International Journal of Research Studies in Science, Engineering and Technology Volume 5, Issue 6, 2018, PP 28-33 ISSN : 2349-476X

Assessment of Energy and of Three Dimensional Quantum Structure (Bulk)

J. Ilouno1, Gesa, F. Newton2,N.Okpara3 1, 3Physics Department, University of Jos, Jos, Nigeria. 2Physics Department, Federal University of Agriculture, Makurdi, Benue State, Nigeria *Corresponding Author: J. Ilouno, Physics Department, University of Jos, Jos, Nigeria.

ABSTRACT Quantum structures are integral part of optoelectronics devices e.g. lasers, modulators, switches etc. In three dimensional quantum structures (bulk), the particles move freely in all directions with zero confinement. The density of state of quantum structure is the possible number of state an excited can occupy per unit volume. The density of state depends on the energy at which the electron moves when excited. In this paper, the energy and density of states of real solids were calculated. The results showed that energy increases with increase in the density of state. But, the density of state does not increase linearly with energy. The density of state has peaks when the energies are at 5eV, 7.5eV and 10eV. The highest peak occurred at 10eV and thereafter, the density of states kept increasing with increase in energy up to 25eV which is the maximum energy under investigation. The findings are in agreement with published literatures. Quantum structure technologies are applicable in measuring devices for IR photo detectors, optics for diode lasers, and micro-electronics for high electron mobility devices. Keywords: Bulk, Confinement, Density of State, Energy of State, Quantum Structure

INTRODUCTION between the valence and conduction bands called inter-band transitions [3]. There are also Quantum structure is a region of a optical transitions between the different electron that has low energy where levels within the quantum structure called the and holes are trapped and whose inter-sub band transitions. The inter-band and properties are controlled by inter-sub band transitions have a smaller energy [1]. The dimension of quantum structure is gap which enable them interact with light in the typically between 1-1000 nm [1, 2]. Quantum mid- to far- infrared part of the spectrum [3, 4]. wells are vital building blocks for semiconductor devices such as LEDs, lasers, A quantum confined structure is classified based detectors etc. Their optical transitions are very on confinement direction as quantum bulk, strong with the capability of tuning the energy. , quantum wire, and quantum dots However, they also rely on optical transitions or nano-crystals as shown in Table 1 [5]. Table 1: classification of quantum confined structures Structure Quantum confinement Number of free dimension Bulk (real solid) 0 3 Quantum well 1 2 Quantum wire 2 1 3 0

A narrow band-gap material is placed between discontinuity in either the valence or conduction wider band-gap materials as shown in Figure 1. band. From Figure 1, if a thin layer of a The decrease of the thickness of the well layer narrower-band gap material 'A' is sandwiched results to decrease in the electron and hole between two layers of a wider-band gap material waveforms in the quantum well by the 'B', then a double hetero-junction is formed. A surrounding layers [6]. single quantum well can also be formed if layer 'A' is sufficiently thin for quantum properties to The band gaps of two different be exhibited [6, 7].The simplest model for can be joined to form a hetero-junction. A absorption between the valence and conduction can be formed from the

