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Nonlinear optical absorption via two-photon process in asymmetrical Gaussian potential quantum wells
Accepted Manuscript Nonlinear optical absorption via two-photon process in asymmetrical Gaussian potential quantum wells Huynh Vinh Phuc, Luong Van Tung, Pham Tuan Vinh, Le Dinh PII: DOI: Reference:
S0749-6036(14)00452-2 http://dx.doi.org/10.1016/j.spmi.2014.11.024 YSPMI 3501
To appear in:
Superlattices and Microstructures
Received Date: Revised Date: Accepted Date:
6 October 2014 7 November 2014 11 November 2014
Please cite this article as: H.V. Phuc, L.V. Tung, P.T. Vinh, L. Dinh, Nonlinear optical absorption via two-photon process in asymmetrical Gaussian potential quantum wells, Superlattices and Microstructures (2014), doi: http:// dx.doi.org/10.1016/j.spmi.2014.11.024
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Nonlinear optical absorption via two-photon process in asymmetrical Gaussian potential quantum wells
Huynh Vinh Phuc a,∗ , Luong Van Tung a , Pham Tuan Vinh a,b , Le Dinh b a Division b Center
of Theoretical Physics, Dong Thap University, Dong Thap, Viet Nam
for Theoretical and Computational Physics, Hue University, Hue, Viet Nam
Abstract In this paper, linear and nonlinear optical absorption spectrum in asymmetrical Gaussian potential quantum wells under the applied magnetic and electric fields are studied via investigating the phonon-assisted cyclotron resonance (PACR) effect. The results are calculated for GaAs and Ga1−x Alx As materials. Our results show that the optical absorption behaviors and the half-width are significantly dependent on the height of the Gaussian potential, the well width, the magnetic field, and the temperature. It is also found that there is a clear monotonic behavior of the resonant peaks and the half-width as functions of the factors mentioned above in both one and two-photon absorption processes. c 2014 Elsevier B.V. All rights reserved. Key words: Nonlinear optics, Phonon-assisted cyclotron resonance, Half-width, Gaussian potential, Electric field.
∗ Corresponding author. Tel.: +84 67 3882919. Email address: [email protected] (Huynh Vinh Phuc).
Preprint submitted to Elsevier
27 November 2014
1
Introduction Nonlinear optical properties in low-dimensional semiconductor structures have
been intensively studied in recent decades because of their novel physical characteristics and possibility of applications in micro-electronic and opto-electronic devices. Therefore, the linear and nonlinear optical absorption effects in these structures have been investigated by a number of researchers [1–21]. Based on the effective mass approximation, the simultaneous effects of the hydrostatic pressure, and temperature on the optical absorption spectrum have been investigated in several systems, such as spherical quantum dots [1,2], quantum wells [3–8], concentric double quantum rings [9,10], and quantum wire [11]. In these works, the authors have found that the energy levels and inter-subband properties can be modified and controlled by the hydrostatic pressure and temperature. In the other hand, with the applied electric and magnetic fields, the linear and nonlinear optical absorption coefficients have been investigated in a square quantum well [12], V-shaped quantum well [13], double inverse parabolic quantum well [14], parabolic quantum well [15], GaAs/GaAlAs asymmetric double quantum wells [16], cylindrical quantum wires [17], quantum disk with flat cylindrical geometry [18], quantum dot [19,20], and in parabolic two-dimensional quantum rings [21]. These works results showed that the optical absorption coefficients depend not only on the structure of the system but also on the strength of the static magnetic and/or electric fields. The confined potential used in this paper is Gaussian potential. It is known that although the parabolic potential are often used to display the confined potential in low-dimensional semiconductor structures [14,15,22–25], it is not perfectly suitable to describe the experimental results [26,27]. Therefore, the parabolic potential should be replaced by Gaussian potential [28]. Based on the Gaussian potential, there are a number of works have been reported to investigate the optical proper-
2
ties [27,29–32]. However, in most of these studies, the optical absorption has been only investigated by one-photon absorption, while the two-photon absorption process has not been done. In this paper, we use the Gaussian potential to investigate the linear and nonlinear optical absorption via two-photon absorption process in GaAs and Ga1−x Alx As quantum wells. Our results show that the optical absorption behaviors and the half-width are significantly dependent on the height of the Gaussian potential, the well width, the magnetic field, and the temperature. We also found that there is a clear monotonic behavior of the resonant peaks and the half-width as functions of the factors mentioned above in both one and two-photon absorption processes. The paper is organized as follows: In Section 2, the theoretical framework and analytical results are presented. Section 3 is dedicated the numerical results and discussion. Finally, our conclusion is given in Section 4.
