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Electron states and electron Raman scattering in an asymmetrical double quantum well: External electric field
Physica B: Condensed Matter 545 (2018) 215–221
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Physica B: Condensed Matter journal homepage: www.elsevier.com/locate/physb
Electron states and electron Raman scattering in an asymmetrical double quantum well: External electric field
T
L. Ferrer-Galindoa, Ri. Betancourt-Rierab, Re. Betancourt-Rierab,∗, R. Rierac a
Departamento de Física, Universidad de Sonora, Blvd. Luis Encinas y Rosales S/N, Apartado Postal 1626, Col. Centro C.P. 83000, Hermosillo, Sonora, Mexico Instituto Tecnológico de Hermosillo, Avenida Tecnológico S/N, Col. Sahuaro, C.P. 83170, Hermosillo, Sonora, Mexico c Departamento de Investigación en Física, Universidad de Sonora, Blvd. Luis Encinas y Rosales S/N, Apartado Postal 5-88, Col. Centro, C.P. 83000 Hermosillo, Sonora, Mexico b
A R T I C LE I N FO
A B S T R A C T
Keywords: Electron states Raman scattering Quantum well
In this paper studies an electron Raman scattering process for a semiconductor, coupled and asymmetrical double quantum well. Then, the presence of an electron in a single conduction band is considered. In addition, the system is subjected to an external electric field. To carry out this study, the net Raman gain and the differential cross section are calculated. The emission spectra are interpreted and discussed. For this, we obtain the exact solutions of the electron states considering the envelope function approximation and a single parabolic conduction band, which is split into a sub-bands system due to confinement. Furthermore, the effect of the electric field on the electron states and in the differential cross section is studied. To illustrate our findings, we have considered a system growing in a GaAs / Al x Ga1 − x As matrix.
1. Introduction The structures based on multiple quantum wells have been studied for more than three decades because of their optical and electronic properties used in the development of applications such as solar cells, high efficiency multiband detectors and lasers [1–3]. Furthermore, it has been proposed to use them as a source of coherent mid-infrared radiation based on electron Raman scattering [4–9]. Recently, the use of Raman scattering as a control method in the growth of multiple quantum well systems has gained interest [10]. Thus, Raman scattering is a highly efficient tool, both theoretically and experimentally, for research of electronic and optical properties of low dimensional semiconductor structures [11–23]. GaAs-based semiconductors have a low macroscopic polarization, unlike those based on wurtzite nitrides (as the group-III nitride semiconductor compounds) where the macroscopic polarization is strong [24]. The magnitude of the built-in electric field induced by the piezoelectricity and the spontaneous polarization in the wurtzite nitrides semiconductors is estimated in the order of MV/cm [24–28]; however, in the GaAs-based semiconductors this is negligible. In consequence, this implies that the quantum-confined Stark effect on devices based on GaAs is simpler to control than it is in nitrides. On the other hand, from previous studies, it is known that the electric field effect and the
∗
electronic and optical properties (especially in the Raman net gain and in the Raman emission spectra) are complex in nitrides [29,30]. In this document, the exact solutions of the mathematical expressions of the electron states for a semiconductor, coupled and asymmetrical double quantum well in the presence of an external electric field are obtained. Moreover, the emission spectra and the selection rules obtained when calculating the differential cross-section corresponding to an intra-band electron Raman scattering process are shown. In addition, the Raman net gain is also analyzed in a three-level system. With this system we can analyze the structures that have already been studied, such as the step-quantum well and the double quantum well [29], which allows us an easy comparison with these systems; then, this system generalizes the aforementioned structures. For a better understanding of our results, this article has been divided into several sections: In Section 2, the electron states and the Raman differential crosssection expressions are obtained. In Section 3, the results considering a system that grows in a GaAs / Al x Ga1 − x As matrix are presented, and the effect produced by the electric field and by the change in the system parameters is determined. Finally, in Section 4, the conclusions are presented.
Corresponding author. E-mail addresses: [email protected] (L. Ferrer-Galindo), [email protected] (R. Betancourt-Riera), [email protected] (R. Betancourt-Riera), [email protected] (R. Riera). https://doi.org/10.1016/j.physb.2018.06.017 Received 30 March 2018; Received in revised form 15 June 2018; Accepted 16 June 2018 Available online 19 June 2018 0921-4526/ © 2018 Published by Elsevier B.V.
Physica B: Condensed Matter 545 (2018) 215–221
L. Ferrer-Galindo et al.
2. Model and theory
A2 =
In this section, we show the solution of the Schrödinger equation in the envelope function approximation. Then, we can determine the bound states of a confined electron in a semiconductor, coupled and asymmetrical double quantum well system; with the presence of an external electric field (F), which is constant and uniform. Therefore, the Schrödinger equation takes the following form:
{
∇2 +
A3 = A4 =
}
Q3 (l1) 3 (l ) A + Ξ 3 (l ) B Σ32 2 3 32 2 3
Q2 (l2)
, B3 = − , B4 = −
2 (l ) A + Λ 2 (l ) B Φ23 1 2 23 1 2
Q3 (l1) 3 (l ) A + Λ 3 (l ) B Φ32 2 3 32 2 3
Q2 (l2)
Ai(η2l3) A 4 + Bi(η2l3) B4 Ai(η1l3)
, ,
,
Π22 = Ci(η10 )Ai′ (η20) − α1 Ci′ (η10 )Ai(η20),
(1)
Φijk (z ) = Ai(ηiz )Ai′ (ηjz ) − αk Ai′ (ηiz )Ai(ηjz ), Λijk (z ) = Bi(ηiz )Ai′ (ηjz ) − αk Bi′ (ηiz )Ai(ηjz ),
ℏ2 2 Γ k⊥ − i , 2μ 2
Σijk (z ) = Ai(ηiz )Bi′ (ηjz ) − αk Ai′ (ηiz )Bi(ηjz ), Ξijk (z ) = Bi(ηiz )Bi′ (ηjz ) − αk Bi′ (ηiz )Bi(ηjz ),
being Ez (n) the part of the quasi-bound state energy that corresponds to the confinement and ℏ2k⊥2/2μ the energy that corresponds to the free direction. n = 1,2, …, is the number assigned to the electron states, while Γ is the resonance width found to be positive [31]. A coupled asymmetrical double quantum well was considered with the corresponding layer thickness −∞/ d1/ b/ d2/ +∞. Furthermore, the presence of constant and uniform electric field F is considered. Then, the effective mass (μ) and the confinement potential (Vc ), can be chosen as:
V1, − ∞ < z < 0 V2, 0 ≤ z ≤ l1 V3, l1 < z < l2 V2, l2 ≤ z ≤ l3 V1, l3 < z < + ∞
Qi (z ) = Ai(ηiz )Bi′ (ηiz ) − Ai′ (ηiz )Bi(ηiz ), 0
I1 = ∫−∞ Ci∗ (η1z )Ci(η1z ) dz , l
I2 = ∫0 1 [A2 Ai(η2z ) + B2 Bi(η2z )]∗ [A2 Ai(η2z ) + B2 Bi(η2z )] dz , l
I3 = ∫l 2 [A3 Ai(η3z ) + B3 Bi(η3z )]∗ [A3 Ai(η3z ) + B3 Bi(η3z )] dz , 1 l
I4 = ∫l 3 [A 4 Ai(η2z ) + B4 Bi(η2z )]∗ [A 4 Ai(η2z ) + B4 Bi(η2z )] dz 2 +∞
and I5 = A5∗ A5 ∫l
3
Ai∗ (η1z )Ai(η1z ) dz ,
with
α1 = [μ 2 / μ1]2/3 , α2 = [μ3 / μ 2 ]2/3 , α3 = [μ 2 / μ3]2/3 and α4 = [μ1 / μ 2 ]2/3 .
