Theory and modeling of electrically tunable metamaterial devices using inter-subband transitions in semiconductor quantum wells

Alon Gabbay; Igal Brener

doi:10.1364/OE.20.006584

1. Introduction

Metamaterials (MM) provide for new ways of manipulating light and achieving complex functionality due to the ability to control the spatial distribution of the permittivity and permeability. Examples of this are optical cloaks and other exotic transformation optic devices that have been published recently [1,2]. In particular, planar metamaterials are attractive since a single layer can be used to engineer complex transmission or reflection spectra which previously required optically thick multilayer stacks. These planar metamaterials are also very sensitive to small changes in the local permittivity of the constituent materials, and this has been used to create a number of tunable optical devices at wavelengths where they are currently absent.

Several mechanisms for tuning the metamaterials spectral response have been attempted with various degrees of success. For example, amplitude and phase modulators at THz frequencies were created using metallic MM resonators tuned by depletion of carriers in doped GaAs layers [3–5]. At higher frequencies, (mid-IR) frequency tuning of metamaterial response was achieved by using InSb epilayers with different doping levels [6]. In the near-IR to visible range, frequency tunability was demonstrated by thermally/electrically induced insulator-to-metal phase transition in vanadium dioxide (VO₂) [7] or optical pumping of quantum dots [8]. Of all these tuning mechanisms, only a few lead to electrical control (a modality highly desired for real optical devices) and all of them have a limited range of wavelength operation.

Recently we and others showed that planar metamaterials can interact with intersubband transitions (ISTs) in semiconductor quantum wells (QW) [9,10]; aside from the fundamental interest in this type of phenomena, this interaction could form the basis for a new family of tunable optical devices. Moreover, there is a vast body of work based on ISTs that could be potentially used to manipulate the substrate dielectric function and thus affect the interaction with MM resonators leading to exciting new devices over a wide range of wavelengths. Examples are electrical and optical tunability, second harmonic generation [11–13] and quantum interference [14], to name a few.

In this paper we present a general scheme for electrically tunable MM devices using ISTs in QWs as the substrate on which MM resonators are fabricated. We first develop the formalism for calculating the bias dependent dielectric tensor caused by ISTs in semiconductor heterostructures. We then use numerical modeling to show an example of electrical tuning of the metamaterial response, caused by Stark shifting of the ISTs energies, and subsequent changes in the coupling between MMs and ISTs. Finally we provide an analytic formula predicting the coupling constant as a function of the number of QWs.

2. Intersubband transitions in quantum wells and the dielectric tensor

Inter-subband transitions have been extensively studied in past years. Good reviews can be found in [15–17]. Here we shall supply a brief review of these transitions and discuss their optical properties relevant to this work.

Figure 1(a) shows a schematic diagram of a semiconductor QW. A thin semiconductor layer with a lower bandgap is sandwiched between two layers of a semiconductor having a larger bandgap energy. The resulting band diagram along the growth direction is shown by the black solid lines and corresponds to the minimum (maximum) of the conduction (valence) band. The “potential well” along the growth direction (defined as the z axis) leads to discretization of energy levels both in the conduction and valence bands. These discrete energy levels are referred to as subbands. In this paper we will concentrate on the conduction subbands and focus solely on optical transitions caused by electrons between these subbands.

Fig. 1 (a) A lower bandgap material sandwiched between two higher bandgap materials. The solid lines are the minimum (maximum) of the conduction (valence) band. (b) A schematic diagram of the inplane dispersion of the confined electronic conduction band levels

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In the direction perpendicular to the growth direction (usually referred as the “in-plane” direction) the translational symmetry remains unchanged and electrons still exhibit a quadratic dispersion which can be used to calculate an effective mass, as it is done in the 3D case. Figure 1(b) shows a schematic diagram of the inplane dispersion of the electrons in the first two conduction subbands of the QW. As electrons are introduced into the QW via doping (modulation doping, _$δ$-doping etc…), they occupy the inplane states of the subbands. For a high enough density, the electrons can be treated as free particles and as a non-interacting 2D electron gas (2DEG), at least to first order [18].

