- Email: [email protected]
Effects of strong electric fields in a polyacetylene chain
Journal of Physics and Chemistry of Solids 82 (2015) 17–20
Contents lists available at ScienceDirect
Journal of Physics and Chemistry of Solids journal homepage: www.elsevier.com/locate/jpcs
Effects of strong electric fields in a polyacetylene chain C.R. Muniz a,n, M.S. Cunha b a b
Grupo de Física Teórica (GFT), Universidade Estadual do Ceará, FECLI, Iguatu, Ceará, Brazil Grupo de Física Teórica (GFT), Universidade Estadual do Ceará, CCT, Fortaleza, Ceará, Brazil
art ic l e i nf o
a b s t r a c t
Article history: Received 26 October 2014 Received in revised form 1 February 2015 Accepted 25 February 2015 Available online 26 February 2015
In this work, we study the effects related to the creation of electron/hole pairs via application of an external electric field that acts on a pristine trans-polyacetylene molecular chain at zero-temperature. This phenomenon is termed Schwinger–Landau–Zener (SLZ) effect and arises when a physical system, which can even be the vacuum, is under the action of a strong, static and spatially homogeneous electric field. Initially, we investigate how the electrical conductivity of the polyacetylene changes with the applied field, by considering the carriers production as well as the variation of the interband gap according to certain ab initio models. Next, we analyse the competition between the SLZ effect and another one associated with the incidence of an uniform electric field on one-dimensional crystals – the Bloch oscillations. We evaluate the conditions in which these latter can be destroyed by the particles created through the same field that induces them, and verify the possibility of occurrence of the Bloch oscillations inside the trans-polyacetylene with frequencies equal to or higher than the terahertz scale. & 2015 Elsevier Ltd. All rights reserved.
Keywords: A. Polymers D. Electronic transport D. Electronic band structure
1. Introduction In the early 1950s of the last century, J. Schwinger showed that an intense, static and spatially homogeneous electric field can disrupt the quantum vacuum in such a way to cause the generation of electron/positron pairs [1]. He used the theoretical framework of quantum electrodynamics in order to find the non-perturbative exact amplitude of transition vacuum-to-vacuum, or vacuum persistence, for that physical process. At an instant t and within certain volume V, the correspondent probability is
Pvac = |〈0|U (t) |0〉|2 = exp( − ωVt),
(1)
where U(t) is the time evolution operator and
ω=
(eE)2 4π 3= 2c
∞
∑ n= 1
⎛ nπ E ⎞ 1 cr ⎟. exp ⎜⎜− E ⎟⎠ n2 ⎝
(2)
Here, e is the electronic charge and E is the magnitude of the applied electric field, and
Ecr =
me2 c 3 , =e
(3)
is the electric field needed to create abundant electron/positron pairs in the vacuum, as we will see below. This critical field can be n
Corresponding author. E-mail addresses: [email protected] (C.R. Muniz), [email protected] (M.S. Cunha). http://dx.doi.org/10.1016/j.jpcs.2015.02.009 0022-3697/& 2015 Elsevier Ltd. All rights reserved.
thought of as that one required for realize the work to separate a virtual pair of particle/antiparticle of electronic mass me by a distance equal to its Compton wavelength. It is also possible to calculate the production of pairs per unit time per unit volume, which is given by [2,3]
⎛ πE ⎞ (eE)2 d2N cr ⎟⎟. exp ⎜⎜− = 3 2 dVdt 4π = c ⎝ E ⎠
(4)
This expression is the first term of the sum in Eq. (2). Notice that an applied electric field greater than or equal to the critical one neutralizes the exponential in Eq. (4), with the pair production rate growing approximately as a function of the square of the electric field when E is very high. The value of the critical electric field is Ecr ≈ 1018 V/m and exceeds quite the current technological capability of generation of these fields. The most powerful electric field produced in our laboratories oscillates in certain laser beams, reaching 108 V/m [4], eliminating any hope of presently detecting the Schwinger effect, at least in the vacuum. It is worth notice in here that the Schwinger effect may also occur through action of variable electric fields [5]. However, in the context of condensed matter physics, in semiconductors of low-dimensionality at zero-temperature, the limit velocity of the carriers is much lower than one of the light in vacuum. For example, the Fermi velocity (vF) in the molecular polyacetylene, an one-dimensional system, or in the graphene, a planar crystalline system, is circa 1000 times lower than c [6]. In principle, one could detect an analogue of the Schwinger
