Scanning Near-Field Optical Microscope (SNOM) and TERS

1. Theory

• Definition: Scanning near-field optical microscopy (SNOM) uses a sharp tip to confine light to a nanoscale volume, giving optical contrast below the diffraction limit; tip-enhanced Raman spectroscopy (TERS) adds Raman spectra from the enhanced field under a metal tip (Maissen et al.; Neacsu et al.).
• Scattering-type SNOM: The tip acts as a dipole antenna whose scattered field depends on tip-sample distance; demodulating at higher tapping harmonics isolates the near field (Maissen et al.).
• Field enhancement: A metal tip concentrates the incident field by a factor L, and Raman intensity scales roughly as the fourth power of the field (Neacsu et al.).
• Resolution: Set mainly by tip radius, not wavelength, from visible to THz (Maissen et al.; Neacsu et al.).

2. Working Principle

• Probe: Gold, silver or tungsten tips, or metal-coated AFM tips, with apex radii down to about 10 $\mathrm{nm}$ (Neacsu et al.; Maissen et al.).
• s-SNOM detection: A tapping tip is illuminated by a focused laser; the scattered light is demodulated at the third or fourth harmonic and detected interferometrically for amplitude and phase (Maissen et al.).
• TERS detection: A laser such as 632.8 $\mathrm{nm}$ illuminates the tip-sample gap, and Raman spectra are acquired point by point (Neacsu et al.).

3. Applications

• Carrier and material mapping: Infrared and THz s-SNOM contrast from free carriers and phonons, including dopants and defects in single semiconductor nanoparticles (Maissen et al.; Abate).
• Nanoscale Raman: Phonon modes and symmetry selection in crystals such as BaTiO3 nanorods (Neacsu et al.).
• Molecular sensitivity: Few-molecule to single-molecule Raman detection (Neacsu et al.).
• Probe engineering: Tip shape and length optimized for signal at long wavelengths (Maissen et al.).

Near-Field Scattering

Name	Formula	Related Formulae	Variables/Constants
Tip-scattered field and demodulation	$E_{scat}\propto \left|p\right|,s_n=\int E_{scat}\left(h\left(t\right)\right)e^{in\Omega t}dt$	$h\left(t\right)=A\left(1+\cos \Omega t\right)$	$E_{scat}:scatteredfield$ $p:nettipdipole$ $s_n:n-thharmonicsignal$ $\Omega :tappingfrequency$ $A:tappingamplitude$ $\begin{array}{r}h:tip-sampledistance;Maissenetal.:\\ theseequations,n=3bydefaultand4\\ forsharptips,Aabout50\mathrm{nm},higher\\ harmonicssuppressingbackground,\\ interferometricamplitude-and\\ phase-resolveddetection\end{array}$ $\begin{array}{r}(assumed:\Omega /2\pi =250\mathrm{kHz};computed:n=3\\ demodulatesat750\mathrm{kHz})\end{array}$
Near-field decay	$\left|s_n\left(h_{1/2}\right)\right|=\frac{1}{2}\left|s_n\left(0\right)\right|$	$h_{1/2}\sim r_{tip}(\text{order of magnitude})$	$\begin{array}{r}{h}_{1/2}:decaydistancetohalfamplitude;\\ Maissenetal.:6\mathrm{nm}onNiFeforalltips,\\ 11to16\mathrm{nm}onalumina;Abate:resolution\\ dependingonproberadiusofcurvature;\\ decayscalingwithapexradius(my\\ reading)\end{array}$ $\begin{array}{r}(computed:a50\mathrm{nm}tappingamplitudeis\\ abouteighttimesthe6\mathrm{nm}decaylength)\end{array}$
Resolution independent of wavelength	$\Delta x\approx f\left(r_{tip},A\right)\ll \frac{\lambda }{2NA}$	$\frac{\lambda }{\Delta x}\text{(resolution gain)}$	$\Delta x:lateralresolution$ $\begin{array}{r}NA:numericalaperture;Maissenetal.:\\ resolutionessentiallysetbytipradius\\ independentofwavelength,41\mathrm{nm}(R=25\\ \mathrm{nm})to170\mathrm{nm}(R=750\mathrm{nm})at2.52\mathrm{THz}\\ (119\mu \mathrm{m}),13\mathrm{nm}witha6\mathrm{nm}tungstentipat\\ A=5\mathrm{nm};Neacsuetal.:solelydetermined\\ byapexradius,14\mathrm{nm}onnanotubesand30\\ \mathrm{nm}onBaTiO3withtipsdownto10\mathrm{nm}(see\\ theQ/A)\end{array}$ $\begin{array}{r}(computed:119\mu \mathrm{m}over13\mathrm{nm}isabout9200;\\ at632.8\mathrm{nm}andNA0.9thediffraction\\ limitis352\mathrm{nm},25timesthe14\mathrm{nm}TERS\\ value)\end{array}$