2.1 The interaction Hamiltonian of ISTs

The interaction Hamiltonian of an electron with an electromagnetic field, in the semi-classical approximation, is proportional to:

H_{I} = \frac{e}{2} (A \cdot \overset{\leftrightarrow}{w} \cdot \hat{p} + \hat{p} \cdot \overset{\leftrightarrow}{w} \cdot A)

where A is the vector potential,

\hat{p}

is the momentum operator and

\overset{\leftrightarrow}{w}

is inverse effective mass tensor [15,19]. The only contributing terms in Eq. (1.1) to ISTs are those which include the operator

{\hat{p}}_{z}

[16]. Using the dipole approximation and taking into consideration that

\overset{\leftrightarrow}{w}

is symmetric, Eq. (1.1) can be expressed as:

H_{I} =e (w_{zx} A_{x} {+w}_{zy} A_{y} {+w}_{zz} A_{z}) {\hat{p}}_{z}

Thus it can be seen that if the effective mass tensor principal axes do not coincide with the crystal growth direction principal axes, there are additional contributions to the ISTs optical response and in-plane fields (Ex, Ey

\neq 0

) can induce ISTs.

Equation (1.2) can be rewritten as:

H_{I} =e (\frac{w_{zx}}{w_{zz}} E_{x} + \frac{w_{zy}}{w_{zz}} E_{y} {+E}_{z}) \hat{z}

where

\hat{z}

is the position operator and E is electric field. Thus, the contribution to the optical dipole of the ISTs due to off-diagonal terms in the effective mass tensor is the addition of two terms weighted by a mass ratio according to Eq. (1.3). For example, for a GaAs QW grown in the [0,0,1] direction,

w_{zx} = w_{zy} = 0

, so Eq. (1.3) will contain only one term,

H_{I} {=eE}_{z} \hat{z}

.

Equation (1.3) can be rewritten in terms of an effective dipole moment tensor:

{(H_{I})}_{nm} = - d_{nm} \cdot \frac{1}{w_{zz}} \overset{\leftrightarrow}{w} \cdot E, d_{nm} = - e (0, 0, z_{nm})

2.2 Optical susceptibility of intersubband transitions

In order to simulate the electromagnetic response of the combined MM-IST sample to an incident optical field, we will use a finite difference time domain (FDTD) numerical simulator. In order to calculate the optical response using FDTD we need to include the frequency dependent permittivity tensor of the substrate, which we obtain by first calculating the optical susceptibility of the relevant intersubband transitions.

The linear polarization induced by the electric field (at thermal equilibrium) is approximated by [15,20,21]:

P_{z} = \frac{1}{{Vε}_{0} h} \sum_{M,n,m,k} d_{mn} \cdot \overset{\leftrightarrow}{w} {(H_{I})}_{nm} (f_{m,k} - f_{n,k}) \frac{e^{- iωt}}{ω - ω_{mn} + {iγ}_{mn} / 2}

where e is the electric charge,

ε_{0}

is the vacuum permittivity,

γ_{nm}

is the damping rate,

f_{n,k}

is the Fermi-Dirac distribution function for subband n and state k, V is the volume (QW thickness times the sample surface),

z_{nm} = 〈 n | z | m 〉

is the position matrix element and M is the number of degenerate equal energy ellipsoids in k-space.

Equation (1.5) can be further simplified. If we consider a transition energy which is larger than the average thermal energy $ℏ ω_{ji} > > K_{B} T$ , (long IR and shorter wavelengths) then we can assume that all the electrons are in the first subband. Also, if we restrict this study to semiconductors where the non-parabolicity of the conduction band is negligible, then all the subbands will have the same dispersion and therefore we can treat all vertical transitions as having the same energy. Thus, under the assumption that all electrons are in the first subband, it does not matter which k-state they occupy within that band. These assumptions lead to the following expression for the linear polarization:

P_{z} = - \frac{Ne}{M_{C} ε_{0} ℏ} \sum_{M,m} \sum_{j} \frac{w_{zj}}{w_{zz}} \frac{z_{m1} {(H_{I})}_{1m}}{(ω - ω_{m1} + {iγ}_{m1} / 2)}

where N is volume electron density in the QW and

M_{C}

is the number of ellipsoids under consideration.

