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 $V_{gc}(x)$.

$W_n(x)=\sqrt{\frac{2{\epsilon }_0{\epsilon }_r(\mathrm{\Psi }-V_{gc}(x))}{q{N}_d}}$

$W_n(x{)}^2=\frac{2{\epsilon }_0{\epsilon }_r(\mathrm{\Psi }-V_{gc}(x))}{q{N}_d}$

$\frac{W_n(x{)}^{2}q{N}_d}{2{\epsilon }_0{\epsilon }_r}=\mathrm{\Psi }-V_{gc}(x)$

$V_{gc}(x)=\mathrm{\Psi }-\frac{W_n(x{)}^{2}q{N}_d}{2{\epsilon }_0{\epsilon }_r}$

$\frac{d{V}_{gc}(x)}{d{W}_n(x)}=-\frac{2W_n(x)q{N}_d}{2{\epsilon }_0{\epsilon }_r}$ (1)

The current density in the channel is given by:

$J_n=\sigma \xi$

$I_n(x)=\sigma \xi \cdot W\cdot b(x)$

$I_n(x)=-\sigma \frac{d{V}_{gc}(x)}{dx}W(a-W_n(x))$

where:

$\xi =-\frac{d{V}_{gc}(x)}{dx}$

Therefore,

$I_n(x)=-\sigma aW(1-\frac{W_n(x)}{a})\frac{d{V}_{gc}(x)}{dWn(x)}\frac{dWn(x)}{dx}$

$\underset{0}{\overset{L}{\int }}{I}_n(x)dx=\underset{0}{\overset{L}{\int }}-\sigma aW(1-\frac{W_n(x)}{a})\frac{d{V}_{gc}(x)}{d{W}_n(x)}\frac{d{W}_n(x)}{dx}dx$

$I_n\cdot L=-\sigma aW\underset{Wn(0)}{\overset{W_n(L)}{\int }}(1-\frac{W_n(x)}{a})\frac{d{V}_{gc}(x)}{d{W}_n(x)}d{W}_n(x)$

Substituting from equation 1:

$I_n=\frac{-\sigma aW}{L}\underset{W_n(0)}{\overset{W_n(L)}{\int }}(1-\frac{W_n(x)}{a})(-\frac{2W_n(x)q{N}_d}{2{\epsilon }_0{\epsilon }_r})dWn(x)$

$I_n=\frac{\sigma aW2q{N}_d}{2{\epsilon }_0{\epsilon }_{r}L}\underset{W_n(0)}{\overset{W_n(L)}{\int }}(W_n(x)-\frac{W_n(x{)}^2}{a})dWn(x)$

$I_n=\frac{2\sigma aWq{N}_d}{2{\epsilon }_0{\epsilon }_{r}L}[\frac{W_n^2(x)}{2}-\frac{W_n^3(x)}{3a}{]}_{W_n(0)}^{W_n(L)}$

$I_n=\frac{2\sigma aWq{N}_d}{2{\epsilon }_0{\epsilon }_{r}L}[\frac{W_n^2(L)-W_n^2(0)}{2}-\frac{W_n^3(L)-W_n^3(0)}{3a}]$

$I_n=\frac{2\sigma aWq{N}_d{a}^2}{6L\cdot 2{\epsilon }_0{\epsilon }_r}[\frac{3(W_n^2(L)-W_n^2(0))}{a^2}-\frac{2(W_n^3(L)-W_n^3(0))}{a^3}]$

One defines constant Β as the channel conductance with no depletion. And the work function to deplete the channel W₀₀ [1]:

$W_{00}=\mathrm{\Psi }-V_{to}=\frac{q{N}_d{a}^2}{2{\epsilon }_0{\epsilon }_r}$

$\beta =\frac{\sigma a}{3L{W}_{00}}$

We now define V_to, 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.

$d=\frac{W_n(L)}{a}=\frac{\sqrt{\frac{2{\epsilon }_0{\epsilon }_r(\mathrm{\Psi }-V_{gd})}{q{N}_d}}}{\sqrt{\frac{2{\epsilon }_0{\epsilon }_r(\mathrm{\Psi }-V_{to})}{q{N}_d}}}=\sqrt{\frac{\mathrm{\Psi }-V_{gd}}{W_{00}}}$

$s=\frac{W_n(0)}{a}=\frac{\sqrt{\frac{2{\epsilon }_0{\epsilon }_r(\mathrm{\Psi }-V_{gs})}{q{N}_d}}}{\sqrt{\frac{2{\epsilon }_0{\epsilon }_r(\mathrm{\Psi }-V_{to})}{q{N}_d}}}=\sqrt{\frac{\mathrm{\Psi }-V_{gs}}{W_{00}}}$

Substituting:

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

$I_n=W\cdot \beta W_{00}^2[3(d^2-s^2)-2(d^3-s^3)]$ (2)

Equation 2 is Shockley's expression [2] 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

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

$g_m=3\beta W_{00}(V_{gs}-V_{to})$

$G_{ds}=3\beta W_{00}(V_{gd}-V_{to})$

General power law:

It was found that a general power law provided a better fit for real devices [3].

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

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

$\beta \text{ proportial to }\frac{\sigma aW}{3L{W}_{00}}$

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.

$I_{ds}=\{\begin{array}{cc}0& V_{gs}<V_{to}\\ \beta [(V_{gs}-V_{to}{)}^Q-(V_{gd}-V_{to}{)}^Q]& V_{gs}\le V_{gd}\\ \beta (V_{gs}-V_{to}{)}^Q& V_{gs}>V_{gd}\end{array}$

References

[1] A. E. Parker. Design System for Locally Fabricated Gallium Arsenide Digital Integrated Circuits. PhD thesis, Sydney University, 1990.

[2] W. Shockley. A unipolar field-effect transistor. IEEE Trans/ Electron Devices, 20(11):1365–1376, November 1952.

[3] I. Richer and R.D. Middlebrook. Power-law nature of field-effect transistor experimental characteristics. Proc. IEEE, 51(8):1145–1146, August 1963.