Mid- to long-wavelength infrared plasmonic-photonics using heavily doped n-Ge/Ge and n-GeSn/GeSn heterostructures

doi:10.1364/OE.20.003814

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Mid- to long-wavelength infrared plasmonic-photonics using heavily doped n-Ge/Ge and n-GeSn/GeSn heterostructures

Richard Soref, Joshua Hendrickson, and Justin W. Cleary »View Author Affiliations

Optics Express, Vol. 20, Issue 4, pp. 3814-3824 (2012)
http://dx.doi.org/10.1364/OE.20.003814

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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-_xSn_x 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 10¹⁹ to 10²¹ 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.

1. Introduction

Metallic conductors have traditionally played a key role in plasmonics with the semi-metals of Bi, Sb [1

J. W. Cleary, G. Medhi, R. E. Peale, W. Buchwald, O. Edwards, and I. Oladeji, “Infrared surface plasmon resonance biosensor,” Proc. SPIE 7673, 767306, 767306-11 (2010). [CrossRef]

] and ErAs [2

A. M. Crook, H. P. Nair, D. A. Ferrer, and S. R. Bank, “Suppression of planar defects in the molecular beam epitaxy of GaAs/ErAs/GaAs heterostructures,” Appl. Phys. Lett. 99(7), 072120 (2011). [CrossRef]

] having been introduced recently to plasmonics as alternatives for metals in several situations. However, there is a novel “all-semiconductor plasmonic approach” [3

D. Li and C. Z. Ning, “All-semiconductor active plasmonic system in mid-infrared wavelengths,” Opt. Express 19(15), 14594–14603 (2011). [CrossRef] [PubMed]

] 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

M. Shahzad, G. Medhi, R. E. Peale, W. R. Buchwald, J. W. Cleary, R. Soref, G. D. Boreman, and O. Edwards, “Infrared surface plasmons on heavily doped silicon,” submitted to J. Appl. Phys. (2011).

–6

R. Soref, R. E. Peale, and W. Buchwald, “Longwave plasmonics on doped silicon and silicides,” Opt. Express 16(9), 6507–6514 (2008). [CrossRef] [PubMed]

] and the various metal silicides such as Pd₂Si [6

R. Soref, R. E. Peale, and W. Buchwald, “Longwave plasmonics on doped silicon and silicides,” Opt. Express 16(9), 6507–6514 (2008). [CrossRef] [PubMed]

, 7

J. W. Cleary, R. E. Peale, D. J. Shelton, G. D. Boreman, C. W. Smith, M. Ishigami, R. Soref, A. Drehman, and W. R. Buchwald, “IR permittivities for silicides and doped silicon,” J. Opt. Soc. Am. B 27(4), 730–734 (2010). [CrossRef]

]. 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

G. Sun, R. A. Soref, and H. H. Cheng, “Design of a Si-based lattice-matched room-temperature GeSn/GeSiSn multi-quantum-well mid-infrared laser diode,” Opt. Express 18(19), 19957–19965 (2010). [CrossRef] [PubMed]

]. 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-_xSn_x for alloy compositions from x = 0 (Ge) to x = 0.115 and for doping concentrations that range from 10¹⁹ to 10²¹ cm⁻³. The determined plasma wavelengths for highly doped n-Ge_1-_xSn_x alloys are comparable to those for narrow-gap III-V materials [3

D. Li and C. Z. Ning, “All-semiconductor active plasmonic system in mid-infrared wavelengths,” Opt. Express 19(15), 14594–14603 (2011). [CrossRef] [PubMed]

] 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

R. Soref, R. E. Peale, and W. Buchwald, “Longwave plasmonics on doped silicon and silicides,” Opt. Express 16(9), 6507–6514 (2008). [CrossRef] [PubMed]

, 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.

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

S. Y. Cho and R. A. Soref, “Low-loss silicide/silicon plasmonic ribbon waveguides for mid- and far-infrared applications,” Opt. Lett. 34(12), 1759–1761 (2009). [CrossRef] [PubMed]

]. 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.

Fig. 1 Examples of a CD (a) and DCD (b) waveguide configuration. The conductor is heavily doped Ge or GeSn while the dielectric is undoped Ge or GeSn.

