Phonon scattering limited performance of monolayer MoS₂ and WSe₂ n-MOSFET

Two dimensional (2-D) materials such as the Transition Metal Dichalcogenides (MX₂: M=Mo, W; X=S, Se, Te) have emerged as a prospective channel material for the post-Si CMOS technology.^1,3–6 Among such MX₂ materials so far the MoS₂ and WSe₂ have been successfully realized into MOS devices experimentally.^1–3 Apart from monolayer flakes of such MX₂ materials, few layers thick MX₂ MOSFET have also been the focus of various device fabrication studies.^7–9 

The role of electron-phonon scattering in such MX₂ layers do have a significant impact on the performance of such 2-D channel MOSFETs.^10–12 Recently Guo et al. have shown the effect of phonon scattering in monolayer MoS₂ MOSFETs.¹² According to recent simulations tungsten chalcogenide (e.g. WS₂) FETs are expected to outperform other MX₂ FETs in the ballistic limit.⁶ However in the more realistic case, the devices often operate at the quasi-ballistic limit. In this regard it could be of interest to comparatively study the MoS₂ and WSe₂ MOSFET inclusive of the electron-phonon scattering. In this paper we study difference in performance of 1L-MX₂ (hereafter by MX₂ we refer to MoS₂ and WSe₂ only) channel 2-D planar n-MOSFET in the quasi-ballistic regime. The materials properties of the monolayer MX₂ channel are modeled by ab-initio methods. Thereafter using our in-house simulator based on the non-equilibrium Green’s function (NEGF) formalism, we calculate the performance parameters of the MX₂ n-MOSFETs. It is found that performance of WSe₂ devices are less impacted by phonon scattering compared to the MoS₂ counterparts.

Fig. 1 shows the schematic device structure of the planar 2-D MoS₂ FET considered for our studies. We consider a 1L MoS₂ or WSe₂ as the channel material. As shown in Fig. 1. The 2-D channel is placed over an SiO₂/Si substrate. High-κ HfO₂ of 2.5nm thickness is chosen as the gate dielectric. We consider highly doped (10²⁰ /cm³) n⁺⁺ regions as the source/drain for the n-MOSFET. Such doping concentrations allow for good alignment of the source/drain fermi levels with the conduction band for the monolayer MX₂ FETs.^5,6

In the first step in our study we evaluate the different properties of the MX₂ channel material e.g. bandstructure, electron effective mass, phonon spectra and phonon density of states, by ab-initio simulations. For this purpose we employ density functional theory(DFT) in QuantumWise ATK.¹³ We use a 16 × 16 × 1 Monkhorst-Pack k-grid¹⁵ and employ the Local Density Approximation (LDA) exchange correlation function with the Perdew Zunger(PZ) basis.¹⁶ The DFT simulations are performed by relaxing the structures by optimizing the positions by a Broyden - Fletcher - Goldfarb - Shanno (BFGS)¹⁷ Quasi-Newton optimization method in ATK with maximum force of 0.05 eV/Å. The energy cut-off value in our simulations is set at 75 Hartree.

The phonon dispersion and phonon DOS are calculated in ATK using a supercell based small displacement method also known as the frozen phonon calculations. In this method employing a finite difference scheme the first derivative of the forces are calculated, where the system is displaced by small amounts along each degree of freedom.^10,13 For phonon calculations in ATK we use a 9 × 9 × 1 supercell. 72 small displacements, each of value 0.01Å is applied to the supercell in the x and the y directions. From this the forces are calculated and subsequently the dynamical matrix of the system is found out, which is used in calculating the phonon properties.

Thereafter, we proceed to solve the Poisson and Schrödinger equations self-consistently for our MX₂ FET. The self-consistent solutions are carried out under the Non-Equilibrium Green’s Function (NEGF) formalism^12,18 with our in-house NEGF simulator.¹⁹ The electron-phonon interaction in the quasi-ballistic regime is incorporated into our simulator by self-consistent Born approximation method within the framework of NEGF formalism.^12,18 In our solver, we construct the Green’s function using an effective mass Hamiltonian H and the energy eigenvalue matrix E of the system along with the self-energy matrix of the system Σ.^20,21

In (1)I is the identity matrix and Ø⁺ is an infinitesimally positive quantity. The self-energy matrix Σ consists of Σ_C for the source and drain contacts and Σ_Sc which is the scattering self-energy matrix.

