Temperature dependent excitonic effects in the optical properties of single-layer MoS 2
Alejandro Molina-Sánchez, Maurizia Palummo, Andrea Marini, Ludger Wirtz
TTemperature dependent excitonic effects in the optical properties of single-layer MoS Alejandro Molina-S´anchez, Maurizia Palummo,
2, 3
Andrea Marini, and Ludger Wirtz Physics and Materials Science Research Unit, University of Luxembourg,162a avenue de la Fa¨ıencerie, L-1511 Luxembourg, Luxembourg University of Rome Tor Vergata, Rome, Italy INFN, Laboratori Nazionali di Frascati, Via E. Fermi 40, 00044 Frascati, Italy Istituto di Struttura della Materia of the National Research Council,Via Salaria Km 29.3, I-00016 Monterotondo Stazione, Italy (Dated: April 5, 2016)Temperature influences the performance of two-dimensional materials in optoelectronic devices.Indeed, the optical characterization of these materials is usually realized at room temperature.Nevertheless most ab-initio studies are yet performed without including any temperature effect. Asa consequence, important features are thus overlooked, such as the relative intensity of the excitonicpeaks and their broadening, directly related to the temperature and to the non-radiative excitonrelaxation time. We present ab-initio calculations of the optical response of single-layer MoS ,a prototype 2D material, as a function of temperature using density functional theory and many-body perturbation theory. We compute the electron-phonon interaction using the full spinorial wavefunctions, i.e., fully taking into account effects of spin-orbit interaction. We find that bound excitons( A and B peaks) and resonant excitons ( C peak) exhibit different behavior with temperature,displaying different non-radiative linewidths. We conclude that the inhomogeneous broadening ofthe absorption spectra is mainly due to electron-phonon scattering mechanisms. Our calculationsexplain the shortcomings of previous (zero-temperature) theoretical spectra and match well with theexperimental spectra acquired at room temperature. Moreover, we disentangle the contributions ofacoustic and optical phonon modes to the quasi-particles and exciton linewidths. Our model alsoallows to identify which phonon modes couple to each exciton state, useful for the interpretation ofresonant Raman scattering experiments. I. INTRODUCTION
Ultrathin two-dimensional materials such as grapheneand MoS are appealing candidates for a new gener-ation of opto-electronic devices such as photorespon-sive memories , light-emitting and harvesting devices ,or nano-scale transistors . They are also suitableplatforms for carrying out research on fundamentalphysics phenomena like the valley Hall effect, ultrafastcharge transfer, or valley excitons in two-dimensionalmaterials. Technologically, single-layer MoS is relevantdue to a direct optical gap at 1.8 eV and a high electronmobility. The optical response of MoS is dominated by stronglybound excitons. The same holds for the other groupVI semiconducting single-layer transition-metal dichalco-genides (TMDs) such MX with M = Mo or W and X =S, Se, or Te . This suggests their possible use in opto-electronic devices working at room temperature. Never-theless most of the modern first-principles ground andexcited state simulations are performed at 0 K and thusomit the role of thermal lattice vibrations on the elec-tronic and optical properties.In general, temperature has a capital influence onthe electronic and optical properties of semiconduc-tors determining their application as optoelectronicdevices. It is well known that its drives the band gaprenormalization, and induces changes in the positionand width of the optical peaks. At the same timealso the spectra obtained from other techniques such as angle-resolved photoemission spectroscopy (ARPES) are clearly influenced by the temperature due to the en-hanced mixing of electron and phonon states.The possibility to perform electronic structure cal-culations based on ab-initio approaches including theelectron-phonon (EP) interaction is thus of paramountimportance. Even though many years ago Heine, Allenand Cardona (HAC) pointed out that the EP couplingcan induce corrections of the electronic levels as large asthose induced