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We demonstrate photothermally induced optical switching of ultra-compact hybrid Si-VO2 ring resonators. The devices consist of a sub-micron length ~70nm thick patch of phase-changing VO2 integrated onto silicon ring resonators as small as 1.5μm in radius. The semiconductor-to-metal transition (SMT) of VO2 is triggered using a 532nm pump laser, while optical transmission is probed using a tunable cw laser near 1550nm. We observe optical modulation greater than 10dB from modest quality-factor (~103) resonances, as well as a large –1.26nm change in resonant wavelength Δλ, resulting from the large change in the dielectric function of VO2 in the insulator-to-metal transition achieved by optical pumping.
©2012 Optical Society of America
1. Introduction
Silicon-based photonics has rapidly emerged as the leading candidate platform from which to revolutionize modern computing and communications. By replacing electrical components, such as the copper interconnects, with optical devices and architectures, dramatic improvements to a variety of metrics including bandwidth, speed, loss, cross-talk, and power consumption can be achieved [1]. For the realization of integrated silicon photonics, it is vital to develop compact silicon-based optical modulators that exceed the performance metrics of comparable electronic devices. Micro-ring resonant cavities have already been used to realize compact optical modulators, capable of operating at ~GHz speeds, based on the electro-optic effect in silicon [2]. The micro-ring platform has also been used to demonstrate compact all-optical switching and logic operations on Si [3, 4], and offers a convenient architecture for performing wavelength division multiplexing (WDM) [5]. While the resonator geometry offers substantial reductions in size and energy requirements for optical switching compared to millimeter-scale interferometer approaches [6], such compact all-Si optical modulators still face a number of limitations with respect to speed and performance. The small plasma dispersion effect in silicon requires very narrow band, high Q-factor devices to be utilized. But high Q-factor devices are difficult to implement, both in terms of required fabrication tolerances and in terms of sensitivity to ambient conditions such as temperature fluctuations [1]. Indeed, high Q-factor silicon ring resonators are known to be highly temperature sensitive (30-80 pm K–1) [7, 8], therefore requiring power hungry temperature compensation schemes to maintain temperature tolerances to less than ± 1 °C [9]. Additionally, the switching times of silicon based optical modulators are typically limited by carrier lifetimes for injection based devices (e.g. ~0.35 ns) [2], and RC constants in accumulation or depletion based devices (e.g. ~0.014 ns) [1, 10].
The limitations of Si modulators motivate a search for hybrid material combinations that could satisfy device requirements by incorporating a second material on the silicon platform to control modulation [11–16]. In one recent example, it has been proposed that intensity modulation with speeds up to 1 THz may be feasible with a hybrid polymer-silicon waveguide structure based on the all-optical Kerr effect [14]. Graphene has also recently been integrated onto silicon waveguides and used to demonstrate a high-speed electroabsorption based broadband optical modulator [15]. In a similar fashion, VO2 has been integrated onto silicon waveguides to function as an in-line broadband absorption modulator [16].
Phase-changing VO2 is a particularly attractive candidate hybrid material that is well known for its reversible semiconductor-to-metal transition (SMT) occurring near 67°C, accompanied by a structural change from monoclinic to tetragonal crystal structure [17]. The SMT results in large changes to resistivity, near-infrared transmission, and refractive index (~1.96 to 3.25) [18]. Importantly, the phase transition in VO2 can be triggered by a variety of stimuli aside from temperature including: strain [19], electric field [20, 21], or optical excitation [22]. In the case of pulsed optical excitation, the SMT has been shown to occur on time-scales less than 100 fs [23, 24], perhaps offering the possibility for achieving optical modulation at THz speeds.
Here we report on a silicon-based hybrid optical modulator incorporating VO2 as the optical switching element on an ultra-compact silicon micro-ring resonator. By combining a very small, ~0.28μm2, active area of VO2 on a low-mode volume, ~1μm3, resonator, large changes to resonant wavelength, and thus optical transmission, can be induced by triggering the SMT. This hybrid Si-VO2 resonator lays the foundation for a new class of electro-optic or all-optical modulators utilizing VO2 on Si.