Tip Enhancement and Raman

Name	Formula	Related Formulae	Variables/Constants
Local field enhancement	$E_{loc}\left({\nu }_i\right)=L\left({\nu }_i\right)E\left({\nu }_i\right)$	$G={\left|L\left({\nu }_i\right)L^{\prime }\left({\nu }_s\right)\right|}^2\approx {\left|L\right|}^4$	$\begin{array}{r}L,L^{\prime }:fieldenhancementatincidentand\\ scatteredfrequencies\end{array}$ $\begin{array}{r}G:Ramanenhancement;Neacsuetal.:these\\ relations,I_{T}ERSproportionalto{E}^4,E/E_0\\ about50forgoldand12fortungstenat\\ 630\mathrm{nm},severalhundredat2\mathrm{nm}gap,\\ enhancements1\times 10^{6}to1\times 10^{9}andupto5\times 10^9\\ formalachitegreen,about1\times 10^{4}for\\ nanotubemodes,imagedipoleinthesample\end{array}$ $\begin{array}{r}(computed:L=50gives6.3\times 10^6,L=12\\ gives2.1\times 10^4,L=300gives8.1\times 10^9)\end{array}$
Raman intensity and selection rules	$I\propto {\left|L\left({\nu }_i\right)L^{\prime }\left({\nu }_s\right)E\left({\nu }_i\right)\right|}^2$	$I_s\propto {\left|{\vec{e}}_s\cdot \overleftrightarrow{R}\cdot {\vec{e}}_i\right|}^2$	${\vec{e}}_i,{\vec{e}}_s:incidentandscatteredpolarizations$ $\begin{array}{r}\overleftrightarrow{R}:Ramantensor;Neacsuetal.:these\\ relations,dominantz-polarizedtipfield\\ changingwhichmodesappear,symmetry\\ selectionforcrystallineBaTiO3\end{array}$ $\begin{array}{r}(myreading:amodeforbiddenin\\ backscatteringcanappearunderthetip\\ becausethelocalfieldisalongz)\end{array}$
Near-field to far-field contrast	$C\approx G\frac{A_{nf}}{A_{ff}}$	$A_{ff}\approx \pi {\left(\frac{0.61\lambda }{NA}\right)}^2$	$C:near-fieldtofar-fieldsignalratio$ $\begin{array}{r}{A}_{nf},A_{ff}:near-fieldandfocal-spotareas;\\ area-ratioestimate(myreading);Neacsu\\ etal.:side-oncollectionatNA0.35with\\ about2percentefficiencyagainst27per\\ centatNA0.9,632.8\mathrm{nm}HeNe,5\times 10^{3}to\\ 3\times 10^{4}W/c{\mathrm{m}}^2\end{array}$ $\begin{array}{r}(computed:atNA0.35thespotradiusis\\ 1.1\mu \mathrm{m};a14\mathrm{nm}discgivesanarearatioof\\ 4.0\times 10^{-5},soG=6.3\times 10^{6}givesCofabout\\ 250)\end{array}$

