Effect of intense high-frequency laser field on the linear and nonlinear intersubband optical absorption coefficients and refractive index changes in a parabolic quantum well under the applied electric field

Author's Accepted Manuscript

Effect of intense high-frequency laser field on the linear and nonlinear intersubband optical absorption coefficients and refractive index changes in a parabolic quantum well under the applied electric field U. Yesilgul, F. Ungan, S. Sakiroglu, M.E. MoraRamos, C.A. Duque, E. Kasapoglu, H. Sarı, I. Sökmen www.elsevier.com/locate/jlumin

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S0022-2313(13)00454-7 http://dx.doi.org/10.1016/j.jlumin.2013.07.062 LUMIN12065

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Journal of Luminescence

Received date: 19 February 2013 Revised date: 27 May 2013 Accepted date: 16 July 2013 Cite this article as: U. Yesilgul, F. Ungan, S. Sakiroglu, M.E. Mora-Ramos, C.A. Duque, E. Kasapoglu, H. Sarı, I. Sökmen, Effect of intense high-frequency laser field on the linear and nonlinear intersubband optical absorption coefficients and refractive index changes in a parabolic quantum well under the applied electric field, Journal of Luminescence, http://dx.doi.org/10.1016/j.jlumin.2013.07.062 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Effect of intense high-frequency laser field on the linear and nonlinear intersubband optical absorption coefficients and refractive index changes in a parabolic quantum well under the applied electric field U. Yesilgul1,*, F. Ungan1, S. Sakiroglu2, M. E. Mora-Ramos3, C. A. Duque4, E. Kasapoglu1, H. Sarı1 and I. Sökmen2 1 2 3

Cumhuriyet University, Physics Department, 58140 Sivas, Turkey

Dokuz Eylül University, Physics Department, 35160 Buca, İzmir, Turkey

Facultad de Ciencias Universidad Autonoma del Estado de Morelos, Ave. Universidad 1001, CP. 62209 Cuernavaca, Morelos, Mexico 3

Instituto de Fisica, Universidad de Antioquia, AA 1226 Medellin, Colombia Abstract

The effect of the intense high-frequency laser field on the optical absorption coefficients and the refractive index changes in a GaAs/GaAlAs parabolic quantum well under the applied electric field have investigated theoretically. The electron energy levels and the envelope wave functions of the parabolic quantum well are calculated within the effective mass approximation. The analytical expressions for optical properties are obtained using the compact density-matrix approach. The numerical results show that the intense high-frequency laser field has a great effect on the optical characteristics of these structures. Also we can observe that the refractive index and absorption coefficient changes are very sensitive to the electric field in large well dimension. Thus, this result gives a new degree of freedom in the optoelectronic device applications. Keywords: Parabolic quantum well, Nonlinear optical property, Intense laser field, Electric field. PACS: 73.21.Fg, 78.66.Fd, 78.67.De Corresponding Author: Ünal Yeşilgül E-mail: [email protected] Phone: +90–346–2191010–1523 Fax: +90-346-2191186

1

1. Introduction The nonlinear optical properties in the semiconductor heterostructures can be used for

practical applications in photodetectors and high-speed electro-optical devices. The linear and nonlinear optical absorptions based on intersubband transitions (ISBTs) and the refractive index changes in low-dimensional semiconductor quantum systems have been studied by many researchers in recent years [1-14]. Baskoutas et al. calculated the nonlinear optical rectification coefficients of the quantum dot structure [1]. Karabulut et al. studied only theoretical calculations for the linear and nonlinear optical absorptions associated with ISBTs in asymmetric rectangular quantum well [2]. Karabulut and Baskoutas investigated the influences of impurities, applied electric field, and incident optical intensity on the linear and nonlinear optical absorptions in spherical quantum dots [3]. Chen et al. discussed the linear, third-order nonlinear, and total optical absorptions in the GaAs/GaAlAs double triangular quantum wells [4]. The optical properties of symmetric double semi-parabolic quantum confinement potential have been investigated by Keshavarz and Karimi [5]. The linear and third-order nonlinear optical absorption coefficients for 1s–1p, 1p–1d and 1d–1f transitions of the spherical quantum dot have been studied Yakar et al. [6]. Karabulut investigated the effect of the laser on the nonlinear optical properties of a square quantum well under the applied electric field [7]. Khordad and Khaneghah calculated the intersubband optical absorption coefficients

