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Original research article

Linear and nonlinear magneto-optical absorption in parabolic quantum well Huynh Vinh Phuc a,d,∗ , Doan Quoc Khoa b , Nguyen Van Hieu c , Nguyen Ngoc Hieu a a b c d

Institute of Research and Development, Duy Tan University, Da Nang, Viet Nam Quang Tri Teacher Training College, Quang Tri, Viet Nam Department of Physics, Da Nang University of Education, Da Nang, Viet Nam Division of Theoretical Physics, Dong Thap University, Dong Thap, Viet Nam

a r t i c l e

i n f o

Article history: Received 21 June 2016 Accepted 24 August 2016 Keywords: Magneto-optical properties Transport properties Two-photon process

a b s t r a c t The linear and nonlinear optical absorption via one and two-photon processes in the ﬁnite symmetric parabolic quantum well (FSPQW) under the applied magnetic ﬁeld have been studied numerically for typical GaAs/Gax Al1−x As. The analytical expression of the magnetooptical absorption coefﬁcient (MOAC) is obtained by relating it to the transition probability for the absorption of photons. The effects of the magnetic ﬁeld, the temperature, and the well width on MOAC and full-width at half-maximum (FWHM) are investigated. Results reveal that MOAC and FWHM are monotonic functions of these factors in both one and two-photon absorption processes. Obtained results also show that the magneto-optical properties of FSPQW can be controlled by changing these parameters. This suggests a new capacity for magneto-optical device applications. © 2016 Elsevier GmbH. All rights reserved.

1. Introduction The study of linear and nonlinear optical absorption in low-dimensional semiconductor structures is essential for understanding the potential applications in micro-electronic and optoelectronic devices [1–4]. In the presence of magnetic ﬁeld, the electron–phonon interaction and therefore the optical properties of low-dimensional systems become different because of the quantized energy in the plane perpendicular to the magnetic ﬁeld [5–7]. The linear and nonlinear optical absorption properties, including the nonlinear optical absorption coefﬁcient, in the presence of magnetic ﬁeld have been investigated in quantum wells, which have square [8], parabolic [9], graded [10], and V-shaped [11] potential shapes; in quantum dot [12–14]; in a quantum disk [15]; and in quantum rings [16]. Literature review shows that the optical absorption properties depend not only on the structure of the system, but also on the magnetic ﬁeld. The two-photon absorption process has been presented in several works due to its importance for understanding the response of semiconductors excited by a laser ﬁeld. Prudaev et al. reported the results of experiments on terahertz generation from nitride light-emitting diode heterostructures under two-photon excitation by femtosecond laser pulses [17]. Scheibner et al. studied the two-photon absorption by a quantum dot pair [18]. Tanaka et al. presented a novel conﬁguration of a precision laser distance measurement based on the two-photon absorption photocurrent from a silicon avalanche

∗ Corresponding author at: Dong Thap University, Dong Thap, Viet Nam. E-mail address: [email protected] (H.V. Phuc). http://dx.doi.org/10.1016/j.ijleo.2016.08.070 0030-4026/© 2016 Elsevier GmbH. All rights reserved.

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photodiode [19]. The optical response via two-photon absorption has also been discussed in symmetric quantum well [20], multiple quantum wells [21], quantum wires [22], and quantum dot [23–26]. However, the two-photon absorption process in the presence of electron–phonon scattering in a ﬁnite symmetric parabolic quantum well has not been fully considered in previously published reports. In our previous works, we studied the linear and nonlinear magneto-optical properties of quantum wells with parabolic [9,27] and semi-parabolic [28] inﬁnite potential shapes. It should be noted that the magneto-optical properties of inﬁnite and ﬁnite parabolic quantum wells are signiﬁcantly different. For inﬁnite potential quantum wells, the studies on the linear and nonlinear optical properties are usually based on analytical solution of the stationary Schrödinger equation to ﬁnd the eigenfunctions and the eigenenergies of electrons in the conduction band. Meanwhile, in the ﬁnite potential one, the problem is more complex and there are no analytical solutions of the Schrödinger equation for many types of ﬁnite quantum wells. However, the progress in computational program with high accuracy allows us to ﬁnd these solutions numerically [29–31]; then the results could be used to obtain magneto-optical properties. In this work, we will study the linear and nonlinear magneto-optical properties of ﬁnite symmetric parabolic quantum well (FSPQW) via one and two-photon absorption processes. To do this, ﬁrst, we use the numerical methods to solve the Schrödinger equation for FSPQW. Then the results are used to calculate the magneto-optical absorption coefﬁcient (MOAC). The proﬁle method [32] is used to ﬁnd the full-width of haft-maximum (FWHM) of resonant peaks. We found out that these magneto-optical properties are signiﬁcantly different in the ﬁnite parabolic quantum well in comparison to those in the inﬁnite systems. 2. Theory We consider a ﬁnite symmetric parabolic quantum well structure where electrons move freely in the x–y plane and a uniform static magnetic ﬁeld B = (0, 0, B) is applied to the z-direction. The one-electron eigenfunction |˛ and energy eigenvalue EN,n are given by |˛ =

