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Electromagnetically induced transparency and bound in continuum states in double Aharonov-Bohm coupled rings
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ScienceDirect Materials Today: Proceedings 13 (2019) 1055–1061
www.materialstoday.com/proceedings
ICMES 2018
Electromagnetically induced transparency and bound in continuum states in double Aharonov-Bohm coupled rings T. Mrabtia*, Z. Labdoutib, E. H. El Boudoutib, F. Fethib, O. El Aboutib, and B. Djafari-Rouhanic a
LSTA, Département de Physique, Faculté Polydisciplinaire, Université Abdelmalek Essaadi, Larache, Morocco. LPMR, Département de Physique, Faculté des Sciences, Université Mohamed Premier, 60000 Oujda, Morocco. c IEMN, UMR CNRS 8520, Université de Lille, 59655 Villeneuve d’Ascq, France.
b
Abstract We investigated electromagnetic induced transparency (EIT) resonances and bound in continuum (BIC) states in a simple mesoscopic structure made of double Aharonov-Bohm rings of lengths d1 and d2 threaded by a magnetic flux and coupled by a wire of length d0. These investigations are accomplished through an analysis of the amplitude of the transmission coefficient obtained using the Green’s function method. This structure may exhibit a large band gap and an EIT resonance in the transmission spectrum without introducing any impurity in one arm of the loop. In addition, we show that the shape of the EIT resonances and the width of the band gaps can be tuned by means of the magnetic flux and the geometry of the structure. These results may have important applications for electronic transport in mesoscopic systems. © 2019 Elsevier Ltd. All rights reserved. Peer-review under responsibility of the scientific committee of the International Conference on Materials and Environmental Science, ICMES 2018. Keywords: Green’s function; Transmission spectrum; EIT resonance; Aharonove-Bohm flux; Bound in continuum states.
1. Introduction Electromagnetic induced transparency (EIT) is the phenomenon whereby a sharp transparent window associated with steep dispersion is induced into opaque atomic media [1, 2]. This effect was first observed in Strontium vapor
* Corresponding author. Tel.: +212-636032786; fax: +212-539523961. E-mail address: [email protected] 2214-7853 © 2019 Elsevier Ltd. All rights reserved. Peer-review under responsibility of the scientific committee of the International Conference on Materials and Environmental Science, ICMES 2018.
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by Boller et al. [3]. This phenomenon has received much attention due to its interesting physics and potential applications such as slow light effect [4-6], enhanced nonlinear effect and optical information storage [7, 8]. Recently different studies have demonstrated that the analog of EIT can be also realized in classical system due to the similar interference effects. Different systems have been proposed for this purpose such as: photonic crystal waveguides coupled to cavities [9-11], coupled-microresonator systems [12-14], plasmonic nanostructures and metamaterials [15-17], acoustic waveguides [18-20] and multilayers [21] and photonic circuits [22, 23]. In the low dimensional systems, a rich variety of mesoscopic systems [24] have been proposed in the literature to understand transport phenomena through quantum rings, wires, or dots. These nano-devices have shown several nontrivial effects, such as Aharonov–Bohm conductance oscillations, persistent currents and quantum Hall effect [25]. The A-B effect is one of the most intriguing quantum mechanical phenomenon in which the phase of a charged particle is affected by the vector potential of an electromagnetic field giving rise to charge–particle interference phenomena [26, 27]. Different geometrical mesoscopic devices have been explored in the literature to show essentially another type of resonances called Fano resonances [28, 29]. These latter resonances are characterized by a peak followed by a dip in the transmission spectra as a consequence of a destructive interference of the waves, giving rise to an asymmetric line shape. These resonances have been the subject of intensive study from the theoretical and experimental point of view in single [30-33] and double [34-37] rings placed between two leads. In a recent publication [38], we have treated the possibility of existence of Fano resonances in a double ring structure made by two identical loops separated by a segment. The evolution of the Fano resonance shape and the width of the band gaps with the magnetic field were analyzed in detail. In this communication, we are interested by EIT resonances in such systems. In order to obtain these resonances, we have to choose two loops with different lengths (Fig. 1). The magnetic flux threading the loops can be either similar or different. This paper is organized as follows: in section 2 we give the theoretical model based on the Green’s function formalism and the transmission coefficients through a double ring structure submitted to similar or different magnetic flux. Section 3 gives a summary of the main results of this work and section 4 is devoted to the conclusion. d1
Φ1 d1
d2 d0
Φ2 d2
Fig. 1. Schematic illustration of the Aharonov-Bohm structure made of two rings of lengths 2d1 and 2d2 coupled by a wire of length d0. The whole structure is inserted between two semi-infinite leads.
