Doped semiconductor, donor electron radius

[Yroyathon]

hi folks, almost done my semester of physics. this problem has my goat, can't quite figure it out. Done web searches endlessly, but most of the links are pdf articles that I can't access.

Homework Statement

A donor electron moves in doped semiconductor, for which ε/ε0 = 17.9 and m* = 0.015*m_e. Find the radius of a circular atomic orbit of such an electron in terms of the Bohr radius a_0 (Fig. 43-37). The effective nuclear charge for such a loosely bound electron is Z = 1.
ans= ____ a_0

Homework Equations

the binding energy E=(13.6eV)*(m*/m_e)*(epsilon_0/epsilon)^2

The Attempt at a Solution

(the figure is unimportant: a circle with a dot in it; I didn't include it)

so I've gotten the binding energy, which is pretty small. but I'm unsure how to connect this to the Bohr equations for radius. or, I'm not even sure if that's the right approach to take.

my question is, given this new binding/ionization energy, how can you find the radius?

Tips or suggestions are appreciated.
Thanks.

,Yroyathon

 

[nickjer]

Well you can look up the bohr radius of an electron and proton...

$$a_0 = \frac{4\pi \epsilon_0 \hbar^2}{m_e e^2}$$

For the exciton you get...

$$a_{x0} = \frac{4\pi \epsilon\hbar^2}{m_{r}^* e^2} = \frac{\epsilon}{\epsilon_0}\cdot\frac{m_e}{m_r^*}\cdot a_0$$

Where $$1/m_r^* = 1/m_e^* + 1/m_h^*$$.

P.S.: You might want to check my work.

 

[Yroyathon]

thank you. that was exactly what I needed. I'd seen something similar to this in my web searches, but with the differences in constant notation (epsilon vs. epsilon_0 vs. kappa vs, etc.) I was pretty confused as to what was in the formula.

thanks!