1. Introduction

The properties of photonic crystals (PCs) [1,2] offer unique ways for the control of light. Because of their compatibility with standard semiconductor processing, two-dimensional (2D) PC structures are the most commonly used [3–5]. An important class of 2D-PCs is the photonic crystal slab (PCS) [6–8], which additionally confines photons in the out-of-plane direction with a dielectric slab. Applications of PCSs include micro cavities [9], lasers [10,11] and detectors [12,13].

In a PCS additional eigenmodes can propagate, because of the higher order modes of the slab wave guide. This fact was pointed out in previous works [6,7], but no detailed theoretical or experimental investigation has been shown so far. Analogous to solid-state physics, where electron confinement in a heterostructure creates subbands, the higher order modes, due to optical confinement of photons within the slab, create photonic subbands. For photonic band gap (PBG) devices the unwanted higher order modes reduce the achievable band gap for guided modes [14].

The resonances of a PCS in the mid-infrared region (MIR) can be directly measured by fabricating it from a quantum well infrared photodetector (QWIP) [8,15,16]. In this letter, we present an investigation of PCS-QWIP structures where higher order slab modes are excited and demonstrate a method to identify the order of the slab mode as well as the polarization. This complements characterization methods of PCs such as the photonic band structure mapping technique [4,17,18], which provides information about the PC modes, but not about the slab modes.

2. Design and fabrication

The layer sequence of the used structure is grown by molecular beam epitaxy (MBE) with a 2 µm thick Al_0.85Ga_0.15As sacrificial layer, a 500 nm thick GaAs Si doped (carrier concentration 2 × 10¹⁸ cm⁻³) bottom contact layer, a 1.4 µm thick active region and a 100 nm thick GaAs doped (carrier concentration 2 × 10¹⁸ cm⁻³) top contact layer. The contact layers are used for collection and lateral transport of the photocurrent to the gold contacts. The QWIP consists of 26 periods of 45 nm Al_0.30Ga_0.70As barriers and 4.5 nm GaAs quantum wells. The optical intersubband transition from the bound ground state to the first excited state, here a quasi-bound state, is designed to operate at a wavelength of 8 µm. Electrons in the bound state are supplied by Si delta doping the QW with an equivalent sheet carrier density of 4 × 10¹¹ cm⁻² per period.

The PC with a hexagonal lattice (lattice constant a) of holes (radius r) in a dielectric medium is fabricated by anisotropic deep etching through the active zone into the sacrificial layer. After mesa etching and SiN isolation deposition, the extended top and bottom gold contact pads, which are annealed to obtain ohmic contacts for efficient extraction of the generated photocurrent, are deposited. The last process step is selective underetching with a 24% HCl solution to remove the sacrificial AlGaAs layer creating a free standing PCS of the air-bridge type with a final slab thickness of d = 2 µm (Fig. 1(a) ).

 

Fig. 1 (a) Scanning electron microscope (SEM) image of a cleaved PCS with mode profiles for the 0th and 1st order TE modes of the slab wave guide. (b) Normalized photocurrent spectra of a standard QWIP (dashed line) at 45° wedged illumination and a PCS-QWIP at surface normal incidence (solid line).

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The QWIP is operated at 50 K, well below the background limited performance temperature T_BLIP = 69 K. In a standard QWIP, normal incident light does not generate photocurrent due to intersubband transitions requiring an electric field in the growth direction [16]. Therefore, the photocurrent spectrum of the standard QWIP is obtained by a 45° wedged sample measurement. In the PCS-QWIP, normal incident light can couple into the radiating modes of the PCS, which are located above the light line [19]. Incident photons with insufficient energy to excite electrons from the bound into the quasi-bound state cannot generate photocurrent. Therefore, all resonances of the PCS are designed above the QWIP peak responsivity (1250 cm⁻¹), where photons can excite electrons into the continuum. Due to the transition probability becoming smaller, the generated photocurrent diminishes for higher energetic photons, but the long lifetime of photons in the PCS modes enhances the generated spectral photocurrent. The spectral dependence of the photocurrent is obtained with a broadband glowbar source and a Fourier transform infrared (FTIR) spectroscopy setup. We have previously shown that the PCS enhances the performance of the QWIP [8]. In the range from 1500 to 3000 cm⁻¹, where the QWIP shows a low responsivity (Fig. 1(b), dashed line), sharp peaks are measured at the PCS resonances (Fig. 1(b), solid line).