2. Device fabrication
The Si-VO2 hybrid micro-ring resonator structure was fabricated on a silicon-on-insulator (SOI) substrate with a 220 nm p-type, 14–22 Ω cm resistivity, Si(100) layer and 1 μm buried oxide layer (SOITEC). Electron-beam lithography (JEOL JBX-9300–100kV) was performed using ZEP 520A e-beam resist spun at 6,000 rpm (~300nm thick). After pattern exposure and development in xylenes for 30s followed by an IPA rinse and N2 drying, anisotropic reactive-ion etching was performed (Oxford PlasmaLab 100) using C4F8/SF6/Ar process gases to completely etch the exposed portion of the 220nm Si layer.
A second stage of electron-beam lithography (Raith eLine) was performed to open windows for VO2 deposition, using ZEP 520A spun at 2,000 rpm (~500nm thick) to better facilitate VO2 lift-off. Amorphous VOx was then deposited by electron-beam vaporization of VO2 powder (100 mesh, 99.5% purity) in an Ångstrom Engineering deposition system. After deposition, lift-off in acetone under ultra-sonication was performed, leaving behind amorphous VOx patches across the silicon rings. Samples were then annealed in a vacuum chamber with 250 mTorr of oxygen at 450°C for five minutes.
The fabricated device structure is shown in the scanning electron microscopy (SEM) image in Fig. 1(a) . The radius of the ring, measured to the center of the ring waveguide, is 1.5μm. The ring waveguide width is ~500nm while the bus waveguide width is slightly reduced to ~400nm to promote better phase matching [25]. A coupling gap of ~75nm is employed to achieve near-critical coupling. Note that a top air cladding is used in this structure and all subsequent measurements. A ~560nm long patch of nanocrystalline VO2 with a low-apparent roughness straddles a small portion of the ring. Profilometry measurements revealed the VO2 film thickness to be ~70nm. The quality of the VO2 film was evaluated by temperature-dependent reflectance measurements on witness samples that consisted of VO2 films deposited on Si(100) substrates at the same time and under the same conditions as the device structures. Figure 1(b) reveals the hysteresis behavior in optical reflectivity, measured using a white-light source and silicon photo-detector. As the film temperature is increased beyond a critical temperature, ~63°C, a dramatic increase in reflectivity is observed, indicating initiation of the transition between semiconducting and metallic phases. The steep slope, high contrast, and relatively narrow hysteresis exhibited by these reflectivity measurements indicate the high quality of the deposited VO2 films [26]. Independent X-ray photoelectron spectroscopy (XPS) measurements on similar electron-beam deposited films (not shown) further confirm the near perfect stoichiometry of the as deposited film, measured to be V1.01O2. The high quality VO2, smooth surface structure of the VO2 patch, and the corresponding optical performance are attributed to the short duration annealing conditions, known from a previous work [27].
Fig. 1 (a) SEM image of a hybrid Si-VO2 micro-ring resonator with 1.5μm radius. The lithographically placed VO2 patch is highlighted in false-color. (Inset scale bar is 250nm). (b) White-light reflectivity versus temperature on a thin film VO2 control sample deposited on a Si(100) substrate.
3. Experiment and analysis
3.1 Measurement set-up and passive spectral measurements
In preparation for optical measurements, samples were cleaved across each end of the bus waveguide, several millimeters away from the central device, and mounted on an XY positioning stage. Piezo-controled XYZ stages were used to position and couple light to/from polarization maintaining lensed fibers (OZ Optics Ltd.) as shown in Fig. 2(a) . A tunable cw laser (Santec TSL-510) was used to perform passive transmission measurements, utilizing quasi-TE polarization, over the wavelength range 1500–1630nm.
Fig. 2 (a) Schematic of the optical measurement set-up. (b) Passive spectral measurements of optical transmission on 1.5μm radius micro-ring resonators with (bottom) and without (top) an integrated VO2 patch. For clarity, the top all-Si curve has been offset by + 20dBm.
In Fig. 2(b), the transmission of two 1.5μm radius ring-resonators with and without integration of VO2 are compared. Both devices exhibit a very large free-spectral range (FSR) near 60nm, which is highly desirable for enabling multiplexed photonic architectures [5, 28]. Taking the average FSR values and resonance positions, we estimate group indices of ~4.385 and ~4.415 for the all-Si and hybrid Si-VO2 resonators, respectively. Note that in the case of the hybrid resonator, the measured group index comprises contributions from both the bare-Si and VO2-coated sections. Thus, when taking into account the ~6% VO2 coverage on the ring, we approximate a group index closer to ~4.9 for the VO2 coated section of the ring waveguide. Further, we find that the Si-VO2 hybrid resonator has deeper resonances, which suggest near-critical coupling. This result is deliberately achieved by initially over-coupling the all-Si resonator by the use of the small ~75nm air gap. Introduction of the small VO2 patch increases the round trip loss, primarily due to the modal mismatch between the bare-Si ring waveguide and hybrid VO2-coated waveguide sections. A more detailed discussion can be found in Section 3.2 where we further analyze the device Q-factors.
