A perturbative approach to the polaron shift of excitons in transition metal dichalcogeniedes
AA perturbative approach to the polaron shift of excitons in transition metaldichalcogeniedes
J. C. G. Henriques , and N. M. R. Peres , Department and Centre of Physics, and QuantaLab,University of Minho, Campus of Gualtar, 4710-057, Braga, Portugal and International Iberian Nanotechnology Laboratory (INL),Av. Mestre José Veiga, 4715-330, Braga, Portugal
In this paper we study the phonon’s effect on the position of the 1s excitonic resonance of thefundamental absorption transition line in two-dimensional transition metal dichalcogenides. Weapply our theory to WS a two-dimensional material where the shift in absorption peak position hasbeen measured as a function of temperature. The theory is composed of two ingredients only: i)the effect of longitudinal optical phonons on the absorption peak position, which we describe withsecond order perturbation theory; ii) the effect of phonons on the value of the single particle energygap, which we describe with the Huang Rhys model. Our results show an excellent agreement withthe experimentally measured shift of the absorption peak with the temperature. One of the most prominent and studied types of two-dimensional materials are transition metal dichalcoge-niedes (TMDs) [1]. These, often semi-conducting, ma-terials present remarkable electronic and optical proper-ties, intrinsically related with their excitonic response.An exciton is a quasi-particle corresponding to a boundelectron-hole pair interacting via a Coulomb-like poten-tial. Due to the reduced dielectric screening in two-dimensional materials, these quasi-particles are moretightly bound, and thus more stable, than their three-dimensional analogues. Phonons, the quanta of atomicvibrational energy, are known to have a significant impacton the optical properties of TMDs, especially due to theirinteraction with excitons [2–5]. The exciton-phonon cou-pling influences both the line width and the peak positionof the different absorption resonances associated with theoptically active excitonic states in TMDs. Indeed, thiseffect has been reported recently in Ref.[6], where it wasshown that the s excitonic peak was red-shifted as thetemperature increased, accompanied by a concomitantincreased line width of the resonance. The coupling ofphonons to excitons also affects their radiative lifetime,and allows the access to optically dark states via inter-valley scattering channels [7]. Studies on the tempera-ture dependence of the optical properties of these mate-rials are highly relevant to accurately predict their appli-cability in different technological applications which arerequired to work at room temperature.The problem of electron-phonon interaction is by nomeans a simple one, giving rise to, for example, phonon-mediated supercondutivity and the polaron problem, anelectron dressed with a cloud of phonons. While the for-mer problem can be dealt with an approximate canon-ical transformation, the latter one is, in general, non-perturbative. However, to address the effect of phonon’son the position peak of the absorption resonance aperturbative approach, up to seconder in the electron-phonon interaction, suffices. However, as derived fromtraditional perturbation theory, we end up with a sumover all states of the non-interacting problem, whose ef- fective summation is out of reach simply because it re-quires all the eigenstates of the non-interacting system,which may not be known. Even in the cases where theyare known, the integrals are of insurmountable difficulty.Therefore, we follow a different path for circumventingthe sum over states. We use the Dalgarno-Lewis ap-proach [8] which shifts the sum over states problem tothe solution of an non-homogeneous differential equation.In this procedure, only one eigenstate of the unperturbedtheory is required together with the solution of the afore-mentioned differential equation.In this paper we consider a two dimensionalTMDwhose electrons and holes, interacting via a Coulomb-like potential, may give rise to exciton formation. Inorder to study the effect of temperature on the excitonicproperties, we use a similar model to the one employed inRef. [9], where a Frohlich-like Hamiltonian [10]was usedto characterize the interaction of optical phonons withelectrons and holes in polar crystals. Contrary to Ref.