Quantum Tunneling

incomplete Review Step Potentials and scattering

Using probability flux $j$

$$j_A=\frac{\hslash k}{m}A{A}^{\ast }$$

Negative $\hslash k$ term for B. Applications of Quantum Tunneling

• Scanning tunneling microscopy
• Field emission scanning electron microscopy
• Alpha decay
• Dielectric breakdown
• Ivar Giaever (Nobel Prize winner for quantum tunneling) Also recall photoelectric effect: Note figure on right utilizing quantum tunneling

In class assignment 1: From boundary conditions at x=0 and L, determine the four algebraic equations that relate A,B,C,D,E,F and G

$$A{e}^{ikx}+B{e}^{-ikx}=G{e}^{\kappa x}+F{e}^{-\kappa x}$$

At x=0

$$A+B=F+G$$ $$ik(A-B)=\kappa (-F+G)$$

At x=L

$$G{e}^{\kappa x}+F{e}^{-\kappa x}=C{e}^{ikx}$$ $$-\kappa F{e}^{-\kappa L}+\kappa G{e}^{\kappa L}=ikC{e}^{ikx}$$

Now considering a finite barrier as a scattering problem with a wave coming from the left.

$$@x<0:A{e}^{ikx}+B{e}^{-ikx}$$ $$@0<x<L:F{e}^{\kappa x}+G{e}^{-\kappa x}$$ $$@x>L:C{e}^{ikx}$$

In class Assignment 2:

$${\left(\frac{4k\kappa }{k^2+{\kappa }^2}\right)}^2=16\frac{E}{V_0}\left(1-\frac{E}{V_0}\right)$$

Consider a barrier with an arbitrary shape: With transmission coeff.

$$T\approx C{e}^{-2\sum {\kappa }_{i}dx}\approx C{e}^{-2{\int }_a^{b}dx\sqrt{2m(V_0-E)/\hslash }}$$

There are many situations where we need to consider a barrier that is not rectangular: Recall: Electric field, $E_0$ w/t $V=q{E}_{0}x$ Now,

$$E=W+E-e{E}_0{x}_0$$ $$0=W-e{E}_0{x}_0$$ $$x_0=\frac{w}{e{E}_0}$$ $$\therefore T\approx C{e}^{-2{\int }_0^{x_0}dx\sqrt{2m/\hslash (W-e{E}_{0}x)}}$$

This is known as the Fowler-Nordheim Relationship