Charge Scattering and Mobility in Atomically Thin Semiconductors

The electron transport properties of atomically thin semiconductors such as MoS2 have attracted significant recent scrutiny and controversy. In this work, the scattering mechanisms responsible for limiting the mobility of single layer semiconductors are evaluated. The roles of individual scattering rates are tracked as the 2D electron gas density is varied over orders of magnitude at various temperatures. From a comparative study of the individual scattering mechanisms, we conclude that all current reported values of mobilities in atomically thin transition-metal dichalcogenide semiconductors are limited by ionized impurity scattering. When the charged impurity densities are reduced, remote optical phonon scattering will determine the ceiling of the highest mobilities attainable in these ultrathin materials at room temperature. The intrinsic mobilities will be accessible only in clean suspended layers, as is also the case for graphene. Based on the study, we identify the best choices for surrounding dielectrics that will help attain the highest mobilities.

Two-dimensional (2D) layered crystals such as single layers of transition-metal dichalcogenides represent the thinnest possible manifestations of semiconductor materials that exhibit an energy bandgap.
Such semiconductor layers differ fundamentally from ultrathin heterostructure quantum wells, or thin membranes carved out of three-dimensional (3D) semiconductor materials because there are in principle no broken bonds, and no roughness over the 2D plane. In heterostructure quantum wells, the electron mobility suffers from variations in the quantum-well thickness. A classic 'sixth-power law' due to Sakaki et al. [2] shows that since the quantum-mechanical energy eigenvalues in a heterostructure quantum well of thickness L go as ε~1 / L 2 , variations in thickness ΔL lead to perturbations of the energy Δε~−2ΔL / L 3 . Since the scattering rate depends on the square of Δε , the roughness-limited mobility degrades as µ R~L 6 . When L reduces from ~7 to ~5 nm for example, µ R reduced from ~10 4 to 10 3 cm 2 /Vs in GaAs/AlAs quantum wells at 4.2 K [2]. Though low-temperature mobilities exceeding 10 6 cm 2 /Vs have been achieved in such heterostructures by scrupulous cleanliness and design to reduce roughness scattering, the statistical variations in the quantum well thickness during the epitaxial growth process pose a fundamental limit to electron mobility.
Due to the absence of intrinsic roughness in atomically thin semiconductors, the expectation is that higher mobilities should, in principle, be attainable. However, recent measurements in MoS 2 and similar semiconductors [3][4][5] exhibit rather low mobilities in single layers, which are, in fact, lower than in their multilayer counterparts. Many-particle transport effects can appear in transition-metal dichalcogenides under special conditions due to the contribution of highly localized d-orbitals to the conduction and valence band edge eigenstates. Collective effects have been observed in multilayer structures, such as charge-density waves [6,7] and the appearance of superconductivity at extremely high metallic carrier densities [8] under extreme conditions. We do not discuss such collective phenomena here, and focus the work on single-particle transport in single-layer MoS 2 ; the only many-particle effect included is free-carrier screening. In this work, we perform a comprehensive study of the scattering mechanisms that limit electron mobility in atomically thin semiconductors. The mobility is calculated in the relaxation-time approximation (RTA) of the Boltzmann Transport Equation. The results shed light on the experimentally achievable electron mobility by designing the surrounding dielectrics and lowering the impurity density. The findings thus offer useful guidelines for future experiments.
With the advent of graphene, it was realized that for ultrathin semiconductors, the dielectric environment plays a crucial role in electron transport. It has now been demonstrated that the dielectric mismatch significantly modifies the Coulomb potentials inside a semiconductor thin layer [9][10][11][12].
Thus one can expect the dielectric environment to significantly affect electron transport properties in SL gapped semiconductors. In this work, we take SL MoS 2 as a case study to investigate such effects. The results and conclusions can be extended to other SL gapped semiconductors.
