Producción CientíficaIn this note a one-dimensional band model is proposed based on a periodic Dirac comb having a periodic
mass distribution m(x). The mass function is represented as a Hermitian, non-local separable operator.
Two specific cases–a constant mass model and a sinusoidal mass model–are examined. The lowest
electron and positron bands for the constant mass case are similar to those for the standard relativistic
Kronig-Penney model, suggesting that non-locality has little influence. The results for the sinusoidal case
are consistent with the expectation that at low wavenumber an electron “feels” it has an average constant
mass, but at high wave number, the particle “sees” the periodic mass variation and the band is distorted.Ministerio de Economía, Industria y Competitividad (project MTM2014-57129-C2-1-P)Junta de Castilla y León (project VA057U16

Glasser, M. Lawrence

Physics Letters A

English

arXiv

Repositorio Documental de la Universidad de Valladolid

JID:PLA AID:126277 /SCO Doctopic: Quantum physics [m5G; v1.261; Prn:24/01/2020; 8:44] P.1 (1-4)Physics Letters A ••• (••••) ••••••Contents lists available at ScienceDirectPhysics Letters Awww.elsevier.com/locate/plaA relativistic one dimensional band model with position dependent massM.L. Glasser a,b,∗a Department of Physics, Clarkson University, Potsdam, NY 13699-5820, USAb Departamento de Física Teórica, Atómica y Óptica and IMUVA, Universidad de Valladolid, 47011 Valladolid, Spaina r t i c l e i n f o a b s t r a c tArticle history:Received 13 November 2019Received in revised form 31 December 2019Accepted 17 January 2020Available online xxxxCommunicated by R. WuKeywords:Relativistic Dirac combKronig-Penney modelEnergy band structurePosition-dependent massIn this note a one-dimensional band model is proposed based on a periodic Dirac comb having a periodic mass distribution m(x). The mass function is represented as a Hermitian, non-local separable operator. Two specific cases–a constant mass model and a sinusoidal mass model–are examined. The lowest electron and positron bands for the constant mass case are similar to those for the standard relativistic Kronig-Penney model, suggesting that non-locality has little influence. The results for the sinusoidal case are consistent with the expectation that at low wavenumber an electron “feels” it has an average constant mass, but at high wave number, the particle “sees” the periodic mass variation and the band is distorted.© 2020 Elsevier B.V. All rights reserved.1. IntroductionThe desire for considering spatially dependent electron masses in solid state systems was expressed as long ago as the early 1940’s [1] and was made explicit through the work of Wannier, Slater, Luttinger and Kohn [2] with the development of effective mass theory in the early post-war years. Gora and Williams [3]were, it seems, the first to adapt the kinetic energy operator to this situation, but after noting that their original expression was not Hermitian, they proposed the non-relativistic kinetic energy operatorK= − 14h¯2[m(r)−1/2∇ + ∇m(r)−1/2] (1)which was then derived by others from various physical perspec-tives [4]. O. Von Roos [4] subsequently pointed out that (1) was not unique, being just a special case ofK= − h¯24[ma∇mb∇mc +mc∇mb∇ma], a + b + c = −1. (2)On the basis of Bargmann’s theorem [5] he also argued that (2)was unphysical and proved explicitly that the ambiguity was re-lated to a lack of Galilean invariance, i.e. observers in different * Correspondence to: Department of Physics, Clarkson University, Potsdam, NY 13699-5820, USA.E-mail address: