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Intraband optical absorption in a single quantum ring: Hydrostatic pressure and intense laser field effects
Optics Communications 379 (2016) 41–44
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Intraband optical absorption in a single quantum ring: Hydrostatic pressure and intense laser field effects M.G. Barseghyan a,b,n a b
Department of Solid State Physics, Yerevan State University, Alex Manoogian 1, 0025 Yerevan, Armenia National University of Architecture and Construction of Armenia, Teryan 105, 0009 Yerevan, Armenia
art ic l e i nf o
a b s t r a c t
Article history: Received 24 March 2016 Received in revised form 6 May 2016 Accepted 23 May 2016
The intraband optical absorption in GaAs/Ga0.7Al0.3As two-dimensional single quantum ring is investigated. Considering the combined effects of hydrostatic pressure and intense laser field the energy of the ground and few excited states has been found using the effective mass approximation and exact diagonalization technique. The energies of these states and the corresponding threshold energy of the intraband optical transitions are examined as a function of hydrostatic pressure for the different values of the laser field parameter. We also investigated the dependencies of the intraband optical absorption coefficient as a function of incident photon energy for different values of hydrostatic pressure and laser field parameter. It is found that the effects of hydrostatic pressure and intense laser field lead to redshift and blueshift of the intraband optical spectrum respectively. & 2016 Elsevier B.V. All rights reserved.
Keywords: Intraband absorption Laser field Hydrostatic pressure Quantum ring
1. Introduction Self-assembled semiconductor quantum nanostructures, such as quantum dots (QDs) and quantum rings (QRs) have been investigated extensively given their potential as building blocks for a broad range of novel optoelectronic devices, e.g. nanoemitters, and for quantum information technologies [1–4]. The potential of these nanostructures is based on their remarkable similarity to atomic systems. Furthermore, what makes these nanostructures so attractive is the ability to tune their optoelectronic properties by carefully designing their size, composition, strain and shape. These parameters set the confinement potential of the charge carriers, thus determining the electronic and optical properties of a nanostructure. It is well known that external perturbations are useful tools to manipulate the electronic and optical properties of low-dimensional semiconductor nanostructures. In particular, it was theoretically proved that the nanostructures irradiated by intense laser field (ILF) exhibit characteristics very different from those of a bulk semiconductor and the radiation effects are more pronounced as the carriers’ confinement is increased by dimensionality reduction [5–11]. In addition, studies of the influences of the hydrostatic pressure on the electronic and impurity states have proven to be invaluable in the context of the optical properties of n Correspondence address: Department of Solid State Physics, Yerevan State University, Alex Manoogian 1, 0025 Yerevan, Armenia. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.optcom.2016.05.065 0030-4018/& 2016 Elsevier B.V. All rights reserved.
semiconductor heterostructures [12,13]. For a given structure, the difference in energy between the type-I and type-II transitions can be tuned with external hydrostatic pressure in the continuous and reversible manner. This makes possible an elucidation of the properties of various interband transitions. Simultaneous effects of hydrostatic pressure and ILF on electronic, impurity states and optical properties of semiconductor quantum wells and QDs have been investigated in Refs. [10,14–18]. The studies have shown that the electronic, impurity and optical properties of quantum wells and dots are strongly affected by the mentioned influences. Having this in motivation, the present work aims at the theoretical investigation of the combined influences of the hydrostatic pressure and ILF on one-electron states and intraband optical absorption coefficient in GaAs/Ga0,7Al0.3As twodimensional QR. The paper is organized as follows: in Section 2 we describe the theoretical framework. Section 3 is dedicated to the results and discussion, and our conclusions are given in Section 4.
