New Theoretical Model to Describe Carrier Multiplication in Semiconductors: Explanation of Disparate Efficiency in MoTe2 versus PbS and PbSe

We present a theoretical model to compute the efficiency of the generation of two or more electron–hole pairs in a semiconductor by the absorption of one photon via the process of carrier multiplication (CM). The photogeneration quantum yield of electron–hole pairs is calculated from the number of possible CM decay pathways of the electron and the hole. We apply our model to investigate the underlying cause of the high efficiency of CM in bulk 2H–MoTe2, as compared to bulk PbS and PbSe. Electronic band structures were calculated with density functional theory, from which the number of possible CM decay pathways was calculated for all initial electron and hole states that can be produced at a given photon energy. The variation of the number of CM pathways with photon energy reflects the dependence of experimental CM quantum yields on the photon energy and material composition. We quantitatively reproduce experimental CM quantum yields for MoTe2, PbS, and PbSe from the calculated number of CM pathways and one adjustable fit parameter. This parameter is related to the ratio of Coulomb coupling matrix elements and the cooling rate of the electrons and holes. Large variations of this fit parameter result in small changes in the modeled quantum yield for MoTe2, which confirms that its high CM efficiency can be mainly attributed to its extraordinary large number of CM pathways. The methodology of this work can be applied to analyze or predict the CM efficiency of other materials.


■ INTRODUCTION
The development of a new generation solar cells requires exploring ways to surpass the Shockley−Queisser limit of 33.7%. 1,2One of these ways is to enhance the photocurrent of a solar cell via excitation of two or more electrons by the absorption of a single photon. 3This process of carrier multiplication (CM) can raise the power conversion limit of a single-junction solar cell to 46%. 3,4Apart from solar cells, CM is also of interest to the development of more efficient photodiodes since it enhances the photocurrent.Note that CM has also been referred to as electron−hole pair multiplication (EHPM) or impaction ionization (II) in bulk semiconductors and has been called multiexciton generation (MEG) in quantum confined systems where excitons are formed rather than free charge carriers. 3,5 CM process is schematically shown in Figure 1a.It starts with photoexcitation of a single electron across the band gap of a semiconductor.If the photoexcited electron has sufficient excess energy, exceeding that of the band gap (E g ), it can undergo Coulomb scattering with a valence electron and excite the latter to the conduction band.As depicted in Figure 1b, this leads to the formation of an additional electron−hole pair.Note that the initially photogenerated hole in the valence band can also relax by CM provided its energy is at least 1E g below the top of the valence band, as shown in Figure 1c.Since either the electron or the hole must have an excess energy above E g , CM is possible for photons with energy above twice the band gap, i.e., ℏω ≥ 2E g .In the most ideal case, CM thus has an onset photon energy of 2E g and the number of electron−hole pairs produced per absorbed photon increases with one for each further photon energy increment of E g . 3The number of electron−hole pairs produced per absorbed photon, which is known as the electron−hole pair photogeneration quantum yield, then exhibits a steplike dependence on the photon energy.However, photons with energy equal to a multiple of the band gap will, in general, not transfer their entire excess energy to either an electron or a hole.
In addition, CM occurs in competition with energetic relaxation of electrons and holes by cooling via phonon emission, so that their initial excess energy above the band edges is not fully available for CM.These factors cause the onset energy of CM to be usually larger than twice the band gap and the dependence of the CM quantum yield on photon energy to deviate from the ideal steplike behavior.