and the refractive index changes in a V-groove quantum wire [8]. Kasapoglu et al. studied intense laser field effect on the linear and nonlinear intersubband optical properties of a semiparabolic quantum well [9]. Karabulut and Duque calculated the electron-related nonlinear optical absorption and nonlinear optical rectification in GaAs-GaAlAs asymmetric double quantum wells under applied electric and magnetic field [10]. The exciton-related optical absorption and the change of the refractive index in a GaAs-GaAlAs double quantum well (DQW) as functions of the geometric parameters of the heterostructure have been investigated by Miranda et al. [11]. It is well known that a high-frequency intense laser field (ILF) considerably affects the optical and electronic properties of low dimensional semiconductor heterostructures. It has been reported that the potential of an electronic system irradiated by an ILF is modified which affects significantly the bound state energy levels, a feature that has been observed in transition energy experiments. Therefore, there have been performed many studies investigating the effect of ILF on low-dimensional systems [15–27]. Eseanu et al. investigated

2

the combined effect of the hydrostatic pressure and high-frequency ILF on the binding energy of a hydrogenic impurity in square and parabolic quantum wells (POWs) [15]. Lima et al. studied nonresonant ILF affects the potential well and the corresponding bound states for electrons in a single semiconductor quantum well [22]. Theoretical calculations for the band structure of semiconductor superlattice under intense high-frequency ILF have been investigated by Sakiroglu et al. [24]. In this letter, we investigated the effects of an intense high-frequency laser field on the linear and nonlinear optical properties of a GaAs/GaAlAs PQW under the applied electric field. The paper is organized as follows; In Section 2, we described the theoretical framework, Section 3 is dedicated to results and discussion. Finally, the conclusions are given in Section 4. 2. Theory The method used in the present calculations is based upon a non-perturbative theory that was developed to describe the atomic behavior under intense high-frequency laser field conditions, and it has already been given elsewhere, we will not enter into details here [2428]. Within the framework of the effective mass approximation, the Hamiltonian for the electron in the presence of an intense high-frequency laser field (the laser-field polarization is along the z direction) is given by

H =−

=2 ∂2 + Vb (α 0 , z ) + eFz 2m* ∂z 2

(1) where m* is the effective mass, F is the external applied electric field in the z direction,

α0 =

eF0 is the laser dressing parameter, F0 is the field strength, ω is the frequency of the m*ω 2

laser and Vb (α 0 , z ) is the ‘dressed’ confinement potential which is given by the following expression ;

3

V ⎛

V (α 0 , z ) = 0 ⎜ ArcCos[ b π ⎝

z+L/2

α0

]Θ[α 0 − L / 2 − z ]Θ[α 0 + L / 2 + z ] + ArcCos[

−z + L / 2

α0

]Θ[α 0 + L / 2 − z ]Θ[α 0 − L / 2

+ V0 ( 2 − Θ[α 0 + L / 2 + z ] − Θ[α 0 + L / 2 − z ]) + (1 − Θ[−α 0 − L / 2 − z ] − Θ[−α 0 − L / 2 + z ])

V0 ( z 2 + α 02 / 2) ( L / 2) 2

⎛ V0 ( z 2 + α 02 / 2) ⎞ ⎟ 2 ⎝ π ( L / 2) ⎠

( −Θ[α 0 − L / 2 − z ]Θ[α 0 + L / 2 + z ] − Θ[α 0 + L / 2 − z ]Θ[α 0 − L / 2 + z ]) ⎜ +

( z 2 + α 02 / 2) ⎛ (− L / 2 − z ) (L / 2 − z) ⎞ ] − ArcSin[ ] ⎟ + ( L / 2) ( L / 2 − z ) 2 + α 02 ⎜ ArcSin[ 2π α α 0 0 ⎝ ⎠

