The ratio of the kinetic inductance to the geometric inductance: a key parameter for the frequency tuning of the THz semiconductor split-ring resonator

Jiawei Cong; Binfeng Yun; Yiping Cui

doi:10.1364/OE.21.020363

1. Introduction

Electromagnetic metamaterials (MMs) are artificial media designed to offer a wide range of exotic phenomena, such as anomalous refractive index [1–5], optical beam self-collimation [6, 7], perfect absorption [8, 9] and sub-wavelength focusing [10–12] etc. In particular, MMs are a promising candidate to construct functional components at the THz regime (0.1-10 THz), where rare natural materials effectively respond to the THz radiation. For such applications, it is of great importance to be able to actively control resonant frequencies of MMs. Real-time frequency tuning can be accomplished by applying external stimuli to the reconfigurable elements which are incorporated into the MMs, such as phase-change materials [13, 14], superconductors [15, 16], semiconductors [17–23] and micromechanical components [24, 25]. And the stimulus can be temperature [13, 14, 21, 22], optical-pump [17–19], voltage control [13, 23] and so on.

Several strategies for frequency tuning have been proposed based on the semiconductor-incorporated MMs. Chen et al. [17] reported a scheme of optically controlled frequency agility due to the alteration of the geometric inductance in SRRs. And Shen et al. [18] proposed an improved design based on a photo-induced mode-switching effect. Recently, some all-semiconductor SRRs are proposed for frequency tuning by actively varying the carrier density in the terahertz regime [20–22]. While the tuning characteristics in such structures are numerically investigated, an analytical model of the relative tuning range has not been reported, making it difficult to optimize the structure for the larger tuning range. In this work, we have analytically investigated the resonance frequency shift with the carrier density in SRRs, which are made of III-V semiconductor compound indium antimonide (InSb). The frequency tuning sensitivity is introduced to measure the resonance tuning with the carrier density. And based on the LC circuit model, the explicit expression of the relative tuning range is derived. The analytical model reveals that the relative tuning range is closely related with the RKG, i.e., the ratio of kinetic inductance to the geometric inductance of SRRs. And a larger tuning range can be achieved by increasing the RKG. Based on the above results, a stacked SRR-dimer structure is proposed. Comparing with the single-layer SRR, the stacked SRR-dimer structure can exhibit larger RKG due to the inductive coupling, which effectively increases the relative tuning range. And the analytical results are well reproduced by the FDTD simulation. It is noteworthy that the plasmon resonance of many structures, not limited to the SRRs, can be explained in the framework of the equivalent LC circuit. So the analyzed tuning mechanism may also be extended to other tunable MMs with different topologies and find various applications in actively controlled optical devices, such as modulator, ultrafast switch and biosensors.

2. Structure design and simulation model

The two types of SRR structures studied in this article are presented in Fig. 1. One structure consists of a single SRR in a unit cell while the other consists of a stacked SRR-dimer with a twist angle φ = 0° or φ = 180°, as shown in Figs. 1(a) and 1(b), respectively. And the material of SRRs is InSb, which can be grown on the GaAs or Si substrate by using molecular beam epitaxy (MBE) process [19, 26]. We focus on InSb because its carrier density can be flexibly adjusted by the external stimuli including optical pump [19] and ambient temperature [20]. Moreover, InSb is suitable for sustaining strong localized plasmon in the THz region as a result of the low losses due to its high carrier mobility [19]. The incident THz wave propagates in the direction normal to the SRR plane (along z-axis), with the electric field parallel to the SRR gap (along x-axis). With this configuration, the fundamental mode of the plasmon resonance is excited. At this resonance, the SRR can be viewed as a series RLC circuit, so this mode is also referred as the LC resonance [27, 28]. In order to understand the underlying mechanism of the frequency shift with carrier densities, an analytical treatment is made from the perspective of an equivalent RLC circuit. According to the Kirchhoff voltage rule, we obtain

(L_{g} + L_{k}) \frac{d I}{d t} + R I + \frac{\int I d t}{C} = 0,

where

L_{g}

is the geometric inductance,

L_{k}

is the kinetic inductance, C is the capacitance and R is the resistance of the SRR. Assume that the current has the form of I = I₀exp(-βt-iωt), where β accounts for the damping loss. Then the resonance frequency is obtained from Eq. (1) as

f = \frac{1}{2 π} \sqrt{\frac{1}{(L_{g} + L_{k}) C} - \frac{R^{2}}{4 {(L_{g} + L_{k})}^{2}}} .