−22 Particularly, 2H-TMDCs, where the prefix 2H-denotes a trigonal prismatic phase structure, have gained much attention due to reports of a remarkably high efficiency of bulk CM, 20−22 which highly exceeds that of bulk PbS and PbSe. 7For the 2H−MoTe 2 and 2H−WSe 2 TMDCs, the onset energy of CM was found to be close to a photon energy of 2E g and the quantum yield of electron−hole pairs exhibits an almost ideal steplike dependence on photon energy. 20,21This is very different for bulk PbS and PbSe where the onset energy of CM is as high as ∼6E g and the increase of the CM quantum yield with photon energy is much smaller. 7The origin of the highly efficient CM in these TMDCs has not been rationalized before.A high CM quantum yield requires the rate R CM of CM to be (much) larger than the rate R cool of charge carrier cooling, i.e., their ratio (R CM /R cool ) to be large.In the studies on TMDCs mentioned above, the CM rate was estimated to be an order of magnitude larger than the cooling rate. 20,21A recent theoretical study on TMDCs found that electron−phonon coupling could reduce the band gap upon photoexcitation, which could enable a CM onset even below 2E g and thereby increase the CM efficiency. 23Furthermore, weak carrier-phonon coupling 24 and long-lived optical phonon modes 25 have been observed in 2H−MoTe 2 that could contribute to a low carrier cooling rate.While these are important findings, they do not directly explain why CM in TMDCs is much more efficient than in other bulk materials such as PbS and PbSe.
In this paper, we present a theoretical model that derives the CM efficiency from the number of CM pathways.We apply this model to explain why the experimental CM efficiency of MoTe 2 20,21 is much higher than measured for PbS and PbSe. 7om band structures obtained with density functional theory (DFT), we evaluate all possible Coulomb scattering pathways of photogenerated electrons and holes that lead to the generation of an additional electron−hole pair.The experimental disparate CM quantum yields of MoTe 2 versus PbS and PbSe can be described qualitatively based on the very different numbers of CM decay pathways of initially energetic electrons and holes produced by photoexcitation.We quantitatively reproduce the experimental CM quantum yields as a function of photon energy with only one adjustable parameter for each material, which relates to the ratio of the Coulomb interaction strength involved in CM and the charge carrier cooling rate.Our model provides an explanation for the high CM efficiency in MoTe 2 , as opposed to the much lower CM efficiencies in PbS and PbSe.In our model the Coulomb interaction strength, cooling rate, and optical oscillator strength for photoexcitation were taken to be independent of the electronic states involved.This leads to the conclusion that the variation of the CM rate with photon energy is mainly determined by the number of CM pathways for the decay of electrons and holes.The theoretical framework of this study can be applied to investigate CM efficiencies for other (unexplored) materials.

■ THEORETICAL METHODS
Electronic Band Structure Calculations.Electronic band structures of MoTe 2 , PbS, and PbSe were obtained from DFT calculations with the ABINIT package. 26All calculations were done using optimized norm-conserving Vanderbilt pseudopotentials, 27 a generalized gradient approximation (GGA) exchange−correlation functional, 28 and included spin−orbit coupling.The geometries were based on lattice constants reported literature and are listed in Table S1 in the Supporting Information. 29,30We used a k-point grid of n k = 14 × 14 × 7 crystal momenta for MoTe 2 and n k = 16 × 16 × 16 for PbS and PbS.The convergence condition between two self-consistent field steps was chosen to be a 10 −8 hartree (Ha) absolute difference in total energy.Cutoff energies were chosen to be 30, 25, and 40 Ha for MoTe 2 , PbS, and PbSe, respectively.Band structure diagrams were obtained from k-points along the symmetry lines of the irreducible Brillouin zone, whereas the density of states and the number of CM pathways were

The Journal of Physical Chemistry C
computed from k-points in the full Brillouin zone.The conduction bands were shifted using a scissor operator to match the experimentally determined band gaps reported in the literature, which is further discussed below.
