Mass and potential duality explored via a position-dependent mass ...

MASS AND POTENTIAL DUALITY EXPLORED VIA A POSITION-DEPENDENT MASS QUANTUM APPROACH IHAB H. NAEIM1 , S. ABDALLA2 , J. BATLE3 , A. FAROUK4,5 1

Department of Physics, College of Science, Taibah University, Yanbu, Saudi Arabia E-mail: [email protected] 2 Department of Physics, Faculty of Science, King Abdulaziz University Jeddah, P.O. Box 80203, Jeddah 21589, Saudi Arabia E-mail: [email protected] 3 Departament de F´ısica, Universitat de les Illes Balears, 07122 Palma de Mallorca, Balearic Islands E-mail: [email protected] 4 Faculty of Computer and Information Sciences, Mansoura University, Egypt 5 University of Science and Technology at Zewail City, 12588 Giza, Egypt E-mail: [email protected] Received March 21, 2017

Abstract. The approximation made to quantum systems where a particle possesses an effective mass that depends on the position is relevant to many state-of-the-art compounds, which have been extremely useful in tailoring different quantum physical properties, specially in the field of semiconductors. The approach of a positiondependent mass has proved to be a very effective one in those cases where quantum wells emerge either in one or two dimensions. In the present work we discuss the interplay between m(x) versus the action potential V (x) in the Schr¨odinger equation, the presence of the one excluding the other. As an application, both approaches are compared in the benzene molecule. Key words: Position-dependent mass, mass distribution and potential duality. PACS: 03.65.-w, 03.65.Ge, 73.21.Fg.

1. INTRODUCTION

The study of quantum systems with position-dependent effective masses has received considerable attention in recent years [1–6]. The concomitant Schr¨odinger equation with non-constant provides an interesting and useful model for the description of many physical problems, specially in the case of semiconductor nanostructures. In particular, the most popular of these structures is the semiconductor quantum well, where the Schr¨odinger equation there is effectively one-dimensional. The finite well potential formed by different structures can be modeled so that the wave function is approximately zero at the boundaries, although this should not be strictly the case for otherwise there would not be any transport. Many of the important questions in the theory of solids concern the non-relativistic motion of electrons in periodic Romanian Journal of Physics 62, 121 (2017)

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lattices perturbed by the effect of impurities, which constitutes the paradigm of semiconductors [7] as given by transport phenomena in a position-dependent chemicallycomposite system. These low dimensional carrier systems in semiconductor hetero-structures have gained very much importance due to the potential use of their unique properties in applications ranging from optoelectronics to high speed devices [7–11]. Also, InGaN/GaN quantum-well structures are of particular interest because the band gap can be tailored so that it covers a wide spectral range from red to ultraviolet by changing the In composition. In the present work we shall discuss the dichotomy V (x) − m(x), one excluding the other, in well-know one dimensional quantum systems. We also extend our approach to the case of molecules, the first time that, to our knowledge, is done so. In particular, we shall study the case of benzene, where both descriptions are compared in detail. For the sake of simplicity, we shall study systems in one dimension. Also, atomic units are used throughout the present contribution. The article is structured as follows: Section II describes the mathematical problem of mapping the V (x)-approach into the m(x)-approach. Section III is devoted to finding analytic mass distribution counterparts for well-know quantum potentials. We describe the m(x)−approach to benzene in Section IV. Finally, some conclusions are drawn in Section V. 2. MOTIVATION

To what extend the shape of the mass distribution m(x) and the form of a confining potential V (x) may be related is something that we want to explore in the present contribution. As we shall see, there are cases of practical interest. One has to bear in mind the usual case of nuclear physics where, in the mean field approximation, it is quite reasonable to assume due to the short range interaction between nucleons that the shell-model effective potential V (r) should mirror the radial density distribution ρ(r). Thus, we can propose the opposite and see to what extend the Schr¨odinger equation with a given m(x) and no confinement is related to its potential V (x) counterpart, the usual situation encountered in quantum physics. The most general form for the kinetic operator was proposed by von Roos in [12] 1  −1 (m (~r) p2 + p2 m−1 (~r)) 8 + mα (~r) p mβ (~r) p mγ (~r)  + mγ (~r) p mβ (~r) p mα (~r) , T =

