, Volume 11, Issue 6, pp 1427–1435 | Cite as

Substrate-Independent Lattice Plasmon Modes for High-Performance On-Chip Plasmonic Sensors



We systematically study the lattice plasmon resonance structures, which are known as core/shell SiO2/Au nanocylinder arrays (NCAs), for high-performance, on-chip plasmonic sensors using the substrate-independent lattice plasmon modes (LPMs). Our finite-difference time-domain simulations reveal that new modes of localized surface plasmon resonances (LSPRs) show up when the height-diameter aspect ratio of the NCAs is increased. The height-induced LSPRs couple with the superstrate diffraction orders to generate the substrate-independent LPMs. Moreover, we show that the high wavelength sensitivity and the narrow linewidth of the substrate-independent LPMs lead to the plasmonic sensors with high figure of merit (FOM) and high signal-to-noise ratio (SNR). In addition, the plasmonic sensors are robust in asymmetric environments for a wide range of working wavelengths. Our further study of both far- and near-field electromagnetic distribution in the NCAs confirms the height-enabled tunability of the plasmonic “hot spots” at the sub-nanoparticle resolution and the large field enhancement in the substrate-independent LPMs, which are responsible for the high FOM and SNR of the plasmonic sensors.


Lattice plasmon modes Plasmonic sensors Spectral tunability Figure of merit Hot spots 


Noble metal nanoparticles (NPs) such as Au NPs and Ag NPs support localized surface plasmon resonances (LSPRs), which are the light-coupled coherent oscillations of free electrons confined within the NPs [1, 2, 3, 4, 5]. Various applications such as enhanced spontaneous or stimulated emission [6, 7, 8], solar energy harvesting [9, 10, 11, 12, 13, 14], surface-enhanced Raman spectroscopy, [15, 16, 17, 18] and cancer phototherapy [19, 20] have been investigated. Due to the strong electromagnetic field confinement and enhancement at the nanoscale, LSPRs are highly sensitive to the refractive index (RI) of local environments of the NPs, enabling the development of plasmonic sensors based on the analyte-induced changes in the peak wavelength and/or intensity of LSPRs [21, 22, 23, 24, 25, 26].

According to Drude model, the real part of dielectric function of metal is expressed as [27]
$$ {\varepsilon}_1=1-\frac{{\omega_{\mathrm{p}}}^2}{\omega^2+{\gamma}^2}, $$
where ωp is the plasma frequency and γ is the damping parameter of bulk metal. γ is negligible in visible and infrared wavelength. At resonance wavelength, ε1 = −2εm = −2nm2, where εm and nm are dielectric function and RI of the media surrounding the metal NPs, respectively. Therefore, the peak wavelength of LSPRs is expressed as
$$ {\lambda}_{\mathrm{m}}={\lambda}_{\mathrm{p}}\sqrt{2{n_m}^2+1}, $$
where λp is the wavelength calculated according to the plasma frequency as 2πc/ωp. The nearly linear relationship between the resonance wavelength and RI in surrounding media underpins the plasmonic sensors employing LSPRs. The wavelength sensitivity of the plasmonic sensors that operate under wavelength-shift mode reaches hundreds of nanometers per RI unit (RIU) [28, 29, 30].

The performances of plasmonic sensors can be characterized by their figure of merit (FOM), which is defined as ratio of the wavelength sensitivity and the resonance linewidth. Usually, Ag nanospheres [30] and Au nanospheres [31] as single NPs or uncoupled arrays [32, 33, 34] have FOM between 1 and 3. The FOM is limited by the broad linewidth of LSPRs caused by radiative damping [35, 36, 37, 38]. Immobilizing NPs on substrates as required for on-chip applications also reduces FOM due to the reduced sensing areas from the contact between the NPs and the substrates. Strategies have been developed to improve the FOM of on-chip plasmonic sensors. For example, substrate undercutting has been applied to reduce the NP-substrate contact areas in order to increase the wavelength sensitivity [39]. The design of NPs with sharp tips, like Ag nanotriangles [30] or Au nanostars, [40] has enhanced the FOM up to 5. A significant improvement of FOM has been achieved by breaking the symmetry in two or more NPs (or nanoholes) to introduce the Fano resonances with narrow linewidth, [41, 42] which result from interaction between a discrete state and a continuum of states [43].

