Study of Self-Trapping Using Tight Binding Model

So far, we have learned that the trigger of self-trapping is a short- range type electron-lattice interaction. It would be informative to study self-trapping in various situations with a unified scheme paying attention to the short-range interaction using tight binding models. Although here we will refer to an electron in the conduction band, the following arguments are applicable for a hole in the valence band and a Frenkel exciton.

The Hamiltonian we consider hereafter is

The system consists of an array of sites defined by a lattice vector T, (Fig. 4.7). Each site has an atomic (Wannier) orbital (pt{r) and a local lattice distortion Q, with an elastic constant C. We use bra and ket presentation. Here s, is the site diagonal energy (depends on the site / in general) of an electron which interacts with the lattice distortion Qt by a short-range-type interaction with a coupling constant Д which corresponds to the deformation potential constant £d in the model in Section 4.1. An electron moves in the system with the electron transfer energy tir between |ф,) and |ф['). The kinetic energy of the lattice is omitted since we will take the adiabatic approximation. This Hamiltonian can be applicable for various situations including, pure crystal (Section 4.2.2), low-dimensional system (Section 4.2.3), random system such as alloy or amorphous, confined system as quantum well (Section 4.2.4), and impurity (defect) system (Section 6.1.2). The energy spectrum of he is given in Appendix V.

Sum Rule on the Lowest (Relaxed) States

Our task is to find out the lowest (relaxed) state of the Hamiltonian (4.15) allowing the lattice to deform and to check the possibility of self-trapping. This can be done by calculating the adiabatic potential t/({(?,}) as a function of {(?,} and find its minimum. As for the extremum (minimum or maximum) {

At the extremum dU{{Qi})/dQi = 0, we then have

where xp) = Z,c,|^>,). This rule holds for any form of the electron part in the Hamiltonian, i.e., either for impurity or alloy, and helps us to figure out a gross picture of a free state and self-trapped state (Fig. 4.7). For example,

(a) free state in pure systems: e, = const.

The wave function is extended as much as possible over the crystal and the lattice distortion is negligibly small.

Here N is the number of total sites, e is constant site energy.

(b) self-trapped state:

The wave function is localized at a particular site /, accompanied by a strong lattice distortion at /.

where fLR _ /32/2C is the maximum lattice relaxation energy for a completely localized electron.

Illustration of a free state and self-trapped state. The color density of each circle indicates schematically the amplitude of the trial wave function

Figure 4.7 Illustration of a free state and self-trapped state. The color density of each circle indicates schematically the amplitude of the trial wave function: Eq. (4.22).

Homogeneous Pure System

First, we examine homogeneous pure systems consisting of orthorhombic (2Jx + 1) * (2Jy + 1) x [2]z + 1) lattice sites, where /,• is an integer (Fig. 4.8) [8]. The site Г/s are arrayed along / = (/*, ly, lz),

The site energy s in the Hamiltonian (4.15) is common for all sites, and hence can be set as s = 0. The transfer energy is assumed to be restricted to nearest neighbors:

The model describes various situations, including

(1) infinite isotropic system

exact 3-dimension Jx = ly = lz = oo exact 2-dimension/х = /y = oo, /2 = 0 exact 1-dimension Jx = so, / = /2 = 0

(2) infinite anisotropic system

quasi 2-dimension tx = ty» tz quasi 1-dimension tx = ty«: tz

(3) quantum well of infinite potential barrier

one-dimensional QW (slab) Jx = ly = oo, l2 is finite two-dimensional QW (wire) Jx = oo, ly and lz are finite three-dimensional QW (dot) Jx, /y and lz are finite.

An example of (2J + 1) x (2J + 1) x (2J + 1) array model with ] = 3, J = 2, and ) = 0. The color density of each circle indicates schematically the amplitude of the trial wave function (4.22)

Figure 4.8 An example of (2Jx + 1) x (2Jy + 1) x (2Jz + 1) array model with ]x = 3, Jy = 2, and )z = 0. The color density of each circle indicates schematically the amplitude of the trial wave function (4.22).

Hereafter we search for the minimum of the adiabatic potential = ( where Qt is a local lattice distortion

at site T), with a following trial function

Here c0 is a normalization constant and к/s ((/ = x, y, z)) are variational parameters: к,- = 0 corresponds to an extended state along the z-th direction, while a set of large к/s corresponds to a localized state at the origin. The expecting value of the Hamiltonian (4.15) for this trial function (4.22), which has been extremalized with respect to {(?,} is shown to be

where

Here £lr = /?2/2C. As have done in the previous section, we will minimize U[kx, ку, к2) with respect to к/s. The results are as follows [8].

