1. Introduction

Metal films perforated by hole array patterns, referred to as metal photonic crystals (MPCs) in our work, have been widely used to understand surface plasma waves (SPWs) that result in extraordinary optical transmission (EOT) [1,2]. The SPWs are confined near a metal/dielectric interface, and exponentially decay laterally along and vertically from it. Their spatial variation is characterized with the propagation length and the penetration depth along and from the interface respectively. The SPW propagation length, L_sp, is expressed as [3]:

Most of researches have dealt with the EOT by MPCs of which the lateral dimension is significantly greater than L_sp, implying SPW excitation hardly perturbed by the pattern boundaries. If the hole array is a finite n × n square lattice with n representing a number of holes in a single row of the lattice and a hole period of p, the lateral dimension of the lattice is np. For np sufficiently greater than L_sp, the lattice boundary shouldn't be a matter to the EOT. Then, the lowest n for such full SPW excitation, n_s, can be defined as:

For n < n_s, however, partial or incomplete SPW excitation occurs and any plasmonic effects associated with it must be degraded by finite lattice size and as a result insufficient number of holes. For imagery applications such as focal plane arrays (FPAs), their pixel size is typically 20-30 µm in mid- and long-wavelength infrared (LWIR) range [4]. Ideally, L_sp should be comparable to or less than the pixel size so that the physical boundary of a hole array cannot degrade plasmonic enhancement. Then, the actual L_sp is a critical design parameter of a single pixel for full SPW excitation. Particularly, this is very crucial to the low-dimensional photodetectors such as quantum dot and quantum well infrared photodetectors (QDIPs and QWIPs) which require full SPW excitation for the highest detection enhancement by it. The n_s of Eq. (2) therefore impacts these applications.

Basically, the n_s for full SPW excitation is important in understanding its physics related to the spatial decay and interaction with pattern boundaries. Primarily, this work focuses on SPW propagation depending on lattice size and characterizes the n_s required for full plasmonic excitation with the transmission through n-MPCs. The lattice size is varied from n = 2 to 72 large enough for the n_s measurement. We report how they depend on the dielectric properties with the absorption in the QDIPs integrated by identical n-MPCs. Relied on the n_s's from transmission and absorption, the difference between theoretical and experimental SPW propagation length is extracted and analyzed. The transition from local to propagating plasmonic excitation associated with the number of holes in each lattice is addressed. Finally, the photoresponse enhancement of a QDIP by an MPC that can fit into a single pixel of LWIR FPAs is demonstrated.

2. Design of n-MPCs

For the goal of this work, n × n square lattices of circular holes (n = 2, 4, 6, 8, 12, 24, 36, and 72) with p = 3 µm, fabricated into 90 nm-thick Au films, were employed. They were formed by grouping 5184 (= 72 × 72) holes over a fixed 384 × 384 µm² that results in an M_n × M_n array of lattices (M_n = 36, 18, 12, 9, 6, 3, 2, and 1, matching n in the order to keep M_n = 72/n), referred to as n-MPC for each n. Semi-insulating (SI) GaAs is employed for a dielectric. As an example, a part of the 4-MPC that is an 18 × 18 array of 4 × 4 square lattices with hole diameter ∼ p/2 (white dashed square) on an Au film is schematically illustrated in Fig. 1(a). Figure 1(b) is a scanning electron microscopy (SEM) image of the 4-MPC in bird eyes’ view.

 

Fig. 1. (a) An illustration of n-MPC on SI-GaAs substrate. The Au film at the lower side is for exclusive incidence of the light onto the n-MPC at the top. (b) An SEM image of the 4-MPC in bird eyes' view. A single 4 × 4 lattice [a white-dashed square matching that in (a)] with d₄ = 11 µm is indicated on the Au film. (c) An illustration of n-MPC integrated on QDIP. (d) Layer structure of the QD stack. The ML means monolayer. A part of the 4-MPC (i.e., 4 × 3) was employed in (a) and (c) to simplify the illustrations.

