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S2211-3797(18)30194-3 https://doi.org/10.1016/j.rinp.2018.02.016 RINP 1248

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Results in Physics

Received Date: Revised Date: Accepted Date:

26 January 2018 5 February 2018 7 February 2018

Please cite this article as: Imran, A., Jiang, J., Eric, D., Zahid, M.N., Yousaf, M., Shah, Z., Optical properties of InAs/GaAs quantum dot superlattice structures, Results in Physics (2018), doi: https://doi.org/10.1016/j.rinp. 2018.02.016

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Optical properties of InAs/GaAs quantum dot superlattice structures Ali Imrana, Jianliang Jiang a,*, Deborah Erica, M Noaman Zahida, M Yousafb, ZH Shahc a School of Optoelectronics, Beijing Institute of Technology, Beijing, China b Department of Material Science and Engineering, Peking University, Beijing, China c School of Science, University of Management and Technology, Lahore, Pakistan *Corresponding author: [email protected]

ABSTRACT Quantum dot (QD) has potential applications in modern highly efficient optoelectronic devices due to their band-tuning. The device dimensions have been miniatured with increased efficiencies by virtue of this discovery. In this research, we have presented modified analytical and simulation results of InAs/GaAs QD superlattice (SL). We have applied tight binding model for the investigation of ground state energies using time independent Schrodinger equation (SE) with effective mass approximation. It has been investigated that the electron energies are confined due to wave function delocalization in closely coupled QD structures. The minimum ground state energy can be obtained by increasing the periodicity and decreasing the barrier layer thickness. We have calculated electronics and optical properties which includes ground state energies, transition energies, density of states (DOS), absorption coefficient and refractive index, which can be tuned by structure modification. In our results, the minimum ground state energy of QDSL is achieved to be 0.25eV with a maximum period of 10 QDs. The minimum band to band and band to continuum transition energies are 63meV and 130meV with 2nm barrier layer thickness respectively. The absorption coefficient of our proposed QDSL model is found to be maximum 1.2x104cm-1 and can be used for highly sensitive infrared detector and high efficiency solar cells. Keywords: Quantum dot, superlattice, eigen energy, intermediate band, density of states

1. INTRODUCTION. Recent research advancement in fabrication of nanostructures materials has resulted in the formation of many zero dimensional (0D) heterostructure exhibiting spatial confinement of electronic energies [1-3]. This advancement has created much more opportunities for the controlled modification of carrier states in these entities, and re-engineering of optical, electronic, and thermoelectric properties of many technologically important semiconductor materials [4-6]. The confined structures cause energy confinement which make these QD which are often referred as artificial atoms good candidates for infrared photo detector applications. Recently, these structures have also been applied for high temperature thermoelectric devices. The most 1

advanced fabrication techniques are also used to fabricate QD laser, light storage devices, QD molecules and quantum computers [7-12]. Different kinds of QD structures can be grown through various manufacturing techniques. Among these various nanoscale hetero-structures, the QD array formed by multiple layers of vertically aligned QDs has attracted great interest. In comparison to conventional quantum well or multiple quantum well structures, QD structures which consists of multiple arrays of QD may have many advantages in applications due to its modified DOS and optical selection rules [1315]. For example, due to relaxed intra-band optical selection rules in structure, they are capable of absorbing normally incident radiation while it is not possible in single quantum well structure. So, it is more practical to deal with stacks of semiconductor QDs , which are also referred to QDSL [6,16]. SLs are alternating ultrathin epitaxial layers, in which a 1D potential is superimposed on the crystal potential of the background crystal. The 1D is formed by the band offset of the hetero-layers in a compositional SL or by periodically growing n- and p-doped semiconductor layers, separated by intrinsic layers in the doped SL. While the real crystal potential has the same period as the lattice points, the period of the SL potential is the same as the alternating layers. For convenience, one period of the alternating layers will be called the basis of the SL. In the bulk semiconductor, different bases or primitive cells will result in different crystal potentials, thus different energy bands. Similarly, one may expect that the minibands in the SL will change as the SL basis changes. In the past since its inception, the SL basis has always been barrier-well structure for simplicity of analysis. With modification of the basis, interesting effects are expected. This kind of structure grown by molecular beam epitaxy can be completely random, partially symmetric, such as QDSL with vertical QD site correlation or may have very highly symmetric [17-21]. The self-organization of pyramidal islands during epitaxial growth has resulted in the formation of 3D QD crystals with the dots arranged in a trigonal lattice with a face-centered-cubic-like vertical stacking sequence. Recently, the tuning of lateral and vertical regimentation in selforganized QDSL by changes in the spacer thickness and growth conditions has been demonstrated [20]. Other QD synthesis techniques, such as electrochemical self-assembly, have also led to a high degree of lateral regimentation resulting in hexagonal QD arrays [22,23]. These structures have been studied intensively in theory, experiment, and computation [24-35]. One of the most important issues in this research deals with the induced energy levels and associated wave functions (WF). The ground and excited state spectrums are of basic physical interest and are crucial for designing photoelectric devices. Furthermore, WF overlap results in inter-dot coupling and suggests the possible creation of artificial molecules and a new computing concept [36,37]. The energy levels can be investigated by methods like photoluminescence, photocurrent and photo reflectance [38,39]. The information obtained by these methods, in certain circumstances, can be limited to lower energy states [39].

