Semiconductor devices have been scaled down to the nanoscale region and entered the quasi-ballistic operation mode. Because of the physical limitations of carrier transport, the drift-diffusion model for describing carrier transport becomes less significant in the nanoscale MOSFETs. From the quasi-ballistic toward ballistic region, K. Natori brought up a new insight for modeling the carrier transport phenomenon which is called ballistic theory or backscattering theory in 1994 [33]. Later in 1997, Mark S. Lundstrom developed the complete theorem for explaining the physic of ballistic and quasi-ballistic phenomenon for nanoscale MOSFETs [1]. Besides, in 2002, two experiments for extracting the backscattering coefficient have been reported. One is the current-voltage (I-V) fitting [2] and the other one is temperature dependent method by M. J. Chen [7].
In this decade, the ballistic coefficient of several devices such as bulk-Si, strained-Si, SOI, and etc. have been researched [9]. For SBMOS, the ballistic transport is not easy to explain due to its special carrier transport characteristics we have reported in chapter 3. In this chapter, we applied the mobility view to explain the backscattering phenomenon for SBMOS which has been the first being reported.
4.1 Introduction of Ballistic Theory
In this theory, we treat the moving carriers as the quantum wave. This kind of wave goes through channel from source to drain. When the waves move toward the channel, they bomb into an non-negligible quantum barrier which leads to transmission and reflection in quantum mechanics. In saturation region of MOSFETs, the channel barrier is like the shape of hills; see Fig. 4.1(a). The length l is called critical length or critical distance which means about the energy drops kBT from the top of the channel barrier, where kB is Boltzmann’s constant and T is the temperature in Kelvin coordinates. In the gray area of Fig. 4.1(a), which is called
kBT-layer, carriers pass thermally through the channel barrier and get a lot of scatterings. The scattering process in the channel is due to impurity scattering, lattice vibration and surface roughness. We represent the current flow in the linear region as the following equations [3]:
, , where W is the device width, L is length, υinj is the thermal injection velocity from the maximum of the channel potential, Cox is the oxide capacitance, rlin is the backscattering coefficient in the linear region (VD<kBT/q), VT,lin is the threshold voltage of the linear region of the MOSFET, and λ0 is the mean-free-path in linear region. λ0 is about a few nanometers and is function of the length, gate and drain voltage. While, in the saturation region, we have:
, , where rsat is the backscattering coefficient of the MOSFET in the saturation region, which is also called rc, VT,sat is the threshold voltage in the saturation region, and λ is the mean-free-path.
4.1.1 Derivation of the Backscattering Theory
As aforementioned, we treat the carriers as quantum wave. From the quantum point of view, the carriers transmit or reflect from the barrier. Backscattering rate depends on the shape of the channel barrier. Thus, we illustrate the moving carriers as a carrier flux, which is shown in Fig 4.1(b). The carriers flux incident to the barrier is F, and (1-rc)F is the transmitted flux.
In this situation, rc is a very important parameter which determines the total transmitted flux.
In Fig 4.1(a), the conduction band diagram of conventional MOSFETs is performed, EC1
injected flux from the source to drain, and F－is from drain to source, respectively. We can use the scattering matrix to deduce the backscattering theory [4]. The scattering matrix can be written as where T and T’ represent the fraction of the steady-state right- and left-directed fluxes that transmit across the quantum barrier which are the transmission coefficient of F+ and F－. In equilibrium condition (VD=0), the matrix is symmetrical.
'
T T (4.6) In non-equilibrium condition (VD>0), T’ depends exponential on the barrier encountered lightly. We represent the T’ by TeqV kTD , where T is larger than T0 due to DIBL. The scattering matrix is the following matrix:
1
Thus, the current flow can be described as
0
0
0
where 1R . If the MOSFET operated in the linear region, the operational conditions are TD kT In the source end of the channel, the product of total density of the carrier n(0) and the thermal injection velocity υinj(0) is:
rlin represents R which is the reflection coefficient in the linear region, we have:,
1
,
Besides, by the thermal equilibrium hemi-Maxwellian, the average velocity of source to channel side (υ+) and drain to channel side (υ-) are ,which produces low drain current.On the other hand, the MOSFET operated in the saturation region, the operational conditions are
D kT
V q (4.16) Eq. (4.10) is rewritten and combining with total density of the carrier and thermal injection velocity, i.e.,
(1 )
IDqWF R (4.17a)
0 inj 0
1
n F F R F R (4.17b) From eq. (4.17) and rsat represents R in the saturation region, we have:
,
,
Bsat is the index of the backscattering rate and 0<Bsat<1 and rsat is extracted by eq. (4.4) from [1], which means the reflection probability of the transport carriers.In real devices, we should consider the S/D series resistance and DIBL effect. It will be explained in latter section.
4.1.2 Temperature Dependent Method to Extract Backscattering Coefficient
In the past research, M. J. Chen reported that drain current (ID), thermal injection velocity (υinj), backscattering coefficient (rc) and threshold voltage (VT) are temperature dependent parameters [7]. Therefore, we can apply the devices with both linear and saturation operational condition and extract the backscattering coefficient by difference equation [8]. From eq. (4.18), we apply the log operator on it, and then we differentiate the result by temperature (T). We get
,
where α and η are defined for simplifying the equation, i.e.,
, and λ is the mean-free-path and l is the critical length.
From eq. (4.20) to (4.22) and the experimental result, we can calculate the following parameters in difference equation.