Biexciton - AbsoluteAstronomy.com

Biexcitons are created from two free excitons.

Formation of biexcitons

In quantum information and computation, it is essential to construct coherent combinations of quantum states.
The basic quantum operations can be performed on a sequence of pairs of physically distinguishable quantum bits and, therefore, can be illustrated by a simple four-level system.

In an optically driven system where the

and

states can be directly excited, direct excitation of the upper

level from the ground state

is usually forbidden and the most efficient alternative is coherent nondegenerate two-photon excitation, using

as an intermediate state.

At high exciton

Exciton

An exciton is a bound state of an electron and hole which are attracted to each other by the electrostatic Coulomb force. It is an electrically neutral quasiparticle that exists in insulators, semiconductors and some liquids...

densities in some materials, the formation of exciton

Exciton

molecules called biexcitons is also observed.
This is the equivalent process to the formation of an

molecule from two isolated hydrogen

Hydrogen

Hydrogen is the chemical element with atomic number 1. It is represented by the symbol H. With an average atomic weight of , hydrogen is the lightest and most abundant chemical element, constituting roughly 75% of the Universe's chemical elemental mass. Stars in the main sequence are mainly...

atoms.

Biexcitons have been observed in a number of compound semiconductors, including CdS, ZnSe, ZnO and especially copper chloride.
CuCl has a band gap at 3.40 eV, and the ground state exciton

Exciton

is observed at 3.20 eV, implying that Rx = 0.2 eV.
At high densities, a new feature is observed in the absorption spectrum at 3.18 eV.
This is attributed to biexciton formation.
The energy difference between the two features tells us that the binding energy of the biexciton is 0.02 eV.

Attempts to observe biexcitons in materials like GaAs have been hindered by the nonlinear saturation effects described.

Observation of biexcitons

Three possibilities of observing biexcitons exist:
(a) excitation from the one-exciton

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

Luminescence

Luminescence is emission of light by a substance not resulting from heat; it is thus a form of cold body radiation. It can be caused by chemical reactions, electrical energy, subatomic motions, or stress on a crystal. This distinguishes luminescence from incandescence, which is light emitted by a...

from a biexciton state made up from two free excitons in a dense exciton

Exciton

system.

Binding energy of biexcitons

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

where

is the biexciton energy,

is the exciton

Exciton

energy, and

is the biexciton binding energy.

When a biexciton is annihilated, it disintegrates into a free exciton

Exciton

and a photon. The energy of the photon is smaller than that of the biexciton by the biexciton binding energy,
so the biexciton luminescence

Luminescence

peak appears on the low-energy side of the exciton

Exciton

peak. See Figure 1 in Reference 8.

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

Luminescence

method, increased with decreasing quantum dot

Quantum dot

A quantum dot is a portion of matter whose excitons are confined in all three spatial dimensions. Consequently, such materials have electronic properties intermediate between those of bulk semiconductors and those of discrete molecules. They were discovered at the beginning of the 1980s by Alexei...

size. The data were well fitted by the function

where

is biexciton binding energy,

is the radius of the quantum dots,

is the binding energy of bulk crystal, and

and

are fitting parameters.

A simple model for describing binding energy of biexcitons

In the effective-mass approximation, the Hamiltonian

Hamiltonian (quantum mechanics)

In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

of the system consisting of two electrons (1, 2) and two holes (a, b) is given by

where

and

are the effective masses of electrons and holes, respectively, and

where

denotes the Coulomb interaction between the charged particles

and

(

denote the two electrons and two holes in the biexciton) given by

where

is the dielectric constant of the material.

Denoting

and

are the c.m. coordinate and the relative coordinate of the biexciton, respectively, and

is the effective mass

Effective mass

In solid state physics, a particle's effective mass is the mass it seems to carry in the semiclassical model of transport in a crystal. It can be shown that electrons and holes in a crystal respond to electric and magnetic fields almost as if they were particles with a mass dependence in their...

of the exciton

Exciton

, the Hamiltonian becomes

where

;

and

are the Laplacians with respect to relative coordinates between electron and hole, respectively.
And

is that with respect to relative coordinate between the c. m. of excitons, and

is that with respect to the c. m. coordinate

of the system.

