Metal-oxide-semiconductor-compatible ultra-long-range surface plasmon modes

JOURNAL OF APPLIED PHYSICS 103, 113106 共2008兲

Metal-oxide-semiconductor-compatible ultra-long-range surface plasmon modes C. G. Durfee,1,a兲 T. E. Furtak,1 R. T. Collins,1 and R. E. Hollingsworth2 1

Department of Physics, Colorado School of Mines, Golden, Colorado 80401, USA ITN Energy Systems, Littleton, Colorado 80127, USA

2

共Received 19 March 2008; accepted 29 March 2008; published online 6 June 2008兲 Long-range surface plasmons traveling on thin metal films have demonstrated promising potential in subwavelength waveguide applications. In work toward device applications that can leverage existing silicon microelectronics technology, it is of interest to explore the propagation of surface plasmons in a metal-oxide-semiconductor geometry. In such a structure, there is a high refractive index contrast between the semiconductor 共n ⬇ 3.5 for silicon兲 and the insulating oxide 共typically n ⬇ 1.5− 2.5兲. However, the introduction of dielectrics with disparate refractive indices is known to strongly affect the guiding properties of surface plasmons. In this paper, we analyze the implications of high index contrast in 1D layered surface plasmon structures. We show that it is possible to introduce a thin dielectric layer with a low refractive index positioned next to the metal without adversely affecting the guiding quality. In fact, such a configuration can dramatically increase the propagation length of the conventional long-range mode. While this study is directed at silicon-compatible waveguides working at telecommunications wavelengths, this configuration has general implications for surface plasmon structure design using other materials and operating at alternative wavelengths. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2937191兴 I. INTRODUCTION

Electromagnetic waves bound to metal interfaces, or surface plasmons 共SPs兲, have recently attracted considerable interest for their potential to enable subwavelength integrated optical devices. Components such as two-dimensional reflectors and interferometers, lenses, and mirrors, as well as waveguides, have been demonstrated 共see the review articles in Refs. 1–3兲. These components would be particularly useful if the structures could be built using fabrication methods that have already been perfected for the production of silicon microelectronic systems. More importantly, if the plasmonic structures could also be controlled by conventional metaloxide-semiconductor 共MOS兲 devices, a truly integrated nanoscale optoelectronic technology that takes advantage of the well-developed silicon planar fabrication technology would be possible. For this to be successful, plasmon-based logic devices must be developed. The essential building block of such devices is an electronically controlled modulator within an optical circuit. One of the first considerations in designing such a device is to determine what long-range surface plasmon modes are possible. The insulating oxide in a MOS structure has a high degree of index contrast with the semiconductor 共Si兲 substrate. In this paper, we consider the behavior of long-range surface plasmon modes that can propagate in layered structures with high degrees of refractive index contrast. One of the primary constraints in using surface plasmons for device design has been the intrinsic damping of SPs related to ohmic loss in the metal. This severely limits the propagation length of a plasmonic signal. For this reason it is a兲

Author to whom correspondence should be addressed. Electronic mail: [email protected].

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important to develop devices in which the electric fields associated with the SP are localized outside of the metal. The so-called long-range surface plasmon 共LRSP兲, one of the waveguide modes of an isolated thin metal film, was predicted4 and demonstrated5 nearly 25 years ago. In such a film, surface plasmons on opposite surfaces of the metal interact with their E-fields nominally parallel or antiparallel to create two bound modes. The antiparallel combination leads to an energy density node within the metal, resulting in a lower ohmic loss. LRSPs have been investigated both theoretically6–10 and experimentally5,11,12 in some detail. In this paper we will compare the character of several types of planar structures 共see Fig. 1兲. The initial investigation of LRSPs involved the configuration in which a noble metal film was bounded by identical dielectric media and was explored using light with visible wavelengths.4,5 Later attention expanded to include asymmetric configurations

FIG. 1. Schematic of different layered structures considered in this paper. Central black line: metal layer; gray: low-index dielectric 共e.g., insulating oxide兲; hatched: high index 共e.g., semiconductor兲. Each is labeled with the type of long-range mode supported by the structure, as discussed in the text. The lower two structures could be used in MOS devices, with a potential placed between the metal and the semiconductor layer adjacent to the oxide.