3. Plasmonic properties of heavily doped Ge and GeSn

In this paper we investigate the five alloy compositions of undoped relaxed crystalline Ge₁_xSn_x that were studied by Ragan and Atwater [10

R. Ragan and H. A. Atwater, “Measurement of the direct energy gap of coherently strained Sn_xGe_1-x/Ge(001) heterostructures,” Appl. Phys. Lett. 77(21), 3418–3420 (2000). [CrossRef]

] namely, x = 0.035, 0.05, 0.06, 0.08 and 0.115. These authors present credible results on the direct-bandgap energy E_g of each alloy. Taking E_g = hc/λ_g 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 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_p. This n-GeSn becomes useful as a plasmonic conductor typically at wavelengths longer than λ_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_g. 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_p. Later in this work, we determine that this dual requirement is generally satisfied in the mid-wave and long-wave IR regimes.

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-_xSn_x composition amount x.

Empirical optical constants determined from heavily doped n-type Ge_0.98 Sn_0.02 [11

V. R. D’Costa, J. Tolle, J. Xie, J. Menendez, and J. Kouvetakis, “Transport properties of doped GeSn alloys,” 2008 29th Int. Conf. on the Physics of Semiconductors (AIP, Rio de Janeiro, 2008).

] and Ge_0.97 Sn_0.03 [12

A. V. G. Chizmeshya, C. Ritter, J. Tolle, C. Cook, J. Menendez, and J. Kouvetakis, “Fundamental studies of P(GeH₃)₃, and Sb(GeH₃)₃: practical n-dopants for new group IV semiconductors,” Chem. Mater. 18(26), 6266–6277 (2006). [CrossRef]

] as well as Ge [11

V. R. D’Costa, J. Tolle, J. Xie, J. Menendez, and J. Kouvetakis, “Transport properties of doped GeSn alloys,” 2008 29th Int. Conf. on the Physics of Semiconductors (AIP, Rio de Janeiro, 2008).

] 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

V. R. D’Costa, J. Tolle, J. Xie, J. Menendez, and J. Kouvetakis, “Transport properties of doped GeSn alloys,” 2008 29th Int. Conf. on the Physics of Semiconductors (AIP, Rio de Janeiro, 2008).

, 13

V. R. D’Costa, J. Tolle, J. Xie, J. Kouvetakis, and J. Menendez, “Infrared dielectric function of p-type Ge_0.98 Sn_0.02 alloys,” Phys. Rev. B 80(12), 125209 (2009). [CrossRef]

] 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-_xSn_x 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

ε^{'} = ε_{\infty} [1 - \frac{ω_{p}^{2} τ^{2}}{1 + ω^{2} τ^{2}}],

(1)

and

ε^{″} = \frac{ε_{\infty} ω_{p}^{2} τ}{ω (1 + ω^{2} τ^{2})},

(2)

where ε_∞ is the high frequency dielectric constant. The plasma frequency is determined by

ω_{p} = \frac{2 π c}{λ_{p}} = \sqrt{\frac{N e^{2}}{m_{c} ε_{\infty} ε_{o}}},

(3)

where e is the electron charge, m_c is the conductivity effective mass, and ε_o is the permittivity of free space. The relaxation time is determined by

τ = \frac{μ m_{c}}{e},

(4)

where μ is the carrier mobility whose value depends upon N.

In determining the complex permittivity, it is first necessary to determine the conductivity effective mass of electrons m_c in Ge_1-_xSn_x. GeSn and Ge have 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_c consists of two contributions; one from the anisotropic L-valley masses, m_L, and one from the Γ-valley mass, m_Γ. In GeSn alloys, it is not perfectly clear what the L-valley effective mass is, but for dilute alloys we can say to a first approximation that m_L is the same as the Ge m_L, namely m_L(GeSn) = m_L (Ge) = 0.120 [14

B. Van Zeghbroeck, “Detailed description of the effective mass,” in Principles of Semiconductor Devise, http://ece-www.colorado.edu/~bart/book/ (2004).

