Electron-related optical responses in Gaussian potential quantum wells: Role of intense laser field

Accepted Manuscript Electron-related optical responses in Gaussian potential quantum wells: Role of intense laser field H. Sari, F. Ungan, S. Sakiroglu, U. Yesilgul, E. Kasapoglu, I. Sökmen PII:

S0921-4526(18)30428-9

DOI:

10.1016/j.physb.2018.06.028

Reference:

PHYSB 310936

To appear in:

Physica B: Physics of Condensed Matter

Received Date: 2 April 2018 Revised Date:

23 May 2018

Accepted Date: 22 June 2018

Please cite this article as: H. Sari, F. Ungan, S. Sakiroglu, U. Yesilgul, E. Kasapoglu, I. Sökmen, Electron-related optical responses in Gaussian potential quantum wells: Role of intense laser field, Physica B: Physics of Condensed Matter (2018), doi: 10.1016/j.physb.2018.06.028. 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 proof before it is published in its final 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.

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Electron-related optical responses in Gaussian potential quantum wells: Role of intense laser field H. Saria , F. Unganb,, S. Sakirogluc , U. Yesilgulb , E. Kasapoglud , I. S¨okmenc of Education, Department of Mathematical and Natural Science Education, Sivas Cumhuriyet University, 58140 Sivas, Turkey b Faculty of Technology, Department of Optical Engineering, Cumhuriyet University, 58140 Sivas, Turkey c Faculty of Science, Department of Physics, Dokuz Eyl¨ ˙ ul University, 35160 Buca, Izmir, Turkey d Faculty of Science, Department of Physics, Sivas Cumhuriyet University, 58140 Sivas, Turkey

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a Faculty

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Abstract

A theoretical study of the effects of non-resonant intense laser field on the optical response in a Gaussian potential

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quantum well is performed in the framework of the effective mass approximation. The obtained results indicate that, the depth and range of the considered confinement potential and intense laser field intensity has a significant effect on the optical properties in Gaussian potential quantum wells. Also it was found that the laser field is more effective on the optical response in Gaussian potential quantum wells with smaller range. Keywords: Gaussian quantum well, Optical response, Intense laser field

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PACS: 73.21.Fg, 78.66.Fd, 78.67.De

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1. Introduction

As we all know the low-dimensional semiconductor systems have a great importance in the development of optoelectronic devices, for example, semiconductor lasers, electro-optical modulators, optical switches, photo-detectors,

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semiconductor optical amplifiers, and tera-hertz devices [1–5]. Hence, the nonlinear optical properties of these structures have gained special interest. It has been reported that, due to the motion of the electron is restricted in all three directions, the nonlinear effects in quantum wells (QWs) can be improved more dramatically over those in bulk struc-

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tures. In this context, recently many theoretical studies have been carried out on the nonlinear optical response of semiconductor QWs [6–15]. It should be noted that, the geometry of the confinement potential in QW structures has a significant role in fundamental physical properties of the semiconductor quantum nano-structures [12, 14]. There is no direct experimental method for defining the confinement potential. The confinement potentials are frequently modeled by rectangular potential well or the parabolic potential [16]. It was noted that, the conventional rectangular QW has basic, but unrealistic form owing to the non-parabolic shape at the center of the QW. On the other hand, the parabolic potential is inappropriate to describe the experimental results due to its infinite depth and ∗ Corresponding

Author: F. Ungan Email address: [email protected] (F. Ungan)

Preprint submitted to Physica B

June 23, 2018

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range [16]. Ciurla et al. [17] have suggested the exponential potentials, which are flexible enough to define suitable confinement potentials in quantum dots (QDs). Based on the experimental results, Xie [18] has pointed that the real confinement potential is not parabolic but Gaussian potential which possesses the finite depth and range. Based on the Gaussian well potential, a lot of studies have been conducted investigated optical properties of heterostructures. Zhang

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and co-workers [9] studied the electric field effects on the electron-related optical response in Gaussian potential QWs. They found that the applied external static electric field and the geometric parameters have significant effect on the nonlinear optical rectification in the Gaussian potential well. The electric field dependence of the linear and nonlinear optical properties in Gaussian QWs has been reported by using the density matrix approximation [19]. Their obtained

applied external static electric field and the symmetry of the system.

