## Abstract

Heavily doped n-type Ge and GeSn are investigated as plasmonic conductors for integration with undoped dielectrics of Si, SiGe, Ge, and GeSn in order to create a foundry-based group IV plasmonics technology. N-type Ge_{1-}* _{x}*Sn

*with compositions of 0 ≤ x ≤ 0.115 are investigated utilizing effective-mass theory and Drude considerations. The plasma wavelengths, relaxation times, and complex permittivities are determined as functions of the free carrier concentration over the range of 10*

_{x}^{19}to 10

^{21}cm

^{−3}. Basic plasmonic properties such as propagation loss and mode height are calculated and example numerical simulations are shown of a dielectric-conductor-dielectric ribbon waveguide structure are shown. Practical operation in the 2 to 20 μm wavelength range is predicted.

© 2012 OSA

## 1. Introduction

Metallic conductors have traditionally played a key role in plasmonics with the semi-metals of Bi, Sb [1] and ErAs [2] having been introduced recently to plasmonics as alternatives for metals in several situations. However, there is a novel “all-semiconductor plasmonic approach” [3] that avoids the use of metals and semi-metals. This technique appears significant because it is manufacturable and capable of chip-scale or wafer-scale “plasmonic circuit” integration in a manner analogous to that of waveguide integrated photonics. For the all-semiconductor approach that employs III-V materials, the optimal plasmonic conductors are heavily doped antimonide and arsenide semiconductors. There is also a competing Si-based all-semiconductor approach called “silicon plasmonics” which has its own set of special materials. Initial research on silicon plasmonics identified two key conductors; heavily doped Si [4–6] and the various metal silicides such as Pd_{2}Si [6, 7]. In that technology, plasmonic waveguides consist of a “dielectric” (undoped Si) and either of the two mentioned conductors.

It is known that electrically biased PN and PIN heterostructures and multi-quantum-well structures of GeSn/SiGeSn have gain properties and electronic device properties [8]. Therefore it is likely that a variety of such devices can be monolithically integrated on the above-mentioned plasmon-photonic structures in order to excite, amplify and detect surface plasmon polaritons (SPPs) on a chip or wafer. Active group IV SPP routing switches and modulators also appear feasible for integration. The aforementioned silicon plasmonics may then be generalized into “group IV plasmonics’ by expanding the set of dielectrics to include group IV elements or alloys such Ge, SiGe, or GeSn.

The main goal of this work is to propose and analyze group IV conductors that are practical companions of the aforementioned dielectrics. Specifically, we are proposing and investigating heavily doped n-Ge and n-GeSn alloys as associated integratable conductors. The possibility of plasmonics on these materials is investigated over the 1 μm to 20 μm operational wavelength range. This work is completed utilizing Drude theory and a conductivity effective mass theory for GeSn alloys that takes into account both the *L* and Γ valleys in the conduction band, which is also presented. In particular we determine the plasma wavelengths, the relaxation times, and the complex permittivities of Ge_{1-}* _{x}*Sn

*for alloy compositions from*

_{x}*x*= 0 (Ge) to

*x*= 0.115 and for doping concentrations that range from 10

^{19}to 10

^{21}cm

^{−3}. The determined plasma wavelengths for highly doped n-Ge

_{1-}

*Sn*

_{x}*alloys are comparable to those for narrow-gap III-V materials [3] and enable mid-wave and long-wave infrared plasmonics. The importance of ε′ and ε″ is in SPP waveguiding where these values quantify the tradeoff between propagation loss and how far the mode penetrates into the dielectric [6, 7]. The results presented here allow tailoring of SPP mode confinement and propagation loss at desirable infrared wavelengths as part of a new group IV technology.*

_{x}## 2. Plasmonic and plasmonic-photonic waveguides

A plasmonic waveguide is a channel waveguide elongated along the Z-axis, the direction of surface plasmon propagation. The cross sectional directions of the channel, X and Y, are sub-wavelength or deep sub-wavelength, which is advantageous for plasmonics. Denoting a dielectric as D and a conductor as C, the channel construction is typically CD, CDC, or DCD [9]. When comparing these typical plasmonic waveguide structures to photonic waveguides, the photonic structure relies upon internal reflections from the interfaces between two different dielectrics, one of which could be an intrinsic group IV semiconductor and the other air or insulator.