Calculation of the Number of Carrier Multiplication Pathways.The number of CM pathways can be obtained by forming sets of charge carrier states and counting each set that fulfills the conditions for an allowed CM pathway.As is shown in Figure 1b, a CM event involves four electronic Bloch states.A primary electron in the initial state with index 1 and energy E 1 decays to a final state 1′ at energy E 1′ .This decay is accompanied by exciting a secondary electron from state 2 with energy E 2 in the valence band to state 2′ with energy E 2′ in the conduction band.Conservation of energy applies such that ΔE 1′,2′,1,2 = E 1′ + E 2′ -E 1 -E 2 = 0. Apart from conservation of energy, Coulomb scattering requires the difference of the initial and final crystal momenta k i of the electrons to be a reciprocal lattice vector G, which is zero for Normal-type scattering and nonzero for Umklapp-type scattering. 31,32The conservation condition of crystal momentum is thus Furthermore, in the case an electron initiates CM (Figure 1b), the condition E CBM ≤ E 1′ < E 1 applies for the primary electron in state 1, where E CBM denotes the minimum energy of the lowest conduction band and E 1 (E 1′ ) is the energy of the electron in its initial (final) state.As shown in Figure 1c, CM by relaxation of a hole involves scattering between two valence electrons, where the primary electron 1 remains in the valence band, and thus, the condition E VBM ≥ E 1 > E 1′ must be fulfilled, where E VBM denotes the maximum valence band energy.Finally, the secondary electron in valence band state 2 must be excited to a conduction band state 2′, i.e., E 2 ≤ E VBM < E 2′ , which then along with the aforementioned conditions also implies that the energy loss of the primary electron is greater than or equal to the band gap, The conservation of energy and crystal momentum and the conditions for the initial and final states are used to compute the number of CM pathways.The discrete electronic Bloch states obtained from DFT calculations are characterized by a wave vector k i in the first Brillouin zone, a band index ν i and an energy E i (ν i , k i ).By making all possible combinations of the initial and final states of primary and secondary electronic states and assessing the aforementioned conditions, the number of possible CM pathways for any initial primary carrier state can be counted.This leads to the following expression of the number of CM pathways for the initial primary carrier state 1 where , , ) In the summation over initial and final states, the conservation of crystal momentum is taken into account by a Kronecker delta δ G, k1d ′ + k2d ′ − k1 − k2 .This condition can be satisfied numerically since the DFT calculations provide electronic states on a regular grid of k-points.Conservation of energy of the discrete electronic Bloch states leads to the requirement ΔE 1′,2′,1,2 = 0.However, this constraint is too strict since the DFT calculations provide energies on a grid of a finite number of k-points.In addition, consideration of electronic energy only is insufficient since CM can be assisted by absorption or emission of phonons.Therefore, the energy conservation rule is relaxed by using a tophat function Θ(ΔE, ε tol ) as defined in eq 3, analogous to previous studies. 33,34The parameter ε tol relaxes energy conservation over the tolerance interval 2ε tol and the function Θ(ΔE, ε tol )/2ε tol can be considered as a broadened energy conserving Dirac delta function δ(ΔE).The function I(E 1 , E 1′ , E 2 , E 2′ ) defined in eq 4 checks the energy requirements for the initial and final states of the primary and secondary carriers and therefore distinguishes a CM pathway from any ordinary scattering pathway.The summation in eq 2 over all final states at the k-points (k 1′ , k 2′ ) scales in a trivial manner with the square of the number of k-points (n k ) included in the DFT calculations.Note that the fourth k-point (k 2 ) is fixed due to conservation of crystal momentum and therefore does not contribute to this scaling factor.Therefore, we divide the summation by n k 2 , so that for sufficiently large n k the value of N CM 1 converges to a constant value and results for different sizes of k-point grids can be directly compared.Hence, for each combination of band indices υ i in eq 2 the value of N CM 1 represents the fraction of all n k 2 final states 1′ and 2′ per unit energy that can be reached by CM decay of an electron from the initial state with index 1, which excites another electron from the state with index 2.The unit of N CM 1 is further discussed in the Supporting Information.
Values of N CM 1 were calculated for all initial hole states in the valence bands and electron states in the conduction bands obtained from DFT calculations.In determining the states involved in absorption of a photon with energy ℏω the photon momentum was neglected, so that Δk = 0 for an optical transition.A constant optical oscillator strength was assumed for all optical transitions.Below, we refer to N CM 1 as the density of the CM pathways.
Modeling the Quantum Yield from the Density of Carrier Multiplication Pathways.Below, we describe our model to obtain the CM quantum yield from the density of CM pathways given by N CM 1 in eq 2. The CM quantum yield is defined as the ratio between the number of generated electron− hole pairs and the number of absorbed photons, i.e.