(1)

with the following constraint on the parameters α + β + γ = −1, which clearly makes (c) RJP 62(Nos. 9-10), id:121-1 (2017) v.2.0*2017.11.29#feddfe35

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operator (1) look ambiguous. The first term is added iso as to include the Weyl ordered operator [13], that is, α = γ = 0 in the general expression. If we impose the the usual position-momentum commutation relation in one dimension (from the right in (1)), an additional term is obtained. This term disappears as well as the lack of uniqueness in the definition of (1) by imposing α + γ = 1 = α γ + α + γ. This last condition implies that either {α = 0, γ = 1} or {α = 1, γ = 0} makes the kinetic operator (1) unambiguously defined. Therefore, we are going to choose the following common form for Schr¨odinger equation describing a particle in any dimension with position-dependent effective mass: 1 1 − ∇ ∇|ψi + V (x)|ψi = EN |ψi in Ω, (2) 2 m(x) where Ω is a bounded domain in RN , N ≥ 1 and m (x) is a positive function. This problem has countably many real positive eigenvalues, with finite multiplicity, which diverge to infinity. These eigenvalues are explicitly computable only for a handful of domains. Now, we shall see how the usual description in terms of m = cnt under the action of a certain potential V (x) ψ 00 (x) − 2V (x)ψ(x) + 2An ψ(x) = 0, (3) giving rise to the set of eigenfunctions and eigenvalues {ψn (x), An }, differs from those ψ 00 (x) − m0 (x)/m(x)ψ 0 (x) + 2Bn m(x)ψ(x) = 0 (4) obtained by the sole presence of m(x). We shall illustrate it with relevant, analytical cases. 3. RESULTS 3.1. THE LINEAR POTENTIAL

Let us suppose that we have a potential V (x) = A + Bx (usually a linearly sloping well). The corresponding Schr¨odinger equations reads as 1 d2 ψ(x) + (A + Bx)ψ(x) = Eψ(x) 2 dx2 The corresponding solutions are given by −

(5)

√  3 2(A + Bx − E) ψ(x) = c1 Ai 2 B3 (c) RJP 62(Nos. 9-10), id:121-1 (2017) v.2.0*2017.11.29#feddfe35

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√ 3 + c2 Bi

2(A + Bx − E)

4

 ,

(6) 2 B3 where Ai is the Airy function and Bi is the Airy Bi function. Had we introduced the
1 variable u ≡ (2B) 3 x − E−A , we will be dealing with the original Airy equation. B Due to the divergence at x → ∞ of Bi, only the first one is considered. Also, to satisfy the boundary condition at x = 0 we have to know the zeroes of the Airy function or its derivative. Let us suppose that u0i is the set of all zeroes of the Airy function, and νi0 is the set of all zeros of the derivative of the Airy function, both cases for i = 1, 2, 3, .., ∞. The values for the zeros can be found in handbooks or solved numerically. Even and odd parity solutions are found by imposing Ai0 (u(0)) = 0 and Ai(u(0)) = 0, respectively. 2 1 0 Finally, we have for odd n starting from 1, that En = A − ν(n+1)/2 ( B2 ) 3 , and 2

1

for even n, En = A − u0n/2 ( B2 ) 3 Not let us consider dropping the action of any V (x) in favor of a positiondependent mass m(x) with the same functional form. The Schr¨odinger equation reads as   1 d 1 0 − ψ (x) = Eψ(x), (7) 2 dx A + Bx possessing solutions of the type

ψ(x)

=

0

√ 3

c1 Ai

+ c2 Bi0

√  −2 3 E(A + Bx) 2

√ 3

√ E3  −2 3 E(A + Bx)

, (8) 2 E3 where Ai0 is the derivative of the Airy function and Bi0 is the derivative of the Airy Bi function. For the same reason stated above, we drop the second linearly independent solution. Proceeding a bit further, we shall have a more complex form for the energies En , now involving the first and the second derivative of the Airy function. It is quite remarkable that the m(x)−approach provides i) derivatives of the V (x)−solutions, although with ii) slightly different arguments. We are witnessing a dualistic role that will manifest with later examples. 3.2. THE 1D HARMONIC OSCILLATOR

Consider the quantum harmonic oscillator (HO) in atomic variables, where the Hamiltonian is given in terms of position and momentum operators x ˆ, pˆ, satisfying (c) RJP 62(Nos. 9-10), id:121-1 (2017) v.2.0*2017.11.29#feddfe35