(2) where V0 is the conduction band offset, Θ is the step function and L is the quantum well width. The energy levels E and the corresponding wave functions ψ ( z ) in a PQW can be obtained by solving the Schrödinger equation. The time independent Schrödinger equation in one-dimensional case as Hψ ( z ) = Eψ ( z )

(3)

where ψ ( z ) is the wave function of electron, which is exactly obtained from Schrödinger equation in z-direction. Once the energies and their corresponding wave functions are obtained, the linear and the third-order nonlinear refractive index changes for the ISBTs between two subbands can be calculated as [29]

⎡ ⎤ E21 − =ω 1 Δn(1) (ω ) 2 = 2 M 21 σ v ⎢ 2 2⎥ 2nr ε 0 nr ⎣ ( E21 − =ω ) + (=Γ12 ) ⎦

(4-a)

⎡ σvI ( M 22 − M 11 ) 2 Δn (3) (ω ) μc 2 2 E M = − 3 M 21 × − = − 4( ) ω 21 21 ⎢ 2 nr 4nr ε 0 ( E21 ) 2 + (=Γ12 ) 2 ( ( E21 − =ω )2 + (=Γ12 )2 ) ⎣

{

× ( E21 − =ω ) × [ E21 ( E21 − =ω ) − (=Γ12 ) 2 ] − (=Γ12 ) 2 2( E21 − =ω )}⎤⎦

(4-b)

where nr is the refractive index, σ v is the electron density, μ is the permeability, ε 0 is the permittivity of free space, Γ12 is the relaxation rate for states 1 and 2, and I is the incident optical intensity. E21 = E2 − E1 is the energy of the interval of the final and initial states, and M ij is the matrix element is defined by M ij = ψ i ( z ) ez ψ j ( z ) ,

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(i, j = 1, 2) .

(5)

Then, the total refractive index change can be written as

Δn(ω ) Δn (1) (ω ) Δn (2) (ω ) . = + nr nr nr

(6)

The linear and the third-order nonlinear optical absorption coefficient for the ISBTs between two subbands can be derived as follows [29] 2

β (1) (ω ) = ω

M 21 σ v =Γ12 μ , ε R ( E21 − =ω ) 2 + (=Γ12 ) 2 4

β (3) (ω , I ) = −2ω

M 21 σ v =Γ12 M − M 11 μ ⎛ I ⎞ 1 − 22 ⎜ ⎟× 2 2 2 2 [ ε R ⎝ ε 0 nr c ⎠ [( E21 − =ω ) + (=Γ12 ) ] 2M 21

2 [3E212 − 4 E21=ω + = 2 (ω 2 − Γ12 )] ⎤ × ⎥ 2 2 E21 + (=Γ12 ) ⎦

(7-a) 2

(7-b)

where ε R is the real part of the permittivity and total absorption coefficient α (ω , I ) can be written as

β (ω , I ) = β (1) (ω ) + β (3) (ω , I ) .

(8)

3. Results and discussion

In this study, we have firstly solved numerically the Schrödinger equation to investigate ILF effects on the linear, the third-order nonlinear, and total absorption coefficient and refractive index changes in a GaAs / Al0.3Ga0.7 As PQW under the applied electric field. In the values of the physical parameters used in our calculations are σ v = 3.0 × 1016 cm −3 , nr = 3.2 , T12 = 0.2 ps , Γ12 = 1/ T12 , m* = 0.067 m0 ( m0 is the free electron mass), ε = 12.58 ( static dielectric constant is assumed to be same GaAs and AlGaAs), and the barrier height V0 = 228 meV. In Figures 1a−1d, we show the laser-dressed potential the first two states and squared wave functions related to these first two states in GaAs / Al0.3Ga0.7 As PQW which has the width L = 100 Å for the four different laser-dressing parameters ( α 0 = 25,50, 75 and 100 Å) as a function of the position. Dashed (solid) line corresponds to α 0 = 0 (α 0 ≠ 0) . As the α 0 increases, the width of the well bottom decreases, while the top width increases. Energy levels are closer to each other since ILF creates an additional geometric confinement on the

