying BEs ofthe relative excitonic states, EBñ;‘̃;m̃¼ �Erelñ;‘̃;m̃, obtained bynumerical diagonalization of the matrix resulting from theprojection of the Hamiltonian HrelX as given in eqn (4). Toassign the exciton state according to its primary orbital charac-ter, we labelled them according to the (n, ℓ, m) componentthat produces the highest probability density jΨ relñ;‘̃;m̃ðρ; θ;ϕÞj2.For instance, the 1̃s exciton is mainly dominated by the (1, 0,0) component, while the (2, 1, 0) component largely contrib-utes to the 2p̃0 exciton and the (2, 1, ±1) component contrib-utes significantly to the 2p̃±1 exciton. In the bulk GeS case, asdepicted in the right panel of Fig. 6, μi had a strong anisotropyalong the direction i, while εi shows slight variations. In Fig. 6,we also studied two other different cases: (i) first, the middlepanel represents a transversely isotropic system (such as anuniaxial system). We found that the in-plane reduced massμk ¼εk21εxμxþ 1εyμy ! !�1¼ 0:58 m0 ≠ μz. Similarly, the in-plane (εk ¼ ffiffiffiffiffiffiffiffiεxεyp ¼ 11:11) and out-off-plane (εz) dielectric con-stants are also slightly different. (ii) Second, the left panelshows an isotropic system with a reduced mass of μ̄ = 0.044m0and a dielectric constant of ɛ̄ = 10.98. The comparisonbetween these two cases and the fully anisotropic one (GeS)shows that the anisotropy strongly affects the degeneracy ofthe excitonic states. In fact, for an isotropic system, the resultsof the symmetric Coulomb potential are found to follow −R̄y/n2, where the states are (2n + 1)-fold degenerate. In contrast to3D hydrogenic-like models, the anisotropy clearly lifts thedegeneracy of the different excited excitonic states.Additionally, the excitonic states are apparent in an anomalousenergy level order of the azimuthal quantum number ‘̃ andthe magnetic quantum number m̃. For example, the 2p̃+1 stateslay energetically below the 2s̃ state, due to the asymmetrycaused by the effective mass and the dielectric constant an-isotropy. This means that the radial and angular dependenciesin the Wannier equation can be separated. However, for bulkGeS, the anisotropy reduces the symmetry of the system. Inthis case, the radial and angular degrees of freedom arecoupled, which lifts the degeneracy. Interestingly, a two-folddegeneracy is found for transverse isotropy, so that for thesame ‘̃, positive and negative values of the magnetic quantumFig. 6 (a) BE of exciton states, showcasing the transition from isotropic to anisotropic cases across three panels. Each panel is further subdividedinto columns, with each column denoting a magnetic quantum number m (ranging from 0, ±1, and ±2, from left to right), while the different linecolors represent ℓ states. The left panel illustrates the isotropic case with average values of the reduced exciton mass μ̄ and the dielectric constant ɛ̄.In the middle panel, we consider the transversely isotropic scenario, where μk ≠ μz and εk ≠ εz. The right panel showcases the BE of exciton states inbulk GeS. (b–g) Comparison of the probability density, which represents the squared modulus of the corrected wavefunctions for the ground exci-tonic state projected in real space, presented in contour plots. These plots are shown in the (x, y) plane for figures (b, d and f) and in the (x, z) planefor figures (c, e and g). The unit of Bohr radius is used for all three cases. The figures correspond to the isotropic, transversely isotropic, and aniso-tropic cases, from top to bottom respectively.Paper Nanoscale17022 | Nanoscale, 2023, 15, 17014–17028 This journal is © The Royal Society of Chemistry 2023Open Access Article. Published on 16 October 2023. Downloaded on 12/23/2023 5:53:18 AM.  