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共a-LRSP, different indices of refraction on either side of the film兲, where additional radiation-coupled modes arise.6 It has been shown that adjustments of the index asymmetry can reduce the imaginary part of the mode propagation constant toward zero.13,14 While this type of excitation has been termed an ultra-long-range surface plasmon 共ULRSP兲, we show below that the modes supported by this indexasymmetric structure are too large to be useful for integrated optics applications. Another ultra-long-range configuration we will discuss below has a thin, lower-index dielectric layer placed symmetrically15 on both sides of the metal 共s-ULRSP兲 or asymmetrically on one side of the metal 共a-ULRSP兲. While the discussion in this paper applies to a wide range of materials and wavelengths, our interest in the problem of low-loss SP modes is driven by the challenge of developing light modulator devices in silicon-based metaloxide-semiconductor structures. Designs that can be integrated with silicon-based waveguides have been developed,16 but it would be desirable to have a plasmonsupporting waveguide design that can be directly switched using on-chip structures that exploit existing silicon device technology. In such a silicon-based modulator, electrically induced changes in mobile charge carrier densities17 would modify the characteristics of a LRSP, leading to changes in the mode amplitude and phase. There are three principal design challenges for this architecture: 共1兲 the high refractive index of the silicon 共3.5 for Si, compared to 1.5 for glass兲; 共2兲 the necessity of using a low-index gate oxide next to the metal; and 共3兲 the technological advantage in using metals that are not as favorable for SP propagation as silver. SP losses increase substantially as the refractive index of the surrounding medium increases. Moreover, to produce a charge accumulation or inversion layer next to the metal surface for the purpose of modulation without drawing current with related power loss, a gate oxide must separate the metal from the silicon.18 The symmetric placement of thin dielectric layers 共s-ULRSP兲 has been described in Guo et al.;15 here, we also consider a more general case that allows for a single low-index dielectric next to the metal 共a-ULRSP兲. In previous work,19 it was concluded that a low-index layer near the metal would push the guided modes past cutoff. In fact, there has been a thermo-optic modulator designed to take advantage of this effect.20 In contrast to the conclusions of this previous work, we show in this paper that careful adjustment of the design of a silicon-metal-oxide-silicon structure allows bound long-range modes to exist with propagation lengths greater than 1 mm. The thickness and refractive index of the low-index layer actually offer a measure of control over the SP propagation length, allowing the use of less-than-ideal metals 共such as the MOS-compatible metals Al or Cu兲 in these structures. These ULRSP modes are viable for processing plasmonic signals with an optical wavelength of 1.5 ␮m, and become even more favorable when using longer wavelengths, into the mid-infrared 共midIR兲. A device based on these modes could be fabricated using mature MOS technology, and controlled with integrated switching transistors. To take further advantage of these lowloss modes, we envision a resonator or Fabry-Perot structure

with a transmission that would be strongly sensitive to induced changes in the ULRSP propagation constant. This concept will be explored in a later paper. II. THE ROLE OF INDEX ASYMMETRY IN LONG-RANGE SURFACE PLASMON MODES A. Single-interface and long-range surface plasmon modes

Surface plasmon polaritons 共SPs兲 are electromagnetic excitations that exist at the interface between noble metals and dielectrics.1,21 The simplest form of a surface plasmon polariton mode is a transverse magnetic mode at a single metal-dielectric interface. The propagation constant of a surface plasmon on a planar interface between a dielectric 共␧d兲 and a material with a complex dielectric function ␧m = ␧m ⬘ + i␧m ⬙ is kSP =

␻ ␻ nSP = c c

冑

␧ d␧ m , ␧d + ␧m

共1兲

where ␻ = 2␲c / ␭0 is the SP frequency, ␭0 is the vacuum wavelength, and nSP is the effective index of the mode. The surface plasmon transports energy with a 1 / e propagation length of

⬙ 兲−1 = ␭0/共4␲nSP ⬙ 兲. lSP = 共2kSP

共2兲

The field of the bound mode exponentially decays into the dielectric with a transverse attenuation rate ␣ that is connected to nSP through the imaginary part of the following relation:

