Label-free and labelled SERS: reading a molecule directly, or through a Raman reporter

This is the fourth article in the series. The second, Vibrational spectroscopy, plasmonics and SERS, explains from scratch where Raman bands come from and why a metal nanostructure enhances them; the third, The many faces of Raman, places SERS among the other Raman variants. Where this article needs those ideas it links to the relevant section rather than repeating it. Antibodies and the surface chemistry used to attach them, on which the labelled assays below rely, are covered later in the series in Putting antibodies on an SPR sensor. Numbers in square brackets refer to the references.

01Why we need SERS at all

A Raman spectrum is a fingerprint: each band is a vibration of the molecule, its position fixed by bond strengths and atomic masses, its intensity by how much the vibration modulates the electron cloud (Raman scattering, from first principles). Water scatters weakly, so Raman works in aqueous and biological samples where infrared absorption struggles; one visible or near-infrared laser excites every band at once; and the bands are narrow, 10–20 cm⁻¹ wide.

The price is sensitivity. The light's oscillating electric field \(\mathbf E\) (amplitude \(E_0\), angular frequency \(\omega_0 = 2\pi c/\lambda_0\) for a laser of wavelength \(\lambda_0\)) induces an electric dipole moment \(\mathbf p = \alpha\mathbf E\); because the polarisability depends on the vibrational coordinate, \(\alpha(Q) = \alpha_0 + (\partial\alpha/\partial Q)_0\,Q\), the dipole contains components at \(\omega_0 \pm \omega_v\) whose radiated power scales as

$$I_\text{Raman} \;\propto\; \omega_s^{4}\,\Big|\Big(\frac{\partial\alpha}{\partial Q}\Big)_0\Big|^2 \langle Q^2\rangle\, I_0 .$$

Here \(\mathbf p\) is the induced dipole moment and \(\alpha\) the molecule's polarisability — how easily its electron cloud is distorted by a field. \(Q\) is the normal coordinate of a vibration (the collective displacement of the atoms in that mode), oscillating at the vibrational angular frequency \(\omega_v\); \(\alpha_0\) is the polarisability at the equilibrium geometry and \((\partial\alpha/\partial Q)_0\) how much it changes as the molecule vibrates — a mode is Raman-active only if this derivative is non-zero. \(\omega_s = \omega_0 - \omega_v\) is the frequency of the (Stokes) scattered light, \(\langle Q^2\rangle\) the mean-square amplitude of the vibration, \(I_0\) the incident laser intensity, and \(I_\text{Raman}\) the scattered intensity. The \(\omega_s^4\) factor is the familiar dipole-radiation law (the same one that makes the sky blue).

For an ordinary molecule the differential Raman cross-section is of order 10⁻³⁰–10⁻²⁹ cm² sr⁻¹, whereas a good fluorophore absorbs with a cross-section near 10⁻¹⁶ cm² [5]. Roughly one photon in 10⁷–10⁹ is Raman-scattered. Detecting a few molecules — or the nanomolar and picomolar concentrations of biomarkers in blood — is impossible with normal Raman.

SERS closes that gap. Near gold or silver nanostructures the Raman signal grows by factors of 10⁴–10⁸ on average and 10¹⁰ or more in the best gaps, enough to see single molecules [9, 10]. The effect was first seen as unexpectedly strong pyridine spectra on roughened silver [1] and recognised as a true per-molecule enhancement in 1977 [2, 3]. But the enhancement only acts on molecules within a few nanometres of the metal. Everything in this article follows from that one restriction: either you bring the analyte itself into the near field (label-free), or you put a molecule that is designed to be bright there and connect it to the analyte by molecular recognition (labelled).

02Two ways to read a molecule: direct and indirect

Figure 1

The words in the literature vary. Direct, label-free or intrinsic SERS records the vibrational spectrum of the target molecule itself. Indirect, labelled or extrinsic SERS records a reporter whose presence is tied to the target; the particles that carry the reporter are called SERS tags, SERS nanoprobes or extrinsic Raman labels (ERLs) [23, 24]. A third, intermediate family uses a reporter whose spectrum changes with its surroundings — a pH-sensitive reporter, or a receptor layer whose bands shift on binding — and so reads the environment rather than the presence of a tag [32].

