Effects of Confinement Energies on Lead Sulphide and ...article.aascit.org/file/pdf/9770795.pdfDepartment of Physics, Akwa Ibom State University, Ikot Akpaden, Nigeria Email address

AASCIT Journal of Physics 2017; 3(4): 28-35

http://www.aascit.org/journal/physics

ISSN: 2381-1358 (Print); ISSN: 2381-1366 (Online)

Keywords Confinement Energies,

Quantum Dots (QD),

Brus Equation

Received: October 19, 2017

Accepted: November 1, 2017

Published: November 25, 2017

Effects of Confinement Energies on Lead Sulphide and Indium Phosphide Quantum Dots Within Brus Equation Model

Uduakobong Sunday Okorie

Department of Physics, Akwa Ibom State University, Ikot Akpaden, Nigeria

Email address [email protected]

Citation Uduakobong Sunday Okorie. Effects of Confinement Energies on Lead Sulphide and Indium

Phosphide Quantum Dots Within Brus Equation Model. AASCIT Journal of Physics.

Vol. 3, No. 4, 2017, pp. 28-35.

Abstract Quantum confinement effect in semiconductor quantum dots (QD’s) of Indium

Phosphate and Lead Sulphide has been studied within the framework of Brus Equation,

using the particle-in-a-box model. The two nanocrystals used exhibit a size dependence

phenomenon as predicted by the model used. The results indicate that ground state

confinement energy is inversely proportional to the dot size. As such, when the radius of

the dot increases, its confinement energy decreases without getting to zero. i.e., the

lowest possible energy for the quantum dot sample is never zero. This phenomenon has

made the nano-particles considered more relevant even in today’s world of technology.

1. Introduction

With discovery of physical properties of semiconductor nanostructures, much research

has been carried out to make use of this reduced dimensional structure for noble

applications. The study of low-dimensional semiconductor heterostructure quantum dots

(QDs) is one of the main subjects in condensed matter Physics owing to their application

to optoelectronic devices like light emitting diodes [1] and lasers and solar cells [2].

Quantum dots are semiconductor nanoparticle whose excitons are confined in all three

spatial dimensions. It is essentially a tiny zero-dimensional semiconductor crystal with size

in the order of nanometers, hence, the name “dot or island”. It is often called artificial atom

because of its quantum properties and interactions similar to bulk semiconductor materials.

One of the most important optical and electrical properties of Quantum Dots is the

ability to adjust their bandgap and therefore control their light absorbance and emission

frequencies according to their desired purpose. This is only possible through the

quantization of their energy levels.

The size of the dots greatly affects the optical properties of these nanocrystals. It goes

a long way to change the colour emitted or absorbed by the crystals, as a result of the

energy levels within the crystals.

The dot size has an inverse relationship with the energy level of its band gap; this

phenomenon has effect on the colour and frequency of light emitted. Smaller dots emit

higher energy light that is bluer in colour, whereas larger dots emit lower energy light

which are redder in colour.

The width of the quantum dot band gap depends on its size and chemical composition,

making it easy to tune absorption and emission spectra, which is impossible for atoms,

but desirable for optical properties [3].

29 Uduakobong Sunday Okorie: Effects of Confinement Energies on Lead Sulphide and Indium Phosphide

Quantum Dots Within Brus Equation Model

One of the most fascinating effects of nanoparticles occurs

within the ambient studies of the physics of electrons, atoms

and photons. It’s characteristic effects is observed in dot

particles of various shapes in the range of little nanometers.

Interesting electronic and optical properties have been

acquired by these tiny and unseen nanoparticles.

Quantum dots combination can be controlled sufficiently to

obtain a perfect crystal. This phenomenon is macroscopically

impossible. Louis Brus first determined that an electron and

hole created when the dot absorbs light are bound together

within the confines of a box using perturbation theory [4]. This

led him to the equation called Brus Equation.

QDs are quite interesting as they enable the study of

semiconductors on small length scale. In these materials,

photon of energy greater than band gap causes the maximum

length of separation between an electron and hole at which

they are still linked by Coulombic attraction forces is called

the exciton BohrRadius. Its value varies depending on the

semiconductor material [5]. As the particle size approaches

the exciton Bohr radius, the charge carriers are confined in

three dimensions. This phenomenon known as quantum

confinement causes the continuous band of the bulk to split

into discrete, quantized levels [6].

