## Abstract

This paper proposes one broadly tunable terahertz (THz) slow-light system in a semiconductor-insulator-semiconductor structure. Subject to an external magnetic field, the structure supports in total two surface magneto plasmons (SMPs) bands above and below the surface plasma frequency, respectively. Both the SMPs bands can be tuned by the external magnetic field. Numerical studies show that leveraging on the two tunable bands, the frequency and the group velocity of the slowed-down THz wave can be widely tuned from 0.3 THz to 10 THz and from 1 *c* to 10^{−6} *c*, respectively, when the external magnetic field increases up to 6 Tesla. The proposed method based on the two SMPs bands can be widely used for many other plasmonic devices.

© 2012 OSA

## 1. Introduction

Slow-light technology has attracted a lot of attention in recent years because it introduces a wealth of applications in telecommunications, data processing, and light-matter interactions [1, 2]. Although traditional electronic approaches can efficiently slow down light beams [3–5], it has many constraints such as narrow band-width, limited operating wavelength range, and difficulty in achieving on-chip devices [6]. Photonic approaches utilizing optical resonances have been investigated in recent years [7–9]. Among different photonic approaches, plasmonic technology using metal/dielectric layers is more promising due to its subwavelength confinement of electromagnetic (EM) fields [10–12].

The plasmonic slow-light technology has a big challenge when it is intended to apply in the terahertz (THz) regime because metals cannot support surface plasmons (SPs) in this region [13]. One approach to overcome this challenge is to use spoof SPs [6, 14]. However, the dielectric properties of a metal in the THz regime are similar to those of a perfect electric conductor, and they are difficult to change by external tuning mechanisms [13].

An emerging approach is to use doped semiconductors to replace metals for THz devices [15–19], because doped semiconductors have similar dielectric properties in the THz region to those of metals in the visible region [15]. In addition, they can be designed to achieve tunable characteristics through doping, temperature, and/or external magnetic fields [16–18]. Especially, when an external magnetic field is applied in the Voigt configuration, two tunable surface magneto plasmons (SMPs) bands are available [20]. Not only one band exists below the surface plasma frequency as usual, but also the other appears above the surface plasma frequency. This provides a more flexible and powerful tuning mechanism.

In this paper, we present a magnetic-field tunable THz slow-light system based on SMPs in a semiconductor-insulator-semiconductor (SIS) structure. In this system, both the frequency and the group velocity of the slowed-down THz wave can be tuned. More importantly, due to the existence of two SMPs bands, especially the higher band which has a wider tunable bandwidth, the proposed system has a very broad bandwidth. It is shown through analytical and numerical calculations that when the external magnetic field increases from 0 to 6 Tesla, a monochromatic THz wave in a range of [0.3, 10] THz can be slowed down for a lossless semiconductor. A comprehensive design guideline for the proposed plasmonic showed-down system is also given in the paper.

## 2. The semiconductor-insulator-semiconductor schematic structure and dispersion relation of the surface magneto plasmons

A plasmonic slow-light system is expected to have the same slow-light effect in both forward- and backward-propagating directions. However, due to the nonreciprocal effect, the dispersion relations of these two propagating waves are different in the structure presented in [20]. As a result, the group velocities (defined as *v _{g}* =

*dω*/

*dβ*) are not the same in the two propagating directions. In this paper, we propose a symmetric structure to eliminate the nonreciprocal effect. The schematic structure of the slow-light system is depicted in Fig. 1(a) . The 2D SIS structure is composed of two semiconductor layers with an insulator layer in the middle. The whole structure is invariant in the

*y*-axis. The thickness of the insulator layer along the

*x*-axis is denoted by

*w*. A transverse magnetic (TM)-polarized THz wave (the magnetic field component parallel to the

*y*-axis) is propagating along the

*z*-direction where

*E*(

_{x}**r**,

*ω*),

*E*(

_{z}**r**,

*ω*),

*H*(

_{y}**r**,

*ω*) $\propto $exp[

*i*(

*kz-ωt*)]. An external static magnetic field

*B*is applied uniformly on the whole structure along the -

*y*-axis, in a Voigt configuration.

