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The variational effective mass with respect to the e-p coupling constant for different values of cutoff wave vector is performed in quantum dot. The self-trapping transition of acoustic polaron in quantum dot is reconsidered by character of the effective mass curve varying with the e-p coupling. The holes are determined to be self-trapped in AlN quantum dot systems.

The luminous property of the photoelectric materials has been explored by the trapping electrons for a long time [

It is well-known that the effective mass is corresponding to the carrier mobility, and has more experimental comparability. So that it will be an effective method to count the traps of electron by measuring the effective mass of the acoustic polaron. Effective mass theory had been used for years in several branches of modern physics like nuclear physics or solid-state physics. This theory is a useful tool for studying the motion of carriers in pure crystals and also for the virtual-crystal approximation to the treatment of homogeneous alloys (where the actual one-electron potential is approximated by a periodic potential), as well as of graded mixed semiconductors.

In this paper, we study the interaction of e-LA-p in quantum dots by using a Huybrechts-like variational approach [

Considering the acoustic plaron in nanoscale spherical is confined in three dimensional boxes. The e-LA-p system Hamiltonian in the nanoscale spherical had been written as the following form in our previous work:

where

In case of breathing mode, the e-p coupling function can be given by

where j_{0} is correspondingly the spherical Bessel function of order zero, _{1} is the spherical Bessel function of order one,

The j_{2} in above equation is the spherical Bessel function of order two and can be presented as

Firstly, we carry out two unitary transformations to Equation (1)

where a in above is the variational parameter, which will tend to 0 in the strong coupling limit and 1 in weak coupling limit.

Then let us introduce the linear combination operators of the position and momentum of the electron by the following relations.

where

The Hamiltonian finally transforms to the following form:

Then the ground-state of the acoustic polaron can be calculated by averaging Hamiltonian (6) over the zero- phonon state |0ñ, for which we have

In Unitary transformation U_{1}, a → 0 is corresponding to the strong-coupling limit. Using (6) and (7), by some treatments, the ground-state energy in spherical QD can be obtained as:

Making a variation to a

One can obtain the following equation

And the ratio of effective mass to band mass can be presented as [

In Unitary transformation U_{1}, a → 1 is corresponding to the weak-coupling limit, the ground state energy of the polar on can be obtained as follows:

Then make a variation to a:

The ratio of effective mass to band mass is [

For purpose of comparison facility, we have expressed the energy in units of mc^{2} and the phonon vector in units of mc/ħ in the calculations. As can be seen in

Comparative observation on the numerical effective mass results of the

As can be seen in _{c} ≈ 0.320, for k_{0} = 5, which is called the “phase transition” critical point, where the polaron state transforms from the quasi-free to the self- trapped [_{0} = 10 and 20, the critical points are at α_{c} ≈ 0.111 and 0.062, respectively. It is obviously that the critical point α_{c} shifts toward the weaker e-p coupling with increasing the cutoff wave-vector k_{0}. The character of the critical coupling constant varying with the cutoff wave-vector k_{0} is corresponding to the previous papers [

The α_{c}k_{0} had been used as a criterion for the self-trapping transition of the acoustic polaron. Acoustic polaron can be self-trapped if the material’s αk_{0} larger than the α_{c}k_{0}. In this work the α_{c}k_{0} are close to 1.6, 1.1 and 1.2 for k_{0} = 5, 10 and 20, respectively. The AlN has the αk_{0} closed to 1.16 and 2.79 for light and heavy holes, respectively [

The trapped electron influences luminescence properties of photoelectric material. The self-trapping transition of acoustic polaron in quantum dot has been reconsidered by calculating the effective mass of the acoustic polaron. It can be concluded that the holes in AlN are expected to have the self-trapping transition in quantum dot systems. Some criterion values (α_{c}k_{0}) of the acoustic polaron in spherical QD systems are smaller than those in 3D and 2D systems. Therefore, the self-trapping transition of the acoustic polaron in spherical QD is easier to be realized than that in 3D and 2D systems. The conclusion meets the general sense that the e-p coupling effects will be substantially enhanced in confined quantum structures.

This work is supported under Grant No. 11147159 from the National Natural Science Foundation of China.