Since the above-mentioned 3 dimensional plasmonic waveguides usually have a dielectric/air interface that traps electromagnetic waves via internal reflection, these plasmonic waveguides often have associated photonic waveguiding properties. We could deliberately introduce photonic confinement to supplement that offered by the plasmonic confinement. In other words, we can readily design 3 dimensional channel waveguides that are plasmonic-photonic “hybrids” in terms of mode confinement.

Two examples are given in Fig. 1 that illustrate potential plasmonic-photonic waveguides. The first Si-based CD channel (Fig. 1 a) contains a heavily doped strip of Ge (or GeSn) which gives lateral mode confinement in the X direction as well as mode “attachment” to the conductor in the Y direction. The finite-height (sub-wavelength) undoped strip of Ge or GeSn above the conductor also gives some optical mode confinement at its upper air/dielectric interface. A second hybrid, a DCD channel as shown in Fig. 1 (b), consists of a thin ribbon of doped Ge (or GeSn) that is buried within a dielectric strip of undoped Ge (or GeSn) and here the upper and lower interfaces of the strip give optical internal reflection of the mode which may aid the plasmonic ribbon confinement.

## 3. Plasmonic properties of heavily doped Ge and GeSn

In this paper we investigate the five alloy compositions of undoped relaxed crystalline Ge_{1}* _{x}*Sn

*that were studied by Ragan and Atwater [10] namely, x = 0.035, 0.05, 0.06, 0.08 and 0.115. These authors present credible results on the direct-bandgap energy E*

_{x}_{g}of each alloy. Taking

*E*=

_{g}*hc*/λ

*we determine the corresponding direct gap wavelengths λ*

_{g}*to be 1.55, 1.81, 1.97, 2.02, 2.26, and 2.78 μm for*

_{g}*x*= 0, 0.035, 0.05, 0.06, 0.08 and 0.115 respectively. For

*x*> 0.08 it is noted that this semiconductor is likely “truly direct”.

We can write the complex permittivity of the GeSn as ε = ε′ + i ε″, where both components are functions of the infrared free space wavelength, λ, that varies here from 1 to 20 μm. For heavily doped GeSn with a given doping density N, there is a corresponding plasma frequency ω_{p} and a plasma wavelength λ* _{p}*. The real part of the permittivity, ε′, changes from positive values to negative values near λ

*. This n-GeSn becomes useful as a plasmonic conductor typically at wavelengths longer than λ*

_{p}_{p}. Thus, λ > λ

_{p}is a general criterion for chemical and biological sensing as well as on-chip opto-electronics. Regarding the photonic properties of the undoped “dielectric’ GeSn, we can say this material is transparent to electromagnetic waves at wavelengths longer than λ

*. Therefore, referring to the hybrid-mode structures in Fig. 1, we want an operation wavelength that satisfies both photonic transparency and plasmonic confinement, i.e., λ > λ*

_{g}*and λ > λ*

_{g}*. Later in this work, we determine that this dual requirement is generally satisfied in the mid-wave and long-wave IR regimes.*

_{p}## 4. Determination of ω_{p}, τ, ε′ and ε″ for n-GeSn

The theory presented here indicates that heterostructures of n-GeSn with GeSn are viable for plasmonic devices in the mid-wave and long-wave regions. We shall support that thesis by presenting analytical results for the key plasmonic parameters of λ_{p}, τ, and both ε′ and ε″ over a wavelength range of 1 to 20 μm. The parameters used are carrier concentration N and the Ge_{1-}* _{x}*Sn

*composition amount*

_{x}*x*.

Empirical optical constants determined from heavily doped n-type Ge_{0.98} Sn_{0.02} [11] and Ge_{0.97} Sn_{0.03} [12] as well as Ge [11] show no effects of optical transitions in the IR range of interest. Empirical optical constants determined from p-type Ge_{0.98} Sn_{0.02} on the other hand show intervalence-band transition effects [11, 13] in the wavelength range of 5-12 μm which are not negligible when compared with free carrier response. Since we are discussing n-type Ge_{1-}* _{x}*Sn

*however, optical transitions are negligible and the Drude model suffices to determine the optical constants. The real and imaginary parts of the permittivity are determined by*

_{x}_{∞}is the high frequency dielectric constant. The plasma frequency is determined by

