Insulators — Question 9

Question 9

Insulator tunneling is exponentially sensitive to thickness

For a supplied rectangular-barrier model, leakage current density is $J(d)=J_0e^{-2\kappa d}$ with $\kappa=5.00\,\mathrm{nm^{-1}}$. The prefactor is held fixed while thickness changes.

1. Find the current ratio when d increases by .200 nm.

2. Find the thickness increase needed for a hundredfold current reduction.

Question 9 - Solution

Insulator tunneling is exponentially sensitive to thickness

(a) Method and calculation

$$J(d+\Delta d)/J(d)=e^{-2\kappa\Delta d}.$$$$\boxed{J(d+.200)/J(d)=e^{-2}=.1353}.$$

(b) Interpretation and further calculation

$$e^{-2\kappa\Delta d}=.01\Rightarrow\Delta d=\ln100/(2\kappa).$$$$\boxed{\Delta d=.4605\,\mathrm{nm}}.$$

Physical check

This is a specified tunneling regime, not thermally activated band conduction. Real applied-voltage changes can reshape the barrier and alter the assumed fixed prefactor.