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[ˆ x, pˆ] = i, is given by the formula ˆ = 1 (ˆ H p2 + ω 2 x ˆ2 ), 2 its normalized eigenstates given by

(9)

√ 2 ψn (x) = (ω/π)1/4 (2n n!)−1/2 Hn ( ωx) e−ωx /2 , (10) 2 2 dn −x n x where Hn are Hermite’s polynomials: Hn (x) =
(−1) e dxn e . The corresponding eigenenergies are given by EN = ω N + 12 . Using a mass approach, we have

ψ(x)

  1 0 1 d ψ (x) = Eψ(x) − 2 dx Bx2   √ √ 2  7 3 3 3 B Ex 5 8 8 8 2 √ = c1 2 B E Γ x J− 3 4 4    √ √2 2  1 3 3 3 7 B Ex √ + c 2 2− 8 B 8 E 8 Γ x2 J3 , 4 4 2

(11)

(12)

where Jn (z) is the Bessel function of the first kind. Both solutions are imaginary for x < 0, and |ψ(x)|2 is symmetric around x = 0, being null there for the second solution and different from zero for the first solution. If we want to be consistent inthe sense  that no mass implies null wave func3

tion, then we choose ψ(x) ∝ x 2 J 3 4

√ √ 2 B√ Ex 2

. The roots j 3 ,n increase in a linear 4

fashions with n, and solving for the energy, we obtain that now the dependency is not linear with the main quantum number, but quadratic. Summing up, both approaches possess and infinite spectrum, and in this case the solutions the m(x)−approach do not coincide with the derivatives of those of the V (x)−solution. In point of fact, the m(x) description of the HO resembles that of the infinite square well as far as the spectrum is concerned. Actually, a realistic proposed mass distribution may occur in systems where an infinity square well can be tailored, like semiconductor hetero-junctions, where the position-dependent mass approximation is mandatory. One could propose a mass distribution inside a square well that could imitate the action of a certain potential. In this setting, it is plain that the action of no potential can give rise to a different quantization of energies of the particle if m(x) is not constant. Let us propose a form for m(x) = ax2 + bx + c, with a < 0. The maximum of the corresponding parabola can be tailored to occur in a certain domain [0, L] of the system. By substituting the quadratic functional form of m(x), we obtain a (c) RJP 62(Nos. 9-10), id:121-1 (2017) v.2.0*2017.11.29#feddfe35

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rather complex form for the eigenstates in terms of exponential and hypergeometric functions. The eigenenergies can be obtained then by imposing the eigenstates to be zero at the boundaries, resulting in a highly nonlinear transcendental equations. A particular case when the mass is -zero at the boundaries is given by m(x) =  − x2 + Lx, which is symmetric around x = L/2. The parabolic case is possible to realize in practice using state-of-the-art semiconductor techniques.

Fig. 1 – (Color online) Plot of |ψ0 (x)|2 , ground state solution to (15) versus θ as the parameter A ranges from 1 (inverted parabola) to 10 (peaks). See text for details.

Fig. 2 – (Color online) Spectrum of energies in (15) versus the quantum number for A = 1, 10, 20 and A = 100 (from bottom to top). See text for details.

3.3. THE 1D HYDROGEN ATOM

In this case, there is a necessary change in the sign for the mass cannot be neg1 ative and V (x) = |x| has no bound states. It means that, as opposed to the previous (c) RJP 62(Nos. 9-10), id:121-1 (2017) v.2.0*2017.11.29#feddfe35

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cases where m(x) → V (x), we now contemplate m(x) → −V (x). There is not gene1 ral consensus on taking either V (x) = − |x| or V (x) = − x1 for x ≥ 0 and V (x) = ∞ for x < 0, because in one case the ground state energy is not well-defined. However, the second choice is more amenable to us because the mass cannot take negative values. The eigenfunctions are given in terms of decaying exponentials time the Kummer confluent hypergeometric function 1 F1 . Let us recall that the (negative) spectrum of the 1D hydrogen atom is inversely proportional to the square of the main number.   1 d 1 0 − ψ (x) = Eψ(x). (13) 2 dx 1/x √ √ √ The only solution being bounded is ψ(x) ∝ J0 (2 2 E x), where J0 (z) is the zeroth Bessel function of the first kind. When finding the bound states, the energies quickly go as n2 , the (positive) inverse of those of the 1D hydrogen atom. There is a notable feature in the fact that the mass becomes infinite at x = 0. We shall elaborate on that in further sections. 3.4. THE INVERSE SQUARE ROOT POTENTIAL

