Homework 3 Solutions

Assuming the membrane is permeable only to k and and

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Unformatted text preview: ISFET. Below the threshold, the drain current is approximately: I DS ≈ I0 ⋅ e VGS −VTH kT q where: I0 =1 µΑ, VTH = 0.5 V, T = 300K, and q = 1.6 × 10 ­19 C. Nernst Potential for an ion Az where z is valence of the ion Az € VGS = VG − VS = RT ln [ z ]S zF € [A ] G kT RT = For your reference: q F Plug in VGS into IDS: I DS −VTH ȹ VGS −VTH ȹ VGS ȹ ȹ kT q kT q kT q RT F ȹ ⋅ e ȹ ⋅ e ≈ I0 ⋅ e = ȹ I0 ⋅ e = ȹ I 0 ⋅ e ȹ ȹ ȹ ȹ € ȹ Ⱥ ȹ Ⱥ z RT [ A ] S ȹ z ȹ z ln z 1 ȹ [ A ] S ȹ 1 [ A ]S zF [ A ] −VTH ȹ −VTH ȹ −VTH ȹ ln ln z G ȹ ȹ ȹ z ȹ [ A z ] ȹ z [A ] G = ȹ I0 ⋅ e kT q ȹ ⋅ e RT F = ȹ I0 ⋅ e kT q ȹ ⋅ e = ȹ I0 ⋅ e kT q ȹ ⋅ e ȹ G Ⱥ ȹ ȹ ȹ ȹ ȹ ȹ ȹ Ⱥ ȹ Ⱥ ȹ Ⱥ VGS −VTH kT q −VTH ȹ ȹ = ȹ I0 ⋅ e kT q ȹ ⋅ e ȹ ȹ ȹ Ⱥ ȹ A z ln ȹ z ȹ A ȹ 1/ z [ ]S ȹ ȹ [ ]G ȹ Ⱥ 1z −VTH ȹ ȹ ȹ [A z ]S ȹ = ȹ I0 ⋅ e kT q ȹ ⋅ ȹ z ȹ ȹ ȹ ȹ A ȹ ȹ Ⱥ ȹ [ ]G Ⱥ € 1z Ⱥȹ −VTH ȹ ȹ Ⱥ Ⱥ [A z ]S ȹ Ⱥ log( I DS ) ≈ logȺȹ I0 ⋅ e kT q ȹ ⋅ ȹ z ȹ Ⱥ ȹ ȹ ȹ A ȹ Ⱥ ȹ Ⱥ ȹ [ ]G Ⱥ Ⱥ Ⱥ Ⱥ −VTH kT q 1 = log I0 + + ⋅ log[ A z ]S − log[ A z ]G ln(...
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