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For precise measurement of the SPW propagation, each lattice must be plasmonically independent of others in every n-MPC. The design of n-MPCs, therefore should consider the interaction and crosstalk among neighbor lattices in the spatial distribution of the lattices on a limited area. A critical design parameter is the spacing or the gap between them, d_n. Two important design rules were applied to the n-MPCs. Firstly, d_n [e.g., d₄ in Fig. 1(b)], must be set to avoid strong interactions among lattices which are correlated to p, such as constructive (d_n/p = i_n) and destructive (d_n/p = i_n - 0.5) interference with a positive integer, i_n. In this work, d_n/p = i_n + 1/p or i_n - 1/p, was chosen to avoid them. Secondly, d_n must be maximized on the limited 384 × 384 µm² in each n for the minimal crosstalk between neighbor lattices. For example, d₄ = 11 µm (i₄ = 4) in Fig. 1(b) is the largest spacing available for 4-MPC on the given area.

These design rules can be examined with Fourier transform (FT) structure factor analysis. The structure factor for an n-MPC, F_n, can be written as [5–7]:

 

Fig. 2. (a) Fourier amplitude F₄ of 4-MPC with M₄ = 18 for various d₄'s, and M₄ = 1 (red) (b) Fourier amplitude F₄ of 4-MPC with M₄ = 18 for d₄ = 299 µm (i₄ = 100) (blue), and M₄ = 1 (red) . In each figure, the vertical dashed line indicates k_x = 1/p.

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3. Fabrication and measurement

For transmission, each n-MPC was integrated on a 440 × 440 µm² mesa fabricated into a SI-GaAs substrate, as illustrated in Fig. 1(a). The standard photolithography and electron-beam evaporation were used for their process. As an adhesion promoter, a ∼5 nm-thick Cr film was embedded between the Au film and the dielectric for each n-MPC. The transmission under the normal incidence was characterized with Nicolet 6700 Fourier Transform Infrared (FTIR) spectrometer at room temperature (∼291 K).

For absorption, an n-MPCs identical to that used for the transmission, was integrated on the aperture atop a QDIP processed with the same size mesa for each n, as illustrated in Fig. 1(c). The QDIP was grown on an SI-GaAs substrate by molecular beam epitaxy. Its layer structure, schematically illustrated in Figs. 1(c) and 1(d), is a stack of 30 QD layers having Si-doped InAs QDs and In_0.15Ga_0.85As/GaAs QWs, embedded in each Al_0.1Ga_0.9As layer, which were sandwiched by the n ⁺-GaAs layers for ohmic contact. Their photoresponse was characterized at ∼80 K with the FTIR and an SRS fast Fourier transform 770 network analyzer in the normal incidence from an 800 K blackbody.

Both transmission and absorption were calibrated with another set of devices fabricated on a SI-GaAs substrate and a QDIP, having a single square aperture (= 216 µm × 216 µm² without any hole patterns, opened to incident light) on the same size mesa mentioned above which is equivalent to the effective area of any n-MPCs [= (M_nnp)² with M_nnp = 216 µm, M_nn = 72 and p = 3 µm)]. These are referred to as reference devices. The beam diameter from a Globar source of the FTIR was large enough to cover the whole area of n-MPCs so all the lattices (i.e., all the holes) for each n were assumed to be under uniform irradiance.

4. Results

4.1 Transmission

Figure 3 presents the transmission through the n-MPC/SI-GaAs for some selected n. The primary peaks around 10 µm vary with n in lineshape and intensity. For high n, they maintain Fano lineshape, implying EOT by the fundamental SPW, although it is not clear for low n [2,8,9]. The bold arrows near the bottom point the transmission coupled to higher order SPWs.² Figures 4(a) and 4(b) provide plots of the peak wavelength, λ_n,T, and the transmittance at λ_n,T, T_n, of the fundamental SPW vs. n respectively. The λ_n,T in Fig. 4(a) rapidly redshifts from λ_2,T ∼8.7 µm up to λ_8,T ∼10.4 µm and slightly blushifts to 10.2 µm with increasing n. It becomes invariant for n ${{\lower.3ex\hbox{$\buildrel>\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} }$ 18. For the n-MPCs in this range, their λ_n,T's were identified as the peak wavelengths associated with the fundamental SPW formulated by pIm $\left[ {\sqrt {{\varepsilon_m}{\varepsilon_d}/({{\varepsilon_m} + {\varepsilon_d}} )} } \right]/\sqrt {{i^2} + {j^2}} \; \; $(i ≠ j = ±1 or 0) [2]. Similar lattice size-dependent results have been reported [10,11]. The T_n in Fig. 4(b) is enhanced with n but converges to ∼0.2 for n ${{\lower.3ex\hbox{$\buildrel>\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} }$ 36. These results reveal the SPW excitation is disturbed by lattice boundary or insufficient holes for n < 36. In Fig. 4, λ_n,T is not the same as T_n in the variation with n. It should be noted that λ_n,T doesn’t indicate the resonance wavelength of the fundamental SPW, which is screened by Fano interference in transmission and will be measured more precisely in absorption later [9].

