Excitons in 3D and 2D systems
Electrons and holes in an excited semiconductor create a quasiparticle, named exciton. After separation of centre of gravity and relative orbital motion a series of bound exciton states (n=1,2,3…) are located below the band gap. Solving the Schrödinger equation the binding energy in 3D is
R * E X (3D) ,n 1,2,.... n n2 *e4 Where R* is the Rydberg energy by R* 2 2 2 2 with µ* as reduced effective mass. It 32 0 defines a series of exciton states which can be populated. In analogy to the hydrogen problem the respective orbital wave function for n=1 is
1 r X (r) exp( ) n1 3/ 2 a0 a0
2 X 4 0 With exciton Bohr radius a (3D) and r=re-rh is the 3D relative electron-hole 0 *e2 coordinate. In k-Space the wave function represented by
3/ 2 X 8 a0 n1 (k) 2 2 2 (1 a0 k )
The k-space representation is important to estimate the oscillator strength of excitonic processes.
For discussion of PL excitation spectra and the bleaching of exciton absorption it is important to consider the influence of unbound electron-hole states above the ionization limit (E>0). It is expressed by the Sommerfeld factor
2 R * /( E ) 3D 3D g CEF | E0 (r 0) 1 exp(2 R * /( Eg )
This equation describes the enhancement of absorption above the band gap due to correlated electron-hole pairs. Since equ. Above diverges for Eg one observes a “tooth-like” absorption line.
For the hypothetical case of a 2D QW the excition bonding energy modifies as
R * E X (2D) ,n 1,2,.... n (n 0.5)2
The excition ground state wave function is
X ,2D 2 2 2r X ,2D 2 a0 n1 (r) exp( ) and n1 (k) 2 2 3/ 2 a0 a0 (1 a0 k / 4) Where k is the 2D wave vector in the 2D plane. The Sommerfeld factor becomes
2D 2D 2 CEF | E0 (r 0) 1 exp(2 R * /( Eg )
Comparing the solutions for 3D and 2D one can conclude:
The binding energy in 2D is 4 times larger than in bulk material, consequently in quantum well structures the excition resonances can e observed even at room temperature The 2D excition Bohr radius is 2 times smaller compared to 2D
The oscillator strengths which is proportional to k=1/a0 is strongly enhanced in 2D with respect to 3D The volume in k space is larger for 2D excitons compared to 3D. As consequence the bleaching of of excitonic absorption is more effective for 2D excitons compared to 3D. The 3D Sommerfeld factor explains the tooth-like absorption line, whereas in 2D the enhancement of absorption is structureless and amounts to a factor of 2. Opposite , in 2D states far above the exciton gap can contribute to absorption whereas it decays fast for 3D.
* In 2D effective mass is different for hh and lh holes and described by mhh m/( 1 2 ) and * mhl m/( 1 2 ) with 1,2 are the Luttinger constants. As consequence one observes excitionic transition between quantized states in CB band and quantized hh and lh states in VB, such as e-1hh, 2e-1hh, 2e-2hh,… if hh states at the top of 2D VB. However, under certain conditions hl VB states can come to the top of VB.
The bonding energy of bi-exciton XX state (2e + 2h) below the X ground state is
* EXX 2EX Eb where Eb is the binding energy of second electron and hole. Considering same interactions as know from H2 molecules compared to 2 individual atoms the Hamiltonian is described by
2 2 2 2 2 H * (1 2 ) * (a b ) (V12 V1a V2b V1b V2a V ab) 2me 2mh
Where 1,2, are the electrons and a,b are the holes. To solve Schrödinger equation one has to solve: Coulomb integral, Exchange integral and overlap integral. As for X the binding energy of XX increases for decreasing widths of QW layer. The probability of XX induced recombination increases with excitation energy of the exciting laser .
Considering the similarity of exciton and hydrogen atom/molecule problems one may suppose the existence of other bound excitons such negatively charges excitions, X- , (2e+h) and positively charges + - + excitions, X -(e+2h). (Compared to H2 ground state of H2 molecules is about 2eV and the H2 - molecule is about 16 eV above the ground state of H2).experimentally the binding energy of X is 1.15meV and the X+ is 1.25 meV above the X exciton.
Conclusion : QW structures allow for similar experiments as known from molecule physics.