2
Theoretical framework and analytical results We consider a quantum well where electron is confined in z-direction by an
asymmetric Gaussian potential, which is given by [27,31,32]
U (z) =
⎧ ⎪ ⎪ ⎪ ⎪ ⎨−U0 exp(−z 2 /2L2 )
z ≥ 0,
⎪ ⎪ ⎪ ⎪ ⎩∞
z < 0,
(1)
where U0 is the height of the Gaussian potential, and L is the range of the confinement potential (well width). When the static magnetic B, and electric F fields are applied simultaneously to the z-direction, in the Landau gauge for the vector potential A = (0, Bx, 0), the one electron Hamiltonian reads H=
1 (p + eA)2 + U (z) − eF z, 2m∗
(2)
where p and m∗ are the momentum operator and the effective mass of a conduction electron, respectively. The eigenfunction and eigenvalue corresponding to the 3
Hamiltonian in Eq. (2) are 1 ΨN,n,ky (r) = exp(iky y)ψN (x − x0 )φn (z), Ly
EN,n,ky = N +
1 h ¯ ωc + εn , 2
(3)
N = 0, 1, 2, . . . ,
(4)
where N is the Landau level index, ωc = eB/m∗ is the cyclotron frequency, ψN (x− x0 ) is the harmonic oscillator wave functions, centered at x0 = −a2c ky , Ly and ky are the normalization length and the electron wave vector in the y-direction, respectively, and ac = (¯ h/m∗ ωc )1/2 is the radius of the orbit in the (x, y) plane. The component eigenfunction and eigenvalue in z-direction in Eqs. (3) and (4) are given by [27,31,32]
1
(5)
h ¯ 2 U0 e2 F 2 L2 − U − , n = 0, 1, 2, . . . , 0 m ∗ L2 2U0
(6)
φn (z) = √ e 2n n!az π
3 εn = 2n + 2
z eF L2 − , az az U0
−z 2 /2a2z
Hn
hL)1/2 /(m∗ U0 )1/4 , and Hn (x) is the Hermite polynomials. where az = (¯ We use perturbation theory to derive the expression of absorption power. When an electromagnetic field with frequency Ω is applied to the system, the optical absorption power via two-photon process for the transitions between the states (N, n) and (N , n ) in the case of non-degenerate electron gas can be written as [33–35] P (Ω) = A(ωc )
N,n
N ,n
e−EN,n /kB T
∞ 0
q⊥ dq⊥ |JN N (q⊥ )|2
+∞ −∞
dqz |Fnn (±qz )|2
× NLO δ(p¯hωc + εnn − h ¯ ωLO − h ¯ Ω) ¯ ωLO − h ¯ Ω) + (NLO + 1)δ(p¯hωc + εnn + h 2 a20 q⊥ NLO δ(p¯hωc + εnn − h ¯ ωLO − 2¯ hΩ) 16 ¯ ωLO − 2¯ hΩ) , + (NLO + 1)δ(p¯hωc + εnn + h
+
(7)
where we have denoted p = N −N being an integer, εnn = εn −εn is the subband 4
separation energy, q = (q⊥ , qz ) is the phonon wave vector, NLO is the distribution function of LO-phonon for energy h ¯ ωLO , kB T is the thermodynamic energy, and
√ F 2 χ0 ne V03 e2 h ¯ ωLO a20 1 1 − , λ= e−EN,n /kB T , (8) A(ωc ) = 3 2 16(2π) λ¯hL 0 χ∞ χ0 N,n
|JN N (q⊥ )|2 = Fnn (±qz ) =
N ! −u N −N N −N 2 2 e u LN (u) , u = a2c q⊥ /2, N!
+∞ −∞
(9)
φ∗n (z)e±iqz z φn (z)dz.
(10)
Here, ne is the electron concentration, a0 is the dressing parameter, V0 is volume of the system, 0 is the permittivity of free space, χ∞ and χ0 are the high and low frequency dielectric constants, respectively, and LM N (x) is the associated Laguerre polynomials. After making a straightforward calculation of integral over q⊥ and qz , we obtain the following expression of the optical absorption power in asymmetric Gaussian potential quantum well P (Ω) =
A(ωc ) −EN,n /kB T e |Inn | NLO δ(p¯ hωc + εnn − h ¯ ωLO − h ¯ Ω) a2c N,n N ,n + (NLO + 1)δ(p¯hωc + εnn + h ¯ ωLO − h ¯ Ω) +
a20 (N + N + 1) NLO δ(p¯hωc + εnn − h ¯ ωLO − 2¯ hΩ) 2 8ac
+ (NLO + 1)δ(p¯hωc + εnn + h ¯ ωLO − 2¯ hΩ)
,
(11)
where Inn is the overlap integral, which is calculated as below in the extreme electric quantum limit (n = 0, n = 1) I01 =
+∞ −∞
√
2
|F01 (±qz )| dqz =
π(1 + 4β 2 ) eF L2 √ , β= . az U0 az 2
(12)
Finally, the delta functions in Eq. (11) are replaced by Lorentzians of width Γ± , which are given by Γ2±
¯ ωLO |Inn | V0 e2 h = 2 (2π¯ h) 0 ac L N − N
1 1 − (NLO + 1/2 ± 1/2). χ∞ χ0
5
(13)
In the next section, we will consider the numerical calculation in more detail.
3
Numerical results and discussion We have obtained the expression of the optical absorption power in quantum
well under the applied magnetic and electric fields. In this section, we perform numerical calculations for the GaAs and Ga1−x Alx As quantum wells in the extreme electric quantum limit. For numerical calculations, we take ne = 3 × 1016 cm−3 , F = 4.5 × 105 V/m, and a0 = 5 nm. The other parameters used are displayed in Table 1 [25,36,37]. Table 1 Parameters used in the present numerical calculation. Quantities
GaAs
Ga1−x Alx As
m∗ (m0 )
0.067
0.067 + 0.083x
¯hωLO (meV)
36.25
36.25 + 1.83x + 17.12x2 − 5.11x3
χ0
13.18
13.18 − 3.12x
χ∞
10.89
10.89 − 2.73x
Fig. 1 shows the dependence of the absorption power on photon energy in free standing quantum well for different materials: GaAs (solid), Ga0.7 Al0.3 As (dashed), and Ga0.6 Al0.4 As (dashed-dotted lines). There are two resonant peaks observed in each curve, which present resonance transfer of electrons with absorption of photons accompanied with the absorption or emission of LO-phonons. Using the computational method, we found that in GaAs quantum well (corresponding to the solid line), the peak at h ¯ Ω = 100.87 meV satisfies the condition h ¯Ω = h ¯ ωc + ε01 + h ¯ ωLO , corresponding to the one-photon absorption process 6
P arb.units
— GaAs --- Ga0.7Al0.3As Ga0.6Al0.4As
20
40
60
80
100
120
140
Photon energy meV Fig. 1. (Color online) Dependence of the absorption power on the incident photon energy in different quantum wells at T = 77 K, B = 7 T, L = 5 nm, and U0 = 15 meV.
