Question

A donor atom in a semiconductor has a loosely bound electron. The orbit of this electron is considerably affected by the semiconductor material but behaves in many ways like an electron orbiting a hydrogen nucleus. Given that the electrons has an effective mass of $0.07m_e$ (where $m_e$ is mass of the free electron) and the space in which it moves has a permittivity $13{ϵ}_0$, then the radius of the electron's lowermost energy orbit will be close to (The Bohr radius of the hydrogen atom is $0.53\stackrel{\circ }{A})$.

• A. $0.53\stackrel{\circ }{A}$
• B. $243\stackrel{\circ }{A}$
• C. $10\stackrel{\circ }{A}$
• D. $100\stackrel{\circ }{A}$

Answer 1

Answer: (D)

$100\stackrel{\circ }{A}$

$\frac{m{v}^2}{r}=\frac{K{Q}^2}{r^2}$
and
$mvr=\frac{nh}{2\pi }⟹mv=\frac{nh}{2\pi r}$

and $\frac{m^2{v}^2}{mr}=\frac{K{Q}^2}{r^2}$

$\frac{n^2{h}^2}{4{\pi }^{2}m{r}^3}=\frac{1}{4\pi {ϵ}_0}\frac{Q^2}{r^2}$

$\frac{n^2{h}^2}{4{\pi }^2\left(0.07m_e\right)r}=\frac{1}{4\pi \left(13\right){ϵ}_0}{Q}^2$

$r=\frac{13n^2{h}^2}{4{\pi }^{2}k{Q}^2\left(0.07\right)m_e}$

$r=\frac{13}{0.07}\times r_0$ where $r_0=bohrradius$

$r=\frac{13}{0.07}\times 0.53\stackrel{\circ }{A}$

$r\approx 100\stackrel{\circ }{A}$.