(2)
On the other hand, the differential cross-section for an intra-band electron Raman scattering process in a volume per unit solid angle (Ω) of incoming light with energy (ℏωl ) and scattering light with energy (ℏωs ), is calculated. In Ref. [32] the calculation process to obtain the differential cross section is described; therefore
where l1 = d1, l2 = d1 + b and l3 = d1 + b + d2. The wave function takes the following form:
Ψ (r) =
2 (l ) A + Ξ 2 (l ) B Σ23 1 2 23 1 2
A5 =
This implies that there are not bound states, but quasi-bound states, with energies given by:
⎧ μ1 , ⎪ μ2 , ⎪ μ, Vc = μ3 , ⎨ ⎪ μ2 , ⎪μ , ⎩ 1
Π22 , 2 (0)
B2 = − Q
Π12 = Ci(η10)Bi′ (η20) − α1 Ci′ (η10)Bi(η20),
2μ [E − Vc − e Fz ] Ψ = 0. ℏ2
E (n, k⊥) = Ez (n) +
Π12 , Q2 (0)
exp[i k⊥·r⊥] φn (z ) u 0 (r) Lx L y
(3)
where
∂ 2σ ω = σ0 Γ f2 s ∂ωs ∂Ω ωl
A1 Ci(η1z ),
⎧ −∞ < z < 0 ⎪ A2 Ai(η ) + B2 Bi(η ), 0 ≤ z ≤ l1 2z 2z ⎪ l1 < z < l2 φn (z ) = A3 Ai(η3z ) + B3 Bi(η3z ), ⎨ l2 ≤ z ≤ l3 ⎪ A 4 Ai(η2z ) + B4 Bi(η2z ), ⎪
∑ n, n″
Mf (n, n″)
2 2
[ℏωl − ℏωs + Ez (n″) − Ez (n)] + Γ f2
(4)
where
Mf (n, n″) =
ℏ2 2μ 0 l32
− ∑n′ T (n, n′) T (n′, n″) ⎡ ⎢ ℏωs + Ez (n) − Ez (n′) + iΓa ⎣ 1
Ci(η) = Bi(η) + i Ai(η) and ηjz = −[2μj /(eF ℏ)2]1/3 [Ez (n) − Vc (z )
−
⎤,
1
ℏωl − Ez (n) + Ez (n′) − iΓ b ⎥
(5)
⎦
− e Fz ].
σ0 =
Being u 0 (r) the periodic part of the Bloch function, Ai and Bi the Airy's functions and finally j = 1,2 or 3. Once the boundary conditions are applied, the continuity of the function Ψ and the current (1/ μ)(∂Ψ / ∂z ) at the interface, Ez (n) are determined by the following secular equation:
η (ωs ) e 4 4ℏ ⎛ ⎞ (es ·ez )(el ·ez ) 2 . η (ωl ) ⎝ c ⎠ πμ02 Γf
and in this case 0
⎡1 T (na, nb) = μ0 l3 ⎢ μ ∫ φn∗a (z ) φn′b (z ) dz + 1 ⎣ −∞
2 2 3 4 3 4 [Π12 Σ23 (l1) − Π22 Ξ23 (l1)][Σ32 (l2) Φ21 (l3) − Φ32 (l2) Λ21 (l3)]= 2 2 3 4 3 4 [Π12 Φ23 (l1) − Π22 Λ23 (l1)][Ξ32 (l2) Φ21 (l3) − Λ32 (l2) Λ21 (l3)]
+
while the constants A and B have the following form:
+
A1 = [I1 + I2 + I3 + I4 + I5]−1/2 , A2 = A2 A1 , B2 = B2 A1 , A3 = A3 A1 , B3 = B3 A1 , A 4 = A 4 A1 , B4 = B4 A1 ,
1 μ2
l3
1 μ3
l1
∫ φn∗a (z ) φn′b (z ) dz 0
l2
∫ φn∗a (z ) φn′b (z ) dz l1
∫ φn∗a (z ) φn′b (z ) dz + l2
1 μ2
1 μ1
+∞
⎤
l3
⎦
∫ φn∗a (z ) φn′b (z ) dz⎥
(6)
where μ0 is the electron free mass. η (ωr ) is the refraction index as a function of the radiation with frequency ωr and polarization unit vector er , where r = s (l) indicates the incident (secondary) radiation and c is the speed of light in a vacuum. While Γa, Γb and Γf are the respective lifetimes of the intermediate and final states.
A5 = A5 A1 , where 216
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Finally, it is easy to show that the expression for the net Raman gain [32,33] is proportional to the transition rate: T (n2, n3) T (n3, n1) 2 . In this way, the following electron states were chosen to calculate the net Raman gain: n1 = 1, n2 = 2 and n3 = 3. 3. Results and discussion The structure of an asymmetrical double quantum well is given by three layers with widths d1 (GaAs), b (Al 0.35 Ga0.65 As ) and d2 (GaAs); limited by two barriers of AlAs (see Fig. 1). Then, the potential barriers are V1 = 0.968eV, V2 = 0 and V3 = 0.30eV, while the effective masses are μ1 = 0.124μ0 , μ 2 = 0.096μ0 and μ3 = 0.665μ0 [32]. Fig. 1 shows the wave function (solid line) and energy (dashed line) for an asymmetrical double quantum well with external electric field F = 100.0kV/cm, where: d1 = 30Å, b = 10Å and d2 = 40Å. In addition, the blue line represents the potential barrier. When studying a layered system, it is necessary to take into account the interaction between the layers. In this case, we will define as coupling the interaction between layers d1 and d2 that makes up the active region of system [19,32,34]. The system we have proposed to study can be divided into two regions. The first region composed of electronic states with energy lower than V3, which can be decoupled for a certain value of the barrier (see Ref. [34]). Whereas, the second region is composed of electron states with energy greater than V3 those that are strongly coupled, since the increase in the width of the barrier (b) between the wells is not capable of causing the appearance of independent electron states for each well.
