Electromagnetically induced grating in asymmetric quantum wells via Fano interference

Fengxue Zhou; Yihong Qi; Hui Sun; Dijun Chen; Jie Yang; Yueping Niu; Shangqing Gong

doi:10.1364/OE.21.012249

1. Introduction

Electromagnetically induced grating (EIG) can diffract a probe beam into high-order diffractions by using a standing-wave field to build a spatial absorption or phase grating in a sample. It was first proposed by Ling and coworkers [1], and then experimentally demonstrated in cold [2–4] and hot [5, 6] atomic samples. EIG finds potential applications in atomic/molecular velocimetry [3], realizing optical bistability [7], probing material optical properties [4], all optical switching and routing [5], beam splitting and fanning [8], shaping a biphoton spectrum [9], etc.

So far, most of the schemes of EIG are proposed and demonstrated [10–15] in atomic systems based on electromagnetically induced transparency. Such as, Dutta and Mahapatra [10] showed that the intensities of higher-order diffraction could be enhanced in a three-level ladder system. Xiao and coworkers [11] improved the efficiency of the phase grating via a microwave field in a double-dark state atomic system. Nonlinear modulation was proposed to realize EIG in a four-level N-type atomic ensembles [12–14]. Recently, Schemes of EIG based on spontaneously generated coherence (SGC) [16, 17] were proposed, and it was shown that the diffraction efficiency of phase grating can strikingly enhanced by SGC. However, SGC [18–21] requires two close-lying levels which should satisfy the conditions that these levels are nearly degenerate and that corresponding dipole matrix elements are nonorthogonal. Unfortunately, the rigorous conditions of near-degenerate levels and nonorthogonal dipole matrix elements cannot be simultaneously met in real atomic systems, so that few experiments have been achieved to observe the SGC effect directly. Proposal in solid-state medium was also suggested. Xiao et al. [22] modeled an EIG in a hybrid structure comprised of a semiconductor quantum dot and a metal nanoparticle based on exciton induced transparency.

Great attentions have been paid to the semiconductor quantum well (QW) nanostructures due to their inherent advantages such as large electric dipole moments, high nonlinear optical coefficients. Furthermore, the transition energies, dipole moments, and symmetries can be flexibly engineered on demand by choosing the materials and structure dimensions. The implementation of quantum coherence and interference effects in QWs is much more promising for the practical applications in quantum information processing and quantum networking. Studies [23–29] based on quantum coherence and interference effects especially the Fano interference [30, 31] in QWs have already been done, for examples, tunneling induced transparency (TIT) [24], ultrafast all optical switching [26], enhanced Kerr nonlinearity [27], optical bistability [28], optical precursors [29], and so on. So far, to our best knowledge, studies have not been extended to investigate the EIG phenomenon in the semiconductor QW systems yet.

In this context, we propose a scheme for obtaining an electromagnetically induced grating in semiconductor GaAs/AlGaAs asymmetric QW via Fano interference. We investigate the effects of Fano interference on the forming of the grating and its diffraction efficiency. Our studies show that, by taking advantage of Fano-type interference and the spatial modulation of the control field, a high efficient absorption or phase grating can be created in the sample. Fano interference can greatly enhance the diffraction intensity efficiency. Besides, the influences of the control field intensity, the interaction length, the coupling strength of the tunneling, and the strength of Fano interference on the diffraction efficiency of the grating are also discussed.

2. Model and basic equations

Consider an n-doped asymmetric GaAs/Al_xGa_1−xAs double QW structure as shown in Fig. 1(a). It consists of two wells that are separated by a narrow barrier. At certain bias voltage, the ground state in the shallow well is resonant with the first excited state in the deep well, and due to the coherent coupling via the barrier, they mix into a doublet, the intermediate states |2〉 and |3〉. The splitting (2δ) on resonance is given by the coupling strength and can be tuned by adjusting the height and width of the tunneling barrier with applied bias voltage [32]. Another thin barrier on the right side separates a continuum of energies from the deep well. The decay of electrons in states |2〉 and |3〉 to the continuum by tunneling through the right thin barrier leading to the Fano interference [26, 28]. The ground state |1〉 of the deep well is coupled to the intermediate states by a weak probe field of frequency ω_p:

