NEGF Based Analytical Modeling of Advanced MOSFETs
- Autores: Fabian Hosenfeld
- Directores de la Tesis: François Lime (dir. tes.), Benjamín Iñiguez Nicolau (dir. tes.), Alexander Gunther Klös (dir. tes.)
- Lectura: En la Universitat Rovira i Virgili ( España ) en 2017
- Idioma: español
- Tribunal Calificador de la Tesis: David Jimenez Jimenez (presid.), Antonio Ramon Lázaro Guillén (secret.), Viktor Sverdlov (voc.)
- Programa de doctorado: Programa de Doctorado en Tecnologías para Nanosistemas, Bioingeniería y Energía por la Universidad Rovira i Virgili
- Materias:
- Enlaces
- Tesis en acceso abierto en: hdl.handle.net TDX
- Resumen
Nowadays, the structure sizes of the metal-oxide-semiconductor field-effect transistor (MOSFET) is reduced below 20nm. The miniaturization of this central switching element is forced by the demand for a higher integration density in microelectronics. To overcome the associated short-channel effects, such as the increase in leakage currents and the deterioration of the switching behavior, an introduction of new device structures is required. MOS transistors with multiple gate electrodes (multiple-gate FETs) are of increasing interest worldwide, because they allow a significant reduction of short-channel effects [1]. The further development of complex microelectronic systems requires extensive simulations in the design process. The improvement of new device structures goes hand in hand with the adaption of numerically efficient model equations that analytically describe the device behavior. More complex equations lead to time-consuming simulations and thus have to be minimized. Model equations must not only be time-efficient, but also have to generate stable results for a large number of parameter variations. This degree of freedom allows to simulate the variability of process fluctuations in advance.
An accurate modeling of multiple-gate transistors requires a 2D or even 3D calculation of the electrostatics, especially within the channel area. The continuous shrinking of the geometry leads to the ultra-scaled transistors which are increasingly influenced by quantum effects. Consequently, it is compulsory to consider these quantum effects in the modeling approach [2]. The creation of numerically efficient model equations require closed-form solutions of the physical equations. In order to do so, it is necessary to avoid iterative solutions and make use of approximations. These fundamental requirements complicate the development process of compact models.
There are two types of transistors that are in the focus of the circuit designers. The MOSFET, which is commonly used in circuit design and the tunneling field-effect transistor (TFET) which is traded as the successor of the MOSFET. The current goal in the development of MOSFETs is to shrink the channel length below 10 nm. The source-to-drain tunneling current, which is a quantum mechanical effect, makes it more difficult to achieve the milestone of 10 nm [2]. This effect enables electrons to tunnel through the energy barrier between source and drain and leads to an increase of the leakage current. In order to calculate this effect in a physically based way, it is necessary to include quantum mechanical equations in a compact model.
The TFET is predicted to have a successful future because the device combines two advantages, the CMOS compatibility and a better subthreshold slope than the MOSFET. The device is therefore a steep slope device and consequently allows a supply voltage reduction as well as a reduction of the leakage current. These benefits are particularly important for the development of mobile always-on devices. The steep subthreshold slope is achieved due to the carrier transport caused by the band-to-band (b2b) tunneling current, which is also a quantum mechanical effect.
In 1925, Erwin Schrödinger developed a method to describe the changes over time of a physical system which is influenced by quantum mechanics. Based on these results, the non-equilibrium Green’s function (NEGF) was developed. The method is used to simulate novel devices in a full quantum mechanical treatment [4]. These numerical simulations concern resonant tunneling diodes, silicon nano-wire transistors, TFETs and carbon nanotubes. Based on this formalism, a compact model for double-gate (DG) MOSFETs is derived, which combines classical potential calculation with a quantum-based current calculation. Afterwards, the developed model is applied to derive an analytical TFET model.