Another effective approach towards narrowing the resonance linewidth of LSPRs is to arrange NPs in a highly ordered array so that far-field diffractive coupling occurs [44, 45]. When the diffraction orders change from evanescent to radiative, a strong dipolar interaction occurs, resulting in lattice plasmon modes (LPMs) or collective modes [46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57]. Significant suppression of the radiative damping leads to an ultra-narrow resonance linewidth of LPMs. Both theoretical and experimental studies have shown that the narrow linewidth leads to the enhanced performance in sensing [58, 59, 60, 61]. However, the LPMs have intrinsic drawback for sensing applications due to their requirement of homogeneous environments surrounding the arrays of NPs. The LPMs are suppressed when the NPs are immobilized on substrates because the diffraction orders are cut off at the nanoparticle-substrate interfaces [62]. One can avoid this drawback by either introducing an index-matching layer on the top of NPs to generate a quasi-symmetric environment or increasing the size of the NPs to obtain coupling at the higher polarizability [51, 60, 62]. However, the index-matching layer prevents the interactions between analytes and NPs, and the use of larger NPs reduces the range of working wavelengths, limiting the applications of LPMs in the on-chip sensors [59, 60, 63]. Recently, an extremely high FOM of 108 is reported for LPMs in the Au mushroom arrays [64]. However, the weak coupling in this type of structures leads to a low signal-to-noise ratio (SNR). As a result, the development of LPM-based plasmonic sensors with both high FOM and SNR for a wide range of working wavelengths in the asymmetric environment has remained challenging.

Herein, we introduce an approach towards increasing the FOM of on-chip plasmonic sensors for a wide range of working wavelengths, which synergizes the sub-nanoparticle engineering of plasmonic “hot spots” for high sensitivity and narrow linewidth of the LPMs. Previous work has demonstrated that the height-induced LPMs in the core/shell SiO2/Au nanocylinder arrays (NCAs) [65, 66] shift the plasmonic “hot spots” from the nanoparticle-substrate interface to the nanoparticle-superstrate interface to form the substrate-independent modes. In this study, we focus on evaluating the sensing capability of the substrate-independent LPMs. Our studies are based on finite-difference time-domain (FDTD) simulations, and the far-field sensing performances are supported by the studies of near-field electromagnetic field distribution.

Simulation Details

Figure 1 illustrates the design of our plasmonic sensor, which consists of the core/shell SiO2/Au NCAs on a SiO2 substrate, and the experimental setup for evaluating the sensing performances. The diameter (D) of the SiO2 nanocylinders (NCs) ranges from 50 to 250 nm, while the height (h) is changed from 50 to 400 nm. Since the resonance wavelength of the LSPRs of the core/shell NCs depends on the ratio of the total diameter to the core diameter, we fix the thickness of the outer Au layer at 20 nm to simplify the simulations [67]. The lattice constants are defined with reference to the polarization of the incident light, i.e., a when the direction is parallel to the electric field (Ex) of incident light and a when the direction is vertical to the electric field. The optical constants of Au were taken from Johnson and Christy [68].
Fig. 1

Schematic of the plasmonic sensor consisting of the core/shell SiO2/Au NCAs on a SiO2 substrate and the experimental setup for the performance characterization. Incident light (red arrow) is launched normal onto the sample, and transmission light (blue arrow) is collected. A shift of the resonance wavelength in the transmission spectra is monitored to measure the changes of RI in the superstrate

Results and Discussions

Hot Spots Engineering in Single NC

As a reference to the design of the proposed plasmonic sensors based on the NCAs, we first examine the LSPR modes and FOM of single core/shell SiO2/Au NCs with different height-diameter aspect ratios on SiO2 substrates. As shown in Fig. 2a, the single NC with a low aspect ratio (D = 200 nm and h = 50 nm) exhibits a dipole-bonding mode (D1 mode) at a peak wavelength of 1049 nm when the superstrate RI is 1.0. The peak wavelength of the D1 mode makes a continuous redshift when the superstrate RI is increased from 1.0 to 1.2. Due to the radiative damping in the LSPRs, the linewidth of the scattering spectra is broad. The full width at half maximum (FWHM) for the spectrum with RI = 1.0 is 287 nm. The FWHM increases with RI and reaches 372 nm when RI = 1.2. It has been shown that an increased radiation damping and FWHM can be caused by the increased NP volume [35, 36, 69]. Although the actual volume of the NC remains the same for the different RIs (Fig. 2a), the NC experiences an increased volume because of a decreased effective wavelength of incident light when the surrounding medium has higher RI. Our further study of the near-field electromagnetic (Ex component) distribution at the NC (see upper panel of Fig. 2a) reveals that the LSPR of single NC originates from the propagation interruption at the interface between the NC and the substrate. Since the plasmonic “hot spots” associated with D1 mode interface with both substrate and superstrate, the sensitivity of the LSPR to the superstrate RI is low. From Fig. 2a, we obtain a wavelength sensitivity of 280 nm/RIU and a FOM of <0.98.
Fig. 2

