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Pressure, temperature and plasma frequency effects on the band structure of a 1D semiconductor photonic crystal
Physica E 44 (2012) 773–777
Contents lists available at SciVerse ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Pressure, temperature and plasma frequency effects on the band structure of a 1D semiconductor photonic crystal Luz E. Gonza´lez, N. Porras-Montenegro n Departamento de Fı´sica, Universidad del Valle, A.A. 25360 Cali, Colombia
a r t i c l e i n f o
abstract
Article history: Received 30 July 2011 Received in revised form 3 October 2011 Accepted 17 November 2011 Available online 25 November 2011
In this work using the transfer-matrix formalism we study pressure, temperature and plasma frequency effects on the band structure of a 1D semiconductor photonic crystal made of alternating layers of air and GaAs. We have found that the temperature dependence of the photonic band structure is negligible, however, its noticeable changes are due mainly to the variations of the width and the dielectric constant of the layers of GaAs, caused by the applied hydrostatic pressure. On the other hand, by using the Drude’s model, we have studied the effects of the hydrostatic pressure by means of the variation of the effective mass and density of the carriers in n-doped GaAs, finding firstly that increasing the amount of n-dopants in GaAs, namely, increasing the plasma frequency, the photonic band structure is shifted to regions of higher frequencies, and secondly the appearance of two regimes of the photonic band structure: one above the plasma frequency with the presence of usual Bragg gaps, and the other, below this frequency, where there are no gaps regularly distributed, with their width diminishing with the increasing of the plasma frequency as well as with the appearance of more bands, but leaving a wide frequency range in the lowest part of the spectrum without accessible photon states. Also, we have found characteristic frequencies in which the dielectric constant equals for different applied pressures, and from which to higher or lower values the photonic band structure inverts its behavior, depending on the value of the applied hydrostatic pressure. We hope this work may be taken into account for the development of new perspectives in the design of new optical devices. & 2011 Elsevier B.V. All rights reserved.
1. Introduction After works by Yablonovitch [1] and John [2] in which the photonic crystals (PCs) were proposed, many experimental and theoretical works have been devoted to the understanding of the physical properties of these crystals. PCs are periodic structures characterized by the periodic variation of refractive index and the consequently periodic spacial variation of the dielectric constant, thus allowing the appearance of define frequency ranges and address for which the propagation of the electromagnetic waves are prohibited or permitted. The frequency bands trough which the electromagnetic wave propagation is not permitted are called photonic band gap (PBG) [3]. In essence, a crystal is an artificial periodic structure characterized by a photonic band structure (PBS), which may be modified to control the properties of light, leading to a new era of optical devices [4–6]. As a consequence, the tunability of PCs opens a new perspective in the scientific research and in the technological applications. To obtain a tunable PC, the dielectric constant or the
n
Corresponding author. Tel.: þ57 2 339 4610; fax: þ57 2 3339 32 7. E-mail address: [email protected] (N. Porras-Montenegro).
1386-9477/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2011.11.018
magnetic permeability of one of the constituents materials must depend on some external parameters, such as electric [11] or applied magnetic fields [7–10], temperature and hydrostatic pressure [12], etc., which can modify the response functions of the PC materials. The tunability in PCs has been looked for by applying mechanical force, pressure and stress on the PCs, which modifies the structure of this systems and consequently their optical responses. However, by considering the variations in the intrinsic properties of the PCs materials, taking into account the pressure and temperature dependence of the dielectric and permeability constants, the tunability can be achieved without modifying the PC structure. A lot of work has been devoted to understand the behavior of electromagnetic fields in 1D photonic crystals. The photonic band gap structure as well as the density of photon states of a onedimensional photonic superlattice of alternate layers of air and GaAs characterized by different refractive indexes, which may take on positive as well as negative values, have been theoretically investigated within the Maxwell framework and using a transfer-matrix technique [13–18]. In this work we investigate the effects of temperature and applied hydrostatic pressure on the PBS of 1D photonic crystals.
´lez, N. Porras-Montenegro / Physica E 44 (2012) 773–777 L.E. Gonza
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We also examine the dependence of the PBS with the concentration of n-dopants in GaAs by means of the plasma frequency, taking into account the variation of the effective mass and the carrier density with the applied hydrostatic pressure.