E_{p} (z, t) = \frac{1}{2} E_{p} e^{- i ω_{p} t + i k_{p} z} + c . c .,

where E_p is a slowly varying function of time t and distance z, k_p is the wave vector and c.c. stands for complex conjugate. Meanwhile, a control field of frequency ω_c couples the intermediate states to the upper state |4〉. It is composed of two components overlapping at the sample at an angle, as shown in Fig. 1(b), which form a standing wave along the x dimension

E_{c} (x, z, t) = \frac{1}{2} E_{c} sin (π x / Λ) e^{- i ω_{c} t + i k_{c z} z} + c . c .,

where E_c is assumed to be a real constant for simplicity, and Λ(= π/k_cx) represents the space period and can be controlled by the angle at which the two components intersect, k_cx and k_cz are respectively the x and z components of wave vector k_c of the control field, satisfying

k_{c x}^{2} + k_{c z}^{2} = k_{c}^{2}

.

Fig. 1 (a) Schematic diagram for an asymmetric GaAs/Al_xGa_1−xAs quantum well. (b) Sketch of the probe and control fields propagating through the sample.

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In the interaction picture, the Hamiltonian in the rotating-wave and dipole approximations can be written as

\begin{array}{l} H_{I} & = & - \bar{h} [Ω_{p} | 2 〉 〈 1 | e^{- i (δ + Δ_{p}) t + i k_{p} z} + p Ω_{p} | 3 〉 〈 1 | e^{i (δ - Δ_{p}) t + i k_{p} z} \\ + Ω_{c} sin (π x / Λ) | 4 〉 〈 2 | e^{i (δ - Δ_{c}) t + i k_{cz} z} \\ + q Ω_{c} sin (π x / Λ) | 4 〉 〈 3 | e^{- i (δ + Δ_{c}) t + i k_{cz} z} + h . c .], \end{array}

where 2δ = ω₃₁ − ω₂₁ is the frequency splitting between states |2〉 and |3〉, Δ_p = ω_p − (ω₂₁ + ω₃₁)/2 and Δ_c = ω_c − (ω₄₂ + ω₄₃)/2 are the detunings of corresponding fields, respectively. Ω_p = μ₂₁E_p/2h̄ and Ω_c = μ₄₂E_c/2h̄ are the Rabi frequencies. p = μ₃₁/μ₂₁ and q = μ₄₃/μ₄₂ are the ratio between the intersubband dipole moments of the relevant transition. h.c. represents the Hermitian conjugate. The wave function of the system can be expressed in the form

| ψ (t) 〉 = a_{1} (t) | 1 〉 + a_{2} (t) | 2 〉 + a_{3} (t) | 3 〉 + a_{4} (t) | 4 〉,

where a_i(t)(i = 1, 2, 3, 4) is the probability amplitude of the state |i〉. By utilizing the Schrödinger equation and the following transformations

\begin{array}{l} b_{1} = a_{1}, b_{2} = a_{2} e^{i (δ + Δ_{p}) t - i k_{p} z}, \\ b_{3} = a_{3} e^{- i (δ - Δ_{p}) t - i k_{p} z}, b_{4} = a_{4} e^{i (Δ_{p} + Δ_{c}) t - i (k_{p} + k_{c}) z}, \end{array}

we can obtain the equations of motion of the amplitudes

\begin{array}{l} {\dot{b}}_{1} = i Ω_{p} b_{2} + i p Ω_{p} b_{3}, \\ {\dot{b}}_{2} = i Ω_{p} b_{1} + [i (δ + Δ_{p}) - γ_{2}] b_{2} + κ b_{3} + i Ω_{c} sin (π x / Λ) b_{4}, \\ {\dot{b}}_{3} = i p Ω_{p} b_{1} + κ b_{2} - [i (δ - Δ_{p}) + γ_{3}] b_{3} + i q Ω_{c} sin (π x / Λ) b_{4}, \\ {\dot{b}}_{4} = i Ω_{c} sin (π x / Λ) b_{2} + i q Ω_{c} sin (π x / Λ) b_{3} + [i (Δ_{p} + Δ_{c}) - γ_{4}] b_{4}, \end{array}

where, γ_i(i = 2, 3, 4) is the total electron decay rate of state |i〉 and γ_i = γ_il + γ_id. γ_il is the population decay and can be calculated [33] by solving the effective mass Schrödinger equation with outgoing waves at infinity. γ_id is the dephasing rate originating from intrasubband phonon scattering, electron-electron scattering, and inhomogeneous broadening due to scattering on interface roughness.