2. Theoretical framework Without loss of generality, our system can be considered twodimensional, with the electron confined in the plane z ¼0 [19]: ⎧ if R1 (P ) ≤ x 2 + y 2 ≤ R2 (P ), ⎪ 0, V (x i + y j, P ) = ⎨ 2 2 2 2 ˜ ⎪ ⎩ V (X , P ), if x + y < R1 (P ), or x + y > R2 (P ).
where i and
(1)
j are the unit vectors along the x-axis of laser
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M.G. Barseghyan / Optics Communications 379 (2016) 41–44
Table 1 The parameters considered in the description of the Γ and X energy gap functions. Γ-minimum
X-minimum
aΓ = 1519.36 rmmeV bΓ = 1360 rmmeV cΓ = 220 rmmeV αΓ = 10.7 rmmeV /kbar
aX = 1981 rmmeV bX = 207 rmmeV cX = 55 rmmeV αX = − 1.35 meV/kbar
polarization and y-axis respectively, X˜ is the aluminum concentration, R1 (P ) and R2 (P ) are the pressure dependent inner and outer radii respectively of the QR given by [20]:
R1,2 (P ) = R1,2 (0)[1 − 2 (S11 + 2S12 ) P ]1/2 ,
(2)
where S11 and S12 are the components of the compliance tensor of GaAs [21] ( S11 = 1.16 × 10−3 kbar−1 and S12 = − 3.7 × 10−4 kbar−1). R1 (0) and R2 (0) are the dimensions of the structure at zero hydrostatic pressure. In Eq. (1) V (X˜ , P ) is given by ⎧ E Γ (X˜ , P ) − E Γ (0, P ), if P ≤ P1 (X˜ ), ⎪ g g V (X˜ , P ) = r ⎨ X Γ ˜ ˜ ⎪ Eg (X , P ) − Eg (0, P ) + SΓX (X , P ), if P1 (X˜ ) < P ≤ P 2 (X˜ ). ⎩
(4)
The values of the parameters ai , bi , ci, and αi are shown in the Table 1 [22]. The pressure P1 (X˜ ) is the value corresponding to the crossover between the Γ and X bands minima in Ga1 − X˜ Al X˜ As, and the pressure P2 (X˜ ) is its equivalent with respect to the crossover between the Γ-band minimum at the GaAs well and the X-band minimum at the Ga1 − X˜ Al X˜ As barrier, and the expressions of pressure and aluminum concentration dependent band gaps at Γ and X minima can be found in [21]. Besides,
SΓX (X˜ , P ) = S0
P − P1 (X˜ ) ˜ X P
(5)
is the Γ − X mixing strength coefficient and S0 is the adjustable parameter which fits the experimental measurements [23,24]. We assume the system to be under the action of laser radiation represented by a monochromatic plane wave of frequency ω0. The laser beam is non-resonant with the QR structure, and linearly polarized along a radial direction (oriented along the x-axis). In the high-frequency regime the particle is subjected to the time-averaged potential [5,25]
Vd (x, y , P ) =
ω0 2π
∫0
2π / ω 0
V ((x + α0 sin (ω 0 t )) i + y j, P ) dt
(6)
where α0 is the laser field parameter. Taking into account (Eqs. (1), 3) and (5) one may obtain a closed analytical form of Vd (x, y, P ), as in [26]. Taking for simplicity the same effective mass of the electron inside and outside the QR, the laser-dressed energies are obtained from the time-independent Schrödinger equation [27–29]
⎤ ⎡ = 2 ⎛ ∂2 ∂2 ⎞ ⎢− + ⎜ ⎟ + Vd (x, y , P ) ⎥ ∂y2 ⎠ ⎦ ⎣ 2m (P ) ⎝ ∂x2 Φd (x, y , P ) = Ed (P ) Φd (x, y , P ),
α (=ω) =
16π 2βFS =ω Nif |Mfi |2 δ (Ef − Ei − =ω), ε (P )1/2V
(8)
where ε (P ) is the pressure dependent static dielectric constant of the material [35], V is the volume of the sample per QR (in this work V = 6 × 10−18 cm3) [36], βFS is the fine structure constant, =ω is the incident photon energy and Ef and Ei are the energies of the final and initial states, respectively. Nif = Ni − Nf is the difference between the number of electrons in the initial and final states. Since we consider a one-particle problem, we assume Ni = 1 for the ground state and Nf = 0 for all upper states. Mfi is the matrix element of coordinate and the δ-function is substituted by a Lorentzian profile with a full width at half maximum of 0.8 meV [34,37,38].
