Chapter 2 Theory
2.5 Mesh Grids and the Calculation Details
The mesh grids is important in this simulation since there is two equaled valleys while using a mathematical model. The ground state wave function of a particle confined in the potential
with two balanced valleys is expected to extend into the both potential valleys equivalently.
According to our computation experiences, if the density of the mesh is inadequate or not fol-lowing the reflectional symmetry the calculated wave function of the particle will not reveal such symmetry.
We build the model of the ring using COMSOL multiphysics package (www.comsol.com).
This package can generate meshes automatically and conveniently, but the meshes produced are not always decent and balanced for a full three dimensional calculation. While solving our problem in ideal situation we should have a mesh with infinitesimal distance between every two mesh points so a exact result can be obtained. However, practically due to machine power and memory limitations we can only produce a mesh with distances between every two mesh points as small as possible to reduce the error. In this study we monitor the error which comes from mesh by looking into the terms of (2.20). For example, due to a reflectional symmetry (while there are no structural imperfections) the term
G12= Z
Fe1(re) Fe2(re) Vh1;h1(re)dre= 0. (2.26)
Since Fe1(re) and Vh1;h1(rr) are even functions and Fe2(re) is odd (the potential is an even function due to reflectional symmetry, so the ground state of the particle in the potential is even and the first excited state is odd), ideally the above term should be zero. However, if we use the mesh produced automatically by the package with only 59,711 mesh points the value of the term is in an order of 10−7 eV which is close to the energy diamagnetic shift with an order of 10−6 eV in low magnetic field. Therefore, an error rises while the mesh is not appropriately defined.
There are two possible ways to eliminate the error caused by the mesh. The first one is by increasing the mesh size, thus the distance between mesh points decreases. But at the same time requirements for machine power and memory are getting to physical machine limits. If the mesh is generated with an upper bound of 1.02 nm for the distance between two mesh points in the central part of the simulation domain, the calculated electron and hole wave function can
in total can only be used to calculate electron and hole separately, because of the limitation of the machine memory we can not involve enough large number of electron and hole states to expand the excitonic wave function for our a system. Hence we choose an adaptive mesh grids as shown in Fig. 2.3. The distance between mesh points in the central part within a region 50nm×40nm×20 nm is 1.25nm. This mesh follows the symmetry of the system and minimizes the disbalance in the particles’ wave function with the value of (2.26) reduced to an order of 10−15eV which gives a good accuracy. We shorten the length of the outside domain (relative to the center part) to reduce the requirement of machine power. The particles’ wave functions are localized in the central part and they are more extended in x-direction than in y-direction, thus decreasing the domain length in y-direction causes only negligible errors. For consistency of this work we keep the same mesh upon changing the value of ∆h and ∆V . The simulation is performed within a domain with edges 120nm×80nm×80nm with 92,169 mesh points in total.
In this study we take five electronic states and five hole states to expand the exciton wave function since there are only five electron states confined in the system while the structural im-perfections are not included. We find the convergence by keeping the same number of electron states and increasing the number of hole states. The difference of the exciton ground state en-ergy when we take five hole states or four hole states to expand the exciton wave function is only in an order of 10−7 eV. Therefore, we take five hole states as well. In order to maintain a consistency in this work we keep the same computational configuration while varying the values of ∆h and ∆V .
Figure 2.3: Top view of the meshes. (a) The mesh is generated randomly by COMSOL multi-physics package with 219,009 mesh points. (b) The mesh is controlled and used in our simula-tion with 92,169 mesh points.
Chapter 3
Results and Discussion
We use the parameters for the height profile as suggested in Ref. [12] and c = 0.55 is chosen to be the indium content in the IncGa1−cAs quantum ring. Material parameters such as effective masses, band gaps for electron and hole in strained InAs is taken from Ref. [13]. All values of the parameters are listed in Table 3.1. m0 stands for the electron rest mass and ²0 is the permittivity in vacuum.
