Semiconductors/MESFET Transistors

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MESFET Operation[edit]

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).

\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:

I_n(x)=\sigma \xi\cdot W\cdot b(x)
I_n(x)=-\sigma \frac{dV_{gc}(x)}{dx}W(a-W_n(x))




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)}
I_n= \frac{\sigma aW2qN_d}{2\varepsilon_0\varepsilon_rL}\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_rL}
I_n= \frac{2\sigma aWqN_d}{2\varepsilon_0\varepsilon_rL}
I_n= \frac{2\sigma aWqN_da^2}{6L\cdot

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

\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.



I_n= W\cdot \frac{\sigma a\cdot W_{00}}{3L}
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 [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[edit]

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

General power law:[edit]

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


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.

\beta(V_{gs}-V_{to})^Q & V_{gs}>V_{gd}


[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.