]. In determining m_Γ, we utilize the theory presented in Ref [15

V. R. D’Costa, Y. Fang, J. Mathews, R. Roucka, J. Tolle, J. Menendez, and J. Kouvetakis, “Sn-alloying as a means of increasing the optical absorption of Ge at the C- and L- telecommunications bands,” Semicond. Sci. Technol. 24(11), 115006 (2009). [CrossRef]

], which we re-write as

\frac{m_{o}}{m_{Γ}} = 1 + \frac{4 p}{3} + \frac{2 p}{3} (\frac{E_{g} (x)}{E_{g} (x) + Δ}),

(5)

where

p = \frac{12.6 e V}{E_{g} (x)} {[\frac{a (G e)}{a (x)}]}^{2},

(6)

Δ is the spin-orbit splitting energy, and a(Ge) and a(x) are the cubic lattice parameters of Ge and the GeSn alloys respectively, and m_o 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 x,

a (x) = 5.6575 (1 - x) + 6.489 x + 0.166 x (1 - x),

(7)

E_g values are taken from Ref [10

R. Ragan and H. A. Atwater, “Measurement of the direct energy gap of coherently strained Sn_xGe_1-x/Ge(001) heterostructures,” Appl. Phys. Lett. 77(21), 3418–3420 (2000). [CrossRef]

]. as a function of 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_Γ. Table 1 presents a(x), E_g(x) and m_Γ values as determined directly from Eq. (5). The presented theory determines the trend of m_Γ versus x, however it does not connect accurately to the known Ge mass value. Thus the m_Γ data, determined directly from Eq. (5) and labeled m_Γ[raw]in Table 1, requires an adjustment. To obtain agreement with the actual m_Γ value for Ge, the m_Γ[raw]values are normalized with respect to 0.041m_o [14

B. Van Zeghbroeck, “Detailed description of the effective mass,” in Principles of Semiconductor Devise, http://ece-www.colorado.edu/~bart/book/ (2004).

]. These m_Γ[norm] values, which are also presented in Table 1, will be referred to simply as m_Γ for the remainder of this work.

Table 1 Direct-Gap Mass Considerations for Ge_1-_xSn_x.

x	a(x) [nm]	E_g(x)[meV]	m_Γ/m_o [raw]	m_Γ/m_o[norm]
0.000	0.55658	800	0.0337	0.0410
0.035	0.56922	686	0.0301	0.0366
0.050	0.57069	628	0.0274	0.0333
0.060	0.57168	613	0.0270	0.0329
0.080	0.57362	548	0.0246	0.0299
0.115	0.57700	446	0.0206	0.0251

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_L = 1 and f_Γ = 0. For the case where the Ge_1-_xSn_x alloy composition in which the bandgap becomes truly direct (x = 0.08 as noted earlier), f_L is approximately equal to f_Γ, giving both values of 0.5. A linear extrapolation is used to determine f_L, as well as f_Γ by association, as a function of x. The conductivity effective mass is then found by

\frac{1}{m_{c}} = \frac{f_{L}}{m_{L}} + \frac{f_{Γ}}{m_{Γ}} .

(8)

Table 2 gives all values associated with the conductivity effective mass as a function of x.

Table 2 Conductivity Effective Mass Considerations for Ge_1-_xSn_x.

x	m_L/m_o	m_Γ/m_o	f_L	f_Γ	m_c/m_o
0.000	0.120	0.0410	1.000	0.000	0.1200
0.035	0.120	0.0366	0.781	0.219	0.0801
0.050	0.120	0.0333	0.687	0.313	0.0661
0.060	0.120	0.0329	0.625	0.375	0.0602
0.080	0.120	0.0299	0.500	0.500	0.0478
0.115	0.120	0.0251	0.281	0.719	0.0323

The electron mobility μ is needed here for Ge_1-_xSn_x as a function of N over the investigated range of 10¹⁹ cm⁻³ to 10²¹ cm⁻³. Ref [16

J. Xie, J. Tolle, V. R. D’Costa, C. Weng, A. V. G. Chizmeshya, J. Menendez, and J. Kouvetakis, “Molecular approaches to p- and n- nanoscale doping of Ge_1-ySn_y semiconductors: Structural, electrical and transport properties,” Solid-State Electron. 53(8), 816–823 (2009). [CrossRef]

] 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-_xSn_x here are 400, 200, and 150 cm²/Vs [16

] for N of 10¹⁹, 10²⁰, and 10²¹ cm⁻³ respectively.