*e*is the electron charge,

*m*is the conductivity effective mass, and ε

_{c}*is the permittivity of free space. The relaxation time is determined bywhere μ is the carrier mobility whose value depends upon N.*

_{o}In determining the complex permittivity, it is first necessary to determine the conductivity effective mass of electrons *m _{c}* in Ge

_{1-}

*Sn*

_{x}*. GeSn and Ge have*

_{x}*L*valleys in the conduction band as well as a Γ conduction valley at the k = 0 zone center, the direct bandgap location of the relaxed alloy. The conductivity effective mass

*m*consists of two contributions; one from the anisotropic

_{c}*L*-valley masses,

*m*, and one from the Γ-valley mass,

_{L}*m*. In GeSn alloys, it is not perfectly clear what the

_{Γ}*L*-valley effective mass is, but for dilute alloys we can say to a first approximation that

*m*is the same as the Ge

_{L}*m*, namely

_{L}*m*(GeSn) =

_{L}*m*(Ge) = 0.120 [14]. In determining

_{L}*m*, we utilize the theory presented in Ref [15], which we re-write as

_{Γ}*a*(Ge) and

*a*(

*x*) are the cubic lattice parameters of Ge and the GeSn alloys respectively, and

*m*is the electron mass. For lack of better information, we take Δ as a constant 0.29 eV, the same as Ge [15] across the various dilute alloy compositions. Ref [15]. also provide a good approximation to the lattice parameter as a function of Sn content

_{o}*x*,

*E*values are taken from Ref [10]. as a function of

_{g}*x*. Once

*a*(

*x*),

*a*(Ge) and

*E*(

_{g}*x*) are determined, the

*p*parameters are found using Eq. (6). Equation (5) can then be used with

*p*(

*x*) and

*E*(

_{g}*x*) and Δ to determine the zone-center electron effective masses,

*m*. Table 1 presents

_{Γ}*a*(

*x*),

*E*(

_{g}*x*) and

*m*values as determined directly from Eq. (5). The presented theory determines the trend of

_{Γ}*m*versus

_{Γ}*x*, however it does not connect accurately to the known Ge mass value. Thus the

*m*data, determined directly from Eq. (5) and labeled

_{Γ}*m*[

_{Γ}*raw*]in Table 1, requires an adjustment. To obtain agreement with the actual

*m*value for Ge, the

_{Γ}*m*[

_{Γ}*raw*]values are normalized with respect to 0.041

*m*[14]. These

_{o}*m*[

_{Γ}*norm*] values, which are also presented in Table 1, will be referred to simply as

*m*for the remainder of this work.

_{Γ}For a given GeSn composition, each of the two effective mass contributions is also “weighted” by the number of electrons residing in its valley(s). If there is a total number *N* of electrons in all conduction bands, then we assume there is a fraction of those carriers, *f _{L}*, that is found in the

*L*-valleys, and a fraction,

*f*

_{Γ}, that resides within the Γ-valley, with a further assumption that

*f*+

_{L}*f*

_{Γ}= 1.0. For the case of

*x*= 0, or simply Ge,

*f*= 1 and

_{L}*f*

_{Γ}= 0. For the case where the Ge

_{1-}

*Sn*

_{x}*alloy composition in which the bandgap becomes truly direct (*

_{x}*x*= 0.08 as noted earlier),

*f*is approximately equal to

_{L}*f*

_{Γ}, giving both values of 0.5. A linear extrapolation is used to determine

*f*, as well as

_{L}*f*

_{Γ}by association, as a function of

*x*. The conductivity effective mass is then found by

*x*.

The electron mobility μ is needed here for Ge_{1-}* _{x}*Sn

*as a function of N over the investigated range of 10*

_{x}^{19}cm

^{−3}to 10

^{21}cm

^{−3}. Ref [16] gives measured mobility for phosphorous doped Ge

_{0.98}Sn

_{0.02,}mobility values that agree very well with that of Ge while the mobility is slightly lower for arsenic doping. The slightly higher mobility has the effect of pushing the permittivity negative at lower wavelengths which is ideal for plasmonics. Also in Ref [16] it is noted that the Ge-P bond is stronger than the Ge-As counterpart with the former being stable and the latter decomposing over a short amount of time. For these reasons n-Ge

_{1-x}Sn

_{x}should use phosphorous as its dopant. The μ values used for n-Ge and n- Ge

_{1-}

*Sn*

_{x}*here are 400, 200, and 150 cm*

_{x}^{2}/Vs [16] for N of 10

^{19}, 10

^{20}, and 10

^{21}cm

^{−3}respectively.

