Intrinsic electronic transport and thermoelectric power factor in n-type doped monolayer MoS2

The electronic transport and thermoelectric properties in n-type doped monolayer MoS2 are investigated by a parameter-free method based on first-principles calculations, electron–phonon coupling (EPC), and Boltzmann transport equation (BTE). Remarkably, the calculated electron mobility μ ∼ 47 cm2 V−1s−1 and thermoelectric power factor σS2 ∼ 2.93 × 10−3 W m−1 K−2 at room temperature are much lower than the previous theoretical values (e.g. μ ∼ 130–410 cm2 V−1 s−1 and σS2 ∼ 2.80 × 10−2 W m−1 K−2), but agree well with the most recent experimental findings of μ ∼ 37 cm2 V−1 s−1 and σS2 ∼ 3.00 × 10−3 W m−1 K−2. The EPC projections on phonon dispersion and the phonon branch dependent scattering rates indicate that the acoustic phonons, especially the longitudinal acoustic phonons, dominate the carrier scattering. Therefore, a mobility of 68 cm2 V−1 s−1 is achieved if only the acoustic phonons induced scattering is included, in accordance with the result of 72 cm2 V−1 s−1 estimated from the deformation potential driven by acoustic modes. Furthermore, via excluding the scattering from the out-of-plane modes to simulate the EPC suppression, the obtained mobility of 258 cm2 V−1 s−1 is right in the range of 200–700 cm2 V−1 s−1 measured in the samples with top deposited dielectric layer. In addition, we also compute the lattice thermal conductivity κL of monolayer MoS2 using phonon BTE, and obtain a κL ∼ 123 W m−1 K−1 at 300 K.


Introduction
As a prototypical transition metal dichalcogenide, molybdenum disulfide (MoS 2 ) has drawn a great deal of attention owing to its excellent electronic and optical properties [1][2][3][4]. Similar to graphite, bulk MoS 2 is comprised of stacked layers that are loosely coupled by Van der Waals interaction [5]. This weak interlayer interaction makes fabrication of monolayer MoS 2 be probable by micromechanical exfoliation from the bulk counterpart [5][6][7], same as the synthesis of graphene from graphite [8]. While bulk MoS 2 is an indirect band-gap semiconductor with the gap of about 1.2 eV [9], monolayer MoS 2 has a direct band gap ∼1.9 eV [5,6], which is right in the range of visible frequency, and consequently promise the applications such as photodetector, transistors [7], and electroluminescent devices. In addition, due to the strong electron-phonon coupling (EPC), monolayer MoS 2 also exhibits good conventional superconductivity when the sample is heavily n-type doped [10][11][12][13].
In recent years, studies on electronic transport and thermoelectric properties of monolayer MoS 2 have gained increased interest because characterization of transport properties, especially the intrinsic transport, is Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. crucial to assess and understand its potential significance. For a suspended monolayer MoS 2 , the electron mobility μ is found early to be in the range of 0.5-3 cm 2 V −1 s −1 [8]. Much higher mobilities of 200-700 cm 2 V −1 s −1 can be reached through high-κ gate dielectric engineering to effectively screen the charged-impurities scattering and suppress the electron-phonon scattering [7,[14][15][16]. Theoretically, the phonon-mediated intrinsic mobilities have been calculated to be 130-410 cm 2 V −1 s −1 at room temperature according to the full band Monte Carlo simulations, EPC matrix calculations, or the deformation potential couplings in combination of Fröhlich interactions [17][18][19][20]. However, the deformation potential theory based on only the acoustic phonon scattering mechanism indicates that monolayer MoS 2 possesses an electron mobility of 72 cm 2 V −1 s −1 [21]. In view of further decrease of the mobility by inclusion of optical phonon scattering, this result is in sharp contrast to the theoretical values above 130 cm 2 V −1 s −1 [17][18][19][20]. Moreover, a most recent experimental measure on monolayer MoS 2 shows a small electron mobility of only 37 cm 2 V −1 s −1 at room temperature [22], and a prior Hall measurement exhibits a mobility of only 64 cm 2 V −1 s −1 at 260 K [23]. Besides the electron mobility, the measured thermoelectric power factor σS 2 ∼3.00×10 −3 W m −1 K −2 [22] is also much lower than previous computational results such as 2.80×10 −2 W m −1 K −2 [24]. Therefore, to solve these discrepancies, it is important to reestimate the electron mobility and power factor of monolayer MoS 2 with refined calculations.