κ = \sqrt{γ_{2 l} γ_{3 l}}

is the Fano interference factor representing the cross coupling between states |2〉 and |3〉 arising from the tunneling to the continuum through the thin barrier next to the deep well [28, 34]. The strength or quality of Fano interference is assessed by

η = κ / \sqrt{γ_{2} γ_{3}}

, which can be augmented by reducing the dephasing rate via lowering the temperature. The limits η = 0 and η = 1 correspond to no interference and perfect interference, respectively.

By solving the coupled amplitude equations in steady state in the non-depletion approximation (|b₁|² ≈ 1) and substituting the analytical solutions into the polarization of the medium $P = ε_{0} χ_{p} E_{p} = N μ_{21} (b_{2} b_{1}^{*} + p b_{3} b_{1}^{*})$ , we can arrive at the probe susceptibility

χ_{p} = \frac{N μ_{21}^{2}}{ε_{0} \bar{h}} χ,

where N is the electron density, and

χ = \frac{1}{2} \frac{(p^{2} {\tilde{γ}}_{2} + {\tilde{γ}}_{3}) {\tilde{γ}}_{4} + 2 i p {\tilde{γ}}_{4} κ - {(p - q)}^{2} Ω_{c}^{2} {sin}^{2} (π x)}{({\tilde{γ}}_{2} {\tilde{γ}}_{3} + κ^{2}) {\tilde{γ}}_{4} - (q^{2} {\tilde{γ}}_{2} + {\tilde{γ}}_{3} + 2 i q κ) Ω_{c}^{2} {sin}^{2} (π x)},

where γ̃₂ = Δ_p +δ + iγ₂, γ̃₃ = Δ_p −δ +iγ₃, γ̃₄ = Δ_p + Δ_c + iγ₄. Note that we choose Λ to be the unit of x hereafter. Owing to the resonant tunneling between the ground level of the shallow well and the first excited level of the deep well, the wave functions of states |2〉 and |3〉 are symmetric and antisymmetric combinations of those associated with the isolated deep and shallow wells and we can take p = −1, q = 1 [26]. In the local frame, the dynamic response of the probe field in the sample is described by Maxwell’s wave equation which is reduced to [1, 35]

\frac{\partial Ω_{p}}{\partial z} + i γ_{2} χ Ω_{p},

under the slowly varying envelope approximation and in steady state. As in Refs. [26] and [36], we consider a TM polarized probe incident at an angle of α = 45 degrees with respect to the growth axis (z′) so that the transition dipole moments include a factor

1 / \sqrt{2}

as intersubband transitions are polarized along the growth axis. Here we have eliminated the transverse term in order to focus on the main features of EIG, and z is made dimensionless by setting

z_{0} = 2 \bar{h} γ_{2} ε_{0} / N k_{p} μ_{21}^{2}

as its unit. By solving this equation, the transmission function of the medium at z = L can be derive as

T (x) = e^{- Im [χ] γ_{2} L} e^{i Φ},

where the phase Φ = Re[χ]γ₂L. By the Fourier transformation of T (x), far-field diffraction equation can be obtained as

I_{p} (θ) = {| J_{P} (θ) |}^{2} \frac{{sin}^{2} (M π Λ sin θ / λ_{p})}{M^{2} {sin}^{2} (π Λ sin θ / λ_{p})},

where

J_{P} (θ) = \int_{0}^{1} T (x) exp (- i 2 π Λ x sin θ / λ_{p}) d x

refers to the Fraunhofer diffraction of a single space period, M is the number of spatial periods of the grating illuminated by the probe beam, and θ is the diffraction angle with respect to the z direction. The probe diffraction intensity I_p(θ) is normalized such that if T (x) = 1, then I_p(θ) = 1. The n-order diffraction intensity is determined by Eq. (11) with sinθ = nλ_p/Λ. Since we are mainly interested in the first-order diffraction (n = 1), we calculate the first-order diffraction intensity as

I_{p} (θ_{1}) = {| J_{p} (θ_{1}) |}^{2},

where

J_{p} (θ_{1}) = \int_{0}^{1} T (x) exp (- i 2 π x) d x .

