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Materials Focus Vol. 6, pp. 1–34, 2017 (www.aspbs.com/mat)

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Can Photons Affect the Entropy? P. K. Das1, ∗ , P. Dutta2 , A. Halder3 , R. Bhattacharjee4 , and K. P. Ghatak1 1 Department of Basic Science Institute of Engineering and Management, D-1, Management House, Salt Lake, Sector-V, Kolkata 700091, West Bengal, India 2 Department of Chemistry, Maharaja Manindra Chandra College, Shyambazar, Kolkata, West Bengal, India 3 Department of Chemistry, Presidency University, Kolkata 700073, West Bengal, India 4 Department of Electronics and Communication Engineering, Calcutta Institute of Engineering and Management, Kolkata 700040, West Bengal, India

ABSTRACT

1. INTRODUCTION It is well known that the creation of nanoscience and nanotechnology is based on following two important concepts: i. The symmetry of the wave-vector space of the charge carriers in semiconductors having various band structures is being reduced from a 3D closed surface to a quantized 2D closed surface, quantized non-parabolas and fully quantized wave vector space leading to the formation of 0D systems such as quantum wells (QWs), doping superlattices, inversion and accumulation layers, quantum well super-lattices, carbon nano-tubes, nanowires (NWs), quantum wire super-lattices, magnetic quantization, magneto size quantization, quantum boxes (QBs), magneto inversion and accumulation layers, magneto quantum well super-lattices, magneto NIPIs, quantum dot super-lattices and other field aided nano structured systems. ∗

Author to whom correspondence should be addressed. Email: [email protected] Received: 2 February 2016 Accepted: 3 October 2016

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ii. The advent of modern fabrication methods namely, fine line lithography (FLL), metallo organic chemical vapor deposition (MOCVD), molecular beam epitaxy (MBE), etc. The quantum confined materials have gained much interest in the said arena because of their importance to unlock both new scientific revelations and multi dimensional all together unheard technological applications. In QWs, the quantization of the motion of the carriers in the direction perpendicular to the surface exhibits the two-dimensional carrier motion of the charge carriers and the third direction is being quantized. Another one dimensional structure known as a NW has been proposed to investigate the physical properties in these materials where the carrier gas is quantized in two transverse directions and they can move only in the longitudinal direction. As the concept of quantization increases from 1D to 3D, the degree of freedom of the free carriers decreases drastically and the total density-of-states (DOS) function changes from Heaviside step function to the doi:10.1166/mat.2017.1392

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In this paper an attempt is made to study, the entropy in the presence of intense light waves in heavily doped (HD) III–V and optoelectronic materials on the basis of newly formulated electron dispersion relations within the frame work of k · p formalism. We have also investigated the entropy under magnetic quantization and also in quantum wells (QWs) and nano wires (NWs) respectively. It is found taking HD InSb, InAs, Hg1−x Cdx Te and In1−X Gax Asy P1−y lattice matched to InP as examples III–V, ternary and quaternary compounds that the entropy increases with increasing electron concentration and decreasing film thickness in different spiky manners, since the coincidence of Fermi energy with the sub-band energy leads to the step functional dependence of the density state function and this fact is being reflected in the quantized variations of the entropy with the said variables. The entropy increases with increasing electric field and decreasing alloy composition respectively. The numerical values of entropy with all the physical variables are totally band structure dependent for all the cases. The most striking features are that the presence of poles in the dispersion relation of the materials in the absence of band tails creates the complex energy spectra in the corresponding opto-electronic HD NWs and the effective electron mass exists within the band gap which is impossible without the concept of band tailing. The well-known classical result of entropy for non-degenerate bulk semiconductors having parabolic energy bands has been obtained as a special case of our generalized formulation and thus confirming the compatibility test. The content of this paper finds four important applications in the field of quantum effect devices of nanoscience and nanotechnology. KEYWORDS: Entropy, Photons, Heavily Doped Opto Electronic Materials, Quantized Variations.

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Can Photons Affect the Entropy?

Dirac’s delta function forming QDs which in turn depend on the dispersion relations of the carriers in materials. An enormous range of important applications of such low dimensional structures for modern physics in the quantum regime together with a rapid increase in computing power, have generated considerable interest in the study of the optical properties of quantum effect devices based on various new materials of reduced dimensionality. Examples of such new applications include quantum well and quantum wire transistors, quantum cascade lasers, highfrequency microwave circuits, high-resolution terahertz spectroscopy, advanced integrated circuits, super-lattice photo-oscillator, super-lattice photo-cathodes and transistors, super-lattice coolers, thermoelectric devices, thin film transistors, micro-optical systems, intermediate-band solar cells, high performance infrared imaging systems, optical switching systems, single electron/molecule electronics, nano-tube based diodes, and other nano-electronic devices.1–3 Although many new effects in quantized structures have already been reported, the interest for further researches of other physical aspects of such quantum-confined materials is becoming increasingly important. One such significant concept is the entropy which is a physical phenomenon and occupies a singular position in the whole arena of science and technology in general and whose importance has already been established since the inception of second law of thermodynamics which in recent years finds extensive applications in modern nano structured thermodynamics, characterization and investigation of condensed matter systems, thermal properties of thermal semiconducting devices and related aspects in connection with the investigations of the thermal properties of nanostructures.4–6 It is well known that the entropy is the measure of disorder or uncertainty about a system.4–6 The equilibrium state of a system maximizes the entropy as all the information about the initial conditions except that the conserved variables are lost. According to the second law of thermodynamics the total entropy of any system will not decrease other than by increasing the entropy of some other system. A reduction in the increase of entropy in a specified process, such as a chemical reaction, means that it is energetically more efficient. In accordance to the second law of thermodynamics the entropy of a system that is not isolated may decrease. In mechanics, the second law in conjunction with the fundamental thermodynamic relation places limits on a system’s ability to do useful work. The entropy change of a system at temperature T absorbing an infinitesimal amount of heat q in a reversible way, is given by q/T . Statistical mechanics demonstrates that entropy is governed by probability, thus allowing for a decrease in disorder even in an isolated system. According to Boltzmann’s the entropy is a measure of the number of possible microscopic states (or microstates) of a system in thermodynamic equilibrium. Entropy is the only 2

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quantity in the physical sciences that seems to imply a particular direction of progress, sometimes called an arrow of time. It is important to note that the heavy doping and the carrier degeneracy are the keys to unlock the new properties of technologically important semiconductors and they are especially instrumental in dictating the characteristics of Ohmic contacts and Schottky contacts respectively.7 It is an amazing fact that although the heavily doped semiconductors (HDS) have been investigated in the literature but the study of the carrier transport in such materials through proper formulation of the Boltzmann transport equation which needs in turn, the corresponding heavily doped carrier energy spectra is still one of the open research problems. The band tails are being formed in the forbidden zone of HDS and can be explained by the overlapping of the impurity band with the conduction and valence bands.8 Kane9 and Bruevich10 have independently derived the theory of band tailing for semiconductors having unperturbed parabolic energy bands. Kane’s model11 was used to explain the experimental results on tunneling12 and the optical absorption edges13 14 in this context. Halperin and Lax15 developed a model for band tailing applicable only to the deep tailing states. Although Kane’s concept is often used in the literature for the investigation of band tailing,16 17 it may be noted that this model8 18 suffers from serious assumptions in the sense that the local impurity potential is assumed to be small and slowly varying in space coordinates.17 In this respect, the local impurity potential may be assumed to be a constant. In order to avoid these approximations, we have developed in this paper, the electron energy spectra for HDS for studying the entropy based on the concept of the variation of the kinetic energy8 17 of the electron with the local point in space coordinates. This kinetic energy is then averaged over the entire region of variation using a Gaussian type potential energy. On the basis of the E–k dispersion relation, we have obtained the electron statistics for different HDS for the purpose of numerical computation of the respective entropies. It may be noted that, a more general treatment of many-body theory for the DOS of HDS merges with one-electron theory under macroscopic conditions.8 Also, the experimental results for the Fermi energy and others are the average effect of this macroscopic case. So, the present treatment of the one-electron system is more applicable to the experimental point of view and it is also easy to understand the overall effect in such a case.18 In a HDS, each impurity atom is surrounded by the electrons, assuming a regular distribution of atoms, and it is screened independently.15 18 19 The interaction energy between electrons and impurities is known as the impurity screening potential. This energy is determined by the inter-impurity distance and the screening radius, which is known as the screening length. The screening radius changes with the electron concentration and the effective mass. Furthermore, these entities are important for Mater. Focus, 6, 1–34, 2017

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It is worth remarking that the effects of crossed electric and quantizing magnetic fields on the transport properties of semiconductors having various band structures have relatively been less investigated as compared with the corresponding magnetic quantization, although, the study of the cross-fields are of fundamental importance with respect to the addition of new physics and the related experimental findings in modern quantum effect devices. It is well known that in the presence of electric field (Eo ) along x-axis and the quantizing magnetic field (B) along z-axis, the dispersion laws of the carriers in semiconductors become modified and for which the carrier moves in both the z and y directions respectively. The motion along y-direction is purely due to the presence of E0 along x-axis and in the absence of electric field, the effective electron mass along y-axis tends to infinity indicating the fact that the electron motion along y-axis is forbidden. The effective electron mass of the isotropic, bulk semiconductors having parabolic energy bands exhibits mass anisotropy in the presence of cross fields and this anisotropy depends on the electron energy, the magnetic quantum number, the electric and the magnetic fields respectively, although, the effective electron mass along z-axis is a constant quantity. In 1966, Zawadzki and Lax55 derived the expression of the dispersion relation of the conduction electrons for III–V semiconductors in accordance with the two band model of Kane under cross fields configuration which generates the interest to study this particular topic of solid state science in general.56 In ultrathin films, the restriction of the motion of the carriers in the direction normal to the film (say, the z direction) may be viewed as carrier confinement in an infinitely deep 1D rectangular potential well, leading to quantization [known as quantum size effect (QSE)] of the wave vector of the carrier along the direction of the potential well, allowing 2D carrier transport parallel to the surface of the film representing new physical features not exhibited in bulk semiconductors.57 The low-dimensional hetero structures based on various materials are widely investigated because of the enhancement of carrier mobility.58 These properties make such structures suitable for applications in quantum well lasers,59 hetero junction FETs,60 high-speed digital networks,61 high-frequency microwave circuits,62 optical modulators,63 optical switching systems,64 and other devices. The constant energy 3D wave-vector space of bulk semiconductors becomes 2D wave-vector surface in ultrathin films or quantum wells due to dimensional quantization. Thus, the concept of reduction of symmetry of the wave-vector space and its consequence can unlock the physics of low dimensional low dimensional structures. With the advent of nano-photonics, there has been a considerable interest in studying the optical processes in semiconductors and their nanostructures.65 It appears from the literature, that the investigations have been carried out 3

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HDS in characterizing the semiconductor properties20 21 and the modern electronic devices.15 22 The works on Fermi energy and the screening length in an n-type GaAs have already been initiated in the literature,23 24 based on Kane’s model. Incidentally, the limitations of Kane’s model,8 16 as mentioned above, are also present in their studies. At this point, it may be noted that many band tail models are proposed using the Gaussian distribution of the impurity potential variation.9 16 Our method is not at all related with the DOS technique as used in the aforementioned works. From the electron energy spectrum, one can obtain the DOS but the DOS technique, as used in the literature cannot provide the E–k dispersion relation. Therefore, our study is more fundamental than those in the existing literature, because the Boltzmann transport equation Boltzmann transport equation, which controls the study of the charge transport properties of the semiconductor devices, can be solved if and only if the E–k dispersion relation is known. It is worth remarking that in nano wires (NWs), the restriction of the motion of the carriers along two directions may be viewed as carrier confinement by two infinitely deep 1D rectangular potential wells, along any two orthogonal directions leading to quantization of the wave vectors along the said directions, allowing 1D carrier transport.25 With the help of modern fabricational techniques, such one dimensional quantized structures have been experimentally realized and enjoy an enormous range of important applications in the realm of nano science in the quantum regime. They have generated much interest in the analysis of nano structured devices for investigating their electronic, optical, and allied properties.26–28 Examples of such new applications are based on the different transport properties of ballistic charge carriers which include quantum resistors,29–31 resonant tunneling diodes and band filters,32 33 quantum switches,34 quantum sensors,35–37 quantum logic gates,38 39 quantum transistors and sub tuners,40–42 hetero junction FETs,43 high-speed digital networks,44 high-frequency microwave circuits,45 optical modulators,46 optical switching systems,47 48 and other devices.49–51 It is important to note that the effects of quantizing magnetic field (B) on the band structures of compound Semiconductors are most striking than that of the parabolic one and are easily observed in experiments. A number of interesting physical features originate from the significant changes in the basic energy wave vector relation of the carriers caused by the magnetic field. The valuable information could also be obtained from experiments under magnetic quantization regarding the important physical properties such as Fermi energy and effective masses of the carriers, which affect almost all the transport properties of the electron devices52 of various materials having different carrier dispersion relations.53 54