2. Model and formalism : Transmission coefficient The mesoscopic structure described in Fig. 1 is composed of two mesoscopic rings, each composed of wires of lengths 2d1 and 2d2 in the presence of an Aharonov–Bohm (A–B) flux. These rings are coupled by a wire of length d0; the whole structure is inserted between two semi-infinite leads. The expression giving the inverse of the Green’s function g-1(MM) of the whole system (Fig. 1) can be obtained from the juxtaposition of the inverse Green’s functions of the different constituents. For more detail about the method of calculation of the Green’s functions from the Schrödinger equation, we refer the reader to Ref. [38]. Note that the magnetic field is applied only inside the rings. Within the interface space M{0, d1, d1+d0, d1+d0+d2}, the inverse of the Green’s function of the whole system (Fig.1) is given by [39].
T. Mrabti et al / Materials Today: Proceedings 13 (2019) 1055–1061
2C1 j S1 2C '1 S 1 g 1 ( MM ) F 0 0
2C '1 S1 2C1 C0 S1 S0 1 S0 0
0 1 S0
2C 2 C0 S2 S0 2C '2 S2
0 2C '2 S2 2C 2 j S2
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0
(1)
where, Si = sin(kdi), Ci = cos(kdi) (i = 0, 1, 2), F = (ħ²/2m)k, where ħ, k and m refer respectively to the reduced Planck constant, the wave vector of the constituting medium and the effective mass of the electron. C’i = cos(fi) and fi i 0 (i = 1, 2) is the ratio of the flux i to the quantum flux 0.0 = hc/e is the quantum flux associated with a single charge of the electron e and c the speed of light in vacuum. The transmission coefficient through the structure given in Fig. 1 is defined by [39] t = 2jFg(0, d1+d0+d2) where g(0, d1+d0+d2) is obtained by inverting the matrix given in Eq. (1), namely
t
8C '1 C '2 S1S2 1 j 2
(2)
where
1 ( S1S 2 4C1C2 ) 8C2C '12 (2C2 S0 C0 S 2 ) 8C1C '22 (2S0C1 C0 S1 ) 16S0C '12 C '22 and with
2 2( S1C2 C1S 2 ) 4S1C '22 (2 S0C1 C0 S1 ) 4 S 2C '12 (2C2 S0 C0 S 2 )
4S0C1C2 S1S2 S0 2C0 (C1S2 S1C2 ) .
(3) (4) (5)
From the expression of t (Eq. (2)), one can deduce the transmission rate
T
64C '12 C '12 S12 S 22 . 12 22
(6)
3. Results and discussion 3.1. BIC states, Transmission gaps and EIT resonance Figure 2(b) gives an example of the transmission spectrum versus the dimensionless wave vector kd0for two loops with specific lengths slightly detuned: d1 = 1/3+0.025 and d2 = 1/3-0.025. All the lengths are given in units of d0. The magnetic flux inside the ring is chosen such that: i.e., f1 = f2 = 0.2). One can notice the existence of two transmission zeros indicated by solid circles on the abscissa of Fig. 2(b) and a full transmission (transparence window) around kd0 This is a characteristic of an EIT resonance. Now, if the lengths of the two loops are chosen identical such that d1 = d2 = d = 1/3, then the EIT resonance collapses giving rise to the socalled trapped state or BIC state (Fig. 2 (a)). The position of this state is indicated by a solid circle on the abscissa of Fig. 2(a) at kd0/2
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Fig. 2. (a) Transmission coefficient versus the dimensionless wave vector kd0/2π for identical rings (d1=d2=d and f1=f2=0.2). (b) The same as in (a) but for the lengths dand d2=1/3-
3.2. Effect of the magnetic field on transmission gaps and EIT resonances 3.2.1. Case of two loops with similar fluxes In this section, we studied the effect of the magnetic flux on the transmission gaps and EIT resonances for the same geometrical structure as in Figs. 2(a) and (b). Figure 3(a) shows the effect of the magnetic flux on the width of the gaps for the same geometry as in Fig 2(a). One can notice that the width of the gap increases when the magnitude of the magnetic flux increases and transforms into a large band gap. In addition, without the magnetic flux (i.e., f1=f2 = 0) the gap disappears (continuous curve). These results clearly show the possibility to tune the band gaps by means of the magnetic flux. Also, just two loops are sufficient to have large gaps (see dotted curves for f1=f2=0.3) without repeating the structure periodically as it is usually the case [39]. In Fig. 3(b) we have plotted the effect of the magnetic flux on the shape of the EIT resonance. We can see that the width of the EIT resonance increases when the magnetic flux increases, its shape remains almost symmetrical and reaches unity (i,e., T=1) at kd0The resonance disappears in the absence of the magnetic flux (continuous curve). Also, the EIT resonance remains squeezed between the same transmission zeros (indicated by solid circles on the abscissa of Fig. 3(b)) induced by the two different rings.