Using the *m _{c}* and μ values, we immediately find the τ values using Eq. (4). After that, the λ

*are determined from Eq. (3) using the calculated*

_{p}*m*and an ε

_{c}_{∞}that varies with

*x*. ε

_{∞}is taken to be 16 for Ge and 24 for Sn. The high frequency dielectric constant is then taken to be a linear function of the composition amount

*x*using ε

_{∞}= 16 + 8

*x*, with the values being given in Table 3 . Table 3 also summarizes our findings of

*λ*and τ at three selected values of N while Fig. 2 pictorially presents λ

_{p}*as a function of N, for n-Ge*

_{p}_{1-}

*Sn*

_{x}*, as well as n-Ge and n-Si. An increasing ratio of Sn/Ge in the GeSn alloys gives blue-shifting plasmon wavelengths and decreasing of relaxation times. As an example with*

_{x}*N*= 10

^{19}cm

^{−3}, n-Ge has a plasmon wavelength of 14.7 μm which shifts to 7.82 μm for Ge

_{0.885}Sn

_{0.115}. At higher doping, the difference in plasmon wavelength with Sn alloying of Ge decreases although in each case λ

*for*

_{p}*x*= 0.115 is approximately double that of n-Ge.

It is worthwhile to compare these new results with results representing the silicon plasmonics branch of group IV plasmonics, so additional results are presented here for n-Si. For Si, the conductivity effective mass of 0.26 m_{o} [14] was employed, along with ε_{∞} = 11.7, and μ = 114, 95, and 92.4 cm^{2}/Vs for N = 10^{19}, 10^{20}, and 10^{21} cm^{−3} respectively [17]. Our estimates of the Si response are presented alongside the GeSn results below. The n-Ge has λ_{p} at a wavelength which is ~80% of that for n-Si. This comparison supports our assertion that n-Ge is slightly better suited for plasmonics than n-Si due to the ability to excite plasmons at a particular wavelength with a lower doping, thus relaxing fabrication constraints. It should be noted however that this relaxed doping constraint in n-Ge versus that of n-Si, likely comes with a tradeoff of an increase in the difficulty of fabrication processes.

Figure 3
presents complex infrared permittivity values for n-GeSn alloys compared with n-Ge (*x* = 0) and n-Si plotted from 1 to 20 μm as calculated by Eqs. (1) and 2. As expected from Table 3, the lower doping has real permittivities that cross zero near the middle of this infrared range. The zero crossing, which doesn’t occur exactly at the Drude *λ _{p}* utilized here due to the effects of τ as can be seen in Eq. (1), blue shifts with increasing

*x*and

*N*as expected. It must be kept in mind that the direct bandgap wavelength, determined from Eq in Table 1, increases to ~2.8 μm for x = 0.115. So even though the plasmon wavelength is pushed into the NIR for 10

^{21}cm

^{−3}, at this large

*x*the absorption due to the direct bandgap will dominate which practically rules out these GeSn alloys for NIR plasmonics. Although in those cases of large

*x*, MIR operation still remains quite feasible.

The decreasing effective mass with increasing Sn concentration gives a ε′ that reaches a negative limit in the investigated IR wavelengths. This limit for the real part of the permittivity is

_{∞}as the μ, N or

*m*decreases. However, ε″ has no such limit as it constantly increases with decreasing ω according to the Drude model. So while ε′ goes negative at shorter wavelengths for larger Sn amounts and larger doping, ε′ approaches the limit determined by Eq. (9) while ε″continues increasing. Increasing ε″ at long wavelengths with a constant ε′ is indicative of broadening plasmon resonance features.

