Supporting Information AFM-IR: Technology and applications in nanoscale infrared spectroscopy and chemical imaging Alexandre Dazzi 1 * and Craig B. Prater 2 1 Laboratoire de Chimie Physique, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay cedex, France 2 Anasys Instruments, 325 Chapala St., Santa Barbara CA 93101 USA *corresponding author: alexandre.dazzi@u-psud.fr 1. AFM-IR theoretical overview In this section, we outline an analytical approach to describe key physical phenomena involved in the detection of infrared absorption by the AFM-IR technique. We will underline the major parameters involved and demonstrate that the oscillation amplitude of the cantilever is proportional to optical absorption of the sample. We will also compare different types of laser excitation: single pulse (laser repetition rate lower than the time of cantilever relaxation), synchronous pulsing resonance (laser repetition rate equal to a resonance of the cantilever), and synchronous modulation (laser modulation equal to the resonance of the cantilever). The AFM-IR technique is based on the detection of IR absorption by the tip of an atomic force microscope (AFM). The sample is illuminated by a pulsed tunable laser that can be tuned to absorption 1
wavelengths in one or more regions of the sample. The absorption of infrared photons leads to a rapid local heating of the absorbing regions of the sample. The AFM tip detects this thermal expansion of the IR absorbing regions with a spatial resolution far below the conventional diffraction limit. The spatial resolution can exceed the diffraction limit because the thermal expansion of the sample varies on the nanoscale according to local variations in IR absorbance due to any variations in the local chemical composition. Thus the AFM tip can measure and map the thermal expansion limited only by either the radius of the AFM tip or in some cases the spreading of heat from absorbing regions. Once a sample region absorbs IR radiation and heats up, the heated region expands on very close to the same timescale as the temperature change. This rapid thermal expansion causes a force impulse on the tip of the AFM than induces an oscillatory response of the AFM cantilever. In the case of a low repetition rate of the laser pulses, the cantilever s response to the force impulse is a transient decaying oscillation, i.e. a ring-down. As we will discuss later, it is also possible to pulse or modulate the laser synchronously with the cantilever oscillation to provide constant resonant excitation of the cantilever. In either case, the amplitude of the cantilever oscillation is directly proportional to the absorption of the sample. 1.1. Optical absorption Optical absorption in the infrared is linked to vibrational modes of molecules. Molecular bonds have discrete vibration modes at different frequencies. When these molecular bonds are illuminated by IR radiation at the same frequencies, a portion of the IR light excites the vibrational modes and when the molecules return to their ground vibrational state, a portion of the vibrational energy is dissipated as heat. It is the dissipated heat which causes a thermal expansion of the sample that is detected by AFM- IR. From an optical point of view, materials are defined by their complex optical refractive index : 2
=+ (eq.1) where λ is the wavelength, n(λ) is the real component of the refractive index and κ the imaginary component of the index. The principle of spectroscopy is to estimate the energy of the radiation transmitted through a sample as a function of the wavelength. In the infrared domain, it is most common to use the wavenumber (σ=1/λ) instead of the wavelength parameter. The Beer-Lambert law describes the intensity transmitted through a sample of thickness d as a function of the incident intensity: = (eq.2) where, I t is the transmitted intensity, I inc the incident intensity and σ the wavenumber (expressed in cm - 1 ). The transmittance coefficient T is defined by the ratio of the transmitted and incident intensity. The absorbance coefficient, the parameter often used to represent spectra, is then expressed by: = = (eq.3) This expression shows clearly that absorption spectra are in fact the variation of the extinction coefficient, κ, multiplied by the wavenumber σ. The power absorbed by the sample is obtained by the difference between the incoming and outgoing pointing vector: 1 3
= I (eq.4) where V is the volume of the sample, λ the wavelength, E loc the electric field inside the sample, and Im indicates the extraction of the imaginary part of the function. The amount of energy absorbed depends strongly on the local field inside the sample. The calculation or estimation of the electric field is not so easy analytically. But in the cases we are interested, the sample size is smaller than the wavelength, assuming that the electric field inside the sample is constant. Also we can express this field as a function of the incident electric field. In our study, we assume that the absorption is weak : n 2 >>κ 2. Using these assumptions, we can rewrite the expression of the P abs in a simpler way: = (eq.5) where α opt contains all the constants and optical parameters, depending only on the refractive index and speed of light. The dispersive effect of the sample is weak and considered negligible. We clearly see that in this equation, the power absorbed is proportional to the absorbance parameter (eq.3) as it is proportional to κ. 4