(3)
Here, r (¼0.6) is the fraction of the band gap discontinuity, X˜ is the aluminum concentration. The energy gap function at the i-point (i = Γ , X ) of the conduction band is given by 2 Egi (X˜ , P ) = ai + bi X˜ + ci X˜ + αi P .
diagonalization technique [30–32]. The light absorption process can be described as an optical transition that takes place from an initial state to a final one assisted by a photon. The optical absorption calculations for the intraband transitions are based on Fermi's golden rule derived from time-dependent perturbation theory [33,34]:
(7)
where m(P) is the pressure dependent effective mass of the electron [21]. The laser-dressed energy eigenvalues Ed(P) and eigenfunctions Φd (x, y, P ) may be calculated by solving Eq. (6) with a 2D
3. Results and discussion In the calculations the aluminum concentration has been taken as X˜ = 0.3. It is worth to note that the appropriate choices of pressure-dependent energy gaps, structure dimensions, electron effective mass and static dielectric constant of the structure used in numerical calculations in this work, guarantee good agreement between theoretical calculations and experimental measurements [22,24]. In Figs. 1(a)–(c) are shown the dependencies on hydrostatic pressure of the ground and few excited state energies for different values of the laser field parameter α0. We see that all energies are decreasing functions of pressure and such a behavior can be explained as follows: for P ≤ P1 the radial-confining potential height is constant (see Ref. [21]) and the energy decrease is only due to the increase of the electron effective mass with pressure, but if P1 < P < P2 there will be a faster decrease in the electron energy with pressure, since in addition to the decrease associated to the growth in the value of the electron effective mass, now there is an extra reduction in the energy, that is associated with the fall in the radial-potential barrier height. On the other hand, Fig. 1(a) shows that in the absence of the laser field (α0 = 0), the first two excited levels are twice degenerated. The laser field removes this degeneracy by breaking the initial axial symmetry (see Figs. 1(b) and (c)). It should be mentioned that with the increase of laser field the effective length (the dressing effect of the laser ”reshapes” the confinement potential by enlarging the QR) of the confinement potential (for low lying states) along the laser field polarization (x-direction) decreases. For this reason, in all investigated cases the laser field brings the increment of the dressed energies. Fig. 2(a)–(c) contains the threshold energy associated with the transition between the ground state and excited states varying as a function of hydrostatic pressure, for different values of the laser field parameter α0. We can see that the threshold energy is a decreasing function of the hydrostatic pressure for all transitions except for 1 → 2 transition in the case of α0 = 5 nm , where it has both decreasing and increasing behaviors. This is associated with the change in the rate of decreasing of the ground and first excited state energies, which can be seen in Fig. 1(c). Meanwhile, the threshold energy of 1 → 3 transition increases with the augment of the laser field parameter, which can also be understood from the energy spectrum shown in Fig. 1. Note that, by analysing the parities of the laser dressed wave
M.G. Barseghyan / Optics Communications 379 (2016) 41–44
43
Fig. 1. The energy levels of electron as a function of hydrostatic pressure P. The results are presented for R1 (0) = 5nm and R2 (0) = 25nm. Several values of the laser field parameter α0 are considered.
Fig. 2. Dependencies of threshold energies on hydrostatic pressure P. The results are presented for R1 (0) = 5 nm and R2 (0) = 25 nm . Several values of laser field parameter α0 are considered.
functions, the selection rules have been obtained in Refs. [26,34]. It has been observed that in the case x-polarization of light, the transitions from the ground state to the second excited state (1 → 3) are allowed. Considering this, in Fig. 3, the incident photon energy dependent intraband absorption coefficient is presented for various values of the laser field parameter α0 and hydrostatic pressure P, and only x-polarization of the light is considered. Here the increment of hydrostatic pressure creates the redshift of absorption spectrum. This happens because with the increase of the hydrostatic pressure, the threshold energy of intraband 1 → 3 transition decreases (see Fig. 2). In contrast to it, the increase of laser field parameter a blueshift is observed, due to the corresponding behavior of the threshold energy.
