Quantum Confinement in Si-channel Nanowire MOSFETs

Saurabh Sant SemiVi LLC Zurich, Switzerland. saurabh.sant@semivi.ch

Abstract Gate-all-around nanowire MOSFETs exhibit excellent gate control of the channel. However, quantum confinement in Silicon channel moves peak electron concentration away from oxide/Silicon interface. This reduces on-state current of the MOSFETs. Quantum effects are not readily taken into account in the drift-diffusion framework. Various models have been developed to include these effects in the framework. SemiVi drift-diffusion simulator models quantum confinement effects using Van-Dort, modified local density approximation, and density gradient models.

Index Terms Nanowire MOSFETs, Silicon, quantum confinement, mobility degradation.

I. Introduction

Silicon channel FinFET scaling has reached its limit. Further scaling can be achieved by changing the device geometry from FinFET to a Gate-all-around (GAA) nanowire FET. In this work, one such GAA-FET is studied by TCAD simulations using SemiVi drift-diffusion solver. Quantum confinement in the channel moves peak electron (hole) concentration in n-FET (p-FET) away from oxide/Silicon interface. This degrades gate control of the channel and also reduces On-current. This work models quantum confinement using three different models available in the literature - Van Dort model, Modified Local Density Approximation (MLDA) model, and Density Gradient method.

II. Modeling quantum confinement

A. Van Dort model

In this model, quantum potential is defined as follows.

(1)

where ϵ_r relative permittivity of the semiconductor, ϵ₀ is permittivity of free space, kT thermal voltage, F_⊥ is local electric field normal to the nearest oxide/semiconductor interface. The parameters β and F_crit are calibration parameters. f(||) is a function of the distance of the vertex from the interface, given by.

(2)

B. MLDA model

In this model, quantum potential is defined at each vertex in the region as follows.

(3)

Here, d_⊥ is the distance of the vertex from nearest insulator/semiconductor interface and λ_n∕p is thermal wavelength given by,

(4)

Here, m_n∕p is the quantum confinement mass which is taken as a fitting parameter.

C. Density Gradient Equation

In this approach, quantum potential is defined by a function of the gradient of carrier density, hence the name density gradient. In this approach, quantum potential is defined as,

Λn	= -	(5)
	= -	(6)

Here, m_n∕p is the quantization effective mass, m_e is free electron mass, n is electron density. In second part of Eq. 5, the quantized effective mass is replaced by a fitting parameter γ.

Defining electron density with Boltzmann distribution (n = n_i exp), Eq. 5 can be reformulated as follows.

(7)

where u = . Notice, that the solution term Λ_n exists on both sides of Eq. 7. Additionally, RHS of Eq. 7 depends on logarithm of electron density n, whereas n depends on Λ_n. Hence, the above equation must be solved together with Poisson and carrier continuity equation.

III. Simulation setup

A. Device structure

The Si nanowire MOSFETs simulated in this work were fabricated using a CMOS-compatible self-aligned replacement-metal-gate process. The fabrication process is described in  [?] and  [?]. The process is designed to fabricate ultra-thin-body MOSFETs with variable widths. Here, we have selected the MOSFETs with 500nm width for the simulation. Since the MOSFET structure is invariant along the width, 2D cross-section of the MOSFET with LG = 13nm, 100nm, and 300nm are generated and meshed by using SemiVi structure generator and mesher  [1]. The device structure and the doping profiles of LG = 13nm MOSFET are shown in Fig. mbox III-Ambox . Channel region of the MOSFET is intrinsically n-doped with doping of 2 × 10¹⁶cm^-3, whereas raised- source and drain regions are heavily n-doped. The gate-oxide is a high-k oxide with an equivalent oxide thickness (EOT) of 8.5Å.

The Si nanowire p- and n-MOSFETs are simulated in this work using cylindrical coordinate system. 2D radial cross-sections of the devices with LG = 300nm are created using SemiVi structure generator and mesher  [1]. These devices are simulated using SemiVi drift-diffusion solver  [2] with the Cylindrical coordinate system. The device structure and the doping profiles of LG = 300nm n-MOSFET are shown in Fig. mbox III-Ambox . Channel-region of both p- and n-MOSFET is n-doped with doping of 2 × 10¹⁶cm^-3, whereas raised-source and drain regions are heavily n- or p-doped in n- or p-MOSFETs, respectively. Gate-oxide is modeled as SiO₂ with an equivalent oxide thickness (EOT) of 8.5Å.

B. Simulation parameters

In this work, bulk electron and hole mobility in Silicon is set to 1450 cm²/Vs and 450 cm²/Vs, respectively. Channel mobility degradation is modeled using “Lombardi model”. Electron and hole lifetimes in the material are set to 1μsec.

Quantum confinement moves electron density away from the GOX/Silicon interface. This effect is modeled in the drift-diffusion simulations using three different models - Van Dort model, modified local density approximation (MLDA), and density gradient model.

IV. Results

Comparison of the simulated transfer characteristics of the nanowire p- and n-MOSFETs with the measurements is shown in Fig. 2(a) and Fig. 2(b), for n- and p-MOSFETs of LG = 300nm, respectively. The figures show that on-state current is reduced when quantum confinement is taken into account. This is due to redistribution of carrier density due to quantum effects. In the on-state, electron (hole) density in nMOSFET (pMOSFET) shifts away from GOX/Si interface. This is shown in Fig. 4. Transfer characteristics are plotted on semilog-y scale in Fig. 3(a) and Fig. 3(b) to highlight the subthreshold region.

(a) n-MOSFET

(b) p-MOSFET

Fig. 2.  Simulated transfer characteristics of Silicon nanowire (a) n-MOSFET, and (b) p-MOSFET with LG =300nm showing the quantum effects modeled by the three different models.

(a) n-MOSFET

(b) p-MOSFET

Fig. 3.  Simulated transfer characteristics of Silicon nanowire (a) n-MOSFET, and (b) p-MOSFET showing the quantum effects in the subthreshold region.

References