This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.View Article Onlinehttp://creativecommons.org/licenses/by/3.0/http://creativecommons.org/licenses/by/3.0/https://doi.org/10.1039/d3nr03168cnumber m̃ lead to the same energy. This degeneracy isremoved in GeS, producing, for example, a 0.2 meV separationbetween 2p̃+1 and 2p̃−1 states. In the transverse anisotropycase, the exciton Hamiltonian exhibits a uniaxial symmetry.Consequently, it is possible to diagonalize the matrices separ-ately for even and odd ℓ and for different m. In the angularmomentum space, this special characteristic results in a block-diagonal eigenvalue problem. In contrast, for GeS, the an-isotropy mixes states with the same ℓ but different m.In addition to lifting the degeneracy, we can clearly noticethat the exciton BE strongly depends on the reduced mass andthe dielectric constant and hence on the anisotropic para-meters A, B, and C. Strong anisotropy (A ≠ B ≠ C) leads to anincrease (EB/R̄y = 1.5) in the exciton BE compared to the isotro-pic case (A = B = C). In the EMA, we found that the 1̃s-excitonBE and Bohr radius of the bulk GeS were ∼7.6 meV and13 nm, respectively. This is consistent with the BSE method,which indicates that the exciton BE is ∼10 meV. In fact, thesmall BE (largest Bohr radius) is due to the highest effectivedielectric function ɛ̄ and the smaller reduced mass μ̄, wherethe reduced mass out-of-plane is significantly smaller thanthose in-plane. These values account for the swift dis-appearance of exciton emission at high temperatures, sincetheir magnitudes are lower than the thermal energy (kBT ). Infact, in section A of the ESI† using temperature-dependent PL,we estimated the quenching of the exciton emission at a temp-erature of 130 K under 2.41 eV excitation, compared to 190 Kunder 1.88 eV resonant excitation.To better understand the impact of anisotropy on excitonwavefunctions, we compared the probability densities (squaredmoduli of the excitonic wavefunctions) for the ground-stateexcitonic wavefunction 1̃s for the three previous cases in Fig. 6(b–g). The isotropic case produces the well-known hydrogen-like wavefunctions. For the transversal isotropic case, asexpected, the excitonic wavefunction for the in-plane com-pound has a spherical symmetry, while the orbital becomes adisk stretched along the z-direction, since the reduced mass inthe z-direction is much smaller than the average in-planereduced mass. For GeS, the anisotropy modifies the groundstate from a spherically symmetric s (isotropic) to a squeezedwavefunction with a slightly peanut shape. In fact, we foundthat the 1̃s state stretched in the three directions; however, thedistortions (squeezing) are more significant in the out-of-planedirection due to the heavier mass of this direction. Thesmaller area across which the probability density extends inthe in-plane direction for the anisotropic case as compared tothe isotopic one is due to the higher BE, which leads to morelocalized wavefunctions.Fig. 7(b) illustrates the impact of anisotropic parameters (A,B, C) on the ground state (1̃s) and highlights the strong depen-dence of the exciton energy on the degree of anisotropy. Byadjusting these parameters, we can shift from the strong an-isotropy case to the isotropy one. In fact, depending on thevalues of A and B, we found three limiting regimes. If A = B = C= 1, eqn (4) and (5) become the well-known equation for theisotropic case, whose eigenvalues are −R̄y/n2, with the well-known degree of degeneracy n2 so that EB1s = R̄y, as clearlyshown in Fig. 7(c). Then, with the decrease of A and B, thedegree of perturbation induced by the anisotropy increases,leading to an increase in the exciton BE. Indeed, for A = B ≥0.1, the excitonic ground state tends to the well-known 2D-hydrogenic energy EB1̃s ¼ 4R̄y due to the asymmetry in theCoulomb potential caused by the strong perturbation inducedby the anisotropy in this regime. Interestingly, for A = B ≥ 1,we found that the 1s̃-exciton energy is lower than the Rydbergenergy EB1̃s ¼ R̄y.From the knowledge of the exciton energies and wavefunc-tions as functions of the anisotropic parameters, we cancompute the OS and hence the PL signal for these excitonstates. Indeed, the OS is a dimensionless quantity that givesthe relative strength of a particular optical transition. Here, weconsider only the most common case of direct allowed opticaltransitions between the valence and conduction bands. Forinstance, the OS of the optical interband transition for excitonstates isf