1

EN,n =

Ly

exp(iky y)

N+

1 2

N (x

ωc + n ,

− x0 )n (z),

(1)

N = 0, 1, 2, . . .,

(2)

where N is the Landau level index, ωc = eB/m* is the cyclotron frequency with m* = m0 (0.067 + 0.083x) [33] the effective mass of a conduction electron, where m0 and x are the free electron mass and the alloy concentration, N (x − x0 ) is the harmonic oscillator wave functions, centered at x0 = −˛2c ky , Ly and ky are the normalization length and the electron wave vector in 1/2

the y-direction, respectively, and ˛c = (/m∗ ωc ) is the radius of the orbit in the (x, y) plane. The envelope wave function n (z) and its corresponding eigenenergies n are the solutions of the one-dimensional stationary Schrödinger equation −

2 d 2 n (z) + U(z)n (z) = n n (z), 2m∗ dz 2

(3)

where U(z) is the conﬁning potential introduced as

U(z) =

4U0 z 2 /L2 ,

|z| < L/2

U0 ,

|z| ≥ L/2,

(4)

where U0 is the potential height between GaAs and Alx Ga1−x As, and L is the well width. After the energies and their corresponding wave functions are obtained, the magneto-optical absorption coefﬁcient (MOAC) due to photon absorption with simultaneous absorption and emission of phonon is given by [34,35] K() =

1 W± f (1 − f˛ ), ˛,˛ ˛ V0 (I/)

(5)

˛,˛

where V0 = SL is the volume of the system, I/ is the injected number of photons, energy , per unit area per second with the optical intensity I = nr c 0 2 A20 /2. Here, nr is the refractive index of the material, c and 0 are the speed of light and the permittivity in free space, respectively, and A0 is the amplitude of the vector potential for the optical electric ﬁeld. In Eq. (5), f˛ and f˛ are the electron distribution functions in the initial and ﬁnal states. The transition matrix element per unit area for electron–photon–phonon interaction of the 2D carrier [27,36], including -photon absorption process [37,38], is given by Born second-order golden rule [39] W± = ˛,˛

2 ± 2 rad 2 (˛0 q)2 |M˛,˛ | M˛,˛ ı(E˛ − E˛ − ± ω0 ), 2 2 (!) 22 q,

(6)

H.V. Phuc et al. / Optik 127 (2016) 10519–10526

10521

where the upper (+) and lower signs (−) refer to the emission and absorption processes of phonons, respectively, ˛0 is the dressing parameter. The electron–phonon matrix element M± ˛,˛, is given by 4e2 ∗ ω0 |JNN (u)|2 |Jn n (qz )|2 N0± ık ,ky ±qy , y 0 V0 q2

|M± |2 = ˛,˛

(7)

−1 where ∗ = (−1 ∞ − 0 ) with ∞ = 10.89 − 2.73x and 0 = 13.18 − 3.12x [33,40] being the high- and low-frequency dielectric constant, respectively. Also q = (q⊥ , qz ) is the bulk phonon wave vector, N0± = (N0 + 1/2 ± 1/2) for the Bose factor N0 which presents the number of phonon with energy ω0 ; and

|JNN (u)|2 =

N2 ! −u N1 −N2 N1 −N2 [LN (u)]2 , e u 2 N1 ! +∞

Jn n (qz ) = −∞

(8)

n∗ (z)e±iqz z n (z)dz,

(9)