 

Fig. 3. Transmission of the n-MPCs for some selected n at room temperature. Inset: Transmission of the 2-MPC. The arrows with a dashed line indicate the wavelength range [or wavelength (inset)] of the fundamental SPW. The bold arrows identify higher order SPW excitations, which are not clear for n = 2 and 4.

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Fig. 4. (a) A plot of wavelength of the primary peak (λ_n,T) vs. n from transmission. (b) A plot of peak transmission at λ_n,T (T_n) vs. n. Inset: A log-log plot of t_n (= T_n/M_n²) vs. n. The dashed line indicates t_n ∝ n². The solid red line was from the least square fit (∼n^2.6) for n < 36.

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The inset of Fig. 4(b) is the variation of the peak transmittance of a single n × n lattice, t_n, (= T_n/M_n²). As expected, t_n is proportional to n² (total number of holes in a given lattice) for n ${{\lower.3ex\hbox{$\buildrel>\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} }$ 36 where T_n is saturated. However, it drops down from the n² dependence (dashed line) for n < 36, and decreases faster as n is lowered further (∼n^2.6 along the red solid line). This is experimental evidence for partial SPW excitation at the given range of n. Then, the n_s of transmission is n_s,T = 36 for full excitation of the fundamental SPW, which provides the L_sp in transmission, L_sp,T, ∼108 µm from Eq. (2).

4.2 Absorption

The L_sp,T from Figs. 3 and 4 is evidently greater than the size of a single pixel typically used in LWIR FPAs. This work concentrates on the 8-MPC (24 × 24 µm²) of which the lattice is comparable to it but still has insufficient number of holes for full SPW excitation. The absorber of the QDIP shown in Figs. 1(c) and 1(d) highlights the dielectric properties underneath the 8-MPC. The component of the SPW near-fields vertical to the QD stack is coupled to the absorber, satisfies the selection rule of the field polarization for the photoexcitation of electrons, and plasmonically enhances the detection performance [12,13]. Its penetration depth near the wavelength of the fundamental SPW is ∼10 µm, greater than the absorber thickness ∼1.8 µm in Fig. 1(d) for the highest coupling [3,13]. However, its decay from the MPC/QDIP interface is accelerated by the presence of the absorber [14]. As seen later, L_sp is affected in the same way.

Figure 5 shows spectral response from the reference and the 8-MPC device. The reference device (dark yellow) exhibits two peaks, one strong at ∼ 8.7 µm and the other weak and relatively broad at ∼5.5 µm. These are identified as the transitions of an electron from the ground state to the first excited state and to the continuum state above the QD/QW potential barriers in the absorber respectively [12]. The 8-MPC device (violet) shows the photoresponse radically different from the reference device. It has the highest peak at 9.0 µm (= λ_8,A, with a subscript A for absorption) corresponding to the absorption coupled to the fundamental SPW, and two additional peaks at 6.8 µm, and 4.9 µm, which result from the coupling to higher order SPWs (bold arrows), like those in Fig. 2 [13]. In Fig. 5, the peak responsivity of the reference device is 3.0 mA/W, which is increased to 11.2 mA/W at the 8-MPC device. If β_n is the peak responsivity ratio of the n-MPC-integrated QDIP (n-MPC device) to the reference device, it means enhancement of responsivity by plasmonic coupling to the 8-MPC and β₈ ∼3.6.