18
C.R. Muniz, M.S. Cunha / Journal of Physics and Chemistry of Solids 82 (2015) 17–20
effect in such condensed matter systems, where the external electric field would generate electron/hole pairs in a way quite similar to the Landau–Zener effect in semiconductors. In this phenomenon, the presence of an external field induces the transition via quantum tunneling of an electron from the valence band to the conduction one [7]. Thus, in the graphene sheet, the rate of pair production per unit area per unit time is given by [8]
⎛ πE ⎞ (eE)3/2 d2N cr ⎟⎟. = exp ⎜⎜− dAdt E 4π 2= 3/2c 1/2 ⎝ ⎠
(5)
In the case of one-dimensional systems (polyacetylene, for instance), one gets the rate of pair production per length unit per time unit [18]
⎛ πEcr ⎞ d2N eE ⎟. = exp ⎜− dLdt 2π = ⎝ E ⎠
(6)
In Eqs. (5) and (6), the critical electric field Ecr is yet given by Eq. (3), with the change c → vF , and then we have Ecr ∼ 1011 V/m , a value that is not so far from our current technological possibilities. The abundance of pairs produced by this electric field would be about 1024 per meter per second in the molecular chain. The polyacetylene is a long molecular chain, a conjugated polymer that presents a behavior of semiconductor. When it is richly doped, presents the properties of a metal, but the transition to metallic state occurs in an unconventional way due to its lowdimensionality [9]. Its simpler form trans-polyacetylene has alternating single and double bonds which have lengths of 1.44 and 1.36 Å, respectively, which confer high stability to the polymer [10,11]. Th energy gap of the polyacetylene can be written as
Δ = 2me vF2,
(7)
which is equivalent to the minimal energy needed to create an electron/positron pair in the vacuum, if vF → c . Some values for the polyacetylene gap energy are found in [12]. Bloch oscillations are another interesting phenomenon associated with the incidence of electric fields on one-dimensional crystalline lattices. F. Bloch demonstrated in 1928 the existence of electronic oscillations within one-dimensional crystals when these latter are subjected to a static and spatially homogeneous electric field generated externally [19]. Such oscillations occur with a period inversely proportional to both the field magnitude and the distance between the atoms that form the polymer (lattice parameter). The Bloch oscillations have been tested in some laboratories after its theoretical elucidation, but they are very difficult to observe due to their short duration, of the order of 10 12 s, because thermal and acoustic disturbances, as well as impurities and defects in the crystals, tend to scatter the electrons within them, undoing the coherence of the oscillations. It is worth point out that the occurrence of the Bloch oscillations is not exclusive of 1-D systems. They may exist in 2-D lattices, with the concomitant transport of the electronic wave packet, as it was theoretically shown in [14]. The observation of those two effects combined, namely Landau–Zener and Bloch oscillations, in photonic 2-D lattices was reported in [21]. Bloch oscillations were also observed in semiconductor superlattices [22,23]. But it remains an unsolved question whether electronic Bloch oscillations can occur in a bulk crystal [24]. In this paper, we will analyze the behavior of the electrical conductivity of the undoped trans-polyacetylene molecular chain while a static and spatially homogeneous electric field is applied on the polymer. In this analysis, the production of electron/hole pairs via Schwinger–Landau–Zener (SLZ) effect will be studied and how the generated carriers affect the polyacetylene electrical conductivity will be investigated, considering the modification of
the gap width caused by the electric field action according to an ab initio model found in the literature. In our study, we will neglect the possibility of appearance in such molecular chain the bound electron/hole pairs (excitons) and their possibility of self-trapping [25]. We will also study how the Bloch oscillations induced by the external electric field are disturbed by the carriers produced via SLZ effect, investigating the possibility of the induction of Bloch oscillations with terahertz frequency or beyond within the undoped trans-polyacetylene and indicating a possible mechanism for their protection. The paper is organized as follows: in Section 2, we do the analysis of the polyacetylene conductivity under the action of the electric field in light of the SLZ effect. In Section 3, we study the effects of the SLZ dynamics on the Bloch oscillations within the polymer and in Section 4 we discuss the results.