by the electronic correlation, the numberof works based on first-principles simulations address-ing this problem is still very limited and mainly dedi-cated to traditional bulk compounds. The inclusion ofEP couplings considerably broadens the scope of first-principles electronic-structure calculations beyond thestudy of temperature effects. It opens the way to thestudy of many interesting phenomena such as polaronformation in crystals and transport properties. As for MoS , theoretical ab-initio studies including theEP interaction have addressed specific aspects such asphonon-limited mobility, thermal conductivity, elec-tron cooling, or electron transport. Tongay et. al. have performed a theoretical and experimental study ofthe band gap dependence on temperature for multi-layerMoSe and MoS . They have attributed all temperatureeffects to lattice renormalization induced by the thermalexpansion. They capture correctly the band gap trendonly for high temperatures (above 300 K). Below roomtemperature, the electron-phonon interaction plays a cru-cial role but it is ignored by Tongay et. al. In Ref. 11Qiu et. al. have studied temperature effects by includ- a r X i v : . [ c ond - m a t . m e s - h a ll ] A p r ing the quasi-particle linewidths. However, they ignoredthe energy renormalization and the accurate calculationof the linewidths across all the Brillouin zone. Here weexplore using a fully ab initio approach how the EP in-teraction induces changes in the electronic structure andoptical properties of the MoS single-layer. This alsoenables us to address photoluminescence, ARPES, and resonant Raman scattering experiments. Differently from most of the recent works on bulkmaterials we do not limit our study to the band gaprenormalization but we extend our investigation to thefull band structure, with special attention to the electronstates of interest for opto-electronic applications. Start-ing from previous studies which established the existenceof several kind of excitonic states in this low-dimensionalmaterial, whose behavior we characterize as a func-tion of the temperature. We calculate the shift of thebinding energy and the non-radiative linewidths of ex-cited states.It is worth to underline that in our approach we usethe full spinorial nature of the wave functions throughall ground and excited state calculations. This is quiteimportant because it is well known that spin-orbit cou-pling determines the valley polarization dynamics and isfundamental to understand the optical properties of allTMDs. II. THE THEORETICAL APPROACH
Our calculations start with density-functional theory(DFT) to obtain a first estimate of the electronic bands.We use density-functional perturbation theory (DFPT)to calculate the phonon modes and the electron-phononcoupling matrix elements. With the latter we calculatethe change of the electronic bands due to the latticevibrations. Afterwards, we solve the temperature de-pendent Bethe-Salpeter equation . We thus explore thechange in the optical spectra, and in the exciton energiesand linewidths when temperature increases.The ground state properties of single-layer MoS ,eigenvalues and wave functions, are calculated with theQUANTUM ESPRESSO code within the local densityapproximation (LDA) for the exchange-correlation po-tential. We use DFPT to obtain the phonon modes aswell as the first and second order electron-phonon ma-trix elements. As mentioned in the introduction thespin-orbit interaction is essential to correctly describe ex-citons in MoS , for this reason also the electron-phononmatrix elements are calculated taking into account thefull spinor wave functions.We study the temperature effects on the electronicstates and on the excitons by merging DFT/DFPT withmany-body perturbation theory. Within this framework,two self-energy diagrams, which correspond to the lowestnon vanishing terms of a perturbative treatment, have tobe evaluated. The Fan self-energy, related to first orderterms Σ F ann, k ( ω, T ) = i (cid:88) n (cid:48) q λ | g q λnn (cid:48) k | N q ×× (cid:20) N q ( T ) + 1 − f n (cid:48) k − q ω − ε n (cid:48) k − q − ω q λ − i + (cid:21) × (cid:20) N q ( T ) + f n (cid:48) k − q ω − ε n (cid:48) k − q + ω q λ − i + (cid:21) , (1)where