Using the m_c and μ values, we immediately find the τ values using Eq. (4). After that, the λ_p are determined from Eq. (3) using the calculated m_c and an ε_∞ 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 + 8x, with the values being given in Table 3 . Table 3 also summarizes our findings of λ_p and τ at three selected values of N while Fig. 2 pictorially presents λ_p as a function of N, for n-Ge_1-_xSn_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 N = 10¹⁹ 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_p for x = 0.115 is approximately double that of n-Ge.

Table 3 High frequency dielectric constant, plasmon resonance wavelength, and relaxation time for Ge_1-_xSn_x at three different n-type doping concentrations in units of cm⁻³. λ_p and τ are in units of μm and fs respectively. n-Si values are shown for comparison.

x	ε_∞	λ_p	τ	λ_p	τ	λ_p	τ
		10¹⁹	10¹⁹	10²⁰	10²⁰	10²¹	10²¹
0.000	16.0	14.7	27.3	4.64	13.7	1.47	10.2
0.035	16.3	12.1	18.2	3.82	9.12	1.21	6.84
0.050	16.4	11.0	15.1	3.48	7.53	1.10	5.65
0.060	16.5	10.5	13.7	3.33	6.86	1.05	5.14
0.080	16.6	9.43	10.9	2.98	5.44	0.94	4.08
0.115	16.9	7.82	7.36	2.47	3.68	0.78	2.76
n-Si	11.7	18.4	16.9	5.83	14.1	1.85	13.7

Fig. 2 Plasma wavelengths for n-Ge_1-_xSn_x, n-Ge and n-Si as a function of carrier concentration.

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

B. Van Zeghbroeck, “Detailed description of the effective mass,” in Principles of Semiconductor Devise, http://ece-www.colorado.edu/~bart/book/ (2004).

] was employed, along with ε_∞ = 11.7, and μ = 114, 95, and 92.4 cm²/Vs for N = 10¹⁹, 10²⁰, and 10²¹ cm⁻³ respectively [17

C. Jacoboni, C. Canali, G. Ottaviani, and A. Alberigi Quaranta, “A review of some charge transport properties of silicon,” Solid-State Electron. 20(2), 77–89 (1977). [CrossRef]

]. 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 10²¹ 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.

Fig. 3 Real (left) and imaginary parts (right) of the permittivity of n-Ge_1-_xSn_x, n-Ge and n-Si. The carrier concentrations from top to bottom are 10¹⁹, 10²⁰ and 10²¹ cm⁻³.

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

{ε^{'}}_{\lim ω \to 0} = ε_{\infty} [1 - ω_{p}^{2} τ^{2}] = ε_{\infty} [1 - \frac{N μ^{2} m_{c}}{ε_{\infty} ε_{o}}],

(9)

which agrees with the limits observed in Fig. 3 (left). This limit approaches ε_∞ as the μ, N or m_c 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.

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

L_{d, c} = {[\frac{ω}{c} Re \sqrt{\frac{- ε^{2}_{d, c}}{ε_{d} + ε_{c}}}]}^{- 1},

(10)

while the plasmon intensity propagation loss in dB/cm can be determined by

L_{z} = 8.68 [d B] [\frac{ω}{c} Im \sqrt{\frac{ε_{d} ε_{c}}{ε_{d} + ε_{c}}}] .

(11)

The conducting permittivities, ε_c, used are those calculated in this work for doped Ge₁_xSn_x, Ge, and Si as were shown in Fig. 3. The dielectric material with permittivity, ε_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_p [(ε_∞ + 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_p [2

]^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-xSn_x being presented for clarity. At a doping of N = 10¹⁹ cm⁻³ (Fig. 4 top), only Ge_1-_xSn_x with the x > 0.06 appear to enable plasmonic activity below 20 μm. Doped Si and Ge with N = 10¹⁹ 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 ~10²⁰ cm⁻³ may be required. As an example at a wavelength of 10 μm with N = 10²⁰ cm⁻³, the propagation loss for the n-Ge_0.885Sn_0.115 (~10⁴ 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_0.885Sn_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

R. Soref, S.-Y. Cho, W. Buchwald, R. E. Peale, and J. Cleary, “Silicon plasmonic waveguides,” in Silicon Photonics for Telecommunications and Biomedical Applications, S. Fathpour and B. Jalali, eds. (Taylor and Francis, UK, 2011).

] in this wavelength regime.