The linear susceptibility is defined by $P_{z} = \sum_{j=x,y,z} χ_{zj} E_{j}$ [21] and can therefore be expressed as:

χ_{zj} (ω) = - \frac{{Ne}^{2}}{M_{C} ε_{0} ℏ} \sum_{M,m} \sum_{v=x,y,z} (\frac{w_{zj} w_{zv}}{w_{zz}^{2}}) \frac{{| z_{m1} |}^{2}}{(ω - ω_{m1} + {iγ}_{m1} / 2)}

The next step is to construct the effective dielectric function tensor. Using the fact that

ε_{ij} (ω) = ε_{ij}^{b} (ω) + χ_{ij} (ω)

, where

ε_{ij}^{b} (ω)

is the background dielectric tensor [22]:

ε (ω) = {\overset{\leftrightarrow}{ε}}_{b} (ω) + (\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ χ_{zx} (ω) & χ_{zy} (ω) & χ_{zz} (ω) \end{matrix})

A good reference to obtain the inverse effective mass tensor components for different crystal orientations, for general

Γ

, X, and L – valley semiconductors, is [17]. In the following we will provide 2 examples for the effective dielectric tensor considering only the two lowest subbands. For example, the permittivity tensor for a GaAs QW grown in the [0,0,1] direction is:

ε_{GaAs}^{QW} (ω) = (\begin{matrix} ε_{GaAs}^{b} (ω) & 0 & 0 \\ 0 & ε_{GaAs}^{b} (ω) & 0 \\ 0 & 0 & ε_{GaAs}^{b} (ω) \end{matrix}) + (\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}) χ_{0} (ω)

where

χ_{0} (ω) = - \frac{{Ne}^{2}}{ε_{0} ℏ} \frac{{| z_{21} |}^{2}}{(ω - ω_{21} + {iγ}_{21} / 2)}

It is known that IST transitions in GaAs-based QWs require in-plane light propagation selection rules. These selection rules become evident when examining Eq. (1.9). Only a field component in the z direction can induce an optical transition due to ISTs. There are many derivations of the dielectric tensor in semiconductor heterostructures (see for example [23,24]), but our final expressions account fully for anisotropy and can be readily used in electromagnetic modeling software.

A second example of a dielectric tensor in semiconductor heterostructures is the tensor for a GaSb QW grown in the [0,0,1] direction. The QW is narrow enough, so that the lowest state in the conduction band is in the L-valley. Thus, the principal axes of the inverse mass tensor do not coincide with the crystal principal axes [25].

ε_{GaSb}^{QW(L<4nm)} = (\begin{matrix} ε_{GaSb}^{b} (ω) & 0 & 0 \\ 0 & ε_{GaSb}^{b} (ω) & 0 \\ 0 & 0 & ε_{GaSb}^{b} (ω) \end{matrix}) + (\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0.2 & 0.2 & 1 \end{matrix}) χ_{0} (ω)

In Eqs. (1.9) and (1.11)

χ_{0}

is defined in Eq. (1.10) and effective mass values are quoted from [17]. The additional components in the ISTs susceptibility tensor (right matrix in Eq. (1.11)) enable normal incidence excitation of the IST (“breaking” of the in-plane selection rules). This effect has been observed by several groups and is highly advantageous for normal incidence infrared detectors. Although the additional non-diagonal matrix elements are smaller than

χ_{zz}

, the interaction strength with planar metamaterials is also determined by the electric field amplitude at the location of the QW; therefore, a strong electric field can give rise to a large interaction strength.

3. Planar metamaterials

Metallic MM resonators have been extensively studied in recent years [26,27]. These subwavelength metallic elements exhibit a resonant response to electromagnetic radiation at certain frequencies, depending on their geometry, constituent materials and scale. One of the advantages of planar metamaterials in general is that by fabricating a single sheet of resonator arrays on a substrate, we obtain effectively an optical cavity effect, i.e. a selective enhancement of light in a certain region in space.