(linear). Similarly, the peak at h ¯ Ω = 50.44 meV satisfies the condition 2¯ hΩ = h ¯ ωc + ε01 + h ¯ ωLO , describing the two-photon absorption process (nonlinear). Besides, we can also see that the peak value in the nonlinear process is about 15% those in the linear case. This means that the nonlinear process is strong significantly so that it can be observed. In Ga1−x Alx As quantum wells, the phenomenon occurs similarly, but the resonant peaks are observed to give a red-shift and the peak values are reduced. This is due to the electron effective mass and LO-phonon energy, which are modified in Ga1−x Alx As materials. For instance, with x = 0.3 (corresponding to the dashed-line), the effective mass m∗ = 0.092m0 leading to the cyclotron energy h ¯ ωc = 8.87 meV, phonon energy h ¯ ωLO = 38.20 meV, and ε01 = 44.78 meV. Therefore, the peak at h ¯ Ω = 91.85 meV satisfies the condition h ¯Ω = h ¯ ωc + ε01 + h ¯ ωLO . This value of the resonant peak is smaller than that in GaAs (100.87 meV), which explains the red-shift behavior of the absorption spectrum in Ga0.7 Al0.3 As quantum well in comparison with that in the GaAs case. The other peaks of the dashed and dashed-dotted lines can be explained in the same ways. 7
P arb.units
— U0 10 meV --- U0 15 meV U0 20 meV
20
40
60
80
100
120
140
Photon energy meV Fig. 2. (Color online) Absorption power in GaAs quantum well is shown as a function of photon energy for three values of the height of Gaussian potential quantum well. Here T = 77 K, B = 7 T, and L = 5 nm.
Fig. 2 illustrates the optical absorption power in GaAs quantum well as a function of the incident photon energy for three values of U0 . We can see from the figure that there is a blue-shift behavior of the optical absorption spectrum when the height of the Gaussian potential quantum well increases. This result is in good agreement with previous papers for the optical absorption coefficient [27], the optical rectification coefficients [31], and the second-harmonic generation coefficients [32] in the absence of the magnetic field. The physical meaning of this behavior is that when the height of potential increases, the subband separation energy ε01 become larger leading to the increase of the value of absorbed photon energies, which results in the blue-shift. Besides, we can also see from the figure that with the increase of U0 , the relative intensity of the resonant peaks decreases significantly, being in good agreement with previous papers [27,31,32]. This is due to the decrease of the overlap between different states when the height of potential increases. Indeed, from Eq. (11), we can obtain the result that when U0 is changed, the intensity of the resonant peak is mainly decided by the overlap integral I01 via the quantity β as shown in Eq. 8
(12). Therefore, when the height of the Gaussian potential increases, the quantum confinement effect is enhanced, which leads to the decrease of the overlap between different states [27]. This explains the decrease behavior of the relative intensity of
Halfwidth meV
the resonant peaks when U0 increases. 8 6 4
, Ga0.7Al0.3As , GaAs
2 0
10
15
20
25
30
U0 meV Fig. 3. (Color online) Dependence of the half-width on the height of the Gaussian potential at T = 77 K, L = 5 nm, and B = 7 T. The filled and empty lines correspond to the oneand two-photon absorption processes, respectively.
For the half-width (half-width at half maximum), using profile method [38], we obtain the dependence of the half-width on U0 as shown in Fig. 3. It can be found that the half-width in both GaAs and Ga1−x Alx As (with x = 0.3) decrease slightly with the increase of U0 . This is accompanied with the decrease of the overlap integral I01 when U0 increases. Besides, the half-widths in Ga0.7 Al0.3 As quantum well are greater than that in GaAs quantum well in both one- and two-photon absorption processes. It means that the probability of electron−phonon scattering becomes larger in Ga0.7 Al0.3 As material in comparison with that in GaAs. In addition to this, we can also see that the half-width in the nonlinear absorption process is smaller than that in the linear case in both Ga0.7 Al0.3 As and GaAs quantum wells. In Fig. 4, the optical absorption power in GaAs quantum well is shown as functions of photon energy for several values of well width. When L increases, a 9
P arb.units
— L 5 nm --- L 10 nm L 15 nm
20
40
60
80
100
120
140
Photon energy meV Fig. 4. (Color online) Absorption power in GaAs quantum well is shown as a function of photon energy for three values of well width. Here T = 77 K, B = 7 T, and U0 = 15 meV.
