The Pinch-off Voltage Vp (JFET)

The Pinch-off Voltage V_p

• The Fig. 3.5.1 shows a single-ended-geometry junction FET. Here, the substrate is of p-type material onto which an n-type channel is epitaxially grown. A p-type gate is then diffused into the n-type channel. In this device, a slab of n-type semiconductor is sandwiched between two layers of p-type material, forming two p-n junctions.

• Let us assume that the p-type region is doped with NA acceptors per cubic meter, n-type region is doped with ND donors per cubic meter and that the junction formed is abrupt.

• We know that, if NA » ND we have Wp « Wn and for the space-charge width, WR(x) = W(x) at a distance x along the channel in Fig. 3.5.2.

where

Ɛ : Dielectric constant of channel material

q : Magnitude of electric charge

V_O : Junction contact potential at x

V(x) : Applied potential across space-charge region at x and is a negative number for an applied reverse bias

a - b(x) : Penetration W(x) of depletion region into channel at a point x along channel

• When the drain current is zero, b(x) and V(x) are independent of x and b(x) = b. By substituting b(x) = b = 0 we have

• Now solving for V with assumption |V| <<  |V| we obtain the pinch-off voltage Vp (the diode reverse voltage that removes   all the free charge from the channel) as

|V_P| = qN_D / 2E a² … (3.5.3)

If we substitute V_GS for V_O – V(x) and b(x) = b in equation (3.5.1) we get,

From equation (3.5.3) we have

2E / qND = a² / V_P … (3.5.5)

Substituting equation (3.6.5) in equation (3.6.4) we get,

This equation shows that the voltage V_GS (the reverse bias across the gate junction) is independent of distance along the channel if I_D = 0.

Ex. 3.5.1 For an n-channel silicon FET, find the pinch-off voltage and the channel half-width.

Assume : ɑ = 3 × 10^-6 m, N_D = 10²¹ electrons/m², V_GS = ½ Vpv ID = 0 and relative dielectric constant of silicon = 12.

Sol. : Since relative dielectric constant of silicon is 12 we have Ɛ = 12 Ɛ₀. Using values of q and Ɛ₀ we have

Ex. 3.5.2 For an n-channel silicon FET with a = 3×l0^-4 cm and Nd = 1015 electrons/cm-3. Find (a) the pinch off voltage and (b) the channel half-width for V_GS = 0.5 Vp.

AU : June-16, Marks 6

Sol. : a) Note that :

Ɛ₀ = 8.85 × 10^-14 F/cm

q = 1.6 × 10^-19C

Dielectric constant of silicon = 12

Ex. 3.5.3 Assume that the p+n junction of a uniformly doped silicon n channel JFET at T = 300 K has doping concentrations of Nɑ = 10¹⁸ cm^-3 and Nd = 10¹⁶ cm^-3. Assume that the metallurgical channel thickness a is 0.7 pm Calculate the pinch off voltage. Also calculate pinch-ojf voltage if the metallurgical channel thickness a is 0.75 µm

AU : May-15, Dec.-15, Marks 2

Sol. : Here all quantities are in centimeters instead of meters. Channel thickness a= 0.7 ^ti= 0.7 xlO 4 cm. The internal pinch off voltage is given by

Review Question

1. Derive the expresionfor pinch-off voltage.

AU : Dec.-18