Klein Paradox and Disorder-Induced Delocalization of Dirac Quasiparticles in One-Dimensional Systems

Transcription

1 Commun. Theor. Phys. (Beijing, China 54 (200 pp c Chinese Physical Society and IOP Publishing Ltd Vol. 54, No. 6, December 5, 200 Klein Paradox and Disorder-Induced Delocalization of Dirac Quasiparticles in One-Dimensional Systems YUAN Jian-Hui ( ï, CHENG Ze ( Ä, YIN Miao (ì, ZENG Qi-Jun (É, and ZHANG Jun-Pei ( Department of Physics, Huazhong University of Science and Technology, Wuhan , China (Received March 3, 200; revised manuscript received April 9, 200 Abstract Dirac particle penetration is studied theoretically with Dirac equation in one-dimensional systems. We investigate a one-dimensional system with N barriers where both barrier height and well width are constants randomly distributed in certain range. The one-parameter scaling theory for nonrelativistic particles is still valid for massive Dirac particles. In the same disorder sample, we find that the localization length of relativistic particles is always larger than that of nonrelativistic particles and the transmission coefficient related to incident particle in both cases fits the form T exp( αl. More interesting, massless relativistic particles are entirely delocalized no matter how big the energy of incident particles is. PACS numbers: Fz, 7.55.Jv, Gg, 7.23.An Key words: Anderson localization, disordered structures, quantum transport, localized states Introduction The Klein paradox is presented firstly dating back to the investigation of Klein by calculating Dirac particle penetration to a step potential. [] It is an ordinary issue in the relativistic quantum mechanics. In past decades, the physics behind Klein paradox was always a very interesting topic for the theoretical physical scientists. [ 7] At one time, the interpretation of Klein paradox resorted to the notion of»hole¼ in the negative-energy sea, though the notion of»hole¼ predicted successfully the existence of the antiparticle. As most paradoxes in physics, Klein s paradox was resolved dating back to the investigations of Sauter. [2] Especially, this problem had been solved satisfactorily with the building of the quantum field theory (QFT. The resulting explanations are based on the effects of spontaneous production of particle-antiparticle pairs when the electrons pass through a high barrier. [3] However, it is still interesting that application of Dirac particles is addressed in disordered systems. Since the notion of localization¼had been presented firstly in the pioneering work of Anderson, [8] the transport properties of electron in disordered systems have drawn a large amount of attention. [8 3] Anderson in his study firstly gave a criteria for transport in relation to the condition of localization. Along the direction, the scaling theory was presented by Thouless [9] and Abrahams et al. [0] There is a very famous result that arbitrary weak disorder in one-dimensional systems can lead to localization of all of electron states. [9 ] We have to mention equitably the method of Landauer [2] for treating the simplest onedimensional case, though he acknowledged that certain results of his are faulty in a fairly subtle way. His simple model makes us easily realize that the wavefunction of electron decays exponentially. Based on quantum scattering theory, Anderson [3] et al. generalized the method of Landauer and an exact scaling theory for general onedimensional case is given with many channels transverse to the unique dimension. A precondition, however for the above results is that particles have a low energy and can be described by the Schrödinger equation. Recently, Dirac quasiparticles have been found in honeycomb lattices, such as electron in the graphene, [4] three-dimensional topological insulator, [5] and cold atoms in the optical lattices. [6] The existences of Dirac quasiparticles in condensed matter naturally give us an opportunity to investigate their properties. For example, Bai et al. [4] and Abedpour et al. [4] studied the conductance of a graphene superlattice, respectively. And the topic of anisotropic behaviours of Dirac fermions in graphene under