The role of temperature in plasmon sensors in physical and biological research

The plasmon resonance method is attractive in that it has a suffi ciently high sensitivity to changes in concentration (the most widespread use) of the medium under study (analyte). But this method can be no less attractive for research other properties. For example, the effects of temperature during laser irradiation. In such sensors, on the one hand, the temperature can signifi cantly affect the frequency position of the plasmon resonance, which cannot be ignored. On the other hand, the "temperature effect" itself can be the basis for measuring physical quantities that are sensitive to temperature.


Introduction
Recently, sensors that use the phenomenon of plasmon resonance have been widely used [1]. In this case, biosensors are of particular interest [2].
The plasmon resonance method is attractive in that it has a suffi ciently high sensitivity to changes in concentration (the most widespread use) of the medium under study (analyte). But this method can be no less attractive for research other properties. For example, the effects of temperature during laser irradiation. In such sensors, on the one hand, the temperature can signifi cantly affect the frequency position of the plasmon resonance, which cannot be ignored. On the other hand, the "temperature effect" itself can be the basis for measuring physical quantities that are sensitive to temperature.
It is the effect of temperature on plasmon sensors, including biological ones, which will be discussed in this short review.
The radiation enters through a glass prism (in the fi gure above) onto the surface of a metal layer (most often gold) up to 40 nm thick.
After passing through the metal layer, at a certain angle of incidence , a plasmon wave is excited on the lower surface (at the metaldielectric interface), which absorbs the input wave, and the intensity of the refl ected beam from the interface metal-dielectric decreases signifi cantly. Such a decrease is a sensory effect, since the plasmon frequency depends signifi cantly on the dielectric constant of the dielectric [3]: Here pr  is the plasmon resonance frequency, 4 p e e e n m    is the plasma frequency of the electron subsystem of the metal (e is the electron charge, m e is its mass in the metal, n e is the bulk density of electrons),  is the dielectric constant of the dielectric. As can be seen from formula (1), a change in  causes a change in pr  , which causes a change in the angle at which the resonance occurs. That is why (due to the change in angle) this method is quite sensitive.

Infl uence of temperature
Since the penetration depth of radiation for gold exceeds 400 nm [4], a layer up to 40 nm thickness will, on the one hand, be suffi ciently transparent for electromagnetic radiation, and, on the other hand, it is known [5] that absorption will be quite noticeable in the near-surface layer of such thickness and will cause an increase a temperature in it according to equality: Here q is the radiation fl ux, is the pulse duration (or exposure time),  is the bulk density of the material, C is its specifi c heat capacity,  c is the thermal conductivity coeffi cient, k is the absorption index (dimension 1/m) associated with the dimensionless absorption coeffi cient k by the ratio , the near-surface layer of gold will be heated by 2°. But the applied fl uxes of radiation and pulse durations can be much larger. So the heating of the gold layer can reach 60°. Due to the processes of heat transfer, the near-surface layer of the dielectric will also warm up to the same temperature.
This, in any case, cannot be ignored if "sensitivity" € to temperature is signifi cant.

Dependence of the dielectric constant of conventional dielectrics on temperature
For conventional (traditional) isotropic dielectrics Clausius and Massotti independently established the relationship: Here  is the static dielectric constant; n is the bulk density of molecules, that are polarized,  is the polarizability of an individual molecule.
In relation (3), the bulk density n molecules should depend on temperature. This is quite obvious from the following considerations. With increasing temperature, thermal expansion of the material occurs. That is, the effective volume increases by one molecule. But the bulk density n of molecules is reciprocal to this volume. That is, it decreases with increasing temperature.
By direct differentiation of relation (3) with respect to temperature, after some transformations using the relation (3) itself, one can obtain the differential equation: is the coeffi cient of volume expansion, which in a wide temperature range (up to the melting point) also depends on temperature, but to increase the sensitivity of the method, the dependence  on T should be taken into account for all temperatures. The resulting equation unambiguously demonstrates a decrease in dielectric constant with temperature, since its right side is negative. From formula (1) it follows that in this case the plasmon frequency increases.

Dependence of the dielectric constant of proteins and other organic dielectrics on the temperature
In the case when the dielectric layer (in Figure 1 -its lower part) is formed by protein molecules, one speaks about biosensors [6,7]. Then, instead of relation (3), one should use the Langevin-Debye relation: