[<< wikibooks] 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
          
        
      
    
    {\displaystyle 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
          
            g
            c
          
        
        (
        x
        )
      
    
    {\displaystyle V_{gc}(x)}
  .

  
    
      
        
          W
          
            n
          
        
        (
        x
        )
        =
        
          
            
              
                2
                
                  ε
                  
                    0
                  
                
                
                  ε
                  
                    r
                  
                
                (
                Ψ
                −
                
                  V
                  
                    g
                    c
                  
                
                (
                x
                )
                )
              
              
                q
                
                  N
                  
                    d
                  
                
              
            
          
        
      
    
    {\displaystyle W_{n}(x)={\sqrt {\frac {2\varepsilon _{0}\varepsilon _{r}(\Psi -V_{gc}(x))}{qN_{d}}}}}
  
  
    
      
        
          W
          
            n
          
        
        (
        x
        
          )
          
            2
          
        
        =
        
          
            
              2
              
                ε
                
                  0
                
              
              
                ε
                
                  r
                
              
              (
              Ψ
              −
              
                V
                
                  g
                  c
                
              
              (
              x
              )
              )
            
            
              q
              
                N
                
                  d
                
              
            
          
        
      
    
    {\displaystyle W_{n}(x)^{2}={\frac {2\varepsilon _{0}\varepsilon _{r}(\Psi -V_{gc}(x))}{qN_{d}}}}
  
  
    
      
        
          
            
              
                W
                
                  n
                
              
              (
              x
              
                )
                
                  2
                
              
              q
              
                N
                
                  d
                
              
            
            
              2
              
                ε
                
                  0
                
              
              
                ε
                
                  r
                
              
            
          
        
        =
        Ψ
        −
        
          V
          
            g
            c
          
        
        (
        x
        )
      
    
    {\displaystyle {\frac {W_{n}(x)^{2}qN_{d}}{2\varepsilon _{0}\varepsilon _{r}}}=\Psi -V_{gc}(x)}
  
  
    
      
        
          V
          
            g
            c
          
        
        (
        x
        )
        =
        Ψ
        −
        
          
            
              
                W
                
                  n
                
              
              (
              x
              
                )
                
                  2
                
              
              q
              
                N
                
                  d
                
              
            
            
              2
              
                ε
                
                  0
                
              
              
                ε
                
                  r
                
              
            
          
        
      
    
    {\displaystyle V_{gc}(x)=\Psi -{\frac {W_{n}(x)^{2}qN_{d}}{2\varepsilon _{0}\varepsilon _{r}}}}
  
  
    
      
        
          
            
              d
              
                V
                
                  g
                  c
                
              
              (
              x
              )
            
            
              d
              
                W
                
                  n
                
              
              (
              x
              )
            
          
        
        =
        −
        
          
            
              2
              
                W
                
                  n
                
              
              (
              x
              )
              q
              
                N
                
                  d
                
              
            
            
              2
              
                ε
                
                  0
                
              
              
                ε
                
                  r
                
              
            
          
        
      
    
    {\displaystyle {\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
          
        
        =
        σ
        ξ
      
    
    {\displaystyle J_{n}=\sigma \xi }
  
  
    
      
        
          I
          
            n
          
        
        (
        x
        )
        =
        σ
        ξ
        ⋅
        W
        ⋅
        b
        (
        x
        )
      
    
    {\displaystyle I_{n}(x)=\sigma \xi \cdot W\cdot b(x)}
  
  
    
      
        
          I
          
            n
          
        
        (
        x
        )
        =
        −
        σ
        
          
            
              d
              
                V
                
                  g
                  c
                
              
              (
              x
              )
            
            
              d
              x
            
          
        
        W
        (
        a
        −
        
          W
          
            n
          
        
        (
        x
        )
        )
      
    
    {\displaystyle I_{n}(x)=-\sigma {\frac {dV_{gc}(x)}{dx}}W(a-W_{n}(x))}
  where:

  
    
      
        ξ
        =
        −
        
          
            
              d
              
                V
                
                  g
                  c
                
              
              (
              x
              )
            
            
              d
              x
            
          
        
      
    
    {\displaystyle \xi =-{\frac {dV_{gc}(x)}{dx}}}
  Therefore,

  
    
      
        
          I
          
            n
          
        
        (
        x
        )
        =
        −
        σ
        a
        W
        
          
            (
          
        
        1
        −
        
          
            
              
                W
                
                  n
                
              
              (
              x
              )
            
            a
          
        
        
          
            )
          
        
        
          
            
              d
              
                V
                
                  g
                  c
                
              
              (
              x
              )
            
            
              d
              W
              n
              (
              x
              )
            
          
        