The dielectric constant of the semiconductor is a tensor in the Voigt configuration under an external magnetic field [21]:

In the lossless case, the parameters in Eq. (1) have the expressions of *ε _{xx}* = 1 –

*ω*

_{p}^{2}/ (

*ω*

^{2}–

*ω*

_{c}^{2}),

*ε*= –

_{xz}*iω*

_{p}^{2}

*ω*/ [

_{c}*ω*(

*ω*

^{2}–

*ω*

_{c}^{2})], and

*ε*= 1 –

_{yy}*ω*

_{p}^{2}/

*ω*

^{2}, where

*ω*is the angular frequency of the incident wave,

*ω*is the plasma frequency of the semiconductor,

_{p}*ε*is the high-frequency permittivity, and

_{∞}*ω*=

_{c}*eB*/

*m** is the cyclotron frequency.

*e*and

*m** are the charge and the effective mass of electrons, respectively.

*B*is the applied external magnetic field. With the consideration of the characteristics of the surface mode and the continuity of

*H*and

_{y}*E*at the interfaces of the semiconductor/insulator, the dispersion relation of the waveguide is given by [22]

_{z}*ε*is the dielectric constant of the insulator layer;

_{d}*κ*

_{1}and

*κ*

_{2}are respectively defined as

*κ*

_{1}

^{2}=

*β*

^{2}–

*k*

_{0}

^{2}

*ε*,

_{d}*κ*

_{2}

^{2}=

*β*

^{2}–

*k*

_{0}

^{2}

*ε*,

_{V}*β*is the propagation constant, and

*ε*=

_{V}*ε*

_{xx}+ ε_{xz}^{2}

*/ε*is the Voigt dielectric constant of the semiconductor.

_{xx}*ε*can be considered as the bulk permittivity of the semiconductor if the polarization of the EM wave is along the

_{V}*x*or

*z*directions [20]. It can be clearly seen that the nonreciprocal effect disappears in Eq. (2) for the dispersion equations are the same for +

*β*and –

*β*. In Fig. 1(b), we plot the dispersion curves of the SMPs in the structure when the external magnetic field is applied with intensities of 0, 0.5, 1 and 2 Tesla. Without loss of generality, here we consider

*InSb*and air as the semiconductor and insulator materials, respectively, in the calculation. The corresponding parameters of

*InSb*at room temperature are given by

*m** = 0.014

*m*

_{0}(

*m*

_{0}is the free electron mass in vacuum),

*ω*= 12.6THz, and

_{p}*ε*

_{∞}= 15.68 [17]; and

*ε*=

_{xx}*ε*= –47.6,

_{yy}*ε*= 0 for B = 0T and

_{xz}*f*= 1THz;

*ε*= 36.8,

_{xx}*ε*= –47.6,

_{yy}*ε*= 42.2

_{xz}*i*for B = 1T and

*f*= 1THz. The width of the waveguide is

*w*= 0.1

*λ*= 15

_{p}*μm*(~1/20

*λ*of 1 THz wave), where

*λ*is the

_{p}*InSb*plasma wavelength defined as 2

*πc*/

*ω*, and

_{p}*c*is the light velocity in vacuum. Without the magnetic field being applied, the dispersion relation has only one curve (the black dashed line). It starts from the zero point, then rises along with the light line in vacuum

*ω*=

*k*

_{0}

*c*(the red line), and extends to infinity at a frequency of 1.95 THz. However, when a magnetic field is applied, two SMP bands are excited, and the dispersion curves are seen by the blue, the pink dash-dotted and the green dotted lines. The lower band curves (indicated by the blue shadow region) are similar to the black dashed line, converging to infinity at a lower frequency. The higher band region (indicated by the pink shadow) starts at a hybrid cyclotron-plasma frequency

*ω*=

_{H}*sqrt*(

*ω*

_{c}^{2}+

*ω*

_{p}^{2}), which is defined by

*ε*= 0 [23]. The higher band curves then converge to infinity at a higher cut-off frequency. Because