03The physics both share

The SERS intensity equation

For \(N\) molecules in the laser spot, each with Raman cross-section \(\sigma_\text{ads}\) in its adsorbed state, illuminated with laser intensity \(I_L\), the detected SERS signal can be written [4, 5]

$$I_\text{SERS} = \eta_\text{det}\, N\, \sigma_\text{ads}\, \langle G\rangle\, I_L ,$$

where \(\eta_\text{det}\) is the fraction of scattered photons that are collected and counted (optics and detector efficiency), \(\sigma_\text{ads}\) is a cross-section — an effective area such that the scattered power equals \(\sigma\) times the incident intensity, a convenient measure of how strongly one molecule scatters — and \(\langle G\rangle\) is the enhancement averaged over the molecules. Each factor is a design lever: label-free work can only change \(N\) (by getting more analyte to the surface) and \(G\) (better substrates); labelled work also chooses \(\sigma_\text{ads}\), by picking a reporter with a large, possibly resonant, cross-section, and packs \(N\) as a dense monolayer of reporter on every tag.

Electromagnetic enhancement

The dominant mechanism is electromagnetic. A localized surface plasmon concentrates the field near the particle (localized plasmons); the molecule is driven by the enhanced local field at the laser frequency, and its Raman-shifted dipole radiates through the particle at the scattered frequency. Both steps scale with the intensity enhancement, giving the well-known fourth-power law derived in the spectroscopy article (SERS section):

$$G_\text{EM} \approx \frac{|E_\text{loc}(\omega_0)|^2\,|E_\text{loc}(\omega_s)|^2}{|E_0|^4} \approx \Big|\frac{E_\text{loc}}{E_0}\Big|^4 .$$

Here \(E_0\) is the amplitude of the incident field, \(E_\text{loc}\) the amplitude of the local field at the molecule, \(\omega_0\) the laser frequency and \(\omega_s\) the Raman-scattered frequency; \(G_\text{EM}\) is the electromagnetic enhancement of the Raman intensity. For a small sphere the field factor is set by \(g = (\varepsilon - \varepsilon_m)/(\varepsilon + 2\varepsilon_m)\), where \(\varepsilon(\omega)\) is the complex dielectric function (relative permittivity) of the metal and \(\varepsilon_m\) that of the surrounding medium (1.78 for water). \(g\) is large when \(\mathrm{Re}\,\varepsilon \approx -2\varepsilon_m\) and the imaginary (absorptive) part \(\mathrm{Im}\,\varepsilon\) is small — the reason silver and gold dominate. In the gap between two particles the fields of both add and \(|E/E_0|\) can exceed 100, so \(G\) exceeds 10⁸; these hot spots are tiny and dominate the signal (section 12).

Chemical enhancement

A molecule bonded to the metal is not the same molecule as in solution. Bonding mixes its orbitals with the metal's states and can open a charge-transfer transition between the metal Fermi level and the molecular orbitals; when the laser is near that transition, the scattering becomes partly resonant. Lombardi and Birke showed that plasmon, charge-transfer and molecular resonances can be written in one expression whose numerator couples them through Herzberg–Teller vibronic terms [7]; schematically,

$$R(\omega) \;\propto\; \frac{\mu_\text{PR}\,\mu_\text{K}\,\mu_\text{CT}\,h_\text{IK}}{\big[(\varepsilon_1(\omega)+2\varepsilon_m)^2+\varepsilon_2^2\big]\,\big(\omega_\text{FK}^2-\omega^2+\gamma_\text{FK}^2\big)\big(\omega_\text{IK}^2-\omega^2+\gamma_\text{IK}^2\big)} ,$$