Fig. 1. The wave function (solid line) and energy (dashed line) for an asymmetrical double quantum well with external electric field F = 100.0kV/cm, where: d1 = 30Å, b = 10Å, and d2 = 40Å. Whereas, the blue line represents the potential barrier. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
Fig. 2. Electron states and transition rate for an asymmetrical double quantum well; where the solid red line corresponds to F = 0.0kV/ cm, the solid black line corresponds to F = 10.0kV/ cm, the dashed red line corresponds to F = 40.0kV/ cm and the dashed black line corresponds to F = 100.0kV/ cm. Then: (a) electron states energies (Ez ) for b = 30Å and d2 = 60Å, (b) electron states energies (Ez ) for d1 = 60Å and b = 30Å, (c) transition rate for b = 30Å and d2 = 60Å, (d) transition rate for d1 = 60Å and b = 30Å. Whereas, the dotted blue line represents the energy barrier V3. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) 217
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Fig. 3. Electron states and transition rate for an asymmetrical double quantum well with b = 30Å; where the solid red line corresponds to F = 0.0kV/ cm, the solid black line corresponds to F = 10.0kV/ cm and the dashed black line corresponds to F = 100.0kV/ cm. Then: (a) electron states energies (Ez ) for l3 = 60Å, (b) electron states energies (Ez ) for l3 = 100Å, (c) transition rate for l3 = 60Å, (d) transition rate for l3 = 100Å. Whereas, the dotted blue line represents the energy barrier V3. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
as a result of which the two maximum values of gain appear. However, it is very remarkable that the presence of the electric field does not “break” the symmetry but rather causes its “displacement” and causes the two maximums of gain to separate. When the width of the first well changes, see Fig. 2(c), the increase of the electric field causes that the point where the transition rate is zero moves to values where d1 < d2, being fixed d2. If instead it is the width of the second well that changes, see Fig. 2(d), the increase in the electric field causes the point where the transition rate is zero to move to values where d1 > d2, being fixed d1. That is, for a zero gain value to be obtained, it must happen that d1 ≤ d2. It can be observed in Fig. 2(c) that the increase in the electric field causes the intensity of the first maximum to decrease significantly, while the intensity of the second maximum increases slightly. Also, it slightly increases the separation between the two gain maximums. On the other hand, Fig. 2(d) shows that the increase in the electric field causes the intensity of the first maximum to increase, while the intensity of the second maximum decreases rapidly. It is also observed that the separation between the two gain maximums increases drastically. Fig. 3 shows the electron states and the transition rate obtained for an asymmetrical double quantum well. To obtain this figure b = 30Å and l3 are fixed, that is, the width of the barrier and the width of the system are fixed respectively. Moreover, the solid red line corresponds to F = 0.0kV/ cm, the solid black line corresponds to F = 10.0kV/ cm and the dashed black line corresponds to F = 100.0kV/ cm. Whereas, the dotted blue line represents the energy barrier V3. Finally, we have to: (a) electron states energies (Ez ) for l3 = 60Å, (b) electron states energies (Ez )
Fig. 2 shows the results obtained for an asymmetrical double quantum well. The solid red line corresponds to F = 0.0kV/cm, the solid black line corresponds to F = 10.0kV/cm, the dashed red line corresponds to F = 40.0kV/ cm and the dashed black line corresponds to F = 100.0kV/ cm. Whereas, the dotted blue line represents the energy barrier V3. For this case, we have to: (a) electron states energies (Ez ) for b = 30Å and d2 = 60Å, (b) electron states energies (Ez ) for d1 = 60Å and b = 30Å, (c) transition rate for b = 30Å and d2 = 60Å, (d) transition rate for d1 = 60Å and b = 30Å. This figure shows the effect on the electron states and on the transition rate, of increasing the width of one of the two wells that make up the system. As it can be observed in Fig. 2(a) and (b), the electron states do not cut, which is related to the coupling of the system; that is, if the wells are coupled, there are no cuts between the electron states. Otherwise, if the coupling breaks the electron states can be cut between them (see Ref. [29]). When changing the width of the first quantum well, the electron states associated to both quantum wells are strongly affected. This implies that the electric field produces an increase in the coupling of the system, being stronger when the width of the first quantum well (d1) changes, since it is observed that the electron states tend to mix. This is due to the electric field direction since when we change the width of the second quantum well (d2) it is observed that the states of the electrons tend to mix less. On the other hand, in Fig. 2(c) and (d), the transition rate is shown. It can be noted immediately that when the electric field is zero (solid red line) and d1 = d2, the gain for the states we have chosen is zero. This occurs because when the system is symmetrical, the electronic states are even or odd and optical transitions between states of equal parity are forbidden, 218
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Fig. 4. The transition rate corresponding for an asymmetrical double quantum well with b = 30Å, then: (a) d1 = 30Å (dotted black line), d1 = 40Å (dashed black line), d1 = 50Å (solid black line) and d2 = 60Å; (b) d1 = 70Å (solid black line), d1 = 80Å (dashed black line), d1 = 90Å (dotted black line) and d2 = 60Å; (c) d1 = 60Å, d2 = 30Å (dotted black line), d2 = 40Å (dashed black line), d2 = 50Å (solid black line); and (d) d1 = 60Å, d2 = 70Å (solid black line), d2 = 80Å (dashed black line), d2 = 90Å (dotted black line). The solid red line represents the transition rate for a symmetrical double quantum well with d1 = 60Å, b = 30Å and d2 = 60Å. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
Fig. 5. Raman emission spectra for an asymmetrical double quantum well with d1 = 30Å, b = 10Å and d2 = 20Å, then: (a) differential cross-section for ℏωl = 0.90eV, where the solid line corresponds to F = 10.0kV/ cm and the dashed line corresponds to F = 100.0kV/ cm; and (b) shows the positions of the resonant singularities.
for l3 = 100Å, (c) transition rate for l3 = 60Å, (d) transition rate for l3 = 100Å. This figure shows the effect on the system to adjust the position of the barrier, if the barrier width and the system width are fixed. When comparing Fig. 3(a) and (b) we can observe that the increase of the width of the system (l3) causes the increase of the number of electron states inside the well and the decrease of the energy of those states
already inside, while the change of position of the barrier causes relatively small variations of the energy of electron states, as if it oscillated. Furthermore, as in Fig. 2(a) and (b), it is observed that the increase of the electric field increases the energy of the electron states. In Fig. 3(c) and (d) the effect of the system width and the electric field on the transition rate is shown. As we can observe the increase of the electric 219
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field causes the decrease of the transition rate, this is due to the chosen configuration since the effect of the electric field on the system is complex (as will be discussed in the next figure). It is again verified that the symmetry causes the transition rate to be zero; and that the electric field in this case causes the displacement of the zero gain point towards smaller values of the width, this was discussed in the previous figure. In addition, increasing the width of the system increases the maximum value of the transition rate by more than double, due to the gap of the electron states is smaller. It is interesting to note that this system is more efficient than the step-quantum well, which is reproduced when d1 = 0 or d2 = 0, see Ref. [29]. Fig. 4 shows the transition rate corresponding to the asymmetrical double quantum well, being b = 30Å and: (a) d1 = 30Å (dotted black line), d1 = 40Å (dashed black line), d1 = 50Å (solid black line) and d2 = 60Å; (b) d1 = 70Å (solid black line), d1 = 80Å (dashed black line), d1 = 90Å (dotted black line) and d2 = 60Å; (c) d1 = 60Å, d2 = 30Å (dotted black line), d2 = 40Å (dashed black line), d2 = 50Å (solid black line); and (d) d1 = 60Å, d2 = 70Å (solid black line), d2 = 80Å (dashed black line), d2 = 90Å (dotted black line). The solid red line represents the transition rate for a symmetrical double quantum well with d1 = 60Å, b = 30Å and d2 = 60Å. This figure shows the effect of the electric field on