where b a Tð 7a, 7bÞ ¼ M 2 7 M1 7 ¼ 2 2
P ¼ cos 2. Theoretical framework In Fig. 1 we present the 1D photonic superlattice studied in this work composed of alternating layers of GaAs and air. In the theoretical treatment we have considered the propagation of an in-plane linearly polarized electromagnetic field of the ^ along the z-axis. By using Maxwell’s form ~ Eðz,tÞ ¼ EðzÞeiot x, equation for linear and isotropic media, it is demonstrated that the amplitude EðzÞ of the electric field satisfies [19] d 1 dEðzÞ nðzÞ o2 ¼ EðzÞ, ð1Þ dz nðzÞZðzÞ dz ZðzÞ c2 pffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi where nðzÞ ¼ EðzÞ mðzÞ and ZðzÞ ¼ mðzÞ= EðzÞ are the refraction index and impedance, respectively, of each layer material. For a photonic crystal composed of alternating layers of two different materials, Eq. (1) must be solved by assuming the continuity of both the electric field EðzÞ and of the expression ð1=nðzÞZðzÞÞð@E=@zÞ which means that the two-component function: 0 1 Ez @ ð2Þ cðzÞ ¼ 1 dE A nZ dz is continuous through the photonic structure. Following Cavalcanti et al. [13], this condition may be conveniently written by means of a transfer matrix as
cðzÞ ¼ Mi ðzz0 Þcðz0 Þ,
ð3Þ
where 0 B M i ðzÞ ¼ @
cos
o9ni 9 c
z
9n 9 o o9ni 9 sin z nii cZ c i
ni cZ i 9ni 9 o
o9ni 9
1
sin c z C A, o9n 9 cos c i z
one may write that aþb ¼ Tð 7a, 7 bÞcð0Þ, c 7 2
ð4Þ
ð5Þ
Q¼
7Q
7R
S
! ,
bk2 ak1 Z 2 9n1 9 n2 bk2 ak1 cos sin , sin 2 2 Z 1 n1 9n2 9 2 2
9n1 9 cZ 1 bk2 ak1 n2 cZ 2 bk2 ak1 sin þ cos , cos sin 2 2 2 2 n1 o 9n2 9 o
R¼
9n2 9 o bk2 ak1 9n1 9 o bk2 ak1 cos sin , sin cos 2 2 2 2 n2 cZ 2 n1 cZ 1
S ¼ cos
bk2 ak1 Z 1 9n2 9 n1 bk2 ak1 sin cos sin , 2 2 Z 2 n2 9n1 9 2 2
ð6Þ
ð7Þ
ð8Þ
ð9Þ
ð10Þ
with k1 ¼ ðo=cÞ9n1 9, k2 ¼ ðo=cÞ9n2 9, noting that PSQR ¼ 1. By the periodicity of the photonic crystal we may use the Bloch condition:
cðz þ dÞ ¼ eiqd cðzÞ,
ð11Þ
with q chosen within the first Brillouin zone (BZ) of the photonic superlattice, i.e., p=d rq r p=d, obtaining the secular equation: PSð1lÞ2 QRð1 þ lÞ2 ¼ 0,
ð12Þ
with l ¼ eiqd , which leads to the two following equivalent relations: qd ¼ QR, ð13Þ sin2 2 cos2
qd ¼ PS 2
ð14Þ
by means of which we may obtain the dispersion relationship or photonic band structure of the periodic superlattice, o ¼ oðqÞ. In this study, as we are interested in the optical response of the photonic crystal due to the effects of external probes on the GaAs layer, we have assumed that the dielectric constant as well as the thickness of the air layer do not vary under pressure. In order to include the variation of the thickness of the GaAs layer and its dielectric constant as a function of pressure and temperature, we follow the works by Samara [20] and Elabsy [21]: bðPÞ ¼ b0 ½1ðS11 þ 2S12 ÞP,
d
P
ð15Þ
where b(P) is the thickness of the GaAs layer as a function of 1 1 pressure, S11 ¼ 1:16 103 kbar and S12 ¼ 3:7 104 kbar are the elastic constants of GaAs, and b0 is the original thickness at atmospheric hydrostatic pressure. The expression that describes the behavior of the dielectric constant under both the pressure and temperature is
2
1
b/2 -d/2
Eb ðP,TÞ ¼ ðE0 þ AeT=T 0 ÞeaP ,
2
b/2 -a/2
0
a/2
d/2
Fig. 1. Pictorial view of the 1D photonic crystal studied, composed of alternating layers of GaAs and air. In (a) the origin is located at the center of a first slab (with dielectric constant E1 and magnetic permeability m1 ) of width a where b is the slab width of the second material (with dielectric constant E2 and magnetic permeability m2 ) as shown in (b).
ð16Þ
where E0 ¼ 12:446, A ¼0.21125, T 0 ¼ 240:7 K, and a ¼ 1 0:00173 kbar . In order to consider dispersive effects on the optical response of the system, we have used the frequency dependent dielectric constant of the GaAs, given by [22] !
E0b ¼ Eb ðP,TÞ 1
o2p , o2
ð17Þ
where op is the plasma frequency, a function of the effective mass mG D and carrier density D, which for this stage of the work we assume to be only determined by the n-dopants, and
´lez, N. Porras-Montenegro / Physica E 44 (2012) 773–777 L.E. Gonza
775
consequently weakly dependent on the temperature:
op ¼
DðPÞe2
E0b ðP,TÞE0 mGD ðPÞ
:
ð18Þ
To analyze the pressure effects on the plasma frequency, we have used the data given in the work by Monroe [23] who has reported that for an arbitrary free-electron density D and pressure P, the electron effective mass in GaAs is given by " # 1 ð1 þ 7:4 103 P kbar Þ G mn =m0 ¼ 0:067 ð19Þ ð13:9 1015 D2=3 cm2 Finally, the variation of carrier density with pressure is given by DðPÞ ¼ D0 e½1ðbðPÞ=b0 Þ ,
ð20Þ
which expresses the density of carriers in terms of the carrier density for the initial pressure D0 , the thickness of the layer ofGaAs as a function of pressure b(P), and b0 the initial thickness of the GaAS layer.