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Fig. 3. (a) The same as in figure 2 (a) but for f1 = f2 taken between 0 and 0.3.). (b) The same as in figure 2(b) but for f1 = f2 taken between 0 and 0.3.
3.2.2. Case where the field f1 is fixed and f2 is variable In this section, we consider the magnetic field f1 fixed while f2 is variable. Figure 4(a) shows the transmission spectrum as a function of the dimensionless wave vector kd0/2for the same geometries as in Figs. 2(a) and 3(a). We suppose that the magnetic field f1 is fixed in the first ring (f1=0.2) whereas the magnetic field f2 is variable in the second ring. One can see that the gap becomes larger when f2 increases followed by a reduction in the transmission spectrum in the allowed band at the vicinities of the gap. On other hand, the effect of the magnetic field on EIT resonances is illustrated in Fig. 4(b) for the same geometry as in Fig. 2(b). One can see that the magnitude of magnetic flux f2 affects considerably the shape of the EIT resonance. In addition, it is remarkable that the EIT resonance undergoes a small shift to higher wave vectors accompanied by an asymmetric shape and an increase in its width when f2 increases. Indeed, for f2 < f1 = 0.2, we can notice a shift of the EIT resonance to the right side of kd0=3 its width increases and its shape becomes asymmetrical when the magnetic flux decreases. For f2 > f1 = 0.2, the EIT resonance moves to the left of kd0=3, its width decreases and its shape becomes symmetrical. Also, in both cases the amplitude of the resonance decreases from unity. 3.2.3. Case of two loops with different fluxes In this section, we consider the magnetic field f1 and f2 are variable. Figure 5 (a) shows the evolution of the transmission under the effect of the magnetic fields for the same geometries as in Fig 2(a), one can notice that the transmission spectrum exhibits a deformation accompanied with a decrease in its amplitude at the vicinity of the gap. In Fig. 5(b), we have presented the effect of the magnetic flux on the EIT resonance for the same geometry as in Fig. 2(b). One can notice that starting from the case f1=f2 = 0.2, when f1 decreases and f2 increases the EIT resonance shifts to the left side of kd0=3 its shape becomes asymmetrical and its amplitude decreases. The same results are obtained in the opposite (i.e., when f1 increases and f2 decreases) case but the resonances shift to the right side of kd0=3.
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Fig. 4. (a) Same as in Fig. 2 (a) but for f1 = 0.2 and f2 is variable. (b) Same as in Fig. 2 (b) but for f1 = 0.2 and f2 is variable.
Fig. 5. (a) Same as in Fig. 2 (a) but both f1 and f2 are variable. (b) Same as in Fig. 2 (b) but both f1 and f2 are variable.
T. Mrabti et al / Materials Today: Proceedings 13 (2019) 1055–1061
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4. Conclusion In this work, we have investigated the effect of the magnetic flux (Aharonov-Bohm effect) on the band gaps and EIT resonances in a simple mesoscopic structure made of two loops separated by a segment. This study is performed by analyzing the evolution of the transmission spectra. The analytical expression of the transmission coefficient is obtained by means of the Green’s function method. We have clearly demonstrated the existence of large band gaps in the transmission spectrum which are attributed to the Aharonov-Bohm effect; these band gaps depend considerably on the magnitude of the magnetic flux. Besides the transmission gaps, we have shown the possibility of existence of EIT resonances and BIC states inside the forbidden bands by tailoring the lengths of the arms constituting the rings. The evolution of the EIT resonance shape with the magnetic flux is