2. SLZ Effect and the Polyacetylene Electric Conductivity In certain ab initio models, the gap width of the polyacetylene vanishes when one applies a static and spatially homogeneous electric field sufficiently strong [13,20]. According to these models, the gap varies quadratically with the applied field as (in SI units)
Δ = a − bE 2,
(8)
where a = 1.3 × 10−18 J and b = 2.5 × 10−38 J/V2. This expression must yield the gap value compatible with Eq. (7) when the electric field is turned off. On the other hand, the gap energy vanishes for a field magnitude equal to Edb ∼ 1010 V/m , meaning that at this point the state of dielectric breakdown (db) is reached, and the transition insulator–metal occurs without the need to dope it. Notice that the value of the field required for the dielectric breakdown is one order of magnitude below that necessary for generating abundant electron/hole pairs via SLZ effect. Let us now write just the electron creation rate, according to (3) and (5), as a function of the gap width (7) and integrated in the time, which yields
ne =
⎛ ⎛ 4π m 2 v 4 ⎞ eEt eEt πΔ2 ⎞ e F ⎟⎟, ⎟⎟ = exp ⎜⎜− exp ⎜⎜− 4π = ⎝ 4vF =eE ⎠ ⎝ 4vF =eE ⎠ 4π=
(9)
allowing to analyze the production of free electrons in terms of the gap variation with the applied field, according to Eq. (7). ne is the linear density of these carriers. We can express the one-dimensional electric current density as
jx = ene vF =
⎛ e2vF Et πΔ2 ⎞ ⎟⎟, exp ⎜⎜− 4π = 4 v ⎝ F =eE ⎠
(10)
where we remark the presence of the factor that determines the linear-response, Et, which does not lead in any field regime, even considering the variation of the gap width with the electric field, according to Eq. (7). This feature seems to be typical of lowdimensional systems, since the same occurs, for example, in the graphene, at least in the context of the SLZ effect [16]. The polyacetylene is an ohmic material, just presenting a nonohmic behavior when is highly doped, as, for example, with iodine [15]. First we are analysing here the non-doped polyacetylene, then we can find its classical electrical conductivity s, by dividing Eq. (10) by EvFt, arriving at
σ=
⎛ πΔ2 ⎞ e2 ⎟. exp ⎜− 4π = ⎝ 4vF =eE ⎠
(11)
If Δ has the form given in Eq. (8), then one can show that the conductivity, starting from zero, reaches a maximum value when
C.R. Muniz, M.S. Cunha / Journal of Physics and Chemistry of Solids 82 (2015) 17–20
E db
σ
=
1.0
19
⎛ πEcr ⎞ 4π = exp ⎜ ⎟. ⎝ E ⎠ eEt
(12)
It can be easily seen that the spacing decreases with time, since more and more electrons are being produced. It is assumed that the electron oscillating under the field effect only interacts with the produced electrons when it is very close to them. One knows that the Bloch amplitude AB of the electronic oscillation is
0.8
0.6
0.4
AB = 0.2
me vF2 Δ = , 2eE eE
(13)
and the Bloch frequency is given by
0.5
0.6
0.7
0.8
E
νB =
eEa , 2π =
(14)
Fig. 1. Polyacetylene electrical conductivity, in units of e2/4π= , as a function of the applied electric field, in units of 109 V/m.
where a is the lattice parameter [29]. We consider that the oscillating electrons will be scattered when 2A B is equal to, and the time for this to occur is, therefore
the gap width vanishes, i.e., when E = Edb = (a/b)1/2. From this point, the conductivity does no longer varies with the applied electric field, since that the gap energy cannot be negative, which is also compatible with studies of the ac-conductivity at low frequencies [17]. In Fig. 1, we depict s as a function of the applied field E. Notice that, initially, the variation of the electric conductivity is very low, after raising up to reach the plateau for the critical electric field equal to Edb = 7.2 × 109 V/m , correspondent to the transition insulator–metal (dielectric breakdown).
ts =
3. SLZ effect versus Bloch oscillations Currently, there are new technologies involving the growth of crystals free of impurities at low temperatures, and it is known that in nano-structured semiconductors (superlattices), Bloch oscillations with frequencies at terahertz scale have been reached [26,27]. We have seen that the undoped trans-polyacetylene, starting from a critical value for the applied electric field, behaves as an one-dimensional metal. At the same time, this field must induce Bloch oscillations within the crystal. The problem that can be seen by examining the generation of Bloch oscillations inside one-dimensional (or quasi-one-dimensional) condensed matter systems, for example, in polyacetylene or carbon nanoribbons [28], is that the SLZ mechanism also interferes in the process, since both phenomena are caused by an electrical field generated externally. As mentioned in the previous section, depending on the intensity of the applied field, this can generate a non-negligible production of electron/hole pairs, which certainly will interact with the oscillating electrons, scattering them and destroying their periodic motion, besides the perturbation due to thermal and crystal malformation effects. Establishing the theoretical conditions in which an effect is compromised or even nullified by the other is, therefore, of fundamental importance for the research related to the technological applications of the Bloch oscillations. In two-dimensional condensed matter systems (graphene, for instance), these two combined effects are studied in [16]. The proposal here is to estimate the survival time t of the Bloch oscillations in trans-polyacetylene, considering the simultaneous pair production via SLZ effect. Transient times will be disregarded. Taking the inverse of the density of the generated electrons calculated from Eq. (6), we have the average spacing between them
⎛ πEcr ⎞ 2π = ⎟⎟. exp ⎜⎜ me vF2 ⎝ E ⎠
(15) 2
We can verify that the Bloch oscillations last about 10 s when one applies an electric field E = Edb on the polyacetylene, and just 10 14 s if E = Ecr , for a ≈ 10−10 m and vF ≈ 106 m/s. One also notices from Eq. (13) that there is a theoretical lower limit for the duration of the oscillations, which is t s = tmin = 2π= / me2 vF ≈ 10−15 s, when E → ∞. However, if we consider the variation of the interband gap with the applied electric field as in the previous section, a numerical analysis of Eq. (13), in which the gap width appears in both the exponential argument and the factor that multiplies it, shows us that the survival time of the Bloch oscillations, which is theoretically infinity when E ¼0, falls off to a minimum at E ≃ 6.3 × 109 V/m , from which increases with the field, again tending to infinity when E → Edb .