the asymmetrical double quantum well. In Fig. 4(a) the width of the first quantum well decreases, where d1 < d2, while keeping the width of the second quantum well constant. The result of this the zero transition rate point is moved towards greater values of the electric field, as the difference between d1 and d2 increases; while the maximum main transition rate slightly decreases. In Fig. 4(b) instead, the width of the first quantum well is increased, being now d1 > d2, while keeping the width of the second quantum well constant. As a consequence, there is a displacement of the graph towards lower values of the electric field; so the transition rate describes a decreasing curve, which is smaller the greater the difference between d1 and d2. On the other hand, for Fig. 4(c) the width of the first quantum well is kept constant and the width of the second quantum well decreases (d1 > d2); the result is similar to that obtained for Fig. 4(b), although it is not the same since this system is not symmetrical due precisely to the electric field. Finally, in Fig. 4(d), the width of the first quantum well is kept constant and the width of the second quantum well increases (d1 < d2). So that the displacement of the point of zero transition rate towards greater values of the electric field is observed, as in Fig. 4(a); however, the main maximum of gain slightly increases. In Fig. 5, the emission spectra for an asymmetrical double quantum well with d1 = 30Å, b = 10Å and d2 = 20Å, is observed. For this case, we have to: (a) differential cross-section for ℏωl = 0.90eV, where the solid line corresponds to F = 10.0kV/ cm and the dashed line corresponds to F = 100.0kV/ cm ; and (b) shows the positions of the resonant singularities. Moreover, with the aim of simplifying our calculations and analysis [23,33], we have chosen that the lifetimes be Γa = Γb = Γf = 1.0meV for all cases. For the analysis of Raman emission spectra, all singularities present in the differential cross-section must be located. So it is possible to observe two different types of singularities [29,32]: resonant and non-resonant. Then, resonant singularities are obtained from Eq. (5) are characterized for being non-dependent on the incident radiation, so
the electron states involved in the transition that originates the peak. If we compare with the step-quantum well, it is noted that the coupled quantum well has much higher intensities. 4. Conclusions In this paper, the electron states and intra-band electron Raman scattering in a semiconductor, coupled and asymmetrical double quantum well with the presence of an external electric field are determined. This system is developed on a GaAs / Al x Ga1 − x As matrix in this case with three layers. So, we have studied the effect of the electric field on the system showing that any change in the configuration of the system leads to changes in the intensity and frequency of the secondary radiation. Besides, we obtained the electron Raman scattering intensities and show the performance of the transition rate as a measure of the net Raman gain. Furthermore, we demonstrate that the energy sub-bands structure of a coupled asymmetrical double quantum well can be studied through the structure of the emission spectra of the electron Raman scattering, as shown in the figures. In addition, in comparison with the step-quantum well it was determined that the coupled asymmetrical double quantum well is more efficient, it is also more efficient than the asymmetrical double quantum well (see Refs. [29] and [34]). References [1] G. Ariyawansa, P.V.V. Jayaweera, A.G.U. Perera, S.G. Matsik, M. Buchanan, Z.R. Wasilewski, H.C. Liu, Normal incidence detection of ultraviolet, visible, and mid-infrared radiation in a single GaAs/AlGaAs device, Opt. Lett. 34 (2009) 2036. [2] J. Nishinaga, A. Kawaharazuka, Y. Horikoshi, High absorption efficiency of AlGaAs/ GaAs superlattice solar cells, Jpn. J. Appl. Phys. 54 (2015) 052301. [3] L. Schrottke, X. Lü, G. Rozas, K. Biermann, H.T. Grahn, Terahertz GaAs/AlAs quantum-cascade lasers, Appl. Phys. Lett. 108 (2016) 102102. [4] G. Sun, J.B. Khurgin, L. Friedman, R.A. Soref, Tunable intersubband Raman laser in GaAs/AlGaAs multiple quantum wells, J. Opt. Soc. Am. B 15 (1998) 648. [5] B.H. Wu, J.C. Cao, G.Q. Xia, Simulation of semiconductor intersubband Raman laser, J. Appl. Phys. 94 (2003) 5710. [6] S.M. Maung, S. Katayama, Theory of intersubband Raman laser in modulationdoped asymmetric coupled double quantum wells, J. Phys. Soc. Jpn. 73 (2004) 2562. [7] M. Miura, S. Katayama, Electric-field effects on intersubband Raman laser gain in modulation-doped GaAs/AlGaAs coupled double quantum wells, Sci. Technol. Adv. Mater. 7 (2006) 286. [8] G. Sun, J.B. Khurgin, R.A. Soref, Design of a GaN/AlGaN intersubband Raman laser electrically tunable over the 3-5 μm atmospheric transmission window, J. Appl. Phys. 99 (2006) 033103. [9] M. Scheinert, H. Sigg, S. Tsujino, M. Giovannini, J. Faist, Intersubban Raman laser from GaInAs/AlInAs double quantum wells, Appl. Phys. Lett. 91 (2007) 131108. [10] Taegeon Lee, Heesuk Rho, Jin Dong Song, Won Jun Choi, Raman scattering from GaAs/AlGaAs, Curr. Appl. Phys. 17 (2017) 398. [11] V.I. Belitsky, M. Cardona, Polaron effect in resonant Raman scattering from quantum wells in a high magnetic field: decompensation of electron and hole contributions, Phys. Rev. B 47 (1993) 13003. [12] Q.H. Zhong, C.H. Liu, Y.Q. Zhang, H.C. Sun, One-phonon-assisted resonant electron Raman scattering of GaAs quantum dots in an AlAs matrix, Phys. Lett. A 372 (2008) 2103. [13] W. Xie, Electron Raman scattering of a two-dimensional pseudodot system, Phys. Lett. A 376 (2012) 1657. [14] Q. Zhong, Y. Xuehua, Electron Raman scattering in a cylindrical quantum dot, J. Semiconduct. 33 (2012) 052001. [15] X.F. Zhao, C.H. Liu, Electron Raman scattering in quantum well wires, Physica B 392 (2007) 11. [16] S.H. Rezaiee, H.R. Askari, M. Rahimi, A. Fatemidokht, The electron Raman scattering in the spherical parabolic quantum dot in the present of magnetic field, Optic Laser. Technol. 49 (2013) 325. [17] L. Lu, W. Xie, H. Hassanabadi, Q. Zhong, The effect of intense laser field on the electronic Raman scattering of shallow donor impurities in quantum dots, Superlattice. Microst. 50 (2011) 501. [18] N. Zamani, A. Keshavarz, M.J. Karimi, Electronic Raman scattering in double semiparabolic quantum well, Chin. Phys. B 22 (2013) 057802. [19] X.F. Zhao, C.H. Liu, One-phonon resonant Raman scattering in cylindrical quantum wire, Physica E 36 (2007) 34. [20] G. Rezaei, M.J. Karimi, H. Pakarzadeh, Magnetic field effects on the electron Raman scattering in coaxial cylindrical quantum well wire, J. Lumin. 143 (2013) 551. [21] F. Lu, C.H. Liu, Z.L. Guo, One phonon-assisted electron Raman scattering in a wurtzite cylindrical quantum wel wire, Physica B 407 (2012) 165. [22] T.G. Ismailov, B.H. Mehdiyev, Electron Raman scattering in a cylindrical quantum dot in a magnetic field, Physica E 31 (2006) 72.
ℏωs (na, nb) → ℏωs = Ez (na) − Ez (nb), while the non-resonant singularities are obtained from Eq. (4) and are characterized by being dependent on incident radiation, unlike resonant singularities, then
ℏωs = ℏωl + Ez (na) − Ez (nb). As can be seen in Fig. 5(a) the electric field causes changes in both the intensity and the position of the singularities, these changes are different in all cases, so there is no general effect that affects all peaks of the system. It should be noted that the position of the singularities, see Fig. 5(b), varies almost linearly, since this depends on the gap between 220
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[29] Ri Betancourt-Riera, Re Betancourt-Riera, L.A. Ferrer-Moreno, J.M. Nieto Jalil, Electron states and electron Raman scattering in semiconductor step-quantum well: electric field effect, Superlattice. Microst. 104 (2017) 428. [30] M.J. Karimi, Intense laser field effects on the electron Raman scattering in a strained InGaN/GaN quantum well, Physica E 66 (2015) 18. [31] D. Ahn, S.L. Chuang, Exact calculations of quasibound states of an isolated quantum well with uniform electric field: quantum–well Stark resonance, Phys. Rev. B 34 (1986) 9034. [32] L.A. Ferrer-Moreno, Ri Betancourt-Riera, Re Betancourt-Riera, R. Riera, Electron Raman scattering in semiconductor step-quantum well, Phys. B 477 (2015) 87. [33] A. Tiutiunnyk, V. Akimov, V. Tulupenko, M.E. Mora-Ramos, E. Kasapoglu, A.L. Morales, C.A. Duque, Electron and donor-impurity-related Raman scattering and Raman gain in triangular quantum dots under an applied electric field, Eur. Phys. J. B 89 (2016) 107. [34] R. Betancourt-Riera, R. Rosas, I. Marín-Enriquez, R. Riera, J.L. Marín, Electron Raman scattering in asymmetrical multiple quantum well, J. Phys. Condens. Matter 17 (2005) 4451.
