1.
Introduction

Multi-quantum wells (MQWs) with favorable optical properties at mid-infrared and far-infrared wavelength range are widely used in the development of the optoelectronic devices such as infrared lasers^{[1, 2]} and quantum cascade lasers^{[3, 4]}, which need the incident light at a certain wavelength and a wider absorption spectrum. In recent decades, the optical properties of GaAs/Al_xGa_1?xAs MQWs have always been a hot issue for researchers because of the highly adjustable band gaps and preferable energy band structure of these systems^[5].



In the early years, Mei et al. presented the polarization dependence of p-type and n-type infrared photodetectors of GaAs/AlGaAs MQWs with different doping concentrations^[6]. Recently, Dakhlaoui et al. found that the total optical absorption coefficient produces a red shift first and then a blue shift by changing the doping position from the quantum well (QW) to the left barrier in GaAs/Al_0.2Ga_0.8As with a fixed doping concentration of 5 × 10¹⁶ cm^?3 and refractive index changes (RCIs) appear as red shifts first then blue shifts with an increase of the distance from the doping position to the right barrier^[7]. Solaimani et al. indicated that by increasing the number from 1 to 3 of QWs with a fixed total width, the peak of absorption has a red shift, whereas, if the number is greater than 3 the peak has a blue shift. Furthermore, they pointed out that the critical number (the transition point from a red shift to a blue one) for RICs is 6^[8]. Early on, Kong et al. revealed that there is a certain considerable variation of the RICs when there is one more or less than 16 monolayers in an asymmetric coupled QWs under a ?50 kV/cm electric field without doping^[9]. However, Herrera et al. recently confirmed that manipulating the Al composition changes the nonlinear phenomena in MQWs, which is composed of 40 layers in experiments by fixing the width to 1.2 μm without the consideration of the size effect^[10]. On the other hand, Bilel et al. obtained that an optimized doping value of 5 × 10¹⁷ cm^?3 leads the transition energy to a blue shift at 1.55 μm by self-consistent calculation for uncoupled double QWs^[11]. Choi et al. calculated the intersubband optical absorption coefficients (IOACs) in a typical infrared photo-detector composed of 19 periodic GaAs/Al_0.22Ga_0.78As QWs with a certain Al composition and 5 × 10¹⁷ cm^?3 doping, then comparing the results with experiments^[12]. Unfortunately, in most of these works, the effects of the Al component on IOACs and RICs were considered without an external electric field, which should have had an obvious modulation on the optoelectronic property of relative devices.



In this work, we investigate the effects of size, ternary mixed crystal (TMC), doping concentration N_D, and temperature T on IOACs and RICs in GaAs/Al_xGa_1?xAs MQWs by introducing a lateral electric field (LEF). Firstly, we use the method of effective mass approximation to solve self-consistently the Schr?dinger equation and Poisson equation to obtain the energy levels and wave functions of electrons in the conduction band. Secondly, the linear, and third-order nonlinear IOACs and RICs influenced by the intersubband transition of electrons are discussed. Finally, we calculate the IOACs of MQWs and compare the results with experiments.

2.
Model and theory

Let us consider the transition of an electron from the ground state to the first excited state in a simplified model with three AlGaAs/GaAs periodic structures in MQWs as shown in Fig. 1. The MQWs are constituted by three QWs of GaAs with the same width L_w, two side barriers of AlGaAs with width L_b, and two middle barriers of AlGaAs with the same width L_b0 in a 30 kV/cm LEF. Without losing generality, it can be assumed that the outsides of the two side barriers are vacuum layers. The origin point of the z axis is to the far left of the left side barrier, and is perpendicular to the direction of the substrate surface, which is in the x–y plane. The motion of an electron along the z axis is quantized since it is affected by the additional periodic potential, whereas the motion of the electron in the x–y plane is a plane wave. As a result, one can simplify the electron transition only by considering the transition along the z direction.






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Figure1.
(Color online) A simplified schematic of MQWs with three periodic Al_xGa_1?xAs/GaAs containing the ground states (dashed lines), first excited states (dotted lines), second excited states (dash-dotted lines), and Fermi energy level (dash-dotted-dotted lines between the ground state and first excited state) in a 30 kV/cm LEF. The widths of the wells, side barriers, and middle barriers are denoted by L_w, L_b, and L_b0, respectively. The energy band offset between the well and barrier is denoted by ΔE_g. The arrows in the wells indicate the transition of electrons.