We first study the effect of the dielectric environment on Coulomb scattering of carriers from charged impurities located inside the MoS 2 single layer. Figure 1 in-plane 2D wave vector, and  ρ is the in-plane location vector of the electron from the point charge. The dielectric mismatch between the MoS 2 (relative dielectric constant ε s ) layer and its environment ( ε e ) creates an infinite array of image charges at points z n = na , where n = ±1, ±2... [9,10,20]. The n th point charge has a magnitude of eγ n , where γ = (ε s − ε e ) (ε s + ε e ) . These image charges contribute to the net electric potential seen by the electron, which is given by Figure 1 shows the net unscreened Coulomb potential contours in the dielectric/MoS 2 /dielectric system with three different ε e . The Coulomb interaction is strongly enhanced for low-κ dielectric environment and is damped for the high-κ case.
When a point charge is located inside a 3D semiconductor, its Coulomb potential is lowered by the dielectric constant of the semiconductor host alone. For thin semiconductor layers, the Coulomb potential is determined by the dielectric constants of both the semiconductor itself as well as the surrounding dielectrics. When a high density of mobile carriers is present in the semiconductor, the Coulomb potential is further screened. For atomically thin semiconductors, understanding the dielectric mismatch effect on the free-carrier screening of scattering potentials is necessary. At zero temperature, screening by the 2D electron gas is captured by the Lindhard function [21]: where q is the 2D scattering wave vector, Π is the polarizability function at zero temperature [22], where Θ[...] is the Heaviside unit-step function. The function Φ 1 is the form factor, and Φ 2 is the dielectric mismatch factor, which are defined by the equations [23] where χ ± = dz exp(±qz) ∫ χ 2 (z) . The free-carrier screening is taken into account by dividing the unscreened scattering matrix elements by ε 2d . Eq. (2) can be re-cast as the Thomas-Fermi formula: ε 2d = 1+ q TF eff q in analogy to the case in the absence of dielectric mismatch. Here q TF eff corresponds to the Thomas-Fermi screening wave vector q TF 0 without dielectric mismatch. Figure 2 (a) shows the ratio q TF eff q TF 0 that captures the effect of the dielectric mismatch on screening at zero temperature. The 2D electron density is n s~1 0 12 cm -2 in this figure. As can be seen, the free-carrier screening is weakened by a high-κ dielectric, and is enhanced in a low-κ case. This dependence is opposite to the effect of the dielectric environment on the net unscreened Coulomb interaction.
The momentum relaxation rate (τ m ) −1 due to elastic scattering mechanisms is evaluated using Fermi's golden rule in the form where M kk ' is the matrix element for scattering from state k to k ' , θ is the scattering angle, E k and E k ' are the electron energies for states k and k ' , respectively. For the charged impurity scattering momentum relaxation rate (τ m c ) −1 , the scattering matrix element is evaluated as The reduction of (τ m c ) −1 for a high-κ environment is much enhanced for high n s , as indicated in For finite temperatures, following Maldague, the polarizability function is [22,24,25]: where E F is the Fermi energy, and k B is the Boltzmann constant. Figure 3 (a) shows the calculated temperature-dependent polarizability normalized to the zero-temperature value at different n s . The electron gas is less polarizable at higher temperatures and lower n s . Polarizability is caused by the spatial re-distribution of the electron gas induced by the Coulomb potential, thus it is proportional to n s .
As temperature increases, the thermal energy randomizes the electron momenta, accelerating the transition of the electron system back into an equilibrium distribution, consequently weakening the polarization. The decrease of polarizability reduces the free-carrier screening. Figure 3 (b) shows the temperature-dependent Coulomb-scattering-limited mobility ( µ imp ) at two different n s . The dielectric mismatch effect is more significant for low n s , because of the fast decrease of the polarizability with increasing temperature. For high n s on the other hand, the dielectric mismatch effect is not as drastic.
The shape of the temperature-dependent µ imp curve is highly dependent on the polarizability and n s .
Consequently, if the electron transport is dominated by impurity scattering, one can infer n s from the shape of the temperature dependence of the electron mobility.