3. Photonic band structure calculation

To design the response peaks of the PCS-QWIP, the photonic band structure of the PC has to be modeled. The band structure is calculated with the revised plane wave expansion method (RPWEM) [20,21] with an effective refractive index approximation to account for wave guiding in the slab. In an ideal 2D-PC, the modes can be separated into pure transversal electric (TE) and transversal magnetic (TM) polarized waves. Due to the missing translational invariance in the out-of-plane direction, the PCS modes are not purely TE or TM. However, since they strongly resemble TE and TM modes of an ideal 2D-PC, they are classified as TE-like and TM-like [22]. In contrast to the pure TE modes in an ideal 2D-PC, the TE-like modes of a PCS generate photocurrent in the QWIP, because of mode-mixing [17] and a non-zero electric field component along the growth direction.

For the band structure calculation the PCS is split into a 2D-PC and a uniform slab wave guide problem. The effective refractive index n_eff of the slab wave guide is then introduced as frequency dependent permittivity of the PC material. Depending on the ratio of slab thickness to PC lattice constant, the slab-to-PC ratio d/a, higher order modes of the slab (1st order, 2nd order, etc.) are excited and additional response peaks are contributed to the photocurrent spectrum of the PCS-QWIP. Since higher order modes have a lower effective refractive index, the PCS resonances are located at higher frequencies compared to the fundamental mode.

The effective refractive index for TE and TM modes of a slab waveguide consisting of the slab material n_s surrounded by air cladding material n_c = 1 can be solved analytically (Fig. 2 ). As an approximation for the heterostructure of the slab, an average refractive index of n_s = 3.12 is used. For large filling factors F the perturbation due to the PC air holes can be neglected for this calculation. By introducing the slab normalized frequency $kd=d/\lambda =\omega d/2\pi c_0$ (analogous to the PC normalized frequency $ka$) the cutoff frequency of the m ^th order mode is given by

 

Fig. 2 Analytical solution to the effective refractive index n_eff for TM (solid) and TE (dashed) modes of a slab waveguide. Depending on the slab-to-PC ratio d/a higher order slab modes have to be considered when designing PCS devices (gray axes insets). A measured peak (indicated by arrow, see Fig. 1(b) in the photocurrent spectrum can correspond to several slab modes.

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The slab normalized frequency is related to the PC normalized frequency by the slab-to-PC ratio: $kd=ka*d/a$. Because of the scaling property of the electromagnetic field PCSs with identical slab-to-PC ratios exhibit the same band structure regardless of their absolute dimensions. In Fig. 2 the insets of the PC normalized frequencies axes for d/a = 0.63 and d/a = 0.45 illustrate which slab modes have to be considered for a given PC normalized frequency range. For example, the arrow on the lower axis indicates the 3rd pronounced PC peak in the measured photocurrent spectrum of Fig. 1(b), which can correspond to three different slab mode orders.

With the effective refractive index for the 0th order and 1st order slab mode the band structure of the TM-like (Figs. 3(a) and 3(b)) and TE-like (not shown) PCS modes are computed with the RPWEM for d/a = 0.45 and d/a = 0.63. For increasing slab-to-PC ratio, the PCS approaches the ideal two-dimensional PC with infinite extension into the third dimension (d/a→∞). Reducing the slab-to-PC ratio shifts the resonances of the PCS towards higher PC normalized frequencies. The effective refractive index of higher order slab modes is more sensitive to a change in d/a. Therefore, the PCS resonances of higher order slab modes exhibit a larger shift, as indicated by the arrows in Fig. 3. Since the effective refractive index of TE slab modes differs from TM slab modes, the polarization can also be determined. From the magnitude of the resonance shift in the photocurrent spectra we deduce the slab mode, as well as the polarization of measured resonances.