Fig. 4 Plasmon intensity propagation loss (left) and plasmon mode height (right) n-Ge_1-_xSn_x, n-Ge and n-Si. The dielectric region is undoped Ge_1-_xSn_x, Ge and Si respectively. The carrier concentrations from top to bottom are 10¹⁹, 10²⁰ and 10²¹ cm⁻³.

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-_xSn_x 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

S. T. Lim, C. E. Png, E. A. Ong, and Y. L. Ang, “Single mode, polarization-independent submicron silicon waveguides based on geometrical adjustments,” Opt. Express 15(18), 11061–11072 (2007). [CrossRef] [PubMed]

]. 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_1-_xSn_x waveguide structures.

Fig. 5 DCD ribbon modes found by FDTD software. The waveguide was lossy Ge utilizing the software material library. The ribbon is Ge_0.94Sn_0.06 with N = 10²¹ cm⁻³. The free space wavelength is 8 μm. The ribbon is 80nm thick and 8 μm wide while the waveguide is 1.6 μm high and 16 μm wide. Shown are X-Y cross sections of E_y at 1, 4, 10, and 16 μm from the source from left to right respectively.

A physical realization would begin with an epitaxial growth of an undoped relaxed crystalline layer of Ge_1-_xSn_x 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_1-_xSn_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_1-_xSn_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_1-_xSn_x with thickness H/2 is epitaxially grown upon the first Ge_1-_xSn_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.

Lumerical finite-difference time-domain (FDTD) 3D Maxwell solver was used to model DCD structures of Ge_1-_xSn_x 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_0.94Sn_0.06 with N = 10²¹ 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_y 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_1-_xSn_x based plasmonic-photonic hybrid structures with minimized loss and maximized mode confinement.

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-xSn_x over the 10¹⁹ to 10²¹ 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 ~10²⁰ 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

D. Li and C. Z. Ning, “All-semiconductor active plasmonic system in mid-infrared wavelengths,” Opt. Express 19(15), 14594–14603 (2011). [CrossRef] [PubMed]

]. 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.