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results demonstrate that the magnitude of the second harmonic generation coefficient significantly depends on the

In previous studies it has been mentioned that, the non-resonant intense laser field (ILF) has significant effects on

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the optical and electronic properties of the semiconductor heterostructures [20–31]. In the presence of high-frequency laser field, the confinement of the electronic structure is altered which influences the subband energy states, a feature that has been observed in intersubband optical transitions. Therefore, the laser field dependence of the confinement potential and corresponding bound energy states is an important issue to focus on. Radu et al. [29] investigated the theoretical feasibility of tuning a conventional QW laser by using a high-frequency laser field dressing, and they reported that a significant tunability of the electronic states in a QW is obtainable by changing the ILF. The tunability of the electronic states means an alteration of the bound energy states and corresponding electronic wave functions

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of the confined electrons under the high-frequency ILF [30, 31]. Recently, we investigated the effects of electric and

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magnetic fields on the electron-related optical responses in the n-type δ-doped QW under ILF [32]. We concluded that externally applied fields have substantial effects on the optical properties of doped QWs. Although many studies carried out on the electronic and optical response of semiconductor QWs, the laser field

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affect on the absorption coefficients (ACs) and refractive index (RI) changes in Gaussian potential QWs has not been studied so far. By considering the importance of the ILF, the depth and range of the considered confinement potential on the investigation of the optical response in Gaussian QWs, in this work, we calculate the effects of these factors on

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the optical ACs and RICs in the GaAs/Ga1−x Alx As QW with the symmetric Gaussian potential. The present study is arranged as follows: the theoretical calculations are given in Sec. 2. The discussion and presentation of the obtained results are shown in Sec. 3. The conclusion of the present study is presented in Sec. 4.

2. Theory

The considered system is a symmetric Gaussian potential QW under the high-frequency ILF, which is polarized linearly along the z-axis. The Hamiltonian for a single electron in a Gaussian potential QW has the form H=−

~2 d2 + V(z), 2m∗ dz2 2

(1)

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where ~ is the Planck constant, m∗ is the conduction-band effective mass and V(z) is the Gaussian QW which is defined as follows [9, 17]: V(z) = −V0 exp(−z2 /L2 ),

(2)

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where V0 and L are the depth and effective well width of the potential, respectively.

The approach used in this work is based on the nonperturbative theory previously developed to describe the atomic states under the presence of a THz non-resonant ILF of frequency Ω [33–35]. In the framework of the high-frequency → − of Floquet theory, the corresponding vector potential is A(t) = A0 cos(Ωt)ˆz. The average laser dress potential given by [24]: Z

2π/Ω

V(z + α0 sin Ωt)dt.

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Ω hV(z, α0 )i = 2π

0

(3)

where α0 = eA0 /m∗ Ω is the laser-dressing parameter, and A0 is the vector potential of the ILF. In the high-frequency

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limit [33, 36], the laser-dressed eigenstates are solutions of the following time independent one-dimensional Schr¨odinger equation:

# " ~2 d2 − ∗ 2 + hV(z, α0 )i ψ(z) = Eψ(z). 2m dz

(4)

In order to calculate the first-order linear, the third-order nonlinear and total RI changes and ACs for the intersubband transitions between quantized states within the same band, we assume that the structure is simultaneously irradiated by non-resonant monochromatic ILF and the incident probe electromagnetic wave which is linearly polar-

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ized along the z-direction with frequency ω. The bound energy states and corresponding electronic wave functions are

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numerically calculated by using diagonalization method. The method used here is described in detail in our previous studies [37, 38]. After finding the energies and their corresponding envelope wave-functions, the first-order linear ACs and RI changes are obtained [39, 40]:

r

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β(1) (ω) = ω

µ |M12 |2 σv ~Γ12 , εr (∆E − ~ω)2 + (~Γ12 )2

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" # ∆n(1) (ω) σv |M12 |2 ∆E − ~ω = , nr 2n2r ε0 (∆E − ~ω)2 + (~Γ12 )2

and the third-order nonlinear ACs and RI changes are given to be [39, 40] ! r µ I |M12 |4 σv ~Γ12 |M22 − M11 |2 β(3) (ω, I) = −2ω [1 − 2 2 2 εr ε0 nr c [(∆E − ~ω) + (~Γ12 ) ] |2M10 |2 (∆E − ~ω)2 − (~Γ12 )2 + 2(∆E)(∆E − ~ω) × ], (∆E)2 + (~Γ12 )2 ∆n(3) (ω, I) nr

σv I µc|M12 |2 3 4nr ε0 [(∆E − ~ω)2 + (~Γ12 )2 ]2 (M22 − M11 )2 ×[4(∆E − ~ω)|M12 |2 − {(∆E − ~ω) (∆E)2 + (~Γ12 )2 ×[(∆E)(∆E − ~ω) − (~Γ12 )2 ] − (~Γ12 )2 (2(∆E) − ~ω)}],

(5)

(6)

(7)

= −

3

(8)

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where nr is the refractive index, ε0 is the dielectric permittivity of the vacuum, µ is the magnetic permeability, σv is the electronic density, I is the optical intensity of the incident probe electromagnetic field, ∆E = E2 − E1 is the energy interval between the ground and the first excited states, Γ12 is called as the relaxation rate of 1th state and 2th state, is

The total ACs and RI changes are given as follows β(ω, I) = β(1) (ω) + β(3) (ω, I).