Different from the commonly employed LC model [27], where the resonance frequency solely depends on the inductance and capacitance of SRRs, the derived RLC model also takes into account the resistance of the semiconductor. For InSb with the low resistance (i.e.,

R^{2} / 4 {(L_{g} + L_{k})}^{2} ≪ 1 / (L_{g} + L_{k}) C

), the resistance has negligible influence on the resonance frequency. So in the subsequent analysis, the LC model (Eq. (3)) is employed for simplicity.

f = \frac{1}{2 π \sqrt{(L_{g} + L_{k}) C}} .

However, if SRRs are made of some highly resistive semiconductors (such as GaAs and Si), the effect of the resistance on the resonance frequency cannot be neglected. And the resistance is expected to reduce the resonance frequency relative to that calculated from the LC model. For the SRR shown in Fig. 1(a), the carrier kinetic inductance

L_{k}

, the geometric inductance

L_{g}

, the total capacitance C (the sum of the gap capacitance

C_{g}

and surface capacitance

C_{s}

) take the following form [28]:

L_{k} = \frac{2 π r - g}{ε_{0} w h ω_{p}^{2}} = \frac{(2 π r - g) m^{*}}{w h e^{2}} \cdot \frac{1}{N},

L_{g} = μ_{0} r (L o g (\frac{8 r}{w + h}) - 0.5),

C_{g} = \frac{ε_{0} w h}{g} + ε_{0} (w + h + g),

C_{s} = \frac{2 ε_{0}}{π} (w + h) L o g (\frac{4 r}{g}),

C = C_{g} + α C_{s},

R = \frac{γ}{ε_{0} ω_{p}^{2}} \cdot \frac{2 π r - g}{w h} .

Here

ε_{0}

is the vacuum permittivity,

ω_{p}^{}

is the plasma frequency and

γ

is the damping constant of the semiconductor. α is a correction factor to the surface capacitance, since Eq. (4d) delivers a smaller result than the true value for a thin SRR [28]. With regards to the SRR sizes under investigation,

α = 2.25

ensures good agreement between the analytical and simulation results.

Fig. 1 Schematic of an array of InSb metamaterial composed of (a) a single SRR and (b) two coaxially aligned SRRs in each unit cell. The geometric parameters are initially chosen as h = 2 μm, r = 15 μm, w = 4 μm, g = 6 μm, P_x = P_y = 54 μm and s = 3 μm.

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Also the finite difference time domain (FDTD) numerical simulation [29] is implemented to testify the analytical results. In the simulation, the periodic boundary condition along the x and y axis and the absorption boundary in the z direction are adopted. A uniform grid spacing is set as $Δ x = Δ y = Δ z = 0.02 μ m$ which is sufficient for the convergence of the FDTD results. The complex permittivity of InSb can be described by the Drude model: $ε (f) = ε_{\infty} - \frac{ω_{p}^{2}}{{(2 π f)}^{2} + i 2 π γ f}$ [19–21]. Here, $ε_{\infty}$ is the high-frequency permittivity and the plasma frequency $ω_{p}^{} = \sqrt{N e^{2} / ε_{0} m^{*}}$ depends on the carrier density (N), the electron charge (e), the vacuum permittivity (ε₀) and the carrier effective mass (m*). The values of these parameters are taken as $ε_{\infty} = 15.68$ , $γ = 2 π \times 0.05 THz$ , m* = 0.015m_e [20, 21, 30], where m_e is the electron’s rest mass. These parameters allow for a good fitting of the InSb permittivity around 1 THz when the carrier density is below N = 1 × 10¹⁷ cm⁻³ [31]. The substrate is taken as vacuum in the following analysis for simplicity without loss of generality, given that the material choice of these media would only slightly shift the resonances but not affect the tuning mechanism [22].