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on the assumption that the carrier energy spectra are invariant quantities in the presence of intense light waves, which is not fundamentally true. The physical properties of semiconductors in the presence of light waves which change the basic dispersion relation have relatively less investigated in the literature.66–68 In this appendix we shall study the DR in HD III–V, ternary and quaternary semiconductors on the basis of newly formulated electron dispersion law under external photo excitation. In Section 2.1 of the theoretical background 2, we have formulated the entropy of the conduction electrons of HD III–V, ternary and quaternary materials in the presence of light waves whose unperturbed electron energy spectrum is described by the three-band model of Kane in the absence of band tailing. The III–V compounds find applications in infrared detectors,72 quantum dot light emitting diodes,73 quantum cascade lasers,74 QWs wires,75 optoelectronic sensors,76 high electron mobility transistors,77 etc. The electron energy spectrum of III–V semiconductors can be described by the three-and two-band models of Kane,78 79 together with the models of Stillman et al.,80 Newson and Kurobe81 and, Palik et al.82 respectively. In this context it may be noted that the ternary and quaternary compounds enjoy the singular position in the entire spectrum of optoelectronic materials. The ternary alloy Hg1−x Cdx Te is a classic narrow gap compound. The band gap of this ternary alloy can be varied to cover the spectral range from 0.8 to over 30 m83 by adjusting the alloy composition. Hg1−x Cdx Te finds extensive applications in infrared detector materials and photovoltaic detector arrays in the 8–12 m wave bands.84 The above uses have generated the Hg1−x Cdx Te technology for the experimental realization of high mobility single crystal with specially prepared surfaces. The same compound has emerged to be the optimum choice for illuminating the narrow subband physics because the relevant material constants can easily be experimentally measured.85 Besides, the quaternary alloy In1−x Gax Asy P1−y lattice matched to InP, also finds wide use in the fabrication of avalanche photodetectors,86 hetero-junction lasers,87 light emitting diodes88 and avalanche photodiodes,89 field effect transistors, detectors, switches, modulators, solar cells, filters, and new types of integrated optical devices are made from the quaternary systems.90 In the same section, we have studied the entropy for the said HD materials in the presence of external photoexcitation when the unperturbed energy spectra are defined by the two band model of Kane and that of parabolic energy bands in the absence of band tails respectively for the purpose of relative comparison. In Section 2.2, we have studied the opto entropy in the said HD materials under magnetic quantization. In Section 2.3, we have studied the opto entropy in the presence of crossed electric and quantizing magnetic fields. In Section 2.4, we have studied the opto entropy in QWs in HD Kane type semiconductors. 4

2. THEORETICAL BACKGROUND 2.1. The Entropy in the Presence of Light Waves in HD III–V, Ternary and Quaternary Semiconductors ˆ of an electron in the presence of The Hamiltonian (H) light wave characterized by the vector potential A can be written following70 as 2 /2m + V r Hˆ = Pˆ + eA ¯

(1)

in which, pˆ is the momentum operator, V r ¯ is the crystal potential and m is the free electron mass. (1) can be expressed as (2) Hˆ = Hˆ 0 + Hˆ where, Hˆ 0 = Pˆ 2 /2m + V r ¯ and Hˆ =

e ¯ A · Pˆ 2m

(3)

The perturbed Hamiltonian Hˆ can be written as −ie ¯ Hˆ = A · (4) 2m √ where, i = −1 and Pˆ = i . of the monochromatic light of The vector potential A plane wave can be expressed as A = A0 s coss0 · r − t

(5)

where A0 is the amplitude of the light wave, s is the polarization vector, s0 is the momentum vector of the incident photon, r is the position vector, is the angular frequency of light wave and t is the time scale. The matrix element of r r and final state n k Hˆ nl between initial state, 1 q in different bands can be written as Hˆ nl =

e nkA · pl ˆ q 2m

(6)

Using (4) and (5), we can re-write (6) as −ieA0 is0 ·r lqe ˆ Hnl = s · nke −i t 4m is0 ·r lqe + nke i t

(7)

The first matrix element of (7) can be written as is0 ·r lq nke r s0 −k· = eiq+ iqu ∗n k · rul q r d 3 r r ∗ s0 −k· rul q + eiq+ un k r d 3 r

(8)

The functions u∗n ul and u∗n ul are periodic. The integral over all space can be separated into a sum over unit cells times an integral over a single unit cell. It is assumed that the wave length of the electromagnetic wave is sufficiently Mater. Focus, 6, 1–34, 2017

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large so that if k and q are within the Brillouin zone, is not a reciprocal lattice vector. q + s0 − k Therefore, we can write (8) as is0 −r lq nke 3 2 nl = iq ¯ q + s0 − k ∗ 3 + q + s0 − kun k ru1 q r d r cell

23 rul q = u∗n k r d 3 r (9) q + so − k cell where, is the volume of the unit cell and ∗ rul q nl = 0, since n = l. un k r d 3 r = q − k The delta function expresses the conservation of wave vector in the absorption of light wave and s0 is small compared to the dimension of a typical Brillouin zone and we set q = k. From (8) and (9), we can write,

Hˆ nl =

eA0 · pˆ nl k 2m s

ak± ≡ Eg0 − 0k± 2 Eg0 − 1/2 Eg0 + −1/2 ≡ 6Eg0 + 2/3Eg0 + /1/2 1k ∓ Eg0 1/2 2 2 ≡ 6Eg0 + Eg0 + 4 0k± ≡ 21k + 1/2 mc I11 E 1k = Ec k − Ev k = Eg0 1 + 2 1 + mv Eg0

(11)

2 ≡ Eg0 −1

where, = s cos t. When a photon interacts with a semiconductor, the carriers (i.e., electrons) are generated in the bands which are followed by the inter-band transitions. For example, when the carriers are generated in the valence band, the carriers then make inter-band transition to the conduction band. The transition of the electrons within the same band i.e., Hˆ nk is neglected. Because, in such a case, = nk Hˆ nn i.e., when the carriers are generated within the same bands by photons, are lost by recombination within the aforementioned band resulting zero carriers. Therefore, =0 Hˆ nk (12) nk

X , Y , and Z are the p-type atomic orbital’s in the primed coordinates, ↑ indicates the spin-up function in the primed coordinates, bk± ≡ 0k± , ≡ 42 /31/2 , ck± ≡ t0k± and t ≡ 6Eg0 + 2/32 /1/2 . We can, therefore, write the expression for the optical matrix element (OME) as

With n = c stands for conduction band and l = v stand for valance band, the energy equation for the conduction electron can approximately be written as 2 2 2 av k eA0 /2m2 · pˆ cv k I11 E = + 2mc − Ev k Ec k (13) where, I11 E ≡ EaE + 1bE + 1/cE + 1, a ≡ 1/Eg0 , a ≡ 1/Eg0 , Eg0 is the un-perturbed bandgap, b ≡ 1/Eg0 + , c ≡ 1/Eg0 + 2/3, and · 2 av represents the average of the square of the pˆcv k optical matrix element (OME). For the three-band model of Kane, we can write,

SpS ˆ = XpX ˆ = Y pY ˆ = ZpZ ˆ =0

2 − Ev k = Eg0 + Eg0 2 k2 /mr 1/2 1k = Ec k

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(14)

= u1 k rpu r ˆ 2 k OME = pˆ cv k

(17)

Since the photon vector has no interaction in the same band for the study of inter-band optical transition, we can therefore write

and XpY ˆ = Y pZ ˆ = ZpX ˆ =0 There are finite interactions between the conduction band (CB) and the valance band (VB) and we can obtain SpX ˆ = iˆ · pˆ = iˆ · pˆ x SpY ˆ = jˆ · pˆ = jˆ · pˆ x SpZ ˆ = kˆ · pˆ = kˆ · pˆ x where, iˆ, jˆ and kˆ are the unit vectors along x, y and z axes respectively. 5

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eA0 q − k cos t · pˆ k (10) 2m s nl ∗ = −i un u1 d 3 t = u∗n K r d 3 r rpu ˆ l k where pˆ nl k Therefore, we can write Hˆ nl =

where, mr is the reduced mass and is given by −1 m−1 + m−1 r = mc v , and mv is the effective mass of the heavy hole at the top of the valance band in the absence of any field. r and The doubly degenerate wave functions u1 k 81 u2 k r can be expressed as − iY r = ak is ↓ + bk X √ + ck+ Z ↓ ↑ u1 k + + 2 (15) and X + iY u2 k r = ak− is ↑ − bk− ↓ + ck− Z ↑ √ 2 (16) s is the s-type atomic orbital in both unprimed and primed coordinates, ↓ indicates the spin down function in the primed coordinates,

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It is well known49 that −i/2

cos/2 ei/2 sin/2 e ↑ ↑ = ↓ −e−i/2 sin/2 ei/2 cos/2 ↓ and ⎡

X

⎤

⎡

cos cos

⎢ ⎥ ⎢ ⎢ Y ⎥ = ⎢ − sin ⎣ ⎦ ⎣ sin cos Z

cos sin − sin cos

⎤⎡

X

⎤

⎥⎢ ⎥ ⎥⎢Y ⎥ ⎦⎣ ⎦ Z cos

sin sin

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From above, we can write = u1 k rpu r ˆ 2 k PˆCV k X − iY = ak+ iS ↓ + bk+ ↑ √ 2 + ck+ Z ↓ Pˆ ak− iS ↑ − bk−

X + iY √ 2

↓ +ck− Z ↑

Using above relations, we get = u1 k rPˆ u2 k r pˆCV k =

bk+ ak− √ X − iY Pˆ iS↑↑ 2 + ck ak Z Pˆ iS↓↑ +

−

ak+ bk− − √ iSPˆ X + iY ↓ ↓ 2 + ak+ ck− iSPˆ Z ↓ ↑ (18) We can also write X − iY Pˆ iS

= X Pˆ iS − iY Pˆ iS = i u∗X + Pˆ S − −iu∗Y + Pˆ iux = iX Pˆ iS − Y Pˆ iS

From the above relations, for X , Y and Z , we get

X = cos cos X + cos sin Y − sin Z Thus, X Pˆ S = cos cos X Pˆ S + cos sin Y Pˆ S − sin ZPˆ S = Pˆ rˆ1 6

Y = − sin X + cos Y + 0Z Thus, ˆ Y Pˆ S = − sin XPˆ S + cos Y Pˆ S + 0ZPS = Pˆ rˆ2

0

Besides, the spin vector can be written as S = /2 , where, 0 1 0 −i 1 0 x = y = and z = 1 0 i 0 0 −1

where, rˆ1 = iˆ cos cos + jˆ cos sin − kˆ sin

where rˆ2 = −iˆ sin + jˆ cos . So that X − iY Pˆ S = Pˆ irˆ1 − rˆ2 Thus, ak− bk+ ak− bk+ irˆ1 − rˆ2 ↑ ↑ √ X − iY Pˆ S↑ ↑ = √ 2 2 (19) Now since, ˆ ˆ − SPY ˆ = Pˆ irˆ1 − rˆ2 iSPX + iY = iSPX

We can write, ak+ bk− − √ iSPˆ X + iY ↓ ↓ 2 ak+ bk− = √ Pˆ irˆ1 − rˆ2 ↓ ↓ 2

(20)

Similarly, we get Z = sin cos X + sin sin Y + cos Z So that, Z Pˆ S ˆ ˆ = iZ Pˆ S − iP sin cos iˆ + sin sin jˆ + cos k = iPˆ rˆ3 where, rˆ3 = iˆ sin cos + jˆ sin sin + kˆ cos . Thus, ck+ ak− Z Pˆ S↓ ↑ = ck+ ak− iPˆ rˆ3 ↓ ↑

(21)

similarly, we can write, ck+ ak− iSPˆ Z ↓ ↑ = ck− ak+ iPˆ rˆ3 ↓ ↑

(22)

Therefore, we obtain ak− bk+ √ X + iY Pˆ S↑ ↑ 2 ak− bk+ ˆ − √ + iY ↓ ↓ iSPX 2 Pˆ = √ −ak+ bk− ↓ ↓ + ak− bk+ ↑ ↑ irˆ2 − rˆ2 2

(23)

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Also, we can write,

Similarly, we can write

ck+ ak− Z Pˆ iS↓↑ + ck− ak+ iSPˆ Z ↓ ↑ = iPˆ ck+ ak− + ck− ak+ rˆ3 ↓ ↓

(24)

Combining (23) and (24), we find = √pˆ irˆ1 − rˆ2 bk ak ↑ ↑ pˆ CV k + − 2 − bk ak ↓ ↓ + iPˆ rˆ3 ck ak −

+

+

Using the above results, we can write

−

− ck− ck+ ↓ ↑

(25)

From the above relations, we obtain, ↑ = e−i/2 cos/2 ↑ +ei/2 sin/2 ↓

1 1 ↑ ↑ = iˆ sin cos + jˆ sin sin + kˆ cos = rˆ3 2 2 and 1 ↓ ↓ = − rˆ3 2 pˆ CV k pˆ = √ irˆ2 − rˆ2 ak− bk+ ↑ ↑ − bk− ak+ ↓ ↓ 2 + iPˆ rˆ3 ck+ ak− − ck− ak+ ↓ ↑ ak− bk+ bk− ak+ Pˆ = rˆ3 irˆ1 − rˆ2 √ + √ 2 2 2 ˆ P + rˆ3 irˆ2 − rˆ2 ck+ ak− + ck− ck+ 2