References and links

1.	J. W. Cleary, G. Medhi, R. E. Peale, W. Buchwald, O. Edwards, and I. Oladeji, “Infrared surface plasmon resonance biosensor,” Proc. SPIE 7673, 767306, 767306-11 (2010). [CrossRef]
2.	A. M. Crook, H. P. Nair, D. A. Ferrer, and S. R. Bank, “Suppression of planar defects in the molecular beam epitaxy of GaAs/ErAs/GaAs heterostructures,” Appl. Phys. Lett. 99(7), 072120 (2011). [CrossRef]
3.	D. Li and C. Z. Ning, “All-semiconductor active plasmonic system in mid-infrared wavelengths,” Opt. Express 19(15), 14594–14603 (2011). [CrossRef] [PubMed]
4.	M. Shahzad, G. Medhi, R. E. Peale, W. R. Buchwald, J. W. Cleary, R. Soref, G. D. Boreman, and O. Edwards, “Infrared surface plasmons on heavily doped silicon,” submitted to J. Appl. Phys. (2011).
5.	J. C. Ginn, R. L. Jarecki Jr, E. A. Shaner, and P. S. Davids, “Infrared plasmons on heavily-doped silicon,” J. Appl. Phys. 110(4), 043110 (2011). [CrossRef]
6.	R. Soref, R. E. Peale, and W. Buchwald, “Longwave plasmonics on doped silicon and silicides,” Opt. Express 16(9), 6507–6514 (2008). [CrossRef] [PubMed]
7.	J. W. Cleary, R. E. Peale, D. J. Shelton, G. D. Boreman, C. W. Smith, M. Ishigami, R. Soref, A. Drehman, and W. R. Buchwald, “IR permittivities for silicides and doped silicon,” J. Opt. Soc. Am. B 27(4), 730–734 (2010). [CrossRef]
8.	G. Sun, R. A. Soref, and H. H. Cheng, “Design of a Si-based lattice-matched room-temperature GeSn/GeSiSn multi-quantum-well mid-infrared laser diode,” Opt. Express 18(19), 19957–19965 (2010). [CrossRef] [PubMed]
9.	S. Y. Cho and R. A. Soref, “Low-loss silicide/silicon plasmonic ribbon waveguides for mid- and far-infrared applications,” Opt. Lett. 34(12), 1759–1761 (2009). [CrossRef] [PubMed]
10.	R. Ragan and H. A. Atwater, “Measurement of the direct energy gap of coherently strained Sn_xGe_1-x/Ge(001) heterostructures,” Appl. Phys. Lett. 77(21), 3418–3420 (2000). [CrossRef]
11.	V. R. D’Costa, J. Tolle, J. Xie, J. Menendez, and J. Kouvetakis, “Transport properties of doped GeSn alloys,” 2008 29th Int. Conf. on the Physics of Semiconductors (AIP, Rio de Janeiro, 2008).
12.	A. V. G. Chizmeshya, C. Ritter, J. Tolle, C. Cook, J. Menendez, and J. Kouvetakis, “Fundamental studies of P(GeH₃)₃, and Sb(GeH₃)₃: practical n-dopants for new group IV semiconductors,” Chem. Mater. 18(26), 6266–6277 (2006). [CrossRef]
13.	V. R. D’Costa, J. Tolle, J. Xie, J. Kouvetakis, and J. Menendez, “Infrared dielectric function of p-type Ge_0.98 Sn_0.02 alloys,” Phys. Rev. B 80(12), 125209 (2009). [CrossRef]
14.	B. Van Zeghbroeck, “Detailed description of the effective mass,” in Principles of Semiconductor Devise, http://ece-www.colorado.edu/~bart/book/ (2004).
15.	V. R. D’Costa, Y. Fang, J. Mathews, R. Roucka, J. Tolle, J. Menendez, and J. Kouvetakis, “Sn-alloying as a means of increasing the optical absorption of Ge at the C- and L- telecommunications bands,” Semicond. Sci. Technol. 24(11), 115006 (2009). [CrossRef]
16.	J. Xie, J. Tolle, V. R. D’Costa, C. Weng, A. V. G. Chizmeshya, J. Menendez, and J. Kouvetakis, “Molecular approaches to p- and n- nanoscale doping of Ge_1-ySn_y semiconductors: Structural, electrical and transport properties,” Solid-State Electron. 53(8), 816–823 (2009). [CrossRef]
17.	C. Jacoboni, C. Canali, G. Ottaviani, and A. Alberigi Quaranta, “A review of some charge transport properties of silicon,” Solid-State Electron. 20(2), 77–89 (1977). [CrossRef]
18.	R. Soref, S.-Y. Cho, W. Buchwald, R. E. Peale, and J. Cleary, “Silicon plasmonic waveguides,” in Silicon Photonics for Telecommunications and Biomedical Applications, S. Fathpour and B. Jalali, eds. (Taylor and Francis, UK, 2011).
19.	S. T. Lim, C. E. Png, E. A. Ong, and Y. L. Ang, “Single mode, polarization-independent submicron silicon waveguides based on geometrical adjustments,” Opt. Express 15(18), 11061–11072 (2007). [CrossRef] [PubMed]

OCIS Codes
(130.3060) Integrated optics : Infrared
(130.5990) Integrated optics : Semiconductors
(160.4670) Materials : Optical materials
(230.7370) Optical devices : Waveguides
(240.6680) Optics at surfaces : Surface plasmons

ToC Category:
Integrated Optics

History
Original Manuscript: October 14, 2011
Revised Manuscript: December 5, 2011
Manuscript Accepted: January 10, 2012
Published: February 1, 2012

Citation
Richard Soref, Joshua Hendrickson, and Justin W. Cleary, "Mid- to long-wavelength infrared plasmonic-photonics using heavily doped n-Ge/Ge and n-GeSn/GeSn heterostructures," Opt. Express 20, 3814-3824 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-3814