3. Results and discussion In what follows, firstly we investigate the effects of temperature and applied hydrostatic pressure on the PBS of 1D photonic crystals taking into account the dependence of the dielectric function on these probes, and secondly we examine the dependence of the PBS with the concentration of n-dopants in GaAs by means of the plasma frequency, considering the variations of the dielectric constant with the applied hydrostatic pressure and temperature, the modification of the electron effective-mass with pressure, and the carrier density which we assume as that determined by the n-dopants. In order to analyze the effects of both the temperature and applied hydrostatic pressure on the PBS we have followed and used the data given in the works by Samara [20] and Elabsy [21], for the GaAs dielectric constant as a function of pressure and temperature, as established in Eq. (16), without considering the modifications in the plasma frequency. Next, we give the values of the GaAS dielectric constant for different values of pressure and temperature, obtained by using Eq. (16), which are used in the present calculations:
Eð0 kbar,0 KÞ ¼ 12:66, Eð40 kbar,0 KÞ ¼ 11:81, Eð0 kbar,340 KÞ ¼ 13:31, Eð40 kbar,340 KÞ ¼ 12:42. It is observed that the pressure effect on the dielectric constant is higher in comparison to that due to temperature. In Fig. 2 we present in three panels the effects of temperature and applied hydrostatic pressure on the PBS in a 1D periodic structure made of alternating layers of air and GaAs. Panel (a) is devoted to show the temperature effect for a given value of the applied hydrostatic pressure ðP ¼ 40 kbarÞ. Panel (b) is to contrast the effect of the applied hydrostatic pressure for a given value of the temperature ðT ¼ 0Þ. Panel (c) is to compare the combined effects of temperature and applied hydrostatic pressure. It is noticeable in panel (a) that the temperature effect on the PBS results in a shift to lower frequencies, effect that is less apparent when compared with the result in panel (b), where the PBS is shifted to regions of higher frequencies as pressure increases. The resulting variation occurs in panel (c) where it is clearly observed that the pressure effect on the PBS is greater than that of temperature, shifting the PBS to higher frequencies. As expected the dielectric constant decreases with pressure, shifting the PBS to
Fig. 2. Photonic band structure o vs q, of a superlattice (period d) with equal alternate layer of air (with thickness a ¼ 1 mm) and GaAs (thickness b ¼ 1 mm). Panel (a) is for P ¼ 40 kbar with T¼0 (black line) and with T ¼ 340 K (red line), curves which coincide. Panel (b) is for T¼ 0 with P ¼0 (blue line) and with P ¼ 40 kbar (purple line). Panel (c) is for T¼ 0 with P¼ 0 (blue line) and for T ¼ 340 K with P ¼ 40 kbar (red line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Photonic band structure o vs q with alternating layers of air and GaAs. Panel (a) is for a ¼ b ¼ 1 mm; Panel (b) is for a ¼ 0:8 mm and b ¼ 1:2 mm, and Panel (c) is for a ¼ 0:5 mm and b ¼ 1:5 mm. All panels are performed with T ¼ 340 K and P ¼ 40 kbar.
higher frequencies, result which is in agreement with the electromagnetic variational theorem [3]. According with the data above on temperature and pressure dependence of dielectric function and results in Fig. 2 on the PBS in the 1D air-GaAs PC, which show that the shifting in the PBS is mainly due to the applied hydrostatic pressure instead of the temperature, in what follows we are only devoted to analyze the optical response of the system caused by the parameters which depend on the applied hydrostatic pressure like the electron effective-mass, carrier density, layer widths, and the dielectric function, which modify the plasma frequency. Modifications introduced in the PBS, when different a and b layer widths are used, with d ¼a þb constant, are displayed in Fig. 3. One clearly sees that, for layers of air ða ¼ 0:5 mmÞ narrower than the GaAs width ðb ¼ 1:5 mmÞ, the corresponding dispersion curves o vs q become flatter and the band gaps at the
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BZ edge are quite different than in the case of equal layer widths. Also, it is observed a PBS shifting to lower frequencies. Although several studies have been performed to look for the PBS of 1D semiconductor photonic crystals using the Drude model, authors have not included the modifications of the dielectric response due to variations in the carrier density and effective mass through the plasma frequency. In order to reach this goal, we have studied the case of a photonic structure with alternating homogeneous layers of two different materials with na positive and independent of frequency (layer a is air with na ¼ 1), and layer b with a frequency-dependent dielectric constant, which gives a refraction index, nb ðoÞ, negative, positive or pure imaginary, depending on the chosen range of frequency. As expected, the system optical response is essentially due to the geometry of the structure and the parameters of layer b, like EðoÞ, the dielectric permittivity, which we have taken as a Drude-type response, as given by Eq. (17). In Fig. 4 it is observed the refractive index nb as a function of frequency, with mb ¼ 1 and Eb ðoÞ described by the Eq. (17). This figure shows a positive branch obtained when both E and m are positive, an imaginary portion in the case when E and m have opposite signs. The point where the refractive index vanishes, corresponds to the point where the plasma frequency ðop Þ and the frequency ðoÞ coincide. Results on the study of the effects on the optical response of the PC made of air and n-doped GaAs, taking into account the dispersive dielectric function defined in Eq. (17) are presented in Fig. 5. In this case, we have used for the plasma frequency the values 0.5, 0.785, 0.942, 1.1, 1.3 and 1.5 THz, respectively. It is observed that for values of o around the plasma frequency the gap tends to close as the plasma frequency increases. The blue dashed line indicates the values of n0 ¼ o0 =2p corresponding to those of the plasma frequency listed above, that is, the frequency value for which the refractive index of the n-doped GaAs is null. Observe that there exists two regimes for the PBS, one above the plasma frequency where the refractive index is real with the presence of usual Bragg gaps, and the other below the plasma frequency, where there are not gaps regularly distributed in the PBS. It is observed that as the plasma frequency increases, there appears more bands with diminishing band gaps, but leaving a wide frequency range in the lowest part of the spectrum without accessible photon states. This result is in agreement with the fact that with the increasing of the plasma frequency the electromagnetic fields with lower frequency do not resonate with those of the plasma. On the other hand, it is observed as expected, the shift of the PBS to higher frequencies due to the decrease of the dielectric constant which diminishes with the carrier density. In order to analyze the effects of pressure on the PBS through the variations of the plasma frequency, see Eq. (18), we have followed and used the data given in the work by Monroe [23], who has reported the effective mass as a function of the
8 4 0 -4 0.00
0.05
0.10
0.15
0.20
0.25
0.30
Fig. 4. Frequency-dependent refractive index according to the Drude’s effective dielectric permitivity.