In this work, we investigate the electronic and thermoelectric transport of n-type doped monolayer MoS 2 by a parameter-free first-principles technique. The calculated mobility μ∼47 cm 2 V −1 s −1 and power factor σS 2 ∼2.93×10 −3 W m −1 K −2 at 300 K show good agreement with the most recent experimental findings [22]. With the scattering induced by only the acoustic phonons, the obtained mobility of 68 cm 2 V −1 s −1 accords with the result of 72 cm 2 V −1 s −1 estimated from the acoustic phonon-mediated deformation potential [21]. Via excluding the scattering from the out-of-plane vibrations to simulate the EPC suppression, the estimated mobility of 258 cm 2 V −1 s −1 is comparable to the experimental results measured in the samples with top deposited dielectric layer. The calculated lattice thermal conductivity κ L ∼123 W m −1 K −1 at 300 K is also consistent with the previous calculations.

Methodology
Our electronic and thermoelectric transport properties of n-type doped monolayer MoS 2 are calculated from the electron Boltzmann transport equation (BTE) and the first-principles density functional theory as implemented in the QUANTUM-ESPRESSO package [25]. The norm-conserving pseudopotential is used to model the ion cores, and a plane-wave basis set with the cutoff energy of 80 Ry and the exchange-correlation functional of local-density approximation (LDA) with Perdew-Zunger parametrization [26] is chosen to simulate the valance electrons. A vacuum space of 15 Å is taken to eliminate the interactions between MoS 2 layer and its periodic images. To calculate the phonon dispersion and EPC matrix, the density functional perturbation theory and the Wannier interpolation technique are applied [27]. The electronic structure and dynamical matrix are both initially calculated within a 36×36×1 grid, and then uniformly interpolated into a dense mesh of 216×216×1 by using the maximally localized Wannier functions as coded in the EPW package [28,29]. With the interpolated dense EPC matrix elements, the k-dependent electron lifetime can be obtained, which is taken as the input quantity for the electron BTE.
From the electron Boltzmann theory, we can write the electrical conductivity σ and thermopower S along the α axis as follows [30,31]  where Ω BZ is the volume of the full BZ, w n q and n n q are the phonon frequency and equilibrium Bose-Einstein distribution function of phonon mode nñ |q , and is the EPC matrix element induced by mode nñ |q between states + ñ |mk q and ñ |nk . In equation (5), M is the atomic mass, ¶ n V q is the derivative of the Kohn-Sham potential with respect to atomic displacement associated with the phonon mode nñ |q . Following the calculation of electrical conductivity σ, the mobility μ is obtained by where n is the doping concentration computed from the integral of electron density of states (EDOS) by with ε F0 as the Fermi energy of the undoped sample. The lattice thermal conductivity κ L is obtained by solving the phonon BTE [32], where the anharmonic interatomic force constants (IFCs) are generated within 5×5 × 1 supercells by the finite-difference approach. The interaction up to fifth-nearest neighbors is included in the calculations of anharmonic IFCs, and a q-mesh of 40×40×1 is used to ensure the convergence of κ L .

Results and discussion
The energetically stable monolayer MoS 2 is the H 2 phase with the symmetry D h 3 , in which a molybdenum (Mo) layer is sandwiched between two sulfur (S) layers, as sketched in figure 1(a). Since the largest effective thermoelectric power factor σS 2 is captured around a small electron doping concentration n∼6.0×10 12 cm −2 (corresponding to about 0.005e per unit cell) [22] and the band structure and phonon dispersion have almost no change induced by such weak doping [13,19], the rigid band model of the undoped monolayer MoS 2 is sufficient to study the EPC and carrier scattering rates (SRs). Hence, unless otherwise stated, we take the undoped H 2 phase and the rigid band model to study the transport properties of n-type doped monolayer MoS 2 in this paper. The doping concentration n at temperature T is calculated according to equation (7) by shifting Fermi level ε F .