analyzed in detail. These results may have important applications for electronic transport in mesoscopic systems. References [1] S. Harris, L. Hau, Phys. Rev. Lett. 82 (1999) 4611–4614. [2] A. Mouadili, E.H. El Boudouti, A. Soltani, A. Talbi, A. Akjouj, B. Djafari Rouhani, J. Appl. Phys. 113 (2013) 164101–10. [3] K.J. Boller, A. Imamoglu, S.E. Harris, Phys. Rev. Lett. 66 (1991) 2593–2596. [4] Y. Wang, Z. Li and F. Hu, J. Phys. D: Appl. Phys.51 (2018) 025103. [5] Y. Huang, C. Min, G. Veronis, App. Phys. Lett. 99 (2011) 143117–3. [6] M. Manjappa, S.Y. Chiam, L.Q. Cong, A.A. Bettiol, W.L. Zhang, R. Singh, Appl. Phys. Lett. 106 (2015) 181101–5. [7] C. Liu, Z. Dutton, C.H. Behroozi, L.V. Hau, Nature 409 (2001) 490–493. [8] M. Fleischhauer, A. Imamoglu, J.P. Marangos, Rev. Mod. Phys. 77 (2005) 633–673. [9] S. Fan, J.D. Joannopoulos, Phys. Rev. B 65 (2002) 235112–8. [10] X. Yang, M. Yu, D.L. Kwong, C.W. Wong, Phys. Rev. Lett. 102 (2009) 173902–4. [11] Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, S. Noda, Nat. Photon. 6 (2012) 56–61. [12] Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, M. Lipson, Phys. Rev. Lett. 96 ( 2006) 123901–4. [13] K. Totsuka, N. Kobayashi, M. Tomita, Phys. Rev. Lett. 21 (2007) 213904–4. [14] H.M. Li, S.B. Liu, S.Y. Liu, S.Y. Wang, G.W. Ding, H. Yang , Z.Y. Yu, H.F. Zhang, Appl. Phys. Lett. 106 (2015) 083511–4. [15] S. Zhang, D.A. Genov, Y. Wang, M. Liu, X. Zhang, Phys. Rev. Lett. 101(2008) 047401–4. [16] R. Taubert, M. Hentshel, J. Kastel, H. Giessen, Nano. Lett. 12 (2012) 1367–1371. [17] N. Papasimakis, V.A. Fedotov, N. Zheludev, S. Prosvirnin, Phys. Rev. Lett. 101 (2008) 253903–4. [18] E.H. El Boudouti, T. Mrabti, H. Al-Wahsh, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. Phys.: Condens. Matter 20 (2008) 255212–10. [19] A. Santillan, S.I. Bozhevolnyi, Phys. Rev. B 84 (2011) 064304–5. [20] A. Merkel, G. Theocharis, O. Richoux, V. Romero-Garcìa, V. Pagneux, Appl. Phys. Lett. 107 (2015) 244102–4. [21] I. Quotane, E.H. El Boudouti, B. Djafari-Rouhani, Phys. Rev. B 97 (2018) 024304–17. [22] A. Mouadili, E.H. El Boudouti, A. Soltani, A. Talbi, A. Akjouj, B. Djafari-Rouhani, J. Appl. Phys. 113 (2013) 164101–11. [23] A. Mouadili, E.H. El Boudouti, A. Soltani, A. Talbi, B. Djafari- Rouhani, A. Akjouj, K. Haddadi, Phys. Condens. Matter 26 (2014) 505901–12. [24] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, Cambridge, 1995. [25] Y. Imry, Introduction to mesoscopic physics, Oxford University Press, 1997. [26] Y. Aharonov, D. Bohm, Phys. Rev. 115 (1959) 485–491; R.G. Chambers, Phys. Rev. Lett. 5 (1960) 3–5. [27] M. Peshkin, A. Tonomura, Springer The Aharonov-Bohm Effect, Lecture Notes in Physics, 1989. [28] U. Fano, Phys. Rev. 124 (1961) 1866–1878. [29] A.E. Miroshnichenko, S. Flach, Y.S. Kivshar, Rev. Mod. Phys. 82 (2010) 2257–2298. [30] K. Kobayashi, H. Aikawa, S. Katsumoto, Y. Iye, Phys. Rev. Lett. 88 (2002) 256806–4. [31] J. Göres, D. Goldhaber-Gordon, S. Heemeyer, M.A. Kastner, H. Shtrikman, D. Mahalu, U. Meirav, Phys. Rev. B 62 (2000) 2188–2194. [32] M.L. Ladrón de Guevara, F. Claro, P.A. Orellana, Phys. Rev. B 67 (2003) 195335–6. [33] H. Al-Wahsh, E.H. El Boudouti, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski. Phys. Rev. B 75 (2007) 125313–11. [34] J. Yi, J.H. Wei, J. Hong, S.I. Lee, Phys. Rev. B 65 (2001) 033305–4. [35] C. Jiang, Y.S. Zheng, Solid State Commun. 212 (2015) 14–18. [36] S.K. Maiti, Phys. Lett. A 373 (2009) 4470–4474. [37] P. Dutta, S.K. Maiti, S.N. Karmakar, Solid State Commun. 150 (2010) 1056–1061. [38] T. Mrabti, Z. Labdouti, O. El Abouti, E.H. El Boudouti, F. Fethi , B. Djafari-Rouhani, Phys. Lett. A 382 (2018) 613–620. [39] J. Vasseur, A. Akjouj, L. Dobrzynski, B. Djafari-Rouhani, E.H. El Boudouti, Surf. Sci. Rep. 54 (2004) 1–156.