M (u) are the associated Laguerre polynomials. In Eq. (6), the steadywhere u = ˛2c q2 /2, N1 = max(N, N ), N2 = min(N, N ), and LN state matrix element for carrier–photons interaction is calculated as follows

eA0 B˛˛ , 2

Mrad ˛,˛ =

(10)

where we have denoted

√ √ B˛˛ = [x0 ıN ,N + (˛c / 2)( NıN ,N−1 + N + 1ıN ,N+1 )]ık ,ky . y

(11)

The transition matrix element in Eq. (6) contains a contribution of -photon process, however, in this work, we only restrict ourselves to consider the two-photon process, i.e., =1, 2. In this case, the MOAC via two-photon process is found to be Se4 ˛20 ∗ ω0

K() =

+ where

323 nc c 20 L˛6c 2 N,n N ,n

˛20 8˛2c

|B˛˛ |2 fN,n (1 − fN ,n )|Fn n |{N0− ı(X1− ) + N0+ ı(X1+ )

(N + N + 1)[N0− ı(X2− ) + N0+ ı(X2+ )]},

(12)

+∞

Fn n = −∞

|Jn n (qz )|2 dqz ,

X± = (N − N)ωc + n n − ± ω0 , ( = 1, 2),

(13) (14)

with n n = n − n being the energy separation between the two states. The delta functions in Eq. (12) are replaced by 2 2 Lorentzians of width ± , namely ı(X± ) = ( ± /)/[(X± ) + ( ± ) ] [41], where 2

( ± ) =

e2 ∗ ω0 |Fn n |N0± . 42 0 (N − N)

(15)

In the next section, we will consider the numerical calculation in more details. 3. Results and discussion In this section, we perform numerical calculations for the magneto-optical properties in a GaAs/Ga0.7 Al0.3 As quantum well (corresponding to Al concentration x = 0.3), which leads to U0 = 228 meV. The material parameters used are as follows [33]: nr = 3.2, ne = 3 ×1016 cm−3 , ˛0 = 10 nm, and ω0 = 36.25 meV. The following results are obtained in the quantum limit, where N = 0, and N = 1. In Fig. 1, the ﬁrst two wave functions of FSPQW are plotted as a function of the growth (z) direction for two values of the well width. As seen in this ﬁgure, the overlap between 1 (z) and 2 (z) increases with the increasing well width, which leads to the increase of the overlap integral F21 . This results in the increase of MOAC with well width as shown in Fig. 4. The energy of the ground state and ﬁrst excited state are shown in Fig. 2 as a function of the well width. The result shows that ground state energy 1 , the ﬁrst excited one 2 , and the inter-subband transition energy 21 all decrease when the well width increases due to the decrease of electron quantum conﬁnement. Fig. 3 shows the dependence of MOAC on photon energy at L = 10 nm, T = 77 K, and B = 10 T. There are four resonant peaks, which correspond to the PACR for inter-subband transitions. These PACR peaks satisfy the condition = ωc + 21 ± ω0 , representing the resonance transfer of electrons with an absorption of -photons accompanied by the emission/absorption of LO-phonons. In speciﬁc terms, the peaks labeled 1 and 3 in Fig. 3 occur at a photon energy of = 52.16 and 104.32 meV,

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H.V. Phuc et al. / Optik 127 (2016) 10519–10526

0.20

Wave function

0.15 0.10 0.05 0.00 − 0.05 − 0.10 − 0.15 − 15

− 10

−5

0

5

10

15

z (nm) Fig. 1. The ground state (solid lines) and the ﬁrst excited state (dashed lines) wave function 1 (z) and 2 (z) for L = 10 nm (black lines) and L = 15 nm (red lines). (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

400

Energy (meV)

350

2

300

1

250

21

200 150 100 50 5

10

15

20

L (nm) Fig. 2. The ground state 1 , the ﬁrst excited state 2 , and the energy separation between the subband energies 21 are shown as a function of the well width L.