International Journal of Research Studies in Science, Engineering and Technology V5 ● I6 ● 2018 28 Assessment of Energy and Density of States of Three Dimensional Quantum Structure (Bulk) bands in a bulk semiconductor can be seen when energy at which the electron moves when an electron is raised from the valence band to a excited. It is the first derivative of the number of state of essentially the same momentum in the state with respect to the energy. It is given conduction band by the absorption of a photon. mathematically as [11]. 푑 ϕ(E) g (E) = (2) 푑퐸 where g(E) is the density of state and ϕ is number of states. Density of electron states in bulk, 2D, 1D and 0D semiconductor structure is shown in Figure 2 [2, 12]. 0D structures has very well defined and quantized energy levels. The quantum confinement effect corresponding to the size of the nanostructure can be estimated via a simple Figure 1: Energy band diagram of quantum structure effective-mass approximation model. This simple model is presumed that all transitions have identical strength, although they will have different energies corresponding to the different energies for such vertical transitions [8, 9]. Hence, the optical absorption spectrum follows directly from the density of states in energy, and in bulk semiconductors the result is an absorption edge that rises as the square root of energy. In bulk semiconductors excitonic effect is neglected [10, 11]. Quantum structure could be finite, infinite or super lattice [12]. In finite quantum structure, the particle in the box exhibit tunneling penetration while in an infinite Figure 2: Density of electron states of a quantum structure the particle is seen in the box semiconductor as a function of dimension only. Whereas, in super lattice quantum structure the wells are so close that the wave- RESEARCH METHODOLOGY functions couple to give “mini bands” [9]. Energy and Density of States of 3- The number of states attained by a quantum Dimensional Cubic Quantum Structure (Real system is the possible number of available Solid) states. It is given mathematically as [2]: Let us consider a real semiconductor crystal as a 푉 system cubic real solid in 3- dimensional system with φ (E) = 푥 푵 (1) 푉 푠푖푛푔푙푒 −푠푡푎푡푒 sides of length l, using the Schrodinger where equation; 2푚 (퐸−푉) φ(E) is number of states, V is volume of ∇2훹 푥, 푦, 푧 + 훹 푥, 푦, 푧 = 0 system ħ2 whole system (sphere, circle, line), V single is (3) volume of single state of that system, and N is the number of atoms in the crystal. Each But, 휕2 휕2 휕2 quantum state has unique wave function. ∇2= + + 휕푥2 휕푦2 휕푧2 Density of state is the possible number of state an electron when excited can occupy per unit volume [10]. The density of state depends on the Thus, equation 3 becomes

휕2훹 푥,푦,푧 휕2훹 푥,푦,푧 휕2훹 푥,푦,푧 2푚 (퐸−푉) + + + 훹 푥, 푦, 푧 = 0 (4) 휕푥 2 휕푦 2 휕푧 2 ħ2 But for potential energy equal zero then,

29 International Journal of Research Studies in Science, Engineering and Technology V5 ● I6 ● 2018 Assessment of Energy and Density of States of Three Dimensional Quantum Structure (Bulk)

휕2 휕2 휕2 2푚퐸 훹 푥, 푦, 푧 + 훹 푥, 푦, 푧 + 훹 푥, 푦, 푧 + 훹 푥, 푦, 푧 = 0 (5) 휕푥 2 휕푦 2 휕푧 2 ħ2 휕훹 푑훹 휕훹 푑훹 휕훹 푑훹 Also, = 훹 푦 훹 푧 , = 훹 푥 훹 푧 , and = 훹 푥 훹 푦 휕푥 푑푥 휕푦 푑푦 휕푧 푑푧 휕2훹 푑2훹 = 훹 푦 훹 푧 휕푥 2 푑푥 2 휕2훹 푑2훹 = 훹 푥 훹 푧 (6) 휕푦 2 푑푦 2 휕2훹 푑2훹 = 훹 푥 훹 푦 휕푧 2 푑푦 2 Putting equation 6 into equation 5, yields:

푑2훹 푥 푑2훹 푦 푑2훹 푧 2푚퐸 훹 푦 훹 푧 + 훹 푥 훹 푧 + 훹 푥 훹 푧 + 훹 푥 훹 푦 훹 푧 = 0 (7) 푑푥 2 푑푦 2 푑푦 2 ħ2 Divide through by 훹 푥 훹 푦 훹 푧 gives:

1 푑2훹 푥 1 푑2훹 푦 1 푑2훹 푧 2푚퐸 + + + = 0 (8) 훹 푥 푑푥 2 훹 푦 푑푦 2 훹 푧 푑푧2 ħ2

2푚퐸 2푚퐸 Let = 퐾2, then K = ħ2 ħ2

1 푑2훹 푥 1 푑2훹 푦 1 푑2훹 푧 + + +퐾2 = 0 (9) 훹 푥 푑푥 2 훹 푦 푑푦 2 훹 푧 푑푧2 where K is a constant.