1.2. Photothermal expansion effects To describe the temperature change in the sample we can use the Fourier heat equation where the source of heating is directly equal to the energy absorbed: = + Δ (eq.6) where ρ is the density of the sample, C p the heat capacity, k th the thermal conductivity, t p the duration of the laser pulse, represents the Laplacian and Π is a rectangle function of length t p. The full equation resolution, taking into account the laser pulse duration and the shape of the sample, has been treated by Dazzi et al. 2 To summarize, the behavior of the temperature changes inside the sample when the laser pulse is shorter than the thermal diffusion time of the sample can be expressed by the following equations and is illustrated in Figure S1: Δ= 0 Δ= (eq.7) where = and = and k eff is the external thermal conductivity associated with the environment of the sample (air, prism), and a is the size of the sample (radius for a sphere, edge length for a cube). 5
Figure S1. Characteristic behavior of the sample temperature (red) illuminated by a laser pulse (blue) in the condition where the thermal relaxation time is longer than the laser illumination. The temperature increases linearly during the laser pulse time and then decays exponentially at a rate dependent on the thermal properties of the sample and the sample substrate. The key result here is that the maximum temperature increase is proportional to the power absorbed by the sample which in turn is proportional to the optical absorbance of the sample. The temperature increase induces an internal stress resulting in thermal expansion of the heated region. 3 In our case for a simple shape (sphere, cube, etc.), the expression of the expansion can be given simply by: = Δ (eq.8) where a is a characteristic size of the heated region, G is a constant depending on the geometry, α T is the sample thermal expansion coefficient, and Δ is the temperature increase of the absorbing region. As expected, the thermal expansion follows the time-dependent change in temperature. There is a simple linear chain of dependence so far: The power absorbed is proportional to the optical absorption coefficient multiplied by sigma ( ~ ); the temperature increase is proportional to the power absorbed (Δ~ ) and the thermal expansion is proportional to the temperature change (~Δ). So the end result is that the thermal expansion signal is also proportional to the optical absorption 6
coefficient multiplied by sigma (~). Thus if we can measure the thermal expansion as a function of wavenumber, we can create absorption spectra of the sample. 1.3. Cantilever response The local expansion of the sample is measured with the tip of an AFM cantilever. The motion of the cantilever is usually described by the Euler-Bernoulli equation 4-5 : + + =, (eq.9) where E is the cantilever Young modulus, I the area moment of inertia, ρ the density, S cross section of the cantilever, γ the damping and W the external excitation. The general solution of this equation is the sum over the eigenmodes of the cantilever, given by:,= h (eq.10) where P n is amplitude coefficient of the mode n, g n is the spatial distribution of the mode n and h(t) is the temporal behavior. The mode shape g n is given by the resolution of the boundary conditions (Figure S2) and depends strongly to the β wave vector deduced from the eigenvalue equation. This can be done by many different approaches (analytical or finite element) and has been extensively studied in the case of acoustic AFM. 6 One can note that this theoretical approach based on contact resonance is also valid for tapping mode, but in this particular case the mode shape g n has a different expression and the lateral stiffnesses k x and k z are null. 7
Figure S2. AFM cantilever scheme. The lever is embeded at x=0 and the length is L. The tip is positioned at x =L-δx. The contact stiffness is represented by two springs, one for the vertical displacement and one for the lateral. The source term W(x,t) describes the variation of force induced by the thermal expansion impulse on the tip. We consider that the tip is rigid and propagates the induced force instantly on the cantilever. The tip contact is modelled by the Hertz theory or other more complex contact mechanics models. In the limit of a small thermal expansion, a linear approximation can be employed using a contact stiffness parameter k z that depends on the tip geometry and the mechanical properties of the sample. In this limit, the tip-sample force can be approximated by Hooke s law: = (eq.11) This force is applied at the tip apex which is not located under the extremity of the lever and moreover the surface can be tilted, compared to the lever (Figure S3). Under these conditions, the force applied is the combination of the expansion (normal force) and the lever bending (lateral force). The final expression of the source term can be given by:,=+ (eq.12) where =cos+ sin, L the length of the cantilever and δx the position of the tip. 8