In the notation of [7], \(R(\omega)\) is the SERS intensity contribution of a given vibrational mode at laser frequency \(\omega\); \(\mu_\text{PR}\), \(\mu_\text{K}\) and \(\mu_\text{CT}\) are the transition dipole moments of the plasmon resonance, the molecular transition (from the ground state I to an excited state K) and the charge-transfer transition (between the metal Fermi level F and the molecule); \(h_\text{IK}\) is the Herzberg–Teller coupling, which lets the vibration borrow intensity from these transitions. In the denominators, \(\varepsilon_1\) and \(\varepsilon_2\) are the real and imaginary parts of the metal's dielectric function, \(\varepsilon_m\) that of the medium, \(\omega_\text{FK}\) and \(\omega_\text{IK}\) the charge-transfer and molecular transition frequencies, and the \(\gamma\) terms their damping (line widths). Each bracket becomes small — and \(R\) large — near its resonance: one for the plasmon, one for the charge-transfer transition and one for the molecular transition. The practical lessons are that chemical enhancement (typically 10–100) depends on how the molecule binds, varies between molecules on the same substrate, and can change relative band intensities. For label-free work it is a nuisance to be understood; for reporters it is one more reason to choose molecules that bind covalently and reproducibly.

Surface selection rules

The enhanced field at a metal surface is mostly normal to it. Vibrations whose polarisability changes along the surface normal (\(\alpha_{zz}\)) are therefore enhanced more than those polarised parallel to it [8]. The same molecule standing up or lying flat gives different relative intensities. This is why a label-free SERS spectrum need not match the reference Raman spectrum of the pure compound, and why band assignment should use SERS references measured under similar conditions.

04Distance: the rule that shapes every design

The dipolar near field of a sphere of radius \(r\) falls as \((r/(r+d))^3\) with distance \(d\) from its surface; the intensity as the sixth power; and the SERS enhancement, the square of the intensity enhancement, as the twelfth [4]:

$$\frac{G(d)}{G(0)} = \Big(\frac{r}{r+d}\Big)^{12}, \qquad d_{1/2} = r\,\big(2^{1/12} - 1\big) \approx 0.059\,r .$$

Here \(G(d)\) is the enhancement for a molecule at distance \(d\) from the surface, \(G(0)\) its value at contact, \(r\) the radius of the sphere (or of curvature of the feature), and \(d_{1/2}\) the distance at which half the enhancement is lost. For a 20 nm radius of curvature half the enhancement is gone after only 1.2 nm. Real surfaces have a fast short-range component and a slower long-range tail: atomic-layer-deposited alumina spacers on silver film-over-nanosphere substrates showed both, with signal still measurable through about 3 nm of oxide [12]. Sharp tips and gaps decay faster still.

Figure 2 · Interactive

Three design rules follow directly.

• Label-free sensing needs contact. The analyte must adsorb on the metal or sit within a nanometre or two of it. A receptor layer that holds the analyte far away — a 10 nm antibody — leaves little enhancement for the analyte's own bands. Label-free biosensing therefore uses small receptors (short thiols, aptamers) or none.
• Reporters belong right on the metal. A SERS tag puts its reporter as a covalently bound monolayer at \(d \approx 0\), where the enhancement is largest, and only then adds shells and antibodies on the outside, where distance no longer matters.
• Shells can be thin and still useful. A 2 nm pinhole-free silica or alumina shell isolates the metal from the sample while keeping most of the enhancement — the idea behind shell-isolated nanoparticle-enhanced Raman spectroscopy, SHINERS [13].

05Label-free SERS I — getting the molecule onto the metal

In label-free SERS the surface chemistry is the selectivity. Whatever binds to the metal is seen; whatever does not, is not. Three kinds of interaction decide this.

• Covalent-like chemisorption. Thiols and disulfides bond to gold and silver through sulfur with a strength of roughly 40–50 kcal mol⁻¹; isothiocyanates, selenols and some N-heterocycles also bind strongly. Molecules with these groups (thiol drugs, thiopurines, many pesticides) are easy targets.
• Coordination. Lone pairs of amines and pyridine-type nitrogens, carboxylates and phosphates coordinate to surface atoms or adsorbed ions. Pyridine on silver — the molecule of the discovery [1] — binds this way.
• Electrostatics and hydrophobicity. Colloids are usually stabilised by a charged layer: citrate makes gold and silver negative. Cations approach easily, anions are repelled. Hydrophobic molecules may adsorb from water simply to escape it.