        
          
            
              d
              W
              n
              (
              x
              )
            
            
              d
              x
            
          
        
      
    
    {\displaystyle I_{n}(x)=-\sigma aW{\bigg (}1-{\frac {W_{n}(x)}{a}}{\bigg )}{\frac {dV_{gc}(x)}{dWn(x)}}{\frac {dWn(x)}{dx}}}
  
  
    
      
        
          ∫
          
            0
          
          
            L
          
        
        
          I
          
            n
          
        
        (
        x
        )
        
        d
        x
        =
        
          ∫
          
            0
          
          
            L
          
        
        −
        σ
        a
        W
        
          
            (
          
        
        1
        −
        
          
            
              
                W
                
                  n
                
              
              (
              x
              )
            
            a
          
        
        
          
            )
          
        
        
          
            
              d
              
                V
                
                  g
                  c
                
              
              (
              x
              )
            
            
              d
              
                W
                
                  n
                
              
              (
              x
              )
            
          
        
        
          
            
              d
              
                W
                
                  n
                
              
              (
              x
              )
            
            
              d
              x
            
          
        
        
        d
        x
      
    
    {\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}
  
  
    
      
        
          I
          
            n
          
        
        ⋅
        L
        =
        −
        σ
        a
        W
        
          ∫
          
            W
            n
            (
            0
            )
          
          
            
              W
              
                n
              
            
            (
            L
            )
          
        
        
          
            (
          
        
        1
        −
        
          
            
              
                W
                
                  n
                
              
              (
              x
              )
            
            a
          
        
        
          
            )
          
        
        
          
            
              d
              
                V
                
                  g
                  c
                
              
              (
              x
              )
            
            
              d
              
                W
                
                  n
                
              
              (
              x
              )
            
          
        
        
        d
        
          W
          
            n
          
        
        (
        x
        )
      
    
    {\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)}
  Substituting from equation 1:

  
    
      
        
          I
          
            n
          
        
        =
        
          
            
              −
              σ
              a
              W
            
            L
          
        
        
          ∫
          
            
              W
              
                n
              
            
            (
            0
            )
          
          
            
              W
              
                n
              
            
            (
            L
            )
          
        
        
          
            (
          
        
        1
        −
        
          
            
              
                W
                
                  n
                
              
              (
              x
              )
            
            a
          
        
        
          
            )
          
        
        
          
            (
          
        
        −
        
          
            
              2
              
                W
                
                  n
                
              
              (
              x
              )
              q
              
                N
                
                  d
                
              
            
            
              2
              
                ε
                
                  0
                
              
              
                ε
                
                  r
                
              
            
          
        
        
          
            )
          
        
        
        d
        W
        n
        (
        x
        )
      
    
    {\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)}
  
  
    
      
        
          I
          
            n
          
        
        =
        
          
            
              σ
              a
              W
              2
              q
              
                N
                
                  d
                
              
            
            
              2
              
                ε
                
                  0
                
              
              
                ε
                
                  r
                
              
              L
            
          
        
        
          ∫
          
            
              W
              
                n
              
            
            (
            0
            )
          
          
            
              W
              
                n
              
            
            (
            L
            )
          
        
        
          
            (
          
        
        
          W
          
            n
          
        
        (
        x
        )
        −
        
          
            
              
                W
                
                  n
                
              
              (
              x
              
                )
                
                  2
                
              
            
            a
          
        
        
          
            )
          
        
        
        d
        W
        n
        (
        x
        )
      
    
    {\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)}
  
  
    
      
        
          I
          
            n
          
        
        =
        
          
            
              2
              σ
              a
              W
              q
              
                N
                
                  d
                
              
            
            
              2
              
                ε
                
                  0
                
              
              
                ε
                
                  r
                
              
              L
            
          
        
        
          
            [
          
        
        
          
            
              
                W
                
                  n
                
                
                  2
                
              
              (
              x
              )
            
            2
          
        
        −
        
          
            
              
                W
                
                  n
                
                
                  3
                
              
              (
              x
              )
            
            
              3
              a
            
          
        
        
          
            
              ]
            
          
          
            
              W
              
                n
              
            
            (
            0
            )
          
          
            
              W
              
                n
              
            
            (
            L
            )
          
        
      
    
    {\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)}}
  
  
    
      
        
          I
          
            n
          
        
        =
        
          
            
              2
              σ
              a
              W
              q
              
                N
                
                  d
                
              
            
            
              2
              
                ε
                
                  0
                
              
              
                ε
                
                  r
                
              
              L
            
          
        
        
          
            [
          
        
        
          
            
