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Finite element simulations of laser-induced heating in scanning probe microscopy are presented. The electromagnetic field is first simulated for a variety of tip and substrate materials, and for air and aqueous environments. This electromagnetic field, in the end of the tip and substrate under the tip, produces Joule heating. Using this Joule heat source, steady state thermal simulations are performed. As a result of the large enhancement of optical power by the tip-substrate cavity, predicted temperature rises can be over 3 orders of magnitude higher than the values predicted without a tip present, but the optical signal can be enhanced by over 10 orders. Gold tips and substrates are predicted to give the highest optical signal for a given temperature increase.
©2006 Optical Society of America
1. Introduction
Scanning probe microscopy (SPM) is a powerful technique for nanotechnology, giving molecular resolution images, but its main drawback is its inability to perform local chemical characterization. Tip-enhanced optical microscopy [1,2] can overcome this problem by analyzing the light scattered from the region beneath the tip. Fluorescence mapping and spectroscopy can be performed, but the majority of recent interest has been in tip-enhanced Raman scattering (TERS) [3–5]. Raman spectroscopy is able to identify any type of condensed matter from the resulting vibrational spectrum.
Tip-enhanced optical microscopy relies on the enhancement of the electric field in the end of the tip, in the region of the substrate directly beneath the tip, and the gap in-between. This increases the optical scattering from an object between tip and substrate by a factor measured as 6×106 [3], or predicted to be as high as 1012 [6]. Despite this massive enhancement, Raman scattering is still relatively weak — with cross sections per molecule of around 10-30 cm2 [7], long acquisition times are required. However, as the optical power density has been enhanced by a large factor, this also means that laser-induced heating will be locally enhanced. Indeed, in practice laser intensities in tip-enhanced optical microscopy have to be limited to around 1 mW µm-2 [8,9] to avoid damage to the sample.
Several electromagnetic simulations of tip-enhanced optical scattering have been published [6,8–13]. Only two simulations have been published, both containing only a small aspect of thermal analysis. The first [14] considers 3-photon absorption with femto-second pulses of infra-red illumination. The second [15] only considers an isolated gold tip of radius 5 nm in water illuminated with near infra-red light, for use as nano-optical tweezers.
2. Methods
Using a commercial finite element package (ANSYS Multiphysics), we developed an electromagnetic simulation to calculate the electric field in the vicinity of the tip apex, applying an optical power density of 1 mW µm-2 (equivalent to an electric field of 8.7×105 V m-1). The model consists of around 3×104 elements, with the highest density being located in the tip-sample gap. The size of the model is 168 nm in diameter, and 280 nm in the vertical direction [6,12].
3. Results
The electric field simulations produce a highly enhanced spot under the tip giving a large enhancement, E, of the electric field amplitude. So, the incident optical intensity is enhanced by a factor of E2, and by symmetry the radiated optical intensity is also enhanced by E2. This means that the two-stage scattering process of illumination and radiation enhances the optical intensity by a factor of E4. The electric field is plotted in Fig. 1(a), for a gold tip and gold substrate, giving an electric field enhancement, E, of 504. It is worth noting that the predicted electric field is over 3×108 Vm-1, which is close to the range of 109 Vm-1 required to dissociate molecules with laser pulses [16]. Desorption of molecules on surfaces can also occur, at a lower field strength.
Fig. 1. Gold tip (20 nm radius) and gold substrate, separated by 2 nm, illuminated with 1 mW µm-2 of 533 nm p-polarized radiation incident along the arrow, at 45 degrees to the substrate plane. (a) Electric field simulations, in units of V m-1. (b) Resultant heat generation, in units of W m-3. (c) Temperature simulations, in Celsius. Dotted lines display the positions of line profiles used in Figs 3 and 4.
The finite element package is also able to calculate Joule heating caused by absorption of light, which is proportional to the imaginary part of the dielectric function [17], ε(λ)=εr+iεim=n2, n being the complex refractive index. The Joule heating is plotted in Fig. 1(b), being concentrated at the end of the tip, and the region of the substrate directly under the tip apex. This Joule heating was passed onto a separate steady state thermal simulation. Temperature dependent values of thermal conductivity and heat capacity (from a variety of reference sources) were used for the tip, substrate and surrounding medium. The emissivity of the tip or substrate material is also included for radiation. Boundary conditions must be applied in order to reach a meaningful solution — in this case the top and bottom of the model are fixed at room temperature.
Figure 1(c) shows the results of the thermal simulations. A boundary condition must be applied to the model — the temperature is fixed at 20 °C at the top and bottom. The hottest part of the model is the very end of the tip, which coincides with the maximum in Joule heat production portrayed in Fig. 1b. The substrate is substantially cooler than the tip, despite the similar amount of Joule heat production. There is no heat generation in the gap, but heat is transferred from the tip and substrate by conduction and radiation.
We modeled the same geometry without the tip, and the results are presented in Fig. 2. In all thermal simulations, the temperature rise is proportional to the incident optical power density (although this relation is not exact thanks to temperature dependent material properties). Replacing the gold surface by one of BK7 glass, the temperature rise is significantly lower due to the lower optical absorption.
Fig. 2. Temperature profiles normal to a gold surface, below air or water, with no tip present. Illumination is with 1 mW µm-2 of p-polarized radiation incident at 45 degrees to the vertical tip axis, at a wavelength of 533 nm. The temperature is fixed at 20 °C at the top and bottom of the model: at -40 and +240 nm.
