Fano signatures between intersubband and ponderomotive responses in MQW structures

Diogo De Moura Pedroso; Johannes Schmidt; Angelo Passaro; Manfred Helm; Harald Schneider

doi:10.1364/OE.26.024054

1. Introduction

Intersubband transitions and band-gap engineering of semiconductor nanostructures have been extensively investigated for the last decades. In addition to the development of relevant applications, such as Quantum Well Infrared Photodetectors [1] (QWIP) and Quantum Cascade Lasers [2,3] (QCL), they have been used in the study of interesting fundamental physical phenomena, such as gain without inversion [4], Autler-Townes effect [5,6], coherent control [7,8], cavity polaritons [9], Bloch gain [10], and Rabi flopping [11]. Fano-resonance spectra [12] have also been looked into, employing Wannier-Stark ladders and Floquet-state excitons in biased quantum well (QW) superlattices [13,14], excited and virtual states of self-assembled quantum dots [15], intersubband-phonon coupling in quantum wells [16] and quantum wires [17], quantum dots in quasi one-dimensional waveguides [18], and asymmetric coupled quantum wells [19–21].

The physical origin of the Fano-resonance spectra mentioned above [12–22] is the interaction between a continuum and a discrete set of quantum states. In contrast to these examples [12–22], a Fano signature in multiquantum well (MQW) structures was demonstrated a few years ago, coming out as a result of optical interference due to the induced phase changes by the presence of a ponderomotive current in addition to the usual polarization associated with intersubband transitions [23]. In this case, the presence of a wiggling motion of carriers due to an oscillating electromagnetic field induces a Fano lineshape in the differential transmission spectra obtained experimentally by phase sensitive terahertz time domain spectroscopy of multiquantum well structures [23].

The ponderomotive current is associated with the wiggling motion of carriers, which in the case of a layer with negligible absorption and scattering effects generates the simplistic response in the dipole approximation: $χ_{A} (ω) \approx - ω_{P L}^{2} / ω^{2}$ , where $ω_{P L}$ is the three dimensional plasma frequency [23]. In this case, $χ_{A} (ω)$ approximates a purely real susceptibility and the corresponding Fano signature does not manifest itself in the absorption but only in the differential transmission spectrum [23,24].

Additional experimental investigations have shown that the Fano-signature associated with the ponderomotive current only appears in samples containing a bulk-like region in addition to the QW layers [24], similar to the structure exemplified in Fig. 1. The bulk-like region refers to a layer in which quantization effects are negligible, presenting a three dimensional density of states. However, further details regarding the necessary conditions to the observation of this signature with respect to the structural parameters have not been addressed yet in the literature. Despite the good fitting of the experimental transmission data, the theoretical approach as previously presented [23] does not distinguish the contributions of the different layers of the structure. Hence, in order to discuss the role of the structural parameters in the observation of these Fano signatures we need to move the theoretical approach one step forward.

Fig. 1 Generic conduction band edge profile for a QW-based structure.

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The present work aims at going beyond the previous approach [23] by the inclusion of the effects of the whole structure. As a result, we quantify the relation between the parameters of both QW and bulk-like regions which allows us to predict the Fano-signature in the differential transmission spectra. In this paper, the response of each layer of the structure is addressed within the density matrix formalism and then encompassed into an optical transfer matrix, providing naturally a description of the optical interferences. Theoretical results are in a good agreement with experimental results. Different numerical studies on the tuning of parameters are presented, including comparisons of the Fano signatures with a well-known fitting formula.

2. Theoretical approach

In order to investigate numerically the observation of the Fano signatures associated with the ponderomotive motion, we calculate the differential transmission spectra in the structure, allowing a direct comparison with the experimental results obtained by Time Domain Spectroscopy. Hereafter, we consider an experiment focused on probing intersubband transitions in the conduction band of one-dimensional multiquantum well structures. In this experiment, the difference between two transmitted optical fields is recorded by phase sensitive terahertz time domain spectroscopy, aiming to be sensitive to the investigated small effects. The transmission through the structure without electrons in the bulk-like region and in the MQW region is acquired as a reference. In respect to this spectrum, another transmitted optical field is subtracted, where the bulk-like and the active region are filled with carriers by using either optical pumping or applied gate voltage. Acquiring these both transmitted fields including their phase and calculating the difference between them allows us to obtain the phase change of electric field caused by photon-electron interaction. The experiment is assumed to be performed in a linear regime using a low intensity probe beam which, in principle, may present an arbitrary polarization, although only the component of polarization along the growth axis couples to the intersubband transition.