Figure S3. Scheme of the AFM tip in contact with the surface. The tilt angle between the cantilever and the surface is α. The tip height is H and the position of the tip is shifted by δx from the extremity of the cantilever. Substituting the expression of the source term (eq.12) into the equation (eq.9) and by using Fourier transformation and modal orthogonality properties, we can find the expression of the amplitude coefficients P n and the Fourier expression of h(t). 7-8. = h= (eq.13) where = and Γ= Most AFMs measure the end slope of the cantilever using optical lever detection, rather than directly measuring the cantilever deflection. The AFM detector signal Z (t) can be obtained from the cantilever z(x,t) using properties of the AFM optical lever detection system, as shown below: = h= (eq.14) 9
(This expression assumes diameter D of the focused laser diode spot is small compared to the cantilever length L.) The final equation shows that the signal obtained on the detector is the convolution of the expansion, u(t), with the transfer function of the cantilever (product of sine and exponential function). 1.4. Comparison with different laser excitations 1.4.1. Optical Parametric Oscillator pump by a Nd:YAG This type of laser produces a pulse of 5-20 ns with a repetition rate of 1 khz maximum. The duration of the pulse t p and the relaxation time τ relax associated with the thermal diffusion (for sub-micron size samples) are smaller than the response time of the cantilever (10-50 µs). The repetition rate of 1 khz corresponds to an excitation every 1 ms on the cantilever. Considering the damping on the surface, the cantilever usually goes back to its static state after 0.3-0.5 ms, meaning each pulse acts independently on the cantilever (no accumulation). Using these assumptions the Fourier Transform of the expansion for each pulse can be calculated: 2 = = + (eq.15) The time duration of the expansion is so short the cantilever feels it like a delta function. Replacing (eq.15) in (eq.13) the final solution can be written: = sin (eq.16) The signal detected is the composition of all cantilever modes weighted by a coefficient associated with the slope of the mode shape and the response time multiplied by the temperature maximum reached by the sample. This last parameter is proportional to the power absorbed (eq.7) and the power absorbed is 10
proportional to the absorbance (eq.5). This demonstrates that measuring the AFM deflection signal is a direct way to access and obtain the local sample absorbance. 9-11 1.4.2. Quantum cascade laser The quantum cascade laser (QCL) is quite attractive as an excitation source for AFM-IR, because of its ability to have high repetition rate (up to 1 MHz or higher), even though the wavenumber tunability range is relatively limited compared to the OPO technology. The major advantage of this technology is the ability to synchronize the repetition rate with one or more of the resonance frequencies of the cantilever. Under this condition, the cantilever will oscillate continuously at the frequency of the selected mode that the QCL repetition rate is matched to. This provides very efficient excitation of the cantilever and can improve the detection sensitivity. The expansion process can be described by a succession of short pulses, considering that the relaxation time, τ relax, is shorter than the period of repetition T 0 : = (eq.17) Replacing the Fourier Transform of u(t) (eq.17) in (eq.13), assuming that the mode is equal to the repetition pulse rate =2π/T 0 (assuming the sum over n vanishes) and only m=0,1 give a non-zero solution from the sum of a delta function, the final expression is obtained by doing the inverse Fourier transform : = sin (eq18) where Q n is the quality factor associated with the n mode (ω 0 =ω n, Q n >>1) and t p is the duration of one pulse. 11
In this case, we clearly see that the signal is no longer damped by an exponential function and that the amplitude is increased by a factor Q/2π compared to the single pulse excitation (eq.16). Experimentally, it is easy to gain one order of magnitude, using the QCL laser, making this approach sensitive enough to measure signals on nanometer-thick samples. 