(26)

↓ = e−i/2 sin/2 ↑ +ei/2 cos/2 ↓ Therefore, ↓ ↑ x = −sin/2cos/2↑↑x

+e−i cos2 /2↓↑x −ei sin2 /2↑↓x (27)

Thus,

But we know from above that ↑↑x = 0

1 ↓↑ = 2

↓↑x =

1 2

and ↓↓x = 0

(29)

we can write that,

Thus, we get

rˆ1 = rˆ2 = rˆ2 = 1

1 ↓ ↑ x = e−i cos2 /2 − ei sin2 /2 2 1 = cos − i sin cos2 /2 2 − cos + i sin sin2 /2

bk− pˆ pˆ CV k = rˆ3 irˆ2 − rˆ2 ak+ √ + ck− 2 2 bk+ +ak− √ + ck+ 2

1 cos cos − i sin 2 Similarly, we obtain =

1 ↓ ↑y = i cos + sin cos and 2 1 ↓ ↑z = − sin 2 ˆ ↑ ˆ ↑ y + k↓ ↓ ↑ = iˆ↓ ↑ x + j↓

Therefore, 1 cos cos − i sin iˆ 2 ˆ +icos + sin cos jˆ − sin k 1 ˆ = cos cos iˆ + sin cos jˆ − sin k 2 + i −iˆ sin + jˆ cos 1 1 = rˆ1 + irˆ2 = − iirˆ1 − rˆ2 2 2 Mater. Focus, 6, 1–34, 2017

also, Pˆ rˆ3 = Pˆx sin cos iˆ + Pˆy sin sin jˆ + Pˆz cos kˆ

(28)

where, Pˆ = SP X = SP Y = SP Z, SP X = u∗C 0 rPˆ uVX 0 r d 3 r = PˆCVX 0 and SPˆ Z = PˆCVZ 0 Thus, Pˆ =PˆCVX 0 = PˆCVY 0 = PˆCVZ 0 = PˆCV 0 where, PˆCV 0 = u∗C 0 r PˆuV 0 rd 3 r ≡ Pˆ For a plane polarized light wave, we have the polarizaˆ when the light wave vector is traveling tion vector s = k, along the z-axis. Therefore, for a plane polarized lightˆ wave, we have considered s = k. Then, from (29) we get ˆ = k · P rˆ3 irˆ1 − rˆ2 Ak+B · pˆ CV k kcos t 2 and

⎫ b ⎪ = ak √k+ + ck ⎪ Ak ⎪ ⎬ − + 2 ⎪ b ⎪ = ak √k− + ck ⎪ ⎭ Bk + − 2

(30)

(31)

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+ sin/2cos/2↓↓x

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Thus,

Thus,

Pˆ 2 2 + Bk 2 cos t · pˆ CV k = k · rˆ3 irˆ1 − rˆ2 2 Ak 2 =

1 + Bk 2 cos2 t Pˆ cos 2 Ak 4 z

(32)

2 for a plane polarized So, the average value of · pˆ CV k light-wave is given by 2 av · pˆ CV k

2 2 + Bk 2 = Pˆz 2 Ak d cos2 sin d 4 0 0 1 2 + Bk 2 Pˆ 2 Ak × (33) 2 3 z

where, Pˆz 2 = 1/2k · pˆ cv 02 and

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k · pˆ cv 02 =

m2 Eg0 Eg0 + 4mr Eg0 + 2/3

(34)

and Bk in terms of constants of We shall express Ak the energy spectra in the following way: and Bk Substituting ak± , bk± , ck± and 0k± in Ak in (31) we get Eg0 2 Ak = t + √ Eg0 + 0k+ 2 Eg0 − 1/2 2 2 −0k (35) + 0k− Eg0 + Eg0 2 Bk = t + √ Eg0 + 0k− 2 Eg0 − 1/2 2 2 −0k (36) + 0k− Eg0 + in which, 2 0k +

Eg0 + 1 ≡ = 1− 21k + 2 1k + 1k − Eg0

and 2 0k = −

1k + Eg0 21k

+

Substituting x ≡ 1k + in

≡

Eg0 − 1 1− 2 1k +

2 0k , ±

we can write,

Eg0 Eg0 + 1 Ak = t + √ 1− Eg0 + 2 x 2 Eg0 + 1 Eg0 − − 1 − 4 Eg0 + x Eg − 1/2 × 1− 0 x 8

= Ak

1/2 a 2a 1 − 0 + 21 t+ √ 2 x x 2

where, a0 ≡ Eg20 + 2 Eg0 + −1 and a1 ≡ Eg0 − 2 . After tedious algebra, one can show that 1/2 1 1 = t + √ Eg − − Ak 0 2 1k + Eg0 + 2 Eg0 + 1/2 1 × (37) 1k + Eg0 − 2 Similarly, from (36), we can write, Eg0 Eg0 − 1 = t + √ Bk 1 + Eg0 + 2 x 2 Eg + 1 Eg0 − − 1− 0 4 Eg0 + x Eg − 1/2 × 1+ 0 x So that, finally we get, Eg − = t + √ Bk 1+ 0 2 1k + 2

(38)

Using (33), (34), (37) and (38), we can write

2 2 av eA0 · pˆ CV k − Ev k 2m Ec k 2 2 2 eA0 2 2 k · pˆ CV 0 = t+ √ 2m 3 4 2 Eg0 − 1 1+ + Eg0 − × 1k 1k + 1/2 1 1 × − 1k + Eg0 + Eg0 + 1/2 2 1 × − (39) 1k + Eg0 − 2

Following Nag,71 it can be shown that A20 =

I 2 √ sc 0

2 2 c 3

(40)

where, I is the light intensity of wavelength , 0 is the permittivity of free space and c is the velocity of light. Thus, the simplified electron energy spectrum in III–V, ternary and quaternary materials in the presence of light waves can approximately be written as 2 k2 = 0 E 2mc

(41)

Mater. Focus, 6, 1–34, 2017

Das et al.


where, 0 E ≡ I11 E − 0 E ,

where,

e2 I 2 Eg0 Eg0 + 2 0 E ≡ √ 96mr c 3 sc 0 Eg0 + 2/3 4 Eg0 − 2 1 × t+ √ 1+ 0 E + 2 0 E 1/2 1 1 + Eg0 − − 0 E + Eg0 + Eg0 + 1/2 2 1 × − 0 E + Eg0 − 2 and

(44)

2 k2 = t1 E + t2 E 2 − 2mc

(45)

2 k2 = t1 E − 2mc

(46a)

I 2 Eg0 1 e2 B0 E ≡ √ 384c 3mr sc 0 1 E 2 Eg0 1 1 − × 1+ + Eg0 0 E 1 E Eg0 (2) 1 E ≡ Eg0 1 + 2mc /mr aE1 + aE 1/2 . For relatively wide band gap semiconductors, one can write, a → 0, b → 0, c → 0 and I11 E → E Thus, from (42), we get, (43)

C0 BJ t + A A 2 I Eg0 Eg0 + 2 2 e2 1 + C0 = √ √ 96c 3 mr sc 0 Eg0 + 2/3 4 2 m∗ A = Eg0 B = 1 + mV 2B BC − G = A + 3 A + U = 1 +

=

C = Eg0 + −1 + Eg0 + Eg0 − −2 A + −1 C P = 0 J J = D + 2Eg0 − f A 2Eg0 − D = 1 + A + 1 1 + − C f = A + 2 Eg0 − 2 1 3mc t1 = 1 + = mr Eg0 =

e2 I 2 √ 96mr c 3 sc 0

and t2 = t1

It is well known that the band tails are being formed in the forbidden zone of HDS and can be explained by the overlapping of the impurity band with the conduction and valence bands.91 Kane92 and Bonch Bruevich93 have independently derived the theory of band tailing for semiconductors having unperturbed parabolic energy bands. Kane’s model92 was used to explain the experimental results on tunneling94 and the optical absorption edges95 96 in this context. Halperin and Lax97 developed a model for band tailing applicable only to the deep tailing states. Although Kane’s concept is often used in the literature for the investigation of band tailing,98 99 it may be noted that this model92 100 suffers from serious assumptions in the sense that the local impurity potential is assumed to 9

ARTICLE

(42)

where, 0 E ≡ E1 + aE − B0 E ,

Mater. Focus, 6, 1–34, 2017

2 k2 = U I11 E − P 2mc

where

Thus, under the limiting condition k → 0, from (41), we observe that E = 0 and is positive. Therefore, in the presence of external light waves, the energy of the electron does not tend to zero when k → 0, where as for the un-perturbed three band model of Kane, I11 E = 2 k2 /2mc in which E → 0 for k → 0. As the conduction band is taken as the reference level of energy, therefore the lowest positive value of E for k → 0 provides the increased band gap (Eg ) of the semiconductor due to photon excitation. The values of the increased band gap can be obtained by computer iteration processes for various values of I and respectively. Special Cases: (1) For the two-band model of Kane, we have → 0. Under this condition, I11 E → E1 + aE = 2 k2 /2mc . Since, → 1, t → 1, → 0, → 0 for → 0 from (41), we can write the energy spectrum of III–V, ternary and quaternary materials in the presence of external photo-excitation whose unperturbed conduction electrons obey the two band model of Kane as

2 k2 = 0 E 2mc

−3/2 e2 I 2 2mc 1 + aE √ 96C 3 mr sc 0 mr

The (41), (42) and (43) can approximately be written as

and

m I E 1/2 0 E ≡ Eg0 1 + 2 1 + c 11 mv Eg0

2 k2 = 0 E 2mc

0 E ≡ E −

Das et al.

ARTICLE


be small and slowly varying in space coordinates.99 In this respect, the local impurity potential may be assumed to be a constant. In order to avoid these approximations, we have developed in this book, the electron energy spectra for HDS for studying the EP based on the concept of the variation of the kinetic energy91 99 of the electron with the local point in space coordinates. This kinetic energy is then averaged over the entire region of variation using a Gaussian type potential energy. It may be noted that, a more general treatment of many-body theory for the DOS of HDS merges with one-electron theory under macroscopic conditions.91 Also, the experimental results for the Fermi energy and others are the average effect of this macroscopic case. So, the present treatment of the oneelectron system is more applicable to the experimental point of view and it is also easy to understand the overall effect in such a case.101 In a HDS, each impurity atom is surrounded by the electrons, assuming a regular distribution of atoms, and it is screened independently.98 100 102 The interaction energy between electrons and impurities is known as the impurity screening potential. This energy is determined by the inter-impurity distance and the screening radius, which is known as the screening length. The screening radius changes with the electron concentration and the effective mass. Furthermore, these entities are important for HDS in characterizing the semiconductor properties103 104 and the modern electronic devices.98 105 The works on Fermi energy and the screening length in an n-type GaAs have already been initiated in the literature,106 107 based on Kane’s model. Incidentally, the limitations of Kane’s model,92 99 as mentioned above, are also present in their studies. The Gaussian distribution F V of the impurity potential is given by92 93 F V = g2 −1/2 exp−V 2 /g2

(46a )

where, g is the impurity scattering potential. It appears from Eq. (46a ) that the variance parameter g is not equal to zero, but the mean value is zero. Further, the impurities are assumed to be uncorrelated and the band mixing effect has been neglected in this simplified theoretical formalism. Under the condition of heavy doping, using the method of averaging the kinetic energy of the electron, the HD dispersion relations in this case in the presence of light waves can be written as 2 k2 = T1 E g 2mc 2 k2 = T2 E g 2mc 2 k2 = T3 E g 2mc

2 b E g 1 + Erf E/g c 0 c + bc − b 1 0 E g + + 1− c2 c c b 1 E × 1− 1 + Erf c 2 g 1 b 2 − √ exp−u22 1− 1− c c c cg exp−p2 /4 sinhpu2 × p p=1 1 1 + cE 1 b≡ c≡ u2 ≡ Eg + Eg + 2/3 cg 1 b 2 1− 1− T32 E g ≡ 1 + Erf E/g c c c √ × exp−u22 cg 0 E g g E 1 2 E −E 2 2 + + 2E 1 + Erf = √ exp g 2 g 4 g 2 0 E g √ E = g exp−E 2 /g2 2 −1 + 1 + Erf E/g 2 T2 E g = U1 3 E g + t2 20 E g × 1 + Erf E/g −1 − 2 3 E g ≡ E g 1 + Erf E/g 0

and T3 E g = t1 3 E g −

(47)

The DOS functions for (46b), (47) and (48) can, respectively, be written as 2mc 3/2 NL E = 4gv T1 Eg T1 Eg h2 (48b) 3/2 2mc NL E = 4gv T2 Eg T2 Eg h2 (48c) 3/2 2mc T3 Eg T3 Eg NL E = 4gv h2 (48d)

(48a)

The EEM can be expressed in this case by using (46), (47) and (48a).