3. Results and discussion

We firstly explore the single-layer SRRs to capture the physics underlying the tuning effects and introduce the corresponding analytical model. Then we extend the analytical treatment to the case of stacked SRR-dimers. The resonance frequency of the single-layer SRRs undergoes a continuous blue shift from 0.28 THz to 0.97 THz when the carrier density N is increased from 5 × 10¹⁵ cm⁻³ to 1 × 10¹⁷ cm⁻³, as shown in Fig. 2(a). For N = 5 × 10¹⁵ cm⁻³ and 1 × 10¹⁶ cm⁻³, the higher order modes are also excited which will not be discussed in this work. Two features can be seen in Fig. 2. First, the transmission dip becomes pronounced with the increment of N, which indicates that the resonance strength is gradually enhanced. This is because that InSb transits from an insulator to a conductor when considerable free carriers are thermally or optically excited. And a more conductive SRR is able to induce the stronger LC resonance [32]. Second, the growth rate of resonance frequencies is gradually reduced with an increment of carrier densities, as shown in Fig. 2(b). The reason will be demonstrated below.

Fig. 2 (a) Transmission spectra of the single-layer InSb SRRs with different carrier density N. The transmission dips indicated by triangles represent the LC resonance mode. (b) The resonance frequency and tuning sensitivity as a function of N. The red and black solid lines show the analytical results and the blue dots show the FDTD simulation results.

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We can see from Eqs. (3) and (4) that only the kinetic inductance and the resistance vary with the carrier density. But the resistance of InSb SRR has negligible influence on the resonance frequency. So it is reasonable to ascribe the resonance shift with the carrier density to the variation of the carries kinetic inductance of SRRs. Based on the LC model, the increment of N will reduce $L_{k}$ , which in turn leads to the higher resonance frequency. An excellent agreement between the calculated resonance frequency from the LC model (red line) and the FDTD simulation results (blue dot line) verifies the validity of the LC model.

The resonance enhancement and blue shift with increasing carrier densities can also be explained from the perspective of the varying complex conductivity of InSb. It is necessary to understand the variation of the conductivity with N since it can be directly measured from the experiment [16] while the kinetic inductance cannot. From the experimental data of the conductivity, one can further obtain other important parameters such as the resistance and the kinetic inductance. The complex conductivity is described as $σ = i ε_{0} ω_{p}^{2} / (ω + i γ)$ [33]. Obviously, both the real part and the imaginary part of the conductivity are proportional to the carrier density N. And it is known that the resistance R and the kinetic inductance L_k are related with the complex conductivity by $R = Re (\frac{1}{σ}) \cdot \frac{2 π r - g}{w h}$ and $L_{k} = \frac{i}{ω} \cdot Im (\frac{1}{σ}) \cdot \frac{2 π r - g}{w h}$ , respectively [33]. So both the resistance and the kinetic inductance are inversely proportional to the carrier density. Therefore, as N increases, the resistance is reduced, which leads to the stronger resonance. Meanwhile, the reduction of the kinetic inductance is responsible for the resonance blue shift.

In order to evaluate the resonance frequency shift with the carrier density, the frequency tuning sensitivity S is introduced by us as:

S = \frac{\partial f}{\partial N} .

Based on Eq. (3) and Eq. (4), the tuning sensitivity is derived as

S = \frac{L_{k}}{4 π N {(L_{k} + L_{g})}^{3 / 2} (C_{g} + α C_{s})} = \frac{f}{2 (L_{k} + L_{g})} \frac{L_{k}}{N} .

Figure 2(b) shows that as N increases, the tuning sensitivity is reduced (black line), indicating that the kinetic inductance and hence the resonance frequency is less sensitive to the increasing carrier density. Therefore, the resonance frequency undergoes a slow growth with N and will finally saturate to the value

1 / (2 π \sqrt{L_{g} (C_{g} + α C_{s})})

when

L_{k}

is negligible compared to

L_{g}

.