12-13 1.4.3. Sinusoidal modulation The basis of the AFM-IR technology is the detection of fast expansion inducing the motion of the cantilever. If we assume that the continuous laser period of modulation T 0 is slower than the thermal diffusion relaxation (τ relax ) allowing the temperature change following the laser modulation, we can imagine that the expansion of the sample can be expressed as: = = 1cos (eq.19) To obtain the expression of h(t), we calculate the Fourier Transform of (eq.19) replace it in (eq.13), simplify the expression considering that ω 0 =ω n and finally take the inverse Fourier transform. Substituting this expression into the final expression and assuming that Q n >>1, we can find : = sin (eq.20) This expression shows that using of a continuous modulation is really more efficient than a repeated pulse with the same increase of temperature. The gain is simply the ratio of the period T n /2 over (t p /2+τ relax ) and can be around 10 times for a contact resonance around 200 khz with sample size under 100 nm. Given these considerations, the AFM deflection signal will have the same shape as for the QCL laser (eq.18). Figure S4 illustrates the Q(t) signal for the different laser sources. In either case, all the expressions of Z (t) obtained with the different type of laser excitation are proportional to the absorbance. 12
1.5. General expression of resonance mode amplitude Usually the absorption evaluation is done two different ways, giving experimentally the same information. As seen in the previous paragraph, the absorption can be evaluated by the measurement of the oscillation amplitude but equivalently obtained by Fourier analysis. The Fourier transform of Z (t) at frequency gives directly the expression of the mode resonance amplitude of order n: =F = (eq.21) with = + ; = + ; = Independent of the laser excitation method, the amplitude of the contact resonance A n is proportional to the absorbance (via T max ). The only difference is the relative intensity lead by the factor c k that translates the efficiency to drive the cantilever with the laser excitation. In conclusion of this section, we have demonstrated that the signal detected by the four-quadrant detector of the AFM allows us to directly have an estimation of the local absorbance. All expressions, whatever the type of laser source, are proportional to the maximum temperature of heating (T max, eq.7) which is proportional to power absorbed (P abs, eq.5) which in turn is proportional to the absorbance (A, eq.3). All physical phenomenon involve in this detection technique are linear and lead to a perfect measurement of the spectra without any optical artifacts (dispersion or scattering). The driving of the cantilever motion by tuning the repetition rate to the contact resonance is a really efficient way to dramatically increase the sensitivity of the technique. This has been confirmed by the experimental results obtained by Belkin et al. on organic molecules monolayers. 12-13 Looking at the sinusoidal modulation expression (eq.20) it seems even promising that further improvements in sensitivity of up to one or two orders of magnitude may be possible. 13
Figure S4. AFM deflection signal (top) of the cantilever oscillations as a function of the laser excitation type (bottom). Short, low repetition rate pulses (bottom, in red, e.g. an OPO laser) induces a decaying ringdown of the cantilever. High repetition rate pulse (bottom in blue, e.g a QCL with pulses synchronized to a contact resonance of the cantilever result in continuous wave oscillations (top, in blue). Sinusoidal excitation (bottom in green) provoke similar continuous wave oscillatary behavior as the QCL. References (1) Born, M. A. X.; Wolf, E.; Born, M. A. X.; Wolf, E., Principals of Optics, Chapter 1 - Basic Properties of the Electromagnetic Field. Cambridge University Press: Cambridge, 1980. (2) Dazzi, A., Theory of Infrared Nanospectroscopy by Photothermal Induced Resonance. J. Appl. Phys. 2010, 107 (12), 124519. (3) Nowacki, W., Thermoelasticity. Pergamon, London, 1962, pp20-21. (4) Boussinesq, J., Notes Complémentaires. Gauthier Villars Paris 1885, pp 435-435. (5) Stockey, W. F., Shock and Vibration Handbook. 2nd Edition ed.; McGraw-Hill: New York, 1976. (6) Rabe, U.; Arnold, W., Acoustic Microscopy by Atomic Force Microscopy. Appl. Phys. Lett. 1994, 64 (12), 1493-1493. (7) Yuya, P. A.; Hurley, D. C.; Turner, J. A., Contact-Resonance Atomic Force Microscopy for Viscoelasticity. Journal of Applied Physics 2008, 104 (7), 074916-7. (8) Rabe, U.; Janser, K.; Arnold, W., Vibrations of Free and Surface-Coupled Atomic Force Microscope Cantilevers: Theory and Experiment. Rev. Sci. Instrum. 1996, 67 (9), 3281-3281. (9) Dazzi, A.; Glotin, F.; Carminati, R., Theory of Infrared Nanospectroscopy by Photothermal Induced Resonance. J. Appl. Phys. 2010, 107, 124519. (10) Dazzi, A.; Prater, C. B.; Hu, Q.; Chase, D. B.; Rabolt, J. F.; Marcott, C., Afm-Ir: Combining Atomic Force Microscopy and Infrared Spectroscopy for Nanoscale Chemical Characterization. Appl. Spectrosc. 2012, 66 (12), 1365-1384. (11) Lahiri, B.; Holland, G.; Centrone, A., Chemical Imaging Beyond the Diffraction Limit: Experimental Validation of the Ptir Technique. Small 2013, 9, 439-445. (12) Lu, F.; Belkin, M. A., Infrared Absorption Nano-Spectroscopy Using Sample Photoexpansion Induced by Tunable Quantum Cascade Lasers. Opt. Express 2011, 19, 19942-19947. (13) Lu, F.; Jin, M.; Belkin, M. A., Tip-Enhanced Infrared Nanospectroscopy Via Molecular Expansion Force Detection. Nat. Photonics 2014, 8 (4), 307-312. 14
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