Optical Properties and Plasmonic Performance of Titanium Nitride

Titanium nitride (TiN) is one of the most well-established engineering materials nowadays. TiN can overcome most of the drawbacks of palsmonic metals due to its high electron conductivity and mobility, high melting point and due to the compatibility of its growth with Complementary Metal Oxide Semiconductor (CMOS) technology. In this work, we review the dielectric function spectra of TiN and we evaluate the plasmonic performance of TiN by calculating (i) the Surface Plasmon Polariton (SPP) dispersion relations and (ii) the Localized Surface Plasmon Resonance (LSPR) band of TiN nanoparticles, and we demonstrate a significant plasmonic performance of TiN.

The most popular plasmonic metals are gold and silver. However, their melting point is low, especially when in nanoparticle form [71][72][73], making them unstable for photothermal and hot electron devices; their conduction electron mobility is very low, and, consequently, they exhibit high conduction electron losses [74,75]. TiN can overcome all these obstacles due to its high electron conductivity and mobility [14,19,30,31], high melting point and low work function, and due to the compatibility of its growth with CMOS technology, which enables its easy integration and upscaling in realistic, mainstream electronic devices.
In this work, we review the fundamental properties of TiN and its dielectric function spectra extracted from an extended literature survey of reported ellipsometry and reflectivity spectra. As a result, we critically compare the reported electronic properties (such as resistivity/mean free path, and conduction electron density-via the unscreened plasma energy). We evaluate the plasmonic performance of TiN by calculating (i) the SPP dispersion relations at planar TiN/dielectric interfaces; and (ii) the LSPR band of TiN nanoparticles based on the dielectric functions of bulk TiN. Finally, we demonstrate that the plasmonic response of nanoparticles of TiN is equivalent, if not superior, to that of nanoparticles of noble metals.

Fundamental Features and Methods of Growth
TiN belongs to the transition metals of the group IVb of the periodic table of elements. As such, it has four valence electrons and their configuration is 3d 2 4s 2 . Ti forms bonds with N atoms (valence electronic configuration 2s 2 2p 3 ). While various nitride phases such as the Ti2N [76,77] and Ti3N4 [78,79] were reported, the most stable and durable nitrides of these metals are those in the cubic rocksalt B1-TiN structure, also called δ-TiN. The B1-TiN is characterized by gold-like yellow color [10][11][12][13], high hardness [1][2][3][4], electrical conductivity [15,19], and refractory character. Its electronic conductivity is due to the partially filled valence Ti-3d orbitals that are not completely hybridized with the N-2p electrons [80][81][82].

