SUPPLEMENTARY INFORMATION - Nature Research€¦ · estimate that the absorption coefficient of the monolayer at CH. 2-wagging mode absorption peak is α. abs = 6×10. 3. cm-1. which

NATURE PHOTONICS | www.nature.com/naturephotonics 1

SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHOTON.2013.373Manuscript # NPHOT-2013-06-00690B

1

Supplementary Information for the manuscript entitled “Tip-enhanced infrared nanospectroscopy

via molecular expansion force detection” by Feng Lu, Mingzhou Jin and Mikhail A. Belkin

I. COMSOL simulations of sample heating

a. General parameters

COMSOL 4.3a was used to simulate local electrical field intensity enhancement and sample

heating. The simulated sample consisted of a 2-nm-thick molecular monolayer on top of a 40-nm-thick

layer of gold on top of a 2-µm-thick layer of epoxy. The gold AFM tip was simulated to be conical with

half cone angle of 17o and 25 nm radius of curvature of the tip apex, in agreement with the scanning

electron microscopy image of an actual tip used in experiments. The tip length was taken to be 10 µm,

which is close to the actual length of the illuminated part of the tip in our experiments. A λ=8 µm p-

polarized plane wave was incident upon the sample at an angle of 75o to the surface normal. The electric

field amplitude in the wave was chosen to correspond to a 100-µm-radius beam with 500 mW power in

free space. For the monolayer, thermal conductivity κth, material density ρ, heat capacity C, and thermal

expansion coefficient αth were assumed to be the same as those of a bulk polymer material in Ref. [16].

Specifically, we used κth ≈ 0.1 Wm−1K-1 [S1], ρ ≈ 1.2×103 kg·m−3, C ≈ 1.2×103 J·kg−1, and αth = 10-4 K-1

[S2]. The values of κth, ρ, C, and αth were assumed to be the same for the epoxy layer for simplicity.

b. Real and imaginary parts of the refractive index of monolayer

The real part of the refractive index of a monolayer is taken as n=1.5 [S3]. The imaginary part of

the refractive index κ varies significantly with wavelength. In order to have a typical value of κ for

simulations, we focus on the CH2-wagging mode which is present for both EG6-OH and PEG spectra in

the paper. By comparing the strength of different vibrational modes of EG6-OH in Ref. [S4] and [20], we

estimate that the absorption coefficient of the monolayer at CH2-wagging mode absorption peak is

αabs = 6×103 cm-1 which corresponds to κ = 0.38. The refractive index for gold was taken from Ref. [S5]

as 8.5+i×46.4.

c. Simulation of sample heating

Simulated local field intensity enhancement under the tip is shown in Fig. 1(b). Simulated

temperature distribution in the sample at the end of the laser pulse is shown in Fig. 4(a) and Fig. S1(a).

Figure S1(b) plots the temperature increase in the monolayer at different times during and after the laser

pulse. The results indicate, in particular, that sample heating and cooling time is much smaller than the

laser pulse duration and that the sample maintains the same temperature during most the laser pulse.

© 2014 Macmillan Publishers Limited. All rights reserved.

2 NATURE PHOTONICS | www.nature.com/naturephotonics

SUPPLEMENTARY INFORMATION DOI: 10.1038/NPHOTON.2013.373

Manuscript # NPHOT-2013-06-00690B

2

Figure S1. Simulation of temperature increase in the monolayer during and after a single 160 ns

pulse which arrives at t=0 ns. (a) Distribution of temperature increase at the end of the pulse (at t=160

ns). (b) Temperature variation along the white dash line in (a) during (13 ns, 160 ns) and after (180 ns, 5

µs) the pulse. The sample is cooled to the room temperature before the next pulse arrives (assuming the

repetition frequency of laser pulses is 200 kHz).