[9], where 3D systems were considered, we will focus onexcitons on 2D materials, leading to a difference in theform of the interaction term [11]. Moreover, contrary tothe aforementioned work where only the case of T = 0 Kwas considered, our calculations cover any temperaturevalue. The Hamiltonian of the considered system in thecenter of mass frame of the electron-hole pair reads: H = H + H + H + H + H , (1)where H = p µ + V ( r ) , H = (cid:88) q (cid:126) ω q a † q a q H = − UA / (cid:88) q i √ q a q e i q · ( m h /M ) r + h.c.H = UA / (cid:88) q i √ q a q e − i q · ( m e /M ) r + h.c.H = 12 M (cid:32) K − (cid:88) q (cid:126) q a † q a q (cid:33) . a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b The term H is the Hamiltonian of the exciton, with m e/h the electron/hole effective mass, µ − = m − e + m − h thereduced mass of the electron-hole pair, p their relativemomentum, r their relative position vector and V ( r ) theelectron-hole interaction potential which we model usingthe Rytova-Keldysh potential [12, 13] V ( r ) = − e π(cid:15) π r (cid:20) H (cid:18) κrr (cid:19) − Y (cid:18) κrr (cid:19)(cid:21) , (2)where e is the elementary charge, (cid:15) is the vacuum per-mittivity, κ is the mean dielectric constant of the mediaabove and below the TMD monolayer, r is a material-dependent screening length (which is microscopically re-lated to the material’s polarizability), H is the Struvefunction of the first kind and Y the Bessel function of thesecond kind, both of order zero. In the total Hamiltonian,the term H describes the thermally excited phononsand a † q / a q refers to the creation/annihilation operator ofa phonon with momentum q and energy (cid:126) ω q . In thisterm we only consider the contribution originating fromlongitudinal-optical (LO) phonons. Also, we will considerthat the energy (cid:126) ω q is independent of momentum andequal to a constant value (cid:126) ω LO when numerical resultsare computed. The terms H and H correspond to theinteraction between the phonons and the electrons andholes, A is the area of the 2D monolayer, M = m e + m h and U is the coupling potential defined as [11] U = (cid:126) ω LO (cid:16) √ πα (cid:17) / (cid:18) (cid:126) m ω LO (cid:19) / , (3)with m the bare electron mass, α a dimensional couplingconstant, which we will consider as a fitting parameter,but whose typical value is between 2-5 [11]. Finally, theterm H depends on the center of mass momentum K . It is not in general expected that a term which depends onthe center of mass momentum will play a significant rolein the system’s internal dynamics, thus hereinafter, justlike in Ref. [9], we neglect its contribution to H , that is H ≈ H + H + H + H . (4)Now, in order to compute the effects of the couplingof the excitons to the LO phonons we will follow a per-turbative approach, taking H + H as the unperturbedHamiltonian and H + H as the perturbative term. Fromsecond order perturbation theory, we write the energycorrection to the system’s ground state as ∆ E = (cid:88) ν X ν ph (cid:12)(cid:12)(cid:12) (cid:104) s ; n ph ( T ) | H + H | ν X ; ν ph (cid:105) (cid:12)(cid:12)(cid:12) E GS − E ν X ν ph , (5)where ν X and ν ph refer to the states of the excitonand the phonons, respectively, with a combined energy E ν X ν ph , s refers to the most tightly bound excitonicstate and n ph ( T ) corresponds to the phonon distributionat a temperature T , with a combined energy E GS . Thesum runs over all the ν X and ν ph except { ν X , ν ph } = { s, n ph ( T ) } . A direct evaluation of Eq. (5) would un-doubtedly be a daunting task, with little probability ofsuccess, since all the excitonic wave functions would berequired and an infinite number of matrix elements wouldhave to be evaluated. As an alternative route, one canfollow the Dalgarno-Lewis approach [8] in order to by-pass the sum over states. This ingenious approach con-sists on the introduction of an operator, defined througha differential equation, which when inserted in Eq. (5)allows the sum over states to be removed. The problemof computing ∆ E is then reduced to the evaluation ofa single matrix element. More specifically, it is possibleto show that Eq. (5) can be written as the sum of fourcontributions ∆ E = ∆ E + ∆ E + ∆ E + ∆ E , (6)where ∆ E = U A (cid:88) ν X (cid:88) q q (cid:26) n ph ( q , T ) + 1 E s − E ν X − (cid:126) ω q (cid:104) s | e − i q · ( m e /M ) r | ν X (cid:105)(cid:104) ν X | e i q · ( m e /M ) r | s (cid:105) + n ph ( q , T ) E s − E ν X + (cid:126) ω q (cid:104) s | e i q · ( m e /M ) r | ν X (cid:105)(cid:104) ν X | e − i q · ( m e /M ) r | s (cid:105) (cid:27) , (7)and ∆ E = − U A (cid:88) ν X (cid:88) q q (cid:26) n ph ( q , T ) + 1 E s − E ν X − (cid:126) ω q (cid:104) s | e − i q · ( m e /M ) r | ν X (cid:105)(cid:104) ν X | e − i q · ( m h /M ) r | s (cid:105) + n ph ( q , T ) E s − E ν X + (cid:126) ω q (cid:104) s | e − i q · ( m h /M ) r | ν X (cid:105)(cid:104) ν X | e − i q · ( m e /M ) r | s (cid:105) (cid:27) , (8)with E s and E ν X the energies of the s and ν X excitonic states, respectively, and n ph ( q , T ) the Bose-Einstein dis-tribution function for phonons with energy ω q at a tem-perature T . Here we still consider the phonon energyas a function of the momentum in order to present gen-eral expressions, but later we will consider ω q = ω LO when numerical results are computed. The expressionsfor ∆ E and ∆ E follow directly from these two by re-placing m e ( m h ) with m h ( m e ) . Each of the four contri-butions is made up of two terms with distinct physicalorigin: one originating from phonon emission and theother from phonon absorption. In the limit of vanish-ing temperature only the former contributes due to theprocess of spontaneous phonon emission. As we just mentioned, to forego the sum over states, wewill make use of Dalgarno and Lewis’ formulation of per-turbation theory. In order to evaluate ∆ E we introducetwo operators F ± which obey to the relations ([ F ± , H ] ± (cid:126) ω q F ± ) | s (cid:105) = e ∓ i q · r m e /M | s (cid:105) . (9)Now, we apply these to Eq. (7), remove the sum overstates and introduce three complete sets of plane waves.Doing so, and taking advantage of the orthogonality re-lation between plane waves, one finds the following ex-pression for ∆ E : ∆ E = − µ (cid:126) U A (cid:88) q (cid:88) k q [ n ph ( q , T ) + 1] (cid:104) s | k (cid:105)(cid:104) k | s (cid:105) q (cid:0) m e M (cid:1) + 2 k · q m e M + 2 µω q / (cid:126) − µ (cid:126) U A (cid:88) q (cid:88) k q n ph ( q , T ) (cid:104) s | k (cid:105)(cid:104) k | s (cid:105) q (cid:0) m e M (cid:1) + 2 k · q m e M − µω q / (cid:126) . (10)Comparing Eqs. (7) and (10), we note that with theapproach of Dalgarno and Lewis the problem of com-puting ∆ E was drastically modified. While in Eq. (7)the knowledge of all the excitonic states was required,in (10) only the Fourier transform of the s state wavefunction is needed. Moreover, the complexity of the cal-culations was greatly reduced, since now we need onlycompute two sums over the momenta q and k while pre-viously, the computation of an infinite number of matrixelements was required. Finally we note that in Eq. (10)the operators F ± are absent, since we do not need themexplicitly, but rather the expression for their matrix ele-ment between plane waves, which can be computed fromEq. (9). In order to progress analytically, we follow avariational path to describe the wave function of the s excitonic state. Our variational ansatz, inspired by theHydrogen atom, reads [14] ψ s ( r ) = a (cid:114) π e − r/a , ψ s ( k ) = 2 a √ π (1 + a k ) / , (11)where a is a variational parameter determined from theminimization of the expectation value of H , and can beroughly interpreted as the excitonic Bohr radius. Thischoice of the trial wave function produces wave functionsin good agreement with the ones found using exact nu-merical methods. A more sophisticated ansatz combiningtwo exponential functions [15] could also be used. Thisoption would produce results in perfect agreement withthe numerical ones, with the cost of more involved calcu-lations, without a simple analytical solution. In any case,the choice of one of the ansätze over the other should pro-duce only minimal changes in the final result. To con-tinue with the calculation of ∆ E the sums over q and k must be converted into two-dimensional integrals. From this point onward we explicitly consider that ω q = ω LO ;as a consequence the terms n ph ( q , T ) become momen-tum independent and can be taken out of the integrals.Writing the integrals in polar coordinates, one finds thefollowing angular integral (cid:90) π dθσ ± + k cos θ = (cid:40) sign σ ± π √ σ ± − k , | σ ± | > , | σ ± | < (12)where σ ± = (cid:0) q ± µω LO (cid:1) / q . When | σ ± | < the prin-cipal value of the integral should be considered. We nownote that for the second term in Eq. (10), the term asso-ciated with absorption of phonons, σ − = ± dependingon the value of q . The same does not apply to the othercontribution, where σ + > ∀ q ≥ . As a consequenceof