4. Conclusions In this article we have studied the combined influence of hydrostatic pressure and intense laser field on the intraband optical absorption in GaAs/Ga0.7Al0.3 two-dimensional single quantum ring. Our results show that the behavior of the threshold energy and as follows the position of the maximum of the intraband optical absorption, caused by the transitions from ground state to the corresponding excited state strongly depends on the hydrostatic pressure and intense laser field parameter. Additionally, we have found that the optical absorption is affected by the considered influences. The present results can be useful to understand the electro-optical properties of ring-like structures under the
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M.G. Barseghyan / Optics Communications 379 (2016) 41–44
References
Fig. 3. Dependence of the intraband optical absorption coefficient on incident photon energy in single QR. The results are presented for R1 (0) = 5 nm and R2 (0) = 25 nm . Several values of hydrostatic pressure P and laser field parameter α0 are considered.
influences of hydrostatic pressure and intense laser field.
Acknowledgments The author thanks A.A. Kirakosyan and H.M. Baghramyan for useful discussions. The work was supported by the Armenian State Committee of Science Project (no. 15T-1C331) and Armenian National Science and Education Fund (ANSEF Grant no. 4199).
[1] T. Chakraborty, Quantum Dots, Elsevier, Amsterdam, 1999. [2] P. Michler, A. Kiraz, C. Becher, W.V. Schoenfeld, P.M. Petroff, L. Zhang, E. Hu, A. Imamoglu, Science 290 (2000) 2282. [3] X. Li, Y. Wu, D. Steel, D. Gammon, T.H. Stievater, D.S. Katzer, D. Park, C. Piermarocchi, L.J. Sham, Science 301 (2003) 809. [4] V.M. Fomin, Physics of Quantum Rings, Springer-Verlag, Berlin, Heidelberg, 2014. [5] Qu Fanyao, A.L.A. Fonseca, O.A.C. Nunes, Phys. Rev. B 54 (1996) 16405. [6] H.S. Brandi, A. Latgé, L.E. Oliveira, Solid State Commun. 107 (1998) 31. [7] H. Sari, E. Kasapoglu, I. Sökmen, M. Güneş, Phys. Lett. A 319 (2003) 211. [8] E.C. Niculescu, L.M. Burileanu, A. Radu, Superlattices Microstruct. 44 (2008) 173. [9] A. John Peter, J. Comput. Theor. Nanosci. 6 (2009) 1702. [10] C.A. Duque, E. Kasapoglu, S. Sakiroglu, H. Sari, I. Sökmen, Appl. Surf. Sci. 256 (2010) 7406. [11] A. Radu, E.C. Niculescu, Phys. Lett. A 374 (2010) 1755. [12] M. Chandrasekhar, H.R. Chandrasekhar, High Pressure Res. 9 (1992) 57. [13] M. Chandrasekhar, H.R. Chandrasekhar, Phil. Mag. B 70 (1994) 369. [14] N. Eseanu, Phys. Lett. A 374 (2010) 1278. [15] U. Yesilgul, F. Ungan, E. Kasapoglu, H. Sari, I. Sökmen, Chin. Phys. Lett. 28 (2011) 077102. [16] F. Ungan, U. Yesilgul, E. Kasapoglu, H. Sari, I. Sökmen, Opt. Commun. 285 (2012) 373. [17] F. Ungan, U. Yesilgul, S. Sakiroglu, M.E. Mora-Ramos, C.A. Duque, E. Kasapoglu, H. Sari, I. Sökmen, Opt. Commun. 309 (2013) 158. [18] E.C. Niculescu, Phys. E 63 (2014) 105. [19] T. Chakraborty, Adv. Solid State Phys. 43 (2003) 79. [20] F.J. Culchac, N. Porras-Montenegro, A. Latgé, J. Appl. Phys. 105 (2009) 094324. [21] H.M. Baghramyan, M.G. Barseghyan, A.A. Kirakosyan, R.L. Restrepo, M.E. MoraRamos, C.A. Duque, J. Lumin. 