Another important consideration for electrically tunable devices is that all the resonators should be electrically interconnected. There is a large library of MM resonator geometries that could be used for this type of work and Fig. 2(a) shows one particular example that we use in this paper. These resonators have been successfully used in previous tunable MM work and their symmetry cancels out any magnetic response: they consist of two single gap split-ring resonators (SRR), which are mirror images of each other, and located side by side. The bottom part of the resonator acts as a bus line. In the simulation we chose Au as the metamaterial metal. The complex dielectric function of evaporated Au was measured using an IR variable angle spectral ellipsometer [28]. Figure 2(b) shows the transmission spectra of this element on top of a GaAs substrate for both linear polarizations of the incident beam (E_x and E_y), calculated using a commercial FDTD simulator [29] and the following parameters: L = 570 nm, G = 86 nm, W = 103.5 nm and P = 1725 nm. The dip in the transmission spectra for the E_y polarization corresponds to the frequency of the lowest mode of this resonator. Figures 2(c)–2(h) were calculated for the frequency denoted by the arrow in Fig. 2(b) (33.1 THz) and for the E_y incident polarization. Figures 2(c)–2(e) show the amplitude distribution of the different electric field components, E_x, E_y and E_z, respectively. The amplitudes are relative to the incident field E₀, and they are shown in a plane located 15nm away from the MM resonator into the sample. It can be seen that 15nm into the substrate, the maximum enhancement of E_z and E_y is about 5 times the incident field, although the amplitude of E_z is distributed on a much larger area than that of E_y. It is interesting to compare the E_z field distribution between the SRR geometry considered here and the geometry considered in [9], in which the E_z field is mainly concentrated at the edge of the SRR (near the gap) and its maximal value is lower by a factor of 2 than the case considered here (Fig. 2(e)). The maximum enhancement of E_x is about 3 times the incident field. The amplitudes of the different field components as a function of z are shown in Fig. 2(f)–2(h). These plots correspond to a single point in the plane of the MM resonator and are denoted by a + in Fig. 2(f)–2(h). According to these plots the decay of the fields into the substrate is approximately exponential. When trying to maximize the interaction between optical dipoles (such as the ones located in QWs or quantum dots (QDs)) and MM resonators, one needs to analyze the fields resulting from the MM resonator along with the selection rules of the optical dipole.

Fig. 2 (a) Schematic diagram of the split-ring resonator (SRR) used with the dimension definitions. (b) Calculated transmission spectrum of the SRR in (a) on top of a GaAs substrate designed to resonate at 33 THz, for both linear polarizations. The arrow indicates the frequency at which plots (c-h) were calculated. (c-e) The enhanced field amplitude (relative to the incident field E₀, of the three components E_x, E_y and E_z, respectively. The plots correspond to a plane located 15nm below the interface. (f-h) The dependence of the field components on z (the growth direction). (f-h) were drawn along a single line passing through the points denoted in (c-e) by a + sign.

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4. Simulation and results

4.1 Sample parameters

In the following we will consider the specific configuration shown in Fig. 3(a) . Two asymmetric coupled GaAs QWs, 5.7nm and 2.5nm wide, are separated by a 1nm layer of Al_0.5Ga_0.5As and sandwiched between two Al_0.5Ga_0.5As layers. The energy levels of this coupled well design show a strong dependence on an applied electrical bias compared to a single QW [12,15] and thus provide a good platform for controlling the MM behavior with voltage. This figure also shows the first three subbands and their wavefunctions (modulus).

Fig. 3 (a) The potential (solid black lines) of a two asymmetric coupled GaAs QWs with Al_0.5Ga_0.5As barriers. (b) The potential under a bias of −75 kV/cm. (c) The dependence of three lowest ISTs on the applied bias.

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The potential energies in Fig. 3(a), 3(b) were calculated using a self-consistent calculation of the Schrödinger and Poisson equations [30]. A uniform distribution of dopants was assumed in the wider well. The 2D doping density was N = 8x10¹¹ cm⁻²; _{$γ_{21}$} was 3 THz (12 meV) [12].