The structural imperfections are included in this work by considering ∆V = -0.2, -0.1, 0, 0.1, 0.2 (we use (2.8) while ∆V =0 and (2.10) when ∆V 6= 0 ) and ∆h = -20%, -10%, 0%, 10%, 20% with appropriate sets of {ain, aout} listed in Table 3.2. While varying ∆h both the geometry and the confinement potential alter (see Fig. 3.1 and 3.2). Changes in the confinement potential due to changes of ∆V is depicted in Fig. 3.2(d) and 3.2(e). In Fig. 3.3 we present the electronic confinement potentials with both ∆h and ∆V varying.
From our simulation we found that when the potential is symmetric about x=0 plane the exciton wave function are distributed equally within two valleys of the potential at the ring’s rim. However, when we induce small variations in either the ring’s geometry or the potential the wave function collapses into one of the valleys, as a consequence, the spread of the probability distribution of the exciton shrinks.
Fig. 3.4 demonstrates the probability density of the excitons confined in the quantum ring.
While there is no wobbling asymmetry the wave function extends to both valleys equally ( see Fig. 3.4(a) ). If we impose a structural imperfection either by decreasing ∆h or ∆V with 10%
of their original value the exciton tends to be stay in the valley on the negative x side ( Fig.
3.4(b) ). If we increase each of ∆h or ∆V the wave function of the exciton collapses into another valley ( Fig. 3.4(c) ).
Table 3.1: Values of parameters used in this study.
†Parameters inside the IncGa1−cAs/GaAs ring are taken by interpolation.
Figure 3.1: The ring’s height profiles with different ∆h. The ring’s heights are unchanged along the negative x-direction.
We know that the diamagnetic shift coefficient (γ) is proportional to the effective area spanned by the exciton wave function, therefore, if the extension of the exciton wave function decreases the coefficient drops as well. We define γ(∆h, ∆V ) as a function of the parameters
∆h and ∆V and present with the ratio of γ(∆h, ∆V ) to γ(0, 0). We calculate γ with varying only one parameter for the structural imperfection and the ratio is shown in table 3.3. The result demonstrates that with either a small variation in the wobbling asymmetry or a small disbal-anced potential lead to decrease of γ. In table 3.4 we show the ratio while both the wobbling asymmetry and the disbalanced potential are considered, and we obtain the same tendency for γ.
Figure 3.2: Projections of the electronic confinement potential onto x-z plane, the positive x direction is shown. (a): potential of a symmetrically wobbled ring; (b): ∆h=20%, ∆V =0; (c):
∆h=-20%, ∆V =0; (d): ∆h=0%, ∆V =0.2; (e): ∆h=0%, ∆V =-0.2.
Figure 3.3: Projections of electronic potential on x-z plane changing both ∆h and ∆V . (a):
∆h = ∆V = 0; (b): ∆h=20%, ∆V =0.2; (c): ∆h=-20%, ∆V =-0.2.
Figure 3.4: Probability densities for the excitons confined in nano-rings, projected onto the x-y plane with (a) ∆h = ∆V = 0, (b) ∆h = -10% and ∆V = -0.1 and (c) ∆h = 10% and ∆V = 0.1.
(d) the height profile of a symmetrically wobbled ring and gives a reference for the location of the exciton probability densities. The dashed lines indicate the position at x = 11.5nm and x = -11.5nm.
Table 3.3: Normalized coefficients of energy diamagnetic shift as a function on ∆h and ∆V .
Table 3.4: Normalized coefficients of energy diamagnetic shift while changing both ∆h and
∆V .