4. Concluding remarks We have analyzed some electronic properties of the undoped trans-polyacetylene subject to the action of an external electric field, static and spatially homogeneous, at zero-temperature. Firstly, we studied the variation of the electric conductivity with the applied field, taking into account the SLZ effect and assuming that the gap is a quadratic function of the electric field, according to the ab initio model given in [13]. In this analysis, it became evident the transition insulator–metal for a critical value of the applied field, for which the dc-conductivity reaches a maximum and from there stays constant. We also showed that, in the fielddependent expression found for the classical electric current density of the undoped trans-polyacetylene, the linear-response term does not leader under any field regime, the same happening with the graphene [16]. We have also focused on the Bloch oscillations when they compete with the SLZ effect within the polyacetylene, by calculating the duration of the former effect while the oscillating electrons are not scattered by those ones generated from the latter one. We remark that Bloch oscillations with a frequency up to 10 times the Terahertz scale, which are interesting of a technological point of view, survive for a very long time. However, if one takes into account Bloch frequencies with 100 times the Terahertz scale, at the point in which the polymer turns into a metal, the survival time of the oscillations drops to fractions of second, probably meaning that the theoretical limit for the production of Bloch
20
C.R. Muniz, M.S. Cunha / Journal of Physics and Chemistry of Solids 82 (2015) 17–20
oscillations in undoped one-dimensional crystals will remain in that scale. On the other hand, if the gap width varies with the applied field according to Eq. (7), the survival time of the Bloch oscillations decreases from infinity, when E ¼0, to 5.1 × 1018 s, therefore a very large duration, for E = 6.3 × 109 V/m , again tending to infinity when E = Edb = 7.2 × 109 V/m . Thus, we draw the important conclusion that the variation of the gap width with the external field protects these oscillations against the continuous generation of electron/hole pairs through this same field, and all the frequencies are, in principle, allowed.
References [1] [2] [3] [4] [5] [6] [7]
J. Schwinger, Phys. Rev. 82 (1951) 664. T.D. Cohen, D.A. McGady, Phys. Rev. D 78 (2008) 036008. A.I. Nikshov, Nucl. Phys. B 21 (1970) 346. A. Ringwald, Phys. Lett. B 510 (2001) 107. D.B. Blaschke, et al., Eur. Phys. J D55 (2009) 341. P. Avouris, Z. Chen, V. Perebeinos, Nat. Nanotech. 2 (2007) 605. O. Morandi, L. Demeio, A Wigner-function approach to Landau–Zener transitions based on the multiband-envelope-function model, in: Annals of 20th International Conference on Transport Theory, Obninsk, Russia, 2007. [8] D. Allor, T.D. Cohen, Phys. Rev. D 78 (2008) 096009. [9] E.J. Mele, M.J. Rice, Phys. Rev. B23 (1981) 5397.
[10] R. Peierls, The Quantum Theory of Solids, Oxford University Press, Oxford, 1955. [11] W.P. Su, J.R. Schrieffer, A.J. Heegr, Phys. Rev. Lett. 42 (25) (1979) 1698. [12] K. Roiciszewski, B. Oles, J. Phys.: Condens. Matter 5 (39) (1993) 7289. [13] Y. Yuanfeng, M. Zhanga, J. Zhao, J. Mol. Str. 822 (2007) 1. [14] W. Liu, Y. Li, Z. Sun, D.S. Liu, S.J. Xie, Phys. Lett. A372 (2008) 4315. [15] H. Chaabane, E. Ettlinger, W. Schoepe, Z. Phys. B: Condens. Matter 64 (1983) 19. [16] B. Dora, R. Moessner, Phys. Rev. B 81 (2010) 165431. [17] P.M. Grant, M. Krounbi, Solid State Commun. 36 (1980) 291. [18] T.D. Cohen, D.A. McGady, Phys. Rev. D 78 (2008) 036008. [19] F. Bloch, Z. Phys. 52 (1928) 555. [20] J.M. Zhang, W.M. Liu, Phys. Rev. A 82 (2010) 025602. [21] H. Trompeter, et al., Phys. Rev. Lett. 96 (2006) 053903. [22] K. Leo, P.H. Bolivar, F. Brüggemann, R. Schwedler, Solid State Commun. 84 (10) (1992) 943. [23] H. Kurz, H.G. Roskos, T. Dekorsy, K. Koehler, Phil. Trans. R. Soc. Lond. A 354 (1996) 2295. [24] K.F. Renk, Basics of Laser Physics—For Students of Science and Engineering, Springer-Verlag, Berlin, Heidelberg, 2012. [25] V.M. Agranovich, G. Czajkowski, Excitations in Organic Solids, Oxford University Press, Oxford, 2009. [26] Dekorski, et al., Semicond. Sci. Technol. 9 (1994) 1959. [27] T. Hyart, N.V. Alexeeva, J. Mattas, K.N. Alekseev, Phys. Rev. Lett. 102 (2009) 140405. [28] C.R. Muniz, R.N. Costa Filho, Solid State Commun. 150 (2010) 1457–1459. [29] M.B. Dahan, E. Peik, J. Reichel, Y. Castin, C. Salomon, Phys. Rev. Lett. 76 (24) (1996) 4508.