_{c}## 5. n-GeSn plasmonic and plasmonic-photonic waveguide properties

The plasmonic mode confinement and propagation length are important in determining the usefulness of any conducting material to plasmonic applications. As such, we present here first calculations of these parameters for a purely plasmonic mode at a conductor-dielectric interface. The 1/e plasmon field mode height into the dielectric or conductor can be determined by

The conducting permittivities, ε_{c}, used are those calculated in this work for doped Ge_{1}* _{x}*Sn

*, Ge, and Si as were shown in Fig. 3. The dielectric material with permittivity, ε*

_{x}_{d}, is the same material without doping. Thus ε

_{d}is the same as ε

_{∞}, which has also been calculated earlier for Ge, GeSn and Si. The more specific optical wavelength limit for exciting “ideal” surface plasmons in a given structure is found by

*λ*>

*λ*[(ε

_{p}_{∞}+ ε

_{d}) / ε

_{∞}]

^{1/2}. Since ε

_{d}= ε

_{∞}for these calculations, this limit is the same as the well-known plasmon limit of

*λ*>

*λ*[2]

_{p}^{1/2}, which is where we limit our calculations for clarity purposes, although this may not be the exact limit for plasmon aided confinement of photonic structures.

Figure 4
presents calculations of the plasmon intensity propagation loss and plasmon mode heights with only two compositions of Ge_{1-x}Sn* _{x}* being presented for clarity. At a doping of N = 10

^{19}cm

^{−3}(Fig. 4 top), only Ge

_{1-}

*Sn*

_{x}*with the*

_{x}*x*> 0.06 appear to enable plasmonic activity below 20 μm. Doped Si and Ge with N = 10

^{19}cm

^{−3}does not appear to be useful for plasmonics in the wavelength range under discussion. Higher doping pushes all materials to become plasmonic at lower wavelengths as expected. For mid- to long-wave IR plasmonics, doping of at least ~10

^{20}cm

^{−3}may be required. As an example at a wavelength of 10 μm with N = 10

^{20}cm

^{−3}, the propagation loss for the n-Ge

_{0.885}Sn

_{0.115}(~10

^{4}dB/cm) is 6 times smaller than that of n-Si while the mode height into the dielectric is 2.5 times larger (~1 μm) for n- Ge

_{0.885}Sn

_{0.115}than n-Si. All plasmonic materials under investigation here however still have very large propagation losses when compared to metal silicides and noble metals [18] in this wavelength regime.

With the knowledge of basic plasmon properties for these materials, we now investigate plasmonic-photonic waveguides via the DCD ribbon structure (Fig. 1 b). Let us first discuss how this structure may be physically realizable. The width and height of our hybrid Ge_{1-}* _{x}*Sn

*strip waveguide is denoted as W and H, respectively. If the free space wavelength of operation is λ, then W will be approximately λ/3, with the corresponding H being approximately λ/5. This satisfies the single-dielectric-mode condition as determined by Fig. 5b of Ref [19]. That work however investigated sub-wavelength silicon waveguides in the short-wave infrared, so work in the optimization of these dimensions remains before actual implementation of such Ge*

_{x}_{1-}

*Sn*

_{x}*waveguide structures.*

_{x}A physical realization would begin with an epitaxial growth of an undoped relaxed crystalline layer of Ge_{1-}* _{x}*Sn

*of thickness H/2 upon a Si wafer, with the dislocations due to the lattice mismatch pinned at the lower interface of this layer. Next, using photomasking on the top of the Ge*

_{x}_{1-}

*Sn*

_{x}*, the doping impurity phosphorous is ion implanted through the mask to form a shallow implant of depth much less than λ in the Ge*

_{x}_{1-}

*Sn*

_{x}*. The doping stripe would have a width of approximately width λ /4 and may or may not be driven inward by heat. Following this ion implantation, a second undoped crystal film of Ge*

_{x}_{1-}

*Sn*

_{x}*with thickness H/2 is epitaxially grown upon the first Ge*

_{x}_{1-}

*Sn*

_{x}*layer. Then the composite H layer with buried mid-level n-doped stripe is photolithographically etched into the plasmonic-photonic DCD structure having a cross-section of W x H.*

_{x}Lumerical finite-difference time-domain (FDTD) 3D Maxwell solver was used to model DCD structures of Ge_{1-}* _{x}*Sn

*similar to that described above and shown in Fig. 1 (b). For this example, the free space wavelength is chosen to be 8 μm, the conducting ribbon is chosen to be Ge*