Figure 3(b) shows the calculated energies and wavefunctions under a bias of −75 kV/cm. Figure 3(c) shows the dependence of the IST energies on the applied electrical field across the sample. For example, across a 1 $μm$ thick sample, one would need to apply 7.5V to obtain the potential and wavefunctions shown in Fig. 3(b). As can be seen in Fig. 3(c), E2-E1 (and E3-E2) show a strong dependence on bias and thus can be used as a “voltage tunable dielectric” to tune and manipulate the interaction with the MM resonators.

4.2 FDTD simulations

We simulate the electromagnetic response of the coupled MM-IST system using a commercial FDTD simulator [29]. The dielectric tensor for a single QW unit cell was obtained using Eqs. (1.9) and (1.10) for several applied bias voltages. For the simulation, the QWs were modeled as 10nm layers, each with the corresponding calculated dielectric tensor. The QWs were separated by 20nm of Al_0.5Ga_0.5As. The structure was sandwiched between a GaAs substrate and a 30nm GaAs cap layer (this is all shown schematically in Fig. 4 ). The double gap SRR was placed on top of the cap layer, and the following dimensions were used: L = 570 nm, G = 86 nm, W = 103.5 nm and P = 1725 nm (according to the notations followed in Fig. 2(a)). A bus line of the same line thickness is included too and it is important for biasing the entire region underneath all the MM resonators. With these geometrical parameters the lowest-frequency electromagnetic resonance of the MM is at 33.1 THz, which is close to the zero bias IST transition E1-E2 frequency.

Fig. 4 A schematic diagram of the layer sequence used for the FDTD calculation. The first (lowest) layer is a GaAs substrate. The following layers are a repetition of a unit cell, which is composed of a 10nm layer (associated with the calculated ISTs susceptibility) sandwiched between two 20nm Al_0.5Ga_0.5As layers. The upper layer is a 25nm cap GaAs layer. The top most structure is a gold SRR.

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4.3 Results (N_QW = 25)

An example of the in-plane component of the permittivity tensor for the coupled QW system is shown in Fig. 5(a) , for a bias of 23.1 kV/cm. We then vary the bias voltage and calculate the transmission spectra; a sequence of spectra are shown in Fig. 5(b), where we used 25 coupled-QW layers. For a bias voltage of 23.1 kV/cm, the IST and the MM resonances coincide and the spectra shows two dips. For clarity, a single transmission spectrum for this bias is shown in Fig. 5(c) for both exciting polarizations (as defined in Fig. 2(a)). Also shown is the transmission spectrum of a reference sample (dashed line) in which the coupled-QWs are undoped and therefore do not contribute to the susceptibility. The transmission for the reference sample is denoted by $T_{ref}$ .

Fig. 5 (a) The calculated susceptibility of the IST (2→1) for an applied bias of 23.1 kV/cm used for the FDTD calculation. (b) The calculated transmission spectra for several applied biases. The red curve shows the resonant condition where the IST energy coincides with the SRR resonance. (c) The calculated transmission spectra at resonance (red curve in (b)) of both polarizations (solid curves). The dashed curve is the calculated transmission specra for the same structure where the QW layers were replaced by GaAs layers. (d) The energy minima of each transmission spectrum (Ey polarization) as a function of the applied bias. Anti-crossing behavior is clearly observed, indicating strong coupling between the MM resonator and the IST.

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When the incident polarization is in the x direction, the spectra for the coupled-QW sample and bulk GaAs are almost identical. However, when the incident beam is polarized in the y-direction, the transmission spectra from these 2 sample cases are very different. The coupled-QW sample spectrum exhibits 2 dips which are displaced in energy from the GaAs sample spectrum dip. The minimum energies of the dips from all the coupled-QWs sample spectra (E_y polarization) are plotted in Fig. 5(d) and as a function of the applied bias (blue circles).

