In traditional semiconductor devices, carriers are frequently scattered from phonons, ionized impurities and surface roughness. In the traditional devices, the backscattering mean free path λ is much shorter than the device channel. So, drift-diffusion approach is used to describe the carrier transport. However, as devices downscale to nanometer regime, backscattering mean free path become comparable to transistor dimensions. When the backscattering mean free path becomes much larger than the transistor channel length, scattering can be totally ignored. In this situation, a nano-MOSFET behaves like a vacuum tube. In practical devices, scatterings are unavoidable in semiconductor devices. Therefore, modern devices operate in quasi-ballistic mode which is between drift-diffusion and ballistic regimes. Put in other words, drift-diffusion theory is no longer strictly valid as well as ballistic treatment. Hence, modern device engineer must familiar with both approaches. Then, the nano-MOSFET studied in this paper is applied in implementing logical NOT transistor level circuit     .
2. Theory and Methodology
Silicon (Si) MOSFETs currently operate between the ballistic and diffusive limits; the scattering model provides a conceptual model for transport in this quasi-ballistic regime. In this scattering model, the most important scatterings occur in the low-field region near the beginning of the channel at source side. Carrier scattering in the channel reduces the current and can be described by ballistic efficiency. Scattering model predicts that the drain current is close to the ballistic limit under high drain bias than under low drain bias, and the on-state current in strong inversion is limited by a small portion of the channel near the source, that is the top region of sub-band potential barrier.
Figure 1. Structural DG Nano-MOSFET used in nanoMOS simulation tool.
Table 1. Double gate nano-MOSFET device simulation parameter.
The on-state current of the nano-MOSFET is controlled by a short low-field region close to the source end of the channel. The length l of this area is called critical length which is defined as the distance from the peak of the potential barrier to the point
where the potential reduces by. is a numerical factor ≥1. This factor has a
value of 1 for non-degenerate case and slightly greater than 1 for degenerate case. In this paper, take. is the backscattering mean free path. Then, the backscattering coefficient r is given by
The ballistic efficiency B is given by
where electron mobility at ballistic transport in Silicon is cm2/Vs. The thermal velocity is given by
where and T = 300 K. The critical length is given by
Since lower bound for is used at diffusive transport and upper bound for is used at ballistic transport, is used at quasi-ballistic transport.
In studying the theoretical part of this paper, the following Fermi-Dirac integrals are used:
is the average energy between source and drain in sub-band energy profile whereas is the energy level at the center of the device. Next, the following expression is used to analyze the drain current per micron of width:
After considering the ballistic efficiency B,
is the gate oxide capacitance per unit area
is the average energy between source and drain in sub-band energy profile whereas is the energy level at the region around top of the potential barrier. This region limits on-state current because scatterings mostly occur in this region. In analyzing Equation (10) and Equation (11), the following Fermi-Dirac integrals are used:
The on-line current-voltage (I-V) simulation result of NanoMOS is compared with theoretical calculation using Equation (11).
In order to calculate resistance RLoad of nano-MOSFET at quasi-ballistic limit, uses
Since digital logic gates operate at linear portion of I-V curve. This RLoad is used in analyzing rise time of transistor loaded NOT gate circuit. On the other hand, the following expression is used to obtain on-state channel resistance Rchannel at on-state which is used in fall time analysis.
= electron mobility at ballistic = 1200 cm2/Vs.
= Oxide capacitance per unit area.
Transistor loaded NOT gate as shown in Figure 2 is simulated using WinSpice. The simulated rise time and fall time extracted from timing diagram are compared with theoretical calculated rise time and fall time  -  .
Since the nano-MOSFET operates at quasi-ballistic condition:
From Figure 3,
Figure 2. Transistor loaded NOT gate circuit.
Figure 3. Capacitance models in nano-MOSFET device.
From  , subthreshold swing S = 75 mV/V and drain induced barrier lowering DIBL = 80 mV/dec. So, CG, CS and CD can be calculated.
Total Capacitance of NOT gate = Gate Capacitance + Source Capacitance + Drain Capacitance + Area Capacitance + Sidewall Capacitance.
Rise time constant gate total capacitance.
Rise time, it takes 6.1 times duration to pass logic 1 than logic 0 through an n-channel MOS pass-transistor.
Fall time constant gate total capacitance.
Maximum signal frequency
3. Results and Discussion
Figure 4 shows the energy sub-band profile along the channel for nano-MOSFET studied in this paper. Drain-to-source voltage, VDS lowers the sub-band potential at the drain side by 0.60 eV    .
From Equation (3), the backscattering mean free path is
From Equation (5), the critical length is
From Equation (1), the backscattering coefficient is
From Equation (2), the ballistic efficiency is
Figure 4. The sub-band energy profile along the channel for Nano-MOSFET.
In order to analyze the NanoMOS simulation result of Figure 5, Equation (10) and Equation (11) are needed. Take VDS = 0.60 V.
Then, by using Equation (10),
After considering the ballistic efficiency B and using Equation (11),
Simulated result with NanoMOS, as in Figure 5, has From theoretical calculation of Equation (11), These two results are
87.3% closely matched. In Figure 5, drain current in saturation region is sloping because electron scattering is considered in Figure 5 and at high drain bias, scattering model in nano-MOSFET exhibits drain current closer to the ballistic limit than under low drain bias.
At region above threshold, the Fermi-Dirac integrals in Equation (11) can be simplified to exponential terms as in equation below.
Sub-band potential at drain side is lower by, therefore
Then Equation (19) becomes
Figure 5. Drain current versus drain voltage for nano-MOSFET by NanoMOS simulation.
After analysis, Equation (19) and Equation (20) both has the same value.
To implement transistor level NOT gate circuit as in Figure 2, the nano-MOSFET should operate in the linear region which is the region for digital logic operation. From Figure 5, linear region is from VDS = 0.00 V until 0.20 V. Use Equation (11) to calculate the drain current at this linear region and then apply Equation (17) to calculate RLoad at quasi-ballistic limit. From Equation (11),
In order to calculate the resistance of nano-MOSFET at quasi-ballistic limit, use Equation (17) since digital logic gates operate at linear portion of I-V curve. Using Vth =
0.20 V, and from device dimension W = 125 nm, RLoad = 748.8
Ω. The resistance value is used in analyzing theoretical value of rise time in NOT gate circuit. On the other hand Equation (18) is used to obtain the resistance needed in analyzing theoretical value of fall time in NOT gate circuit. Finally, the NOT gate circuit in Figure 2 is simulated using WinSpice. The timing diagram result are shown in Figure 6(a) and Figure 6(b).
Low output voltage VOL of NOT transistor level circuit in Figure 2 is given by
From WinSpice simulation timing diagram Figure 6(b),
By comparing Equation (21) and Equation (22),
From theoretical modeling and also WinSpice simulation, VOH = 0.4 V. Nano- MOSFET at the bottom is at off state and thereby at high impedance state. Threshold voltage lost 0.20 V occurs at top side nano-MOSFET load which acts as pass transistor.
Table 2 tabulates the result of this investigation. The theoretical and simulated result are almost matched each other.
Modern MOSFET semiconductor devices operate in quasi-ballistic transport. Quasi- ballistic transport is the carrier transport between drift-diffusion and ballistic regimes.
Figure 6. (a). WinSpice input signal with period 8 ps to NOT gate; (b) WinSpice output signal of NOT gate.
Table 2. Theoretical and simulated result comparison table.
Theoretical calculations and simulation results about this transport have been done in this paper and this paper shows that theoretical calculation values and simulation results are closely matched. Logic NOT circuit level has been implemented using nano- MOSFET and correct logical operation has been achieved.