The optimized lattice constant, thickness, and Mo-S bond length are 3.14, 3.12, and 2.39 Å, respectively, which are in accordance with other reported results [13,[33][34][35]. The calculated band structure predicts a direct band gap of 1.86 eV between the valence band maximum (VBM) and conduction band minimum (CBM) at the K point in the BZ, as shown in figure 1(b). For the conduction band, we also detect the existence of the second energy minima (the so-called Q valley), which is about 82 meV higher than the CBM and locates approximately at the halfway point of the ΓK line, as shown in figures 1(b) and (c). The band dispersion around the energy minima in the conduction band is almost quadratic and thus can be described by the effective mass approximation. The calculated longitudinal and transverse effective masses for the Q valley are  [5,6,10,13,18,36,37]. The electron linewidth, which is an indicator of the carrier scattering, is also projected on the band structure. It exhibits that there is relatively small linewidth and thus weak scattering around the VBM and the Q and K valleys. To illustrate total scattering, the carrier SRs in the whole BZ as a function of energy at 150 and 300 K are shown in figure 1(d).
Overall, raising the temperature T increases the carrier SRs almost uniformly. The SRs at the same T follow closely the EDOS since the accessible phase space reflects the magnitude of the EDOS. The small SRs at the band edges correspond to the small linewidth projections around the VBM and the Q and K valleys, which leads to a largest electron lifetime of e.g. 30.7 fs for the conduction bands at 300 K.
The calculated electron mobilities μ and electrical conductivities σ as functions of the doping concentration n for the n-type doped monolayer MoS 2 at 150-300 K are plotted in figure 2. The conductivity σ is re-scaled according to equation (1) by a factor of c/h, where c is the vacuum space of 15 Å and h is the effective thickness ∼6.5 Å of the monolayer MoS 2 sheet [38]. As shown in figure 2, both the mobility μ and conductivity σ decrease with the increasing T at the same n, consistent with the change of SRs versus T. At the same T, in contrast to the decrease of mobility μ versus n, the conductivity σ increases with n due to the enhancement of EDOS. At room temperature, the calculated maximum value of the electron mobility is μ∼47 cm 2 V −1 s −1 , which is much lower than the theoretical values of 130-410 cm 2 V −1 s −1 [17][18][19][20]. Compared with the experimental data, this result is in agreement with the most recent finding of 37 cm 2 V −1 s −1 [22], although it is much higher than the early data in the range of 0.5-3 cm 2 V −1 s −1 [8] and much lower than the values of 200-700 cm 2 V −1 s −1 [7,[14][15][16]. It is worth noting that the experimental data of 200-700 cm 2 V −1 s −1 are achieved by top deposited high-κ gate dielectric engineering, in which the electron-phonon scattering in the interlayer MoS 2 sheet would be effectively screened by the top deposited dielectric layers and substrate, and thus results in high mobilities [7,[14][15][16]. However, the recent finding of 37 cm 2 V −1 s −1 is measured in a perfect monolayer MoS 2 on the SiO 2 substrate [22], which lacks of the screening of electron-phonon scattering from the top deposited dielectric layers, and gives rise to a mobility much more close to the intrinsic one. Considering the reduction of mobility resulted from the screenings of electron-phonon scattering from the SiO 2 substrate in [22], our calculated intrinsic value of μ∼47 cm 2 V −1 s −1 agrees well with the finding of 37 cm 2 V −1 s −1 . In addition, the analyses in the experimental works in [39,40] have revealed that the coupling capacitance between top and bottom gates is ignored in the mobility measurements based on the two-point method in the thin-film MoS 2 transistor channel, which results in an overestimation of the mobility by a factor of about 14. With this effect in consideration, the modified data from the experiments show the mobility values being much smaller than the theoretical predictions of 130-410 cm 2 V −1 s −1 . A subsequent Hall measurement also exhibited a small mobility of about 64 cm 2 V −1 s −1 at 260 K [23], with which our result of μ∼68 cm 2 V −1 s −1 at 250 K shows good agreement, as shown in figure 2(a).