␣=

␻冑 2 ␧d − nSP . c

共3兲

Note that the real part of Eq. 共3兲, ␣⬘, corresponds to the transverse traveling wave component of the wave. The sign of ␣⬘ determines the direction of power flow, e.g., toward the metal in a bound mode in the presence of ohmic losses 共with 兩␣⬘兩 ⬍ 兩␣⬙兩兲, and away from the metal 共with 兩␣⬘兩 ⬇ 兩nSP ⬘ 兩兲 in the case of radiation losses. Surface plasmons can propagate when ␧m ⬘ is negative and ␧m ⬙ is small, a condition satisfied in noble metals in an appropriate range of frequencies. Silver 共␧m ⬇ −134− i 3.4 at ␭0 = 1.5 ␮m兲 supports SPs very effectively. At a metal-air interface lSP is in the range 20− 200 ␮m in silver for visible wavelengths. Gold is nearly as favorable. Aluminum and copper, which are compatible with silicon technology, can also be used for the propagation of surface plasmons using near- and mid-IR wavelengths. Generally, the losses of these metals decrease for longer wavelengths. To design practical silicon-based devices for signal processing of SP modes, it is important to appreciate the optical role of the bounding dielectric. The mode attenuation increases dramatically as the dielectric’s refractive index increases. For ␭0 = 1.5 ␮m the propagation length of a singleinterface SP on silver changes from larger than 1200 ␮m for a Ag-air interface to 25 ␮m at a Ag-Si interface. An expansion of Eq. 共2兲 in terms of ␧m ⬙ / 兩␧m⬘ 兩 and ␧d / 兩␧m⬘ 兩, both of which are small, shows that lSP scales with n−3 d . Qualitatively, as nd increases the E-field is forced into the metal, which leads to

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FIG. 2. Panels 共a兲 Real part nSP ⬘ and 共b兲 imaginary part nSP ⬙ of the effective index at ␭0 = 1.55 ␮m of surface plasmon modes vs metal 共Ag兲 layer thickness 共hm兲 for different structures. Solid thin: 共Si兲/共Ag兲/共Si兲, LRSP; Dotted: 共Si兲 / 共7.5 nm TiO2兲 / 共Ag兲 / 共7.5 nm TiO2兲 / 共Si兲, s-ULRSP. Long dash: 共Si兲 / 共15 nm TiO2兲 / 共Ag兲 / 共Si兲, a-ULRSP. The lower 共upper兲 curves correspond to the H-symmetric, sb 共H-antisymmetric, ab兲 solutions. Panels 共c兲 and 共d兲 correspond to nSP ⬘ and nSP ⬙ for the structure 共TiO2兲 / 共Ag兲 / 共Si兲, a-LRSP. The thin, black dashed curve is for the exponentially growing leaky solution sl.

a greater loss. This increase in the propagation loss with a high-index substrate is one of the challenges in working with a silicon-compatible configuration. To explore guided modes in a multilayer system, we used a transfer matrix approach.22 The complex propagation constants are found from an eigenvalue equation derived from this matrix. Since the manipulation of the matrix equations is done symbolically with the program Mathematica 共Wolfram Research兲, the eigenvalue equations we obtain are equivalent to analytic expressions. The roots in the positive quadrant of the complex plane correspond to lossy forward propagating modes. As part of the solution, a choice must be made for the sign of the transverse wavenumber in the outer dielectric 关Eq. 共3兲兴. This choice determines whether the envelope of the field is exponentially damped 共bound兲 or increasing 共leaky兲 away from the interface. In general, there are four solutions, corresponding to the combinations of symmetric/antisymmetric and bound/leaky 共see for example Ref. 14兲. The behavior of several configurations is illustrated in Fig. 2, which shows the real and imaginary parts of the bound modes of the effective mode index 共nSP = 共c / ␻兲kSP兲 as a function of the thickness 共hm兲 of the metal layer. Schematics of these configurations are shown in Fig. 1. The solid curves in Figs. 2共a兲 and 2共b兲 show the behavior of the modes of a conventional LRSP propagating on a silver layer embedded in silicon with ␭0 = 1.55 ␮m. For simplicity, we show only the bound modes. For large hm there is a single value of nSP, the same as what one finds for SPs at a single interface. As hm decreases, the mode splits into high-loss 共ab兲 and low-

loss 共sb兲 branches. For the sb branch the size of nSP ⬙ goes to zero 共and its range goes to infinity兲 only in the limit of hm = 0. Depending on the method of fabrication, there is a practical limit to how thin the metal can be made while still forming a conducting layer that is necessary in a MOS device. B. Index-asymmetric long-range surface plasmon modes