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In this article, we have analyze the band structure of a 3D InAs/GaAs QDSL using timeindependent SE. The aim of this study is to investigate the confinement of electron energies in QDSL structures and the variation by changing the periodicity of the SL. This research work has key importance in the studies of novel intermediate band (IB) solar cells structures. The effect of change in the barrier layer width has also been discussed. We have assumed the structure in ideal 3D symmetry where the total WF consist of three WF in all 3D. The ground and first excited state energies are calculated using simulation approach while transition energies and DOS are investigated later sections of this article.

2. THEORY AND METHOD. We have supposed 3D cubic QD arrays as shown in Figure: 1(a). All QDs have same size along x, y and z axis embedded in matrix of host material having cubical dimension of 3nm. The width of BR material is taken to be lb. The length L=lb+ lqd is a measure of one periodic (combined unit of QD and BR material). The effective mass of QD is take to be m*qd = 0.023m for InAs and m*b=0.063m for GaAs, where m stands for electron mass. The potential values are taken to be zero inside QD and 0.067eV in BR material [40-42].

Figure: 1 (a) Schematic diagram of 3D QDSL (b) one dimensional potential of QD and BR material.

The total energy can be calculated by time-independent SE in 3D.

(1)

where the position in 3D can be taken as, (2) The potential can be divided into 3 potentials in each dimension, (3) 3

Similarly the WF can be divided in three sub WFs for each dimension, (5) The solution of the SE equation gives the energy of this system which can be given as, =

(5)

where n is the Bloch band index. The solution of the equation can be found by simplifying the system to 1D by finite SLs or multiple quantum wells method, where the electron WFs are delocalized and can be in any of the quantum wells. These states are referred as Bloch states [43]. The envelop function in the QD region can be written as (6) and in the BR region as (7) where (8) In case of E > V, the value of kb is positive integer. The WF for a travelling wave state of the form exp (ikx) can be given as, (9) So if the SL is infinite, then a particle is likely to be found in any well, which means that its WF must be periodic with the lattice, (10) then (11) (12) by using this periodic form and applying the first Ben Daniel Duke (BDD) boundary condition at x = L. (13) Using the WFs of equations (6) and (7), then: (14) by employing the same periodicity condition and applying the second BDD boundary condition also at x = L.

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(15)

(16) Now the first BDD boundary condition at x = lqd. (17) and the second BDD boundary condition at x = lqd. (18) Equations 14, 16, 17 and 18 can be written in the matrix form as

(19)

which implies that, for a solution other than the trivial A, B, C and D are zero, the determinant of this matrix must be zero,

=0

(20)

evaluating the determinant gives (21) if E < V then kb becomes imaginary which can be written as i , where (22)

substituting for kb into equation and using the following identities:

(23)

we get

3. RESULTS

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(24)

We have applied above discussed method in COMSOL Multiphysics to calculate the Eigen energies in periodic QD structures in 1D. We have used the experimentally verified parameters as basement of these calculations [41]. We have applied BDD boundary conditions on the interface between the QD and BR interface, while Drichlet boundary conditions on the outer boundaries. The potentials in the QD and BR regions taken in our simulation are shown in Figure 2. Our results are more accurate compared with results in literature [44]. Figure 2(a) is a representation of potential profile of QD and BR material along x axis. We have made two kinds of investigation, in the first part we have varied the periodicity of system, i.e. the Bloch band index. This causes the increase in the band width with increasing order of periodicity and results in lowering the ground state energy of SL system. We have taken three values 1nm, 2nm and 3nm of BR width. The minimum ground state energy is obtained at 1nm which can be seen in Figure 2(c). The reason for this is the delocalization of electron WF and coupling with neighbor electrons. The coupling has to obey the Pauli Exclusion Principle and electron energy levels split into bonding and antibonding states with one having lower energy and the other with higher [4553]. Hence the overall ground energy state of this system is shifted towards lower energies. Table.1: Material parameters used for simulation [41,42] Name