The Langmuir isotherm

If adsorption and desorption balance on a fixed number of equivalent sites, with rate constants \(k_a\) and \(k_d\) and surface coverage \(\theta\),

$$k_a\,C\,(1-\theta) = k_d\,\theta \quad\Longrightarrow\quad \theta = \frac{K C}{1 + K C}, \qquad K = \frac{k_a}{k_d} = \frac{1}{c^\circ}\,e^{-\Delta G^\circ_\text{ads}/RT} .$$

Here \(C\) is the analyte concentration in solution (M), \(\theta\) the fraction of surface sites occupied (0 to 1), \(K\) the adsorption equilibrium constant (M⁻¹; \(1/K\) is the concentration at which half the sites are filled), \(\Delta G^\circ_\text{ads}\) the standard free energy of adsorption (negative when adsorption is favourable), \(c^\circ = 1\) M the standard concentration, \(R = 8.314\) J mol⁻¹ K⁻¹ the gas constant and \(T\) the absolute temperature. Since the SERS signal \(I\) is proportional to the number of adsorbed molecules, \(I \propto \theta\): linear at \(C \ll 1/K\), saturating at \(C \gg 1/K\). A free energy of adsorption of −34 kJ mol⁻¹ gives \(K \approx 10^6\ \mathrm{M^{-1}}\) at room temperature and a linear range ending near 100 nM. This is why label-free SERS calibration curves bend over and why they span only a few decades.

Competition with the matrix

Real samples contain other molecules that adsorb. For an analyte A and a competitor B on the same sites,

$$\theta_A = \frac{K_A C_A}{1 + K_A C_A + K_B C_B} .$$

Here \(\theta_A\) is the fraction of sites holding the analyte, \(C_A\) and \(C_B\) the concentrations of analyte and competitor, and \(K_A\) and \(K_B\) their adsorption constants. At low analyte concentration the sensitivity falls by the factor \(1 + K_B C_B\). Serum proteins at mg mL⁻¹ concentrations, salts that change the surface, and the citrate itself can win the competition and hide the analyte completely. The next figure lets you see how fast this happens.

Figure 3 · Interactive

06Label-free SERS II — surfaces, aggregation and partition layers

Colloids and why they are aggregated

The workhorse label-free substrate is a silver or gold colloid, often made by citrate reduction [34]. Isolated spheres give only modest enhancement; the strong signal comes from gaps between particles. Adding salt or an aggregating agent compresses the electrical double layer — for a 1:1 electrolyte in water at 25 °C the Debye length — the distance over which the ions in solution screen a charged surface — is \(\kappa^{-1} \approx 0.304\ \mathrm{nm}/\sqrt{I/\mathrm{M}}\), where \(I\) is the ionic strength of the solution (\(I = \tfrac12\sum_i c_i z_i^2\) over all ions of concentration \(c_i\) and charge number \(z_i\); equal to the salt concentration for NaCl). It is about 1 nm at 0.1 M — so van der Waals attraction wins and particles cluster, trapping analyte in the junctions. The price is that aggregation is a process: the signal rises, peaks and falls as clusters grow and sediment, so measurements must be made at a fixed, reproducible time.

Cleaning the surface

Citrate, its oxidation products and adventitious carbon occupy sites and add their own bands. Iodide adsorbs very strongly on silver, displaces citrate and leaves a clean, uniform surface. With iodide-modified silver nanoparticles and Mg²⁺ added to aggregate them and bring the negatively charged DNA to the surface, Xu and co-workers recorded reproducible label-free spectra of DNA sensitive to single-base changes [16].

Partition layers: making a weak adsorber stick

Glucose is the textbook difficult analyte: a small Raman cross-section and almost no affinity for bare silver [14]. Van Duyne and co-workers solved this by coating the silver with a self-assembled monolayer that partitions glucose into the first nanometre or so of the surface, much like the stationary phase in chromatography. A mixed decanethiol/mercaptohexanol layer on silver film-over-nanosphere substrates allowed reversible, real-time glucose measurement in bovine plasma, with multivariate calibration [15]. The analyte is still read by its own spectrum — the method is label-free — but a designed layer now supplies selectivity. Thiol monolayer chemistry is described in the antibody article.

Shell isolation

SHINERS particles carry an ultrathin, pinhole-free silica or alumina shell [13]. The shell keeps the metal from reacting with or perturbing the sample, stops the particles from fusing, and lets the same particles be spread over any surface — electrodes, crystals, leaves, cells — to record the spectrum of what lies beneath. It is label-free in that the sample's own spectrum is recorded, but by a probe that never touches the sample.