To reveal the internal electronic transport mechanism, the phonon dispersion, Eliashberg spectral function, and intervalley scattering schematic are shown in figures 3(a) and (c). For phonon dispersion, there are three   acoustic branches: out-of-plane acoustic (ZA), transverse acoustic (TA), and longitudinal acoustic (LA) branches, which are separated by a gap of about 5.73 meV below the nonpolar transverse optical (TO 1 ) and longitudinal optical (LO 1 ) modes. The TO 2 , LO 2 , and ZO 1 modes represent three polar optical branches, while the ZO 2 branch is homopolar with a nondispersive behavior and a breathing mode eigenvector. which represents the density of EPC strength in phonon frequency space, is calculated at the n-type doping concentration n∼6.0×10 12 cm −2 , at which the largest effective thermoelectric power factor σS 2 is experimentally captured [22]. The EPC projections on phonon dispersion are also computed at this doping concentration. Evidently, the peaks in Eliashberg spectral function a w ( ) F 2 corresponds to the EPC projections on phonon dispersion. These results indicate that the TA, LA, LO 2 , and ZO 2 phonons around the M point give rise to the main EPC and thus govern the electron scattering, as shown in figure 3(a). This is due to the fact that the phonons around the M point can induce electron intervalley scattering between the K and Q′ valleys, as sketched in figure 3(c). In addition, around the Γ point, due to small phonon wave vector, the presence of intravalley scattering results in some EPC projections on the ZO 2 and three acoustic branches; near the K point, the electron transition from the Q to Q′ valleys will lead to the projections on the TA and LA branches.
In contrast to optical phonons dominating the electron scattering [17], our results show that the acoustic modes, especially the LA and TA phonons around the M point, play significant roles in the scattering, as shown in figure 3(a), which is consistent with the analysis in [18][19][20]. Via including the electron scattering induced by only the acoustic phonons, our calculated mobility is μ∼68 cm 2 V −1 s −1 , as shown in figure 3(b), which accords well with the value of 72 cm 2 V −1 s −1 estimated from the acoustic phonon-mediated deformation potential [21]. In fact, the EPC strength captured in the acoustic phonon-mediated deformation potential also originate mainly from the LA and TA phonons [21], because the ZA modes are relatively insensitive to the tiny deformation. If the electron scattering from only the LA and TA phonons is taken into account, the obtained mobility is μ∼76 cm 2 V −1 s −1 , as shown in figure 3(b), which also shows a good agreement with the result of 72 cm 2 V −1 s −1 in [21]. In addition, the electronic transport caused by only the optical modes is also computed to mimic the result without inclusion of the scattering from the acoustic modes in [17]. Among the six optical branches, only the LO 2 and ZO 2 modes induce considerable EPC, as shown in figure 3(a). If only the electronphonon scattering arising from the LO 2 and ZO 2 modes is considered, the obtained mobility is μ∼310 cm 2 V −1 s −1 , as shown in figure 3(b), which is comparable to the result of 410 cm 2 V −1 s −1 in [17].
The experimental works that exhibit the mobilities of 200-700 cm 2 V −1 s −1 at room temperature are achieved by top deposited high-κ gate dielectric engineering [7,[14][15][16]. In these experiments, monolayer MoS 2 is the interlayer of the sandwich structure and its charged-impurities scattering and the electron-phonon scattering would be effectively screened by the substrate and dielectric layers. Before estimating the mobility of this monolayer MoS 2 , we firstly show the vibration modes of the three acoustic and the LO 2 and ZO 2 optical branches at the M point in figure 3(d). Evidently, the vibrations along the out-of-plane direction, such as ZA, LA, ZO 2 modes at the M point, for the MoS 2 interlayer in the sandwich structure are easily quenched by top deposited dielectric layer and substrate. As a result, only the in-plane vibration modes induced EPC dominates the electron scattering. As shown in figures 3(a) and (d), the in-plane vibration modes resulting in strong EPC are only the TA and LO 2 phonons around the M point. With the scattering driven by only the TA and LO 2 phonons, we obtain a mobility μ∼258 cm 2 V −1 s −1 at 300 K, which is just in the range of 200-700 cm 2 V −1 s −1 [7,[14][15][16]. Although the method of excluding the EPC contributions from the out-of-plane phonon modes is not very accurate to simulate the EPC suppression quantitatively, our results indicate that the mobility can be sharply enhanced once the out-of-plane modes are quelled by top deposition and substrate.