periodic potentials was studied by Park et al. [4] Thus, Dirac particles have again attracted a significant amount of attention. [7 22] As we known, Klein s paradox of relativistic particles can induce some new phenomena that is different from the nonrelativistic particles. It is interesting to us how relativistic effect influences on the state of quasiparticle in one-dimensional disordered systems. Now, the relativistic effect on electron motion in disordered systems can be well described by relativistic Dirac equation in such systems. The investigation of relativistic particles in disordered systems, however is less active than that of nonrelativistic particles because of the complexity of relativistic particles. In fact, up to now the nature of relativistic electron states has been treated only in onedimensional disordered systems. [9 20] For example, Roy Supported by the National Natural Science Foundation of China under Grant Nos and Corresponding author, zcheng@mail.hust.edu.cn

2 30 YUAN Jian-Hui, CHENG Ze, YIN Miao, ZENG Qi-Jun, and ZHANG Jun-Pei Vol. 54 et al. [9] reported Dirac electrical conduction in a onedimensional disordered system based on generalization of a nonrelativistic approach of Landauer for electrical resistance. Recently, Zhu et al. [20] have discussed the localization of Dirac particle states in one-dimensional disordered systems with N barriers, and a scheme has been presented to simulate the Dirac particles with developed techniques in the cold atomic systems. Varying from Ref. [20], both barrier height disorder and well width disorder are taken into account in our work. In this paper, we report the phenomenon of Klein paradox for Dirac particle penetrating into a rectangular potential barrier. Also Anderson localization of relativistic systems by using the transfer-matrix technique and the generalization of Landauer for electrical resistance. An eccentric phenomenon called Klein paradox is revealed that it is very different from the nonrelativistic particles. The results show that Dirac particles can pass through the high potential barrier in Klein region even if the potential barrier approaches infinity. As well as the phenomenon can induce the difference of Anderson localization between Dirac particles and nonrelativistic particles in onedimensional systems with the same disorder strength. We find that the localization length of relativistic particles is always larger than that of nonrelativistic particles in the same disorder sample. More interesting, a necessary consequence of Klein paradox is that massless relativistic particles in one-dimensional disordered systems are entirely delocalized no matter how big the energy of incident particles is, which breaks down the famous conclusion that arbitrary weak disorder can cause the localization of nonrelativistic particles in one-dimensional systems. In Sec. 2, we need to introduce relativistic Dirac equation to describe the essential features of Dirac particles. In Sec. 3, the numerical analysis to our important analytical issues are reported. Finally, a brief summary is given in Sec Theoretical Background 2. One-Dimensional Dirac Equation Our analysis begins with the relativistic Dirac equation for the relativistic treatment of electron motion in one-dimensional systems. Considering a Dirac particle of mass m and energy E penetrated to a square barrier of width a and height V, the relativistic Dirac equation of Dirac particle can be written as follows: [9 20] ( d i cσ x dx + mc2 σ z + V (x φ(x = Eφ(x, ( where σ x, σ z are the components of the Pauli spin matrix, c denotes the velocity of light, and φ(x represents a twocomponent spinor. A general solution of Eq. ( is given by [9 20] φ(x = A ( ( e iκx + B e iκx, (2 ν ν where κ 2 = (ε V (ε V + 2mc2 ( c 2, cκ ν = ε + 2mc 2 V, ε = E mc2. (3 Here the coefficients A and B denote the amplitudes of the spinor moving along the positive x-axis and its opposite direction, respectively. ε is a positive that denotes the energy of the moving particle. 