In the units of the exciton

Exciton

Rydberg and Bohr radius

Bohr radius

The Bohr radius is a physical constant, approximately equal to the most probable distance between the proton and electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an atom...

, the Hamiltonian can be written in dimensionless form

where

with neglecting kinetic energy operator of c. m. motion. And

can be written as

To solve the problem of the bound states of the biexciton complex, it is required to find the wave functions

satisfying the wave equation

If the eigenvalue

can be obtained, the binding energy of the biexciton can be also acquired

where

is the binding energy of the biexciton and

is the energy of exciton

Exciton

Binding energy in nanotubes

Biexcitons with bound complexes formed by two excitons are surprisingly stable for carbon nanotube

Carbon nanotube

Carbon nanotubes are allotropes of carbon with a cylindrical nanostructure. Nanotubes have been constructed with length-to-diameter ratio of up to 132,000,000:1, significantly larger than for any other material...

in a wide diameter range.
Thus, a biexciton binding energy exceeding the inhomogeneous exciton

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

, except perhaps for the smallest values of

The actual biexciton binding energy is inversely proportional to the physical nanotube radius.

Binding energy in CuCl QDs

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

(meV)

where

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.

Lifetime of biexcitons and excitons

In solid state physics, the recombination

Carrier generation and recombination

In the solid state physics of semiconductors, carrier generation and recombination are processes by which mobile charge carriers are created and eliminated. Carrier generation and recombination processes are fundamental to the operation of many optoelectronic semiconductor devices, such as...

of biexcitons (XX), which can be regarded as an entity of two weakly bound excitons (X), is often discussed in a simple way of argumentation:
The recombination of any of the two excitons forming the biexciton will result in its disappearance.
Therefore one would expect that the biexciton lifetime

Lifetime

Lifetime may refer to:*Life expectancy, the length of time a person is expected to remain alive*Mean lifetime, a certain number that characterizes the rate of reduction of a particle of an assembly...

, is about half of that of the exciton

Exciton

.

However, some elementary points of view should be taken into account.
For example, a direct radiative recombination into the ground state of the crystal, allowed for the exciton

Exciton

state, is optically forbidden for singlet biexcitons.
Furthermore, the spin structure and the Coulomb interaction of the electrons and the holes forming the excitons and the biexcitons has to be considered.

In a CdSe/ZnSe quantum dot

Quantum dot

, the lifetime of biexciton is measured as 310 ps, and that of exciton

Exciton

is about 290 ps.
The ratio of the two lifetimes

LifeTimes

LifeTimes is an album by Diana Hubbard, released in 1979 by Waterhouse Records 8. In addition to Diana Hubbard, the album includes musical contributions from Chick Corea, Stanley Clarke, John Goodsall, Michael Boddicker, and Patrick Moraz. Diana is the daughter of Scientology founder L. Ron Hubbard...

is not about 2, but about 1.

The final state of the biexciton recombination is an optically allowed (bright) exciton

Exciton

state.
Because of exchange interaction, the excitons are split into optically allowed states (bright,

) and optically forbidden states (dark,

).
The spin flip time between

and

states,

, is expected to be in the range of several ns, i.e., much longer than the bright exciton

Exciton

lifetime.

The time evolution of the probabilities for the population of the biexciton and exciton

Exciton

states,

and

, respectively:

with

, where

and

are the

and the XX radiative recombination times, respectively.

Solving this equation system, the X and XX dynamics can be described quantitatively.
As a result, while decay of XX is monoexponential, the onset of the X line is significantly delayed, resulting in a plateaulike characteristics of the exciton

Exciton

decay curve.

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