In using the MOS structure to guide surface plasmons, a challenge is presented by the insulating gate oxide layer that is present next to the metal. Since insulating oxides typically have refractive indices that are substantially smaller than that of silicon, the role of refractive index asymmetry on LRSP propagation must be considered. This discussion is important in view of previous claims that it is possible to employ index asymmetry to decrease propagation losses,13,14 and that a low-index layer next to the metal will push the SP mode beyond cutoff.19,20 Consider Figs. 2共c兲 and 2共d兲, which show the dependence of the 共sb兲 mode solutions on metal thickness for the case where silicon is on one side 共nhi ⬇ 3.5兲 of a Ag film and a representative low-index material is on the other 共nlow ⬇ 2.5兲. Also shown are the symmetric leaky solutions 共sl兲 共dashed兲. Recall that the designations “bound” and “leaky” refer to whether the modal fields decay exponentially away from the interfaces, or if there is an exponentially growing component. As asymmetry in the refractive indices is introduced, the 共sb兲 branch becomes primarily associated with the

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lower-index interface, and the more lossy 共ab兲 is associated with the higher-index interface. While the symmetry of the structure is of course no longer present, the nomenclature continues to refer to the mode that does 共a兲 or does not 共s兲 have a node in the H-field. In order to discuss the nature of the long-range bound mode, we will focus our attention on the symmetric solutions. For this layer structure, note that both the bound and leaky symmetric solutions are nearly degenerate in nSP ⬘ , but have dramatically different values of nSP ⬙ . Since for these modes nSP ⬘ ⬍ nhi, they will be coupled with traveling waves in the silicon substrate. In the leaky case, the power flow is away from the metal; in the bound case it is toward the metal. The high value of nSP ⬙ for the leaky case shows that this mode has a very short range. In contrast to what is observed with a symmetric structure, nSP ⬙ for the 共sb兲 branch decreases to zero at a critical value of metal thickness hm that is somewhat greater than zero.6,14,23 For hm smaller than this critical value this low-loss mode is cut off. Since this asymmetric structure yields a mode with a long propagation length 关see Eq. 共2兲兴, it appears to be useful for integrated optic applications, but as we will show, this is not the case. To explore the character of the modes that are allowed in this index-asymmetric configuration, let us consider launching a mode at z = 0 that is well-localized near the interface. As the wave propagates along the metal, it is damped both from ohmic losses and from radiation into the higher-index substrate. The radiation propagates away at an angle ␪0 that is characteristic of the phase velocity of the mode: sin ␪0 = nSP ⬘ / nhi. Owing to this radiation angle, the exponential decrease of the field along the metal maps to an exponential increase in the field away from the metal. This scenario corresponds to the leaky wave, which is found by solving for a mode that exponentially increases in the high-index substrate. The bound solution, with an exponential decrease into the substrate, also has a traveling-wave character in the silicon, but the wave power is directed toward the interface. In this mode, the overall attenuation rate is less than for the leaky mode since the ohmic losses are compensated by energy drawn from the high-index side. Again, the radiation propagates toward the metal layer at the same characteristic coupling angle as for the leaky mode. Launching this mode in practice would be very similar to the familiar Kretschmann prism coupling geometry,24 except that, instead of a plane wave input, the transverse profile of the wave would be exponentially damped away from the metal, so that there would be zero reflection. Although this is formally a long-range mode in the sense that there is a transverse mode shape that remains constant during propagation, the mode size is necessarily extremely large since there is the same mapping of the longitudinal damping rate to the transverse damping rate as in the leaky case. In fact, the mode size is of the same order of magnitude as the propagation length, tens to hundreds of microns. This extremely large spatial extent severely limits the utility of this mode for integrated optics.