We assume a sample comprised of the usual AlGaAs/GaAs material system in which the Al concentrations are low enough to ensure direct band gap. Bulk-like regions and QW layers are n-type doped, so the modulation of the carrier concentration is obtained by an appropriate time-dependent applied gate voltage. We consider samples in which carrier concentration in the QWs is about ~10¹⁰ cm² and relevant intersubband transition energies about ~100 meV, allowing us to neglect many-body effects, such as depolarization shift [25]. The dipole approximation applies considering wavelength much longer than quantum wells.

The transfer matrix formalism is based on an ordered multiplication of interface and propagation matrixes according to the layers of the structure [26]. The interface, I, and propagation, P, matrixes of the i-th layer are given by:

I_{i - 1, i} = \frac{1}{t_{i - 1, i}} (\begin{matrix} 1 & r_{i - 1, i} \\ r_{i - 1, i} & 1 \end{matrix})_{} a n d_{} P_{i} = (\begin{matrix} e^{- i ϕ_{i}} & 0 \\ 0 & e^{i ϕ_{i}} \end{matrix}),

where

r_{i - 1, i}

and

t_{i - 1, i}

are the Fresnel reflection and transmission coefficients at the interface between two layers,

ϕ_{i} = G n_{i} L_{i} \cos α_{i}

is the change of phase due to the propagation through the layer, where

\vec{G}

is the wave vector of the vector potential,

n

is the refractive index,

L

is the layer width, and

α

is the incidence angle with respect to the growth direction. The optical transfer matrix, M, is then given by the product [26]:

M = I_{01} P_{1} I_{12} P_{2} I_{23} P_{3} ....

When considering the modulation of carrier concentration due to the applied gate voltage, we should calculate the induced change of refractive index $Δ n$ as well as the corresponding change of phase $Δ ϕ$ for each layer of the structure. The parameter $Δ n$ can be obtained from the values of susceptibility [27–29], while the change of phase in the i-th layer is given by $Δ ϕ = G (n_{i} + Δ n_{i}) L_{i} \cos α_{i}$ . The values of $Δ ϕ$ are encompassed in the optical transfer matrix of the structure, more specifically in the propagation matrix. The anisotropic characteristic of the refractive index in QWs layers is included by using an angle-dependent average [30]. Once the optical transfer matrix is created, calculations of the transmission probability spectra [26], $T$ , are performed considering the situations with and without carriers in order to reproduce the carrier modulation process employed in the experimental setup of the phase-sensitive terahertz time domain spectroscopy [23,24], see Fig. 2. The link between theoretical and experimental results is obtained by means of the differential transmitted electric field $Δ E = E_{T} - E_{r e f}$ , in which $E_{T}$ and $E_{r e f}$ denote the transmitted fields with and without electrons, respectively. Hence:

\frac{Δ E}{E_{r e f}} = \frac{E_{T} - E_{r e f}}{E_{r e f}} = \frac{T_{c a r r i e r s} - T_{r e f}}{T_{r e f}},

where

T_{c a r r i e r s} = E_{T} / E_{i n}

and

T_{r e f} = E_{r e f} / E_{i n}

are the field transmission probabilities calculated in the situations with and without electrons, respectively.

Fig. 2 Schematic representation of the modulation of carrier concentration in the sample with doped cap layer reported by Baudisch et al. [24]. Different colours indicate different values of complex refractive index.

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We now need to address the susceptibility of each layer of the structure which is employed to obtain the induced change of refractive index $Δ n$ . The electromagnetic response of the QW and cap layer includes the effects introduced by the dynamics of the carriers inserted in the system by the gate voltage modulation in addition to the background response. Hence, the frequency-dependent susceptibility of the each layer of the system can be obtained by means of [31]:

χ (ω) = χ_{B} + \frac{i J (ω)}{ω ε_{0} E (ω)},

where

χ_{B}

denotes the background susceptibility,

ω

is the angular frequency, and

E

is the electric field of the incident beam.

The induced current $\vec{J}$ can be obtained by means of:

\vec{J} = \sum_{k', k} {\vec{j}}_{k' k} ρ_{k' k},

where

{\vec{j}}_{k' k}

are the matrix elements of the single-particle current operator [32] and

ρ_{k' k}

are the elements of the density matrix [33], whose dynamics is obtained from the Liouville-von Neumann equation of motion:

\frac{\partial \hat{ρ}}{\partial t} = - \frac{i}{ℏ} [\hat{H}, \hat{ρ}] + {\frac{\partial \hat{ρ}}{\partial t} |}_{s c},

in which scatterings and dephasing mechanisms are treated within the relaxation time approximation.