ScienceDirect Materials Today: Proceedings 13 (2019) 1055–1061
www.materialstoday.com/proceedings
ICMES 2018
Electromagnetically induced transparency and bound in continuum states in double Aharonov-Bohm coupled rings T. Mrabtia*, Z. Labdoutib, E. H. El Boudoutib, F. Fethib, O. El Aboutib, and B. Djafari-Rouhanic a
LSTA, Département de Physique, Faculté Polydisciplinaire, Université Abdelmalek Essaadi, Larache, Morocco. LPMR, Département de Physique, Faculté des Sciences, Université Mohamed Premier, 60000 Oujda, Morocco. c IEMN, UMR CNRS 8520, Université de Lille, 59655 Villeneuve d’Ascq, France.
b
Abstract We investigated electromagnetic induced transparency (EIT) resonances and bound in continuum (BIC) states in a simple mesoscopic structure made of double Aharonov-Bohm rings of lengths d1 and d2 threaded by a magnetic flux and coupled by a wire of length d0. These investigations are accomplished through an analysis of the amplitude of the transmission coefficient obtained using the Green’s function method. This structure may exhibit a large band gap and an EIT resonance in the transmission spectrum without introducing any impurity in one arm of the loop. In addition, we show that the shape of the EIT resonances and the width of the band gaps can be tuned by means of the magnetic flux and the geometry of the structure. These results may have important applications for electronic transport in mesoscopic systems. © 2019 Elsevier Ltd. All rights reserved. Peer-review under responsibility of the scientific committee of the International Conference on Materials and Environmental Science, ICMES 2018. Keywords: Green’s function; Transmission spectrum; EIT resonance; Aharonove-Bohm flux; Bound in continuum states.
1. Introduction Electromagnetic induced transparency (EIT) is the phenomenon whereby a sharp transparent window associated with steep dispersion is induced into opaque atomic media [1, 2]. This effect was first observed in Strontium vapor
* Corresponding author. Tel.: +212-636032786; fax: +212-539523961. E-mail address: [email protected] 2214-7853 © 2019 Elsevier Ltd. All rights reserved. Peer-review under responsibility of the scientific committee of the International Conference on Materials and Environmental Science, ICMES 2018.
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by Boller et al. [3]. This phenomenon has received much attention due to its interesting physics and potential applications such as slow light effect [4-6], enhanced nonlinear effect and optical information storage [7, 8]. Recently different studies have demonstrated that the analog of EIT can be also realized in classical system due to the similar interference effects. Different systems have been proposed for this purpose such as: photonic crystal waveguides coupled to cavities [9-11], coupled-microresonator systems [12-14], plasmonic nanostructures and metamaterials [15-17], acoustic waveguides [18-20] and multilayers [21] and photonic circuits [22, 23]. In the low dimensional systems, a rich variety of mesoscopic systems [24] have been proposed in the literature to understand transport phenomena through quantum rings, wires, or dots. These nano-devices have shown several nontrivial effects, such as Aharonov–Bohm conductance oscillations, persistent currents and quantum Hall effect [25]. The A-B effect is one of the most intriguing quantum mechanical phenomenon in which the phase of a charged particle is affected by the vector potential of an electromagnetic field giving rise to charge–particle interference phenomena [26, 27]. Different geometrical mesoscopic devices have been explored in the literature to show essentially another type of resonances called Fano resonances [28, 29]. These latter resonances are characterized by a peak followed by a dip in the transmission spectra as a consequence of a destructive interference of the waves, giving rise to an asymmetric line shape. These resonances have been the subject of intensive study from the theoretical and experimental point of view in single [30-33] and double [34-37] rings placed between two leads. In a recent publication [38], we have treated the possibility of existence of Fano resonances in a double ring structure made by two identical loops separated by a segment. The evolution of the Fano resonance shape and the width of the band gaps with the magnetic field were analyzed in detail. In this communication, we are interested by EIT resonances in such systems. In order to obtain these resonances, we have to choose two loops with different lengths (Fig. 1). The magnetic flux threading the loops can be either similar or different. This paper is organized as follows: in section 2 we give the theoretical model based on the Green’s function formalism and the transmission coefficients through a double ring structure submitted to similar or different magnetic flux. Section 3 gives a summary of the main results of this work and section 4 is devoted to the conclusion. d1
Φ1 d1
d2 d0
Φ2 d2
Fig. 1. Schematic illustration of the Aharonov-Bohm structure made of two rings of lengths 2d1 and 2d2 coupled by a wire of length d0. The whole structure is inserted between two semi-infinite leads.