ε n, k are the LDA eigenvalues, ω q ,λ the phononfrequencies, f n, k and N q ( T ) are the Fermi and Bosedistribution functions of electrons and phonons, respec-tively. The self-energy associated to an electron state( n, k ) is the sum over all the electron states n (cid:48) andphonon modes λ , where N q is the number of q vectors inthe Brillouin zone. Conservation of momentum is explic-itly enforced. The first order electron-phonon matrix el-ements g q λnn (cid:48) k represents the amplitude for the scatteringprocess | n k (cid:105) → | n (cid:48) k − q (cid:105) ⊗ | q λ (cid:105) . We have a similar ex-pression for the Debye-Waller (DW) self-energy, relatedto the second order terms,Σ DWn, k ( T ) = 1 N q (cid:88) q λ Λ q λ, − q λnn k (2 N q λ ( T ) + 1) , (2)where Λ q λ, q (cid:48) λ (cid:48) nn (cid:48) k , insted, represents the amplitude for thesecond–order scattering process | n k (cid:105) → | n (cid:48) , k − q − q (cid:48) (cid:105)⊗| q λ (cid:105) ⊗ | q (cid:48) λ (cid:48) (cid:105) . In both self-energy terms, tempera-ture enters via the phonon population. In polar semi-conductors the electron-phonon interaction strength be-comes larger when including the Fr¨ohlich polar-couplingterm. Nonetheless, the LO-TO splitting in MoS israther small, 3 cm − , and we do not expect significantchanges in single-layer.The fully interacting electron propagator (accountingfor the electron-phonon interaction) is G n k ( ω, T ) = (cid:0) ω − ε n k − Σ F ann k ( ω, T ) − Σ DWn k ( T ) (cid:1) − . (3)The complex poles of this equation define the electronicexcitations of the interacting system. If the quasi-particleapproximation (QPA) is valid, and assuming a smoothfrequency dependence, the electron-phonon self-energycan be expanded up to the first order around the bareenergies ( (cid:15) nk ). In this case the temperature dependentquasi-particle energies are defined as, E n k ( T ) = ε n k + Z n k ( T ) (cid:2) Σ F ann k ( ε n k , T ) + Σ DWn k ( T ) (cid:3) . (4)It is clear that the quasi-particle energies depend ontemperature and are complex numbers, where the realparts are the quasi-particle energies and the imaginaryparts, Γ n k ( T ), correspond to the quasi-particle widths.The renormalization factor Z n k represents the quasi-particle charge. Therefore the QPA makes sense when Z n k takes a value close to 1. If the QPA holds, thespectral function, A n k ( ω, T ), the imaginary part of theGreen’s function, is a single peak Lorentzian functioncentred at E n k and with width Γ n k ( T ). The narrowerthe spectral function is the weaker the electron-latticeinteraction is. When the electron-lattice interaction be-comes strong enough, the QPA breaks down, the spectralfunctions does no longer consist of Lorentzian peaks butspan a wide energy range. III. TEMPERATURE DEPENDENTELECTRONIC STRUCTURE OF SINGLE-LAYERMOS The calculation of the electronic structure of MoS hasbeen done in a plane-wave basis using norm-conservingpseudopotentials and a kinetic energy cutoff of 80 Ryand a k -grid of 12 × ×
1. On top of self-consistentDFT simulations, electron-phonon matrix elements areobtained by DFPT in the local-density approximation.From the explicit expressions of the self-energy terms(Eqs. (1) and (2)), it becomes clear that a careful con-vergence over the number of bands n (cid:48) and the numberof transferred phonon momenta q to evaluate the inte-gral over the Brillouin zone, is required. From our studywe have found that, the spectral functions of MoS con-verge using a set of 400 randomly distributed q pointsand 36 bands (18 occupied bands - we do not take intoaccount Mo-semicore electrons - and 18 empty bands).The calculations are converged with respect to numberof q -points and bands. We have checked this on the pro-file of the spectral function which is a more stringent testthan checking the eigenvalue correction. We have used aLorentzian broadening of 60 meV. Recent works on dia-mond and silicon required a much larger number of bandsand q -points to reach convergence. . The rapid conver-gence with the number of q -points we have found here ismainly due to the two-dimensional nature of the materialunder investigation.Figure 1 shows the spectral function of single-layerMoS