145 (2014) 676. [22] E. Reyes-Gómez, N. Raigoza, L.E. Oliveira, Phys. Rev. B 77 (2008) 115308. [23] A.M. Elabsy, Phys. Scr. 48 (1993) 376. [24] A.M. Elabsy, J. Phys.: Condens. Matter 6 (1994) 10025. [25] E.C. Niculescu, L.M. Burileanu, A. Radu, A. Lupaşcu, J. Lumin. 131 (2011) 1113. [26] A. Radu, A.A. Kirakosyan, D. Laroze, H.M. Baghramyan, M.G. Barseghyan, J. Appl. Phys. 116 (2014) 093101. [27] Q. Fanyao, P.C. Morais, Phys. Lett. A 310 (2003) 460. [28] L.M. Burileanu, A. Radu, Opt. Commun. 284 (2011) 2050. [29] M.G. Barseghyan, C.A. Duque, E.C. Niculescu, A. Radu, Superlattices Microstruct. 66 (2014) 10. [30] S. Gangopadhyay, B.R. Nag, Phys. Stat. Sol. (b) 195 (1996) 123. [31] A. Tiutiunnyk, V. Tulupenko, M.E. Mora-Ramos, E. Kasapoglu, F. Ungan, H. Sari, I. Sökmen, C.A. Duque, Physica E 60 (2014) 127. [32] H.M. Baghramyan, M.G. Barseghyan, D. Laroze, A.A. Kirakosyan, Physica E 77 (2016) 81. [33] M. Şahin, K. Köksal, Semicond. Sci. Technol. 27 (2012) 125011. [34] A. Radu, A.A. Kirakosyan, D. Laroze, M.G. Barseghyan, Semicond. Sci. Technol. 30 (2015) 045006. [35] M.G. Barseghyan, Alireza Hakimyfard, S.Y. López, C.A. Duque, A.A. Kirakosyan, Physica E 42 (2010) 1618. [36] A. Lorke, R.J. Luyken, A.O. Govorov, J.P. Kotthaus, J.M. Garcia, P.M. Petroff, Phys. Rev. Lett. 84 (2000) 2223. [37] P.K. Basu, Theory of Optical Processes in Semiconductors, Clarendon Press, Oxford, 1997. [38] S.V. Gaponenko, Optical Properties of Semiconductor Nanocrystals, Cambridge University Press, Cambridge, UK, 2005.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Intraband optical absorption in a single quantum ring: Hydrostatic pressure and intense laser field effects M.G. Barseghyan a,b,n a b
Department of Solid State Physics, Yerevan State University, Alex Manoogian 1, 0025 Yerevan, Armenia National University of Architecture and Construction of Armenia, Teryan 105, 0009 Yerevan, Armenia
art ic l e i nf o
a b s t r a c t
Article history: Received 24 March 2016 Received in revised form 6 May 2016 Accepted 23 May 2016
The intraband optical absorption in GaAs/Ga0.7Al0.3As two-dimensional single quantum ring is investigated. Considering the combined effects of hydrostatic pressure and intense laser field the energy of the ground and few excited states has been found using the effective mass approximation and exact diagonalization technique. The energies of these states and the corresponding threshold energy of the intraband optical transitions are examined as a function of hydrostatic pressure for the different values of the laser field parameter. We also investigated the dependencies of the intraband optical absorption coefficient as a function of incident photon energy for different values of hydrostatic pressure and laser field parameter. It is found that the effects of hydrostatic pressure and intense laser field lead to redshift and blueshift of the intraband optical spectrum respectively. & 2016 Elsevier B.V. All rights reserved.