The relative tuning range of the resonance frequency is an important factor of tunable metamaterials and widely used to measure the frequency tuning performance [15, 17, 18, 25]. We will now derive the relative tuning range based on the developed LC model. Assume the InSb SRRs are excited by external stimulus, e.g., optical pump or temperature control, such that the carrier density increases from N₁ to N₂. As a result, the resonance undergoes a blue shift Δf, which is related with the tuning sensitivity by $Δ f = \int_{N_{1}}^{N_{2}} S (N) d N$ . The relative tuning range, defined as Δf / f (N₁), is then derived as Eq. (7). The last term of Eq. (7) is obtained by converting $L_{k 1}$ to $L_{k 2}$ using the relationship $L_{k 1} = L_{k 2} N_{2} / N_{1}$ according to Eq. (3):

\frac{Δ f}{f (N_{1})} = \frac{\int_{N_{1}}^{N_{2}} S (N) d N}{f (N_{1})} = \frac{f (N_{2}) - f (N_{1})}{f (N_{1})} = \sqrt{\frac{(L_{g} + L_{k 1}) (C_{g} + α C_{s})}{(L_{g} + L_{k 2}) (C_{g} + α C_{s})}} - 1 = \frac{\sqrt{1 + \frac{N_{2}}{N_{1}} \frac{L_{k 2}}{L_{g}}}}{\sqrt{1 + \frac{L_{k 2}}{L_{g}}}} - 1.

It indicates that for a fixed carrier density ratio $N_{2} / N_{1}$ , the relative tuning range is solely related with the RKG at the carrier density N₂, i.e., $L_{k 2} / L_{g}$ . As $N_{2} / N_{1}$ is larger than unity, the relative tuning range is monotonically increased with RKG. A larger RKG $(L_{k 2} / L_{g})$ contributes to a larger relative tuning range. In the limit case where the magnitude of kinetic inductance is far beyond the geometric inductance, i.e., $L_{k 2} / L_{g} > > 1$ , the relative tuning range approaches $\sqrt{N_{2} / N_{1}} - 1$ . Otherwise if $L_{k 2} / L_{g} < < 1$ , the relative tuning range becomes very limited. Therefore, for a given variation of carrier densities, it is necessary to investigate how to increase RKG so as to improve the relative tuning range. According to the LC model, RKG is dependent on the kinetic inductance and the geometric inductance, which are both closely related to the geometric parameters of SRRs (see Eqs. (4a) and (4b)). So it is feasible to increase RKG via optimizing the structural design so as to improve the relative tuning range. Compared with the gap (g) of SRRs, the radius (r) is expected to have more significant influence on RKG because $2 π r > > g$ in the expression of the kinetic inductance. And the thickness (h) and width (w) of SRRs are mathematically equivalent in Eqs. (4a) and (4b), so we discuss only one of the both parameters, i.e., the thickness (h). Therefore, we focus on the effect of the radius and thickness on RKG.

Based on the LC model, RKG is derived as a function of the thickness (h) and radius (r) of SRR at N = 1 × 10¹⁶ cm⁻³ (Fig. 3(a)) and 1 × 10¹⁷ cm⁻³ (Fig. 3(b)). As shown from a comparison between Figs. 3(a) and 3(b), the dependence of RKG on the geometric sizes (h and r) is similar at the two carrier densities. But RKG is larger at the lower N because the kinetic inductance is inversely proportional to the carrier density according to Eq. (4a). Figures 3(a) and 3(b) further show that for a fixed carrier density, RKG is reduced with an increasing h. So under the same variation of N, a lower relative tuning range is expected at a larger h. This is verified in Fig. 3(c), where the relative tuning range (N₁ = 1 × 10¹⁶ cm⁻³ and N₂ = 1 × 10¹⁷ cm⁻³) as a function of h is obtained. The analytical result (red solid line) shows that the relative tuning range decreases from 170% to 130% as h increases from 1 μm to 3 μm, which agrees well with the simulation results (blue dots). The decrease of RKG and hence the relative tuning range with h can be explained by the LC model. An examination of Eqs. (4a) and (4b) reveals that the inductance $L_{k}$ and $L_{g}$ are both reduced as h is increased. Nonetheless, $L_{k}$ undergoes a more substantial change than $L_{g}$ . Therefore, RKG is reduced as the thickness of SRRs grows, which then leads to a reduction of the relative tuning range. Besides, the resonance shifts to higher frequencies with increasing h at the same time, as shown in Fig. 3(e). Although the capacitance $C_{g}$ and $C_{s}$ both increase with h, which is expected to reduce the resonance frequency according to Eq. (3), the simultaneous reduction of the inductance makes a more significantly competing effect, eventually increasing the resonance frequency.