(46b)

where T1 E g = U T31 E g + iT32 E g − P 10

T31 E g ≡

m∗ EFHDL g = mc Real part of T1 EFHDL g

(49)

Mater. Focus, 6, 1–34, 2017

Das et al.


m∗ EFHDL g = mc T2 EFHDL g m∗ EFHDL g = mc T3 EFHDL g

(50) (51a)

The electron concentration can be written as gv 2mc 3/2 Real part of T1 EFHDL g 3/2 n0 = 3 2 2 s 3/2 + LrT1 EFHDL g (51b) r=1

n0 =

gv 2mc 3/2 T2 EFHDL g 3/2 3 2 2 s 3/2 + LrT2 EFHDL g

(51c)

r=1

n0 =

3/2 gv 2mc T3 EFHDL g 3/2 3 2 2 s 3/2 + LrT3 EFHDL g

n0 =

(51d)

r=1

The EEM can be expressed in this case by using (46), (47) and (48a). m∗ EFL = mc 0 EFL

(53a)

m∗ EFL = mc 0 EFL

(53b)

m∗ EFL = mc 0 EFL

(53c)

where EFL is the Fermi energy in the absence of band— tails and in the presence of light waves. The electron concentration can be written as 3/2 gv 2mc 0 EFL 3/2 n0 = 3 2 2 s + Lr0 EFL 3/2 (54a)

2r and in this case Lr = 2kB T 1−2r 2r 2r /EFL . where kB is the Boltzmann constant, T is the tempeture and 2r is the Zeta function of order 2r.108 In the absence of light waves and band tails we get 2mc 3/2 N E = 4gv I E I11 E (55a) 11 h2 3/2 2mc N E = 4gv 1+2E E1+E (55b) h2 3/2 √ 2mc N E = 4gv E (55c) h2

Equations (57b) and (57c) are well-known in the literature. The EEM can be expressed in this case by using (46), (47) and (48a). m∗ EF = mc I11 EF

+

s r=1

Mater. Focus, 6, 1–34, 2017

Lr0 EFL 3/2

(54b)

(56a)

∗

m EF = mc 1 + 2EF

(56b)

m∗ EF = mc

(56c)

where EF is the Fermi energy in the absence of band—tails and light waves. Equations (56b) and (56c) are well-known in the literature. The electron concentration can be written as gv 2mc 3/2 n0 = I11 EF 3/2 3 2 2 s (57a) + LrI11 EF 3/2 n0 =

gv 3 2

r=1

3/2 2mc EF 1 + EF 3/2 2 +

s

LrEF 1 + EF 3/2

(57b)

r=1

n0 = Nc F1/2

(57c)

where NC ≡ gv 22mc kB T /h , ≡ EF /kB T and Fj is the one parameter Fermi-Dirac integral of order j which can be written as,109 1 Fj = y j 1 + expy − −1 dy j + 1 0 2 3/2

j > −1

r=1

gv 2mc 3/2 0 EFL 3/2 n0 = 3 2 2

(54c)

r=1

(58a)

where j + 1 is the complete Gamma function or for all j, analytically continued as a complex integral around the negative axis 0+ Fj = Aj y j 1 + exp−y − −1 dy (58b) 0

11

ARTICLE

where gv is the valley degeneracy and Lr = 2r 2kB T 1−2r 2r 2r /EFHDL The DOS functions in the absence of heavy doping can respectively be written (41), (42) and (43) as 3/2 2mc 0 E 0 E (52a) NL E = 4gv 2 h 2mc 3/2 NL E = 4gv 0 E 0 E (52b) h2 2mc 3/2 0 E 0 E (52c) NL E = 4gv h2

gv 2mc 3/2 0 EFL 3/2 3 2 2 s + Lr0 EFL 3/2


√ in which Aj ≡ −j/2 −1 and in this case Lr = 2kB T 1−2r 2r 2r /EF2r . The Eq. (57c) is well-known in the literature. Under the conditions Eg or Eg together with the inequality EF 1, the electron concentration in bulk specimens of III–V, ternary and quaternary semiconductors whose energy band structures are defined by two band model of Kane can be expressed as 15akB T F3/2 (59a) n0 = Nc F1/2 + 4

ARTICLE

The entropy per unit volume (S0 ) can be written as S0 = − (59b) T E=EF in which is the thermodynamic potential which, in turn, can be expressed in accordance with the Fermi-Dirac statistics as EF − E0 (59c) ln 1 + exp = −kB T kB T where the summation is carried out over all the possible 0 states. Thus, combining (59b) and (59c) and following Tsidilkovskii,110 under the condition of heavy doping we get n0 (59d) S¯0 = 2 kB2 T /3 EF − E where E is the energy with zero free motion part. It should be noted that being a thermodynamic relation, the entropy as expressed by (59d), in general, is valid for electronic materials having arbitrary dispersion relations and their nanostructures. In addition to bulk materials, (59d) is valid under one, two and three dimensional quantum confinement of the charge carriers (such as quantum wells in ultrathin films, nipi structures, inversion and accumulation layers, quantum well superlattices, carbon nanotubes, quantum wires, quantum wire superlattices, quantum dots, magneto inversion and accumulation layers, quantum dot superlattices, magneto nipis, quantum well superlattices under magnetic quantization, ultrathin films under magnetic quantization, etc.). The formulation of entropy requires the relation between electron statistics and the corresponding Fermi energy which is basically the band structure dependent quantity and changes under different physical conditions. It is worth remarking to note that the number 2 /3 has occurred as a consequence of mathematical analysis and is not connected with the well-known Lorenz number. For quantum wells in ultrathin films, nipi structures, inversion and accumulation layers, quantum well superlattices, magneto inversion and accumulation layers, magneto nipis, quantum well superlattices under magnetic quantization and magneto size quantization, the carrier concentration is measured per unit area 12

Das et al.

and therefore entropy should be measured per unit area. For quantum wires, quantum wires under magnetic field, quantum wire superlattices and such allied systems, the electron concentration is measured per unit length and thus the entropy should be measured per unit length. Besides, for bulk materials under strong magnetic field, quantum dots, quantum dots under magnetic field, quantum dot superlattices and quantum dot superlattices under magnetic field, the carrier concentration is expressed per unit volume and therefore the entropy should be measured per unit volume. Using (54a), (54b) and (54c) together with (59d) we can study the entropy in HD bulk specimens of III–V, ternary and quaternary compounds in the presence of light waves whose conduction electrons obey unperturbed three and two band models of Kane together with parabolic energy bands. Using (57a), (57b) and (57c) together with (59d) we can study the entropy in bulk specimens of III–V, ternary and quaternary compounds in the absence of light waves whose conduction electrons obey unperturbed three and two band models of Kane together with parabolic energy bands. For bulk semiconductors having parabolic energy bands in the absence of any field and under the condition of non-degeneracy, the entropy assumes the well known form as 2 kB n0 S0 = (59e) 3 2.2. The Entropy Under Magnetic Quantization in HD Kane Type Semiconductors in the Presence of Light Waves (i) Using (46b), the magneto-dispersion law, in the absence of spin, for HD III–V, ternary and quaternary semiconductors, in the presence of photo-excitation, whose unperturbed conduction electrons obey the three band model of Kane, is given by 2 kz2 1 T1 E g = n + (60) 0 + 2 2mc Using (66), the DOS function in the present case can be expressed as max gv e 2mc n Dg E g = T1 E g 2 2 2 n=0 −1/2 1 × T1 E g − n + 0 2 × HE − El1 (61) where, El1 is the Landau sub-band energies in this case and is given as 1 T1 El1 g = n + (62) 0 2 Mater. Focus, 6, 1–34, 2017

Das et al.


The EEM in this case assumes the form m∗ EFHDLB g = mc Real part of T1 EFHDLB g

(63)

where, EFHDLB is the Fermi energy under quantizing magnetic field in the presence of light waves as measured from the edge of the conduction band in the vertically upward direction in the absence of any quantization. The electron concentration is given by nmax gv e 2mc Real part of n0 = T1 EFHDLB g 22 n=0 1/2 r=s 1 − n+ + Lr T1 EFHDLB g 0 2 r=1 1/2 1 0 (64) − n+ 2

Using (66), the DOS function in the present case can be expressed as max gv e 2mc n 0 E 0 E Dg E = 2 2 2 n=0 −1/2 1 − n+ HE − Ei10 (66) 0 2 where, Ei10 is the Landau sub-band energies in this case and is given as 1 (67) 0 0 El10 g = n + 2 The EEM in this case assumes the form m∗ EFLB = mc 0 EFLB

(68)

where, EFLB is the Fermi energy under quantizing magnetic field in the presence of light waves and band tails as measured from the edge of the conduction band in the vertically upward direction in the absence of any quantization. The electron concentration is given by gv e 2mc n0 = 2 2 1/2 nmax 1 0 EFLB − n+ 0 × 2 n=0 1/2 r=s 1 0 (69) + Lr 0 EFLB − n+ 2 r=1 Mater. Focus, 6, 1–34, 2017

where EFB is the Fermi energy in this case. Using (70) and (59d) we can study the entropy in this case. (ii) Using (47), the magneto-dispersion law, in the absence of spin, for HD III–V, ternary and quaternary semiconductors, in the presence of photo-excitation, whose unperturbed conduction electrons obey the two band model of Kane, is given by 2 kz2 1 (71) 0 + T2 E g = n + 2 2mc Using (71), the DOS function in the present case can be expressed as Dg E g max gv e 2mc n = T2 E g T2 E g 2 2 2 n=0 −1/2 1 − n+ 0 HE − El2 (72) 2 where, El2 is the Landau sub-band energies in this case and is given as 1 (73) T2 El2 g = n + 0 2 The EEM in this case assumes the form m∗ EFHDLB g = mc T2 EFHDLB g

(74)

The electron concentration is given by gv e 2mc n0 = 22 1/2 nmax 1 × T2 EFHDLB g − n + 0 2 n=0 r=s + Lr T2 EFHDLB g r=1

1/2 1 − n+ 0 2

(75)

Thus using (75) and (59d) we can study the entropy in this case. 13

ARTICLE

Thus using (64) and (59d) we can study the entropy in this case. Using (41), the magneto-dispersion law, in the absence of spin and band tails for III–V, ternary and quaternary semiconductors, in the presence of photo-excitation, whose unperturbed conduction electrons obey the two band model of Kane, is given by 2 kz2 1 0 E = n + (65) 0 + 2 2mc

Thus using (69) and (59d) we can study the entropy in this case. In the absence of light waves and heavy doping, the electron concentration can be written as 1/2 max gv e 2mc n 1 n0 = I11 EFB − n + 0 22 2 n=0 1/2 r=s 1 0 (70) + Lr I11 EFB − n + 2 r=1

Das et al.


Using (42), the magneto-dispersion law, in the absence of spin and band tails for III–V, ternary and quaternary semiconductors, in the presence of photo-excitation, whose unperturbed conduction electrons obey the two band model of Kane, is given by 2 kz2 1 0 E = n + (76) 0 + 2 2mc

ARTICLE

Using (76), the DOS function in the present case can be expressed as max gv e 2mc n E 0 E Dg E = 2 2 2 n=0 0 −1/2 1 − n+ HE − Ei20 (77) 0 2 where, Ei20 is the Landau sub-band energies in this case and is given as 1 0 El20 g = n + (78) 0 2 The EEM in this case assumes the form m∗ EFLB = mc 0 EFLB

(79)

The electron concentration is given by gv e 2mc n0 = 22 1/2 nmax 1 × 0 EFLB − n+ 0 2 n=0 1/2 r=s 1 (80) + Lr 0 EFLB − n+ 0 2 r=1 Thus using (80) and (59d) we can study the entropy in this case. In the absence of light waves and band tails, the electron concentration for two band model of Kane in the presence of magnetic quantization can be written as gv e 2mc n0 = 22 1/2 nmax 1 0 EFB 1 + EFB − n + × 2 n=0 r=s + Lr EFB 1 + EFB r=1

1/2 1 0 − n+ 2

where B1 = 0 /kB T , a01 = 1 +n +1/2 0 , b01 = 1 + n + 1/2 0/a01 and ¯ B1 = EFB − b01 /kB T . The Eq. (82) is well-known in the literature.79 Thus using (82) and (59d) we can study the entropy in this case. (iii) Using (48a), the magneto-dispersion law, in the absence of spin, for HD III–V, ternary and quaternary semiconductors, in the presence of photo-excitation whose unperturbed conduction electrons obey the parabolic energy bands is given by 2 kz2 1 0 + (83) T3 E g = n + 2 2mc Using (71), the DOS function in the present case can be expressed as max gv e 2mc n Dg E g = T3 E g 2 2 2 n=0 −1/2 1 × T3 E g − n + 0 2 × HE − El3 (84) where, El3 is the Landau sub-band energies in this case and is given as 1 0 T3 El3 g = n + (85) 2 The EEM in this case assumes the form m∗ EFHDLB g = mc T3 EFHDLB g

(86)

The electron concentration is given by gv e 2mc n0 = 22 1/2 nmax 1 × T3 EFHDLB g − n + 0 2 n=0 r=s + Lr T3 EFHDLB g r=1

(81)

Thus using (81) and (59d) we can study the entropy in this case. 14

Under the condition EFB 1, the electron concentration in this case can be expressed as nmax 1 3 n0 = Nc B1 1 + b01 √ a01 2 n=0 3 × F−1/2 ¯ B1 + kB T F1/2 ¯ B1 (82) 4

1/2 1 − n+ 0 2

(87)

Using (87), (51e) and (51f) we can study the entropy in this case. Mater. Focus, 6, 1–34, 2017

Das et al.