Contents lists available at ScienceDirect
Journal of Physics and Chemistry of Solids journal homepage: www.elsevier.com/locate/jpcs
Effects of strong electric fields in a polyacetylene chain C.R. Muniz a,n, M.S. Cunha b a b
Grupo de Física Teórica (GFT), Universidade Estadual do Ceará, FECLI, Iguatu, Ceará, Brazil Grupo de Física Teórica (GFT), Universidade Estadual do Ceará, CCT, Fortaleza, Ceará, Brazil
art ic l e i nf o
a b s t r a c t
Article history: Received 26 October 2014 Received in revised form 1 February 2015 Accepted 25 February 2015 Available online 26 February 2015
In this work, we study the effects related to the creation of electron/hole pairs via application of an external electric field that acts on a pristine trans-polyacetylene molecular chain at zero-temperature. This phenomenon is termed Schwinger–Landau–Zener (SLZ) effect and arises when a physical system, which can even be the vacuum, is under the action of a strong, static and spatially homogeneous electric field. Initially, we investigate how the electrical conductivity of the polyacetylene changes with the applied field, by considering the carriers production as well as the variation of the interband gap according to certain ab initio models. Next, we analyse the competition between the SLZ effect and another one associated with the incidence of an uniform electric field on one-dimensional crystals – the Bloch oscillations. We evaluate the conditions in which these latter can be destroyed by the particles created through the same field that induces them, and verify the possibility of occurrence of the Bloch oscillations inside the trans-polyacetylene with frequencies equal to or higher than the terahertz scale. & 2015 Elsevier Ltd. All rights reserved.
Keywords: A. Polymers D. Electronic transport D. Electronic band structure
1. Introduction In the early 1950s of the last century, J. Schwinger showed that an intense, static and spatially homogeneous electric field can disrupt the quantum vacuum in such a way to cause the generation of electron/positron pairs [1]. He used the theoretical framework of quantum electrodynamics in order to find the non-perturbative exact amplitude of transition vacuum-to-vacuum, or vacuum persistence, for that physical process. At an instant t and within certain volume V, the correspondent probability is
Pvac = |〈0|U (t) |0〉|2 = exp( − ωVt),
(1)
where U(t) is the time evolution operator and
ω=
(eE)2 4π 3= 2c
∞
∑ n= 1
⎛ nπ E ⎞ 1 cr ⎟. exp ⎜⎜− E ⎟⎠ n2 ⎝
(2)
Here, e is the electronic charge and E is the magnitude of the applied electric field, and
Ecr =
me2 c 3 , =e
(3)
is the electric field needed to create abundant electron/positron pairs in the vacuum, as we will see below. This critical field can be n
Corresponding author. E-mail addresses: [email protected] (C.R. Muniz), [email protected] (M.S. Cunha). http://dx.doi.org/10.1016/j.jpcs.2015.02.009 0022-3697/& 2015 Elsevier Ltd. All rights reserved.
thought of as that one required for realize the work to separate a virtual pair of particle/antiparticle of electronic mass me by a distance equal to its Compton wavelength. It is also possible to calculate the production of pairs per unit time per unit volume, which is given by [2,3]
⎛ πE ⎞ (eE)2 d2N cr ⎟⎟. exp ⎜⎜− = 3 2 dVdt 4π = c ⎝ E ⎠
(4)
This expression is the first term of the sum in Eq. (2). Notice that an applied electric field greater than or equal to the critical one neutralizes the exponential in Eq. (4), with the pair production rate growing approximately as a function of the square of the electric field when E is very high. The value of the critical electric field is Ecr ≈ 1018 V/m and exceeds quite the current technological capability of generation of these fields. The most powerful electric field produced in our laboratories oscillates in certain laser beams, reaching 108 V/m [4], eliminating any hope of presently detecting the Schwinger effect, at least in the vacuum. It is worth notice in here that the Schwinger effect may also occur through action of variable electric fields [5]. However, in the context of condensed matter physics, in semiconductors of low-dimensionality at zero-temperature, the limit velocity of the carriers is much lower than one of the light in vacuum. For example, the Fermi velocity (vF) in the molecular polyacetylene, an one-dimensional system, or in the graphene, a planar crystalline system, is circa 1000 times lower than c [6]. In principle, one could detect an analogue of the Schwinger
18
C.R. Muniz, M.S. Cunha / Journal of Physics and Chemistry of Solids 82 (2015) 17–20
effect in such condensed matter systems, where the external electric field would generate electron/hole pairs in a way quite similar to the Landau–Zener effect in semiconductors. In this phenomenon, the presence of an external field induces the transition via quantum tunneling of an electron from the valence band to the conduction one [7]. Thus, in the graphene sheet, the rate of pair production per unit area per unit time is given by [8]
⎛ πE ⎞ (eE)3/2 d2N cr ⎟⎟. = exp ⎜⎜− dAdt E 4π 2= 3/2c 1/2 ⎝ ⎠
(5)
In the case of one-dimensional systems (polyacetylene, for instance), one gets the rate of pair production per length unit per time unit [18]
