Assume an N channel MESFET with uniform doping and sharp depletion
region shown in figure 1.
The depletion region
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
.
![{\displaystyle W_{n}(x)={\sqrt {\frac {2\varepsilon _{0}\varepsilon _{r}(\Psi -V_{gc}(x))}{qN_{d}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d732cd7f60d97d1390ff21f469a343a5fed30f8)
![{\displaystyle W_{n}(x)^{2}={\frac {2\varepsilon _{0}\varepsilon _{r}(\Psi -V_{gc}(x))}{qN_{d}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c469ceaea6f9421795a8506ea208dea04147edff)
![{\displaystyle {\frac {W_{n}(x)^{2}qN_{d}}{2\varepsilon _{0}\varepsilon _{r}}}=\Psi -V_{gc}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2eb01d1f29c71a231a34db14c9d63d39b08a8c5f)
![{\displaystyle V_{gc}(x)=\Psi -{\frac {W_{n}(x)^{2}qN_{d}}{2\varepsilon _{0}\varepsilon _{r}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb121cada209e27fdfd5362c5f9a02f892fa0bc4)
(1)
The current density in the channel is given by:
![{\displaystyle J_{n}=\sigma \xi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/922f57611176ad53c7edd3bfbd850c5456c7b4ef)
![{\displaystyle I_{n}(x)=\sigma \xi \cdot W\cdot b(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cef101476fc1f6c5515380d706616dcaeaa81e0)
![{\displaystyle I_{n}(x)=-\sigma {\frac {dV_{gc}(x)}{dx}}W(a-W_{n}(x))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0c9a00531efb9be232abb622af8aa9243c74ffa)
where:
![{\displaystyle \xi =-{\frac {dV_{gc}(x)}{dx}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4554053af085533eca6a939d3dc64942b1c1ed2)
Therefore,
![{\displaystyle I_{n}(x)=-\sigma aW{\bigg (}1-{\frac {W_{n}(x)}{a}}{\bigg )}{\frac {dV_{gc}(x)}{dWn(x)}}{\frac {dWn(x)}{dx}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99d65137d4cb1b7fcf3633ac0578dec7cde4d964)
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/321f0e205a05428424456492a49aea348a273483)
![{\displaystyle 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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/136d6801bedd8e2b229e75216f531a302d63e58e)
Substituting from equation 1:
![{\displaystyle 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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48aa71613cdb59ba2ff376ecc935eeaeb422ea59)
![{\displaystyle 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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d94e4a0bc281534876a20ec3d3fcd126fb036278)
![{\displaystyle 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)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4210e417c8fa374b5aec036d232566c15881d378)
![{\displaystyle 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 ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f75c8a3fbad1eb8f46d8050cd9990a7a73b901d)
![{\displaystyle 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 ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08eb7e4698c89a0a5a1d7d66f46648c99f6705c7)
One defines constant Β as the channel conductance with no
depletion. And the work function to deplete the channel
W00 [1]:
![{\displaystyle W_{00}=\Psi -V_{to}={\frac {qN_{d}a^{2}}{2\varepsilon _{0}\varepsilon _{r}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e71c77cc9f3b96cb276a85c7d80bb9ed61d95552)
![{\displaystyle \beta ={\frac {\sigma a}{3LW_{00}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfa7110a50c008a1ad5393080d05f2fe48614cc9)
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.
![{\displaystyle 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}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c440eda6c1ab8d6e8ff08cd7964dfc471220e87)
![{\displaystyle 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}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdf082ff8a27004a2520d9224ef57e94c58a333a)
Substituting:
![{\displaystyle I_{n}=W\cdot {\frac {\sigma a\cdot W_{00}}{3L}}{\big [}3(d^{2}-s^{2})-2(d^{3}-s^{3}){\big ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/417a332a4b763ae7926b36e06a51998a61a1ab39)
(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:
![{\displaystyle I_{sat}=\beta W_{00}^{2}(1-3s^{2}+2s^{3})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c4339c7e8ed46ced3de110a3419e49c48797227)
![{\displaystyle g_{m}=3\beta W_{00}(s-1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3132afc06426ce8ad60f9b481239b18ce1347605)
![{\displaystyle G_{DS}=3\beta W_{00}(1-d)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac2133aa979bd3806d6d4557ca55a61a2b5ff559)
![{\displaystyle 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 ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c576c7ae1fc1a5cfe558fbc8e77dec2ff6c4d502)
![{\displaystyle g_{m}=3\beta W_{00}(V_{gs}-V_{to})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/861c5a9e25802b81bbaf2220ceeba09a4fa71934)
![{\displaystyle G_{ds}=3\beta W_{00}(V_{gd}-V_{to})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/676d5cf6eb37598dadf02a9431e5d74e80df19b1)
It was found that a general power law provided a better fit for real devices [3].
![{\displaystyle I_{ds}=\beta {\big [}(V_{gs}-V_{to})^{Q}-(V_{gd}-V_{to})^{Q}{\big ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd8f44444879cc70985d90cbe8f8bd6f46a8d59)
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
![{\displaystyle \beta {\mbox{ proportial to }}{\frac {\sigma aW}{3LW_{00}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66a7eb67e2f1cb5c2d5966471ed9c1cdd42989e5)
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.
![{\displaystyle I_{ds}=\left\{{\begin{matrix}0&V_{gs}<V_{to}\\\beta {\big [}(V_{gs}-V_{to})^{Q}-(V_{gd}-V_{to})^{Q}{\big ]}&V_{gs}\leq V_{gd}\\\beta (V_{gs}-V_{to})^{Q}&V_{gs}>V_{gd}\end{matrix}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16f2b5a61ba07035e0eb13724bb5b217ab97a8ec)
[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.