II. Cantilever mechanics

a. Cantilever force constant

The rectangular cantilever used in the experiments has a static (first bending mode) force constant

of kc = 0.2 N/m as specified by the manufacturer. The force constant of the second bending mode k2 can be

linked to kc using an equivalent point-mass model as described in Ref. [S6]. We obtain 2 / 40.5ck k = and

k2 = 8.1 Nm-1. This number will later be used to calculate the tip deflection.

b. Expansion force

The Derjaguin-Muller-Toporov (DMT) model of sample-tip interaction [25,26] gives the force F

on a tip as:

* 1/2 3/24 23

F E R Rwδ π= − , (S1)

where E* is the reduced Young’s modulus, R is the tip apex radius, δ is the indentation depth, and 2πRw is

the pull-off force due to sample adhesion. The values of E* and 2πRw for our samples are assumed to be

the same as those measured in Ref. [28] for CH3(CH2)17SH monolayer sample: E*= 5 GPa and 2πRw=10

nN. Note that tip radius used in our experiments (~25 nm) is similar to that used in Ref. [28] (~20 nm).

Equation (S1) then gives the sample indentation of δ ≈ 0.7 nm for F= 10 nN the contact mode force used

in our experiments.




3

Assuming the tip-sample distance change Δδ is much smaller than δ, the photoexpansion-induced

mechanical force Fabs on the tip can be derived from Eq. (S1) to be

δδ Δ≈ 2/12/1*2 REFabs . (S2)

The value of Δδ is calculated to be ≈ 3.2 pm for our experiments when the laser wavelength

coincides with the CH2-wagging mode absorption peak (see Fig. 4(c,d)). We then calculate from Eq. (S2)

that Fabs = 0.13 nN.

III. Cantilever oscillation amplitude

a. Experimental value

Experimentally, we excite cantilever oscillations in the second bending mode in contact with the

sample. The cantilever oscillation amplitude z2 may be determined from the amplitude of the PSPD

detector output VPSPD as:

22 η

PSPDVz = , (S3)

where η2 is the calibration coefficient that links VPSPD with the physical cantilever deflection amplitude z2.

The value of VPSPD can be determined by direct measurement of the PSPD output with an oscilloscope; it

is also linked to the lock-in amplifier voltage output as 2PSPDinlock VgV =− where g is the lock-in gain

coefficient.

To determine η2, we compare the slope of the cantilever end section for the first free bending

mode with the slope of the cantilever end section for the second bending mode in contact with the sample,

see Fig. S2(a). Cantilever bending is described by Euler–Bernoulli equation and we quote its spatial

solution from Ref. [S7]:

0 cos cosh( ) ((cos cosh ) (sin sinh ))sin sinh

n nn n n n n n

n n

L Lz x z x x x xL L

β ββ β β β

β β+

= − − −+

, (S4)

where zn(x) refers to the shift from the rest position at distance x from the clamped end, 𝛽𝛽n is the

wavenumber of the n-th bending mode, and L=450 µμm is the cantilever length in our experiments.

For the first free bending mode of the cantilever, the cantilever shape is given by Eq. (S4) with

β1L = 1.875 [S7]. For the second bending mode in contact with the sample we need to take into account

the force constant of the tip-sample interaction k*. The physical picture is presented in Fig. S2(b). The

force constant of the sample-tip interaction is found to be k*≈193 N/m from the shift in the cantilever



SUPPLEMENTARY INFORMATION DOI: 10.1038/NPHOTON.2013.373Manuscript # NPHOT-2013-06-00690B

4

Figure S2. Cantilever shape in bending modes. (a) Cantilever deflection in force-distance

measurements (black line) and in the second bending mode in contact with the sample. (b) The tip-

sample interaction can be approximated with a spring k* connecting them.

resonance frequency in contact with the sample (90 kHz shifted to 200 kHz). The value of β2L is then

obtained to be 7.006 as described in Ref. [S7]. Since the PSPD signal is determined only by the slope of

the cantilever end (namely “optical sensitivity” [S8]), we obtain:

22

11

zdxdzKV

zdxdzKV

LxPSPD

LxPSPD

×=

×=

=

= , (S5)

where K is the proportionality constant between PSPD voltage and the slope of the end of a cantilever.