Fig. 5. Photonic band structure n vs q, with n ¼ o=2p of a superlattice composed of alternate layer of air and GaAs. We presented result for different values of op . All panels are for a ¼ b ¼ 1 mm. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
10 P =10 kbar P =40kbar
5
0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
Fig. 6. Dielectric constant behavior in the case in that the plasma frequency is varied with the pressure.
free-electron density and pressure for GaAs, as established in Eq. (19). In addition, we have used the expression already known for the behavior of the carrier density under pressure, Eq. (20). In our calculation, the initial carrier density was taken as n0 ¼ 1:64231 1020 m3 that corresponds to the plasma frequency op ¼ 0:785 THz. According with Eq. (18) we have found that increasing pressure, the plasma frequency decreases, resulting 0.7748 and 0.7284 THz for P¼10 kbar and P¼40 kbar, respectively. In Fig. 6 it is shown the behavior of the dielectric constant E2 ðP,T, oÞ as a function of the plasma frequency, which is modified by pressure according to Eq. (17). Note that for frequency values n ¼ 0:1159 THz and n ¼ 0:1233 THz, for P¼40 kbar and P¼10 kbar, respectively, the dielectric constant of the doped GaAs vanishes, i.e., its refractive index is null. On the other hand for a frequency value n ¼ 0:22 THz, the dielectric constant for P¼10 kbar and P¼40 kbar are the same; and for higher values of
´lez, N. Porras-Montenegro / Physica E 44 (2012) 773–777 L.E. Gonza
Fig. 7. Photonic band structure n vs q, with n ¼ o=2p of a superlattice composed of alternate layer of air (with thickness a ¼ 1 mm) and GaAs (thickness b ¼ 1 mm). This figure displays the behavior of the photonic band structure when varying the plasma frequency with pressure. Solid and dashed-dotted lines correspond to P ¼ 40 kbar and P ¼ 10 kbar, respectively. In the panel (b) the same results are zoomed indicating with black dashed-lines the values of n0 in which the refractive index of the n-doped GaAs is null.
777
function of the GaAs layers. We have found that the temperature dependence of the PBS is negligible as it has been found in 2D photonic crystals and that its changes are due to the variation of the width and the dielectric constant of the GaAs layers, due mainly to the applied hydrostatic pressure. On the other hand, by using the Drude’s model, we studied the effects of the hydrostatic pressure by means of the variation of the effective mass and density of the carriers in n-doped GaAs, finding that increasing the amount of the latter, that is, increasing the plasma frequency, the PBS is shifted to regions of higher frequencies, appearing two regimes of the photonic band structure, one above the plasma frequency with the presence of usual Bragg gaps, and the other, below this frequency, where there are no gaps regularly distributed, with their width diminishing with the increasing of the plasma frequency as well as with the appearance of more bands, but leaving a wide frequency range in the lowest part of the spectrum without accessible photon states. Also, we have found characteristic frequencies in which the dielectric constant equals for different applied pressures, and from which to higher or lower values, the PBS inverts its behavior, depending on the value of the applied hydrostatic pressure. We do hope these findings may be taken into account for future applications in the development of photonic devices.
Acknowledgments frequency, the dielectric constant for P¼10 kbar is greater than that for P ¼40 kbar, while for lower frequencies this situation inverts. As expected, the pressure dependence of the dielectric constant is apparent in the PBS, as it is shown in Fig. 7. In the region around the frequency for which the n-doped GaAs refractive index is null, n ¼ 0:1233 THz and n ¼ 0:1159 THz the gap tends to close, behavior that is also observed in Fig. 5. In a frequency n ¼ 0:22 THz , the energy bands practically coincide, frequency in which the dielectric constant for both pressure values is the same, as it was pointed out in Fig. 6. Finally, it is worth to mention that for values of frequency higher than n ¼ 0:22 THz, where the dielectric constants for P¼10 kbar and P¼ 40 kbar are reversed (see Fig. 6), the PBS corresponding to P¼10 kbar is shifted to higher energies than for P¼40 kbar, while for frequencies lower than n ¼ 0:22, the dielectric constant is higher for P ¼40 than for P¼10 kbar, and the PBS moves to lower frequencies, results that clearly obeys de electromagnetic variational principle. In other words, this inversion of the PBS with pressure obeys the variations of the dielectric constant with the plasma frequency (see Fig. 6), which depends on the applied hydrostatic pressure through the electron effective-mass, the dielectric layer width, and the carrier density according with Eq. (18).
4. Conclusions Within the transfer-matrix technique, we have studied the photonic band structures of a 1D photonic crystal consisting of a superlattice with two alternating layers of air and GaAs. We have examine the PBS dependence on both the applied hydrostatic pressure and temperature, taking into account the variations caused by this parameters on the width and the dielectric
This research was partially supported by Vicerrectora de Investigaciones at Universidad del Valle and CENM with funds of Patrimonio Autnomo Fondo Nacional de Financiamiento para la Ciencia, la Tecnologa y la Innovacin Francisco Jos de Caldas, grant RC - No. 275-2011.