Using (43), the magneto-dispersion law, in the absence of spin and band tails for III–V, ternary and quaternary semiconductors, in the presence of photo-excitation whose unperturbed conduction electrons obey the parabolic energy bands is given by 2 kz2 1 0 E = n + (88) 0 + 2 2mc Using (88), the DOS function in the present case can be expressed as max gv e 2mc n Dg E = 0 E 2 2 2 n=0 −1/2 1 × 0 E − n + 0 2 × HE − Ei21 (89)

The EEM in this case assumes the form m∗ EFLB = mc 0 EFLB

(91)

The electron concentration is given by gv e 2mc n0 = 22 1/2 nmax 1 × 0 EFLB − n+ 0 2 n=0 1/2 r=s 1 + Lr 0 EFLB − n+ 0 (92) 2 r=1 Thus using (92) and (59d) we can study the entropy in this case. In the absence of light waves and band tails, the electron concentration for isotropic parabolic energy bands can be written under magnetic quantization as n0 = Nc B1

nmax

F−1/2 ¯ B2

(93)

n=0

where ¯ B2 =

EFB − n + 1/2 0 kB T

The Eq. (93) is well-known in the literature.79 Thus using (93) and (59d) we can study the entropy in this case. Mater. Focus, 6, 1–34, 2017

T1 E g 1 kz E2 E0 = n+ − ky T1 E g 0 + 2 2mc B 2 2 mc E0 T1 E g (94) − 2B 2 The use of (91) leads to the expressions of the EEMs’ along z and y directions as m∗z EFBLHDC nE0 B = Real part of mc T1 EFBLHDC g +

mc E02 T1 EFBLHDC g T1 EFBLHDC g B2

(95)

m∗y EFBLHDC n E0 B 2 B T1 EFBLHDC g −1 = Real part of E0 1 0 × T1 EFBLHDC g − n + 2 mc E02 T1 EFBLHDC g + 2B 2 T1 EFBLHDC g × T1 EFBLHDC g 2 1 × T1 EFBLHDC g − n + 0 2 mc E02 T1 EFBLHDC g 2 + 2B 2 mc E02 T1 EFBLHDC g +1 + (96) B2 where EFBLHDC is the Fermi energy in this case. The Landau energy (Enl41 ) can be written as T1 Enl41 g mc E02 T1 Enl41 g 2 1 = n+ 0 − 2 2B 2

(97)

The electron concentration in this case can be expressed as n0 =

nmax 2gv B 2mc Real part of M161 EFBLHDC nE0 B 2 2 3Lx E0 n=0 +N161 EFBLHDC nE0 B

(98) 15

ARTICLE

where, Ei21 is the Landau sub-band energies in this case and is given as 1 0 El21 g = n + (90) 0 2

2.3. The Entropy Under Crossed Electric and Quantizing Magnetic Fields in HD Kane Type Semiconductors in the Presence of Light Waves (i) The electron dispersion law in the present case is given by

Das et al.


where

+

M161 n EFBL 1 = T1 EFBLHDC n E0 B − n + 0 2 −

mc E02 T1 EFBLHDC n E0 B 2 2B 2

+1 +

3/2

1 − T1 EFBLHDC n E0 B − n + 0 2 3/2 m E2 − c 20 T1 EFBLHDC n E0 B 2 2B

and

ARTICLE

N161 n EFBL =

s

2B 2 mc E02 0 EFBLC

(103)

B2

0 Enl411 mc E02 0 Enl411 2 1 0 − = n+ (104) 2 2B 2

+eE0 Lx T1 EFBLHDC n E0 B

1 T1 EFBLHDC n E0 B

where, EFBLC is the Fermi energy in this case. The Landau energy Enl411 can be written as

×

mc E02 0 EFBLC 2

(99)

The electron concentration in this case can be expressed as max 2gv B 2mc n n0 = M E n E0 B 2 2 3Lx E0 n=0 1612 FBLC +N1612 EFBLC n E0 B

LrM161 n EFBL

(100)

(105)

where

r=1

M1612 n EFBL 1 ≡ 0 EFBLC n E0 B − n + 0 2

Thus using (98) and (59d) we can study the entropy in this case. The electron dispersion law in the present case in the absence of band tails is given by 1 kz E2 E0 − ky 0 E 0 E = n+ 0 + 2 2mc B mc E02 0 E 2 (101) − 2B 2

−

1 0 − 0 EFBLC n E0 B − n + 2 3/2 mc E02 2 − 0 EFBLC n E0 B 2B 2

m∗z EFBLC n E0 B = mc 0 EFBLC +

×

B2

m∗y EFBLC n E0 B

B 2 0 EFBLC −1 E0 1 0 × 0 EFBLC − n + 2 mc E02 0 EFBLC + 2B 2 0 EFBLC × 0 EFBLC 2 1 × 0 EFBLC − n + 0 2

=

16

(102)

3/2

+eE0 Lx 0 EFBLC n E0 B

The use of (101) leads to the expressions of the EEMs’ along z and y directions as

mc E02 0 EFBLC 0 EFBLC

mc E02 0 EFBLC n E0 B 2 2B 2

1 0 EFBLC n E0 B

and N1612 n EFBL ≡

s

LrM1612 n EFBL

r=1

Thus using (105) and (59d) we can study the entropy in this case. (ii) The electron dispersion law in the present case is given by 1 kz E2 T2 E g = n + 0 + 2 2mc E − 0 ky T2 E g B mc E02 T2 E g 2 − (106) 2B 2 Mater. Focus, 6, 1–34, 2017

Das et al.


The use of (123) leads to the expressions of the EEMs’ along z and y directions as m∗z EFBLHDC nE0 B = mc T2 EFBLHDC g +

B2

(107)

1 T2 EFBLHDC n E0 B

(111)

N162 n EFBL =

s

LrM162 n EFBL

(112)

r=1

(113)

m∗z EFBLC n E0 B = mc 0 EFBLC (108)

T2 Enl412 g mc E02 T2 Enl412 g 2 1 0 − (109) = n+ 2 2B 2 The electron concentration in this case can be expressed as max 2gv B 2mc n n0 = M E n E0 B 3Lx 2 2 E0 n=0 162 FBLHDC +N162 EFBLHDC n E0 B

(110)

where M162 n EFBL 1 0 = T2 EFBLHDC n E0 B − n + 2

3/2

+eE0 Lx T2 EFBLHDC n E0 B 1 − T2 EFBLHDC n E0 B − n + 0 2

+

mc E02 0 EFBLC 0 EFBLC B2

(114)

m∗y EFBLC n E0 B 2 B 0 EFBLC −1 0 EFBLC = E0 mc E02 0 EFBLC 1 − n+ 0 + 2 2B 2 0 EFBLC × E 0 EFBLC 2 0 FBLC mc E02 0 EFBLC 2 1 − n+ 0 + 2 2B 2 mc E02 0 EFBLC +1 + (115) B2 The Landau energy Enl4112 can be written as 0 Enl4113 mc E02 0 Enl4113 2 1 0 − = n+ (116) 2 2B 2 The electron concentration in this case can be expressed as max 2gv B 2mc n n0 = M E n E0 B 2 2 3Lx E0 n=0 1614 FBLC +N1614 EFBLC n E0 B

(117) 17

ARTICLE


The Landau energy Enl412 can be written as

Mater. Focus, 6, 1–34, 2017

×

3/2

Thus using (110) and (59d) we can study the entropy in this case. The electron dispersion law in the present case in the absence of band tails is given by 1 kz E2 0 E = n+ 0 + 2 2mc E mc E02 0 E 2 − 0 ky 0 E − B 2B 2

m∗y EFBLHDC n E0 B 2 B T2 EFBLHDC g −1 = E0 1 0 × T2 EFBLHDC g − n + 2 mc E02 T2 EFBLHDC g + 2B 2 T2 EFBLHDC g × T2 EFBLHDC g 2 1 × T2 EFBLHDC g − n + 0 2 mc E02 T2 EFBLHDC g 2 + 2B 2 mc E02 T2 EFBLHDC g +1 + B2



and

mc E02 T2 EFBLHDC g T2 EFBLHDC g

−

−

Das et al.


1 × T3 EFBLHDC g − n + 0 2 mc E02 T3 EFBLHDC g 2 + 2B 2 mc E02 T3 EFBLHDC g +1 + B2

where M1614 n EFBL 1 ≡ 0 EFBLC n E0 B − n + 0 2 −


3/2

+eE0 Lx 0 EFBLC n E0 B

The Landau energy Enl413 can be written as

1 − 0 EFBLC n E0 B − n + 0 2 3/2 mc E02 2 − 0 EFBLC n E0 B 2B 2 1 × 0 EFBLC n E0 B

ARTICLE


s

T3 Enl413 g mc E02 T3 Enl413 g 2 1 = n+ (121) 0 − 2 2B 2 The electron concentration in this case can be expressed as max 2gv B 2mc n M E n E0 B n0 = 2 2 3Lx E0 n=0 163 FBLHDC +N163 EFBLHDC n E0 B

LrM1614 n EFBL

r=1

M163 n EFBL 1 ≡ T3 EFBLHDC n E0 B − n + 0 2 −

m∗y EFBLHDC n E0 B

B 2 T3 EFBLHDC g −1 = E0 1 0 × T3 EFBLHDC g − n + 2 mc E02 T3 EFBLHDC g + 2B 2 T3 EFBLHDC g × T3 EFBLHDC g 2

18

× (119)

3/2

1 0 − T3 EFBLHDC n E0 B − n + 2 3/2 m E2 − c 20 T3 EFBLHDC n E0 B 2 2B

m∗z EFBLHDC nE0 B = mc T3 EFBLHDC g

B2


+eE0 Lx T3 EFBLHDC n E0 B


mc E02 T3 EFBLHDC g T3 EFBLHDC g

(122)

where

Thus using (117) and (59d) we can study the entropy in this case. (iii) The electron dispersion law in the present case is given by 1 kz E2 0 + T3 E g = n + 2 2mc E − 0 ky T3 E g B mc E02 T3 E g 2 − (118) 2B 2

+

(120)

1 T3 EFBLHDC n E0 B

(123)

and N163 n EFBL =

s

LrM163 n EFBL

(124)

r=1

Thus using (122) and (59d) we can study the entropy in this case. The electron dispersion law in the present case in the absence of band tails is given by 1 kz E2 E0 0 E = n+ 0 + − ky 0 E 2 2mc B 2 2 mc E0 0 E − (125) 2B 2 Mater. Focus, 6, 1–34, 2017

Das et al.


The use of (119) leads to the expressions of the EEMs’ along z and y directions as m∗z EFBLC n E0 B = mc 0 EFBLC +

mc E02 0 EFBLC 0 EFBLC

(126)

B2

The Landau energy Enl4114 can be written as 0 Enl4114 mc E02 0 Enl4114 2 1 0 − (128) = n+ 2 2B 2 The electron concentration in this case can be expressed as max 2gv B 2mc n n0 = M E n E0 B 3Lx 2 2 E0 n=0 1615 FBLC + N1615EFBLC n E0 B

(129)

where M1615 n EFBL 1 ≡ 0 EFBLC n E0 B − n + 0 2 −


3/2

+eE0 Lx 0 EFBLC n E0 B 1 − 0 EFBLC n E0 B − n + 0 2 3/2 mc E02 2 − 0 EFBLC n E0 B 2B 2 ×

1 0 EFBLC n E0 B


s r=1

Mater. Focus, 6, 1–34, 2017

LrM1615 n EFBL

−

E0 m E 2 I E 2 ky I11 E − c 0 112 B 2B

(130)

The use of (126) leads to the expressions of the EMMs’ along the z and y directions as mz E¯FB nE0 B m E 2 I E¯ I E¯ = mc I11 E¯FB + c 0 11 FB 2 11 FB (131) B m∗y E¯FB n E0 B 2 1 B 1 ¯ E n + = 0 I E0 I11 E¯FB 11 FB 2 m E 2 I E¯ 2 − I11 E¯FB + c 0 11 2 FB I11 E¯FB 2B I11 E¯FB 2 1 m E 2 I E¯ 2 − n+ 0 + c 0 11 2 FB +1 2 2B mc E02 I11 E¯FB 2 + (132) B2 where E¯FB is the Fermi energy in this case. The Landau energy (E¯ n1) can be written as 1 mc E02 I11 E¯n1 2 ¯ I11 En1 = n + (133a) 0 − 2 2B 2 Equation (133) at the Fermi energy EF B generates a cubic equation in B, the real single root of which when combined with 1 1 1 = − (133b) B Bn+1 Bn The electron concentration in this case assume the forms max 2gv B 2mc n T n E¯FB + T44 n E¯FB (134) n0 = 3Lx 2 2 E0 n=0 43 where T43 n EFB 1 ≡ I11 E¯FB − n + 0 2 m E2 − c 20 I11 E¯FB 2 + eE0 Lx I11 E¯FB 2B

3/2

19

ARTICLE

m∗y EFBLC n E0 B 2 B −1 0 EFBLC 0 EFBLC = E0 mc E02 0 EFBLC 1 − n+ 0 + 2 2B 2 0 EFBLC × E 0 EFBLC 2 0 FBLC mc E02 0 EFBLC 2 1 − n+ 0 + 2 2B 2 mc E02 0 EFBLC +1 + (127) B2

Thus using (129) and (59d) we can study the entropy in this case. (iv) In the absence of light waves and heavy doping the dispersion relation in III–V semi-conductors whose energy band structures are defined by the three band model of Kane can be written in the presence of cross -fields configuration as 1 kz E2 I11 E = n+ 0 + 2 2mc

Das et al.