red-shift behavior of the absorption spectrum is observed, being in good agreement with previous results [27,31,32]. This is because that with the augmentation of well width, the quantum confinement effect gets weakened, leading to a decrease in the subband separation energy. Therefore, the values of the absorbed photon energies satisfying the resonant condition reduce and result in the red-shift. Moreover, the increase behavior of relative intensity of the resonant peaks can be also explained from the increase of the overlap between different states when the well width increases. In Fig. 5, we depict the dependence of the half-width on L. It can be seen from the figure that the half-width decreases rapidly with the well width from L = 4 nm to about L = 10 nm in both GaAs and Ga0.7 Al0.3 As quantum wells. At the larger well width, the decrease behavior of the half-width continues, but at a slighter rate. It can be said that the quantum confinement effect is observed distinctly only in a narrow quantum well with L ≤ 10 nm. The physical meaning of this can be explained as follows: when the well width increases, the quantum confinement effect reduces leading to the decrease of electron−phonon scattering, and so does 10
Halfwidth meV
12
, Ga0.7Al0.3As 10 , GaAs 8 6 4 2 0 5 10 15 20 L nm
Fig. 5. (Color online) Dependence of the half-width on well width at T = 77 K, U0 = 15 meV, and B = 7 T. The filled and empty lines correspond to the one- and two-photon absorption processes, respectively.
the half-width.
P arb.units
—B5T --- B 7 T B 9 T
20
40
60
80
100
120
140
Photon energy meV Fig. 6. (Color online) Absorption power in GaAs quantum well is shown as a function of photon energy for three values of magnetic field. Here T = 77 K, L = 5 nm, and U0 = 15 meV.
In Fig. 6, we show the dependence of the optical absorption power on photon energy for various values of magnetic field. The blue-shift behavior of the absorption spectrum associated with the increase of magnetic field is observed. This result is in good agreement with previous results [34,35]. This is because that when the 11
magnetic field increases, the Landau level separation becomes larger. Therefore, the values of the absorbed photon energies satisfying the resonant condition increase. Besides, we can also see from the figure that with the increase of the magnetic field, the relative intensity of the resonant peaks increases significantly. This is due to the decrease of the cyclotron radius, ac , when the magnetic field increases. Indeed, from Eq. (11), we can see that the intensity of the resonant peak is mainly decided by the quantity ac when magnetic field is changed. Therefore, when the magnetic field increases, the cyclotron frequency, ωc , is enhanced, which leads to the decrease of the cyclotron radius. This explains the increase behavior of the relative intensity of the resonant peaks when magnetic field increases. , Ga0.7Al0.3As , GaAs 10 8 6 4 2 0 0 5 10 15 20
Halfwidth meV
12
B T Fig. 7. (Color online) Dependence of half-width on magnetic field at T = 77 K, L = 5 nm, and U0 = 15 meV. The filled and empty lines correspond to the one- and two-photon absorption processes, respectively.
Fig. 7 shows the dependence of the half-width on B. It can be seen from the figure that the half-width increases with the increase of the magnetic field in both GaAs and Ga0.7 Al0.3 As quantum wells, as well as in the one- and two-photon absorption processes. The present result is in good agreement with the results of previous papers [33,34,39–43]. This increase behavior of the half-width is explained from the increase of the probability of electron−phonon interaction when magnetic 12
field increases.
P arb.units
— T 77 K --- T 150 K T 300 K
20
40
60
80
100
120
140
Photon energy meV Fig. 8. (Color online) Absorption power in GaAs quantum well is shown as a function of photon energy for different values of temperature. Here B = 7 T, L = 5 nm, and U0 = 15 meV.
In Fig. 8, we display the optical absorption power in GaAs quantum well as a function of photon energy for different values of temperature. From the figure, we can see that the resonant peaks are located at the same position but their intensity are seen to increase with temperature. This temperature-dependent behavior of the resonant peaks value is similar to that of the electron−phonon scattering rate [44], and the behavior of the absorption coefficient [45] in the presence of the magnetic field in square quantum well. This results from the enhancement of electron−phonon interaction when temperature increases. In Fig. 9, we describe the dependence of the half-width on temperature from T = 50 K to 350 K. We can see from the figure that the half-width increases nonlinearly with the increase of the temperature for both GaAs and Ga0.7 Al0.3 As quantum wells, as well as for one and two-photon absorption cases. The augmentation behavior of the half-with is due to the increase of the electron−phonon interaction when temperature increases. 13
Halfwidth meV
12
, Ga0.7Al0.3As 10 , GaAs 8 6 4
2 0
50
100
150
200
250
300
350
T K Fig. 9. (Color online) Dependence of half-width on temperature at B = 7 T, L = 5 nm, and U0 = 15 meV. The filled and empty lines correspond to the one- and two-photon absorption processes, respectively.
4
Conclusion In this paper, we have discussed the the linear and nonlinear optical absorp-
tion in GaAs and Ga1−x Alx As asymmetric Gaussian potential quantum wells with applied magnetic and electric fields via investigating the phonon-assisted cyclotron resonance effect. The calculations mainly focus on the dependence of the optical absorption power and the half-width on the height of the Gaussian potential quantum well, the well width, the magnetic field, and the temperature in both one- and two-photon absorption processes. The obtained results show that with the increase of U0 , B, and L, the optical absorption spectrum give the blue-shift and/or redshift, respectively, whereas, the position of the resonant peaks are independent of the change of the temperature. Besides, the half-width increases with the increase of magnetic field and temperature, but decreases with the increase of the height of the Gaussian potential and the well width in both one and two-photon cases. These results may be useful to understand the nonlinear optical properties in quantum wells under applied magnetic and electric fields. 14
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17
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Linear and nonlinear optical absorption spectra in AGPQWs have been investigated. The results are calculated for GaAs and Ga1-xAlxAs materials. The two-photon absorption process has been included. There is a clear monotonic behavior of the resonant peaks and the half-width in both one and twophoton absorption processes.