Optical Properties of TiN Films
The optical properties of B1-TiN, which are relevant to plasmonic applications, have been also a subject of intense experimental research using mostly spectroscopic ellipsometry (SE) in all of its varieties, and spectral reflectivity measurements usually at normal incidence [14][15][16]18,19,82,. SE measures the ellispometric angles Ψ-Δ, which are associated with the ratio ρf of the Fresnel reflection coefficients for s-and p-polarization [134]: Then, the real and imaginary parts of the dielectric function ̃ can be analytically calculated from ρf and the angle of incidence ϕ(usually in the range 55°-75°) [134]: On the other hand, the real and imaginary parts of the complex refractive index can be extracted from the spectral reflectivity measurements at normal incidence R via the relations [135]: where θ is the phase change due to reflectivity, which is determined by Kramers-Kronig integration [135]: Alternatively, the n, k can be also determined by fitting the spectral reflectivity curves using specific dispersion relation models, as we will discuss below. The complex dielectric function and the complex refractive index are equivalent and interchangeable, as they are analytically correlated: In the case of a thin film grown on a bulk substrate, the measured spectra by SE or spectral reflectivity accounts the effect of the substrate and film's thickness, in addition to the film's optical properties. Opaque TiN films should be thicker than 100 nm in order to measure directly their complex dielectric function, without any contribution from the substrate; else, a three-phase model (air/homogeneous film/semi-infinite substrate) [136] is required to determine the complex dielectric function.
In this work, we retrieved and critically review the optical data of TiN from . Table 1 lists the samples grown and the spectra measured by numerous groups worldwide regarding the optical properties of TiN. It includes information relevant to the growth conditions, morphology, and optical measurement conditions, as well as the corresponding references . In order to have a unified picture and compare all the available data in the literature, when the n, k values are reported in the original references, we calculated the ε1, ε2 values using Equation (5). The ε1, ε2 from [108][109][110][111][112][113][114][115][116][117][118][119][120][121][122][123] are presented in Figure 1. There is a substantial variation and scattering of the reported experimental spectra due to the sensitivity of TiN's optical performance to various factors, such as stoichiometry ([N]/[Ti] ratio), impurities (residual oxygen in sputter growth or chlorine in CVD growth, post growth oxidation), grain size (which affects the mean free path of the conduction electrons, as we will discuss in more detail below) and density/porosity (which affect the conduction electron density [120]).
The ε1 spectra have attracted particular attention, and are considered to be characteristic of the TiN's chemical and structural quality [117]. The first spectral region (up to 3 eV) is characterized by negative values of ε1 due to the interaction of light with the conduction electrons of TiN, i.e., due to intraband absorption. In particular, and of major importance, is the screened plasma energy Eps = ħωps, which is the spectral energy at which TiN's ε1 = 0. Eps may be affected by both the intraband and interband characteristics and ranges from 2 to 2.95 eV. Eps was reported to be an indicator of the stoichiometry of TiNx, and gets the value 2.65 eV for x = 1 [117]; despite of the efficiency of this phenomenological observation, Eps does not have any other solid physical meaning but it is critical for the plasmonic behavior of TiN-based SPP devices, as we will show below.  Beyond 3 eV, we observe positive values of ε1, which are associated with the interband absorption; these bands are also observed in the ε2 spectra. The interband absorption, which has the form of two distinct peaks, would be solidly identified, if we considered the electron density of states (EDOS) of TiN. Figure 2 presents EDOS calculations for TiN from [82], which are based on the linear augmented plane wave (LAPW) method within the density functional theory using the Wien2k software [137] and using the Generalized Gradient Approximation (GGA) in the form given by Perdew-Burke-Ernzerhof (PBE96) [138].
The observed wide interband absorption originates from the states located 2.5-5.5 eV below the Fermi level, which are predominantly due to N-p electrons. A N-p→Ti-d(t2g) transition is in accordance with the selection rules for photonic excitation (Δl = 0, ±1) and the spectral separation between the N-p states and the Fermi level that intercept the t2g conduction band of the metals. The cut off energy (the maximum energy below the Fermi level, where the partial EDOS of the N-p states gets non-zero values) is around 2.5 eV and defines the threshold at which the dielectric losses contribute to the optical repsonse of TiN. These dielectric losses have a severe impact on the optical reflectivity spectra of TiN at normal incidence, as shown in Figure 3.
At around 3.5 eV below the Fermi level, there is the local minimum E1 in the N-p band; this is in excellent agreement with most of the experimental obsrevations presented in Figure 1b,c. The global maximum of ε2, which is defined as the spectral separation between the EDOS maximum of the N-p and the Fermi level defines the second absorption band E2, which is manifested over 5 eV.  Optical reflectivity spectra of TiN at normal incidence reported by several groups. The absolute values vary due to TiN film's thickness or normalization standards; however, they all share the same qualitative features. In particular, above 2.5 eV all spectra exhibit reduced reflectivity values due to dielectric losses.
The dielectric function of TiN films can be analyzed, via appropriate modeling in order to take into account contributions of intraband and interband transitions as described by a Drude term and several Lorentz oscillators, respectively [120,139]: In Equation (2) ε∞ is a background constant, larger than unity, which is due to higher-energy contributions (beyond the experimental spectral range) referring to transitions that are not taken into account by the Lorentz term(s). Each of the Lorentz oscillators is located at an energy position Eoj = ħoj, with strength fj and damping (broadening) factor j.
The Drude term is characterized by the unscreened plasma energy pu pu ω E   and the damping factor ΓD Epu depends on the concentration of the conduction electrons in the film and is defined by the relation [120,139]: where N is the conduction electron density, e is the electron charge, εo is the permittivity of free space and m * is the electron effective mass, in SI units. Since Epu is directly correlated with the conduction electron density, it can be used to quantify the metallic character of the TiN. ΓD is due to the scattering of electrons and according to the free-electron theory it is related to the electron relaxation time through the relation [120,139]: The ωpu and τD are also closely related with the electrical resistivity (ρ. The relaxation time of conduction electrons is associated with the film resistivity through the relation [120,140]: Changes of ωpu may be attributed, according to Equation (7), either to changes in free electron density or to the electron effective mass [19]. In any case, the increase of ωpu is strongly correlated with the enhancement of TiN metallic character expressed in terms of electrical resistivity.
Based on the above, the relaxation time can be used to calculate the mean free path (MFP) of the conduction electrons in TiN. The MFP for nanocrystalline TiN or for TiN nanostructures and nanoparticles is mostly due to scattering of electrons at the particle surfaces. MFP can be expressed, in terms of the relaxation time and the velocity ( v F) at the Fermi surface [141]: where τbulk is the electron relaxation time of the bulk TiN. The velocity at the Fermi surface is calculated according to the free electron model through the relation [120,140]: The velocity of thermal movement of electrons (vT = (kBT/2m * ) 1/2 = 4.77 × 10 6 cm/s) is smaller by two orders of magnitude than vF and thus it can be reasonably neglected in the calculation of MFP: The Γbulk for TiN was determined to be 0.13 eV by ab initio calculations [82]; note that in [120], the MFP was determined using the crude assumption Γbulk = 0 eV, because at that time there were no relevant data available for Γbulk, therefore, the reported values in Table 2 are more realistic approximations of the real conduction electron MFP for TiN.
In this work we fitted most of the spectra presented in Figure 1 by Equation (6) using the Drude term and two Lorentz oscillators (m = 2), which simulate the contributions of the two interband transitions, E01, E02 defined in Figure 2, in order to provide a quantitative comparison; we call this model D2L. Spectra of non identified thickness or reference optical data of the substrate are excluded from the comparison.
Representative experimental dielectric function spectra along with the corresponding D2L fit results and the convolution to the individual contributions of the Drude term and the two Lorentz oscillators for two TiN films exhibiting high (blue symbols and lines) and low (red symbols and lines) Eps values are presented in Figure 4. The results of the fits, as well as the values of resistivity and MFP [according to Equations (9) and (12)], are presented in Table 2. Note that the presented resistivity values are consistent with the reported resistivity acquired by electrical measurements [120,[142][143][144][145].  Comparing the values of the various parameters as a result of the D2L models, we can conclude the following results: (1) The maximum MFP acquired for the reported TiN films is less than 50 nm; this value, compared with equivalent values for noble metals, can provide a measure of the size limitations for single-crystalline TiN nanoparticles, which might exhibit LSPR, (2) Eps, which is associated with the SPP performance of TiN as we will show below, is associated only with Epu (see also Figure 5) and there is no explicit relation of Eps with any other D2L parameter. The meaning of this observation is that, despite that the screening of the plasma energy in TiN is due to the existence of the interband transitions, similar to Au, the numerical variations of these interband transitions are not strong enough to affect Eps. (3) Epu, which is associated with the conduction electron density, is above 6.9 eV for stoichiometric and highly conductive TiN; the same films exhibit Eps close to 2.65 eV as observed by Logothetidis et al. [117].  Table 2 (the line is a guide to the eye).