1 − I11 E¯FB − n + 0 2 3/2 mc E02 1 2 ¯ − I11 EFB 2 2B I11 E¯FB T44 n E¯ FB =

s

and

1 − E¯FB 1 + E¯ FB − n + 0 2 3/2 mc E02 2 ¯ − 1 + 2EFB 1 + 2E¯ FB −1 2B 2 T46 n E¯FB =

LrT43 n E¯FB

Thus using (134) and (59d) we can study the entropy in this case. (b) Under the condition Eg , (126) assumes the well known from1 1 E 0 − 0 ky 1 + 2E E1 + E = n + 2 B mc E02 kz E2 2 1 + 2E + 2B 2 2mc

(135)

The use of (135) leads to the expressions of the EMMs’ along x and y directions as

ARTICLE

m∗ E¯FB n E0 B 2mc E02 1 + 2E¯ FB = mc 1 + 2E¯ FB + (136) B2 m∗ E¯FB n E0 B 2 1 B E¯ 1 + E¯ FB = E0 1 + 2E¯ FB FB 1 m E 2 1 + 2E¯ FB 2 − n+ 0 + c 0 2 2B 2 1 −2 ¯FB 1 + E¯ FB n + E × 0 2 1 + 2E¯ FB 2 m E 2 1 + 2E¯ FB 2 2mc E02 + c 0 + 1 + (137) 2B 2 B2 The Landau energy (E¯ n2 ) can be written as 1 m E2 ¯ ¯ 0 − c 20 1 + 2E¯ n2 2 En2 1 + En2 = n + 2 2B (138) The expression for n0 in this case assume the forms max 2gv B 2mc n n0 = T n E¯FB + T46 n E¯FB (139) 2 2 3Lx E0 n=0 45 where T45 n E¯FB 1 ≡ E¯FB 1 + E¯ FB − n + 0 2 + eE0 Lx 1 + 2E¯ FB − 20

LrT45 n E¯FB

r=1

r=1

−

s

and

mc E02 1 + 2E¯ FB 2 2B 2

(c) For parabolic energy band → 0, and we can write 2 1 kz E2 1 E E0 E = n+ − mc − 0 ky 0 − 2 2mc 2 B B (140) Using (140) the expressions of the EMM’s along the y and z directions can be written as m∗z E¯FB n E0 B = mc m∗y E¯FB n E0 B 2 1 mc E02 B ¯ EFB − n + = 0 + E0 2 2B 2 The Landau energy (E¯ n3 ) can be written as 1 m E2 E¯n3 = n + 0 − c 20 2 2B

(141)

(142)

(143)

The electron concentration in this case can, be expressed as max kB T n n0 = Nc gv F − F1/2 2 (144) eE0 Lx n=0 1/2 1 where ≡ 0 /kB T , 1 ≡ E¯FB − ¯ 1 /kB T , ¯ 1 ≡ n + 1/2 0 + 1/2m∗ E0 /B2 − eE0 Lx , 2 ≡ E¯FB − ¯ 2 /kB T and ¯ 2 ≡ ¯ 1 + eE0 Lx . 2.4. The Entropy in QWS of HD Kane Type Semiconductors in the Presence of Light Waves (i) The 2D DR in QWs of HD III–V, ternary and quaternary materials, whose unperturbed band structure is defined by the three band model of Kane, in the presence of light waves, can be expressed as 2 nz 2 2 ks2 + = T1 E g (145) 2mc 2mc dz The sub band energies Enl7HD can be written as T1 Enl7HD g =

2 n /dz 2 2mc z

(146)

The expression of the EEM in this case is given by 3/2

m∗ EF2DLHD nz = mc Real part of T1 EF2DLHD g

(147)

Mater. Focus, 6, 1–34, 2017

Das et al.


where, EF2DLHD is the Fermi energy in the present case as measured from the edge of the conduction band in the vertically upward direction in absence of any quantization. The DOS function can be written as N E g nzmax mc gv = T E g HE − Enl7HD dz 2 nz =1 1

(148)

The electron concentration can be written as nzmax mc gv n0 = Real part of T1 EF2DLHD nz 2 dz nz =1 s 2 nz 2 − + LrT1 EF2DLHD nz 2mc dz r=1 (149)

The sub band energies Enl7HD can be written as 0 Enl7HD =

2 n /dz 2 2mc z

(151)

where EFs is the Fermi energy in this case. It is worth noting that the EMM in this case is a function of Fermi energy alone and is independent of size quantum number. The total density-of-states function can be written as nzmax mc gv I E HE − Enz (157) NT E = 2 dz nz =1 11 where, the sub band energies Enz can be expressed as 2

I11 Enz = 2

n /dz 2 2mc z 2

The carrier concentration assumes the form n0 =

nzmax mc gv T E n + T54 EFs nz dz 2 nz =1 53 Fs z

The sub band energies Enl7HD an be written as

m EF2DL nz = mc 0 EF2DL

(152)

where, EF2DL is the Fermi energy in the present case as measured from the edge of the conduction band in the vertically upward direction in absence of any quantization. The DOS function can be written as nzmax mc gv N E = E HE − Enl7 (153) 2 dz nz =1 0 The electron concentration can be written as nzmax 2 mc gv 2 nz n0 = 0 EF2DL nz − dz 2 nz =1 2mc dz s + Lr0 EF2DL nz (154) r=1

In the absence of band tails and light waves and for isotropic three band model of Kane, the 2D electron dispersion relation in this case can be written as 2 2 ks2 + n /dz 2 = I11 E 2mc 2mc z Mater. Focus, 6, 1–34, 2017

(159)

where T53 EFs nz ≡ I11 EFs − 2 /2mc nz /dz 2 and T54 EFs nz ≡ sr=1 LrT53 EFs nz . (ii) The 2D DR in QWs of HD III–V, ternary and quaternary materials, whose unperturbed band structure is defined by the two band model of Kane, in the presence of light waves can be expressed as 2 nz 2 2 ks2 + = T2 E g (160) 2mc 2mc dz

The expression of the EEM in this case is given by ∗

(158)

(155)

T2 Enl7HD g =

2 n /dz 2 2mc z

(161)

The expression of the EEM in this case is given by m∗ EF2DLHD nz = mc T2 EF2DLHD g

(162)

where, EF2DLHD is the Fermi energy in the present case as measured from the edge of the conduction band in the vertically upward direction in absence of any quantization. The DOS function can be written as N E g nzmax mc gv = T E g HE − Enl7HD 2 dz nz =1 2

(163)

The electron concentration can be written as mc gv n0 = dz 2 nzmax 2 nz 2 T2 EF2DLHD nz − × 2mc dz nz =1 s + LrT2 EF2DLHD nz (164) r=1

21

ARTICLE

The 2D DR in QWs of III–V, ternary and quaternary materials in the absence of band tails, whose unperturbed band structure is defined by the three band model of Kane, in the presence of light waves, can be expressed as 2 ks2 2 nz = 0 E + (150) 2mc 2mc dz

Using (184), the EMM in x–y plane for this case can be written as (156) m∗ EFs = mc I11 EFs

Das et al.


The 2D DR in QWs of III–V, ternary and quaternary materials absence of band tails, whose unperturbed band structure is defined by the two band model of Kane, in the presence of light waves can be expressed as 2 2 nz 2 ks2 + = 0 E (165) 2mc 2mc dz The sub band energies Enl7HD an be written as 0 Enl7HD =

2 n /dz 2 2mc z

(166)

The expression of the EEM in this case is given by m∗ EF2DL nz = mc 0 EF2DL

(167)

ARTICLE

The DOS function can be written as nzmax mc gv N E = E HE − Enl7 (168) 2 dz nz =1 0 The electron concentration can be written as nzmax mc gv 2 nz 2 n0 = 0 EF2DL nz − dz 2 nz =1 2mc dz s + Lr0 EF2DL nz (169) r=1

In the absence of light waves and heavy doping, the 2D electron dispersion relation for isotropic two band model of Kane can be written as 2 ks2 2 nz 2 E1 + E = + (170) 2mc 2mc dz

nzmax mc kB Tgv 1 + 2Enz n0 = 3 dz 2 nz =1

×F0 n1 + 2kB T F1 n1

where, n1 ≡ EFs − Enz /kB T . 3 The Eq. (175) is well-known in the literature.109 (iii) The 2D DRin QWs of HD III–V, ternary and quaternary materials, whose unperturbed band structure is defined by the parabolic energy bands in the presence of light waves, can be expressed as 2 nz 2 2 ks2 + = T3 E g (176) 2mc 2mc dz The sub band energies Enl7HD1 can be written as T3 Enl7HD1 g =

2 n /dz 2 2mc z

m∗ EF2DLHD nz = mc T3 EF2DLHD g

nzmax mc gv 1 + 2EHE − Enz 3 dz 2 nz =1

(171)

(172)

where, the sub-band energy (Enz can be expressed as 3

2 n /dz 2 = Enz 1 + Enz 3 3 2mc z

The DOS function can be written as N Eg nzmax mc gv = T Eg HE −Enl7HD1 dz 2 nz =1 3

n0 = 22

mc gv dz 2

nz =1 Enz3

1 + 2E dE 1 + expE − EFs /kB T

(179)

The electron concentration can be written as nzmax mc gv n0 = T3 EF2DLHD nz 2 dz nz =1 2 s 2 nz − + LrT3 EF2DLHD nz 2mc dz r=1

The 2D DR in QWs of III–V, ternary and quaternary materials absence of band tails, whose unperturbed band structure is defined by the parabolic energy band in the presence of light waves, can be expressed as 2 nz 2 2 ks2 + = 0 E (181) 2mc 2mc dz The sub band energies Enl73 an be written as 0 En173 =

2 n /dz 2 2mc z

(182)

The expression of the EEM in this case is given by (173)

The electron statistics can be written as nzmax

(178)

(180)

Thus, we see that the EMM in the present case is a function of Fermi energy only due to the presence of band non-parabolicity. The total density-of-states function assumes the form NT E =

(177)

The expression of the EEM in this case is given by

The EMM in this case can be written as m∗ EFs = mc 1 + 2EFs

(175)

(174)

m∗ EF2DL nz = mc 0 EF2DL

(183)

The DOS function can be written as nzmax mc gv E HE − Enl73 (184) N E = 2 dz nz =1 0 Mater. Focus, 6, 1–34, 2017

Das et al.


The electron concentration can be written as nzmax 2 mc gv 2 nz E n − n0 = 0 F2DL z dz 2 nz =1 2mc dz s + Lr0 EF2DL nz (185) r=1

In the absence of light waves and heavy doping for isotropic parabolic energy band the 2D electron dispersion relation can be written as 2 ks2 2 nz 2 E= + (186) 2mc 2mc dz The EMM in this case can be written as m∗ EFs = mc

(187)

nzmax mc gv NT E = HE − Enz 33 dz 2 nz =1

(188)

where, the sub-band energy (Enz can be expressed as

T3L1 E ny nz g 2 nz /dz 2 2 ny /dy 2 + = T1 E g − 2mc 2mc 1/2 2m × 2c The sub-band energy (E3HDNWL1 ) in this case can be expressed as

2 nz /dz 2 2 ny /dy 2 2 kx2 + + = T1 E g 2mc 2mc 2mc (193) The EEM in this case is given by m∗ EF1HDNWL1 g = mc Real part of T1 EF1HDNWL1 g

The electron statistics per unit length can be written as n0 =

nymax nzmax 2gv T3L1 EF1HDNWL1 ny nz g Real part of ny =1 nz =1

+

33

2 n /dz 2 = Enz 33 2mc z

(189)

The electron statistics can be written as m k Tg F n0 = c B 2 v dz nz =1 0 n11 nzmax

(190)

2.5. The Entropy in NWS of HD Kane Type Semiconductors in the Presence of Light Waves (a) The 1D DR in NWs of HD III–V, ternary and quaternary materials, whose unperturbed band structure is defined by the three band model of Kane in the absence of any field, in the presence of light waves can be expressed as 2 nz /dz 2 2 ny /dy 2 2 kx2 + + = T1 E g 2mc 2mc 2mc (191) The DOS function in this case assumes the form nymax nzmax 2gv T E ny nz g ny =1 nz =1 3L1

Mater. Focus, 6, 1–34, 2017

LrT3L1 EF1HDNWL1 ny nz g

(195)

r=1

The 1D DR for NWs of III–V materials whose energy band structures are defined by the three-band model of Kane in the absence of band tailing assumes the form 2 nz /dz 2 2 ny /dy 2 2 kx2 + + = 0 E (196) 2mc 2mc 2mc

N1D L E

33

The Eq. (190) is well-known in the literature.109

× HE − E3HDNWL1

r=s

The DOS function can be expressed as

where, n11 ≡ EFs − Enz /kB T .