Confinement in quantum dots can be seen arise from

reduction of the dot’s dimension and doping of the dot

material, in which the resultant effect is the increase of the

dot’s confinement energy [7].

Most times, Quantum confinementnormally results in the

enlargement of the band-gap. This in-turn decreases the size of

the quantum dots [8]. This confinement results in properties

that are not seen in bulk form of materials. A typical example

is silicon which is known to be a poor light emitter in its bulk

form due to its indirect band gap. When it is confined as

quantum dot, it emits light [9]. Two fundamental factors

contribute to the variance observed between quantum dot

properties and its bulk counterpart. Firstly, there exist a larger

surface to volume ratio in nanoparticles; Secondly, QDs have a

tunableband-gap as shown in figure 1 below:

Figure 1. Splitting of valence band and conduction band into discrete

energy levels as a result of quantum confinement effect [8].

Figure 2. Changes in the photoluminescence colour of colloidal solutions of

CdSe QDs [10].

The change in colour of an optically clear solution of

Cadmium Selenide quantum dots with variation in particle

size is shown in Figure 2 above [10].

Semiconductor QDs areattracting growing interest from

the sensor research.

One of these lie within the ambient of advanced IR image

sensors and THz detectors. This has been viewed recently as

a potential solution inaddressing challenges in diagnostics

and therapeutics [11]. In the manufacturing processes, the dot

size can be rebranded to obtain a nanocrystal suitable for

optical imaging [12].

Quantum dot technology has been used recently to

manufacture a start up device called Store Dot which is used

to revive dead phone batteries back to life within a very short

period. These dots are peptides that are altered to possess

optical properties and the ability of generating charges for

optimum operation of device being used. The Store Dot uses

nanocrystal solution in the place of electrolyte, being used in

traditional batteries to generate electrons.

Quantum Dot application has yielded much interest in

structural and functional imaging to study the interactions

between cells and between a cell and its environment in

diseased tissues [11], in cancer diagnosis [13], in lymph-node

mapping during biopsy and surgery [14] and in biomedical

applications [15].

Several theoretical methods have been used to investigate

this concept. This includes: Tight-Binding Approach (TBA)

[16], the K.P. method [17], Effective-Mass Approximation

(EMA) [4] and most recently, the Finite-Depth Square-Well

Effective-Mass Approximation (FWEMA) model [18],

Potential-Morphing Method (PMM) [19] and Single Band

Toy Model (SBTM) [20].

Baskoutas et al. [21] calculated the exciton energy of the

narrow band gap colloidal PbS, PbSe and InAs QD using the

PMM, using an assumption of a single dependent dielectric

function.

Kumar et al. [22] also used k.p. model to calculate the

shape and size dependent electronic properties of

GaAs/AlGaAs QD’s. This model was adopted due to its

accuracy for modeling the band structure near the first

AASCIT Journal of Physics 2017; 3(4): 28-35 30

Brillouin zone [23].

Ekong and Osiele [24] employed a quantum confinement

model to study different shapes of nanocrystalline silicon

(nc–Si) QD, within the limits of an effective diameter of 3nm.

This research seeks to demonstratehow the Brus equation

can be used to obtain the confinement energy at various dots

radii in other to deduce the confinement nature associated

with the individual dot understudy, which are Lead Sulfide

(Pbs) and Indium Phosphide (InP). The theoretical

framework of this research is presented in section 2, results

and discussion in section 3, and finally conclusion in section 4.

2. Theoretical Framework

The theoretical framework adopted for this discussion was

first proposed by Brus [4]. This framework relies on

“Effective mass Approximation”, where an exciton confined

to a spherical volume of the crystallite is put into

consideration with the mass of electron and hole being

replaced with effective masses ( em and hm ) to define the

wave function:

2 2

2 * * 2

1 1 1.786( )

8 4g bulk

e h o r

h eE qd E

R m m Rπε ε

= + + −

(1)

Here, h , e , R , *

em , *

hm , oε , oε , are Planck’s constant,

electron charge, radius of quantum dot, Effective Mass of

excited electron, Effective mass of excited hole, Permittivity

of vacuum, and Relative permittivity respectively.