_{x}_{0.94}Sn

_{0.06}with N = 10

^{21}cm

^{−3}, and the waveguide surrounding the ribbon utilized the software material library for lossy Ge for simplicity. The Lumerical mode solver was utilized to choose a propagating mode with a single intensity peak nearest to the center of the structure. The incident mode is mostly TM polarized with calculated propagation loss of ~3 dB/cm in the plain lossy Ge. To illustrate photonic modes being confined by added plasmonic effects, the ribbon was chosen to be 80 nm high (λ/100) and 8 μm wide (λ) with the waveguide being 16 μm wide (2λ) and 1.6 μm high (λ/5). Figure 5 presents

*E*intensity cross sections in the X-Y plane, or along the ribbon, at multiple distances from the excitation point. Near the source, the observed mode is nearly that of the photonic mode, which was confirmed by separate simulations propagating through the same structure with no ribbon. At distances of multiple microns away from the source, it is apparent that the mode becomes increasingly asymptotically compressed which is indicative of a plasmon like mode. In the direction of propagation, the mode has high loss which, when compared to the case with no ribbon, further indicating the features observed may be plasmonic in nature. For this material, the basic plasmon propagation intensity loss is estimated at 2000 dB/cm, from Fig. 4, which equates to an order of magnitude loss over a distance of 5 μm. While high loss is observed in Fig. 5, less than a half of an order of magnitude of loss is observed over 16 μm simulation region, which is indicative of this mode being a plasmonic-photonic hybrid as opposed to purely plasmonic. Simulation results using a ribbon with height ~1/4 of those shown in Fig. 5, indicates that thinning of the ribbon can indeed decrease loss of the hybrid mode. Future work may investigate the optimum ribbon thickness and other structure parameters necessary for Ge

_{y}_{1-}

*Sn*

_{x}*based plasmonic-photonic hybrid structures with minimized loss and maximized mode confinement.*

_{x}## 6. Summary

The purpose of this work is to expand the repertoire of silicon plasmonics into a wider and more capable silicon-based “group IV plasmonics” that is readily manufacturable and avoids the use of both metals and semi-metals. The idea is to employ heavily doped Ge, GeSn, and SiGeSn as conductors that join easily with undoped “dielectrics” of Si, Ge, GeSn and SiGeSn to form composite plasmonic structures such as channel waveguides. These group IV material composites can utilize a combination of surface plasmon mode confinement and internal-reflection photonic mode confinement in hybrid-mode “plasmonic-photonics”. Alternatively, purely plasmonic structures can couple readily to photonic structures.

Some key applications include on-chip opto-electronics such as chemical and biological sensing, signal processing, and interconnection. For all these it is necessary to know the negative permittivity spectral regions of the n-GeSn as well as its detailed ε′ + i ε″ properties over the particular infrared wavelengths of interest from 1 to 20 μm. In this paper we have determined the plasma wavelengths, relaxation times, and the detailed real-and-imaginary permittivity responses of n-type Ge_{1-x}Sn_{x} over the 10^{19} to 10^{21} cm^{−3} doping density range for alloy compositions ranging from x = 0 to x = 0.115.

Plasma wavelengths determined in this work are found to be in the 1 to 5 μm region for very high doping, or N greater than ~10^{20} cm^{−3}, and in the 7 to 15 μm range for more accessible doping. We find that the λ_{p} for n-GeSn are in fact similar to those found in heavily doped samples of GaAs, InP, GaSb and AlSb [3]. As the Sn content of GeSn increases, the direct bandgap narrows and the gaps examined in this paper are similar to those of GaSb and its ternary alloys. We also find λ_{p} for n-Ge is 1.3x shorter than that of n-Si for the same doping. With increasing wavelength, the ε′ are found to reach an asymptotic value for *x* approaching 0.1, and this saturation is pronounced for very heavy doping. We find that ε′ is generally comparable-in-magnitude to ε″ in the mid-wave infrared region. The ε′ and ε″ determined, as well as the basic conductor-dielectric plasmon properties such as propagation loss and mode height shown here, will all enable the specific design of a variety of plasmonic devices. Numerical simulations presented here are indicative of a hybrid plasmonic-photonic mode, although future work will optimize such hybrid waveguides.

## Acknowledgments

This work is supported in part by the Air Force Office of Scientific Research, Gernot Pomrenke, Program Manager, under Grant Number FA9550-10-1-0417 and LRIR award numbers 09RY01COR and 12RY10COR.

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