⎛ πEcr ⎞ d2N eE ⎟. = exp ⎜− dLdt 2π = ⎝ E ⎠
(6)
In Eqs. (5) and (6), the critical electric field Ecr is yet given by Eq. (3), with the change c → vF , and then we have Ecr ∼ 1011 V/m , a value that is not so far from our current technological possibilities. The abundance of pairs produced by this electric field would be about 1024 per meter per second in the molecular chain. The polyacetylene is a long molecular chain, a conjugated polymer that presents a behavior of semiconductor. When it is richly doped, presents the properties of a metal, but the transition to metallic state occurs in an unconventional way due to its lowdimensionality [9]. Its simpler form trans-polyacetylene has alternating single and double bonds which have lengths of 1.44 and 1.36 Å, respectively, which confer high stability to the polymer [10,11]. Th energy gap of the polyacetylene can be written as
Δ = 2me vF2,
(7)
which is equivalent to the minimal energy needed to create an electron/positron pair in the vacuum, if vF → c . Some values for the polyacetylene gap energy are found in [12]. Bloch oscillations are another interesting phenomenon associated with the incidence of electric fields on one-dimensional crystalline lattices. F. Bloch demonstrated in 1928 the existence of electronic oscillations within one-dimensional crystals when these latter are subjected to a static and spatially homogeneous electric field generated externally [19]. Such oscillations occur with a period inversely proportional to both the field magnitude and the distance between the atoms that form the polymer (lattice parameter). The Bloch oscillations have been tested in some laboratories after its theoretical elucidation, but they are very difficult to observe due to their short duration, of the order of 10 12 s, because thermal and acoustic disturbances, as well as impurities and defects in the crystals, tend to scatter the electrons within them, undoing the coherence of the oscillations. It is worth point out that the occurrence of the Bloch oscillations is not exclusive of 1-D systems. They may exist in 2-D lattices, with the concomitant transport of the electronic wave packet, as it was theoretically shown in [14]. The observation of those two effects combined, namely Landau–Zener and Bloch oscillations, in photonic 2-D lattices was reported in [21]. Bloch oscillations were also observed in semiconductor superlattices [22,23]. But it remains an unsolved question whether electronic Bloch oscillations can occur in a bulk crystal [24]. In this paper, we will analyze the behavior of the electrical conductivity of the undoped trans-polyacetylene molecular chain while a static and spatially homogeneous electric field is applied on the polymer. In this analysis, the production of electron/hole pairs via Schwinger–Landau–Zener (SLZ) effect will be studied and how the generated carriers affect the polyacetylene electrical conductivity will be investigated, considering the modification of
the gap width caused by the electric field action according to an ab initio model found in the literature. In our study, we will neglect the possibility of appearance in such molecular chain the bound electron/hole pairs (excitons) and their possibility of self-trapping [25]. We will also study how the Bloch oscillations induced by the external electric field are disturbed by the carriers produced via SLZ effect, investigating the possibility of the induction of Bloch oscillations with terahertz frequency or beyond within the undoped trans-polyacetylene and indicating a possible mechanism for their protection. The paper is organized as follows: in Section 2, we do the analysis of the polyacetylene conductivity under the action of the electric field in light of the SLZ effect. In Section 3, we study the effects of the SLZ dynamics on the Bloch oscillations within the polymer and in Section 4 we discuss the results.
2. SLZ Effect and the Polyacetylene Electric Conductivity In certain ab initio models, the gap width of the polyacetylene vanishes when one applies a static and spatially homogeneous electric field sufficiently strong [13,20]. According to these models, the gap varies quadratically with the applied field as (in SI units)
Δ = a − bE 2,
(8)
where a = 1.3 × 10−18 J and b = 2.5 × 10−38 J/V2. This expression must yield the gap value compatible with Eq. (7) when the electric field is turned off. On the other hand, the gap energy vanishes for a field magnitude equal to Edb ∼ 1010 V/m , meaning that at this point the state of dielectric breakdown (db) is reached, and the transition insulator–metal occurs without the need to dope it. Notice that the value of the field required for the dielectric breakdown is one order of magnitude below that necessary for generating abundant electron/hole pairs via SLZ effect. Let us now write just the electron creation rate, according to (3) and (5), as a function of the gap width (7) and integrated in the time, which yields
ne =
⎛ ⎛ 4π m 2 v 4 ⎞ eEt eEt πΔ2 ⎞ e F ⎟⎟, ⎟⎟ = exp ⎜⎜− exp ⎜⎜− 4π = ⎝ 4vF =eE ⎠ ⎝ 4vF =eE ⎠ 4π=
(9)
allowing to analyze the production of free electrons in terms of the gap variation with the applied field, according to Eq. (7). ne is the linear density of these carriers. We can express the one-dimensional electric current density as