S0749-6036(14)00452-2 http://dx.doi.org/10.1016/j.spmi.2014.11.024 YSPMI 3501
To appear in:
Superlattices and Microstructures
Received Date: Revised Date: Accepted Date:
6 October 2014 7 November 2014 11 November 2014
Please cite this article as: H.V. Phuc, L.V. Tung, P.T. Vinh, L. Dinh, Nonlinear optical absorption via two-photon process in asymmetrical Gaussian potential quantum wells, Superlattices and Microstructures (2014), doi: http:// dx.doi.org/10.1016/j.spmi.2014.11.024
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.
Nonlinear optical absorption via two-photon process in asymmetrical Gaussian potential quantum wells
Huynh Vinh Phuc a,∗ , Luong Van Tung a , Pham Tuan Vinh a,b , Le Dinh b a Division b Center
of Theoretical Physics, Dong Thap University, Dong Thap, Viet Nam
for Theoretical and Computational Physics, Hue University, Hue, Viet Nam
Abstract In this paper, linear and nonlinear optical absorption spectrum in asymmetrical Gaussian potential quantum wells under the applied magnetic and electric fields are studied via investigating the phonon-assisted cyclotron resonance (PACR) effect. The results are calculated for GaAs and Ga1−x Alx As materials. Our results show that the optical absorption behaviors and the half-width are significantly dependent on the height of the Gaussian potential, the well width, the magnetic field, and the temperature. It is also found that there is a clear monotonic behavior of the resonant peaks and the half-width as functions of the factors mentioned above in both one and two-photon absorption processes. c 2014 Elsevier B.V. All rights reserved. Key words: Nonlinear optics, Phonon-assisted cyclotron resonance, Half-width, Gaussian potential, Electric field.
∗ Corresponding author. Tel.: +84 67 3882919. Email address: [email protected] (Huynh Vinh Phuc).
Preprint submitted to Elsevier
27 November 2014
1
Introduction Nonlinear optical properties in low-dimensional semiconductor structures have
been intensively studied in recent decades because of their novel physical characteristics and possibility of applications in micro-electronic and opto-electronic devices. Therefore, the linear and nonlinear optical absorption effects in these structures have been investigated by a number of researchers [1–21]. Based on the effective mass approximation, the simultaneous effects of the hydrostatic pressure, and temperature on the optical absorption spectrum have been investigated in several systems, such as spherical quantum dots [1,2], quantum wells [3–8], concentric double quantum rings [9,10], and quantum wire [11]. In these works, the authors have found that the energy levels and inter-subband properties can be modified and controlled by the hydrostatic pressure and temperature. In the other hand, with the applied electric and magnetic fields, the linear and nonlinear optical absorption coefficients have been investigated in a square quantum well [12], V-shaped quantum well [13], double inverse parabolic quantum well [14], parabolic quantum well [15], GaAs/GaAlAs asymmetric double quantum wells [16], cylindrical quantum wires [17], quantum disk with flat cylindrical geometry [18], quantum dot [19,20], and in parabolic two-dimensional quantum rings [21]. These works results showed that the optical absorption coefficients depend not only on the structure of the system but also on the strength of the static magnetic and/or electric fields. The confined potential used in this paper is Gaussian potential. It is known that although the parabolic potential are often used to display the confined potential in low-dimensional semiconductor structures [14,15,22–25], it is not perfectly suitable to describe the experimental results [26,27]. Therefore, the parabolic potential should be replaced by Gaussian potential [28]. Based on the Gaussian potential, there are a number of works have been reported to investigate the optical proper-
2
ties [27,29–32]. However, in most of these studies, the optical absorption has been only investigated by one-photon absorption, while the two-photon absorption process has not been done. In this paper, we use the Gaussian potential to investigate the linear and nonlinear optical absorption via two-photon absorption process in GaAs and Ga1−x Alx As quantum wells. Our results show that the optical absorption behaviors and the half-width are significantly dependent on the height of the Gaussian potential, the well width, the magnetic field, and the temperature. We also found that there is a clear monotonic behavior of the resonant peaks and the half-width as functions of the factors mentioned above in both one and two-photon absorption processes. The paper is organized as follows: In Section 2, the theoretical framework and analytical results are presented. Section 3 is dedicated the numerical results and discussion. Finally, our conclusion is given in Section 4.