Fig. 3 Contour plot of RKG (L_k /L_g) as a function of SRR thickness h and radius r at (a) N = 1 × 10¹⁶ cm⁻³ and (b) N = 1 × 10¹⁷ cm⁻³. The red to blue colors represent high to low values. The relative tuning range in dependence with (c) thickness and (d) radius. The resonance frequency as a function of N at different (e) thickness and (f) radius. In (c)-(e), the dots represent the simulation results and solid curves show the analytical results from LC model.

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In addition to the altering of the SRR thickness, Fig. 3 shows that tailoring the radius r provides another means to adjust the RKG. Similar to the situation of h, an increment of r also results in reduced RKG, which then reduces the relative tuning range. However, $L_{k} / L_{g}$ is less sensitive to the radius r than to the thickness h, as shown in the contour plots. And another notable difference can be seen by comparing Fig. 3(e) with Fig. 3(f). Figure 3(e) shows that at a given N the resonance frequency increases with the thickness, as mentioned above. While the resonance frequency is reduced with an increasing radius because both the inductance and capacitance are increased. In other words, by reducing h, an increased relative tuning range is achieved and accompanied by the reduction of the resonance frequency. In contrast, by reducing r, the relative tuning range Δf / f(N₁) and the resonance frequencies f(N₁) are both increased. Consequently, as r is reduced the absolute resonance shift Δf is also increased. For example, as N rises from 1 × 10¹⁶ cm⁻³ to 1 × 10¹⁷ cm⁻³, the resonance frequency of SRRs with r = 17 μm increases from 0.36 THz to 0.86 THz, with a relative tuning range of 139% and the absolute resonance shift Δf = 0.50 THz. While for r = 9 μm, the resonance shifts from 0.58 THz to 1.47 THz, with a relative tuning range of 153% and Δf = 0.89 THz. So the adjustment of the radius of SRRs can result in the simultaneous increment of the relative tuning range and absolute resonance shift. In Fig. 3(f), there is a slight discrepancy between the analytical and simulated resonance frequency. The reason is that Eq. (4b) takes the expression of the geometric inductance of a closed ring, while the simulated SRR structure has a gap. Furthermore, the geometric inductance of a SRR can be approximated with Eq. (4b) only when the condition $2 π r > > g$ is well satisfied [28]. So with the same gap size g, the larger the radius r, the smaller discrepancy between the analytical and simulation results can be achieved, which is shown as in Fig. 3(f).

Although the relative tuning range can be increased by reducing the thickness and/or radius of SRRs, there are some constraints at the same time. First, as the thickness is reduced, the LC resonance strength is weakened [34]. Consequently, the transmission dip at resonance is greatly decreased and even vanishes (not shown here), which is undesirable for practical applications. Second, since the RKG is not sensitive to the radius, increasing the tuning range via reducing the radius is not very effective [Figs. 3(b) and 3(d)]. Therefore, an alternative approach is in demand which can effectively increase the relative tuning range without weakening the resonance strength.

In the aforementioned discussion of the geometric inductance, only the self-inductance is considered while mutual inductance (M) arising from the inductive coupling between SRRs [35–37] is not taken into the LC model. Under a large separation between neighboring SRRs (20 μm as chosen above), such an approximation is reasonable because the inter-SRR inductive coupling is sufficiently weak compared with the self-inductance. It is verified by the good agreement between the analytical and simulation results. However, if SRRs are densely spaced, the inductive coupling among SRRs is substantially enhanced. So the mutual inductance has to be taken into account. Subsequently, an extended LC model which includes the effects of the mutual inductance will be introduced. And it is shown that the relative tuning range can be increased by adjusting the mutual inductance.

The sign of mutual inductance can be either positive or negative, depending on the relative direction of the magnetic field induced by coupled SRRs. Mutual inductance is negative if the magnetic field generated by the coupled SRRs cancels each other, and becomes positive otherwise. In the former case with M<0, the net geometric inductance is reduced from $L_{g}$ to $L_{g}^{'} = L_{g} - | M |$ . Correspondingly, RKG is increased from $L_{k} / L_{g}$ to $L_{k} / (L_{g} - | M |)$ . In the latter case with a positive mutual inductance, the net geometric inductance increases from $L_{g}$ to $L_{g}^{'} = L_{g} + | M |$ and RKG drops from $L_{k} / L_{g}$ to $L_{k} / (L_{g} - | M |)$ . In addition to the inductive coupling, there also exists capacitive coupling between two neighboring SRRs, which results in the mutual capacitance. Assume the total capacitance is C_tot, which takes into account both the self capacitance and the mutual capacitance. Then the resonance frequency is obtained as:

f = \frac{1}{2 π \sqrt{(L_{g} \mp | M | + L_{k}) C_{t o t}}} .