\hat{H}

denotes the single-particle Hamiltonian of the system, including the potential of the band-edge profile, the Hartree potential [34], and the classical light-matter interaction [31]:

H_{A} = - \frac{q}{m} \vec{A} \cdot \vec{p} + \frac{q^{2} A^{2}}{2 m},

where

q

is the fundamental charge,

m

is the parabolic effective mass,

\vec{p}

is the linear momentum operator. The vector potential

\vec{A} (t)

is addressed classically in the Coulomb gauge,

\vec{\nabla} \cdot \vec{A} = 0

[31] and for the considered range of wavelength, the dipole approximation can be assumed in the direction of the growth axis (z-direction), as indicated in Fig. 1. The electric field of the incident beam links to the vector potential by means of

\vec{E} = - \partial \vec{A} / \partial t

. The first term in Eq. (7) generates a linear position- and time-dependent variation of the band-edge potential. The second term in Eq. (7) describes a time-dependent offset of the potential which depends on the square amplitude of the vector potential. The ponderomotive motion stems from the combination of these two terms, while intersubband transitions come only from the first term.

At this point it is worth mentioning that the second term in Eq. (7) should not be neglected even for low amplitude vector potential. This term is directly associated with the description of the ponderomotive motion as demonstrated in the next derivation steps. Additionally, since all terms of the electron-photon interaction are considered, the derivation is gauge independent as well as the given physical interpretation.

For thick enough barriers, we can neglect interactions between the cap layer and multiquantum well (MQW) layers, allowing us to evaluate separately the dynamics of the bulk-like regions and MQW layers by using Eq. (6). In the real space, the elements of the basis ${| ξ_{ν, k | |} 〉}$ , obtained by means of $H_{0} | ξ_{ν, k | |} 〉 = ε_{ν, k | |} | ξ_{ν, k | |} 〉$ , can be written as:

〈 r | ξ_{ν, k | |} 〉 = \frac{1}{\sqrt{S}} e^{i {\vec{k}}_{| |} \cdot {\vec{r}}_{| |}} φ_{ν} (z),

for QW layers, and:

〈 r | ξ_{ν, k | |} 〉 = \frac{1}{\sqrt{S}} e^{i \vec{k} \cdot \vec{r}},

for bulk-like regions, in which

S

is the transverse area,

ν

is the subband index,

\vec{k} = (k_{| |}, k_{z})

and

{\vec{r}}_{| |} = (x, y)

.

Since we are interested in probing the intersubband transition, only the current in the growth axis is considered. Replacing the basis ${| ξ_{ν, k | |} 〉}$ in Eq. (5) and considering that the modulated applied gate voltage does not generate current in the growth direction, we obtain the following expression of current for the QW layers:

J_{z} = - \frac{i ℏ q}{2 m} \sum_{ν' \neq ν} ρ_{ν ν'} \int d z [φ_{ν'}^{*} \frac{\partial φ_{ν}}{\partial z} - φ_{ν} \frac{\partial φ_{ν'}^{*}}{\partial z}] - \frac{q^{2} A (t) N}{m},

and for the bulk-like regions, we obtain:

J_{z} = - \frac{q^{2} A (t) N}{m},

where

ρ_{ν' ν} = \sum_{k_{| |}} ρ_{ν' k_{| |} ν k_{| |}}

and

N = T r {ρ}

is the carrier concentration in the corresponding layer.

The z-component of the linear response can now be obtained from Eq. (4) for the QW and bulk-like regions. The vector potential can be modelled as a composition of complex harmonic plane waves assuming the dipole approximation, $\vec{A} (t) = \sum_{ω} {\vec{a}}_{ω} e^{i (ω + i δ) t}$ , in which a phenomenological damping parameter $δ$ is included in order to take into account possible absorption and scattering effects.