αqñ;‘̃;m̃¼ 2m0ℏω0� Fñ;‘̃;m̃q;c;v��� ���2;where ω0 is the angular frequency of the optical transitionwith energy ℏω0 ¼ EXñ;‘̃;m̃. The quantity Fñ;‘̃;m̃q;c;v ¼ αq �h1 eiq�rp�� ��ζXñ;‘̃;m̃i is the OME between the crystal ground state|∅〉 and the excited states jζñ;‘̃;m̃X i corresponding to the directexciton in bulk GeS. Here,ζXñ;‘̃;m̃ðre; rhÞ ¼ ϒ ñ;‘̃;m̃ðRCM; ρÞuc;keðreÞu*v;khðrhÞ, and αq denotesthe photon polarization unit vector. For the scenario of uncon-fined exciton COM motion in bulk GeS, the OS is expressed asfollows (see section F in the ESI† for more details)67fαqñ;‘̃;m̃¼ 2Vm0EXñ;‘̃;m̃Xn;‘;mCn;‘;mΦn;‘;mðρ ¼ 0Þ����������2�Mc;vi;i ðkÞ: ð7ÞThe OME and the relative wavefunction give rise to differenttypes of selection rules. Indeed, only allowed transitions occurwithin the excitonic state wherein the relative wavefunctionΨ relñ;‘̃;m̃ðρ ¼ 0Þ = 0. For the OME, the selection rules comemainly from the interband coupling term, which depends onthe nature of the Bloch function.Fig. 7(a) presents the calculated normalized OS of the threelow-lying s-exciton states 1s̃, 2s̃, and 3s̃ in bulk GeS (blackline). For comparison, we also plotted the OS for the isotropic(blue line) and transversal isotropic (red line) cases. It is seenthat the OS decay ratio in the anisotropic case shows an anom-alous behavior compared to that in the isotopic case. In GeS,the intensity ratio f AC1̃s =f ACñs exhibits different values comparedto a bare isotropic hydrogen-like Coulomb potential, where theexciton OS of states ñs decays as f ACñs ¼ f AC1̃s =n3. Specifically, theratio is approximately 15 for 1̃s/2̃s and 55 for 1̃s/3̃s, while it isequal to 8 and 27, respectively, in the isotropic case. Thisfinding is consistent with the results reported by T. Shubinaet al. for InSe.68 Indeed, this difference is due to the fact that,in contrast to the isotropic case, the perturbation induced byanisotropy generates linear combinations of basis functions,Nanoscale PaperThis journal is © The Royal Society of Chemistry 2023 Nanoscale, 2023, 15, 17014–17028 | 17023Open Access Article. Published on 16 October 2023. Downloaded on 12/23/2023 5:53:18 AM.  This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.View Article Onlinehttp://creativecommons.org/licenses/by/3.0/http://creativecommons.org/licenses/by/3.0/https://doi.org/10.1039/d3nr03168cso different orbitals with different weights, Cn,ℓ,m, contributein each anisotropic excitonic transition. Therefore, the s-, p-and even d-state shell excitons are mixed due to the reductionof symmetry induced by the anisotropy of GeS. This mixingmakes the p-shell states optically active. Hence, both s-shelland p-shell excitons are active in both one- and two-photonprocesses, providing an efficient mechanism of second-harmo-nic generation. The discernible characteristic may be observedthrough the perturbed matrix elements. Moreover, this illus-trates that states with different angular momentum com-ponents are indistinguishable in GeS, which well describes themixing of excitonic states with different parities, particularlythe s-shell and p-shell states. These non-linear contributionsare similar to those related to the magneto-Stark effect or theelectric-field-induced mixing of excitons in ZnO69 and GaAs.70The non-linear contributions to the exciton emission arebeyond the scope of this work, as the one-photon emission isinvestigated. In addition, we found that the ñs-state BEs forthe transversal isotropic and full anisotropic cases are almostsimilar. This is also shown in Fig. 6, in which the exciton BEsof the ñs-state for these two cases are almost equal. In fact,due to the effective light mass along the z direction comparedto the in-plane effective mass, the total effective mass in theanisotropic case (μ̄) is almost equal to the transversal isotopiccase (μk), which produces a comparable Rydberg energy (R̄y)and Bohr radius (āb) in both cases.The OS per unit volume can be related to the probability ofemission of a photon over all the photon modes using the fol-lowing formula:PñsαqðωqÞ ¼ðdqf ñsαqLðℏωq � EXñsÞ ð8Þwhere the Lorentzian Lðℏωq � EXñsÞ ¼ γñs=π½ðℏωq � EXñsÞ2 þ γ2ñs�expresses the energy conservation taking into account the stateof the linewidth broadening extracted from our experiment, γ1̃s= 8 meV. Notably, the OS of exciton states is generally differentbetween one- and two-photon processes. In Fig. 7(d), weplotted the PL spectra for different excitonic states, and in thesame plot we show the experimental PL spectrum of the freeexciton at low temperature. We found that the exciton groundFig. 