With the effective mass approximation, the time-independent one-dimensional Schr?dinger equation of an electron in MQWs in the z direction can be written as

$left{ { - frac{{{hbar ^2}}}{2}frac{partial }{{partial z}}left( {frac{{ m{1}}}{{m_{ m{i}}^{ m{*}}(z)}}} ight)frac{partial }{{partial z}} + Vleft( z ight) + {V_{{ m{H}},{ m{i}}}}left( z ight) + eFleft( z ight)z} ight}{phi _{ m{i}}}left( z ight) = {E_{ m{i}}}{phi _{ m{i}}}left( z ight),$

(1)

where $hbar $ is the Planck constant, m^*(z) the effective mass of the electron, e the absolute value of the electronic charge, and F(z) the strength of the LEF, respectively. V(z) is the confinement potential, which equals 0 in the wells, and V₀ in the barriers. V_{H, i}(z) is the effective Hartree potential corresponding to the ith eigen-state and can be solved by the Poisson equation. According to the offset principle of conduction bands being 60 : 40^[13], V₀ can be written as

${V_0} = 60% times left( {{E_{{ m{g}},{ m{AlGaAs}}}} - {E_{{ m{g}},{ m{GaAs}}}}} ight).$

(2)

In Eq. (2) the TMC band gap E_{g, AlGaAs} can be obtained by an interpolation between the band gaps E_{g, GaAs} of the two binary materials with the Al component x^[14]

${E_{{ m{g}},{ m{AlGaAs}}}} = x{E_{{ m{g}},{ m{AlAs}}}} + (1 - x){E_{{ m{g}},{ m{GaAs}}}}.$

(3)

The effective Hartree potential V_{H, i}(z) in Eq. (1) arises from the ionization generated by the interaction of electrons and conforms to the generalized Poisson equation^[15]

$frac{{ m d}}{{{ m d}z}}left[ {{varepsilon _0}{varepsilon _{ m{r}}}left( z ight)frac{{ m d}}{{{ m d}z}}{V_{{ m{H}},{ m{i}}}}left( z ight)} ight] = - {e^2}left[ {N_{ m{D}}^ + left( z ight) + {n_{2{ m{D}},{ m{i}}}}left( z ight)} ight],$

(4)

where ε₀ and ε_r(z) are the permittivities?of?the vacuum and the relative one of the position dependence. N_D⁺(z) is the concentration of ionized dopants, n_{2D, i}(z) denotes the two-dimensional electron gas concentration of the ith subband. One can have the following relation

$N_{ m{D}}^ + left( z ight) = {N_{ m{D}}}left( {frac{1}{{1 + 2exp left[ {left( {{E_{ m{f}}} - {E_{ m{D}}}} ight)/{k_{ m{B}}}T} ight]}}} ight),$

(5)

and

${n_{2{ m{D}},{ m{i}}}}left( z ight) = sumlimits_n^{} {frac{{{m^{ m{*}}}{k_{ m{B}}}T}}{{pi {hbar ^2}}}{{left| {{phi _{ m{i}}}left( z ight)} ight|}^2}ln left[ {1 + exp left( {frac{{{E_{ m{F}}} - {E_{ m{n}}}}}{{{k_{ m{B}}}T}}} ight)} ight]} ,$

(6)

in which N_D is the Si doping concentration and donor energy level E_D = E_{g, GaAs} ? 0.03 eV^[14]. k_B is the Boltzmann constant. E_f and E_F are the Fermi energy at T = 0 K and a finite T, respectively. They satisfy^[16]

${E_{ m{f}}} = pi {hbar ^2}N/{m^{ m{*}}},$

(7)

and

${E_{ m{F}}} = {k_{ m{B}}}Tln left[ {exp left( {{E_{ m{f}}}/{k_{ m{B}}}T} ight) - 1} ight].$

(8)

In Eq. (7) N = 2 × 10¹⁵ m^?2[17], which is the electron density in the ith eigen-state at T = 0 K.