Semiconductor Contrast and Probe Design

Name	Formula	Related Formulae	Variables/Constants
Carrier plasma frequency	${\omega }_p^2=\frac{n{e}^2}{{\epsilon }_0{\epsilon }_{\infty }{m}^{\ast }}$	$n\propto {\omega }_p^2$	${\omega }_p:plasmafrequency$ $\begin{array}{r}n:free-carrierdensity,{\epsilon }_{\infty }:\\ high-frequencypermittivity,m^{\ast }:\\ effectivemass;Wangetal.:Drude\\ relationwith{\epsilon }_{s}for{\epsilon }_{\infty },permittivity\\ changingsignnear{\omega }_p;Abate:near-field\\ imagesgivingfree-carrierconcentration;\\ Maissenetal.:2.52\mathrm{THz}nanoscopy\end{array}$ $\begin{array}{r}(Wangetal.:Si,{\epsilon }_s=11.7,m^{\ast }=0.27m_0;\\ computed:n=1\times 10^{19}\mathrm{c}{\mathrm{m}}^{-3}gives533\mathrm{c}{\mathrm{m}}^{-1},and\\ 2.52\mathrm{THz}(84\mathrm{c}{\mathrm{m}}^{-1})matchesnofabout\\ 2.5\times 10^{17}\mathrm{c}{\mathrm{m}}^{-3})\end{array}$
Tip antenna length	$L_{tip}\approx \frac{\lambda }{2}(\text{antenna resonance})$	$\frac{s_n(L=60\mu \mathrm{m})}{s_n(L=20\mu \mathrm{m})}\approx 4$	$\begin{array}{r}{L}_{tip}:tipshaftlength;Maissenetal.:\\ lengtheningtoabout60\mu \mathrm{m}raisedsignal\\ almostfourtimes,noclearresonanceat\\ halfwavelengthattributedtoalignment\\ uncertainty,largerradiusraisingsignal\\ overanorderofmagnitude,enhancement50\\ onNiFeand7onAl2O3\end{array}$ $(computed:\lambda /2=59.5\mu \mathrm{m}at2.52\mathrm{THz})$
Illumination and detection budget	$P=I_0\pi w^2$	$\frac{{\eta }_{ff}}{{\eta }_{TERS}}=\frac{27}{2}$	$P:poweronthefocus$ $I_0:intensity$ $w:spotradius$ $\begin{array}{r}\eta :collectionefficiency;Neacsuetal.:\\ 5\times 10^{3}to3\times 10^{4}W/c{\mathrm{m}}^2,1sspectrafor\\ malachitegreen,10msperpixelon\\ BaTiO3,signal-to-noiseabove40forabout\\ 100molecules,27and2percent\\ efficiencies\end{array}$ $\begin{array}{r}(assumed:w=1.1\mu \mathrm{m};computed:3\times 10^{4}W/c{\mathrm{m}}^2\\ is1.1\mathrm{mW};efficiencyratio13.5)\end{array}$

Important Questions

Ques: How does s-SNOM separate the near field from background light?

By tapping the tip and demodulating the detector signal at higher harmonics. Maissen et al. model the tip as a dipole whose scattered field $E_{scat}\propto |p|$ varies with height $h(t)=A(1+\cos \Omega t)$, and extract $s_n=\int E_{scat}{e}^{in\Omega t}dt$. Because the near-field interaction decays within nanometres, 6 $\mathrm{nm}$ to half amplitude on NiFe, it is strongly nonlinear in h, while background scattering from the shaft and sample varies slowly; harmonics with n above 2 therefore suppress background, and they use n = 3 by default and 4 for sharp tips. A 50 $\mathrm{nm}$ amplitude is about eight times the decay length (computed). Interferometric detection gives amplitude and phase and suppresses multiplicative background (Maissen et al.). Reference: C. Maissen, S. Chen, E. Nikulina, A. Govyadinov and R. Hillenbrand, ‘Probes for ultra-sensitive THz nanoscopy’, arXiv:2005.05722 (fetch-tool summary).

Ques: Is s-SNOM and TERS resolution set solely by the tip radius?

Mostly, but not linearly. Neacsu et al. state that lateral resolution is solely determined by the apex radius, reporting 14 $\mathrm{nm}$ on nanotubes with tips down to 10 $\mathrm{nm}$, and Maissen et al. say essentially by tip radius independent of wavelength. Yet Maissen’s data show 41 $\mathrm{nm}$ for a 25 $\mathrm{nm}$ tip and 170 $\mathrm{nm}$ for 750 $\mathrm{nm}$, a 30-fold radius increase giving only about four times poorer resolution (computed), and 13 $\mathrm{nm}$ required reducing the tapping amplitude to 5 $\mathrm{nm}$ with a 6 $\mathrm{nm}$ tip. The tip radius is therefore the main factor, while tapping amplitude and demodulation order also matter, so the strict word solely is less likely to hold (my reading). Either way, 119 $\mu \mathrm{m}$ light resolving 13 $\mathrm{nm}$ is a gain of about 9200 (computed). Reference: C. C. Neacsu, S. Berweger and M. B. Raschke, ‘Tip-enhanced Raman imaging and spectroscopy: sensitivity, symmetry and selection rules’, Nanobiotechnology (2007), arXiv:0803.4464 (fetch-tool summary); C. Maissen, S. Chen, E. Nikulina, A. Govyadinov and R. Hillenbrand, ‘Probes for ultra-sensitive THz nanoscopy’, arXiv:2005.05722 (fetch-tool summary).

Ques: Why is TERS enhancement quoted anywhere from $1\times 10^4$ to $1\times 10^9$?