07Label-free SERS III — what it can read, and what goes wrong

What it reads well

• Small molecules with an anchoring group — drugs, dyes, pesticides, explosives, metabolites — at trace levels, usually against a simple matrix.
• Nucleic acids. The bases dominate; adenine's ring-breathing band near 736 cm⁻¹ is especially strong. Barhoumi and Halas replaced adenine with 2-aminopurine in the capture strand so that hybridisation with a target could be read from the reappearance of the adenine band, without labels [17]. With clean surfaces, differences of a single base are detectable [16].
• Bacteria and cells. Multivariate analysis of SERS spectra of bacteria on silver colloids discriminated species and, for E. coli, strains [18]. The spectrum is a sum over whatever adsorbs from the cell envelope and its secretions, so discrimination is statistical rather than a band-by-band assignment.
• Proteins, with caveats. Large proteins expose only a small part of themselves to the metal; the spectrum shows the residues in contact (aromatic side chains, disulfides, the amide backbone) and can change as the protein unfolds on the surface. The bands that matter in biological spectra are summarised in reading a biological spectrum.

What goes wrong

• Selectivity is borrowed from adsorption. The molecules with the highest \(K\) and the largest cross-section dominate, not the ones you want. In serum, proteins form a corona on the particles within minutes.
• Photochemistry on the surface. Hot electrons and local heating drive reactions. The classic example: 4-aminothiophenol, long assigned on the basis of “b₂ modes” enhanced by charge transfer, is oxidised during the measurement to 4,4′-dimercaptoazobenzene, whose bands were being observed; defocusing the laser avoided the reaction [19]. Any label-free spectrum that changes with laser power or time should be suspected of this.
• Spectra differ from references, because of binding, orientation and selection rules (section 3).
• Reproducibility and quantification are limited by hot-spot statistics (section 12) and by aggregation kinetics. Most applications therefore combine label-free SERS with chemometrics and machine learning, which brings its own risks: models must be validated on independent samples, substrates and days, not on spectra from the same batch [20, 22].

08Labelled SERS I — anatomy of a SERS tag

A SERS tag turns the weaknesses of label-free SERS around. Instead of hoping that the analyte binds the metal and scatters strongly, it places a molecule chosen to do both on a particle chosen to enhance it, and leaves selectivity to biology [23].

Figure 4

The core

Gold is chosen for stability and biocompatibility and for red and near-infrared excitation; silver, or a silver shell on gold (Au@Ag), for higher enhancement in the visible. Anisotropic cores (stars, rods) put hot spots on the particle itself. The most reproducible tags build the gap inside the particle: gold nanoparticles with a 1 nm interior gap, bridged by DNA, with the reporter placed in the gap, gave enhancement factors above 10⁸ with a narrow particle-to-particle distribution [29]. Such gap-enhanced Raman tags do not need aggregation, which removes the largest source of variation in label-free SERS.

The shell

A silica shell (grown by Stöber chemistry after a silane primer) or a dense PEG layer stops reporter desorption and exchange, keeps proteins and salts from reaching the metal, prevents aggregation, and gives a surface for attaching recognition molecules with standard chemistry — amine coupling and EDC/NHS on carboxylated shells (EDC/NHS chemistry), maleimide–thiol, or streptavidin–biotin. In vivo tags of this kind have been reported to be more than 200 times brighter than near-infrared quantum dots [27].

The recognition element

Antibodies are the usual choice; their structure and how to orient them are covered in the antibody article. Aptamers are smaller and chemically defined; peptides and small ligands (folate, RGD) target receptors on cells. Because the reporter is already at the metal, the size of this layer no longer costs signal — but it does decide non-specific binding.

09Labelled SERS II — the chemistry of Raman reporters

A good reporter has (i) a large Raman cross-section — aromatic, conjugated, and ideally resonant with the laser; (ii) a group that binds the metal strongly and in a defined orientation — thiol, disulfide or isothiocyanate; (iii) a few strong, narrow bands that do not overlap those of other reporters or the sample; (iv) photostability and no surface photochemistry; and (v) for multiplexing, a family of related molecules whose bands can be shifted by design [23].