for temperature 0 K (left panel) and 300 K (rightpanel). An animated represenation of the band struc-tures for temperature ranging from 0 to 1000 K in step of100 K can be found in the Supplementary Informations.Dotted black lines represent the LDA band structure(without electron-phonon interaction). We have markedwith squares some important points in the band struc-tures, which will be discussed in more detail below. Thebands are no longer a line and they acquire a broadening.This broadening is directly related to the linewidths ofeach quasiparticle state. Considering that lifetimes areinversely proportional to linewidths, narrow lines are re-lated to long lifetimes, i.e., stable states. On the opposite,broader states have a stronger interaction with phononsand they have more non-radiative recombination paths,meaning a shorter lifetime.We identify very narrow line shapes at the band edges like the valence band states at K and Γ points. In theconduction band we find narrow line shapes at K and atthe minimum between K and Γ. Temperature tends toreduce the quasi-particle energy but it does not changesignificantly the spectral function, it only moves the max-imum to lower energies. The increase of temperature re-sults in a shrinking of the gap. Even at 0 K the gapis diminished by 75 meV (with respect to its value cal-culated without electron-phonon coupling). This is aneffect of the zero-point vibrations of the atoms.The spectral functions of quasi-particle states far fromthe band edges have a different aspect. Close to cross-ings, the bands become blurred, making it difficult todistinguish individual bands. For instance, the conduc-tion band around Γ and the crossing close to M have anoticeable broadening, even at 0 K. The M point showsalso broader bands than the Γ and the K point. Theincreasing of temperature blurs even more the reminis-cence of the LDA band dispersion. In areas close to Γ,the band index becomes almost obsolete and we observea wide spectral range. It is worth to note that quasi-particle states are not necessarily broadened peaks cen-tered at the renormalized electron energy. They can alsobe mixed states which can have a structure very differentfrom the superposition of the electron and hole densityof states. We also expect important consequences on theoptical properties. Excitons from states in these rangeof energies (like the resonant or van-Hove exciton )should be affected by the increasing of temperature muchmore that those coming from band edges.Figure 2 represents the spectral function of the quasi-particle states marked with squares in Fig. 1. We haveselected three temperatures, 0 (dotted), 300 (dashed)and 1000 (solid) K. The arrow indicates the LDA en-ergy. Even though 1000 K is a very high temperature forcommon experiments, it can help us conceptually to dis-cuss the nature of the electron-phonon interaction. Pan-els (a) and (b) of Fig. 2 show the conduction and va-lence band extrema at K . We observe a shift of thepeaks with a slight broadening, but always conservingthe Lorentzian shape. Regarding the spin-orbit interac-tion, there is no electron-phonon mediated spin mixing,neither of the conduction nor of the valence band statesat K . Our calculation rules out the possibility of inter-valley scattering from the point K to K (cid:48) at the VBM. Forthe conduction band states, the electron-phonon interac-tion conserves the spin degeneracy. However, the valenceband states are splitted due to the spin-orbit interaction.Comparison with MoS ARPES data collected at 80 Kis a delicate issue. Experimental broadening is not ex-clusively related to electron-phonon decay. Nevertheless,the measured spin-orbit splitting of 145 meV agrees verywell with our calculation of 135 meV.Fig.2 (c) shows the spectral function of the state in thelocal minimum between K and Γ. This spectral functionhas a similar behavior as the cases (a) and (b) but itsasymmetry is stronger. This result is compatible withthe exposition of Ref. 39, in which transitions are possibleFIG. 