Because it scales as the fourth power of local field, which varies strongly with tip, gap and sample. Neacsu et al. write $G=|L({\nu }_i)L^{\prime }({\nu }_s){|}^2$, so with $L\approx 50$ for gold at 630 $\mathrm{nm}$, G is $6.3\times 10^6$, and with 12 for tungsten only $2.1\times 10^4$ (computed). At a 2 $\mathrm{nm}$ gap the field enhancement reaches several hundred, and L = 300 gives $8.1\times 10^9$ (computed), consistent with their $1\times 10^9$ to $5\times 10^9$ for malachite green. Nanotube modes showed only about $1\times 10^4$, partly because the extended tube is not confined to the hot spot and partly because of selection rules (Neacsu et al.; my reading). Enhancements of $1\times 10^6$ to $1\times 10^9$ across tips show the strong tip-to-tip variation they report. Reference: C. C. Neacsu, S. Berweger and M. B. Raschke, ‘Tip-enhanced Raman imaging and spectroscopy: sensitivity, symmetry and selection rules’, Nanobiotechnology (2007), arXiv:0803.4464 (fetch-tool summary).

Ques: How much does the near field actually outshine the far field in TERS?

Less than the enhancement factor suggests, because the near-field area is tiny. A useful estimate is $C\approx G{A}_{nf}/A_{ff}$, the enhancement scaled by the ratio of near-field to focal-spot area (my reading). With side-on collection at NA 0.35 and 632.8 $\mathrm{nm}$, the focal spot radius is about 1.1 $\mu \mathrm{m}$; a 14 $\mathrm{nm}$ near-field disc covers only $4.0\times 10^{-5}$ of it, so G = $6.3\times 10^6$ gives a contrast of about 250 (computed). Neacsu et al. also note collection efficiency of about 2 per cent at NA 0.35 against 27 per cent at NA 0.9, 13.5 times lower (computed). Their intensities of $5\times 10^3$ to $3\times 10^4$ $W/c{\mathrm{m}}^2$ correspond to about 1.1 $\mathrm{mW}$ on that spot (computed), so exposure must balance signal against bleaching, which they observed at the higher intensity. Reference: C. C. Neacsu, S. Berweger and M. B. Raschke, ‘Tip-enhanced Raman imaging and spectroscopy: sensitivity, symmetry and selection rules’, Nanobiotechnology (2007), arXiv:0803.4464 (fetch-tool summary).

Ques: Why do selection rules change under a TERS tip?

Because the tip field is mainly polarized along its axis, normal to the sample. Neacsu et al. give the scattering intensity as $I_s\propto |{\vec{e}}_s\cdot \overleftrightarrow{R}\cdot {\vec{e}}_i{|}^2$, so the polarization directions select which Raman tensor elements contribute. In normal-incidence far-field backscattering the fields lie in the sample plane, whereas the enhanced tip field is along z, so modes weak or forbidden in the far field can dominate the TERS spectrum, as they show for crystalline BaTiO3 nanorods with 30 $\mathrm{nm}$ resolution (Neacsu et al.; my reading). They also describe an image dipole in the sample that strengthens the gap field. Rotating the incident polarization therefore changes which phonon modes appear, giving a way to confirm that a band comes from the near field rather than the focal spot (my reading). Reference: C. C. Neacsu, S. Berweger and M. B. Raschke, ‘Tip-enhanced Raman imaging and spectroscopy: sensitivity, symmetry and selection rules’, Nanobiotechnology (2007), arXiv:0803.4464 (fetch-tool summary).

Ques: How can s-SNOM map free carriers in semiconductors?