If an incident photon carries at least two band gap multiples of energy (ℏω ≥ 2E g ), it becomes energetically possible that the photogenerated electron or hole induces CM to yield a second electron−hole pair.At a photon energy of three band gap multiples or more (ℏω ≥ 2E g ), a primary carrier may have an excess energy above ≥2E g , so that more CM steps are possible and three or more electron−hole pairs can be produced.Due to computational limitations, these subsequent CM events are not taken into account.Therefore, for photon absorption event i, the quantum yield is determined by the probabilities of CM for the initially photogenerated electron (P CM e i ) and hole (P CM h i ) only, and can have a value of 1 up to 3, i.e.

The Journal of Physical Chemistry C
where e i and h i , respectively, denote the conduction and valence band states occupied by the photogenerated primary carriers (electron and hole).We assume that directly after photoexcitation a generated carrier can follow only one of two paths: (1) it induces CM by Coulomb scattering and yields an additional electron−hole pair or (2) it relaxes by phonon emission without generating an additional electron−hole pair.Carrier cooling is thus modeled as a single step, and a possible sequence of cooling and CM steps is neglected.From these assumptions, the probability of CM for carrier i is then defined as the ratio between the rate of CM and the sum of the CM and carrier cooling rates where R CM i and R cool i are the total CM and carrier cooling rates, respectively, for a primary carrier in initial state i.
−47 In these methods, Fermi's Golden Rule is used, which requires Coulomb matrix elements of all initial and final states.−50 Hence, to a good approximation, the Coulomb matrix elements can be kept at a constant value, thereby assuming that each single scattering event occurs at an equal rate.As described in the Supporting Information, the total CM rate for primary carrier in initial state i is then directly proportional to the density of CM decay pathways, N CM i , in eq 2, according to with the prefactor F CM taking into account Coulomb coupling, which is assumed to be independent of the initial and final states.
Using eq 8 for the CM rate, the probability of CM in eq 7 can be rearranged to The Journal of Physical Chemistry C This equation describes the probability of CM for a carrier in the initial state i.The quantum yield for photon absorption event i, that generates the primary initial electron state e i and hole state h i is obtained by substitution of eq 9 for the CM probabilities into eq 7, yielding A photon with energy ℏω can induce different optical transitions from the valence band to conduction band and thereby can result in the generation of different initial states of a pair of an electron and a hole, e i and h i , with different values for N CM e i and N CM h i .Assuming a constant optical oscillator strength, the quantum yield at photoexcitation with energy in a small range between ℏω and ℏω + Δℏω is the average quantum yield, Φ(ℏω), of the total number of possible optical photoexcitations, N ℏω , so that ( ) Finally, substituting eq 10 into eq 11 results in the CM quantum yield As discussed below, we calculate N CM using band structures obtained from DFT and fit eq 12 to experimental quantum yields with F CM /R cool as an adjustable parameter.
The equations above are valid for the physically realistic case of a nonzero cooling rate.However, to obtain insights into the effect of cooling, we also consider the CM quantum yield that would be obtained in absence of cooling, i.e., R cool →0.In that case, the quantum yield is given by with P CM i = 1 if N CM e i > 0, so that CM is possible, and P CM e i = 0 if N CM e i = 0, and analogously for the hole.Hence, the second and third terms in eq 13 are the fractions of the initially photoexcited electrons and holes that decay by the CM, respectively.
■ RESULTS AND DISCUSSION Calculated Band Structures and Density of States.The band structures and density of states obtained from the DFT calculations are shown in Figure 2. The calculated band gaps for MoTe 2 , PbS, and PbSe are 0.72, 0.30, and 0.32 eV, respectively.A scissor operator was applied to match the experimental band gaps of 0.85 eV for MoTe 2 20 and 0.42 and 0.26 eV for PbS 51 and PbSe, 51

The Journal of Physical Chemistry C
From the band structures, it is clear that the bands of MoTe 2 are relatively flat compared to the more curved bands of the lead chalcogenides.Moreover, MoTe 2 has a larger number of bands within the shown energy range, which results in a density of states that is larger than that of the lead chalcogenides.It can also be seen that many of the electronic bands of MoTe 2 appear as "bundles", where multiple bands overlap or lie closely together at many values of k.The relatively flat and bundled bands of MoTe 2 result in large peaks in the density of states.For PbS and PbSe, at a low photon energy, the excitation occurs around the L-point.Due to the similar curvature of the valence and conduction bands near the L-point, the excess photon energy is divided almost equally over the hole and the electron.As a result, the energy onset of CM is much higher than 2E g .