Keywords: Intraband absorption Laser field Hydrostatic pressure Quantum ring
1. Introduction Self-assembled semiconductor quantum nanostructures, such as quantum dots (QDs) and quantum rings (QRs) have been investigated extensively given their potential as building blocks for a broad range of novel optoelectronic devices, e.g. nanoemitters, and for quantum information technologies [1–4]. The potential of these nanostructures is based on their remarkable similarity to atomic systems. Furthermore, what makes these nanostructures so attractive is the ability to tune their optoelectronic properties by carefully designing their size, composition, strain and shape. These parameters set the confinement potential of the charge carriers, thus determining the electronic and optical properties of a nanostructure. It is well known that external perturbations are useful tools to manipulate the electronic and optical properties of low-dimensional semiconductor nanostructures. In particular, it was theoretically proved that the nanostructures irradiated by intense laser field (ILF) exhibit characteristics very different from those of a bulk semiconductor and the radiation effects are more pronounced as the carriers’ confinement is increased by dimensionality reduction [5–11]. In addition, studies of the influences of the hydrostatic pressure on the electronic and impurity states have proven to be invaluable in the context of the optical properties of n Correspondence address: Department of Solid State Physics, Yerevan State University, Alex Manoogian 1, 0025 Yerevan, Armenia. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.optcom.2016.05.065 0030-4018/& 2016 Elsevier B.V. All rights reserved.
semiconductor heterostructures [12,13]. For a given structure, the difference in energy between the type-I and type-II transitions can be tuned with external hydrostatic pressure in the continuous and reversible manner. This makes possible an elucidation of the properties of various interband transitions. Simultaneous effects of hydrostatic pressure and ILF on electronic, impurity states and optical properties of semiconductor quantum wells and QDs have been investigated in Refs. [10,14–18]. The studies have shown that the electronic, impurity and optical properties of quantum wells and dots are strongly affected by the mentioned influences. Having this in motivation, the present work aims at the theoretical investigation of the combined influences of the hydrostatic pressure and ILF on one-electron states and intraband optical absorption coefficient in GaAs/Ga0,7Al0.3As twodimensional QR. The paper is organized as follows: in Section 2 we describe the theoretical framework. Section 3 is dedicated to the results and discussion, and our conclusions are given in Section 4.
2. Theoretical framework Without loss of generality, our system can be considered twodimensional, with the electron confined in the plane z ¼0 [19]: ⎧ if R1 (P ) ≤ x 2 + y 2 ≤ R2 (P ), ⎪ 0, V (x i + y j, P ) = ⎨ 2 2 2 2 ˜ ⎪ ⎩ V (X , P ), if x + y < R1 (P ), or x + y > R2 (P ).
where i and
(1)
j are the unit vectors along the x-axis of laser
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M.G. Barseghyan / Optics Communications 379 (2016) 41–44
Table 1 The parameters considered in the description of the Γ and X energy gap functions. Γ-minimum