 

Fig. 5. Absorption of the 8-MPC and the reference device for 3 V at ∼80 K. Inset: Absorption from some selected n-MPC device. The arrows with a dashed line indicate the peak wavelength range (inset) or the peak wavelength of the absorption coupled to the fundamental SPW. The bold arrows identify the peaks coupled to higher order SPW excitations, which are not clear for n = 2 and 4.

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Other n-MPCs were also individually integrated on the QDIPs fabricated with the same process used for the 8-MPC device. The inset of Fig. 5 presents spectral response of selected n-MPC device. Figures 6(a) and 6(b) are plots of the peak wavelength, λ_n,A, and peak responsivity at λ_n,A, A_n, of n-MPC device vs. n. In Figs. 5 and 6(a), λ_2,A = 8.3 µm is slightly lower than the 8.7 µm for the reference device (horizontal blue dashed line) which is comparable to λ_4,A = 8.6 µm, and λ_24,A = λ_72,A = 9.5 µm [15]. In a word, λ_n,A shifts from 8.3 µm to 9.5 µm with increasing n up to 24 and becomes almost invariant beyond it. Obviously, it is free from Fano interference and corresponds to the resonance wavelength of the fundamental SPW [9]. Unlike the transmission of Fig. 4, therefore, λ_n,A is similar to A_n in the variation with n.

 

Fig. 6. (a) A plot of wavelength of the primary peak (λ_n,A) vs. n from absorption. (b) A plot of peak absorption at λ_n,A (A_n) vs. n. The horizontal dashed line in each figure was from the reference device. Inset of (b): A log-log plot of a_n (= A_n/M_n²) vs. n. The dashed line indicates a_n ∝ n². The solid red line was from the least square fit (∼n^2.6) for n < 24.

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The A_n in Fig. 6(b) shows the saturation with n, similar to the T_n in Fig. 4(b). The inset of Fig. 6(b) presents the variation of the peak responsivity of a single n × n lattice, a_n (= A_n/M_n²). Like the t_n in the inset of Fig. 4(b), a_n is proportional to n² (dashed line) for n ${{\lower.3ex\hbox{$\buildrel>\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} }$ 24 but rapidly drops away from the n² dependence (solid red line ∼n^2.6) with decreasing n for n < 24. On the other hand, both λ_n,A and a_n in Fig. 6 imply the n_s and corresponding L_sp for full excitation in absorption, n_s,A ∼24 and L_sp,A ∼72 µm by Eq. (2), different from those of the transmission where it happens at n_s,T ∼36 and L_sp,T ∼108 µm [16]. This experimentally verifies the SPW excitation perturbed by dielectric properties of the absorber as well as the boundaries and number of holes in each lattice.

5. Discussion

The QD stack is a dominant absorber in the QDIP as confirmed by Fig. 5. The L_sp,A reduced from L_sp,T is due to the absorption in the QDIP caused by the change from the lossless SI-GaAs at the given wavelength.⁵ An analytical approximation of the L_sp from Eq. (1) with the absorption coefficient of the dielectric, α_d, is expressed as [14]:

At λ ∼10 µm, $\varepsilon_d^{\prime} \cong 10.69$ and $\varepsilon_m{ }^{\prime}+i \varepsilon_m{ }^{\prime \prime} \sim-4230+i 1440$ for Au [17,18]. Applying these material parameters to Eq. (4) for the n-MPC/GaAs with the α_d negligible at the given wavelength (∼0), results in L_sp ∼570 µm, which is significantly longer than L_sp,T ∼108 µm. The SPW propagation decays faster in the metal film with a hole pattern than in a blanket metal film without it [19]. It should be noted that Eq. (4) assumes a blanket metal film. The uniformity of hole patterns and Au/dielectric interfaces, and the quality of Au films affect SPW propagation. The Cr film embedded between each n-MPC and the SI-GaAs substrate or the QDIP for better adhesion of the Au film also causes fast decay of the SPWs along their interfaces. Although in-depth analysis of the L_sp variation is beyond the scope of this work, large difference between L_sp,T and the ideal L_sp of Eq. (4) implies those factors seriously degrade the SPW propagation and accelerate its decay along the metal/dielectric interfaces. Then, minor difference of L_sp,A from L_sp,T (L_sp,A/L_sp,T = 0.66) means the absorption by the QD stack is a sub-major contribution to its further decay. The ratio of L_sp to L_sp,T implies the experimental propagation length of the n-MPC from the transmission is reduced to ∼20% of that predicted from Eq. (4).

