A simulation study of short channel effects with a QET model based on Fermi–Dirac statistics and nonparabolicity for high-mobility MOSFETs

In this paper, the quantum conﬁnement and short channel effects of Si, Ge, and In 0 . 53 Ga 0 . 47 As n-MOSFETs are evaluated. Both bulk and double-gate structures are simulated using a quantum energy transport model based on Fermi–Dirac statistics. Nonparabolic band effects are further considered. The QET model allows us to simulate carrier transport including quantum conﬁnement and hot carrier effects. The charge control by the gate is reduced in the Ge and In 0 . 53 Ga 0 . 47 As bulk n-MOSFETs due to the low effective mass and high permittivity. This charge control reduction induces the degradation of short channel effects. In double-gate structures, different improvements of drain induced barrier lowering (DIBL) and subthreshold slope (SS) are seen. The double-gate structure is effective in the suppression of DIBL for all channel materials. The SS degradation depends on channel materials even in double-gate structure.


Introduction
The scaling of conventional bulk Si-MOSFET approaches the fundamental limit due to the increase of off-leakage current and short channel effects [1]. Further performance B Shohiro Sho shohiro@cas.cmc.osaka-u.ac.jp 1 improvements require new channel materials such as Ge and III-V compound semiconductors [2] and new device structures such as FinFETs [3] and nanowire gate-all-around structures [4]. Performance analysis of single and multigate MOSFETs on high mobility substrates and Si is an important issue. A number of authors have focused on numerical and theoretical studies of such devices, using self-consistent Poisson/Monte Carlo simulations [5,6], comprehensive semiclassical multisubband Monte Carlo simulations [7], self-consistent solutions of Schrödinger/Poisson equations [6,8,9], a quantum-corrected Monte Carlo simulations [10], and an atomistic Schrödinger/Poisson equations in the non-equilibrium Green's function formalism [11].
This paper describes performance analysis of Si, Ge, and In 0.53 Ga 0.47 As n-MOSFETs using a quantum energy transport (QET) model based on Fermi-Dirac statistics and nonparabolicity. The QET model is viewed as one of the hierarchy of the quantum hydrodynamic models [12], which allows simulations of carrier transport including quantum confinement and hot carrier effects [13]. The simulation study focuses on the analysis of quantum confinement and short channel effects. Both bulk and double-gate n-MOSFETs are simulated.
The paper is organized as follows: In Sect. 2, we describe a four-moments QET model based on Fermi-Dirac statistics and nonparabolicity. Section 3 presents numerical simulations of the QET model. The results are further compared with those calculated by quantum drift diffusion (QDD) and classical energy transport (ET) models. The analysis of short channel and quantum confinement effects of Si, Ge, and In 0.53 Ga 0.47 As n-MOSFETs for bulk and double-gate structures is presented. The dependence of short channel effects on channel materials is discussed. Section 4 concludes this paper.