푑2훹 푥 푑2훹 푦 푑2훹 푧 + + + 퐾2훹 푥 훹 푦 훹 푧 (10) 푑푥 2 푑푦 2 푑푧2 The solution to the equation 10 is given as: 훹 푥 = 퐴 푠푖푛 퐾푥 + 퐵 cos 퐾푥 훹 푦 = 퐴 푠푖푛 퐾푦 + 퐵 cos 퐾푦 훹 푧 = 퐴 푠푖푛 퐾푧 + 퐵 cos 퐾푧 Thus for a metal of length l along each axis; 훹 푥 = 0 at x=0 0=A sin (0) +B cos (0) Therefore, B=0 Hence,훹 푥 = 퐴 푠푖푛 퐾푥 Taking the boundary conditions at 훹 푥 = 0 ; 푥 = 푙 0 = A sin Kl But, A≠0 Sin Kl = 0 Thus, Kl = sin -1(0) = n휋 n휋 Therefore, K= 푙 푛 휋푥 푛 휋 Substituting back, 훹 푥 = 퐴 푠푖푛 푥 and 퐾 = 푥 푙 푥 푙 푛 휋푦 푛 휋 훹 푦 = 퐴 푠푖푛 푦 and 퐾 = 푦 푙 푦 푙 푛 휋푧 푛 휋 훹 푧 = 퐴 푠푖푛 푧 and 퐾 = 푧 푙 푧 푙 푛 휋 푛 휋 푛 휋 Thus, K= 퐾 + 퐾 + 퐾 = 푥 + 푦 + 푧 푥 푦 푧 푙 푙 푙 푛 휋 Normalizing the wave function, 훹 푥 훹 푥 ∗ = 퐴2푠푖푛2 푥 = 1 푙

2 Therefore, A= and 푙

International Journal of Research Studies in Science, Engineering and Technology V5 ● I6 ● 2018 30 Assessment of Energy and Density of States of Three Dimensional Quantum Structure (Bulk)

2 푛 휋푥 2 푛 휋푦 2 푛 휋푧 훹 푥, 푦, 푧 = 푠푖푛 푥 푠푖푛 푦 푠푖푛 푧 (11) 푙 푙 푙 푙 푙 푙

8 푛 휋푥 푛 휋푦 푛 휋푧 Thus, the wave function of such a system is given by: 훹 푥, 푦, 푧 = 푠푖푛 푥 푠푖푛 푦 푠푖푛 푧 푙3 푙 푙 푙 To obtain the energy of such a system we use the effective mass approximation model, such that

ħ2퐾2 E = (12) 2푚 ∗ 2푚 ∗퐸 n휋 Thus, 퐾2= = ( )2 ħ2 푙 The energy for a 3-dimensional system is therefore given as:

ħ2휋2 E= (푛 2 + 푛 2 + 푛 2) (13) 2푚 ∗푙2 푥 푦 푧 Therefore,

2푚 ∗푙2퐸 푛 2 + 푛 2 + 푛 2 = (14) 푥 푦 푧 휋2ħ2 where

푛1,푛2,푛3, 푛4…………………n = 1,2,3,4 …………………….. 2푚 ∗푙2퐸 But,푛 2 + 푛 2 + 푛 2 is equivalent to the radius R for the K- space. 푅2 = 푥 푦 푧 휋2ħ2 Hence,