By adopting the method of density matrix, linear IOACs α⁽¹⁾(ω), and the third-order nonlinear IOACs α⁽³⁾(I, ω) can be given as follows^{[14, 15, 18, 19]}:

${alpha ^{(1)}}left( omega ight) = omega sqrt {frac{mu }{{{varepsilon _{ m{R}}}}}} {left| {{M_{21}}} ight|^2}frac{{sigma hbar varGamma }}{{{{left( {{E_{21}} - hbar omega } ight)}^2} + {{left( {hbar varGamma } ight)}^2}}},$

(9)

and

$begin{split} & {alpha ^{(3)}}left( {I,omega } ight) !=! - omega sqrt {frac{mu }{{{varepsilon _{text{R}}}}}} frac{I}{{{text{2}}{varepsilon _{text{0}}}{n_{text{r}}}c}}frac{{{{left| {{M_{21}}} ight|}^2}sigma hbar varGamma }}{{{{left[ {{{left( {{E_{21}} - hbar omega } ight)}^2} + {{left( {hbar varGamma } ight)}^2}} ight]}^2}}}{ Bigggr[{22} {{text{4}}left| {{M_{21}}} ight|}^2} & quad - left. {frac{{{{left| {{M_{22}} - {M_{11}}} ight|}^2}left{ {{{({E_{21}} - hbar omega )}^2} - {{(hbar varGamma )}^2} + 2{E_{21}}({E_{21}} - hbar omega )} ight}}}{{{{({E_{21}})}^2} + {{(hbar varGamma )}^2}}}} ight], end{split} $

(10)

where I and ω are the strength and frequency of the incident light. μ is the vacuum permeability. ε_R is the real part of permittivity ε_R = n_r²ε₀, here n_r is the refractive index^[20], c the speed of light in a vacuum, σ the carrier density, and $varGamma $ the relaxation rate. In Eqs. (9) and (10), ${M_{21}} = int {phi _2^*left( z
ight)left| e
ight|} z{phi _1}left( z
ight)operatorname{d} z$^[21] and E₂₁ = E₂ ? E₁ are the overlap integral and energy level difference from the ground state to the first excited state, respectively. σ is related to the electron numbers n_i = σL_w in the ith subband

${n_{ m{i}}} = int_0^L {frac{{{m^{ m{*}}}{k_{ m{B}}}T}}{{pi {hbar ^2}}}{{left| {{phi _{ m{i}}}left( z ight)} ight|}^2}ln left[ {1 + exp left( {frac{{{E_{ m{F}}} - {E_{ m{i}}}}}{{{k_{ m{B}}}T}}} ight)} ight]} { m{d}}z.$

(11)

Then, the total IOACs α(I, ω) can be written as

$alpha (I,omega ) = {alpha ^{left( 1 ight)}}left( omega ight) + {alpha ^{left( 3 ight)}}left( {I,omega } ight).$

(12)

The linear RICs Δn⁽¹⁾(ω)/n_r and third-order nonlinear RICs Δn⁽³⁾(I, ω)/n_r can be written as^{[22, 23]}

$frac{{Delta {n^{left( 1 ight)}}(omega )}}{{{n_{ m{r}}}}} = frac{{sigma {{left| {{M_{21}}} ight|}^2}}}{{2n_{ m{r}}^{ m{2}}{varepsilon _0}}}left[ {frac{{{E_{21}} - hbar omega }}{{{{left( {{E_{21}} - hbar omega } ight)}^2} + {{left( {hbar varGamma } ight)}^2}}}} ight],$

(13)

and

$begin{split}& frac{{Delta {n^{left( 3 ight)}}(omega,I )}}{{{n_r}}} = & frac{{sigma {{left| {{M_{21}}} ight|}^2}}}{{4n_r^3{varepsilon _0}}}frac{{mu cI}}{{{{left[ {{{left( {{E_{21}} - hbar omega } ight)}^2} + {{left( {hbar varGamma } ight)}^2}} ight]}^2}}}left[ {4left( {{E_{21}} - hbar omega } ight)} ight.{left| {{M_{21}}} ight|^2} & - frac{{{{left( {{M_{22}} - {M_{11}}} ight)}^2}}}{{{{left( {{E_{21}}} ight)}^2} + {{left( {hbar varGamma } ight)}^2}}}left{ {left( {{E_{21}} - hbar omega } ight)left[ {{E_{21}}left( {{E_{21}} - hbar omega } ight) - {{left( {hbar varGamma } ight)}^2}} ight]} ight. hfill & left. {left. { - ,{{left( {hbar varGamma } ight)}^2}left( {2{E_{21}} - hbar omega } ight)} ight}} ight]. end{split} $