respectively, correspond to the phonon absorption transitions; whereas the peaks labeled 2 and 4 at 88.41 meV and 176.82 meV correspond to the emission one. The phonon absorption transition (peaks 1 and 3) is much smaller in comparison with the phonon emission one since the occupation number of optical phonons, N0 , is also much smaller than 1. Indeed, for the one-photon absorption process (peaks 3 and 4), the peak value due to phonon absorption (peak 3) is about 15% of that of phonon emission (peak 4). This rate is about 17% for the two-photon absorption process. Besides, the two-photon absorption process makes a signiﬁcant contribution to the total. In the phonon emission (absorption) case, the magnitude of the two-photon peak is about 40% (43%) of that of the one-photon case. This rate is bigger in comparison with those in the inﬁnite parabolic quantum well [9]. Fig. 4 shows the dependence of MOAC on photon energy at different values of the well width. With the increasing of well width, a red-shift behavior of the absorption spectrum is observed. This is because with the increase of well width, the quantum conﬁnement effect gets weakened, leading to a decrease in the subband separation energy, 21 . Therefore, the values of the absorbed photon energies satisfying the resonant condition reduce and result in the red-shift. Moreover, the increased behavior of relative intensity of MOAC can be also explained by the increase of the overlap between two states 1 (z) and 2 (z) when the well width increases as shown in Fig. 1. An interesting dependence on well width is seen for FWHM. From the curves in Fig. 4, using the proﬁle method [32], we obtain the well width dependence of FWHM for phonon emission transitions as shown in Fig. 5. The result shows that the 2.5

K (10 4 /m)

2.0 1.5

4

1.0 0.5

2

3

1 0.0 50

100

150

200

Photon energy (meV) Fig. 3. MOAC is shown as a function of photon energy at T = 77 K, B = 10 T, and L = 10 nm.

H.V. Phuc et al. / Optik 127 (2016) 10519–10526

2.5

L L L

K 104 m

2.0

10523

10 nm 12 nm 14 nm

1.5 1.0 0.5 0.0 50

100

150

200

Photon energy meV Fig. 4. MOAC is shown as a function of photon energy at different values of well width. The results are calculated at T = 77 K and B = 10 T.

one photon two photon

FWHM meV

20 15 10 5 5

10

15

20

25

30

L nm Fig. 5. Dependence of FWHM for phonon emission transitions on well width at T = 77 K and B = 10 T. The ﬁlled and empty lines correspond to the one- (peak 4 in Fig. 3) and two-photon (peak 2 in Fig. 3) absorption processes, respectively.

FWHM decreases with the increase of L for both one- and two-photon absorption processes. The result is consistent with that shown in some previous works for square [42,43] and Gaussian potential [44] quantum wells. We ﬁnd that FWHM seems proportional to the inverse square root of L (i.e. FWHM ∼L−1/2 ). The best ﬁts are given by FWHM [meV]= 52.08/ L[nm]

in the case of one-photon process and FWHM [meV]= 13.02/ L[nm] for two-photon process, respectively. The physical meaning of this can be explained by the decrease of electron–phonon scattering with the well width. To illustrate clearly magnetic ﬁeld effect on magneto-optical absorption properties in FSPQW, in Fig. 6, we show the variation of MOAC as a function of the incident photon energy at different values of magnetic ﬁeld. The MOAC increases in the magnitude and gives a blue-shift with the increasing magnetic ﬁeld, agreeing with the previous works for square [43,45,46] and inﬁnite parabolic [27,47] quantum wells. This results from the increase of cyclotron energy with magnetic ﬁeld. Besides, the inﬂuence of magnetic ﬁeld leads to a signiﬁcant change in the FWHM of the PACR peaks. A detailed evaluation of the FWHM is illustrated in Fig. 7. Fig. 7 shows the magnetic ﬁeld dependence of FWHM at L = 10 nm at a temperature T = 77 K. It can be seen that the FWHM increases with the increasing magnetic ﬁeld for both one and two-photon processes. The FWHM is found to be proportional to the square root of B, agreeing with the results shown in some previous works for quantum well [27] and graphene [48–50]. The best ﬁts are given by FWHM [meV]= (8.23 + 2.60 B[T]) in the case of one-photon process and

3.0

B= 8T B = 10 T B = 12 T

K (10 4 /m)

2.5 2.0 1.5 1.0 0.5 0.0 50

100

150

200

Photon energy (meV) Fig. 6. MOAC is shown as a function of photon energy at different values of magnetic ﬁeld. The results are calculated at T = 77 K and L = 10 nm.