N1DHD L E g =

(194)

(192)

nymax nzmax 2gv = f E ny nz HE − E3L1 ny =1 nz =1 12L1

(197)

where f12L1 E ny nz 2 nz /dz 2 2 ny /dy 2 = 0 E − + 2mc 2mc 1/2 2m × 2c and E3L1 is the sub-band energy in this case which can be obtained from the following equation

2 nz /dz 2 2 ny /dy 2 2 kx2 + + = 0 E3L1 2mc 2mc 2mc (198) 23

ARTICLE

Thus, we see that the EMM in the present case is a constant. The total density-of-states function assumes the form

where

Das et al.


The EEM in this case is given by (199)

The 1D DR, for NWs of III–V materials whose energy band structures are defined by the two-band model of Kane in the absence of band tailing assumes the form

where EF1NWL2 is the Fermi energy in this case. The electron statistics per unit length can be written as

2 nz /dz 2 2 ny /dy 2 2 kx2 + + = 0 E (206) 2mc 2mc 2mc

m∗ EF1NWL2 g = mc 0 EF1NWL2

nymax nzmax 2gv n0 = n n f E ny =1 nz =1 12L1 F1NWL2 y z r=s + Lrf12L1 EF1NWL2 ny nz

The DOS function can be expressed as N1D L E = (200)

× HE − E3L2

r=1

ARTICLE

Using (59d) and (200) we can find the entropy in this case. (b) The 1D DR in NWs of HD III–V, ternary and quaternary materials, whose unperturbed band structure is defined by the two band model of Kane in the absence of any field, in the presence of light waves can be expressed as 2 nz /dz 2 2 ny /dy 2 2 kx2 + + = T2 E g 2mc 2mc 2mc (201) The DOS function in this case assumes the form N1DHD L E g =

nymax nzmax 2gv f E ny nz ny =1 nz =1 12L2

nymax nzmax 2gv T E ny nz g ny =1 nz =1 3L2 × HE − E3HDNWL2

(202)

(207)

where f12L2 E ny nz 2 nz /dz 2 2 ny /dy 2 = 0 E − + 2mc 2mc 1/2 2m × 2c and E3L2 is the sub-band energy in this case which can be obtained from the following equation

2 nz /dz 2 2 ny /dy 2 2 kx2 + + = 0 E3L2 2mc 2mc 2mc (208) The EEM in this case is given by

where T3L2 E ny nz g 2 nz /dz 2 2 ny /dy 2 = T2 E g − + 2mc 2mc 1/2 2m × 2c In (202), E3HDNWL2 is the sub-band energy in this case which can be expressed as

2 nz /dz 2 2 ny /dy 2 + = T2 E3HDNWL2 g 2mc 2mc (203) The EEM in this case is given by m∗ EF1HDNWL2 g = mc T2 EF1HDNWL2 g (204) The electron statistics per unit length can be written as n0 =

nymax nzmax 2gv T E n n ny =1 nz =1 3L2 F1HDNWL2 y z g

+

r=s

LrT3L2 EF1HDNWL2 ny nz g (205)

r=1

Using (59d) and (205) we can find the entropy in this case. 24

m∗ EF1NWL21 g = mc 0 EF1NWL21

(209)

where EF1NWL21 is the Fermi energy in this case. The electron statistics per unit length can be written as nymax nzmax 2gv f E n n n0 = ny =1 nz =1 12L2 F1NWL21 y z

+

r=s

Lrf12L2 EF1NWL21 ny nz

(210)

r=1

Using (59d) and (210) we can find the entropy in this case. (c) The 1D DR in NWs of HD III–V, ternary and quaternary materials, whose unperturbed band structure is defined by the parabolic energy bands in the absence of any field, in the presence of light waves can be expressed as 2 nz /dz 2 2 ny /dy 2 2 kx2 + + = T3 E g 2mc 2mc 2mc (211) The DOS function in this case assumes the form N1DHD L E g =

nymax nzmax 2gv T E ny nz g H ny =1 nz =1 3L3 × E − E3HDNWL3

(212)

Mater. Focus, 6, 1–34, 2017

Das et al.


where

The electron statistics per unit length can be written as

T3L3 E ny nz g 2 nz /dz 2 2 ny /dy 2 + = T3 E g − 2mc 2mc 1/2 2m × 2c In (212), E3HDNWL3 is the sub-band energy in this case which can be expressed as

2 nz /dz 2 2 ny /dy 2 + = T3 E3HDNWL3 g 2mc 2mc (213) The EEM in this case is given by m∗ EF1HDNWL3 g = mc T3 EF1HDNWL3 g (214)

r=1

Using (59d) and (215) we can find the entropy in this case. The 1D DR for NWs of III–V materials whose energy band structures are defined by the parabolic energy bands in the absence of band tailing assumes the form 2 nz /dz 2 2 ny /dy 2 2 kx2 + + = 0 E (216) 2mc 2mc 2mc The DOS function can be expressed as N1D L E =

nymax nzmax 2gv f E ny nz ny =1 nz =1 12L3 × HE − E3L3

(217)

where f12L3 E ny nz 2 nz /dz 2 2 ny /dy 2 = 0 E − + 2mc 2mc 1/2 2m × 2c E3L3

and is the sub-band energy in this case which can be obtained from the following equation. ny /dy nz /dz + + = 0 E3L3 2mc 2mc 2mc (218) The EEM in this case is given by 2

2

2

2

2 kx2

m∗ EF1NWL3 g = mc 0 EF1NWL3 Mater. Focus, 6, 1–34, 2017

(219)

×

r=s

Lrf12L3 EF1NWL3 ny nz

(220)

r=1

The content of this paper finds the following three important applications. A. Heat Capacity The heat capacity of the carriers in semiconductors is a very important quantity. The total heat capacity ct is given by ct = ce + cl

(221)

where ce is the electronic heat capacity and is given by54 f E −E EF 0 F + ce = − E −EF N EdE (222) E T T 0 Thus (227) can be written as EF N EF 2 kB2 T N EF 1+T +cl (223) ct = 3 T N EF Thus by using different expressions of the DOS function which can be formulated from the content of this paper we can study the total heat capacity of different quantized 1D materials. b. Normalized Hall Coefficient The normalized Hall coefficient in a semiconductor can be expressed as109 2 kB T 2 RT 1 N E = 1+ (224) R0 3 N E E E=EFT where RT is the Hall coefficient at T , R0 is the same at T → 0 and EFT is the Fermi energy at T . Thus by using different expressions of the DOS function which can be formulated from the content of this paper we can study the normalized Hall coefficient of different quantized 1D materials. C. Reflection Coefficient In the presence of light waves, the reflection coefficient R is given by110 R=

x − 12 x + 12

(225)

where x = sc − n0 e2 2 /m∗ c 2 , m∗ is the effective electron mass and c is the velocity of light. Thus by using different expressions of the effective masses which can be formulated from the content of this paper we can study the normalized reflection coefficient of different quantized 1D materials. 25

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The electron statistics per unit length can be written as nymax nzmax 2gv T E n n n0 = ny =1 nz =1 3L3 F1HDNWL3 y z g r=s + LrT3L3 EF1HDNWL3 ny nz g (215)

nymax nzmax 2gv f E n n n0 = ny =1 nz =1 12L3 F1NWL3 y z


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3. RESULTS AND DISCUSSION Using the appropriate equations together with the values of the energy band constants as given in Table I,52 we have plotted the normalized entropy as a function of electron concentration per unit volume, in the presence of light wave for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) in accordance with three band model of Kane where I = 10−4 nWm−2 and = 660 nm. It appears from Figure 1 that the entropy increases with increasing electron statistics for all the materials with different values of the energy band constants. The Figure 2 shows the plot of the normalized entropy as a function of light intensity for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cd x Te (green) and In1−x Gax Asy P1−y (black) in accordance with three band model of Kane where no = 1022 m−3 and = 660 nm. From Figure 2 it appears that the entropy increases with increasing light intensity. The Figure 3 shows the plot of the normalized entropy as a function of wavelength for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) in accordance with three band model of Kane where no = 1022 m−3 and I = 10−4 nWm−2 . It appears from Figure 3 that with decreasing wave length, the entropy also decreases. The Figure 4 exhibits the plot of the normalized entropy as a function of alloy composition x in the presence of light wave for HD samples of Hg1−x Cdx Te (blue) and In1−x Gax Asy P1−y (black) in accordance with three band model of Kane where no = 1022 m−3 , I = 10−4 nWm−2 and = 660 nm. It appears from Figure 4 that the entropy increases with increasing alloy composition. In Figure 5 we have plotted of the normalized entropy as a function of inverse magnetic field for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and

Fig. 1. Plot of the normalized entropy as a function of electron concentration in the presence of light wave for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) in accordance with three band model of Kane where I = 10−4 nWm−2 and = 660 nm.

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Fig. 2. Plot of the normalized entropy as a function of light intensity for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) in accordance with three band model of Kane where no = 1022 m−3 and = 660 nm.

In1−x Gax Asy P1−y (black) in the presence of magnetic field and light in accordance with three band model of Kane where I = 10−4 nWm−2 , = 660 nm and no = 1023 m−3 . It appears from both the figures that the entropy is an oscillatory function of the inverse quantizing magnetic field. The oscillatory dependence is due to the crossing over of the Fermi level by the Landau sub-bands in steps resulting in successive reduction the number of occupied Landau levels as the magnetic field is increased. For each coincidence of a Landau level, with the Fermi level, there would be a discontinuity in the density-of-states function resulting in a peak of oscillation. Thus the peaks should occur whenever the Fermi energy is a multiple of energy separation between the two consecutive Landau levels and it may

Fig. 3. Plot of the normalized entropy as a function of wavelength for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) in accordance with three band model of Kane where no = 1022 m−3 and I = 10−4 nWm−2 . Mater. Focus, 6, 1–34, 2017

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Fig. 4. Plot of the normalized entropy as a function of alloy composition x in the presence of light wave for HD samples of Hg1−x Cdx Te (blue) and In1−x Gax Asy P1−y (black) in accordance with three band model of Kane where no = 1022 m−3 , I = 10−4 nWm−2 and = 660 nm.


Fig. 6. Plot of the normalized entropy as a function of wavelength in the presence of magnetic field and light in accordance with three band models of Kane for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) where 1/B = 01 tesla−1 , no = 1023 m−3 and I = 10−4 nWm−2 .

It appears that the entropy under magnetic quantization increases with increasing wave length for all the materials. Figure 7 exhibits the plot of the normalized entropy as a function of light intensity in the presence of magnetic field and light in accordance with three band models of Kane for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cd x Te (green) and In1−x Gax Asy P1−y (black) where 1/B = 01 tesla−1 , no = 1023 m−3 = 660 nm. We observe from Figure 7 that the entropy increases with increasing light intensity for all the materials. The Figure 8 shows the plot of the normalized entropy as a function of electron concentration in the presence of magnetic field and light in

Fig. 5. Plot of the normalized entropy as a function of inverse magnetic field for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) in the presence of magnetic field and light in accordance with three band model of Kane where I = 10−4 nWm−2 , = 660 nm and no = 1023 m−3 .

Fig. 7. Plot of the normalized entropy as a function of light intensity in the presence of magnetic field and light in accordance with three band models of Kane for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) where 1/B = 01 tesla−1 , no = 1023 m−3 = 660 nm.

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be noted that the origin of oscillations in the entropy is the same as that of the Subhnikov-de Hass oscillations. With increase in magnetic field, the amplitude of the oscillation will increase and, ultimately, at very large values of the magnetic field, the conditions for the quantum limit will be reached when the entropy will be found to decrease monotonically with increase in magnetic field. In Figure 6 exhibits the plot of the normalized entropy as a function of wavelength in the presence of magnetic field and light in accordance with three band models of Kane for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) where 1/B = 01 tesla−1 , no = 1023 m−3 and I = 10−4 nWm−2 .


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Fig. 8. Plot of the normalized entropy as a function of electron concentration in the presence of magnetic field and light in accordance with three band models of Kane for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) where 1/B = 01 tesla−1 , I = 10−4 nWm−2 and, = 660 nm.

accordance with three band models of Kane for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) where 1/B = 01 tesla−1 , I = 10−4 nWm−2 and, = 660 nm. From Figure 8 it appears that the entropy oscillates with increasing electron concentration and the root of oscillation in this case is the SdH effect. The Figure 9 shows the plot of the normalized entropy as a function of alloy composition in the presence of magnetic field and light in accordance with three band models of Kane for HD samples of Hg1−x Cdx Te (blue) and In1−x Gax Asy P1−y (black) where 1/B = 01 tesla−1 , I = 10−4 nWm−2 , = 660 nm and

Fig. 9. Plot of the normalized entropy as a function of alloy composition in the presence of magnetic field and light in accordance with three band models of Kane for HD samples of Hg1−x Cdx Te (blue) and In1−x Gax Asy P1−y (black) where 1/B = 01 tesla−1 , I = 10−4 nWm−2 , = 660 nm and no = 1023 m−3 .