(14)

The total RICs Δn(I, ω)/n_r can be written as following

$frac{{Delta n(omega,I )}}{{{n_{ m{r}}}}} = frac{{Delta {n^{left( 1 ight)}}(omega )}}{{{n_{ m{r}}}}} + frac{{Delta {n^{left( 3 ight)}}(omega,I )}}{{{n_{ m{r}}}}}.$

(15)

3.
Results and discussions

The parameters used in the computational process are as follows^[26]: n_r = 3.2, $varGamma $ = 1/0.14 ps, and I = 4 MW/cm², the others needed are shown in Table 1.

Parameter	AlAs	GaAs	AlxGa1?xAs
Eg (eV)	3.09a	1.519a	3.09x + 1.519(1 ? x)b
mi*(m0)	0.15a	0.067a	0.15x + 0.067(1 ? x)b
εr (ε0)	10.8a	13.18a	10.8x + 13.18(1 ? x)b
a Ref. [24]. b Ref. [25].

Table1.
Parameters used in numerical calculations.



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Parameter	AlAs	GaAs	AlxGa1?xAs
Eg (eV)	3.09a	1.519a	3.09x + 1.519(1 ? x)b
mi*(m0)	0.15a	0.067a	0.15x + 0.067(1 ? x)b
εr (ε0)	10.8a	13.18a	10.8x + 13.18(1 ? x)b
a Ref. [24]. b Ref. [25].

In Fig. 2, the corresponding wave functions of the ground state, first excited state, and second excited state in a single QW, three, and five QWs, as well as MQWs are calculated by self-consistently solving the Schr?dinger equation and Poisson equation in GaAs/Al_0.26Ga_0.74As MQWs with L_w = 5 nm, N_D = 5 × 10¹⁷ cm^?3, and T = 77 K in a 30 kV/cm LEF. As is well known, the electrons distribute symmetrically in a single QW without an LEF. As the number of QWs increases the electrons still distribute symmetrically over a wider range in different QWs. There is a negligible effect to break the symmetry on the ground state of electrons in a single QW because the energy level of the ground state is deeper in the well. However, the energy levels of excited states rise from the bottom of the conduction band, making an enhanced penetration of electrons into the barriers due to the tilt of QWs in an LEF. The symmetry of the distribution is broken as shown in Fig. 2(a). Figs. 2(b)–2(d) indicate that the effect of the LEF increases the right-side barriers of MQWs, however, electrons do not concentrate in the left well except on the ground states, whereas the distribution of electrons moves to the right gradually for higher excited states. Fig. 3 shows the linear IOACs α⁽¹⁾(ω) as functions of incident light wavelength λ for a single QW, double, three, four, and five QWs, as well as MQWs. It can be seen that the peak of α⁽¹⁾(ω) is sharp and locates at a shorter λ for a single QW. As the number of QWs increases, the peaks of α⁽¹⁾(ω) decrease, and they move to a longer λ. The half width of each α⁽¹⁾(ω) is broadened since the ground states and first excited states in different QWs split off to form energy bands due to the Pauli exclusion principle. Electrons can transition between different sub-levels. On the other hand, this effect results in the peak positions vibrating with increasing the number of QWs.






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Figure2.
(Color online) Electronic wave functions at the ground states (dashed curves), first excited states (dash-dotted curves), and second excited states (dotted curves) in conduction bands (full lines) of (a) a single QW, (b) three QWs, (c) five QWs, and (d) MQWs as a function of z in a 30 kV/cm LEF at N_D = 3 × 10¹⁷ cm^?3, respectively.






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Figure3.
(Color?online) Linear IOACs α⁽¹⁾(ω) as a function of incident light wavelength λ in a single (short dotted curve) QW, double (dash-dotted-dotted curve), three (dotted curve), four (dash-dotted curve), five (full curve) QWs, and MQWs (dashed curve), respectively.