4-moments QET model based on Fermi-Dirac statistics
For the simulations of quantum confinement transport with hot carrier effects, we develop a four-moments QET model in [13]. This model is viewed as one of the hierarchy of the quantum hydrodynamic models [12]. In classical hydrodynamic simulations, a four-moments energy transport model is proposed in [14] for simulations of thin body MOSFETs. In this work, Fermi-Dirac statistics and nonparabolic corrections are further included for the performance analysis of MOSFETs on high mobility substrates. In fact, high mobility materials such as III-V compound semiconductors have strong degeneracy, low density of state, and nonparabolic band structures [15]. Numerical implementation of Fermi-Dirac statistics is discussed in [16] for QDD models and in [17] for QET models. The electron density n is approximated by introducing the band parameter ω n as where ϕ, ϕ n , and T n are the electrostatic potential, quasi-Fermi-level, and electron temperature, respectively. n i , q, and k are the intrinsic carrier density, electronic charge, and Boltzmann constant, respectively. The quantum potential γ n is described as where m andh are the effective mass and Plank constant. The band parameter ω n is determined as where N c is the density of states in the conduction band, and G 1 2 is the inverse Fermi function of order 1/2 defined with The carrier density n including nonparabolic band effects is given by [18] where η = (E f − E c )/kT is the normalized Fermi level. The parameter α is a coefficient of nonparabolicity that can be calculated as where g is the normalized band gap (= E c −E v /kT ) and m 0 is the free electron rest mass. The simple analytical approximation of the inverse Fermi function is given in [19] for a weak degenerate case (η < 10). For high η, we apply Sommerfeld's approximation to calculate the inverse Fermi function. Both approximations are linearly interpolated. By employing the expression (1) in the QET model, we obtain the current density where μ n is the electron mobility. From (2), the quantum potential equation is obtained as where b n =h 2 12qm . The root-density ρ n is written as ρ n = √ n = √ n i ex p(u n ) by a variable u n = q kT n ( (ϕ+γ n +ω n −ϕ n ) 2 ) in (1). As shown in [16], under Fermi-Dirac statistics, (8) is replaced by the equivalent form b n ∇ · (ρ n ∇u n ) − kT n q ρ n u n = − ρ n 2 (ϕ + ω n − ϕ n ).
If the variable u n is uniformly bounded, the electron density is maintained to be positive. This approach provides a numerical advantage for developing an iterative solution method. For electrons, the four-moments QET model based on Fermi-Dirac statistics is described as follows: where p, and C imp are the hole density, the permittivity of semiconductor, and the ionized impurity density, respectively. T L and τ are the lattice temperature and energy relaxation time. The ratio μ n /μ s selected here is 0.8 [20]. For holes, similar expressions are obtained.

Mobility model
For the energy dependence of the mobility, we apply the model of the Baccarani et al. [21], In the homogeneous case, this model is equivalent to the Hänsch mobility model [20] μ n (T n ) where v s is the saturation velocity. As mentioned in [22,23], the Hänsch mobility model is consistent with the high-field mobility model with the parameters ξ = 1/2 and β = 2.
To account for the mobility reduction due to the ionized impurity scattering, we use the formula of Caughey and Thomas [24] for the low-field mobility μ L F in this work: The model parameter values [25,26] are summarized in Table 1. In this work, numerical simulations are performed by using the Baccarani's mobility model (16) for the QET model and the high-field mobility model (18) for the QDD model, respectively. The effects of interface traps and surface roughness scattering are not included in this work.

Device structures
The schematic views of simulated devices are shown in Fig. 1. Si, Ge, and In 0.53 Ga 0.47 As n-MOSFETs with high-k/metal gates are examined. Selected material parameters are listed in Table 2. The relative dielectric permittivity considered here is 22, and the value is known as "HfO 2 ". The equivalent oxide thickness (EOT) is 0.6 nm. The threshold of all devices is obtained by the adjustment of the gate work function, which is selected for each semiconductor material to meet a common threshold voltage of 0.2 V. The threshold voltage is defined as the gate voltage when the drain current is 10 µA/µm. The channel length of simulated devices is varied from 35 to 16 nm. The S/D doping is N S D =    Figure 2a and b demonstrates the I D -V G characteristics of 50 and 20 nm Si bulk n-MOSFETs at V d = 0.05V and 0.8 V, which are calculated by the QET and QDD models. The same work function is used for both models. In a long channel device, the I D -V G characteristics calculated by two models are almost identical, as shown in Fig. 2a. For the ultra-short channel device, two models exhibit the different results of I D -V G characteristics due to the non-local transport effects and the reduction of the quantum confinement effects. The QET model provides higher drain current. The subthreshold of electrons toward the bulk in the channel and hence in ultrashort channel devices, a significant difference between two models is induced. The results clearly indicate that the quantum confinement effect in the ultra-short channel is reduced by the enhanced diffusion due to the high electron temperature. Figure 4a and b shows the electron density distributions calculated by the QET and QDD models for long and short channel devices. The results are plotted at the center of the channel. In the long channel device, the electron density distributions calculated by two models are almost identical at the surface. For the ultra-short channel device, due to the hot carrier effects, the electron density distribution calculated by the QET model is spread towards the bulk. This result in the reduction of charge control calculated by the QET model.