Notice that when we decrease either ∆h or ∆V , the ratio γ(∆h, ∆V )/γ(0, 0) approaches 0.3. Because for those changes the excitonic wave functions are located at the unchanged potential valley, the effective areas spanned by the excitonic wave functions are similar no matter which parameter we decrease. The same results appear while we decrease both the parameters simultaneously (see table 3.4), which supports our arguments. However, if ∆h and ∆V continue increasing γ keeps dropping. When we enhance both ∆h and ∆V to 20%
and 0.2 respectively the ratio γ(∆h, ∆V )/γ(0, 0) approaches 0.23 and the calculated excitonic energy diamagnetic shift coefficient reaches 10 µeV /T2. The reason for the minification of γ is that the confinement is stronger when imposing two kinds of imperfections than considering only one of them, thus the extension of the exciton wave function is more restricted inside the valley which the structural imperfections are applied. The result conforms with the experimental measurement of a averaging value of 6.8 µeV /T2 [9].
The calculated excitonic ground state energies are shown in Fig. 3.5 and Fig. 3.6. With a symmetrically wobbled ring and balanced confinement potential the excitonic ground state
en-ergy is about 1360 meV which is higher compared with the energies when imperfections are ap-plied. The coulombic interaction between electron and hole is smaller due to a wider extension of excitonic wave function when the excitonic wave function is distributed into both potential valleys, therefore the excitonic ground state energy is higher. If we impose structural imper-fections the excitonic ground state energy would go down because now the exciton is located in one of the potential valleys and the coulombic interaction is stronger because of a reduced extension of the excitonic wave functions. While we decrease either ∆h or ∆V the energies reach 1350 meV (see Fig. 3.5 and 3.6). Because the excitons stay now in the unchanged poten-tial valley, the energies saturate while the degree of the imperfections become larger. However, when the imperfections are imposed involving a raise of ∆V the exciton ground state energies continue dropping. Since in this situation the exciton locates in the valley where the potential depth is decreasing, the excitonic energies are dropping as ∆V increasing. The calculated ex-citonic ground state energies range from 1300 meV to 1340 meV while disbalanced potentials are imposed and the results consist with the experimental measurement of 1320 meV [9].
Figure 3.5: The excitonic transition energies with different (a) ∆h and (b) ∆V . The dashed lines indicate the experimental measurement of excitonic energy: 1320 meV.
Figure 3.6: The excitonic transition energies when changing simultaneously both ∆h and ∆V . The dashed line indicates the experimental measurement of excitonic energy: 1320 meV.
Chapter 4
Conclusion and Future Work
4.1 Conclusion
In conclusion, we calculate the excitonic diamagnetic shift coefficients of wobbled and singly-connected InAs/GaAs semiconductor nano-rings using the mapping and exact-diagonalization methods. Beside a mathematically model with a reflectional symmetry in the confinement po-tential we introduce two possible structural imperfections - wobbling asymmetry and disbal-anced potentials - in our simulation. Within these approaches and taking an appropriate mesh we are able to describe this system. From our simulation we found that with small deviations in the geometries and confinement potentials the excitonic diamagnetic shift coefficient decreases greatly. Our results are in a good agreement with the experimental measurings.
4.2 Previous Works
In Ref. [18] we calculated the excitonic and biexcitonic energies of InAs/GaAs nano-rings with geometry suggested in Ref. [8] and radius 7 nm. The problem was solved self-consistently using our mapping method, and we obtained preliminary results about this nano structure. After that we implemented the diagonalization method and calculated the magnetic response of rings in a weak magnetic field with an asymmetry in the geometry, this result was published in Ref.
[19]. The magnetic susceptibility of wobbled semiconductor nano-rings were calculated in Ref. [20], and we found that the averaged susceptibility shows small temperature effect. The mapping method was used to calculate the energy of concentric triple nano-rings as well, and the calculated inhomogeneous broadening of the excitonic energy peaks was found in a good agreement with the experimental data [21].
4.3 Future Work
It is interesting to figure out the effect of structural imperfections on the diamagnetic shift coefficient of biexciton, and it can be compared with experimental measurement [9]. The reac-tion of charged excitons such as X+ and X− (exciton with an extra electron or a hole) in the system with applying the imperfections is worth studying as well.
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