References [1] E. Yablonovitch, Physical Review Letters 58 (1987) 2059. [2] S. John, Physical Review Letters 58 (1987) 2486. [3] J.D. Joannopoulos, et al., Photonic Crystals: Molding the Flow of Light, Princeton University Press, Princeton, 1995. [4] H. Kosaka, et al., Physical Review B 58 (1998) R10096. [5] I.V. Konoplev, et al., Journal of Applied Physics 97 (2005) 073101. [6] I.V. Konoplev, et al., Applied Physics Letters 87 (2005) 121104. [7] C.A. Duque, N. Porras-Montenegro, S.B. Cavalcanti, L.E. Oliveira, Journal of Applied Physics 105 (2009) 034303. [8] P.S. Pershan, Journal of Applied Physics 38 (1967) 1482. [9] C.R. Pidgeon, in: M. Balkanski (Ed.), Handbook on Semiconductors, vol. 2, North-Holland, Amsterdam, 1980, p. 223. [10] C. Xu, X. Hu, Y. Li, X. Liu, R. Fu, J. Zi, Physical Review B 68 (2003) 193201. [11] K. Busch, S. John, Physical Review Letters 83 (1999) 967. [12] N. Porras-Montenegro, C.A. Duque, Physica E 42 (2010) 1865–1869. [13] S.B. Cavalcanti, M. de Dios-Leyva, E. Reyes-Go´mez, L.E. Oliveira, Physical Review B 74 (2006) 153102. [14] S.B. Cavalcanti, M. de Dios-Leyva, E. Reyes-Go´mez, L.E. Oliveira, Physical Review E 75 (2007) 026607. [15] L-.L. Lin, Z-.Y. Li, K-.M. Ho, Journal of Applied Physics 94 (2003) 811. [16] J. Li, Z-.Y. Li, D-.Z. Zhang, Physical Review B 77 (2008) 195127. [17] M.S. Vasconcelos, et al., Physical Review B 76 (2007) 165117. [18] Peter Markos, Costas M. Soukoulis, Wave Propagation: From Electrons to Photonic Crystals and Left-Handed Materials, Princeton University Press, Princeton, 2008. [19] J. Li, et al., Physical Review Letters 90 (2003) 083901. [20] G.A. Samara, Physical Review B 27 (1983) 3494. [21] A.M. Elabsy, Physica Scripta 48 (1993) 376. [22] N. Ashcroft, D. Mermin, Introduction to Solid State Physics, 5th ed., Wiley, New York, 1976. [23] D. Monroe, Applied Physics Letters 59 (1991) 2293.
Contents lists available at SciVerse ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Pressure, temperature and plasma frequency effects on the band structure of a 1D semiconductor photonic crystal Luz E. Gonza´lez, N. Porras-Montenegro n Departamento de Fı´sica, Universidad del Valle, A.A. 25360 Cali, Colombia
a r t i c l e i n f o
abstract
Article history: Received 30 July 2011 Received in revised form 3 October 2011 Accepted 17 November 2011 Available online 25 November 2011
In this work using the transfer-matrix formalism we study pressure, temperature and plasma frequency effects on the band structure of a 1D semiconductor photonic crystal made of alternating layers of air and GaAs. We have found that the temperature dependence of the photonic band structure is negligible, however, its noticeable changes are due mainly to the variations of the width and the dielectric constant of the layers of GaAs, caused by the applied hydrostatic pressure. On the other hand, by using the Drude’s model, we have studied the effects of the hydrostatic pressure by means of the variation of the effective mass and density of the carriers in n-doped GaAs, finding firstly that increasing the amount of n-dopants in GaAs, namely, increasing the plasma frequency, the photonic band structure is shifted to regions of higher frequencies, and secondly the appearance of two regimes of the photonic band structure: one above the plasma frequency with the presence of usual Bragg gaps, and the other, below this frequency, where there are no gaps regularly distributed, with their width diminishing with the increasing of the plasma frequency as well as with the appearance of more bands, but leaving a wide frequency range in the lowest part of the spectrum without accessible photon states. Also, we have found characteristic frequencies in which the dielectric constant equals for different applied pressures, and from which to higher or lower values the photonic band structure inverts its behavior, depending on the value of the applied hydrostatic pressure. We hope this work may be taken into account for the development of new perspectives in the design of new optical devices. & 2011 Elsevier B.V. All rights reserved.
1. Introduction After works by Yablonovitch [1] and John [2] in which the photonic crystals (PCs) were proposed, many experimental and theoretical works have been devoted to the understanding of the physical properties of these crystals. PCs are periodic structures characterized by the periodic variation of refractive index and the consequently periodic spacial variation of the dielectric constant, thus allowing the appearance of define frequency ranges and address for which the propagation of the electromagnetic waves are prohibited or permitted. The frequency bands trough which the electromagnetic wave propagation is not permitted are called photonic band gap (PBG) [3]. In essence, a crystal is an artificial periodic structure characterized by a photonic band structure (PBS), which may be modified to control the properties of light, leading to a new era of optical devices [4–6]. As a consequence, the tunability of PCs opens a new perspective in the scientific research and in the technological applications. To obtain a tunable PC, the dielectric constant or the
n
Corresponding author. Tel.: þ57 2 339 4610; fax: þ57 2 3339 32 7. E-mail address: [email protected] (N. Porras-Montenegro).
1386-9477/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2011.11.018
magnetic permeability of one of the constituents materials must depend on some external parameters, such as electric [11] or applied magnetic fields [7–10], temperature and hydrostatic pressure [12], etc., which can modify the response functions of the PC materials. The tunability in PCs has been looked for by applying mechanical force, pressure and stress on the PCs, which modifies the structure of this systems and consequently their optical responses. However, by considering the variations in the intrinsic properties of the PCs materials, taking into account the pressure and temperature dependence of the dielectric and permeability constants, the tunability can be achieved without modifying the PC structure. A lot of work has been devoted to understand the behavior of electromagnetic fields in 1D photonic crystals. The photonic band gap structure as well as the density of photon states of a onedimensional photonic superlattice of alternate layers of air and GaAs characterized by different refractive indexes, which may take on positive as well as negative values, have been theoretically investigated within the Maxwell framework and using a transfer-matrix technique [13–18]. In this work we investigate the effects of temperature and applied hydrostatic pressure on the PBS of 1D photonic crystals.