Calculated Density of CM Pathways and Experimental Quantum Yields.The density of CM pathways was computed from eq 2 with energies of the valence and conduction band states from DFT calculations covering the full Brillouin zone.All electronic states that can be generated by photons with energy up to 4 eV were included.Energy conservation was ensured within a range of ε tol = 30 meV in eq 2, which is suitable for the convergence of the results, see the left panel of Figure S1.
Figure 3a shows the summed average density of CM pathways, , as a function of the band gap multiple of the photon energy (ℏω/ E g ).In literature, the rate of impact ionization has been described analytically in the form of R = A(E − E 0 ) P . 37,39,40,53e fit the average density of CM pathways similarly, according to where A and P are fitting parameters, and E th is the lowest energy for which we find that N cm > 0. The dashed lines in Figure 3a are the results of fits of eq 14 to N̅ CM .E th and the optimal values of A and P are given in Table 1.The fit reproduces the data for N̅ CM very well.It is clear from Figure 3a and the values of A and P that N̅ CM increases much more rapidly with photon energy for MoTe 2 than for the lead chalcogenides.Figure 3b shows experimental quantum yields obtained from photoinduced bleaching (PIB), intraband absorption (PIA), or terahertz (THz) conductivity measurements. 20,21Comparing Figure 3a,b shows that the calculated onset of the rise of N̅ CM agrees very well with the experimental onsets of CM, which are close to 2.8E g for MoTe 2 , 5.5E g for PbS, and 7.0E g for PbSe.The calculated trend of the slopes of N̅ CM in Figure 3a reflects that of the experimental CM quantum yields in Figure 3b, with the slope drastically decreasing in the order MoTe 2 , PbS, and PbSe.Hence, the much higher experimental CM quantum yield for MoTe 2 can be explained qualitatively by the higher density of CM pathways than for PbS and PbSe.
Quantitative Modeling of Experimental Quantum Yields.We used our model outlined above to get insights into the origin of the much higher experimental CM quantum yields reported for MoTe 2 , 20,21 as compared to PbS and PbSe. 7We fit eq 12 to the experimental quantum yields with F CM /R cool as the only adjustable parameter.To get insights into the effect of cooling, we also calculated the upper limit of the quantum yield that is reached in the absence of cooling according to eq 13.In the latter case, the quantum yield is solely determined by the presence or absence of at least one CM pathway for each initial state of a carrier.
The experimental CM quantum yields for MoTe 2 obtained from transient absorption spectroscopy (PIB and PIA) by Kim et al. 20 and THz conductivity measurements by Zheng et al. 21re shown in Figure 4a.The quantum yields obtained from PIB are similar to those from the THz conductivity measurements, while PIA gives higher CM quantum yields.The discrepancy between the yields obtained from PIB and THz measurements and those from PIA could be due to a smaller signal-to-noise ratio for the latter, as reflected in the larger error bars for the PIA data.The larger effect of noise for PIA may stem from the fact that the change in transmission during PIA experiments was about 1 order of magnitude smaller than for PIB. 20The experimental CM quantum yields for PbS and PbSe in Figure 4b,c have been obtained from THz conductivity measurements by Pijpers et al. 7 The results of fitting eq 12 to experimental quantum yields are shown in Figure 4a,c.The best fit of our model to the experimental data is shown as red dots and the one standard deviation interval as the shaded area.The open diamonds in Figure 4a−c represent the quantum yield calculated in the absence of cooling (R cool → 0) according to eq 13. Figure 4a shows the best fit of our model to the experimental PIB and THz data for MoTe 2 . 20,21The fit agrees well with the measured results.Much like the THz data 21 the modeled quantum yield of MoTe 2 exhibits a step-like increase near 2.5 eV.The higher CM quantum yields obtained from PIA measurements cannot be reproduced with our model, even in case the maximum quantum yield of CM in the absence of cooling (R cool → 0) is calculated.We therefore did not include the PIA data in our best fit and fitted our model only to the PIB and THz data.As mentioned above, this may be (in part) due to an uncertainty of the PIA data, and it will be of interest to repeat PIA and PIB experiments on MoTe 2 .Figure 4b,c shows that our model also reproduces the CM quantum yields for PbS and PbSe very well.