the two possible signs that originate from the angu-lar integration, when the integrals over dk and dq arecomputed the phonon absorption contribution from ∆ E (and ∆ E after the roles of m e and m h are switched)vanishes. Computing the remaining integrals, we find ∆ E = − m h π (cid:126) U a [ n ph ( ω LO , T ) + 1] × (13) (cid:20) π ( χ − χ + 3)32 a β χ + f + (cid:21) , (14)with χ + = 1 + a β , β = 2 µω LO / (cid:126) and f + = (cid:90) ∞ q aζ + ( q )] a (cid:2) a ζ ( q ) (cid:3) / dq ≈ πa , (15)where ζ + ( q ) = (cid:0) q + β (cid:1) / q . The value of f + is roughlyless that one half of the the term with which it is summed.As we have already mentioned, the contribution ∆ E isobtained from ∆ E by substituting m h with m e .Now that ∆ E and ∆ E were computed, we turn ourfocus to the contributions ∆ E and ∆ E . The processto compute these terms is very much alike the one de-scribed for the other two. Similarly to before, we startby defining the operators F ± which obey to the relation ([ F ± , H ] ± (cid:126) ω q F ± ) | s (cid:105) = e − i q · ( m e/h /M ) r | s (cid:105) . The in-troduction of these operators allows us to remove thesum over states. After the plane wave basis have beenintroduced, and their orthogonality relations employed,we arrive at the following expression: ∆ E = − µ (cid:126) U A (cid:88) q (cid:88) k q (cid:26) [ n ph ( q , T ) + 1] (cid:104) s | k (cid:105)(cid:104) k + q | s (cid:105) q m e + m h ] m h M + 2 k · q m h M − µω q / (cid:126) + n ph ( q , T ) (cid:104) s | k (cid:105)(cid:104) k + q | s (cid:105) q m h + m e ] m e M + 2 k · q m e M + 2 µω q / (cid:126) (cid:27) . (16)Comparing this result with Eq. (10), we note that inthe present case the Fourier transforms of the s wavefunction are evaluated at different momenta. This modi-fication significantly increases the complexity of the inte-grals that must be computed, preventing the existence ofa simple analytical solution. The angular integrals can,however, be computed analytically, yielding: (cid:90) π dθ [ ξ ± + cos θ ] [ δ + cos θ ] / == − ξ ± ) E (cid:16) δ (cid:17) − ( δ − Π (cid:18) ξ ± (cid:12)(cid:12)(cid:12)(cid:12) δ (cid:19) (1 + ξ ± ) √ δ + 1 ( δ −
1) ( δ − ξ ± ) , (17)where E ( x ) and Π ( x | y ) are elliptic integrals of the secondand third kind, respectively, δ = (cid:0) a − + k + q (cid:1) / kq > and ξ ± = (cid:18) q [ m h/e + m e/h ] M ± m h/e ω LO (cid:126) (cid:19) / kq . This so-lution is valid for both | ξ ± | > and | ξ ± | < . For the lat-ter case, the principal value of the integral should be con-sidered. The remaining integrals over dk and dq do notyield analytical solutions, and as a consequence must beevaluated numerically, taking the principal value of theintegral when necessary. When doing so, one must pro-ceed carefully, starting by determining the points wherepoles and branch cuts appear. These points correspondto the ones where the the conditions ξ − = − and ξ ± = 1 are satisfied; in addition care must be exercised when δ − ξ ± .Up to this point we have described and given equationsthat characterize the exciton-phonon interaction. As anapplication of the results so far derived we will producea theoretical description of the experimental data pre-sented in Ref. [6], where the shift of the s excitonicresonance was measured as a function of the tempera-ture. To accurately describe this effect we must considersomething so far ignored. As the temperature increasestwo distinct effects take place. On the one hand, theexciton-phonon interaction, dominated by LO phonons,will modify the exciton binding energy, shifting the exci-tonic peak; this is the effect we have theoretically de- scribed. On the other hand, the band gap decreasesas the temperature increases, also contributing to theexcitonic resonance shift. To describe this effect theVarshni empirical model [16] is commonly used, howeverwe choose to consider the vibronic model of Huang andRhys [17], which takes into account the effect of acousticphonons: E g ( T ) = E g (0) − S (cid:104) (cid:126) ω A (cid:105) (cid:20) coth (cid:104) (cid:126) ω A (cid:105) k B T − (cid:21) , (18)where E g ( T ) is the band gap magnitude at a temperature T , (cid:104) (cid:126) ω A (cid:105) is the mean energy of the acoustic phonons(about meV [18]), k B is the Boltzmann constant, and S is a fitting parameter of the order of 1, describing theelectron-acoustic-phonon coupling. The expression forthe s resonance position as a function of T , which welabel as E X ( T ) , relatively to its position at T = 0 K,reads: E X ( T ) − E X (0) = − S (cid:104) (cid:126) ω A (cid:105) (cid:20) coth (cid:104) (cid:126) ω A (cid:105) k B T − (cid:21) + E B ( T ) − E B (0) , (19)where E B ( T ) < is the s state binding energy at atemperature T . This quantity can be obtained using [9] E B ( T ) = E ab initio + ∆ E ( T ) + 2 α (cid:126) ω LO , (20)where E ab initio is the s binding energy of the unper-turbed system, that is, when no phonons are present, ∆ E ( T ) = ∆ E + ∆ E + ∆ E + ∆ E and α (cid:126) ω LO isthe sum of the free electron and hole polarons (wherewe assumed that α is approximately the same for elec-trons and holes). The value of E ab initio can be foundnumerically or with semi-analytical methods, however,since this is a temperature independent value, it vanishesfrom E X ( T ) − E X (0) . Using Eq. 19 we performed a fitto the experimental data of Ref. [6]. The comparisonbetween our theoretical description and the experimen-tal results is depicted in Figure 1; the parameters arepresented in Table I. Inspecting Figure 1 we observe anexcellent agreement between our theoretical description κ r m e m h a (cid:126) ω LO α (cid:104) (cid:126) ω A (cid:105) S κ corresponds to the mean dielectric constant ofvacuum and SiO . The value of r was taken from Ref. [19],and the values of m e and m h from Ref. [20]. The value of a was obtained from the minimization of H using the param-eters just mentioned. The value for the LO phonons energywas taken from Ref. [18]. The values of α , (cid:104) (cid:126) ω A (cid:105) and S wereobtained from the fit of the theoretical model to the data ofRef. [6]. and the experimental data-points. At room temperaturethe gap renormalization is responsible for a shift of ap-proximately 65meV while the polaron shift contributeswith approximately 15meV, in rough agreement with thevalues found in Ref. [7]. Analyzing the content of TableI we note that the fitting parameters, α , (cid:104) (cid:126) ω A (cid:105) and S are in agreement with previous results found in the liter-ature. The value of α lies inside the interval between 0and 5 indicated in Ref. [11]. The value of (cid:104) (cid:126) ω A (cid:105) matchesthe one obtained in Ref. [18] and used in Ref. [21], whereMoSe was studied. In Ref. [18] a value of S = 1 . wasfound for MoSe . Using the fact that this parameter,which characterizes the coupling to phonons, should beproportional to the square root of the effective masses,and noting that the effective masses in MoSe differ fromthose in WS approximately by a factor of . [20]we canestimate that the value of S in WS should be around 1.3,in total agreement with the value we found.In summary, using second order perturbation theorywe have successfully described the effect of temperatureon aset of experimental data-points on the position ofthe fundamental absorption line peak of the 1s excitonictransition (corresponding to the more tightly bound exci-ton) in WS [6]. The experiment shows a red shift of theabsorption line when the temperature increases. We wereable to describe, in quantitative terms, the observed shiftconsidering two different effects: the polaron shift andthe renormalization of the single particle gap with tem-perature. Both effects were shown to produce a sizableeffect to the overall red shift, and should be accountedfor in theoretical descriptions of this phenomenom. Westress that both effects are due to different set of phonons:the longitudinal optical phonons in the intrinsic red shift of the absorption line and the acoustic phonons in themodification of the single particle gap. A follow up ofthis work will focus on the calculation of line shape ofthe absorption peak as a function of temperature whichrequires solving the Bethe-Salpeter equation in the pres-ence of the phonon’s field.N.M.R.P acknowledges support by the PortugueseFoundation for Science and Technology (FCT) in theframework of the Strategic Funding UIDB/04650/2020. R e l . p e a k s h i f t ( m e V ) Huang-Rhys modelPolaron shiftTotalRaja et al. Figure 1. Comparison between the fit obtained with Eq. (19)and the experimental data of Ref. [6]. An excelent agree-ment between the theoretical description and the experimen-tal points is evident. Also depicted are the isolated contri-butions of the polaron shift, and the gap renormalization de-scribed with the Huang-Rhys model. The value of the fittingparameters is given in Table I.
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