Resonance: SERRS

If the laser falls inside the reporter's electronic absorption, the Raman cross-section itself rises by resonance — typically by factors of 10²–10⁶ — on top of the plasmonic enhancement. This is surface-enhanced resonance Raman scattering, SERRS. The fluorescence that would normally swamp a resonant dye is quenched by the metal. With dye-labelled oligonucleotides and two excitation wavelengths, Faulds, Graham and co-workers detected five DNA sequences quantitatively in one mixture, without separation [26].

The silent region

Biological samples have almost no Raman bands between about 1800 and 2800 cm⁻¹: no common bond in proteins, lipids or nucleic acids vibrates there (why the silent region is silent). A triple bond does: C≡C near 2100–2200 cm⁻¹, C≡N near 2220 cm⁻¹, and C–D bonds a little lower. Reporters with such bands appear against a flat background, even inside cells. Changing the substituents on a 4-ethynylbenzenethiol core shifts the alkyne band in steps of a few tens of cm⁻¹, producing a “palette” of distinct single-band reporters for interference-free multiplex cell imaging [30].

The reporter as a sensor

Some reporters change with their environment, and are used for that. The carboxylate band of 4-MBA grows and the C=O band of the acid fades as pH rises; the ratio, measured on silver nanoparticles taken up by cells, reported intracellular pH over roughly pH 6–8 [32]. The tag is then reading a property of its surroundings — the intermediate case between label-free and labelled.

10Labelled SERS III — assay formats and imaging

Sandwich immunoassays

A capture antibody on a gold or glass surface binds the antigen; a SERS tag carrying a second antibody binds another epitope; the reporter's band is read. Grubisha, Porter and co-workers used gold particles labelled with a nitro reporter and an antibody to detect prostate-specific antigen down to about 1 pg/mL in human serum [25]. The tag also adds the plasmonic coupling between the particle and the gold surface below it — a particle-on-mirror gap — which brightens the signal further. The capture surface is built exactly as on an SPR chip (immobilisation chemistries).

The binding of antigen to the surface follows the same Langmuir form as section 5, now with the antibody dissociation constant \(K_D\). With a background \(S_\text{blank}\) from tags that stick non-specifically,

$$S(C) = S_\text{blank} + S_\text{max}\,\frac{C}{K_D + C}, \qquad \mathrm{LOD}: \; S(C_\text{LOD}) = S_\text{blank} + 3\sigma_\text{blank} .$$

Here \(C\) is the antigen concentration, \(S(C)\) the reporter signal, \(S_\text{max}\) the extra signal when all capture antibodies are occupied, \(S_\text{blank}\) the mean signal of samples without antigen, \(\sigma_\text{blank}\) the standard deviation of repeated blank measurements, and \(C_\text{LOD}\) the limit of detection — the lowest concentration whose signal stands clear of the blank's noise by three standard deviations (≈ 99.9 % one-sided confidence for normally distributed noise). Solving,

$$C_\text{LOD} = K_D\,\frac{3\sigma_\text{blank}}{S_\text{max} - 3\sigma_\text{blank}} \;\approx\; \frac{3\sigma_\text{blank}}{S_\text{max}/K_D} = \frac{3\sigma_\text{blank}}{\text{slope}} .$$

The slope is the calibration sensitivity, the signal change per unit concentration at low concentration, \(\mathrm dS/\mathrm dC = S_\text{max}/K_D\). The detection limit is set by the noise of the blank, which comes mostly from non-specific tag binding (non-specific binding), not by how bright a single tag is. Brighter tags help only until the background they create dominates.

Figure 5 · Interactive

Lateral-flow and point-of-care tests

Replacing the gold particles of a lateral-flow strip with SERS tags keeps the simple format while reading the test line with a Raman reader instead of the eye; the narrow bands let several analytes share one line. The same caution applies: the detection limit is decided by the blank of the strip.

Imaging in cells and tissue

Tags are bright, photostable and multiplexable, which makes them attractive for imaging. Antibody-fragment tags with a near-infrared resonant reporter targeted tumours in living mice [27]. Gold–silica tags combining MRI, photoacoustic and Raman contrast delineated brain-tumour margins in mice, the Raman signal resolving residual tumour at the edge of the resection [28]. Combining SERS with spatially offset Raman spectroscopy (SESORS) recovered signals of four kinds of tags through about 20 mm of tissue [33].

