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The ground-state energy and its derivate of the acoustic polaron in free-standing slab are calculated by using the Huybrechts-like variational approach. The criteria for presence of the selftrapping transition of the acoustic polaron in free-standing slabs are determined qualitatively. The critical coupling constant for the discontinuous transition from a quasi-free state to a trapped state of the acoustic polaron in free-standing slabs tends to shift toward the weaker electronphonon coupling with the increasing cutoff wave-vector. Detailed numerical results confirm that the self-trapping transition of holes is expected to occur in the free-standing slabs of wide-bandgap semi-conductors.

The electron mobility is important because it is a parameter which associates microscopic electron motion with macroscopic phenomena such as current-voltage characteristics. The mobility will be changed markedly if electron state transforms from the quasi-free to the self-trapped. Moreover, many physical properties of photoelectric material are also influenced by the electron state. The self-trapping of an electron is due to its interaction with acoustic phonons. The polaron problem had also gained interest in explaining the high-

Various calculations for the ground-state energy of the acoustic polaron as a function of the e-p coupling strength have led to a discontinuous transition from a quasi-free state to a trapped state [

It is determined in our previous works [

In this work, a new Hamiltonian describing the deformation potential interaction between the electron and the acoustic phonon in free-standing slab systems will be derived. The self-trapping transition of the Q2D acoustic polaron will be discussed.

The interaction between the electron and the longitudinal acoustic phonon (e-LA-p) in free-standing slab is given by [

where

In the free-standing slab, the displacements can be taken as the form:

where

where

The

Inserting Equations (3) and (4) into (5), one can obtain the following relation

where

Inserting Equations (6), (3) and (2) into (1), the e-LA-p coupling Hamiltonian is then written as

Here the e-p coupling function

Then the e-LA-p system Hamiltonian in the free-standing slab is written as

where

In this section, a Huybrechts-like variational approach [

Firstly we carry out a unitary transformation

where a is a variational parameter and will tend to 0 in the strong coupling limit and 1 in the weak coupling limit. Therefore, the Hamiltonian turns into

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

And

where

Inserting (12a) and (12b) into (11) and performing the second unitary transformation

The Hamiltonian finally becomes the following form:

Here we have omitted the multi-phonon processes, which contribute less to the polaronic energy.

The displacement amplitude in the second unitary transformation is determined as

by the diagonalization of the vital important part of_{ }

The ground-state energy can be calculated by averaging Hamiltonian (14) over the zero-phonon state

By some standard treatments, the variational energy of the polaronic ground-state can be obtained as follows

The e-LA-p coupling constant is given by

In Equation (18) the variational parameters

The variational calculations for the ground-state energies of the acoustic polaron in free-standing slabs are numerically performed for different thickness of the slab

As can be seen in

It is worth noting the critical values of the e-p coupling constant increase with the increasing thickness of the slab. For example, when the cutoff wave-vector

The

Now we use the criterion of the _{ }(0.24 for GaN and 0.57 for AlN) can get the same order of magnitude as

Holes have larger effective masses than electrons and must be easier to be self-trapped. For GaN, which has the light and heavy-hole masses 0.37 and 0.39 [

but larger than that in slab. Therefore the light-hole in AlN can be self-trapped only in the slab with

The critical coupling constant for the self-trapping transition of the acoustic polarons in free-standing slab systems is determined by calculating the ground-state energies and the derivates of the acoustic polaron. The value of the criterion

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