1: Spectral functions of single-layer MoS for temperatures 0 K (left panel) and 300 K (right panel). The colorsquares denotes the points T c (red), T c (cid:48) (blue), K c (purple), K v (green). Spectral function are normalized. The colorscale bar indicates the maximum value in black and the minimum in white.from this local minima to the point K .We have found a signature of a potential breakdown ofthe quasi-particle approximation for some states abovethe band gap. Figure 2 (d) shows a drastic change of thespectral function due to temperature effects. We havechosen a band close to Γ, relevant for describing the ex-citon C . The shape even at 300 K becomes asymmetricand when we reach 1000 K a secondary peak emerges.Notice that the high energy peak is separated from thelow energy one by an energy far larger than any phononin MoS . This latter peak appears at higher energy ofthe LDA energy, contrary to the others spectral func-tions. This is a proof of the many-body character of thenew state and of the breakdown of the quasi-particle ap-proximation. The new states cannot be interpreted anymore as an independent sum of electrons and phononreplica. The energy separation between the shoulder andthe lower peak is bigger than any phonon frequency. Fol-lowing Ref. 41, the electron is fragmented in severalentangled electron-phonon states, as a result of virtualtransitions not bound to respect the energy conserva-tion. This explains the appearance of these structuresin a wide energy range. We follow the analysis of the temperature dependentelectronic structure by investigating the band gap renor-malization. We have seen that at 0 K the band gap ofsingle-layer MoS shrinks. The reason is the uncertaintyprinciple and this is known as the zero-point motionrenormalization (ZPR) effect. At 0 K, atoms cannot beat rest and have zero velocity, there is a minimum quan-tum of energy which supplies the vibration which makespossible the electron-phonon interaction. Table I showsthe ZPR for several semiconductors, calculated in pre-vious works. Single-layer MoS exhibits a smaller ZPReffect, especially in comparison with Diamond. The wavefunction of the conduction and valence band state at K are mostly concentrated around the molybdenum atoms. FIG. 2: Spectral functions for selected band states inFig. 1, T c (red), T (cid:48) c (blue), K c (magenta) and K v (green), and for T = 0 (dotted), 300 (dashed) and 1000(solid) K.The large mass of molybdenum reduces the phonon am-plitude with the consequence of a smaller correction.In order to shed light on which phonon modes con-tribute to the electron-phonon interaction it is useful tocalculate the Eliashbergh functions. g F ( ω ) = (cid:88) λ q (cid:34) (cid:80) n (cid:48) | g q λnn (cid:48) k | N − q ε n k − ε n (cid:48) k (cid:48) (cid:35) δ ( ω − ω q λ ) − (cid:88) λ q (cid:34) (cid:80) n (cid:48) Λ q λnn (cid:48) k N − q ε n k − ε n (cid:48) k (cid:48) (cid:35) δ ( ω − ω q λ ) . (5) ZPR (meV)MoS SiC 223 Si 123 TABLE I: Zero-point motion renormalization ofsingle-layer MoS , Diamond, SiC and Si.Figure 3 shows phonon dispersion of single-layerMoS (panel a), together with the phonon density ofstates (panel b) and the Eliashbergh functions (pan-els (c),(d),(e)) calculated for the quasi-particle statesmarked with the same color in Figs. 1 and 2. In Fig.3 (a) the color of the phonon dispersion curves indicatethe vibration mode direction: vibrations in-plane are rep-resented by red dots and out-of-plane vibrations are rep-resented by blue dots.A common trend to all the Eliashberg functions isthe absence of acoustic-phonon contributions close to Γ.In calculations without the DW term (not shown here),there is a finite contribution of the Fan Eliasbergh func-tion, which is removed once the DW is added. Therefore,even though the DW term has a small contribution, it isimportant in order to achieve accurate results. The statesat K (panel (d)) have similar Eliashberg functions, al-most symmetric. They have opposite sign which resultsin the shrinking of the band gap. The main contribu-tions come from phonons at the edge of the Brillouinzone around 200 cm − and from optical phonons around400 cm − . In the case of the phonons of the state T c (e) we find a similar Eliashberg function. The high fre-quency contribution is almost identical to the one of thestate K c .The Eliashberg function for the state close to Γ (panelc) has a different shape