1. Hydrogen Atom Review of Quantum Mechanics
We will discuss hydrogenic model of impurities
 Useful to first review the QM of the hydrogen (H) atom.
Start with Schrdinger Equation for H atom:
[-()2/(2mo) -(e2/r)] = E
Results (That you should know):
Energies: Rydberg series:
En = - (e4mo)/(22n2) , n=1,2,3,…
En = - (13.6 eV)/n2
Ave. electron-proton distance (in ground state, n=1): Bohr radius:
ao= 2/(e2mo) = 0.53 Å (Angstrms)

2. Hydrogen Atom Review of Quantum Mechanics
Results:
Wavefunctions:
nlm(r,,)= Rnl(r)Ylm(,)
Rnl(r) = radial function
Ylm(,) = spherical harmonic
n = 1,2,3,.. ; 0  l  n-1 -l  m  l
Ground State (1s) n=1, l=0, m=0
R10(r)=  exp[-r/(2ao)]
|100|2  exp[-r/ao]
Bohr Radius ao A measure of average spatial separation of electron & proton.

3. Shallow (Hydrogenic) Levels Overview. More details later!
Near the bandgap (BZ center or not), the bands look ~ like:
Bandgap: forbidden energy region for the perfect solid

4. Shallow (Hydrogenic) Levels
To perfect crystalline solid, add a defect or impurity: New one e- Hamiltonian is:
H = Ho +V
Ho = perfect crystal Hamiltonian  gives the bandstructures.
V defect potential (discussed in detail later)
Now, must solve new Schrdinger Eq. :
H = (Ho +V)= E

5. Shallow (Hydrogenic) Levels
New Schrdinger Eq. :
H = (Ho +V)= E
TRANSLATIONAL SYMMETRY IS BROKEN!
Strictly speaking k is no longer a good quantum number & Bloch’s Theorem is not valid!
Defect potential V may produce energy levels in bandgap of perfect solid.
We seek these levels. That is, given V, we seek eigenvalue solutions E which are in the bandgap of the perfect solid.

6. Shallow (Hydrogenic) Donor
Add substitutional impurity with valence = host atom valence + 1 (DONOR)
Si: host valence = 4; impurity valence = 5 (Group V atom, say As for definiteness)
sp3 bonded materials:
 4 electrons taken up in bonds
 5th electron is not tightly bound & thus is relatively “free” to move around the lattice.
5th electron attracted to As core (which has charge +e) by screened Coulomb potential.

7. Shallow (Hydrogenic) Donor Qualitative.
Schematically looks like:
When ionized, extra electron goes to conduction band (valence band states are full)
So mo  me (CB effective mass)

8. Shallow (Hydrogenic) Donor Qualitative.
Single As donor atom in Si
Extra e- + extra ionic charge
Form of impurity potential is screened Coulomb ( dielectric const.):
V  -e2/(r)
Screened Coulomb potential. NOTE: Long ranged!
1 electron + Coulomb potential
 Problem reduces to “Hydrogen Atom” problem with
“Proton Charge”  e/½
“Electron Mass”  me = effective mass at CB bottom.
Can use Rydberg formula for energy eigenvalues with these changes!

9. Shallow (Hydrogenic) Donor Qualitative & semiquantitative
Energy eigenvalues given by Rydberg series:
En = EC - (e4me)/(222n2) n=1,2,3,… or,
En = EC -(13.6 eV)(me/mo)/(2n2)
En = EC - (EH)n (me/mo)/2
(EH)n true H atom energy
Effective Bohr radius (ave. distance of e- from donor ion core in n=1 state) has form:
a = ( 2)/(mee2) or,
a = ao (mo/me) = (0.5 A)  (mo/me)
ao  true H Bohr radius

10. Shallow (Hydrogenic) Donor Qualitative & semiquantitative
Single hydrogenic donor
Energy levels in bandgap:
En = EC - (13.6 eV)(me/mo)/(2n2)
En = EC - (EH)n (me/mo)/2
(EH)n true H atom energy
For Si (& other materials),  ~ 10
Typically, me~ 0.07mo (definitely, me<<mo)
En - EC ~ a few meV! (Shallow!)
Donor effective Bohr radius :
a = (0.5 A)  (mo/me)
a = ao  (mo/me)
ao  true H Bohr radius
 a ~ several 10’s of Angstroms! (large!)