laryg@clarkson.edu.https://doi.org/10.1016/j.physleta.2020.1262770375-9601/© 2020 Elsevier B.V. All rights reserved.inertial frames would measure different results for physical prop-erties of such a system. Thus, Von Roos maintained that the very concept of position dependent mass should be avoided as was possible by returning to more fundamental principles. Neverthe-less the position dependent mass (PDM) concept has been popular and (2) (usually with c = a) has been the basis of a plethora of calculations over the last 30 years. A representative set of papers where PDM is applied to various simple quantum systems, both non-relativistic [6–9] and relativistic [10–18].In the relativistic case, though Bargmann’s theorem, that phys-ical wave packets cannot be constructed from components corre-sponding to different masses, still cannot be avoided, at least it can be argued that non-uniqueness is less of a problem since the mass in the Dirac equation occurs simply as a scalar operator, sim-ilar to the potential [16–18]. The aim here is to take advantage of this by introducing a class of exactly solvable relativistic Kronig-Penney models (i.e. where the lattice potential is a periodic Dirac comb with amplitude v0) relative to a fixed frame where the mass is replaced by the non-local separable linear operatorm(x)ψ(x) =m0N/2∑l=−N/2μ(x− la)∞∫−∞dx′μ(x′ − la)ψ(x′), (3)(a is the lattice spacing, N → ∞ the number of unit cells). As will be seen, to construct a particular model one need only specify the Fourier coefficients of μ(x), the mass profile in a unit cell. This note examines the caseJID:PLA AID:126277 /SCO Doctopic: Quantum physics [m5G; v1.261; Prn:24/01/2020; 8:44] P.2 (1-4)2 M.L. Glasser / Physics Letters A ••• (••••) ••••••Fig. 1. Unit cell mass profile for b = .1,a = 5.μ(x) = 12[(1+b)+(1−b) cos(2πx/a)]θ(a/2−|x|), (0 ≤ b < 1)(4)illustrated in Fig. 1 for b = .1. For b = 1 the cosine component is eliminated and μ(x) = 1.The Fourier coefficients for (4) areμˆn = 22π2(b + 1) − a2bk2nkn(4π2 − a2k2n)sin(kna/2) (5)kn = k + Kn and Kn = 2πn/a is the n-th reciprocal lattice vector.The constant mass case corresponds to b = 1.We present the details of the calculation in the next section, followed by the numerical examination of the specific case (4); the results are discussed in the concluding section.2. CalculationThe one-dimensional Dirac equation can be writtenih¯cφ′1 = [E −m(x)c2 − V (x)]φ2(x) (6a)ih¯cφ′2 = [E +m(x)c2 − V (x)]φ1(x) (6b)and after inserting the Bloch form of the wave function compo-nents,φ j(x) =∑nC jneiknx, (7)where k is the crystal momentum (for simplicity we write kn =k + Kn) one has∑n[h¯cknC1n + EC2n ]eiknx=∑l,nC2neiknla{v0δ(x− la) + u0μ(x− la)μˆn} (8a)∑n[h¯cknC2n + EC1n ]eiknx=∑l,nC1neiknla{v0δ(x− la) − u0μ(x− la)μˆn} (8b)where we write u0 =m0c2. For any integrable function g(x) definegˆn =∞∫dx′eiknx′ g(x′). (9)−∞Next, we multiply each of (8a), (8b) by exp[−ikmx] and integrate over x, noting that∞∫−∞dxei(kn−km)x = Naδn,m (10a)∑lei(kn−km)la = N (10b)to findh¯ckmC1m + EC2m =1a∑n[C2n [v0 + u0μˆnμˆ∗m] (11a)h¯ckmC2m + EC1m =1a∑n[C1n [v0 − u0μˆnμˆ∗m] (11b)These can be writtenMC =(v0D2 + u0d2μˆ∗mv0D1 − u0d1μˆ∗m)(12)withD j = 1a∑nC jn, d j = 1a∑nC jnμˆn, (13a)M=(h¯ckm EE h¯ckm), C =(C1mC2m). (13b)Therefore, by matrix inversionC1m =1m(k){h¯ckm[v0D2+u0d2μˆ∗m]− E[v0D1−u0d1μˆ∗m]} (14a)C2m =1m(k){h¯ckm[v0D1 − u0d1μˆ∗m] − E[v0D2 + u0d2μˆ∗m]}(14b)where m(k) = h¯2c2k2m − E2. So, in terms of the quantitiesAr1 =∑mkrmm(k), Ar2 =∑mkrm|μˆm|2m(k), Ar3 =∑mkrmμˆ∗mm(k)(15)one has(1+ Ev0A01)D1 − h¯cv0A11D2 − Eu0A03d1 − h¯cu0A13d2 = 0h¯cv0A11D1 − (1+ Ev0A01)D2 − h¯cu0A13d1 − Eu0A03d2 = 0Ev0A03D1 + h¯cv0A13D2 − (1− Eu0A03)d1 + h¯cu0A12d2 = 0h¯cv0A13D1 − Ev0A03D2 + h¯cu0A12d1 − (1+ Eu0A02)d2 = 0(16)Consequently, the band structure is given by the roots E(k) of the 4 × 4 determinantD(k, η) =∣∣∣∣∣∣∣∣∣1+ Ev0A01 −h¯cv0A11 −Eu0A03 −h¯cu0A13−h¯cv0A11 1+ Ev0A01 h¯cu0A13 Eu0A03−Ev0A03 −h¯cv0A13 1− Eu0A03 −h¯cu0A12−h¯cv0A13 Ev0A03 −h¯cu0A12 1+ Eu0A02∣∣∣∣∣∣∣∣∣(17)A word about units is appropriate here. Lengths are given in Bohr radii, a0 and we set h¯c = 1 so η = E/h¯c, v0 and u0 are all reciprocal lengths.The six series Ari can be evaluated analytically; for example:A01 =a2h¯2c2aη[sin(aη)cos(aη) − cos(ka)](18a)A11 =a2h¯2c2[sin(ka)cos(aη) − cos(ka)](18b)JID:PLA AID:126277 /SCO Doctopic: Quantum physics [m5G; v1.261; Prn:24/01/2020; 8:44] P.3 (1-4)M.L. Glasser / Physics Letters A ••• (••••) •••••• 3Fig. 2. Lowest particle band: a = 1, b = .3 (lower curve), b = 1 (upper curve) v0 = 2, u0 = 1000.Fig. 3. First excited particle band: a = 1, b = .3 (initially upper curve), b = 1 (lower curve) v0 = 2, u0 = 1000.3. Numerical exampleAs a test case, we set: a = 1, v0 = 2, u0 = 1000 and b = 0.3. Since the ratio u0/v0 = 500 is relatively small, this model may be considered mildly relativistic. The choice a = 1, which was selected for convenience, means that the lattice is rather dense and may emphasize anomalies. For comparison, we also examine the con-stant mass version: b = 1.The energy levels for the lowest electron band in each case, were found by plotting the determinant (17), with k specified, as a function of η and recording the lowest positive zero. We note that there may be zeros corresponding to vanishing energy denom-inators m(k) = 0, i.e. free, zero-mass bands, which are spurious. Lengths are measured in Bohr radii and we have set h¯c = 1, so that energies have units of reciprocal length. For model (3) with b = 1, 0.3, the lowest band E0(k), is shown in Fig. 2. Both bands are continuous and roughly parabolic.In Fig. 3 we show the next higher band for the two cases. Here again, that for b = 1 lies above the one for b = 0.3.4. DiscussionSince in the constant mass case b = 1 the lowest band resem-bles that for the ordinary relativistic Kronig-Penney model [19–21], it appears that the non-local nature of the mass produces no anomalies. However, at the edge of the BZ, k = π/a, the energy is driven toward zero, as indicated by a vertical line. The relative Fig. 4. Antiparticle band: a = 1, b = .3 (initially upper curve), b = 1 (initially lower curve) v0 = 2, u0 = 1000.nature of the two band structures in Fig. 2 can be heuristically ex-plained as follows: At low wave vector k, for the sinusoidal case the particle has long wave length and “senses” only tan average mass, so the bands nearly merge, but separate at higher k value where the particle wave length is small enough that the mass vari-ation is detectable. In which case the energy is depressed. At the zone boundary the particle and mass wavelengths are in resonance and the particle velocity is reduced.As pointed out in [18] the upper and lower components of the two-component spinor here are particle and anti-particle wave functions, resp. Hence the negative roots of (17) correspond to positron energies. The fundamental bands of the two cases con-sidered here are shown in Fig. 4.In conclusion, a version of the relativistic Kronig-Penney model has been constructed which may be suitable for exploring the ef-fects of variation of effective masses in semiconductor structures. It incorporates the unusual replacement of the mass as a non-local operator, but this does not appear to introduce anomalies in the band structure where the