´lez, N. Porras-Montenegro / Physica E 44 (2012) 773–777 L.E. Gonza
774
We also examine the dependence of the PBS with the concentration of n-dopants in GaAs by means of the plasma frequency, taking into account the variation of the effective mass and the carrier density with the applied hydrostatic pressure.
where b a Tð 7a, 7bÞ ¼ M 2 7 M1 7 ¼ 2 2
P ¼ cos 2. Theoretical framework In Fig. 1 we present the 1D photonic superlattice studied in this work composed of alternating layers of GaAs and air. In the theoretical treatment we have considered the propagation of an in-plane linearly polarized electromagnetic field of the ^ along the z-axis. By using Maxwell’s form ~ Eðz,tÞ ¼ EðzÞeiot x, equation for linear and isotropic media, it is demonstrated that the amplitude EðzÞ of the electric field satisfies [19] d 1 dEðzÞ nðzÞ o2 ¼ EðzÞ, ð1Þ dz nðzÞZðzÞ dz ZðzÞ c2 pffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi where nðzÞ ¼ EðzÞ mðzÞ and ZðzÞ ¼ mðzÞ= EðzÞ are the refraction index and impedance, respectively, of each layer material. For a photonic crystal composed of alternating layers of two different materials, Eq. (1) must be solved by assuming the continuity of both the electric field EðzÞ and of the expression ð1=nðzÞZðzÞÞð@E=@zÞ which means that the two-component function: 0 1 Ez @ ð2Þ cðzÞ ¼ 1 dE A nZ dz is continuous through the photonic structure. Following Cavalcanti et al. [13], this condition may be conveniently written by means of a transfer matrix as
cðzÞ ¼ Mi ðzz0 Þcðz0 Þ,
ð3Þ
where 0 B M i ðzÞ ¼ @
cos
o9ni 9 c
z
9n 9 o o9ni 9 sin z nii cZ c i
ni cZ i 9ni 9 o
o9ni 9
1
sin c z C A, o9n 9 cos c i z
one may write that aþb ¼ Tð 7a, 7 bÞcð0Þ, c 7 2
ð4Þ
ð5Þ
Q¼
7Q
7R
S
! ,
bk2 ak1 Z 2 9n1 9 n2 bk2 ak1 cos sin , sin 2 2 Z 1 n1 9n2 9 2 2
9n1 9 cZ 1 bk2 ak1 n2 cZ 2 bk2 ak1 sin þ cos , cos sin 2 2 2 2 n1 o 9n2 9 o
R¼
9n2 9 o bk2 ak1 9n1 9 o bk2 ak1 cos sin , sin cos 2 2 2 2 n2 cZ 2 n1 cZ 1
S ¼ cos
bk2 ak1 Z 1 9n2 9 n1 bk2 ak1 sin cos sin , 2 2 Z 2 n2 9n1 9 2 2
ð6Þ
ð7Þ
ð8Þ
ð9Þ
ð10Þ
with k1 ¼ ðo=cÞ9n1 9, k2 ¼ ðo=cÞ9n2 9, noting that PSQR ¼ 1. By the periodicity of the photonic crystal we may use the Bloch condition:
cðz þ dÞ ¼ eiqd cðzÞ,
ð11Þ
with q chosen within the first Brillouin zone (BZ) of the photonic superlattice, i.e., p=d rq r p=d, obtaining the secular equation: PSð1lÞ2 QRð1 þ lÞ2 ¼ 0,
ð12Þ
with l ¼ eiqd , which leads to the two following equivalent relations: qd ¼ QR, ð13Þ sin2 2 cos2
qd ¼ PS 2
ð14Þ
by means of which we may obtain the dispersion relationship or photonic band structure of the periodic superlattice, o ¼ oðqÞ. In this study, as we are interested in the optical response of the photonic crystal due to the effects of external probes on the GaAs layer, we have assumed that the dielectric constant as well as the thickness of the air layer do not vary under pressure. In order to include the variation of the thickness of the GaAs layer and its dielectric constant as a function of pressure and temperature, we follow the works by Samara [20] and Elabsy [21]: bðPÞ ¼ b0 ½1ðS11 þ 2S12 ÞP,
d
P
ð15Þ
where b(P) is the thickness of the GaAs layer as a function of 1 1 pressure, S11 ¼ 1:16 103 kbar and S12 ¼ 3:7 104 kbar are the elastic constants of GaAs, and b0 is the original thickness at atmospheric hydrostatic pressure. The expression that describes the behavior of the dielectric constant under both the pressure and temperature is
2
1
b/2 -d/2
Eb ðP,TÞ ¼ ðE0 þ AeT=T 0 ÞeaP ,
2
b/2 -a/2
0
a/2
d/2
Fig. 1. Pictorial view of the 1D photonic crystal studied, composed of alternating layers of GaAs and air. In (a) the origin is located at the center of a first slab (with dielectric constant E1 and magnetic permeability m1 ) of width a where b is the slab width of the second material (with dielectric constant E2 and magnetic permeability m2 ) as shown in (b).
ð16Þ
where E0 ¼ 12:446, A ¼0.21125, T 0 ¼ 240:7 K, and a ¼ 1 0:00173 kbar . In order to consider dispersive effects on the optical response of the system, we have used the frequency dependent dielectric constant of the GaAs, given by [22] !
E0b ¼ Eb ðP,TÞ 1
o2p , o2
ð17Þ
where op is the plasma frequency, a function of the effective mass mG D and carrier density D, which for this stage of the work we assume to be only determined by the n-dopants, and
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consequently weakly dependent on the temperature:
op ¼
DðPÞe2
E0b ðP,TÞE0 mGD ðPÞ
:
ð18Þ
To analyze the pressure effects on the plasma frequency, we have used the data given in the work by Monroe [23] who has reported that for an arbitrary free-electron density D and pressure P, the electron effective mass in GaAs is given by " # 1 ð1 þ 7:4 103 P kbar Þ G mn =m0 ¼ 0:067 ð19Þ ð13:9 1015 D2=3 cm2 Finally, the variation of carrier density with pressure is given by DðPÞ ¼ D0 e½1ðbðPÞ=b0 Þ ,
ð20Þ
which expresses the density of carriers in terms of the carrier density for the initial pressure D0 , the thickness of the layer ofGaAs as a function of pressure b(P), and b0 the initial thickness of the GaAS layer.