11. Shallow (Hydrogenic) Donor Qualitative & semiquantitative
Single H-like donors in Ge and Si. Measured n=1 donor levels.
En = EC - (EH)n (me/mo)/2
 En - EC ~ a few meV! (Shallow!)
Donor effective Bohr radius :
a = ao  (mo/me)
 a ~ several 10’s of Å ! (Large!)
 Spatial extent of donor electron in ground (1s, n=1 state) is large (many lattice spacings)! |100|2  exp[-r/a]

12. Shallow (Hydrogenic) Donor Qualitative & semiquantitative
Single H-like donors GaAs.
En = EC - (EH)n (me/mo)/2
me= 0.07 mo ,  = 11
 En=1 - EC = - (13.6 eV)(.07)/(121)
En=1 - EC = meV
a = ao  (mo/me)
 a = (0.5 Å) (11)/(0.07)
a = 78.6 Å
 Spatial extent of electron in ground (1s, n=1 state) is huge!
|100|2  exp[-r/a]

13. Shallow (Hydrogenic) Acceptor
Add substitutional impurity with valence = host atom valence - 1 (ACCEPTOR)
Si: host valence = 4; impurity valence = 3 (Group III atom, say B for definiteness)
sp3 bonded materials:
 Bonds want 4 e- but impurity has only 3 to give
 Gets 4th e- from somewhere else (another bond, etc.)
Causes B atom to be like negatively charged B- ion.
Causes missing e- or hole (e+) to occur where 4th e- originally was.
Positively charged hole negative B- by screened Coulomb potential.

14. Shallow (Hydrogenic) Acceptor Qualitative.
Schematically looks like:
When ionized, resulting hole occurs in valence band
So mo  mh (VB effective mass)

15. Shallow (Hydrogenic) Acceptor Qualitative.
Single B acceptor atom in Si
Hole e+ + negative ionic charge
Form of impurity potential is screened Coulomb ( dielectric const.):
V  -e2/(r)
Screened Coulomb potential. NOTE: Long ranged!
1 hole + Coulomb potential
 Problem reduces to “Hydrogen Atom” problem with
“Proton Charge”  -e/½ (<0 !)
“Electron Mass”  mh = effective mass at CB bottom.
Positive “electron charge”
Use Rydberg formula for energy eigenvalues with these changes!

16. Shallow (Hydrogenic) Acceptor Qualitative & semiquantitative
Energy eigenvalues given by Rydberg series:
En = EV + (e4mh)/(222n2) n=1,2,3,… or,
En = EV +(13.6 eV)(mh/mo)/(2n2)
En = EV + (EH)n (mh/mo)/2
(EH)n true H atom energy
Effective Bohr radius (ave. distance of e+ from acceptor ion core in n=1 state) has form:
a = ( 2 )/ (mhe2) or,
a = ao (mo/me) = (0.5 A)  (mo/mh)
ao  true H Bohr radius

17. Shallow (Hydrogenic) Acceptor Qualitative & semiquantitative
Single hydrogenic acceptor
Energy levels in bandgap:
En = EV + (13.6eV)(mh/mo)/(2n2)
En = EV + (EH)n (mh/mo)/2
(EH)n true H atom energy
For Si (& other materials),  ~ 10
Typically, mh~ 0.3mo (definitely, me <mo)
En - EV ~ few 10’s meV! (Shallow!)
Acceptor effective Bohr radius :
a = (0.5 A)  (mo/mh)
a = ao  (mo/mh)
ao  true H Bohr radius
 a ~ several 10’s of Å (large!)

18. Shallow (Hydrogenic) Acceptor Qualitative & semiquantitative
Single H-like acceptors in Ge and Si. Measured n=1 acceptor levels.
En = EV + (EH)n (mh/mo)/2
 En - EV ~ few 10’s of meV! (Shallow!)
Acceptor effective Bohr radius :
a = ao  (mo/mh)
 a ~ several 10’s of Å ! (Large!)
 Spatial extent of acceptor hole in ground (1s, n=1 state) is large (many lattice spacings)! |100|2  exp[-r/a]

19. Shallow (Hydrogenic) Acceptor Qualitative & semiquantitative
Single H-like acceptors in GaAs.
En = EV + (EH)n (mh/mo)/2
me= 0.5 mo ,  = 11
 En=1 - EV = (13.6 eV)(0.5)/(121)
En=1 - EC = 56.1 meV
a = ao  (mo/mh)
 a = (0.5 Å) (11)/(0.5)
a = 11.0 Å
 Spatial extent of hole in ground (1s, n=1 state) is huge!
|100|2  exp[-r/a]