In order to clarify the discrepancies between our mobilities and the results of 130 and 150 cm 2 V −1 s −1 obtained by the similar method to ours [18,20], we show the zoomed-in view of the total electron SRs near the CBM in figure 4. The SRs originating from the LA, TA, LO 2 , and ZO 2 phonons are also shown in figure 4 (for the electron SRs of each phonon branch, see figure A1 in appendix). Obviously, the LA phonons contribute the main SRs of the monolayer MoS 2 , being consistent with the EPC projections in figure 3(a). At 82 meV above the CBM, which is just the energy of the so-called Q and Q′ valleys, the electron scattering between Q and Q′ valleys gets involved, which, as a result, leads to an additional distribution for the total electron SRs and the SRs of each phonon branch. This is not distinguished clearly in figure 3(a), because the carrier concentration n∼6.0×10 12 cm −2 used in the EPC projection calculations corresponds to the electron doping of only about 20 meV above the CBM. For the EPC projections at heavy doping, see figure 3(a) in [13]. Strikingly, around the CBM, the calculated carrier SRs are about three times higher than those in the previous calculations (see figures 7 and 8 in [20]). It is based on these higher carrier SRs of monolayer MoS 2 that our calculated mobilities of 47 cm 2 V −1 s −1 at 300 K and 68 cm 2 V −1 s −1 at 250 K are in good agreement with the most recent experimental finding of 37 cm 2 V −1 s −1 at 300 K [22] and the prior Hall measurement result of 64 cm 2 V −1 s −1 at 260 K [23]. Moreover, as analyzed above, the electron mobility estimated from the acoustic phonon-mediated deformation potential in [21], the mobility induced by only the optical phonons in [17], and a high mobility comparable to the values measured in the samples with top deposited dielectric layer in [7,[14][15][16] can be rightly simulated based on these higher carrier SRs through excluding part of electron-phonon scattering according to their respective physical mechanisms. These results support that our calculated carrier SRs are more rational than those in the previous works [18,20]. It is worth noting that our carrier SRs are obtained from the relationship  t = S 1 2 n n k k el , where the electron linewidth S nk is outputted directly by the EPW package [28,29], while the specific computational process of the linewidth S nk in [20] is unknown. To test the validity of our predictions for carrier SRs, we also calculate the electron SRs of crystalline Si, as shown in figure A2 in appendix. The calculated SRs of Si at 300 K is identical to the result in the EPW examples (see figure 8 in [29]), indicating the validity of our computing method.
The calculated thermopower S and thermoelectric power factor σS 2 for the n-type doped monolayer MoS 2 at 150-300 K are plotted in figure 5. Owing to the minus value of S in monolayer MoS 2 with the n-type doping, the −S is plotted for convenience. As shown in figure 5(a), the −S increases with temperature T at the same doping concentration n, and decreases with n at the same T, similar to the tendency observed in majority of semiconducting thermoelectric materials. The values of the calculated −S are fairly large, although they are smaller than previous computational results [24]. For instance, our calculations give the −S of about 0.37 mV K −1 at T=300 K and n∼10 12 cm −2 while a value of 0.52 mV K −1 is obtained at the same conditions in [24]. Obviously, both of the two values exceed the range of 0.20-0.30 mV K −1 which is the typical thermopower for a good thermoelectric material. As shown in figure 5(b), the largest power factor σS 2 at each temperature has a value exceeding 2.93×10 −3 W m −1 K −2 , which is comparable to that of Bi 2 Te 3 . At 300 K, the largest power factor σS 2 ∼2.93×10 −3 W m −1 K −2 can be achieved at n∼10 13 cm −2 , and a power factor of about 2.80×10 −3 W m −1 K −2 is obtained at n∼6.0×10 12 cm −2 . These power factor values show good agreement with the most recent experimental finding of σS 2 ∼3.00×10 −3 W m −1 K −2 around n∼6.0×10 12 cm −2 [22], although they are about one order of magnitude smaller than the result of 2.80×10 −2 W m −1 K −2 in [24].
