NMOS Transistor

A Metal-Oxide-Semiconductor Field-Effect Transistor(MOSFET) is a four terminal device whose terminals are named as Gate(G), Drain(D), Source(S) and Bulk(B).

A cross-sectional view of n-channel enhancement mode transistor is shown in Figure 1.

In an n-channel enhancement-mode device, a conductive channel does not exist naturally within the transistor, and a positive gate-to-source voltage is necessary to create one. The positive voltage attracts free-floating electrons within the body towards the gate, forming a conductive channel. But first, enough electrons must be attracted near the gate to counter the dopant ions added to the body of the FET; this forms a region free of mobile carriers called a depletion region, and the voltage at which this occurs is referred to as the threshold voltage of the FET. Further gate-to-source voltage increase will attract even more electrons towards the gate which are able to create a conductive channel from source to drain; this process is called inversion.

From square law model of an n-channel MOS transistor, drain to source current is given by

$\begin{eqnarray*} I_{D} = \begin{cases} 0 & V_{GS} \leq V_{T} \text{, Cut-off } \\ \frac{1}{2} \mu_n C_{ox} \frac{W}{L} (V_{GS}-V_{T})^2 & \text{ }V_{DS} \geq V_{GS}-V_{T} \text{, Saturation } \\ \mu_n C_{ox} \frac{W}{L} [(V_{GS}-V_{T})V_{DS}-{V_{DS}^2 \over 2}] & \text{ }V_{DS} \leq V_{GS}-V_{T} \text{, Triode region}\end{cases} \end{eqnarray*}$

Assumptions:

Carrier velocity is proportional to lateral electric field in the channel

Transconductance of an n-channel MOS trasistor operating in saturation region is given by

$\begin{eqnarray*}g_m=\frac{\partial{I_{\tiny D}}}{\partial{V_{\tiny GS}}} = \begin{cases} \mu_n C_{ox} \frac{W}{L} (V_{\tiny GS}-V_{\tiny T})\\ \frac{2I_{\tiny D}}{(V_{\tiny GS}-V_{\tiny T})}\\ \sqrt{2 {I_{\tiny D}} \mu_n C_{ox} \frac{W}{L}}\end{cases} \end{eqnarray*}$

Secondary effects

Body effect

The threshold voltage of a transistor depends on source to bulk voltage and is given by

$\begin{equation*} V_{T} = V_{T0} + \gamma(\sqrt{2\Phi_F + V_{SB}} - \sqrt{2\Phi_F}) \end{equation*}$

where,
$\gamma$ : body-effect coefficient
$V_{TO}$ : flatband voltage without substrate bias, or threshold voltage at the $V_{SB}=0$
$\Phi_F$ : Fermi potential, $\Phi_F = {kT \over q} \ln{N_A \over n_i}$
$k$ : Boltzmann contant
$T$ : absolute temperature in kelvin
$q$ : charge of electron
At $V_{SB}=0$ , $V_T = V_{T0}$ . With increasing $V_{SB}$ , $V_T$ also increases.

Channel length modulation

When $V_{ds}>V_{gs}-V_T$ , the channel is pinched off and the effective length of the channel is reduced. In square law model this effect is accounted through $\lambda$ in saturation region.

(1) $\begin{equation*} I_d = \frac{1}{2} \mu_n C_{ox} \frac{W}{L} (V_{GS}-V_{\tiny T})^2 (1+\lambda V_{DS}) \end{equation*}$

Mobility degradation

Drain current depends on mobility of the surface carriers. In short channel devices the electron or hole mobility ( $\mu_n$ or $\mu_p$ ) is constant at very low field strengths, and are reduced with the vertical or transverse component of the electric field. This reduces the drain current even at moderate electric fields.

An empirical relation

(2) $\begin{equation*} \mu = {\mu_o \over {1 + \eta (V_{GS}-V_T)}} \end{equation*}$

$\mu_o$ : carrier mobility at low electric fields
$\eta$ : constant

Eq.(2) indicates that mobility decreases with increasing $V_{GS}$

Velocity saturation

The carrier velocity( $v$ ) is proportional to lateral electric field only under low field conditions. The average field of the channel is given by $E = {V_{DS}\over L}$

Technology scaling scales down channel length, and increases the field across the channel. Increasing the field beyond critical field ( $E_c$ ) does not help in increasing the velocity of the carrier, and said to be as velocity saturation of the carriers.

(3) $\begin{eqnarray*} v = \frac{\mu_o E }{1+\frac{E}{E_c}} \begin{cases} \text{If } E << E_c, v=\mu_o E \text{, linear dependence} \\ \text{If } E >> E_c, v = v_{sat} \approx \mu_o E_c \text{, velocity saturation} \end{cases} \end{eqnarray*}$

References

Principles of Semiconductor Devices

Analog/RF IntgCkts

A RFIC designer's notes