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FWHM (meV)

20

15 one−photon two−photon

10

5 0

5

10

15

20

B (T) Fig. 7. Dependence of FWHM for phonon emission transitions on magnetic ﬁeld at T = 77 K and L = 10 nm.

T = 77 K T = 150 K T = 300 K

K (104 /m)

2.5 2.0 1.5 1.0 0.5 0.0 50

100

150

200

Photon energy (meV) Fig. 8. MOAC is shown as a function of photon energy at different values of temperature. The results are calculated at B = 10 T and L = 10 nm.

√ FWHM [meV]= (2.06 + 0.65 B[T]) for two-photon process, respectively. The B-dependent FWHM can be explained physically by the broadening of the Landau levels [51]. This results from the increase in the possibility of electron–phonon scattering with the magnetic ﬁeld. The dependence of MOAC on photon energy at different values of temperature is shown in Fig. 8. We found that temperature does not affect the position of the resonant peaks but the peak values, especially in the case of phonon absorption. The higher temperature is, the higher peak values arise. While the MOAC magnitude mainly depends on temperature due to the temperature dependence of phonon populations, N0 , its position is decided through the selection rules as indicated by the delta functions in Eq. (12). As the increase of temperature, the phonon populations increase, which in turn leads to the increase in MOAC magnitude. Whereas, since the temperature independence of the delta functions, the absorbed photon energies are maintained with the change of the temperature. Besides, we can see that the phonon absorption transition is strongly dependent on temperature in comparison with the emission one. This can be shown more clearly in the following when we consider the dependence of FWHM on the temperature. In Fig. 9, we show the dependence of FWHM on temperature up to the room temperature. We found that FWHM increases non-linearly with the increasing temperature for both one and two-photon processes. The result is consistent with that shown in previous works for the optically detected magnetophonon [46] and electro-phonon [52] resonance line-widths. Besides, FWHM for phonon absorption increases faster with the increasing temperature but has smaller values than it does in

FWHM meV

15

phonon emission phonon absorption

10

5

50

100

150

200

250

300

T K Fig. 9. Dependence of FWHM on temperature at B = 10 T and L = 10 nm. The ﬁlled and empty lines correspond to the one- and two-photon absorption processes, respectively.

H.V. Phuc et al. / Optik 127 (2016) 10519–10526

10525

comparison with the phonon emission. Therefore, in FSPQW, the electron–phonon scattering in the case of phonon emission is always stronger than that in the case of phonon absorption. Moreover, the thermal broadening FWHM can be written as [53] FWHM(T ) = 0 + ω0 /(eω0 /kB T − 1),

(16)

where 0 and ωLO are low-temperature and a measure of the phonon broadening, respectively. Using Eq. (16), in the one-photon process, we obtain a best ﬁt with 0 = 16.45 meV and ω0 = 7.59 meV for phonon emission transition and

0 = 2.78 meV and ω0 = 23.06 meV for phonon absorption transition. These parameters would be useful to orientate the experimental investigation in the future. 4. Conclusion In the present work, we have presented the magneto-optical transport properties of the ﬁnite symmetric parabolic quantum well. The eigenenergies and eigenfunctions of FSPQW are found by numerically solving the Schrödinger equation using the effective mass approximation. The solutions are employed to study MOAC and FWHM due to one and two photon absorption processes. The results are calculated for a speciﬁc GaAs/Ga0.7 Al0.3 As FSPQW. Our results reveal that, MOAC, and the FWHM depend signiﬁcantly on the change of magnetic ﬁeld, temperature and well width. The resonant peaks of MOAC shifts toward higher energies as the magnetic ﬁeld increases, moves downward lower energies as the well width increases, but is unchanged with temperature. Also, FWHM increases with the magnetic ﬁeld and temperature, but decreases with the well width in both one and two-photon absorption processes. Besides, the temperature effects on phonon absorption transitions are stronger in comparison with that on phonon emission one. Finally, our results on the dependence of FWHM on the well width, magnetic ﬁeld, and temperature would be useful to orientate the experimental observation in the future. These results could be important for device applications based on the nonlinear optical properties due to two-photon absorption process. Acknowledgements This research is funded by Ministry of Education and Training of Vietnam under Grant number B2016.DNA.13. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39]

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