2.2 Barrier Penetration and Klein Paradox We now look into the transmission for a square barrier as shown in Fig. (a. The wave function in the different regions can be written in terms of incident and reflect waves. In every region, the wave function needs to meet the relation of the Eq. (2. Denoting the amplitudes of the spinor across the barrier, we can obtain a relation between the amplitudes based on the continuity of the wave function, ( A B = M ( A B, (4 where M is the transfer matrix of the barrier and its elements are obtained by M = (cos(κ a + i ν2 + ν2 sin(κ a e iκa, 2νν M 2 = i ν2 ν2 2νν sin(κ ae iκa, M 2 = M 2, M 22 = M, det(m =, (5 where κ, ν are in relation to the barrier V (x = V in the region 0 < x < a; κ, ν are in relation to the barrier V (x = 0 in the region x > a or x < 0. Considering no reflection coefficients in region, so the transmission coefficients can be given by M 2 = + (ν /ν ν/ν 2 sin 2 (κ a/4. (6 Considering the relation of Eq. (3, Eq. (6 can be simplified as follows: [6] + [m 2 V 2 /(κ 2 κ2 4 ] sin 2 (κ a. (7 In the non-relativistic limit, κ 2 = 2m(ε V / 2 and κ 2 = 2mε/ 2. So the transmission coefficients in the nonrelativistic limit can be obtained by + s[v 2 /4ε(ε V ] sin 2 (κ a, (8 which is in agreement with that from the Schrödinger equation and s = sign(ε V. Actually, the precondition for the validity from Eq. (2 to Eq. (8 is that κ > 0. When the condition above is not met, the equation is able to solve these issues only replacing κ by iκ. Compared with Eq. (8, the behavior of transmission coefficients of Eq. (7 is remarkably different when the

3 No. 6 Klein Paradox and Disorder-Induced Delocalization of Dirac Quasiparticles in One-Dimensional Systems 3 barrier height V approaches to infinity and the energy of particle is restricted in the Klein region E < V mc 2 or ε < V 2mc 2. As the barrier V and Dirac particles penetrate into it, ν =, so that the transmission coefficients can be given by + (ν /ν 2 sin 2 (κ a/4 0, (9 where ν = ε/(ε + 2mc 2. From Eq. (8, however, T s approaches to zero exponentially even the barrier height is only a litter bigger than the energy of the incident particle. Furthermore, if κ a = nπ (n =, 2,..., T becomes one and relativistic tunneling occur. More intriguingly, the transmission coefficients T is always equal to one for massless limit where ν = ± and ν = ±, so the barrier is totally transparent. The results above are some manifestations of the Klein paradox and do not occur for the nonrelativistic particles. A reasonable interpretation is the electron-positron pair creation process from vacuum. [7,22] For massless limit, the phenomenon is just a manifestation of the chirality of the electron (or positron. [20] Fig. Schematic representation of the system. (a A rectangular potential barrier. (b Model of N rectangular potential barriers. 3 Model and Method Let us consider a particle penetrating into a onedimensional chain with N rectangular barriers as shown in Fig. (b. According to the generalization of Landauer for electrical resistance, [3] we can treat the structure as a stack of layers, and compute the stack transmission amplitudes (T n = T n t n /( R n r n and reflection amplitudes (R n = r n + R n t 2 n /( R n r n by recursion. [3,23,25] Here, R n, T n denote the amplitudes of the reflection and transmission of a stack of n layers, r n, t n denote the amplitudes of the reflection and transmission of the n-th layer, and ÃB denotes a phase factor between A and B. Thus, the conductance through the N barriers corresponding to the transport properties is given by Landauer formula G = (2πe 2 / T N 2 = (2πe 2 / g where g denotes the dimensionless conductance. [2 3,20] For quantitative analysis, some assumptions are given as follows: (a There is no shape disorder; (b The reflection coefficients between two barriers are stochastically unrelated to each other. It implies that the mean interval between two barriers is much larger than the plane wave length (λ incident normally on the random sample, that is to say, the phase θ of R n r n is restricted to the region [0, 2π]; (c V N is a constant randomly distributed in the region [ δ, δ]. At zero temperature, the dimensionless localization length ξ is defined as the reciprocal of the Lyapunov exponent γ, [20,23 24] where γ /ξ lim L ln T (N /2L and denotes the averaging over the disorder. Here, L denotes the size of the sample (L = N(a + d where d is the mean interpotential distance and a is the barrier width. So the definition above implies that a state is a localized state if ξ is finite and is a delocalized state if ξ is divergent. 