# Semiconductors/MESFET Transistors

## MESFET Operation

Assume an N channel MESFET with uniform doping and sharp depletion region shown in figure 1.

The depletion region $W_{n}$  is given by the depletion width for a diode. Where the voltage is the voltage from the gate to the channel, where the channel voltage is given for a position x along the channel as $V_{gc}(x)$ .

$W_{n}(x)={\sqrt {\frac {2\varepsilon _{0}\varepsilon _{r}(\Psi -V_{gc}(x))}{qN_{d}}}}$
$W_{n}(x)^{2}={\frac {2\varepsilon _{0}\varepsilon _{r}(\Psi -V_{gc}(x))}{qN_{d}}}$
${\frac {W_{n}(x)^{2}qN_{d}}{2\varepsilon _{0}\varepsilon _{r}}}=\Psi -V_{gc}(x)$
$V_{gc}(x)=\Psi -{\frac {W_{n}(x)^{2}qN_{d}}{2\varepsilon _{0}\varepsilon _{r}}}$
${\frac {dV_{gc}(x)}{dW_{n}(x)}}=-{\frac {2W_{n}(x)qN_{d}}{2\varepsilon _{0}\varepsilon _{r}}}$  (1)

The current density in the channel is given by:

$J_{n}=\sigma \xi$
$I_{n}(x)=\sigma \xi \cdot W\cdot b(x)$
$I_{n}(x)=-\sigma {\frac {dV_{gc}(x)}{dx}}W(a-W_{n}(x))$

where:

$\xi =-{\frac {dV_{gc}(x)}{dx}}$

Therefore,

$I_{n}(x)=-\sigma aW{\bigg (}1-{\frac {W_{n}(x)}{a}}{\bigg )}{\frac {dV_{gc}(x)}{dWn(x)}}{\frac {dWn(x)}{dx}}$
$\int _{0}^{L}I_{n}(x)\,dx=\int _{0}^{L}-\sigma aW{\bigg (}1-{\frac {W_{n}(x)}{a}}{\bigg )}{\frac {dV_{gc}(x)}{dW_{n}(x)}}{\frac {dW_{n}(x)}{dx}}\,dx$
$I_{n}\cdot L=-\sigma aW\int _{Wn(0)}^{W_{n}(L)}{\bigg (}1-{\frac {W_{n}(x)}{a}}{\bigg )}{\frac {dV_{gc}(x)}{dW_{n}(x)}}\,dW_{n}(x)$

Substituting from equation 1:

$I_{n}={\frac {-\sigma aW}{L}}\int _{W_{n}(0)}^{W_{n}(L)}{\bigg (}1-{\frac {W_{n}(x)}{a}}{\bigg )}{\bigg (}-{\frac {2W_{n}(x)qN_{d}}{2\varepsilon _{0}\varepsilon _{r}}}{\bigg )}\,dWn(x)$
$I_{n}={\frac {\sigma aW2qN_{d}}{2\varepsilon _{0}\varepsilon _{r}L}}\int _{W_{n}(0)}^{W_{n}(L)}{\bigg (}W_{n}(x)-{\frac {W_{n}(x)^{2}}{a}}{\bigg )}\,dWn(x)$
$I_{n}={\frac {2\sigma aWqN_{d}}{2\varepsilon _{0}\varepsilon _{r}L}}{\bigg [}{\frac {W_{n}^{2}(x)}{2}}-{\frac {W_{n}^{3}(x)}{3a}}{\bigg ]}_{W_{n}(0)}^{W_{n}(L)}$
$I_{n}={\frac {2\sigma aWqN_{d}}{2\varepsilon _{0}\varepsilon _{r}L}}{\bigg [}{\frac {W_{n}^{2}(L)-W_{n}^{2}(0)}{2}}-{\frac {W_{n}^{3}(L)-W_{n}^{3}(0)}{3a}}{\bigg ]}$
$I_{n}={\frac {2\sigma aWqN_{d}a^{2}}{6L\cdot 2\varepsilon _{0}\varepsilon _{r}}}{\bigg [}{\frac {3(W_{n}^{2}(L)-W_{n}^{2}(0))}{a^{2}}}-{\frac {2(W_{n}^{3}(L)-W_{n}^{3}(0))}{a^{3}}}{\bigg ]}$

One defines constant Β as the channel conductance with no depletion. And the work function to deplete the channel W00 :

$W_{00}=\Psi -V_{to}={\frac {qN_{d}a^{2}}{2\varepsilon _{0}\varepsilon _{r}}}$
$\beta ={\frac {\sigma a}{3LW_{00}}}$

We now define Vto, the voltage such that the channel is pinched off. d is the ratio of channel depletion to maximum depletion for the drain. s the ratio of channel depletion to maximum depletion for the source.

$d={\frac {W_{n}(L)}{a}}={\frac {\sqrt {\frac {2\varepsilon _{0}\varepsilon _{r}(\Psi -V_{gd})}{qN_{d}}}}{\sqrt {\frac {2\varepsilon _{0}\varepsilon _{r}(\Psi -V_{to})}{qN_{d}}}}}={\sqrt {\frac {\Psi -V_{gd}}{W_{00}}}}$
$s={\frac {W_{n}(0)}{a}}={\frac {\sqrt {\frac {2\varepsilon _{0}\varepsilon _{r}(\Psi -V_{gs})}{qN_{d}}}}{\sqrt {\frac {2\varepsilon _{0}\varepsilon _{r}(\Psi -V_{to})}{qN_{d}}}}}={\sqrt {\frac {\Psi -V_{gs}}{W_{00}}}}$

Substituting:

$I_{n}=W\cdot {\frac {\sigma a\cdot W_{00}}{3L}}{\big [}3(d^{2}-s^{2})-2(d^{3}-s^{3}){\big ]}$
$I_{n}=W\cdot \beta W_{00}^{2}{\big [}3(d^{2}-s^{2})-2(d^{3}-s^{3}){\big ]}$  (2)

Equation 2 is Shockley's expression  for drain current in the linear region. When the device enters saturation, one end is pinched off(normally the drain). Thus $d=1$ and one may derive the equation for the saturation region:

$I_{sat}=\beta W_{00}^{2}(1-3s^{2}+2s^{3})$
$g_{m}=3\beta W_{00}(s-1)$
$G_{DS}=3\beta W_{00}(1-d)$

## Simpler Model

$I_{ds}={\frac {3}{2}}\beta W_{00}^{2}{\bigg [}{\frac {(V_{gs}-v_{to})^{2}}{W_{00}^{2}}}-{\frac {(V_{gd}-v_{to})^{2}}{W_{00}^{2}}}{\bigg ]}$
$g_{m}=3\beta W_{00}(V_{gs}-V_{to})$
$G_{ds}=3\beta W_{00}(V_{gd}-V_{to})$

### General power law:

It was found that a general power law provided a better fit for real devices .

$I_{ds}=\beta {\big [}(V_{gs}-V_{to})^{Q}-(V_{gd}-V_{to})^{Q}{\big ]}$

Where Q is dependent on the doping profile and a good fit is usually obtained for Q between 1.5 and 3. A general power law is approximately equal to Shockley's equation for Q = 2.4. Β is also empirically chosen and is proportion to the previous Β

$\beta {\mbox{ proportial to }}{\frac {\sigma aW}{3LW_{00}}}$

Modelling the various regions is done though model binning. This however infers that a sharp transition exists from one region to another, which may not be accurate.

$I_{ds}=\left\{{\begin{matrix}0&V_{gs}V_{gd}\end{matrix}}\right.$