2
Theoretical framework and analytical results We consider a quantum well where electron is confined in z-direction by an
asymmetric Gaussian potential, which is given by [27,31,32]
U (z) =
⎧ ⎪ ⎪ ⎪ ⎪ ⎨−U0 exp(−z 2 /2L2 )
z ≥ 0,
⎪ ⎪ ⎪ ⎪ ⎩∞
z < 0,
(1)
where U0 is the height of the Gaussian potential, and L is the range of the confinement potential (well width). When the static magnetic B, and electric F fields are applied simultaneously to the z-direction, in the Landau gauge for the vector potential A = (0, Bx, 0), the one electron Hamiltonian reads H=
1 (p + eA)2 + U (z) − eF z, 2m∗
(2)
where p and m∗ are the momentum operator and the effective mass of a conduction electron, respectively. The eigenfunction and eigenvalue corresponding to the 3
Hamiltonian in Eq. (2) are 1 ΨN,n,ky (r) = exp(iky y)ψN (x − x0 )φn (z), Ly
EN,n,ky = N +
1 h ¯ ωc + εn , 2
(3)
N = 0, 1, 2, . . . ,
(4)
where N is the Landau level index, ωc = eB/m∗ is the cyclotron frequency, ψN (x− x0 ) is the harmonic oscillator wave functions, centered at x0 = −a2c ky , Ly and ky are the normalization length and the electron wave vector in the y-direction, respectively, and ac = (¯ h/m∗ ωc )1/2 is the radius of the orbit in the (x, y) plane. The component eigenfunction and eigenvalue in z-direction in Eqs. (3) and (4) are given by [27,31,32]
1
(5)
h ¯ 2 U0 e2 F 2 L2 − U − , n = 0, 1, 2, . . . , 0 m ∗ L2 2U0
(6)
φn (z) = √ e 2n n!az π
3 εn = 2n + 2
z eF L2 − , az az U0
−z 2 /2a2z
Hn
hL)1/2 /(m∗ U0 )1/4 , and Hn (x) is the Hermite polynomials. where az = (¯ We use perturbation theory to derive the expression of absorption power. When an electromagnetic field with frequency Ω is applied to the system, the optical absorption power via two-photon process for the transitions between the states (N, n) and (N , n ) in the case of non-degenerate electron gas can be written as [33–35] P (Ω) = A(ωc )
N,n
N ,n
e−EN,n /kB T
∞ 0
q⊥ dq⊥ |JN N (q⊥ )|2
+∞ −∞
dqz |Fnn (±qz )|2
× NLO δ(p¯hωc + εnn − h ¯ ωLO − h ¯ Ω) ¯ ωLO − h ¯ Ω) + (NLO + 1)δ(p¯hωc + εnn + h 2 a20 q⊥ NLO δ(p¯hωc + εnn − h ¯ ωLO − 2¯ hΩ) 16 ¯ ωLO − 2¯ hΩ) , + (NLO + 1)δ(p¯hωc + εnn + h
+
(7)
where we have denoted p = N −N being an integer, εnn = εn −εn is the subband 4
separation energy, q = (q⊥ , qz ) is the phonon wave vector, NLO is the distribution function of LO-phonon for energy h ¯ ωLO , kB T is the thermodynamic energy, and
√ F 2 χ0 ne V03 e2 h ¯ ωLO a20 1 1 − , λ= e−EN,n /kB T , (8) A(ωc ) = 3 2 16(2π) λ¯hL 0 χ∞ χ0 N,n
|JN N (q⊥ )|2 = Fnn (±qz ) =
N ! −u N −N N −N 2 2 e u LN (u) , u = a2c q⊥ /2, N!
+∞ −∞
(9)
φ∗n (z)e±iqz z φn (z)dz.
(10)
Here, ne is the electron concentration, a0 is the dressing parameter, V0 is volume of the system, 0 is the permittivity of free space, χ∞ and χ0 are the high and low frequency dielectric constants, respectively, and LM N (x) is the associated Laguerre polynomials. After making a straightforward calculation of integral over q⊥ and qz , we obtain the following expression of the optical absorption power in asymmetric Gaussian potential quantum well P (Ω) =
A(ωc ) −EN,n /kB T e |Inn | NLO δ(p¯ hωc + εnn − h ¯ ωLO − h ¯ Ω) a2c N,n N ,n + (NLO + 1)δ(p¯hωc + εnn + h ¯ ωLO − h ¯ Ω) +
a20 (N + N + 1) NLO δ(p¯hωc + εnn − h ¯ ωLO − 2¯ hΩ) 2 8ac
+ (NLO + 1)δ(p¯hωc + εnn + h ¯ ωLO − 2¯ hΩ)
,
(11)
where Inn is the overlap integral, which is calculated as below in the extreme electric quantum limit (n = 0, n = 1) I01 =
+∞ −∞
√
2
|F01 (±qz )| dqz =
π(1 + 4β 2 ) eF L2 √ , β= . az U0 az 2
(12)
Finally, the delta functions in Eq. (11) are replaced by Lorentzians of width Γ± , which are given by Γ2±
¯ ωLO |Inn | V0 e2 h = 2 (2π¯ h) 0 ac L N − N
1 1 − (NLO + 1/2 ± 1/2). χ∞ χ0
5
(13)
In the next section, we will consider the numerical calculation in more detail.
3
Numerical results and discussion We have obtained the expression of the optical absorption power in quantum
well under the applied magnetic and electric fields. In this section, we perform numerical calculations for the GaAs and Ga1−x Alx As quantum wells in the extreme electric quantum limit. For numerical calculations, we take ne = 3 × 1016 cm−3 , F = 4.5 × 105 V/m, and a0 = 5 nm. The other parameters used are displayed in Table 1 [25,36,37]. Table 1 Parameters used in the present numerical calculation. Quantities
GaAs
Ga1−x Alx As
m∗ (m0 )
0.067
0.067 + 0.083x
¯hωLO (meV)
36.25
36.25 + 1.83x + 17.12x2 − 5.11x3
χ0
13.18
13.18 − 3.12x
χ∞
10.89
10.89 − 2.73x
Fig. 1 shows the dependence of the absorption power on photon energy in free standing quantum well for different materials: GaAs (solid), Ga0.7 Al0.3 As (dashed), and Ga0.6 Al0.4 As (dashed-dotted lines). There are two resonant peaks observed in each curve, which present resonance transfer of electrons with absorption of photons accompanied with the absorption or emission of LO-phonons. Using the computational method, we found that in GaAs quantum well (corresponding to the solid line), the peak at h ¯ Ω = 100.87 meV satisfies the condition h ¯Ω = h ¯ ωc + ε01 + h ¯ ωLO , corresponding to the one-photon absorption process 6
P arb.units
— GaAs --- Ga0.7Al0.3As Ga0.6Al0.4As
20
40
60
80
100
120
140
Photon energy meV Fig. 1. (Color online) Dependence of the absorption power on the incident photon energy in different quantum wells at T = 77 K, B = 7 T, L = 5 nm, and U0 = 15 meV.