Potential of TiN for SPP Applications
After the extensive review of the optical properties of TiN thin films, we proceed with the evaluation of the potential of TiN surfaces for applications involving SPP. SPP are electromagnetic excitations propagating across a planar conductor/dielectric interface, as a direct consequence of the coupling between the electromagnetic field of light and the resonant plasma oscillation of the conduction electrons of the conductor. Therefore, the optical and electrical properties of the conductor will have severe effects on the SPP behavior at a conductor-dielectric interface. As we have reported in the previous paragraphs, TiN exhibits a unique tunability of its conductivity offering great tailoring potential for SPP devices.
In order to evaluate the SPP at interfaces involving TiN as a conductor, we initially consider the TiN/AlN interface. We base this choice upon the structural and chemical compatibility of the conducting fcc B1-TiN (111) and the dielectric hcp w-AlN(0001) (note that the fundamental gap of AlN is 6.2 eV [146]) surfaces that result in stable and sharp TiN/AlN interfaces [147][148][149][150][151][152][153]. The most characteristic description of the SPP across the interface is via the dispersion relation that correlates the frequency of light ω with the wave vector kx in the direction of propagation of SPP. The general equation that defines the aforementioned relation is the following [32]: where ̃ and ̃ are the complex dielectric functions of TiN and the dielectric (AlN in our case), respectively, and c is the speed of light in vacuum. Given that the dielectric functions are inherently complex, Equation (13) should be modified to: when optical attenuation in the conductor exists. A more convenient way of writing the dispersion relation in order to be more easily compared to the experiments is: Figure 6 presents the calculated (via Equation (15)) SPP dispersion relations at the TiN/AlN interfaces using the dielectric function spectra reported by various groups (Table 1) and exhibiting a wide range of Eps values. The first significant observation is that in all cases the dispersion relation is characteristic of a lossy metal and getting finite extreme values of kx [32] due to conduction electron (due to finite ΓD values in Table 2) and dielectric (due to the existence of interband transitions) losses. This has the consequence of the existence of quasi-bound modes at which the slope of the dispersion relation is inversed. Secondly, the characteristic surface Plasmon Esp is scaled to the screened plasma energy Eps of TiN (i.e., the energy where the actual ε1 gets zero value) and not to the unscreened plasma energy Epu, which is the exclusive characteristic of the conduction electrons. This observation points out the significance of Eps not only as a diagnostic tool [117] but also as a critical parameter for the design of plasmonic devices. In addition, Figure 7 demonstrates the SPP dispersion relations at TiN/Dielectric interfaces for various popular dielectrics and for a highly conductive, low-loss TiN film. Air can hardly sustain a SPP mode, while the SPP is stronger and gets bigger extreme kx values with increasing the dielectric constant of the dielectric material; of special importance is the exceptional SPP performance across the TiN/AlN and TiN/GaN interfaces which are exceptionally stable and sharp [18,[147][148][149][150][151][152][153]. Note, however, that SPP-based devices using the TiN/GaN interface are limited to a photon energy range below 3.45 eV, which is the direct fundamental gap of GaN.