In a linear regime, the off-diagonal terms of the density-matrix in Eq. (10) can be obtained analytically by perturbation methods [29,35,36]. Hence, for QW layers, the z-component of susceptibility can be written as:

χ_{Q W} (ω) \approx χ_{B} - \frac{q^{2}}{m^{2} ω^{2} ε_{0}} \sum_{ν' \neq ν} {| p_{ν' ν} |}^{2} \frac{(ρ_{ν' ν'}^{0} - ρ_{ν ν}^{0})}{ε_{ν} - ε_{ν'} + ℏ (ω - γ_{ν' ν})} - \frac{ω_{P L}^{2}}{ω (ω + i δ)},

where

ω_{P L} = \sqrt{q^{2} N / ε_{0} m}

and

γ_{ν' ν}

denotes a dephasing constant. The upper index zero indicates non-perturbed density matrix elements which in our case correspond simply to the thermal distribution. The linear response for the bulk-like regions does not include the term of intersubband polarization and can be given by:

χ_{c a p} (ω) \approx χ_{B} - \frac{ω_{P L}^{2}}{ω (ω + i δ)} .

The last term in Eq. (12) and Eq. (13) denotes the ponderomotive response, resembling a Drude-like response. It is worth noticing that in the experimental results previously reported [23,24] the limiting case $δ ≪ ω$ is observed, leading the ponderomotive term to an approximate purely real response $χ_{A} (ω) \approx - ω_{P L}^{2} / ω^{2}$ .

3. Results

Numerical results presented in this section were obtained using two similar modulation doped samples, previously reported by Baudisch et al. [24]. The samples present nine periods of 8.5 nm thick GaAs QW, separated by 32 nm thick Al_0.25Ga_0.75As barriers. The surface carrier concentration in the QW layers is about 4x10¹⁰ cm⁻², obtained by means of delta doping in the barriers on both sides of the QW with 2x10¹⁷ cm⁻³. The MQW region was grown on top of a 300 nm thick doped buffer layer (N = 5x10¹⁷ cm⁻³) which as a contact. One of the samples presents an additional 200 nm thick doped cap layer with nominal carrier concentration N = 10¹⁷ cm⁻³. The actual carrier concentration in this doped cap layer is higher due to the carriers from the delta doping in the barriers. For both samples, the carrier concentration in the structure is modulated by an applied gate voltage which depletes the QWs and the cap layer. The carrier concentration in the buffer layer of both samples is not modulated. The trapezoidal facets of the samples with and without the additional cap layers are polished in 70-deg and 38-deg, respectively [24]. The results were calculated using 10K and assuming the limiting case $δ ≪ ω$ .

Figure 3(a) shows the normalized differential transmission considering the samples with and without the additional doped cap layer, assuming a typical value of 400 fs for the dephasing time. Results of the structure with the additional doped cap layer (solid lines) demonstrate a clear contribution from the ponderomotive motion at the lower frequency region between 18 and 25 THz, which is in the same order of magnitude as the main peak of the spectrum related to the intersubband transition. Regarding the observation of the Fano signature in the differential transmission spectra, the carrier concentration in the bulk-like region must also be modulated by the applied gate voltage in conjunction with the quantum wells. Doped buffer layers of both samples, whose carrier concentration is not modulated, do not provide any contribution. For the sample without the cap layer (dashed line), the peak in the low-frequency region of the spectrum is two orders of magnitude lower than the main peak related to the intersubband excitation and comes uniquely from the ponderomotive current of carriers in the QW layer.

Fig. 3 (a) Theoretical results of differential transmission for samples with and without an additional doped cap layer. (b) Change of refractive index in the QW layer as a function of frequency considering different mechanisms of contribution.

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Figure 3(b) shows the change of refractive index in the QW layer, illustrating the competition between the effects related to the ponderomotive current and the intersubband polarization. The result in Fig. 3(b) refers uniquely to the QW layers, meaning that the presence or not of the doped cap layer does not alter it. Changes of refractive index due to the ponderomotive motion (solid line with symbols) are always negative and its magnitude decreases with the frequency. The contribution of the ponderomotive motion in the value of $Δ n$ can be inferred by the deviation of the total response (solid line without symbols) from the results associated only with the intersubband polarization (dashed line). Nevertheless, $Δ n$ is strongly influenced by the mechanism of intersubband polarization mainly for frequencies above 21 THz, which can be verified by the small differences between the dashed line and the solid line without symbols. As a consequence, the ponderomotive current considering only carriers from the QWs does not provide significant Fano signatures, as illustrated in Fig. 3(a).

In fact, intersubband polarization is the dominant mechanism in the determination of $Δ n$ in the QW layers. Hence, the required opposite change of phase of the transmitted optical field has to originate from another region of the structure without intersubband transitions, which explains the need for a bulk-like region with non-zero carrier concentration.