[23] M.J. Karimi, G. Rezaei, H. Pakarzadeh, Electron Raman scattering in single and multilayered spherical quantum dots: effects of hydrogenic impurity and geometrical size, Phys. Lett. A 377 (2013) 2164. [24] F. Lu, C.-H. Liu, Effect of built in electric field on electron Raman scattering in InGaN/GaN coupled quantum wells, Superlattice. Microst. 50 (2011) 582. [25] O. Mayrock, H.J. Wünsche, F. Henneberger, Polarization charge screening and indium surface segregation in (In, Ga)N/GaN single and multiple quantum well, Phys. Rev. B 62 (2000) 16870. [26] S.F. Chichibu, T. Azuhata, T. Sota, T. Mukai, S. Nakamura, Localized quantum well excitons in InGaN single-quantum-well amber light-emitting diodes, J. Appl. Phys. 88 (2000) 5153. [27] R. Cingolani, A. Botchkarev, H. Tang, H. Morkoc, G. Traetta, G. Coli, M. Lomascolo, A.D. Carlo, F.D. Sala, P. Lugli, Spontaneous polarization and piezoelectric field in GaN/Al0.15Ga0.85N quantum well: impact on the optical spectra, Phys. Rev. B 61 (2000) 2711–2715. [28] A. Liu, S.L. Chuang, C.Z. Ning, Piezoelectric field-enhanced second-order nonlinear optical susceptibilities in wurtzite GaN/AlGaN quantum well, Appl. Phys. Lett. 76 (2000) 333–335.
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Physica B: Condensed Matter journal homepage: www.elsevier.com/locate/physb
Electron states and electron Raman scattering in an asymmetrical double quantum well: External electric field
T
L. Ferrer-Galindoa, Ri. Betancourt-Rierab, Re. Betancourt-Rierab,∗, R. Rierac a
Departamento de Física, Universidad de Sonora, Blvd. Luis Encinas y Rosales S/N, Apartado Postal 1626, Col. Centro C.P. 83000, Hermosillo, Sonora, Mexico Instituto Tecnológico de Hermosillo, Avenida Tecnológico S/N, Col. Sahuaro, C.P. 83170, Hermosillo, Sonora, Mexico c Departamento de Investigación en Física, Universidad de Sonora, Blvd. Luis Encinas y Rosales S/N, Apartado Postal 5-88, Col. Centro, C.P. 83000 Hermosillo, Sonora, Mexico b
A R T I C LE I N FO
A B S T R A C T
Keywords: Electron states Raman scattering Quantum well
In this paper studies an electron Raman scattering process for a semiconductor, coupled and asymmetrical double quantum well. Then, the presence of an electron in a single conduction band is considered. In addition, the system is subjected to an external electric field. To carry out this study, the net Raman gain and the differential cross section are calculated. The emission spectra are interpreted and discussed. For this, we obtain the exact solutions of the electron states considering the envelope function approximation and a single parabolic conduction band, which is split into a sub-bands system due to confinement. Furthermore, the effect of the electric field on the electron states and in the differential cross section is studied. To illustrate our findings, we have considered a system growing in a GaAs / Al x Ga1 − x As matrix.
1. Introduction The structures based on multiple quantum wells have been studied for more than three decades because of their optical and electronic properties used in the development of applications such as solar cells, high efficiency multiband detectors and lasers [1–3]. Furthermore, it has been proposed to use them as a source of coherent mid-infrared radiation based on electron Raman scattering [4–9]. Recently, the use of Raman scattering as a control method in the growth of multiple quantum well systems has gained interest [10]. Thus, Raman scattering is a highly efficient tool, both theoretically and experimentally, for research of electronic and optical properties of low dimensional semiconductor structures [11–23]. GaAs-based semiconductors have a low macroscopic polarization, unlike those based on wurtzite nitrides (as the group-III nitride semiconductor compounds) where the macroscopic polarization is strong [24]. The magnitude of the built-in electric field induced by the piezoelectricity and the spontaneous polarization in the wurtzite nitrides semiconductors is estimated in the order of MV/cm [24–28]; however, in the GaAs-based semiconductors this is negligible. In consequence, this implies that the quantum-confined Stark effect on devices based on GaAs is simpler to control than it is in nitrides. On the other hand, from previous studies, it is known that the electric field effect and the
∗
electronic and optical properties (especially in the Raman net gain and in the Raman emission spectra) are complex in nitrides [29,30]. In this document, the exact solutions of the mathematical expressions of the electron states for a semiconductor, coupled and asymmetrical double quantum well in the presence of an external electric field are obtained. Moreover, the emission spectra and the selection rules obtained when calculating the differential cross-section corresponding to an intra-band electron Raman scattering process are shown. In addition, the Raman net gain is also analyzed in a three-level system. With this system we can analyze the structures that have already been studied, such as the step-quantum well and the double quantum well [29], which allows us an easy comparison with these systems; then, this system generalizes the aforementioned structures. For a better understanding of our results, this article has been divided into several sections: In Section 2, the electron states and the Raman differential crosssection expressions are obtained. In Section 3, the results considering a system that grows in a GaAs / Al x Ga1 − x As matrix are presented, and the effect produced by the electric field and by the change in the system parameters is determined. Finally, in Section 4, the conclusions are presented.
Corresponding author. E-mail addresses: [email protected] (L. Ferrer-Galindo), [email protected] (R. Betancourt-Riera), [email protected] (R. Betancourt-Riera), [email protected] (R. Riera). https://doi.org/10.1016/j.physb.2018.06.017 Received 30 March 2018; Received in revised form 15 June 2018; Accepted 16 June 2018 Available online 19 June 2018 0921-4526/ © 2018 Published by Elsevier B.V.
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2. Model and theory
A2 =
In this section, we show the solution of the Schrödinger equation in the envelope function approximation. Then, we can determine the bound states of a confined electron in a semiconductor, coupled and asymmetrical double quantum well system; with the presence of an external electric field (F), which is constant and uniform. Therefore, the Schrödinger equation takes the following form:
{
∇2 +
A3 = A4 =
}
Q3 (l1) 3 (l ) A + Ξ 3 (l ) B Σ32 2 3 32 2 3
Q2 (l2)
, B3 = − , B4 = −
2 (l ) A + Λ 2 (l ) B Φ23 1 2 23 1 2
Q3 (l1) 3 (l ) A + Λ 3 (l ) B Φ32 2 3 32 2 3
Q2 (l2)
Ai(η2l3) A 4 + Bi(η2l3) B4 Ai(η1l3)
, ,
,
Π22 = Ci(η10 )Ai′ (η20) − α1 Ci′ (η10 )Ai(η20),
(1)
Φijk (z ) = Ai(ηiz )Ai′ (ηjz ) − αk Ai′ (ηiz )Ai(ηjz ), Λijk (z ) = Bi(ηiz )Ai′ (ηjz ) − αk Bi′ (ηiz )Ai(ηjz ),
ℏ2 2 Γ k⊥ − i , 2μ 2
Σijk (z ) = Ai(ηiz )Bi′ (ηjz ) − αk Ai′ (ηiz )Bi(ηjz ), Ξijk (z ) = Bi(ηiz )Bi′ (ηjz ) − αk Bi′ (ηiz )Bi(ηjz ),
being Ez (n) the part of the quasi-bound state energy that corresponds to the confinement and ℏ2k⊥2/2μ the energy that corresponds to the free direction. n = 1,2, …, is the number assigned to the electron states, while Γ is the resonance width found to be positive [31]. A coupled asymmetrical double quantum well was considered with the corresponding layer thickness −∞/ d1/ b/ d2/ +∞. Furthermore, the presence of constant and uniform electric field F is considered. Then, the effective mass (μ) and the confinement potential (Vc ), can be chosen as:
V1, − ∞ < z < 0 V2, 0 ≤ z ≤ l1 V3, l1 < z < l2 V2, l2 ≤ z ≤ l3 V1, l3 < z < + ∞
Qi (z ) = Ai(ηiz )Bi′ (ηiz ) − Ai′ (ηiz )Bi(ηiz ), 0
I1 = ∫−∞ Ci∗ (η1z )Ci(η1z ) dz , l
I2 = ∫0 1 [A2 Ai(η2z ) + B2 Bi(η2z )]∗ [A2 Ai(η2z ) + B2 Bi(η2z )] dz , l
I3 = ∫l 2 [A3 Ai(η3z ) + B3 Bi(η3z )]∗ [A3 Ai(η3z ) + B3 Bi(η3z )] dz , 1 l
I4 = ∫l 3 [A 4 Ai(η2z ) + B4 Bi(η2z )]∗ [A 4 Ai(η2z ) + B4 Bi(η2z )] dz 2 +∞
and I5 = A5∗ A5 ∫l
3
Ai∗ (η1z )Ai(η1z ) dz ,
with
α1 = [μ 2 / μ1]2/3 , α2 = [μ3 / μ 2 ]2/3 , α3 = [μ 2 / μ3]2/3 and α4 = [μ1 / μ 2 ]2/3 .
