Biexciton

In the effective-mass approximation, the Hamiltonian of the system consisting of two electrons (1, 2) and two holes (a, b) is given by

${\displaystyle H_{XX}=-{\frac {\hbar ^{2}}{2m_{e}^{*}}}({\nabla _{1}}^{2}+{\nabla _{2}}^{2})-{\frac {\hbar ^{2}}{2m_{h}^{*}}}({\nabla _{a}}^{2}+{\nabla _{b}}^{2})+V}$

where ${\displaystyle m_{e}^{*}}$ and ${\displaystyle m_{h}^{*}}$ are the effective masses of electrons and holes, respectively, and

${\displaystyle V=V_{12}-V_{1a}-V_{1b}-V_{2a}-V_{2b}+V_{ab}}$

where ${\displaystyle V_{ij}}$ denotes the Coulomb interaction between the charged particles ${\displaystyle i}$ and ${\displaystyle j}$ (${\displaystyle i,j=1,2,a,b}$ denote the two electrons and two holes in the biexciton) given by

${\displaystyle V_{ij}={\frac {e^{2}}{\epsilon |\mathbf {r} _{i}-\mathbf {r} _{j}|}}}$

where ${\displaystyle \epsilon }$ is the dielectric constant of the material.

Denoting ${\displaystyle \mathbf {R} }$ and ${\displaystyle \mathbf {r} }$ are the c.m. coordinate and the relative coordinate of the biexciton, respectively, and ${\displaystyle M=m_{e}^{*}+m_{h}^{*}}$ is the effective mass of the exciton, the Hamiltonian becomes

${\displaystyle H_{XX}=-{\frac {\hbar ^{2}}{4M}}{\nabla _{R}}^{2}-{\frac {\hbar ^{2}}{M}}{\nabla _{r}}^{2}-{\frac {\hbar ^{2}}{2\mu }}({\nabla _{1a}}^{2}+{\nabla _{2b}}^{2})+V}$

where ${\displaystyle 1/\mu =1/{m_{e}^{*}}+1/{m_{h}^{*}}}$; ${\displaystyle {\nabla _{1a}}^{2}}$ and ${\displaystyle {\nabla _{2b}}^{2}}$ are the Laplacians with respect to relative coordinates between electron and hole, respectively. And ${\displaystyle {\nabla _{r}}^{2}}$ is that with respect to relative coordinate between the c. m. of excitons, and ${\displaystyle {\nabla _{R}}^{2}}$ is that with respect to the c. m. coordinate ${\displaystyle \mathbf {R} }$ of the system.

In the units of the exciton Rydberg and Bohr radius, the Hamiltonian can be written in dimensionless form

${\displaystyle H_{XX}=-({\nabla _{1a}}^{2}+{\nabla _{2b}}^{2})-{2\sigma }{(1+\sigma )^{2}}{\nabla _{r}}^{2}+V}$

where ${\displaystyle \sigma ={m_{e}^{*}}/{m_{h}^{*}}}$ with neglecting kinetic energy operator of c. m. motion. And ${\displaystyle V}$ can be written as

${\displaystyle V=2\left({\frac {1}{r_{12}}}-{\frac {1}{r_{1a}}}-{\frac {1}{r_{1b}}}-{\frac {1}{r_{2a}}}-{\frac {1}{r_{2b}}}+{\frac {1}{r_{ab}}}\right)}$

To solve the problem of the bound states of the biexciton complex, it is required to find the wave functions ${\displaystyle \psi }$ satisfying the wave equation

${\displaystyle H_{XX}\psi =E_{XX}\psi }$

If the eigenvalue ${\displaystyle E_{XX}}$ can be obtained, the binding energy of the biexciton can be also acquired

${\displaystyle E_{b}=2E_{X}-E_{XX}}$

where ${\displaystyle E_{b}}$ is the binding energy of the biexciton and ${\displaystyle E_{X}}$ is the energy of exciton.[5]

The diffusion Monte Carlo (DMC) method provides a straightforward means of calculating the binding energies of biexcitons within the effective mass approximation. For a biexciton composed of four distinguishable particles (e.g., a spin-up electron, a spin-down electron, a spin-up hole and a spin-down hole), the ground-state wave function is nodeless and hence the DMC method is exact. DMC calculations have been used to calculate the binding energies of biexcitons in which the charge carriers interact via the Coulomb interaction in two and three dimensions,[6] indirect biexcitons in coupled quantum wells,[7][8] and biexcitons in monolayer transition metal dichalcogenide semiconductors.[9][10][11]

Biexcitons with bound complexes formed by two excitons are predicted to be surprisingly stable for carbon nanotube in a wide diameter range. Thus, a biexciton binding energy exceeding the inhomogeneous exciton line width is predicted for a wide range of nanotubes.

The biexciton binding energy in carbon nanotube is quite accurately approximated by an inverse dependence on ${\displaystyle r}$, except perhaps for the smallest values of ${\displaystyle r}$.

${\displaystyle E_{XX}\approx {\frac {0.195eV}{r}}}$

The actual biexciton binding energy is inversely proportional to the physical nanotube radius.[12] Experimental evidence of biexcitons in carbon nanotubes was found in 2012. [13]

The binding energy of biexcitons increase with the decrease in their size and its size dependence and bulk value are well represented by the expression

${\displaystyle {\frac {78}{{a^{*}}^{2}}}+{\frac {52}{a^{*}}}+33}$ (meV)

where ${\displaystyle a^{*}}$ is the effective radius of microcrystallites in a unit of nm. The enhanced Coulomb interaction in microcrystallites still increase the biexciton binding energy in the large-size regime, where the quantum confinement energy of excitons is not considerable.[14]