Simulated scattering spectra of single core/shell SiO2/Au NC on SiO2 substrate with D = 200 nm. ah = 50 nm, superstrate RI changes from 1.0 to 1.2, and the real part of Ex is illustrated for RI = 1.0. The inset shows the enlarged scattering spectra in the spectral range indicated by the dash box. b RI = 1.0, the height of NC (i.e., h) changes from 50 to 400 nm, and the real part of Ex is illustrated for Q mode, D2 mode, and D1 mode for NC with h = 400 nm

We hypothesize that one can enhance the wavelength sensitivity of the LSPRs of single NCs to the superstrate RI (i.e., analytes) by shifting the plasmonic “hot spots” away from the nanoparticle-substrate interface to maximize their interactions with analytes. To test this hypothesis, we achieve the spatial control of the plasmonic “hot spots” within NCs at the sub-nanoparticle resolution by tuning the height-diameter aspect ratio of the NCs. Figure 2b shows the simulated scattering spectra and the near-field electromagnetic fields of single NCs with different heights for superstrate RI = 1.0. We observe a strong dependence of the LSPRs on the height of the NCs. Firstly, the D1 mode makes a slight redshift with the increased height. Secondly, a new dipole-bonding mode (D2 mode) shows up at the shorter wavelength of the D1 mode. With the “hot spots” located at the side of the NC (see upper panel of Fig. 2b), the D2 mode makes a redshift with the increased height. Lastly, a quadrupole mode (Q mode) appears at the higher energy side of the D2 mode when the height of the NC is further increased (i.e., h > 300 nm). This higher-order mode has the “hot spots” both within the core at the bottom of the NC and at the outer part of the Au shell on the top of the NC. Compared with the FOM of D1 mode (<0.98) in Fig. 2a, the D2 mode and Q mode have larger FOM, which are 3.15 (h = 200 nm; RI = 1) and 5.95 (h = 400 nm; RI = 1) (not shown here). The higher FOM for the D2 and Q modes justifies our hypothesis that the shift of the “hot spots” away from the nanoparticle-substrate interface increases the FOM. We focus on the height-induced D2 mode because it provides the opportunity to couple with the superstrate diffraction orders to generate the substrate-independent LPMs in 2D NCAs, which enables the development of the high-FOM plasmonic sensors.