(2)
On the other hand, the differential cross-section for an intra-band electron Raman scattering process in a volume per unit solid angle (Ω) of incoming light with energy (ℏωl ) and scattering light with energy (ℏωs ), is calculated. In Ref. [32] the calculation process to obtain the differential cross section is described; therefore
where l1 = d1, l2 = d1 + b and l3 = d1 + b + d2. The wave function takes the following form:
Ψ (r) =
2 (l ) A + Ξ 2 (l ) B Σ23 1 2 23 1 2
A5 =
This implies that there are not bound states, but quasi-bound states, with energies given by:
⎧ μ1 , ⎪ μ2 , ⎪ μ, Vc = μ3 , ⎨ ⎪ μ2 , ⎪μ , ⎩ 1
Π22 , 2 (0)
B2 = − Q
Π12 = Ci(η10)Bi′ (η20) − α1 Ci′ (η10)Bi(η20),
2μ [E − Vc − e Fz ] Ψ = 0. ℏ2
E (n, k⊥) = Ez (n) +
Π12 , Q2 (0)
exp[i k⊥·r⊥] φn (z ) u 0 (r) Lx L y
(3)
where
∂ 2σ ω = σ0 Γ f2 s ∂ωs ∂Ω ωl
A1 Ci(η1z ),
⎧ −∞ < z < 0 ⎪ A2 Ai(η ) + B2 Bi(η ), 0 ≤ z ≤ l1 2z 2z ⎪ l1 < z < l2 φn (z ) = A3 Ai(η3z ) + B3 Bi(η3z ), ⎨ l2 ≤ z ≤ l3 ⎪ A 4 Ai(η2z ) + B4 Bi(η2z ), ⎪
∑ n, n″
Mf (n, n″)
2 2
[ℏωl − ℏωs + Ez (n″) − Ez (n)] + Γ f2
(4)
where
Mf (n, n″) =
ℏ2 2μ 0 l32
− ∑n′ T (n, n′) T (n′, n″) ⎡ ⎢ ℏωs + Ez (n) − Ez (n′) + iΓa ⎣ 1
Ci(η) = Bi(η) + i Ai(η) and ηjz = −[2μj /(eF ℏ)2]1/3 [Ez (n) − Vc (z )
−
⎤,
1
ℏωl − Ez (n) + Ez (n′) − iΓ b ⎥
(5)
⎦
− e Fz ].
σ0 =
Being u 0 (r) the periodic part of the Bloch function, Ai and Bi the Airy's functions and finally j = 1,2 or 3. Once the boundary conditions are applied, the continuity of the function Ψ and the current (1/ μ)(∂Ψ / ∂z ) at the interface, Ez (n) are determined by the following secular equation:
η (ωs ) e 4 4ℏ ⎛ ⎞ (es ·ez )(el ·ez ) 2 . η (ωl ) ⎝ c ⎠ πμ02 Γf
and in this case 0
⎡1 T (na, nb) = μ0 l3 ⎢ μ ∫ φn∗a (z ) φn′b (z ) dz + 1 ⎣ −∞
2 2 3 4 3 4 [Π12 Σ23 (l1) − Π22 Ξ23 (l1)][Σ32 (l2) Φ21 (l3) − Φ32 (l2) Λ21 (l3)]= 2 2 3 4 3 4 [Π12 Φ23 (l1) − Π22 Λ23 (l1)][Ξ32 (l2) Φ21 (l3) − Λ32 (l2) Λ21 (l3)]
+
while the constants A and B have the following form:
+
A1 = [I1 + I2 + I3 + I4 + I5]−1/2 , A2 = A2 A1 , B2 = B2 A1 , A3 = A3 A1 , B3 = B3 A1 , A 4 = A 4 A1 , B4 = B4 A1 ,
1 μ2
l3
1 μ3
l1
∫ φn∗a (z ) φn′b (z ) dz 0
l2
∫ φn∗a (z ) φn′b (z ) dz l1
∫ φn∗a (z ) φn′b (z ) dz + l2
1 μ2
1 μ1
+∞
⎤
l3
⎦
∫ φn∗a (z ) φn′b (z ) dz⎥
(6)
where μ0 is the electron free mass. η (ωr ) is the refraction index as a function of the radiation with frequency ωr and polarization unit vector er , where r = s (l) indicates the incident (secondary) radiation and c is the speed of light in a vacuum. While Γa, Γb and Γf are the respective lifetimes of the intermediate and final states.
A5 = A5 A1 , where 216
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Finally, it is easy to show that the expression for the net Raman gain [32,33] is proportional to the transition rate: T (n2, n3) T (n3, n1) 2 . In this way, the following electron states were chosen to calculate the net Raman gain: n1 = 1, n2 = 2 and n3 = 3. 3. Results and discussion The structure of an asymmetrical double quantum well is given by three layers with widths d1 (GaAs), b (Al 0.35 Ga0.65 As ) and d2 (GaAs); limited by two barriers of AlAs (see Fig. 1). Then, the potential barriers are V1 = 0.968eV, V2 = 0 and V3 = 0.30eV, while the effective masses are μ1 = 0.124μ0 , μ 2 = 0.096μ0 and μ3 = 0.665μ0 [32]. Fig. 1 shows the wave function (solid line) and energy (dashed line) for an asymmetrical double quantum well with external electric field F = 100.0kV/cm, where: d1 = 30Å, b = 10Å and d2 = 40Å. In addition, the blue line represents the potential barrier. When studying a layered system, it is necessary to take into account the interaction between the layers. In this case, we will define as coupling the interaction between layers d1 and d2 that makes up the active region of system [19,32,34]. The system we have proposed to study can be divided into two regions. The first region composed of electronic states with energy lower than V3, which can be decoupled for a certain value of the barrier (see Ref. [34]). Whereas, the second region is composed of electron states with energy greater than V3 those that are strongly coupled, since the increase in the width of the barrier (b) between the wells is not capable of causing the appearance of independent electron states for each well.