than the others. There is nocontribution from mid-frequency phonons. The main in-teraction is due to optical phonons, around 380 cm − .In this case, the Eliashberg function changes the sign,crossing several times the zero axis (dotted lines). TheEliasberg functions shown in Fig. 3 (d) are centered closeto the frequency of mode A g , in a frequency range ex-clusively populated by out-of-plane phonons. In contrast,the function in panel (c) is built from in-plane phononsclose to the frequency of the phonon mode E g Thesefindings seem to support the conclusion reached in a re-cent work by Carvalho et. al on the base of pure sym-metry arguments. Measuring Raman spectra of a MoS monolayer, they concluded that exciton A couples withphonon mode A g while exciton C couples with phononmode E g .Since excitons and phonons are particularly complex inTMDs, some deviations can occur as excitons are builtfrom many electron and hole states of different electronmomentum k and we are just examining a few states.We have analyzed the Eliashberg functions at different k points (around K and Γ) observing a small energy shiftsbut not drastic changes under small changes of k . Fromthis result we can affirm that the identification of Ref. 28correspond to the excitons A and C. A definitive proofof this statement would consist in calculating the Ramantensor (in resonant conditions) but this is out of the scopeof our work. IV. FINITE TEMPERATURE EXCITONICEFFECTS ON THE OPTICAL ABSORPTION
As mentioned in the introduction it is well known thattemperature not only affects the energies but also thewidths of the peaks in the optical spectra of materials.Up to now a systematic study of the behaviour of the ab-sorption spectrum of MoS (both as single-layer or bulk)on temperature is still missing. Only the low energyA exciton has been measured in photoluminescence atdifferent temperature. Measurements of the absorption(reflectance) spectra of Mo S are usually done at roomtemperature.From the theoretical point of view, if electron-phononinteraction is not taken into account, the ab-initio opticalspectra are restricted to the use of a homogeneous ad-hoc broadening. Here, following Ref. 31, we solve thetemperature-dependent Bethe-Salpeter equation, wherethe corresponding excitonic Hamiltonian is: H F Aee (cid:48) hh (cid:48) = ( E e + ∆ E e ( T ) − E h − ∆ E h ( T )) δ eh,e (cid:48) h (cid:48) + ( f e − f h )Ξ ee (cid:48) hh (cid:48) , (6) E e and E h stand for electron and hole energies, f e and f h are the occupations and Ξ ee (cid:48) hh (cid:48) is the Bethe-Salpeter(BS) kernel. The BS kernel is the sum of the directand exchange electron-hole scattering. In a temperatureindependent formulation, we would calculate the ener-gies and the BS kernel from DFT with the correspondingGW corrections to take care of the bandgap underesti-mation inherent to DFT. In this work we have used ascissor operator of 0.925 eV and a stretching factor of 1.2for the conduction and valence bands. These values wereobtained by comparison of the DFT-LDA band structurewith a GW calculation for single-layer MoS . In thetemperature-dependent approach we use the QP eigen-values obtained from Eq. 4, which now are complex num-bers and depend on temperature. As said above, the fi-nite linewidth is given as the imaginary part of the eigen-value correction. The temperature-dependent BS Eq. 6contains a non-hermitian operator. The excitonic stateswill thus have a complex energy E X ( T ) depending ontemperature and the imaginary part represents the non-radiative linewidth of the exciton. Moreover, tempera-ture not only changes the energy of the excitonic states,adding the imaginary term for the linewidth. In systemswith a strong electron-lattice interaction, temperature-dependent excitonic states are a mixture of the excitonic
Γ M K Γ0100200300400500 ω ( c m − ) (a) E ′ A ′ LA TAZA (b) DOS (c) (d) (e)