Fig. 4 shows α⁽¹⁾(ω), α⁽³⁾(I, ω), α(I, ω) and Δn⁽¹⁾(ω)/n_r, Δn⁽³⁾(I, ω)/n_r, Δn(I, ω)/n_r as a function of Al components x from 0.1 to 0.4 for the three QWs with L_w = 5 nm, N_D = 3 × 10¹⁷ cm^?3, and T = 77 K in a 30 kV/cm LEF. As x increases, all of the IOACs do not change obviously, but the peaks shift to the long wavelength direction, as shown in Fig. 4(a). It can be seen clearly from Fig. 4(b) that the saturation points of the RICs shift to a longer λ. However, without an LEF, the electron energy levels rise from the bottom of the conduction band as x increases, the energy level difference also increases, and the peaks of the IOACs shift to a shorter λ. When an LEF is applied, the electrons in the right QWs tunnel into the left ones to show the coupling of the wells as x increases, the barrier potential increases to reduce the tunneling of electrons. As a competition result, the effect of the LEF on the MQWs reduces so does the energy level difference, the peaks of the IOACs appear as a red shift, and the RICs have similar properties. It can also be seen that, as the incident light intensity becomes stronger the total IOACs trigger a light saturation because an equilibrium will appear between the number of the electrons transition from the ground state to the first excited one, and that from the first excited state to the ground state according to the relationship of K-K^[27].






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Figure4.
(Color online) Under a 30 kV/cm LEF with N_D = 3 × 10¹⁷ cm^?3, (a) α⁽¹⁾(ω) (dash-dotted curves), α⁽³⁾(I, ω) (dashed curves), and α(I, ω) (full curves), (b) Δn⁽¹⁾(ω)/n_r (dash-dotted curves), Δn⁽³⁾(I, ω)/n_r (dashed curves), and Δn(I, ω)/n_r (full curves) as functions of incident light wavelength λ with Al components x = 0.1, 0.23, and 0.39, respectively.




In Fig. 5, the effects of (a) L_b (while L_w = 5 nm) and (b) L_w (while L_b = 30 nm) on IOACs are given with x = 0.26, N_D = 5 × 10¹⁷ cm^?3 and T = 77 K in a 30 kV/cm LEF. It can be seen from Fig. 5(a) that with increasing L_b from 20 to 50 nm, the peaks of IOACs increase, and at the same time they shift to a shorter λ. The half widths of α⁽¹⁾(ω) and α⁽³⁾(I, ω) become narrower, and α(I, ω) saturates more obviously since the contribution from α⁽³⁾(I, ω) is bigger. The IOACs exhibit similar properties as L_w increases from 4 to 6 nm, as shown in Fig. 5(b). They are very sensitive to the size effect. With an increase of L_b, the tunneling of electrons from the left well to the left side barrier becomes weaker, and the energy levels rise up from the bottom of the conduction band to result in a blue shift of the peak. The increase of L_b on both sides leads to less separation of energy levels so as to narrow the half widths of IOACs. As L_w increases, the peaks appear with a distinct blue shift since the coupling of the wells and the LEF increases the separation between the ground states and excited states. This effect is shown in Fig. 5(b), and the RICs have similar properties, as shown in Fig. 6. As discussed above, it can be seen that the widths of L_b and L_w in QWs are important parameters to influence nonlinear optical properties.






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Figure5.
(Color online) Under a 30 kV/cm LEF with N_D = 3 × 10¹⁷ cm^?3, α⁽¹⁾(ω) (dash-dotted curves), α⁽³⁾(I, ω) (dashed curves), and α(I, ω) (full curves) as functions of incident light wavelength λ with (a) L_b = 20, 25, 30, 35 nm, and (b) L_w = 4, 4.5, 5, 5.5 nm, respectively.






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Figure6.
(Color online) Under a 30 kV/cm LEF with N_D = 3 × 10¹⁷ cm^?3, Δn⁽¹⁾(ω)/n_r (dash-dotted curves), Δn⁽³⁾(I, ω)/n_r (dashed curves), and Δn(I, ω)/n_r (full curves) as functions of incident light wavelength λ with (a) L_b = 20, 30, 35 nm, and (b) L_w = 4, 5 nm, respectively.




Fig. 7 demonstrates that IOACs and RICs vary with Si doping concentration N_D from 10¹⁷ to 10¹⁹ cm^?3 at x = 0.26, L_w = 5 nm, and T = 77 K in a 30 kV/cm LEF. As N_D increases, the peak amplitudes of IOACs change insignificantly, since it does not affect the overlap of the electronic wave functions at different states, however, the peaks shift to a longer λ in general because N_D decreases the separation of the energy levels at different states. On the one hand, N_D influences the Fermi energy to weaken the LEF electron transition. It must be pointed out that the larger the N_D is, the more obvious it affects the red shift. On the other hand, N_D can smear the red shift when it takes a larger value^[26]. Also, RICs have similar properties.