After a similar mathematical deduction to obtain Eq. (7), the relative tuning range taking mutual inductance into account is derived as Eq. (9). It takes the similar form with Eq. (7), but with

L_{g} .

replaced by

L_{g}^{'}

, which is defined as

L_{g}^{'} = L_{g} - | M |

for M<0 and

L_{g}^{'} = L_{g} + | M |

for M>0.

\frac{Δ f}{f (N_{1})} = \frac{\int_{N_{1}}^{N_{2}} S (N) d N}{f (N_{1})} = \sqrt{\frac{(L_{g}^{'} + L_{k 1}) C_{t o t}}{(L_{g}^{'} + L_{k 2}) C_{t o t}}} - 1 = \sqrt{1 + \frac{N_{2}}{N_{1}} \frac{L_{k 2}}{L_{g}^{'}}} / \sqrt{1 + \frac{L_{k 2}}{L_{g}^{'}}} - 1.

Equation (9) reveals that the mutual inductance may affect the relative tuning range by altering RKG. While the mutual capacitance, no matter how large it is, has no effect on the relative tuning range. A comparison between Eq. (7) and (9) shows that under the same carrier density variation, an introduction of negative mutual inductance will increase RKG and hence improve the relative tuning range. On the contrary, a positive mutual inductance is expected to reduce the relative tuning range.

In order to test this hypothesis, a stacked SRR-dimer structure is proposed due to the strong inductive coupling between the two identical SRRs, which are coaxially aligned in the same unit cell. The structure is shown in Fig. 1(b), with the same SRR size parameters as those in Fig. 1(a), except for an introduction of separation s and twist angle φ. This so-called stereo-metamaterial has been investigated in terms of the plasmon coupling between the two stacked SRRs [37]. But as we know, it has not yet been employed for the frequency tuning. It is known that when the two SRRs are brought in close proximity, the original LC resonance of each SRR splits into two new plasmon modes: the bonding mode with the loop currents of two SRRs oscillating in phase and the anti-bonding mode with the loop currents 180° out of phase [37]. Figure 4 displays the spatial distribution of the magnetic field H_z at the anti-bonding mode for φ = 180° (Fig. 4(a)) and the bonding mode for φ = 0° (Fig. 4(b)), respectively. The magnetic fields induced by the two SRRs interferes destructively (constructively) at the anti-bonding (bonding) mode, which gives rise to the negative (positive) mutual inductance.

Fig. 4 The magnetic field H_z of the stacked SRR-dimer at different resonance modes. (a) H_z of the anti-boning mode for the stacked SRR-dimer MM with φ = 180°. (b) H_z of the boning mode for the stacked SRR-dimer MM with φ = 0°. The white arrows represent the direction of loop currents. The color scale represents the direction and the magnitude of H_z, with red color meaning the maximum value in the z direction and blue color the maximum value in the -z direction.

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The effect of the negative mutual inductance, which arises from the anti-bonding resonance (see Fig. 4(a)), on the frequency tuning is explored in an array of stacked SRR-dimers with φ = 180°. Under the incident configuration as exhibited in Fig. 1, the bonding mode cannot be effectively excited and only the anti-bonding mode couples to the incident wave, which is manifested by the dip in the transmission spectrum. The simulated transmission spectra of the stacked SRR-dimers (solid lines) and single-layer SRRs (dash lines) are presented in Fig. 5. The carrier density ranges from 1 × 10¹⁶ cm⁻³ (black) to 1 × 10¹⁷ cm⁻³ (pink). For the same carrier density, the stacked SRR-dimer structure exhibits the higher resonance frequency than the single-layer SRRs. It can also be explained with the LC model. On one hand, a negative mutual inductance is induced in the stacked SRR-dimer MM. So the stacked SRRs have a lower geometric inductance $(L_{g} - | M |)$ than the single-layer SRRs. On the other hand, the capacitive coupling results in reduced total capacitance than that of single-layer SRRs. Therefore, the resonance frequency of the stacked SRRs blue shifts relative to the single-layer SRR. As N is increased from 1 × 10¹⁶ cm⁻³ to 1 × 10¹⁷ cm⁻³, the resonance frequency of single-layer SRR (stacked SRR-dimer) shifts from 0.39 THz (0.41 THz) to 0.97 THz (1.12 THz), with a relative tuning range of 149% (172%). So the improvement of relative tuning range due to the negative mutual inductance is observed, which meets the expectation from the analytical model as proposed above. Moreover, different from reducing the SRR thickness, the SRR-dimer structure effectively increases the relative tuning range with no expense of weakening the resonance strength.

