Biexciton

Three possibilities of observing biexcitons exist:^[3]

(a) excitation from the one-exciton band to the biexciton band (pump-probe experiments);

(b) two-photon absorption of light from the ground state to the biexciton state;

(c) luminescence from a biexciton state made up from two free excitons in a dense exciton system.

The biexciton is a quasi-particle formed from two excitons, and its energy is expressed as

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

where ${\displaystyle E_{XX}}$ is the biexciton energy, ${\displaystyle E_{X}}$ is the exciton energy, and ${\displaystyle E_{b}}$ is the biexciton binding energy.

When a biexciton is annihilated, it disintegrates into a free exciton and a photon. The energy of the photon is smaller than that of the exciton by the biexciton binding energy, so the biexciton luminescence peak appears on the low-energy side of the exciton peak.

The biexciton binding energy in semiconductor quantum dots has been the subject of extensive theoretical study. Because a biexciton is a composite of two electrons and two holes, we must solve a four-body problem under spatially restricted conditions. The biexciton binding energies for CuCl quantum dots, as measured by the site selective luminescence method, increased with decreasing quantum dot size. The data were well fitted by the function

${\displaystyle B_{XX}={\frac {c_{1}}{a^{2}}}+{\frac {c_{2}}{a}}+B_{bulk}}$

where ${\displaystyle B_{XX}}$ is biexciton binding energy, ${\displaystyle a}$ is the radius of the quantum dots, ${\displaystyle B_{bulk}}$ is the binding energy of bulk crystal, and ${\displaystyle c_{1}}$ and ${\displaystyle c_{2}}$ are fitting parameters.^[4]

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 in a quantum dot decreases with size. In CuCl, the biexciton's 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]