Quantum confinement effects
The dependence of quantum confinement effects on channel materials is investigated in Figs. 5 and 6. The results under Fermi-Dirac statistics and Boltzmann statistics are  Fig.  5a, the inversion layer electrons in Ge and In 0.53 Ga 0.47 As n-MOSFETs spread into the bulk at the source end of the channel due to the low effective mass and high permittivity. Figure 5b reveals that in all devices the quantum confinement effect is further reduced by the enhanced diffusion due to the high electron temperature. These properties degrade the short channel effects of Ge and In 0.53 Ga 0.47 As devices when compared with Si devices, as discussed later. Since N c (= 2.64 × 10 17 cm −3 ) of In 0.53 Ga 0.47 As is low, the inversion layer electrons in the In 0.53 Ga 0.47 As n-MOSFET are further decreased due to the high degeneracy material.
In Fig. 7a and b, we compare the electron density distributions calculated by the QET model based on Fermi-Dirac statistics for 20 nm Si, Ge, and In 0.53 Ga 0.47 As double-gate n-MOSFETs, respectively. The double-gate structures having a film thickness of 8 nm are simulated. The results are plot- ted at the source and drain ends of the channel. The devices are simulated at V g = 0.8 V and V d = 0.8 V. The inversion layer electrons in Ge and In 0.53 Ga 0.47 As n-MOSFETs spread into the center of the channel due to the low effective mass and high permittivity. In analogy to the results of bulk n-MOSFETs, the single inversion layer electrons in the In 0.53 Ga 0.47 As n-MOSFET are further decreased due to the high degeneracy material. In Fig. 7a, Si and Ge n-MOSFETs exhibit two inversion layers at the source end of the channel. In the film thickness of 8 nm, Ge n-MOSFETs form a single inversion layer due to the hot electron effects at the drain end of the channel as shown in Fig. 7b.  of the Si n-MOSFET due to smaller electron effective mass and higher permittivity. It is shown in Fig. 8b that the DIBL of In 0.53 Ga 0.47 As n-MOSFET is suppressed because of the low S/D doping concentration. Fig. 9a and b shows the dependence of SS and DIBL on the channel length for Si, Ge, and In 0.53 Ga 0.47 As double-gate n-MOSFETs. The short channel effects are suppressed in the multi-gate structure. A different improvement between SS and DIBL is seen. In all devices, the DIBL effect is reduced in double-gate n-MOSFETs. In the In 0.53 Ga 0.47 As n-MOSFET, the DIBL effect is significantly reduced due to the low S/D doping concentration. This is because the thin film suppresses an extension of drain electric field into the channel. The SS improvement depends on the channel material even in double gate structure. The SS of Ge and In 0.53 Ga 0.47 As n-MOSFETs are larger than that of Si n-MOSFET due to the low effective mass and high permittivity as well as the results of bulk n-MOSFETs. both bulk and double-gate n-MOSFETs are shown. The V T roll-off of the In 0.53 Ga 0.47 As n-MOSFET is almost the same as that of Si n-MOSFET in the double gate structure because of the low S/D doping concentration. The Ge n-MOSFET shows the worst short channel effects.

Conclusion
The quantum confinement and short channel effects of Si, Ge, and In 0.53 Ga 0.47 As n-MOSFETs have been evaluated using a 4-moments QET model based on Fermi-Dirac statistics and nonparabolicity. The dependence of quantum confinement effects on channel materials has been clarified. The charge control by the gate is reduced in Ge and In 0.53 Ga 0.47 As n-MOSFETs due to the low effective mass and high permittivity. This results in the degradation of short channel effects. The double-gate structure is effective in the suppression of DIBL for all channel materials. The SS degradation depends on channel materials even in double-gate structure.