Fig. 5 The transmission spectra of a single-layer (dashed lines) and stacked SRR-dimers with φ = 180° (solid lines) at different carrier densities. The black to pink color represents an increasing carrier density from 1 × 10¹⁶ cm⁻³ to 1 × 10¹⁷ cm⁻³. The structure sizes are the same with those in Fig. 1.

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Since the strength of the inductive coupling is dependent on the separation between SRRs [35], the mutual inductance can be tailored via adjusting the separation s. The mutual inductance is evaluated from Eq. (10), which is widely used to calculate the mutual inductance between two coaxially aligned circular coils [38]. Because the circumference of the SRR is much longer than the gap width, i.e., $2 π r > > g$ , Eq. (10) can describe the mutual inductance of stacked SRRs quite well.

| M | = \frac{2 μ_{0} r (K (x) - E (x))}{\sqrt{x}} .

E and K are the complete elliptic integrals of the first and second kind with the geometry-dependent parameter

x = \frac{\sqrt{4 r^{2} + {(h + s)}^{2}} - (h + s)}{\sqrt{4 r^{2} + {(h + s)}^{2}} + (h + s)}

. The mutual inductance, which is normalized to the self-inductance for convenience of comparison, is calculated as a function of separation s and shown in the red line of Fig. 6(a). Not surprisingly, M decreases with increased s due to weakened inductive coupling between SRRs. And the decreased mutual inductance leads to the decline of

RKG = L_{k} / (L_{g} - | M |)

when the two SRRs of a dimer are gradually separated (blue line in Fig. 6(a)). Combining the expression of mutual inductance with Eq. (9), the relative tuning range of the stacked SRR-dimers as a function of the separation is derived and shown in the upper branch of Fig. 6(b). The stacked SRR-dimer structure shows larger relative tuning range than the single-layer SRRs of ~150% (green dashed line) under the same variation of carrier density (N increases from 1 × 10¹⁶ cm⁻³ to 1 × 10¹⁷ cm⁻³). And the improvement of the relative tuning range is especially pronounced at a smaller separation due to the stronger inductive coupling, which is responsible for a larger negative mutual inductance. As the separation s grows, the relative tuning range of the stacked SRR-dimers is reduced as a result of the decreased mutual inductance. But even at a large separation of s = 8 μm, the relative tuning range of stacked SRRs is still larger than that of the single-layer structure. The analytical results are well reproduced by the simulation data, indicating that the LC model effectively captures the main physics involved in the frequency tuning.

Fig. 6 (a) RKG and the normalized mutual inductance (red line) at different separation s. The blue (black) line presents RKG of stacked SRR-dimers at the anti-bonding (bonding) mode under the carrier density N = 1 × 10¹⁷ cm⁻³. The green dashed line shows RKG of the single-layer SRRs. (b) The s dependent relative tuning range of stacked SRR-dimers at the anti-bonding mode (upper branch) and the bonding-mode (lower branch). Also the green dashed line shows the relative tuning range of the single-layer SRRs.