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References

J. W. Cleary, G. Medhi, R. E. Peale, W. Buchwald, O. Edwards, and I. Oladeji, “Infrared surface plasmon resonance biosensor,” Proc. SPIE7673, 767306, 767306-11 (2010). [CrossRef]
A. M. Crook, H. P. Nair, D. A. Ferrer, and S. R. Bank, “Suppression of planar defects in the molecular beam epitaxy of GaAs/ErAs/GaAs heterostructures,” Appl. Phys. Lett.99(7), 072120 (2011). [CrossRef]
D. Li and C. Z. Ning, “All-semiconductor active plasmonic system in mid-infrared wavelengths,” Opt. Express19(15), 14594–14603 (2011). [CrossRef] [PubMed]
M. Shahzad, G. Medhi, R. E. Peale, W. R. Buchwald, J. W. Cleary, R. Soref, G. D. Boreman, and O. Edwards, “Infrared surface plasmons on heavily doped silicon,” submitted to J. Appl. Phys. (2011).
J. C. Ginn, R. L. Jarecki, E. A. Shaner, and P. S. Davids, “Infrared plasmons on heavily-doped silicon,” J. Appl. Phys.110(4), 043110 (2011). [CrossRef]
R. Soref, R. E. Peale, and W. Buchwald, “Longwave plasmonics on doped silicon and silicides,” Opt. Express16(9), 6507–6514 (2008). [CrossRef] [PubMed]
J. W. Cleary, R. E. Peale, D. J. Shelton, G. D. Boreman, C. W. Smith, M. Ishigami, R. Soref, A. Drehman, and W. R. Buchwald, “IR permittivities for silicides and doped silicon,” J. Opt. Soc. Am. B27(4), 730–734 (2010). [CrossRef]
G. Sun, R. A. Soref, and H. H. Cheng, “Design of a Si-based lattice-matched room-temperature GeSn/GeSiSn multi-quantum-well mid-infrared laser diode,” Opt. Express18(19), 19957–19965 (2010). [CrossRef] [PubMed]
S. Y. Cho and R. A. Soref, “Low-loss silicide/silicon plasmonic ribbon waveguides for mid- and far-infrared applications,” Opt. Lett.34(12), 1759–1761 (2009). [CrossRef] [PubMed]
R. Ragan and H. A. Atwater, “Measurement of the direct energy gap of coherently strained SnxGe1-x/Ge(001) heterostructures,” Appl. Phys. Lett.77(21), 3418–3420 (2000). [CrossRef]
V. R. D’Costa, J. Tolle, J. Xie, J. Menendez, and J. Kouvetakis, “Transport properties of doped GeSn alloys,” 2008 29th Int. Conf. on the Physics of Semiconductors (AIP, Rio de Janeiro, 2008).
A. V. G. Chizmeshya, C. Ritter, J. Tolle, C. Cook, J. Menendez, and J. Kouvetakis, “Fundamental studies of P(GeH3)3, and Sb(GeH3)3: practical n-dopants for new group IV semiconductors,” Chem. Mater.18(26), 6266–6277 (2006). [CrossRef]
V. R. D’Costa, J. Tolle, J. Xie, J. Kouvetakis, and J. Menendez, “Infrared dielectric function of p-type Ge0.98 Sn0.02 alloys,” Phys. Rev. B80(12), 125209 (2009). [CrossRef]
B. Van Zeghbroeck, “Detailed description of the effective mass,” in Principles of Semiconductor Devise, http://ece-www.colorado.edu/~bart/book/ (2004).
V. R. D’Costa, Y. Fang, J. Mathews, R. Roucka, J. Tolle, J. Menendez, and J. Kouvetakis, “Sn-alloying as a means of increasing the optical absorption of Ge at the C- and L- telecommunications bands,” Semicond. Sci. Technol.24(11), 115006 (2009). [CrossRef]
J. Xie, J. Tolle, V. R. D’Costa, C. Weng, A. V. G. Chizmeshya, J. Menendez, and J. Kouvetakis, “Molecular approaches to p- and n- nanoscale doping of Ge1-ySny semiconductors: Structural, electrical and transport properties,” Solid-State Electron.53(8), 816–823 (2009). [CrossRef]
C. Jacoboni, C. Canali, G. Ottaviani, and A. Alberigi Quaranta, “A review of some charge transport properties of silicon,” Solid-State Electron.20(2), 77–89 (1977). [CrossRef]
R. Soref, S.-Y. Cho, W. Buchwald, R. E. Peale, and J. Cleary, “Silicon plasmonic waveguides,” in Silicon Photonics for Telecommunications and Biomedical Applications, S. Fathpour and B. Jalali, eds. (Taylor and Francis, UK, 2011).
S. T. Lim, C. E. Png, E. A. Ong, and Y. L. Ang, “Single mode, polarization-independent submicron silicon waveguides based on geometrical adjustments,” Opt. Express15(18), 11061–11072 (2007). [CrossRef] [PubMed]

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