Potential of TiN for LSPR Applications
Another field of potential applications of TiN in plasmonics is that of localized surface Plasmon resonance (LSPR) of TiN nanoparticles. The production of TiN nanoparticles was reported [141]. The use of TiN nanoparticles might be beneficial for applications such as solar harvesting and biosensing via Surface-Enhanced Raman Scattering. The optical response of TiN nanoparticle assemblies can be reasonably described by the Maxwell-Garnett Effective Medium Approximation (MG-EMA), which describes the effective optical response of a composite material consisting of a host (matrix) phase and isolated inclusions of the secondary phase [154]: ε ε ε 2ε ε ε ε 2ε (16) where fi is the volume filling ratio of the inclusions, εi is the reference dielectric function of the TiN inclusions and εm is the reference dielectric function of the host (matrix). Given that the MFP of conduction electrons of TiN films is less than 250 nm (according to the data presented in Table 2), we used directly the reported dielectric functions of TiN in MG-EMA; we also considered εm = 1 assuming air as the host medium. The ε2 spectra of TiN nanoparticle assemblies in air with filling ratio fTiN = 10% vol calculated by MG-EMA are presented in Figure 8. In particular, Figure 8a,b present results determined using dielectric functions retrieved from literature and covering a wide range of Eps and Epu, or retrieved from [120], where a variety of stoichiometric TiN films of varying MFP (and consequently grain size) are reported, respectively. In all cases, strong LSPR bands are observed. The narrower and stronger band is located just above 500 nm and resembles the far field LSPR behavior of gold, as it is clearer presented in Figure 9; in particular, Figure 9 shows the spectra of (a) the real ε1 and imaginary ε2 parts of the dielectric function, (b) the refractive index n and the extinction coefficient k, which are associated with ε1 and ε2 via the relations ε1 = n 2 -k 2 and ε2 = 2 nk; and (c) the absorption coefficient α = 4 πk/λ, of nanoparticle assemblies of TiN (from [120]) and 40 nm Gold. Gold exhibits higher quality factor of LSPR [λ/Δλ (FWHM)] but higher dielectric losses at short wavelengths than TiN.
The spectral variation of the LSPR band of TiN nanoparticles extends to an exceptional range, which covers almost the whole visible range in expense, however, of the strength of the LSPR bands. This observation confirms and supports the great potential and perspectives of TiN plasmonic nanostructures reported recently [155]. The variation of the maximum value of ε2 at the LSPR wavelength vs. the LSPR wavelength exhibits a distinct maximum above 500 nm ( Figure 10, red symbols), in contrast to the monotonous increase of max-ε2 with the LSPR wavelength observed for Ag nanoparticles [156]. Finally, the importance of Eps is also demonstrated for the case of LSPR, as well. Indeed, the spectral location of LSPR of TiN nanoparticles is straightly scaled with Eps as it is demonstrated in Figure 10    Comparison of the spectra of (a) the real ε1 and imaginary ε2 parts of the dielectric function; (b) the refractive index n and the extinction coefficient k; and (c) the absorption coefficient of nanoparticle assemblies of TiN (from [120]) and 40 nm Gold. Figure 10. ε2 value at LSPR wavelength (red symbols) and the Eps parameter (blue symbols) vs. the LSPR wavelength. All reported values are deduced by the ε2 spectra of TiN nanoparticles in air calculated by MG-EMA using TiN dielectric functions retrieved from the literature (triangles), and [120] (disks) and presented in Figure 8. The lines are guides to the eye.