2푚 ∗푙2퐸 1 R = ( )2 휋2ħ2 4 But the volume of the sphere for 3-dimensional K-space = 휋푅3 =V system (sphere), 3 ∗ 2 3 4 3 4 2푚 푙 퐸 V = 휋푅 = 휋( )2 (15) 3 3 휋2ħ2 휋 휋 휋 휋3 휋3 Volume of the single state cubic in K-space = ( ) ( ) ( ) = ( ) = V single state = 푎 푏 푐 푉 푉 1 1 1 1 1 1 Number of atom for a crystal at position (0,0,0) and ( , , ) in 3-dimension = 2 x ( 푥 푥 ) 2 2 2 2 2 2 푉 system (sphere ) Hence, the number of state ϕ E = 푥 푁푢푚푏푒푟 표푓 푎푡표푚푠 푉 푠푖푛푔푙푒 −푠푡푎푡푒 3 4 2푚 ∗푙2퐸 2 휋( 2 2 ) 1 1 1 ϕ E = 3 휋 ħ x2 x 푥 푥 but V = 푎 푥 푏 푥 푐 = 푙3 휋3 2 2 2 푉 3 3 3 3 3 4 2푚 ∗푙2퐸 4 휋푙 3 2푚 ∗ 4 휋 푙3 2푚 ∗ 2 2 2 2 2 휋( 2 2 ) 1 3 ( 2 ) 퐸 1 3 ( 2 ) 퐸 1 ϕ E = 3 휋 ħ 푥 = 3 휋 ħ 푥 = 3 휋 ħ 푥 휋3 4 휋3 4 휋3 4 푙3 푙3 푙3 1 2푚 ∗ 3 3 ϕ E = ( )2 퐸2 (16) 3휋2 ħ2 Thus, the density of states is obtained as:

3 3 3 1 푑 ϕ E 3 1 2푚 ∗ −1 1 2푚 ∗ g(E) = = 푥 ( )2 퐸2 = ( )2 퐸2 3퐷 푑퐸 2 3휋2 ħ2 2휋 2 ħ2 1 2푚 ∗ 3 1 g(E) = ( )2 퐸2 (17) 3퐷 2휋2 ħ2 Equation 17 gives the expression for the density Data Analysis of state for a 3-dimensional system and shows The following parameters were used in the that it is directly proportional to the square root calculations of the energy and density of states of the energy E of the system.

31 International Journal of Research Studies in Science, Engineering and Technology V5 ● I6 ● 2018 Assessment of Energy and Density of States of Three Dimensional Quantum Structure (Bulk) of real solids in three dimensions and result L = dimension of well and line assumed =10 A0 tabulated in Table 2. = 10 x 10-10 m h ∗ -31 ħ = = 1.054 x 10-34 J. S m = mass of the electron = 9.11 x 10 kg 2π e = charge of electron = 1.6 x 10-19 C π =22/7 = 3.142 RESULTS Table2: Results Data of the Energy and Density of States for Three Dimensional Energy Levels ENERGY ENERGY ENERGY DENSITY OF DENSITY OF LEVELS (JOULES) (eV) STATES STATES -3 -1 -3 -1 nx, ny, nz (m J ) (m eV ) 1, 1, 1 1.8060x 10-19 1.12875 4.522x 1046 1.130x 1056 1, 1, 2 3.6120x 10-19 2.25750 6.395x 1046 1.599x 1056 1, 2, 2 5.4100x 10-19 3.38625 7.832x 1046 1.958x 1056 1, 2, 3 8.4280 x 10-19 5.26750 9.768x 1046 2.442x 1056 2, 2, 3 1.0234 x 10-18 6.39625 1.076x 1047 2.691x 1056 3, 1, 1 6.6220 x 10-19 4.13875 8.658x 1046 2.165x 1056 3, 3, 3 1.6254 x 10-19 10.1588 1.357x 1047 3.391x 1056 4, 2, 1 1.2642 x 10-18 7.90125 1.196x 1047 2.991x 1056 4, 2, 2 1.4880 x10-18 9.03000 1.279x 1047 3.197x 1056 4, 2, 3 1.7458 x 10-18 10.9113 1.406x 1047 3.515x 1056 4, 4, 4 2.8896 x 10-18 18.0600 1.809x 1047 4.522x 1056 4, 1, 1 1.0836 x 10-18 6.77250 1.108x 1047 2.769x 1056 4, 3, 1 1.5652 x 10-18 9.78250 1.331x 1047 3.328x 1056 5, 1, 1 1.6250 x 10-18 10.1588 1.357x 1047 3.391x 1056 5, 2, 1 1.8060 x 10-18 11.2875 1.430x 10467 3.575x 1056 5, 3, 1 2.1070 x 10-18 13.1688 1.544x 1047 3.860x 1056 5, 3, 2 2.2876 x 10-18 14.2975 1.610x 1047 4.023x 1056 5, 3, 3 2.5886 x 10-18 16.1788 1.712x 1047 4.280x 1056 5, 4, 1 2.5280 x 10-18 15.8025 1.692x 1047 4.230x 1056 5, 4, 2 2.7090 x 10-18 16.9313 1.751x 1047 4.378x 1056 5, 4, 3 3.0100 x 10-18 18.8125 1.846x 1047 4.615x 1056 5, 4, 4 3.4314 x 10-18 21.4463 1.971x 1047 4.927x 1056 5, 4, 5 3.9732 x 10-18 24.8325 1.120x 1047 5.300x 1056