jx = ene vF =
⎛ e2vF Et πΔ2 ⎞ ⎟⎟, exp ⎜⎜− 4π = 4 v ⎝ F =eE ⎠
(10)
where we remark the presence of the factor that determines the linear-response, Et, which does not lead in any field regime, even considering the variation of the gap width with the electric field, according to Eq. (7). This feature seems to be typical of lowdimensional systems, since the same occurs, for example, in the graphene, at least in the context of the SLZ effect [16]. The polyacetylene is an ohmic material, just presenting a nonohmic behavior when is highly doped, as, for example, with iodine [15]. First we are analysing here the non-doped polyacetylene, then we can find its classical electrical conductivity s, by dividing Eq. (10) by EvFt, arriving at
σ=
⎛ πΔ2 ⎞ e2 ⎟. exp ⎜− 4π = ⎝ 4vF =eE ⎠
(11)
If Δ has the form given in Eq. (8), then one can show that the conductivity, starting from zero, reaches a maximum value when
C.R. Muniz, M.S. Cunha / Journal of Physics and Chemistry of Solids 82 (2015) 17–20
E db
σ
1.0
19
⎛ πEcr ⎞ 4π = exp ⎜ ⎟. ⎝ E ⎠ eEt
(12)
It can be easily seen that the spacing decreases with time, since more and more electrons are being produced. It is assumed that the electron oscillating under the field effect only interacts with the produced electrons when it is very close to them. One knows that the Bloch amplitude AB of the electronic oscillation is
0.8
0.6
0.4
AB = 0.2
me vF2 Δ = , 2eE eE
(13)
and the Bloch frequency is given by
0.5
0.6
0.7
0.8
E
νB =
eEa , 2π =
(14)
Fig. 1. Polyacetylene electrical conductivity, in units of e2/4π= , as a function of the applied electric field, in units of 109 V/m.
where a is the lattice parameter [29]. We consider that the oscillating electrons will be scattered when 2A B is equal to
the gap width vanishes, i.e., when E = Edb = (a/b)1/2. From this point, the conductivity does no longer varies with the applied electric field, since that the gap energy cannot be negative, which is also compatible with studies of the ac-conductivity at low frequencies [17]. In Fig. 1, we depict s as a function of the applied field E. Notice that, initially, the variation of the electric conductivity is very low, after raising up to reach the plateau for the critical electric field equal to Edb = 7.2 × 109 V/m , correspondent to the transition insulator–metal (dielectric breakdown).
ts =
3. SLZ effect versus Bloch oscillations Currently, there are new technologies involving the growth of crystals free of impurities at low temperatures, and it is known that in nano-structured semiconductors (superlattices), Bloch oscillations with frequencies at terahertz scale have been reached [26,27]. We have seen that the undoped trans-polyacetylene, starting from a critical value for the applied electric field, behaves as an one-dimensional metal. At the same time, this field must induce Bloch oscillations within the crystal. The problem that can be seen by examining the generation of Bloch oscillations inside one-dimensional (or quasi-one-dimensional) condensed matter systems, for example, in polyacetylene or carbon nanoribbons [28], is that the SLZ mechanism also interferes in the process, since both phenomena are caused by an electrical field generated externally. As mentioned in the previous section, depending on the intensity of the applied field, this can generate a non-negligible production of electron/hole pairs, which certainly will interact with the oscillating electrons, scattering them and destroying their periodic motion, besides the perturbation due to thermal and crystal malformation effects. Establishing the theoretical conditions in which an effect is compromised or even nullified by the other is, therefore, of fundamental importance for the research related to the technological applications of the Bloch oscillations. In two-dimensional condensed matter systems (graphene, for instance), these two combined effects are studied in [16]. The proposal here is to estimate the survival time t of the Bloch oscillations in trans-polyacetylene, considering the simultaneous pair production via SLZ effect. Transient times will be disregarded. Taking the inverse of the density of the generated electrons calculated from Eq. (6), we have the average spacing
⎛ πEcr ⎞ 2π = ⎟⎟. exp ⎜⎜ me vF2 ⎝ E ⎠
(15) 2
We can verify that the Bloch oscillations last about 10 s when one applies an electric field E = Edb on the polyacetylene, and just 10 14 s if E = Ecr , for a ≈ 10−10 m and vF ≈ 106 m/s. One also notices from Eq. (13) that there is a theoretical lower limit for the duration of the oscillations, which is t s = tmin = 2π= / me2 vF ≈ 10−15 s, when E → ∞. However, if we consider the variation of the interband gap with the applied electric field as in the previous section, a numerical analysis of Eq. (13), in which the gap width appears in both the exponential argument and the factor that multiplies it, shows us that the survival time of the Bloch oscillations, which is theoretically infinity when E ¼0, falls off to a minimum at E ≃ 6.3 × 109 V/m , from which increases with the field, again tending to infinity when E → Edb .