Note that in the absence of cooling (R cool → 0) we do not obtain the dashed step-like increase of the quantum yield with photon energy, which implies that only part of all initially photogenerated electrons and holes can decay by CM.This is due to the restrictions of conservation of crystal momentum and energy.Interestingly for MoTe 2 the quantum yield without cooling is not much larger than the results from the fit.This means that CM is much faster than cooling in MoTe 2 , in agreement with the conclusion by Zheng et al. that the rate for the first CM step is nearly an order of magnitude larger than the cooling rate. 21By contrast, in the absence of cooling, the quantum yields of PbS and PbSe are much higher than those obtained from the best fits; see Figure 4b,c.The latter implies that cooling significantly competes with CM in PbS and PbSe. Figure 2 shows that the electronic bands of MoTe 2 are relatively flat compared to the more curved bands of PbS and PbSe.In the case of more flat bands, a change of the k-vector of an electron within a band due to electron−phonon scattering does not Table 1.Values of A and B from Fit of eq 14 to the N̅ CM Data in Figure 3a material A (eV The Journal of Physical Chemistry C modify its energy much, in contrast to the situation for the more curved bands of PbS and PbSe.As a result, cooling could be slower in MoTe 2 and compete less effectively with CM than in the case for PbS and PbSe.
The best fitted values of F CM /R cool are 349 ± 315 eV for MoTe 2 , 766 ± 393 eV for PbS, and 83.9 ± 28.7 eV for PbSe.The relatively large uncertainty of F CM /R cool of MoTe 2 can be attributed to its highly increasing N̅ CM as a function of photon energy.As can be seen in Figures S2 and S3, the values of N̅ CM for MoTe 2 increase so rapidly with photon energy that adjusting F CM /R cool from 1 eV to 10 6 eV has a much smaller impact on the quantum yield than is the case for PbS and PbSe.This therefore leads to a larger uncertainty in the parameter F CM /R cool for MoTe 2 .Moreover, Figure S2 also shows that the CM efficiency in MoTe 2 can predominately be attributed to its large and rapidly increasing number of CM pathways, as large changes in F CM /R cool only slightly affect the quantum yield.
The large uncertainty of F CM /R cool in the materials indicates that quantitative comparison must be done with caution.The large uncertainty of the value of F CM /R cool for MoTe 2 makes it unmeaningful to compare it with those of PbS and PbSe.However, the F CM /R cool values of PbS and PbSe do show a significant difference that cannot be attributed solely to the fitting uncertainty and therefore requires further discussion.The value of F CM /R cool of PbSe is smaller than that of PbS because the value of N̅ CM is larger in PbSe than in PbS at any given photon energy, whereas the quantum yield of both materials is similar.This is illustrated in Figure S3a,b.Note that the difference between Figures S3 and 3 is the horizontal axis, which is defined as photon energy in eV in Figure S3 rather than band gap multiple ℏω/E g in Figure 3. Since the value of N̅ CM of PbSe is greater than that of PbS while the quantum yields are similar, the F CM /R cool value of PbSe must be lower to compensate for its larger N̅ CM value.This implies that PbSe exhibits a larger R cool and/or a smaller F CM than PbS.F CM is inversely proportional to the square of the dielectric function, i.e., F q CM 1 ( ) 2 . 33,34The dielectric constants of PbS and PbSe are 169 and 210, The effective ratio of CM and carrier cooling rates for electrons, holes, and the average of both carriers obtained from eq 15 with the results from fitting eq 12 to the experimental quantum yield.Note that the fluctuations in the calculated quantum yield will in part be due to variations in N CM due to the use of a finite k-point grid, see right panel of Figure S1.This effect is largest at low photon energies, at which only a few optical transitions are possible.