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∆n(ω, I) ∆n(1) (ω) ∆n(3) (ω, I) = + . nr nr nr

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the inverse of the inverse relaxation time τ21 and Mi j = hψi (z)|ez|ψ j (z)i is the dipole moment matrix element.

(10)

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3. Results and discussion

(9)

In this study, for our numerical calculations, we have used the following parameters which are suitable for GaAs/Ga1−x Alx As materials: m∗ = 0.067m0 (m0 = 9.10956 × 10−31 kg), µ = 4π × 10−7 Hm−1 , τ12 = 0.2 ps (where τ12 = 1/Γ12 ), σv = 1.0 × 1023 m−3 , nr = 3.2, V0 = 200 meV (which corresponding to x = 0.27 for the Aluminum concentration), and I = 0.05 MWcm−2 [32, 37, 41].

In Fig. 1, we present the change of the confinement potential, and absolute square of the wave functions corresponding to the first two bound states as a function of the position for three different values of the dressing parameter,

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V0 = 200meV and L = 100, 200 Å. As seen in Figs. 1(a) and (b), the depth of the potential reduces as the effective

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length of the confinement potential increases with the α0 . The increment of the effective well width leads a decrement in the energy levels of the electron and an increment of the overlap between the electronic wave functions. By comparing Fig. 1 (a) with (b) one can see that, the laser field dependence of the confinement potential is more pronounced

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at small L values. From these results, it should be noted that, the evolution of the confinement potential form directly affects the localization of the electron and this will affect the electron-related optical response of the structure based on the intersubband transitions as expected.

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The squared electronic wave functions corresponding to related energy levels are given in Fig. 2 for different values of the laser field amplitude, α0 = 0, 50 and 100 Å. As clearly seen in Figs. 2(a) and (b), the probability densities expand to larger range with α0 . This is because the effective range of the gaussian potential QW increases with the raise of the ILF intensity. Therefore, the electron which is almost entirely localized around the center of the QW gets to spreads over a larger range as the laser dressing parameter increases. Moreover, the decrement of the electron confinement leads to a reduction in the subband energy states of the electron. Also, the overlap between the electronic wave functions will become bigger for large laser dressing parameter values, so the nonlinear effects may be increased with α0 . The reason is that the electron confinement weakens with the increase of the effective width of the well. 4

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The change of the energy interval ∆E and the dipole moment matrix elements in the Gaussian QW as a function of the α0 for different three L values and V0 = 200meV is depicted in Fig. 3(a) and (b), respectively. From Fig. 3(a), it is clearly seen that as a consequence of the change in the confinement potential with laser field intensity, the energy difference ∆E rapidly decreases as α0 increases for L = 100 Å and L = 150 Å, whereas for L = 200 Å, it first increases

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in small amount, reaches its maximum for a specific value of α0  50 Å and then begins to decrease. As seen from Fig. 3(b), the dipole matrix elements exhibit a behavior completely contrary to that of ∆E. The physical origin of this feature is that, the overlap between the electronic wave functions gets to be larger (smaller) as the energy difference between the corresponding states decreases (increases), since the overlap between wave functions is proportional to

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the effective length of the confinement potential. These results are compatible with those of reported by Yuan et al. [19]. They reported in their work that; for symmetrical Gaussian QWs, the overlap between the first two states will be bigger for the larger effective well-width, therefore improved nonlinear effects expected.

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In Fig. 4 we draw the change of the ACs as a function of the incident photon energy for different values of the dressing parameter and V0 = 200meV for (a) L = 100 Å and (b) L = 200 Å. In Figs. 4(a) and (b) it is clearly seen that, as a result of the reduction in ∆E with the α0 , the peak position of the AC shows a red-shift as the α0 increases. On the other hand, although the matrix elements increase with α0 (see Fig. 3(b)), because of the fact that the energy difference ∆E is the predominant term in the magnitude of the AC decreases with the α0 . By comparing Fig. 4(a) with Fig. 4(b), one can see that the ILF is more effective on the ACs for smaller L values. This behavior is in consistent with the change of ∆E with the α0 for different L values (see Fig. 3(a)).