The derivatives can be calculated from Eq. (S4). Comparing the derivatives in Eq. (S5), we then obtain

that z1 cantilever deflection in the first bending mode produces the same PSPD signal as the second

bending mode with z2≈z1/35 deflection.

Force-distance measurements on a hard surface (e.g. silicon) gave us VPSPD=23 mV·nm-1 × z1,

where z1 is the cantilever deflection in [nm] for the first bending mode. The maximum amplitude of the

PSDP signal at the second bending mode frequency in our experiments was approximately 200 mV. We

then obtain the maximum amplitude of cantilever oscillation in the second bending mode in our

experiments as:

zmax=200 mV / 23 mV·nm-1/35 ≈ 0.25 nm




5

b. Theoretical calculations of expected cantilever deflection amplitude

For resonant cantilever excitation, we can simplify the theoretical model with a damped harmonic

oscillator rather than solving the Euler–Bernoulli equation: 2

20 02

( )2d z dz f tzdt dt m

ζω ω+ + = , (S6)

where z(t) is the deflection amplitude, ζ = (2Q)-1 is the damping coefficient with Q being the quality-

factor of the mode, ω0 is the resonant angular frequency, m = k/ω02 with k being the force constant of

the cantilever in contact with the sample, and f(t) is the applied external force. Since the sample heating

and cooling happen over time scale much smaller than the cantilever response time in our experiments

(see Fig. S1(b)), the force on the cantilever may be represented as a train of delta functions:

00

( ) ( )n

f t I t nTδ∞

=

= −∑ , (S7)

where ∫= dttFI abs )(0 is impulse from the absorption-induced mechanical force on the tip and T =

2π/ω0 is laser pulse repetition period. By applying Laplace transform L{...} to both sides of Eq. (S6) and

assuming z(0)=0, we obtain

202 2

0 0

1( ) (1 ...)2

sT sTIz s e em s sςω ω

− −= ⋅ ⋅ + + ++ +

, (S8)

where z(s)=L {z(t)}. To obtain time dependent cantilever deflection z(t) we perform inverse Laplace

transform:

1 202 2

0 0

1( ) (1 ...)2

sT sTIz t e em s sςω ω

− − −⎧ ⎫= ⋅ ⋅ + + +⎨ ⎬

+ +⎩ ⎭L , (S9)

Using Convolution theorem we obtain

1 1 202 2

0 00

1( ) {(1 ...)} '2

tsT sTIz t L L e e dt

m s sςω ω− − − −⎧ ⎫

= ⋅ + + +⎨ ⎬+ +⎩ ⎭

∫ , (S10)

Since

0 01 20 02 2 2

0 0 00

1 1 1{ } sin( 1 ) sin( )2 1

t te t e ts s

ςω ςωω ς ωςω ω ωω ς

− −− = ⋅ − ≈+ + −

L ,

and

1 2

0{(1 ...)} ( )sT sT

ne e t nTδ

∞− − −

=

+ + + = −∑L ,

we obtain from Eq. (S10)



SUPPLEMENTARY INFORMATION DOI: 10.1038/NPHOTON.2013.373


6

0 ( ')00

00 0

( ) ( ' ) sin( ( - ')) 't

t t

n

Iz t t nT e t t dtm

ςωδ ωω

∞− −

=

= ⋅ −∑∫ , (S11)

Noticing that we have ω0nT=2nπ, we obtain for sufficiently large t≥T/ ζ

( ) ( ) ( )0 ( )0 0 00 0 02

00 0

21 1( ) sin sin sint nT

n

I I QIz t t e t tm T m T k

ζωω ω ωω ω ζ

∞− −

=

= = =∑ , (S12)

Equation (S12) allows us to calculate the amplitude of the cantilever deflection if we know

resonant frequency ω0, Q-factor, force constant k and the I0. The latter can be estimated as I0=Fabs𝜏𝜏, where

𝜏𝜏 is the laser pulse duration. Using Fabs = 0.13 nN, τ =160 ns, T=5 µs, experimentally-measured Q=93,

and k = 8 Nm-1 as derived in Section II, we obtain oscillation amplitude z≈0.1 nm, which is close to the

experimentally-measured z≈0.25 nm. The discrepancy between theory and experiment is likely stemming

from uncertainty in the temperature change and photoexpansion of the sample as well as in the Young’s

modulus of the sample.