Conclusions
In this work, we reviewed the optical properties of TiN films grown by various techniques and on various substrates as reported by several research groups worldwide. The electronic conductivity of TiN is due to the semi filled d-band of Ti. Its optical properties are affected by the co-existence of Ti-d conduction (free) electrons and the N-p to Ti-d interband transitions. Consequently, the dielectric function spectra of TiN can be fitted by a combined model which consists of a Drude term that describes the conduction electrons, and two Lorentz oscillators that describe the interband transitions. The phenomenological parameter Eps, which has no explicit physical meaning, is associated mostly with the unscreened plasma energy Epu of the conduction electrons of TiN and determines the SPP and LSPR characteristics of TiN plasmonic devices. In particular, the SPP performance of TiN/AlN and TiN/GaN planar interfaces is remarkable. In addition, varying the composition and microstructural characteristics of TiN results in variations of Eps, which in its turn can be used to tailor the spectral position of LSPR emerging from assemblies of TiN nanoparticles. However, the spectral variation of TiN's LSPR comes at expense of the LSPR strength. The strongest LSPR observed for TiN nanoparticles is located above 500 nm and resembles that of gold

Author Contributions
Panos Patsalas decided the scope and contents of the paper, had the overview of the work and wrote the paper; Nikolaos Kalfagiannis and Spyros Kassavetis gathered and evaluated the optical spectra from the literature; Nikolaos Kalfagiannis performed the modeling and did the quantitative analysis of the optical spectra.

Conflicts of Interest
The authors declare no conflict of interest.