Carbide and Nitride Based MXene Substrates for SERS—Theoretical Consideration

A promising area of MXenes’ application is molecular sensing by surface enhanced Raman scattering (SERS). For the most widely studied MXene - Ti₃C₂T_xthe first experimental result published in [1] demonstrated SERS enhancement factor (EF) of the order of 10⁶. As to the theoretical consideration of SERS EF in MXene to the best our knowledge there are no published results. The problem is that when the analyte molecule in solution appears at a distance of no more than 0.4-0.5 nm from the MXene substrate, numerical calculation of contribution of image effect (one of the dominant mechanisms of SERS) faces great difficulties. In order to overcome this obstacle, we model the molecule as a point polarizable dipole located near the apex of MXene spheroid. This model allows along with image effect accounting for lightening rod effect and plasmonic resonances. Further we modified the approach developed in [2] for the analytical consideration of SERS introducing the frequency dependence of polarizability of the point dipole as a probe molecule near the MXene surface.<br/>Consider ensemble of MXene flakes with known frequency dependent dielectric function ε(ω), which is subject to the action of the incident EM field. The modified (because of nanoparticles) field acting within the system is denoted as <b>E</b>(<b>r</b>,ω). We introduce at the point <b>r₀</b> a probe molecule much smaller than the nanoparticle size and interparticle distances. Next, this molecule (with known polarizability α(ω)) can be approximated as a point dipole, which due to the resulting electric field <b>E_total</b> (<b>r</b>₀,ω) acquires a dipole moment <b>p</b>(ω) causing the charge redistribution in the nanoparticles. As a result, the total field acting on the molecule differs from the field <b>E</b>(<b>r,ω</b>) that acts at the point <b>r</b>₀ in the absence of the molecule. Appeared additional field <b>E</b>(<b>r</b>,ω) is known in the theory of dielectric susceptibility as ”reaction field”. In order to calculate the SERS EF, one needs to determine the total field acting on the molecule exploiting the principle of superposition <b>E</b>_total(<b>r</b>₀,ω) =<b>E</b>(<b>r</b>₀,ω)+<b>E</b>_r (<b>r</b>₀,ω). It follows from the superposition principle that <b>E</b>_r(<b>r</b>₀,ω) is proportional to the dipole moment <b>p</b>(ω) that produces the reaction field, i.e. <b>E</b>_r(<b>r</b>₀,ω)=γ<b>p</b>(ω). Introducing a small parameter of the theory y=<i>l/</i>ρ, is the curvature radius at the vertex of spheroid, <i>l</i> is the distance from the apex to the dipole, we hold the terms up to y². Carrying out the calculation of the electric field of the image dipole acting on the dye molecule under mentioned conditions we obtain explicit expressin for the parameter γ, containing complex dielectric function of spheroid material. Thus, using obtained expression for γ we calculate the enhancement factor according to the well known definition EF=│<b>E</b>_total│⁴/│<b>E₀</b>│⁴. Further calculations show that when the probe molecule is separated by <i>l</i>=0.7 nm from the spheroidal Ti₃C₂T_x nanoparticle with major and minor semiaxes e.g. 40nm and 7 nm (the corresponding curvature radius is 0.83) the parameter γ=1.56. Finally, we obtain for the maximum value of EF = 4x10⁵for th ewavelngth of the incident light 600 nm. This result is close to the experimentally measured in [1] value. Thus our analytical approach allows to estimate realistically the SERS efficiency of MXene substrate.<br/>Applying the same approach, we calculated as well the EF for Ti₂N, using dielectric function of this MXene calculated in the frame of DFT [3]. We found that the EF of SERS in this MXene is close to that of Ti₃C₂T_x.<br/>1.A. Sarycheva et al. JPCC 121, 19983-19988 (2017).<br/>2. J. Gersten, A, Nitzan. J. Chem. Phys. 73, 3023-3037 (1980).<br/>3. H. Lashgari, et al. Sol. State Comm. 195, 61-69 (2014).