DISCUSSION change in the energy level and hence the density of state of the system studied. From Table 2 and Figure 3, it is clearly seen that the energy of state of quantum bulk increases 30 with the density of states non-linearly. The 25 density of state has peaks when the energy is at 5eV, 7.5eV and 10eV. The highest peak occurs 20 at 10eV and after which the density of states 15 kept increasing with increase in energy up to 10 25eV which is the maximum investigated. The 5 results obtained in this work may slightly differ from real life situations because of the following 0 reasons: 0 10 20 30 I. In the calculation, the electronic mass of the electron was used throughout and not the Energy (eV) effective mass of the electron, which varies in Figure3: Variation of the density of states with the reciprocal space lattice of the solid. energy of real solids II. The dimension of the well was kept constant CONCLUSION 0 at a value of 10 A irrespective of the dimension The assessment of energy and density of states under consideration. Changes with the of quantum bulk has a tremendous impact in dimension of the box will certainly cause a opto-electronics with diverse applications. In

International Journal of Research Studies in Science, Engineering and Technology V5 ● I6 ● 2018 32 Assessment of Energy and Density of States of Three Dimensional Quantum Structure (Bulk) this paper, the energy and density of states for a [6] Vijay, A.S., Ranjan, V., & Manish, K. (1999). three dimensional cubic quantum structure were Semiconductor quantum dots: Theory & calculated and analyzed. The result showed phenomenology. Indian academy sciences, maximum and minimum peaks for the non- (22)3, 563-569. linear relationship between energy and density [7] Vijay, A.S., Ranjan, V., & Manish, K. (1999). of states of quantum bulk. Semiconductor quantum dots: Theory & phenomenology. Indian academy sciences, REFERENCES (22)3, 563-569 [1] Paul, H. (2011). Quantum Wells, Wires and [8] Richard, D.T. (2008). Synthesis and Dots: Theoretical and Computational Physics of applications of nano particles and quantum Semiconductor Nanostructures. Wiley. dots. Chemistry in New Zealand, pg 146-150. [2] Arthur, E. and Gabriel, K.O. (2015). [9] Bimberg, D. (1999). Quantum dot hetero Fundamentals of Solid State Physics. structures. London, United Kingdom, Wiley- Department of Physics, Delta State University Blackwell. Abraka, Nigeria. [10] Brennan, K.F. (1997). The Physics of [3] Liu, H.C. (2000). Inter sub band Transitions in Semiconductors. New York, Cambridge Quantum Wells: Physics and Device University Press. Applications II.vol.65 of Semiconductors and [11] Calvin R.K. (2005). Density of States: 2D, 1D, Semimetals, Academic press. and 0D. Georgia Institute of Technology ECE [4] Rosencher, E. and B. Vinter, B. (2002). 6451. Optoelectronics. Cambridge University Press. [12] Brennan, K.F. (1997). The Physics of [5] Alastair, R. (1998). Quantum Mechanics.3rd Ed, semiconductors. New York, Cambridge IOP Publishing. University press.

Citation: J. Ilouno, Gesa, F. Newton, N.Okpara, " Assessment of Energy and Density of States of Three Dimensional Quantum Structure (Bulk)",International Journal of Research Studies in Science, Engineering and Technology, vol. 5, no. 6, pp. 28-33, 2018.

Copyright: © 2018 J. Ilouno, Gesa, F. Newton, N.Okpara, This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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