X-minimum
aΓ = 1519.36 rmmeV bΓ = 1360 rmmeV cΓ = 220 rmmeV αΓ = 10.7 rmmeV /kbar
aX = 1981 rmmeV bX = 207 rmmeV cX = 55 rmmeV αX = − 1.35 meV/kbar
polarization and y-axis respectively, X˜ is the aluminum concentration, R1 (P ) and R2 (P ) are the pressure dependent inner and outer radii respectively of the QR given by [20]:
R1,2 (P ) = R1,2 (0)[1 − 2 (S11 + 2S12 ) P ]1/2 ,
(2)
where S11 and S12 are the components of the compliance tensor of GaAs [21] ( S11 = 1.16 × 10−3 kbar−1 and S12 = − 3.7 × 10−4 kbar−1). R1 (0) and R2 (0) are the dimensions of the structure at zero hydrostatic pressure. In Eq. (1) V (X˜ , P ) is given by ⎧ E Γ (X˜ , P ) − E Γ (0, P ), if P ≤ P1 (X˜ ), ⎪ g g V (X˜ , P ) = r ⎨ X Γ ˜ ˜ ⎪ Eg (X , P ) − Eg (0, P ) + SΓX (X , P ), if P1 (X˜ ) < P ≤ P 2 (X˜ ). ⎩
(4)
The values of the parameters ai , bi , ci, and αi are shown in the Table 1 [22]. The pressure P1 (X˜ ) is the value corresponding to the crossover between the Γ and X bands minima in Ga1 − X˜ Al X˜ As, and the pressure P2 (X˜ ) is its equivalent with respect to the crossover between the Γ-band minimum at the GaAs well and the X-band minimum at the Ga1 − X˜ Al X˜ As barrier, and the expressions of pressure and aluminum concentration dependent band gaps at Γ and X minima can be found in [21]. Besides,
SΓX (X˜ , P ) = S0
P − P1 (X˜ ) ˜ X P
(5)
is the Γ − X mixing strength coefficient and S0 is the adjustable parameter which fits the experimental measurements [23,24]. We assume the system to be under the action of laser radiation represented by a monochromatic plane wave of frequency ω0. The laser beam is non-resonant with the QR structure, and linearly polarized along a radial direction (oriented along the x-axis). In the high-frequency regime the particle is subjected to the time-averaged potential [5,25]
Vd (x, y , P ) =
ω0 2π
∫0
2π / ω 0
V ((x + α0 sin (ω 0 t )) i + y j, P ) dt
(6)
where α0 is the laser field parameter. Taking into account (Eqs. (1), 3) and (5) one may obtain a closed analytical form of Vd (x, y, P ), as in [26]. Taking for simplicity the same effective mass of the electron inside and outside the QR, the laser-dressed energies are obtained from the time-independent Schrödinger equation [27–29]
⎤ ⎡ = 2 ⎛ ∂2 ∂2 ⎞ ⎢− + ⎜ ⎟ + Vd (x, y , P ) ⎥ ∂y2 ⎠ ⎦ ⎣ 2m (P ) ⎝ ∂x2 Φd (x, y , P ) = Ed (P ) Φd (x, y , P ),
α (=ω) =
16π 2βFS =ω Nif |Mfi |2 δ (Ef − Ei − =ω), ε (P )1/2V
(8)
where ε (P ) is the pressure dependent static dielectric constant of the material [35], V is the volume of the sample per QR (in this work V = 6 × 10−18 cm3) [36], βFS is the fine structure constant, =ω is the incident photon energy and Ef and Ei are the energies of the final and initial states, respectively. Nif = Ni − Nf is the difference between the number of electrons in the initial and final states. Since we consider a one-particle problem, we assume Ni = 1 for the ground state and Nf = 0 for all upper states. Mfi is the matrix element of coordinate and the δ-function is substituted by a Lorentzian profile with a full width at half maximum of 0.8 meV [34,37,38].
(3)
Here, r (¼0.6) is the fraction of the band gap discontinuity, X˜ is the aluminum concentration. The energy gap function at the i-point (i = Γ , X ) of the conduction band is given by 2 Egi (X˜ , P ) = ai + bi X˜ + ci X˜ + αi P .