The high power factor hints a good thermoelectric performance if the thermal conductivity is lower enough. For monolayer MoS 2 , the lattice thermal conductivity is reported to be in the range of 18-140 W m −1 K −1 at room temperature when an effective thickness of 6.5 Å is used [41][42][43][44]. Here we calculate the lattice thermal conductivity κ L of the monolayer MoS 2 by solving the phonon BTE with the harmonic and anharmonic IFCs as input quantities, which has been demonstrated to be an accurate and predictive method for studying the thermal transport performance [45][46][47][48][49]. The calculated κ L of the naturally occurring sample as a function of T is plotted in figure 6. Similar to the conductivity σ, the κ L is also re-scaled by the factor of c/h, where the effective thickness h is set to be 6.5 Å. The iterative solutions (ITS) of the phonon BTE and the relaxation time approximation (RTA) results are both shown, which indicates a difference smaller than 30% for the κ L above 300 K. At 300 K,  [41][42][43][44]. This relatively high κ L combined with the high power factor σS 2 indicates that two-dimensional monolayer MoS 2 is not suitable for design of thermoelectric generator (figure of merit ZT<0.01 at 300 K), but holds promise for in-plane thin-film Peltier coolers, in which a high power factor and a high thermal conductivity are both required [24]. In addition, the κ L of the doped MoS 2 would be modified if more scattering processes are  included. It is reported that the EPC plays important role in lattice thermal transport in heavy doping samples. For instance, the lattice thermal conductivity of crystalline Si may reduce 45% at a heavy p-type doping of about 10 21 cm −3 [50,51]. The four-phonon scattering also has a significant influence on lattice thermal transport, e.g. four-phonon scattering reduces the intrinsic thermal conductivity of graphene and decreases the contributions from flexural phonons [52]. The calculated κ L of the n-type doped MoS 2 sheet may be reduced significantly if these effects are included, and thus the figure of merit ZT would also be modified.
In addition, we have also investigated the cumulative thermal conductivity κ C with respect to the allowed phonon frequency and phonon maximum mean-free path (MFP), respectively, as shown by the insets in figure 6. The κ C versus the phonon frequency gives the summed contribution from all phonon modes below the specified frequency, which exhibits that almost all of the κ L (∼99%) is dominated by the acoustic phonon modes. The κ C versus the phonon maximum MFP reveals the size dependence of the κ L , which indicates that the phonons with a MFP shorter than 100 nm dominate about 20% of the total κ L , implying the nanostructures with a characteristic length of the phonon MFP smaller than 100 nm are required to reduce the κ L drastically. Furthermore, if the nanostructures with the MFP characteristic length shorter than 10 nm is applied, the κ L may reduced to 1.20 W m −1 K −1 , and thus a figure of merit ZT∼0.73 can be obtained at 300 K.

Conclusion
In summary, we have used a parameter-free method based on first-principles technique, EPC, and BTE to investigate the electronic transport and thermoelectric properties in n-type doped monolayer MoS 2 . Due to the higher carrier scattering rates (SRs) than those of the previous calculations, our calculated electron mobilities μ of 47 cm 2 V −1 s −1 at 300 K and 68 cm 2 V −1 s −1 at 250 K show good agreement with the experimental findings of 37 cm 2 V −1 s −1 at 300 K and 64 cm 2 V −1 s −1 at 260 K. The EPC projections on phonon dispersion and the phonon branch dependent SRs indicate that the acoustic phonons, especially the longitudinal acoustic phonons, dominate the carrier scattering. Thus, the obtained acoustic phonon induced mobility is only about 68 cm 2 V −1 s −1 , which accords well with the result of 72 cm 2 V −1 s −1 estimated from the deformation potential driven by acoustic modes. Meanwhile, the mobility μ∼310 cm 2 V −1 s −1 induced by only the optical modes is close to the value of 410 cm 2 V −1 s −1 without inclusion of the scattering from the acoustic phonons. Furthermore, via excluding the scattering from the out-of-plane modes to simulate the EPC suppression, the calculated mobility μ∼258 cm 2 V −1 s −1 is right in the range of 200-700 cm 2 V −1 s −1 measured in the samples with top deposited dielectric layer. Due to the rationality of the electronic transport properties, our calculated thermoelectric power factor σS 2 ∼2.93×10 −3 W m −1 K −2 is also consistent with the experimental value of σS 2 ∼3.00×10 −3 W m −1 K −2 . In addition, the computed lattice thermal conductivity κ L ∼123 W m −1 K −1 at 300 K by the phonon BTE also shows consistency with the previous works. These results clarify the discrepancies between the experimental results and previous theoretical values, and shed further light on the electronic and thermoelectric transport of monolayer MoS 2 . Figure A1. The phonon branch decomposed electron scattering rates (SRs) above the CBM at 300 K. The energy of the CBM is set to be zero. Note the difference of the vertical coordinate for each panel. Figure A2. Total electron scattering rates (SRs) of crystalline Si at 300 K. The electronic density of states (EDOS) is also plotted.