_{xx}*ω*is always larger than

_{H}*ω*=

_{sp}*sqrt*[

*ω*

_{p}^{2}/(1 +

*ε*)] [19], the SMPs can propagate at a frequency above the surface plasma frequency in the higher band. Another important feature from Fig. 1(b) is that with the increase of the magnitude of the external magnetic field, the two bands move toward opposite directions (as indicated by the blue arrow in Fig. 1(b)). Furthermore, the higher band dispersion curve moves ‘faster’ than the lower band dispersion curve. When the magnetic field increases from 0 to 2T, the cut-off frequency of the lower band changes 1.1 THz (from 1.9 to 0.8 THz), while that of the higher band changes 2.9 THz (from 1.9 to 4.8 THz).

_{d}The excitation of the two propagating bands can be explained by the effect of the applied magnetic field on the oscillation of the free electrons in semiconductors. When no magnetic field is applied, there is only one responding oscillation of the free electrons, which is along the polarization of the incident wave. However, when a magnetic field is applied, another electron oscillation perpendicular to the incident polarization is excited due to the hybrid Lorentz force [21]. Therefore, two surface modes with different frequencies can be supported by the two different kinds of free electron oscillations. In a consequence, the bulk Voigt permittivity *ε _{V}* also changes due to the external magnetic field. Without an external magnetic field, there is only one frequency region for

*ε*<0 in [0, 2] THz, as indicated by the region A in Fig. 2(a) . While, when a 0.5 Tesla magnetic field is applied, a singular point in the

_{V}*ε*-frequency curve appears around 2.2 THz. Thus, two frequency regions appear for

_{V}*ε*<0, indicated by regions B and C. This shows that two propagating bands could be supported when the magnetic field is applied. In Figs. 2(b) and 2(c), the normalized electric field intensities of the SMPs modes in the lower (for an incident frequency at 1THz) and the higher (for an incident frequency at 3THz) bands are plotted under a magnetic field of 1Tesla. It is noticed that with the magnetic field, the mode distribution becomes asymmetric. For the SMPs mode in the lower band, most of the energy is confined on the lower surface of the dielectric layer, while the energy is mostly confined on the upper surface in the higher frequency band.

_{V}## 3. Tuning of the slow-light system

We are now studying the slowed-down characteristics of the structure. Since the analytical expression of the group velocity cannot be obtained from Eq. (2), we calculate *dω*/*dβ* numerically by fitting the curves of Fig. 1(b). In Fig. 3(a)
, the normalized group velocity *v _{g}*/

*c*of the SMPs mode as a function of the incident frequency is plotted when the external magnetic fields are 0, 0.5, 1, and 2 Tesla, respectively. It shows that

*v*can achieve ~10

_{g}^{−6}

*c*in the structure for both the lower and higher bands. The position of the slow-light region changes obviously with the magnetic fields. When the magnetic field increases from 0 to 2 T, the corresponding frequency for

*v*10

_{g}<^{−6}

*c*moves from 1.9 to 0.8 THz for the lower band, while that of the higher band moves from 1.9 to 4.8 THz. To verify these results, finite-difference time-domain (FDTD) simulations are conducted for

*B*= 0 and 1 Tesla [24], respectively, as shown in Fig. 3(a) (see the triangular points). The group velocities of the simulation results agree well with the analytical results from Eq. (2). This verifies the slow-light effects of the system. It should be noted that the slow-light frequency moves continuously with the increase of the magnetic field. Therefore, any frequency in the region of [0.8, 4.8] THz can be slowed down in the structure by applying an appropriate magnetic field between [0, 2] Tesla.

