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 Schrdinger Equation for H atom: [-()2/(2mo) -(e2/r)] = E Results (That you should know): Energies: Rydberg series: En = - (e4mo)/(22n2) , 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 (Angstrms)
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.
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
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 Schrdinger Eq. : H = (Ho +V)= E
Shallow (Hydrogenic) Levels New Schrdinger 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.
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.
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)
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!
Shallow (Hydrogenic) Donor Qualitative & semiquantitative Energy eigenvalues given by Rydberg series: En = EC - (e4me)/(222n2) 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
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!)
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]
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 = -7.87 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]
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.
Shallow (Hydrogenic) Acceptor Qualitative. Schematically looks like: When ionized, resulting hole occurs in valence band So mo mh (VB effective mass)
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!
Shallow (Hydrogenic) Acceptor Qualitative & semiquantitative Energy eigenvalues given by Rydberg series: En = EV + (e4mh)/(222n2) 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
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!)
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]
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]