4 Results and Discussions In the following, we will study Anderson localization of a particle penetrating into a one-dimensional disordered system with N barriers. We assume that the mean interval of interbarriers d 0λ and the mass of quasiparticles m = m 0 with m 0 being the mass of a nuclear in both relativistic case and nonrelativistic case. [20] We choose b = a + d as the unit of length. We firstly demonstrates the localization length ξ in Fig. 2(a and the corresponding Lyapunov exponent γ in Fig. 2(b as a function of barrier width in both the relativistic case and the nonrelativistic case. A superscript letter D is corresponding with the relativistic case and S is in relation to the nonrelativistic case. We find that the localization length of relativistic particles is always larger than that of nonrelativistic particles and the length of localization decreases monotonously with the increase of the width of barriers. A justifiable explanation is that the transmission coefficient varies inversely with the barrier width as quasiparticles penetrate into a disorder system with N barriers. Also, the localization length ξ is finite in the whole range of a, which implies that the state of massive particle for both relativistic case and nonrelativistic case is a localized state. The results are similar to those of Ref. [20, 26]. The same results can be found in Fig. 2(b. Furthermore, The variation of ln g corresponding to dimensionless conductance as a function of size of sample where the barrier width a = 0.2 is shown in Fig. 2(c in both the relativistic case and the nonrelativistic case. It is seen easily that lng in both the relativistic case and the nonrelativistic case varies linearly with the size of the sample, namely, lng αl. The

4 32 YUAN Jian-Hui, CHENG Ze, YIN Miao, ZENG Qi-Jun, and ZHANG Jun-Pei Vol. 54 slope of the curve, however, related to the relativistic particle is smaller than that of the nonrelativistic particle, namely, α D < α S, According to the definition of the Lyapunov exponent γ, we can easily find that the transmission coefficients of a stack of N layers in disordered systems fits T exp( αl where /α denotes the strength of localization for disorder sample, so one can see why the localization length for relativistic quasiparticles is larger than that for nonrelativixtic quasiparticles in one-dimensional disordered systems with the same sample. Consequently, one can easily see that when a particle penetrates into a disordered system with the size of sample L, the wave amplitudes of the incident particle will decay exponentially. Fig. 2 (a The localization length ξ and (b The corresponding Lyapunov exponent γ as a function of the barrier width in both the relativistic case and the nonrelativistic case for N = 000 layers. (c The variation of lng corresponding to dimensionless conductance as a function of size of sample where the barrier a = 0.2. The other parameters are ε = 0.05 and σ [ 2,2] with the energy units of mc 2. Now, in Fig. 3 we check the validity of a single-parameter scaling equation, namely, β = d lng /d lnl = f(g. Compared with the result of nonrelativistic case, our results for relativistic case are similar to that of Ref. [26]. Thus, this assumption is still correct for massive Dirac quasiparticles in one-dimensional systems, provided that a proper choice for g is made. From Fig. 3, one sees the form β ln g. More interesting, massless Dirac quasiparticles are totally delocalized penetrating into a disordered system no matter how big the energy of incident particles is. An intuitive interpretation of the above result can be done with Klein paradox because the transmission coefficients T is always equal to one for massless limit, that is to say, the barrier is totally transparent. An essential interpretation is the chiral symmetry of Dirac particles. [20] For massless Dirac particle, we