7 Anisotropy effect on OS for low-lying exciton states in GeS. (a) Normalized OS for the first low-lying ñs state for different cases. (b) 1̃s-exciton BE as a function of anisotropy parameters. The plot presents three critical regions (isotropic, weak anisotropic, and strong anisotropic),which are separated by a dashed cyan line. (c) Theoretical prediction of the angle-resolved PL spectrum for GeS. (d) Direct comparison betweenexperimental and theoretical PL for low-lying exciton states.Paper Nanoscale17024 | Nanoscale, 2023, 15, 17014–17028 This journal is © The Royal Society of Chemistry 2023Open Access Article. Published on 16 October 2023. Downloaded on 12/23/2023 5:53:18 AM.  This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.View Article Onlinehttp://creativecommons.org/licenses/by/3.0/http://creativecommons.org/licenses/by/3.0/https://doi.org/10.1039/d3nr03168cstate 1̃s located at around 1.772 eV dominates the PL spec-trum, which is expected since it has the largest OS. When com-paring our experimental results, it is evident that the excitedexcitonic states, such as 2̃s and 3̃s, are not clearly visible in ourmeasurements. This is due to different factors: (i) the largestlinewidth broadening (γ1̃s), which is significantly greater thanthe energy separation between different excitonic s̃ states(δE1̃s;ñs ¼ EXñs � EX1̃s) and (ii) the value of the OS (which isdirectly proportional to the PL intensity) that corresponds tothe excited exciton ñs states is lower than the 1̃s state. Thus,the nearly energetic ñs states overlap and assemble into asingle peak in the PL spectra, which appears to be broadenedand asymmetric with a lower slope on the lower-energy side. Itis important to note that this peak’s asymmetry may also bedue to phonon-assisted processes.71Using eqn (8), we calculated the angle-resolved PL at lowtemperature in Fig. 7(d). The PL signal is strongly dependenton the polarization angle. In fact, the exciton signal ismaximum along the AC direction, then the signal intensitydecreases until it disappears in the ZZ direction. This demon-strates the anisotropic optical signature in GeS, and agreeswell with the experimental results in Fig. 1.Our experimental results reveal the existence of multipleemission peaks at low temperature in GeS encapsulated withh-BN, in addition to the free neutral exciton peak, these peaksare located in the energy range of approximately 60–100 meVbelow the neutral exciton. Despite thorough investigations, thenature of these low-energy emission remains unknown, bothfrom theoretical and experimental perspectives. During thisstudy, to identify these peaks, we qualitatively studied differentscenarios, including biexciton, optical phonon-assisted excitonrecombination, emission of charged excitons (trions) and theirfine structure, and exciton plus a localized exciton. Moredetails about the nature of these peaks can be found in sec-tions A and B of the ESI;† however, a deep quantitative studyof these low-energy emissions is beyond the scope of thepresent paper.5. ConclusionsAnisotropy of the low-symmetry orthorhombic crystal structureof GeS offers possibilities for the manipulation of physical pro-perties along different crystal directions. We presented adetailed study of the anisotropic properties of GeS using temp-erature- and polarization-dependent PL techniques supportedby ab initio calculations. Temperature-dependent PL experi-ments enabled us to investigate the excitonic impact on theoptical properties of GeS, whilst polarization-dependent PLmeasurements revealed the anisotropic nature of the material.Ab initio calculations were used to theoretically predict theelectronic