Through the Drude response near the plasma frequency. Free carriers give a permittivity that changes sign near ${\omega }_p=\sqrt{n{e}^2/({\epsilon }_0{\epsilon }_{\infty }{m}^{\ast })}$, so near-field amplitude and phase change strongly with density (Wang et al.; Abate). For silicon with ${\epsilon }_s$ = 11.7 and $m^{\ast }$ = 0.27 $m_0$ (Wang et al.), $1\times 10^{19}$ $\mathrm{c}{\mathrm{m}}^{-3}$ gives 533 $\mathrm{c}{\mathrm{m}}^{-1}$ in the mid-infrared, while the 2.52 $\mathrm{THz}$ (84 $\mathrm{c}{\mathrm{m}}^{-1}$) source of Maissen et al. matches about $2.5\times 10^{17}$ $\mathrm{c}{\mathrm{m}}^{-3}$ (computed), since $n\propto {\omega }_p^2$. THz s-SNOM therefore suits moderately doped regions and infrared s-SNOM heavily doped ones (my reading). Maissen et al. show that larger tip radius raises THz signal more than an order of magnitude, at the cost of resolution, 41 against 170 $\mathrm{nm}$. Reference: M. Wang et al., ‘Mid- and far-infrared localized surface plasmon resonances in chalcogen-hyperdoped silicon’, arXiv:2210.03373: epsilon = epsilon_s (1 - omega_p^2/(omega^2 + i omega Gamma)), omega_p^2 = n e^2/(epsilon_s epsilon_0 m*), epsilon_s about 11.7 and m* = 0.27 m_e for heavily doped n-Si, real permittivity changing sign near omega_p, omega_p controlled by doping (fetch-tool summary); Y. Abate, ‘Near-Field Spectroscopy and Imaging of Single Nanoparticles’, Dekker Encyclopedia of Nanoscience and Nanotechnology, 3rd ed., Taylor and Francis 2014 (DTIC copy): s-SNOM resolution depending on probe radius of curvature, higher-harmonic demodulation suppressing background, near-field images giving free-carrier concentration, SiO2 phonon-polariton contrast near 1000 cm-1, dopants and defects in single semiconductor nanoparticles, near-field penetration 30 to 50 nm (fetch-tool summary); C. Maissen, S. Chen, E. Nikulina, A. Govyadinov and R. Hillenbrand, ‘Probes for ultra-sensitive THz nanoscopy’, arXiv:2005.05722 (fetch-tool summary).

Ques: How do tip length and shape affect long-wavelength s-SNOM signal?

Longer and blunter tips collect more signal, at a resolution cost. Maissen et al. found that lengthening the tip to about 60 $\mu \mathrm{m}$ raised the 2.52 $\mathrm{THz}$ signal almost four times compared with 20 $\mu \mathrm{m}$, near the half-wavelength length of 59.5 $\mu \mathrm{m}$ (computed), though they saw no clear antenna resonance and attribute this to interferometer alignment uncertainty. Increasing the radius from 25 to 750 $\mathrm{nm}$ raised the signal more than an order of magnitude, while resolution fell from 41 to 170 $\mathrm{nm}$. They report near-field enhancement of about 50 on NiFe and 7 on alumina, and decay lengths of 6 $\mathrm{nm}$ on NiFe against 11 to 16 $\mathrm{nm}$ on alumina. For metals the near-field signal is therefore larger and more confined (Maissen et al.). Reference: C. Maissen, S. Chen, E. Nikulina, A. Govyadinov and R. Hillenbrand, ‘Probes for ultra-sensitive THz nanoscopy’, arXiv:2005.05722 (fetch-tool summary).

References

1. C. Maissen, S. Chen, E. Nikulina, A. Govyadinov and R. Hillenbrand, “Probes for ultra-sensitive THz nanoscopy,” preprint arXiv:2005.05722. Available: https://arxiv.org/pdf/2005.05722
2. C. C. Neacsu, S. Berweger and M. B. Raschke, “Tip-enhanced Raman imaging and spectroscopy: sensitivity, symmetry and selection rules,” Nanobiotechnology, 2007; preprint arXiv:0803.4464. Available: https://arxiv.org/pdf/0803.4464
3. Y. Abate, “Near-field spectroscopy and imaging of single nanoparticles,” in Dekker Encyclopedia of Nanoscience and Nanotechnology, 3rd ed. Taylor and Francis, 2014; DTIC AD1079219. Available: https://apps.dtic.mil/sti/pdfs/AD1079219.pdf
4. M. Wang, Y. Yu, S. Prucnal, Y. Berencen, M. S. Shaikh, L. Rebohle, M. B. Khan, V. Zviagin, R. Huebner, A. Pashkin, A. Erbe, Y. M. Georgiev, M. Grundmann, M. Helm, R. Kirchner and S. Zhou, “Mid- and far-infrared localized surface plasmon resonances in chalcogen-hyperdoped silicon,” arXiv:2210.03373. Available: https://arxiv.org/pdf/2210.03373

Related pages in this vault: Atomic Force Microscope (AFM), Scanning Microwave Impedance Microscope (sMIM), Raman Spectrometer, Confocal Laser Scanning Microscope, Scanning Capacitance Microscope (SCM), Scanning Thermal Microscope (SThM), Piezoresponse Force Microscope (PFM), Terahertz Time-Domain Spectrometer.