The Journal of Physical Chemistry C respectively. 54We therefore estimate the F CM value of PbS to be roughly 1.5 times larger than that of PbSe.Details of the physical meaning and unit of F CM are discussed in the Supporting Information.Stewart et al. estimated that the energy-loss rate due to Froḧlich-type electron−phonon coupling is two times larger in PbS than in PbSe, due to a higher longitudinal optical phonon energy and stronger electron−phonon polar coupling. 55he cooling rate R cool is therefore expected to be twice as large in PbS.The combined estimates of the relative values of F CM and R cool predict that the F CM /R cool of PbS is 1.5/2.0= 0.75 times that of PbSe, which is clearly in disagreement with our findings.
The cause of the discrepancy between the relative values of F CM /R cool of PbS and PbSe obtained by fitting and the abovementioned estimation is unclear.As shown in Figure 2, PbS and PbSe have a similar band structure with the main difference being that PbSe has a band gap of 0.26 eV and PbS a band gap of 0.42 eV. 51The difference in band gap will play a minor role, since the actual CM onset for both is at much higher energy near 2 eV; see Figure S2.Understanding the different fitted values of F CM /R cool for PbS and PbSe asks for calculating F CM , which involves calculating the Coulomb matrix elements with the initial and final state wave functions, and measuring or calculating R cool .This is, however, out of the scope of this work.
From the fitted CM probability, we can back calculate the effective ratio of the CM and cooling rates for electrons and holes separately, with the following expression for CM of an electron in state i = e i , or a hole in state i = h i , see eqs 6 and 7. Note that P CM i depends on the density of CM pathways and this will be reflected in the effective ratio R CM eff /R cool .Figure 4d shows the effective R CM eff /R cool for electrons, holes, and the average R CM eff /R cool of both carriers as a function of band gap multiple.It can be seen that the slope of R CM eff /R cool is steepest for MoTe 2 , and then, it is steepest for PbS and PbSe, respectively.Equation 15shows that when the ratio R CM eff /R cool for electrons and holes each reaches unity, P̅ CM = 0.5 for both, and so the total probability to generate a secondary electron−hole pair is 1, resulting in a total quantum yield of 2. As can be seen from comparing Figure 4a−d, this point is reached at a photon energy of 3.5E g (3.0 eV) for MoTe 2 , while this occurs at a much higher energy of 8.3E g (3.5 eV) for PbS and at 13.8E g (3.6 eV) for PbSe.This variation of R CM eff /R cool with ℏω/E g in Figure 4d reflects the trends of the experimental quantum yields in Figure 4a−c.The high CM quantum yield as a function of ℏω/E g for MoTe 2 as compared to PbS and PbSe is thus due to the relatively high density of CM pathways (see Figure 3a), yielding a high CM rate that effectively outcompetes relatively slow cooling.
In addition, Figure 4d shows that R CM eff /R cool for MoTe 2 above 3.2E g (2.7 eV) is larger for holes than for electrons.This is caused by a difference in the distribution of the density of the CM pathways, N CM , of the initial electron and hole states.The distribution of N CM for all photogenerated initial states (i.e., all possible combinations of initial electron and hole states at a certain photon energy) is shown in Figure S4.From the upper panel for MoTe 2 , it can be seen that while the mean value of N CM is typically similar for electrons and holes, above 3 eV the N CM distribution for holes becomes narrower, and the median becomes higher than that of the electrons.This means that at photon energies above 3 eV, a larger fraction of photogenerated holes have a high number of CM pathways than is the case for electrons.This makes CM decay more likely for holes than for electrons above 3 eV.For PbS and PbSe, the N CM distribution is similar for electrons and holes, which leads to comparable CM rates for both carriers.
According to the theoretical analysis above, the high CM efficiency of MoTe 2 is mainly due to an extraordinarily high density of CM pathways that goes paired with a beneficial relatively slow cooling rate.These effects can be attributed to the band structure diagram of MoTe 2 (Figure 2), which shows a large density of relatively flat bands as opposed to the case of PbS and PbSe.The flatness of the bands of MoTe 2 reduces cooling, and their large density results in many CM pathways with conservation of electronic energy and crystal momentum.By contrast, the bands in PbS and PbSe are more curved leading to faster intraband cooling of electrons and holes and the smaller density of states leads to less CM pathways.