2. Model and formalism : Transmission coefficient The mesoscopic structure described in Fig. 1 is composed of two mesoscopic rings, each composed of wires of lengths 2d1 and 2d2 in the presence of an Aharonov–Bohm (A–B) flux. These rings are coupled by a wire of length d0; the whole structure is inserted between two semi-infinite leads. The expression giving the inverse of the Green’s function g-1(MM) of the whole system (Fig. 1) can be obtained from the juxtaposition of the inverse Green’s functions of the different constituents. For more detail about the method of calculation of the Green’s functions from the Schrödinger equation, we refer the reader to Ref. [38]. Note that the magnetic field is applied only inside the rings. Within the interface space M{0, d1, d1+d0, d1+d0+d2}, the inverse of the Green’s function of the whole system (Fig.1) is given by [39].
T. Mrabti et al / Materials Today: Proceedings 13 (2019) 1055–1061
2C1 j S1 2C '1 S 1 g 1 ( MM ) F 0 0
2C '1 S1 2C1 C0 S1 S0 1 S0 0
0 1 S0
2C 2 C0 S2 S0 2C '2 S2
0 2C '2 S2 2C 2 j S2
1057
0
(1)
where, Si = sin(kdi), Ci = cos(kdi) (i = 0, 1, 2), F = (ħ²/2m)k, where ħ, k and m refer respectively to the reduced Planck constant, the wave vector of the constituting medium and the effective mass of the electron. C’i = cos(fi) and fi i 0 (i = 1, 2) is the ratio of the flux i to the quantum flux 0.0 = hc/e is the quantum flux associated with a single charge of the electron e and c the speed of light in vacuum. The transmission coefficient through the structure given in Fig. 1 is defined by [39] t = 2jFg(0, d1+d0+d2) where g(0, d1+d0+d2) is obtained by inverting the matrix given in Eq. (1), namely
t
8C '1 C '2 S1S2 1 j 2
(2)
where
1 ( S1S 2 4C1C2 ) 8C2C '12 (2C2 S0 C0 S 2 ) 8C1C '22 (2S0C1 C0 S1 ) 16S0C '12 C '22 and with
2 2( S1C2 C1S 2 ) 4S1C '22 (2 S0C1 C0 S1 ) 4 S 2C '12 (2C2 S0 C0 S 2 )
4S0C1C2 S1S2 S0 2C0 (C1S2 S1C2 ) .
(3) (4) (5)
From the expression of t (Eq. (2)), one can deduce the transmission rate
T
64C '12 C '12 S12 S 22 . 12 22
(6)
3. Results and discussion 3.1. BIC states, Transmission gaps and EIT resonance Figure 2(b) gives an example of the transmission spectrum versus the dimensionless wave vector kd0for two loops with specific lengths slightly detuned: d1 = 1/3+0.025 and d2 = 1/3-0.025. All the lengths are given in units of d0. The magnetic flux inside the ring is chosen such that: i.e., f1 = f2 = 0.2). One can notice the existence of two transmission zeros indicated by solid circles on the abscissa of Fig. 2(b) and a full transmission (transparence window) around kd0 This is a characteristic of an EIT resonance. Now, if the lengths of the two loops are chosen identical such that d1 = d2 = d = 1/3, then the EIT resonance collapses giving rise to the socalled trapped state or BIC state (Fig. 2 (a)). The position of this state is indicated by a solid circle on the abscissa of Fig. 2(a) at kd0/2
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Fig. 2. (a) Transmission coefficient versus the dimensionless wave vector kd0/2π for identical rings (d1=d2=d and f1=f2=0.2). (b) The same as in (a) but for the lengths dand d2=1/3-
3.2. Effect of the magnetic field on transmission gaps and EIT resonances 3.2.1. Case of two loops with similar fluxes In this section, we studied the effect of the magnetic flux on the transmission gaps and EIT resonances for the same geometrical structure as in Figs. 2(a) and (b). Figure 3(a) shows the effect of the magnetic flux on the width of the gaps for the same geometry as in Fig 2(a). One can notice that the width of the gap increases when the magnitude of the magnetic flux increases and transforms into a large band gap. In addition, without the magnetic flux (i.e., f1=f2 = 0) the gap disappears (continuous curve). These results clearly show the possibility to tune the band gaps by means of the magnetic flux. Also, just two loops are sufficient to have large gaps (see dotted curves for f1=f2=0.3) without repeating the structure periodically as it is usually the case [39]. In Fig. 3(b) we have plotted the effect of the magnetic flux on the shape of the EIT resonance. We can see that the width of the EIT resonance increases when the magnetic flux increases, its shape remains almost symmetrical and reaches unity (i,e., T=1) at kd0The resonance disappears in the absence of the magnetic flux (continuous curve). Also, the EIT resonance remains squeezed between the same transmission zeros (indicated by solid circles on the abscissa of Fig. 3(b)) induced by the two different rings.
