Bohr Model applied to Excitons

[adamaero]

An exciton is a bound electron-hole pair (in a semiconductor). For this problem, think of an exciton as a hydrogen-like atom, with a negatively charged electron and positively charged hole orbiting each other.

The permittivity of free space (ε₀) is replaced with permittivity of the semiconductor (ε = 12).
The mass of the electron is replaced with the effective mass of the electron-hole pair.

1. Homework Statement (bold below is what I really need help on)
A) Estimate the radius in nm and the ground state energy in eV for an exciton in Si.

B) Approximately how large is the separation between atoms in a crystal of silicon? How does the radius compare with this number?

C) Silicon atoms have an average kinetic energy of T*k_B. How does the exciton binding energy (E₁) compare with this number? What does this mean?

D) All this is about electrostatic potential energy. Prove that it's reasonable to neglect the gravitational potential energy.

m_e = 9.1*10^-31
eV = 1.602×10⁻¹⁹ J (N*m)
h = 6.626*10^-34
ħ = 1.055*10^-34
a₀ = 0.0529 nm
ε*ε₀ = 1.0359*10^-10

permittivity of silicon = ε_Si = k_Siε₀ where k = dielectric constant

effective masses
m_e^* = 0.26m_e
m_h^* = 0.36m_e

Homework Equations

r = m_ek²e⁴/(πħ³)

m_e^*m_h^*/(m_e^* + m_h^*) = 0.15m_e

r = n²h²*1.0359*10^-10/(z*π*m_effectivee²)

The Attempt at a Solution

A)

12*a₀/0.15 = 4.2nm (n²/z)

R_y = -13.6eV

B) I do not remember chemistry much. How is the separation between silicon atoms found?

C) 300*1.38*10^-23 m²kg/s² = 4.14*10^-23
E₁ = ?

D) PE_{electrostatic} = kqQ/r
images.slideplayer.com/22/6421287/slides/slide_3.jpg
F_E = qE
F_g = mg
∴ qE = mg
& as long as qE/m is much larger than g, gravity can be ignored.

Sources
http://www.course-notes.org/book/export/html/10891
https://upload.wikimedia.org/wikipedia/commons/9/97/Exciton_energy_levels.jpg

 

[DrClaude]

R_y = -13.6eV

You get the same Rydberg constant as for hydrogen?

B) I do not remember chemistry much. How is the separation between silicon atoms found?

Google There is no way to find that by first principles, so just Google "silicon interatomic distance".

C) 300*1.38*10^-23 m²kg/s² = 4.14*10^-23
E₁ = ?

That should be the binding energy. Do you remember how to find that for hydrogen?

 

[adamaero]

Oh yes, sorry, with ε:
The energy (for part A) is m_ee⁴/[8(h*1.0359*10^-10)²]
= (9.1*10^-31)(e^4)/[8*((6.626*10^-34)*(1.0359*10^-10))^2]
= 1.32*10⁵⁷
Was that suppose to be a different mass?

Thank you. I did Google it. Although, I either searched the wrong phrase or thought it wasn't simple from the results that came up earlier.
atomic radius = 0.132 nm
lattice parameter = 0.543 nm
nearest neighbor distance = 0.235 nm
Looks like the calculation before (4.2 nm) is wrong?

The binding energy of an electron to the nucleus in the hydrogen atom is 13.6 eV.
So the binding energy is the absolute value of Rydberg energy??

 

[adamaero]

BE=(m_p+m_e-m_H)*c²
BE = (938MeV/c² + 0.511MeV/c² - 938.3MeV/c²)*c² =
(938.484+ 0.511 - 938.783)*10^6
= 212keV

938.783 MeV (mass of hydrogen)
1.673e-27 kg (mass of proton)
>> 1 eV = 1.602e-19 J <<
>> 1 J = m³ kg / s² <<
[(1.673e-27)/(1.602e-19)]*c²
= 9.3989e8
= 939.89e6
= 939.89 MeV

 

[adamaero]

sol.

A) The 4.2nm (n²/z) is correct. The energy is found this way: E = m^*e⁴/(8h²ε²), but
B) 0.235 nm or 0.543 nm means that there are multiple atoms in-between each
C) This is the same E₁, but can be modified from more general E₁ = me⁴/(8h²ε₀²) = -13.6 eV... E = E₁*(m^T/m)/[(ε/ε₀)²]