Enhanced FOM in the Core/Shell NCAs

To study the height-induced LPMs in the 2D NCAs, we choose the NCs with the diameter D = 200 nm and the height h = 400 nm, which exhibit strong D2 mode. When light is incident on the 2D NCAs, photonic diffraction orders occur under specific conditions. The diffraction orders are calculated by treating the out-of-plane wave number as zero, which is given by
$$ {k}_{i,j}^{\perp }=\sqrt{{\left(n\frac{2\pi }{\lambda}\right)}^2-{k_{i,j}^{\left\Vert \right.}}^2}=2\pi \sqrt{\frac{n^2}{\lambda^2}-\frac{i^2}{{a_x}^2}-\frac{j^2}{{a_y}^2}}, $$
where n is the RI of the media surrounding the NCAs, λ is the wavelength of the incident light, and ki,j is the in-plane wave number, which is determined by the lattice constants of the NCAs ax and ay (corresponding to a and a in Fig. 1). The different diffraction orders along the x and y axes are determined by the integers i and j, respectively. When orthogonal coupling (i.e., the electric field Ex of incident light is vertical to a as shown in Fig. 1) is considered, the wavelength of the (0, j) diffraction orders is simplified as
$$ {\lambda}_{0,j}=na\perp /\left|j\right|, $$
Therefore, the coupling can be controlled by tuning the lattice constants of the NCAs. When the lattice constants of the NCAs are set as 500 nm for both x and y directions, the diffraction orders have no overlap with the LSPRs of the NCs and no photonic-plasmonic coupling occurs. Figure 3a shows a series of transmission spectra of the NCAs (D = 200 nm and h = 400 nm) when the superstrate RI changes from 1.0 to 1.2. Without the coupling, the transmission spectra of the NCAs are similar to that of the single NC with the same diameter, height, and RI (Fig. 2b). It should be noted that the transmission spectra, which arise from both light absorption and backward scattering by the NCAs, are different from the scattering spectra in Fig. 2b. However, both types of spectra can reveal the characteristics of the various modes of the LSPRs. As shown in Fig. 3a, the D1 mode shows up in the transmission spectra of the NCAs as dips around the wavelength of 1000 nm. Similarly, the D1 mode has a relatively low sensitivity to the changes of the superstrate RI (133 nm/RIU). The dips at 660 and 750 nm (referring to the spectrum for RI = 1 in Fig. 3a) correspond to the Q mode and D2 mode (also see Fig. 2b), respectively [65]. The wavelength sensitivity of the Q mode is calculated as 411 nm/RIU, which is higher than that of D1 mode. Like single NC, the enhanced wavelength sensitivity of the Q mode is attributed to the “hot spots” located at the sides of the NCs. The FWHM of the Q mode is calculated as 50 nm for RI = 1, leading to a high FOM of 8.2. The D2 mode merges with other modes at the higher superstrate RI, which makes it challenging to quantify the corresponding FOM.
Fig. 3

Simulated transmission spectra of the core/shell SiO2/Au NCAs of two types of a on the SiO2 substrates as a function of the superstrate RI, which changes from 1.0 to 1.2. The NCAs have D = 200 nm, a = 500 nm, h = 400 nm, and aa = 500 nm, ba = 900 nm. The inset of a shows the enlarged transmission spectra in the spectral range indicated by the dash box

To enable the plasmonic-photonic coupling for the LPMs in the NCAs, we tune the lattice constants to make the diffraction orders overlap with the LSPRs, in particular, the D2 mode. We focus on the D2 mode because its plasmonic “hot spots” located at the sides of the NCs can couple with the superstrate diffraction waves to generate the substrate-independent LPMs for the enhanced sensing performance. Figure 3b shows a series of transmission spectra of the NCAs (D = 200 nm, h = 400 nm, and a = 900 nm), which exhibit the coupling and the LPMs, when the superstrate RI changes from 1.0 to 1.2. A strong dip at the wavelength of 937 nm in the spectrum (for RI = 1) is assigned to the LPM that arises from the coupling between the (0, ±1) superstrate diffraction orders and the D2 mode. This LPM makes a continuous redshift when the RI is increased, leading to a sensitivity of 733 nm/RIU. The increased sensitivity of the LPM is attributed to the high sensitivity of the superstrate diffraction orders to the changes of superstrate RI. As radiative damping is suppressed in this collective mode, the LPM has a linewidth that is much smaller than that of the LSPRs of single NCs. Even if we take an intermediate FWHM value from the spectrum with RI = 1.08, a high FOM of 17.9 is obtained. In contrast, the LPM that arises from the coupling between the (0, ±1) substrate diffraction orders and the D1 mode is not sensitive to the superstrate RI. Such a LPM is identified as a dip around the wavelength of 1258 nm in the spectra of Fig. 3b. When the superstrate RI changes from 1.0 to 1.2, the dip wavelength of this substrate-related LPM has almost remained unchanged.

To better understand the two types of LPMs in Fig. 3b, we describe the (0, ±1) diffraction orders as λ0,j = na/|j| according to Eq. 4. We can see that the resonance wavelength is proportional to the RI of media surrounding the NCAs (i.e., n in Eq. 4) and to the lattice constants that are perpendicular to the external electric field Ex of incident light. Since n can be the RI of either superstrate or substrate, two sets of diffraction orders exist. For the LPM that arises from the coupling between the D2 mode and the (0, ±1) superstrate diffraction orders, the increase of superstrate RI makes a redshift in both the superstrate diffraction orders and the D2 mode, leading to the redshift of the LPM accordingly. The Fano-like LPM can be explained by the coupling between the discrete state (diffraction orders) and a continuum of states (LSPRs), and therefore, the coupling wavelength is slightly deviated from the (0, ±1) superstrate diffraction orders. This deviation is responsible for the difference in the wavelength sensitivity between the LPM and the (0, ±1) superstrate diffraction orders, i.e., 733 nm/RIU versus 900 nm/RIU (according to Eq. 4).