A theory-experiment comparison of the differential transmission spectrum for the sample with the additional doped cap layer is shown in Fig. 4. Results with the presented model were calculated assuming 400 fs for the dephasing time and 2.3x10¹⁷ cm⁻³ for the carrier concentration in the cap layer. The increment in the carrier concentration of the cap layer with respect to the nominal value is justified due to additional carriers from the delta doping in the barrier layers and from the gate. The experimental result with the Fano signature [24] is appropriately reproduced by the present model, which provides correctly the relation between the secondary (due to ponderomotive current) and main (due to intersubband polarization) peaks of the spectrum, $r_{p} = Δ E_{p o n d} / Δ E_{int e r s u b}$ . Differences in the comparison of results might be related to effects of the delta doping in the barrier layers or even slight misalignments in the experimental setup.

Fig. 4 Theory-experiment comparison of differential transmission spectra as well as the fit using the Fano formula for the sample with the additional doped cap layer.

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A fitting of the well-known Fano formula [16] is also presented in Fig. 4 for comparison. It allows obtaining two common characterization parameters of the Fano signatures, named Fano factor and linewidth. This formula is expressed by [16]:

| \frac{Δ E}{E_{r e f}} | = \frac{{(η + \frac{ω - ω_{0}}{Δ ω})}^{2}}{1 + {(\frac{ω - ω_{0}}{Δ ω})}^{2}},

where

η

is the Fano factor,

Δ ω

is the linewidth parameter, and

ω_{0}

is the resonance frequency. Least square fitting of Eq. (14) to the experimental data results in

η \approx 0.65

and

Δ ω \approx 0.18 T H z

. The spectrum obtained from the Fano formula presents the same behaviour as the curve calculated with the model described in this paper. A constructive interference is expected in the high frequency region above 27 THz which is not observed in the experimental data. In fact, the high-frequency tail of the experimental curve is considerably shorter when compared to results obtained from a similar sample previously reported in [23], which might indicate the presence of some experimental artefact.

Hereafter, the influence of the carrier concentration and width of each region of the structure on the characteristics of the Fano-like signatures is evaluated numerically. To this end, parameters of the cap layer are assessed in comparison with those of the multiquantum well layers. Therefore, $r_{L} = L_{c a p} / (L_{Q W} * n_{Q W})$ is the ratio of lengths and $r_{N} = N_{c a p} / N_{Q W}$ is the ratio of carrier concentrations, where $n_{Q W}$ is the number of QW periods. All calculations were performed based on the characteristics of the sample with additional doped cap layer described before, including the reference field shown in Fig. 4.

The Fano factor $η$ decreases with the increase of $r_{L}$ and $r_{N}$ , Fig. 5(a). These values are obtained from least square fitting of Eq. (14) to theoretical results from the model described in this work. Negative values of $η$ indicate that the peak related to the ponderomotive current is higher than the one associated with the intersubband polarization. Likewise, the linewidth parameter $Δ ω$ decreases with the increase of the dephasing time and $r_{N}$ , considering a fixed value of $r_{L} = 2.6$ , Fig. 5(b). As expected, $Δ ω$ is related to the dephasing time of the system, yet not linearly. For comparison, experimental points are shown in Fig. 5(a)-5(b).

Fig. 5 (a) Fano factor as function of the ratio of lengths between the doped cap layer and MQW layers, considering dephasing time 400 fs. The experimental point refers to Ref [24]. (b) Linewidth parameter $Δ ω$ as a function of the dephasing time, assuming r_L = 2.6.

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The hatched area in Fig. 6(a) refers to values of $r_{p} = Δ E_{p o n d} / Δ E_{int e r s u b}$ ranging from 0.1 to 0.8. This range of $r_{p}$ denotes that one in which the secondary peak of the spectrum remains sharp and lower than the main peak. Theoretical calculations are performed considering two typical values of dephasing time and considering the reference spectrum given in Fig. 4. One can observe that the peak ratio in the differential transmission spectrum $r_{p}$ presents a linear dependence on the ratio of lengths $r_{L}$ for all considered values of $r_{N}$ . It is also verified that larger values of $r_{N}$ and shorter dephasing times lead to steeper linear dependences of $r_{p}$ on $r_{L}$ . For comparison, the experimental point obtained with the sample with the additional doped cap layer using dephasing time 400 fs is presented in Fig. 6(a).

Fig. 6 (a) r_P as a function of r_L for different values of r_N and dephasing times. The experimental point refers to Ref [24]. (b) The hatched area denotes the region in which r_P ranges between 0.1 and 0.8 considering r_N = 1.