Expression

Value

Description

Lqd

3[nm]

3E−9 m

Width of QD

lb

3[nm]

3E−9 m

Width of BR

meqd

0.026 *me_const

2.09E-32 kg

Electron effective mass of QD

meb

0.067 *me_const

6.10E-32 kg

Electron effective mass of BR

mhqd

0.40 *me_const

3.64E-32 kg

Hole effective mass of QD

mhb

0.45 *me_const

4.09E-32 kg

Hole effective mass of BR

Egqd

0.471[V]

0.471 V

Band gap of QD

Egb

1.403[V]

1.403 V

Band gap of BR

CBO

0.544[V]

0.544 V

Conduction band offset

VBO

Egb - Egw - CBO

0.388 V

Valence band offset

hmax

0.1[nm]

1E−10 m

Max mesh element size

Figure 2(b) shows the potential profile with variation in the BR width with slowly decreasing to zero. It can be observed in Figure 2(d) that the energy starts splitting into minibands. The reason is the same as discussed earlier. The crystal field starts growing with due to hetrostructure crystal potential effect. The neighboring coulomb forces becomes so strong which results in band splitting. The number of the new formed minibands depends on the DOS. As the BR width decreases, the atoms get more closely packed and the electronic clouds get shared resulting in increased DOS. This increment increases band width resulting in decrease in bandgap transition energies. In our case we have taken two eigen energy states i.e. the ground state e0 and the first excited state e1 for a system of two QDs. In this least case, the QDs boundaries are supposed to

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be touching. In this case the two bands overlap and the system behaves as metallic in character. This is possible theoretically but difficult to achieve in practical.

Figure: 2(a) Eigen energy values of single QD. (b) Eigen energy values with third periodicity level (c) Ground state energy with increasing periodicity (d) Band split due to delocalized electronic WF coupling

Figure 3(a) is the transition energy of new SL system. The gap energy is very small at lower values of inter-dot spacing, but increases exponentially towards constant values with increase in BR width and at very large distance behaves as single QD. The reason is that at larger distance the coulomb forces became inactive for QD materials. So instead of SL structure, it behaves as individual QD. Figure 3(b) shows that the DOS in QDSL has sharp values while changing the dimensions of the system. This is a unique behavior on nanostructured SL system that there are zero DOS in certain energy ranges, unlike the bulk materials where we have continuous changes over a higher range of energies. This feature makes this material system very attractive for highly sensitive infrared detectors and IB based solar cells. Hence the DOS can be controlled during fabrication process by controlling the width of the BR material accurately. This DOS greatly affects the photovoltaic effect in IB solar cells where multiple photon absorption process occurs. The higher DOS in IB compared to valence and conduction band may lead to much higher efficiencies in solar cells. 7

Figure: 3 (a) Transition energy as function of BR width for SL system (b) variation in DOS as function of BR width.

The absorption coefficient highly depends on the transition energy of SL material, and the transition energies are indirectly proportional to the width of BR material and the size of QD. We have calculated the absorption coefficient at 2nm, 4nm, 6nm and 8nm respectively. It can be seen in Figure 4(a) that the absorption process starts at much lower energy values at 2nm BR width. The trend leads to decrease in the absorption compared with higher width values. The dimensional variation highly affects the overall electronic energy of system. This also affects the optical response of the SL system when exposed to incident radiation, and hence the refractive index. The variation in the refractive index with corresponding BR width is shown in Figure 4(b). We have applied herve’s method to calculate refractive index [54].

Figure: 4 (a) Absorption coefficient different width of BR for QDSL (b) Refractive index of QDSL

In summary, we have investigated the electronic and optical properties of InAs/GaAs QDSL system in which we have theoretically observed that the ground and excited state energies of 8

system highly depend on the dimensional size and width of BR material. The proposed model provides very strong basement for the experimentalists which can result in highly sensitive and efficient future optoelectronic devices. 4. CONCLUSION The band structure of a simplified InAs/GaAs QDSL has been investigated in this research. We have applied 1D time-independent SE with non-parabolic effective mass approximation. It has been investigated in our analysis that the minimum ground state energy for QDSL can be obtained up to 0.25eV with a system of 10 QD in a 1D lattice. The minimum band to band and band to continuum transition energies are calculated to be 63meV and 130meV respectively. The maximum achieved absorption coefficient is 1.2x104 cm-1 and the refractive index up to 4.16. This research article provides extensive details for the mini band formation in QDSL structures and their optoelectronic properties. References. [1] [2]

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