Assuming the same degradation in the n-MPC/QDIP, Eq. (4) provides the α_sp,Q ∼2.3 × 10⁻³ µm^-1 with the ideal SPW propagation length ∼360 (= 5 × 72) µm calibrated from the actual L_sp,A by the correction. This is not very different from the 1.2 × 10⁻³ µm^-1 obtained by the SPW penetration with a different QDIP in magnitude [20]. These imply QDIPs originally have at most α_d ∼10⁻⁴ µm^-1 without MPC, speculated from the responsivity enhancement by plasmonic coupling ∼10 × seen later. Such a moderate absorption coefficient is expected from low fill factors of the QDs in self-assembled epitaxy. There could be SPW reflection at the lattice boundaries and multiple pass effects which affect the QDIP absorption. Rigorous theoretical approach to a given device structure counting those secondary enhancements is necessary for accurate calculation of α_sp,Q. Nonetheless, the experimental L_sp,A that includes such additional factors and provides the α_sp,Q consistent with the result of our previous work, supports the correction to the theoretical SPW propagation length extracted above [20]. It also confirms that the further decay of the SPWs by the dielectric properties of the given QDIP absorber is ∼30% with its reduction from L_sp,T.

From the n-MPCs with low n in Fig. 6(a), the λ_1,A ∼8 µm for n = 1, plasmonic excitation with a single hole, is speculated as a crude extrapolation. This could be one of the local plasmonic modes of which the propagation is supported by the hole periodicity in an MPC [21,22]. The resonance energy of such a mode is critically affected by hole geometry such as shape, lateral dimension, and depth, and lowered by the interaction with nearby identical holes through broken degeneracy when they approach each other. For more number of those holes, the lower bound resonance energy is decreased further and λ_n,A shows larger red shift with increasing n, as observed in Fig. 6(a). This is analogous to the variation of an energy level isolated in a single atom as other atoms approach it (i.e., transition from a discrete energy level in a single atom to a Bloch wave in the energy band of a solid). Then, the red shift in Fig. 6(a) is interpreted as the delocalization of a plasmonic excitation isolated in individual holes to the propagating SPWs by gathering them to form an n × n lattice. Only a few articles reported the surface plasmonic excitation of a single hole and an array of holes together but with no results on the transition between them discussed above [23,24].

The transmission and the absorption associated with a single hole in each n × n lattice, t_n,1 and a_n,1, was extracted from the t_n and the a_n in the insets of Figs. 4(b) and 6(b) divided by n², the number of holes in an n × n lattice. Figure 7 shows the plots of t_n,1 and a_n,1, normalized by t_72,1 and a_72,1 respectively, vs. n², where both are roughly proportional to n^0.6 ∼ $\sqrt[3]{{{n^2}}}$, and saturated beyond n_s,T and n_s,A. They can be expressed as:

 

Fig. 7. A log-log plot of t_n,1 [ = t_n/(n²t₇₂)] and a_n,1 [ = a_n/(n²a₇₂)] vs. n². The dashed line corresponds to $\sqrt[3]{{{n^2}}}$, which is not curve fitting but only for eye guiding, and therefore belongs neither t_n,1 nor a_n,1.

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Here, t_n,1 = t_n/(n²t₇₂), a_n,1 = a_n/(n²a₇₂), corresponding to the normalized transmission and absorption by a single hole in a given n-MPC. In Fig. 7, despite some fluctuations and different d_n's, both consistently show noticeable tendency, parallel to the dashed line which simply indicates $\sqrt[3]{{{n^2}}}$ dependence without any fitting. This implies the finite crosstalk predicted in the second design rule with Fig. 2 doesn't affect the experiment seriously. The plot confirms n_s,A ∼24 < n_s,T ∼36, by the a_n,1 higher than t_n,1 in data distribution although they are almost identically dependent on n².