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electronic states in the PQW. Therefore, the figure clearly shows the reduction of the energy difference E2 − E1 as the laser-dressing parameters α 0 increases. Furthermore, for α 0 values which satisfy the condition α 0 > L / 2 PQW potential turns into double quantum well (DPQW) potential. The arising of a DPQW enables one to create controllable resonant states located in the well material. This obviously does not need any growth of conventional DPQWs, which are more difficult to tune to the desired resonance states. The linear, the third-order nonlinear and total refractive index changes as a function of the photon energy for the four different laser dressing parameters ( α 0 = 0,50, 75 and 100

Å), L = 100 Å, F = 0 and I = 0.4 MW/cm2 are given Fig. 2. We observe from this figure that linear, the third-order nonlinear refractive index change is very sensitive to the ILF. As the α 0 increases, the magnitude of the changes in the resonant peaks of both the linear and the third-order nonlinear refractive index increase, and also shift towards lower energies. The main reason for this result is the reduction in the energy difference E2 − E1 (see Fig. 1), when the α 0 increases. We can see that the changes in the total refractive index depend on the linear and nonlinear refractive index change. It is clearly seen that the magnitude of the changes in the resonant peaks of the total refractive index increases and also shifts towards lower energies with the increasing the α 0 . These results are in agreement with the results obtained for previous studies [7, 9, 30-32]. The results for the calculated linear, third-order nonlinear and total relative changes of the refractive index in the system are presented in Fig. 3 for the four different laser

dressing parameters, F = 25 kV/cm, L = 100 Å and I = 0.4 MW / cm 2 . In comparison to the Fig.2, the refractive index is not very affected from electric field for α 0 = 0,50, 75 Å, since the geometric confinement is predominant. As known, the refractive index is very sensitive to the electric field in large well dimensions. Also the electric field induces a blue shift,

mainly for α 0 = 100 Å, and reduces the magnitude of the resonant peaks. This is a consequence of the energy interval, E2 − E1 increases and the dipole matrix element (M21) decreases with increasing the electric field intensity (see Fig. 8). These effects are only visible when we have large values of α 0 . Fig.4, the total refractive index change is the plotted as a function of the incident the photon energy for the four different laser dressing parameters, two different values of

the electric field ( F = 0 and 25 kV/cm), L = 200 Å and I = 0.4 MW/cm2 . We observe

6

from this figure that the total refractive index change is very sensitive to the electric field with respect to the L = 100 Å. For α 0 = 0,50, 75 Å, the presence of the electric field,

the total refractive index changes shift toward lower energies. The reason for the redshift is due to the smaller transition energy E2 − E1 for a stronger electric field (see Fig. 8). Furthermore, for α 0 = 100 Å, it can be clearly seen that the electric field induces a blue shift, and reduces the magnitude of the resonant peaks, as the electric field increases. In Fig. 5 shows the results of the linear, the third-order nonlinear and the total absorption coefficients changes as a function of the photon energy for the four different

laser dressing parameters ( α 0 = 0,50, 75 and 100 Å), L = 100 Å, F = 0 and I = 0.1 MW/cm2 . The magnitudes of the resonant peaks of the linear and the third-order nonlinear absorption coefficients depend on the values of the energy interval E21 and the dipole matrix element M21. The energy interval decreases and the dipole matrix element increases with the increasing the α 0 (see Fig. 8). For the linear absorption coefficient, the variation of E21 has an important role with respect to the variation of M21. The case for third-order nonlinear absorption coefficient is more complicated. In this case, for the lower values of the α 0 , the effects of M21 are dominant, elsewhere the dominant term is E21. As a result of these, the magnitude of the changes in the resonant peaks of the total optical absorption coefficient decreases and also shifts towards lower energies with the increasing the α 0 . Furthermore, as seen in this figure, the resonant peak splits up into two separate peaks with the increasing the