T. Mrabti et al / Materials Today: Proceedings 13 (2019) 1055–1061
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Fig. 3. (a) The same as in figure 2 (a) but for f1 = f2 taken between 0 and 0.3.). (b) The same as in figure 2(b) but for f1 = f2 taken between 0 and 0.3.
3.2.2. Case where the field f1 is fixed and f2 is variable In this section, we consider the magnetic field f1 fixed while f2 is variable. Figure 4(a) shows the transmission spectrum as a function of the dimensionless wave vector kd0/2for the same geometries as in Figs. 2(a) and 3(a). We suppose that the magnetic field f1 is fixed in the first ring (f1=0.2) whereas the magnetic field f2 is variable in the second ring. One can see that the gap becomes larger when f2 increases followed by a reduction in the transmission spectrum in the allowed band at the vicinities of the gap. On other hand, the effect of the magnetic field on EIT resonances is illustrated in Fig. 4(b) for the same geometry as in Fig. 2(b). One can see that the magnitude of magnetic flux f2 affects considerably the shape of the EIT resonance. In addition, it is remarkable that the EIT resonance undergoes a small shift to higher wave vectors accompanied by an asymmetric shape and an increase in its width when f2 increases. Indeed, for f2 < f1 = 0.2, we can notice a shift of the EIT resonance to the right side of kd0=3 its width increases and its shape becomes asymmetrical when the magnetic flux decreases. For f2 > f1 = 0.2, the EIT resonance moves to the left of kd0=3, its width decreases and its shape becomes symmetrical. Also, in both cases the amplitude of the resonance decreases from unity. 3.2.3. Case of two loops with different fluxes In this section, we consider the magnetic field f1 and f2 are variable. Figure 5 (a) shows the evolution of the transmission under the effect of the magnetic fields for the same geometries as in Fig 2(a), one can notice that the transmission spectrum exhibits a deformation accompanied with a decrease in its amplitude at the vicinity of the gap. In Fig. 5(b), we have presented the effect of the magnetic flux on the EIT resonance for the same geometry as in Fig. 2(b). One can notice that starting from the case f1=f2 = 0.2, when f1 decreases and f2 increases the EIT resonance shifts to the left side of kd0=3 its shape becomes asymmetrical and its amplitude decreases. The same results are obtained in the opposite (i.e., when f1 increases and f2 decreases) case but the resonances shift to the right side of kd0=3.
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Fig. 4. (a) Same as in Fig. 2 (a) but for f1 = 0.2 and f2 is variable. (b) Same as in Fig. 2 (b) but for f1 = 0.2 and f2 is variable.
Fig. 5. (a) Same as in Fig. 2 (a) but both f1 and f2 are variable. (b) Same as in Fig. 2 (b) but both f1 and f2 are variable.
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4. Conclusion In this work, we have investigated the effect of the magnetic flux (Aharonov-Bohm effect) on the band gaps and EIT resonances in a simple mesoscopic structure made of two loops separated by a segment. This study is performed by analyzing the evolution of the transmission spectra. The analytical expression of the transmission coefficient is obtained by means of the Green’s function method. We have clearly demonstrated the existence of large band gaps in the transmission spectrum which are attributed to the Aharonov-Bohm effect; these band gaps depend considerably on the magnitude of the magnetic flux. Besides the transmission gaps, we have shown the possibility of existence of EIT resonances and BIC states inside the forbidden bands by tailoring the lengths of the arms constituting the rings. The evolution of the EIT resonance shape with the magnetic flux is