J. Appl. Phys. 103, 113106 共2008兲

C. Ultra-long-range SP modes: Addition of a low-index layer

Returning to Fig. 2, consider the introduction of a lowindex layer 共the oxide兲 on one 共dotted兲 or both 共dashed兲 sides of a metal film surrounded by Si. In these calculations, the index of the oxide is nox = 2.5 共corresponding to TiO2 at 1.55 ␮m兲, and the oxide thickness 共hox兲 is held constant at 15 nm. First consider the asymmetric placement of this oxide layer 共dotted curves兲. For large hm, the modes are uncoupled. In this case nSP ⬘ and nSP ⬙ lie at values corresponding to the different interfaces. For this configuration it is possible to reach zero losses in the H-symmetric 共sb兲 mode at a finite value of hm. We use the term ultralong to describe this mode because the propagation length for this mode surpasses that of the conventional LRSP for a given metal thickness. The ULRSP behavior can also be obtained in the case where there is a symmetric placement of two low-index layers 共dashed curves兲, which has recently been examined by Guo and Adato.15 It is important to stress that, for the entire range shown in this figure, the s-ULRSP and a-ULRSP modes are tightly bound, and not coupled to radiation. As we saw above, the guided mode size is crucial to determining the utility of the modes for integrated optical applications. It is well understood that as the propagation length of a LRSP mode is increased, by thinning the metal layer for example, the transverse mode size increases. For an ideal metal ␧m ⬙ = 0, and there is good 共sub-␭兲 confinement with no losses. For a real 共lossy兲 metal, losses are decreased by pushing the field out of the metal, which also reduces mode confinement. As seen in Fig. 2共a兲, decreasing the metal thickness results in a decrease in nSP ⬘ toward that of the surrounding dielectric. This serves also to lower the value of nSP ⬘ , and increases the transverse attenuation length 关see Eq. 共3兲兴. A figure of merit 共FOM兲 for a guided mode can be defined as the ratio of the propagation length to the mode size.25 Since asymmetry in the index structure leads to asymmetry in the mode profiles, we calculate the mode size 共wrms兲 as the root-mean-square deviation of the power from the power centroid. For a symmetric double-exponential profile, wrms = lt = 1 / 关2 Im共␣⬙兲兴. Figure 3 shows the variation of the FOM with hox for the s-ULRSP 共dashed兲 and a-ULRSP 共solid兲 modes. These reduce to the LRSP case at hox = 0, where it is seen that the FOM increases for thinner hm. As hox is increased, the propagation length increases until cutoff, where the lines terminate. For the s-ULRSP configuration, this increase in propagation length comes at no cost to the FOM. For the a-ULRSP configuration, there is a moderate decrease in the FOM that results from the lower value of nSP ⬘ for this mode seen in Fig. 2共a兲. The radiation-coupled a-LRSP modes discussed above have a FOM ⬇2. The ULRSP structures have a FOM 2–3 orders of magnitude higher than this and are therefore much better-suited for integrated optics. Figure 4 shows the profile of 兩H共x兲兩2 for the low-loss mode at different values of the low-index layer thickness. As the low-index oxide thickness is increased, the field is drawn toward that side. There is little variation of the field within the oxide layer. The value of nSP ⬘ decreases as more of the

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FIG. 3. Figure of merit 共FOM ⫽ propagation length/transverse mode size兲 vs low-index oxide layer 共hox兲 for two different silver thickness, hm = 25 nm 共thin兲, hm = 40 nm 共thick兲. Solid lines correspond to the a-ULRSP structure; dashed lines correspond to the s-ULRSP structure. Lines terminate at cutoff.

mode overlaps the lower-index layer; this in turn puts more of the field into the surrounding cladding, decreasing the loss. Clearly, the oxide thickness hox and refractive index nox are both involved in giving ultra-long-range propagation. This parameter space is best described by defining the range of values of nox and hox for which a guided mode can propagate. While the presence of radiative and ohmic losses makes the cutoff boundary somewhat difficult to precisely define, we can estimate the cutoff condition by considering the case where the metal is perfect, ␧m = −兩␧m ⬘ 兩 共no losses兲. Here, the cutoff condition is where nSP = nSi. Inserting this cutoff condition into the analytic expression for the quantization condition given in Stegeman,19 we can write an expression relating hox and nox at cutoff,

FIG. 4. H-field intensity profiles for constant 25 nm Ag thickness and several oxide thicknesses as indicated. Solid vertical lines show Ag boundaries; dashed vertical line shows a representative oxide/Si boundary. The propagation lengths, lSP, defined in Eq. 共2兲 for the 0, 10, 15, and 18 nm oxide thicknesses are 0.26, 0.39, 0.82, and 2.75 mm, respectively. Inset shows same fields over a larger horizontal scale.

FIG. 5. Low-index layer thickness 共hox兲 at cutoff vs refractive index of the layer nox for the metal thicknesses indicated. The metal is a no-loss representation of Ag, with ␧m = −133. Modes are guided in the parameter space below the curves.

tanh关k0hox冑nSi − nox兴 =

2 冑 2 兩nm兩2 + nSi nox

兩nm兩2冑nSi − nox

2 ⫻tanh关k0hm冑兩nm兩2 + nSi 兴.