The closed-form analytical potential model for a DG MOSFET is composed of three separately calculated components [5]. The first and most significant component consists of a closed-form solution of the 2D Laplace equation in the 4-corner channel region. The boundary conditions for the channel area are separated into constant, linear and parabolically shaped potentials. A further simplification of the potential calculation is done by separating the channel area from a 4-corner problem into two single 2-corner problems. The resulting source and drain related case of the channel area allows to make use of the Schwarz-Christoffel transformation, which maps the device in a less complex plane. Within this plane the potential is calculated in a closed-form. So far, the Laplace equation neglects channel charges and consequently is only accurate in the off-state [6]. In order to extend the potential model to above threshold operation the mobile charge density is estimated using a closed-form approach. Until now, the focus has been on the potential within the channel area, because this potential has a significant influence on the device current. However, the tunneling current from both the MOSFET and the TFET depends on the tunneling length. Therefore, it is necessary to consider additionally the potential shape in the source and drain regions. This third component continuously extends the potential solution into the source and drain regions by a parabolically shaped quasi-2D approach. Confinement from gate to gate, which occurs at a channel thickness below 5 nm and strongly influences the device behavior, is considered by an adaption of the flat-band voltage. Due to the accurate potential calculation it is possible to calculate the current without iterations by using the NEGF formalism.
Based on the 2D potential model, 1D profiles of the conduction band are calculated for slices along the x-axis. Each slice is used as an individual energy profile at a specific y-position. The 1D NEGF formalism, based on the effective mass approximation and the energy profile, is applied to calculate the resulting ballistic current density. The open boundary conditions connect the device with the contacts. The contacts inject states, which are described by wave mechanics, into the device. Furthermore, the connection leads to a level broadening of the discrete energy levels into a continuous density of states. Filling up the states by the fermi function, the 1D electron density is obtained. The electron density finally leads to the 1D current density at a specific energy. Since a calculation of the current for each infinitesimal energy and position is far too complex, the computational effort is reduced to a minimum by an interpolation based on some specific energies and geometric positions. The ballistic NEGF formalism is quite accurate for considering ultra-short devices having a channel length below 10 nm but when considering a longer channel length the deviation increases. To adapt the model for longer channel lengths, backscattering effects are considered in a closed-form [7].
By adapting the analytical potential model of the DG MOSFET, it is also suitable for modeling a DG TFET. This mainly includes a change of the source doping which results in a modification of the built-in potential. Conventional TFET compact models calculate the tunneling current using the Wentzel-Kramers-Brillouin (WKB) approximation [8]. The WKB method applies small triangular profiles to approximate the tunneling barrier. With shrinking dimensions the approximation of the potential profile is less accurate than considering the origin potential profile. Since the b2b tunneling current is a quantum mechanical effect, the current can also be calculated with the help of the NEGF formalism [9]. The formalism avoids the approximations of triangular potential profiles and allows to use the original potential profile. The model introduces the physics-based calculation of the b2b tunneling current by using the device’s band structure. In order to avoid the computational burden of a two band structure, the b2b tunneling is mapped into direct tunneling within a single band. This is achieved by merging the conduction- and valence bands to a quasi-conduction band. Performing this for different energies results in an exact tunneling distance and an accurate barrier shape. Once again, the computational effort is reduced by calculating the tunneling current at certain energies. The subsequently interpolation by using an analytical function leads to a time-efficient current calculation.
The conduction band of the DG MOSFET was verified with the numerical nanoMOS TCAD [10] simulation data for a channel length from 6 nm to 30 nm at the channel’s center and surface position. The band structure of the DG TFET was compared to numerical TCAD Sentaurus [11] data at the channel surface for a channel length of 22 nm. An accurate potential calculation could be proven for both devices. The next verification step focuses on the current calculation. The ballistic device current of the analytical DG MOSFET model were compared to those received by the fare more time-consuming numerical nanoMOS TCAD simulator. It was pointed out that the model correctly computes the source-to-drain tunneling current and is consequently accurate to a minimum channel length down to 6 nm. The calculation of backscattering makes the model equally accurate for a device with a channel length of up to 30 nm. The model also allows to perform physics-based investigations such as the appearance of subthreshold slope degradation and leakage current influences of ultra-scaled devices. By taking quantization effects within the channel area into account, the model is verified for a channel thickness from 2 nm to 5 nm.
The analytical DG TFET model was able to correctly predict the b2b tunneling current because the quantum mechanical charge transport is solved by using the NEGF formalism. The transfer and output current characteristics were compared to numerical TCAD Sentaurus simulation data and showed accurate results of the device’s on-state.
References:
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