3.2 Photothermally induced optical modulation
To characterize the optical response of the hybrid Si-VO2 micro-ring resonator in both states of the SMT, a 532nm cw pump laser (New Focus 3951-20) is focused onto the device with a 20x objective as shown in Fig. 2(a). An infrared (IR) camera was used for alignment purposes. IR imaging at maximum exposure and contrast settings was used to determine an upper bound for the Gaussian beam size, w0 ≈90μm. Given that nearly 100% of the total power (Ptotal) is contained within the radius 2w0, we estimate the average intensity to be approximately 15 W/cm2. The peak on-axis intensity is therefore ~30 W/cm2. In our experiments we found that precise positioning of the pump beam, to within approximately 10 microns, was required to trigger the SMT, suggesting that the peak on-axis intensity may be a more reasonable indicator of the laser intensity near the Si-VO2 micro-ring. We note that photothermal VO2 switching has previously been demonstrated with threshold intensities below 10 W/cm2 for thin-films on glass substrates while under continuous optical pumping at 532nm [29]. However, the threshold intensity and switching dynamics depend strongly on the VO2 film thickness, and will further vary strongly with the properties of the substrate (e.g., thermal conductivity and diffusivity) and the particular geometry employed.
In Fig. 3 , we present the optical transmission of the 1.5μm radius hybrid Si-VO2 ring resonator as measured before and after triggering the SMT with the 532nm pump laser. After unblocking the 532nm laser, the system is given a few minutes to reach thermal equilibrium before again measuring the optical transmission with the tunable laser. These measurements reveal a sizeable shift Δλ = –1.26nm in the resonance wavelength, coinciding with an optical modulation greater than 10dB at the initial resonance position, λ = 1568.78nm. Because VO2 exhibits a dramatically reduced refractive index in the metallic state, a blue-shift in resonance frequency is naturally expected to arise from triggering the SMT. However, additional effects are also expected to be present during this experiment, including dependence on: (1) the thermo-optic (TO) effect in silicon, Δn/ΔT = + 1.86 × 10−4/K [30], and (2) the free-carrier index (FCI) [6]. These two effects are weaker than the much larger optical response of the VO2 and, in this experiment, also carry opposite signs. Thus, the photothermal approach for triggering VO2’s SMT enables silicon’s TO and FCI refractive index contributions to be used against each other so that the optical signature of VO2 can be more readily distinguished. We verified that the dominant contribution to this resonance shift was indeed coming from the VO2 by performing a control experiment (not shown) on an all-Si micro-ring with the same dimensions, revealing a Δλ = + 0.938nm net red-shift in resonance wavelength. This indicates that in the absence of TO or FCI effects, the achievable resonance blue-shift arising solely from VO2’s SMT is even larger than the Δλ = –1.26nm value reported from this experiment. A further analysis of the interplay between these effects can be found in Section 3.2.
Fig. 3 Optical transmission of the 1.5μm radius hybrid Si-VO2 ring resonator as a function of wavelength, before and after triggering the SMT with a 532nm pump laser. The lines are Lorentzian fits. Inset: IR camera images revealing vertical radiation at a fixed probe wavelength, λ = 1568.78nm (dashed line).
Figure 4 illustrates the time-dependent optical response at the fixed probe wavelength λ = 1568.78nm. When VO2 is in the semiconducting state, this wavelength corresponds to being ‘on-resonance’, and thus the initial optical transmission is very low. Illuminating the device with the 532nm pump laser results in an immediate increase in the optical transmission followed by a ~15s decay toward low transmission and then an increase toward high transmission lasting about 3 min (Fig. 4(a)). This time-dependent optical response can be explained entirely in the context of laser-induced heating. The initial spike to high transmission results from rapid heating of the silicon ring and a thermo-optic dominated shift to longer resonance wavelength; this was confirmed by repeating the experiment at slightly red-shifted probe wavelength (not shown). However, once the SMT threshold temperature is approached, the resonance immediately begins to blue-shift back toward its original position and beyond, effectively sweeping across the resonance and producing a dip and rise in transmission. Here we emphasize that the photothermal approach for triggering the SMT, taken in this experiment (Fig. 2(a)), can be tuned to reduce response time by over three orders of magnitude by increasing the pump intensity [29]. Further, the photothermal technique is very robust, especially when compared to substrate heating methods. Substrate heating/cooling is relatively slow and also leads to unwanted thermal expansion or contraction of the substrate and heating stage, making it very difficult to maintain consistent coupling on- and off-chip. Localized photothermal excitation overcomes these problems and further exhibits an optical configuration in which future nanosecond and even ultrafast all-optical switching measurements could be carried out.
