Resumen de Two-Dimensional Analytical Modeling of Tunnel-FETs
Nowadays the MOSFET is the most technologically advanced device of the transistor variety. Over decades the semiconductor industry was improving its performance, not only by scaling but also by e.g. introducing high-k dielectrics, applying strain technologies and recently by using multiple gate structures. During this scaling process, the supply voltage needed to be decreased as well, which lead to an inevitable problem. Due to the thermionic-emission based current transport mechanism of the MOSFET, it has a physically limited subthreshold slope (S) of 60 mV/dec at room temperature. Therefore, a decreasing supply voltage is always resulting in a significant increase in off-current, hence the on/off ratio reduces. In order to solve this problem and enable low voltage operation regimes, a device has to be found which combines a fast switching behavior with a sufficient on-current.
Based on a band-to-band (b2b) current transport mechanism, the Tunnel-FET (TFET) is able to overcome this physical subthreshold slope limitation of 60 mV/dec [1]. Therefore, it has become one of the most promising devices to be the successor of the classical MOSFET in the last few years. Recently fabricated devices show steep switching behaviors with a minimum S of 30 mV/dec (50 mV/dec over 2 decades) and drive currents Ion of 2.2 uA/um @Vd = -0.75 V and Vg = 1 V [1]. These values are achieved because the device technology development has come a long way using hetero-structure nanowire TFETs with high-k dielectrics, suppressing the ambipolar current with gate underlaps and introducing pocket dopings to boost the on-current [1]. Despite the fast improvement of the TFET in all areas, there are still some challenges to meet. The strongest negative influences on the device performance are caused by two physical effects which have their origin in the fabrication process of the device. After the source/drain doping process, the dopants are diffusing into the undoped channel region during the annealing step, thus they leave doping profiles at the junctions of the channel region. These profiles lead to a reduced electrostatic gate influence on the potential at the channel junctions, therefore they aggravate the subthreshold slope and the maximum on-current. The other restriction is the trap-assisted-tunneling (tat) effect. Due to mid-gap traps localized at the channel junctions of the device, carriers are able to perform Fowler-Nordheim tunneling by emitting to mid-gap traps and subsequently tunneling into the channel region. This effect plays a major role particularly in the off-state of the device, where band-to-band tunneling is restrained and the tat current is predominant. Therefore, the minimum off-current fully depends and the subthreshold slope highly depends, on the tat effect. So far basic inverters based on TFETs are published [2] and other small circuits are under investigation. This shows the growing need of compact models for this new kind of device, which can be integrated in circuit simulators.
This thesis describes all necessary steps to analytically calculate the device current of the TFET. The physics-based calculations offer the opportunity for a deeper TFET understanding regarding working principle, versatile effect influences and parameter dependencies. The first step in modeling TFETs belays in capturing all important physical effects that occur within the device and model these as accurate as possible.
The TFET is basically a gate-controlled p-i-n diode. Due to a symmetrical geometry and the possibility to model Fin-FET structures, the model is designed for two-dimensional double-gate devices. For an increased influence of the gate electrodes on the channel region the gate oxide consists of the high-k material HfO2. In an n-conducting device the source region is highly p-doped, the drain area has a slightly reduced n-doping and the channel stays intrinsic. The drain doping reduction is an easy way to suppress the ambipolar behavior of the device. This ambipolarity is caused by the symmetrical structure of the TFET. In the on-state a b2b current at the source/channel junction is dominating, whereby in the off-state the tat current is predominant. In the ambipolar-state a reduced b2b current at the drain/channel junction exists. The reduced drain doping results in an increased tunneling distance at the drain/channel junction, which suppresses the ambipolar-current.
From the modeling point of view there are certain objectives to fulfill in order to describe the current TFET development stage mathematically. The foundation belays in an exact electrostatics solution of the whole device including source and drain region. This is important for modeling hetero-structures and calculating an exact tunneling length. A two-dimensional solution in the channel region is needed in order to capture short-channel effects, which are important considering the current technological capabilities. For the potential extensions in source and drain region a parabola based approach is used.