The energy minima exhibit an anticrossing behavior, indicative of the coupling between the MM resonator and the IST in the QW. The coupling between the IST and the MM resonator can be described by a simple coupled oscillators model (COM) using a 2x2 matrix:

(\begin{matrix} ν_{MM} - {iγ}_{MM} & Ω / 2 \\ Ω^{*} / 2 & ν_{IST} - {iγ}_{IST} \end{matrix})

where

ν_{MM}

(

ν_{IST}

) is the MM (IST) resonant frequency,

γ_{MM}

(

γ_{IST}

) is the damping term for each element, and

Ω / 2

is the coupling strength. Diagonalization of Eq. (1.12) gives the coupled system eigenvalues.

ν_{\pm} = \frac{ν_{MM} + ν_{IST} + i (γ_{MM} + γ_{IST})}{2} \pm \sqrt{\frac{1}{4} Ω^{2} + {[ν_{MM} - ν_{IST} + i (γ_{MM} - γ_{IST})]}^{2}}

Here, we used $ν_{MM} = 32.5$ THz, $ν_{IST} {=ν}_{21}$ (shown in Fig. 3(c)), $γ_{MM}$ and $γ_{21}$ were extracted from Fig. 5(a), 5(c) (4.5 THz and 3 THz, respectively). By fitting Eq. (1.13) to the modeled data we find $Ω = 2$ THz. The results of the COM are shown in Fig. 5(d) (black solid lines).

For modulator devices, an important parameter to study is the modulation depth as a result of the applied bias. We define the degree of amplitude modulation as:

M (ν) = \frac{T_{QW} (ν) - T_{ref} (ν)}{T_{ref} (ν)}

where

T_{QW} (ν)

(

T_{ref} (ν)

) is the transmission through the QW (reference) sample for a certain frequency,

ν

. Figure 6(a) shows

M (ν)

as a function of the frequency,

ν

, for different bias voltages. The spectra are offset in the y- direction, so the y-axis labels are correct only for the first spectrum (−72.1 kV/cm). Figure 6(b) shows

M (ν)

for

ν = 32.7 T H z

(9.2

μm

) as a function of the applied electric field. It can be seen that the peak modulation (45%) occurs at a bias of 23.1 kV/cm, which is expected as the largest level splitting occurs at the same bias (Fig. 5(c)).

Fig. 6 (a) The degree of amplitude modulation, defined in Eq. (1.14), for several applied biases. The spectra are offset from each other in the y-axis direction and therefore the labels refer only to the first spectrum. (b) The modulation degree at a certain frequency (32.7 THz) as function of the applied biases. The data corresponds to the dashed red line in (a).

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4.4 Level splitting as a function of N_QW

It is interesting to calculate what the minimal number of QWs should be, in order to achieve the largest level splitting and thus modulation depth. It is well known that for QWs embedded in microcavity structures, the interaction strength depends on the square root of the number of QWs, where it is assumed that the QWs are interacting with the same field amplitude [31,32].

In the case studied here, the field is decaying as we move away from the SRR (Fig. 2). We expect the level splitting to show a square root dependence for a small number of QWs and finally saturate due to the decay of the field strength; QWs that are too far from the surface will not interact at all with the metamaterial resonators. In reality, the situation is more complex since the field distribution is not uniform (Fig. 2(e)).

Figure 7(a) shows the calculated spectra at resonance and as a function of the number of QWs. We see that there is very little change in the spectra for $N_{QW} > 8$ . Figure 7(b) shows the level splitting (black circles) as a function of the number of QWs, extracted from the spectra of Fig. 7(a) and using a two Lorentzian fit. The dependence of the level splitting on the number of QWs follows our intuitive description.

Fig. 7 (a) calculated transmission spectra when a different number of QWs were used in the FDTD simulation. (b) The spectra in (a) were fitted by two Lorentzians. The peak frequencies of the Lorentzians are shown by the black circles and indicate the level splitting. The solid blue curve is a plot of Eq. (1.15) when using A = 1 and $α_{avg} ΔZ = 0.1$ .

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For a more mathematical approach we follow [31], and calculate an effective number of QWs, due to the variation of the field amplitude in different QW locations. The derivation appears in the appendix. There we will show that the dependence of the coupling strength on the number of QWs is given by the following equation:

Ω = A \sqrt{\frac{1 - e^{- {2α}_{avg} {ΔZN}_{QW}}}{1 - e^{- 2 α_{avg} ΔZ}}}

where

α_{avg}

is the average absorption coefficient, A is a constant (defined in the appendix) which reflects the averaged field-QW interaction, and

ΔZ

is the distance between QWs.

