In 2002, J. Guo and Mark S. Lundstrom reported that the backscattering characteristics of SBMOS with different Schottky-barrier height [5]. The results show that the ballistic theory is failed for SBMOS, which is due to most carriers are tunneling through the channel barrier instead of emitting over the barrier. Under this argument, the backscattering theory is suitable for SBMOS just when Schottky-barrier height is negative.
In recent years, some companies demonstrated the better performance of DSI-SBMOS than conventional SBMOS and bulk-Si MOSFETs [25-26]. With dopant segregation implantation (DSI) technique, carriers are easy to tunnel through the thin Schottky barrier in the source end that induced a large drain current. The conduction band edge of the DSI-SBMOS is shown in Fig. 4.4(b), and is compared with conventional MOSFETs; see Fig.
4.4(a). When carriers inject from source to channel, most of them tunnel through the Schottky barrier, which is more than 90% carriers tunneled the barrier in ON-state, and then emit over the channel barrier, as we have mentioned in chapter 3. Under this condition, the channel barrier leads to a backscattering situation for carriers. So, it is important for devices designer to re-examine in the backscattering theory of DSI-SBMOS.
A latest researches show the injection velocity of the DSI-SBMOS [35], but they do not explain the relations between injection velocity and ballistic coefficient. They describe that the injection velocity of SBMOS is larger than conventional MOSFET, where their injection velocity is the carrier average velocity in the channel instead of thermal injection velocity on the source side. In the following sections, we will demonstrate the ballistic coefficient in low field region of the channel barrier with a new observable method and the high field ballistic coefficient in the next chapter.
4.3.1 Effective Ballistic Mobility
Early in 1980s, Michael S. Shur developed the effective ballistic mobility (also called apparent mobility) for evaluating the limitation of high electron mobility transistor (HEMT) [36]. For HEMTs, in general, are composed of AlGaAs and GaAs for attaching high performance. In this structure, there is a barrier with the shape of spike between the AlGaAs/GaAs heterojunction. For SBMOS, there is a barrier between the S/D to channel due to metal-semiconductor junction. This barrier is like heterojunction. Therefore, we can launch the effective ballistic mobility on SBMOS. In addition, somebody used it for researching mobility degradation effect in very short devices. This target is similar to our experiment.
In MOSFETs, carriers propagate in the channel with a randomly oriented thermal velocity (υth) for non-degenerate condition or Fermi velocity (υF) for degenerate condition. In low electric fields, the current is proportional to the electric field and the electron concentration, just like in the collision-dominated case. We illustrate the drift velocity versus electric field plot in Fig. 4.5(b) [15]. The slope in low field region presents the constant mobility (μ). When carriers accelerate with thermal velocity by the field across the channel, the limitation time is L/υth. We defined the effective ballistic mobility as the following equation. For non-degenerate condition, the Fermi integral () is canceled. Then, we get
B *
T
qL
m
(4.37) Eq. (4.37) is also called Shur’s expression. The unidirectional thermal velocity is
2k TB
(4.38)
And, the thermal average speed is Finally, the effective ballistic mobility is expressing as
* where m* is the effective mass, q is basic electron charge, kB is Boltzmann’s constant, T is the temperature and L is the effective length of the devices. This mobility is treated as carriers directly transmit through the channel. In quasi-ballistic situation, carriers will produce a lot of collision which results in different scattering such as impurity, phonon and surface roughness scattering. We applied the Mathiessen’s rule for describing the carriers transport in nanoscale MOSFETs. Thus, the effective mobility is defined as
0
1 1 1
eff B
(4.41) where μeff is the effective mobility we are measured in MOSFETs in low field region and μ0 is the effective mobility in very long channel devices which is scattering- dominate without ballistic condition and is also named μ∞. Usually, we choose the channel length longer than 10μm. However, with channel length scaled down, the effective mobility will decrease drastically. This is due to effective ballistic mobility is decreasing with down-scaling and pulling down the effective mobility. We illustrated the picture with the carriers directly overshoot the channel which contributes ballistic mobility, and other scattering mechanisms present the scattering-dominate mobility, see Fig 4.5(a). Because of easily passing through the channel without collision, the former is very small due to small length (eq. (4.40)) of the device in the nanoscale devices that decrease the mobility. Under this inference, the effective ballistic mobility (μB) becomes the target for predicting device’s ballistic rate.
4.3.2 The Links between Ballistic Theory, Effective Ballistic Mobility and Drift-Diffusion Model
we expect to link the drift-diffusion model with effective ballistic mobility [6].
From eq. (4.30), and the Einstein’s relation,
0 n
B
q D
k T (4.42) Where Dn=υTλ/2 is the diffusion coefficient and kB is Boltzmann’s constant. The long channel mobility is
Combining eq. (4.37) with (4.43), we have
0 Under low drain bias, the ballistic current can be expressed in terms of ballistic mobility,
0 DS Therefore, we treat the effective mobility as0
This equation is the same as eq. (4.41). So, we derived that the effective ballistic mobility is suitable for drift-diffusion model.
Then, recalling the ballistic theory, the eq. (4.30) describes the ballistic coefficient in linear region. Thus, linking the eq. (4.30) and (4.44), the reflection coefficient can be expressed as So, the relations between ballistic theory, effective ballistic mobility and drift-diffusion model are performed.
On the other hand, when the MOSFET operates in the high field, which is related to degeneracy condition, the drain current should add the Fermi integral (n( )F ). This will complicate the extraction method [6]. In order to simplify the extracting process, we focus on the other significant parameter which is velocity component. In the high field region, the significance of saturation velocity is larger than effective mobility, especially short channel devices, which someone called mobility saturation. Thus, effective ballistic mobility is valid only in the low field region. The inferential process will be introduced in the later chapter.
Before our experiment, we have to make sure why we will try to use the effective ballistic mobility to evaluate linear backscattering coefficient. As the barrier variation in Fig.
3.16, the tunneling barrier is the same with different channel length and the channel barrier is the dominant factor of driven current in DSI-SBMOS, further, the measured mobility depends on channel barrier. Hence, the effective ballistic mobility (μB) can extract the channel barrier backscattering coefficient. On the other hand, we cannot use temperature dependent method we have introduced in section 4.1.2. This is due to this method is established on thermionic emission of carriers. But in DSI-SBMOS, thermionic emission is not the dominant transport method. Under these inferences, we practice the effective ballistic mobility on DSI-SBMOS in low field region.