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Fig. 10. Plot of the normalized entropy as a function of dz in the presence of light wave in QWs in accordance with three band models of Kane for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) where I = 10−4 nWm−2 , = 660 nm and no = 1017 m−2 .

no = 1023 m−3 . From Figure 9 we observe that the entropy increases with increasing alloy composition. In Figure 10 we have plotted the normalized entropy as a function of dz in the presence of light wave in QWs in accordance with three band models of Kane for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) where I = 10−4 nWm−2 , = 660 nm and no = 1017 m−2 . The influence of quantum confinement is immediately apparent from all the curves of Figure 10, since, the entropy depends strongly on the nano-thickness, which is in direct contrast with the corresponding bulk specimens which is also the direct signature of quantum confinement. It appears from the said figure that the entropy increases with the increasing film thickness in oscillatory manners as considered here, although the numerical values vary widely and determined by the constants of the energy spectra. The oscillatory dependence is due to the crossing over of the Fermi level by the size quantized levels. For each coincidence of a size quantized level with the Fermi level, there would be a discontinuity in the density-of-states function resulting in a peak of oscillations. With large values of film thickness, the height of the steps decreases and the entropy increases with increasing film thickness in non-oscillatory manner and exhibit monotonic increasing dependence. The rate of increment is totally dependent on the band structure. The appearance of the humps of the respective curves is due to the redistribution of the electrons among the quantized energy levels when the quantum numbers corresponding to the highest occupied level changes from one fixed value to the others. In Figure 11 we have plotted the normalized entropy as a function of electron concentration per unit area in the presence of light wave in Mater. Focus, 6, 1–34, 2017

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Fig. 12. Plot of the normalized entropy as a function of light intensity in the presence of light wave in QWs in accordance with three band models of Kane for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) where dz = 10 nm, no = 1017 m−2 and = 660 nm.

QWs in accordance with three band models of Kane for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) dz = 10 nm, I = 10−4 nWm−2 and = 660 nm, with varying electron concentration, a change is reflected in the entropy through the redistribution of the electrons among the quantized levels. It appears that the entropy increases with increasing carrier degeneracy and also reflects the signature of the 1D confinement through the spiky dependence with the 2D electron statistics. This oscillatory dependence will be less and less prominent with increasing carrier concentration and ultimately, for bulk specimens of the same material, the entropy will be found to increase continuously with increasing electron statistics under the condition of nondegeneracy in a non-oscillatory manner. The numerical values of the entropy depend on the energy band constants of different materials. The Figure 12 shows the plot of the normalized entropy as a function of light intensity in the presence of light wave in QWs in accordance with three band models of Kane for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) where dz = 10 nm, no = 1017 m−2 and = 660 nm. It appears from Figure 12 that the entropy increases with increasing light intensity for all the cases. In Figure 13 we have plotted the normalized entropy as a function of wavelength in the presence of light wave in QWs in accordance with three band models of Kane for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) where dz = 10 nm, no = 1017 m−2 and I = 10−4 nWm−2 . The Figure 13 exhibits that the entropy decreases with decreasing wave length. In Figure 13 we have plotted the normalized entropy as a function of wavelength in the presence of light wave in QWs in accor-

dance with three band models of Kane for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) where dz = 10 nm, no = 1017 m−2 , = 660 nm and I = 10−4 nWm−2 . We note that the entropy increases with increasing wave length. In Figure 14 we have plotted the normalized entropy as a function of alloy composition in the presence of light wave in QWs in accordance with three band models of Kane for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) where dz = 10 nm, no = 1017 m−2 , = 660 nm and I = 10−4 nWm−2 . From Figure 14 we observe that the entropy decreases with decreasing alloy composition. In Figure 15 we have plotted

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Fig. 13. Plot of the normalized entropy as a function of wavelength in the presence of light wave in QWs in accordance with three band models of Kane for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) where dz = 10 nm, no = 1017 m−2 and I = 10−4 nWm−2 .

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Fig. 11. Plot of the normalized entropy as a function of concentration in the presence of light wave in QWs in accordance with three band models of Kane for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) dz = 10 nm, I = 10−4 nWm−2 and = 660 nm.

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Fig. 14. Plot of the normalized entropy as a function of alloy composition in the presence of light wave in QWs in accordance with three band models of Kane for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) where dz = 10 nm, no = 1017 m−2 , = 660 nm and I = 10−4 nWm−2 .

Fig. 16. Plot of the normalized entropy as a function of electron concentration per unit length in the presence of light wave in NWs in accordance with three band models of Kane for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) where = 660 nm, I = 1 nWm−2 , dz = 10 nm and dy = 10 nm.

the normalized entropy as a function of dz in the presence of light wave in NWs in accordance with three band models of Kane for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) where = 660 nm, I = 1 nWm−2 , no = 108 m−1 and dy = 10 nm. It appears that when the Fermi energy touches the sub-band energy there occurs a peak for a particular value of film thickness after which the entropy decreases rapidly with increasing film thickness for whole range of thicknesses as considered here. In Figure 16 we have plotted the normalized entropy as a function of electron concentration per unit length in the presence of

light wave in NWs in accordance with three band models of Kane for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) where = 660 nm, I = 1 nWm−2 , dz = 10 nm and dy = 10 nm. The entropy increases with increasing electron concentration for relatively low values of the electron statics and when the Fermi energy touches the sub-band energy for a particular value of the electron concentration, the entropy jumps to the next value due to quantum effect and increases again with increasing electron concentration. The Figure 17 exhibits the plot of the normalized entropy as a function of light intensity in the presence of light wave in

Fig. 15. Plot of the normalized entropy as a function of dz in the presence of light wave in NWs in accordance with three band models of Kane for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) where = 660 nm, I = 1 nWm−2 , no = 108 m−1 and dy = 10 nm.

Fig. 17. Plot of the normalized entropy as a function of light intensity in the presence of light wave in NWs in accordance with three band models of Kane for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) where = 660 nm, dz = 10 nm, dy = 10 nm and no = 108 m−1 .

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Fig. 18. Plot of the normalized entropy as a function of wavelength in the presence of light wave in NWs in accordance with three band models of Kane for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) where = 660 nm, I = 1 nWm−2 , dz = 10 nm, dy = 10 nm and no = 108 m−1 .

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Another important point in this context is the existence of the effective mass within the forbidden zone, which is impossible without the formation of band tails. It is an amazing fact that the study of the carrier transport in HD materials in the presence of light waves through proper formulation of the Boltzmann transport equation which needs in turn, the corresponding HD carrier energy spectra is still one of the open research problems. We have not considered other types of HD materials or external physical variables for numerical computations in order to keep the presentation brief. With different sets of energy band constants, we shall get different numerical values of the entropy though the nature of variations of the entropy as shown here would be similar for other cases and the simplified analysis of this paper exhibits the basic qualitative features of the entropy for such HD materials. We must note that the study of transport phenomena and the formulation of the electronic properties of HD materials are based on the dispersion relations in their constituent materials. The theoretical results of our paper can be used to determine the entropy from quantized HD materials and the constituent heavily-doped bulk materials in the absence of size effects. It is worth remarking that this simplified formulation exhibits the basic qualitative features of entropy from such quantized structures. The basic objective of this paper is not solely to demonstrate the influence of quantum confinement on the entropy for quantized structures of HD non-parabolic materials but also to formulate the appropriate electron statistics in the most generalized form, since the transport and other phenomena in HD quantized structures having different band structures and the derivation of the expressions of many important electronic properties are based on the temperaturedependent electron statistics in such quantum confined materials. Our method is not at all related to the DOS technique as used in the literature. From the E–k dispersion relation, we can obtain the DOS, but the DOS technique as used in the literature cannot provide the E–k dispersion relation. Therefore, our study is more fundamental than those of the existing literature because the Boltzmann transport equation, which controls the study of the charge transport properties of semiconductor devices, can be solved if and only if the E–k dispersion relation is known. We wish to note that we have not considered the many body effects in this simplified theoretical formalism due to the lack of availability in the literature of proper analytical techniques for including them for the generalized systems as considered in this paper. Our simplified approach will be useful for the purpose of comparison when methods of tackling the formidable problem after inclusion of the many body effects for the present generalized systems appear. It is worth remarking in this context that from our simple theory under certain limiting conditions we get the well-known result of the entropy from wide gap materials having parabolic energy bands. The inclusion of the 31

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NWs in accordance with three band models of Kane for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) where = 660 nm, dz = 10 nm, dy = 10 nm and no = 108 m−1 . From Figure 17 we observed that the entropy increases with increasing light intensity. The Figure 18 shows the plot of the normalized entropy as a function of wavelength in the presence of light wave in NWs in accordance with three band models of Kane for HD samples of n-InSb (blue), n-InAs (red), Hg1−x Cdx Te (green) and In1−x Gax Asy P1−y (black) where = 660 nm, I = 1 nWm−2 , dz = 10 nm, dy = 10 nm and no = 108 m−1 . We observe in Figure 18 that the entropy increases with increasing wave length. Under certain limiting conditions, all the results for all the models as derived here get simplified to have transformed into the well-known expressions of the entropy. This indirect test not only exhibits the mathematical compatibility of the present formulation but also shows the fact that our simple analysis is a more generalized one, since one can obtain the corresponding results for relatively wide gap bulk materials having parabolic energy bands under certain limiting conditions from the present generalized analysis. A direct research application of the quantized materials is in the area of band structure. The theoretical results as derived in this paper can be used to determine the entropy of the constituent bulk materials in the absence of quantum effects and this simplified formulation exhibits the basic qualitative features of entropy for different quantum confined materials. Finally, it is logical to conclude that the numerical values of the entropy are totally different in all cases which exhibit the signature of the respective band structure of HD materials under different physical conditions and the rates of variation are again totally energy spectrum dependent.


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said effects would certainly increase the accuracy of the results, although the qualitative features of the entropy as discussed in this paper would not change in the presence of the aforementioned effects. The influence of energy band models and the various band constants on the entropy for different quantized superlattices can also be studied from all the figures of this paper. One important concept of this paper is the presence of poles in the finite complex plane in the dispersion relation of the constituent materials in the absence of band tails creates the complex energy spectrum in the corresponding HD samples. Besides, from the DOS function in this case, it appears that a new forbidden zone has been created in addition to the normal band gap of the semiconductors. If the basic dispersion relation in the absence of band tails contains no poles in the finite complex plane, the corresponding HD energy band spectrum will be real, although it may be the complicated functions of exponential and error functions and deviate considerably from that in the absence of band tailing. The numerical results presented in this paper would be different for other materials but the nature of variation would be unaltered. The theoretical results as given here would be useful in analyzing various other experimental data related to this phenomenon. Finally, we can write that the analysis as presented in this paper can be used to investigate the Burstein Moss shift, the carrier contribution to the elastic constants, the specific heat, the screening length, activity coefficient, reflection coefficient, Hall coefficient, plasma frequency, various scattering mechanisms and other different transport coefficients of modern HD materials operated under different external conditions having varying band structures.

4. CONCLUSION In this paper an attempt is made to study, the entropy in the presence of light waves in bulk samples, bulk samples under quantized magnetic field, quantum wells and nano-wires of heavily doped (HD) III–V and optoelectronic materials on the basis of newly formulated electron dispersion relations within the frame work of k · p formalism. It is found taking HD materials of InSb, InAs, Hg1−x Cdx Te and In1−X Gax Asy P1−y lattice matched to InP as examples III–V, ternary and quaternary compounds that the entropy increases with increasing electron concentration, decreasing film thickness in different spiky manners, since the coincidence of Fermi energy with the sub-band energy leads to the step functional dependence of the density state function and this fact is being reflected in the quantized variations of the entropy with the said variables. The entropy increases with increasing wave length, intensity and alloy composition in all the cases respectively. The numerical values of entropy with all the physical variables are totally band structure dependent for all the cases. The most striking features are that the presence of poles in the dispersion relation of the materials in the absence of band 32

tails creates the complex energy spectra in the corresponding opto-electronic HD materials and the effective electron mass exists within the band gap which is impossible without the concept of band tailing. The well-known classical result of entropy for non-degenerate bulk semiconductors having parabolic energy bands has been obtained as a special case of our generalized formulation and thus confirming the compatibility test. The content of this paper finds four important applications in the field of quantum effect devices of nanoscience and nanotechnology. Acknowledgments: The authors are grateful to Professor S. Chakrabarti, Director of the Institute of Engineering Management, Salt lake Kolkata, India and Professor B. Chatterjee, the Registrar Principal of the University of engineering and Management for help and inspiration.

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