FIG. 3: Phonon band structure and density of states of single-layer MoS . Eliasberg functions of the band states T c , T (cid:48) c , K c and K v (see definitions in the text).states from the temperature-independent regime. For in-stance, in hexagonal boron nitride, temperature changesdramatically the oscillator strength of the excitons andone observes temperature-driven transition from dark tobright exciton. Certainly, the temperature effect willalso depend on the kind of excitons as we will see below.It is also worth to mention that although at T = 0 K theBS Eq. does not reduce to the frozen-atom approxima-tion due to the zero-point vibrations. We only recover theBS equation within the frozen-atom approximation whenthe terms ∆ E e ( T ) and ∆ E h ( T ) are explicitly removed.The dielectric function depends explicitly on thetemperature ε ( ω, T ) ∝ (cid:88) X | S X ( T ) | (cid:61) (cid:18) ω − E X ( T ) (cid:19) , (7)where S X ( T ) is the oscillator strength of each exciton.The broadening of the excitonic peaks is introduced nat-urally as the imaginary part of the exciton energy, with-out introducing any damping parameter. It is worth tonotice than the linewidth associated with the electron-electron interaction is negligible in the energy range inwhich we study the optical spectra.Figure 4 shows the Bethe-Salpeter spectra calculatedwithout electron-phonon interaction (black dashed line),at 0, 100, 200 and 300 K (red, green, magenta, and cyanrespectively), and with dots the experimental data atroom temperature from Ref. 26. The change on the elec-tronic states due to temperature has a repercussion onthe excitons and on the optical spectra. We have calcu-lated the Bethe-Salpeter spectra in a 30 × × k − grid,for 4 conduction band states and 2 valence band states.The rest of convergence criteria can be found elsewhere. The temperature correction to the quasi-particle stateshave been done in the same k -grid and following the pre-vious convergence criteria with respect to the number of . . . . . .
5E (eV) ε ( ω ) A B C . . . . FIG. 4: Optical spectra of single-layer MoS as afunction of temperature. Dashed line represents opticalspectra without temperature effects, and the solid line,in increasing intensity temperature of 0, 100, 200 and300 K (red, green, magenta, cyan). Dots areexperimental data from Ref. 26.bands and q points.First, the A and B excitons are shifted down in en-ergy but the intensity is rather constant. The A peakis slightly narrower than the B peak, in agreement withthe experiments. The B excitons is build mainly fromthe second valence band maximum, which has more non-radiative paths for recombination than the A exciton.The behavior of the C exciton is drastically different tothat of A and B excitons. The C exciton comes fromtransitions close to the Γ. In this region of the bandstructure the electron-phonon interaction alters substan-tially the electronic states but not the energy. The inten-sity drops remarkably from the BS spectrum in absence ofelectron-phonon interaction. It is worth to notice than wehave used an homogeneous broadening of 50 meV for theBS spectra without electron-phonon interaction. The in-creasing of temperature reduces C exciton intensity witha faster pace than in the case of the others excitons andalso increases the width. Another effect is the collapseof the multi-peak structure at the LDA spectra in onebroad peak. The result is consistent with the spectralfunctions of Figs. 1 and 2.In order to see in a clearer way the temperature effectson the exciton energies, Figure 5 shows the exciton en-ergy as function of temperature. The dashed area standsfor the width of every excitonic state. The dashed linerepresents the exciton energy without electron-phononinteraction. All the excitonic states decreases their en-ergy with increasing temperature but not with the samepace. The biggest correction to the energy of the exci-tons A and B is made by the ZPR, being very similar toboth states (75 meV). We can see that the width of thesestates is almost constant with the increasing of tempera-ture, being only slightly bigger for the B exciton (44 meVvs. 