(linear). Similarly, the peak at h ¯ Ω = 50.44 meV satisfies the condition 2¯ hΩ = h ¯ ωc + ε01 + h ¯ ωLO , describing the two-photon absorption process (nonlinear). Besides, we can also see that the peak value in the nonlinear process is about 15% those in the linear case. This means that the nonlinear process is strong significantly so that it can be observed. In Ga1−x Alx As quantum wells, the phenomenon occurs similarly, but the resonant peaks are observed to give a red-shift and the peak values are reduced. This is due to the electron effective mass and LO-phonon energy, which are modified in Ga1−x Alx As materials. For instance, with x = 0.3 (corresponding to the dashed-line), the effective mass m∗ = 0.092m0 leading to the cyclotron energy h ¯ ωc = 8.87 meV, phonon energy h ¯ ωLO = 38.20 meV, and ε01 = 44.78 meV. Therefore, the peak at h ¯ Ω = 91.85 meV satisfies the condition h ¯Ω = h ¯ ωc + ε01 + h ¯ ωLO . This value of the resonant peak is smaller than that in GaAs (100.87 meV), which explains the red-shift behavior of the absorption spectrum in Ga0.7 Al0.3 As quantum well in comparison with that in the GaAs case. The other peaks of the dashed and dashed-dotted lines can be explained in the same ways. 7
P arb.units
— U0 10 meV --- U0 15 meV U0 20 meV
20
40
60
80
100
120
140
Photon energy meV Fig. 2. (Color online) Absorption power in GaAs quantum well is shown as a function of photon energy for three values of the height of Gaussian potential quantum well. Here T = 77 K, B = 7 T, and L = 5 nm.
Fig. 2 illustrates the optical absorption power in GaAs quantum well as a function of the incident photon energy for three values of U0 . We can see from the figure that there is a blue-shift behavior of the optical absorption spectrum when the height of the Gaussian potential quantum well increases. This result is in good agreement with previous papers for the optical absorption coefficient [27], the optical rectification coefficients [31], and the second-harmonic generation coefficients [32] in the absence of the magnetic field. The physical meaning of this behavior is that when the height of potential increases, the subband separation energy ε01 become larger leading to the increase of the value of absorbed photon energies, which results in the blue-shift. Besides, we can also see from the figure that with the increase of U0 , the relative intensity of the resonant peaks decreases significantly, being in good agreement with previous papers [27,31,32]. This is due to the decrease of the overlap between different states when the height of potential increases. Indeed, from Eq. (11), we can obtain the result that when U0 is changed, the intensity of the resonant peak is mainly decided by the overlap integral I01 via the quantity β as shown in Eq. 8
(12). Therefore, when the height of the Gaussian potential increases, the quantum confinement effect is enhanced, which leads to the decrease of the overlap between different states [27]. This explains the decrease behavior of the relative intensity of
Halfwidth meV
the resonant peaks when U0 increases. 8 6 4
, Ga0.7Al0.3As , GaAs
2 0
10
15
20
25
30
U0 meV Fig. 3. (Color online) Dependence of the half-width on the height of the Gaussian potential at T = 77 K, L = 5 nm, and B = 7 T. The filled and empty lines correspond to the oneand two-photon absorption processes, respectively.
For the half-width (half-width at half maximum), using profile method [38], we obtain the dependence of the half-width on U0 as shown in Fig. 3. It can be found that the half-width in both GaAs and Ga1−x Alx As (with x = 0.3) decrease slightly with the increase of U0 . This is accompanied with the decrease of the overlap integral I01 when U0 increases. Besides, the half-widths in Ga0.7 Al0.3 As quantum well are greater than that in GaAs quantum well in both one- and two-photon absorption processes. It means that the probability of electron−phonon scattering becomes larger in Ga0.7 Al0.3 As material in comparison with that in GaAs. In addition to this, we can also see that the half-width in the nonlinear absorption process is smaller than that in the linear case in both Ga0.7 Al0.3 As and GaAs quantum wells. In Fig. 4, the optical absorption power in GaAs quantum well is shown as functions of photon energy for several values of well width. When L increases, a 9
P arb.units
— L 5 nm --- L 10 nm L 15 nm
20
40
60
80
100
120
140
Photon energy meV Fig. 4. (Color online) Absorption power in GaAs quantum well is shown as a function of photon energy for three values of well width. Here T = 77 K, B = 7 T, and U0 = 15 meV.