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Figure7.
(Color online) Under a 30 kV/cm LEF with L_w = 5 nm, (a) α⁽¹⁾(ω) (dash-dotted curves), α⁽³⁾(I, ω) (dashed curves), and α(I, ω) (full curves), (b) Δn⁽¹⁾(ω)/n_r (dash-dotted curves), Δn⁽³⁾(I, ω)/n_r (dashed curves), and Δn(I, ω)/n_r (full curves) as functions of incident light wavelength λ with N_D = 3 × 10¹⁷, 1 × 10¹⁸, 5 × 10¹⁸, and 8 × 10¹⁸ cm^?3, respectively.




In Fig. 8, the variations of the linear IOACs with T increasing from 77 to 300 K are shown when x = 0.26, L_w = 5 nm, and N_D = 5 × 10¹⁷ cm^?3 in a 30 kV/cm LEF. On the one hand, as T increases, the peaks of IOACs have blue shifts first and then red ones since the Fermi energy decreases to compensate for the separation of the energy levels at different states. On the other hand, the amplitudes of IOACs increase first and then decrease because the Fermi energy level decreases to enhance the electronic transition, which is a competition between the electron tunneling and the transition as T increases.






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Figure8.
(Color online) α⁽¹⁾(ω) as a function of incident light wavelength λ with L_w = 5 nm in a 30 kV/cm LEF at temperature T = 77 (full curves), 200 (dashed curves), and 300 K (dotted curves), respectively.




Fig. 9 gives the effect of an LEF on α(I, ω) with x = 0.26, L_w = 5 nm, T = 77 K, and N_D = 5 × 10¹⁷ cm^?3. One can find that when an LEF is less than 30 kV/cm, the peaks of IOACs do not show an obvious shift because at this moment the doping is the main reason to influence the potential of QWs. When the LEF increases from 30 kV/cm, the peaks of IOACs appear to have blue shifts since an LEF induces a tilt of potentials to increase the separation of electronic energy levels between different states.






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Figure9.
(Color online) α(I, ω) as a function of incident light wavelength λ for LEF = 0 (full curve), 35 (dotted curve), 65 (dash-dotted-dotted curve), 80 (dash-dotted curve), and 100 (short dotted curve) kV/cm, respectively.




Finally, for comparison, IOACs for two MQWs are calculated. A graph illustrating 20 periods of AlGaAs/GaAs QWs is shown in Fig. 10(a). The peak position by our theoretical results is 1.7% higher than that of the experiment^[28], while Fig. 10(b) gives the results of 20 periods of AlGaAs/GaAs/InGaAs/GaAs QWs. It is shown that our theoretical value is 8% higher than that by the same experiment. Our results agree with the experiment. It can be seen that the half width of IOACs by the above experiment is greater than our theoretical value and there are secondary wave peaks in the long wavelength range. The former is related to the relaxation time of phonons and the latter is the excitonic absorption. This is expected since their effects have not been considered in our model.






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Figure10.
Comparison of IOACs calculated (solid curves) in (a) AlGaAs/GaAs MQWs, and (b) AlGaAs/GaAs/InGaAs MQWs as a function of wavelength λ with that of experimental values (dotted curves).

4.
Conclusion

We computed the energy levels and wave functions by solving the self-consistent Schr?dinger equation and Poisson equation and discussed the effects of size, TMC, N_D, and T on linear, nonlinear IOACs and RICs due to the electron transition between the ground states and first excited states based on Fermi’s golden rule. Our results demonstrated that the amplitudes and ranges of IOACs and RICs can be modulated by the above effects, and more importantly the influence from an LEF on the above effects is exhibited. On the other hand, the transition energy and relaxation time can also be designed by choosing an appropriate size. Our theoretical results show that the absorption spectrum can be broadened by increasing the numbers or decreasing the size of the MQWs, whereas the detection range can be moved to a longer wavelength by increasing the aluminum component and doping concentration. Furthermore, the RICs can be reduced by decreasing the size of the system or increasing the incident light strength. These modulations are helpful to optimize the optical properties of optoelectronic devices consisting of MQWs.