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Having demonstrated that the negative mutual inductance improves the relative tuning range, we proceed to study the effect of the positive mutual inductance on the relative tuning range. The positive mutual inductance originates from the bonding resonance mode of a stacked SRR-dimer structure, as shown in Fig. 4(b). Note that the bonding mode is considered in the stacked SRR-dimers with φ = 0°, rather than the structure with φ = 180°.The reason is that the bonding mode is a dark mode for φ = 180°, which weakly couples to the incident beam and so can hardly be discerned in the transmission spectrum [39]. However for φ = 0°, the bonding mode can be effectively excited by the incident THz wave. At the bonding resonance, the magnetic field along z axis (H_z) induced by the two SRRs of the dimer oscillates in phase, giving rise to a positive mutual inductance and a net geometric inductance $L_{g}^{'} = L_{g} + | M |$ . Therefore, the SRR-dimer has a larger geometric inductance and hence lower RKG than the single-layer SRR, as shown in Fig. 6(a). Based on Eqs. (9) and (10), the relative tuning range is calculated and shown as the lower red line in Fig. 6(b), which is well supported by the simulation results. As we expected above, the stacked SRR-dimer at the bonding mode actually exhibits a lower relative tuning range than the single-layer SRRs. When the separation s grows, the mutual inductance becomes smaller due to the gradually weakened inductive coupling. Consequently, RKG increases with s until it approaches the value of the single-layer SRRs. Correspondingly, the relative tuning range increases with s and asymptotically approaches the relative tuning range of the single-layer SRRs.

By means of a stacked SRR-dimer metamaterial, we have verified the proposed hypotheses, i.e., the relative tuning range of SRRs can be adjusted through the incorporation of the inductive coupling. The relative tuning range is improved by the negative mutual inductance originating from the destructive inductive coupling, but reduced by the positive mutual inductance arising from the constructive inductive coupling. We would like to mention that the contribution from the inductive coupling to the tuning performance is not limited to the multilayer vertically-stacked SRRs. Similar phenomena are also expected in the planar SRR-dimer configuration [40], where the inductive coupling exists as well. It will be investigated in the future work.

4. Conclusion

In summary, we investigate the resonance frequency tuning in the InSb SRRs by adjusting the carrier density at the THz regime. An analytical model based on the LC equivalent circuit is presented, which reveals that the continuous resonance shift with carrier densities is caused by the variation of the carrier kinetic inductance in SRRs. By introducing the frequency tuning sensitivity, the relative tuning range is analytically derived. According to the LC model, the frequency relative tuning range can be improved by increasing the RKG, i.e., the ratio of kinetic inductance to the geometric inductance of SRRs. The analytical model also demonstrates that the RKG and hence the relative tuning range can be increased by reducing the thickness and/or the radius of the single-layer SRRs, which is evidenced by the FDTD simulation results. To further increase the relative tuning range, a stacked SRR-dimer metamaterial is studied. As predicted by the LC model, the RKG in such a structure can be improved compared with the single-layer SRRs due to the inductive coupling among SRRs, which results in increased relative tuning range correspondingly. And the improvement of relative tuning range is especially notable as the separation between the two SRRs of the dimer is small. The relative tuning range as high as ~170% is achieved via the stacked SRR-dimers when the carrier density ranges from 1 × 10¹⁶ to 1 × 10¹⁷ cm⁻³. It should be noted that in the analytical model, the capacitance is assumed to be constant and independent of the carrier density. It is known that the SRR capacitance is closely related with the charge distribution on the gap surface and the dielectric inside the gap which is constant in this work. While the inductance strongly dependents on the carrier density inside the whole SRR. Therefore, the inductance undergoes the more significant variation than the capacitance as the carrier density changes. And the resonance shift mainly results from the variation of the inductance. In the frequency band (~1 THz) and the range of carrier densities (1 × 10¹⁶ cm⁻³-1 × 10¹⁷ cm⁻³) of interest, the analytical model agrees well with the simulation results. However, if more accurate results are required, the effect of the carrier density on the capacitance has to be taken into consideration. The presented method may offer a guideline for designing continuously tunable THz devices such as modulators and sensors.

Acknowledgment

This work was supported by the National Science Foundation of China under Grant No.60907025 and the Fundamental Research Funds for the Central Universities.

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The ratio of the kinetic inductance to the geometric inductance: a key parameter for the frequency tuning of the THz semiconductor split-ring resonator

Abstract

1. Introduction

2. Structure design and simulation model

3. Results and discussion

4. Conclusion

Acknowledgment

References and links

Cited By

Figures (6)

Equations (15)

Optics Express