Fig. 1. The wave function (solid line) and energy (dashed line) for an asymmetrical double quantum well with external electric field F = 100.0kV/cm, where: d1 = 30Å, b = 10Å, and d2 = 40Å. Whereas, the blue line represents the potential barrier. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
Fig. 2. Electron states and transition rate for an asymmetrical double quantum well; where the solid red line corresponds to F = 0.0kV/ cm, the solid black line corresponds to F = 10.0kV/ cm, the dashed red line corresponds to F = 40.0kV/ cm and the dashed black line corresponds to F = 100.0kV/ cm. Then: (a) electron states energies (Ez ) for b = 30Å and d2 = 60Å, (b) electron states energies (Ez ) for d1 = 60Å and b = 30Å, (c) transition rate for b = 30Å and d2 = 60Å, (d) transition rate for d1 = 60Å and b = 30Å. Whereas, the dotted blue line represents the energy barrier V3. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) 217
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Fig. 3. Electron states and transition rate for an asymmetrical double quantum well with b = 30Å; where the solid red line corresponds to F = 0.0kV/ cm, the solid black line corresponds to F = 10.0kV/ cm and the dashed black line corresponds to F = 100.0kV/ cm. Then: (a) electron states energies (Ez ) for l3 = 60Å, (b) electron states energies (Ez ) for l3 = 100Å, (c) transition rate for l3 = 60Å, (d) transition rate for l3 = 100Å. Whereas, the dotted blue line represents the energy barrier V3. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
as a result of which the two maximum values of gain appear. However, it is very remarkable that the presence of the electric field does not “break” the symmetry but rather causes its “displacement” and causes the two maximums of gain to separate. When the width of the first well changes, see Fig. 2(c), the increase of the electric field causes that the point where the transition rate is zero moves to values where d1 < d2, being fixed d2. If instead it is the width of the second well that changes, see Fig. 2(d), the increase in the electric field causes the point where the transition rate is zero to move to values where d1 > d2, being fixed d1. That is, for a zero gain value to be obtained, it must happen that d1 ≤ d2. It can be observed in Fig. 2(c) that the increase in the electric field causes the intensity of the first maximum to decrease significantly, while the intensity of the second maximum increases slightly. Also, it slightly increases the separation between the two gain maximums. On the other hand, Fig. 2(d) shows that the increase in the electric field causes the intensity of the first maximum to increase, while the intensity of the second maximum decreases rapidly. It is also observed that the separation between the two gain maximums increases drastically. Fig. 3 shows the electron states and the transition rate obtained for an asymmetrical double quantum well. To obtain this figure b = 30Å and l3 are fixed, that is, the width of the barrier and the width of the system are fixed respectively. Moreover, the solid red line corresponds to F = 0.0kV/ cm, the solid black line corresponds to F = 10.0kV/ cm and the dashed black line corresponds to F = 100.0kV/ cm. Whereas, the dotted blue line represents the energy barrier V3. Finally, we have to: (a) electron states energies (Ez ) for l3 = 60Å, (b) electron states energies (Ez )
Fig. 2 shows the results obtained for an asymmetrical double quantum well. The solid red line corresponds to F = 0.0kV/cm, the solid black line corresponds to F = 10.0kV/cm, the dashed red line corresponds to F = 40.0kV/ cm and the dashed black line corresponds to F = 100.0kV/ cm. Whereas, the dotted blue line represents the energy barrier V3. For this case, we have to: (a) electron states energies (Ez ) for b = 30Å and d2 = 60Å, (b) electron states energies (Ez ) for d1 = 60Å and b = 30Å, (c) transition rate for b = 30Å and d2 = 60Å, (d) transition rate for d1 = 60Å and b = 30Å. This figure shows the effect on the electron states and on the transition rate, of increasing the width of one of the two wells that make up the system. As it can be observed in Fig. 2(a) and (b), the electron states do not cut, which is related to the coupling of the system; that is, if the wells are coupled, there are no cuts between the electron states. Otherwise, if the coupling breaks the electron states can be cut between them (see Ref. [29]). When changing the width of the first quantum well, the electron states associated to both quantum wells are strongly affected. This implies that the electric field produces an increase in the coupling of the system, being stronger when the width of the first quantum well (d1) changes, since it is observed that the electron states tend to mix. This is due to the electric field direction since when we change the width of the second quantum well (d2) it is observed that the states of the electrons tend to mix less. On the other hand, in Fig. 2(c) and (d), the transition rate is shown. It can be noted immediately that when the electric field is zero (solid red line) and d1 = d2, the gain for the states we have chosen is zero. This occurs because when the system is symmetrical, the electronic states are even or odd and optical transitions between states of equal parity are forbidden, 218
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Fig. 4. The transition rate corresponding for an asymmetrical double quantum well with b = 30Å, then: (a) d1 = 30Å (dotted black line), d1 = 40Å (dashed black line), d1 = 50Å (solid black line) and d2 = 60Å; (b) d1 = 70Å (solid black line), d1 = 80Å (dashed black line), d1 = 90Å (dotted black line) and d2 = 60Å; (c) d1 = 60Å, d2 = 30Å (dotted black line), d2 = 40Å (dashed black line), d2 = 50Å (solid black line); and (d) d1 = 60Å, d2 = 70Å (solid black line), d2 = 80Å (dashed black line), d2 = 90Å (dotted black line). The solid red line represents the transition rate for a symmetrical double quantum well with d1 = 60Å, b = 30Å and d2 = 60Å. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
Fig. 5. Raman emission spectra for an asymmetrical double quantum well with d1 = 30Å, b = 10Å and d2 = 20Å, then: (a) differential cross-section for ℏωl = 0.90eV, where the solid line corresponds to F = 10.0kV/ cm and the dashed line corresponds to F = 100.0kV/ cm; and (b) shows the positions of the resonant singularities.