The first term in the right hand side of Equation (1)

represents the band gap energy of bulk materials, which are

the characteristics of the material. The second additive term

of the equation represents the additional energy due to

quantum confinement having a dependence on the band gap

energy (also known as ground state confinement energy).

The third subtractive term stands for the columbic interaction

energy exciton.

Neglecting the coulombic interaction energy exciton due

to high dielectric constant of the semiconductor material, the

overall equation for calculating the emission energy is given

as:

++=∆

**2

2 11

8)()(

he

gmmR

hRERE (2)

=∆E Emission energy

=gE Band gap energy

3. Results and Discussion

Lead sulfide (Pbs) and Indium Phosphide (InP) quantum

dots, in addition to its necessary parameters as shown in the

tables below were used for this computation.

Table 1. Showing material parameter used for the computation of the

confinement energies at various radii which is less than the Bohr radius aB

[27].

Quantum Dot InP Pbs

*

em 0.08 om 0.11 om

*

hm 0.6 om 0.9 om

bulkE at 300k 1.344eV 0.41eV

Ba (Bohr radius) 15nm 20nm

Table 2. Showing confinement and emission energies obtained at different

dot radii for InP semiconductor quantum dot.

Dots Radius (nm) Confinement Energy

(eV)

Emission Energy

(eV)

0.5 21.125 22.469

0.7 10.8125 12.1565

1.0 5.2850 6.6291

1.4 2.6964 4.0404

1.95 1.3898 2.7338

2.30 0.9991 2.3431

2.80 0.6741 2.0181

3.50 0.4314 1.7754

4.40 0.2729 1.6169

5.60 0.1685 1.5125

6.50 0.1251 1.4691

7.00 0.1079 1.4519

Table 3. Showing confinement and emission energies obtained using

different dot radii for Pbs semiconductor quantum dot.

Dot Radius (nm) Confinement

Energy (eV)

Emission Energy

(eV)

0.6 1.0588 1.4688

0.85 0.5273 0.9375

1.00 0.3812 0.7912

1.45 0.1813 0.5913

1.90 0.1056 0.5156

2.40 0.0662 0.4762

3.30 0.0350 0.4450

4.10 0.0227 0.4326

4.90 0.0158 0.4258

5.70 0.0117 0.4217

6.40 0.0093 0.4193

7.20 0.0073 0.4174



Figure 3. A plot of Confinement Energy (eV) against Radius (nm) for InP Quantum Dot.

Figure 4. A plot of Emission Energy (eV) against Radius (nm) for InP Quantum Dot.


Figure 5. A plot of Confinement Energy (eV) against Radius (nm) for Pbs Quantum Dot.

Figure 6. A plot of Emission Energy (eV) against Radius (nm) for Pbs Quantum Dot.



Figure 7. A Plot of Confinement Energies for InP and Pbs Quantum Dots.

Figure 8. A plot of Emission Energies for InP and Pbs Quantum Dots.

The graphs of ground state confinement energy against

size (radius) for lead sulfide (Pbs) and Indium phosphide

(InP) semiconductor quantum dots in Figures 3 and 5

respectively shows the dependence of confinement on the

size of quantum dots. The resulthere shows an

inverseproportionality ratio between the ground state

confinement energy and the dot size (radius). The graphs are

asymptotic to the radius (horizontal) axis. Thus, as one

increases the radius (size), the confinement energy decreases,

gradually approaching zero.


The confinement energy is observed in quantum dots

through an increase in the energy of the band gap.

Confinement begins when radius of the quantum dot sample

is comparable to the order of the exciton Bohr radius, aB (15

nm for Indium Phosphate and 20 nm for Lead sulfide). In

order words, the size is comparable to 2 aB that is (doubles

the exciton Bohr radius). The confinement energy increases

as the size of the quantum dot is gradually reduced until the

cluster and magic number limit for the particular crystal is

reached. At this limit, Brus Equation no longer holds, hence,

the crystal losses its stability. We can say here that the energy

spectrum is discrete rather than continuous in the

confinement regime. As such, only certain energies are

allowed for a quantum dot of a given size. The confinement

region is subdivided into strong confinement regime and

weak confinement regime. It must however be noted that in

the weak confinement regime, the energy levels form a near

continuum. In Figure 3, sharp increase in confinement energy

begins at r = 1.40 nm. Thus, the limit of strong confinement

for Indium Phosphide is at size 1.95 nm, which corresponds

to confinement energy of about 1.389 eV. Beyond this limit,

the discrete nature of the energy spectrum becomes more

apparent until one gets to the cluster and magic number limit.