BS and optical properties of GeS and to interpret theexperimental results. By applying both the IPA and GW + BSEmethods, we calculated the dielectric functions and showedthat including exciton effects through the BSE method mod-ifies the optical absorption spectra in bulk GeS. Using theEMA, we showed that compared to an isotropic system, the an-isotropy in GeS breaks the degeneracy, causes mixing of stateswith the same quantum number but different orbital andangular quantum numbers, and increases the BE. We alsofound a relatively low BE, on the order of ∼10 meV, and a quitelarge exciton Bohr radius, on the order of 13 nm, whichexplains the observed rapid decrease in the exciton peak as thetemperature increases. This result provides an essential under-standing of the complexities of anisotropic materials and mayhave ramifications for the design of certain technologicalapplications.6. Methods6.1. SampleThe studied sample is composed of a thin layer of GeS, charac-terized by a thickness of about 60 nm, encapsulated in h-BNflakes and placed on a Si/SiO2 substrate. The GeS crystal,which was used for the preparation of the investigated sample,was purchased from HQ graphene. Thin GeS flakes weredirectly exfoliated on a 285 nm SiO2/Si substrate in an inertgas glovebox (O2 < 1 ppm, H2O < 1 ppm). Then we used a poly(bisphenol A carbonate)/polydimethylsiloxane stamp on aglass slide to pick up top h-BN, thin flakes of GeS, and bottomh-BN at 80 °C with the assistance of the transfer stage in theglove box. Finally, the stack was released on a Si/SiO2 substrate.The thicknesses of the GeS flakes were first identified byoptical contrast and then measured more precisely with anatomic force microscope.6.2. Experimental techniquesPL spectra were recorded at different illumination wavelengthswith a series of continuous wave (CW) laser diodes: λ = 488 nm(2.54 eV), λ = 515 nm (2.41 eV), λ = 561 nm (2.21 eV), λ =633 nm (1.96 eV), and λ = 660 nm (1.88 eV). PLE experimentwas carried out using a supercontinuum light source coupledwith a monochromator as an excitation source. For the RCstudies, the only difference in the experimental setup withrespect to the one used to record the PL and PLE signals con-cerned the excitation source, which was replaced by a tungstenhalogen lamp. The studied samples were placed on a coldfinger in a continuous-flow cryostat mounted on x–y motorizedpositioners. The excitation light was focused by means of a100× long working distance objective with a 0.55 numericalaperture that produced a spot of about 1/4 μm diameter in thePL/RC measurements. The signal was collected via the samemicroscope objective, sent through a 0.75 m monochromator,and then detected using a cooled charge-coupled devicecamera cooled with liquid nitrogen. The excitation powerfocused on the sample was kept at 100 μW during all measure-ments to avoid local heating. Polarization-resolved PL and RCspectra were recorded using a motorized half-wave platemounted on top of the microscope objective and a fixed linearpolarizer placed in the detection path, which provides simul-Nanoscale PaperThis journal is © The Royal Society of Chemistry 2023 Nanoscale, 2023, 15, 17014–17028 | 17025Open Access Article. Published on 16 October 2023. Downloaded on 12/23/2023 5:53:18 AM.  This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.View Article Onlinehttp://creativecommons.org/licenses/by/3.0/http://creativecommons.org/licenses/by/3.0/https://doi.org/10.1039/d3nr03168ctaneous rotation of the co-linearly polarized excitation lightand detected signal.6.3. Computational details6.3.1. Ground-state calculations from first-principles.Structural relaxation and electronic properties were investi-gated using the QUANTUM ESPRESSO72,73 (QE) package withDFT74 based on the GGA of the Perdew–Burke–Ernzerhof (PBE)exchange–correlation functional75 within the plane-waveexpansion. The vdW correction of the PBE functional was per-formed using the Grimme DFT-D2 method.76 The optimizednorm-conserving pseudopotentials77 from the QE repositorywere used to describe the core-valence interaction. To evaluatethe