The spectral density A(E) is expressed as

the Green’s correlation function for the n-type and p-type carriers exclusively can be expressed as Gⁿ and G^p, and spectral density can also be expressed in terms of these two as¹⁸ 

The broadening Γ is obtained from the self-energy as

in terms of the in-scattering and out-scattering self-energies of the system it takes the form

For electron-phonon scattering we consider Gⁿ, which is found as¹⁸ 

while G^p, which is found as

In the self-consistent Born approximation approach the in and the out scattering self-energies are evaluated as

here Δ represents the spin relaxation tensor. The number density of the electrons in a 2-D system of cell volume V_xy is

similarly the number density for holes is given by

g_s and g_v being the spin and the valley degeneracies, r being the positional co-ordinate. This carrier (electron) density is evaluated self-consistently within the Poisson-Schrodinger solver.

The Green’s functions G, G^n,p the in and out scattering self energies Σ^in,out are all evaluated using a recursive algorithm with the self-consistent Born approximation approach, as described in detail by Nikonov et al.¹⁸ The drain current is evaluated as^4,19

m_t being the carrier effective mass in the transverse direction, k_B is Boltzmann constant, T is temperature, E_k,x the energy of the conducting level, F_−1/2 is the Fermi integral of order −1/2. η₁ and η₂ are the chemical potentials of the source and drain respectively. ℑ(E) is the transmission matrix given as

the subscripts 1 and 2 designating the source and the drain contacts. For calculating currents purely ballistic in nature, the scattering self-energy matrix Σ_Sc in (2) is considered zero and the calculations for such a case is described in detail by Sengupta et al.¹⁹ Among the measured parameters the ballisticity is expressed as a ratio between the drain currents including scattering I_D and the purely ballistic drain current I_D,Bal. The ON current I_ON for the devices are calculated at V_G = V_D = 0.4 V. The intrinsic delay time τ is expressed as τ = (C_gg + C_f) × V_DD/I_ON, where C_gg is the device gate capacitance expressed as series combination of geometrical oxide capacitance C_ox and the channel quantum capacitance C_Q, C_f is the fringing capacitance.^20,21V_DD is the supply voltage (0.4 V).

Fig. 2 shows the simulated bandstructure of the 1L MX₂ sheets. Our DFT simulations show a direct band gap at the K point of the hexagonal Brillouin zone for both the 1L-MoS₂ and 1L-WSe₂ sheets with band-gap values of 1.82 and 1.62 eV respectively. Such values are agreeable to ab-initio results in recent reports.¹⁴ The effective masses obtained in our studies (Table I) are also consistent with ab-initio simulation results reported elsewhere.^4,6,12

In Fig. 3, we show the calculated phonon bandstructures for the layered MX₂ sheets under consideration. The in-plane acoustic vibration modes of the MX₂, namely the longitudinal acoustic (LA) and the transverse acoustic (TA) modes have higher energy compared to the out-of plane acoustic mode (ZA). These acoustic phonon modes in MX₂ are clearly separated from the optical phonon modes. The two lowest branches of the optical modes are non-polar (NP) in nature, followed by the transverse optical (TO) and the longitudinal optical (LO) modes. The mode(s) with very low dispersion at energies 50 meV in MoS₂ and 30 meV in WSe₂ are known as the homopolar (HP) modes. Fig. 4 shows the calculated phonon density of states of the monolayer MoS₂ and WSe₂.

Among the different modes we can consider the LA, TA and ZA modes to contribute to acoustic scattering, non-polar optical and TO and HP modes to contribute to the optical phonon scattering, while the polar LO modes are responsible for the Frölich interaction mechanism of electron-phonon scattering. The phonon energies of the different modes at the CB minima of the electronic bandstructure are considered in order to evaluate the electron-phonon interaction matrix and subsequently the scattering self-energy matrix Σ_S in the channel.¹⁸ 

In Fig. 5, we show the effect of the electron-phonon scattering on the output characteristics of the device. A gate voltage V_G = 0.4 V is considered for these simulations. A channel length L_Ch = 25 nm is taken so as to show the phonon scattering contributions more clearly in the plot. We see for purely ballistic currents (shown in black) the drain current I_D for 1L-WSe₂ is higher than that for the 1L-MoS₂ channel. The optical phonon scattering (comprising of non-polar, TO and HP modes) contributes the highest in the degradation of drain current. This is followed by that due to the Frölich interaction and the acoustic phonon contributions (due to LA, TA and ZA modes).