3. Results and discussion In what follows, firstly we investigate the effects of temperature and applied hydrostatic pressure on the PBS of 1D photonic crystals taking into account the dependence of the dielectric function on these probes, and secondly we examine the dependence of the PBS with the concentration of n-dopants in GaAs by means of the plasma frequency, considering the variations of the dielectric constant with the applied hydrostatic pressure and temperature, the modification of the electron effective-mass with pressure, and the carrier density which we assume as that determined by the n-dopants. In order to analyze the effects of both the temperature and applied hydrostatic pressure on the PBS we have followed and used the data given in the works by Samara [20] and Elabsy [21], for the GaAs dielectric constant as a function of pressure and temperature, as established in Eq. (16), without considering the modifications in the plasma frequency. Next, we give the values of the GaAS dielectric constant for different values of pressure and temperature, obtained by using Eq. (16), which are used in the present calculations:
Eð0 kbar,0 KÞ ¼ 12:66, Eð40 kbar,0 KÞ ¼ 11:81, Eð0 kbar,340 KÞ ¼ 13:31, Eð40 kbar,340 KÞ ¼ 12:42. It is observed that the pressure effect on the dielectric constant is higher in comparison to that due to temperature. In Fig. 2 we present in three panels the effects of temperature and applied hydrostatic pressure on the PBS in a 1D periodic structure made of alternating layers of air and GaAs. Panel (a) is devoted to show the temperature effect for a given value of the applied hydrostatic pressure ðP ¼ 40 kbarÞ. Panel (b) is to contrast the effect of the applied hydrostatic pressure for a given value of the temperature ðT ¼ 0Þ. Panel (c) is to compare the combined effects of temperature and applied hydrostatic pressure. It is noticeable in panel (a) that the temperature effect on the PBS results in a shift to lower frequencies, effect that is less apparent when compared with the result in panel (b), where the PBS is shifted to regions of higher frequencies as pressure increases. The resulting variation occurs in panel (c) where it is clearly observed that the pressure effect on the PBS is greater than that of temperature, shifting the PBS to higher frequencies. As expected the dielectric constant decreases with pressure, shifting the PBS to
Fig. 2. Photonic band structure o vs q, of a superlattice (period d) with equal alternate layer of air (with thickness a ¼ 1 mm) and GaAs (thickness b ¼ 1 mm). Panel (a) is for P ¼ 40 kbar with T¼0 (black line) and with T ¼ 340 K (red line), curves which coincide. Panel (b) is for T¼ 0 with P ¼0 (blue line) and with P ¼ 40 kbar (purple line). Panel (c) is for T¼ 0 with P¼ 0 (blue line) and for T ¼ 340 K with P ¼ 40 kbar (red line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Photonic band structure o vs q with alternating layers of air and GaAs. Panel (a) is for a ¼ b ¼ 1 mm; Panel (b) is for a ¼ 0:8 mm and b ¼ 1:2 mm, and Panel (c) is for a ¼ 0:5 mm and b ¼ 1:5 mm. All panels are performed with T ¼ 340 K and P ¼ 40 kbar.
higher frequencies, result which is in agreement with the electromagnetic variational theorem [3]. According with the data above on temperature and pressure dependence of dielectric function and results in Fig. 2 on the PBS in the 1D air-GaAs PC, which show that the shifting in the PBS is mainly due to the applied hydrostatic pressure instead of the temperature, in what follows we are only devoted to analyze the optical response of the system caused by the parameters which depend on the applied hydrostatic pressure like the electron effective-mass, carrier density, layer widths, and the dielectric function, which modify the plasma frequency. Modifications introduced in the PBS, when different a and b layer widths are used, with d ¼a þb constant, are displayed in Fig. 3. One clearly sees that, for layers of air ða ¼ 0:5 mmÞ narrower than the GaAs width ðb ¼ 1:5 mmÞ, the corresponding dispersion curves o vs q become flatter and the band gaps at the
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BZ edge are quite different than in the case of equal layer widths. Also, it is observed a PBS shifting to lower frequencies. Although several studies have been performed to look for the PBS of 1D semiconductor photonic crystals using the Drude model, authors have not included the modifications of the dielectric response due to variations in the carrier density and effective mass through the plasma frequency. In order to reach this goal, we have studied the case of a photonic structure with alternating homogeneous layers of two different materials with na positive and independent of frequency (layer a is air with na ¼ 1), and layer b with a frequency-dependent dielectric constant, which gives a refraction index, nb ðoÞ, negative, positive or pure imaginary, depending on the chosen range of frequency. As expected, the system optical response is essentially due to the geometry of the structure and the parameters of layer b, like EðoÞ, the dielectric permittivity, which we have taken as a Drude-type response, as given by Eq. (17). In Fig. 4 it is observed the refractive index nb as a function of frequency, with mb ¼ 1 and Eb ðoÞ described by the Eq. (17). This figure shows a positive branch obtained when both E and m are positive, an imaginary portion in the case when E and m have opposite signs. The point where the refractive index vanishes, corresponds to the point where the plasma frequency ðop Þ and the frequency ðoÞ coincide. Results on the study of the effects on the optical response of the PC made of air and n-doped GaAs, taking into account the dispersive dielectric function defined in Eq. (17) are presented in Fig. 5. In this case, we have used for the plasma frequency the values 0.5, 0.785, 0.942, 1.1, 1.3 and 1.5 THz, respectively. It is observed that for values of o around the plasma frequency the gap tends to close as the plasma frequency increases. The blue dashed line indicates the values of n0 ¼ o0 =2p corresponding to those of the plasma frequency listed above, that is, the frequency value for which the refractive index of the n-doped GaAs is null. Observe that there exists two regimes for the PBS, one above the plasma frequency where the refractive index is real with the presence of usual Bragg gaps, and the other below the plasma frequency, where there are not gaps regularly distributed in the PBS. It is observed that as the plasma frequency increases, there appears more bands with diminishing band gaps, but leaving a wide frequency range in the lowest part of the spectrum without accessible photon states. This result is in agreement with the fact that with the increasing of the plasma frequency the electromagnetic fields with lower frequency do not resonate with those of the plasma. On the other hand, it is observed as expected, the shift of the PBS to higher frequencies due to the decrease of the dielectric constant which diminishes with the carrier density. In order to analyze the effects of pressure on the PBS through the variations of the plasma frequency, see Eq. (18), we have followed and used the data given in the work by Monroe [23], who has reported the effective mass as a function of the
8 4 0 -4 0.00
0.05
0.10
0.15
0.20
0.25
0.30
Fig. 4. Frequency-dependent refractive index according to the Drude’s effective dielectric permitivity.