3. Results and discussions

In this section, we will present our main results and discuss the effects of Fano interference on EIG in a semiconductor QW structure and analyze the controllability of the diffraction efficiency of the grating via tuning the parameters of the QW system. To illustrate the main feature of the semiconductor QW structure, we will compare cases with and without Fano interference. Note that the structural parameters adopted in this context are not compulsory and the results here are also obtained for other parameter choices. Without loss of generality, an example of the asymmetric double-quantum-well structure is outlined as follows: the widths of both the shallow and deep wells are 7.5 nm, the thickness of the barrier between the two wells is 3.2 nm, the thickness of the thin barrier next to the deep well is 2.0 nm, the composition ratios of Al for the barrier, the shallow well, the deep well and the continuum are 0.4, 0.18, 0, and 0.2, respectively. The decay rates, the splitting energy, and the eigenenergy of the subbands can be estimated from this structural design.

The standing-wave pattern of the control field can lead to intensity-dependent periodic variation of probe susceptibility. By using Eq. (8), we plot the absorption and dispersion profiles with and without the resonant control field in Fig. 2. At the nodes of the standing wave (Ω_c = 0), the absorption almost vanishes at the probe resonance, i.e., TIT [24] occurs [solid black line in Fig. 2(a)]. As shown, the transparent window is greatly deepened due to the existence of Fano interference, while without Fano interference (κ = 0), the absorption spectrum is the incoherent sum of two Lorentzian functions. Further, the dispersion shown in Fig. 2(b) becomes strong and steep due to Fano interference. At the antinodes of the standing-wave (Ω_c ≠ 0), the former transparent window is split into two since the control field results in the nonlinear absorption of the probe field causing an absorption peak located at the line center. It can be proved that the central absorption peak is mainly due to the nonlinear absorption of the probe field, which is caused by the presence of the control field, while the side band absorption peaks are mainly caused by the linear absorption of the probe field. The basic equations can be found in Ref. [29], and we do not show them here. Figure 2(c) also shows that Fano interference can deepen the transparent windows and reduces the nonlinear absorption of the probe field to a certain degree. The corresponding dispersion profiles are shown in Fig. 2(d).

Fig. 2 Intersubband absorption spectrum and dispersion spectrum with (a), (b) Ω_c = 0 meV and (c), (d) Ω_c = 3.0 meV. The other parameters are γ_2l = 1.8 meV, γ_3l = 1.6 meV, γ_4l = 0.4 meV, γ_2d = 0.4 meV, γ_3d = 0.4 meV, γ_4d = 0.1 meV, Δ_c = 0 meV, δ = 4.7 meV.

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Generally speaking, the medium within the TIT window is nearly transparent to the probe fields at nodes while being opaque at antinodes. This will lead up to an amplitude modulation across the probe field. Moreover, when the control field is from off to on, the properties of the dispersion in the line center is modified from normal to anomalous. This can result in a phase modulation to the probe field. Thus, an absorption grating or a phase grating can be created in the sample. When the probe field propagates perpendicularly to the standing wave, it will be diffracted.

In the following, we will investigate the diffraction pattern of the probe field by using equations outlined in the last section. First start with a case in which Δ_p = 0 meV and Δ_c = 0 meV. Under this condition, as shown in Figs. 2(b) and 2(d), the refraction is nearly eliminated, so there is no phase modulation of the probe field [dash-dotted blue line in Fig. 3(a)]. The amplitude of the transmission function is also presented there. The spatially modulated control field causes the alternating regions of high probe transmission and absorption, which diffracts the resonant probe field, as shown in Fig. 3(b), namely an absorption grating is formed here.