共4兲

Figure 5 shows the cutoff thickness versus oxide refractive index for several choices of metal thickness. We use ␧m = Re共␧Ag兲. Guided modes exist for any value of hox below the curves. Clearly, if the index contrast is high, for example, in the case of a thin air gap considered by Stegeman,19 the mode will be cutoff unless the layer is extremely small. In the thermo-optic modulator considered by Nikolajsen,20 the region of lower index surrounding the metal was determined by thermal diffusion and could not be controlled as precisely as is necessary. With modern microelectronic processing techniques, oxides can be deposited or grown with sufficient thickness control to introduce a low-index layer next to the metal without pushing the mode into cutoff. Moreover, the propagation length can actually be controlled in this way. Owing to the relatively weak confinement of any longrange mode, it is important to consider potential constraints on materials and structure fabrication. For example, curvature in the substrate 共as opposed to curvature of a finitewidth waveguide in the plane of the metal兲 might be expected to lead to tunneling losses for a mode that is near cutoff. By expressing the wave equation in cylindrical coordinates and expanding for a large bend radius ␳,26 it can be shown that the effect of a large radius of curvature is to add an overall linear component to the refractive index profile: 2 x / ␳. Here, nSP is the effective index of the un⌬n2b共x兲 = 2nSP perturbed mode. The field penetrates into the cladding and will tunnel through at a distance xt, where the cladding index increases to match that of nSP. WKB analysis shows that tunneling will be important when 3k0␣⬙xt / 2 ⬇ 1, where ␣⬙ is the transverse attenuation rate for the flat waveguide 关Eq. 共3兲兴. Thus, if the bend radius is smaller than ␳min 2 ⬇ 2k20nSP / ␣⬙3, the radiative losses will equal the ohmic losses. Consider, for example, the a-ULRSP mode shown in the dashed line of Fig. 4, which has a propagation length of

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2.75 mm, and a transverse attenuation length into the substrate of 6 ␮m. In this case ␳min ⬇ 100 cm. This flatness requirement is 3 orders of magnitude within limits of fabrication technology, so bending losses are not expected to be a serious limitation to the propagation length of these modes. Surface roughness and scattering from imperfect dielectric materials are likely to be more of a limiting factor experimentally. III. IMPLICATIONS OF THE ULTRA-LONG-RANGE MODE FOR DEVICE APPLICATIONS A. MOS-compatible structures

We have seen that control of the refractive index near the metal film has a strong effect on the losses and the character of the mode. In the a-LRSP case, the imaginary component of the propagation constant can be reduced to zero, but the bound mode was very large, sustaining its propagation length by drawing energy from the transversely extended field on the high-index side. A localized, end-coupled wave would drive the leaky mode and decay sharply as it radiates. In contrast, when a thin, low-index oxide layer is introduced next to the metal film 共either on one or both sides兲, the loss of the mode is reduced without any coupling to radiation. Only these a- and s-ULRSP modes are useful for integrated plasmonic applications. The integration of photonics with electronics will see wide use if existing device fabrication techniques and materials can be employed. Our ULRSP design, with a single oxide layer adjacent to the metal, is similar to the conventional MOS transistor structure. This allows a voltage bias to be applied between the metal and the substrate with a low power loss. We anticipate that the resulting electrically induced free-carrier modulation of the dielectric constant in the Si adjacent to the oxide would lead to a modulation of the mode characteristics of the ULRSP, making possible the design and construction of novel, high-speed, Si-compatible electro-optic devices such as modulators.3 It is important to note that the field energy is strongest on the side of the structure with the oxide, the same side where carrier density can be electrically modulated in the MOS structure 共see Fig. 4兲. It is also important to note that the field energy is lowest on the side with deposited silicon, which will reduce the losses associated with the higher defect density in deposited silicon compared to the single-crystalline substrate. An important issue is the metal used for the gate. While Ag has very favorable properties for surface plasmon propagation, other materials such as Cu and Al are more common in Si technology. An important consequence of the ULRSP modes discussed in this paper is that adjustment of the thickness of the oxide layer 共hox兲 can be used to produce the desired attenuation length even for metals that are not as favorable for SP propagation. This effect is illustrated in Fig. 6, which shows the variation of the power attenuation coefficient 1 / lSP with hox 共at nox = 2.0兲 for Ag, Au, Cu, and Al using ␭0 = 1.5 ␮m. The range of hox in the figure is compatible with typical oxide thicknesses in Si MOS structures. We note that surface roughness and material purity may experimentally place an upper bound on the propagation length,