Returning to the tip-enhanced case, we modeled a variety of materials which are common choices in SPM and which are most likely to be used in tip-enhanced optical microscopy. Gold is the preferred choice for tip material as it has a large enhancement of the electric field within the visible spectrum. Silver has a larger enhancement, but the width of the resonance in the visible spectrum — at around 5 nm [10] — is far sharper than that of gold, which has a resonance width of a few tens of nanometers [12]. In Raman or fluorescence spectroscopy, the scattered light is red shifted with respect to the incident light — typically by several tens of nanometers. The overall enhancement is given by (E(λinc))2(E(λrad))2 , λinc and λrad being the incident and radiated wavelengths respectively. So silver is considered a poor choice because if λinc is chosen to match the resonance, then E(λrad) will be close to zero and the overall enhancement will be very low. SPM often requires atomically flat surfaces, so mica and silicon are good choices, but the highest enhancement is predicted for a gold surface.
Line profiles through the tip axis, gap, and substrate are presented in Fig. 3, for a variety of tip and substrate combinations in air or water — following the vertical dotted line in Fig. 1(c). When the tip and substrate material are the same, then the tip is significantly hotter than the substrate : the temperature rise in the tip is around 20 times that in the substrate. This can be ascribed to the one dimensional nature of the tip, which has a small half cone angle of only 10 degrees. For improved conduction, this cone angle should be increased. The substrate, by contrast, can conduct heat away in three dimensions from the source.
Fig. 3. Temperature profiles for various tip and substrate materials, and surrounding media. The line profiles (displayed in Fig. 1c) are in the vertical direction, from the tip (left), crossing the 2 nm gap to the substrate (right). Illumination is 1 mW µm-2 of p-polarized radiation incident at 45 degrees to the vertical tip axis, and the wavelength is chosen to give the maximum electric field.
When a gold tip is used above a mica substrate, there is negligible heat generation in the mica due to its lack of optical absorption (εim is zero), yet the mica heats up due to conduction.
Aqueous environments were also considered, given both its suitability for scanning with biological systems, and its likelihood of conducting away more heat than air. However, this is not observed to be the case. It is not easy to assign this to one single material property; the specific heat of water is 4 times that of air, the thermal conductivity is 22 times higher, and its density is 830 times higher.
Lateral line profiles are presented in Fig. 4, along the horizontal dotted line in Fig. 1(c). Discrete jumps are due to individual elements, which are more pronounced than in the vertical direction. The concern with temperature rises from an experimental point of view is not necessarily the temperature within the tip or substrate, but the material or film of interest on top of the substrate. Temperature rises of 100 °C are acceptable for many inorganic materials, but many biological samples will barely tolerate rises of 10 °C. The width (fwhm) of the thermal distribution is around 20 nm, which is much larger than the 6 nm wide spot of the optical (E4) distribution [6].
Fig. 4. Temperature profiles for various tip and substrate materials, and surrounding media. The line profiles (displayed in Fig. 1c) are in the horizontal direction, on a plane midway between the tip apex and substrate. Illumination is 1 mW µm-2 of p-polarized radiation incident at 45 degrees to the vertical tip axis, and the wavelength is chosen to give the maximum electric field.
A silicon tip and substrate were also modeled, as the material has a much reduced value of εim. Its enhancement is not as high as that calculated with gold tips, but it is more suited to the near infra-red region of the spectrum and has a wide resonance. Indeed, the observed temperature rise is far lower than the gold tip and substrate, but so is the optical enhancement and therefore the amount of Joule heat. To take this into account, Table 1 shows both the overall optical enhancements and temperature maxima. The right hand column indicates the amount of scattered light for a given temperature rise, in the limit λinc=λrad. It is clear that the gold tip and substrate in air perform better than any other combination of materials. The addition of a gold tip gives an improvement over far field studies on a BK7 glass surface by a factor of 213.
For Raman and fluorescence studies — when λrad>λinc — the silicon tip and substrate combination will be even more advantageous as it has a broader resonance than any of the others presented [6], giving a value of (E(λinc))2(E(λrad))2 which is close to (E(λinc))4.
Table 1. Thermal and optical characteristics for tip-enhanced and far field optical microscopy
We lastly turn to issues of thermal expansion, which is not considered in the finite element model. The linear expansion is given by δL=α L δT, L being the length and δT being the temperature rise. The linear thermal expansion coefficient, α is 1.4×10-5 °C-1 for gold and 3×10-6 °C-1 for silicon. L can be approximated as around 100 nm within the tip, and 20 nm within the substrate. For a temperature rise of 100 °C, this corresponds to an expansion of 0.14 nm for a gold tip, or 0.03 nm for a silicon tip. In the substrate these figures are 0.03 nm and 0.006 nm respectively. Unless tunneling feedback is being employed, these figures are not problematic. In any case, if the laser power is constant instead of pulsed, this expansion will only occur once and the laser power can be increased at a slower rate than the feedback response.
4. Conclusion
We have performed finite element electromagnetic and thermal simulations on a variety of materials used in SPM. The absorption of light causes heating, which is localized at the tip apex and in the substrate, beneath the tip. However, the temperature maximum is located well away from the tip apex, in the surrounding medium. Silicon tips and substrates are predicted to give the highest optical signal for a given temperature rise.
Acknowledgments
This work was supported by an award from The Medical Research Council, UK (G0401498).
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