Fig. 3 (a) The amplitude of the transmission function |T(x)| with κ ≠ 0 (solid black line) and κ = 0 (dashed red line), the phase of the transmission function Φ/π with κ ≠ 0 (dotted blue line) and κ = 0 (dash-dotted magenta line). (b) The diffraction pattern I_p(θ) as a function of sin(θ) with κ ≠ 0 (solid black line) and κ = 0 (dashed red line). The parameters are Δ_p = 0 meV, Ω_c = 3.0 meV, L = 4.0z₀, Λ/λ_p = 4.0, M = 5. Other parameters are the same as in Fig. 2.

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In the presence of Fano interference, the destructive interference induces transparency at nodes of the standing wave while the control field causes the nonlinear absorption at the antinodes. As a result, the transmission of probe filed is higher and the absorption modulation depth is much larger [solid black line in Fig. 3(a)] than that in the case without Fano interference [dashed red line in Fig. 3(a)]. Additionally, Fano interference also reduces nonlinear absorption at the antinodes, as verified in Fig. 2(c), which contributes to the increase of the average transmissivity of the probe field, too. Therefore, the diffraction intensity, as shown in Fig. 3(b), is greatly enhanced due to Fano interference. Of course, the intensity of the first-order diffraction (around sinθ = 0.25) of this grating is also promoted. As can be seen in Fig. 3(b), the first-order diffraction intensity with Fano interference is about two times as large as that without Fano interference.

In Figs. 4(a) and 4(b), by using Eq. (13), we display I_p(θ₁) as a function of the intensity of the control field Ω_c and the interaction length L, respectively. With the increasing of Ω_c, the absorption modulation depth increases, which leads to the improvement of the first-order diffraction intensity first. However, after peaking at certain Ω_c, it fades away. This is because of the decreased average transmission of the probe field due to the strongly increased nonlinear absorption. Likewise, it has the same trend for the interaction length L and the reason is similar to the case of the control filed Ω_c. For small L, the absorption modulation depth increases with L which can be easily demonstrated and is not shown here, then leading to the linearly increase of I_p(θ₁) with L. For large L, the absorption is dominant, then the diffraction efficiency decreases.

Fig. 4 The first-order diffraction intensity I_p(θ₁) as a function of Ω_c (a) and L (b), respectively. Other parameters are the same as in Fig. 3.

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As well known, absorption gratings in general have low diffraction efficiency. In order to increase the efficiency of the first-order diffraction, a phase grating is pressingly expected. We adjust the system parameters to reach one. Detuned fields are applied and some numerical results are illustrated as follows. Figure 5 displays the amplitude |T (x)|, phase Φ/π, and the diffraction pattern with Δ_c = −11.0 meV and Δ_p = −1.5 meV. In the absence of Fano interference, although a deep phase modulation can be attained [dashed red line in Fig. 5(b)], the amplitude of transmission is so small [dashed red line in Fig. 5(a)] that most of the probe field is absorbed. Hence, the diffraction efficiency is very low [dashed red line in Fig. 5(c)]. In the presence of Fano interference, the amplitude of transmission is highly improved [solid black line in Fig. 5(a)]. Besides, the phase modulation is deepened due to the enhanced refraction by Fano interference [27, 37] and a phase modulation on the order of π is induced across the probe field [solid black line in Fig. 5(b)]. More light is delivered to high order diffraction, as is proved in Fig. 5(c), and the diffraction efficiency is greatly improved. The phase modulation transfers energy from zero-order to high-order diffraction, which is demonstrated in Fig. 6, where the Fraunhofer diffractions of |T(x)| and exp(iΦ) are plotted. It is clearly shown in Fig. 6(a) that, Fano interference promotes the diffraction intensity by reducing the absorption of the probe field in forming an absorption grating. On the other hand, Fano interference favors the dispersion of light into the high-order diffractions in producing a phase grating, as shown in Fig. 6(b), Fano interference contributes to lowering the zero-order diffraction and raising the first-order diffraction. It is not hard to achieve the conclusion that Fano interference plays a significant role in generating an EIG in the asymmetric QW structure.

Fig. 5 (a) The amplitude of the transmission function |T(x)|. (b) The phase of the transmission function Φ/π. (c) The diffraction intensity I_p(θ) as a function of sinθ. The parameters are Δ_c = −11.0 meV, Δ_p = −1.5 meV, Ω_c = 5.0 meV, L = 10.5z₀. Other parameters are the same as in Fig. 3.