α 0 . This is known as bleaching effect. These changes are consistent with data from previous studies [7, 9, 30-32]. In Fig.6, for the four different laser dressing parameters, I = 0.1 MW/cm2 , L = 100 Å, and F = 25 kV/cm, the linear, the third-order nonlinear and the total absorption coefficients changes as a function of the photon energy are shown. As known, the absorption coefficients are not very sensitive to the electric field in narrow well dimension. Therefore, for α 0 = 0,50, 75 Å, in comparison to the Fig.5, the absorption coefficients are not very affected from electric field. For α 0 = 100 Å, it is clearly seen from this figure that

the magnitude of the changes in the resonant peaks of the total optical absorption coefficient increases and also shifts towards higher energies with the increasing electric field. Fig. 7 shows the total absorption coefficient as a function of the photon energy for the four different laser dressing parameters, two different values of the electric field (

F = 0 and 25 kV/cm), L = 200 Å and I = 0.1 MW/cm2 . We observe from this figure that 7

the total absorption coefficient is very sensitive to the electric field with respect to the

L = 100 Å. It is clearly seen from this figure, the resonant peak of the total absorption coefficient suffers an obvious red-shift the increasing the electric field, for α 0 = 0,50, 75 Å. However, the magnitude of the changes in the resonant peaks of the total optical absorption coefficient increases and also shifts towards higher energies with the increasing the electric field for α 0 = 100 Å. 4. Conclusion

In this present work, we have theoretically investigated how the linear, the third-order nonlinear and the total absorption and the refractive index changes in a PQW under the applied electric field are affected by an intense high-frequency laser field. Our results reveal that the peak position of the absorption coefficient is sensitive to ILF, the resonant peaks of the absorption shifts towards lower energies for increasing the ILF value. We can observe that the total refractive index change is very sensitive to ILF. We have found that the magnitude of the changes in the resonant peaks of the linear and nonlinear refractive index increases and also shifts towards lower energies as the ILF strength increases. Also we can observe that the refractive index and absorption coefficient changes are very sensitive to the electric field in large well dimension.

In summary, we can conclude, in general, ILF and electric field the important roles in the optical absorption coefficients and the refractive index. Thus, this gives a new degree of freedom in device applications, such as photo-detectors, electro-optical modulators, and all optical switches.

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Figure Captions Fig.1 The variation of the laser-dressed potential, the first two states and squared wave

functions related to these first two states in GaAs / Al0.3Ga0.7 As parabolic quantum well which has the width L = 100 Å as a function of the position. The results are for (a) α 0 = 25 Å, (b)

α 0 = 50 Å, (c) α 0 = 75 Å and (d) α 0 = 100 Å. Dashed (solid) line corresponds to α 0 = 0 (α 0 ≠ 0) . Fig. 2 The refractive index changes as a function of the photon energy for the four

different laser dressing parameters ( α 0 = 0,50, 75 and 100 Å), L = 100 Å, F = 0 and I = 0.4 MW/cm2 . The results are for (a) linear, (b) nonlinear, (c) total. Fig. 3 The refractive index changes as a function of the photon energy for the four

different laser dressing parameters ( α 0 = 0,50, 75 and 100 Å), L = 100 Å, F = 25 kV/cm and

I = 0.4 MW/cm2 . The results are for (a) linear, (b) nonlinear, (c) total Fig. 4 The total refractive index changes as a function of the photon energy for the four

different laser dressing parameters ( α 0 = 0,50, 75 and 100 Å), two different values of the electric field ( F = 0 and 25 kV/cm), L = 200 Å and I = 0.4 MW/cm2.