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Fig. 5 shows the variations of the RI changes as the function of the photon energy for three different values of the

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dressing parameter and V0 = 200meV for (a) L = 100 Å and (b) L = 200 Å. As can be seen from Figs. 5(a) and (b), although the peak position of the RI changes mostly exhibits a same behavior with the ACs described above, but the magnitude of the RI changes increases with the increase of dressing parameter value. The different behavior of RI

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changes according to the ACs owing to the additional terms in the matrix elements that appear in equations (6, 8) and compensate the decrement in the difference between energy levels. The obtained results show that the intensity of the ILF, and range of the confinement potential can modify the

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subband energy spectrum of the bound states, the overlap between the electronic wave functions which have a considerable impact on the optical response in Gaussian QWs. This specific feature may be used in the practical applications such as electronic and optoelectronics devices based on the QW structures.

4. Conclusions

In the present study, we have studied the non-resonant monochromatic ILF effects on the optical response in the GaAs/Ga1−x Alx As QW with the symmetric Gaussian potential. Within the effective mass and envelope wave function approximation, the bound states and related eigenfunctions of the system have been calculated by solving the time independent one-dimensional Schr¨odinger equation in the gaussian QW by using diagonalization method. Our results 5

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indicate that, the optical ACs and RI changes are significantly affected by non-resonant ILF intensity, and range of the considered confinement potential. Also it was found that the ILF is more effective on the optical response in Gaussian potential QWs with smaller range. Our numerical results may contribute to the studies on the designing of the adjustable optoelectronic devices based on QWs. In addition, by considering the effect of high-frequency laser

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field on the geometry of the confinement potential it is possible to obtain the ideal well depth and range of Gaussian QWs by applying the laser field along the growth direction.

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0

0 0

= 200 meV

V

0

-50

-50

0

0

= 50Å

Energy (meV)

0

-100

= 0

= 100Å

-150

-200

0

= 100Å

-150

-200

-250 -500

0

-100

= 0

= 50Å

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Energy (meV)

0

= 200 meV

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V

-250

-375

-250

-125

0

125

250

375

z (Å)

-500

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(a)

500

-375

-250

-125

0

125

250

375

500

z (Å)

(b)

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Figure 1: The variations of the gaussian potential profile for different ILF values, V0 = 200meV and for (a) L = 100 Å, (b) L = 200 Å.

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0

0

V

0

0

= 200 meV

= 0

(b)

= 50Å

V

0

(z)

L = 100Å

0

= 100Å

-30 Energy (meV)

-100

-150

0

= 200 meV

-60

-90

-200

-120

-500

-375

-250

-125

0

125

250

375

500

-500

-375

-250

-125

= 0 = 50Å

0

125

250

= 100Å

375

500

z (Å)

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z (Å)

0

0

L = 100Å

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Energy (meV)

-50

0

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(a)

(a)

(b)

Figure 2: The squared wave functions related to the (a) first and (b) second energy level in the gaussian potential QW for values α0 = 0, 50 and

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100 Å for V0 = 200meV and L = 100 Å.

60

55 (b)

V = 200 meV 0

55

(a)

L = 100Å

50

50

L = 150Å

V = 200 meV 0

L = 100Å L = 150Å

40 35

M12(Å)

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45

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E (meV)

L = 200

30

45 40 35

25 20

0

20

40

60

0

80

30

100

0

20

40

60

(Å)

0

(a)

80

100

(Å)

(b)

Figure 3: The variations of (a) the energy difference between ground and first excited states (b) the dipole matrix elements associated to the transitions between electron energy levels in gaussian potential QW as a function of the laser parameters α0 .

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5000

5000 = 0

4000

0

V = 200 meV 0

) , I) (cm

-1

3000 2000

3000 2000

0

1000

0

20

40

60

80

0

100

0

20

40

60

= 100Å

80

100

Photon energy (meV)

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Photon energy (meV)

0

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1000

= 0 = 50Å

0

(

) -1

0

L = 200Å

(

, I) (cm

0

V = 200 meV

4000

= 100Å

0

L = 100Å

(b)

= 50Å

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0

(a)

(b)

Figure 4: The variation of the total AC as a function of the incident photon energy for three different values of the dressing parameter in the gaussian

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potential QW, V0 = 200meV and for (a) L = 100 Å, (b) L = 200 Å.

0,4

0,0

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-0,1

r

0

0,2

0

, I)/ n

0,1

= 0

= 100Å

0

-0,3

20

30

40

= 100Å

-0,1

-0,3

-0,4

0

= 0 = 50Å

0,0

-0,2

10

0

0,1

-0,2

0

L = 200Å

= 50Å

n(

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r

, I)/ n

0

0

0,3

L = 100Å

0,2

V

(b)

0

0,3

n(

0,4

V

(a)

-0,4 50

60

70

80

0

Photon energy (meV)

10

20

30

40

50

60

70

80

Photon energy (meV)

(a)

(b)

Figure 5: The variation of the total relative RIC as a function of the incident photon energy for three different values of the dressing parameter in the gaussian potential QW, V0 = 200meV and for (a) L = 100 Å, (b) L = 200 Å.

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