11The mathematics of multiplexing

Why can SERS read many labels at once when fluorescence struggles beyond four or five? Compare widths in the same units. A fluorophore's emission band is typically 30–60 nm wide; at 650 nm a width \(\Delta\lambda\) corresponds to

$$\Delta\tilde\nu = \frac{\Delta\lambda}{\lambda^2} = \frac{50\ \mathrm{nm}}{(650\ \mathrm{nm})^2} \approx 1200\ \mathrm{cm^{-1}},$$

Here \(\lambda\) is the wavelength, \(\Delta\lambda\) the band width in wavelength and \(\Delta\tilde\nu\) the same width in wavenumbers (\(\tilde\nu = 1/\lambda\), the unit of Raman shifts); to get cm⁻¹, write the lengths in cm (50 nm = 5 × 10⁻⁶ cm). A Raman band, by contrast, is 10–20 cm⁻¹ wide. Within the 400–2400 cm⁻¹ window covered by one spectrometer, many reporters can be told apart, all excited by one laser.

Linear unmixing

If the reporters do not interact, the measured spectrum is a weighted sum of their reference spectra plus a background and noise. Writing the spectrum as a vector \(\mathbf y\) (one entry per wavenumber), the reference spectra as the columns of \(\mathbf S\), and the background as a few smooth basis functions \(\mathbf B\),

$$\mathbf y = \mathbf S\,\mathbf c + \mathbf B\,\mathbf b + \boldsymbol\varepsilon .$$

Here \(\mathbf c\) is the vector of unknown tag amounts (one entry per reporter), \(\mathbf b\) the coefficients of the background functions, and \(\boldsymbol\varepsilon\) the noise at each wavenumber. Least squares — choosing \(\mathbf c\) and \(\mathbf b\) to minimise the sum of squared differences between the measured and modelled spectra — gives the amounts directly:

$$\hat{\mathbf c} = (\mathbf A^\mathsf T\mathbf A)^{-1}\mathbf A^\mathsf T\mathbf y, \qquad \mathbf A = [\,\mathbf S\;\mathbf B\,], \qquad \mathrm{Cov}(\hat{\mathbf c}) = \sigma^2(\mathbf A^\mathsf T\mathbf A)^{-1} .$$

Here \(\mathbf A\) is the matrix holding all reference and background spectra side by side, \(\mathbf A^\mathsf T\) its transpose, \((\cdot)^{-1}\) the matrix inverse, the hat in \(\hat{\mathbf c}\) marks an estimate, \(\sigma^2\) is the variance of the noise at each point, and \(\mathrm{Cov}(\hat{\mathbf c})\) the covariance matrix of the estimates (its diagonal holds their variances). The covariance shows what limits multiplexing: when two reference spectra overlap, \(\mathbf A^\mathsf T\mathbf A\) approaches singularity and noise in the estimates grows. Because amounts cannot be negative, non-negative least squares (constrained \(\mathbf c \ge 0\)) is usually used; the figure removes negative components and refits.

Figure 6 · Interactive

12Quantification: hot spots, internal standards and enhancement factors

Hot-spot statistics

Enhancement is not shared fairly. Fang, Seong and Dlott measured the distribution of site enhancements on a silver film-over-nanosphere substrate and found a power law, \(p(\eta) \propto \eta^{-1.75}\) — where \(\eta\) is the enhancement factor of an individual site and \(p(\eta)\) the probability density of finding a site with that enhancement — spanning 2.8 × 10⁴ to 4.1 × 10¹⁰; the 63 hottest sites in a million produced 24 % of the signal [11]. For such a heavy-tailed distribution the variance is dominated by rare, very bright sites, so the relative standard deviation (RSD: standard deviation divided by the mean) of a sum of \(N\) molecules falls much more slowly than the \(1/\sqrt N\) of ordinary statistics. At low coverage the signal from one spot to the next can vary several-fold — and in the single-molecule limit it blinks on and off [9, 10].