Figure 7(b) shows a plot of Eq. (1.15) as a function of the number of QWs (solid line). The values $A = 1$ and $α_{avg} ΔZ \equiv 0.1$ were extracted from the fit. The good fit of Eq. (1.15) to the results obtained from the FDTD calculations suggests that the derivation presented in the appendix is correct in principle, but requires a more detailed analysis for quantitative agreement.

5. Discussion and conclusions

Summarizing, we presented in this paper a theoretical study of the interaction between metamaterial (MM) resonators and intersubband transitions (ISTs) in quantum wells (QWs) and its dependence on an applied electrical bias. Understanding the interaction of MM resonators with these types of engineered optical transitions is interesting scientifically and important from a practical point of view since this could lead to a new way of tuning the spectral response of planar MMs. For most semiconductor QWs, the effective mass principal axes do not coincide with the QW principal axes, and this leads to a non-diagonal effective dielectric function in the form of a second order rank tensor (Eqs. (1.7) and (1.8)). Even when direct optical excitation of ISTs is forbidden by selection rules, as in the case of GaAs QWs, the presence of MM resonators provides a coupling mechanism to ISTs that can be exploited for active tuning, if the ISTs can be designed to vary with an applied bias. However, these selection rules impose a limit to the maximum coupling strength since only one of the three electric field components contributes to this interaction. Strategies for increasing the oscillator strength using different semiconductors could be exploited to maximize this interaction strength even under the limitation of these in-plane selection rules (for example, using semiconductors with different effective masses for electrons). Despite these limitations, using GaAs QWs we calculated amplitude modulation depths of 45% when using the definition of Eq. (1.14).

Another option to increase the interaction strength between planar MM’s and ISTs and thus tunability is to find a semiconductor combination that breaks the usual in-plane selection rules that appear in most III-V QWs. As was discussed in section 2.2 and shown in Eq. (1.11), thin QWs of GaSb allow for normal incidence excitation of ISTs [33–36]. If we compare the two cases, QWs of GaAs and GaSb, and assume dipole moments of similar orders of magnitude, then we can see the benefits of using GaSb QWs (or any other L-valley semiconductor): there are two extra contributions to the interaction with the MM (Eq. (1.11)). Furthermore, since the strongest field component of most MM resonators lies in the plane of the QWs, the contribution to the interaction can be enhanced significantly.

Finally, the interaction strength between MM resonators and ISTs depends on the number of QWs and the distance between the QWs and the metallic resonators similar to the case of coupling between MMs and phonons [28]. For a constant electric field, this interaction grows as the square root of the number of QWs. We derived an analytic formula that provides the dependence of the level splitting on the number of QWs. This dependence saturates (Fig. 7(b)) and thus shows that there is an optimal number of QWs or more generally that there is a limit to the distance between the MM and the active layer (QW) beyond which QWs do not contribute to this interaction. This issue needs to be balanced with practical device requirements that tend to distance the QWs from the surface, such as cap layers or semiconductor barriers that need to be included when top electrodes will be used for biasing the MM-QW combined samples.

Appendix: level splitting as a function of the number of QWs

Following [31], we calculate an effective number of QWs, due to the variation of the field amplitude in different QW locations; the level splitting is then given by $Ω_{0} \sqrt{N_{eff}}$ . The aim is to express the dipoles from all QWs in a new basis state, the normalization of which will give the effective number of QWs. We use the same notations as [31], but the specific basis we choose is different.