IV. Spectra normalization

The measured spectra shown in the main text (Figs. 2 & 3(c)) were normalized by the spectrum taken on a

clean TSG substrate, which originates from the expansion of the substrate and an AFM tip due to residual

broadband absorption of mid-IR light by gold. Fig. S3(a) shows a good agreement between the cantilever

deflection spectrum and the laser power spectrum measured by a properly-aligned mercury cadmium

telluride (MCT) detector.

Figure S3. Comparison of the background TSG spectrum (black solid line) with the QCL

power spectra taken by a MCT detector (red dashed lines). Slight differences between the two

curves may be explained by differences in the MCT detector alignment compared to the tip

position and slight beam steering during QCL tuning.



SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHOTON.2013.373


7

V. SAM sample characterization

Thickness of uniform EG6-OH and NTP monolayers on gold was measured by removing

monolayer from part of the sample. To do that, the sample was partly covered by a piece of silicon and

exposed to O2 plasma. AFM topographic scan was then performed across the interface of SAM and

exposed gold, see Fig. S4(a). The thickness of EG6-OH monolayer was measured to be about 1.5 nm, as

shown in Fig. S4(b). We were not able to measure the thickness of NTP monolayer in this way as its

thickness was below the topographic detection level of our system (0.5 nm).

Figure S4. Topographic measurement of monolayer samples. (a) EG6-OH after partial O2

plasma etching. The bright regions are EG6-OH monolayer while the dark regions are exposed

gold. (b) Line-scan averaged within the red box in (a). (c) PEG monolayer islands self-assembled

on gold after a short immersion time. Inset: line-scan along the blue line.

Typical topography of PEG monolayer islands obtained in AFM tapping mode is shown in Fig.

S4(c). The island height is about 2.5 nm.



SUPPLEMENTARY INFORMATION DOI: 10.1038/NPHOTON.2013.373Manuscript # NPHOT-2013-06-00690B

8

Supplemental References

S1. Wang, R. Y., Segalman, R. A. & Majumdar, A. Room temperature thermal conductance of

alkanedithiol self-assembled monolayers. Appl. Phys. Lett. 89, 173113 (2006).

S2. Mark, J. E. ed., Physical Properties of Polymers Handbook, 2nd ed. (Springer, New York, 2007).

S3. Hu, Z. G., Prunici, P., Patzner, P. & Hess, P. Infrared spectroscopic ellipsometry of n-alkylthiol (C5-

C18) self-assembled monolayers on gold. J. Phys. Chem. C 110, 14824–14831 (2006).

S4. Marshall, G. M., Bensebaa, F. & Dubowski, J. J. Observation of surface enhanced IR absorption

coefficient in alkanethiol based self-assembled monolayers on GaAs(001). J. Appl. Phys. 105, 094310

(2009).

S5. Palik, E. D. Handbook of Optical Constants of Solids, (Academic Press, Boston, 1985).

S6. Melcher, J., Hu, S. & Raman, A. Equivalent point-mass models of continuous atomic force

microscope probes. Appl. Phys. Lett. 91, 053101 (2007).

S7. Rabe, U., Janser, K. & Arnold, W. Vibrations of free and surface-coupled atomic force microscope

cantilevers: Theory and experiment. Rev. Sci. Instrum. 67, 3281–3293 (1996).

S8. Garcia, R. & Herruzo, E. T. The emergence of multifrequency force microscopy. Nature Nanotech. 7,

217–26 (2012).


Documents

SUPPLEMENTARY INFORMATION - Nature Research€¦ · estimate that the absorption coefficient of the monolayer at CH. 2-wagging mode absorption peak is α. abs = 6×10. 3. cm-1. which