Fig. 4 Optical transmission as a function of time at λ = 1568.78nm, in the 1.5μm radius hybrid Si-VO2 device, when turning the 532nm pump laser (a) ON and (b) OFF. (c) Single-frame images from the IR camera movie Media 1, highlighting the delayed probe response observed in (b).
In Figs. 4(b) and 4(c) we examine the optical response, again at λ = 1568.78nm, after turning off (i.e. blocking) the 532nm pump laser. When this occurs, the device immediately begins cooling off, ultimately resulting in a return to the VO2 semiconducting state and an ‘on-resonance’ level of low transmission. However, an immediate drop in transmission is not observed; rather, turning off the laser coincides with a very small (~0.5dB) increase in transmission followed by a ~2s delay before a dramatic, ~2-5dB, drop in transmission. The initial increase, which is repeatably observed during multiple experiments, might possibly be attributed to either a small TO shift from the cooling silicon ring waveguide or to the recombination of photo-generated carriers in the silicon, which would eliminate free-carrier absorption effects. Most importantly however, we attribute the ~2s delay to device cooling at temperatures above the threshold SMT temperature. Once the threshold temperature is reached, the transition between metallic and semiconducting states is triggered. The ~2s delay between the pump shut-off and large probe response is similarly observed using the IR camera (Fig. 4(c), Media 1), and is strong evidence that a complete SMT indeed takes place prior to blocking the pump laser beam. After crossing the metal-to-semiconductor threshold temperature, the device requires several more minutes to cool completely back to room temperature. We note that this cooling time-scale is not fundamental to the phase-transition, but is rather a function of our experimental configuration and the absence of well-designed heat dissipation components in this proof-of-concept experiment.
3.3 Influence of ring-size and VO2 fractional coverage
In addition to the 1.5μm radius hybrid Si-VO2 resonator, we also fabricated and tested devices with larger radii of 5μm and 10μm. For comparison purposes, these devices were made with VO2 patches of the same dimensions as the 1.5μm radius structure (~560nm long). In Fig. 5(a) we plot the observed resonance shift, Δλ, normalized to the initial resonance position, after photothermally triggering the SMT. Interestingly, these results show a trend of decreasing blue-shift with increasing ring radius. As mentioned in Section 3.1, there are a number of effects contributing to the net resonance shift, primarily including: the SMT in the VO2, and the TO and FCI effects in silicon. The net contribution of these effects to the effective index change Neff of the hybrid micro-ring can be approximated by
where ΔNSMT represents the change in effective index of the VO2 coated waveguide section, which has a length LVO2, R is the ring radius, ΓSi the field confinement factor within the Si portion of the waveguide [31], and ΔnTO and ΔnFCI are the refractive index changes arising from the TO and FCI effects in silicon, respectively. Here the effective index N is distinguished from the real refractive index n by capitalization. Although this model is somewhat simplified, the fractional VO2 coverage on the hybrid micro-ring is taken into account with the term LVO2/2πR. Three-dimensional, finite-difference time-domain (FDTD) mode calculations for a 220nm x 500nm SOI waveguide coated with ~70nm of VO2 on top reveal Nsemi = 2.49 and Nmetal = 2.42 near a wavelength of 1568nm, resulting in ΔNSMT = −0.07. Experimentally, we estimate ΔNSMT = −0.055 after applying Eq. (1), where the measured resonant response is related to the change in effective index by Δλ/λ = ΔNeff/Neff and Neff is estimated by mode calculations to be ~2.3 for the ring waveguide. The change in effective index due to the TO and FCI effects is estimated to be ΔNSi = + 4.95 × 10−4 in the control experiment on the ultra-compact 1.5μm radius ring resonator. Overall, there is good agreement between simulation and experiment for the change of effective index, ΔNSMT, induced by the SMT of VO2. The experimentally determined value of ΔNSMT = −0.055 is very large, especially when compared to conventional silicon based modulators where ΔNSi is typically on the order of 10−4 [1], enabling significant changes to the overall effective index of the ring resonator, Neff, to be realized.Fig. 5 (a) Normalized resonance shift after photothermally triggering the SMT for hybrid Si-VO2 micro-ring resonators of different radii and the same VO2 patch length LVO2 = 560nm. (b) Corresponding Q-factor for these devices before and after triggering the SMT. Inset illustrates the various decay channels affecting total Q-factor and an IR camera image showing vertical radiation from a 10μm radius hybrid resonator when ‘on-resonance’ in the semiconducting state.