The phonon scattering seems to degrade carrier transport in MoS₂ more than that in WSe₂ devices. The reason behind this is the stronger electron–phonon coupling in MoS₂, than the WSe₂ monolayer. This is quantitatively represented by the higher values of the deformation potentials (DP) for the various phonon modes for MoS₂ compared to WSe₂ as obtained by Kaasbjerg et al.¹⁰ and Jin et al.²² For the L_Ch = 25 nm device, the net reduction in I_D due to electron-phonon scattering effects are 18.6% for WSe₂, and 26.8% for 1L-MoS₂ devices. The variation of purely ballistic and the quasi-ballistic (inclusive of phonon scattering) ON currents with the device channel length is shown in Fig. 6. The ON currents are evaluated at V_G = V_D = 0.4 V. It is observed that the performance of the WSe₂ device is less impacted by phonon scattering, showing a ballisticity of 83% for 1L-WSe₂ FET for channel length of 10 nm. Though 1L-MoS₂ FET of similar dimension shows a ballisticity of 75%. As the channel length increases, the electrons tend to encounter more number of scattering events and therefore the difference between the ballistic and quasi-ballistic currents tend to increase with the channel length. Here it must be mentioned that in our work we have included only the additional effect of electron-phonon scattering to the ballistic transport in MX₂ FET. Also we looked to focus our study on mostly short channel lengths as MX₂ as a channel material is essentially a candidate for post-Silicon sub decananometer CMOS technology. In experimentally fabricated devices the channel dimensions vary from several hundred nanometer^1,3 to even a few microns² with gate dielectric thickness of the order of few tens of nanometers.^1–3 In case of such fabricated devices, a vast number of non-idealities like surface roughness, defects and impurity scattering, intrinsic ripples in 2-D membranes, interface strain, processing related defects etc. apart from electron-phonon scattering, come into play. Owing to these effects there is a large difference between experimental results^1–3 and simulation studies that are being conducted by different groups.^4–6,12 For theoretical studies conducted for nanoscale FET of MX2 material, the ON currents are in the range of 500 μA/μm for HfO₂ top gated Schottky barrier MoS2 SBFET of L_Ch = 15 nm.⁴ For HfO₂ top-gated MOSFET in the scaling limit of L_Ch = 5 nm the simulated ON current is reported as 238 μA/μm by Alam et al.⁵ In this context our simulated results for L_Ch = 10 nm, top gated HfO₂ dielectric doped contact MoS2 MOFET shows ON current of 666 μA/μm which is consistent with other theoretical results based on similar structures and device geometry and gating.⁴ 

The calculated values for the geometrical oxide capacitance C_ox is 0.22fF/μm, the channel quantum capacitance C_Q at V_G = 0.4V is 0.16fF/μm and the fringing capacitance C_f is 0.025fF/μm. The capacitances are calculated by the methods described by Alam et al.⁵ For the 25 nm device, τ in the ballistic limit was calculated to be 0.67 ps for MoS₂ and 0.53 ps for WSe₂ FET. In case of MoS₂ FETs scattering seems to affect delay time much more severely than the WSe₂ FETs. As electron-phonon scattering is included the delay time increases significantly and the values increase to 0.91 ps and 0.64 ps for the MoS₂ and the WSe₂ FETs respectively.

The transconductance (g_m) vs V_G characteristics in Fig. 7 of a 25nm device show a the ballistic values of g_m for the MX₂ are rather closely grouped together, while in the quasi-ballistic case the g_m-V_G curve are more spread out. This trend was earlier observed in the ON currents for ballistic and quasi-ballistic cases as well. The reason behind being the disparity in the amount of electron-phonon scattering among the two different MX₂ materials. The values of peak g_m for the MX₂ n-MOSFETs vary between 1226–1359μS/μm for the ballistic cases.

The energy and position resolved current spectra is shown in Fig. 8. It is clear that the WSe₂ MOSFETs show a higher current density compared to the MoS₂ ones. Also the extension of the region of highest current density within the devices show that, for the 1L-WSe₂ this region (marked red in the colorbar) extends to a significant portion of the channel length. For 1L-MoS₂ the length of this region diminishes gradually due to higher degree of electron phonon scattering in the channel for those structures. In all the cases there is no evidence of direct tunnelling between the source and drain, as is expected for channel lengths of L_Ch = 25 nm.

In this paper we study the effect of phonon scattering on the performance of monolayer (1L) MoS₂ and WSe₂ n-MOSFETs. Material properties of the channel are evaluated using the local density approximation (LDA) in density functional theory (DFT) using the Perdew–Zunger (PZ) exchange correlation. Thereafter using our in-house non-equilibrium Green’s function (NEGF) simulator, we study the effect of phonon scattering on the performance of such devices. Individual contributions of the different phonon modes to the scattering and the effect of varying channel length of the device on performance parameters such as ON current, ballisticity, intrinsic delay time and transconductance are studied in detail. It is found that performance of WSe₂ devices are less impacted by phonon scattering compared to the MoS₂ counterparts.

Dr. A. Sengupta thanks DST, Govt. of India, for the DST INSPIRE Faculty award. (Grant No. DST/INSPIRE/04/2013/000108).