diagonalization technique [30–32]. The light absorption process can be described as an optical transition that takes place from an initial state to a final one assisted by a photon. The optical absorption calculations for the intraband transitions are based on Fermi's golden rule derived from time-dependent perturbation theory [33,34]:
(7)
where m(P) is the pressure dependent effective mass of the electron [21]. The laser-dressed energy eigenvalues Ed(P) and eigenfunctions Φd (x, y, P ) may be calculated by solving Eq. (6) with a 2D
3. Results and discussion In the calculations the aluminum concentration has been taken as X˜ = 0.3. It is worth to note that the appropriate choices of pressure-dependent energy gaps, structure dimensions, electron effective mass and static dielectric constant of the structure used in numerical calculations in this work, guarantee good agreement between theoretical calculations and experimental measurements [22,24]. In Figs. 1(a)–(c) are shown the dependencies on hydrostatic pressure of the ground and few excited state energies for different values of the laser field parameter α0. We see that all energies are decreasing functions of pressure and such a behavior can be explained as follows: for P ≤ P1 the radial-confining potential height is constant (see Ref. [21]) and the energy decrease is only due to the increase of the electron effective mass with pressure, but if P1 < P < P2 there will be a faster decrease in the electron energy with pressure, since in addition to the decrease associated to the growth in the value of the electron effective mass, now there is an extra reduction in the energy, that is associated with the fall in the radial-potential barrier height. On the other hand, Fig. 1(a) shows that in the absence of the laser field (α0 = 0), the first two excited levels are twice degenerated. The laser field removes this degeneracy by breaking the initial axial symmetry (see Figs. 1(b) and (c)). It should be mentioned that with the increase of laser field the effective length (the dressing effect of the laser ”reshapes” the confinement potential by enlarging the QR) of the confinement potential (for low lying states) along the laser field polarization (x-direction) decreases. For this reason, in all investigated cases the laser field brings the increment of the dressed energies. Fig. 2(a)–(c) contains the threshold energy associated with the transition between the ground state and excited states varying as a function of hydrostatic pressure, for different values of the laser field parameter α0. We can see that the threshold energy is a decreasing function of the hydrostatic pressure for all transitions except for 1 → 2 transition in the case of α0 = 5 nm , where it has both decreasing and increasing behaviors. This is associated with the change in the rate of decreasing of the ground and first excited state energies, which can be seen in Fig. 1(c). Meanwhile, the threshold energy of 1 → 3 transition increases with the augment of the laser field parameter, which can also be understood from the energy spectrum shown in Fig. 1. Note that, by analysing the parities of the laser dressed wave
M.G. Barseghyan / Optics Communications 379 (2016) 41–44
43
Fig. 1. The energy levels of electron as a function of hydrostatic pressure P. The results are presented for R1 (0) = 5nm and R2 (0) = 25nm. Several values of the laser field parameter α0 are considered.
Fig. 2. Dependencies of threshold energies on hydrostatic pressure P. The results are presented for R1 (0) = 5 nm and R2 (0) = 25 nm . Several values of laser field parameter α0 are considered.
functions, the selection rules have been obtained in Refs. [26,34]. It has been observed that in the case x-polarization of light, the transitions from the ground state to the second excited state (1 → 3) are allowed. Considering this, in Fig. 3, the incident photon energy dependent intraband absorption coefficient is presented for various values of the laser field parameter α0 and hydrostatic pressure P, and only x-polarization of the light is considered. Here the increment of hydrostatic pressure creates the redshift of absorption spectrum. This happens because with the increase of the hydrostatic pressure, the threshold energy of intraband 1 → 3 transition decreases (see Fig. 2). In contrast to it, the increase of laser field parameter a blueshift is observed, due to the corresponding behavior of the threshold energy.
4. Conclusions In this article we have studied the combined influence of hydrostatic pressure and intense laser field on the intraband optical absorption in GaAs/Ga0.7Al0.3 two-dimensional single quantum ring. Our results show that the behavior of the threshold energy and as follows the position of the maximum of the intraband optical absorption, caused by the transitions from ground state to the corresponding excited state strongly depends on the hydrostatic pressure and intense laser field parameter. Additionally, we have found that the optical absorption is affected by the considered influences. The present results can be useful to understand the electro-optical properties of ring-like structures under the
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M.G. Barseghyan / Optics Communications 379 (2016) 41–44
References
Fig. 3. Dependence of the intraband optical absorption coefficient on incident photon energy in single QR. The results are presented for R1 (0) = 5 nm and R2 (0) = 25 nm . Several values of hydrostatic pressure P and laser field parameter α0 are considered.
influences of hydrostatic pressure and intense laser field.
Acknowledgments The author thanks A.A. Kirakosyan and H.M. Baghramyan for useful discussions. The work was supported by the Armenian State Committee of Science Project (no. 15T-1C331) and Armenian National Science and Education Fund (ANSEF Grant no. 4199).
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