10

Fig. 5 The absorption coefficients changes as a function of the photon energy for the four

different laser dressing parameters ( α 0 = 0,50, 75 and 100 Å), L = 100 Å, F = 0 and I = 0.1 MW/cm2 . The results are for (a) linear, (b) nonlinear, (c) total. Fig. 6 The absorption coefficients changes as a function of the photon energy for the four

different laser dressing parameters ( α 0 = 0,50, 75 and 100 Å), L = 100 Å, F = 25 kV/cm and

I = 0.1 MW/cm2 . The results are for (a) linear, (b) nonlinear, (c) total. Fig. 7 The total absorption coefficient change as a function of the photon energy for the

four different laser dressing parameters ( α 0 = 0,50, 75 and 100 Å), two different values of the electric field ( F = 0 and 25 kV/cm), L = 200 Å and I = 0.4 MW/cm2. Fig. 8 a) The energy difference between the first excited and ground states as a function of the laser field for two different well widths and electric field values. b) Variation of the main involved intersubband matrix element as a function of the laser field for two different well widths and electric field values.

250

(b)

200

150

i=1

i

E ( meV )

i=2

100

L = 100 αo= 50

50

0 -200

11

-150

-100

-50

0

z( )

50

100

150

200

250

(d)

200

i=2 i=1

Ei( meV )

150

L = 100 αo = 100

100

50

0 -200

-150

-100

-50

z(

0

)

Fig. 1 0.20

(a)

L = 100 αo = 0 αo= 50 αo = 75 αo= 100 F=0

(1)

Δ n /nr

0.10

0.00

-0.10

-0.20 0.00 0.40

0.05

0.10

0.15

0.20

0.10

0.15

0.20

Photon Energy ( eV)

(b)

(3)

Δ n /nr

0.20

0.00

-0.20

-0.40 0.00

0.05

Photon Energy ( eV )

12

50

100

150

200

0.40

(c)

0.20

Δ n /nr

F=0 0.00

-0.20

-0.40 0.00

0.05

0.10

Fig.2

13

0.15

0.20

0.40

(b)

(3)

Δ n /n r

0.20

0.00

-0.20

-0.40 0.00 0.20

0.05

0.10

0.15

0.20

(c)

Δn/nr

0.10

0.00

-0.10

-0.20

0.00

0.05

0.10

Fig.3

14

0.15

0.20

0.06

L = 200 F=0 F = 25 kV/cm

0.04

Δ n/n r

0.02

0.00

-0.02

-0.04

-0.06 0.00

0.02

0.04

0.06

0.08

Photon Energy ( eV )

0.10

0.12

Fig.4 1600

(a)

1200

800

(1)

β ( cm-1)

F=0

400

0 0.00

0.05

0.10

0.15

Photon Energy( eV )

15

0.20

0.25

0.14

0

(b)

β ( cm -1)

-400

(3)

-800

-1200

F=0 -1600 0.00

1600

0.05

0.10

0.15

0.20

0.25

(c)

1200

-1 β (cm )

F=0 800

400

0 0.00

0.05

0.10

0.15

0.20

0.25

Photon Energy ( eV ) 1600

Fig.5 (a)

L = 100

(1)

-1

β ( cm )

1200

800

400

0 0.00

0.05

0.10

Photon Energy ( eV )

16

0.15

0.20

0

(b)

(3)

-1

β ( cm )

-200

-400

-600 0.00

0.05

0.10

0.15

0.20

Photon Energy ( eV ) 1200

(c)

-1

β (cm )

800

400

0 0.00

0.05

0.10

Photon Energy ( eV )

Fig.6

17

0.15

0.20

1400

L = 200 F=0 F = 25 kV/cm

1200

800 600 400 200 0 0.00

0.04

0.08

0.12

Photon energy ( eV ) Fig.7

150

(a)

F=0 F = 25 kV/cm

125

L = 100 100

75

21

E ( meV )

β ( cm-1)

1000

L = 200 50

25

18

0 0

20

40

αo (

)

60

80

100

70

(b)

F=0 F = 25 kV/cm

M21 (

)

60

50

40

L = 200

30

L = 100

20 0

20

40

α o(

)

60

80

100

Fig.8 Research Highlights •

The ILF have a great effect on the optical properties of these structures.

•

The total absorption coefficients increased as the ILF increases.

•

The RICs increased as the ILF increases.

19