 

Fig. 3 TM-like photonic band structure between the Γ-K symmetry points of a PCS for the (a) 0th order and (b) 1st order slab mode. Reducing the slab-to-PC ratio from d/a = 0.63 (filled circles) to d/a = 0.45 (hollow circles) shifts the resonances towards higher PC normalized frequencies. Modes inside the gray area are guided modes, whereas modes on the outside are radiating. The boundary between radiating and guided modes is marked by the light line.

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4. Experimental results and discussion

Variation of the slab-to-PC ratio can be achieved by either changing the slab thickness or the lattice constant. For different slab thicknesses, various samples would have to be grown. Variation of the lattice constant can be achieved on the same sample, which also means that processing variations will be the same for all devices. Therefore, we used the latter approach, fabricating a series of PCSs with lattice constants ranging from a = 3.2 to 4.4 µm (d/a = 0.63 to 0.45) and a hole radius of r = 0.22a (F = 0.82, measured with SEM) from the same QWIP sample with a slab thickness of d = 2 µm.

Using normal incident light, the shift of the resonances in the Γ-point can be observed in the measured spectra with PC normalized frequency axes (Fig. 4 ). Since the QWIP absorption stays constant, it will shift on the PC normalized frequency axis when changing the lattice constant. Comparing the shift magnitude of the PCS resonances with the RPWEM results, the pronounced peaks can be distinguished into 3 groups according to the 0th order TM-like, 1st order TM-like and 0th order TE-like PCS modes.

 

Fig. 4 Measured spectra for normal incident light of a series of PCSs with varying lattice constant a and constant slab thickness d. The overlaid frames illustrate the peak shift due to a changing slab-to-PC ratio d/a. The 1st order TM-like modes (dotted frame) shift stronger than 0th order TM-like modes (dashed frame), whereas 0th order TE-like modes (dash-dotted frame) exhibit the smallest peak shift.

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The coupling efficiency into the PCS depends on the in-plane symmetry-matching of the PCS mode to the incident wave [22], hence not all theoretically possible modes of the band structure are measured with the PCS-QWIP.

Detailed comparison of the RPWEM simulation to the measured spectra (Fig. 5 ) shows good agreement of the location of the peaks as well as the magnitude of their shift. For the lowest 1st order TM-like PCS mode at a = 3.8 µm the calculated resonance is located at ka = 0.698 and the measured peak is at 0.706 (1% error). The calculated slope is Δ(ka) / Δa = 0.114 µm⁻¹, whereas the measured value is 0.118 µm⁻¹ (4% error). At higher frequencies the simulation fits better to the measurement, whereas for lower frequencies the offset becomes larger. This can be attributed to an increasing error of the effective refractive index approximation and the evanescent field of the slab mode coupling into the substrate at low frequencies.

 

Fig. 5 Comparison of the measured peak shift (symbols) to the RPWEM simulation (lines) by variation of d/a. Symmetry unmatched modes with low coupling efficiency and hence less pronounced peaks are not plotted. The axis break on the left shows the simulated resonances for an ideal 2D-PC (d/a = ∞).

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5. Conclusion

We have presented a method to identify the slab mode order of PCS resonances by using a QWIP photodetector for direct resonance frequency measurement. The slab wave guide in the out-of-plane direction shifts the PCS modes to higher frequencies, compared to the ideal 2D-PC. Variation of the slab-to-PC ratio by processing a series of devices with different lattice constants allowed us to deduce the order of the slab mode as well as the polarization of resonances by analyzing the magnitude of their shift. The locations of the peaks as well as the shift magnitude in the measured photocurrent spectra are in good agreement with the results of RPWEM simulations with an effective refractive index approximation for the slab.

Using the RPWEM as a fast and reliable design tool, we envision fine-tuning of PCS detectors by post-fabrication thinning (e.g. etching of the top contact layer) or thickening (e.g. deposition of a dielectric material) of the slab to achieve a desired resonance shift. Further, the presented identification technique and knowledge about higher order slab modes will aid in the design of advanced PCS devices.