mass has purely sinusoidal variation. In a future report we hope to consider the more usual case where the mass is constant but its value varies from cell to unit cell. Fi-nally, it is worth pointing out that the model is easily extended to two and three dimensions.Declaration of competing interestI declare I have no competing financial interests or personal re-lationship that could have appeared to influence the work reported in this paper.AcknowledgementsThe author thanks Profs. L.M. Nieto and M. Gadella for in-troducing him to this problem and many helpful suggestions. He also acknowledges the financial support of MINECO (Project MTM2014-57129-C2-1-P) and Junta de Castilla y León (VA057U16).References[1] L. Pekar, J. Phys. (USSR) 10 (1946) 431.[2] G.H. Wannier, Phys. Rev. 52 (1937) 192;J.C. Slater, Phys. Rev. 76 (1949) 1592;J.M. Luttinger, W. Kohn, Phys. Rev. 97 (1955) 869.[3] T. Gora, F. Williams, Phys. Rev. 177 (1969) 1179.[4] O. Von Roos, Position dependent effective masses in semi-conductor theory, Phys. Rev. B 27 (1985) 7547–7552;A. Trabelsi, F. Madouri, A. Matar, Classification scheme for kinetic energy oper-ators with PDM, arXiv:1302 .3763v1, 2013.[5] F.A. Kaempffer, Concepts in Quantum Mechanics, Academic Press, New York, 1965, Appendix 7.JID:PLA AID:126277 /SCO Doctopic: Quantum physics [m5G; v1.261; Prn:24/01/2020; 8:44] P.4 (1-4)4 M.L. Glasser / Physics Letters A ••• (••••) ••••••[6] J-M. Levy-Leblonde, Elementary quantum models with PDM, Eur. J. Phys. 13 (1992) 316–318.[7] J. Yu, S-H. Dong, Exactly solvable potentials for the Schroedinger equation with spatially dependent mass, Phys. Lett. A 325 (2004) 194–198.[8] J. Yu, S-H. Dong, G-H. Sun, Series solution of the Schroedinger equation with position dependent mass for the Morse potential, Phys. Lett. A 322 (2004) 290–297.[9] S-H. Dong, M. Lozada-Cassou, Solutions of the Schroedinger equation with position dependent mass for a hard-core potential, Phys. Lett. A 337 (2005) 313–320.[10] R. Koc, H. Tütüncülen, Exact solution of the PDM Schroedinger equation by supersymmetric quantum mechanics, arXiv:quant -ph /0410088, 2004.[11] S. Cruz y Cruz, J. Negro, L.M. Nieto, On PDM harmonic oscillators, J. Phys. Conf. Ser. 128 (2008) 012053.[12] S. Cruz y Cruz, O. Rosas-Ortiz, PDM oscillators and coherent states, arXiv:0902 .2029v1, 2009.[13] Ju Guo-Xing, C.A.I. Chang-Ying, R.E.N. Zhang-Zhou, Generalized harmonic oscil-lators and the Schroedinger equation with PDM, Commun. Theor. Phys. (Bei-jing) 51 (2009) 797–802.[14] H. Panahi, Z. Bakhshi, Solvable potentials with PDM and constant mass, Acta Phys. Pol. B 41 (2010) 1–21.[15] S. Cunha, H.R. Christiansen, Analytic results in the PDM Schroedinger equation, arXiv:1306 .0933v1, 2013.[16] A.D. Alhaidari, Solution of the Dirac equation with PDM in the Coulomb field, Phys. Lett. A 322 (2004) 72–77.[17] O. Panella, S. Biondini, A. Andra, New exact solution of the one-dimensional Dirac equation for the Woods-Saxon potential with the effective mass case, J. Phys. A, Math. Theor. 43 (2010) 325902.[18] M. Eshghi, H. Mehraban, Dirac Poschl-Teller problem with PDM, Eur. J. Sci. Res. 54 (2011) 22–28.[19] F. Dominguez Adame, Am. J. Phys. 55 (1987) 1004.[20] P. Strange, Relativistic Quantum Mechanics, Cambridge Univ. Press, 1998, Chap. 9.[21] M.L. Glasser, A class of one-dimensional relativistic band models, Am. J. Phys. 51 (1983) 938.

A relativistic one dimensional band model with position dependent mass

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A relativistic one dimensional band model with position dependent mass

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