FIG. 6. Attenuation coefficient 共1 / lSP兲 vs oxide layer thickness 共hox兲 for a-ULRSP modes at ␭0 = 1.5 ␮m with a 共Si兲/共oxide兲/共40 nm metal兲/共Si兲 structure. The oxide index is nox = 2.0. Dotted: Ag. Thin solid: Au. Dashed: Al. Thick solid: Cu.

but the ULRSP modes are expected to have a much longer range than conventional LRSPs for reasonable metal layer thickness. A second issue is the choice of the dielectric for the oxide layer. Our calculations indicate that, as nox is decreased, the value of hox at cutoff also decreases. A dielectric with a low nox, such as SiO2, can be used with hox in the 5–10 nm range. The push in the industry to change to highdielectric constant gate oxide materials such as HfO2 is consistent with our proposed ULRSP structure, because the higher values of nox allow thicker oxide layers for ULRSP behavior. The configuration with a single oxide layer presented here may be easier to implement than a symmetric structure.15 For a given propagation length the single-layer configuration allows the use of somewhat thicker oxide layers than the symmetric configuration. B. Wavelength scaling of the ULRSP modes

While a wavelength near 1.5 ␮m is important in communications applications, we find that the ULRSP modes have application in other parts of the spectrum, and in particular in the mid-infrared 共2 − 6 ␮m兲. While the mode size increases roughly as the square of the wavelength, the intrinsic propagation length increases faster 共approaching macroscopic values, ⬎10 mm兲, resulting in a larger FOM for large ␭0. Compare as an example the conventional LRSP mode for a 25 nm thick silver layer surrounded by silicon for wavelengths of 4 and 1.5 ␮m. Moving to the longer wavelength, the propagation length increases by 17⫻ to 4 mm, the mode size increases by 7.7⫻ to 2.1 ␮m, yielding an increase in the FOM by 2.2⫻ to 1850. The increase in propagation length and mode size results from two principal effects. First, losses of the metals are reduced as ␭0 increases. The second factor is the explicit k-dependence in both lSP and the transverse attenuation length 关see Eqs. 共2兲 and 共3兲兴. As seen at shorter wavelength, a thin, low-index layer can be added without much cost to the mode FOM. While Ag is best, Al, which is a well-established material in MOS fabrication, also performs very well in this regime. Additionally, changes in car-

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rier density in a MOS configuration will yield an increasingly pronounced effect on the refractive index as this cutoff wavelength is approached.

that are routinely used in Si devices, such as Cu and Al, can also be fabricated into structures that support ULRSP behavior. ACKNOWLEDGMENTS

IV. CONCLUSION

We have analyzed a design that enables the implementation of low-loss, bound SP waves on metal films in MOS structures in a way that is compatible with well-established silicon processing technology. The propagation characteristics of these modes, and their dependence on the asymmetry of the structure, have been determined. In contrast to the conclusions of earlier work, we have shown that it is possible to add a thin layer of low refractive index on one or both sides of a metal film while maintaining a guided surface plasmon mode. Up to a maximum oxide thickness discussed in the text, the propagation length of the surface plasmon mode is extended beyond that of the conventional LRSP. In accordance with the same behavior seen with LRSP modes, the increased propagation length comes with a corresponding increase in the transverse mode size. However, the figure of merit for guiding 共the ratio of these two兲 is not substantially affected by the addition of the low-index layer共s兲. In addition to the general interest in the design of surface plasmon structures, these ultra-long-range modes should find applications in plasmonic devices: the extra layer can be added to passively compensate for the low LRSP guiding length found in high index dielectrics, or the ULRSP structure can be used in an active MOS device. For example, we envision a SP modulator built around a MOS capacitor. The voltage on the capacitor would change the refractive index in a thin layer near the oxide, which would modulate the propagation of the mode. The extension of the propagation length of surface plasmons would allow an increase in the sensitivity of the device to the small carrier-induced index changes. Finally, we emphasize that, in addition to noble metals, other metals

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