Reducing the fractional coverage of VO2 on the ring (e.g., by increasing R or decreasing LVO2) diminishes the effect of the SMT on the overall effective index change. This results in a corresponding reduction in resonant response, as observed in Fig. 5(a). This sort of behavior would be present even in the absence of TO or FCI effects. Further, the observed change of sign in the resonant response (from blue-shift to red-shift) for the largest ring radii, is an expected result, occurring when the TO effect in Si dominates the overall effective index change. When adapting this device structure for operation under different conditions (e.g., triggering the SMT by electrical or all-optical excitation) it is important to recognize that the achievable resonance shift can be precisely tuned by varying the fractional VO2 coverage on the resonator and probably by the volume of the VO2 patch. Furthermore, the resonant response could be adjusted by employing alternate waveguide geometries, such as the slot waveguide or pinch waveguide [31, 32], which could simultaneously maximize ΔNSMT and minimize ΓSi. In considering alternate waveguide geometries, however, it will be important to take account of the trade-off between increased resonant response, Δλ/λ, and the potential for increased losses, or reduced Q-factor.
Figure 5(b) shows the Q-factor of the different radii hybrid resonators, both before and after photothermally triggering the SMT, as determined by fitting the resonances to a Lorentzian lineshape. In both cases, we observe a trend of increasing Q-factor with increasing ring radius. An exponential increase in Q-factor is generally expected to occur with increasing ring radius, owing to a corresponding exponential reduction in bending losses [25]. While we do observe an exponential increase in Q-factor when increasing the radius from 1.5μm to 5μm, this increase appears to plateau at larger ring radii. This result can be explained by decomposing the total Q-factor, Qtot, into the primary decay channels of coupling loss and propagation loss through the relation:
We can further isolate the sources of propagation loss, as coming from the Si ring or the VO2 patch [16], by:The dashed trendlines shown in Fig. 5(b) were obtained by combining Eqs. (2) and (3) and assuming Qring to have an exponential dependence on ring radius. The trends show that for low ring radii the Q-factor is primarily limited by bending loss, while for larger ring radii the Q-factor is limited by loss induced from the presence of the VO2 patch. IR camera imaging of the 10μm radius hybrid resonator (inset Fig. 5(b)) further confirms that vertical radiation losses are primarily coming from the location of the VO2 patch. Furthermore, we note that transitioning from the semiconducting to metallic state is expected to result in a decrease of QVO2 owing to the increased absorption of the metallic VO2 film. However, because we have employed a short ~560nm long patch of VO2, the added round trip losses due to absorption in the metallic state are expected to be very low [16], and the change in QVO2 should be relatively small. This matches our observations, as we find a decrease of QVO2 from ~8,800 to ~7,800. The SMT would yield a more dramatic change to the Q-factor if a larger VO2 fractional coverage were utilized. Photogenerated carriers in the Si, under ~15 W/cm2 532nm illumination, are another potential source of increased losses within the ring, and in the present experiment may be generated at significant rate, ~1023 cm−3/s. However, past work has shown that while carrier concentrations on the order of ~1017 cm−3 (which could be achieved in the present experiment assuming a ~1μs effective carrier lifetime) are large enough to produce large, nm scale, resonance blueshifts, the added losses are still several times smaller than the intrinsic scattering losses of a conventional ring resonator with a 5μm radius [3]. Therefore, the absorption loss induced from photogenerated carriers is not expected to account for a significant fraction of the total losses, especially when compared to the bending losses at low ring radii or radiation losses induced from the VO2 patch. We emphasize the experiments reported here demonstrate that a small fractional coverage of VO2 (~6%) on a silicon ring is sufficient to produce a large resonant response and almost no effect to the total Q-factor in the case of the ultra-compact 1.5μm radius device. Further, the modest Q-factor (~103) results in a very short cavity lifetime (< 1ps), opening the possibility for future ultrafast operation.4. Outlook