Researches on the MPCs having finite number of holes are rare but a similar relation from a one-dimensional hole array which reveals a nonlinear dependence on n was reported [25]. In a 2D hole array, although it was not about transmission, another nonlinear dependence on n² associated with plasmonic excitation was demonstrated for low n [26]. Also, it has been interpreted as an array-enhancement factor [11]. Such a nonlinearity of the transmission on n² for the n less than n_s is not observable from optical gratings at the same dimension where SPW excitation is absent and each hole constantly maintains its transmission through it so the intensity of the zeroth order diffraction is linearly proportional to the number of holes regardless of the proximity to their neighbor holes at a given wavelength. For this reason, Eq. (5) provides another experimental evidence that Figs. 4 and 6 are due to the plasmonic excitation by the holes in each n-MPC whereas the results for n = 2 and 4 are not very clear in lineshape and peak identification. While the $\sqrt[3]{{{n^2}}}$ dependence in Eq. (5) is not understood yet, it clearly reveals a transition from local to propagating surface plasmonic excitation mentioned above. The formula of the SPW resonance wavelength for an MPC used earlier for peak identification may not be applicable in low n where the characteristics of the local plasmonic excitation is prominent. It is evident that so-called edge effect meaning the hole rows near the edge of the lattice, where the SPW propagation is abruptly decayed by the discontinuity of the pattern, is another major reason for the drop from the linearity on n² and the $\sqrt[3]{{{n^2}}}$ dependence of a single hole in both transmission and absorption. Further study to understand the transition from local to propagating plasmonic excitation that is revealed in Eq. (5) is presently under way.

Finally, β₈ ∼3.6 is a result of front-side illumination (light incident on n-MPC). In Fig. 6(b), A₇₂ reaches 30.3 mA/W and β₇₂ is increased to ∼10, which corresponds to full plasmonic enhancement in responsivity. It has been reported that the plasmonic enhancement of QDIPs becomes more than double at back-side illumination (light incident on substrate) [27,28]. Since commercial FPAs use this scheme, 2β₈ ∼7 × would be the minimal enhancement achieved by the 8-MPC on a pixel from the given absorber. Although this is approximately one third of the full plasmonic enhancement, the 8-MPC fit into a single pixel demonstrates noticeable upgrade of photoresponse in spite of its incomplete SPW excitation and would be practically useful for the FPAs depending on the purpose of its applications. Higher α_QDIP means lower L_sp. Arithmetically, full SPW excitation in a single pixel for infrared spectral imagery is achievable if α_QDIP is increased up to ∼0.08 µm^-1 from Eq. (4). Ultimately, radical improvement of α_QDIP by enhancing areal density of QDs and number of QD layers without strain relaxation therefore should be the primary issue on the QDIP performance for FPAs.

6. Summary and conclusions

The propagation of the SPWs excited in n-MPCs has been investigated. Conserving the total number of holes and grouping them to form an array of lattices on the fixed area of the Au film provide the transmission through individual lattices and the absorption coupled to them that vary with n clearly. Their convergence with n reveals the SPW excitation significantly affected by the number of holes and physical boundary of the lattice. The fundamental SPW excited in the n-MPC/SI-GaAs and the n-MPC/QDIP has the propagation length of ∼108 µm for n ${{\lower.3ex\hbox{$\buildrel>\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} }$ 36 and ∼72 µm for n ${{\lower.3ex\hbox{$\buildrel>\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} }$ 24, respectively. The lower propagation of the SPWs coupled to the QDIP is interpreted by their interaction with the absorber which causes fast decay of the SPW near-fields along the interface. The comparison of the experimental SPW propagation lengths with the theoretical predictions provides a correction factor of ∼5, calibrating their difference. The transition from local to propagating plasmonic excitation associated with the variation of the hole numbers has been addressed with the characteristics of a single hole, nonlinearly depending on n² in both transmission and absorption, for each lattice. An 8 × 8 lattice (24 × 24 µm²), far below the SPW propagation length in lateral dimension but comparable to a single pixel size of commercial FPAs, has demonstrated plasmonic enhancement up to ∼7 × the QDIP performance.