red-shift behavior of the absorption spectrum is observed, being in good agreement with previous results [27,31,32]. This is because that with the augmentation of well width, the quantum confinement effect gets weakened, leading to a decrease in the subband separation energy. Therefore, the values of the absorbed photon energies satisfying the resonant condition reduce and result in the red-shift. Moreover, the increase behavior of relative intensity of the resonant peaks can be also explained from the increase of the overlap between different states when the well width increases. In Fig. 5, we depict the dependence of the half-width on L. It can be seen from the figure that the half-width decreases rapidly with the well width from L = 4 nm to about L = 10 nm in both GaAs and Ga0.7 Al0.3 As quantum wells. At the larger well width, the decrease behavior of the half-width continues, but at a slighter rate. It can be said that the quantum confinement effect is observed distinctly only in a narrow quantum well with L ≤ 10 nm. The physical meaning of this can be explained as follows: when the well width increases, the quantum confinement effect reduces leading to the decrease of electron−phonon scattering, and so does 10
Halfwidth meV
12
, Ga0.7Al0.3As 10 , GaAs 8 6 4 2 0 5 10 15 20 L nm
Fig. 5. (Color online) Dependence of the half-width on well width at T = 77 K, U0 = 15 meV, and B = 7 T. The filled and empty lines correspond to the one- and two-photon absorption processes, respectively.
the half-width.
P arb.units
—B5T --- B 7 T B 9 T
20
40
60
80
100
120
140
Photon energy meV Fig. 6. (Color online) Absorption power in GaAs quantum well is shown as a function of photon energy for three values of magnetic field. Here T = 77 K, L = 5 nm, and U0 = 15 meV.
In Fig. 6, we show the dependence of the optical absorption power on photon energy for various values of magnetic field. The blue-shift behavior of the absorption spectrum associated with the increase of magnetic field is observed. This result is in good agreement with previous results [34,35]. This is because that when the 11
magnetic field increases, the Landau level separation becomes larger. Therefore, the values of the absorbed photon energies satisfying the resonant condition increase. Besides, we can also see from the figure that with the increase of the magnetic field, the relative intensity of the resonant peaks increases significantly. This is due to the decrease of the cyclotron radius, ac , when the magnetic field increases. Indeed, from Eq. (11), we can see that the intensity of the resonant peak is mainly decided by the quantity ac when magnetic field is changed. Therefore, when the magnetic field increases, the cyclotron frequency, ωc , is enhanced, which leads to the decrease of the cyclotron radius. This explains the increase behavior of the relative intensity of the resonant peaks when magnetic field increases. , Ga0.7Al0.3As , GaAs 10 8 6 4 2 0 0 5 10 15 20
Halfwidth meV
12
B T Fig. 7. (Color online) Dependence of half-width on magnetic field at T = 77 K, L = 5 nm, and U0 = 15 meV. The filled and empty lines correspond to the one- and two-photon absorption processes, respectively.
Fig. 7 shows the dependence of the half-width on B. It can be seen from the figure that the half-width increases with the increase of the magnetic field in both GaAs and Ga0.7 Al0.3 As quantum wells, as well as in the one- and two-photon absorption processes. The present result is in good agreement with the results of previous papers [33,34,39–43]. This increase behavior of the half-width is explained from the increase of the probability of electron−phonon interaction when magnetic 12
field increases.
P arb.units
— T 77 K --- T 150 K T 300 K
20
40
60
80
100
120
140
Photon energy meV Fig. 8. (Color online) Absorption power in GaAs quantum well is shown as a function of photon energy for different values of temperature. Here B = 7 T, L = 5 nm, and U0 = 15 meV.
In Fig. 8, we display the optical absorption power in GaAs quantum well as a function of photon energy for different values of temperature. From the figure, we can see that the resonant peaks are located at the same position but their intensity are seen to increase with temperature. This temperature-dependent behavior of the resonant peaks value is similar to that of the electron−phonon scattering rate [44], and the behavior of the absorption coefficient [45] in the presence of the magnetic field in square quantum well. This results from the enhancement of electron−phonon interaction when temperature increases. In Fig. 9, we describe the dependence of the half-width on temperature from T = 50 K to 350 K. We can see from the figure that the half-width increases nonlinearly with the increase of the temperature for both GaAs and Ga0.7 Al0.3 As quantum wells, as well as for one and two-photon absorption cases. The augmentation behavior of the half-with is due to the increase of the electron−phonon interaction when temperature increases. 13
Halfwidth meV
12
, Ga0.7Al0.3As 10 , GaAs 8 6 4
2 0
50
100
150
200
250
300
350
T K Fig. 9. (Color online) Dependence of half-width on temperature at B = 7 T, L = 5 nm, and U0 = 15 meV. The filled and empty lines correspond to the one- and two-photon absorption processes, respectively.
4
Conclusion In this paper, we have discussed the the linear and nonlinear optical absorp-
tion in GaAs and Ga1−x Alx As asymmetric Gaussian potential quantum wells with applied magnetic and electric fields via investigating the phonon-assisted cyclotron resonance effect. The calculations mainly focus on the dependence of the optical absorption power and the half-width on the height of the Gaussian potential quantum well, the well width, the magnetic field, and the temperature in both one- and two-photon absorption processes. The obtained results show that with the increase of U0 , B, and L, the optical absorption spectrum give the blue-shift and/or redshift, respectively, whereas, the position of the resonant peaks are independent of the change of the temperature. Besides, the half-width increases with the increase of magnetic field and temperature, but decreases with the increase of the height of the Gaussian potential and the well width in both one and two-photon cases. These results may be useful to understand the nonlinear optical properties in quantum wells under applied magnetic and electric fields. 14
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17
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Linear and nonlinear optical absorption spectra in AGPQWs have been investigated. The results are calculated for GaAs and Ga1-xAlxAs materials. The two-photon absorption process has been included. There is a clear monotonic behavior of the resonant peaks and the half-width in both one and twophoton absorption processes.