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 Density of States 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 electron 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 transistor 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 semiconductor that has low energy where levels within the quantum structure called the electrons and holes are trapped and whose inter-sub band transitions. The inter-band and properties are controlled by quantum mechanics 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 well, 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 Quantum dot 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 semiconductors be exhibited [6, 7].The simplest model for can be joined to form a hetero-junction. A absorption between the valence and conduction potential well 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 ∗ 2 2 푛푥 휋 Normalizing the wave function, 훹 푥 훹 푥 = 퐴 푠푖푛 = 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 …………………….