From prior estimates on the energy density required to thermally switch thin-film VO2, ~102 J/cm3 [16], the minimum energy required to thermally switch our hybrid device is expected to be ~3 × 10−12 J. For faster and lower threshold switching, either electric-field assisted switching or all-optical excitation should be employed. Given the measured threshold laser fluence of ~0.25 mJ/cm2 for nanoscale VO2 pumped at 1550 nm, and an active area ~0.3μm2 of VO2 [33], the minimum energy required for ultra-fast all-optical switching in the hybrid Si-VO2 ring resonator is predicted to be of order ~190 fJ/bit (~750 fJ × 1/4; accounting for the likelihood of a 0-1 transition in a random signal) [34]. Similar switching energies are expected for electrically driven devices, although the SMT would occur on slower timescales, on the order of 10−8 s compared to 10−13 s for all-optical excitation [16]. For comparison, forward biased electro-optic ring-resonator modulators with switching times below 10−9 s have been shown to use ~300 fJ/bit after accounting for temperature stabilization [1, 35]. Notably, the low mode-volume resonator geometry employed in this work enables >10dB optical modulation to be achieved with approximately 1/10th the active area of VO2 that would be required to achieve similar modulation depths in a single-pass broadband absorption modulator [16], thereby promoting reduced power requirements as well as a compact device footprint.
Unlike the athermal and ultrafast ~10−13 s optically driven SMT of VO2, the structural phase transition (SPT) — from the tetragonal, metallic crystal structure back into the monoclinic, semiconducting state — relies on carrier-phonon interactions taking place on longer ~10−9 s timescales. Improving the relaxation time is currently a topic of significant research interest. A number of factors are known or expected to influence this relaxation time, including the active VO2 volume, crystallite domain size, doping, and excitation fluence. For example, measurements of the THz signal show that up to a threshold of about 3 mJ/cm2 in a thin VO2 film pumped by a femtosecond 800 nm pulse, the recovery from the metallic state takes only about 1 ps [24]. For the nanoscale VO2 patches needed for the ring resonator, the fluence needed to reach the threshold should be substantially lower, particularly for near band-edge pump wavelengths. It may also be possible to speed up the relaxation time by incorporating an electrical bias or diode configuration to rapidly sweep away excess carriers. Critically important to resolving this issue are measurements of the time-dependent dielectric function as a function of fluence, and measurements of the SMT threshold and SPT relaxation for near band-edge pump wavelengths.
While there are still a number of challenges and opportunities associated with the details of the SMT of VO2, these experiments demonstrate that the hybrid Si-VO2 resonator offers an attractive platform for robust optical modulation and reconfigurable optical routing. Further, this platform may form the basis for future electro-optic or all-optical modulators utilizing a hybrid Si-VO2 geometry.
5. Summary
We have demonstrated photothermally induced optical switching of hybrid Si-VO2 micro-ring resonators. Triggering the SMT in VO2 results in a large reduction in refractive index and a correspondingly large blue-shift in resonant frequency. Optical modulation greater than 10dB from a low mode volume (~1μm3) silicon-based micro-ring device is found to require only a very small (~0.28μm2) active area of VO2. Combined with a large FSR, modest Q-factor, short cavity lifetime, and the potential for ultrafast operation, the hybrid Si-VO2 micro-ring platform presents a robust framework for next-generation optical switching.
Acknowledgments
This work was supported in part by the Air Force Office of Scientific Research under Grant FA9550-10-1-0366. The authors further acknowledge support from equipment grants W911-NF-10-1-0319 and DMR-0957701 from the ARO and NSF, respectively. Portions of this work were performed at the Vanderbilt Institute of Nanoscale Science and Engineering, using facilities renovated under NSF ARI-R2 DMR-0963361, and the Center for Nanophase Materials Sciences, which is sponsored at Oak Ridge National Laboratory by the Division of Scientific User Facilities, U.S. Department of Energy. The authors thank C. Kang, P. Markov, and T. Whittle for useful discussions. R.E.M. was supported by a research assistantship provided by a grant from the Office of Science, U. S. Department of Energy (DE-FG02-01ER45916). J.D.R. acknowledges support from a NSF Graduate Research Fellowship.
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