4. Concluding remarks We have analyzed some electronic properties of the undoped trans-polyacetylene subject to the action of an external electric field, static and spatially homogeneous, at zero-temperature. Firstly, we studied the variation of the electric conductivity with the applied field, taking into account the SLZ effect and assuming that the gap is a quadratic function of the electric field, according to the ab initio model given in [13]. In this analysis, it became evident the transition insulator–metal for a critical value of the applied field, for which the dc-conductivity reaches a maximum and from there stays constant. We also showed that, in the fielddependent expression found for the classical electric current density of the undoped trans-polyacetylene, the linear-response term does not leader under any field regime, the same happening with the graphene [16]. We have also focused on the Bloch oscillations when they compete with the SLZ effect within the polyacetylene, by calculating the duration of the former effect while the oscillating electrons are not scattered by those ones generated from the latter one. We remark that Bloch oscillations with a frequency up to 10 times the Terahertz scale, which are interesting of a technological point of view, survive for a very long time. However, if one takes into account Bloch frequencies with 100 times the Terahertz scale, at the point in which the polymer turns into a metal, the survival time of the oscillations drops to fractions of second, probably meaning that the theoretical limit for the production of Bloch
20
C.R. Muniz, M.S. Cunha / Journal of Physics and Chemistry of Solids 82 (2015) 17–20
oscillations in undoped one-dimensional crystals will remain in that scale. On the other hand, if the gap width varies with the applied field according to Eq. (7), the survival time of the Bloch oscillations decreases from infinity, when E ¼0, to 5.1 × 1018 s, therefore a very large duration, for E = 6.3 × 109 V/m , again tending to infinity when E = Edb = 7.2 × 109 V/m . Thus, we draw the important conclusion that the variation of the gap width with the external field protects these oscillations against the continuous generation of electron/hole pairs through this same field, and all the frequencies are, in principle, allowed.
References [1] [2] [3] [4] [5] [6] [7]
J. Schwinger, Phys. Rev. 82 (1951) 664. T.D. Cohen, D.A. McGady, Phys. Rev. D 78 (2008) 036008. A.I. Nikshov, Nucl. Phys. B 21 (1970) 346. A. Ringwald, Phys. Lett. B 510 (2001) 107. D.B. Blaschke, et al., Eur. Phys. J D55 (2009) 341. P. Avouris, Z. Chen, V. Perebeinos, Nat. Nanotech. 2 (2007) 605. O. Morandi, L. Demeio, A Wigner-function approach to Landau–Zener transitions based on the multiband-envelope-function model, in: Annals of 20th International Conference on Transport Theory, Obninsk, Russia, 2007. [8] D. Allor, T.D. Cohen, Phys. Rev. D 78 (2008) 096009. [9] E.J. Mele, M.J. Rice, Phys. Rev. B23 (1981) 5397.
[10] R. Peierls, The Quantum Theory of Solids, Oxford University Press, Oxford, 1955. [11] W.P. Su, J.R. Schrieffer, A.J. Heegr, Phys. Rev. Lett. 42 (25) (1979) 1698. [12] K. Roiciszewski, B. Oles, J. Phys.: Condens. Matter 5 (39) (1993) 7289. [13] Y. Yuanfeng, M. Zhanga, J. Zhao, J. Mol. Str. 822 (2007) 1. [14] W. Liu, Y. Li, Z. Sun, D.S. Liu, S.J. Xie, Phys. Lett. A372 (2008) 4315. [15] H. Chaabane, E. Ettlinger, W. Schoepe, Z. Phys. B: Condens. Matter 64 (1983) 19. [16] B. Dora, R. Moessner, Phys. Rev. B 81 (2010) 165431. [17] P.M. Grant, M. Krounbi, Solid State Commun. 36 (1980) 291. [18] T.D. Cohen, D.A. McGady, Phys. Rev. D 78 (2008) 036008. [19] F. Bloch, Z. Phys. 52 (1928) 555. [20] J.M. Zhang, W.M. Liu, Phys. Rev. A 82 (2010) 025602. [21] H. Trompeter, et al., Phys. Rev. Lett. 96 (2006) 053903. [22] K. Leo, P.H. Bolivar, F. Brüggemann, R. Schwedler, Solid State Commun. 84 (10) (1992) 943. [23] H. Kurz, H.G. Roskos, T. Dekorsy, K. Koehler, Phil. Trans. R. Soc. Lond. A 354 (1996) 2295. [24] K.F. Renk, Basics of Laser Physics—For Students of Science and Engineering, Springer-Verlag, Berlin, Heidelberg, 2012. [25] V.M. Agranovich, G. Czajkowski, Excitations in Organic Solids, Oxford University Press, Oxford, 2009. [26] Dekorski, et al., Semicond. Sci. Technol. 9 (1994) 1959. [27] T. Hyart, N.V. Alexeeva, J. Mattas, K.N. Alekseev, Phys. Rev. Lett. 102 (2009) 140405. [28] C.R. Muniz, R.N. Costa Filho, Solid State Commun. 150 (2010) 1457–1459. [29] M.B. Dahan, E. Peik, J. Reichel, Y. Castin, C. Salomon, Phys. Rev. Lett. 76 (24) (1996) 4508.