(a) infinite isotropic system

In the rigid lattice {Qt = 0}, the electron eigenstates and energies are

Here a is the lattice constant and the full width of the electron energy band is 2B = 4t x 6 (б-dimension: d = 1, 2, 3) centered at £. Hence the lowest energy is e - В at k = 0.

Figure 4.9 shows the total energy U as a function of the localization index «, = 1 - exp(-/c,), which varies 0 < cr, < 1. The index a,- = 0 corresponds to an extended state along the z'-th direction, and ax= ay = az= 1 corresponds to a completely localized state at the center. Here the coupling constant g is defined as

The total energy U[= E) as a function of the localization index ai(i = x,y, z) for exact (a) three-, (b) two- and (c) one-dimension [8]

Figure 4.9 The total energy U[= E) as a function of the localization index ai(i = x,y, z) for exact (a) three-, (b) two- and (c) one-dimension [8].

It is found self-trapping occurs approximately when g > 1. To find the exact critical coupling constant, one needs to include a finite energy gain in the self-trapped state by an electron transfer to neighboring sites.

(a) 3-dim

Extended F state (ax = ay = az = 0) is always stable or at least metastable with respect to distortions. As the coupling constant 9 = Elr/B increases, S-state (ax = ay = az ~ 1) first appears as metastable for g > 0.605 and then becomes stable for g > 0.906; the F-S transition is the first order. A potential barrier always exists which separates F and S.

(b) 2-dim

Extended F state (ax = ay = 0) is at first stable for g < 0.706. Self-trapped state S appears first as metastable for 0.706 < g < 0.840 and then becomes stable for 0.840 < g. Finally, F becomes unstable for $ > 2.00.

(c) 1-dim

Extended state (ax = 0) is unstable for nonzero coupling constant g. The ground state changes its character from F to S continuously as g increases.

Infinite Anisotropic System

In discussing infinite anisotropic systems, it would be sufficient to study tx = ty * tz case and introduce the degree of anisotropy,

which varies between 0 and oo. If we compare the present tight binding model with the effective mass approximation, it can be related with X = mz/mx, where m, is the electron effective mass along the /-th direction.

X «: 1 corresponds to quasi one-dimensional (chain-like) systems.

X » lcorresponds to quasi two-dimensional (layered) systems.

X = 1 to the isotropic three-dimensional systems.

Figure 4.10 shows (a) the phase diagram of the ground state of an electron with respect to the anisotropy X and the coupling constant g and (b) the anisotropy dependence of the energies of the F state, the S state and the potential barrier height for g = 1 case. It is found that the character of F-S transition is qualitatively the same as in isotropic three-dimension, and there always exists a potential barrier between F and S. But The barrier height decreases as the anisotropy increases (Я 0, oo).

(a) The phase diagram of a relaxed electron with respect to the anisotropy X and the coupling constant g

Figure 4.10 (a) The phase diagram of a relaxed electron with respect to the anisotropy X and the coupling constant g. (b) the anisotropy dependence of the energies of the F state, the S state and the potential barrier height for S(=£lr) = В constant {g = 1) case [8].

Quantum Well

Here, we discuss the effect of quantum confinement to self-trapping, only in a case of isotropic transfer (tx = ty = tz = t), for simplicity. In order to avoid any confusion, we will name here as follows: d-dimensional quantum well means that an electron is confined with respect to {/-dimension, where it moves free in the rest 3-d dimension.

(a) one-dimensional QW (slab):

Figure 4.11a shows the phase diagram of an electron in onedimensional QW (slab) with respect to the well size Jz and the e-L coupling g = ELR/B = 61] is the half band width in perfect 3-dim.). The character of F-S transition is qualitatively the same as in isotropic 3-dim., but as the well width becomes thin, i.e., when Jz = 0 it behaves as in the exact 2-dim. described above.

(b) two-dimensional QW (wire):

Figure 4.11b shows the phase diagram of an electron in two- dimensional QW (wire) with respect to the well size Jz and the e-L coupling g = ELR/B = 61) is the half band width in perfect 3-dim.). The character of F-S transition is qualitatively the same as in isotropic 3-dimension, but as the well width becomes thin, i.e., when Jz = 0, it behaves as in the exact 1-dim., where the transition occurs continuously.

(a) The phase diagram of a ground state of an electron in deformable lattice of

Figure 4.11 (a) The phase diagram of a ground state of an electron in deformable lattice of (a) 1-dim. QW (slab) and (b) 2-dim. QW (wire), with respect to the well size and the coupling constant g. A metastable state, if it exists, is shown in the parentheses [8].

 
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