36 meV). In the case of the C excitons we have a sur-prising behaviour. We expected a larger ZPR correction,proportional to the width. While the width is alreadylarge, 88 meV) at 0 K and 132 meV at 300 K, the tem-perature increasing does not imply a strong correctionof the excitonic energy, which remains rather constant.From the spectral functions of Figs. 1 and 2(d) we cansee that the states close to Γ exhibit a remarkably in-creasing of the broadening but it seems that more or lesscentred at the same energy. We have added the photolu-minescence results of Ref. 44, representing the full widthat half-maximum (FWHM) with the gray dashed area.The energies show a good agreement at low temperaturesand diverge slightly starting at 200 K. The main causes ofthis disagreement is the thermal expansion, not includedin our calculations. Regarding the widths, both experi-mental and theoretical values increase with temperaturebut the experimental to a larger extend. This suggeststhe contribution of more processes like the radiative re-combination, has larger lifetimes than the carrier-phononscattering processes described here. From the theoreti-cal results we can infer that the band-gap dependenceon temperature is dictated mainly by EP interaction,whereas linewidths are influenced by other processes likeradiative recombination or defects scattering. The comparison with the experimental data is rathersatisfactory. We can explain the broadening of the Cpeak as the coupling of electron with lattice vibrations.On the another side, the A and B peaks compare alsowell. The approximation made for modeling the C ex-citon seems to be valid, at least to give a qualitativeexplanation of the spectral width. It is worth to notethat modelling temperature effects on the optical prop-erties cannot rely only on the thermal expansion, espe-cially at temperatures below 200 K, where the thermalexpansion is small. Moreover, only by taking into ac- . . . . .
00 50 100 150 200 250 300Temperature (K)1 . . . . . . E x c i t o n e n e r g y ( e V ) ABC
FIG. 5: Exciton energy as a function of temperature(solid lines). Shadow region shows the width of eachexcitons. Dashed line marks the exciton energy withoutelectron-phonon interaction. The photoluminescencedata of Ref. 44 have been represented by black dots andthe FWHM with the gray area.count the electron-phonon interaction we can calculatenon-radiative linewidths and interpret some data fromphotoluminescence spectra such as the FWHM or thebroadening of the optical spectra.
V. CONCLUSIONS
We have presented, to our knowledge, the first cal-culations of temperature effects on the electronic struc-ture and the optical properties of a 2D material in thepresence of spin-orbit coupling. For this purpose, wehave calculated the electron-phonon matrix elements andthe temperature-dependent spectral function using fullspinorial wave functions. We have used single-layer MoS as a test material and we expect that this work is a goodbasis for studies in other monolayer TMDs and in multi-layer MoS . The electron-phonon interaction serves alsoto understand the behaviour of the resonant Raman spec-troscopy. The Eliashberg functions evaluate the exciton-phonon coupling and we can identify which excitons willcouple to each phonon mode. We have also discovered adifferent behaviour with temperature for the two kind ofexcitons existing in MoS2. First, excitons from the bandedges (bound excitons A and B) are down-shifted in en-ergy when temperature increases and the small linewidthdoes not change significantly. In the case of resonant ex-citons (C exciton) the situation is more complex. Ourcalculations show that bands far from the bandgap havemore non-radiative paths available for decaying, as theelectron states occupy a wider energy range. Conse-quently, the non-radiative linewidth is strongly affectedby the increase of temperature. The overall result is anoptical absorption with a characteristic inhomogenousbroadening, with a C peak much broader than A andB peaks. Our theoretical spectra agrees well with re-cent experimental measurements. With this contributionwe show the importance of temperature effects, deter-mined by electron-phonon coupling, for a more realisticapproach to the optical properties of semiconductor 2Dmaterials. VI. ACKNOWLEDGEMENTS
A. M.-S. and L.W. acknowledge supportby the National Research Fund, Luxembourg(Projects C14/MS/773152/FAST-2DMAT and IN- TER/ANR/13/20/NANOTMD). M. Palummo acknowl-edges the support received from the European ScienceFoundation (ESF) for the activity entitled ’AdvancedConcepts in
Ab-initio
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