analyzed in detail. These results may have important applications for electronic transport in mesoscopic systems. References [1] S. Harris, L. Hau, Phys. Rev. Lett. 82 (1999) 4611–4614. [2] A. Mouadili, E.H. El Boudouti, A. Soltani, A. Talbi, A. Akjouj, B. Djafari Rouhani, J. Appl. Phys. 113 (2013) 164101–10. [3] K.J. Boller, A. Imamoglu, S.E. Harris, Phys. Rev. Lett. 66 (1991) 2593–2596. [4] Y. Wang, Z. Li and F. Hu, J. Phys. D: Appl. Phys.51 (2018) 025103. [5] Y. Huang, C. Min, G. Veronis, App. Phys. Lett. 99 (2011) 143117–3. [6] M. Manjappa, S.Y. Chiam, L.Q. Cong, A.A. Bettiol, W.L. Zhang, R. Singh, Appl. Phys. Lett. 106 (2015) 181101–5. [7] C. Liu, Z. Dutton, C.H. Behroozi, L.V. Hau, Nature 409 (2001) 490–493. [8] M. Fleischhauer, A. Imamoglu, J.P. Marangos, Rev. Mod. Phys. 77 (2005) 633–673. [9] S. Fan, J.D. Joannopoulos, Phys. Rev. B 65 (2002) 235112–8. [10] X. Yang, M. Yu, D.L. Kwong, C.W. Wong, Phys. Rev. Lett. 102 (2009) 173902–4. [11] Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, S. Noda, Nat. Photon. 6 (2012) 56–61. [12] Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, M. Lipson, Phys. Rev. Lett. 96 ( 2006) 123901–4. [13] K. Totsuka, N. Kobayashi, M. Tomita, Phys. Rev. Lett. 21 (2007) 213904–4. [14] H.M. Li, S.B. Liu, S.Y. Liu, S.Y. Wang, G.W. Ding, H. Yang , Z.Y. Yu, H.F. Zhang, Appl. Phys. Lett. 106 (2015) 083511–4. [15] S. Zhang, D.A. Genov, Y. Wang, M. Liu, X. Zhang, Phys. Rev. Lett. 101(2008) 047401–4. [16] R. Taubert, M. Hentshel, J. Kastel, H. Giessen, Nano. Lett. 12 (2012) 1367–1371. [17] N. Papasimakis, V.A. Fedotov, N. Zheludev, S. Prosvirnin, Phys. Rev. Lett. 101 (2008) 253903–4. [18] E.H. El Boudouti, T. Mrabti, H. Al-Wahsh, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. Phys.: Condens. Matter 20 (2008) 255212–10. [19] A. Santillan, S.I. Bozhevolnyi, Phys. Rev. B 84 (2011) 064304–5. [20] A. Merkel, G. Theocharis, O. Richoux, V. Romero-Garcìa, V. Pagneux, Appl. Phys. Lett. 107 (2015) 244102–4. [21] I. Quotane, E.H. El Boudouti, B. Djafari-Rouhani, Phys. Rev. B 97 (2018) 024304–17. [22] A. Mouadili, E.H. El Boudouti, A. Soltani, A. Talbi, A. Akjouj, B. Djafari-Rouhani, J. Appl. Phys. 113 (2013) 164101–11. [23] A. Mouadili, E.H. El Boudouti, A. Soltani, A. Talbi, B. Djafari- Rouhani, A. Akjouj, K. Haddadi, Phys. Condens. Matter 26 (2014) 505901–12. [24] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, Cambridge, 1995. [25] Y. Imry, Introduction to mesoscopic physics, Oxford University Press, 1997. [26] Y. Aharonov, D. Bohm, Phys. Rev. 115 (1959) 485–491; R.G. Chambers, Phys. Rev. Lett. 5 (1960) 3–5. [27] M. Peshkin, A. Tonomura, Springer The Aharonov-Bohm Effect, Lecture Notes in Physics, 1989. [28] U. Fano, Phys. Rev. 124 (1961) 1866–1878. [29] A.E. Miroshnichenko, S. Flach, Y.S. Kivshar, Rev. Mod. Phys. 82 (2010) 2257–2298. [30] K. Kobayashi, H. Aikawa, S. Katsumoto, Y. Iye, Phys. Rev. Lett. 88 (2002) 256806–4. [31] J. Göres, D. Goldhaber-Gordon, S. Heemeyer, M.A. Kastner, H. Shtrikman, D. Mahalu, U. Meirav, Phys. Rev. B 62 (2000) 2188–2194. [32] M.L. Ladrón de Guevara, F. Claro, P.A. Orellana, Phys. Rev. B 67 (2003) 195335–6. [33] H. Al-Wahsh, E.H. El Boudouti, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski. Phys. Rev. B 75 (2007) 125313–11. [34] J. Yi, J.H. Wei, J. Hong, S.I. Lee, Phys. Rev. B 65 (2001) 033305–4. [35] C. Jiang, Y.S. Zheng, Solid State Commun. 212 (2015) 14–18. [36] S.K. Maiti, Phys. Lett. A 373 (2009) 4470–4474. [37] P. Dutta, S.K. Maiti, S.N. Karmakar, Solid State Commun. 150 (2010) 1056–1061. [38] T. Mrabti, Z. Labdouti, O. El Abouti, E.H. El Boudouti, F. Fethi , B. Djafari-Rouhani, Phys. Lett. A 382 (2018) 613–620. [39] J. Vasseur, A. Akjouj, L. Dobrzynski, B. Djafari-Rouhani, E.H. El Boudouti, Surf. Sci. Rep. 54 (2004) 1–156.