Much interest exists in using atomically thin semiconductors as possible channel materials for electronic devices, in which such layers are in close proximity to dielectrics. To that end, we investigate both the intrinsic and extrinsic phonon scattering in SL MoS 2 . Kaasbjerg et al. [26] have predicted the theoretical intrinsic phonon-limited mobility ( µ i− ph ) of SL MoS 2 from first principles using a densityfunctional-based approach. They estimated a room temperature upper-limit for the experimentally achievable mobility of ~410 cm 2 /Vs, which weakly depended on n s . Their estimate did not include the effects of free-carrier screening and dielectric mismatch. In light of the strong effect of these factors on the Coulomb scattering, we evaluate µ i− ph in MoS 2 in the Boltzmann transport formalism with the modified free-carrier screening. The material parameters for SL MoS 2 were obtained from Ref. [27]. The momentum relaxation rate due to quasi-elastic scattering by acoustic phonon is given by where ρ s is the areal mass density of SL MoS 2 , v s is the sound velocity, and Ξ ac is the acoustic deformation potential. For inelastic electron-optical phonon interactions, the momentum relaxation rate in the RTA is obtained by summing the emission and absorption processes, where ω op ν is the frequency of the νth optical-phonon mode. The momentum relaxation rates with superscript '+' and '-' are associated with phonon emission and absorption, respectively. For optical deformation potentials (ODP) [26], where D is the optical deformation potential, N q = 1 / [exp(ω / k B T ) −1] is the Bose-Einstein distribution for optical phonons of energy ω , and the subscript 0 and 1 denote the zero-and first-order ODP, respectively.
The scattering rate by polar optical (LO) phonons is given by the Fröhlich interaction [28], where ε ∞ is the high frequency relative dielectric constant, and Φ 1 is the form factor defined by Eq. (4). . This is in agreement with the previous predictions (320~410 cm 2 /Vs) [26,29]. However, the screened increases sharply with increasing n s . As can be seen in Fig. 4 (a), introducing a high-κ dielectric leads to a reduction of µ i− ph ; the highest values of µ i− ph reduce from 3100 to 1500 cm 2 /Vs as ε e increases from ~7.6 to ~20. The strong dependence of µ i− ph on the dielectric environment is entirely due to the dielectric-mismatch effect on free-carrier screening, since the unscreened phonon scattering matrix element is not affected by ε e .
Over the entire range of n s , longitudinal optical phonon scattering is dominant. This finding is different from previous works on multi-layer MoS 2 transport where the room temperature µ i− ph was determined by homopolar phonon scattering [30][31][32]. Kaasbjerg et. al. [27] have argued that the LA mode of ADP can be treated as screened by the longwavelength dielectric function, while the screening of TA mode ADP by free carriers can be neglected.  Fig 4(a). For the plot in Fig 4 (b), we have screened the polar optical and LA phonon scattering as in Fig. 4 (a), and leave the TA and ODP interactions unscreened. The highest µ i− ph reached by free carrier screening effect is reduced to ~750 cm 2 /Vs by not screening the DP modes. The mobility is dominated by the polar optical phonon interaction at low carrier density and by TA and ODP at moderate and high density. The scattering of electrons due to piezoelectric phonons is not considered because it is relevant only at very low temperatures and there are still uncertainties in the piezoelectric coefficients of SL MoS 2 [27,35].
In both cases, the calculated room temperature µ i− ph are much higher than reported experimental values, implying that there is still a large room for improvement of mobilities in atomically thin semiconductors. For the rest of this work, we use the fully screened intrinsic phonon scattering as shown in Fig. 4 (a). To pinpoint the most severe scattering mechanisms limiting the mobility in current samples, we discuss an extrinsic phonon scattering mechanism at play in these materials, again motivated by similar processes in graphene.
Electrons in semiconductor nanoscale membranes can excite phonons in the surrounding dielectrics via long-range Coulomb interactions, if the dielectrics support polar vibrational modes. Such 'remote phonon' or 'surface-optical' (SO) phonon scattering has been investigated recently for graphene and found to be far from negligible [15][16][17]. SO phonon scattering can severely degrade electron mobility; however this process has not been studied systematically in atomically thin semiconductors.