Internal standards

Some variations multiply all signals in a spot equally: laser power, focus, local particle density, aggregation state. Dividing the analyte band by a band from an internal standard measured in the same spot cancels them:

$$\frac{I_A}{I_\text{IS}} = \frac{N_A\,\sigma_A\,\langle G\rangle_A}{N_\text{IS}\,\sigma_\text{IS}\,\langle G\rangle_\text{IS}} .$$

Here \(I_A\) and \(I_\text{IS}\) are the intensities of an analyte band and an internal-standard band, \(N\) the number of molecules of each in the spot, \(\sigma\) their Raman cross-sections and \(\langle G\rangle\) the enhancement averaged over the sites each occupies. The ratio is reliable only if \(\langle G\rangle_A \approx \langle G\rangle_\text{IS}\), i.e. if the standard samples the same hot spots. That is why the best internal standards are placed inside the nanostructure — core–shell particles with an embedded reporter [31] — or are isotopically labelled versions of the analyte, which adsorb identically. A standard that sits on different sites still cancels the shared factors but not the hot-spot lottery: when a few molecules in the best gaps dominate, the analyte and the standard draw different winners, and the ratio scatters as much as the raw signal. The figure shows both cases, and the alternative of making the hot spots uniform in the first place.

Figure 7 · Interactive

Calibration in practice

• Average over many spots or scan the laser, so that \(N\) in the statistics is large.
• Use the ratio to an internal standard, ideally embedded or isotopic [31].
• Fit the calibration with the isotherm, not a straight line, outside the low-concentration limit.
• Report the detection limit from the blank's standard deviation, measured on independent substrates.

Enhancement factors: a worked example

Two definitions are common [6, 20]. The analytical enhancement factor compares signals per unit concentration, \(\mathrm{AEF} = (I_\text{SERS}/c_\text{SERS})/(I_\text{RS}/c_\text{RS})\); it is what matters for a detection method; \(I_\text{SERS}\) and \(I_\text{RS}\) are the SERS and normal Raman intensities of the same band, measured under identical conditions, and \(c_\text{SERS}\), \(c_\text{RS}\) the concentrations used for each. The substrate enhancement factor compares signals per molecule, \(\mathrm{SSEF} = (I_\text{SERS}/N_\text{surf})/(I_\text{RS}/N_\text{vol})\), where \(N_\text{surf}\) is the number of adsorbed molecules in the SERS laser spot and \(N_\text{vol}\) the number of molecules in the scattering volume of the normal Raman measurement; it describes the substrate. An illustrative calculation:

13Advantages, disadvantages and how to choose

A short decision guide

• You need to know what the molecule is, or to follow a chemical change: label-free.
• The target binds the metal and is present at more than about 10⁻⁸ M in a clean matrix: label-free, probably with an internal standard.
• The target is a protein, at pM in serum: labelled, almost always. Its own spectrum would be weak, far from the metal behind an antibody, and outcompeted by albumin.
• You need many targets at once, or imaging in cells and tissue: labelled, with a reporter family designed for separation (silent-region or resonant).
• A weak adsorber in a complex matrix, with no good receptor: label-free with a partition layer, or a receptor that changes its own spectrum when it binds.
• Unknown samples, screening, diagnostics without a defined marker: label-free with multivariate analysis — validated across substrates and days.

14Good practice and troubleshooting

The 2020 consensus guidance on quantitative SERS [20] and recent reviews [21, 22] converge on a short list that applies to both modes:

• Characterise every substrate batch (extinction spectrum, electron microscopy) and report the excitation wavelength, power density, acquisition time and objective.
• Always measure blanks: substrate alone, matrix alone, and for tags, the full assay without target.
• Report how spectra were selected — averaged over the whole area, not the brightest spots.
• State which enhancement factor is quoted and how \(N\) was estimated.
• Validate chemometric and machine-learning models on samples, substrates and days kept out of training.
• Check that spectra do not change with laser power or exposure time.

15References

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The distance law, isotherms, assay model, hot-spot statistics and unmixing are the textbook models given in the text; all spectra and numbers in the figures are synthetic and illustrative, and band positions in the reporter table are approximate (they shift with substrate and environment). Corrections and questions are welcome at rakesh_n@cpqct.iitr.ac.in.

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