find the form [σ x, H] 0, so ν = ± in Eq. (2. Thus, we can see that the transfer matrix is a diagonalized one, namely, M = diag {e iϕ, e iϕ }, where ϕ is a pure phase factor. So one sees that the localization length ξ D is infinite from the definition of ξ D. The result breaks down the famous conclusion that arbitrary weak disorder can induce the localization of all of particle states in the one-dimensional system. Fig. 3 The scaling function β vs. ln g corresponding to the relativistic case for N = 000 layers. The other parameters are ε = 0.05, σ [ 2, 2] with the energy units of mc 2 and a well-proportioned scale is chosen in there. 5 Conclusion In conclusion, we report the phenomena of Klein paradox for Dirac particles penetrating into a rectangular potential barrier. Also Anderson localization of relativistic systems. We investigate a one-dimensional system with

5 No. 6 Klein Paradox and Disorder-Induced Delocalization of Dirac Quasiparticles in One-Dimensional Systems 33 N barriers that the barrier height and well width are constants randomly distributed in certain range. An eccentric phenomenon called Klein paradox is revealed that it is very different from the nonrelativistic particles. The results show that Dirac particles can pass through the high potential barrier in Klein region, even the potential barrier approaches infinity. Also the localization of Dirac systems compared with the nonrelativistic particles. We find that (i As the same with nonrelativistic case, the one-parameter scaling theory for massive Dirac particle is valid in the disordered system; (ii The localization length of the relativistic particles is always larger than that of the nonrelativistic particles in the same disorder sample; (iii The transmission coefficient related to incident particle fits the form T exp( αl for both relativistic case and nonrelativistic case in one-dimensional disordered system; (iv More interesting, massless relativistic particles are entirely delocalized no matter how big the energy of incident particles is. References [] O. Klein, Z. Phys. 53 ( [2] F. Sauter, Z. Phys. 69 ( [3] A. Hansen and F. Ravndal, Phys. Scripta 23 ( [4] S.D. Leo and P.P. Rotelli, Phys. Rev. A 73 ( [5] B.H.J. McKellar and G.J. Stephenson, Phys. Rev. A 36 ( [6] B.H.J. McKellar and G.J. Stephenson, Phys. Rev. C 35 ( [7] R.K. Su, G.G. Siu, and X. Chou, J. Phys. A: Math. Gen 26 ( [8] P.W. Anderson, Phys. Rev. 09 ( [9] D.J. Thouless, Phys. Rep. 3 ( [0] E. Abrahama, P.W. Anderson, D.C. Licciardello, and T.V. Ramakerishnan, Phys. Rev. Lett. 42 ( [] N.F. Mott and W.D. Twose, Adv. Phys. 24 ( [2] R. Landauer, Phil. Mag. 2 ( [3] P.W. Anderson, D.J. Thouless, E. Abrahama, and D.S. Fisher, Phys. Rev. B 22 ( [4] A.H. Castro Neto, et al., Rev. Mod. Phys. 8 ( ; K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M. Katsnelson, V. Grigorieva, S.V. Dubonos, and A.A. Firsov, Nature (London 438 ( ; C.X. Bai and X.D. Zhang, Phys. Rev. B 76 ( ; N. Abedpour, A. Esmailpour, R. Asgari, and M.R.R. Tabar, Phys. Rev. B 79 ( ; C.H. Park, L. Yang, Y.W. Son, M.L. Cohen, and S.G. Louie, Nature Phys. 4 ( [5] Y.L. Chen, et al., Science 325 ( [6] S.L. Zhu, B.G. Wang, and L.M. Duan, Phys. Rev. Lett. 98 ( ; C.J. Wu and S.D. Sarma, Phys. Rev. B 77 ( ; G. Juzeliünas, et al., Phys. Rev. A 77 ( (R. [7] C. Chamon, et al., Phys. Rev. B 77 ( [8] L. Lamata, et al., Phys. Rev. Lett. 98 ( [9] C.L. Roy and C. Basu, Phys. Rev. B 45 ( ; P.K. Mahapatra, et al., Phys. Rev. B 58 ( ; C.L. Roy, J. Phys. Chem Solids 57 ( [20] S.L. Zhu, D.W. Zhang, and Z.D. Wang, Phys. Rev. Lett. 02 ( ; C.R. de Olivra and R. Prado, J. Phys. A 38 (2005 L5. [2] W. Greiner, B. Müller, and J. Reflski, Quantum Electrodynamics on Strongfields, Berlin, Springer (985; M. Soffel, B Müller, and W. Greiner, Phys. Pep. 85 ( [22] Krekora, Q. Su, and R. Grobe, Phys. Rev. Lett. 93 ( ; A. Hansen and F. Ravndal, Phys. Scripta. 23 ( [23] A.A. Asatryan, et al., Phys. Rev. Lett. 99 ( [24] I.M. Lifshitz, S.A. Gredeskul, and L.A. Pastur, Introduction to the Theory of Disordered Systems, Wiley, New York (989. [25] V. Baluni and J. Willemsen, Phys. Rev. A 3 ( [26] E.N. Economou and C.M. Soukoulis, Phys. Rev. Lett. 46 (98 68.