The electron-SO phonon interaction Hamiltonian is [15,17,18]: where a q ν + ( a q ν ) represents the creation (annihilation) operator for the νth SO phonon mode. Neglecting the dielectric response of the atomically thin MoS 2 layer in lieu of the surrounding media, the electron-SO phonon coupling parameter F ν is: where ε ox ∞ ( ε ox 0 ) is the high (low) frequency dielectric constant of the dielectric hosting the SO phonon, and ε ox ' ∞ is the high frequency dielectric constant from the dielectric on the other side of the membrane.
The frequency of the SO phonon ω SO ν is [17,36] where ω TO ν is the νth bulk transverse optical-phonon frequency in the dielectric. The scattering rate due to SO phonon is then given by Table I summarizes the parameters for some commonly used dielectrics. Figure 5 shows the room-temperature electron mobility for various dielectric environments for two representative temperatures, 100 K and 300 K. N I and n s are both ~10 13 cm -2 . The solid lines show the net mobility by combining the scattering from charged impurities, intrinsic and SO phonons, whereas the dashed lines show the cases neglecting the SO phonons. When SO phonon scattering is absent, the electron mobility is limited almost entirely by µ imp , which increases with ε e due to the reduction of Coulomb scattering by dielectric screening. The addition of the SO phonon scattering does not change things much at 100 K except for the highest ε e case (HfO 2 /ZrO 2 ). But it drastically reduces the electron mobility at room temperature, as is evident in Fig. 5. For instance, neglecting SO phonon scattering, one may expect that by using HfO 2 /ZrO 2 as the dielectrics instead of SiO 2 /air, the RT mobility µ imp should improve from ~45 to ~80 cm 2 /Vs. However, when the SO phonon scattering is in action, the mobility in HfO 2 /MoS 2 /ZrO 2 structure is actually degraded to ~25 cm 2 /Vs, even lower than the SiO 2 /air case. Thus, SL MoS 2 layers suffer from enhanced SO phonon scattering if they are in close proximity to high-κ dielectrics that allow low-energy polar vibrational modes.
To calibrate our calculations, we study the temperature-dependent electron mobility for SL MoS 2 embedded between SiO 2 and HfO 2 , and compare the calculations with reported experimental results. This structure is often used in top-gated MoS 2 field effect transistors (FETs), thus understanding the transport in it provides a pathway to understand the device characteristics. In Fig. 6 (a) [4]. N I and n s necessary to fit the data are indicated in Fig. 6 (a).
At low temperature, the experimental electron mobility in SL MoS 2 is entirely limited by µ imp . This is really not unexpected; it took several decades of careful epitaxial growths and ultraclean control to achieve the high mobilities in III-V semiconductors at low temperatures. Based on this study, we predict that the low-temperature mobilities in atomically thin semiconductors can be significantly improved by lowering the impurity density. The room-temperature mobility in III-V semiconductors is limited by intrinsic polar-optical phonon scattering. For comparison, we find that for SL MoS 2 , the roomtemperature mobility is considerably degraded by SO phonon scattering, even with N I as high as 6×10 12 cm -2 , as shown in Fig. 6. When SO phonon scattering is absent, the room temperature mobility is expected to be ~130 cm 2 /Vs with N I =6×10 12 cm -2 , but the measured values are typically lower (~50 cm 2 /Vs). Consequently, using HfO 2 as gate dielectrics can modestly improve µ imp . However the strong SO phonon scattering that comes with HfO 2 can severely decrease the high-temperature electron mobility in clean MoS 2 with low charged impurity densities.
An important question then is: which dielectric can help in improving the room-temperature electron mobility in SL MoS 2 ? To answer that question, in Fig. 6 (b), we plot the room-temperature (intrinsic+SO) phonon-limited electron mobility ( µ ph ) in SL MoS 2 surrounded by different dielectrics.