Fig. 5. Photonic band structure n vs q, with n ¼ o=2p of a superlattice composed of alternate layer of air and GaAs. We presented result for different values of op . All panels are for a ¼ b ¼ 1 mm. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
10 P =10 kbar P =40kbar
5
0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
Fig. 6. Dielectric constant behavior in the case in that the plasma frequency is varied with the pressure.
free-electron density and pressure for GaAs, as established in Eq. (19). In addition, we have used the expression already known for the behavior of the carrier density under pressure, Eq. (20). In our calculation, the initial carrier density was taken as n0 ¼ 1:64231 1020 m3 that corresponds to the plasma frequency op ¼ 0:785 THz. According with Eq. (18) we have found that increasing pressure, the plasma frequency decreases, resulting 0.7748 and 0.7284 THz for P¼10 kbar and P¼40 kbar, respectively. In Fig. 6 it is shown the behavior of the dielectric constant E2 ðP,T, oÞ as a function of the plasma frequency, which is modified by pressure according to Eq. (17). Note that for frequency values n ¼ 0:1159 THz and n ¼ 0:1233 THz, for P¼40 kbar and P¼10 kbar, respectively, the dielectric constant of the doped GaAs vanishes, i.e., its refractive index is null. On the other hand for a frequency value n ¼ 0:22 THz, the dielectric constant for P¼10 kbar and P¼40 kbar are the same; and for higher values of
´lez, N. Porras-Montenegro / Physica E 44 (2012) 773–777 L.E. Gonza
Fig. 7. Photonic band structure n vs q, with n ¼ o=2p of a superlattice composed of alternate layer of air (with thickness a ¼ 1 mm) and GaAs (thickness b ¼ 1 mm). This figure displays the behavior of the photonic band structure when varying the plasma frequency with pressure. Solid and dashed-dotted lines correspond to P ¼ 40 kbar and P ¼ 10 kbar, respectively. In the panel (b) the same results are zoomed indicating with black dashed-lines the values of n0 in which the refractive index of the n-doped GaAs is null.
777
function of the GaAs layers. We have found that the temperature dependence of the PBS is negligible as it has been found in 2D photonic crystals and that its changes are due to the variation of the width and the dielectric constant of the GaAs layers, due mainly to the applied hydrostatic pressure. On the other hand, by using the Drude’s model, we studied the effects of the hydrostatic pressure by means of the variation of the effective mass and density of the carriers in n-doped GaAs, finding that increasing the amount of the latter, that is, increasing the plasma frequency, the PBS is shifted to regions of higher frequencies, appearing two regimes of the photonic band structure, one above the plasma frequency with the presence of usual Bragg gaps, and the other, below this frequency, where there are no gaps regularly distributed, with their width diminishing with the increasing of the plasma frequency as well as with the appearance of more bands, but leaving a wide frequency range in the lowest part of the spectrum without accessible photon states. Also, we have found characteristic frequencies in which the dielectric constant equals for different applied pressures, and from which to higher or lower values, the PBS inverts its behavior, depending on the value of the applied hydrostatic pressure. We do hope these findings may be taken into account for future applications in the development of photonic devices.
Acknowledgments frequency, the dielectric constant for P¼10 kbar is greater than that for P ¼40 kbar, while for lower frequencies this situation inverts. As expected, the pressure dependence of the dielectric constant is apparent in the PBS, as it is shown in Fig. 7. In the region around the frequency for which the n-doped GaAs refractive index is null, n ¼ 0:1233 THz and n ¼ 0:1159 THz the gap tends to close, behavior that is also observed in Fig. 5. In a frequency n ¼ 0:22 THz , the energy bands practically coincide, frequency in which the dielectric constant for both pressure values is the same, as it was pointed out in Fig. 6. Finally, it is worth to mention that for values of frequency higher than n ¼ 0:22 THz, where the dielectric constants for P¼10 kbar and P¼ 40 kbar are reversed (see Fig. 6), the PBS corresponding to P¼10 kbar is shifted to higher energies than for P¼40 kbar, while for frequencies lower than n ¼ 0:22, the dielectric constant is higher for P ¼40 than for P¼10 kbar, and the PBS moves to lower frequencies, results that clearly obeys de electromagnetic variational principle. In other words, this inversion of the PBS with pressure obeys the variations of the dielectric constant with the plasma frequency (see Fig. 6), which depends on the applied hydrostatic pressure through the electron effective-mass, the dielectric layer width, and the carrier density according with Eq. (18).
4. Conclusions Within the transfer-matrix technique, we have studied the photonic band structures of a 1D photonic crystal consisting of a superlattice with two alternating layers of air and GaAs. We have examine the PBS dependence on both the applied hydrostatic pressure and temperature, taking into account the variations caused by this parameters on the width and the dielectric
This research was partially supported by Vicerrectora de Investigaciones at Universidad del Valle and CENM with funds of Patrimonio Autnomo Fondo Nacional de Financiamiento para la Ciencia, la Tecnologa y la Innovacin Francisco Jos de Caldas, grant RC - No. 275-2011.
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