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In general, the dependence of the Fano signatures on $r_{L}$ and $r_{N}$ should be understood in terms of the competition between the phase change induced in the doped cap layer $Δ ϕ_{c a p}$ and MQW layers $Δ ϕ_{Q W}$ . Increasing values of $r_{L}$ and $r_{N}$ result in larger values of $Δ ϕ_{c a p}$ , which sharpens the contribution of the peak associated with the ponderomotive current. Conversely, lower values of $r_{L}$ and $r_{N}$ provide larger $Δ ϕ_{Q W}$ , which makes the main peak more pronounced. In this context, a case of interest refers to the situation that the carrier concentration in the cap layer is equal to the concentration in the quantum well, that is, $r_{N} = 1$ . The bottom edge (bottom solid line) of the hatched area in Fig. 6(b) denotes the minimum value of $r_{L}$ which provides $r_{p} = 0.1$ . Likewise, the top edge (top solid line) of the hatched area is associated with $r_{p} = 0.8$ . A linear behaviour with respect to the increase of dephasing time is observed for the edges of the hatched area. The corresponding range of $r_{L}$ increases with the dephasing time in agreement with the results of Fig. 6(a).

Numerical results presented here aim to provide guidance for a clear observation of the contribution of the ponderomotive current in the differential transmission spectrum within a regime of low-intensity incident field. For instance, considering the reference field shown in Fig. 4, one could expect a pronounced secondary (ponderomotive) peak but lower than the main one when the cap layer presents the same carrier concentration of the quantum well layers ( $r_{N} \approx 1$ ) and length between three and five times larger than the total length of the quantum wells layers ( $3 \leq r_{L} \leq 5$ ).

When considering generation of carriers by means of optical pumping, the relation of carrier concentrations between MQW and the cap layer depends considerably on the reference field. As a consequence, the parameters of the Fano signature might vary, as can be seen in the experimental results [23,24] obtained from a sample with $r_{L} \approx 0.6$ , whose value of $r_{p}$ varies from 0.2 to 0.6, approximately, for three different reference fields.

Theoretical results obtained with the presented model are in full concordance with the experimental verification that only samples with an additional cap layer present a significant Fano signature. The role of the cap layer in this process is to provide a change of phase during the propagation of the beam through this layer which combined with the one of the QW layers results in the destructive interference characteristic of the Fano signature. This description is supported by the fact that if we keep all structural parameters fixed and change only the length of the cap layer a considerable variation of the linewidth and Fano factor is observed. Within the formalism of the transfer matrix, the length of the cap layer affects uniquely the change of phase provided by this layer.

4. Conclusions

In this paper, we have presented an in-depth discussion on the Fano signatures observed in the differential terahertz transmission spectra of multiquantum well structures. These signatures stem from the optical interference related to the sequential change of phase of the transmitted field when passing through the structure and provide evidence of ponderomotive currents already at weak electromagnetic fields. In fact, ponderomotive current in bulk-like regions and intersubband polarization in QW layers cause different changes in the refractive indexes which lead to distinct changes of phase throughout the structure. The model described here, based on density matrix formalism and transfer matrix method, is used to predict quantitavely the weight of the ponderomotive and intersubband contributions in the composition of the differential transmission spectra. It is shown that one can tune the characteristics of the Fano signatures by setting the length and carrier concentration of the cap layer and MQW layers, counting on an appropriate compromise between them. It is verified that thicker cap layers and larger carrier concentrations result in a more pronounced peak of the Fano signature rising from the contribution of the ponderomotive current. Likewise, the main peak related to the effect of the intersubband polarization becomes higher and narrower when considering larger carrier concentrations in the QW layers, keeping the same dephasing time. The increase of the number of quantum wells in the structure also results in a more pronounced main peak, which can be explained by the increase in the contribution of the intersubband polarization in the transmission spectrum.

At last, since the phase change of the transmitted field depends on the length, the incidence angle, and on the refractive index of the layers, two distinct samples with the same intersubband transition energy and carrier concentration in the QWs do not necessarily present the same parameters for the Fano signature. In this context, the presented theory predicts the resultant spectrum for different band edge profiles, allowing the engineering of the frequency dependence of the differential phase and, as a consequence, a further control of the Fano signature.

Funding

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) (program “Science without Borders”, No. 88881.132666/2016-01); Bundesministerium für Bildung und Forschung (BMBF) (No. 05K14CRA).

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Fano signatures between intersubband and ponderomotive responses in MQW structures

Abstract

1. Introduction

2. Theoretical approach

3. Results

4. Conclusions

Funding

References

Cited By

Figures (6)

Equations (14)

Optics Express