We shall proceed here as in the case of the hydrogen atom. The solution to V (x) = − √1x has been found quite recently [14]. To a very high degree of precision,
2 1 −3 the spectrum is given by En = − 12 n − 2π for n ≥ 1. Solutions are given as √ √ √ √ − 2ny−δ·x/2 ψn (x) = e (Hn (y) − 2nHn−1 (y)), with y = 2n − δ · x, which are quasi-polynomial functions that do not vanish at the origin. Using m(x) = √1x , which is again infinite at the origin, we obtain the following solution:     √ √ 4√ √ 3 4√ √ 3 4 4 ψ(x) = c1 xJ− 1 2 Ex 4 + c2 xJ 1 2 Ex 4 . (14) 3 3 3 3 Because the mass is divergent at x = 0, we choose the wave function to be null there. Thus, we take the second solution. When finding the energies, the spectrum will be positive and going as n2 , which is by no means related to the V (x) = − √1x counterpart. 4. APPLICATION: APPROXIMATION TO THE BENZENE MOLECULE

In the benzene molecule C6 H6 , six carbon atoms are placed in the vertices of a regular hexagon [15]. Each carbon atom has six electrons which only four participate (c) RJP 62(Nos. 9-10), id:121-1 (2017) v.2.0*2017.11.29#feddfe35

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Fig. 3 – (Color online) Series of the first five eigenstates |ψ(θ)|2 solution to (16) for A = 1 (top) and A = 10 (bottom). See text for details.

in the chemical bond (the other two are too strongly tied to the core); two of the four are engaged, forming a link (approximately covalent) with the two neighboring carbon atoms, whereas another one is linked to the adjacent hydrogen atom. Thus, the system has six electrons under the action of the potential of the ion core C6 H6+ 6 . The distance between carbons is approximately around 1.39 Angstrom. This molecule has been widely studied in quantum chemistry, one of the reason being the high degree of symmetry of the system. In point of that, this is the reason why we shall study this system in the m(x)−approach. The six electrons in benzene belong to a six-dimensional space spanned by those localized states on one of the six carbons. Therefore we have the tensor product of a 6-dimensional space, the total dimension being 66 = 46656. This amounts to saying that in order to obtain the eigenenergies we should diagonalize matrix 46656× 46656. The problem is then simplified by assuming that electrons are independent. If we assume that the six electrons do not interact, and are free to move on a ring of radius 1.39 angstrom, we can check how good the approximation is. In fact, the longest wavelength absorption in the benzene spectrum can be estimated from ∆E = 2R1 2 (22 −12 ), which results in 210 nm, whereas the experimental absorption is around 270 nm. Thus, the free-particle approach (of potential-free) is not essentially erroneous after all. In the present approximation, we are going to assume the non-interacting particles to move on a ring of the same size, but under the action of a potential V (θ) that takes into account the attraction with each carbon atom. The explicit form will be V (θ) = A2 (1 − cos 6θ). Parameter A controls the height of the barrier between the six equally spaced positions of the carbon atoms. The corresponding Schr¨odinger (c) RJP 62(Nos. 9-10), id:121-1 (2017) v.2.0*2017.11.29#feddfe35