The z- component of the electric field inside the sample is taken to be (this is a general behavior observed in most plasmonic structures, provided that there is no metallic backplane):

E_{z} (z) = E_{0} e^{- αz}

_{$d_{i}$} is the individual dipole moment of each QW. These dipole moments span a vector space. This vector space can be spanned using a different basis which is defined as:

d^{(μ)} = \sum_{i = 1}^{N_{QW}} C_{i}^{(μ)} d_{i}

where $μ = 1... N$

C_{i}^{(1)} = C_{0} e^{- {αZ}_{i}}, C_{0} = {[\sum_{i}^{N_{QW}} e^{- {2αZ}_{i}}]}^{- \frac{1}{2}}

where $α$ is the decay coefficient of the field and $Z_{i}$ is coordinate of the i^th QW. The other N-component vectors, $C^{(μ)}$ , are by definition, orthogonal to $C^{(1)}$ :

\sum_{i}^{} C_{i}^{(μ) *} C_{0} e^{- {αZ}_{i}} = δ_{μ,1}

The coupling constant between the Electric field and the dipole is proportional to:

d^{(μ)} E = \sum_{i = 1}^{N_{QW}} C_{i}^{(μ)} d_{i}^{} E_{0} e^{- {αZ}_{i}} = {dE}_{0} \sum_{i = 1}^{N_{QW}} C_{i}^{(μ)} e^{- {αZ}_{i}} = \frac{{dE}_{0}}{C_{0}} δ_{μ,1}

So only $d^{(1)}$ interacts with the electric field. The coupling constant is proportional to $C_{0}^{- 1}$ .

Assuming equal spacing, $Δ Z$ , between the QWs:

C_{0}^{- 2} = (\sum_{i = 1}^{N_{QW}} {| e^{- {αz}_{1}} |}^{2}) = e^{- 2 {αz}_{1}} \sum_{i = 0}^{N_{QW} - 1} e^{- 2 αΔZ \cdot i} = e^{- 2 {αz}_{1}} \frac{1 - e^{- 2 {αΔZN}_{QW}}}{1 - e^{- 2 αΔZ}}

The effective number of QWs equals the square root of $C_{0}^{- 2}$ :

\sqrt{N_{eff}} = e^{- {αz}_{1}} \sqrt{\frac{1 - e^{- 2 {αΔZN}_{QW}}}{1 - e^{- 2 αΔZ}}}

As stated above, the actual situation is more complex since the field distribution is not uniform, and the QWs have a finite thickness. Thus a more accurate theory should incorporate an averaging algorithm of the fields at all locations, but this is out of the scope of this paper.

As a final step of our simplified model, we can assume a level splitting of the following form:

ν_{+} - ν_{-} = A \sqrt{\frac{1 - e^{- 2 α_{avg} {ΔZN}_{QW}}}{1 - e^{- 2 α_{avg} ΔZ}}}

where A is a constant which depends on the average QW-field interaction discussed in the last paragraph.

Acknowledgments

Parts of the simulation work were funded by DARPA/MTO's CEE program under DOE/NNSA Contract DE-AC52-06NA25396. This work was performed, in part, at the Center for Integrated Nanotechnologies, a U.S. Department of Energy, Office of Basic Energy Sciences user facility. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

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Theory and modeling of electrically tunable metamaterial devices using inter-subband transitions in semiconductor quantum wells

Abstract

1. Introduction

2. Intersubband transitions in quantum wells and the dielectric tensor

2.1 The interaction Hamiltonian of ISTs

2.2 Optical susceptibility of intersubband transitions

3. Planar metamaterials

4. Simulation and results

4.1 Sample parameters

4.2 FDTD simulations

4.3 Results (N_QW = 25)

4.4 Level splitting as a function of N_QW

5. Discussion and conclusions

Appendix: level splitting as a function of the number of QWs

Acknowledgments

References and links

Cited By

Figures (7)

Equations (23)

Optics Express

Abstract

1. Introduction

2. Intersubband transitions in quantum wells and the dielectric tensor

2.1 The interaction Hamiltonian of ISTs

2.2 Optical susceptibility of intersubband transitions

3. Planar metamaterials

4. Simulation and results

4.1 Sample parameters

4.2 FDTD simulations

4.3 Results (NQW = 25)

4.4 Level splitting as a function of NQW

5. Discussion and conclusions

Appendix: level splitting as a function of the number of QWs

Acknowledgments

References and links

Cited By

Figures (7)

Equations (23)

Optics Express

4.3 Results (N_QW = 25)

4.4 Level splitting as a function of N_QW