Electromagnetic Field Distribution and SNR

To gain a deeper insight into the photonic-plasmonic coupling and the field enhancement, we further study the electromagnetic field distribution in the NCAs. A large field enhancement is highly desired to achieve the plasmonic sensors with large extinction efficiency and thus high SNR. Figure 4 shows the field intensity distribution at an NC within the NCAs (D = 200 nm, a = 500 nm, h = 400 nm, and superstrate RI = 1) with different a and λ. Figure 4a shows the electric field distribution of the D2 mode without the coupling when a = 500 nm and λ = 750 nm (see Fig. 3a for the spectrum). We can see that the D2 mode arises from the bonding interaction between the inner and outer dipoles and a weak electric field also exists inside the SiO2 core of the NC. Once the D2 mode couples with the (0, ±1) superstrate diffraction orders to generate the LPM when a = 900 nm and λ = 937 nm (see Fig. 3b for the spectrum), the inner electric field disappears and the outer electric field is significantly enhanced (Fig. 4b). Therefore, the LPM arises from the far-field radiative interaction between the D2 mode on the NCs, which is the collective behavior caused by the electric field from both incident light and other NCs in the structures.
Fig. 4

|Ex|2 in the xz plane and |Hz|2 in the yz plane through the center of an NC in the NCAs (also see Fig. 1). The NCAs have D = 200 nm, a = 500 nm, h = 400 nm, and RI = 1. aa = 500 nm, λ = 750 nm, ba = 900 nm, λ = 937 nm, ca = 900 nm, λ = 897 nm (bare SiO2 NCAs), and da = 900 nm, λ = 900 nm

To further examine the plasmonic enhancement of the field in the diffraction orders, we simulate the magnetic field distributions for both the bare SiO2 NCAs (without Au shells) and the core/shell SiO2/Au NCAs. The orthogonal diffraction orders lead to the horizontal propagating magnetic field, i.e., Hz component propagating along y axis (see Fig. 1) [65, 66]. Figure 4c shows that, for the bare SiO2 arrays, the superstrate diffraction order of a low intensity is extended from the sides of the NC into the superstrate with two lobes parallel to z axis. The similar lobes with weaker intensity exist in the substrate, which arises from the imaginary part of Hz rather than the diffraction orders. When the plasmonic-photonic coupling occurs in the core/shell NCAs, the Hz intensity is enhanced with a factor of 20∼30, indicating an energy transfer from the plasmonic mode to the photonic diffraction modes (Fig. 4d).

In order to understand the RI-dependent extinction intensity (i.e., the extinction intensity increases and then decrease when the RI changes from 1.0 to 1.2) of the height-induced LPMs in Fig. 3b, we study both near-field and far-field intensity distributions in the NCAs as a function of the superstrate RI. As shown in Fig. 5, the coupling becomes stronger when RI is increased from 1.0 to 1.04 due to the increased near-field electric intensity (Figs. 4b vs. 5a) and far-field magnetic intensity (Figs. 4d vs. 5c). However, the coupling strength becomes weaker when the RI is further increased. The decrease in both near-field electric intensity and far-field magnetic intensity (Fig. 5b, d vs. Fig. 5a, c) confirms the decreased coupling strength when RI reaches 1.2. The RI-dependent field intensity matches the dependence of the extinction intensity on the RI (Fig. 3b). This RI dependence indicates the possibility to design the intensity-dependent plasmonic sensors. For the wavelength-dependent sensors, the understanding of the RI dependency helps guide on the design of the parameters of the NCAs to achieve a high extinction ratio (and a high SNR) for target analytes with RI in a specific range.
Fig. 5

|Ex|2 in the xz plane and |Hz|2 in the yz plane through the center of an NC in the NCAs (also see Fig. 1). The NCAs have D = 200 nm, a = 500 nm, a = 900 nm, and h = 400 nm. a RI = 1.04, λ = 967 nm; b RI = 1.2, λ = 1087 nm; c RI = 1.04, λ = 941 nm; and d RI = 1.2, λ = 1087 nm. The different λ for a and c is due to the deviation between the discrete state (diffraction orders) and a continuum of states (LSPRs) in the Fano-like coupling of the LPMs