From the overall trend, µ ph decreases with increasing ε e , and suspended SL MoS 2 shows the highest potential electron mobility (over 10,000 cm 2 /Vs). It is worth noting that if the scattering of electrons by intrinsic phonons is only partially screened, as shown in Fig. 4 (b), the highest achievable mobility in SL MoS 2 will be an order lower (~1000 cm 2 /Vs). However these high values are attainable in suspended SL MoS 2 . Because µ ph for MoS 2 surrounded by high-κ materials is dominated by SO phonon scattering, the values do not vary much. The critical impurity densities ( N cr ) corresponding to µ imp = µ ph are shown in Fig. 6 (c). As long as N I ≥ N cr , µ imp completely masks µ ph . When N I < N cr , the electron mobility starts to be dominated by phonons and moves towards the upper-limit. High µ ph indicates a greater potential for attaining higher electron mobilities. However, it also needs the sample to be highly pure. In high-κ environments that support low-energy polar vibrational modes, there is not as much room for improving the electron mobility as is in low-κ structures. A compromise is seen for AlN and BN based dielectrics, which by virtue of the light atom N allows high-energy optical modes in spite of their polar nature. From Fig. 6 (b) Fig. 7 (a). Within this window, high-κ dielectrics can improve the mobility, but only very nominally because the unscreened mobilities are already quite low. When N I is lowered below ~10 12 cm -2 , a low-κ environment shows higher electron mobility. For most of the dielectric environments, when N I >10 12 cm -2 , the mobility fits to the following empirical impurity-scattering-dominated relationship: µ ≈ 4200 / [N I /10 11 cm −2 ] cm 2 /Vs, as shown by dashed line in Fig. 7 (a). Using this expression, one can estimate N I from measured electron mobility for high n s . As n s decreases, electron mobility in different dielectric environments starts to separate from each other, as shown in Fig. 7 (c). In this case, the electron mobilities can fit to the following relationship: µ ≈ 3500 N I /10 11 cm −2 A(ε e ) + (  Fig. 7 (c). High-κ dielectrics with low energy phonons (HfO 2 , ZrO 2 ) severely degrade the electron mobility over the entire N I range because of the dominant effect of SO phonon scattering. Note that the dielectric mismatch effect can be slightly overestimated here since we have assumed the thickness of the dielectric to be infinite [25]. In top-gated FETs, the top dielectric could be very thin. Thus the capability of improving electron mobility by high-κ dielectrics can be even less significant. Since most applications require high mobilities, high n s , and high ε e to be present simultaneously in the same structure for achieving the highest conductivities, AlN/Al 2 O 3 or BN/BN encapsulation emerge as the best compromises among the dielectric choices considered here. One can also perceive of dielectric heterostructures, with a few BN layers closest to MoS 2 to damp out the SO phonon scattering, followed by higher-κ dielectrics to enhance the gate capacitance for achieving high carrier densities. All this however requires ultraclean MoS 2 to start with, with N I well below 10 12 cm -2 to attain the high roomtemperature mobilities ~1000 cm 2 /Vs. The presence of high impurity densities will always mask the intrinsic potential of the materials, and is the most important challenge moving forward.
In conclusion, carrier transport properties in atomically thin semiconductors are found to be highly dependent on the dielectric environment, and on the impurity density. For current 2D crystal materials, electron mobilities are mostly dominated by charged impurity scattering. Remote phonons play a secondary role at high temperature depending on the surrounding dielectrics. The major point is that the mobilities achieved till date are far below the intrinsic potential in these materials. High-κ gate dielectrics can increase the electron mobility only for samples infected with very high impurity densities. Clean samples with low-κ dielectrics show much higher electron mobilities. AlN and BN based dielectrics offer the best compromise if a high mobility and high gate capacitance are simultaneously desired, as is the case in field effect transistors. The truly intrinsic mobility limited by the atomically thin semiconductor itself can only be achieved in ultraclean suspended samples, just as is the case for graphene.