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equation reads as 1 d2 A ψ(θ) + (1 − cos 6θ)ψ(θ) = Eψ(θ). (15) 2 2 2R dθ 2 The (not normalized) solution to (15) with the proper symmetry is given by the sine
2 2 A AR2 elliptic odd Mathieu function ψ(θ) = S − 9 R ( 2 − E), − 18 , 3θ . The energy is 2 quantized because, for non-zero − AR 18 , the Mathieu functions are only periodic for certain values of − 29 R2 ( A2 − E). These particular values are given by b2(n−1) , n being the number of nodes of the wave function ∈ [0, 2π). These values b2(n−1) de2 pend in turn on − AR 18 . Unfortunately, these b2(n−1) −values are not uniquely defined (they are multivaluated). This fact implies that one cannot be guided by handbook of formulas. Instead, we have solved (15) numerically for different A−values. In Fig. 1 we depict the ground state wave function for A ∈ [1, 10] versus θ. Starting from almost half a circumference, the wave function gradually peaks at the position of the carbon atoms as the height A between them increases. The form for the spectrum is rather particular. In Fig. 2 the energies for A = 1, 10, 20 and A = 100 (from bottom to top are depicted). As the barrier height A increases, an extraordinary behavior for the energy is unveiled. There appear states which almost have the same energy, 2 changing abruptly in a step-like fashion. The straight line n55 in Fig. 2 shows how the energies of the excited states tend to those of and infinite square well. The absorption spectra of benzene, corresponding to electronics transitions in the ultraviolet range, is rather characteristic, ranging from 220 nm to 270 nm every 10 nm. When compared to our approach, even very low barriers for A cannot account for the exact values for these energy transitions, but at least the order of magnitude is very similar to experiment, as well as the separation between difference in energy levels. −

The fact that the particles are confined to a ring give us freedom to propose the existence of an effective mass of the whole system distributed in the same way that the interaction with electrons occurs. Thus, we shall investigate, with m(θ) = A 2 (1 − cos 6θ), the solutions to   1 d 1 0 − 2 ψ (x) = Eψ(x). (16) 2R dx A2 (1 − cos 6θ) Luckily, equation (16) is analytically solvable. Taking into account the parity of the wave functions, the solution (up to a normalization constant) is given by  1 2 1 2 ψ(x) = C (2 + A)ER , AER , 3θ , (17) 9 18 (c) RJP 62(Nos. 9-10), id:121-1 (2017) v.2.0*2017.11.29#feddfe35 0



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where C 0 is the derivative of the cosine elliptic even Mathieu function. The same arguments for solving the eigenenergies as in the V (θ)−case hold here, but the fact that the two arguments in 17 depend on E makes the analytical solution via handbook of tables virtually impossible. In any case, we have resorted to a numerical solution of (16). The first series of five eigenenergy solutions for A = 1 are given by {0.01207, 0.048218, 0.10818, 0.19111, 0.293126}, and {0.003011, 0.011947, 0.0264272, 0.045385, 0.06545} for A = 10. Increasing considerably the height A in the mass makes the individual energies change by more or less the same factor. We notice, then, that the more massive the carbon atoms become, the less bounded the states are. In the m(θ)−approach, we cannot explain the experimental absorption spectra quantitatively, only qualitatively. Furthermore, the behavior with A of the eigenenergies is opposite to that of the V ( theta)−approach. As far as the wave functions are concerned, they look very similar for low values of A as compared to the ones in the usual Schr¨odinger equation. However, and very much surprisingly, they barely change. In Fig. 3 we depict the first five eigenstates probability densities for A = 1 (top), as well as for A = 10 (bottom). Increasing the height A does not affect much the shape, which is very much particular. As in previous cases, we have to acknowledge once again that the solutions to the m(θ)−description are the first derivatives of those of the V (θ)−description, with differences in some parameters. 5. DISCUSSION

We have discussed to what extend the description of a quantum system under the action of a potential V (x) can be described in terms of the absence of any interaction, solely driven by a position-dependent mass m(x). The dependency of the mass of a particle has great importance in describing effective interactions in chemicallytailored compounds like semiconductors. Our leitmotiv consists in reverting the usual argument in nuclear physics that assumes that the potential is the mirror image of the density distribution, and applied it to the solution of the Schr¨odinger equation. Thus, given the potential V (x), we define the mass distribution m(x) and then compare both approaches as far as wave functions and energies are concerned. We have seen a especial functional relation between the eigenfunctions if the V (x) scenario and their first derivative in their m(x) counterpart. As an physically motivated application, we have approached the benzene molecule with satisfactory qualitative results. Finally, the existence of mass distributions m(x) that are divergent at the origin surprisingly provide analytic 1 solutions naturally invites us to discuss the very same validity of the − 21 ∇ m(x) ∇ (c) RJP 62(Nos. 9-10), id:121-1 (2017) v.2.0*2017.11.29#feddfe35

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formalism. Acknowledgements. J. Batle acknowledges fruitful discussions with J. Rossell´o, Maria del Mar Batle and Regina Batle, as well as Mohammed A.H. Khalafalla.

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