Working Wavelengths of Plasmonic Sensors

To generalize the concept of the height-induced LPMs for the high-performance, on-chip plasmonic sensors, we study the sensing characteristics of the core/shell SiO2/Au NCAs with different structural parameters as summarized in Fig. 6. We set the lattice constants of the NCAs as a = 550, 700, and 1000 nm for the NCs with diameters D = 50, 100, and 250 nm in order to tune the (0, ±1) superstrate diffraction orders to overlap with the resonance wavelength of the D2 modes to achieve the plasmonic-photonic coupling. We choose the smaller a to tune the (±1, 0) substrate diffraction orders to the shorter wavelength side of the LSPRs of the NCs to avoid the interference from the occurring of the parallel coupling. From Fig. 6, we can see that the coupling between the D1 mode and (0, ±1) substrate diffraction orders disappears when the diameter of the NCs becomes small. Only small dips are observed in the spectra for D = 50 nm at RI = 1.2 and for D = 100 nm at RI = 1.16 and 1.2, indicating that the refractive-index mismatch between substrate and superstrate suppresses the LPMs when the polarizability of the NCs is not large enough (due to the small diameter of the NCs) to support the plasmonic-photonic coupling. In contrast, the height-induced LPM still exists even when the diameter of the NCs is reduced down to 50 nm. Therefore, different from the previously studied LPMs in the arrays of NPs with low aspect ratio where the plasmonic-photonic coupling and thus the LPMs are suppressed by the presence of substrates due to the asymmetric environments, [52, 62] the height-induced LPMs in the NCAs of high aspect ratio are robust under the asymmetric environments, which are frequently experienced in the on-chip sensing applications.
Fig. 6

Transmission spectra of the core/shell SiO2/Au NCAs on the SiO2 substrates. The height is set as h = 400 nm. aD = 50 nm, a = 300 nm, a = 550 nm, bD = 100 nm, a = 400 nm, a = 700 nm, and cD = 250 nm, a = 500 nm, a = 1000 nm. The superstrate RI changes from 1.0 to 1.2. d The dependence of the coupling wavelength and the FWHM of the core/shell SiO2/Au NCAs on the diameters of the NCs and the superstrate RI

Figure 6d summarizes the FWHM and the coupling wavelength for the core/shell NCAs with the different structural parameters and superstrate RIs. The FWHM decreases when the RI increases due to the weakening coupling strength. By taking the FWHM for the NCAs with strong plasmonic-photonic coupling (i.e., RI = 1.12, 1.08, and 1.04 for D = 50, 100, and 250 nm), we calculate the FOM as 23.7, 22.4, and 13.2, respectively. The highest FOM obtained at D = 50 nm suggests that reducing the FWHM is critical in enhancing the FOM of the LPM-based plasmonic sensors. The high FOM for both large and small NCAs is also supported by theoretical analysis. According to Eq. 4, the wavelength sensitivity is determined by a, which decreases for the arrays with smaller NCs. However, the FWHM also decreases for the smaller NCs, which maintains the high FOM for the NCAs.


In summary, the substrate-independent LPMs, which arise from the coupling between superstrate diffraction orders and the height-induced modes of LSPRs in the lattice plasmon resonance structures of high aspect ratio (e.g., SiO2/Au core/shell NCAs), presents a tremendous opportunity for the development of the high-performance, on-chip plasmonic sensors. The narrow linewidth and the high wavelength sensitivity of the substrate-independent LPMs lead to the sensors of high FOM for a wide range of working wavelengths. The high SNR of the plasmonic sensors is enabled by the LPM-associated large field enhancement in the NCAs. The proposed structures can be fabricated with low-cost and high-throughput nanofabrication techniques, including nanoimprinting lithography. With the high FOM, high SNR, and robustness of the substrate-independent LPMs, the on-chip plasmonic sensors will find a wide range of applications in molecular analysis, biomedicine, and environmental protection.



The authors acknowledge the financial support of the Beckman Young Investigator Program. We also thank the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing HPC resources that have contributed to the research results reported within this paper. URL: We thank B. Bangalore Rajeeva, X. Peng, M. Wang, and Z. Wu for their helpful discussions on the simulation results and proofreading the manuscript.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Mechanical Engineering, Materials Science & Engineering Program, and Texas Materials InstituteThe University of Texas at AustinAustinUSA

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