              
                W
                
                  n
                
                
                  2
                
              
              (
              L
              )
              −
              
                W
                
                  n
                
                
                  2
                
              
              (
              0
              )
            
            2
          
        
        −
        
          
            
              
                W
                
                  n
                
                
                  3
                
              
              (
              L
              )
              −
              
                W
                
                  n
                
                
                  3
                
              
              (
              0
              )
            
            
              3
              a
            
          
        
        
          
            ]
          
        
      
    
    {\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 ]}}
  
  
    
      
        
          I
          
            n
          
        
        =
        
          
            
              2
              σ
              a
              W
              q
              
                N
                
                  d
                
              
              
                a
                
                  2
                
              
            
            
              6
              L
              ⋅
              2
              
                ε
                
                  0
                
              
              
                ε
                
                  r
                
              
            
          
        
        
          
            [
          
        
        
          
            
              3
              (
              
                W
                
                  n
                
                
                  2
                
              
              (
              L
              )
              −
              
                W
                
                  n
                
                
                  2
                
              
              (
              0
              )
              )
            
            
              a
              
                2
              
            
          
        
        −
        
          
            
              2
              (
              
                W
                
                  n
                
                
                  3
                
              
              (
              L
              )
              −
              
                W
                
                  n
                
                
                  3
                
              
              (
              0
              )
              )
            
            
              a
              
                3
              
            
          
        
        
          
            ]
          
        
      
    
    {\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 ]}}
  One defines constant Β as the channel conductance with no
depletion.  And the work function to deplete the channel
W00 [1]:

  
    
      
        
          W
          
            00
          
        
        =
        Ψ
        −
        
          V
          
            t
            o
          
        
        =
        
          
            
              q
              
                N
                
                  d
                
              
              
                a
                
                  2
                
              
            
            
              2
              
                ε
                
                  0
                
              
              
                ε
                
                  r
                
              
            
          
        
      
    
    {\displaystyle W_{00}=\Psi -V_{to}={\frac {qN_{d}a^{2}}{2\varepsilon _{0}\varepsilon _{r}}}}
  
  
    
      
        β
        =
        
          
            
              σ
              a
            
            
              3
              L
              
                W
                
                  00
                
              
            
          
        
      
    
    {\displaystyle \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
        =
        
          
            
              
                W
                
                  n
                
              
              (
              L
              )
            
            a
          
        
        =
        
          
            
              
                
                  2
                  
                    ε
                    
                      0
                    
                  
                  
                    ε
                    
                      r
                    
                  
                  (
                  Ψ
                  −
                  
                    V
                    
                      g
                      d
                    
                  
                  )
                
                
                  q
                  
                    N
                    
                      d
                    
                  
                
              
            
            
              
                
                  2
                  
                    ε
                    
                      0
                    
                  
                  
                    ε
                    
                      r
                    
                  
                  (
                  Ψ
                  −
                  
                    V
                    
                      t
                      o
                    
                  
                  )
                
                
                  q
                  
                    N
                    
                      d
                    
                  
                
              
            
          
        
        =
        
          
            
              
                Ψ
                −
                
                  V
                  
                    g
                    d
                  
                
              
              
                W
                
                  00
                
              
            
          
        
      
    
    {\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}}}}}
  
  
    
      
        s
        =
        
          
            
              
                W
                
                  n
                
              
              (
              0
              )
            
            a
          
        
        =
        
          
            
              
                
                  2
                  
                    ε
                    
                      0
                    
                  
                  
                    ε
                    
                      r
                    
                  
                  (
                  Ψ
                  −
                  
                    V
                    
                      g
                      s
                    
                  
                  )
                
                
                  q
                  
                    N
                    
                      d
                    
                  
                
              
            
            
              
                
                  2
                  
                    ε
                    
                      0
                    
                  
                  
                    ε
                    
                      r
                    
                  
                  (
                  Ψ
                  −
                  
                    V
                    
                      t
                      o
                    
                  
                  )
                
                
                  q
                  
                    N
                    
                      d
                    
                  
                
              
            
          
        
        =
        
          
            
              
                Ψ
                −
                
                  V
                  
                    g
                    s
                  
                
              
              
                W
                
                  00
                
              
            
          
        
      
    
    {\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}}}}}
  Substituting:

  
    
      
        
          I
          
            n
          
        
        =
        W
        ⋅
        
          
            
              σ
              a
              ⋅
              
                W
                
                  00
                
              
            
            
              3
              L
            
          
        
        
          
            [
          
        
        3
        (
        
          d
          
            2
          
        
        −
        
          s
          
            2
          
        
        )
        −
        2
        (
        
          d
          
            3
          
        
        −
        
          s
          
            3
          
        
        )
        
          
            ]
          
        
      