Similarly, Figure 5 shows that a sharp increase in

confinement energy for Lead Sulfide begins from size 1 nm

up to the cluster and magic number limit. Thus, the limit or

threshold for strong confinement is at 1.45 nm which

corresponds to energy of about 0.181 eV.

Figures 4 and 6 also show the size dependence of these

dots on emission energy. It is observed vividly that quantum

dots used also demonstrate an inverse dependence

characteristic on the emission energy. For comparison, the

plots showing the confinement energy and the emission

against the dot radius in Figures 7 and 8 shows that both the

confinement and emission energies is higher in InP quantum

dots than Pbs quantum dots. In other words, it can be said

that the smaller the size or radius of the dot, the higher or

more effective the confinement energy, hence, the more

efficiency of the electronic device it will be applied to. When

comparing with results of similar worksof [25] and [26]

using Brus Equation, it was found that the nanocrystals

exhibit the size dependence predicted by the particle-in-a-box

model and that the confinement energy exhibits an inverse

proportionality phenomenon with the dot radius. Hence, the

theoretical model considered here are in perfect agreement

with the experimental observation of the QD’s size

dependence on the confinement energy.

4. Conclusion

The simple models obtained for the two different

semiconductor nanocrystals exhibits the size dependence

predicted by the particle-in-a-box model. Also the

confinement energy exhibits an inverse proportionality

phenomenon with the dot radius. Hence, the theoretical

model considered here are in perfect agreement with the

experimental observation of the QD’s size dependence on the

confinement energy.

The level of confinement is discovered to be stronger in

Indium Phosphide, as compared to Lead Sulfide. The

confinement of electrons in semiconductor quantum dots

tends to increase as the dot size decreases. It is found that the

tunable range of the QD is solely dependent on the size of the

exciton Bohr radius. Finally, the replacement of the

continuum observed in the conduction band and valence band

in the case of bulk materials with discrete atomic like energy

levels as the particles’ size decreases tends to add more value

to the dot materials, making it for relevant in today’s world

of technology.

References

[1] Martyniuk, P. and Rogalski, A. (2008). “Quantum-Dot Infrared Photodetectors: Status and Outlook,” Progress in Quantum Electronics, 32, 3-4, 89-120.

[2] Schuler, M.; Python M.; Valle del Olmo; and De Chambrier, E. (2007). Quantum dot containing nano composite thin films for photoluminescent solar concentrators, Solar Energy 81, 1159-1165.

[3] Wang, C.; Shim, M.; and Guyot-Sionnest, P. (2001). “Electrochromic Nanocrystal Quantum Dots,” Science, 291, 5512, 2390-2392.

[4] Brus, L. E. (1984). Electron-Electron and Electron-Hole Interactions in Small semiconductor Crystallites: The Size Dependence of the Lowest Excited Electronic State. J. Chem. Phys., 80, 4403.

[5] Jacqueline, TanedoSiy-Ronquillo (2010). Low Temperature Growth and Dissolution of Colloidal Cdse Nanocrystal Quantum Dots. Ph.D. Thesis.

[6] Revaprasadu, N., Mlondo, S. N. (2006). Use of metal complexes to synthesize semiconductor nanoparticles. Pure Appl. Chem, 78, 1691-1702.

[7] Michler, P. (2003). “Single Quantum Dots: Fundamentals, Applications and New Concept, Physics and Astronomy Classification Scheme (PACS),” Springer-Verlag, Berlin.

[8] Bera, D.; Qian L.; Tseng T. K.; Holloway P. H. (2010). Quantum Dots and Their Multimodal Applications: A review. Materials, 3, 2260-2345.

[9] Pavesi, L. Negro, L D, Mazzoleni, C. Franzo, G. Priolo, F. (2000). Optical gain in silicon nanocrystals. Nature, 408, 440-444.