role of spin–orbit interaction, calculations were performedusing the fully relativistic version of the same pseudopoten-tials from the QE repository. The atomic positions and latticeconstants were fully relaxed by DFT. It is assumed that therelaxation of the structure has reached convergence when themaximum component of the residual ionic forces is less than10−10 Ry per Bohr. The Broyden–Fletcher–Goldfarb–Shanno78method was used for structural optimization. After conver-gence tests, the energy cutoff for the plane-wave expansion ofthe wavefunction was set to 816 eV for all calculations and anappropriate Monkhorst–Pack k-point sampling79 in the BZ wascentered with 18 × 16 × 4, 18 × 16 × 4, and 24 × 22 × 6 meshesfor geometry optimizations, self-consistent calculation, andprojected density of states, respectively. The cutoff energy andk-point sample were tested with PBE-GGA calculation in theconvergence study to ensure numerical stability. The criteriafor the convergence of forces and total energy in optimizationwere set to 10−4 eV Å−1 and 10−5 eV, respectively.6.3.2. Excitation energies and exciton wavefunctions fromfirst-principles. Self and non-self-consistent DFT calculationswere performed to obtain KS eigenvalues and eigenfunctionsto be used in the many-body perturbation theory80–82 by usingthe YAMBO code.83,84 YAMBO interfaced with QE, whichallows for the calculation of the optical response starting fromthe previously generated KS wavefunctions and energies in aplane-wave basis set. The YAMBO code was used to calculateQP adjustments at G0W0 and evGW, which was then used tocalculate the optical excitation energies and optical spectra bysolving the BSE. For GW simulations, the inverse of the micro-scopic dynamic dielectric function, εG,G′−1, was obtainedwithin the plasmon-pole approximation.84,85 After we calcu-lated the convergence test of the parameters, we set the follow-ing parameters as the starting point for our calculation. Wehave used 200 bands, a 45 Ry energy cutoff for the self-energyexchange component (the number of G-vectors in theexchange) and a 18 Ry cutoff for the correlation part (energycutoff in the screening) or response block size. To speed upconvergence with respect to empty states, we adopted the tech-nique described in ref. 86 as implemented in the YAMBOcode. QP BSs were then used to build up the excitonicHamiltonian and to solve the BSE. We obtained convergedexcitation energies considering, respectively, six empty statesand six occupied states in the excitonic Hamiltonian, the irre-ducible BZ being sampled up to a 24 × 22 × 4 k-point mesh.We used the Tamm–Dancoff approximation80,87 for the Bethe–Salpeter Hamiltonian and took into account the local fieldeffects.Author contributionsZ. C. and M. K. fabricated the samples; K. W. andT. T. provided the hBN crystals; M. A., N. Z., A.B., and M. R. M.carried out the experiments; N. Z. and M. R. M performed theexperimental data analysis; M. A. and S. A. performed thetheoretical study; M. A. carried out DFT, GW, BSE and EMAcalculations; M. R. M. designed the project; M. A., N. Z., S. A.,S. J, and M. R. M. prepared the manuscript with contributionfrom all other co-authors.Conflicts of interestThere are no conflicts to declare.AcknowledgementsThis work was supported by the National Science Centre,Poland (Grant No. 2017/27/B/ST3/00205 and 2018/31/B/ST3/02111), the Ministry of Education (Singapore) through theResearch Centre of Excellence Program (grant EDUN C-33-18-279-V12, I-FIM) and under its Academic Research Fund Tier 2(MOE-T2EP50122-0012), and the Air Force Office of ScientificResearch and the Office of Naval Research Global under awardnumber FA8655-21-1-7026. K. W. and T. T. acknowledge thesupport from the JSPS KAKENHI (Grant Numbers 20H00354and 23H02052) and World Premier International ResearchCenter Initiative (WPI), MEXT, Japan.References1 R. Mas-Balleste, C. Gomez-Navarro, J. Gomez-Herrero andF. Zamora, Nanoscale, 2011, 3, 20–30.2 G. R. Bhimanapati, Z. Lin, V. Meunier, Y. Jung, J. Cha,S. Das, D. Xiao, Y. Son, M. S. Strano, V. R. 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