Then the next question is how to determine the intensity of the magnetic field for a certain frequency? Although direct relation of the magnetic field intensity *B* and the incident angular frequency *ω* is difficult to obtain, it is found that the slow light region is always close to the cut-off frequency of the SMPs modes, which can be derived analytically. From the non-retarded limit [23], the cut-off frequencies of the lower and the higher modes can be expressed by 1 + *ε _{xx}* +

*iε*and1 +

_{xz}*ε*, respectively. Then from the expressions of

_{xx}−iε_{xz}*ε*and

_{xx}*ε*in Eq. (1), we have

_{xz}*ω*<

*ω*, ‘

_{sp}*−*’ denotes

*ω*>

*ω*. From Eq. (3), one can calculate the magnetic field needed to slow down the THz waves. In Fig. 3(b), the calculated magnetic fields versus the frequencies of the incident THz waves are plotted. When the magnetic field increases up to 6 Tesla, the group velocities of waves in the region of [0.3, 10] THz can all be reduced. Therefore, a broad band tuning plasmonic slow-light system is achieved.

_{sp}From the curves in Fig. 3(a), it can be inferred that for a monochromatic THz wave, the group velocity can also be tuned by changing the intensity of the magnetic field. Without loss of generality, we choose 1 THz and 3 THz in the lower and higher bands, respectively. We then calculate their corresponding group velocities as a function of *B*, which are shown in Fig. 4
. The results between the FDTD simulations and the analytical model agree very well. Because of the counter moving directions of the frequency-*v _{g}* curves in the lower band and higher band when increasing the magnetic field, it is found that

*v*in the lower band decreases with the magnetic field, while

_{g}*v*in the higher band increases. When the magnetic field increases from 0 to 1.36 Tesla, the group velocity of 1 THz frequency deceases from 0.65

_{g}*c*to 10

^{−3}

*c*. For frequency at 3 THz, the required magnetic field is from 1.04 to 0.88 Tesla in order to slow down the wave from 0.23

*c*to 0.048

*c*. Therefore, in addition to the broad tuning range, the group velocity can also be flexibly tuned in the proposed slow-light system.

At last, we discuss the effects of the material loss on the tuning properties of the slow-light system. It is reported that the slowdown factor may be limited by the material losses [12]. This phenomenon is also observed with the proposed structure. In a semiconductor, the loss is mainly caused by the collision of charge carriers. If the loss is considered, the elements in the permittivity tensor of Eq. (1) become complex values, which can be expressed as *ε _{xx}* = 1–

*ω*

_{p}^{2}(

*ω*+

*i*/

*τ*)/{

*ω*[(

*ω*+

*i*/

*τ*)

^{2}–

*ω*

_{c}^{2})]},

*ε*= –

_{xz}*iω*

_{p}^{2}

*ω*/{

_{c}*ω*[(

*ω*+

*i*/

*τ*)

^{2}–

*ω*

_{c}^{2})]}, and

*ε*= 1–

_{yy}*ω*

_{p}^{2}/[

*ω*(

*ω*+

*i*/

*τ*)] [21], respectively. Here,

*τ*is the collision time of the carriers. If the loss is low, we have 1/

*τ*= 0.01

*ω*, the group velocity

_{p}*v*cannot be as low as 10

_{g}^{−6}

*c*but in the order of 10

^{−2}~10

^{−1}

*c*. This value increases with the increase of the loss. In addition, owing to the loss, the group velocity no longer decreases monotonously with the frequency, but increases to

*c*sharply at a resonant frequency after reaching the minimum value [12]. The physical origin of this resonant frequency is related to the loss but it is complicated and needs further quantitative investigations. Even though the group velocity increases in a lossy system, the tuning mechanism of the proposed structure (including the tuning of the lower and higher bands, as well as the group velocity) is still valid. It is also found that the group velocity curves of the lossy cases become closer to the lossless group velocity curves when the loss is decreased, which suggests that the material loss affects the tunable range, and it could be effectively compensated by using active materials [12], designing doping levels, etc.

## 4. Conclusions

In conclusion, we present a broadly tunable plasmonic slow-light system based on SMPs in a semiconductor-insulator-semiconductor structure. Since a higher propagating band of SMPs can be excited when an external magnetic field is applied perpendicularly to the propagating direction of the incident wave, the slow-light frequency can be tuned from 0.3 to 10 THz when the magnetic field increases from 0 to 6 Tesla. We also show that the group velocity can be tuned by the magnetic field. It is believed that the broad tuning mechanism by employing the higher band can also be applied to other tunable THz devices.

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