for l3 = 100Å, (c) transition rate for l3 = 60Å, (d) transition rate for l3 = 100Å. This figure shows the effect on the system to adjust the position of the barrier, if the barrier width and the system width are fixed. When comparing Fig. 3(a) and (b) we can observe that the increase of the width of the system (l3) causes the increase of the number of electron states inside the well and the decrease of the energy of those states
already inside, while the change of position of the barrier causes relatively small variations of the energy of electron states, as if it oscillated. Furthermore, as in Fig. 2(a) and (b), it is observed that the increase of the electric field increases the energy of the electron states. In Fig. 3(c) and (d) the effect of the system width and the electric field on the transition rate is shown. As we can observe the increase of the electric 219
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field causes the decrease of the transition rate, this is due to the chosen configuration since the effect of the electric field on the system is complex (as will be discussed in the next figure). It is again verified that the symmetry causes the transition rate to be zero; and that the electric field in this case causes the displacement of the zero gain point towards smaller values of the width, this was discussed in the previous figure. In addition, increasing the width of the system increases the maximum value of the transition rate by more than double, due to the gap of the electron states is smaller. It is interesting to note that this system is more efficient than the step-quantum well, which is reproduced when d1 = 0 or d2 = 0, see Ref. [29]. Fig. 4 shows the transition rate corresponding to the asymmetrical double quantum well, being b = 30Å and: (a) d1 = 30Å (dotted black line), d1 = 40Å (dashed black line), d1 = 50Å (solid black line) and d2 = 60Å; (b) d1 = 70Å (solid black line), d1 = 80Å (dashed black line), d1 = 90Å (dotted black line) and d2 = 60Å; (c) d1 = 60Å, d2 = 30Å (dotted black line), d2 = 40Å (dashed black line), d2 = 50Å (solid black line); and (d) d1 = 60Å, d2 = 70Å (solid black line), d2 = 80Å (dashed black line), d2 = 90Å (dotted black line). The solid red line represents the transition rate for a symmetrical double quantum well with d1 = 60Å, b = 30Å and d2 = 60Å. This figure shows the effect of the electric field on the asymmetrical double quantum well. In Fig. 4(a) the width of the first quantum well decreases, where d1 < d2, while keeping the width of the second quantum well constant. The result of this the zero transition rate point is moved towards greater values of the electric field, as the difference between d1 and d2 increases; while the maximum main transition rate slightly decreases. In Fig. 4(b) instead, the width of the first quantum well is increased, being now d1 > d2, while keeping the width of the second quantum well constant. As a consequence, there is a displacement of the graph towards lower values of the electric field; so the transition rate describes a decreasing curve, which is smaller the greater the difference between d1 and d2. On the other hand, for Fig. 4(c) the width of the first quantum well is kept constant and the width of the second quantum well decreases (d1 > d2); the result is similar to that obtained for Fig. 4(b), although it is not the same since this system is not symmetrical due precisely to the electric field. Finally, in Fig. 4(d), the width of the first quantum well is kept constant and the width of the second quantum well increases (d1 < d2). So that the displacement of the point of zero transition rate towards greater values of the electric field is observed, as in Fig. 4(a); however, the main maximum of gain slightly increases. In Fig. 5, the emission spectra for an asymmetrical double quantum well with d1 = 30Å, b = 10Å and d2 = 20Å, is observed. For this case, we have to: (a) differential cross-section for ℏωl = 0.90eV, where the solid line corresponds to F = 10.0kV/ cm and the dashed line corresponds to F = 100.0kV/ cm ; and (b) shows the positions of the resonant singularities. Moreover, with the aim of simplifying our calculations and analysis [23,33], we have chosen that the lifetimes be Γa = Γb = Γf = 1.0meV for all cases. For the analysis of Raman emission spectra, all singularities present in the differential cross-section must be located. So it is possible to observe two different types of singularities [29,32]: resonant and non-resonant. Then, resonant singularities are obtained from Eq. (5) are characterized for being non-dependent on the incident radiation, so
the electron states involved in the transition that originates the peak. If we compare with the step-quantum well, it is noted that the coupled quantum well has much higher intensities. 4. Conclusions In this paper, the electron states and intra-band electron Raman scattering in a semiconductor, coupled and asymmetrical double quantum well with the presence of an external electric field are determined. This system is developed on a GaAs / Al x Ga1 − x As matrix in this case with three layers. So, we have studied the effect of the electric field on the system showing that any change in the configuration of the system leads to changes in the intensity and frequency of the secondary radiation. Besides, we obtained the electron Raman scattering intensities and show the performance of the transition rate as a measure of the net Raman gain. Furthermore, we demonstrate that the energy sub-bands structure of a coupled asymmetrical double quantum well can be studied through the structure of the emission spectra of the electron Raman scattering, as shown in the figures. In addition, in comparison with the step-quantum well it was determined that the coupled asymmetrical double quantum well is more efficient, it is also more efficient than the asymmetrical double quantum well (see Refs. [29] and [34]). References [1] G. Ariyawansa, P.V.V. Jayaweera, A.G.U. Perera, S.G. Matsik, M. Buchanan, Z.R. Wasilewski, H.C. Liu, Normal incidence detection of ultraviolet, visible, and mid-infrared radiation in a single GaAs/AlGaAs device, Opt. Lett. 34 (2009) 2036. [2] J. Nishinaga, A. Kawaharazuka, Y. Horikoshi, High absorption efficiency of AlGaAs/ GaAs superlattice solar cells, Jpn. J. Appl. Phys. 54 (2015) 052301. [3] L. Schrottke, X. Lü, G. Rozas, K. Biermann, H.T. Grahn, Terahertz GaAs/AlAs quantum-cascade lasers, Appl. Phys. Lett. 108 (2016) 102102. [4] G. Sun, J.B. Khurgin, L. Friedman, R.A. Soref, Tunable intersubband Raman laser in GaAs/AlGaAs multiple quantum wells, J. Opt. Soc. Am. B 15 (1998) 648. [5] B.H. Wu, J.C. Cao, G.Q. Xia, Simulation of semiconductor intersubband Raman laser, J. Appl. Phys. 94 (2003) 5710. [6] S.M. Maung, S. Katayama, Theory of intersubband Raman laser in modulationdoped asymmetric coupled double quantum wells, J. Phys. Soc. Jpn. 73 (2004) 2562. [7] M. Miura, S. Katayama, Electric-field effects on intersubband Raman laser gain in modulation-doped GaAs/AlGaAs coupled double quantum wells, Sci. Technol. Adv. Mater. 7 (2006) 286. [8] G. Sun, J.B. Khurgin, R.A. Soref, Design of a GaN/AlGaN intersubband Raman laser electrically tunable over the 3-5 μm atmospheric transmission window, J. Appl. Phys. 99 (2006) 033103. [9] M. Scheinert, H. Sigg, S. Tsujino, M. Giovannini, J. Faist, Intersubban Raman laser from GaInAs/AlInAs double quantum wells, Appl. Phys. Lett. 91 (2007) 131108. [10] Taegeon Lee, Heesuk Rho, Jin Dong Song, Won Jun Choi, Raman scattering from GaAs/AlGaAs, Curr. Appl. Phys. 17 (2017) 398. [11] V.I. Belitsky, M. Cardona, Polaron effect in resonant Raman scattering from quantum wells in a high magnetic field: decompensation of electron and hole contributions, Phys. Rev. B 47 (1993) 13003. [12] Q.H. Zhong, C.H. Liu, Y.Q. Zhang, H.C. Sun, One-phonon-assisted resonant electron Raman scattering of GaAs quantum dots in an AlAs matrix, Phys. Lett. A 372 (2008) 2103. [13] W. Xie, Electron Raman scattering of a two-dimensional pseudodot system, Phys. Lett. A 376 (2012) 1657. [14] Q. Zhong, Y. Xuehua, Electron Raman scattering in a cylindrical quantum dot, J. Semiconduct. 33 (2012) 052001. [15] X.F. Zhao, C.H. Liu, Electron Raman scattering in quantum well wires, Physica B 392 (2007) 11. [16] S.H. Rezaiee, H.R. Askari, M. Rahimi, A. Fatemidokht, The electron Raman scattering in the spherical parabolic quantum dot in the present of magnetic field, Optic Laser. Technol. 49 (2013) 325. [17] L. Lu, W. Xie, H. Hassanabadi, Q. Zhong, The effect of intense laser field on the electronic Raman scattering of shallow donor impurities in quantum dots, Superlattice. Microst. 50 (2011) 501. [18] N. Zamani, A. Keshavarz, M.J. Karimi, Electronic Raman scattering in double semiparabolic quantum well, Chin. Phys. B 22 (2013) 057802. [19] X.F. Zhao, C.H. Liu, One-phonon resonant Raman scattering in cylindrical quantum wire, Physica E 36 (2007) 34. [20] G. Rezaei, M.J. Karimi, H. Pakarzadeh, Magnetic field effects on the electron Raman scattering in coaxial cylindrical quantum well wire, J. Lumin. 143 (2013) 551. [21] F. Lu, C.H. Liu, Z.L. Guo, One phonon-assisted electron Raman scattering in a wurtzite cylindrical quantum wel wire, Physica B 407 (2012) 165. [22] T.G. Ismailov, B.H. Mehdiyev, Electron Raman scattering in a cylindrical quantum dot in a magnetic field, Physica E 31 (2006) 72.
ℏωs (na, nb) → ℏωs = Ez (na) − Ez (nb), while the non-resonant singularities are obtained from Eq. (4) and are characterized by being dependent on incident radiation, unlike resonant singularities, then
ℏωs = ℏωl + Ez (na) − Ez (nb). As can be seen in Fig. 5(a) the electric field causes changes in both the intensity and the position of the singularities, these changes are different in all cases, so there is no general effect that affects all peaks of the system. It should be noted that the position of the singularities, see Fig. 5(b), varies almost linearly, since this depends on the gap between 220
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