[10] Kalasad, M. N.; Rabinal, M. K. and Mulimani, B. G. (2009). Ambient Synthesis and characterization of High- Quality CdSe Quantum Dots by an Aqueous Route. Langmuir, 25 (21), 12729-12735.

[11] Iyer, G., Xu, J., and Weiss, S. (2011). Single step conjugation of Antibodies to Quantum dots for labeling cell surface Receptors in mammalian cells. Methods of Mol. Biol., 751, 553-563.

[12] Bagher, A. M. (2016). Quantum dots Applications. Sensors and Transducer, 198, 3, 37-43.

[13] Peng, C. and Li, Y. (2010). Application of Quantum dot – based Biotechnology in cancer diagnosis: Current status and future perspectives, J. of Nanomaterials, 2010, 676839.



[14] Zhang, H, Douglas Y, and Wang, C. (2008). Quantum Dot for cancer diagnosis and therapy: Biological and Clinical Perspectives, Nanomedicine (Lond.), 3, 1, 83-91.

[15] Smith, A. M., Nie, S. (2009). Next Generation Quantum Dots. Nature biotechnology, 27, 8, 732-733.

[16] Delerue, C., Allen, G., Lannoo, M. (1993). Theoretical Aspect of the Luminescence of Porous Silicon. Physical Review B, 48, 11024.

[17] Fu, H., wang, L. W., Zunger, A. (1998). Applicability of k. P. Method to the Electronics Structure of Quantum Dots. Physical Review B, 57, 9971.

[18] Nanda, K. K., Kruis, F. E., Fissan, H. (2004). Effective Mass Approximation for Two extreme Semiconductors: Band gap of PbS and CuBr Nanoparticles. Journal of Applied Physics, 95, 5035.

[19] Baskoutas, S., Schommers, W., Terzis, A. F., Rieth, M., Kapaklis, V., Politis, C. (2003). Stability of an Exciton Bound to an Ionized Donor in Quantum Dots. Physics Letters A, 308, 219.

[20] Zhang, X., Gharbi, M., Sharma, P. and Johnson, H. T. (2009). Quantum fieldinduced strains in Nanostructures and prospects for optical actuation. International journal of Solids and Structures, 46, 3810-3824.

[21] Baskoutas, S., Terzis, A. F., Schommers, W. (2006). Size Dependent Exciton Energy of Narrow band Gap Colloidal Quantum Dots in the Finite Depth Square well Effective Mass

Approximation. Journal of Computational and Theoretical Nanoscience, 3, 269-271.

[22] Kumar, D., Negi, C. M. S., Gupta, K. S., and Kumar, J. (2012). Shape and Size dependent Electronic Properties of GaAs/AlgaAs Quantum Dots. Bonfring International Journal of Power Systems and Integrated Circuits, 2, 3.

[23] Schliwa, A., Winkelnkemper, M., Bimberg, D. (2007). Impact of Size, Shape, and Composition on Piezoelectric Effect and electronic Properties of In(Ga)As/ GaAs Quantum Dots. Physical Review B, 76, 205324.

[24] Ekong, S. A and Osiele, M. O. (2016). A Quantum Confinement Study of the Electronic Energy of some Nanocrystalline Silicon Quantum- Dots. International Letters of Chemistry, Physics and Astronomy, 63, 106-110.

[25] Chukwuocha, E. and Onyeaju, M. (2012). Effect of Quantum Confinement on the Wavelength of CdSe, ZnS and GaAs Quantum Dots (QDs). International Journal of Scientific and technology Research, 1, 7, 21-24.

[26] Chukwuocha, E. O., Onyeaju, M. C. and Harry, S. T. (2012). Theoretical Studies on the Effect ofConfinement on Quantum Dots using the Brus Equation. World Journal of condensed Matter Physics, 2, 96-100.

[27] Sinclair, J. and Dagotto (2009). An Introduction to Quantum Dots: components, Synthesis, Artificial Atoms and Applications. Solid State II Lecture Notes, University of Tennessee, Knoxville.

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Effects of Confinement Energies on Lead Sulphide and ...article.aascit.org/file/pdf/9770795.pdfDepartment of Physics, Akwa Ibom State University, Ikot Akpaden, Nigeria Email address