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Fig. 6 Fraunhofer diffractions of |T(x)| (a) and exp(iΦ) (b) as functions of sinθ. The parameters are the same as in Fig. 5.

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To understand controllability of the first-order diffraction intensity I_p(θ₁), in Fig. 7 and Fig. 8, we display the variation of I_p(θ₁) as a function of Ω_c or L under various parametric conditions. It is shown that an optimum value of Ω_c or L exists for which the first-order diffraction intensity is dramatically increased. For large Ω_c or L, the nonlinear absorption restrains the diffraction efficiency, and the first-order diffraction intensity is highly decreased. How the energy splitting δ between sates |2〉 and |3〉 modifies the first-order diffraction intensity can be obtained in Fig. 7. With the increasing of δ, I_p(θ₁) can be increased, especially for a medium with given thickness L [for example, highlighted in Fig. 7(b) with magenta line]. The reason is that, the larger the splitting, the more light can be transmitted through the medium, the more light would be diffracted towards the first-order diffraction. As mentioned above, the splitting on resonance is given by the coupling strength of the tunneling and can be controlled by adjusting the height and width of the tunneling barrier with applied bias voltage. Therefore, the diffraction efficiency of the phase grating can be technically tuned by appropriately adjusting the splitting.

Fig. 7 The first-order diffraction intensity I_p(θ₁) for different energy splitting δ as a function of Ω_c (a) and L (b), respectively. Other parameters are the same as in Fig. 5.

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Fig. 8 The first-order diffraction intensity I_p(θ₁) for different interference strength η as a function of Ω_c (a) and L (b), respectively. Other parameters are the same as in Fig. 5.

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The effects of the strength of the interference on the diffraction efficiency can be clearly seen from Fig. 8. We can see the increasing of first-order diffraction intensity as we go from η = 0.75 to η = 0.85. The reason is similar to the case of the energy splitting. With the increasing of η, the absorption for the probe field on the intersubband transitions |1〉 → |2〉 and |1〉 → |3〉 of QW can be reduced [26], then more light can be transferred to the first-order diffraction. Although, it is not easy to achieve the perfect interference due to the unwanted dephasing effects in our structure, as we already know, η can be augmented by decreasing the temperature to get the reduced dephasing rates [24, 38]. Therefore, it is possible to optimally attain the desired diffraction efficiency in the sample via properly controlling the strength of Fano interference.

4. Conclusions

In conclusion, we have theoretically investigated the electromagnetically induced grating devices in semiconductor GaAs/AlGaAs asymmetric QW structures. The effects of Fano interference on the forming of an absorption or phase grating and its diffraction efficiency are studied. On the one hand, it is found that, Fano interference can deepen the depth of the absorption or phase modulation, and thereby the diffraction intensity of the grating is greatly improved. On the other hand, Fano interference can help in diffracting more light to the first-order diffraction, and therefore the first-order diffraction efficiency can be increased. There are two reasons for this. One is that, Fano interference decreases the nonlinear absorption caused by the control field, thereby the total transmission of the probe field is greatly promoted and more light is available to be diffracted. The more important one is that, it helps the phase grating to transfer more light from the zero-order to the high-order diffraction.

In addition, analyzes show that, by appropriately controlling the system parameters of the intensity of the control field, the interaction length, the coupling strength of the tunneling, the strength of Fano interference, etc., the increasing of the diffraction efficiency of the grating can be achieved as well. We hope that our results will find potential applications in developing novel photonic devices, e.g., all-optical switching and routing at low light levels.

Acknowledgments

The research is supported by the National Natural Science Foundation of China (Grant Nos. 11274112, 11074263), the Shanghai Rising-Star Program of Grant No. 11QA1407400, the Fundamental Research Funds for the Central Universities ( WM1114024), and National Laboratory for Infrared Physics, CAS(201101). H. S. would like to acknowledge the support from the NSF-China under Grant No. 11104176.

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Electromagnetically induced grating in asymmetric quantum wells via Fano interference

Abstract

1. Introduction

2. Model and basic equations

3. Results and discussions

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (8)

Equations (14)

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