■ CONCLUSIONS
We calculated the density of CM decay pathways, N CM , for the bulk materials 2H−MoTe 2 , PbS, and PbSe, using electronic band structures from DFT calculations.The variation of N CM with photon energy averaged for electrons and holes, already qualitatively describes the much more efficient CM in 2H− MoTe 2 , as compared to PbS and PbSe.We take into account cooling of charge carriers in competition with CM, by introducing a fit parameter.In this way, our model quantitatively reproduces the disparate experimental CM quantum yields for the above-mentioned materials in a quantitative manner.The relatively small changes induced by adjusting the fit parameter confirms that the high CM efficiency of 2H−MoTe 2 can be mainly attributed to its high density of CM pathways.Our calculations predict a higher probability of CM for holes than for electrons in MoTe 2 at photon energies above 3 eV, while these are found to be similar for PbS and PbSe.The theoretical model can be applied to screen new materials for the prospects of efficient CM.

* sı Supporting Information
The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpcc.4c00383.The code that was used for the calculation of N CM , cmscript.py, is available upon reasonable request Derivation of eq 8 with definition of the prefactor N CM , discussion of convergence of density of CM decay pathways with the value of energy tolerance ε tol and the number of k-points, and box plots showing distributions of density of electron and hole decay pathways as a function of energy (PDF)

Figure 1 .
Figure 1.CM is illustrated in a band structure diagram.(a) Photoexcitation of a valence band electron with energy ℏω leads to the generation of an electron−hole pair.In this example, both photogenerated carriers can scatter to yield multiple electron−hole pairs, which is illustrated as a subsequent processes in the following subpanels.(b) The photogenerated electron in state 1 transfers energy to a valence band electron in state 2, resulting in the two final states 1′ and 2′ to be occupied by the electrons.(c) Scattering between two valence band electrons initially in states 1 and 2 results in the relaxation of the electron from state 1 to a lower empty valence band state 1′, this is equivalent to the relaxation of a hole from state 1′ to state 1.The simultaneous excitation of an electron from state 2 to conduction band state 2′ ensures that energy is conserved.(d) The two scattering processes in (b) and (c) finally yield three electron−hole pairs and thus a quantum yield of 3.

Figure 2 .
Figure 2. Band structures, density of states, and the Brillouin Zones of 2H−MoTe 2 , PbS, and PbSe.Brillouin Zone illustrations taken with permission from ref 52.Licensed under a Creative Commons Attribution License.

Figure 3 .
Figure 3. (a) Computed average density of CM pathways, = + = [ + ] = N N N N N N i N CM CM e CM h 1 1 CM e CM h

Figure 4 .
Figure 4.Quantum yields from experiment, best fits of eq 12, and calculated for R cool → 0 according to eq 13 for (a) MoTe 2 , (b) PbS, and (c) PbSe.The shaded areas represent the one standard deviation interval of the fit, obtained from the variance of the parameter estimate (see SciPy documentation scipy.optimize.curve_fit).The dashed step-like curves illustrate the case of ideal CM.(d) The effective ratio of CM and carrier cooling rates for electrons, holes, and the average of both carriers obtained from eq 15 with the results from fitting eq 12 to the experimental quantum yield.Note that the fluctuations in the calculated quantum yield will in part be due to variations in N CM due to the use of a finite k-point grid, see right panel of FigureS1.This effect is largest at low photon energies, at which only a few optical transitions are possible.
20,21ctively.It should be noted that Zheng et al. and Kim et al. reported slightly different band gaps for MoTe 2 , where Kim et al. found an indirect band gap of 0.85 eV and Zheng et al.an excitonic band gap of 0.9 eV.20,21This difference may be due to different sample thicknesses, which were 16.4 and 5 nm for Kim et al. and Zheng et al., respectively.Since our calculations were done on bulk 2H−MoTe 2 , the band gap reported by Kim et al. was used.