In the first step of the potential solution the boundary conditions are defined. On the far source and drain side the respective built-in potentials are used. The definition of the channel boundaries is more complex. At the gate electrodes a constant gate voltage is assumed, whereas at the channel junctions a parabolic boundary condition has to be applied. The calculation of this boundary condition is based on an effective built-in potential model [ 3]. The resulted 4-corner structure with mathematical complex boundaries is decomposed into four 2-corner structures as shown in [4]. Two source- and two drain-sided cases for which one has a constant boundary at the channel junction and the other one a parabolic boundary (without the constant part) respectively. The gate boundary is attended to only in the source-sided case with a constant boundary (which has to be subtracted at the channel junction in the drain-sided case for a constant boundary for a proper decomposition).
After the decomposition, the structures are conformally mapped into a plane, where a potential solution can be obtained much easier using the Schwarz-Christoffel transformation [4]. The potential solution in the mapped plane is calculated by using Poisson’s integral with the mapped boundary conditions [4]. Due to the conformal mapping, the potential solution is directly linked to the original plane. With the help of the channel potential and the built-in potentials in source/drain region, parabolic expression for the potential extensions in source and drain region are found [4]. At this point a model is introduced to capture potential differences in the device caused by doping profiles at the channel junctions [5]. This last adjustment completes the device potential solution. Based on this solution the band-structure is evaluated, taking into account doping concentration-based bandgap-narrowing and material depending hetero-structures [4].
Afterwards, the band-to-band tunneling length is calculated for every point in the channel region, where b2b tunneling is possible. For the estimation of the two different tunneling probabilities, a quasi two-dimensional WKB approach is introduced. With the help of the electric field in the channel region an individual triangular barrier is formed for each tunneling position in the channel region, taking into account the different tunneling distances for tat and b2b tunneling [4]. Through applying the WKB approach on these barriers, the tunneling probabilities are calculated [4]. Landauer’s transmission theory is used to calculate the device current for each operation regime [6]. The calculations had to be adjusted for the tat calculation to include the trap concentrations and carrier emission rates. With this step the device current calculation is complete and the model can be compared to TCAD Sentaurus simulation data and measurements.
With a single fitting the model is able to predict an accurate potential solution in the complete channel region and an electric field solution within the channel region. Due to a well fitting band-structure, the current transfer characteristics are modeled accurately. One upside of a physics-based two-dimensional model is the capability to predict the current even for parameter changes with the same fitting. The model capabilities are tested with geometry variations of the channel length, channel thickness and insulator thickness as well as parameter variations in the standard deviation of the doping profiles and different trap concentrations. The obtained results show a good prediction of the simulation data with a few exceptions. Performance enhancements are investigated by reducing the drain doping concentration to suppress the ambipolar-behavior and hetero-junction devices are simulated. Here, the model also shows a good fit with the simulation data. The performance evaluation has shown that a combination of doping profiles and mid-gap traps at the channel junctions are the main performance determining effects within the TFET. The device can be optimized by realizing steep doping profiles with a low trap concentration. Further improvement can be achieved by introducing hetero-junction devices with a three-dimensional nanowire geometry.
The vast improvement of general transistor performance the last few years is mainly due to device miniaturization. Recently fabricated devices already reach channel lengths down to 14 nm. In these dimensions discretization effects gain influence on the device performance. In this thesis the random dopant fluctuation (rdf) influence on the device threshold voltage is investigated and modeled. The general rdf model is able to capture the rdf influence on the gate voltage in TFET devices for a specific operating regime. A for MOSFET devices optimized rdf model is able to predict the rdf-based threshold voltage variation on the potential barrier for various channel lengths, channel doping concentrations and doping profile standard deviations at the channel junctions. By further investigating the rdf dependencies in both devices, the MOSFET current varies less compared to a TFET device. In the MOSFET rdf only has a direct influence on the potential barrier, which limits the device current. In TFETs, however, a variation of the potential near the channel junctions leads to more complex variations in tunneling distance and band-overlap position, and with that, a higher variation in device current. Optimization possibilities are steep doping profiles at the channel junctions, as well as low channel doping concentrations regardless of the device type.
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