    
    {\displaystyle 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
        ⋅
        β
        
          W
          
            00
          
          
            2
          
        
        
          
            [
          
        
        3
        (
        
          d
          
            2
          
        
        −
        
          s
          
            2
          
        
        )
        −
        2
        (
        
          d
          
            3
          
        
        −
        
          s
          
            3
          
        
        )
        
          
            ]
          
        
      
    
    {\displaystyle 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
          
            s
            a
            t
          
        
        =
        β
        
          W
          
            00
          
          
            2
          
        
        (
        1
        −
        3
        
          s
          
            2
          
        
        +
        2
        
          s
          
            3
          
        
        )
      
    
    {\displaystyle I_{sat}=\beta W_{00}^{2}(1-3s^{2}+2s^{3})}
  
  
    
      
        
          g
          
            m
          
        
        =
        3
        β
        
          W
          
            00
          
        
        (
        s
        −
        1
        )
      
    
    {\displaystyle g_{m}=3\beta W_{00}(s-1)}
  
  
    
      
        
          G
          
            D
            S
          
        
        =
        3
        β
        
          W
          
            00
          
        
        (
        1
        −
        d
        )
      
    
    {\displaystyle G_{DS}=3\beta W_{00}(1-d)}
  


== Simpler Model ==

  
    
      
        
          I
          
            d
            s
          
        
        =
        
          
            3
            2
          
        
        β
        
          W
          
            00
          
          
            2
          
        
        
          
            [
          
        
        
          
            
              (
              
                V
                
                  g
                  s
                
              
              −
              
                v
                
                  t
                  o
                
              
              
                )
                
                  2
                
              
            
            
              W
              
                00
              
              
                2
              
            
          
        
        −
        
          
            
              (
              
                V
                
                  g
                  d
                
              
              −
              
                v
                
                  t
                  o
                
              
              
                )
                
                  2
                
              
            
            
              W
              
                00
              
              
                2
              
            
          
        
        
          
            ]
          
        
      
    
    {\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 ]}}
  
  
    
      
        
          g
          
            m
          
        
        =
        3
        β
        
          W
          
            00
          
        
        (
        
          V
          
            g
            s
          
        
        −
        
          V
          
            t
            o
          
        
        )
      
    
    {\displaystyle g_{m}=3\beta W_{00}(V_{gs}-V_{to})}
  
  
    
      
        
          G
          
            d
            s
          
        
        =
        3
        β
        
          W
          
            00
          
        
        (
        
          V
          
            g
            d
          
        
        −
        
          V
          
            t
            o
          
        
        )
      
    
    {\displaystyle 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
          
            d
            s
          
        
        =
        β
        
          
            [
          
        
        (
        
          V
          
            g
            s
          
        
        −
        
          V
          
            t
            o
          
        
        
          )
          
            Q
          
        
        −
        (
        
          V
          
            g
            d
          
        
        −
        
          V
          
            t
            o
          
        
        
          )
          
            Q
          
        
        
          
            ]
          
        
      
    
    {\displaystyle 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

  
    
      
        β
        
          
             proportial to 
          
        
        
          
            
              σ
              a
              W
            
            
              3
              L
              
                W
                
                  00
                
              
            
          
        
      
    
    {\displaystyle \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
          
            d
            s
          
        
        =
        
          {
          
            
              
                
                  0
                
                
                  
                    V
                    
                      g
                      s
                    
                  
                  <
                  
                    V
                    
                      t
                      o
                    
                  
                
              
              
                
                  β
                  
                    
                      [
                    
                  
                  (
                  
                    V
                    
                      g
                      s
                    
                  
                  −
                  
                    V
                    
                      t
                      o
                    
                  
                  
                    )
                    
                      Q
                    
                  
                  −
                  (
                  
                    V
                    
                      g
                      d
                    
                  
                  −
                  
                    V
                    
                      t
                      o
                    
                  
                  
                    )
                    
                      Q
                    
                  
                  
                    
                      ]
                    
                  
                
                
                  
                    V
                    
                      g
                      s
                    
                  
                  ≤
                  
                    V
                    
                      g
                      d
                    
                  
                
              
              
                
                  β
                  (
                  
                    V
                    
                      g
                      s
                    
                  
                  −
                  
                    V
                    
                      t
                      o
                    
                  
                  
                    )
                    
                      Q
                    
                  
                
                
                  
                    V
                    
                      g
                      s
                    
                  
                  >
                  
                    V
                    
                      g
                      d
                    
                  
                
              
            
          
          
        
      
    
    {\displaystyle I_{ds}=\left\{{\begin{matrix}0&V_{gs}V_{gd}\end{matrix}}\right.}
  


== 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.